Optimization of Shortest Path of Multiple Transportation Model Based on Cost Analyses

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1 Optmzaton of Shortest Path of Multple Transportaton Model Based on Cost Analyses Yang Yang 1,2 Ruyng Wang 1 Qanqan Zhang 1 1 Chna Unversty of Mnng & Technology (Bejng), School of Management, Bejng, , Chna, 2 Bejng Wuz Unversty,Insttute of Logstcs, Bejng, , Chna, Abstract In transportaton process, normally there are several modes to delver goods and sometmes there s crosstransportaton or multmodal transport mode. Meanwhle, n the transportaton process, transportaton costs, tme and rss vary wth tmeframe. Due to the characterstcs of transportaton networ, transportaton costs and transt tme wll vary wth dfferent startng tme. Transportaton costs can be dvded nto fxed costs, road transportaton costs and transt fees consderng multple nfluence factors and varous optons, for nstance, ralway, arlne and waterlne. Ths artcle llustrates n detal that optmzaton of shortest path of multple transportatons based on cost analyses and ths mode has been verfed by genetc algorthm. Keywords - multmodal transport; route optmzaton; shortest path; genetc algorthm I. INTRODUCTION Wth the development of E-Commerce, there s an ncreasng boomng of orders. Sngle transportaton s far less able to meet nternet consumers need, especally durng the rapd development of offshore E-Commerce. Offshore E-Commerce logstcs are normally requred two or more modes of transportaton from startng ponts to ts destnatons whch s multple transportaton. In multple transportaton process, optmzaton of shortest path are most lely decded by the type of goods, transportaton dstance, clents need, relablty and flexblty of transportaton servces and transportaton nfrastructure. Due to the ncreasng needs of multple transportaton, scholars have pad a lot of attenton to the multple transportaton mode. There were varous theores regardng ths matter. Among those, Angelca Lozano (2001) has studed shortest path n multple transportaton mode and verfed by sequental algorthm. A straght path under multple transportaton mode has been fully studed by Dezh Zhang (2002).Other scholars, such as Modest, Scomachen (1998) and Angelcalozano, Govann Storch (2002) have all been researched on shortest path of multple transportaton. However, those researches have not taen nternet nto account. In realty, tme factor has some mpact on the targets consdered durng transportaton, such as tme, cost, etc. About the subject of the shortest crcut under the tmevaryng condton, t's manly concentrated n obtanng the shortest crcut of the shortest transport tme under the condton of sngle mode of transportaton. Else, d. m., Han s.m., (1998) studed the shortest crcut n the shortest tme random tme-varyng networ, proposed and compared two dfferent algorthm.else, d. m., Han s.m (2003) put forward some prncples of comparng dfferent paths under random tme-varyng networ. Athanasso z., Dmt ros. and Han, s. m. (1997) analyzed the applcaton of consderng the shortest crcut of the shortest tme under the condton of the tme-varyng n ntellgent transportaton system. Danele p. (2000) studed the shortest crcut consderng the shortest tme under the condton of dscrete random tme-varyng hypergraph networ, and gave a soluton. Else, d. m. (2001) studed the shortest crcut of the mnmum expected travel tme under the condton of the tme-varyng random networs, and gave a algorthm to solve the mnmum expected travel tme. Sathaporn Opasanon, Else Mller - Hoos (2006) studed the shortest crcut problem of mult-rules under the condton of the tme-varyng stochastc networ, and gave a solvng algorthm. Zhang Janyong etc. (2002) started from the prncple of mnmzng the total cost, and set up a nd of optmal allocaton model of multmodal transport networ, n order to analyze the reasonable organzaton pattern of the multmodal transport system from the pont of quanttaton. Wang tao, etc. (2005) put forward a transportaton mode combnaton