Research Article Entropy-Based Weighting for …downloads.hindawi.com/journals/mpe/2015/608325.pdfResearch Article Entropy-Based Weighting for Multiobjective Optimization: An Application

Research ArticleEntropy-Based Weighting for Multiobjective OptimizationAn Application on Vertical Turning

Luiz Ceacutelio Souza Rocha1 Anderson Paulo de Paiva1 Pedro Paulo Balestrassi1

Geremias Severino2 and Paulo Rotela Junior1

1 Institute of Production Engineering andManagement Federal University of Itajuba BPS Avenue 1303 37500-903 ItajubaMG Brazil2Mahle Metal Leve SA Itajuba MG Brazil

Correspondence should be addressed to Paulo Rotela Junior paulorotelagmailcom

Received 15 May 2015 Revised 13 July 2015 Accepted 22 July 2015

Academic Editor Giovanni Falsone

Copyright copy 2015 Luiz Celio Souza Rocha et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In practical situations solving a given problem usually calls for the systematic and simultaneous analysis of more than one objectivefunction Hence a worthwhile research question may be posed thus In multiobjective optimization what can facilitate for thedecision maker choosing the best weighting Thus this study attempts to propose a method that can identify the optimal weightsinvolved in a multiobjective formulationThe proposedmethod uses functions of Entropy and Global Percentage Error as selectioncriteria of optimal weights To demonstrate its applicability this method was employed to optimize the machining process forvertical turning maximizing the productivity and the life of cutting tool and minimizing the cost using as the decision variablesfeed rate and rotation of the cutting toolThe proposed optimization goals were achieved with feed rate = 037mmrev and rotation= 250 rpm Thus the main contributions of this study are the proposal of a structured method differentiated in relation to thetechniques found in the literature of identifying optimal weights for multiobjective problems and the possibility of viewing theoptimal result on the Pareto frontier of the problem This viewing possibility is very relevant information for the more efficientmanagement of processes

1 Introduction

As companies today face fierce and constant competitionmanagers are under increasing pressure to reduce their costsand improve the quality standards of products and processesAs part of their response to this demand managers havepursued rigorous methods of decision making includingoptimization methods [1]

In many practical fields such as engineering designscientific computing social economy and network commu-nication there exist a large number of complex optimizationproblems [2] and because of this optimization techniques inrecent years have evolved greatly finding wide application invarious types of industriesThey are now capable thanks to anew generation of powerful computers of solving ever largerand more complex problems

According to [1] optimization is the act in any givencircumstance of obtaining the best result In this context

the main purpose of decision making in industrial processesis to minimize the effort required to develop a specific taskor to maximize the desired benefit The effort required or thebenefit desired in any practical situation can be expressed asa function of certain decision variables This being the caseoptimization can be defined as the process of finding theconditions that give the maximum or minimum value of afunction [1]This function is known as the objective function

In practical situations however solving a given problemusually calls for the systematic and simultaneous analysis ofmore than one objective function giving rise tomultiobjectiveoptimization [3]

In multiobjective problems it is very unlikely that allfunctions are minimized simultaneously by one optimalsolution119909lowast Indeed themultiple objectives (119898) have conflictsof interest [1] What becomes of great relevance to thesetypes of problems according to [1] is the concept of aPareto-optimal solution also called a compromise solution

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 608325 11 pageshttpdxdoiorg1011552015608325

2 Mathematical Problems in Engineering

The author in [1] refers to a feasible solution 119909lowast as Pareto-

optimal if no other feasible solution 119911 exists such that 119891119894(119911) le

119891119894(119909lowast

) 119894 = 1 2 119898 with 119891119895(119911) lt 119891

119894(119909lowast

) in at leastone objective 119895 In other words a vector 119909lowast is said to bePareto-optimal if no other solution 119911 can be found thatcauses a reduction in the objective function without causinga simultaneous increase in at least one of the other objectivesPareto-optimal solutions occur because of the conflictingnature of the objectives where the value of any objectivefunction cannot be improved without impairing at least oneof the others In this context a trade-off represents giving upone objective to improve another [4]

The purpose ofmultiobjective optimizationmethods is tooffer support and ways to find the best compromise solutionPlaying important roles in this are a decision maker andhis preference information [4] A decision maker accordingto [4] is an expert in the domain of the problem underconsideration and who typically is responsible for the finalsolution In order to define the relative importance of eachobjective function the decision maker must assign themdifferent weights

Because a characteristic property of multiobjective opti-mization is the objective function weighting problem how adecision maker is involved with the solution of this problemis the basis for its classification According to [5 6] the classesare as follows (1) no-preference methods methods where noarticulation of preference information is made (2) a priorimethods methods where a priori articulation of preferenceinformation is used that is the decision maker selects theweighting before running the optimization algorithm (3)interactive methods methods where progressive articulationof preference information is used that is the decision makerinteracts with the optimization programduring the optimiza-tion process and (4) a posteriori methods methods where aposteriori articulation of preference information is used thatis no weighting is specified by the user before or during theoptimization process However as no classification can becomplete these classifications are not absolute Overlappingand combinations of classes are possible and some methodscan be considered to belong to more than one class [5] Thispaper considers the a posteriori method in consonance withgenerate-first-choose later approach [7]

A multiobjective problem is generally solved by reducingit to a scalar optimization problem and hence the termscalarization Scalarization is the converting of the problemby aggregation of the components of the objective functionsinto a single or a family of single objective optimizationproblems with a real-valued objective function [5] Theliterature reports different scalarization methods The mostcommon is the weighted sum method

Theweighted summethod is widely employed to generatethe trade-off solutions for nonlinearmultiobjective optimiza-tion problems According to [9] a biobjective problem isconvex if the feasible set119883 is convex and the functions are alsoconvex When at least one objective function is not convexthe biobjective problem becomes nonconvex generatinga nonconvex and even unconnected Pareto frontier Theprincipal consequence of a nonconvex Pareto frontier is that

points on the concave parts of the trade-off surface will notbe estimated [10] This instability is due to the fact that theweighted sum is not a Lipshitzian function of the weight 119908[11] Another drawback to the weighted sums is related tothe uniform spread of Pareto-optimal solutions Even if auniform spread of weight vectors is used the Pareto frontierwill be neither equispaced nor evenly distributed [10 11]

Given its drawbacks the weighted sum method is notused in this paper Instead the paper employs a weightedmetric method as cited by [12] in association with a normalboundary intersection method (NBI) as proposed by [13]Reference [13] proposed the NBI method to overcome thedisadvantages of the weighted sum method showing thatthe Pareto surface is evenly distributed independent of therelative scales of the objective functions It is for this featurethat this study uses the NBI method to build the Paretofrontier

