Research ArticleA Two-Sided Matching Decision Model Based onUncertain Preference Sequences

Xiao Liu and Huimin Ma

School of Management Huazhong University of Science and Technology Wuhan 430074 China

Correspondence should be addressed to Huimin Ma huimin mamailhusteducn

Received 28 February 2015 Revised 23 May 2015 Accepted 28 May 2015

Academic Editor Kyandoghere Kyamakya

Copyright copy 2015 X Liu and H Ma This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two-sided matching is a hot issue in the field of operation research and decision analysis This paper reviews the typical two-sided matching models and their limitations in some specific contexts and then puts forward a new decision model based onuncertain preference sequences In this model we first design a data processingmethod to get preference ordinal value in uncertainpreference sequence then compute the preference distance of each matching pair based on these certain preference ordinal valuesset the optimal objectives as maximizing matching number and minimizing total sum of preference distances of all the matchingpairs under the lowest threshold constraint of matching effect and then solve it with branch-and-bound algorithm Meanwhilewe take two numeral cases as examples and analyze the different matching solutions with one-norm distance two-norm distanceand positive-infinity-norm distance respectively We also compare our decision model with two other approaches and summarizetheir characteristics on two-sided matching

1 Introduction

Two-sided matching is an important research branch inthe field of operation research and decision analysis whichhas been applied into many aspects of engineering andeconomics such as commerce trading [1 2] work assignment[3 4] and resource allocation [5 6] Two-sided matchingdecision problem derives from Gale and Shapleyrsquos researchon stable marriage matching and college admission problemin 1962 [7] Based on the pioneering work of Gale andShapley Roth first gives an accurate conception of two-sidedmatching ldquotwo-sidedrdquo refers to the fact that agents in suchmarkets belong to one of two disjoint sets for examplefirms or workers that are specified in advance and matchingrefers to the bilateral nature of exchange in these markets forexample if I am employed by the University of Pittsburghwhen the University of Pittsburgh employs me [8] In manyactual two-sided matching cases due to the difficulty ofinformation acquisition and fuzziness of the informationidentification it is much easier for a decision maker toacquire the preference sequence of each element than otherkinds of information such as the weight value of connection

between two elements in two disjoint sides so preferencesequence usually may be the essential and even the onlybasis for decision making However when the scale of datagrows rapidly on one hand it is nearly impossible to collectthe complete preference sequence and on the other handtwo or more elements in the preference sequence cannot bedistinguished which ranks higher or lower because they havethe same preference degree for the preference subject

Now we define some conceptions about the preferencesequence If preference sequence of an element in one set tothe other disjoint set includes all the elements in the latter setwe call it a complete preference sequence otherwise we nameit as an incomplete preference sequence If any two elementsin the preference sequence of an element in one set to theother disjoint set do not have the same preference degree wecall it a strong preference sequence otherwise we name it as aweak preference sequenceWe define the preference sequenceof an element in one set as a certain preference sequence onlywhen all the elements in the other set have direct and certainordinal value in accordance with this sequence otherwise itis an uncertain preference sequence Obviously a preferencesequence is certain only when it is complete and strong

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 241379 10 pageshttpdxdoiorg1011552015241379

2 Mathematical Problems in Engineering

at the same time otherwise it is an uncertain preferencesequenceThe decisionmodel constructed in this paper is justagainst this background the only information given for two-sided matching is the uncertain preference sequences

2 Research on Two-Sided Matching

In general we can categorize two-sided matching probleminto three typical kinds of models in terms of differentdecision objectives stable matching maximum cardinalitymatching andmaximumweightmatching In the firstmodelthe objective is to seek a stable matching solution and wecount a solution as stable matching only when there doesnot exist any alternative pairing (119860 119861) in which 119860 and119861 are individually better off than they would be with theelement currently matched Gale and Shapley put forward anapproach also named Gale-Shapley algorithm to get a stablematching solution in the perspective of mathematics andgame theory which symbolizes the beginning of two-sidedmatching research and enlightens the subsequent scholarsto pay more attention to this topic In the second modelthe objective is to seek a solution in which the number ofmatching pairs is maximized This kind of problem has beenwidely applied to graph theory and one common solvingapproach is the Hungarian algorithm which was put forwardby Hungarian mathematician Edmonds in 1965 on the basisof the Hall Theorem [9] The key point of the Hungarianalgorithm is to seek an augmenting path and this kind of two-sidedmatching problem is equivalent to maximummatchingin the bipartite graphMaximumweightmatching sometimesalso is called optimal weight matching for minimum weightmatching is easy to transfer to maximum weight matchingand has the same solving approach In this model eachmatching pair consisting of two elements in two disjoint setshas a corresponding weight value and the objective is tomaximize or optimize the total weight sumof all thematchingpairs This matching model can be classified as assignmentproblem in the field of operation research and one commonsolving approach is the Hungarian method which was firstput forward by Kuhn in 1955 on the basis of a mathematicaltheorem found by another Hungarian mathematician Konig[10] It is worth mentioning that the names of approachesput forward by Edmonds and Kuhn are similar because ofldquoHungarianrdquo but they are two totally different methods oralgorithms Meanwhile assignment problem model also canbe regarded as a maximum weight matching problem in thebipartite graph inwhich one common solving approach is theKuhn-Munkres algorithm put forward by Munkres in 1957[11]

Based on the previous scholarrsquos work the current researchon two-sided matching is usually conducted in the follow-ing two ways The first one is to seek a more effectivemethod or analyze some certain algorithms for the typicalmodels especially the stable matching model for exampleRoth puts forward the hospital-resident algorithm regardingmany-to-one matching case [12] Knoblauch researches thecharacteristics of the Gale-Shapley algorithm on the condi-tion of randomly distributed preference ordinal value [13]McVitie and Wilson put forward a new algorithm based on

ldquoBreakmarriage Operationrdquo regarding the situation when thenumber of the two sides is not the same [14] and Teo et alstudy strategic issues in the Gale-Shapley stable marriagemodel [15] The second one is the specific application indifferent decision contexts after all each decision context hasits own characteristics and the decision makers have to takethe distinctive constraints into consideration so they shouldextend or revise the typical two-sided matching model forexample van Raalte andWebers research a two-sided marketwhere the one type of agents needs the service of amiddlemanor matchmaker in order to be matched with the other type[16] and Sarne and Kraus address the problem of agents in adistributed costly two-sided search for pairwise partnershipsin amultiagent system [17]The research content in this paperis just conducted in the second way

In many actual decision situations we cannot easilyclassify most of the two-sided matching cases into thementioned typical matching models For example it is reallyhard to set an appropriate weight value for each matchingpair directly Though we have some quantitative methodsor techniques such as AHP just shown in [18] to help todetermine the weight value they take little effect when thescale of data in the decision background is very huge or theuseful information given is very scarce Many researchersset the stable matching as their most important optimalobjective however stable matching under the uncertain pref-erence sequence has some limitations When the preferencesequence information is incomplete but strong the Gale-Shapley algorithm still takes effect The solution is still stablewhile the number of matching pairs may be reduced Infact when the preference sequence information is incompletebut strong or weak but complete the solutions are bothable to be solved in polynomial time [19] neverthelesswhen the preference sequence information is incomplete andweak at the same time the problem is NP-hard and thecommon solving approach is to release some constraintsor adopt approximation algorithms just shown in [20 21]What is more sometimes decision objective on two-sidedmatching is not to get a stable matching because stablematching not only cannot take the benefit of two sides intoconsideration at the same time but also cannot maximizethe total utility in economics or other perspectives Oneof the common decision objectives of two-sided matchingin rational economic perspective is to maximize the totalutility so a decision maker generally should transfer thepreference sequence information to utility value with somedata processing methods and then use typical maximumweight matching model to solve it Li et al replace the utilitywith satisfactory and try to maximize the total sum of itBased on a hypothesis that satisfactory degree decreases withthe growth of preference ordinal value and the speed of itsdecline is slower and slower Li et al construct a transfor-mation mechanism between preference ordinal value andsatisfactory value [22] However in their research not onlyis the preference sequence complete but also the matchingsolution highly depends on the transfer function betweenpreference ordinal value and satisfactory value and it maybechanges if any parameter in the function or the function itselfchanges Regarding this situation we research the two-sided

Mathematical Problems in Engineering 3

matching on the condition of uncertain preference sequenceinformation and put forward a new decision model whichintegrates typical maximum cardinality model and optimalweight matching model The objectives in the integratedmodel aremaximizing thematching number andminimizingthe distance from ideal status

3 Uncertain Preference Sequencesand Ordinal Value

The two-sided matching model constructed in this papercomputes preference distance of any two elements in twodisjoint sets respectively on the basis of their preferenceordinal value so we first design a data processing methodto get the preference ordinal value in uncertain preferencesequence We denote two disjoint sets by 119883 and 119884 andthe number of elements in them is 119898 and 119899 respectively119883 = 119883

1 1198832 119883

119898 119884 = 119884

1 1198842 119884

119899 so 119883

119894and 119884

119895

are the elements in 119883 and 119884 119894 isin 119868 119895 isin 119869 119868 = [1119898]119869 = [1 119899] We denote the preference sequence of 119883

119894to 119884 by

119875119894(or 119875119894

lowast) Specifically only when the preference sequenceof 119883119894to 119884 is incomplete and 119883

119894would not like to accept

the elements not in its preference sequence do we label thepreference sequence of 119883

119894to 119884 as 119875

119894

lowast otherwise we label thepreference sequence of119883

119894to119884 as 119875

119894 119875119894(or 119875119894

lowast) consists of theelements in 119884 and these elements are sorted in descendingorder according to the preference degree of 119883

119894 If 119883

119894has

the same preference degree as some elements in 119875119894(or 119875119894

lowast)we label them with parentheses For example 119883 and 119884 bothhave 5 elements and the preference sequence of 119883

3 one of

the elements in 119883 to 119884 is labeled as 1198753(or 1198753

lowast) One of itsinstances is 119884

2 (1198841 1198844) 1198845 which refers to the fact that for

1198833 1198842is its first preference item in 119884 119884

1and 119884

4are tied for

the second item and 1198845is the fourth item As the preference

sequence of 1198833is an incomplete preference sequence and 119884

3

is not in it in order to deal with this situation we classifyincomplete preference sequence into two types one is thatafter considering the elements in the preference sequencepreferentially it would like to accept the elements which arenot in its preference sequence the other one is that it wouldnot like to accept the elements not in its preference sequenceat any time To the latter situation we label it with lowast forexample if we denote the preference sequence of 119883

3by 1198753

directly it refers to the fact that 1198833would like to accept 119884

3

as its preference item after considering 1198842 1198841 1198844 and 119884

5

preferentially if we denote it by 1198753

lowast it refers to the fact that1198833would not like to accept 119884

3as its preference item at any

timeWe denote the real ordinal value of 119884

119895in 119875119894(or 119875119894

lowast) by119900119894119895 so when 119884

119895is one of the elements in 119875

119894(or 119875119894

lowast) 119900119894119895is

1 2 119899 and when 119884119895is not in 119875

119894(or 119875119894

lowast) 119900119894119895is Take

1198753

= 1198842 (1198841 1198844) 1198845 mentioned above as an example the

real ordinal value of 1198842 1198841 1198844 and 119884

5is 11990032

= 1 11990031

= 211990034

= 3 and 11990035

= 4 and the real ordinal value of 1198843is 11990033

=

We also denote the preference ordinal value 119884119895in 119875119894(or

119875119894

lowast) by 119903119894119895 according to the data processing method designed

in this paper the transition between 119900119894119895and 119903119894119895is defined as

follows (a) when 119900119894119895is if the preference sequence of 119883

119894is

labeled as 119875119894 119903119894119895is (|119875119894| + 1 + 119899)2 where |119875

119894| is the number

of elements in 119875119894 and else if the preference sequence of 119883

119894

is labeled as 119875119894

lowast 119903119894119895is 119899 + 1 (b) when 119900

119894119895is not if there

does not exist any other element tied with 119884119895in 119875119894(or 119875119894

lowast)119903119894119895equals 119900

119894119895 and else if there exists any other element tied

with 119884119895in 119875119894(or 119875119894

lowast) which we denote including 119884119895by set 1198841015840

119903119894119895equals arithmetic average value of the real ordinal value of

all the elements in 1198841015840 labeled as avg(1198841015840) The definition of 119900

119894119895

and transition between 119900119894119895and 119903119894119895is presented as follows

119900119894119895

=

1 2 3 119899 119884119895isin 119875119894

(or 119875119894

lowast)

119884119895notin 119875119894

(or 119875119894

lowast)