optmzaton model after analyzng the transportaton characterstcs of varous modes of transportaton, and gave a solvng algorthm. Wang Yunpeng etc. (2005) studed the multmodal transport process based on extended Petr net. Mchel Beuthe (2001) studed the subject of transportaton optmzaton of mnmum cost n the multmodal transportaton networ composed of road - ralway - nland marne etc. 10 categores of goods. Lu Cheng etc. (2005) studed the parallel genetc algorthms of the logstcs dstrbuton vehcle routng problem wth soft tme Wndows. We zhong etc. (2006) put forward the shortest tme path transportaton cost model. Hermna I.C alvete (2007) DOI /IJSSST.a ISSN: x onlne, prnt

2 studed the vehcle routng problem wth soft tme Wndows, and put forward the goal programmng method for solvng the problems. Sun Huacan (2008) put forward the concept of reasonable path, and set up an optmzaton model of combned transportaton path contanng the path-ratonalty constrant, and ponted out that except the transportaton beneft maxmzaton, the factors of reasonable sequence of change and the change number must be consdered when choosng and optmze a combned transport route. The studes above have studed systematcally varous combnatons of shortest path and multple-transportaton. The ey of mult-networng model s consdered from varous levels. However, t has ts own lmtaton, some models only consdered tme, transportaton costs or servce as condtons. None of the studes have consdered varous tme whch also have mpact on transportaton costs or the nfluence of transportaton volume. Some studed even neglected exchange goods and delver tme wth the assumpton that transportaton volume cannot be dvded. In realty, f transportaton pont has demand on specfc goods and volume wll change as well. Some research prefer to consder best transportaton path as shortest path whle n the contrary the choces whch have been made are based on fastest way or lowest transportaton costs, normally not based on the shortest path. Thus, ths paper have analyzed fully optmzaton of shortest path of multple transportaton model based on costs analyze. The costs refer to the lowest fee under a certan transport tme, and those two elements are used to construct ths multple transportaton model. In ths artcle, ths model wll be explaned n detal n order to enhance effcency, reduce transportaton fee and optmze transportaton path. II. CONSTRUCTION OF MULTIPLE TRANSPORTATION MODEL A. Net Pont There are many ponts where goods can change ther transportaton method n multple transportaton models. Those ponts were called net ponts. Ground rules of changng transportaton are as follow: If there are more than one ways to transfer transportaton methods, then connect the two ponts wth lnes. One lne s equal to one transportaton method. If there s one pont that goods could be transfer there, then separate them wth nodes. Each way starts and ends wth new nodes. Ths way s named nodes transportaton methods. As can be seen, transportaton net s as follow. There are two lnes between O and A, refer to hghway transportaton and ralway transportaton. Due to transfer at pont A, t s dvded pont A nto 4 nudes. Fg.1 Transformaton Of Multmodal Transport Networ B. Model Hypothess (1) If the dstance between OD s not dvded, then there could be one transportaton methods between OD. (2) There s nether addng nor reducng goods durng transportaton process, or at nodes. C. Explanatons of Symbols For multmodal transport networ G V, E, among them, V s the pont set of multmodal transport networ; V s a node n networ; E s the set of multmodal transport networ edge; e E s an edge n multmodal transport networ; M s collecton for a batch of goods, m M s one nd of these goods; F m s fxed cost when transportng the goods m (RMB); K s the transt transportaton mode for selecton; K, 1 K s a transt way; t, 1 s the tme requred to adopt the mode of transportaton between the nodes and 1 (mn); c m, s the transportaton costs when shppng goods m l wth method (RMB); s s the transtng tme at pont convertng from method to method l (mn); c, l, m s the transmttng cost of goods m from method to l (RMB); T s requred delvery deadlne (mn); P v s a set of multmodal transport mdpont. choose e between and x 1, mod 1, 1 0, other transportaton e turn to l at node y l 1, mod 0, other DOI /IJSSST.a ISSN: x onlne, prnt

