Research Article Allocating Freight Empty Cars in Railway Networks with Dynamic Demands

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1 Discrete Dyamics i Nature ad Society, Article ID , 12 pages Research Article Allocatig Freight Empty Cars i Railway Networks with Dyamic Demads Ce Zhao, Lixig Yag, ad Shukai Li State ey Laboratory of Rail Traffic Cotrol ad Safety, Beijig Jiaotog Uiversity, Beijig , Chia Correspodece should be addressed to Lixig Yag; lxyag@bjtu.edu.c Received 24 March 2014; Accepted 22 July 2014; Published 1 September 2014 Academic Editor: Agacik Zafer Copyright 2014 Ce Zhao et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper ivestigates the freight empty cars allocatio problem i railway etworks with dyamic demads, i which the storage cost, uit trasportatio cost, ad demad i each stage are take ito cosideratio. Uder the costraits of capacity ad demad, a stage-based optimizatio model for allocatig freight empty cars i railway etworks is formulated. The objective of this model is to miimize the total cost icurred by trasferrig ad storig empty cars i differet stages. Moreover, a geetic algorithm is desiged to obtai the optimal empty cars distributio strategies i railway etworks. Fially, umerical experimets are give to show the effectiveess of the proposed model ad algorithm. 1. Itroductio Freight empty cars allocatio aims to distribute the available empty cars from origis to destiatios i the railway etworks so that the demads ca be satisfied with the miimized shipmet costs. Sice the railway freight trasportatio plays a importat role i moder society, reasoable allocatio of the empty cars with systematic optimizatio is actually a crucial issue for the railway compaies. I recet decades, a lot of researchers have ivestigated freight empty cars allocatio problem ad preseted a variety of models ad algorithms.i literature,misra [1] firstly proposed this problem i 1972 ad set up a simple freight empty trai allocatio model i a time period with the miimum cost cosumptio. Trasportatio algorithm ad simplex method were also desiged to search for the optimal solutio of the proposed model. Grai ad Siay [2]figuredoutthefreightemptycar allocatio problem by a liear programmig model, i which the objective was to arrage car flow assigmet to satisfy the maximum proceeds. They adopted trasportatio algorithm adsimplexmethodforthesolutiosofthismodel.haghai [3] coverted the freight empty car allocatio problem ito a ivetory problem i cargo operatio statios i order to allocate the freight cars i these statios. ikuchi [4] structured the problem as a trasshipmet problem uder the cocept that a fleet of sigle-purpose freight cars were pooled at may loadig poits ad emptied at uloadig termials i order to reduce empty car miles ad time. Liu [5] firstly applied the computer techology to solve the freight empty car allocatio problem, which provided techical guidace for the future study. Moreover, Xiog et al. [6] propouded a effective geetic algorithm to solve the large-scale etwork empty cars allocatio problem ad desiged a adapted matrix codig mode, crossover operator, ad mutatio operator to improve the searchig performace of the geetic algorithm. Besides the above metioed researches, recet studiesalwaysformulatedthisproblembasedothestadard trasportatio problem ad traffic flow problem. As for the trasportatio problem based methods, Liu [7] ivestigated both the balaced trasportatio model ad the ubalaced trasportatio model for the problem, which were solved by miimum elemet methods. X. Zhag ad Q. Zhag [8] cosidered the problem based o the balaced trasportatio problem with the ifluece of the costrait coditios. Liag et al. [9] preseted the cocepts of alterative vehicles i ubalaced trasportatio model ad costructed a optimizatio model with maximized profit. Lei et al. [10] itroduced a stochastic programmig model with probability chace-costraits ad desiged a efficiet geetic algorithm to seek a approximate optimal solutio. Narisetty et al. [11] proposed a optimizatio model for the empty car

2 2 Discrete Dyamics i Nature ad Society Problem descriptio Formulatio Table 1: The detailed features of differet studies. Misra [1] ikuchi [4] Lei et al. [10] Joboretal.[17] ThisPaper A simple trasportatio problem A liear programmig model Atrasshipmet problem A improved liear programmig model Abalaced trasportatio problem Astochastic programmig model A etwork flow problem A capacitated etwork desig model Atime-stageetwork flow problem A dyamic etwork flow model Network Physical etwork Physical etwork Physical etwork Time-space etwork Time-space etwork Characteristics Sigle-stage demad Sigle-stage demad Sigle-stage demad Multi-stage demad Algorithm The simplex method GA (geetic GA (geetic TS (tabu search) algorithm) algorithm) assigmet problem based o the customer demad, which was implemeted at Uio Pacific Railroad to assig empty freight cars. The proposed approaches ca further reduce trasportatio costs ad improve delivery rate efficiecy ad customer satisfactio. More recetly, some researchers developed etwork flow models ad optimizatio methods which could be regarded as refereces for the freight empty car allocatio problem. For istace, Wag et al. [12] cosidered freight empty car allocatio problem o the basis of quatity distributio ad etwork matchig flow ad the desiged at coloy algorithm to solve the proposed model. Laporte et al. [13] applied a game theoretic framework to research the railway trasit etwork desig problem. Dorfma ad Medaic [14] developed a discrete evet model for the feedback travel advace strategy, which ca quickly hadle perturbatios i the schedule. Erkut ad Gzara [15] applied the etwork