Research Article Exploring the Impact of Commuter s ...downloads.hindawi.com/journals/mpe/2014/536872.pdf · the density falls as the distance increases. Here, >0 represents the

Research ArticleExploring the Impact of Commuterrsquos ResidentialLocation Choice on the Design of a Rail Transit LineBased on Prospect Theory

Ding Liu

Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

Correspondence should be addressed to Ding Liu stunker126com

Received 7 October 2013 Accepted 18 June 2014 Published 4 August 2014

Academic Editor Chaudry Masood Khalique

Copyright copy 2014 Ding Liu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper explores the impact of prospect theory based commuterrsquos residential location choice on the design problem of a railtransit line located in a monocentric city A closed-form social welfare maximization model is proposed with special considerationgiven to prospect theory based commuterrsquos residential location choice over years Commuters are assumed to make residentiallocation choice by a trade-off between daily housing rent and generalized travel cost tominimize their prospect valuesThe solutionsproperties of the proposed model are explored and compared analytically It is found that overestimation exists for the optimalsolutions of rail line length headway and fare based on traditional utility theory compared with the optimal solutions of theproposed prospect theory based model A numerical example is given to illustrate the properties of the proposed model

1 Introduction

Rail transit lines are being launched in many cities of Chinain recent years due to the rapid development of economy andthe dramatic growth of urban population For instance theShanghaiMunicipalGovernment has commenced the projectof extending railway line 11 about 576 km westwards with atotal of four stations recently InHongKong a newmetro lineconnecting Shatin new town to the central with a total lengthof 17 km and ten stations are also being built which starts in2011 and is expected to finish in 2019

Rail transit lines can alleviate boring traffic congestionand make life more convenient as regards maneuverabilityfor a specific set of people namely those living in thevicinity of the line and new stations to be constructed Hencecommuters prefer living along the candidate rail transit linesso as to enjoy such advantage of rail service

In many areas especially in cities with high popula-tion densities like Shanghai and Hong Kong commutersrsquobehaviour of making residential location choice and railtravel mode choice simultaneously has been identified [1ndash3] In other words commutersrsquo behaviour of simultaneousresidential location choice and rail travel mode choice can

directly affect the performance of the candidate rail transitlineThe output results of the above commuterrsquos simultaneousresidential location choice and rail travel mode choice are thepopulation densities in residential locations

Thediscrete choicemodelswere largely used to determinethe residential location choice with generalized travel costs ofvarious travel mode choices as the determinant factors [4ndash6]The discrete choice models can help estimating populationdensities in residential locations and explain the trade-offscommuters are faced with Nevertheless their use has beencriticized in that most of the discrete choice models wereproposed based on utility theory

Although utility theory was applicable in many contextsit may be inadequate in estimation of population densitiesover a long-term planning horizon Before the reach of rela-tive equilibrium of population densities commuters undergoa relative long-term learning process of residential locationchoice and rail mode choiceThis long-term learning processcan be partially attributed to the existence of perceptionerror and uncertainty on housing rent and generalized travelcost Unfortunately the long-term learning process cannot becaptured by utility theory

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 536872 12 pageshttpdxdoiorg1011552014536872

2 Mathematical Problems in Engineering

Table 1 Modes and reference points in some previous models

Citation Mode Reference pointKatsikopouloset al [10] Car Travel time of the reference route

Senbil andKitamura [11] Commuters Work start time

and preferred arrival timeAvineri [7] Private car Average travel time

Jou et al [12] Commuters Earliest acceptable arrival timeand work start time

Xu et al [9] Travellers Effective reversed time

Prospect theory can be seen as an extension of utilitytheory Comparedwith utility theory which are based onnor-mative preference axioms prospect theory describes lotterieschoices by a two-step process an initial phase of editing and asubsequent phase of evaluation [7 8] Because of the propertyof two-step process prospect theory can be used to describethe above long-term learning process of commuters

Reference point is a key parameter of prospect theoryHowever there are no prefect models to predict the valueof reference point in transportation models [9] Some trans-portation models associated with prospect theory are sum-marized in Table 1 Katsikopoulos et al [10] investigated cardriversrsquo risk preference behaviour on route choice with traveltime of the reference route as reference point Senbil andKitamura [11] explored commutersrsquo departure time choicewithwork start time as reference point in decision frame 1 andpreferred arrival time as reference point in decision frame 2By contrast Jou et al [12] examined commutersrsquo departuretime with the reference point of earliest acceptable arrivaltime and work start time Xu et al [9] modelled driversrsquo routechoice with effective reversed time as reference point

Our goal is to explore the impacts of prospect theorybased commutersrsquo residential location choice on the designof a rail transit line Commuters are assumed to chooseresidential locations along the candidate rail transit line bya trade-off between daily housing rent and generalized travelcost To capture the above long-term learning process of res-idential location choice and rail mode choice simultaneouslytwo reference points are adopted willingness-to-pay on dailyhousing rent and willingness-to-pay on generalized travelcost

Commutersrsquo residential location choice is affected bymany design variables of a rail transit line such as railline length rail station locations (spacing) headway andfare Specifically rail line length is closely concerned withthe coverage area of rail service railway station locations(spacing) have a direct effect on the train operating speeddwelling delays of trains and in-vehicle time of commutersat stations headway could be used to determine the waitingtime of commuters at stations and fare is a component ofgeneralized travel cost

Normally the above four design variables can be distin-guished into two types long-term and short-term decisionvariables Long-term decision variables cannot be changedduring operation stage but short-term variables can be

updated For example rail line length and rail station loca-tions (spacings) should be determined during planning stageand are inflexible to change during operation stage whereasfare and headway are still flexible to unevenly change in actualoperation

Commutersrsquo generalized travel cost consists of fare andvarious travel costs including access time cost waiting timecost and in-vehicle time cost Specifically access time costis closely concerned with rail line length and rail stationlocations (spacing) Waiting time cost depends on head-way In-vehicle time cost is a function of distance betweencommutersrsquo residential locations and central business district(CBD)

In this paper all commuters are assumed to work inthe CBD of a monocentric city and thus homework isa compulsory trip of commuters each day The long-termplanning horizon of rail transit line design is divided intoseveral equal periods In each period rail service can beimproved After the implementation of rail service in eachperiod commutersmake residential location choice by trade-off between daily housing rent and generalized travel cost

The reminder of this paper is organized as followsIn the next section assumptions and notations are givenSection 3 presents model formation Some model propertiesare examined In Section 4 a numerical example is usedto illustrate the insightful findings of the proposed modelsSection 5 concludes this paper

2 Assumptions and Notations

A transportation corridor of 119861 km length is proposed whichextends from the CBD towards the boundary of the city asshown in Figure 1 There is an ordered sequence of stations1 2 119873(120591) + 1 The symbol 119863

119904represents the distance

between station 119904 and the CBD 119873(120591) represents rail stationnumber and 119863

120591

1is the rail line length in period 120591 The

considered designed variables include the combination of railline length119863

120591

1 station location119863

119904or spacing (119863

119904minus1minus119863119904) train

headway ℎ(120591) and fare 119891119904[2]

To facilitate the presentation of the essential ideas with-out loss of generality basic assumptions and notations aremade in this paper as follows

21 Assumptions (A1) Commuters are assumed to be homo-geneous and they have the same preferred arrival time tothe workplace located in the CBD and the same preferreddaily housing rent for each residential location [13] Thisassumption could be extended to multiclass commuters infurther studies

(A2) Commuters are assumed to board trains at thenearest rail station and the trains stop at every station onthe candidate rail transit line This assumption has also beenadopted bymany previousworks such as those ofWirasingheand Ghoneim [14] and Li et al [2]

(A3) In-vehicle crowding cost in trains and moving costsfor commuters from one place to another are not consideredsince the proposed model is for long-term planning purposeof rail transit line Other travel modes are not consideredbecause the main goal of this paper is to explore the prospect

Mathematical Problems in Engineering 3

Corridor boundary

CBD

NN+ 1

Ls

Ds

Lsminus1

Ds+1

D1

B

Gs Es

ss + 1 s minus 1 1

Figure 1 The rail transit line configuration in a monocentric city

theory based commutersrsquo residential location choice on thedesign of a rail transit lineThe situation considered heremayemerge in amonocentric city with highly compact city centreCommuters in this monocentric city live in the dispersedsurrounding suburban area [15]

(A4) The original population density in the monocentriccity is specified as a linear function The original populationdensity at distance 119909 from the CBD in period 120591 is definedas 119892(119909 120591) = 119892

0(120591)(1 minus 119890119909) forall119909 isin [0 119861] where 119892

0(120591)

represents the population density in the CBD of period 120591

and 119890 represents the density gradient describing how rapidlythe density falls as the distance increases Here 119890 gt 0

represents the fact that more commuters live at CBD areawhile 119890 lt 0 represents the fact that more commuters live atsuburban area Smaller value of 119890 means more decentralizedcity Specifically when 119890 equals 0 this linear populationdensity function is reduced to a uniform one With thisassumption the total number of population 119866(120591) in period120591 is given by 119866(120591) = int