The weighting issue has been discussed in literature forat least thirty years Reference [14] proposed an approachto helping to determine proper parameters in linear controlsystem design In the case that the desired specificationsare not given explicitly this approach applies an interactiveoptimizationmethod to search themost suitable weightsTheshortcoming of this proposal is that the decisionmaker needsto make pairwise comparisons of the response curves in eachiteration which becomes a problem when many iterationsare needed Reference [15] determined an optimum locationfor an undesirable facility in a workroom environment Theauthor defined the problem as the selection of a locationwithin the convex region that maximizes the minimumweighted Euclidean distance with respect to all existing facil-ities where the degree of undesirability between an existingfacility and the new undesirable entity is reflected througha weighting factor Reference [16] presented a multicrite-ria decision making approach named Analytic HierarchyProcess (AHP) in which selected factors are arranged ina hierarchic structure descending from an overall goal tocriteria subcriteria and alternatives in successive levelsDespite its popularity this method has been criticized bydecision analysts Some authors have pointed out that Saatyrsquosprocedure does not optimize any performance criterion [17]

In the course of time other methods for deriving thepriority weights have been proposed in literature such asgeometric mean procedure [18 19] methods based on con-strained optimization models [20] trial and error methods[21] methods using grey decision [22ndash24] methods usingfuzzy logic [3 18 19 25 26] and methods using simulatedannealing [26 27]

Recently [28] dealing with multiobjective optimiza-tion of time-cost-quality trade-off problems in construc-tion projects used Shannonrsquos entropy to define the weightsinvolved in the optimization process According to theauthors Shannonrsquos entropy can provide a more reliableassessment of the relative weights for the objectives in theabsence of the decision makerrsquos preferences

In the multiobjective optimization process the decisionmaker as noted above plays an important role for it is thedecision maker that sooner or later obtains a single solutionto be used as the solution to his original multidisciplinary

Mathematical Problems in Engineering 3

decision-making problem Hence a worthwhile researchquestion may be posed thus In multiobjective optimizationwhat can facilitate for the decision maker choosing the bestweighting

In answering such a query it was proposed to usetwo objectively defined selection criteria Shannonrsquos EntropyIndex [29] and Global Percentage Error (GPE) [30] Entropycan be defined as a measure of probabilistic uncertainty Itsuse is indicated in situations where the probability distri-butions are unknown in search of diversification Amongthe many other desirable properties of Shannonrsquos EntropyIndex the following were highlighted (1) Shannonrsquos measureis nonnegative and (2) its measure is concave Property1 is desirable because the Entropy Index ensures nonnullsolutions Property 2 is desirable because it is much easier tomaximize a concave function than a nonconcave one [31]TheGPE as its name declares is an error index In this case theaim was to evaluate the distance of the determined Pareto-optimal solution from its ideal value

Thus this study attempts to propose a method that canidentify the optimal weights involved in a multiobjectiveformulation The proposed method uses both a NormalBoundary Intersection (NBI) approach along with MixtureDesign of Experiments and as selection criteria of optimalweights has the functions of Entropy and Global PercentageError (GPE)

This paper is organized as follows Section 2 presentsthe theoretical background including a discussion of Designof Experiments the NBI approach and optimization algo-rithm GRG Section 3 presents the Metamodeling showingthe proposalrsquos step-by-step procedure Section 4 presents anumerical application to illustrate the adequacy of the workrsquosproposal Finally conclusions are offered in Section 5

2 Theoretical Background

21 Design of Experiments According to [32] an experimentcan be defined as a test or a series of tests in which purposefulchanges are made to the input variables of a process aimingthereby to observe how such changes affect the responsesThegoal of the experimenter is to determine the optimal settingsfor the design variables that minimize or maximize the fittedresponse [33] Design of Experiments (DOE) is then definedas the process of planning experiments so that appropriatedata is collected and then analyzed by statistical methodsleading to valid and objective conclusions

According to [32] the three basic principles of DOE arerandomization replication and blocking Randomization isthe implementation of experiments in a random order suchthat the unknown effects of the phenomena are distributedamong the factors thereby increasing the validity of theresearch Replication is the repetition of the same test severaltimes creating a variation in the response that is used toevaluate experimental error The blocking should be usedwhen it is not possible to maintain the homogeneity of theexperimental conditionsThis technique allows us to evaluatewhether the lack of homogeneity affects the results

The steps of DOE are [32] recognition and problemstatement choice of factors levels and variations selection of

the response variable choice of experimental design execu-tion of the experiment statistical analysis of data conclusionsand recommendations

Regarding the experimental projects the most widelyused techniques include the full factorial design the frac-tional factorial design the arrangements of Taguchi responsesurface methodology and mixture Design of Experiments[32]

For the modeling of the response surface functions themost used experimental arrangement for data collectionis the Central Composite Design (CCD) The CCD for119896 factors is a matrix formed by three distinct groups ofexperimental elements a full factorial 2119896 or fractional 2119896minus119901where 119901 is the desired fraction of the experiment a set ofcentral points (cp) and in addition a group of extreme levelscalled axial points given by 2119896 The number of experimentsrequired is given by the sum 2119896 or (119896minus119901) + cp+ 2119896 In CCD theaxial points are within a distance120572 of the central points being120572 = (2119896)14 [32]

In mixture Design of Experiments the factors are theingredients or components of a mixture and consequentlytheir levels are not independent For example if 119909

1 1199092 119909

119901

indicate the proportions of 119901 components of a mixturethen sum

119901

119894=1 119908119894 = 1 [32] The most used arrangement toplan and conduct the mixture experiments is the simplexarrangements [34]

A disadvantage with the simplex arrangements concernsthe fact that most experiments occur at the borders ofthe array This results in few points of the internal partbeing tested Thus it is recommended whenever possibleto increase the number of experiments by adding internalpoints to the arrangements as the central points and also theaxial points In the case of arrangements of mixtures it isnoteworthy that the central points correspond to the centroiditself

22 Normal Boundary Intersection Approach The normalboundary intersection method (NBI) is an optimizationroutine developed to find Pareto-optimal solutions evenlydistributed for a nonlinear multiobjective problem [13 35]The first step in the NBImethod comprises the establishmentof the payoff matrix Φ based on the calculation of the indi-vidualminimumof each objective functionThe solution thatminimizes the 119894th objective function119891

119894(119909) can be represented

as 119891lowast119894(119909lowast

119894) When it replaces the optimal individual 119909lowast

119894in the

remaining objective functions it becomes119891119894(119909lowast

119894) In matrix

notation the payoff matrixΦ can be written as [36]