flow strategy to hazardous material trasportatio ad proposed a heuristic solutio method to search a stable solutio. I a large-scale railway etwork, Guo et al. [16] proposedthe methods to merge the traffic flow i the mai support statios ad reduce odes i railway etworks, which ca improve the efficiecy of operatios. To uderstad the cotributios of this research clearly, thedetailedfeatureswillbesummarizeditable 1 i compariso with some related works. It is easy to see i the literature that the majority of researches ivestigate this problem i certai eviromets, i which all the parameters are assumed to be static. I practice, sice railway traffic is a complicated system, the demads of empty freight cars are always dyamic i most circumstaces. To clearly state this characteristic, i this study, we particularly treat the demad at each destiatio as a stage-based demad, i which oe stage correspods to a prespecifiedtimeperiod.the,weformulatetheproblemasa mathematical optimizatio model with multistage demads. Also, a geetic algorithm is particularly desiged for the problem to efficietly geerate the approximate optimal solutio of the model. The rest of this paper is orgaized as follows. I Sectio 2, a optimizatio model for freight empty cars allocatio i railway etworks ad some prelimiaries are preseted. I Sectio 3, a geetic algorithm is desiged to fid a optimal solutio for freight empty cars allocatio problem. Some istacesaregivetoshowtheeffectiveessofthemodel ad algorithm i Sectio 4. Coclusiosarefiallymadei Sectio Formulatio of the Problem 2.1. Problem Statemet. Allocatig empty cars is a importat operatio for railway compaies to guaratee the ormal operatios of trasportatio activity. I traditioal operatio modes, empty cars are i geeral required to be trasferred to the destiatio accordig to the prespecified demad that is always treated as a fixed quatity. However, due to the ucertaity of decisio eviromets, the demad of freight empty cars at the destiatio is always chageable i the realworldapplicatios(amely,theeedsvaryidifferettime periods). Thus, the traditioal model will potetially cause a large umber of empty backlogs or supply shortage, leadig to the trasportatio task beig uable to be accomplished effectively. Aimig to provide a framework for practical decisios, i this research we are particularly iterested i the formulatio of the model with dyamic demads ad effective algorithms for the freight empty cars allocatio problem. I dyamic case, the time horizo will be divided ito a variety of time periods, deoted by stages. The, some importat parameters i the problem will be recosidered as the stage-based quatities, icludig the demad, cost, capacity, ad so forth. For differet stages, all of these parameters ca be differet accordig to the practical requiremets, whiletheyareassumedtobefixedquatitieswithieach givestage.toformulatethedyamicemptycarsallocatio problem, we eed to specify the followig three types of costraits. (1) The first costrait cocers the traffic flow volume from supply statios, which is determied by the predicted demads i orgaizatio plas ad trai schedules. I particular, the traffic flow volume is required to be cosistet with the supply capacity of each supply statio. (2) The secod costrait refers to the trasportatio capacities of statios ad railway liks. The total traffic flow o these statios ad liks caot exceed the correspodig capacities. (3) The last costrait is associated with the demad of traffic flow volume.thatis,allthedyamicdemadseedtobesatisfied i each stage. To clearly state the cosidered problem, we shall give a illustratio i Figure 1, where statios 1 ad 2, respectively, represet the supply statio ad demad statio i the

3 Discrete Dyamics i Nature ad Society 3 R=50 D 1 =20 D 2 = Figure 1: A illustratio of delivery plas. 1 3 Figure 2: A illustratio of a simple trasportatio etwork. railway etwork. We here cosider the followig scearios: i day 1, the demad of the destiatio is 20 empty cars, while the demad is 30 empty cars i day 2. Without cosiderig stage-based demads, a total of 50 empty cars will be delivered to statio 2 i day 1, leadig to the waste of resources ad the storage cost for uoccupied cars. O the other had, if the demad is divided ito two stages associated with differet days, that is, D 1 =20ad D 2 =30, the the extra cost ca be avoided if we allocate the empty cars accordig to the stage-based demads. Hereiafter, some assumptios will be give i order to formulate the mathematical model. (A1) I order to characterize the cosidered railway etwork,aabstractgraph,deotedbyg(n, A), will be extracted from the physical etwork (see a illustratio i Figure 2), i which N deotes the set of odes represetig the railway statios, ad A is the set of directed liks betwee adjacet odes. Correspodig to the differet stages, stage-based trasportatio cost will be treated as the weight of each lik. Directed liks idicate the trasportatio directio betwee two odes. (A2) The demad, storage cost, ad uit trasportatio cost i the cosidered time horizo are all dyamic, while these parameters i each stage are treated as costats. (A3) I the railway etwork, the total supply capacity i origial statios is supposed to be greater tha the total demad i destiatios to guaratee the trasportatio activities. (A4)Thedemadieachstageshouldbesatisfied.Uused empty cars ca be left for the ext stage with a extra storage cost. Here, to explicitly demostrate the empty cars allocatio process, a illustrative etwork, which cosists of three ode 5 4 Table 2: Parameters used i formulatio. Parameters Defiitio A The set of liks; N The set of odes; i, j The idex of odes; (i, j) The idex of lik from ode i to ode j; A The total