3 III. OPTIMIZATION MODEL OF MULTIPLE- TRANSPORTATION Text should be produced wthn the dmensons shown on these pages; each column 8.2 cm wde wth 0.6 cm mddle margn, total wdth of 17 cm and In realty of multple transportaton process, some ponts of transmttng have barely occurred (For nstance, from arlne to water transportaton). Therefore, t s taen by nudes. At nudes, t could only select from bacup opton. Based on those condtons, a multple transportaton model has been establshed. The exact model s: l (1) s.t. mn Z F x, 1c( m, ) y c(, l, m) m m m m l x K V 1, (2), 1 y 1, l K, V (3) l l l x x 2y V,, l K (4) 1,1, 1 l l x, 1t, 1 y s T PV,, l K (5) l x 1 0,1 V, y l K,1 V, l K, (6) 0, (7) V, K, m M (8) In the model above, formula (1) represents the cost target functon. Cost functon s manly composed wth three parts: fxed costs, cost, transfer cost. Formula (2) represents the mode of transportaton choce constrant. The constrant between two nodes can only use one mode of transportaton. Formula (3) represents that transt operatons can only transfer to one mode of transportaton. Same node can only be transferred once. Formula (4) represents the mode of transportaton before and after the correspondng constrants. If the transformaton mode turns to l at node, then adopt mode between node 1 to and adopt mode l between node to 1.Formula (5) represents shppng deadlne constrants. Formula (6) and formula (7) represent varable constrant logc. Formula (8) represents varable nonnegatve constrants. IV.TO SOLVE ALGORITHM The nature of the model s to see for the shortest path under those condtons. Current algorthm s able to transfer the questons to shortest path from O to D wth lmted and tmeframe. However, t s rather dffcult to have the best practce or the most satsfed soluton. Thus, ths paper constructs based K how to calculate the shortest path. The man thought s : Startng wth Djstra algorthm to calculate the shortest tme path to end destnaton, then fnd the second shortest tme path, the thrd shortest tme path and so on. Afterwards to search those optons to fnd the best practce based nhert algorthm. A. Fndng All Paths wthn Deadlne wth the Djstra Algorthm When usng the Djstra algorthm to fnd out short crcut problem under the crcumstance of tme-varyng, the symbol of each pont s,, v, T ), s the node before ( the node. s the mode of transportaton before the node (1 represent ralway ; 2 represent hghway; 3 represent waterway; 4 represent arlne; 5 represent transt); v represents the stage number; T represents the tme t taes to reach node ; S s defned as the set of fxed label pont, S s defned as a temporary marng pont set. Accordng to Djstra algorthm, the labels of the vertces n the networ can be dvded nto two categores, fxed label and temporary marng label. The dea of ths algorthm s eep changng temporary label to fxed ndex from the startng pont. Fnd out the shortest path from the startng pont to other ponts. The tme of stage v reachng pont can be expressed as T T T ( m, ). T( m, ) m represents the transport tme of goods m wth transportaton method. The steps to fnd all paths wthn deadlne wth the Djstra algorthm are as follows: Step 1 Accordng to the actual physcal networ; we tae the form of fgure 1to splt networ node. There are no contrandcatons on alternatve transt ways. So we get a weghted drected graph, the weght of each sde represents a certan goods transportaton tme and transportaton cost, the weght on transfer arc represents the transt tme and transt fees. (0) Step 2 v 0, S v 0, v0 represents the startng transport pont of goods, let other ponts n the networ nto S, and gven that T 0, T v0 v, the startng pont s (,,0,0) n