119903119894119895

=

10038161003816100381610038161198751198941003816100381610038161003816 + 1 + 119899

2 119900119894119895

= exist119875119894

119899 + 1 119900119894119895

= exist119875119894

lowast

119900119894119895 119900

119894119895= not tied with 119884

119895in 119875119894

(or 119875119894

lowast)

avg (1198841015840) 119900

119894119895= tied with 119884

119895in 119875119894

(or 119875119894

lowast)

(1)

Similarly we denote the preference sequence of 119884119895to 119883 by

119876119895(or 119876119895

lowast) Specifically we label the preference sequence of119884119895to 119883 as 119876

119895

lowast only when the preference sequence of 119884119895to 119883

is incomplete and 119884119895would not like to accept the elements

not in its preference sequence at any time otherwise we labelit as 119876

119895directly We denote the real ordinal value of 119883

119894in 119876119895

(or119876119895

lowast) by 119904119894119895and the preference ordinal value of119883

119894in119876119895(or

119876119895

lowast) by 119905119894119895 and the transition between 119904

119894119895and 119905119894119895also is similar

to 119900119894119895and 119903119894119895 (a) when 119904

119894119895is if the preference sequence of

119884119895to119883 is labeled as119876

119895 119905119894119895is (|119876119895|+1+119898)2 where |119876

119894| is the

number of elements in119876119895 and else if the preference sequence

of119884119895to119883 is labeled as119876

119895

lowast 119905119894119895is119898+1 (b) when 119904

119894119895is not if

there does not exist any other element tied with 119883119894in 119876119895(or

119876119895

lowast) 119905119894119895equals 119904

119894119895 and else if there exists any other element

tied with119883119894in 119876119895(or119876119895

lowast) which we denote including119883119894by

set 1198831015840 119905119894119895equals the average value of the real ordinal value of

all the elements in1198831015840 labeled as avg(1198831015840) The definition of 119904

119894119895

and the transition between 119904119894119895and 119905119894119895is presented as follows

119904119894119895

=

1 2 3 119898 119883119894isin 119876119895

(or 119876119895

lowast)

119883119894notin 119876119895

(or 119876119895

lowast)

119905119894119895

=

10038161003816100381610038161003816119876119895

10038161003816100381610038161003816+ 1 + 119898

2 119904119894119895

= exist119876119895

119898 + 1 119904119894119895

= exist119876119895

lowast

119904119894119895 119904

119894119895= not tied with 119883

119894in 119876119895

(or 119876119895

lowast)

avg (1198831015840) 119904

119894119895= tied with 119883

119894in 119876119895

(or 119876119895

lowast)

(2)

4 Preference Distance

As we cannot measure the relationship between preferenceordinal value and preference utility exactly no universal

4 Mathematical Problems in Engineering

transition function between them is widely accepted Nomatter whether it is a linear function or a nonlinear functionboth depend on the specific decision background Howeverthe preference ordinal value is a good unit to measure thedistance from ideal status which we name as preferencedistance The larger the ordinal value is the further thedistance is and the less value the matching effect has So ourmodel first computes the preference distances from the idealmatching situation on the basis of preference ordinal valuesand then minimizes the total sum of these distances

Take the elements119883119894and119884119895in119883 and119884 respectively as an

example In 119875119894 the preference sequence of 119883

119894 the preference

ordinal value of 119884119895is 119903119894119895 and in 119876

119895 the preference sequence

of 119884119895 the preference ordinal value of 119883

119894is 119905119894119895 We use (119903

119894119895 119905119894119895)

to represent the matching status between 119883119894and 119884

119895 In the

ideal matching status 119883119894and 119884

119895are both the first preference

items of each other and we label this ideal matching status as(1 1) Preference distance of119883

119894and119884119895is the distance between

the real matching status and ideal matching status of thismatching pair denoted by 119889

119894119895 According to the definition

of distance given by Minkowski the computation of 119889119894119895is

defined as follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+10038161003816100381610038161003816119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(3)

If we take the different importance of two sides into consid-eration the computation mentioned above can be modifiedas follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+120572119902 10038161003816100381610038161003816

119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(4)

And 120572 is the importance factor to balance two sides In thispaper we ignore the difference and assume that two sideshave the same importance so 120572 is always assigned as 1 andthe computation of preference distance is equivalent in (3)and (4) In the computation equation it refers to differentkinds of distance when 119902 varies Theoretically 119902 can be anyreal number from 1 to +infin but the most common value is 12 and +infin

When 119902 = 1 119889119894119895is a one-norm distance also called

Manhattan distance It is a form of geometry in which thedistance between two points is the sum of the absolutedifferences of their Cartesian coordinate value In this case119889119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is 119903

119894119895+ 119905119894119895

minus 2a linear expression of their coordinate value A commoncriterion to evaluate a matching solution is the total sum ofpreference ordinal values of all the matching elements Thetotal sum of preference ordinal values and the total sum ofone-norm distances are linearly equivalent in mathematicsnamely all the feasible solutions of minimizing total sum ofone-norm distances and minimizing total sum of preferenceordinal values are completely the same And both of themhave a negative correlation with matching effect One-normdistance applies to the decision situation where the ranges ofpreference ordinal value of two sides do not have significantdifference and two sides roughly have the same metrics sothat we can use simple additive relationship to represent thewhole matching effect

When 119902 = 2 119889119894119895is a two-norm distance also called

Euclidean distance It is a form of geometry in which the dis-tance between two points is the length of the line segmentconnecting them and is computed by Pythagorean formulaIn this case 119889

119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is

radic(119903119894119895

minus 1)2 + (119905119894119895

minus 1)2 just equaling their Euclidean distanceSince the coordinate value is integer number generally 119889

119894119895

involves floating computing if 119902 = 2 So in the perspectiveof solving efficiency the solving time will increase rapidlywhen the scale of data grows Therefore two-norm distanceapplies to the situation where the dimensions represented intwo sides are independent and the data scale is in an acceptedscope

When 119902 = +infin 119889119894119895is a positive-infinity-norm distance

also called Chebyshev distance It is a form of geometry inwhich the distance between two points is the greatest of theirdifferences along any coordinate dimension In this case 119889

119894119895

between point (119903119894119895 119905119894119895) and point (1 1) equals the larger value

in 119903119894119895minus1 and 119905

119894119895minus1 Inmathematics take one point as the origin

of coordinates the points which have 119889 Chebyshev distancewith origin of coordinates make up a quadrate The originof coordinates is the central point of this quadrate and thelength of its each side is 2119889 meanwhile each side is parallelwith coordinate axes Positive-infinity-norm distance appliesto the situation where the balanced performance of two sidesis important and the difference of two sides should not be toomuch

5 Modeling Construction and Solving

In the typical optimal weight matching model the mainconstraint is that any element in one set can only matchone element in the other set at most and the objective isto optimize the total sum of weight values for examplewe take the preference distance of each matching pair asits weight value the optimal objective is to minimize thetotal sum of preference distances of all the matching pairsBesides matching number is also a constraint though thevalue is obvious in the typical model if the number ofelements in two sides is the same the matching numberjust equals the number of elements in each side otherwisethe matching number equals the less one In the model putforward in this paper in order to avoid the performance ofsome matching pairs being too bad we set a threshold ofmatching performance namely preference distances of allthe matching pairs should not be more than the value set inadvance As this new constraint on matching performancethe matching number is not a constant value any more and itdepends on restraint degree of the threshold Regarding thiscontext we set two optimal objectives in the decision modelone is to maximize the number of all matching pairs and theother one is to minimize the total sum of all matching pairsrsquopreference distances

We use 119909119894119895to denote the matching relationship between

119883119894and 119884

119895 119909119894119895

isin 0 1 If 119909119894119895is 1 it refers to the fact that 119883

119894

and 119884119895match each other and make up a matching pair else if

119909119894119895is 0 it refers to the fact that 119883

119894and 119884

119895do not match each

other and also do notmake up amatching pairThe constraint

Mathematical Problems in Engineering 5

that any element in one set can onlymatch one element in theother set at most is presented as follows

0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

(5)

We denote the maximum value of 119889119894119895

by 119889max and theminimum value by 119889min and the specific value of 119889max and119889min depends on the range of parameter 119902 and the preferencesequences We also denote 120582 as the threshold factor ofmatching performance 120582 isin [0 1] When 120582 = 1 it refers to noconstraint on threshold when 120582 decreases it refers to the factthat the constraint degree on threshold increases and when120582 = 0 it refers to the fact that the constraint on threshold isthe strictest The constraint is presented as follows

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min) forall119894 isin 119868 forall119895 isin 119869 (6)

Since one element should not match an element which is notin the former onersquos preference sequence and meanwhile theformer element also would not like to accept that we shouldadd a new constraint if 119903

119894119895equals 119899+1 119909

119894119895should be zero and

if 119905119894119895equals 119898 + 1 119909

119894119895also should be zero As the maximum

value of 119903119894119895and 119905119894119895is 119899 + 1 and 119898 + 1 respectively and the

second maximum value of 119903119894119895and 119905119894119895is 119899 and 119898 respectively

this constraint can be presented as a linear expression asfollows

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

(7)

Maximizing the matching number and minimizing the totalsum of preference distances are presented as follows

maxsum

119894isin119868

sum

119895isin119869

119909119894119895

minsum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

(8)

As it is a multiobjective optimization we generally have threedifferent ways to deal with it the first one is to transfera multiobjective problem to one single-objective problemsuch as simple linear additive weight method maximal-minimal method and TOPSIS method the critical point ofthis solving way is to ensure that the optimal solution of thenew single-objective problem is also the noninferior solutionof the original multiobjective problem the second one is totransfer it to multiple single-objective problems in a specialorder such as hierarchical method interactive programmingmethod we get the optimal solution of the original problemthrough solving these single-objective problems one by onewhich is also the optimal solution of the last single-objectiveproblem the third way is some nonuniform methods suchas multiplication division method and efficiency coefficientmethod In this paper we hold that maximizing the numberof all matching pairs is the main optimal objective and min-imizing the total sum of preference distances are the second

optimal objective so we use the secondwaymentioned aboveto deal with it Namely we first get the optimal solution of themain objective then add it as a new constraint and finallyget the optimal solution of the second objective These twoobjectives are integrated as a linear equation as follows

max119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895 1198721 ≫ 1198722 (9)

where1198721and119872

2are both positive real numbers and ldquo119872

1≫

1198722rdquo refers to the fact that 119872

1is far greater than 119872

2 We can

also give a specific value to1198721and119872

2if we can measure the

importance of these two objectives exactly but it is not thecontent discussed in this paper In conclusion the decisionmodel constructed in this paper is presented as follows

max 119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

st 0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min)

forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

119909119894119895

= 0 or 1

119889119894119895

= [(119905119894119895minus 1)119902

+ (119903119894119895minus 1)119902

]1119902

120582 isin [0 1]

119902 isin [1 +infin]

1198721 isin 119877+ 1198722 isin 119877

+ 1198721 ≫ 1198722

(10)

In the typical maximum weight matching model as themaximum matching number is constant and obvious themodel is easy to convert to a standard assignment modelIn standard assignment model the number of elements ineach side is the same and every element in one set will bematched with one element in the other set through addingsome zero elements to balance the number of two sidesWhen the scale is not too large the Hungarian method isone common approach to solve this kind of model and itis on basis of the following two theorems (a) if all elementsin one column or row of efficiency matrix are plus or minusa number the optimal solutions of the origin matrix andnew matrix are the same (b) the maximum number ofindependent zero elements in efficiency matrix equals theminimum number of lines which cover all the zero elementsNow so many literatures have researched and promotedthis method and provide programming codes in differentprogramming languages or coding platforms such as CJAVA and MATLAB so we do not discuss it repetitively

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

at the same time otherwise it is an uncertain preferencesequenceThe decisionmodel constructed in this paper is justagainst this background the only information given for two-sided matching is the uncertain preference sequences