umber of liks; N The total umber of odes; k The differet stages, k = 1, 2,..., ; c k The uit trasportatio cost o lik (i, j) i stage ij k; o Origi statio which supplies empty cars, s s = 1, 2,..., S; R s The supply amout of empty cars i statio o s ; d Destiatio statio which requires empty trais, t t = 1, 2,..., T; D t The total demad of empty cars i statio d t ; Q ij The lik capacity o lik (i, j); G i The turover capacity at statio i; e k The uit storage cost for stage k; The demad of empty cars at statio d t for stage k. U k t adthreeliks,isgiveifigure 3. To show the dyamics i the etwork, the time horizo is firstly divided ito four stages. The the right side of Figure 3 depicts the space-time etwork for the empty car allocatig process, where each ode represets the state of a physical ode i each stage (for more detailed descriptio for space-time etworks, iterested readers may refer to Yag ad Zhou [18], Yag et al. [19]). Suppose that odes a ad c, respectively, deote the supply statio ad demad statio. The, two paths ca be employed fortheoperatios,thatis,a b c ad a c, asshowithearrowliks.ithefirststage,therewillbe some empty cars required to be trasferred to statio c from statio a. If the trasferred cars just satisfy the demad at statio c, o storage cost occurs. Otherwise, if the umber of trasferred cars is greater tha the demad at statio c, the the uused empty cars should be delayed at this statio, leadig to a extra storage cost. The dotted arrow liks i Figure 3 represet the waitig arcs for the uused empty cars at statio c. Likewise, i the followig stages, it is required that the summatio of trasferred cars ad stored cars of the previous stages should also satisfy the stage-based demads. The, i the process of allocatig empty cars, the total cost ca be separated ito two parts: the trasportatio cost ad storage cost. The aim is to geerate the optimal allocatio strategies i railway etworks Mathematical Model. I this sectio, we shall formulate the freight empty cars allocatio problem with dyamic demads as a stage-based etwork flow problem below Parameters ad Notatios. To formulate the mathematical model for this problem, some relevat parameters are firstly listed i Table 2.

4 4 Discrete Dyamics i Nature ad Society c should be equal to the outflow cars at each statio. We the have b x k ij = x k ji, (i,j) A (j,i) A i N/{o s,d t,s=1,2,...,s,t=1,2,...,t}, k=1,2,...,. (3) a Stage 1 Stage 2 Stage 3 Stage 4 Directed arc Waitig arc Figure 3: A illustratio of a space-time etwork Decisio Variables. I this problem, the purpose is to geerate a optimal trasportatio pla over the etire time horizo. As the multistage strategy will be cosidered, the decisio variable ca be treated as the amout of empty car flow correspodig to each lik ad stage. The, we have the decisio variable as x k ij : deotes the umber of empty cars delivered o lik (i, j) for stage k, satisfyig x k ij Z+ {0},where Z + deotes the set of positive itegers. With this variable, the freight empty car allocatio problem ca be essetially treated as a traffic flow assigmet process with the followig costrait coditios System Costraits. To guaratee the feasibility of the allocatio plas, we eed to cosider some system costraits i the process of allocatig empty cars. Specifically, there are six types of system costraits i this model, which are preseted as follows. (i) The Supply Capacity Costrait. The available empty cars i supply statio o s should be more tha or equal to the cars thataredispatchedfromthisstatioo s.wethehavethe followig costrait: (o s,j) A k=1 x k o s j R s, s=1,2,...,s. (1) (ii) The Demad Costrait. Whe the flow is delivered at the statio d t, it is required that the delivered flow volume (i,dt ) A k=1 xk id t should meet the total demad D t,which caberepresetedas (i,d t ) A k=1 x k id t D t, t=1,2,...,t. (2) (iii) The Flow Balace Costrait. To keep the balace of the totalflow,itisrequiredthattheamoutoftheiflowcars (iv) The Lik Capacity Costrait. I geeral, the trasportatio capacity o each lik is ot fiite. To show this characteristic i the formulatio, we particularly cosider the lik capacity costrait. That is, for each lik (i, j), the flow volume caot exceed the trasportatio capacity Q ij,which ca be represeted as k=1 x k ij Q ij, (i, j) A. (4) (v) The Turover Capacity Costrait. Accordig to the realworld applicatios, a turover capacity costrait should also be cosidered at each statio. The, at a trasfer statio i,the volume of traffic flow caot exceed the turover capacity G i. The costrait ca be represeted as follows: (i,j) A k=1 x k ij G i, i N/{o s,d t,s=1,2,...,s,t=1,2,...,t}. (vi) The Stage-Based Demad Costrait. As the total demad is separated as stage-based demad, the flow volume delivered to each destiatio ad its stored empty cars should satisfy the demad U k t at statio d t for each stage; amely, if k=1, there will be oe costrait: (5) x 1 id t U 1 t, t=1,2,...,t. (6) (i,d t ) A Ad if k=2, the costraits should be x 1 id t U 1 t, t=1,2,...,t, (i,d t ) A (7) (x 1 id t +x 2 id t ) U 1 t +U2 t, t=1,2,...,t. (i,d t ) A Cosiderig all the stages, we formulate these costraits as follows: t=1,2,...,t, x k id t (i,d t ) A k=1 k=1 U k t, =1,2,...,. With the stage-based demad costrait, we ca esure that thedemadcabesatisfiedieachstage.whe=,itis easy to see that (8) is the same to (2). The,ithefollowig discussio, we will ot explicitly cosider (2). (8)