the networ, label for other nodes s (,,0, ). Step 3: Mae v v 1.Modfy and fx the temporary label ponts attached to the label and calculate the tme of the temporary label ponts, whch 1 s T v mnt, ( ) v T v T s v. Among them, v l represents temporary label; v s represents a fxed number. Calculate value of temporary label node v l and T v. If the l nformaton n the temporary label s T, the label on that pont need to be updated. If n one of the phases (except the frst one), all the labels n the ponts connectng to the fxed ndex pont set S are the ntal labels, then are connected to the temporary label ponts dependng on the fxed before the label nformaton push, fnd the pont 5 to dsable the pont before the transfer arc, deleted from the fxed label set pont S after all has the fxed pont label, and the pont s set to the ntal label all the label Tv DOI /IJSSST.a ISSN: x onlne, prnt

4 nformaton. And then from all ponts wth temporary label, select the tme value of mnmum pont, change the tme mnmum pont to fxed label, and that pont to be ncluded n the fxed label pont set S. Step 4 If the end of the goods has been fxed label, then stop countng, at ths pont can be launched by the end fall under the condton of tme-varyng shortest path, fxed label s n the front, and s the mode of transportaton between that pont and the former pont, so we can get a path that contans transportaton way and the road. If the end of the goods are not fxed label, then go to step 3. Step 5 When gettng the shortest tme path, then fnd the sdes of the shortest tme path contaned, respectvely removed them from the weghted drected graph, so we get some atlas, then n accordance wth steps 2 to 4 respectvely, and then fnd out the shortest tme path n each chld fgure. Step 6 Then for each chld fgure to step 5, f the chld fgure does not exst n the thrd step n the fgure label nformaton that meet Tv T, then stop calculaton. So on, untl fndng all shortest tme path that meet the shppng deadlne. B. Usng Genetc Algorthms K Shortest Path to Fnd Satsfactory Cost Path In ths paper, genetc algorthms use symbol codng method, each chromosome locus symbols have expressed a mode of transport, each locus sequence correspondng to an arc on the gven path. The number of arcs on the path mnus s equal to the length of the chromosome number of nodes on a gven path. For a gven path, each chromosome and a hybrd mode of transport agents may be used n correspondence, ths mode of transport and route wll ln up. Crossover algorthm uses sngle pont crossover, and mutaton operators are usng the basc alleles specfc mplementaton: the value of each allele at chromosome mutaton probablty mutate nto symbols of other modes of transport. Select mode selecton method based on the rato of ftness through roulette ways, n addton to ensure the convergence of the algorthm, the algorthm embedded eltst strategy: f the current -generaton algorthm produced the best ndvdual so far worse than the best ndvdual, then the ndvdual wth the best so far n the random replace a new group of ndvduals. Algorthms shut down condton wth the maxmum number of generatons, teratve algorthm termnates when the condton s not met. Specfc steps above K usng genetc algorthms to fnd satsfactory cost path search path s : Step 1 For a gven path, genetc algorthms nput varous parameters : populaton sze, maxmum number of generatons, as well as crossover and mutaton probabltes ; Step 2 ntal populaton to calculate the ftness of all chromosomes n the populaton, statstcal ndcators ntal populaton ; Step 3 Select the group to perform the operaton, to generate a matchng pool ; Step 4 crossover mplementaton: repeatedly perform the followng operatons untl the new offsprng chromosome number equal to the populaton sze: 1 two chromosomes randomly selected from a pool match ; 2 elected to perform two chromosomes crossover operaton accordng to the crossover probablty ; Step 5 mutaton operaton to acheve : the values of each of the alleles n the populaton of each chromosome