119861

0

1198920(120591)(1 minus 119890119909) 119889119909 [2]

22 Notations Consider the following

119888(119909 119904 120591) commutersrsquo actual generalized travel cost forarriving at CBD from location 119909 from station 119904 bytrain in the 120591th period

119903(119909 120591) commutersrsquo actual housing rent at the location119909 in the 120591th period 119909 isin X andX is the choice set thatis many types of houses exist at location 119909

119888(119909 119904 120591) commutersrsquo perceived generalized travelcost for arriving at CBD from location 119909 from station119904 by train in the 120591th period119903(119909 120591) commutersrsquo perceived housing rent at thelocation 119909 in the 120591th period

119888WTP119909120591 commutersrsquo reference point to decidewhether the generalized travel cost is high or low atthe location 119909 in the 120591th period which is called thecommutersrsquo willingness-to-pay (WTP) on travel cost

119903WTP119909120591 commutersrsquo reference point to decidewhether the housing rent is high or low at thelocation 119909 in the 120591th period which is called thecommutersrsquo willingness-to-pay (WTP) on housingrent

119901(119909 120591) the probability of obtaining low living cost interms of the generalized travel cost and housing rentat location 119909 in the 120591th periodΔ119903119888(119909 119904 120591) the deviation between perceived livingcost and reference points in terms of the generalizedtravel cost and housing rent at location 119909 in the 120591thperiod120587(119901(119909 120591)) the probability weighting function119881(Δ119903119888(119909 119904 120591)) value function of living at location 119909

and travelling to the CBD from station 119904 by train inthe 120591th periodPV119909119904120591

prospect value of living at location 119909 andtravelling to the CBD from station 119904 by train in the120591th period

3 Model Formulation

The design of a rail transit line is considered over a planninghorizon of [0 119879] This horizon is divided into 119872 equalperiods The rail transit line is assumed to be implementedby an operator franchised by government Social welfaremaximization is the decision objective of rail transit linedesign Commuters are assumed to make residential locationchoices as if they are prospect maximizers This question canbe formulated as a mathematically programming model withthe objective of social welfare maximization subjected tothe constraints of prospect theory based residential locationchoice equilibrium condition

31 ProspectTheory BasedResidential LocationChoice Equilib-riumCondition As stated above prospect theory can be usedto capture the learning process of commutersrsquo location choiceover rail design periods in the planning horizonAs inAvineri[7] Wardroprsquos [16] principle of user equilibrium could beextended to prospect theory based equilibrium ldquoEquilibriumunder the condition that no commuter can decrease hisherchoice prospect value by unilaterally switching hisher choicebehaviourrdquo Mathematically this equilibrium condition canbe expressed as119902 (119909 119904 120591) gt 0 997904rArr PV

119909119904120591= min PV

119909119904120591 forall119909 isin [0 119861]

119904 = 1 2 119873 (120591) 120591 isin [1119872]

(1)


where PV119909119904120591

represents prospect value of living at location 119909

and travelling to the CBD from station 119904 by train in the 120591thperiod and 119902(119909 119904 120591) is the peak-hour travel demand densityat location 119909 to the CBD by train through station 119904 in period120591

Prospect value PV119909119904120591

could be calculated by the follow-ing equations

PV119909119904120591

= 120587 (119901 (119909 120591)) 119881 (Δ119903119888 (119909 119904 120591)) (2)

120587 (119901 (119909 120591)) =119901(119909 120591)

120574

[119901(119909 120591)120574

+ (1 minus 119901(119909 120591))1120574

]

(3)

119881 (Δ119903119888 (119909 119904 120591)) = Δ119903119888(119909 119904 120591)

1205721015840

Δ119903119888 (119909 119904 120591) ge 0

minus120582(minusΔ119903119888 (119909 119904 120591))1205731015840

Δ119903119888 (119909 119904 120591) lt 0

(4)

Δ119903119888 (119909 119904 120591) = 119888 (119909 119904 120591) + 119903 (119909 120591) minus 119888WTP119909120591 minus 119903WTP119909120591 (5)

119888 (119909 119904 120591) = 119888 (119909 119904 120591) + 120585 (6)

119903 (119909 120591) = 119903 (119909 120591) + 120595 (7)

where 120585120595 is a random disturbance term reflecting gener-alized travel costhousing rent differences among residentiallocations

The generalized travel cost 119888(119909 119904 120591) consists of rail fareand various travel time cost including access time costwaiting time cost and in-vehicle time cost Specifically it isdefined as

119888 (119909 119904 120591) = 119891119904(120591) + 120601

119906119906119904(119909) + 120601

119904119908119904(120591) + 120601

119905119905119904(120591) (8)

where 119891119904(120591) represents rail fare from station 119904 to the CBD in

period 120591 119906119904(119909) represents commutersrsquo average access time

to station 119904 from location 119909 119908119904(120591) represents commutersrsquo

average waiting time at station 119904 in period 120591 119905119904(120591) represents

commutersrsquo in-vehicle time to the CBD from station 119904 and120601119906 120601119908 120601119905represent commutersrsquo value of time for access time

waiting time and in-vehicle time respectivelyThe commutersrsquo waiting time at station 119904 in period 120591

119908119904(120591) can be given by

119908119904(120591) = 120572ℎ (120591) forall119904 = 1 2 119873 (120591) (9)

where ℎ(120591) represents the headway of railway service inperiod 120591 and 120572 is a calibration parameter which depends onthe distributions of train headway and commuter arrival

The commutersrsquo in-vehicle time from station 119904 to the CBDin period 120591 119905

119904(120591) can be calculated by

119905119904(120591) = 119879

1199041(120591) + 119879

1199042(120591) forall119904 = 1 2 119873 (120591) (10)

where

1198791199041

(120591) =119863119904

119881119905(120591)

forall119904 = 1 2 119873 (120591)

1198791199042

= 1205730(119873 (120591) + 1 minus 119904) forall119904 = 1 2 119873 (120591)

(11)

where 119881119905(120591) represents the average train cruise speed in

period 120591 119863119904represents the station distance from station

119904 to CBD defined as above and 1205730represents the average

train dwelling time at a station which can be calibrated withobserved data [17 18]

To represent the demand-supply relationship of housingrental market the following housing rent is assumed given by

119903 (119909 120591) = 120572119904[1 +

120573119904119875 (119909 120591)

(119867 (119909 120591) minus 119875 (119909 120591))] (12)

where 119867(119909 120591) (in terms of housing unit) denotes potentialhousing supply density at location 119909 in the 120591th period and 120572

119904

and 120573119904are positive scalar parameters that represent the fixed

and demand-dependent components of the rent functionaround station 119904 [19]

In order to determine travel demand density 119902(119909 119904 120591) wefirst define the potential travel demand density at location 119909

in period 120591 which is denoted by 119875(119909 120591) Generally speakingcommutersrsquo destinations are normally distributed along therail line with more concentration close to the CBD ofcourse Denote 120578 by the proportion of trips with CBD beingthe destinations in period 120591 and denote 120588 by the ratio ofpeak-hour flow to the daily average flow and then 120578120588119892(119909)

represents the peak-hour potential travel demand density interms of (A4) We have

119875 (119909 120591) = 1205781205881198920(120591) (1 minus 119890119909)

= 1198750(120591) (1 minus 119890119909) forall119909 isin [0 119861]

(13)

where 1198750(120591) represents the peak-hour potential travel

demand density in the CBD and 1198750(120591) = 120578120588119892

0(120591)

As stated above travel demand density for rail service119902(119909 119904 120591) is closely concerned with several design variablesnamely rail line length rail station or spacing headway andfare in terms of the generalized travel cost To represent sucheffect a negative exponential elastic demand density functionis used as follows

119902 (119909 119904 120591) = 119875 (119909 120591) exp (minus120579119888 (119909 119904 120591))

forall119909 isin [0 119861] 119904 = 1 2 119873 (120591)

(14)

where 120579 (120579 gt 0) represents the sensitivity parameter for thegeneralized travel cost and the perceived generalized travelcost 119888(119909 119904 120591) is given by (6) and (8)

32 Social Welfare of Candidate Rail Transit Line Socialwelfare of the candidate rail transit line can be calculatedby summation of operatorrsquos net profit and consumer surplusof commuters Mathematically the social welfare in theplanning horizon 120591 isin [1119872] SW can be expressed as