2 Research on Two-Sided Matching

In general we can categorize two-sided matching probleminto three typical kinds of models in terms of differentdecision objectives stable matching maximum cardinalitymatching andmaximumweightmatching In the firstmodelthe objective is to seek a stable matching solution and wecount a solution as stable matching only when there doesnot exist any alternative pairing (119860 119861) in which 119860 and119861 are individually better off than they would be with theelement currently matched Gale and Shapley put forward anapproach also named Gale-Shapley algorithm to get a stablematching solution in the perspective of mathematics andgame theory which symbolizes the beginning of two-sidedmatching research and enlightens the subsequent scholarsto pay more attention to this topic In the second modelthe objective is to seek a solution in which the number ofmatching pairs is maximized This kind of problem has beenwidely applied to graph theory and one common solvingapproach is the Hungarian algorithm which was put forwardby Hungarian mathematician Edmonds in 1965 on the basisof the Hall Theorem [9] The key point of the Hungarianalgorithm is to seek an augmenting path and this kind of two-sidedmatching problem is equivalent to maximummatchingin the bipartite graphMaximumweightmatching sometimesalso is called optimal weight matching for minimum weightmatching is easy to transfer to maximum weight matchingand has the same solving approach In this model eachmatching pair consisting of two elements in two disjoint setshas a corresponding weight value and the objective is tomaximize or optimize the total weight sumof all thematchingpairs This matching model can be classified as assignmentproblem in the field of operation research and one commonsolving approach is the Hungarian method which was firstput forward by Kuhn in 1955 on the basis of a mathematicaltheorem found by another Hungarian mathematician Konig[10] It is worth mentioning that the names of approachesput forward by Edmonds and Kuhn are similar because ofldquoHungarianrdquo but they are two totally different methods oralgorithms Meanwhile assignment problem model also canbe regarded as a maximum weight matching problem in thebipartite graph inwhich one common solving approach is theKuhn-Munkres algorithm put forward by Munkres in 1957[11]

Based on the previous scholarrsquos work the current researchon two-sided matching is usually conducted in the follow-ing two ways The first one is to seek a more effectivemethod or analyze some certain algorithms for the typicalmodels especially the stable matching model for exampleRoth puts forward the hospital-resident algorithm regardingmany-to-one matching case [12] Knoblauch researches thecharacteristics of the Gale-Shapley algorithm on the condi-tion of randomly distributed preference ordinal value [13]McVitie and Wilson put forward a new algorithm based on

ldquoBreakmarriage Operationrdquo regarding the situation when thenumber of the two sides is not the same [14] and Teo et alstudy strategic issues in the Gale-Shapley stable marriagemodel [15] The second one is the specific application indifferent decision contexts after all each decision context hasits own characteristics and the decision makers have to takethe distinctive constraints into consideration so they shouldextend or revise the typical two-sided matching model forexample van Raalte andWebers research a two-sided marketwhere the one type of agents needs the service of amiddlemanor matchmaker in order to be matched with the other type[16] and Sarne and Kraus address the problem of agents in adistributed costly two-sided search for pairwise partnershipsin amultiagent system [17]The research content in this paperis just conducted in the second way

In many actual decision situations we cannot easilyclassify most of the two-sided matching cases into thementioned typical matching models For example it is reallyhard to set an appropriate weight value for each matchingpair directly Though we have some quantitative methodsor techniques such as AHP just shown in [18] to help todetermine the weight value they take little effect when thescale of data in the decision background is very huge or theuseful information given is very scarce Many researchersset the stable matching as their most important optimalobjective however stable matching under the uncertain pref-erence sequence has some limitations When the preferencesequence information is incomplete but strong the Gale-Shapley algorithm still takes effect The solution is still stablewhile the number of matching pairs may be reduced Infact when the preference sequence information is incompletebut strong or weak but complete the solutions are bothable to be solved in polynomial time [19] neverthelesswhen the preference sequence information is incomplete andweak at the same time the problem is NP-hard and thecommon solving approach is to release some constraintsor adopt approximation algorithms just shown in [20 21]What is more sometimes decision objective on two-sidedmatching is not to get a stable matching because stablematching not only cannot take the benefit of two sides intoconsideration at the same time but also cannot maximizethe total utility in economics or other perspectives Oneof the common decision objectives of two-sided matchingin rational economic perspective is to maximize the totalutility so a decision maker generally should transfer thepreference sequence information to utility value with somedata processing methods and then use typical maximumweight matching model to solve it Li et al replace the utilitywith satisfactory and try to maximize the total sum of itBased on a hypothesis that satisfactory degree decreases withthe growth of preference ordinal value and the speed of itsdecline is slower and slower Li et al construct a transfor-mation mechanism between preference ordinal value andsatisfactory value [22] However in their research not onlyis the preference sequence complete but also the matchingsolution highly depends on the transfer function betweenpreference ordinal value and satisfactory value and it maybechanges if any parameter in the function or the function itselfchanges Regarding this situation we research the two-sided

Mathematical Problems in Engineering 3

matching on the condition of uncertain preference sequenceinformation and put forward a new decision model whichintegrates typical maximum cardinality model and optimalweight matching model The objectives in the integratedmodel aremaximizing thematching number andminimizingthe distance from ideal status

3 Uncertain Preference Sequencesand Ordinal Value

The two-sided matching model constructed in this papercomputes preference distance of any two elements in twodisjoint sets respectively on the basis of their preferenceordinal value so we first design a data processing methodto get the preference ordinal value in uncertain preferencesequence We denote two disjoint sets by 119883 and 119884 andthe number of elements in them is 119898 and 119899 respectively119883 = 119883

1 1198832 119883

119898 119884 = 119884

1 1198842 119884

119899 so 119883

119894and 119884

119895

are the elements in 119883 and 119884 119894 isin 119868 119895 isin 119869 119868 = [1119898]119869 = [1 119899] We denote the preference sequence of 119883

119894to 119884 by

119875119894(or 119875119894

lowast) Specifically only when the preference sequenceof 119883119894to 119884 is incomplete and 119883

119894would not like to accept

the elements not in its preference sequence do we label thepreference sequence of 119883

119894to 119884 as 119875

119894

lowast otherwise we label thepreference sequence of119883

119894to119884 as 119875

119894 119875119894(or 119875119894

lowast) consists of theelements in 119884 and these elements are sorted in descendingorder according to the preference degree of 119883

119894 If 119883

119894has

the same preference degree as some elements in 119875119894(or 119875119894

lowast)we label them with parentheses For example 119883 and 119884 bothhave 5 elements and the preference sequence of 119883

3 one of

the elements in 119883 to 119884 is labeled as 1198753(or 1198753

lowast) One of itsinstances is 119884

2 (1198841 1198844) 1198845 which refers to the fact that for

1198833 1198842is its first preference item in 119884 119884

1and 119884

4are tied for

the second item and 1198845is the fourth item As the preference

sequence of 1198833is an incomplete preference sequence and 119884

3

is not in it in order to deal with this situation we classifyincomplete preference sequence into two types one is thatafter considering the elements in the preference sequencepreferentially it would like to accept the elements which arenot in its preference sequence the other one is that it wouldnot like to accept the elements not in its preference sequenceat any time To the latter situation we label it with lowast forexample if we denote the preference sequence of 119883

3by 1198753

directly it refers to the fact that 1198833would like to accept 119884

3

as its preference item after considering 1198842 1198841 1198844 and 119884

5

preferentially if we denote it by 1198753

lowast it refers to the fact that1198833would not like to accept 119884

3as its preference item at any

timeWe denote the real ordinal value of 119884

119895in 119875119894(or 119875119894

lowast) by119900119894119895 so when 119884

119895is one of the elements in 119875

119894(or 119875119894

lowast) 119900119894119895is

1 2 119899 and when 119884119895is not in 119875

119894(or 119875119894

lowast) 119900119894119895is Take

1198753

= 1198842 (1198841 1198844) 1198845 mentioned above as an example the

real ordinal value of 1198842 1198841 1198844 and 119884

5is 11990032

= 1 11990031

= 211990034

= 3 and 11990035

= 4 and the real ordinal value of 1198843is 11990033

=

We also denote the preference ordinal value 119884119895in 119875119894(or

119875119894

lowast) by 119903119894119895 according to the data processing method designed

in this paper the transition between 119900119894119895and 119903119894119895is defined as

follows (a) when 119900119894119895is if the preference sequence of 119883

119894is

labeled as 119875119894 119903119894119895is (|119875119894| + 1 + 119899)2 where |119875

119894| is the number

of elements in 119875119894 and else if the preference sequence of 119883

119894

is labeled as 119875119894

lowast 119903119894119895is 119899 + 1 (b) when 119900

119894119895is not if there

does not exist any other element tied with 119884119895in 119875119894(or 119875119894

lowast)119903119894119895equals 119900

119894119895 and else if there exists any other element tied

with 119884119895in 119875119894(or 119875119894

lowast) which we denote including 119884119895by set 1198841015840

119903119894119895equals arithmetic average value of the real ordinal value of

all the elements in 1198841015840 labeled as avg(1198841015840) The definition of 119900

119894119895

and transition between 119900119894119895and 119903119894119895is presented as follows

119900119894119895

=

1 2 3 119899 119884119895isin 119875119894

(or 119875119894

lowast)

119884119895notin 119875119894

(or 119875119894

lowast)

119903119894119895

=

10038161003816100381610038161198751198941003816100381610038161003816 + 1 + 119899

2 119900119894119895

= exist119875119894

119899 + 1 119900119894119895

= exist119875119894

lowast

119900119894119895 119900

119894119895= not tied with 119884

119895in 119875119894

(or 119875119894

lowast)

avg (1198841015840) 119900

119894119895= tied with 119884

119895in 119875119894

(or 119875119894

lowast)

(1)

Similarly we denote the preference sequence of 119884119895to 119883 by

119876119895(or 119876119895

lowast) Specifically we label the preference sequence of119884119895to 119883 as 119876

119895

lowast only when the preference sequence of 119884119895to 119883

is incomplete and 119884119895would not like to accept the elements

not in its preference sequence at any time otherwise we labelit as 119876

119895directly We denote the real ordinal value of 119883

119894in 119876119895

(or119876119895

lowast) by 119904119894119895and the preference ordinal value of119883

119894in119876119895(or

119876119895

lowast) by 119905119894119895 and the transition between 119904

119894119895and 119905119894119895also is similar

to 119900119894119895and 119903119894119895 (a) when 119904

119894119895is if the preference sequence of

119884119895to119883 is labeled as119876

119895 119905119894119895is (|119876119895|+1+119898)2 where |119876

119894| is the

number of elements in119876119895 and else if the preference sequence

of119884119895to119883 is labeled as119876

119895

lowast 119905119894119895is119898+1 (b) when 119904

119894119895is not if

there does not exist any other element tied with 119883119894in 119876119895(or

119876119895

lowast) 119905119894119895equals 119904

119894119895 and else if there exists any other element

tied with119883119894in 119876119895(or119876119895

lowast) which we denote including119883119894by

set 1198831015840 119905119894119895equals the average value of the real ordinal value of

all the elements in1198831015840 labeled as avg(1198831015840) The definition of 119904

119894119895

and the transition between 119904119894119895and 119905119894119895is presented as follows

119904119894119895

=

1 2 3 119898 119883119894isin 119876119895

(or 119876119895

lowast)

119883119894notin 119876119895

(or 119876119895

lowast)

119905119894119895

=

10038161003816100381610038161003816119876119895

10038161003816100381610038161003816+ 1 + 119898

2 119904119894119895

= exist119876119895

119898 + 1 119904119894119895

= exist119876119895

lowast

119904119894119895 119904

119894119895= not tied with 119883

119894in 119876119895

(or 119876119895

lowast)

avg (1198831015840) 119904

119894119895= tied with 119883

119894in 119876119895

(or 119876119895

lowast)

(2)

4 Preference Distance

As we cannot measure the relationship between preferenceordinal value and preference utility exactly no universal

4 Mathematical Problems in Engineering

transition function between them is widely accepted Nomatter whether it is a linear function or a nonlinear functionboth depend on the specific decision background Howeverthe preference ordinal value is a good unit to measure thedistance from ideal status which we name as preferencedistance The larger the ordinal value is the further thedistance is and the less value the matching effect has So ourmodel first computes the preference distances from the idealmatching situation on the basis of preference ordinal valuesand then minimizes the total sum of these distances

Take the elements119883119894and119884119895in119883 and119884 respectively as an

example In 119875119894 the preference sequence of 119883

119894 the preference

ordinal value of 119884119895is 119903119894119895 and in 119876

119895 the preference sequence

of 119884119895 the preference ordinal value of 119883

119894is 119905119894119895 We use (119903

119894119895 119905119894119895)

to represent the matching status between 119883119894and 119884

119895 In the

ideal matching status 119883119894and 119884

119895are both the first preference

items of each other and we label this ideal matching status as(1 1) Preference distance of119883

119894and119884119895is the distance between

the real matching status and ideal matching status of thismatching pair denoted by 119889

119894119895 According to the definition

of distance given by Minkowski the computation of 119889119894119895is

defined as follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+10038161003816100381610038161003816119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(3)

If we take the different importance of two sides into consid-eration the computation mentioned above can be modifiedas follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+120572119902 10038161003816100381610038161003816