5 Discrete Dyamics i Nature ad Society Objective Fuctio. This problem aims to fid the optimal allocatio strategies with the miimized cost. As addressed i the illustratio, the total cost i essece cosists of two parts, amely, delivery cost i the etwork ad storage cost i destiatios. The delivery cost refers to the expeses i the process of deliverig empty cars. Sice i stage k, the flow volume o lik (i, j) is deoted by x k ij,cosiderigallthestagesadliks, we fially formulate the total delivery cost f 1 (X) below: f 1 (X) = k=1 (i,j) A c k ij xk ij. (9) I this problem, the freight empty cars will be delivered i differet stages, which produce the followig two situatios. (1) Oe is that the traffic flow volume just meets the demad. That is, the delivered total empty cars are equal to the umber of cars required i this stage. The, o storage cost will be produced i this case. (2) The other oe is that the umber of empty cars is larger tha that required i this stage. For this case, the redudat cars will be used i the ext stage, resultig i the extra storage cost, which ca be formulated as e 1 ( x 1 id t U 1 t ) (i,d t ) A Mathematical Model. Cosider the objective ad costraits, the mathematical model of this problem ca be formulated as follows: mi s.t. f (X) (o s,j) A k=1 x k o s j R s, x k ij = x k ji, (i,j) A (j,i) A s=1,2,...,s i N/{o s,d t,s=1,2,...,s,t=1,2,...,t}, k=1,2,..., k=1 x k ij Q ij, (i,j) A k=1 x k ij G i, (i, j) A i N/{o s,d t,s=1,2,...,s,t=1,2,...,t} x k id t (i,d t ) A k=1 t=1,2,...,t, k=1 U k t, =1,2,..., +e 2 ( (x 1 id t +x 2 id t ) (U 1 t +U2 t )) (i,d t ) A + +e ( (x 1 id t +x 2 id t + +x id t ) (i,d t ) A (U 1 t +U2 t + +U t )). (10) x k ij Z+ {0}. (14) I this model, the objective is to miimize the total cost i the railway trasportatio system. The first two costraits guaratee the ratioality of the trasportatio activities. The third ad fourth costraits esure that the amout of traffic flow volume caot exceed the capacities of each lik ad each ode, respectively. The fifth costrait esures that the stage-based demad ca be satisfied with the demad costraits. So the total storage cost at destiatio d t cabesummarized as the followig form: =1 e ( (i,d t ) A k=1 x k id t U k t k=1 ). (11) Cosiderig all the destiatios, we fially formulate the total storage cost as T f 2 (X) = e ( t=1 =1 (i,d t ) A k=1 x k id t The, the total cost for these two parts is k=1 U k t ). (12) f (X) =f 1 (X) +f 2 (X). (13) 2.3. Model Aalysis. To discuss the complexity of the proposed formulatio, the followig discussio aims to aalyze thecharacteristicsoftheproposedmodel.wefirstlyfocuso the umber of costraits i the model, which are displayed i Table 3. A illustratio will be give i the followig to demostrate the complexity of the proposed model. Cosider a etwork with 20 odes ad 50 liks. I this etwork, suppose that there are three origi statios ad three destiatio statios. A total of three stages will be take ito cosideratio. With the assumptio above, sice a total of 9 paths eed to be cosidered, the umber of decisio variables should be 150 over the etire etwork. Moreover, a total of 118 costraits eed to be icluded i the model. Note that whe the etwork scale icreases, the umber of costraits ad variables will icrease to a great extet accordigly, which potetially leads to the difficulties i

6 6 Discrete Dyamics i Nature ad Society Table 3: The amout of costraits i each costrait coditio. Begi Costraits Number (1) S (3) ( N S T) (4) A (5) N S T (8) T Total umber ( N S)+ N + A T Geerate the iitial populatio Termiatio coditio No Geetic operatio Yes Ed calculatio. The, i order to simplify the computatioal complexity i the process of allocatig cars, i the ext sectio, we shall trasform the arc-based solutios ito routebased solutios, i which costrait (4) ad costrait (5) cabemergeditooecostrait.thus,itcadecrease the complexity of the solutio methods expectedly. The followig sectio will desig geetic algorithm to search for a approximate optimal solutio. Selectio Crossover Mutatio Calculate fitess Termiatio coditio Yes No Figure 4: The procedure of the geetic algorithm. 3. Geetic Algorithm Ipractice,itisratherdifficulttosolvethelarge-scalefreight empty cars allocatio problem efficietly. Because of this, we aimtodesigaapproximatioalgorithmithissectio, seekig a feasible solutio which ca replace the best solutio approximatively. The approximatio algorithm is also called heuristic algorithms, which iclude tabu search algorithm, simulated aealig algorithm, geetic algorithm, eural etwork algorithm, ad at coloy algorithm. As the geetic algorithm is a kid of direct searchig method which does ot deped o the specific characteristics of problems, it is efficiet ad effective i solvig a variety of real-world problems. The priciple of geetic algorithm is to simulate the mechaism of the livig beigs evolvig ad atural choosig. It was well developed by may researchers. Up to ow, the geetic algorithm has bee successfully used to solve practical optimizatio problems, such as trasportatio problem (Ge et al. [20]), machie schedulig problem (Vallada ad Ruiz [21]), vehicle routig problem (Baker ad Ayechew [22]), etwork optimizatio (Ukkusuri et al. [23]) ad railway operatio problem (Yag et al. [24, 25], Xu et al. [26]). I geeral, the procedure of geetic algorithm is as follows: (1) radomly geerate a certai umber of chromosomes; (2) evaluate the quality of those chromosomes with fitess fuctio accordig to some criteria; (3) obtai fie chromosomes via selectio, crossover, ad mutatio operatios. Figure 4 is the procedure of this algorithm Solutio Represetatio. I solvig the empty car allocatio problem, we eed firstly to determie the solutio represetatio i the geetic algorithm. As described i the model, the solutio is represeted as lik-based flow i each lik. To simplify the complexity of computatio, we here particularly itroduce the path-based solutio represetatio i the geetic algorithm. Sice lik-based flow ca be fially split ito