SW =

119872

sum

120591=1

(119875120591+ CS120591) (15)

where 119875120591and CS

120591are the operatorrsquos net profit and consumer

surplus of commuters in period 120591 respectivelyThe operatorrsquos net profit is closely concerned with rev-

enue from fare and related construction and operation costAccordingly 119875

120591could be calculated by

119875120591= 119877120591minus 119862120591 (16)


where 119877120591is the operatorrsquos revenue in period 120591 and 119862

120591is the

related construction and operation cost in period 120591The operatorrsquos revenue comes from fare It could be cal-

culated by summation of the number of commuters boardingat each station multiplied by the corresponding fare that is

119877120591=

119873

sum

119904=1

119891119904(120591)119876119904120591

(1 + 119894)120591minus1

(17)

where 1(1 + 119894)120591minus1 is the discount factor in period 120591 119894 is the

interest rate and 119876119904120591

is the travel demand of station 119904 inperiod 120591

In terms of (A2) the travel demand of each station comesfrom coverage area of this station that is

119876119904120591

= int

119871119904minus1

119871119904

119902 (119909 119904 120591) 119889119909 forall119904 = 1 2 119873 (120591) (18)

where 119902(119909 119904 120591) is peak-hour travel demand density of railservice at location 119909 in period 120591 given by (14) 119871

119904is commuter

watershed line which is located at the middle point of linesegment (119904 119904 + 1) and the distance of commuter watershedline 119871

119904from the CBD is given by

119871119904=

(119863119904+ 119863119904+1

)

2 forall119904 = 1 2 119873 (120591) (19)

In particular 1198710represents the maximum coverage location

of rail service Beyond this location no one would patronizethe rail service Thus 119871

0holds

119902 (119871120591

0 1 120591) = 0 119871

120591

0isin [119863120591

1 119861] (20)

where 119902(119871120591

0 1 120591) is travel demand density for station 1 at

location 119871120591

0in period 120591 On the basis of (8)ndash(14) the

maximum service coverage 1198710of the rail line can be given

by

1198710=

1

119890 (21)

According to the (A4) (21) implies that railway service isavailable for all the residential people in the consideredcorridor [2] Substituting (14) and (19) into (21) 119876

119904120591can be

rewritten as

119876119904120591

= 1198750(120591) int

119871119904minus1

119871119904

exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

minus 1198901198750(120591) int

119871119904minus1

119871119904

119909 exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

= minus1198750(120591) 119881119905(120591)

120579120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] +

1198901198750(120591) 1198812

119905(120591)

12057921206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(22)

where120582119904(120591) = minus120579119891

119904(120591) minus 120579120601

119906119906119904(119909) minus 120579120601

119904119908119904(120591) minus 120579120601

1199051205730

times (119873 (120591) + 1 minus 119904) + 120579120601119905

119906 (119909)119881119908

119881119905(120591)

minus 120579120585

forall119904 = 1 2 119873 (120591)

(23)

119881119908is denoted as commutersrsquo walking speed from location 119909

to station 119904 and 119906(119909)119881119908is the distance between location 119909

and station 119904The discounted cost 119862

120591 which consists of three cost

components the train operations cost 119862119900120591 rail line cost 119862

119871120591

and rail station cost 119862119904120591 could be expressed as

119862120591= 119862119900120591

+ 119862119871120591

+ 119862119904120591

(24)

The discounted train operating cost 119862119900120591

is given by

119862119900120591

=(120583119900+ 1205831119865 (120591))

(1 + 119894)120591minus1

(25)

where 120583119900is the fixed operating cost 120583

1is the operating cost

per train in each period and 119865 is the fleet size (or the numberof trains) on that line 119865(120591) equals the vehicle round journeytime 119879

119877(120591) divided by the headway 119867(120591) Namely

119865 (120591) =119879119877(120591)

ℎ (120591) (26)

where the round journey time119879119877is composed of the terminal

time line-haul travel time and train dwelling delays at station[20] which could be expressed as

119879119877

= 120589119879119900+ 2 (119879

11+ 11987912) (27)

where119879119900is the constant terminal time on the circular line and

120589 is the number of terminal times on that line11987911and11987912are

respectively the total line-haul travel time and total dwellingdelay for trainrsquos operations from station 1 to CBD given by(11)

The discounted rail line cost 119862119871120591

is the sum of variablecost 120574

11198631(eg land acquisition cost line construction cost)

which is proportional to the rail line length 1198631and the fixed

cost 1205740(eg line overhead cost maintenance cost and labour

cost) discounted to present value terms Namely

119862119871120591

= 12057411198631(1 + 120594)

120591minus1

+1205740

(1 + 119894)120591minus1

(28)

where 1205741is the fixed rail line cost per kilometre in each period

The term 1(1 + 120594)120591minus1 represents the inflation factor It means

that for the same capacity enhancement the fixed rail linecost increases 120594 each period

The discounted rail station cost 119862119904120591

includes a fixedcost (eg station land acquisition cost and design andconstruction cost) and a variable cost (eg station overheadcost operating cost and maintenance cost) discounted topresent value terms Mathematically 119862

119904120591can be expressed as

119862119904120591

= 1205810(1 + 120594)

120591minus1

+1205811(119873 (120591) + 1)

(1 + 119894)120591minus1

(29)


where 1205810is the fixed cost and 120581

1is the operating cost per

station in each periodConsumer surplus measures the difference between what

consumers would be willing to pay for travel and what theyactually pay In order to obtain consumer surplus the inversedemand function is calculated as follows

[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)

with forall119909 isin [0 119861] 119904 = 1 2 119873(120591) The consumer surplusat location 119909 in period 120591 denoted as CS(119909 119904 120591) can becalculated by

CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)

The discounted consumer surplus in period 120591 CS120591 is then

obtained by summing the consumer surplus along the candi-date rail transit line discounted to the present value Namely

CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)

33 Social Welfare Maximization Model As stated above thedesign goal of the rail transit line is social welfare maxi-mization Mathematically this problem can be formulated asfollows

max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)

where D represents the vector of station locations namelyD = (119863

119873(120591) 119863

2 119863120591

1) 119876119904120591

can be determined by (22)The optimal solutions for the rail line length rail station

location headway and fare can be obtained by setting thepartial derivatives of objective function equation (33) withrespect to these decision variables equal to zero and solvingthem simultaneously The following proposition gives theoptimal solutions The proof is given in Appendix A

Proposition 1 With the given population density in a partic-ular period the optimal rail line length rail station locationheadway and fare solutions with the objectives of social welfaremaximization satisfy the systems of equations

120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)

where Δ119904= 1 if 119904 = 1 and 0 otherwise Δ119904

1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)

Proposition 1 presents the partial derivatives of traveldemand 119876

119894120591with respect to railway line length 119863

120591

1and

railway station location 119863119904 There is another alternative

approach to determine these partial derivatives implement-ing equilibrium sensitivity analysis of travel demand withrespect to railway line length and railway station locationDetails on sensitivity analysis approach could be seen inFriesz et al [21] and Yan and Lam [22]

By contrast the closed-form solutions given byProposition 1 can be used to examine the interrelationshipbetween the optimal solutions of rail design variablesdirectly For instance it could be seen that the optimalheadway ℎ(120591) will increase if the railway operating cost pertrain 120583

1increased Li et al [2] proposed a similar closed-form

analysis with the objective of profit maximization based onutility theory and in a static situation

To highlight the difference between the optimal solutionsof the rail design variables based on prospect theory andtraditional utility theory the following proposition is givenThe proof is given in Appendix B

Proposition 2 Overestimation exists for the optimal solutionsof rail line length headway and fare based on traditional utilitytheory compared with prospect theory

The most widely used solution algorithm for solvingconcave problem is the Frank-Wolfe searching algorithmFor solving the prospect theory based residential locationequilibriumproblem (1) this algorithm reduces to a sequence

of shortest path computations and one-dimensional mini-mizations [23] For the optimization of rail design variablesthe heuristic algorithm proposed by Li et al [2] is usedhere which is directly based on the first-order optimalityconditions of the social welfare with respect to the above raildesign variables as shown in Proposition 1

4 Numerical Example

To facilitate the presentation of the essential ideas and con-tributions of this paper an illustrative example is employedSpecifically the difference between the optimal solutions ofrail design variables based on traditional utility theory andprospect theory are compared

The alignment of the rail transit line concerned is shownin Figure 1 The corridor length is fixed as 40 km The timehorizon is 3 years and119872 is 3 Without loss of generality evenstation spacing is set as 10 km Other parameters are given inthe following Table 2

From Table 3 it could be seen that the optimal solutionsof rail line length fare and headway based on prospect theorywere less than those based on utility theory This result wasin accord with Proposition 2 However the social welfarebased on prospect theory was larger than that based onutility theory namely 26669520 gt 24278400 These resultscan be attributed to the long-term learning behaviour ofcommutersrsquo on residential location choice This long-termlearning behaviour reduced the investment of the rail transitline but increased the social welfare

