A Two-layer Time Window Assignment Vehicle Routing Problem

Transcription

1 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 A Two-layer Tme Wndow Assgnment Vehcle Routng Problem Mahd Jallvand Department of Industral Engneerng Shahed Unversty Tehran, Iran mahd.jallvand@shahed.ac.r Mahd Bashr Department of Industral Engneerng Shahed Unversty Tehran, Iran bashr@shahed.ac.r Abstract Ths paper presents a two-layer tme wndow assgnment vehcle routng problem (TL-TWAVRP(. It s assumed that there s a predefned tme wndow determned by the customers whch s called exogenous tme wndow. Also, there are two-layer endogenous tme wndows nsde the prevous one. The outer layer has bgger wdth and the dfference s a volaton varable. These new tme wndows gve a flexblty to career companes for vstng customers after the end of assgned tme wndows to perform servces to more customers. If the vehcle arrves at the customer n her/hs nner layer assgned tme wndow, no penalty s pad but f vehcle volates nner layer, a penalty wll be calculated n the objectve functon. No extra volaton s allowed from the outer layer assgned tme wndow. Ths problem s modeled as a two-stage stochastc problem. The decsons of frst-stage are assgnng nner and outer layers tme wndow to each customer. Then n the next stage, routes wll be planned for each scenaro. Fnally, by comparng ths model wth the TWAVRP, t has been shown that the proposed model has better performance to mnmze total cost wth servng more customers consderng ther exogenous tme wndows. Keywords Vehcle routng, Tme wndow assgnment, Two-layer tme wndow, Stochastc programmng. 1. Introducton In daly goods dstrbuton network, career companes face lots of delvery from supplers to customers that should be cost-effectve. In the classc vehcle routng problem, the total travelng cost s mnmzed as an objectve functon. One of the most studed subjects n ths problem s the vehcle routng problem, consderng customer s tme wndows whch are called VRPTW. In ths knd of problems, each clent must be vsted n a pre-specfed tme wndow. The VRPTW s a problem wth hard constrants although n some practcal applcatons tme wndow constrant can be consdered as a soft constrant and the volatons must be calculated as a penalty n the objectve functon. Ths atttude converts the problem to vehcle routng problem wth soft tme wndow (VRPSTW) whch s studed less than VRPTW. The VRPSTW consders tme wndow but the dfference s that each customer can be vsted at any tme and vehcle must pay penalty for tme wndow volatons. IEOM Socety Internatonal 1315

2 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 In real-world applcatons, the customer and suppler mght agree on a specfc tme nterval for delvery that can consder as an endogenous tme wndow. Also because of some unpredctable condtons, career companes may want to have an extra agreement wth customers to have more flexblty wth payng a penalty. The amount of flexblty may depend on company equpment lke vehcles capacty and uncertan dstrbuton condtons lke traffc condton and etc., so the agreement wth customers s determned by a two-layer assgned tme wndow. It should be noted that mentoned assgned two-layer tme wndows should be nsde of a predefned tme wndow whch s called exogenous tme wndow. As llustrated n fgure 1, all of defntons can be better llustrated. In the real world, demands are usually unknown, hence TWAVRP model s ntroduced for vehcle routng problem wth stochastc demands. exogenous tme wndow endogenous tme wndow Inner layer outer layer Fgure 1: An llustraton of two-layer tme wndow In ths study, more flexblty s consdered by assgnng two-layer endogenous tme wndows. Suppose that a career company assgns a tme wndow for each customer before realzaton of demands and other uncertan parameters. It means that the customer should be vsted durng ts assgned tme wndow whle the customer has defned a specfc wdth for ts assgned tme wndow. In that case extra agreement wth the customer ncludng a penalty wll make dstrbuton more effcent, so n the current study nner and outer layer endogenous tme wndows are assgned n a predefned exogenous tme wndow as the frst stage decson. Then routng decsons are made based on the frst stage decson after realzaton of all uncertan demands. In some cases, payng endogenous volaton penalty cost s more cost-effectve than addng a new vehcle to servce fleet accordng to scanaros. In the proposed model, objectve functon mnmzes the sum of travelng, penalty and fxed vehcle costs. In ths paper, a new model based on two commodty flow formulaton n Baldacc et al. (24) and subtour elmnaton MTZ-nequaltes n Mller et al. (196) s presented. The sectons structured as follows: n the secton 2, a lterature revew s presented. Then the problem s defned and mathematcal formulatons are presented n secton 3. Numercal examples are analyzed n secton 4. Fnally, concluson and future research drectons are consdered n the last secton. 2. Lterature Revew The researches about vehcle routng problem can be categorzed n four man sectons: 1-Determnstc capactated vehcle routng problem(cvrp), 2-Stochastc vehcle routng problem (SVRP), 3-Robust vehcle routng problem(rvrp), 4-Fuzzy vehcle routng problem. Stochastc VRP has been consdered more because of ts more realty. There are some sources of uncertanty that revewed by Oyola et al. (217). For nstance, demands, customers, servce tme, travel tme may be stochastc parameters. VRP manly mnmzes total travelng cost but n most of the cases there are some constrants about the man problem such as budjet, tme, dstance and capacty lmtatons. Another restrcton accordng to real world stuatons s about the servcng tme wndow. Based on the hardness of tme wndow for servcng customers the VRP can be consdered as VRP wth hard tme wndows (volaton from customers tme wndow s not possble), VRP wth soft tme wndow (a penalty cost s pad based on amount of volaton), VRP wth flexble tme wndow (t s lke soft tme wndow wth restrctons on the amount of volatons). In the case of vagueness on the tme wndow, the problem wll be changed to the VRP wth fuzzy tme wndows. In prevous mentoned types of the VRP, tme wndows are predefned by the customers, however n a new type of IEOM Socety Internatonal 1316