Φ =

[

[

[

[

[

[

[

[

[

[

[

[

119891lowast

1 (119909lowast

1 ) sdot sdot sdot 1198911 (119909lowast

119894) sdot sdot sdot 1198911 (119909

lowast

119898)

d

119891119894(119909lowast

1 ) sdot sdot sdot 119891lowast

119894(119909lowast

119894) sdot sdot sdot 119891

119894(119909lowast

119898)

d

119891119898(119909lowast

1 ) sdot sdot sdot 119891119898(119909lowast


lowast

119898(119909lowast

119898)

]

]

]

]

]

]

]

]

]

]

]

]

(1)


The values of each row of the payoff matrixΦ which consistsof minimum and maximum values of the 119894th objective func-tion 119891

119894(119909) can be used to normalize the objective functions

generating the normalized matrixΦ such as [36]

119891 (119909) =

119891119894(119909) minus 119891

119880

119894

119891119873

119894minus 119891119880

119894

119894 = 1 119898 (2)

This procedure is mainly used when the objective functionsare written in terms of different scales or units Accordingto [11] the convex combinations of each row of the payoffmatrix form the convex hull of individualminimum (CHIM)The anchor point corresponds to the solution of singleoptimization problem119891

lowast

119894(119909lowast

119894) [37 38]The two anchor points

are connected by the Utopia line [38]The intersection point between the normal and the near-

est boundary of the feasible region from origin correspondsto the maximization of distance between the Utopia line andthe Pareto frontier Then the optimization problem can bewritten as [36]

Max(x119863)

119863

st Φ119908+119863119899 = 119865 (x)

x isin Ω

(3)

This optimization problem can be solved iteratively fordifferent values of 119908 creating a Pareto frontier uniformlydistributed A common choice for119908was suggested by [13 37]as 119908119899= 1 minus sum119899minus1

119894=1 119908119894The conceptual parameter 119863 can be algebraically elimi-

nated from (3) For bidimensional problems for example thisexpression can be simplified as [36]

Min 1198911 (x)

st 1198911 (x) minus 1198912 (x) + 2119908minus 1 = 0

119892119895(x) ge 0

0 le 119908 le 1

(4)

23 Optimization Algorithm Generalized Reduced Gradi-ent The generalized reduced gradient (GRG) algorithmaccording to [39] is one of the gradients methods thatpresents greater robustness and efficiency which makes itsuitable for solving a wide variety of problems Moreover[40] highlighted the easy access to this algorithm besidesbeing applicable to many nonlinear optimization problemsconstrained or unconstrained it is usually available in com-mercial software as Microsoft Excel

GRG is known as a primal method often called thefeasible direction method According to [41] it has threesignificant advantages if the search process ends before con-firmation of the optimum the last point found is feasible dueto the fact that each point generated is viable and probablyclose to the optimum if the method generates a convergentsequence the limit point reaches at least a local minimum

most of the primal methods are generally absolute notdepending on a particular structure such as the convexityThe GRG algorithm also has as one of its characteristics thefact that it provides adequate global convergence especiallywhen initialized close enough to the solution [42]

The search for the optimal point ends when the mag-nitude of the reduced gradient reaches the desired value oferror (convergence criterion) Otherwise a new search isperformed to find a new point in the direction of the reducedgradient This procedure is repeated until the best feasiblesolution is found

Once the problem has been modeled it can be applied tothe previously proposed optimization system

3 Metamodeling

Many of the techniques used in these strategies rely at least inone of its stages on imprecise and subjective elementsHencethe analysis of weighting methods for multiple responsesdemonstrates that since a large portion of strategies still useelements liable to error significant contributions can still bemade

The effort to contribute to this topic consists of developingan alternative for the identification of optimal weights inproblems of multiobjective optimization Statistical meth-ods based on DOE are important techniques to modelobjective functions Indeed for most industrial processesthe mathematical relationships are unknown The insertionof optimization algorithms takes place during the step ofidentifying optimal solutions for the responses and for theweights after they have been modeled by the statisticaltechniques mentioned above The GRG algorithm is used bythe Excel Solver function The NBI approach is also used inthe search for optimal weights using as selection criteria thefunctions Entropy and Global Percentage Error (GPE)

As shown in Figure 1 to reach the weighting methodol-ogy proposed in this work the following procedures are used

Step 1 (experimental design) It is the establishment of theexperimental design and execution of experiments in randomorder

Step 2 (modeling the objective functions) It is the definitionof equations using the experimental data

Step 3 (formulation of the problem of multiobjective opti-mization) It is the aggregation of the objective functionsinto a multiobjective formulation using a weighted metricmethod

Step 4 (definition of mixtures arrangement) In order to setthe weights to be used in the optimization routine describedin Step 3 a mixtures arrangement is done using Minitab 16

Step 5 (solution of the optimization problem) The opti-mization problem of Step 3 is solved for each experimentalcondition defined in Step 4

Step 6 (calculation of Global Percentage Error (GPE) andEntropy) GPE of Pareto-optimal responses is calculated


Start

(i) Experimental design

(ii) Modeling in theobjective functions

(iii) Formulation of theproblem of mutiobjective

optimization

(iv) Definition of mixturesarrangement

(v) Solution of theoptimization problem

End

Arrangementend

(vii) Modeling of GPE andEntropy

(ix) Defining the optimalweights

(vi) Calculation of GlobalPercentage Error (GPE)

and Entropy

(viii) Formulation of theproblem involving GPE

and Entropy

Yes

No

Figure 1 Research process flowchart

defining how far the point analyzed is from the objectivefunctionrsquos ideal value namely the target

Step 7 (modeling of GPE and Entropy) The canonical poly-nomial mixtures for GPE and Entropy are determined usingas data the results of the calculations from Step 6

Step 8 (formulation of the problem of multiobjective opti-mization involving GPE and Entropy functions) Once theGPE and the Entropy functions have been defined they areaggregated into a formulation ofmultiobjective optimizationusing the NBI methodologyThrough the use of this methodit is possible to define a Pareto frontier with evenly distributedsolutions regardless of the convexity of the functions

Step 9 (defining the optimal weights) To achieve the optimalweights the relation between Entropy and GPE was maxi-mized These parameters are in this proposal the selectioncriteria for optimal weights

Once this procedure has been performed and the optimalweights have been achieved the multiobjective optimizationshould be performed until it reaches the optimal values fordecision variables in the original problem

4 Implementation of the Proposed Method

In order to apply the method proposed in this work theexperimental data presented in [8] was used The authorsaimed to optimize with the application of DOE a processof vertical turning to determine the condition that led to amaximum life of the cutting tool (mm) high productivity(partshour) and minimum cost (US$part)

x2

(0 120572)

(minus1 +1)

(minus120572 0)