5 Conclusions

This paper proposed closed-form models to explore theimpacts of prospect theory based residential location choiceon the design of a rail transit line in a monocentric cityProspect theory was used to model the long-term learningbehaviour of commutersrsquo on residential location choice overa planning horizon Trade-off exists between daily housingrent and generalized travel cost for commuters

The analytical optimal solutions of rail design variableswith social welfare maximization have been given It isconcluded that overestimation exists based on traditionalutility theory compared with prospect theory

This study provides a new avenue for the design of arail transit line Further research is needed in the followingdirections

(1) In this paper amonocentric city is assumedwith onlyoneCBDand several other residential locationsThusthe commutersrsquo mobility between different CBD(s) inlarger cities cannot be explored The city boundaryis not explicitly considered The proposed model canbe extended into polycentric CBD model in a furtherstudy [24ndash26]

(2) All commuters were assumed to be homogenous inthis study However previous studies have shownthat income levels dominated the residential loca-tion choices [27 28] Therefore the proposed modelcan be extended to incorporate the income levels


Table 2 Parameters

Symbol Definition Value119894 Interest rate per period 003120594 Inflation rate per period 0011198910

Fixed component of distance-based fare (HK$) 25120588 Ratio of peak-hour flow to the daily average flow () 80120578 Proportion of trips with the CBD as the destinations () 95119890 Density gradient 004120579 Sensitivity parameters in elastic demand function 0066120585 Commutersrsquo perceived random error for generalized travel cost 119873(0 3)

120601119906120601119905120601119904

Value of time for access time waiting time and in-vehicle respectively (HK$hour) 80120572 Parameter for waiting cost 051205721015840 Parameter for value function 088

1205731015840 Parameter for value function 088

120582 Parameter for value function 225119881119905(120591) Average train operating speed (kmhour) 40

119881119908

Average commuter walking speed from location 119909 to station 119904 (kmhour) 10119906(119909) Average distance between location 119909 to station 119904 (km) 051205730

Average train dwelling time at a station (hour) 0031205830

Fixed train operating cost per period (HK$hour) 13501205831

Variable train operating cost per train in each period (HK$vehicle hour) 540120589 Number of terminal times of train 21198790

Constant terminal time of train (hour) 02120574 Parameter for weighting probability function 0610691205740

Fixed component of rail line cost (HK$hour) 7501205741

Variable component of rail line cost (HK$vehicle hour) 3001205810

Fixed component of rail station cost (HK$hour) 12501205811

Variable component of rail station cost (HK$ station hour) 500120572119904

Parameter representing fixed components of rent function around rail station 119904 80120573119904

Parameter representing demand-dependent components of rent function around rail station 119904 10119867(119909 120591) Potential housing supply density at location 119909 in period 120591 (housing unit) 40000120595 Random disturbance term of daily housing rent differences among houses 119873(0 8)

119888WTP119909120591 Reference point for generalized travel cost (HK$day) 80119903WTP119909120591 Reference point for daily housing rent (HK$day) 100119901(119909 120591) Probability of obtaining low living cost at location 119909 in period 120591 90

Table 3 Optimal solutions of rail design variables

Optimal solutions in year 3 Based onprospect theory

Based on utilitytheory

Rail line length (km) 2350 2500Fare (HK$km) 1151 1241Headway (hour) 012 014Total travel demand(personsday) 12230 14401

Social welfare1 (HK$) 26669520 242784001The parameter 360 (days) was used to convert daily social welfare to yearlysocial welfare

for determining the residential location choices andpopulation density

(3) Only rail travel mode is considered in this paperTo investigate the effects of commutersrsquo travel mode

choice behaviour on the design of a rail transit linemore travel modes should be taken into account forinstance autobus or park-and-ride modes [29ndash31]

Appendices

A Proof of Proposition 1

To obtain the optimal solutions of rail line length and railstation locations the partial derivatives of the objectivefunction SW(sdot) with respect to 119863

119904are set to zero namely

120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ

119904= 1 if 119904 = 1 and 0 otherwise In terms of (18)ndash

(23) 119876119904120591

is a function of 119863119904 120582119904 120582119904minus1

which are functions of119863119904minus1

119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1

) forall119904 = 1 2 119873 (120591) (A2)Thus the following equation holds

120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)

The derivative of int119871120591

0

0

(119902(119908 119904 120591)120579)119889119908with respect to119863119904is

calculated as follows

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)

Substituting (A3) and (A6) into (A1) one immediatelyobtains

119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)

The partial derivative of the objective function SW(sdot) withrespect to headway ℎ(120591) is

120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591

is a function of 119888(119871119904minus1

119904 120591) and 119888(119871119904 119904 120591) and

thus a function of ℎ(120591) In terms of (5) and (15) 119871119904(119904 =

0 119873(120591)) is independent of headway ℎ(120591) With the givenpopulation density at the CBD in period 120591 119875

0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905

times [exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0

(119902(119908 119904 120591)120579)119889119908 with respect to ℎ(120591) is

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0

minus1205721206011199041198750(120591) (1 minus 119890119908) exp (minus120579119888 (119908 119904 120591)) 119889119908

= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)

Combining (A10) and (A13) one immediately obtains

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)

The partial derivative of the objective function SW(sdot) withrespect to flat fare 119891

119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ

3are the same as in (A11) In view of

the above system of equations which consist of (A7) (A11)and (A15) the optimal rail line length rail station location(or spacing) headway and fare can be calculated

B Proof of Proposition 2

In terms of Proposition 1 the rail length 119863120591

1and rail station

location (or spacing) 119863119904can be determined by

119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)

combined with (19)The condition of traditional stochastic user equilibrium

based on utility theory can be expressed as 119888(119909 119904 120591) =

min119888(119909 119904 120591) for locations with 119902(119909 119904 120591) gt 0 Submittingthis condition into (19) and (B1) we could have the optimalsolution of rail line length for traditional stochastic userequilibrium based on utility theory (119863120591

1)lowast shown as follows

(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)

In contrast with the optimal solution of rail line length for theproposed prospect theory based residential location choiceequilibrium (119863120591

1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)

if and only if (119891119904(120591)1198750(120591)2)[exp(minus120579119888(119871

2 119904 120591)) minus exp(minus120579119888(119871

1

119904 120591))] gt 0Under the proposed prospect theory based residential

location choice equilibrium travel cost will be higher forcommuters living far away from the CBD thus

119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)

exists Since 120579 gt 0 and 119891119904(120591)1198750(120591)2 gt 0 we have

119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)

Therefore compared to the proposed prospect theory basedresidential location choice equilibrium overestimation existsfor the optimal solution of rail line length with the traditionalstochastic user equilibrium based on utility theory

Similarly under the traditional stochastic user equilib-rium based on utility theory we have

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)

times exp (minus120579min 119888 (119909 119904 120591))

(B7)

In terms of Proposition 1 we have (ℎ(120591))lowast

lt (ℎ(120591))+ and

(119891119904(120591))lowast

lt (119891119904(120591))+ In conclusion overestimation exists for

the optimal solutions of headway and fare comparing thetraditional stochastic user equilibriumbased on utility theorywith the proposed prospect theory based residential locationchoice equilibrium

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work described in this paper was jointly supportedby a Grant from the Research Grant Council of the HongKong Special Administrative Region (Project no PolyU521509E) and a Postgraduate Studentship from the ResearchCommittee of the Hong Kong Polytechnic University Theauthors would like to thank the anonymous referees for theirvaluable comments

References

[1] T Yip E Hui and R Ching ldquoThe impact of railway on home-moving evidence fromHong Kong datardquo Working Paper 2012

[2] Z Li W H K Lam S C Wong and A Sumalee ldquoDesign of arail transit line for profitmaximization in a linear transportationcorridorrdquo Transportation Research Part E vol 48 no 1 pp 50ndash70 2012

[3] A Ibeas R Cordera L DellrsquoOlio and P Coppola ldquoModellingthe spatial interactions between workplace and residentiallocationrdquo Transportation Research Part A vol 49 pp 110ndash1222013


[4] A Anas ldquoThe estimation of multinomial logit models of jointlocation and travel mode choice from aggregated datardquo Journalof Regional Science vol 21 no 2 pp 223ndash242 1981

[5] J Eliasson and L Mattsson ldquoA model for integrated analysis ofhousehold location and travel choicesrdquo Transportation ResearchPart A vol 34 no 5 pp 375ndash394 2000

[6] A R Pinjari R M Pendyala C R Bhat and P A WaddellldquoModeling the choice continuum an integrated model ofresidential location auto ownership bicycle ownership andcommute tour mode choice decisionsrdquo Transportation vol 38no 6 pp 933ndash958 2011

[7] E Avineri ldquoThe effect of reference point on stochastic networkequilibriumrdquoTransportation Science vol 40 no 4 pp 409ndash4202006

