A Vehicle Routing Optimization Method with Constraints Condition based on Max-Min Ant Colony Algorithm

Transcription

1 Appl. Math. Inf. Sc. 8, No. 1L, (2014) 239 Appled Mathematcs & Informaton Scences An Internatonal Journal A Vehcle Routng Optmzaton Method wth Constrants Condton based on Max-Mn Ant Colony Algorthm We-Cheng Xe 1,, Chang-Mn Chen 1, Song-Song Fan 2 and Lng-Lng L 1 1 School of Electrcal and nformaton Engneerng, Xhua Unversty, Chengdu , Chna 2 School of communcaton and nformaton engneerng UESTC, Chengdu, Chna Receved: 6 May. 2013, Revsed: 31 Aug. 2013, Accepted: 1 Sep Publshed onlne: 1 Apr Abstract: Accordng to the command quantty of customers and lmts of dstrbuton vehcle, n order to obtan more proft, optmzaton study on vehcle routng and quantty s needed. Basc ant colony algorthm has slow convergence speed and s easy to fallng nto local optmum when used to solve ths problem, whle max-mn ant colony algorthm can overcome these shortcomngs and s the best algorthm to solve ths problem at present. In ths paper, vehcle routng optmzaton on constrants condton s smulated wth max-mn ant colony algorthm on MATLAB, and the drecton of vehcle routng and the least number of vehcle can be obtaned from the smulaton results. Keywords: MMAS, the optmzaton of VRP, constrant condtons, MATLAB 1 Introducton Wth the rapd development of scentfc technology and ncrease of economc globalzaton, the world economy s facng unprecedented opportuntes and challenges, and modern logstcs s an mportant part of modern economy. In 2008, Chna s total socal logstcs was 89.9 trllon Yuan, 4.2 tmes ncrease over 2000, an average ncrease of 23%; logstcs ndustry realzed an added value of 2.0 trllon Yuan, 1.9 tmes ncrease over 2000, an average ncrease of 14%. As can be seen from these data, the logstcs ndustry maes great nterests, but the overall level of Chna s logstcs ndustry s low, such as: frst, low operaton effcency of the whole socal, Chna s total logstcs costs to GDP rato of about 1 tmes hgher than developed countres; second, the less demand on socalzaton logstcs and nsuffcent ablty of professonal supply [1,4]. Vehcle routng problem (VRP) s an mportant part of the logstcs system research. In order to reduce the servce provders operatng costs, logstcs system wll be done optmzaton on the vehcle s path. Selectng the approprate transport routes can speed up response tme to customer needs, mprove servce qualty, and enhance customer satsfacton wth the logstcs system [4]. Vehcle routng problem s generally defned as: a seres of delvery pont and/or recevng pont, selected the proper route wth certan constrants orderly through them. To save logstcs costs, two aspects are avalable [5]: Improvng effcency, whch s to reduce the costs of the vehcles from the economc perspectve; the reasonable arrangement of vehcle dstrbuton to mprove the professonal logstcs supply. At present, Max-mn ant colony algorthm s the better algorthm to solve the vehcle path 1-4. On exstng condtons wth two constrants: Frstly, the vehcle maxmum drvng dstances. Secondly, the vehcle s maxmum loadng, can fundamentally reduce the cost of the vehcle to gan greater benefts. 2 The Descrpton of VRP Problem wth Constrants and the Model A. The Descrpton of VRP Problem wth Constrants VRP (vehcle routng problem), the vehcle routng problem can been descrbed as follows1: There are a number of customer ponts, and dstance and demand for Correspondng author e-mal: scxwecheng@alyun.com