(+1 +1)

(0 0) (120572 0)

x1

(0 minus120572)

(minus1 minus1) (+1 minus1)

Figure 2 Central Composite Design for 119896 = 2

For the modeling of the response surface functions theauthors used the CCD As previously mentioned the CCDfor 119896 factors is a matrix formed by three distinct groups ofexperimental elements a full factorial 2119896 or fractional 2119896minus119901where 119901 is the desired fraction of the experiment a set ofcentral points (cp) and in addition a group of extreme levelscalled axial points given by 2119896 The number of experimentsrequired is given by the sum 2119896 or (119896minus119901) + cp + 2119896 In CCDthe axial points are within a distance 120572 of the central pointsbeing 120572 = (2119896)14 Using as the decision variables feedrate (mmrev) and rotation (rpm) of the cutting tool a fullfactorial design 22 was performed with 4 axial points and 5center points as suggested by [43] generating 13 experiments(Table 1) The graphical representation is shown in Figure 2extracted from [32]

According to [32] in running a two-level factorial exper-iment it is usual to anticipate fitting the first-order modelbut it is important to be alert to the possibility that thesecond-order model is really more appropriate Because ofthis a method of replicating center points to the 2119896 designwill provide protection against curvature from second-ordereffects and allow an independent estimate of error to beobtained Besides one important reason for adding thereplicate runs at the design center is that center points do notaffect the usual effect estimates in a 2119896 design

The decision variables were analyzed in a coded way inorder to reduce the variance Only at the end of the analyseswere they converted to their uncoded valuesThe parametersused in the experiments and their levels are shown in Table 2

The analysis of experimental data shown in Table 1 gen-erated the mathematical modeling presented in Table 3 Anexcellent fit can be observed once adjusted 1198772 is greater than90 for all responses

Based on the data presented in Table 3 applying theweighting method for multiobjective optimization proposed


Table 1 CCD for life of the cutting tool productivity and cost [8]

119873

Feed Rotation Life of cutting tool Productivity Cost(mmrev) (rpm) (mm) (parth) (US$part)

1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684

Table 2 Parameters used in the experiments [8]

Factors Levelsminus141 minus1 0 1 141

Feed (mmrev) 031 032 035 038 039Rotation (rpm) 227 235 255 275 283

Table 3 Mathematical models for the objective functions [8]

Terms Life of cutting tool Productivity Cost(mm) (parth) (US$part)

Constant 3003 1713 003626Feed 83 90 minus000293Rotation minus366 75 000054Feed lowast Feed minus638 minus15 000476Rotation lowast Rotation minus300 minus8 000200Feed lowast Rotation minus263 minus5 000236MSE 129119875 value 0003 0000 0000Regression (fullquadratic) 0000 0000 0000

Lack of fit 0926 0136 0279Adjusted 1198772 () 9110 9990 9430Normality of residuals 0600 0511 0552

in this paper was started It is important to mention thatTables 1 and 3 are equivalent to Steps 1 and 2 respectivelyas described in this work

To implement the optimization routine described inStep 3 the payoffmatrix was estimated initially obtaining theresults reported in Table 4

Table 4 Payoff matrix for the objective functions

Life of cutting tool Productivity Cost(mm) (parth) (US$part)3140lowast 1675 0036121524 1850lowast 0042983067 1718 003558lowastlowastOptimum individual

Once the objective functions have been defined they areaggregated into a formulation ofmultiobjective optimizationby a weighted metric method thus [12]

Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)

where 119891(x) is the global objective function 119891lowast119894(x) is the

ideal point or the best result individually possible and thevalues 119891Min

119894and 119891Max

119894are obtained in the payoff matrix The

expression xTx le 1205882 describes the constraint to a region of

spherical solution where 120588 is the radius of the sphereOnce Step 3 was implemented an arrangement of mix-

tures for the weights of each objective function (Step 4) wasdefined Due to the constraint sum119899

119894=1 119908119894 = 1 the use of themixtures arrangement is feasible

Subsequently the solution of the optimization problem ofStep 3 was obtained for each experimental condition definedby the arrangement of mixtures (Step 5)

Based on these results the GPE and the Entropy (Step 6)were calculated and the results are shown in Table 5TheGPEis calculated as shown by [30] through expression

GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)


Table 5 Arrangement of mixtures and calculation of GPE and Entropy

Weights Life of cutting tool Productivity Cost GPE Entropy1199081 1199082 1199083

1 0 0 3140 1675 003612 010974 minus000002075 025 0 2825 1762 003621 016567 056233075 0 025 3122 1697 003572 009254 05623305 05 0 2619 1784 003708 024373 06931405 025 025 2790 1766 003634 017829 10397205 0 05 3115 1700 003567 009139 069314025 075 0 2346 1806 003840 035593 056233025 05 025 2578 1788 003727 026009 103972025 025 05 2751 1771 003650 019253 103972025 0 075 3108 1704 003564 009089 0562330 1 0 1524 1850 004298 072239 minus0000020 075 025 2312 1809 003857 037034 0562330 05 05 2538 1791 003745 027609 0693140 025 075 2711 1775 003667 020782 0562330 0 1 3067 1718 003558 009440 minus0000020333 0333 0333 2707 1776 003669 020946 1098610667 0167 0167 2876 1755 003602 014787 0867560167 0667 0167 2426 1800 003800 032233 0867560167 0167 0667 2801 1765 003630 017413 086756

where 119910lowast119894is the value of the Pareto-optimal responses 119879

119894is

the targets defined119898 is the number of objectivesIn order to diversify the weights of multiobjective opti-

mization Shannonrsquos Entropy Index [29] is calculated usingthe Pareto-optimal responses through the expression

119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)

Figure 3 shows the Pareto frontier obtained It was observedthat life of cutting tool has negative correlation of minus0917significant at 1 with the productivity parameterThis occursbecause to obtain higher productivity it is necessary toincrease the cutting speed in the process exposing the toolto increased wear In a similar way the life of the cutting toolhas a negative correlation of minus0968 significant at 1 withthe cost per part This occurs because the largest portion ofprocess cost is due to the cost of the cutting tool In order tomaximize the life of cutting tool the cost per part is reducedInterestingly the productivity parameter and the cost perpiece have positive correlation of 0787 significant at 1 Theexplanation for this behavior is also based on the fact that thecutting tool is responsible for most of process cost In orderto achieve greater productivity it is necessary to increase thecutting speed compromising tool life Thus the increase inproductivity in the analyzed case is not enough to offset theincreased costs generated by the increased wear of the cuttingtool

The weighted metric method [12] used in Step 5 wasunable to yield an even distribution of the Pareto-optimal

2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)