119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(4)

And 120572 is the importance factor to balance two sides In thispaper we ignore the difference and assume that two sideshave the same importance so 120572 is always assigned as 1 andthe computation of preference distance is equivalent in (3)and (4) In the computation equation it refers to differentkinds of distance when 119902 varies Theoretically 119902 can be anyreal number from 1 to +infin but the most common value is 12 and +infin

When 119902 = 1 119889119894119895is a one-norm distance also called

Manhattan distance It is a form of geometry in which thedistance between two points is the sum of the absolutedifferences of their Cartesian coordinate value In this case119889119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is 119903

119894119895+ 119905119894119895

minus 2a linear expression of their coordinate value A commoncriterion to evaluate a matching solution is the total sum ofpreference ordinal values of all the matching elements Thetotal sum of preference ordinal values and the total sum ofone-norm distances are linearly equivalent in mathematicsnamely all the feasible solutions of minimizing total sum ofone-norm distances and minimizing total sum of preferenceordinal values are completely the same And both of themhave a negative correlation with matching effect One-normdistance applies to the decision situation where the ranges ofpreference ordinal value of two sides do not have significantdifference and two sides roughly have the same metrics sothat we can use simple additive relationship to represent thewhole matching effect

When 119902 = 2 119889119894119895is a two-norm distance also called

Euclidean distance It is a form of geometry in which the dis-tance between two points is the length of the line segmentconnecting them and is computed by Pythagorean formulaIn this case 119889

119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is

radic(119903119894119895

minus 1)2 + (119905119894119895

minus 1)2 just equaling their Euclidean distanceSince the coordinate value is integer number generally 119889

119894119895

involves floating computing if 119902 = 2 So in the perspectiveof solving efficiency the solving time will increase rapidlywhen the scale of data grows Therefore two-norm distanceapplies to the situation where the dimensions represented intwo sides are independent and the data scale is in an acceptedscope

When 119902 = +infin 119889119894119895is a positive-infinity-norm distance

also called Chebyshev distance It is a form of geometry inwhich the distance between two points is the greatest of theirdifferences along any coordinate dimension In this case 119889

119894119895

between point (119903119894119895 119905119894119895) and point (1 1) equals the larger value

in 119903119894119895minus1 and 119905

119894119895minus1 Inmathematics take one point as the origin

of coordinates the points which have 119889 Chebyshev distancewith origin of coordinates make up a quadrate The originof coordinates is the central point of this quadrate and thelength of its each side is 2119889 meanwhile each side is parallelwith coordinate axes Positive-infinity-norm distance appliesto the situation where the balanced performance of two sidesis important and the difference of two sides should not be toomuch

5 Modeling Construction and Solving

In the typical optimal weight matching model the mainconstraint is that any element in one set can only matchone element in the other set at most and the objective isto optimize the total sum of weight values for examplewe take the preference distance of each matching pair asits weight value the optimal objective is to minimize thetotal sum of preference distances of all the matching pairsBesides matching number is also a constraint though thevalue is obvious in the typical model if the number ofelements in two sides is the same the matching numberjust equals the number of elements in each side otherwisethe matching number equals the less one In the model putforward in this paper in order to avoid the performance ofsome matching pairs being too bad we set a threshold ofmatching performance namely preference distances of allthe matching pairs should not be more than the value set inadvance As this new constraint on matching performancethe matching number is not a constant value any more and itdepends on restraint degree of the threshold Regarding thiscontext we set two optimal objectives in the decision modelone is to maximize the number of all matching pairs and theother one is to minimize the total sum of all matching pairsrsquopreference distances

We use 119909119894119895to denote the matching relationship between

119883119894and 119884

119895 119909119894119895

isin 0 1 If 119909119894119895is 1 it refers to the fact that 119883

119894

and 119884119895match each other and make up a matching pair else if

119909119894119895is 0 it refers to the fact that 119883

119894and 119884

119895do not match each

other and also do notmake up amatching pairThe constraint

Mathematical Problems in Engineering 5

that any element in one set can onlymatch one element in theother set at most is presented as follows

0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

(5)

We denote the maximum value of 119889119894119895

by 119889max and theminimum value by 119889min and the specific value of 119889max and119889min depends on the range of parameter 119902 and the preferencesequences We also denote 120582 as the threshold factor ofmatching performance 120582 isin [0 1] When 120582 = 1 it refers to noconstraint on threshold when 120582 decreases it refers to the factthat the constraint degree on threshold increases and when120582 = 0 it refers to the fact that the constraint on threshold isthe strictest The constraint is presented as follows

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min) forall119894 isin 119868 forall119895 isin 119869 (6)

Since one element should not match an element which is notin the former onersquos preference sequence and meanwhile theformer element also would not like to accept that we shouldadd a new constraint if 119903

119894119895equals 119899+1 119909

119894119895should be zero and

if 119905119894119895equals 119898 + 1 119909

119894119895also should be zero As the maximum

value of 119903119894119895and 119905119894119895is 119899 + 1 and 119898 + 1 respectively and the

second maximum value of 119903119894119895and 119905119894119895is 119899 and 119898 respectively

this constraint can be presented as a linear expression asfollows

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

(7)

Maximizing the matching number and minimizing the totalsum of preference distances are presented as follows

maxsum

119894isin119868

sum

119895isin119869

119909119894119895

minsum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

(8)

As it is a multiobjective optimization we generally have threedifferent ways to deal with it the first one is to transfera multiobjective problem to one single-objective problemsuch as simple linear additive weight method maximal-minimal method and TOPSIS method the critical point ofthis solving way is to ensure that the optimal solution of thenew single-objective problem is also the noninferior solutionof the original multiobjective problem the second one is totransfer it to multiple single-objective problems in a specialorder such as hierarchical method interactive programmingmethod we get the optimal solution of the original problemthrough solving these single-objective problems one by onewhich is also the optimal solution of the last single-objectiveproblem the third way is some nonuniform methods suchas multiplication division method and efficiency coefficientmethod In this paper we hold that maximizing the numberof all matching pairs is the main optimal objective and min-imizing the total sum of preference distances are the second

optimal objective so we use the secondwaymentioned aboveto deal with it Namely we first get the optimal solution of themain objective then add it as a new constraint and finallyget the optimal solution of the second objective These twoobjectives are integrated as a linear equation as follows

max119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895 1198721 ≫ 1198722 (9)

where1198721and119872

2are both positive real numbers and ldquo119872

1≫

1198722rdquo refers to the fact that 119872

1is far greater than 119872

2 We can

also give a specific value to1198721and119872

2if we can measure the

importance of these two objectives exactly but it is not thecontent discussed in this paper In conclusion the decisionmodel constructed in this paper is presented as follows

max 119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

st 0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min)

forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

119909119894119895

= 0 or 1

119889119894119895

= [(119905119894119895minus 1)119902

+ (119903119894119895minus 1)119902

]1119902

120582 isin [0 1]

119902 isin [1 +infin]

1198721 isin 119877+ 1198722 isin 119877

+ 1198721 ≫ 1198722

(10)

In the typical maximum weight matching model as themaximum matching number is constant and obvious themodel is easy to convert to a standard assignment modelIn standard assignment model the number of elements ineach side is the same and every element in one set will bematched with one element in the other set through addingsome zero elements to balance the number of two sidesWhen the scale is not too large the Hungarian method isone common approach to solve this kind of model and itis on basis of the following two theorems (a) if all elementsin one column or row of efficiency matrix are plus or minusa number the optimal solutions of the origin matrix andnew matrix are the same (b) the maximum number ofindependent zero elements in efficiency matrix equals theminimum number of lines which cover all the zero elementsNow so many literatures have researched and promotedthis method and provide programming codes in differentprogramming languages or coding platforms such as CJAVA and MATLAB so we do not discuss it repetitively

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

matching on the condition of uncertain preference sequenceinformation and put forward a new decision model whichintegrates typical maximum cardinality model and optimalweight matching model The objectives in the integratedmodel aremaximizing thematching number andminimizingthe distance from ideal status

3 Uncertain Preference Sequencesand Ordinal Value

The two-sided matching model constructed in this papercomputes preference distance of any two elements in twodisjoint sets respectively on the basis of their preferenceordinal value so we first design a data processing methodto get the preference ordinal value in uncertain preferencesequence We denote two disjoint sets by 119883 and 119884 andthe number of elements in them is 119898 and 119899 respectively119883 = 119883

1 1198832 119883

119898 119884 = 119884

1 1198842 119884

119899 so 119883

119894and 119884

119895

are the elements in 119883 and 119884 119894 isin 119868 119895 isin 119869 119868 = [1119898]119869 = [1 119899] We denote the preference sequence of 119883

119894to 119884 by

119875119894(or 119875119894

lowast) Specifically only when the preference sequenceof 119883119894to 119884 is incomplete and 119883

119894would not like to accept

the elements not in its preference sequence do we label thepreference sequence of 119883

119894to 119884 as 119875

119894

lowast otherwise we label thepreference sequence of119883

119894to119884 as 119875

119894 119875119894(or 119875119894

lowast) consists of theelements in 119884 and these elements are sorted in descendingorder according to the preference degree of 119883

119894 If 119883

119894has

the same preference degree as some elements in 119875119894(or 119875119894

lowast)we label them with parentheses For example 119883 and 119884 bothhave 5 elements and the preference sequence of 119883

3 one of

the elements in 119883 to 119884 is labeled as 1198753(or 1198753

lowast) One of itsinstances is 119884

2 (1198841 1198844) 1198845 which refers to the fact that for

1198833 1198842is its first preference item in 119884 119884

1and 119884

4are tied for

the second item and 1198845is the fourth item As the preference

sequence of 1198833is an incomplete preference sequence and 119884

3

is not in it in order to deal with this situation we classifyincomplete preference sequence into two types one is thatafter considering the elements in the preference sequencepreferentially it would like to accept the elements which arenot in its preference sequence the other one is that it wouldnot like to accept the elements not in its preference sequenceat any time To the latter situation we label it with lowast forexample if we denote the preference sequence of 119883

3by 1198753

directly it refers to the fact that 1198833would like to accept 119884

3

as its preference item after considering 1198842 1198841 1198844 and 119884

5

preferentially if we denote it by 1198753

lowast it refers to the fact that1198833would not like to accept 119884

3as its preference item at any

timeWe denote the real ordinal value of 119884

119895in 119875119894(or 119875119894

lowast) by119900119894119895 so when 119884

119895is one of the elements in 119875

119894(or 119875119894

lowast) 119900119894119895is

1 2 119899 and when 119884119895is not in 119875

119894(or 119875119894

lowast) 119900119894119895is Take

1198753

= 1198842 (1198841 1198844) 1198845 mentioned above as an example the

real ordinal value of 1198842 1198841 1198844 and 119884

5is 11990032

= 1 11990031

= 211990034

= 3 and 11990035

= 4 and the real ordinal value of 1198843is 11990033

=

We also denote the preference ordinal value 119884119895in 119875119894(or

119875119894

lowast) by 119903119894119895 according to the data processing method designed

in this paper the transition between 119900119894119895and 119903119894119895is defined as

follows (a) when 119900119894119895is if the preference sequence of 119883

119894is

labeled as 119875119894 119903119894119895is (|119875119894| + 1 + 119899)2 where |119875

119894| is the number

of elements in 119875119894 and else if the preference sequence of 119883

119894

is labeled as 119875119894

lowast 119903119894119895is 119899 + 1 (b) when 119900

119894119895is not if there

does not exist any other element tied with 119884119895in 119875119894(or 119875119894

lowast)119903119894119895equals 119900

119894119895 and else if there exists any other element tied

with 119884119895in 119875119894(or 119875119894

lowast) which we denote including 119884119895by set 1198841015840

119903119894119895equals arithmetic average value of the real ordinal value of

all the elements in 1198841015840 labeled as avg(1198841015840) The definition of 119900

119894119895

and transition between 119900119894119895and 119903119894119895is presented as follows

119900119894119895

=

1 2 3 119899 119884119895isin 119875119894

(or 119875119894

lowast)

119884119895notin 119875119894

(or 119875119894

lowast)

119903119894119895

=

10038161003816100381610038161198751198941003816100381610038161003816 + 1 + 119899

2 119900119894119895

= exist119875119894

119899 + 1 119900119894119895

= exist119875119894

lowast

119900119894119895 119900

119894119895= not tied with 119884

119895in 119875119894

(or 119875119894

lowast)

avg (1198841015840) 119900

119894119895= tied with 119884

119895in 119875119894

(or 119875119894

lowast)

(1)

