Proceedngs of IICMA 2013 Research Topc, pp. xx-xx. SOLVIG CAPACITATED VEHICLE ROUTIG PROBLEMS WITH TIME WIDOWS BY GOAL PROGRAMMIG APPROACH ATMII DHORURI 1, EMIUGROHO RATA SARI 2, AD DWI LESTARI 3 1Department of Mathematcs, Yogyakarta State Unversty, atmn_uny@yahoo.co.d 2Department of Mathematcs, Yogyakarta State Unversty, emnugrohosar@gmal.com 3Department of Mathematcs, Yogyakarta State Unversty, dwlestar@uny.ac.d Abstract. Ths paper presents how to buld multobectve lnear programmng model as soluton of Capactated Vehcle Routng Problem wth Tme Wndows (CVRPTW). We use a goal programmng approach to solve the model. We have dscussed an obectve functon for two man goals: the frst s to mnmze the total number of vehcles and the second s to mnmze the travellng tme of the used vehcles. The proposed model s appled to a problem dstrbuton of Lquefed Petroleum Gas (LPG). Computatonal results of the proposed model are dscussed. Key words and Phrases: CVRPTW, goal programmng, obectve functon. 1. Introducton Several thngs that related to the dstrbuton problems are how many goods are dstrbuted, how the cost of dstrbuton should be spent, the route of dstrbuton, the travellng tme and the number of vehcles are used. Along wth the success of the converson program from kerosene to Lquefed Petroleum Gas (LPG), the needs of LPG are ncreased. PT Pertamna Persero as the company has maxmzed dstrbuton of LPG throughout the terrtory. But LPG shortages stll occurs. Consequently, the dstrbuton problems are mportant to be consdered. They are known as Vehcle Routng Problem (VRP).
ATMII D., EMIUGROHO R.S, AD DWI L. Vehcle Routng Problem (VRP) s a class of problem n whch a set of routes for a fleet of delvery vehcles based at one or several depots must be determned for a number of customers. Man obectve of VRP for servng known customer demands by a mnmum-cost vehcle routng startng and endng at a depot. Vehcle Routng Problem (VRP) whch reducng the cost whle delverng on tme, namely VRP Tme Wndows (VRPTW), has become popular ssue n optmzaton research. On the other hand, f the dstrbuton depends on the capacty of vehcle, t s called Capactated VRP (CVRP) [7]. In ths paper, we dscuss the CVRP wth tme wndows (CVRPTW). It means that the vehcles must arrve to the customer n a restrcted tme nterval whch s gven. It can be stated as follows: fnd a feasble soluton of the vehcles, so that to mnmze both, the number of vehcles used and the total travel tme. To solve the problem, we develop a goal programmng model whch has more than one goal n the obectve functon, even ts goal contradct each other. A goal programmng approach s an mportant technque for modellng and helpng decson-makers to solve mult obectve problems n fndng a set of feasble solutons. The purpose of goal programmng s to mnmze the devatons between the achevement of goals and ther aspraton levels [8]. Based on Az et.al [1], Jola and Aghdagh [4], f VRP has a tme constrans on the perods of the day n whch each customer must be vsted, t s called Vehcle Routng Problem wth Tme Wndow (VRPTW). Belfore, et.al [2], Calvete, et.al [3] and Hashmoto, et al [5] have developed VRP wth tme wndows. Ombuk, et.al [6] have appled a mult obectve genetcs algorthm to solve VRPTW. Sousa,et al [7] have used a mult obectve approach to solve a goods dstrbuton by The Just n Tme Delvery S.A, a dstrbuton company usng Mxed Integer Lnear Programmng. In ths paper, we present a goal programmng model as soluton of Capactated Vehcle Routng Problem wth Tme Wndows (CVRPTW). The obectve functon are two man goals: the frst s to mnmze the total number of vehcles and the second s to mnmze the travellng tme of the used vehcles. Computatonal results of the proposed model wll be dscussed. The paper s organzed as follows, n Secton 2: the mathematcal models of Capactated Vehcle Routng Problem wth Tme Wndows (CVRPTW). In Secton 3, Computatonal results of the proposed model usng a set of data obtaned. Fnally n Secton 4, we descrbe conclusons. 2. Mathematcal Models We formulate CVRPTW model wth 5 nodes (customers locaton) and reasonng as follow: - A fleet of vehcles wth restrcted and known capacty whch s needed to mnmze the number of vehcles. - One depot wth non restrcted and known capacty. 2