Figure 3 Pareto frontier obtained with the weighted metricmethod

points along the frontier This drawback will be overcomewith the use of the NBI method [13]

From the calculation of the GPE and Entropy the math-ematical functions were modeled (Step 7)

GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)


Table 6 Payoff matrix for the functions GPE and Entropy

GPE Entropy0090lowast 06430213 1112lowastlowastOptimum individual

+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2

Entropy = minus 0 0021199081 minus 0 0021199082 minus 0 0021199083

+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)

To implement the routine of multiobjective optimizationdescribed in Step 8 the payoff matrix was estimated initiallyThe results are shown in Table 6

Based on the payoff matrix it was possible to iterativelyimplement (4) choosing 119908 in the range [0 1] Using thisequation and the parameters from [13] 21 points wereachieved and the Pareto frontier was built for the GPE andEntropy functions Figure 4 shows the Pareto frontier builtusing the NBI methodology for the GPE and Entropy func-tions with the optimum point achieved with the weightingmethodology proposed being highlighted

Lastly Step 9 was executed To achieve the optimalweights the following routine was considered

Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)

By the maximization of 120585 described in (9) the optimalweights119908

11199082 and119908

3were foundThe values are as follows

1199081(weight of life of cutting tool) = 048766 119908

2(weight of

productivity) = 003578 and 1199083(weight of cost) = 047656

These optimal weights were used in a multiobjectiveoptimization of life of the cutting tool productivity andcost as (5) reaching the values of 3001 1735 and 003566respectively The optimal values of the decision variables areas follows feed rate = 037mmrev and rotation = 250 rpm

111009080706

022

020

018

016

014

012

010

Entropy

GPE

Figure 4 Pareto frontier by NBI method

2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)

Figure 5 Pareto frontier obtained with the proposal

For each point in the Pareto frontier for the GPE andEntropy functions there is a set of values for the weights1199081 1199082 and 119908

3 In order to build a Pareto frontier these

weights were employed using (5) to optimize the objectivefunctions ldquolife of cutting toolrdquo ldquoproductivityrdquo and ldquocostrdquoTheresults of the optimization process and the Pareto frontierwith the indication of the optimal are presented inTable 7 andFigure 5

The analysis of the data generated with NBI methodshows that the behavior of the correlations between theresults of the parameters life of cutting tool cost per part andproductivity was the same as that obtained previously whenthe weighted metric method was used

However in Figure 5 it was found that the distributionof Pareto-optimal points on the frontier is evenly distributedUsing the NBI method for the selection criteriamdashEntropyand GPEmdashit can reach the parameters for weights thatensure an even distribution when it solves the originalmultiobjective problemusing theweightedmetricmethod asin (5) Moreover using as selection criteria Entropy and GPEin Figure 5 the best fitted point (the highlighted one in thefigure) can be foundWith that proposal the optimal point in


Table 7 Objective functions optimization results

Weights Variable Objective functions1199081 1199082 1199083 Feed Rotation Life of tool Productivity Cost033458 033359 033183 037 262 2707 1776 003669034537 031468 033994 037 261 2721 1774 003663035772 029556 034672 037 261 2736 1773 003656037050 027585 035365 037 260 2752 1771 003650038385 025544 036071 037 260 2768 1769 003643039778 023417 036806 037 259 2786 1767 003636041226 021194 037580 037 259 2804 1765 003629042715 018876 038408 037 258 2824 1762 003621044200 016488 039312 037 257 2846 1759 003613045608 014091 040301 037 256 2869 1756 003605046834 011778 041387 037 255 2892 1753 003597047800 009641 042559 037 254 2915 1750 003589048463 007729 043808 037 253 2938 1746 003582048815 006050 045135 037 252 2961 1742 003576048881 004583 046537 037 251 2983 1738 003570048766 003578 047656 037 250 3001 1735 003566048697 003297 048006 037 250 3007 1733 003565048300 002166 049534 037 248 3033 1728 003561047703 001166 051131 036 246 3064 1720 003559047012 000274 052714 036 243 3103 1707 003563039184 000000 060816 036 241 3112 1702 003566033930 000000 066070 036 241 3111 1703 003565Values in bold represent the optimal defined by the proposed optimization process

the frontier that was at the same time the more diversifiedone and the one with the lowest error when comparing theideal value for each objective function was discovered

5 Conclusions

This work aimed to propose a method that can identify theoptimal weights involved in amultiobjective formulation in anonsubjectivemannerThe lack of works that are proposed tothis end is evidence of this workrsquos relevanceThe definition ofthese weights is also important because this information canbe useful to the decision maker in decision making process

Thus this paper has presented a methodology for defin-ing the optimal weights that by using Design of Experiments(DOE) has generated optimum values for the decisionvariables that can be implemented in the vertical turningprocess analyzed hereinThismethod presents itself as easy toimplement without generating large computational demandsince the tools are available in popular software such as theSolver function of Excel and Minitab

The Entropy and the Global Percentage Error (GPE)function used as a criterion for evaluating Pareto-optimalsolutions were identified as suitable indicators enabling theirmodeling via a polynomial of mixtures that delimited aregion of maximum diversification and minimum error forthe weight combination analyzed

Another finding in this study was the possibility ofconstructing in an easy way an evenly distributed Paretofrontier for more than two objectives With the presentproposal the Entropy and the GPE can be calculated forany number of objective functions and the Pareto frontierand the optimal weights can be reached using the NBImethod as described This is an advantage mainly when thecomputational economy is considered

Thus the main contributions of this study are the pro-posal of a structured method differentiated in relation tothe techniques found in the literature of identifying optimalweights for multiobjective problems and the possibility ofviewing the optimal result along the Pareto frontier of theproblemThis viewing possibility is very relevant informationfor the more efficient management of processes Moreover itcan be stated that the proposed method promotes maximumachievement amongmultiple objectives that is between a setof Pareto-optimal solutions being able to identify the bestoptimal based on the aforementioned selection criteria

As a suggestion for future work the application ofthe presented methodology in other industrial processes isproposed Their application a priori is possible in variouscontexts and with different numbers of objective functionsBesides the proposal is applicable to many studies usingstochastic programming where it is necessary to includeat the same time the mean and variance in the objectivefunction as in the works developed by [33 44]


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 Brazilian Governmentagencies CNPq CAPES FAPEMIG and IFSULDEMINASfor their support Moreover the authors would like tothank the agreement signed between the company IntermecTechnologies Corporation (Honeywell) and the Federal Uni-versity of Itajuba (Process ID 230880004652014-57)

References

[1] S S Rao Engineering Optimization Theory and Practice JohnWiley amp Sons New York NY USA 4th edition 2009