Similarly we denote the preference sequence of 119884119895to 119883 by

119876119895(or 119876119895

lowast) Specifically we label the preference sequence of119884119895to 119883 as 119876

119895

lowast only when the preference sequence of 119884119895to 119883

is incomplete and 119884119895would not like to accept the elements

not in its preference sequence at any time otherwise we labelit as 119876

119895directly We denote the real ordinal value of 119883

119894in 119876119895

(or119876119895

lowast) by 119904119894119895and the preference ordinal value of119883

119894in119876119895(or

119876119895

lowast) by 119905119894119895 and the transition between 119904

119894119895and 119905119894119895also is similar

to 119900119894119895and 119903119894119895 (a) when 119904

119894119895is if the preference sequence of

119884119895to119883 is labeled as119876

119895 119905119894119895is (|119876119895|+1+119898)2 where |119876

119894| is the

number of elements in119876119895 and else if the preference sequence

of119884119895to119883 is labeled as119876

119895

lowast 119905119894119895is119898+1 (b) when 119904

119894119895is not if

there does not exist any other element tied with 119883119894in 119876119895(or

119876119895

lowast) 119905119894119895equals 119904

119894119895 and else if there exists any other element

tied with119883119894in 119876119895(or119876119895

lowast) which we denote including119883119894by

set 1198831015840 119905119894119895equals the average value of the real ordinal value of

all the elements in1198831015840 labeled as avg(1198831015840) The definition of 119904

119894119895

and the transition between 119904119894119895and 119905119894119895is presented as follows

119904119894119895

=

1 2 3 119898 119883119894isin 119876119895

(or 119876119895

lowast)

119883119894notin 119876119895

(or 119876119895

lowast)

119905119894119895

=

10038161003816100381610038161003816119876119895

10038161003816100381610038161003816+ 1 + 119898

2 119904119894119895

= exist119876119895

119898 + 1 119904119894119895

= exist119876119895

lowast

119904119894119895 119904

119894119895= not tied with 119883

119894in 119876119895

(or 119876119895

lowast)

avg (1198831015840) 119904

119894119895= tied with 119883

119894in 119876119895

(or 119876119895

lowast)

(2)

4 Preference Distance

As we cannot measure the relationship between preferenceordinal value and preference utility exactly no universal

4 Mathematical Problems in Engineering

transition function between them is widely accepted Nomatter whether it is a linear function or a nonlinear functionboth depend on the specific decision background Howeverthe preference ordinal value is a good unit to measure thedistance from ideal status which we name as preferencedistance The larger the ordinal value is the further thedistance is and the less value the matching effect has So ourmodel first computes the preference distances from the idealmatching situation on the basis of preference ordinal valuesand then minimizes the total sum of these distances

Take the elements119883119894and119884119895in119883 and119884 respectively as an

example In 119875119894 the preference sequence of 119883

119894 the preference

ordinal value of 119884119895is 119903119894119895 and in 119876

119895 the preference sequence

of 119884119895 the preference ordinal value of 119883

119894is 119905119894119895 We use (119903

119894119895 119905119894119895)

to represent the matching status between 119883119894and 119884

119895 In the

ideal matching status 119883119894and 119884

119895are both the first preference

items of each other and we label this ideal matching status as(1 1) Preference distance of119883

119894and119884119895is the distance between

the real matching status and ideal matching status of thismatching pair denoted by 119889

119894119895 According to the definition

of distance given by Minkowski the computation of 119889119894119895is

defined as follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+10038161003816100381610038161003816119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(3)

If we take the different importance of two sides into consid-eration the computation mentioned above can be modifiedas follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+120572119902 10038161003816100381610038161003816

119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(4)

And 120572 is the importance factor to balance two sides In thispaper we ignore the difference and assume that two sideshave the same importance so 120572 is always assigned as 1 andthe computation of preference distance is equivalent in (3)and (4) In the computation equation it refers to differentkinds of distance when 119902 varies Theoretically 119902 can be anyreal number from 1 to +infin but the most common value is 12 and +infin

When 119902 = 1 119889119894119895is a one-norm distance also called

Manhattan distance It is a form of geometry in which thedistance between two points is the sum of the absolutedifferences of their Cartesian coordinate value In this case119889119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is 119903

119894119895+ 119905119894119895

minus 2a linear expression of their coordinate value A commoncriterion to evaluate a matching solution is the total sum ofpreference ordinal values of all the matching elements Thetotal sum of preference ordinal values and the total sum ofone-norm distances are linearly equivalent in mathematicsnamely all the feasible solutions of minimizing total sum ofone-norm distances and minimizing total sum of preferenceordinal values are completely the same And both of themhave a negative correlation with matching effect One-normdistance applies to the decision situation where the ranges ofpreference ordinal value of two sides do not have significantdifference and two sides roughly have the same metrics sothat we can use simple additive relationship to represent thewhole matching effect

When 119902 = 2 119889119894119895is a two-norm distance also called

Euclidean distance It is a form of geometry in which the dis-tance between two points is the length of the line segmentconnecting them and is computed by Pythagorean formulaIn this case 119889

119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is

radic(119903119894119895

minus 1)2 + (119905119894119895

minus 1)2 just equaling their Euclidean distanceSince the coordinate value is integer number generally 119889

119894119895

involves floating computing if 119902 = 2 So in the perspectiveof solving efficiency the solving time will increase rapidlywhen the scale of data grows Therefore two-norm distanceapplies to the situation where the dimensions represented intwo sides are independent and the data scale is in an acceptedscope

When 119902 = +infin 119889119894119895is a positive-infinity-norm distance

also called Chebyshev distance It is a form of geometry inwhich the distance between two points is the greatest of theirdifferences along any coordinate dimension In this case 119889

119894119895

between point (119903119894119895 119905119894119895) and point (1 1) equals the larger value

in 119903119894119895minus1 and 119905

119894119895minus1 Inmathematics take one point as the origin

of coordinates the points which have 119889 Chebyshev distancewith origin of coordinates make up a quadrate The originof coordinates is the central point of this quadrate and thelength of its each side is 2119889 meanwhile each side is parallelwith coordinate axes Positive-infinity-norm distance appliesto the situation where the balanced performance of two sidesis important and the difference of two sides should not be toomuch

5 Modeling Construction and Solving

In the typical optimal weight matching model the mainconstraint is that any element in one set can only matchone element in the other set at most and the objective isto optimize the total sum of weight values for examplewe take the preference distance of each matching pair asits weight value the optimal objective is to minimize thetotal sum of preference distances of all the matching pairsBesides matching number is also a constraint though thevalue is obvious in the typical model if the number ofelements in two sides is the same the matching numberjust equals the number of elements in each side otherwisethe matching number equals the less one In the model putforward in this paper in order to avoid the performance ofsome matching pairs being too bad we set a threshold ofmatching performance namely preference distances of allthe matching pairs should not be more than the value set inadvance As this new constraint on matching performancethe matching number is not a constant value any more and itdepends on restraint degree of the threshold Regarding thiscontext we set two optimal objectives in the decision modelone is to maximize the number of all matching pairs and theother one is to minimize the total sum of all matching pairsrsquopreference distances

We use 119909119894119895to denote the matching relationship between

119883119894and 119884

119895 119909119894119895

isin 0 1 If 119909119894119895is 1 it refers to the fact that 119883

119894

and 119884119895match each other and make up a matching pair else if

119909119894119895is 0 it refers to the fact that 119883

119894and 119884

119895do not match each

other and also do notmake up amatching pairThe constraint

Mathematical Problems in Engineering 5

that any element in one set can onlymatch one element in theother set at most is presented as follows

0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

(5)

We denote the maximum value of 119889119894119895

by 119889max and theminimum value by 119889min and the specific value of 119889max and119889min depends on the range of parameter 119902 and the preferencesequences We also denote 120582 as the threshold factor ofmatching performance 120582 isin [0 1] When 120582 = 1 it refers to noconstraint on threshold when 120582 decreases it refers to the factthat the constraint degree on threshold increases and when120582 = 0 it refers to the fact that the constraint on threshold isthe strictest The constraint is presented as follows

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min) forall119894 isin 119868 forall119895 isin 119869 (6)

Since one element should not match an element which is notin the former onersquos preference sequence and meanwhile theformer element also would not like to accept that we shouldadd a new constraint if 119903

119894119895equals 119899+1 119909

119894119895should be zero and

if 119905119894119895equals 119898 + 1 119909

119894119895also should be zero As the maximum

value of 119903119894119895and 119905119894119895is 119899 + 1 and 119898 + 1 respectively and the

second maximum value of 119903119894119895and 119905119894119895is 119899 and 119898 respectively

this constraint can be presented as a linear expression asfollows

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

(7)

Maximizing the matching number and minimizing the totalsum of preference distances are presented as follows

maxsum

119894isin119868

sum

119895isin119869

119909119894119895

minsum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

(8)

As it is a multiobjective optimization we generally have threedifferent ways to deal with it the first one is to transfera multiobjective problem to one single-objective problemsuch as simple linear additive weight method maximal-minimal method and TOPSIS method the critical point ofthis solving way is to ensure that the optimal solution of thenew single-objective problem is also the noninferior solutionof the original multiobjective problem the second one is totransfer it to multiple single-objective problems in a specialorder such as hierarchical method interactive programmingmethod we get the optimal solution of the original problemthrough solving these single-objective problems one by onewhich is also the optimal solution of the last single-objectiveproblem the third way is some nonuniform methods suchas multiplication division method and efficiency coefficientmethod In this paper we hold that maximizing the numberof all matching pairs is the main optimal objective and min-imizing the total sum of preference distances are the second

optimal objective so we use the secondwaymentioned aboveto deal with it Namely we first get the optimal solution of themain objective then add it as a new constraint and finallyget the optimal solution of the second objective These twoobjectives are integrated as a linear equation as follows

max119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895 1198721 ≫ 1198722 (9)

where1198721and119872

2are both positive real numbers and ldquo119872

1≫

1198722rdquo refers to the fact that 119872

1is far greater than 119872

2 We can

also give a specific value to1198721and119872

2if we can measure the

importance of these two objectives exactly but it is not thecontent discussed in this paper In conclusion the decisionmodel constructed in this paper is presented as follows

max 119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

st 0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min)

forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

119909119894119895

= 0 or 1

119889119894119895

= [(119905119894119895minus 1)119902

+ (119903119894119895minus 1)119902

]1119902

120582 isin [0 1]

119902 isin [1 +infin]

1198721 isin 119877+ 1198722 isin 119877

+ 1198721 ≫ 1198722

(10)

In the typical maximum weight matching model as themaximum matching number is constant and obvious themodel is easy to convert to a standard assignment modelIn standard assignment model the number of elements ineach side is the same and every element in one set will bematched with one element in the other set through addingsome zero elements to balance the number of two sidesWhen the scale is not too large the Hungarian method isone common approach to solve this kind of model and itis on basis of the following two theorems (a) if all elementsin one column or row of efficiency matrix are plus or minusa number the optimal solutions of the origin matrix andnew matrix are the same (b) the maximum number ofindependent zero elements in efficiency matrix equals theminimum number of lines which cover all the zero elementsNow so many literatures have researched and promotedthis method and provide programming codes in differentprogramming languages or coding platforms such as CJAVA and MATLAB so we do not discuss it repetitively

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

transition function between them is widely accepted Nomatter whether it is a linear function or a nonlinear functionboth depend on the specific decision background Howeverthe preference ordinal value is a good unit to measure thedistance from ideal status which we name as preferencedistance The larger the ordinal value is the further thedistance is and the less value the matching effect has So ourmodel first computes the preference distances from the idealmatching situation on the basis of preference ordinal valuesand then minimizes the total sum of these distances

Take the elements119883119894and119884119895in119883 and119884 respectively as an

example In 119875119894 the preference sequence of 119883

119894 the preference

ordinal value of 119884119895is 119903119894119895 and in 119876

119895 the preference sequence

of 119884119895 the preference ordinal value of 119883

119894is 119905119894119895 We use (119903

119894119895 119905119894119895)

to represent the matching status between 119883119894and 119884

119895 In the

ideal matching status 119883119894and 119884

119895are both the first preference

items of each other and we label this ideal matching status as(1 1) Preference distance of119883

119894and119884119895is the distance between

the real matching status and ideal matching status of thismatching pair denoted by 119889

119894119895 According to the definition

of distance given by Minkowski the computation of 119889119894119895is

defined as follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+10038161003816100381610038161003816119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(3)

If we take the different importance of two sides into consid-eration the computation mentioned above can be modifiedas follows

119889119894119895

= (10038161003816100381610038161003816119903119894119895minus 110038161003816100381610038161003816119902