[8] Z Li and D Hensher ldquoProspect theoretic contributions inunderstanding traveller behaviour a review and some com-mentsrdquo Transport Reviews vol 31 no 1 pp 97ndash115 2011

[9] H L Xu Y Y Lou Y F Yin and J Zhou ldquoA prospect-baseduser equilibrium model with endogenous reference points andits application in congestion pricingrdquo Transportation ResearchPart B vol 45 no 2 pp 311ndash328 2011

[10] K V Katsikopoulos Y Duse-Anthony D L Fisher and S ADuffy ldquoRisk attitude reversals in driversrsquo route choice whenrange of travel time information is providedrdquo Human Factorsvol 44 no 3 pp 466ndash473 2002

[11] M Senbil and R Kitamura ldquoReference points in commuterdeparture time choice a prospect theoretic test of alternativedecision framesrdquo Journal of Intelligent Transportation SystemsTechnology Planning and Operations vol 8 no 1 pp 19ndash312004

[12] R C Jou R Kitamura M C Weng and C C Chen ldquoDynamiccommuter departure time choice under uncertaintyrdquo Trans-portation Research Part A vol 42 no 5 pp 774ndash783 2008

[13] Z Qian and Z H Michael ldquoModeling multi-modal morningcommute in a one-to-one corridor networkrdquo TransportationResearch Part C vol 19 no 2 pp 254ndash269 2011

[14] S C Wirasinghe and N S Ghoneim ldquoSpacing of bus-stops formany to many travel demandrdquo Transportation Science vol 15no 3 pp 210ndash221 1981

[15] Q Tian H J Huang and H Yang ldquoEquilibrium properties ofthe morning peak-period commuting in a many-to-one masstransit systemrdquo Transportation Research Part B vol 41 no 6pp 616ndash631 2007

[16] J G Wardrop ldquoSome theoretical aspects of road trafficresearchrdquo ICE Proceedings Engineering Divisions vol 1 no 3pp 352ndash362 1952

[17] W H K Lam C Y Cheung and Y F Poon ldquoA Study of TrainDwelling Time at theHongKongMass Transit Railway SystemrdquoJournal of Advanced Transportation vol 32 no 3 pp 285ndash2961998

[18] S Chen R Zhou Y Zhou and B Mao ldquoComputation onbus delay at stops in Beijing through statistical analysisrdquoMathematical Problems in Engineering vol 2013 Article ID745370 9 pages 2013

[19] H W Ho and S C Wong ldquoHousing allocation problem in acontinuum transportation systemrdquo Transportmetrica vol 3 no1 pp 21ndash39 2007

[20] Z Li W H K Lam and S C Wong ldquoThe optimal transit farestructure under different market regimes with uncertainty inthe networkrdquo Networks and Spatial Economics vol 9 no 2 pp191ndash216 2009

[21] T L Friesz R L Tobin H-J Cho and N J Mehta ldquoSensitivityanalysis based heuristic algorithms for mathematical programswith variational inequality constraintsrdquoMathematical Program-ming vol 48 no 1ndash3 pp 265ndash284 1990

[22] H Yan andW H K Lam ldquoOptimal road tolls under conditionsof queueing and congestionrdquo Transportation Research Part Avol 30 no 5 pp 319ndash332 1996

[23] Y Sheffi Urban Transportation Networks Equilibrium Analysiswith Mathematical Programming Methods Prentice Hall 1985

[24] W Alonso Location and Land Use Harvard University PressCambridge Mass USA 1964

[25] K F Wieand ldquoAn extension of the monocentric urban spatialequilibriummodel to a multicenter setting the case of the two-center cityrdquo Journal of Urban Economics vol 21 no 3 pp 259ndash271 1987

[26] A Anas and I Kim ldquoGeneral equilibriummodels of polycentricurban land use with endogenous congestion and job agglomer-ationrdquo Journal of Urban Economics vol 40 no 2 pp 232ndash2561996

[27] J Hartwick U Schweizer and P Varaiya ldquoComparative staticsof a residential economy with several classesrdquo Journal ofEconomic Theory vol 13 no 3 pp 396ndash413 1976

[28] Y Kwon ldquoThe effect of a change in wages on welfare in a two-class monocentric cityrdquo Journal of Regional Science vol 43 no1 pp 63ndash72 2003

[29] H Huang ldquoFares and tolls in a competitive system withtransit and highway the case with two groups of commutersrdquoTransportation Research Part E vol 36 no 4 pp 267ndash284 2000

[30] T Liu H Huang H Yang and X Zhang ldquoContinuum model-ing of park-and-ride services in a linear monocentric city withdeterministicmode choicerdquoTransportation Research Part B vol43 no 6 pp 692ndash707 2009

[31] D Liu and W H K Lam ldquoModeling the effects of populationdensity on prospect theory-based travel mode-choice equilib-riumrdquo Journal of Intelligent Transportation Systems TechnologyPlanning and Operations 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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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


Table 1 Modes and reference points in some previous models

Citation Mode Reference pointKatsikopouloset al [10] Car Travel time of the reference route

Senbil andKitamura [11] Commuters Work start time

and preferred arrival timeAvineri [7] Private car Average travel time

Jou et al [12] Commuters Earliest acceptable arrival timeand work start time

Xu et al [9] Travellers Effective reversed time

Prospect theory can be seen as an extension of utilitytheory Comparedwith utility theory which are based onnor-mative preference axioms prospect theory describes lotterieschoices by a two-step process an initial phase of editing and asubsequent phase of evaluation [7 8] Because of the propertyof two-step process prospect theory can be used to describethe above long-term learning process of commuters

Reference point is a key parameter of prospect theoryHowever there are no prefect models to predict the valueof reference point in transportation models [9] Some trans-portation models associated with prospect theory are sum-marized in Table 1 Katsikopoulos et al [10] investigated cardriversrsquo risk preference behaviour on route choice with traveltime of the reference route as reference point Senbil andKitamura [11] explored commutersrsquo departure time choicewithwork start time as reference point in decision frame 1 andpreferred arrival time as reference point in decision frame 2By contrast Jou et al [12] examined commutersrsquo departuretime with the reference point of earliest acceptable arrivaltime and work start time Xu et al [9] modelled driversrsquo routechoice with effective reversed time as reference point

Our goal is to explore the impacts of prospect theorybased commutersrsquo residential location choice on the designof a rail transit line Commuters are assumed to chooseresidential locations along the candidate rail transit line bya trade-off between daily housing rent and generalized travelcost To capture the above long-term learning process of res-idential location choice and rail mode choice simultaneouslytwo reference points are adopted willingness-to-pay on dailyhousing rent and willingness-to-pay on generalized travelcost

Commutersrsquo residential location choice is affected bymany design variables of a rail transit line such as railline length rail station locations (spacing) headway andfare Specifically rail line length is closely concerned withthe coverage area of rail service railway station locations(spacing) have a direct effect on the train operating speeddwelling delays of trains and in-vehicle time of commutersat stations headway could be used to determine the waitingtime of commuters at stations and fare is a component ofgeneralized travel cost

Normally the above four design variables can be distin-guished into two types long-term and short-term decisionvariables Long-term decision variables cannot be changedduring operation stage but short-term variables can be

updated For example rail line length and rail station loca-tions (spacings) should be determined during planning stageand are inflexible to change during operation stage whereasfare and headway are still flexible to unevenly change in actualoperation

Commutersrsquo generalized travel cost consists of fare andvarious travel costs including access time cost waiting timecost and in-vehicle time cost Specifically access time costis closely concerned with rail line length and rail stationlocations (spacing) Waiting time cost depends on head-way In-vehicle time cost is a function of distance betweencommutersrsquo residential locations and central business district(CBD)

In this paper all commuters are assumed to work inthe CBD of a monocentric city and thus homework isa compulsory trip of commuters each day The long-termplanning horizon of rail transit line design is divided intoseveral equal periods In each period rail service can beimproved After the implementation of rail service in eachperiod commutersmake residential location choice by trade-off between daily housing rent and generalized travel cost

The reminder of this paper is organized as followsIn the next section assumptions and notations are givenSection 3 presents model formation Some model propertiesare examined In Section 4 a numerical example is usedto illustrate the insightful findings of the proposed modelsSection 5 concludes this paper

2 Assumptions and Notations

A transportation corridor of 119861 km length is proposed whichextends from the CBD towards the boundary of the city asshown in Figure 1 There is an ordered sequence of stations1 2 119873(120591) + 1 The symbol 119863

119904represents the distance

between station 119904 and the CBD 119873(120591) represents rail stationnumber and 119863

120591

1is the rail line length in period 120591 The

considered designed variables include the combination of railline length119863

120591

1 station location119863

119904or spacing (119863

119904minus1minus119863119904) train

headway ℎ(120591) and fare 119891119904[2]

To facilitate the presentation of the essential ideas with-out loss of generality basic assumptions and notations aremade in this paper as follows