- A set of 5 nodes wth demand of goods, servce tme wndow, travellng tme. We assume that the average velocty of vehcle s constant. Let V = { 1, 2,..., v} represents the fleet of vehcles, = { 0,1,2,..., n + 1} meanng the set of all nodes ncludng depot, each node representng a customer locaton. Every vehcle must depart from depot and termnate at node n+1. We formulate the problem as a graph G(, A ) where A s the set of drected arc (the network connectons between the customer and the depot) [7]. Each connecton s assosated wth a travel tme tme,,,. Tme wndow for customer,,. It means the vehcle must arrve between that nterval tme. paper: The followng of mathematcal symbols and decson varables used n ths - C : the set of n customers. - The connecton between node to usng vehcle k. 1, actve xk,,, k V,, n + 1, 0. 0, not - t k : tme when vehcle k come on node,,, k V. -, : tme wndow for customer, D : demand or quantty of goods for customer,. - - : vehcle capacty, - T : constant parameter to consder n the tme wndow constrant. - : negatve devatonal varable of frst goal - : postve devatonal varable of second goal Goal Programmng Model goals: Goal 1 Goal 2 To generate a goal programmng model for the problem, we set the followng : mnmze the total number of vehcles/ avod underutlzaton of vehcle capacty : mnmze the travellng tme of the used vehcles. So the mathematc formulaton of goal programmng model s 3
ATMII D., EMIUGROHO R.S, AD DWI L. V 1 1k ω + 2 2 k Mn Z = ω d + d (1) subect to V xk tme d + 2 = 0. (2) k V xk = 1, C (3) k x0 k = 1, k V (4) xak xak = 0, C, k V (5) x, n+ 1, k = 1, k V (6) C k 1 k k, D x + d = Q k V (7) ( ) t + tme T 1 x t,,, k V, T + (8) k k k t t t,, k V (9) a k b { } x 0,1,,, k V (10) k t 0, C, k V (11) k Based on the model, Eq.(3) s a constrant to ensure that one vehcle vsts each customer only. Constrans n Eq.(4), Eq.(5), and Eq(6) to guarantee that each vehcle depart from depot, a vehcle unload the goods after comes to a customer and then go to the next customer. Eq.(7) shows the vehcles only maxmum loaded n ts capacty. The constrant n Eq.(8) guarantee feasblty of the tme schedule of a servce tme. Eq.(9) s a constrant that represent the tme wndows for every customer. Decson varables are represented by Eq.(10) and Eq.(11). 3. Computatonal Results In ths secton, we apply the model to the data that obtaned from PT Pertamna. The followng data ncludes tme dstance, demand, and tme wndows for every customer, see Table.1 and Table.2. 4
Table 1. Tme dstance between the nodes (mnutes) Tme unts Depot A B C D E Depot 0 7 8 10 3 8 A 5 0 5 6 6 7 B 7 3 0 5 6 8 C 6 2 3 0 7 6 D 4 5 5 7 0 8 E 7 8 8 6 9 0 Table 2. Demand and tme wndows for each customer Customer A B C D E Demand (unts) 90 200 100 125 100 t a (tme unts) 7 8 10 3 8 t b (tme unts) 37 75 43 45 41 The customers can be descrbed lke shown as follows: 0 2 4 3 1 5 Fgure 1. Depot and Customers A computatonal process s done usng LIGO. The followng result obtaned routes as Fg. 2. 5
ATMII D., EMIUGROHO R.S, AD DWI L. 0 2 3 4 1 5 Fgure 2. Depot and Customers Route Usng Vehcle s Capacty 300 unts 0 1 4 3 2 5 Fgure 3. Depot and Customers Route Usng Vehcle s Capacty 560 unts Fgure 2 Shows that we need three vehcles to satsfy the customer s demand. If we ncrease the capacty of the vehcles to 560 unts, the result can be showed n Fg,3 above. It means, only needs two vehcles to dstrbute the demands. 4. Concluson In ths paper we proposed a mult obectve lnear programmng approach to solve a Capactated Vehcle Routng Problem wth Tme Wndows (CVRPTW). Two goals are to mnmze both, the number of vehcles used and the total travel tme. The model s appled n the dstrbuton of LPG wth fve customers. Tme wndows are very mportant to be consdered. So the travel tme can be mnmzed. As future development we suggest to mprove the method to solve CVRPTW. 6
References [1] Az., Gendreau M., Potvn J. 2007. An exact algorthm for a sngle-vehcle routng problem wth tme wndows and multple routes; European Journal of Operatonal Research. 178 pp 755-766. [2] Belfore, P., Hugo Tsugunobu, Yoshda Yoshzak. 2008. Scatter Search for Vehcle Routng Problem wth Tme Wndows and Splt Delveres. Journal Complaton. I-Tech Educaton and Publshng KG, Venna, Austra. [3] Calvete H.I., Gale C., Olveros M.J. 2007. Valverde B.S., A goal programmng approach to vehcle routng problems wth soft tme wndows; European Journal of Operatonal Research 177; 1720-1733 [4] Jola, F. and Aghdagh, M. 2008. A Goal Programmng Model for Sngle Vehcle Routng Problem wth Multple Routes. Journal of Industral and Systems Engneerng Vol. 2, o. 2, pp 154-163, Summer 2008. [5] Hashmoto H., Ibarak T., Imahor S., Yagura M. 2006. The vehcle routng problem wth flexble tme wndows and travelng tmes; Dscrete Appled Mathematcs 154; 1364-1383 [6] Ombuk B., Ross B.J., Hanshar F. 2006. Mult-Obectve Genetc Algorthms for Vehcle Routng Problem wth Tme Wndows; Appled Intellgence 24; 17 30 [7] Sousa, J.C., Bswas, H.A., Brto, R., and Slvera, A., A Mult Obectve Approach to Solve Capactated Vehcle Routng Problems wth Tme Wndows Usng Mxed Integer Lnear Programmng. Internatonal Journal of Advanced Scence and Technology, Vol. 28, March, 2011. [8] Taha, Hamdy. 2007. Operaton Research 8 th ed. An Introducton. USA: Pearson Prentce hall. 7