+120572119902 10038161003816100381610038161003816

119905119894119895minus 110038161003816100381610038161003816119902

)1119902

(4)

And 120572 is the importance factor to balance two sides In thispaper we ignore the difference and assume that two sideshave the same importance so 120572 is always assigned as 1 andthe computation of preference distance is equivalent in (3)and (4) In the computation equation it refers to differentkinds of distance when 119902 varies Theoretically 119902 can be anyreal number from 1 to +infin but the most common value is 12 and +infin

When 119902 = 1 119889119894119895is a one-norm distance also called

Manhattan distance It is a form of geometry in which thedistance between two points is the sum of the absolutedifferences of their Cartesian coordinate value In this case119889119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is 119903

119894119895+ 119905119894119895

minus 2a linear expression of their coordinate value A commoncriterion to evaluate a matching solution is the total sum ofpreference ordinal values of all the matching elements Thetotal sum of preference ordinal values and the total sum ofone-norm distances are linearly equivalent in mathematicsnamely all the feasible solutions of minimizing total sum ofone-norm distances and minimizing total sum of preferenceordinal values are completely the same And both of themhave a negative correlation with matching effect One-normdistance applies to the decision situation where the ranges ofpreference ordinal value of two sides do not have significantdifference and two sides roughly have the same metrics sothat we can use simple additive relationship to represent thewhole matching effect

When 119902 = 2 119889119894119895is a two-norm distance also called

Euclidean distance It is a form of geometry in which the dis-tance between two points is the length of the line segmentconnecting them and is computed by Pythagorean formulaIn this case 119889

119894119895between point (119903

119894119895 119905119894119895) and point (1 1) is

radic(119903119894119895

minus 1)2 + (119905119894119895

minus 1)2 just equaling their Euclidean distanceSince the coordinate value is integer number generally 119889

119894119895

involves floating computing if 119902 = 2 So in the perspectiveof solving efficiency the solving time will increase rapidlywhen the scale of data grows Therefore two-norm distanceapplies to the situation where the dimensions represented intwo sides are independent and the data scale is in an acceptedscope

When 119902 = +infin 119889119894119895is a positive-infinity-norm distance

also called Chebyshev distance It is a form of geometry inwhich the distance between two points is the greatest of theirdifferences along any coordinate dimension In this case 119889

119894119895

between point (119903119894119895 119905119894119895) and point (1 1) equals the larger value

in 119903119894119895minus1 and 119905

119894119895minus1 Inmathematics take one point as the origin

of coordinates the points which have 119889 Chebyshev distancewith origin of coordinates make up a quadrate The originof coordinates is the central point of this quadrate and thelength of its each side is 2119889 meanwhile each side is parallelwith coordinate axes Positive-infinity-norm distance appliesto the situation where the balanced performance of two sidesis important and the difference of two sides should not be toomuch

5 Modeling Construction and Solving

In the typical optimal weight matching model the mainconstraint is that any element in one set can only matchone element in the other set at most and the objective isto optimize the total sum of weight values for examplewe take the preference distance of each matching pair asits weight value the optimal objective is to minimize thetotal sum of preference distances of all the matching pairsBesides matching number is also a constraint though thevalue is obvious in the typical model if the number ofelements in two sides is the same the matching numberjust equals the number of elements in each side otherwisethe matching number equals the less one In the model putforward in this paper in order to avoid the performance ofsome matching pairs being too bad we set a threshold ofmatching performance namely preference distances of allthe matching pairs should not be more than the value set inadvance As this new constraint on matching performancethe matching number is not a constant value any more and itdepends on restraint degree of the threshold Regarding thiscontext we set two optimal objectives in the decision modelone is to maximize the number of all matching pairs and theother one is to minimize the total sum of all matching pairsrsquopreference distances

We use 119909119894119895to denote the matching relationship between

119883119894and 119884

119895 119909119894119895

isin 0 1 If 119909119894119895is 1 it refers to the fact that 119883

119894

and 119884119895match each other and make up a matching pair else if

119909119894119895is 0 it refers to the fact that 119883

119894and 119884

119895do not match each

other and also do notmake up amatching pairThe constraint

Mathematical Problems in Engineering 5

that any element in one set can onlymatch one element in theother set at most is presented as follows

0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

(5)

We denote the maximum value of 119889119894119895

by 119889max and theminimum value by 119889min and the specific value of 119889max and119889min depends on the range of parameter 119902 and the preferencesequences We also denote 120582 as the threshold factor ofmatching performance 120582 isin [0 1] When 120582 = 1 it refers to noconstraint on threshold when 120582 decreases it refers to the factthat the constraint degree on threshold increases and when120582 = 0 it refers to the fact that the constraint on threshold isthe strictest The constraint is presented as follows

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min) forall119894 isin 119868 forall119895 isin 119869 (6)

Since one element should not match an element which is notin the former onersquos preference sequence and meanwhile theformer element also would not like to accept that we shouldadd a new constraint if 119903

119894119895equals 119899+1 119909

119894119895should be zero and

if 119905119894119895equals 119898 + 1 119909

119894119895also should be zero As the maximum

value of 119903119894119895and 119905119894119895is 119899 + 1 and 119898 + 1 respectively and the

second maximum value of 119903119894119895and 119905119894119895is 119899 and 119898 respectively

this constraint can be presented as a linear expression asfollows

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

(7)

Maximizing the matching number and minimizing the totalsum of preference distances are presented as follows

maxsum

119894isin119868

sum

119895isin119869

119909119894119895

minsum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

(8)

As it is a multiobjective optimization we generally have threedifferent ways to deal with it the first one is to transfera multiobjective problem to one single-objective problemsuch as simple linear additive weight method maximal-minimal method and TOPSIS method the critical point ofthis solving way is to ensure that the optimal solution of thenew single-objective problem is also the noninferior solutionof the original multiobjective problem the second one is totransfer it to multiple single-objective problems in a specialorder such as hierarchical method interactive programmingmethod we get the optimal solution of the original problemthrough solving these single-objective problems one by onewhich is also the optimal solution of the last single-objectiveproblem the third way is some nonuniform methods suchas multiplication division method and efficiency coefficientmethod In this paper we hold that maximizing the numberof all matching pairs is the main optimal objective and min-imizing the total sum of preference distances are the second

optimal objective so we use the secondwaymentioned aboveto deal with it Namely we first get the optimal solution of themain objective then add it as a new constraint and finallyget the optimal solution of the second objective These twoobjectives are integrated as a linear equation as follows

max119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895 1198721 ≫ 1198722 (9)

where1198721and119872

2are both positive real numbers and ldquo119872

1≫

1198722rdquo refers to the fact that 119872

1is far greater than 119872

2 We can

also give a specific value to1198721and119872

2if we can measure the

importance of these two objectives exactly but it is not thecontent discussed in this paper In conclusion the decisionmodel constructed in this paper is presented as follows

max 119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

st 0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min)

forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

119909119894119895

= 0 or 1

119889119894119895

= [(119905119894119895minus 1)119902

+ (119903119894119895minus 1)119902

]1119902

120582 isin [0 1]

119902 isin [1 +infin]

1198721 isin 119877+ 1198722 isin 119877

+ 1198721 ≫ 1198722

(10)

In the typical maximum weight matching model as themaximum matching number is constant and obvious themodel is easy to convert to a standard assignment modelIn standard assignment model the number of elements ineach side is the same and every element in one set will bematched with one element in the other set through addingsome zero elements to balance the number of two sidesWhen the scale is not too large the Hungarian method isone common approach to solve this kind of model and itis on basis of the following two theorems (a) if all elementsin one column or row of efficiency matrix are plus or minusa number the optimal solutions of the origin matrix andnew matrix are the same (b) the maximum number ofindependent zero elements in efficiency matrix equals theminimum number of lines which cover all the zero elementsNow so many literatures have researched and promotedthis method and provide programming codes in differentprogramming languages or coding platforms such as CJAVA and MATLAB so we do not discuss it repetitively

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

that any element in one set can onlymatch one element in theother set at most is presented as follows

0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

(5)

We denote the maximum value of 119889119894119895

by 119889max and theminimum value by 119889min and the specific value of 119889max and119889min depends on the range of parameter 119902 and the preferencesequences We also denote 120582 as the threshold factor ofmatching performance 120582 isin [0 1] When 120582 = 1 it refers to noconstraint on threshold when 120582 decreases it refers to the factthat the constraint degree on threshold increases and when120582 = 0 it refers to the fact that the constraint on threshold isthe strictest The constraint is presented as follows

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min) forall119894 isin 119868 forall119895 isin 119869 (6)

Since one element should not match an element which is notin the former onersquos preference sequence and meanwhile theformer element also would not like to accept that we shouldadd a new constraint if 119903

119894119895equals 119899+1 119909

119894119895should be zero and

if 119905119894119895equals 119898 + 1 119909

119894119895also should be zero As the maximum

value of 119903119894119895and 119905119894119895is 119899 + 1 and 119898 + 1 respectively and the

second maximum value of 119903119894119895and 119905119894119895is 119899 and 119898 respectively

this constraint can be presented as a linear expression asfollows

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

(7)

Maximizing the matching number and minimizing the totalsum of preference distances are presented as follows

maxsum

119894isin119868

sum

119895isin119869

119909119894119895

minsum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

(8)

As it is a multiobjective optimization we generally have threedifferent ways to deal with it the first one is to transfera multiobjective problem to one single-objective problemsuch as simple linear additive weight method maximal-minimal method and TOPSIS method the critical point ofthis solving way is to ensure that the optimal solution of thenew single-objective problem is also the noninferior solutionof the original multiobjective problem the second one is totransfer it to multiple single-objective problems in a specialorder such as hierarchical method interactive programmingmethod we get the optimal solution of the original problemthrough solving these single-objective problems one by onewhich is also the optimal solution of the last single-objectiveproblem the third way is some nonuniform methods suchas multiplication division method and efficiency coefficientmethod In this paper we hold that maximizing the numberof all matching pairs is the main optimal objective and min-imizing the total sum of preference distances are the second

optimal objective so we use the secondwaymentioned aboveto deal with it Namely we first get the optimal solution of themain objective then add it as a new constraint and finallyget the optimal solution of the second objective These twoobjectives are integrated as a linear equation as follows

max119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895 1198721 ≫ 1198722 (9)

where1198721and119872

2are both positive real numbers and ldquo119872

1≫

1198722rdquo refers to the fact that 119872

1is far greater than 119872

2 We can

also give a specific value to1198721and119872

2if we can measure the

importance of these two objectives exactly but it is not thecontent discussed in this paper In conclusion the decisionmodel constructed in this paper is presented as follows

max 119910 = 1198721 sum

119894isin119868

sum

119895isin119869

119909119894119895minus1198722 sum

119894isin119868

sum

119895isin119869

119909119894119895119889119894119895

st 0 le sum

119894isin119868

119909119894119895

le 1 forall119895 isin 119869

0 le sum

119895isin119869

119909119894119895

le 1 forall119894 isin 119868

119909119894119895119889119894119895

le 119889min +120582 (119889max minus119889min)

forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119899+ 1minus 119903119894119895 forall119894 isin 119868 forall119895 isin 119869

0 le 119909119894119895

le 119898+ 1minus 119905119894119895 forall119894 isin 119868 forall119895 isin 119869

119909119894119895

= 0 or 1

119889119894119895

= [(119905119894119895minus 1)119902

+ (119903119894119895minus 1)119902

]1119902

120582 isin [0 1]

119902 isin [1 +infin]

1198721 isin 119877+ 1198722 isin 119877

+ 1198721 ≫ 1198722

(10)

In the typical maximum weight matching model as themaximum matching number is constant and obvious themodel is easy to convert to a standard assignment modelIn standard assignment model the number of elements ineach side is the same and every element in one set will bematched with one element in the other set through addingsome zero elements to balance the number of two sidesWhen the scale is not too large the Hungarian method isone common approach to solve this kind of model and itis on basis of the following two theorems (a) if all elementsin one column or row of efficiency matrix are plus or minusa number the optimal solutions of the origin matrix andnew matrix are the same (b) the maximum number ofindependent zero elements in efficiency matrix equals theminimum number of lines which cover all the zero elementsNow so many literatures have researched and promotedthis method and provide programming codes in differentprogramming languages or coding platforms such as CJAVA and MATLAB so we do not discuss it repetitively

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Regarding nonlinear optimization the common solvingapproach is heuristic algorithms such as genetic algorithmsimulated annealing algorithm and tabu search algorithmHowever if we first compute 119889