21 Assumptions (A1) Commuters are assumed to be homo-geneous and they have the same preferred arrival time tothe workplace located in the CBD and the same preferreddaily housing rent for each residential location [13] Thisassumption could be extended to multiclass commuters infurther studies

(A2) Commuters are assumed to board trains at thenearest rail station and the trains stop at every station onthe candidate rail transit line This assumption has also beenadopted bymany previousworks such as those ofWirasingheand Ghoneim [14] and Li et al [2]

(A3) In-vehicle crowding cost in trains and moving costsfor commuters from one place to another are not consideredsince the proposed model is for long-term planning purposeof rail transit line Other travel modes are not consideredbecause the main goal of this paper is to explore the prospect


Corridor boundary

CBD

NN+ 1

Ls

Ds

Lsminus1

Ds+1

D1

B

Gs Es

ss + 1 s minus 1 1





0(120591)




119861

0

1198920(120591)(1 minus 119890119909) 119889119909 [2]










3 Model Formulation



119909119904120591= min PV

119909119904120591 forall119909 isin [0 119861]

119904 = 1 2 119873 (120591) 120591 isin [1119872]

(1)


where PV119909119904120591





PV119909119904120591

= 120587 (119901 (119909 120591)) 119881 (Δ119903119888 (119909 119904 120591)) (2)

120587 (119901 (119909 120591)) =119901(119909 120591)

120574

[119901(119909 120591)120574

+ (1 minus 119901(119909 120591))1120574

]

(3)

119881 (Δ119903119888 (119909 119904 120591)) = Δ119903119888(119909 119904 120591)

1205721015840

Δ119903119888 (119909 119904 120591) ge 0

minus120582(minusΔ119903119888 (119909 119904 120591))1205731015840

Δ119903119888 (119909 119904 120591) lt 0

(4)


119888 (119909 119904 120591) = 119888 (119909 119904 120591) + 120585 (6)

119903 (119909 120591) = 119903 (119909 120591) + 120595 (7)



119888 (119909 119904 120591) = 119891119904(120591) + 120601

119906119906119904(119909) + 120601

119904119908119904(120591) + 120601

119905119905119904(120591) (8)







119908119904(120591) can be given by

119908119904(120591) = 120572ℎ (120591) forall119904 = 1 2 119873 (120591) (9)




119905119904(120591) = 119879

1199041(120591) + 119879

1199042(120591) forall119904 = 1 2 119873 (120591) (10)

where

1198791199041

(120591) =119863119904

119881119905(120591)

forall119904 = 1 2 119873 (120591)

1198791199042

= 1205730(119873 (120591) + 1 minus 119904) forall119904 = 1 2 119873 (120591)

(11)






119903 (119909 120591) = 120572119904[1 +

120573119904119875 (119909 120591)

(119867 (119909 120591) minus 119875 (119909 120591))] (12)


119904






119875 (119909 120591) = 1205781205881198920(120591) (1 minus 119890119909)


(13)



0(120591)


119902 (119909 119904 120591) = 119875 (119909 120591) exp (minus120579119888 (119909 119904 120591))

forall119909 isin [0 119861] 119904 = 1 2 119873 (120591)

(14)



SW =

119872

sum

120591=1

(119875120591+ CS120591) (15)






119875120591= 119877120591minus 119862120591 (16)



120591is the



119877120591=

119873

sum

119904=1

119891119904(120591)119876119904120591

(1 + 119894)120591minus1

(17)





119876119904120591

= int

119871119904minus1

119871119904

119902 (119909 119904 120591) 119889119909 forall119904 = 1 2 119873 (120591) (18)


119904is commuter



119871119904=

(119863119904+ 119863119904+1

)

2 forall119904 = 1 2 119873 (120591) (19)



0holds

119902 (119871120591

0 1 120591) = 0 119871

120591

0isin [119863120591

1 119861] (20)

where 119902(119871120591


location 119871120591



by

1198710=

1

119890 (21)


119904120591can be

rewritten as

119876119904120591

= 1198750(120591) int

119871119904minus1

119871119904

exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

minus 1198901198750(120591) int

119871119904minus1

119871119904

119909 exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

= minus1198750(120591) 119881119905(120591)

120579120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] +

1198901198750(120591) 1198812

119905(120591)

12057921206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(22)

where120582119904(120591) = minus120579119891

119904(120591) minus 120579120601

119906119906119904(119909) minus 120579120601

119904119908119904(120591) minus 120579120601

1199051205730

times (119873 (120591) + 1 minus 119904) + 120579120601119905

119906 (119909)119881119908

119881119905(120591)

minus 120579120585

forall119904 = 1 2 119873 (120591)

(23)






119871120591


119862120591= 119862119900120591

+ 119862119871120591

+ 119862119904120591

(24)


is given by

119862119900120591

=(120583119900+ 1205831119865 (120591))

(1 + 119894)120591minus1

(25)





119865 (120591) =119879119877(120591)

ℎ (120591) (26)



119879119877

= 120589119879119900+ 2 (119879

11+ 11987912) (27)










119862119871120591

= 12057411198631(1 + 120594)

120591minus1

+1205740

(1 + 119894)120591minus1

(28)







119862119904120591

= 1205810(1 + 120594)

120591minus1

+1205811(119873 (120591) + 1)

(1 + 119894)120591minus1

(29)






[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)


CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)



CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)


max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)


119873(120591) 119863

2 119863120591

1) 119876119904120591




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)


1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Corridor boundary

CBD

NN+ 1

Ls

Ds

Lsminus1

Ds+1

D1

B

Gs Es

ss + 1 s minus 1 1





0(120591)




119861

0

1198920(120591)(1 minus 119890119909) 119889119909 [2]










3 Model Formulation



119909119904120591= min PV

119909119904120591 forall119909 isin [0 119861]

119904 = 1 2 119873 (120591) 120591 isin [1119872]

(1)


where PV119909119904120591





PV119909119904120591

= 120587 (119901 (119909 120591)) 119881 (Δ119903119888 (119909 119904 120591)) (2)

120587 (119901 (119909 120591)) =119901(119909 120591)

120574

[119901(119909 120591)120574

+ (1 minus 119901(119909 120591))1120574

]

(3)

119881 (Δ119903119888 (119909 119904 120591)) = Δ119903119888(119909 119904 120591)

1205721015840

Δ119903119888 (119909 119904 120591) ge 0

minus120582(minusΔ119903119888 (119909 119904 120591))1205731015840

Δ119903119888 (119909 119904 120591) lt 0

(4)


119888 (119909 119904 120591) = 119888 (119909 119904 120591) + 120585 (6)

119903 (119909 120591) = 119903 (119909 120591) + 120595 (7)



119888 (119909 119904 120591) = 119891119904(120591) + 120601

119906119906119904(119909) + 120601

119904119908119904(120591) + 120601

119905119905119904(120591) (8)







119908119904(120591) can be given by

119908119904(120591) = 120572ℎ (120591) forall119904 = 1 2 119873 (120591) (9)




119905119904(120591) = 119879

1199041(120591) + 119879

1199042(120591) forall119904 = 1 2 119873 (120591) (10)

where

1198791199041

(120591) =119863119904

119881119905(120591)

forall119904 = 1 2 119873 (120591)

1198791199042

= 1205730(119873 (120591) + 1 minus 119904) forall119904 = 1 2 119873 (120591)

(11)






119903 (119909 120591) = 120572119904[1 +

120573119904119875 (119909 120591)

(119867 (119909 120591) minus 119875 (119909 120591))] (12)


119904






119875 (119909 120591) = 1205781205881198920(120591) (1 minus 119890119909)


(13)



0(120591)


119902 (119909 119904 120591) = 119875 (119909 120591) exp (minus120579119888 (119909 119904 120591))

forall119909 isin [0 119861] 119904 = 1 2 119873 (120591)

(14)



SW =

119872

sum

120591=1

(119875120591+ CS120591) (15)






119875120591= 119877120591minus 119862120591 (16)



120591is the



119877120591=

119873

sum

119904=1

119891119904(120591)119876119904120591

(1 + 119894)120591minus1

(17)





119876119904120591

= int

119871119904minus1

119871119904

119902 (119909 119904 120591) 119889119909 forall119904 = 1 2 119873 (120591) (18)


119904is commuter



119871119904=

(119863119904+ 119863119904+1

)

2 forall119904 = 1 2 119873 (120591) (19)



0holds

119902 (119871120591

0 1 120591) = 0 119871

120591

0isin [119863120591

1 119861] (20)

where 119902(119871120591


location 119871120591



by

1198710=

1

119890 (21)


119904120591can be

rewritten as

119876119904120591

= 1198750(120591) int

119871119904minus1

119871119904

exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

minus 1198901198750(120591) int

119871119904minus1

119871119904

119909 exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

= minus1198750(120591) 119881119905(120591)