the flow o differet paths betwee differet OD Figure 5: A illustratio of a simple trasportatio etwork. pairs (here, OD pair is a abbreviatio for the origi ad destiatio termials), i this method, potetial paths should be determied firstly, ad the we determie the flow volume o each path to satisfy the demad of each destiatio. A illustratio is give to show the differece betwee lik-based solutio ad path-based solutio, depicted i Figure 5. I Figure 5, eachpathcosistsofthreeodes,amely, a supply statio, a itermediate statio ad a demad statio. Suppose that two-stage demads exist i this etwork. Accordig to the lik-based solutio, there will be 4 variables ad 8 costraits i total. If we set the variables ad costraitsiapathisteadofiliks,therewillbeoly2 variables ad 5 costraits totally, both less tha those i likbased represetatio. I reality, the scale of etwork structure willbemorecomplexthathissimpletrasportatioetwork. So the umber of decisio variables with lik-based represetatio is more tha that with path-based represetatio. I order to reduce the algorithm complexity ad obtai a approximate optimal solutio, the followig discussio particularly adopts the path-based solutio represetatio. I this procedure, a path, deoted by P, isrepreseted by a sequece of ordered odes. For istace, the set P = {1,3,5,9}deotes the followig path: P: (15) I geetic algorithm, all the potetial paths should be geerated betwee each OD pair, which serve as the trasport paths i the process of deliverig empty cars. I the geetic algorithm, each populatio cosists of pop size feasible solutios as chromosomes. Sice we use the path-based solutio i the algorithm, each solutio

7 Discrete Dyamics i Nature ad Society 7 i populatio will be geerated radomly. Moreover, for each geerated solutio, its feasibility will be checked by trasformig it ito lik-based solutio. If all the costraits ca be satisfied, this solutio is a feasible solutio. Otherwise, the solutio will be regeerated radomly. At the begiig of the algorithm, a total of pop size solutios should be produced. I this paper, we set path-based decisio variables as oegative itegers, deotig the umber of empty cars trasferred o the correspodig path Selectio Operatio. I the geetic algorithm, the role of selectio operatio is to produce a ew populatio for the followig crossover ad mutatio operatios. I the iitial populatio, the chromosomes are first raked from the good to the bad accordig to their objectives. Without loss of geerality, the rearraged chromosomes are deoted by X 1,X 2,...,X pop size,wherex 1 ad X pop size deote the best ad the worst chromosomes, respectively. The rak-based fitess of each chromosome ca be calculated by the followig formula: fitess (X 1 )=β, fitess (X l )=(1 β)fitess (X l 1 ), l=2,3,...,pop size, (16) where β (0, 1) is a predetermied parameter. Here, we give the detailed procedure to show the selectio process below. Step 1. Computethefitessofeachidividualipopulatio fitess (X l ), l = 1, 2,..., pop size. Step 2.Computethecumulativeprobabilityq l of each idividual i populatio. Ad q l ca be calculated by the followig formula: q l = l h=1 fitess (X i), l=1,2,...,pop size. pop size (17) h=1 fitess (X i ) Step 3. Geerate a radom umber r i (0, 1). Step 4.Ifr<q 1,selectX 1.Ifq l 1 <r q l, the chromosome X l will be selected as a member of the ew populatio. Step 5. Repeat Step 3 ad Step 4 for pop size times Crossover Operatio. Crossover operatios aim to geerate a ew populatio i the searchig process. Before this operatio, the parets should be specified accordig to the predetermied crossover probability P c.tothised,the followig procedure is desiged. Step 1.Leth=1. Step 2. Radomly geerate a umber r i the iterval (0, 1). Step 3. Ifr<P c, chromosome X h is selected to take part i crossover operatio; otherwise, X h will ot be selected. Step 4.Leth++,ifh pop size,gotostep2. Step 5. Record the selected chromosomes. After the paret chromosomes are selected, ay two chromosomes ca be used for crossover operatio. Suppose that the two paret chromosomes are X 1 ad X 2 ad the childre chromosomes are X 1 ad X 2. The followig process is desiged to perform the crossover operatio. Step 1. Radomly geerate a parameter α i iterval (0, 1),let X 1 =α X 1 +(1 α) X 2 ad X 2 =(1 α) X 1 +α X 2. Step 2. Reset X 1 ad X 2 as iteger vectors by roudig operatio for each elemet. Step 3. Check the feasibility of the two childre chromosomes X 1 ad X 2. If the childre chromosomes are feasible, replace the parets chromosomes by them; Otherwise, remai the parets chromosomes. Step 4.Recordtheewpopulatio Mutatio Operatio. I order to avoid premature covergece, the mutatio operatio is ecessary i the searchig process of geetic algorithm. Like the crossover operatio, we firstly eed to determie the chromosomes for mutatio operatios. The umber of chromosomes selected to perform mutatio operatio is also completely based o the predetermied mutatio probability P m. Step 1.Leth=1. Step 2. Radomly geerate a umber r i the iterval (0, 1). Step 3. Ifr<P m, chromosome X h is selected to take part i mutatio operatio; otherwise, X h is ot the oe. Step 4.Leth++.Ifh pop size, go to Step 2. Step 5. Record the selected chromosomes. For each selected paret chromosome X h,weareaimig to geerate the childre chromosome X h.thefollowig procedure is desiged to perform mutatio operatios. Step 1. Radomly geerate a mutatio directio vector a, i which all of the elemets are geerated i the iterval ( 1, 1). Step 2.LetX h =X h +m a,wherem is a positive iteger. Step 3. Reset X h as a iteger vector by roudig operatio for each elemet. Step 4. Check the feasibility. If X h is feasible, let X h replace X h. Otherwise, set m=m/2,gotostep2.