3 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 problem, the dstrbutor or supplyer may assng the tme wndow accordng to ts lmtatons nstead of customers. In the current study we wll fucos on the Tme Wndow Assgnment Vehcle Routng Problem (TWAVRP). Among researches n recent years there are a few studes on the TWAVRP and t can be seen that t s very common and t s very useful n real dstrbuton cases. In real cases, usually customers prefer to defne tghter tme wndows for ther comfort and for ther own better plannng, but t wll make lots of schedulng and resources dffcultes for the dstrbutor. As a soluton, dstrbutor can assgn tghter tme wndow nsde a predefned exogenous tme wndow by the customer. Ths strategy wll be more benefcal for both parts (customers and dstrbutor). Because of facng to the uncertanty the dstrbutor may fal to serve some customers n ther assgned tme wndows. More flexblty may be benefcal for dstrbutor to solve the mentoned problem as a soluton to encounter uncertanty. It means that the dstrbutor may assgned two layers of tme wndows nsde of the exogenous tme wndow. Internatonal resturaunt delveres, onlne market delveres, school bus schedulng and etc. are some examples n whch may requre to schedule ther dstrbutons by the proposed two layer tme wndow assgnment scheme. In some cases, a customer tme wndow can be volated consderng an approprate penalty. Tas et al. (213) presented a model for vehcle routng problem wth stochastc travel tmes. Each customer has a soft tme wndows and servce costs. Ths model consdered transportaton and servce costs. transportaton cost ncludes sum of dstance traveled, number of vehcles send from depot and all drver's extra workng hours. In ths paper, servce costs are a penalty for early arrval tmes and late arrval tmes. Tas et al. (214) ntrodused a VRP wth flexble tme wndow. In ths model, customers can receve ther servces wth a devaton from ther tme wndow. Ths devaton s determned by a certan tolerance. The man dfference n ths model comparng wth VRPSTW, s the restrcton on arrval tme of the vehcles. In VRPSTW, vehcle can arrve at each customer at any of tme then payng an approprate penalty for latency but n ths model, latency s restrcted by a tolerance. Tas et al. (214) ntroduced a model for the tmedependent vehcle routng problem wth a soft tme wndow and stochastc travel tmes. customers have soft tme wndows and travel tmes are uncertan and dependent to tme and traffc condton. Mothuy et al. (215) proposed a multstage large scale neghbor search for the vehcle routng problem wth soft tme wndows. Ths model frst ntends to mnmze the number of routes then mnmze the number of early arrval and delay at each customer. The man contrbutons of the TWAVRP can be seen n the followng researches. The tme wndow assgnment vehcle routng problem ntroduced by Splet, and Gabor (214) assgns tme wndow to each customer before demand realzaton to mnmze total transportaton cost. For each customer an endogenous tme wndow wth pre-determned wdth s selected wthn the exogenous tme wndow. Ths problem s a specfc knd of vehcle routng problem wth consstency condtons studed by Groër et al. (29). Splet and Desaulners (215) ntroduced a specal case of the TWAVRP, n whch tme wndows should be chosen from the set of defned tme wndows. Splet et al (217) ntroduced a specal knd of the problem n a case of tme-dependent travel tmes. Because of varyng of travel tmes durng a day, t s consdered n assgnng the tme wndow. Dalmejer and Splet (218) presented a branch-and-cut algorthm for the tme wndow assgnment vehcle routng problem (TWAVRP). They showed that ther algorthm has better performance comparng wth the algorthm presented n Splet and Gabor (214) and can solve larger nstances. Fábo Neves-Morera et al. (218) ntroduced a new verson of problem that tme wndows are dependent to products and splt delvery s consdered. Jabal et al. (215) also proposed a model smlar to TWAVRP but n ths case travel tmes are uncertan and a heurstc has been used to solve the model. Based on the presented lterature revew, t s realzed that schedulng of delveres to customers by the tme wndow assgnment n the vehcle routng problem s an mportant ssue whch s consdered by varous researchers. However, t seems that tme wndow assgnment n two layers to gve more flexblty to dstrbutor and more comfort to customers has not been consdered n prevous researches. So n ths study, t s consdered as man characterstc of the current research. Moreover, some career companes may have extra lmtatons for dstrbuton on the dstrbutng tme n a day because of cty envronmental polluton restrctons. It s consdered n ths study, too. 3. Problem defnton In ths secton, a two-layer tme wndow assgnment vehcle routng problem (2L-TWAVRP) s ntroduced. Consder a drected graph G = ( V, A) wth n customers V = {1, 2,... n},wch demonstrates the depot and n + 1 s the same IEOM Socety Internatonal 1317