120579120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] +

1198901198750(120591) 1198812

119905(120591)

12057921206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(22)

where120582119904(120591) = minus120579119891

119904(120591) minus 120579120601

119906119906119904(119909) minus 120579120601

119904119908119904(120591) minus 120579120601

1199051205730

times (119873 (120591) + 1 minus 119904) + 120579120601119905

119906 (119909)119881119908

119881119905(120591)

minus 120579120585

forall119904 = 1 2 119873 (120591)

(23)






119871120591


119862120591= 119862119900120591

+ 119862119871120591

+ 119862119904120591

(24)


is given by

119862119900120591

=(120583119900+ 1205831119865 (120591))

(1 + 119894)120591minus1

(25)





119865 (120591) =119879119877(120591)

ℎ (120591) (26)



119879119877

= 120589119879119900+ 2 (119879

11+ 11987912) (27)










119862119871120591

= 12057411198631(1 + 120594)

120591minus1

+1205740

(1 + 119894)120591minus1

(28)







119862119904120591

= 1205810(1 + 120594)

120591minus1

+1205811(119873 (120591) + 1)

(1 + 119894)120591minus1

(29)






[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)


CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)



CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)


max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)


119873(120591) 119863

2 119863120591

1) 119876119904120591




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)


1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where PV119909119904120591





PV119909119904120591

= 120587 (119901 (119909 120591)) 119881 (Δ119903119888 (119909 119904 120591)) (2)

120587 (119901 (119909 120591)) =119901(119909 120591)

120574

[119901(119909 120591)120574

+ (1 minus 119901(119909 120591))1120574

]

(3)

119881 (Δ119903119888 (119909 119904 120591)) = Δ119903119888(119909 119904 120591)

1205721015840

Δ119903119888 (119909 119904 120591) ge 0

minus120582(minusΔ119903119888 (119909 119904 120591))1205731015840

Δ119903119888 (119909 119904 120591) lt 0

(4)


119888 (119909 119904 120591) = 119888 (119909 119904 120591) + 120585 (6)

119903 (119909 120591) = 119903 (119909 120591) + 120595 (7)



119888 (119909 119904 120591) = 119891119904(120591) + 120601

119906119906119904(119909) + 120601

119904119908119904(120591) + 120601

119905119905119904(120591) (8)







119908119904(120591) can be given by

119908119904(120591) = 120572ℎ (120591) forall119904 = 1 2 119873 (120591) (9)




119905119904(120591) = 119879

1199041(120591) + 119879

1199042(120591) forall119904 = 1 2 119873 (120591) (10)

where

1198791199041

(120591) =119863119904

119881119905(120591)

forall119904 = 1 2 119873 (120591)

1198791199042

= 1205730(119873 (120591) + 1 minus 119904) forall119904 = 1 2 119873 (120591)

(11)






119903 (119909 120591) = 120572119904[1 +

120573119904119875 (119909 120591)

(119867 (119909 120591) minus 119875 (119909 120591))] (12)


119904






119875 (119909 120591) = 1205781205881198920(120591) (1 minus 119890119909)


(13)



0(120591)


119902 (119909 119904 120591) = 119875 (119909 120591) exp (minus120579119888 (119909 119904 120591))

forall119909 isin [0 119861] 119904 = 1 2 119873 (120591)

(14)



SW =

119872

sum

120591=1

(119875120591+ CS120591) (15)






119875120591= 119877120591minus 119862120591 (16)



120591is the



119877120591=

119873

sum

119904=1

119891119904(120591)119876119904120591

(1 + 119894)120591minus1

(17)





119876119904120591

= int

119871119904minus1

119871119904

119902 (119909 119904 120591) 119889119909 forall119904 = 1 2 119873 (120591) (18)


119904is commuter



119871119904=

(119863119904+ 119863119904+1

)

2 forall119904 = 1 2 119873 (120591) (19)



0holds

119902 (119871120591

0 1 120591) = 0 119871

120591

0isin [119863120591

1 119861] (20)

where 119902(119871120591


location 119871120591



by

1198710=

1

119890 (21)


119904120591can be

rewritten as

119876119904120591

= 1198750(120591) int

119871119904minus1

119871119904

exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

minus 1198901198750(120591) int

119871119904minus1

119871119904

119909 exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

= minus1198750(120591) 119881119905(120591)

120579120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] +

1198901198750(120591) 1198812

119905(120591)

12057921206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(22)

where120582119904(120591) = minus120579119891

119904(120591) minus 120579120601

119906119906119904(119909) minus 120579120601

119904119908119904(120591) minus 120579120601

1199051205730

times (119873 (120591) + 1 minus 119904) + 120579120601119905

119906 (119909)119881119908

119881119905(120591)

minus 120579120585

forall119904 = 1 2 119873 (120591)

(23)






119871120591


119862120591= 119862119900120591

+ 119862119871120591

+ 119862119904120591

(24)


is given by

119862119900120591

=(120583119900+ 1205831119865 (120591))

(1 + 119894)120591minus1

(25)





119865 (120591) =119879119877(120591)

ℎ (120591) (26)



119879119877

= 120589119879119900+ 2 (119879

11+ 11987912) (27)










119862119871120591

= 12057411198631(1 + 120594)

120591minus1

+1205740

(1 + 119894)120591minus1

(28)







119862119904120591

= 1205810(1 + 120594)

120591minus1

+1205811(119873 (120591) + 1)

(1 + 119894)120591minus1

(29)






[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)


CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)



CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)


max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)


119873(120591) 119863

2 119863120591

1) 119876119904120591




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)


1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591is the



119877120591=

119873

sum

119904=1

119891119904(120591)119876119904120591

(1 + 119894)120591minus1

(17)





119876119904120591

= int

119871119904minus1

119871119904

119902 (119909 119904 120591) 119889119909 forall119904 = 1 2 119873 (120591) (18)


119904is commuter



119871119904=

(119863119904+ 119863119904+1

)

2 forall119904 = 1 2 119873 (120591) (19)



0holds

119902 (119871120591

0 1 120591) = 0 119871

120591

0isin [119863120591

1 119861] (20)

where 119902(119871120591


location 119871120591



by

1198710=

1

119890 (21)


119904120591can be

rewritten as

119876119904120591

= 1198750(120591) int

119871119904minus1

119871119904

exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

minus 1198901198750(120591) int

119871119904minus1

119871119904

119909 exp(minus120579120601119905

119881119905(120591)

119909 + 120582119904(120591)) 119889119909

= minus1198750(120591) 119881119905(120591)

120579120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))] +

1198901198750(120591) 1198812

119905(120591)

12057921206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(22)

where120582119904(120591) = minus120579119891

119904(120591) minus 120579120601

119906119906119904(119909) minus 120579120601

119904119908119904(120591) minus 120579120601

1199051205730

times (119873 (120591) + 1 minus 119904) + 120579120601119905

119906 (119909)119881119908

119881119905(120591)

minus 120579120585

forall119904 = 1 2 119873 (120591)

(23)






119871120591


119862120591= 119862119900120591

+ 119862119871120591

+ 119862119904120591

(24)


is given by

119862119900120591

=(120583119900+ 1205831119865 (120591))

(1 + 119894)120591minus1

(25)





119865 (120591) =119879119877(120591)

ℎ (120591) (26)



119879119877

= 120589119879119900+ 2 (119879

11+ 11987912) (27)










119862119871120591

= 12057411198631(1 + 120594)

120591minus1

+1205740

(1 + 119894)120591minus1

(28)







119862119904120591

= 1205810(1 + 120594)

120591minus1

+1205811(119873 (120591) + 1)

(1 + 119894)120591minus1

(29)






[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)


CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)



CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)


max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)


119873(120591) 119863

2 119863120591

1) 119876119904120591




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)


1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














[119902(119909 119904 120591)]minus1

(119902 (119909 119904 120591)) = 119888 (119909 119904 120591) + 120585 =1

120579ln 119875 (119909 120591)

119902 (119909 119904 120591)

(30)


CS (119909 119904 120591) = int

119902(119909119904120591)

0

[119902(119909 119904 120591)]minus1

(119908) 119889119908

minus 119902 (119909 119904 120591) (119888 (119909 119904 120591) + 120585) =119902 (119909 119904 120591)

120579

(31)



CS120591=

int1198710

0

CS (119908 119904 120591) 119889119908

(1 + 119894)120591minus1

(32)


max SW (D ℎ (120591) 119891119904(120591))

=

119872

sum

120591=1

119873(120591)

sum

119904=1

(119891119904(120591) 119876119904120591

)

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[1205830+ (1205831ℎ (120591)) (120589119879

0+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))]

(1 + 119894)120591minus1

minus

119872

sum

120591=1

[

[

(1205740+ 1205811(119873 (120591) + 1))