[2] M Chen Y Guo H Liu and C Wang ldquoThe evolutionaryalgorithm to find robust Pareto-optimal solutions over timerdquoMathematical Problems in Engineering vol 2015 Article ID814210 18 pages 2015

[3] H-Z Huang Y-K Gu and X Du ldquoAn interactive fuzzymulti-objective optimization method for engineering designrdquoEngineering Applications of Artificial Intelligence vol 19 no 5pp 451ndash460 2006

[4] P Eskelinen and K Miettinen ldquoTrade-off analysis approachfor interactive nonlinearmultiobjective optimizationrdquoOR Spec-trum vol 34 no 4 pp 803ndash816 2012

[5] C-L Hwang and A S M Masud Multiple Objective DecisionMakingmdashMethods and Applications A State-of-the-Art Surveyvol 164 of Lecture Notes in Economics and MathematicalSystems Springer 1979

[6] K Miettinen Nonlinear Multiobjective Optimization KluwerAcademic 1999

[7] A Messac and C A Mattson ldquoGenerating well-distributedsets of pareto points for engineering design using physicalprogrammingrdquo Optimization and Engineering vol 3 pp 431ndash450 2002

[8] G Severino E J Paiva J R Ferreira P P Balestrassi andA P Paiva ldquoDevelopment of a special geometry carbide toolfor the optimization of vertical turning of martensitic graycast iron piston ringsrdquo The International Journal of AdvancedManufacturing Technology vol 63 no 5ndash8 pp 523ndash534 2012

[9] S Shin F Samanlioglu B R Cho and M M Wiecek ldquoCom-puting trade-offs in robust design perspectives of the meansquared errorrdquo Computers amp Industrial Engineering vol 60 no2 pp 248ndash255 2011

[10] I Das and J E Dennis ldquoA closer look at drawbacks of mini-mizing weighted sums of objectives for Pareto set generation inmulticriteria optimization problemsrdquo Structural Optimizationvol 14 no 1 pp 63ndash69 1997

[11] V Vahidinasab and S Jadid ldquoNormal boundary intersectionmethod for suppliersrsquo strategic bidding in electricity marketsan environmentaleconomic approachrdquo Energy Conversion andManagement vol 51 no 6 pp 1111ndash1119 2010

[12] M K Ardakani and R Noorossana ldquoA new optimizationcriterion for robust parameter designmdashthe case of target is bestrdquoInternational Journal of Advanced Manufacturing Technologyvol 38 no 9-10 pp 851ndash859 2008

[13] I Das and J E Dennis ldquoNormal-boundary intersection a newmethod for generating the Pareto surface in nonlinear multi-criteria optimization problemsrdquo SIAM Journal on Optimizationvol 8 no 3 pp 631ndash657 1998

[14] V Wuwongse S Kobayashi S-I Iwai and A IchikawaldquoOptimal design of linear control systems by an interactiveoptimization methodrdquo Computers in Industry vol 4 no 4 pp381ndash394 1983

[15] E Melachrinoudis ldquoDetermining an optimum location foran undesirable facility in a workroom environmentrdquo AppliedMathematical Modelling vol 9 no 5 pp 365ndash369 1985

[16] T L Saaty ldquoHow to make a decision the analytic hierarchyprocessrdquo European Journal of Operational Research vol 48 no1 pp 9ndash26 1990

[17] A Z Grzybowski ldquoNote on a new optimization based approachfor estimating priority weights and related consistency indexrdquoExpert SystemswithApplications vol 39 no 14 pp 11699ndash117082012

[18] S-P Wan and J-Y Dong ldquoPower geometric operators of trape-zoidal intuitionistic fuzzy numbers and application to multi-attribute group decision makingrdquo Applied Soft Computing vol29 pp 153ndash168 2015

[19] Z-J Wang ldquoConsistency analysis and priority derivation oftriangular fuzzy preference relations based on modal value andgeometric meanrdquo Information Sciences vol 314 pp 169ndash1832015

[20] J H F Gomes A P Paiva S C Costa P P Balestrassi andE J Paiva ldquoWeighted Multivariate Mean Square Error forprocesses optimization a case study on flux-cored arc weldingfor stainless steel claddingsrdquo European Journal of OperationalResearch vol 226 no 3 pp 522ndash535 2013

[21] J S Savier andDDas ldquoLoss allocation to consumers before andafter reconfiguration of radial distribution networksrdquo Interna-tional Journal of Electrical Power and Energy Systems vol 33no 3 pp 540ndash549 2011

[22] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012

[23] J Zhu and K W Hipel ldquoMultiple stages grey target decisionmaking method with incomplete weight based on multi-granularity linguistic labelrdquo Information Sciences vol 212 pp15ndash32 2012

[24] D Luo ldquoDecision-makingmethodswith three-parameter inter-val grey numberrdquo System Engineering Theory and Practice vol29 no 1 pp 124ndash130 2009

[25] L Rubio M De La Sen A P Longstaff and S Fletcher ldquoModel-based expert system to automatically adapt milling forces inPareto optimal multi-objective working pointsrdquo Expert Systemswith Applications vol 40 no 6 pp 2312ndash2322 2013

[26] N H Tran and K Tran ldquoCombination of fuzzy ranking andsimulated annealing to improve discrete fracture inversionrdquoMathematical and Computer Modelling vol 45 no 7-8 pp1010ndash1020 2007

[27] N Narang J S Dhillon and D P Kothari ldquoWeight patternevaluation formultiobjective hydrothermal generation schedul-ing using hybrid search techniquerdquo International Journal ofElectrical Power and Energy Systems vol 62 pp 665ndash678 2014

[28] S Monghasemi M R Nikoo M A Khaksar Fasaee and JAdamowski ldquoA novel multi criteria decision making model for


optimizing time-cost-quality trade-off problems in construc-tion projectsrdquo Expert Systems with Applications vol 42 no 6pp 3089ndash3104 2015

[29] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 no 3 pp 379ndash423 1948

[30] L C S Rocha A P Paiva E J Paiva and P P BalestrassildquoComparing DEA and principal component analysis in themultiobjective optimization of P-GMAW processrdquo Journal ofthe Brazilian Society of Mechanical Sciences and Engineering2015

[31] S-C Fang J R Rajasekera and H-J Tsao Entropy Optimiza-tion and Mathematical Programming Kluwer Academic 1997

[32] D C Montgomery Design and Analysis of Experiments JohnWiley amp Sons 5th edition 2001

[33] I Baba H Midi S Rana and G Ibragimov ldquoAn alternativeapproach of dual response surface optimization based onpenalty function methodrdquoMathematical Problems in Engineer-ing vol 2015 Article ID 450131 6 pages 2015