4 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 depot and acts as a copy of depot. Therefore V = V {, n + 1} denotes all customers and depot spots. Set A s total arcs that s used to connect nodes n network. Arc set A composed total arcs between clents n V and all arcs leavng and enterng to the depot. Set of arcs A s set of arcs A wth two addtonal arcs every drected arc postve cj (, j) A. Furthermore t j s travel tme and c j (, ) n+ and ( 1, ). For s travel cost. Note that travel cost must be, also travel tme must be non-negatve. Each customer, has a endogenous tme wndow that t has two layers, the wdth of the nner layer s w and determned by the customers. The wdth of the outer layer s model determnes ts optmal value. The exogenous tme wndow of the customer also workng hours of startng depot and endng depot are gven by [s,e ] and [s,e ] n+ 1 n+ 1 s gven n an nterval w and the [s,e ], respectvely. It s assumed that there are unlmted dentcal vehcles wth the capacty of Q. As mentoned before, there s a demand uncertanty n a fnte set of scenaros. Each scenaro has related probablty denoted by d tme wndow per unt s that s Q. Each vehcle has a fxed cost of d C d C M p and demand clent V n. A volaton cost of servcng after assgned. Fnally, t s assumed that customers dsaffect to face to an ncreased wdth of outer layer, s tme wndow, so n the model a dssatsfacton rate of C T s consdered for each unt of the outer layer wdth. s the set of ntegral numbers. Our proposed model s a generalzed form of the TWAVRP formulaton n Dalmejer and Splet (218) wth two major dfferences: frst, ths new tme wndows has two layers wth dfferent correspondng characterstc, second penalty cost and vehcle fxed cost s consdered n the objectve functon. Decson varables y : Tme wndow decson varable for V x j : A bnary varable equals 1 f vehcle travels form t : Contnuous varable determnes servce recevng tme of customer z j : Sum of load n vehcle when travels form z j : Remanng capacty when vehcle s travelng from w h to j to j under scenaro. n scenaro. j to V n scenaro. : Contnuous varable determnes the amount of volaton n the nner layer. : Servce tme volaton amount from nner layer of tme wndow for clent under scenaro. V Mathematcal formulaton (1) mn p cj x j + Cd h + CM xj + CT (, j) A V jv V s.t = 1 V, (2) j V n+ 1 x j IEOM Socety Internatonal 1318