8 8 Discrete Dyamics i Nature ad Society 3.5. Procedure of Algorithm. For the completeess of this paper, we shall give the detailed procedure of the algorithm i the followig. Step 1. Radomly geerate pop size chromosomes i the populatio. Step 2. Perform the selectio operatio. Step 3. Perform the crossover operatio. Step 4. Perform the mutatio operatio. Step 5. Repeat Step 2 to Step 4 for a give umber (deoted by Geeratio) of times. Step 6.Outputthebestsolutio. 4. Model Test ad Computatioal Results I this sectio, we give two umerical examples with differet scales to illustrate the effectiveess of the proposed methods, i which the experimets are implemeted by usig C++ softwareiapersoalcomputerwithitel(r)core(tm)i5-3317u 1.70 GHz Small-Scale Istace. I the first set of experimets, we cosider a small-scale etwork, show i Figure 2. This etwork cosists of five odes ad four liks, i which odes 1 ad 2 represet origi statios, ode 3 deotes a itermediate statio, ad odes 4 ad 5 represet destiatio statios.ithisetwork,itiseasytoseethatfourpotetial paths exist betwee differet OD pairs; that is, 1 3 4, 1 3 5, 2 3 4, I this etwork, we cosider two-stage based demads. The detailed data are give as lik 1 3: (3, 3, 65),lik3 4: (2, 5, 80),lik2 3: (2, 2, 55),adlik3 5: (3, 2, 40), where the umbers i brackets deote uit trasportatio cost for stage oe, uit trasportatio cost for stage two, ad trasfer capacity of the lik. Additioally, at trasfer statio 3, the turover capacity is 150. I origi statios 1 ad 2, the umbers of available cars are 110 ad 60, respectively. Besides, it is possible that the storage cost ca occur i the allocatio pla. The, the uit storage cost for differet stages are give i Table 4. It is easy to see i this problem that there are a total of 8 variables ad 13 costraits i the allocatio process. As we formulate this problem as a liear programmig model, we firstly use the Ligo software to calculate this problem, where theoptimalobjectivetursouttobe830adtheoptimal solutioisgiveitable 5. I the followig, we shall ivestigate the performace of the geetic algorithm. To this ed, we particularly list the best solutios ecoutered durig the first 16 geeratios to aalyzethecovergeceofthealgorithm,showitable 6. I Table 6, it is easy to see that all decisio variables ca approach the best solutio gradually i the searchig process, ad the approximate optimal objective ca be quickly achieved i a short time. To further show the covergece Table 4: Storage cost ad demad i destiatio statios. Statio Stage 1 Stage 2 Storage cost Demad Storage cost Demad Table 5: The best solutio solved by Ligo software. Variable x 1 13 x 2 13 x 1 23 x 2 23 x 1 34 x 2 34 x 1 35 x 2 35 Value Table 6: The best solutios ad objective values i differet geeratios. Geeratio x 1 13 x 2 13 x 1 23 x 2 23 x 1 34 x 2 34 x 1 35 x 2 35 Value process of the solutio, Figure 6 gives the detailed variatio of decisio variables x 1 34 ad x2 34 i the first 16 iteratios. It follows from Figure 6 that, for decisio variables x 1 34 ad x 2 34, they ca approach to their optimal values (obtaied by Ligo software) quickly with a relative small udulatio, ad this tedecy is almost kept i the followig geeratios. Additioally,theecouteredoptimalobjectivewillmaitai 840 (see Figure 7) after about 500 geeratio, with +10 error to the best objective obtaied by Ligo software, leadig to a relative small error 1.2%, which shows the effectiveess of geetic algorithm for solvig this problem A Large-Scale Istace i Chia. I the followig, we propose a computatioal experimet derived from real-life istaces of the railway etwork i Beijig, Hebei, ad Shaxi provice of Chia. I this istace, a total of 103 statios are located o the etwork. I particular, we omit some itermediate poits for displayig coveiece i Figure 8, where some importat termial statios are idexed by umbers, correspodig to their positios i the reallife etwork. I this etwork, a path ca be represeted by a sequece of termial statios. Figure 9 gives a illustratio for a path from statio 1 to statio 12, deoted by

9 Discrete Dyamics i Nature ad Society Iterative solutio Best solutio Iterative solutio Best solutio (a) (b) Figure 6: The variatio tedecy of x 1 34 (a) ad x2 34 (b). Objective fuctio value The geeratio of the populatio Objective fuctio Best objective Figure 7: The variatio tedecy of objective fuctio. Zhagjiakou Fagezhuag Shacheg 3 Beixibao Shagbacheg Huairou Datog Zhuolu 7 Qiaa 1 11 orth Chagpig 2 Dashizhuag 14 9 Hudog Fegtai west Supply statios Demad statios Loghua 12 Figure 8: The etwork of the real-life railway etwork I the real case, there still exist a total of 47 itermediate statios o this path (besides origis ad destiatios). I this set of experimets, suppose that there are two origis (ode 1 ad ode 2) ad three destiatios(ode12,ode13,adode14).weusethepathbased solutio for the empty car allocatio process. The, we first determie eleve potetial paths for the solutio process, listed i Table 7. Furthermore, the uit trasportatio cost is also give for differet liks. For istace, the first path cosists of five liks, the (3, 2, 1, 2, 4) represets the correspodig uit trasportatio cost o each lik (here, we adopt ivariat lik uit trasportatio cost over differet stages). I additio, we assume that supply capacities at statios 1 ad 2 are 275 ad 240, respectively. The empty cars allocatig process are divided ito three stages, ad stage-based demads ad storage costs are listed i Table 8,respectively. This set of experimet is implemeted by geetic algorithmic++softwareoapersoalcomputer.totestthe robustess of the proposed algorithm, we perform te experimets with differet parameters i the geetic