[34] J Cornell Experiments with Mixtures Designs Models and theAnalysis of Mixture Data JohnWiley amp Sons 3rd edition 2002

[35] P K Shukla and K Deb ldquoOn finding multiple Pareto-optimalsolutions using classical and evolutionary generating methodsrdquoEuropean Journal of Operational Research vol 181 no 3 pp1630ndash1652 2007

[36] T G Brito A P Paiva J R Ferreira J H F Gomes and P PBalestrassi ldquoA normal boundary intersection approach to mul-tiresponse robust optimization of the surface roughness in endmilling process with combined arraysrdquo Precision Engineeringvol 38 no 3 pp 628ndash638 2014

[37] Z Jia and M G Ierapetritou ldquoGenerate Pareto optimal solu-tions of scheduling problems using normal boundary intersec-tion techniquerdquo Computers and Chemical Engineering vol 31no 4 pp 268ndash280 2007

[38] S V Utyuzhnikov P Fantini and M D Guenov ldquoA methodfor generating a well-distributed Pareto set in nonlinear mul-tiobjective optimizationrdquo Journal of Computational and AppliedMathematics vol 223 no 2 pp 820ndash841 2009

[39] O Koksoy andNDoganaksoy ldquoJoint optimization ofmean andstandard deviation using response surface methodsrdquo Journal ofQuality Technology vol 35 no 3 pp 239ndash252 2003

[40] O Koksoy ldquoA nonlinear programming solution to robustmulti-response quality problemrdquo Applied Mathematics and Computa-tion vol 196 no 2 pp 603ndash612 2008

[41] D G Luenberger and Y Ye Linear and Nonlinear ProgrammingSpringer 3rd edition 2008

[42] G A Gabriele and K M Ragsdell ldquoThe generalized reducedgradient method a reliable tool for optimal designrdquo ASMEJournal of Engineering for Industry vol 99 no 2 pp 394ndash4001977

[43] G E P Box and N R Draper Empirical Model-Building andResponse Surfaces John Wiley amp Sons 1987

[44] M Mlakar T Tusar and B Filipic ldquoComparing solutionsunder uncertainty in multiobjective optimizationrdquo Mathemat-ical Problems in Engineering vol 2014 Article ID 817964 10pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The author in [1] refers to a feasible solution 119909lowast as Pareto-

optimal if no other feasible solution 119911 exists such that 119891119894(119911) le

119891119894(119909lowast

) 119894 = 1 2 119898 with 119891119895(119911) lt 119891

119894(119909lowast

) in at leastone objective 119895 In other words a vector 119909lowast is said to bePareto-optimal if no other solution 119911 can be found thatcauses a reduction in the objective function without causinga simultaneous increase in at least one of the other objectivesPareto-optimal solutions occur because of the conflictingnature of the objectives where the value of any objectivefunction cannot be improved without impairing at least oneof the others In this context a trade-off represents giving upone objective to improve another [4]

The purpose ofmultiobjective optimizationmethods is tooffer support and ways to find the best compromise solutionPlaying important roles in this are a decision maker andhis preference information [4] A decision maker accordingto [4] is an expert in the domain of the problem underconsideration and who typically is responsible for the finalsolution In order to define the relative importance of eachobjective function the decision maker must assign themdifferent weights

Because a characteristic property of multiobjective opti-mization is the objective function weighting problem how adecision maker is involved with the solution of this problemis the basis for its classification According to [5 6] the classesare as follows (1) no-preference methods methods where noarticulation of preference information is made (2) a priorimethods methods where a priori articulation of preferenceinformation is used that is the decision maker selects theweighting before running the optimization algorithm (3)interactive methods methods where progressive articulationof preference information is used that is the decision makerinteracts with the optimization programduring the optimiza-tion process and (4) a posteriori methods methods where aposteriori articulation of preference information is used thatis no weighting is specified by the user before or during theoptimization process However as no classification can becomplete these classifications are not absolute Overlappingand combinations of classes are possible and some methodscan be considered to belong to more than one class [5] Thispaper considers the a posteriori method in consonance withgenerate-first-choose later approach [7]

A multiobjective problem is generally solved by reducingit to a scalar optimization problem and hence the termscalarization Scalarization is the converting of the problemby aggregation of the components of the objective functionsinto a single or a family of single objective optimizationproblems with a real-valued objective function [5] Theliterature reports different scalarization methods The mostcommon is the weighted sum method

Theweighted summethod is widely employed to generatethe trade-off solutions for nonlinearmultiobjective optimiza-tion problems According to [9] a biobjective problem isconvex if the feasible set119883 is convex and the functions are alsoconvex When at least one objective function is not convexthe biobjective problem becomes nonconvex generatinga nonconvex and even unconnected Pareto frontier Theprincipal consequence of a nonconvex Pareto frontier is that

points on the concave parts of the trade-off surface will notbe estimated [10] This instability is due to the fact that theweighted sum is not a Lipshitzian function of the weight 119908[11] Another drawback to the weighted sums is related tothe uniform spread of Pareto-optimal solutions Even if auniform spread of weight vectors is used the Pareto frontierwill be neither equispaced nor evenly distributed [10 11]

Given its drawbacks the weighted sum method is notused in this paper Instead the paper employs a weightedmetric method as cited by [12] in association with a normalboundary intersection method (NBI) as proposed by [13]Reference [13] proposed the NBI method to overcome thedisadvantages of the weighted sum method showing thatthe Pareto surface is evenly distributed independent of therelative scales of the objective functions It is for this featurethat this study uses the NBI method to build the Paretofrontier

The weighting issue has been discussed in literature forat least thirty years Reference [14] proposed an approachto helping to determine proper parameters in linear controlsystem design In the case that the desired specificationsare not given explicitly this approach applies an interactiveoptimizationmethod to search themost suitable weightsTheshortcoming of this proposal is that the decisionmaker needsto make pairwise comparisons of the response curves in eachiteration which becomes a problem when many iterationsare needed Reference [15] determined an optimum locationfor an undesirable facility in a workroom environment Theauthor defined the problem as the selection of a locationwithin the convex region that maximizes the minimumweighted Euclidean distance with respect to all existing facil-ities where the degree of undesirability between an existingfacility and the new undesirable entity is reflected througha weighting factor Reference [16] presented a multicrite-ria decision making approach named Analytic HierarchyProcess (AHP) in which selected factors are arranged ina hierarchic structure descending from an overall goal tocriteria subcriteria and alternatives in successive levelsDespite its popularity this method has been criticized bydecision analysts Some authors have pointed out that Saatyrsquosprocedure does not optimize any performance criterion [17]