5 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 V x j = 1 jv, (3) z + z = ( x + x ) Q (, ), j, j j j j j A (4) ( z j zj ) = 2d V, (5) jv z = d jv jv j V z = x Q n+ 1, j j jv (6) (7) z = x Q d j j jv jv V (8) t + t t x + ( s e )(1 x ) V, j V, (9) j j j j j s + tj t j, j V (1) t + t e V, (11), n+ 1 n+ 1 t y V, (12) t y + w + w V, (13) y s, e w w V (14) h = t ( y + w ) V, (15) h V (16) x (, j) A, (17) j IEOM Socety Internatonal 1319

6 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 z j (, j) A, (18) The objectve functon (1) mnmzes the sum of fxed and varable routng and penalty costs. Penalty cost conssts of two terms ncludng length of assgned outer layer tme wndow and servcng tme volaton from the nner layer tme wndow. Constrants (2) and (3) ensure that each customer s vsted only by one vehcle. Constrants (4) determnes used and left over capacty of a vehcle accordng tots capacty. Constrant (5) determnes the used and left over capacty of a vehcle accordng to served demand of a vsted node. Constrants (6)-(8) determne the requred vehcle load and excess capacty when leavng or returnng to the depot. Constrant (9) to (14) s related to both nner and outer layers tme wndows for each customer. Constrant (9) s MTZ nequaltes that as subtour elmnaton constrants. Constrant (1) ensures that vehcle can serve a customer after leavng the depot and travelng to customer node. Constrant (11) schedules vehcles consderng the depot closng tme. Constrant (12) and (13) guarantees that each customer receve ts demand after start endogenous tme wndow and before end of t. Constrant (14) mples that each endogenous tme wndows must assgn wthn exogenous tme wndow. Constrant (15) calculates the amount of servcng volaton from an assgned nner layer tme wndow for each customer.constrants (16)-(18) defne ype of varables. 4. Numercal expamples and results In ths secton, some nstances ncludnge 1, 15, 2 and 2 customers are consdered and results of last nstance s reported n Table 1. All nstances are run on Intel (R) core (TM) 7-45U 1.8GHz wth 8GB of RAM computer. The model was coded n GAMS verson It s asumed that stochastc demands follow a unform dstrbuton. Table 1 shows that the proposed model has a better performance comparng to classc models. The VRPTW was solved wth all scenaros and avrage of results s reported. Also exogenous tme wndows of TWAVRP nstances are used as customer s tme wndow n VRPTW. On the other hand, the number of requred vehcles decreas n the proposed model n comparson wth model that consdered one layer tme wndow assgnment. Table 1: A comparson of the proposed Two layer tme wndow model wth classc models VRPTW TWAVRP TL-TWAVRP vehcles fxed cost Computaton tme (s) Routng cost vehcles fxed cost Computaton tme (s) Routng cost vehcles fxed cost Computaton tme (s) Routng cost Customers Instance IEOM Socety Internatonal 132

7 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 Fgure2 showed that number of requred vehcles to serve the customers n the frst layer of assgned tme wndow ncreaeses by ncreasng n the penalty cost. 1 Number of vehcles Delay penalty Fgure 2: A senstvty analyss on penalty cost and number of used vehcles Fgure 3 llustrated that pad servcng penalty costs ncrease by ncreasng of travel tme between par of customers. It s because of volatng the assgned tme wndows because of consderable travel tmes. Ttotal penalty cost Fgure 3: Relaton between travel tme total penalty cost In fgure 4 t can be seen that by ncreasng of the travel tme from all nodes to a specfc customer, ts assgned outer layer tme wndow wdth s ncreased. A vehcle should arrve at a customer before end of the exogenous tme wndow and a penalty cost for volatng the frst layer of the tme wndow s pad. Mentoned analyzes confrm the model valdty. Tme wndow wdth Travel tme ncreasng rato Travel tme ncreasng rato Fgure 4: Relevance between travel tme and outer layer assgned tme wndow wdth IEOM Socety Internatonal 1321