mutaton probablty mutate accordng to ; Step 6 Calculate the ftness of the populaton of all chromosomes, and statstcal ndcators at ths populaton; Step 7 Detect the shutdown condton: f the algorthm satsfes the shutdown condton, then go to step 8, otherwse go to step 3; Step 8 For a gven path of some of the best statstcal ndcators of transport and algorthms output algorthm to obtan the le. V.THE EXAMPLE ANALYSIS In order to verfy the above model and algorthm, we use the transportaton networ as shown n fgure 2 to analyze and verfy the model and algorthm. The text represents transportaton exstng between those two nodes n fgure 2. For example, between O and A, there are two transportatons,ralway and hghway. The networ shown n fgure 2 ncludes four nds of mode of transportaton, hghway, ralway, waterway and avaton. o Ralway, hghway ralway Ralway, hghway A F Avaton, Ralway, Ralway, E c Ralway, avaton hghway hghway D Avaton, Ralway hghway G Ralway, waterway hghway Avaton, waterway B H Avaton, hghway Fg.2 Intal Multmodal Transport Networ The networ s as shown n fgure 3 after the actual transport networ nodes splt. Each node s represented by one number for convenent analyss. DOI /IJSSST.a ISSN: x onlne, prnt

5 Fg.3 Multmodal Transport Networ Dagram After Transformaton Accordng to the actual stuaton of the cargo transport and experence, the transt alternatve collectons are {Ralway-hghway, hghway-waterway, hghway-avaton, waterways - hghway}. So the transt way whch s not n the above transt alternatve collectons can be dsabled and deleted,so t can reduce the networ sze to a certan extent after the nodes are splt. In ths case, after some segment are deleted, the optmzed networ s as shown n fgure 4.In the optmzed networ, arc weghts ndcate the transport tme (s) and cost (RMB) between two nodes. And transt arc weghts represent the transfer tme/transfer cost. It s assumed that the transt tme and the transt cost are 0 n the same mode of transportaton n the transt node. Fg.4 The Optmzed Networ Suppose that there are a number of goods to be transported from O to D. Accordng to the above condtons, assumng T=120mn, accordng to the algorthm, frstly, we can obtan all paths wthn the tme gven whch are shown as follows: Wth the path of the genetc algorthm to meet the shppng deadlne set on the mode of transportaton of combnaton to search to fnd satsfacton cost path. The parameters of genetc algorthm: Populaton sze: n=100; The maxmum number of generatons: ger=400; The crossover probablty: pc=0.9; Mutaton probablty: pm=0.01; By usng the MATLAB software to code genetc algorthm, we can obtan the satsfactory cost path wthn the tme gven (T=120mn),the path s So the orgnal path s O F B C D, and the cost s 570 RMB, the transport tme s 106mn, Ralway transport road. In addton, we can also get the optmal cost paths wthn dfferent lmted tme gven, as shown n table I. It can be seen from table I, when the deadlne s 100 mn, the mode of transportaton should be ralway to hghway to avaton, whch has the mnmum cost, now, the transportaton tme s 96 mn, but the cost s 700 RMB; When the deadlne s 120 mn, the mode of transportaton should be ralway to hghway, whch has the mnmum cost, now, the transportaton tme s 106 mn, but the cost s 570 RMB; If there s no deadlne tme, then the optmal path at ths tme s O G H D, the mode of transportaton s hghway to waterway to hghway, the mnmum cost s 520 RMB, but the transportaton tme s 122 mn. TABLE I THE OPTIMAL COST PATHS WITHIN DIFFERENT LIMITED TIME GIVEN tme(mn) Lmted path orgnal path and Transportaton mode tme/mn cost(yuan) O F B C D 96/700 ralway hghway avaton avaton 106/570 ralway hghway hghway hghwa O F B C D y unlmted O G H D 122/520 hghway waterway hghway VI. CONCLUSIONS In the multmodal transport,there are a varety of modes of transportaton between the startng pont and endng pont. But through the networ transformaton, the transfer of multmodal transport networ can be a reasonable representaton. In practce, due to the specal requrements of goods and the ratonalty of the transfer, It s n conformty wth the actual demand to get the optmal path n the multmodal transport based on tme costs or expenses DOI /IJSSST.a ISSN: x onlne, prnt