2 240 W. C. Xe et al: A Vehcle Routng Optmzaton Method wth Constrants... goods between two customers are gven. Each vehcle starts from the dstrbuton staton, and come bac when reachng the lmtatons [6, 8]. Vehcle s maxmum load capacty and drvng range are gven. B. The Model of VRP Problem wth Constrants Record the maxmum load per vehcle for QV. The maxmum drvng range for Lengthmax,the demand of customer for g ( C). C s a collecton of customers. Set up: { 1, If the vehcle along the path(,j), X j = 0, else. Then we obtan the mathematcal model [2]: { } mnz = X j,x 0 j j j s.t j g y QV, (a) D j y j lengthmax, (b) j y = 1, C, x j = y j, j C, x j = y, C, X j S 1,S C ( f), j S S x j,y (0,1),, j C, Of whch: Constrants (a) for the vehcle load lmts; Constrant (b) for the vehcle journey restrctons; Constrant (c) to ensure that every vehcle access to each customer only once; Constrants (d) - (f) to ensure that form a vable crcut; The targets are the least of vehcles number and the mnmum of the total road length. 3 Basc Ant Colony Algorthm (c) (d) (e) Accordng to long-term study on boncs, the ant has no vson, but t can release a specal nd of secreton pheromone on the path when loong for food, so that the later ants can qucly fnd food. They wll randomly choose a path forward and release the pheromone when comng across the path not passed by. The longer path s gong, the less amount of nformaton s released 5-6. The ants comng later wll choose the path of large amount s larger, ths probablty form a postve feedbac mechansm Over tme, the path chosen by most ants the greater the amount of nformaton, whle others gradually reduce the amount of nformaton on the path, eventually the whole colony wll fnd the optmal path. Ant also able to adapt to envronmental changes, when the colony there s an obstacle on the path of movement, the ants can qucly dentfy the optmal path. Result that, n the ant routng process, a sngle ant has the ablty, though lmted selecton, but by the postve feedbac of pheromone the ant colony behavor can have very hgh self-organzaton. Ant path between the exchange of nformaton, and ultmately self-catalyzed by the collectve behavor of ant colony to fnd the optmal path5. Ants search path s shown as Fgure 1. Assumng the ants are startng from the nest, there are four food ponts, each ant needs to reach four food ponts and then bac to the nest, whch s equvalent to a car to four dfferent customer stes and dstrbuton of goods (wth the travelng salesman problem (TSP) related) [9, 12]. Ants depart from the nest, there are four selecton lnes (1, 5, 6, 7), and each route s a two-way. In the fgure the length of 6, 7 were clearly longer than the length of 1, 5, and the pheromone on the route of 6, 7 would less and less over tme, so there wll be more and more ants select the routes 1and 5. And so, eventually the ants wll be based on the strength of the pheromone to fnd the optmal path. Fg. 1: The fgure of ant search path. 4 Max-Mn Ant Colony Algorthm Because the shortcomngs of the basc ant colony algorthm can not be gnored, MMAS can avod premature stagnaton phenomenon 6-8 [13, 17]. At present, Max-mn ant colony algorthm s the better ant colony algorthm to solve TSP problem (ACO) compared wth other algorthms. MMAS lmts possble resdue pheromone lmt of every path to [τ mn,τ max ], and eep the optmum path after one cycle. A. The Improvements over ACO MMAS came drectly from the ACO, manly made the followng mprovements:

3 Appl. Math. Inf. Sc. 8, No. 1L, (2014) / ) After each teratng, only the pheromone on the optmal path s updated; 2) In order to overcomng the ACO s shortcomng of beng easy to fall nto local optmum, MMAS lmt the pheromone of each path to[τ mn,τ max ]. Pheromone wll be forced to set to τ mn or τ max, f t excesses the value. Ths can effectvely avod nformaton of certan path far outwegh the other path, eep all the ants on the same path, mae the algorthm fallng nto the local optmum and not spread the path; 3) Intally, the pheromone on each path s, and when ρ (the pheromone evaporaton coeffcent) tae smaller value, MMAS has a better ablty to fnd better solutons, and τ max and τ mn are determned by followng formula: Pheromone has not been updated: 1 τ max (t)= 2(1 ρ) L best τ mn (t)= τ max(t) 20 Whch L best s the correspondng path length of the global best soluton, after pheromone updated, τ max and τ mn are decded by the followng formula, τ mn s the same as (1). τ max (t)= 1 2(1 ρ) L best + σ L best Where σ s the number of elte ant. 4) In the MMAS, all the pheromone re-ntalzaton when the system s stalled. B. Pheromone Update In ths paper, we use a DC motor as controlled object to analyze the system stablty, the transfer functon of the controlled plant s expressed as follows9-10: Pheromone update has two ways: local update and global update. In ths paper, the global update s decded, the best ant (the top few elte ant n the path constructed) for the pheromone update, the update rules are as follows: τ new j =(1 ρ)τ old σ 1 j + u=1 τ u j+ σ τ j Only when the trajectory (D, Dj) s passng by elte ants µ, the pheromone on the path wll ncrease τ u j ( τu j = (σ µ)/l u ), whch Lu causes the path length of ant µ. The pheromone on the entre best path are ncreased σ τ j (σ τ j = 1/L ), L* ndcates the current best path length. Generally, σ s value between 3 to 6 s more approprate11. C. The mplementaton of MMAS n a mult-constrant VRP 1) parameters ntalzaton, set the ntal amount of nformaton on the path (D, Dj) τ j = tau max, set up dstrbuton ponts for the ntal node, and put the ntal node nto the tabu lst, set the ntal value for the parameter α,β,ρ,ρ; teraton number nc = 0, Ncmax s the maxmum number of teratons; 2) Into the teratve process, nc = 0; 3) Let all not vsted customers nto the set J; 4) If L (C, Cj) + L (Cj, Cj +1) L (Cm-1, Cm) lengthmax or g (, j) + g (j, j +1) g (-1, ) QV, save the current path and re-open a path, whch starts from the startng pont that s the dstrbuton pont, repeat step (3); 5) Pheromone update; 6) If set J s empty, then jump to (8) step; 7) Iteratons nc = nc +1, and nc Ncmax; 8) End and output the results. 5 MMAS s Applcaton of Vehcle Dstrbuton Logstcs on Mult-Constrant Suppose logstcs dstrbuton center dstrbuted goods to 13 supermarets on the outsrts. Every car has certan load and drvng dstance, and each path can not excess the load of vehcle. Each shop s served only by a car. The dstance and demand sze of each shop s on table 2. Suppose that traffc control and congeston was not taen nto account, maxmum load of vehcle was 10 ton, maxmum drvng dstance was 150m. MMAS algorthm s analyzed wth MATLAB and results path are separately shown n fgure 2 and sheet 1. Fg. 2: MATLAB smulaton dagram of the MMAS algorthm. 6 Conclusons Vehcle routng optmzaton problem s a typcal non-lnear combnaton problem, and t occupes a large part n the modern logstcs system s cost control. Max-Mn ant algorthm s a good method to deal wth