8 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, concluson In ths paper, a new model of tme wndow assgnment VRP was proposed consderng two-layer tme wndow assgnment. Varous senstvty analyss confrm the proposed model valdty. It was shown that more penalty cost should be pad when we face to a problem wth more travellng tme. Also t was shown that wdthof the outer layer assgned tme wndow wll be ncreased n cases wth more travel tmes. By the proposed model, t was concluded that less vehcle wll be needed comparng to classc models. As a future study, tme wndow assgnment may be consdered for other varous types of the vehcle routng problem. Moreover, developng heurstc algorthms to solve the largescale nstances of the problem may be a s another drecton for the future study. References Baldacc, R., Hadjconstantnou, E. and Mngozz, A. An Exact Algorthm for the Capactated Vehcle Routng Problem Based on a Two-Commodty Network Flow Formulaton, Operatons Research, vol. 52, no. 5, pp , 24. Dalmejer, K. and Splet, R. A branch-and-cut algorthm for the Tme Wndow Assgnment Vehcle Routng Problem, Computers and Operatons Research, vol 89, pp , 218. Jabal, O., Leus, R., van Woensel, T., de Kok, T., Self-mposed tme wndows n vehcle routng problems, OR Spectrum, vol. 37, no. 2, pp , 215. Mller, C. E., Tucker, A. W. and Zemln, R. A. Integer Programmng Formulaton of Travelng Salesman Problems, Journal of the ACM, vol. 7, no. 4, pp , 196. Mouthuy, S., Massen, F., Devlle, Y., & Van Hentenryck, P. A Multstage Very Large-Scale Neghborhood Search for the Vehcle Routng Problem wth Soft Tme Wndows, Transportaton Scence, vol. 49, no. 2, pp , 215. Neves-Morera, F., Perera da Slva, D., Gumarães, L., Amorm, P., & Almada-Lobo, B. The tme wndow assgnment vehcle routng problem wth product dependent delveres, Transportaton Research Part E: Logstcs and Transportaton Revew, vol. 116, pp , 218. Oyola, J., Arntzen, H. and Woodruff, D. L. The stochastc vehcle routng problem, a lterature revew, part I: models, EURO Journal on Transportaton and Logstcs, vol 6, no. 4, pp , 216. Splet, R., Daba, S. and Van Woensel, T. The Tme Wndow Assgnment Vehcle Routng Problem wth Tme- Dependent Travel Tmes, Transportaton Scence, no. March, pp. 1 16, 217. Splet, R. and Desaulners, G. The dscrete tme wndow assgnment vehcle routng problem, European Journal of Operatonal Research, vol. 244, no. 2, pp , 215. Splet, R. and Gabor, A. F. The Tme Wndow Assgnment Vehcle Routng Problem, Transportaton Scence, vol 49, no 4, pp , 214. Taş, D., Dellaert, N., Van Woensel, T., & De Kok, T. Vehcle routng problem wth stochastc travel tmes ncludng soft tme wndows and servce costs, Computers and Operatons Research, vol. 4, no. 1, pp , 213. Taş, D., Dellaert, N., Van Woensel, T., & De Kok, T The tme-dependent vehcle routng problem wth soft tme wndows and stochastc travel tmes, Transportaton Research Part C: Emergng Technologes, vol. 48, pp , 214. Taş, D., Jabal, O. and Van Woensel, T. A vehcle routng problem wth flexble tme wndows, Computers and Operatons Research, vol. 52, no. PART A, pp , 214. IEOM Socety Internatonal 1322

9 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Washngton DC, USA, September 27-29, 218 Bographes Mahd Jallvand s an M.S. degree student of Industral Engneerng at Shahed Unversty. He receved hs B.Sc. degree from Bu-Al Sna Unversty n 217. Hs research nterests are faclty, transportatons and supply chan. Mahd Bashr s a Professor of Industral Engneerng at Shahed Unversty. He holds a B.Sc. n Industral Engneerng from Iran Unversty of Scence and Technology, M.Sc. and Ph.D. from Tarbat Modarres Unversty. He s a recpent of the 213 young natonal top scentst award from the Academy of Scences of the Islamc Republc of Iran. He serves as the Edtor-n-Chef of Journal of Qualty Engneerng and Producton Optmzaton publshed n Iran and the Edtoral Board member of some reputable academc journals. Hs research nterests are Facltes Plannng, Stochastc Optmzaton, Meta-heurstcs, and Mult-response Optmzaton. He publshed about 1 books and more than 19 papers n reputable academc journals and conferences. IEOM Socety Internatonal 1323