In the course of time other methods for deriving thepriority weights have been proposed in literature such asgeometric mean procedure [18 19] methods based on con-strained optimization models [20] trial and error methods[21] methods using grey decision [22ndash24] methods usingfuzzy logic [3 18 19 25 26] and methods using simulatedannealing [26 27]

Recently [28] dealing with multiobjective optimiza-tion of time-cost-quality trade-off problems in construc-tion projects used Shannonrsquos entropy to define the weightsinvolved in the optimization process According to theauthors Shannonrsquos entropy can provide a more reliableassessment of the relative weights for the objectives in theabsence of the decision makerrsquos preferences

In the multiobjective optimization process the decisionmaker as noted above plays an important role for it is thedecision maker that sooner or later obtains a single solutionto be used as the solution to his original multidisciplinary














1 1199092 119909

119901


119901







119894in the


119894) In matrix


Φ =

[

[

[

[

[

[

[

[

[

[

[

[

119891lowast

1 (119909lowast


119894) sdot sdot sdot 1198911 (119909

lowast

119898)

d

119891119894(119909lowast


119894(119909lowast


119894(119909lowast

119898)

d

119891119898(119909lowast



lowast

119898(119909lowast

119898)

]

]

]

]

]

]

]

]

]

]

]

]

(1)





119891 (119909) =

119891119894(119909) minus 119891

119880

119894

119891119873

119894minus 119891119880

119894

119894 = 1 119898 (2)


lowast

119894(119909lowast




Max(x119863)

119863

st Φ119908+119863119899 = 119865 (x)

x isin Ω

(3)




Min 1198911 (x)

st 1198911 (x) minus 1198912 (x) + 2119908minus 1 = 0

119892119895(x) ge 0

0 le 119908 le 1

(4)






3 Metamodeling











Start




optimization



End

Arrangementend




and Entropy


and Entropy

Yes

No









x2

(0 120572)

(minus1 +1)

(minus120572 0)

(+1 +1)

(0 0) (120572 0)

x1

(0 minus120572)










119873


1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684













Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)



119894and 119891Max








GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)






119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















1 1199092 119909

119901


119901







119894in the


119894) In matrix


Φ =

[

[

[

[

[

[

[

[

[

[

[

[

119891lowast

1 (119909lowast


119894) sdot sdot sdot 1198911 (119909

lowast

119898)

d

119891119894(119909lowast


119894(119909lowast


119894(119909lowast

119898)

d

119891119898(119909lowast



lowast

119898(119909lowast

119898)

]

]

]

]

]

]

]

]

]

]

]

]

(1)





119891 (119909) =

119891119894(119909) minus 119891

119880

119894

119891119873

119894minus 119891119880

119894

119894 = 1 119898 (2)


lowast

119894(119909lowast




Max(x119863)

119863

st Φ119908+119863119899 = 119865 (x)

x isin Ω

(3)




Min 1198911 (x)

st 1198911 (x) minus 1198912 (x) + 2119908minus 1 = 0

119892119895(x) ge 0

0 le 119908 le 1

(4)






3 Metamodeling











Start




optimization



End

Arrangementend




and Entropy


and Entropy

Yes

No









x2

(0 120572)

(minus1 +1)

(minus120572 0)

(+1 +1)

(0 0) (120572 0)

x1

(0 minus120572)










119873


1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684













Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)



119894and 119891Max








GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)






119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119891 (119909) =

119891119894(119909) minus 119891

119880

119894

119891119873

119894minus 119891119880

119894

119894 = 1 119898 (2)


lowast

119894(119909lowast




Max(x119863)

119863

st Φ119908+119863119899 = 119865 (x)

x isin Ω

(3)




Min 1198911 (x)

st 1198911 (x) minus 1198912 (x) + 2119908minus 1 = 0

119892119895(x) ge 0

0 le 119908 le 1

(4)






3 Metamodeling











Start




optimization



End

Arrangementend




and Entropy


and Entropy

Yes

No









x2

(0 120572)

(minus1 +1)

(minus120572 0)

(+1 +1)

(0 0) (120572 0)

x1

(0 minus120572)










119873


1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684













Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)



119894and 119891Max








GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)






119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start




optimization



End

Arrangementend




and Entropy


and Entropy

Yes

No









x2

(0 120572)

(minus1 +1)

(minus120572 0)

(+1 +1)

(0 0) (120572 0)

x1

(0 minus120572)










119873


1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684













Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)



119894and 119891Max








GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)






119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873


1 032 235 2102 1523 0046862 038 235 2853 1712 0036823 032 275 1802 1677 0044744 038 275 1501 1847 0044135 031 255 1652 1555 0050196 039 255 1802 1813 0041177 035 227 2853 1588 0040478 035 283 1952 1807 0039859 035 255 3153 1714 00356210 035 255 3003 1713 00362011 035 255 3303 1716 00350912 035 255 2703 1709 00375513 035 255 2853 1711 003684













Min 119891 (x) =119898

sum

119894=1119908119894sdot [

119891119894(x) minus 119891lowast

119894(x)

119891Max119894

minus 119891Min119894

]

2

st xTx le 1205882

0 le 119908119894le 1

(5)



119894and 119891Max








GPE =119898

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119910lowast

119894

119879119894

minus 1100381610038161003816100381610038161003816100381610038161003816

(6)






119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894is



119878 (119909) = minus

119898

sum

119894=1119908119894ln (119908119894)

st 0 le 119908119894le 1

(7)



2000

003501500

00375

00400

00425

2500

30001850

17001750

1800

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)




GPE = 0 1081199081 + 0 7221199082 + 0 0951199083

minus 0 68911990811199082 minus 0 03911990811199083

minus 0 53711990821199083 + 0 62911990811199082 (1199081 minus1199082)

minus 0 82611990821199083 (1199082 minus1199083)




+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












+ 3 6961199081119908221199083

minus 0 49311990811199082 (1199081 minus1199082)2

minus 0 33811990821199083 (1199082 minus1199083)2


+ 2 74811990811199082 + 2 74811990811199083

+ 2 74811990821199083 + 5 3551199082111990821199083

+ 5 3551199081119908221199083 + 5 35511990811199082119908

23

+ 1 22411990811199082 (1199081 minus1199082)2

+ 1 22411990811199083 (1199081 minus1199083)2

+ 1 22411990821199083 (1199082 minus1199083)2

(8)




Max 120585 =

EntropyGPE

st119899

sum

119894=1119908119894= 1

0 le 119908119894le 1

(9)


11199082 and119908



2(weight of



111009080706

022

020

018

016

014

012

010

Entropy

GPE


2850

003552700

00360

00365

3000

31501775

17001725

1750

Cos

t (U

S$p

art)

Life of tool (mm)

Productivity (Ph)











5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusions










Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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