algorithm. The results are give i Table 9 for aalysis coveiece. As show i Table 9, by usig differet parameters, we ca produce differet approximatio optimal solutios ad objectives. Whe settig pop size =30, P c =0.6,adP m = 0.8, we fially produce the best objective I additio, therelativeerrorsarealsocomputedtoshowtheperformace ofthealgorithm,whichcabecalculatedaccordigtothe followig equatio: error = Curret Objective Best Objecitve Curret Objecitve 100%. (18) Obviously, for differet experimets, the errors of objective values are ot larger tha 1.52%. I particular, the average error ad error variace are oly 0.628% ad %, respectively, which shows the steadiess of the algorithm. To aalyze the searchig process of the proposed algorithm, we especially cosider the computatioal results with parameters P c = 0.6 ad P m = 0.8. Moreover, three experimets with pop size =15,20,ad 30,respectively,are implemeted. Figure 10 gives the covergece of the optimal objectives with respect to differet experimets.

10 10 Discrete Dyamics i Nature ad Society Table 7: The cosidered paths ad uit trasportatio cost o arcs. Number OD Paths Uitcostoarcs (3,2,1,2,4) (3,2,3,2,4) (3, 2, 3, 2, 1, 3) (3, 2, 3, 1, 2, 3) (3, 2, 1, 1, 2) (2,1,2,4) (2,2,2,4) (2, 1, 2, 1, 3) (2, 2, 2, 1, 3) (2, 2, 1, 2, 3) (2, 1, 1, 2) Table 8: The demad ad uit storage cost i differet stages. Stage Timeperiod Node12 Node13 Node14 Storagecost 1 8:00 16: :00 24: :00 8: Table 9: The compariso of the optimal objectives. Pop size P c P m Geeratio Objective value Error (%) Average value Variace It is easy to see from this figure that, with differet scales of populatio, the approximate optimal solutio ca be quickly achieved almost withi 4000 geeratios. As expected, whe we set a larger populatio, the optimal solutio ca be obtaied with less geeratio i the searchig process. Fially, we give a compariso with the sigle-stage model to further show the effectiveess of the proposed approaches i this study. Specifically, the sum of the stage-based demads give i Table 8 will be cosidered as the demad i the sigle-stage model, ad storage cost will be omitted i the process of allocatig empty cars. Uder this cosideratio, the best objective value for allocatig empty cars turs out to be 7130 by usig Ligo optimizatio software, which ehaces the total trasportatio cost by 30% compared with the stage-based model. 5. Coclusios This paper ivestigated a railway freight empty car allocatio problem i the dyamic decisio-makig eviromet, i which the stage-based demads are cosidered, ad the cost is divided ito trasfer cost ad storage cost at destiatios. To characterize this problem mathematically, a iteger liear programmig model was formulated with the miimizatio of total ivolved cost based o the etwork flow optimizatio. To geerate approximate optimal solutios, a geetic algorithm with path-based solutio represetatio is

11 Discrete Dyamics i Nature ad Society itermediate odes 5 itermediate odes 5 itermediate odes 20 itermediate odes Figure 9: A illustratio of a path from statio 1 to statio 12. Objective fuctio value chromosomes 20 chromosomes 15 chromosomes The geeratio of the populatio Figure 10: The compariso of the optimal objectives for differet scales of populatio. desiged to seek the optimal empty car distributio strategies i railway etworks. The umerical experimets are executed to show the performace of the proposed approaches. I particular, compared to the sigle-stage model, the total trasportatio cost i our model ca be reduced by almost 30%, which implies the effectiveess of the proposed approaches. The future research ca be focused o the followig two aspects. (1) The model ca be geeralized to the ucertaity eviromets withi the framework of ucertai programmig methods, sice the real-life decisio systems for railway operatios are essetially i the state of ucertaity (see Yag et al. [19, 25]adLietal.[27]). (2) More exact algorithms ca be expectedly developed to further ehace the quality of the geerated solutios. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets ThisresearchwassupportedbytheNatioalNaturalSciece Foudatio of Chia (o ), Research Foudatio of State ey Laboratory of Rail Traffic Cotrol ad Safety, Beijig Jiaotog Uiversity (o. RCS2014ZT02), the Natioal Basic Research Program of Chia (o. 2012CB725400), ad the Beijig Natural Sciece Foudatio (o ). Refereces [1] S. C. Misra, Liear programmig of empty wago dispositio, Rail Iteratioal,vol.3,o.3,pp ,1972. [2]M.T.F.GraiadM.C.F.D.Siay, Optimaldistributio of empty railroad cars, i Proceedigs of the 1st Iteratioal Cogress i Frace of Idustrial Egieerig ad Maagemet, pp.21 26,EcoleCetraledeParis,Frace,1986. [3] A. E. Haghai, Rail freight trasportatio: a review of recet optimizatio models for trai routig ad empty car distributio, Joural of Advaced Trasportatio,vol.21,o.2,pp , [4] S. ikuchi, Empty freight car dispatchig model uder freight car pool cocept, Trasportatio Research Part B,vol.19,o.3, pp ,1985. [5] M. Liu, The exploratio of railway empty car allocatig problem by usig computer techology, Railway Trasport ad Ecoomy,o.6,pp.29 32,1987. [6]H.Xiog,W.Lu,adH.We, Geeticalgorithmusedi railway empty car allocatig problem, Chia Railway Sciece, vol.23,o.4,pp ,2002. [7] H. Liu, The mathematical model i empty car adjustmet techical plaig, TMIS Egieerig,vol.10,pp.29 31,2002. [8] X. Zhag ad Q. Zhag, Study o the optimizatio method of empty wago distributio based o kowledge costraits, JouraloftheChiaRailwaySociety,vol.25,o.6,pp.14 20, [9] D.Liag,B.Li,H.Ya,adJ.Li, Studyofrailwayemptycar distributio with substitutio of empty car types, Jouralofthe Chia Railway Society,vol.27,o.4,pp.1 5,2005.