119894119895in terms of 119903

119894119895and 119905119894119895 then

input the 119889119894119895matrix and the value of 119889max 119889min and 120582

all the constraints and optimal objectives involving 119909119894119895are

linear expressions so the model we construct is a standard0-1 integer linear programming model which we can usebranch-and-bound method to solve even when the scale ofdata is very large Branch-and-boundmethod is an algorithmdesign paradigm for discrete optimization problems as wellas general real valued problems It is first proposed by Landand Doig in 1960 for discrete programming [23] and hasbecome the most commonly used tool for solving integerprogramming and NP-hard problem It has two proceduresbranching and bounding branching refers to dividing theorigin problem into some subproblems in which the unionset of all solutions covers all the feasible solutions in theorigin problem and bounding refers to computing an upperbound and a lower bound for optimal objective value Themain idea of branch-and-bound algorithm is to increaselower bound and decrease upper bound iteratively and getthe optimal value finally It also can be classified into somespecific types on basis of the different branching searchstrategies Modern linear programming software such asCPLEX solves integer programming model with branch-and-bound algorithm package In this paper we also use thisalgorithm to solve our model and analyze the solution in thefollowing section

6 Numerical Cases and Analysis

We first give a numerical case named Case 1 and each side ofit has 10 elements Set119883 119884 and their preference sequences 119875

and 119876 are shown in Table 1 The real ordinal value of 119884119895in 119875119894

the preference sequence of 119883119894 makes up a matrix labeled as

119874 = [119900119894119895]10times10 and the preference ordinal value alsomakes up

amatrix labeledwith119877 = [119903119894119895]10times10 Similarly the real ordinal

value of 119883119894in 119876119895 the preference sequence of 119884

119895 makes up a

matrix labeled as 119878 = [119904119894119895]10times10 and the preference ordinal

value also makes up a matrix labeled with 119879 = [119905119894119895]10times10

The four matrixes 119874 119877 119878 and 119879 are presented as follows

119874 =

[[[[[[[[[[[[[[[[[[[[[[

[

3 1 2

6 7 1 3 2 4 54 6 5 2 1 3

7 6 1 2 3 4 5 5 2 3 1 4

9 8 10 7 1 3 2 5 6 43 4 2 1

4 6 5 8 1 2 3 710 8 9 7 1 4 2 3 6 58 5 7 4 2 3 1 6

]]]]]]]]]]]]]]]]]]]]]]

]

119878 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 10 6 7 7 9 7 9 6 8 7

8 5 8 3 10

5 6 5 5 54 4 5 6 6 6 4 5 4 4 2 4 4 8

6 3 2 1 1 3 2 1 11 1 3 3 3 1 1 2 32 1 2 2 2 1 3 23 2 7 5 3 4 4

]]]]]]]]]]]]]]]]]]]]]]

]

119877 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 7 7 7 3 1 7 7 2 711 6 7 11 1 25 25 11 4 54 6 5 11 2 11 1 3 11 117 9 6 9 1 2 3 9 45 458 8 5 8 2 3 1 4 8 89 8 10 7 1 3 2 5 6 43 4 75 2 75 1 75 75 75 754 6 5 8 11 1 11 2 3 710 8 9 7 1 4 2 4 6 48 5 7 4 2 3 1 6 95 95

]]]]]]]]]]]]]]]]]]]]]]

]

119879 =

[[[[[[[[[[[[[[[[[[[[[[

[

7 11 10 75 6 7 7 7 9 795 11 9 75 11 7 6 8 7 98 5 8 75 11 3 11 95 10 95 11 6 75 11 7 11 5 5 54 11 4 75 5 7 11 6 6 695 4 5 4 4 2 4 4 8 96 3 2 1 1 7 3 2 1 11 1 2 3 3 1 1 95 2 325 11 2 2 2 7 2 1 35 225 2 7 75 11 7 5 3 35 4

]]]]]]]]]]]]]]]]]]]]]]

]

(11)

After computing matrixes 119874 119877 119878 and 119879 we can get matrix119889119894119895according to (3) when 119902 = 1 119902 = 2 or 119902 = +infin

We first set 120582 as 1 namely without threshold constraint theresults solved through CPLEX based on branch-and-boundalgorithm are as follows when 119902 = 1 the optimal solutionlabeled as Solution 1 is that 119909

16 11990927 11990931 119909410 11990953 11990965 11990974

11990989 11990998 and 119909

102 are 1 and the rest are 0 when 119902 = 2 theoptimal solution labeled as Solution 2 is that 119909

15 11990927 11990931

119909410 11990953 11990966 11990974 11990989 11990998 and 119909

102 are 1 and the rest are 0when 119902 = +infin the optimal solution labeled as Solution 3 isthat 119909

11 11990929 11990932 119909410

11990953 11990967 11990974 11990986 11990995 and 119909

108 are 1and the rest are 0 Now we compare these three solutions inthe following six different criteria

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 1

119883119894

119875 119884119894

119876

1198831

1198751= 1198846 1198849 1198845 119884

11198761= 1198838 (1198839 11988310

) 1198835 1198834 1198837 1198831 1198833

1198832

1198752

lowast= 1198845 (1198847 1198846) 1198849 11988410

1198842 1198843 119884

21198762

lowast= 1198838 11988310

1198837 1198836 1198833

1198833

1198753

lowast= 1198847 1198845 1198848 1198841 1198843 1198842 119884

31198763= (119883

9 1198837 1198838) 1198835 1198836 1198834 11988310

1198833 1198832 1198831

1198834

1198754= 1198845 1198846 1198847 (1198849 11988410

) 1198843 1198841 119884

41198764= 1198837 1198839 1198838 1198836

1198835

1198755= 1198847 1198845 1198846 1198848 1198843 119884

51198765

lowast= 1198837 1198839 1198838 1198836 1198835 1198831

1198836

1198756= 1198845 1198847 1198846 11988410

1198848 1198849 1198844 1198842 1198841 1198843 119884

61198766= 1198838 1198836 1198833

1198837

1198757= 1198846 1198844 1198841 1198842 119884

71198767

lowast= 1198838 1198839 1198837 1198836 11988310

1198832 1198831

1198838

1198758

lowast= 1198846 1198848 1198849 1198841 1198843 1198842 11988410

1198844 119884

81198768= 1198839 1198837 11988310

1198836 1198834 1198835 1198831 1198832

1198839

1198759= 1198845 1198847 (1198848 1198846 11988410

) 1198849 1198844 1198842 1198843 1198841 119884

91198769= 1198837 1198838 (1198839 11988310

) 1198834 1198835 1198832 1198836 1198831 1198833

11988310

11987510

= 1198847 1198845 1198846 1198844 1198842 1198848 1198843 1198841 119884

1011987610

= 1198837 1198839 1198838 11988310

1198834 1198835 1198831

Criterion 1 (concordance rate on the first item labeled as1198881) We define the number of matching pairs in which one

element matches its first preference item (including tiedranking and the same below) as 119873

1and the number of all

matching pairs as 119873 1198881is the ratio between 119873

1and 119873

Obviously 1198881has a positive correlation with matching effect

the higher the rate value the better the matching effect TakeSolution 1 as an example to set 119883 only 119883

1and 119883

6match

their first preference items so concordance rate of 119883 is 20to set 119884 only 119884

4and 119884

8match their first preference items

so concordance rate of 119884 is also 20 Hence the averageconcordance rate on the first item of Solution 1 is 20Similarly the average concordance rate on the first item ofSolutions 2 and 3 is 10 and 20 respectively

Criterion 2 (concordance rate on the first three items (includ-ing tied ranking and the same below) labeled as 119888

2)Wedefine

the number of matching pairs in which one element matchesone of its first three preference items as119873

3 and the number of

all matching pairs is still denoted by119873 1198882is the ratio between

1198733and 119873 119888

2also has a positive correlation with matching

effect the higher the rate value the better thematching effectIn Solution 1 to set 119883 119883

1 1198832 1198836 1198837 1198838 and 119883

9match

elements in their first three preference items so concordancerate of 119883 is 60 to set 119884 119884

2 1198844 1198848 and 119884

9match elements

in their first three preference items so concordance rate of 119884is 40 Hence the average concordance rate on the first threeitems of Solution 1 is 50 Similarly the average concordancerate on the first three items of Solutions 2 and 3 is 60 and40 respectively

Criterion 3 (concordance rate on all the items labeled as1198883) We define the number of matching pairs in which one

element matches the item in its preference sequence as 1198730

and the number of allmatching pairs is still denoted by119873 1198883is

the ratio between 1198730and 119873 119888

3also has a positive correlation

with matching effect the higher the rate value the better thematching effect In Solution 1 to set 119883 all elements in 119883

match elements in their preference sequence so concordancerate of 119883 is 100 to set 119884 all elements except 119884

6match

elements in their preference sequences so concordance rateof 119884 is 90 Hence the average concordance rate on thefirst three items of Solution 1 is 95 Similarly the average

concordance rate on all items of Solutions 2 and 3 is 100and 95 respectively

Criterions 4 and 5 (average preference ordinal value andthe standard deviation of it labeled as 119888

4and 1198885 resp) In

Solution 1 preference ordinal value of the elements in set 119883

to set 119884 is 1 25 4 45 5 1 2 3 4 and 5 so the averagepreference ordinal value and standard deviation are 32 and153 preference ordinal value of the elements in set 119884 to set119883 is 7 6 8 5 4 4 1 2 1 and 2 so the average preferenceordinal value and standard deviation are 4 and 249 take sets119883 and 119884 as a whole the average preference ordinal valueand standard deviation are 36 and 206 Similarly we can getthe average preference ordinal value and standard deviationin Solutions 2 and 3 Obviously 119888

4has a positive correlation

with matching effect while 1198885has a negative correlation with

matching effect

Criterion 6 (the total sum of disagreements of two sideslabeled as 119888

6) It equals the total sum of all the absolute

differences between two sidesrsquo preference ordinal valueswhich measures the difference of two sides So 119888

6is defined

as follows

1198886 = sum

119894isin119868

sum

119895isin119869

10038161003816100381610038161003816119905119894119895minus 119903119894119895

10038161003816100381610038161003816119909119894119895 forall119894 isin 119868 forall119895 isin 119869 (12)

Criterion 6 also has a negative correlation with matchingeffect the lower the total sum of disagreements the betterthe matching effect The values on Criterion 6 in Solutions1 2 and 3 are 26 21 and 125 respectively

The value on these six criteria in three solutions is shownin Table 2 and we can see that no solution performs betterthan the rest on all the criteria Solution 1 performs betteron Criterions 1 and 4 for it has a higher concordance rateon the first item and the least average preference distance iftaking 119883 and 119884 as a whole Solution 3 performs better onCriterions 1 and 6 for it has the same performance as Solution1 on Criterion 1 and the least total sum of disagreementsof two sides Solution 2 performs better on Criterions 23 and 5 for it has a higher concordance rate on the firstthree items and concordance rate on all the items and theleast standard deviation no matter whether it is on set 119883 orset 119884 or sets 119883 and 119884 together What is more Solution 2

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 Solutions and the value on six criteria based in Case 1

Solution 1 (q = 1) Solution 2 (q = 2) Solution 3 (q = +infin)

Solution (120582 = 1)1198831sim1198846

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198845

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198845

1198832sim1198847

1198833sim1198841

1198834sim11988410

1198835sim1198843

1198836sim1198846

1198837sim1198844

1198838sim1198849

1198839sim1198848

11988310

sim1198842

1198831sim1198841

1198832sim1198849

1198833sim1198842

1198834sim11988410

1198835sim1198843

1198836sim1198847

1198837sim1198844

1198838sim1198846

1198839sim1198845

11988310

sim1198848

Criterion 1 X 20 Y 20 Avg 20 X 0 Y 20 Avg 10 X 20 Y 20 Avg 20Criterion 2 X 60 Y 40 Avg 50 X 60 Y 60 Avg 60 X 40 Y 40 Avg 40Criterion 3 X 100 Y 90 Avg 95 X 100 Y 100 Avg 100 X 90 Y 100 Avg 95Criterion 4 X 320 Y 400 X and Y 360 X 360 Y 370 X and Y 365 X 385 Y 390 X and Y 388Criterion 5 X 153 Y 249 X and Y 206 X 105 Y 245 X and Y 184 X 221 Y 218 X and Y 214Criterion 6 26 21 125