(1 + 119894)120591minus1

minus (1 + 120594)120591minus1

times (1205741119863120591

1+ 1205810) +

int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

(1 + 119894)120591minus1

]

]

(33)


119873(120591) 119863

2 119863120591

1) 119876119904120591




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119904+1

sum

119894=119904minus1

119891119904(120591)

(1 + 119894)120591minus1

120597119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891119904(120591)sum119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(34)


1 Δ1199042 and Δ

3are

given by

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591))

minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1205791206012

119905+ 119890120579120601

119905120601119904119881119905(120591) + 119890120579120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(35)

and 120597119876119894120591120597119863119904are given by

120597119876119904minus1120591

120597119863119904

= minus1198750(120591)

2exp (minus120579119888 (119871

119904minus1 119904 120591)) +

1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) exp (minus120579119888 (119871

119904minus1 119904 120591))]

forall119904 = 2 119873 (120591)


1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205971198761120591

120597119863120591

1

= minus1198750(120591)

2exp (minus120579119888 (119871

1 119904 120591))

+1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

exp (minus120579119888 (1198711 119904 120591))

120597119876119904120591

120597119863119904

=1198750(120591)

2[exp (minus120579119888 (119871

119904minus1 119904 120591))


1198901198750(120591) 119881119905(120591)

2120579120601119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+1

2) sdot exp (minus120579119888 (119871

119904minus1 119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 2 119873 (120591)

120597119876119904+1120591

120597119863119904

=1198750(120591)

2exp (minus120579119888 (119871

119904+1 119904 120591)) minus

1198901198750(120591) 119881119905(120591)

2120579120601119905

sdot [(120579120601119905

119881119905(120591)

119871119904+

1

2) exp (minus120579119888 (119871

119904 119904 120591))]

forall119904 = 1 119873 (120591) minus 1

(36)



120591

1and










4 Numerical Example




5 Conclusions







Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 Parameters



120601119906120601119905120601119904




119881119908





















Appendices




120597SW (sdot)

120597119863119904

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597119863119904

minus Δ119904

times

119872

sum

120591=1

[

[

(1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)


+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A1)where Δ


(23) 119876119904120591



119863119904and 119863

119904+1 namely

119876119904120591

= 119876119904120591

(119863119904minus1

119863119904 119863119904+1


120597119876119894120591

120597119863119904

= 0 forall119904 = 119904 minus 1 119904 119904 + 1 (A3)


0

0



120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=

120597int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

+

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

(A4)Since

120597 int119863119904

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

=119902 (119863119904 119904 120591)

120579

forall119904 = 1 2 119873 (120591)

120597 int119871120591

0

119863120591

1

(119902 (119908 119904 120591) 120579) 119889119908

120597119863120591

1

=1

120579[119902 (1198710 1 120591)

120597119871120591

0

120597119863120591

1

minus 119902 (119863120591

1 1 120591)] = minus

119902 (119863120591

1 1 120591)

120579

120597 int119871120591

0

119863119904

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= minus119902 (119863119904 119904 120591)

120579

forall119904 = 2 3 119873 (120591)

(A5)we have

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119863119904

= 0 forall119904 = 1 2 119873 (120591) (A6)


119891119904(120591)

(1 + 119894)120591minus1

sum119904+1

119894=119904minus1120597119876119894120591

120597119863119904

minus Δ119904(

1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

forall119904 = 1 119873 (120591)

(A7)


120597SW (sdot)

120597ℎ (120591)

=

119872

sum

120591=1

119891119904(120591)

(1 + 119894)120591minus1

120597sum119873

119894=1119876119894120591

120597ℎ (120591)

+

119872

sum

120591=1

[

[

(1205831ℎ2

(120591)) (1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

(1 + 119894)120591minus1

+1

(1 + 119894)120591minus1

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

]

]

= 0

forall119904 = 1 2 119873 (120591)

(A8)

From (13) 119876119904120591


119904 120591) and 119888(119871119904 119904 120591) and



0(120591) we have

120597119876119904120591

120597ℎ (120591)=

1205721198750(120591) 120601119904119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1205721198901198750(120591) 1206011199041198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

(A9)


0

0


120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597ℎ (120591)

= int

119871120591

0

0


= minus1205721206011199041198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1205721206011199041198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))


times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1205791206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A10)


ℎ (120591) = radic1205831(1205891198790+ (2119863

120591

1119881119905(120591)) + 2120573

0119873(120591))

120572120601119904[Δ3minus 119891sum

119873(120591)

119894=1(Δ119904

1minus Δ119904

2)]

(A11)

where

Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

times [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

Δ3= 1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A12)


119904(120591) is

120597SW (sdot)

120597119891119904(120591)

=

119872

sum

120591=1

1

(1 + 119894)120591minus1

times (

119873(120591)

sum

119894=1

119876119894120591

+ 119891119904(120591)

120597sum119873(120591)

119894=1119876119894120591

120597119891119904(120591)

+

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

) = 0

(A13)

where

120597119876119904120591

120597119891119904(120591)

=1198750(120591) 119881119905(120591)

120601119905

[exp (minus120579119888 (119871119904minus1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))]

minus1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

sdot [(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

120597 int119871120591

0

0

(119902 (119908 119904 120591) 120579) 119889119908

120597119891119904(120591)

= int

119871120591

0

0


= minus1198750(120591) exp (minus120579119888 (119908 119904 120591))

times [1 +1198901206011199041198812

119905(120591)

1205791206012

119905

(120579120601119905

119881119905(120591)

119908 + 1)]

100381610038161003816100381610038161003816100381610038161003816

119871120591

0

0

= minus1198750(120591) [exp (minus120579119888 (119871

120591

0 119873 (120591) 120591))

times1206012

119905+ 119890120601119905120601119904119881119905(120591) + 119890120601

1199041198812

119905(120591)

1206012

119905

minus exp (minus120579119888 (0 0 120591))1205791206012

119905+ 1198901206011199041198812

119905(120591)

1205791206012

119905

]

(A14)

Thus

119891119904(120591) =

sum119872

120591=1sum119873(120591)

119894=1119876119894120591

minus sum119872

120591=1Δ3

sum119872

120591=1sum119873(120591)

119894=1(Δ119904

2minus Δ119904

1)

(A15)

where Δ119904

1 Δ1199042 and Δ





1and rail station


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))]

+ 119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

sdot (120579120601119905

119881119905(120591)

1198711+

1

2) exp (minus120579119888 (119871

1 119904 120591))


minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741) = 0

(B1)

exp (minus120579119888 (119871119904+1

119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591)) = 0

forall119904 = 2 119873 (120591)

(B2)





(119863120591

1)lowast

= minus119881119905(120591)

120579120601119905

times ln((1

(1 + 119894)120591minus1

21205831

ℎ (120591) 119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891(1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times120579120601119905

119881119905(120591)

1198711+

1

2)

minus1

)

minus119881119905(120591)

120601119905

(119891119904(120591) + 120601

119906119906119904(1198711) + 120601119904120572ℎ (120591) + 120601

1199051205730119873(120591) + 120585)

0 lt (1

(1 + 119894)120591minus1

21205831

ℎ (120591)119881119905(120591)

+ (1 + 120594)120591minus1

1205741)

times (119891119904(120591) (

1198901198750(120591) 119881119905(120591) (119881

119905(120591) minus 120579120601

119905)

212057921206012

119905

minus1198901198750(120591) 119881119905(120591)

2120579120601119905

)

times (120579120601119905

119881119905(120591)

1198711+

1

2))

minus1

lt 1

(B3)


1)+ we could have

(119863120591

1)lowast

lt (119863120591

1)+

(B4)


2 119904 120591)) minus exp(minus120579119888(119871

1



119888 (1198712 119904 120591) minus 119888 (119871

1 119904 120591) lt 0 (B5)


119891119904(120591) 1198750(120591)

2[exp (minus120579119888 (119871

2 119904 120591)) minus exp (minus120579119888 (119871

1 119904 120591))] gt 0

(B6)



Δ119904

1=

1198750(120591) 119881119905(120591)

120601119905


119904 120591)) minus exp (minus120579119888 (119871119904 119904 120591))] = 0

Δ119904

2=

1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

[(120579120601119905

119881119905(120591)

119871119904minus1

+ 1) exp (minus120579119888 (119871119904minus1

119904 120591))

minus (120579120601119905

119881119905(120591)

119871119904+ 1) exp (minus120579119888 (119871

119904 119904 120591))]

=1198901198750(120591) 1198812

119905(120591)

1205791206012

119905

120579120601119905

119881119905(120591)

(119871119904minus1

minus 119871119904)


(B7)


lt (ℎ(120591))+ and

(119891119904(120591))lowast





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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Research Article Exploring the Impact of Commuter s ...downloads.hindawi.com/journals/mpe/2014/536872.pdf · the density falls as the distance increases. Here, >0 represents the