12 12 Discrete Dyamics i Nature ad Society [10] Z. Lei, S. He, R. Sog, ad J. Cai, Stochastic chace-costraied model ad geetic algorithm for empty car distributio i railway trasportatio, JouraloftheChiaRailwaySociety, vol.27,o.5,pp.1 5,2005. [11] A.. Narisetty, J. P. Richard, D. Ramchara, D. Murphy, G. Miks, ad J. Fuller, A optimizatio model for empty freight car assigmet at Uio Pacific Railroad, Iterfaces,vol.38,o. 2, pp , [12] D. Wag, H. Ya, ad Y. Ta, Network ode of railway refrigerator empty car adjustmet i at coloy algorithm, Chia Railway Sciece,vol.29,o.2,pp ,2008. [13] G. Laporte, J. A. Mesa, ad F. Perea, A game theoretic framework for the robust railway trasit etwork desig problem, Trasportatio Research Part B: Methodological, vol.44,o.4, pp , [14] M. J. Dorfma ad J. Medaic, Schedulig trais o a railway etwork usig a discrete evet model of railway traffic, Trasportatio Research Part B: Methodological,vol.38,o.1,pp.81 98, [15] E. Erkut ad F. Gzara, Solvig the hazmat trasport etwork desig problem, Computers ad Operatios Research, vol.35, o.7,pp ,2008. [16] P. Guo, B. Li, ad Y. Yu, Sectio ceter optimizatio method i large-scaled etwork car deploymet, Chia Railway Sciece,vol.22,o.2,pp ,2001. [17] M. Jobor, T. G. Craiic, M. Gedreau,. Holmberg, ad J. T. Ludgre, Ecoomies of scale i empty freight car distributio i scheduled railways, Trasportatio Sciece,vol.38,o.2,pp , [18] B. L. Yag ad X. Zhou, Costrait reformulatio ad lagragia relaxatio-based solutio algorithm for a least expected time path problem, Trasportatio Research Part B, vol. 59, pp , [19] L. Yag, X. Zhou, ad Z. Gao, Credibility-based reschedulig model i a double-track railway etwork: a fuzzy reliable optimizatio approach, Omega,vol.48,pp.75 93,2014. [20] M. Ge, F. Altiparmak, ad L. Li, A geetic algorithm for twostage trasportatio problem usig priority-based ecodig, OR Spectrum,vol.28,o.3,pp ,2006. [21] E. Vallada ad R. Ruiz, A geetic algorithm for the urelated parallel machie schedulig problem with sequece depedet setup times, Europea Joural of Operatioal Research, vol. 211, o. 3, pp , [22] B. M. Baker ad M. A. Ayechew, A geetic algorithm for the vehicle routig problem, Computers & Operatios Research, vol. 30, o. 5, pp , [23] S. V. Ukkusuri, T. V. Mathew, ad S. T. Waller, Robust trasportatio etwork desig uder demad ucertaity, Computer-Aided Civil ad Ifrastructure Egieerig, vol.22, o.1,pp.6 18,2007. [24] L. Yag,. Li, Z. Gao, ad X. Li, Optimizig trais movemet o a railway etwork, Omega,vol.40,o.5,pp ,2012. [25]L.Yag,Z.Gao,ad.Li, Railwayfreighttrasportatio plaig with mixed ucertaity of radomess ad fuzziess, Applied Soft Computig Joural, vol. 11, o. 1, pp , [26] X. Xu,. Li, L. Yag, ad J. Ye, Balaced trai timetablig o a sigle-lie railway with optimized velocity, Applied Mathematical Modellig,vol.38,o.3,pp ,2014. [27] S. Li, L. Yag,. Li, ad Z. Gao, Robust sampled-data cruise cotrol schedulig of high-speed trai, Trasportatio Research Part C,vol.46,pp ,2014.

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