Table 3 Matching number rate when 120582 and 119902 vary in Case 1

119902 120582 isin [60 1] 120582 = 50 120582 = 40 120582 = 30 120582 = 20 120582 = 10 120582 = 0q = 1 100 100 90 60 40 30 10q = 2 100 90 60 40 40 30 10q = +infin 100 70 40 40 30 10 10

is also the only one in which the elements matched areall in their preference sequences When 119902 varies from 12 and +infin the value on Criterion 4 increases while thevalue on Criterion 6 decreases namely the performance onCriterions 4 and 6 moves in an opposite direction howeverthe value on Criterion 5 goes down at first and then risesup Among Criterions 4sim6 each solution performs better onone criterion and worse on the other two criteria So whichkind of preference distancewill be used depends on the actualdecision background for example the decisionmaker shouldfirst balance the importance of each evaluation criterion andthen choose an appropriate solution in terms of its differentperformance on these criteria

When 120582 varies from 0 to 1 as the influence of thresholdconstraint matching number also will vary Matching num-ber rate is the ratio between the actual number of matchingpairs in the optimal solution and the maximum possiblenumber of matching pairs and the latter one equals thesmaller value between119898 and 119899 obviouslyThe rate is shown inTable 3 when 120582 and 119902 have a different value We can see thatthe range scope of 120582 which has effect on matching numberrate is wider in high normdistance than in lownormdistancethat is to say 120582 is more sensitive to matching number when119902 grows This has a guiding significance to set an appropriatevalue of 120582 in the actual decision situation the constraint onthreshold is stricter when 119902 grows so if we want to havethe same level of matching number rate in different normdistance we should set 120582 of a larger value with the increasingof 119902

We also take the numerical case in [22] as an exampleand name it as Case 2 for it compares its own approach andthe approach put forward in [14] The preference sequencesare complete but the number of elements in two sides isnot equal and the information of matching sets and theirpreference sequences are shown in Table 4 We label thesolving approaches when 119902 = 1 119902 = 2 and 119902 = +infin

as Approaches 1 2 and 3 and label the approaches in [22]

and [14] as Approaches 4 and 5 respectivelyWe use these fiveapproaches to solve two-sided matching problem in Table 4and get three solutions labeled as Solutions119860 119861 and119862 Eachsolution and its value on the six criteria mentioned above arejust presented in Table 5

When comparing these three solutions is there a solu-tion which performs better than the rest of the solutionsdefinitely Unfortunately we cannot find it when not givinga determinate evaluation system of decision making It isobvious that each approach makes the value in its ownoptimal objective better than other approaches for exampleApproach 1 is to minimize the total sum of one-normdistances which is equivalent to minimizing the total sumof preference ordinal values so Approach 1 has the minimumtotal sum of preference ordinal values compared with otherapproaches Approaches 2 and 3 are similar to Approach 1they perform better in the total sum of two-norm distancesor positive-infinity-norm distances Approach 4 has a bettersatisfactory degree which highly depends on transition func-tion put forward by itself Approach 5 guarantees that thesolution is a stable matching So we list some other criteriawhich are different from their original optimal objectives toevaluate the matching effect Three solutions have the sameperformance onCriterion 3 because the preference sequencesare complete Solutions119860 and 119861 perform better than Solution119862 on Criterion 2 but worse than that on Criterion 1 AndSolution 119860 has the least average preference ordinal valueand Solution 119861 has the least standard deviation of preferenceordinal value if taking sets 119883 and 119884 together MeanwhileSolution 119861 also has the least total sum of disagreements oftwo sides It is worth mentioning that when the number ofmatching pairs is a constant value Criterion 4 is linearlyequivalent with the total sum of one-norm distances Aftercomparing on these criteria we can find that Solutions119860 119861 and 119862 solved with five different approaches havetheir own advantages Before giving more specific evaluationinformation we cannot regard any of them to be better than

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 4 Matching sets 119883 and 119884 and preference sequences 119875 and 119876 in Case 2

119883119894

119875 119884119894

119876

1198831

1198751= 1198841 (1198843 1198844) 1198842 1198848 1198845 1198846 1198847 119884

11198761= (119883

2 1198836) 1198831 1198835 1198833 1198834

1198832

1198752= 1198841 (1198844 1198843) 1198842 1198847 1198846 1198848 1198845 119884

21198762= 1198831 1198833 1198832 1198835 1198836 1198834

1198833

1198753= 1198844 1198841 1198842 (1198843 1198845) 1198847 1198846 1198848 119884

31198763= 1198835 1198834 (1198832 1198836) 1198833 1198831

1198834

1198754= 1198843 1198841 1198844 1198845 1198846 1198842 1198847 1198848 119884

41198764= 1198833 1198832 1198834 1198831 1198835 1198836

1198835

1198755= 1198843 1198844 1198842 1198841 1198847 1198848 1198846 1198845 119884

51198765= 1198832 (1198833 1198836) 1198835 1198831 1198834

1198836

1198756= (119884

2 1198846) 1198841 1198844 1198845 1198847 1198848 1198843 119884

61198766= 1198831 1198832 1198835 1198833 1198834 1198836

1198847

1198767= 1198836 1198833 1198832 1198831 (1198834 1198835)

1198848

1198768= 1198833 (1198835 1198836) 1198831 1198834 1198832

Table 5 Solutions and the value on six criteria in Case 2

Solution A Solution B Solution C

Solution (120582 = 1) 1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198843

1198835sim1198848

1198836sim1198847

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198846

1198835sim1198843

1198836sim1198845

1198831sim1198842

1198832sim1198841

1198833sim1198844

1198834sim1198845

1198835sim1198843

1198836sim1198846

Approach Approach 1 Approaches 2 3 and 5 Approach 4Criterion 1 X 50 Y 50 Avg 50 X 50 Y 50 Avg 50 X 67 Y 67 Avg 67Criterion 2 X 67 Y 100 Avg 83 X 67 Y 83 Avg 75 X 67 Y 67 Avg 67Criterion 3 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100 X 100 Y 100 Avg 100Criterion 4 X 317 Y 150 X and Y 233 X 283 Y 200 X and Y 242 X 208 Y 275 X and Y 242Criterion 5 X 248 Y 063 X and Y 193 X 204 Y 158 X and Y 179 X 150 Y 252 X and Y 201Criterion 6 13 6 10

the rest We should still choose an appropriate approachaccording to the specific decision background though somedifferent approaches have the same one solution just likeSolution 119861

7 Conclusion

Two-sided matching based on preference sequences hasdifferent solutions under different optimal objectives We listsix criteria to evaluate the matching solutions concordancerate on the first item concordance rate on the first threeitems concordance rate on all the items average preferenceordinal value standard deviation of preference ordinal valueand total sum of disagreements of two sides All of thesesix criteria are determined by preference ordinal value ofthe elements in matching sets As no approach performsbetter than other approaches on all or most of the criteriawe think that no approach mentioned above applies for alltwo-sided matching decision situations In fact if we regardthese six criteria as the decision objectives Solutions 1sim3or Solutions 119860 sim 119862 are all noninferior solutions We havethree ways to deal with multiobjective optimal problem justmentioned in Section 4 As we have no further informationabout weight value of each objective Approaches 1sim5 are allconducted in the second way The model we construct holdsthat maximizing the matching number and minimizing thetotal sum of preference distances are the most important andappropriate objectives for two-sided matching problem onthe condition that uncertain preference sequences are theonly decision information source

Compared with other approaches the model we con-struct extends preference sequence from a complete andstrong to an incomplete andweak one and adds the constrainton threshold to guarantee matching effect of each matchingpair is not too bad Through adjusting 120582 from 0 to 1 wecan strengthen or slack the threshold constraint We alsoadopt different norms to represent different kinds of distanceand promote the applicability in the decision backgroundThrough adjusting 119902 from 1 2 and 119902 = +infin we can usethis universal decision model in different decision groundwhich is our advantage to other approaches Besides themodel we construct is succinct and explicit which is classifiedinto 0-1 integer programming problem in the field of linearoptimization and can be solved with branch-and-boundalgorithm We think it is a better method to solve two-sided problemswith uncertain preference sequences in actualdecision situation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable and insightful comments for further improvingthe quality of this work The authors also would like to thankDr Wu Xiang for his great and sincere help This researchwas financially supported by Key Projects in the National

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Science amp Technology Pillar Program during the TwelfthFive-Year Plan Period of China (2012BAJ05B06)

References

[1] M Janssen and A Verbraeck ldquoComparing the strengths andweaknesses of Internet-based matching mechanisms for thetransport marketrdquo Transportation Research Part E Logistics andTransportation Review vol 44 no 3 pp 475ndash490 2008

[2] M Sorensen ldquoHow smart is smart money A two-sided match-ing model of venture capitalrdquo Journal of Finance vol 62 no 6pp 2725ndash2762 2007

[3] H-T Lin ldquoA job placement intervention using fuzzy approachfor two-way choicerdquo Expert Systems with Applications vol 36no 2 pp 2543ndash2553 2009

[4] D K Huang H N Chiu R H Yeh and J H ChangldquoA fuzzy multi-criteria decision making approach for solvinga bi-objective personnel assignment problemrdquo Computers ampIndustrial Engineering vol 56 no 1 pp 1ndash10 2009

[5] C Haas S O Kimbrough S Caton and C WeinhardtldquoPreference-based resource allocation using heuristics to solvetwo-sided matching problems with indifferencesrdquo in Economicsof Grids Clouds Systems and Services vol 8193 of Lecture Notesin Computer Science pp 149ndash160 Springer Berlin Germany2013

[6] A Abdulkadiroglu P A Pathak and A E Roth ldquoTheNewYorkcity high school matchrdquoAmerican Economic Review vol 95 no2 pp 364ndash367 2005

[7] D Gale and L S Shapley ldquoCollege admissions and the stabilityof marriagerdquo The American Mathematical Monthly vol 69 no1 pp 9ndash15 1962

[8] A E Roth ldquoCommon and conflicting interests in two-sidedmatching marketsrdquo European Economic Review vol 27 no 1pp 75ndash96 1985

[9] J Edmonds ldquoMaximum matching and a polyhedron with01-verticesrdquo Journal of Research of the National Bureau ofStandards vol 69 no 1-2 pp 125ndash130 1965

[10] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

[11] J Munkres ldquoAlgorithms for the assignment and transportationproblemsrdquo Journal of the Society for Industrial and AppliedMathematics vol 5 no 1 pp 32ndash38 1957

[12] A E Roth ldquoOn the allocation of residents to rural hospitalsa general property of two-sided matching marketsrdquo Economet-rica vol 54 no 2 pp 425ndash427 1986

[13] V Knoblauch ldquoMarriage matching and gender satisfactionrdquoSocial Choice and Welfare vol 32 no 1 pp 15ndash27 2009

[14] D G McVitie and L B Wilson ldquoThe stable marriage problemrdquoCommunications of the Association for Computing Machineryvol 14 no 7 pp 486ndash490 1971

[15] C-P Teo J Sethuraman and W-P Tan ldquoGale-Shapley stablemarriage problem revisited strategic issues and applicationsrdquoManagement Science vol 47 no 9 pp 1252ndash1267 2001

[16] C van Raalte and H Webers ldquoSpatial competition withintermediated matchingrdquo Journal of Economic Behavior andOrganization vol 34 no 3 pp 477ndash488 1998

[17] D Sarne and S Kraus ldquoManaging parallel inquiries in agentsrsquotwo-sided searchrdquo Artificial Intelligence vol 172 no 4-5 pp541ndash569 2008

[18] I Korkmaz H Gokcen and T Cetinyokus ldquoAn analytichierarchy process and two-sided matching based decisionsupport system formilitary personnel assignmentrdquo InformationSciences vol 178 no 14 pp 2915ndash2927 2008

[19] K Iwama D Manlove S Miyazaki and Y Morita ldquoStablemarriage with incomplete lists and tiesrdquo in Automata Lan-guages and Programming vol 1644 of Lecture Notes in ComputerScience pp 443ndash452 Springer Berlin Germany 1999

[20] D F Manlove R W Irving K Iwama S Miyazaki and YMorita ldquoHard variants of stable marriagerdquo Theoretical Com-puter Science vol 276 no 1-2 pp 261ndash279 2002

[21] M Halldorsson K Iwama S Miyazaki and Y Morita ldquoInap-proximability results on stable marriage problemsrdquo in LATIN2002 Theoretical Informatics vol 2286 of Lecture Notes inComputer Science pp 554ndash568 Springer Berlin Germany2002

[22] M Li Z Fan and Y Liu ldquoA method for two-sided matchingbased on preference ordinal informationrdquo Operations Researchand Management Science vol 21 no 4 pp 112ndash118 2012(Chinese)

[23] A H Land and A G Doig ldquoAn automatic method of solvingdiscrete programming problemsrdquo Econometrica vol 28 no 3pp 497ndash520 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of