6 cost. In the model bult of ths study, the restrctons on the optmal path under the condton of tme-varyng are consdered. In the selectng of transfer mode, we delete the transfer modes that are mpractcal, that s to say, we can only choose from the alternatve set, whch can reduce the sze of the networ.consderng the constrants to the tme lmt, we establshed a cost orented dynamc path optmzaton model of multmodal transport, and proposed the correspondng algorthms, fnally, we proved the avalablty of the model through analyss on a case of multmodal transport networ. ACKNOWLEGDMENT Ths study s supported by the Fundamental Research Funds for the Central Unverstes. (project number: 2014 QG01). REFERENCES [1] Bonteonng YM, Machars C, Trp J J. Is a new appled transportaton research feld emergng? A revew of ntermodal ral2truc freght transport lterature, Transportaton Research Part A, 38 : 1~9,2004. [2] Lozano A, Storch G. Shortest vable path algorthm n multmodal networs, Transportaton Research Part A,35 : 225~241,2001. [3] Zhang Janyong Guo Yaohuang. A optmal allocaton model of multmodal transport networ study, Journal of ralway, 24 (4) : 114 ~ 116,2002. [4] Wang tao, wang gang. A way of multmodal transport networ transport model of combnatoral optmzaton study, Journal of Chna engneerng scence,7 (1) : 46 ~ 50,2005. [5] Wang Yunpeng Wang Zhanzhong Zhao Yng, etc. Multmodal transport process based on extended Petr study, Industral technology economy,24 (4) : 77 ~ 79,2005. [6] Beuthe M. Freght transportaton demand elastctes: a geographc multmodaltransportaton networs analyss, TransportatonResearch PartE.,37 (4): ,2001. [7] Cheng, Chen Zhya, seal all. Logstcs dstrbuton vehcle routng problem wth soft tme Wndows of parallel genetc algorthm, Journal of systems engneerng, 23 (10) : 72-11,2005. [8] We the ShenJnSheng, etc. Tthe shortest tme path - transport multmodal transport cost model to study, Chna engneerng scence, 8 (8),2006. [9] Calvete H I. Agoal programmng approach to vehcle routng problemwth soft tmewndows. European Journal of OperatonalResearch, 177(3): ,2007. [10] Sun Huacan L Xuhong, Chen Dawe, etc., Reasonable paths n comprehensve transportaton networ optmzaton model, Journal of southeast unversty, 42 (5),2008. [11] Success, LngChunYu. Varous mode of transportaton of combnatoral optmzaton model and algorthm, Journal of changsha ralway nsttute, 20 (4) : 71-75,2002. [12] Angelca Lozano, Govann Storch. Shortestvable path algorthm n multmodal networs, Transportaton Research, Part A,35 : ,2001. [13] Modest, Scomachen. A utlty measure forfndng mult -objectve shortest paths n urban multmodal transportaton networs, European Journal of Operatonal Research,111 : ,1998. [14] Angelca Lozano, Govann Storch, Shortest vable hyperpath n multmodal networs, Transportaton Research,Part B, 36 : ,2002. [15] Else D. M., Han S. M.. Least possble tme paths n stochastc, tme-varyng networs, Computers and Operatons Research, 25 (12) : ,1998. [16] Else D. M., Han S. M.. Path comparsons for apror and tmeadaptve decsons n stochastc,tme varyng networs, European Journal of Operatonal Research,146 : 67-82,2003. [17] Athanasso Z., Dmtros K., Han S. M. Desgnand mplementaton of parallel tme- dependentleast tme path algorthm for ntellgent transportaton systems applcatons, Transportaton Research, Part C,5 (2) :95-107,1997. [18] Danele P.. A drected hypergraph model forrandom tme dependent shortest path, European Journal of Operatonal Research, 123 : ,2000. [19] Else D. M.. Adaptve Least Expected TmePaths n Stochastc, Tme- Varyng Transportaton and Data Networs, Networs,37 (1) : ,2001. [20] Sathaporn Opasanon, Else D. M. Multcrteraadaptve paths n stochastc, tme-varyngnetwors, European Journal of Operatonal Research, 173 : 72-91,2006. DOI /IJSSST.a ISSN: x onlne, prnt