4 242 W. C. Xe et al: A Vehcle Routng Optmzaton Method wth Constrants... Table 2: The specfc crcumstances of each supermaret Dstr-buton Dstance between ponts Demands Table 1: Vehcle routng optmzaton results Vehcle Drvng Drectons these problems from the angle of cost savngs. Two constrants are added on ths base n the paper, whch optmze the path and have a great economc value n solvng the vehcle routng optmzaton. Acnowledgement Ths wor was supported by the project 10ZA098 of the Scentfc Research Fund of Schuan Provncal Educaton Department, the project Ycjj of the graduate student nnovaton Fund of Xhua Unversty and the Open Research Fund of the Key Laboratory of Sgnal and Informaton Processng, Xhua Unversty (No. szjj ). References [1] B. Yu, Z. Z. Yang, B. Z. Yao. An mproved ant colony optmzaton for vehcle routng problem, European Journal of Operatonal Research, 196, (2009). [2] S. H. Huang, P. C. Ln. A modfed ant colony optmzaton algorthm for mult-tem nventory routng problems wth demand uncertanty, Transportaton Research Part E: Logstcs and Transportaton Revew, 46, (2010). [3] L. Santos, J. Coutnho-Rodrgues, J. R. Current. An mproved ant colony optmzaton based algorthm for the capactated arc routng problem, Transportaton Research Part B: Methodologcal, 44, (2010). [4] B. Yu, Z. Z. Yang. An ant colony optmzaton model: The perod vehcle routng problem wth tme wndows, Transportaton Research Part E: Logstcs and Transportaton Revew, 47, (2011). [5] Ha-bn Duan. Ant Colony Algorthms: Theory and Applcatons. Bejng: Scence Press, (2005). [6] C. M. Chen, W. C. Xe, S. S. Fan, et al. The logstcs vehcle routng optmzaton method and mplementaton based on ant colony algorthm, Journal of Computatonal Informaton Systems, 8, (2012). [7] John E. Bell, Patrc R. McMullen. Ant colony optmzaton technques for the vehcle routng problem, Advanced Engneerng Informatcs, 18, (2004). [8] Q. Chen, B. Nng. Applcaton of MMAS on vehcle routng problem wth tme wndow. Journal of Jangsu Unversty of Scence and Technology (Natural Scence Edton), 23, [9] M. Trpath, G. Kurger, H. D. Wan. An Ant Based Smulaton Optmzaton for Vehcle Routng Problem wth Stochastc Demands, Proceedng of the 2009 Wnter Smulaton Conference, (2009). [10] S. R. Balsero, I. Loseau, J. Ramonet. An Ant Colony algorthm hybrdzed wth nserton heurstcs for the Tme Dependent Vehcle Routng Problem wth Tme Wndows, Computers & Operatons Research, 38, (2011). [11] D. M. Zhao, L. Luo, K. Zhang. An mproved ant colony optmzaton for the communcaton networ routng problem, Mathematcal and Computer Modellng, 52, (2010). [12] C. Garcła-Martłnez, O. Cordn, F. Herrera. A taxonomy and an emprcal analyss of multple objectve ant colony optmzaton algorthms for the b-crtera TSP, European Journal of Operatonal Research, 180, (2007). [13] Y. Yang, Mohamed S. Kamel. An aggregated clusterng approach usng mult-ant colones algorthms, Pattern Recognton, 39, (2006). [14] C. Rajendran, H. Zegler. Ant-colony algorthms for permutaton flow shop schedulng to mnmze mae span/total flow tme of jobs, European Journal of Operatonal Research, 155, (2004).

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