Integer Programming Formulations for the Uncapacitated Vehicle Routing p-hub Center Problem

Transcription

1 21st Internatonal Congress on Modellng and Smulaton, Gold Coast, Australa, 29 No to 4 Dec Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem Z. Kartal a and A.T. Ernst b a Anadolu Unersty, Department of Industral Engneerng, 26555, Esksehr, Turkey b CSIRO, Prate Bag 33, Clayton South, Vc. 3169, Australa Emal: zkartal@anadolu.edu.tr Abstract: Hub facltes are used n many-to-many transportaton networks such as passenger arlnes, parcel delery, and telecommuncaton system networks. In these networks, the flow that s nterchanged between the demand centers s routed a the hubs to prode dscounted transport. Many parcel delery frms sere on a hub based system where the flows from dfferent demand nodes are concentrated, sorted and dssemnated at the hub centers n order to transfer them to the destnatons ponts. The man purpose of hub locaton problem s to decde the locaton of hub facltes and to allocate demand nodes to the hubs. In hub based systems, as an alternate way of serng each orgn destnaton node drectly, the flow s accumulated at the hub facltes n order to explot the substantal economes of scale. Hub locaton problems can be categorzed n terms of the obecte functon of the mathematcal models. In the lterature, hub locaton problems wth total transportaton cost obectes (medan), mn-max type obectes (center) and coerng type obectes are well studed. In the hub locaton problem lterature, t s assumed that only one ehcle seres between each demand center and hub. The ehcles are not permtted to st more than one cty. The need to desgn a network of combned hub locatons and ehcle routes arses n arous applcatons. For example, n cargo delery systems, sendng separate ehcles between each demand center and hub s rather costly n terms of nestment on the total number of ehcles. Instead, f the ehcles are allowed to follow a route by stng dfferent demand nodes n each stop, the total nestment cost may decrease consderably. In arlne companes, smlarly, f a separate arcraft and separate ar staff are assgned for each destnaton, they ncur hgh nestment and operatng costs. Also, traffc congeston occurs at arports and n ar networks. In the lght of aboe-mentoned real lfe consderatons, the ehcle routng hub locaton problem has been receng ncreased attenton from researchers. Ths problem s to decde the locaton of hubs, the allocaton demand centers to the hubs and the assocated routng structure wth multple stopoers and allowng ehcles to make a tour so as to mnmze total transportaton cost. In addton to the cost, parallel to the ncrease of the competton n the market, companes tend to promse to the customers next day delery or delery wthn 24 h guarantees. Howeer, the hub locaton and ehcle routng problem, whch consder both the flows and dstances, may sometmes lead to delays from non-smultaneous arrals at hubs, when worst-case route lengths for ehcles are excessely large. Although classcal hub locaton problems prode one opton when orgn-destnaton dstances are huge, they become less approprate when ehcle routng s requred and delery tme s a maor concern. In ths study, we ntroduce the uncapactated ehcle routng p hub center problem to the lterature. The am of our model s to fnd the locaton of the hubs, assgn demand centers to the hubs and determne the routes of ehcles for each hub such that the maxmum dstance or trael tme between orgn-destnaton pars s mnmzed. We propose mathematcal programmng formulatons for ths problem wth O (n2 ) and O (n4 ) arables. The formulatons trade off tghtness aganst formulaton sze. The computatonal results on standard data sets from the lterature allow ths trade off to be ealuated emprcally and prode an ndcaton of the challenge of solng these combned ehcle routng hub locaton problems. Keywords: p-hub center problems, hub locaton, hub locaton and routng problem, p-hub center and routng problems 1724

2 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem 1 INTRODUCTION In most many-to-many dstrbuton systems, goods are collected from arous orgns to be dstrbuted to arous destnatons. Howeer, sendng the goods drectly from each orgn to each destnaton wth a dedcated ehcle s often not economcal, snce the flow rates for a substantal fracton of the orgn-destnaton pars can sometmes be so small. Hub network systems are used to consoldate the flows between orgn-destnaton pars to elmnate expense drect connecton. The am of the classcal sngle allocaton p-hub medan problem s to mnmze the total cost of transportaton whle establshng p nodes (hubs) among a gen set of nodes wth parwse traffc demands and allocate each non-hub node to the hubs. Whle n the p-hub center problem, the man obecte s mnmzng the maxmum dstance/cost between orgn-destnaton pars. In the hub locaton problems, the flow from to must be routed on at least one hub f both the nodes and are allocated the same hub; otherwse at most two hubs, f node and are allocated to dfferent hubs. There s a path n the route as k l f node s allocated to hub k and node s allocated to hub l. The flow traersng the hub lnk k l s the flow from nodes allocated to hub k to nodes allocated to hub l. Commonly a dscount factor apples on the nter-hub connectons snce the transportaton cost on these lnks are lower due to economes of scale. Sureys on arous dfferent hub locaton problems can be found n Campbell and O Kelly [2012] and Alumur and Kara [2008]. The fundamental assumpton of hub locaton problem s that there s a drect connecton between each non-hub node and ts assgned hub. Howeer, more recently, ths unrealstc assumpton hae been adusted and adapted to deal wth some transportaton problems and logstcs applcatons n hub networks where nodes do not hae adequate demand to connect them drectly wth hubs [Rodrguez-Martn et al., 2014]. In ths paper, we ntroduce the uncapactated ehcle routng p-hub center problem (UVRpHCP) to the lterature, relaxng the man hub locaton problem assumpton, where there s a separate ehcle whch seres between any non-hub node and ts assgned hub. To the best of authors knowledge, no preous study has nestgated the combnaton of p-hub center and ehcle routng problems. The am of the problem s to mnmze maxmum dstance/cost between any orgn-destnaton par, whle locatng p hubs, assgnng the non-hub nodes to exactly one hub and formng the routng structures of the gen ehcles for each hub. The outlne of the paper s as follows. In the next secton, we ntroduce three dfferent mathematcal formulatons of the UVRpHCP, whose complextes n terms of arables are O (n 2 ) and O (n 4 ). In the thrd secton, we ge some numercal tests wth regard to all formulatons. Fnally, n the last secton concludng remarks are presented. 2 MATHEMATICAL FORMULATIONS The UVRpHCP determnes the locaton of the hubs, the allocaton of non-hub nodes to hubs, and the assocated routng structure between non-hub nodes and hubs wth multple stopoers so as to mnmze maxmum dstance/cost between orgn-destnaton pars. Ths problem can be defned on a complete, symmetrc network; G = (N, A) wth node set N = {1, 2,..., n} and arc set A. Each arc (, ) N has a nonnegate cost (dstance), d = d satsfyng the trangle nequalty. The economy of scale s modeled by a dscount factor α (0, 1) between hub nodes. The costs can be used to capture trael tme, path length or monetary costs. We assume that a lmted fleet of homogenous uncapactated ehcles V operates on the routes. The number of aalable ehcles for each hub to deploy; n() s known a pror. These ehcles should st at least two nodes ncludng the hub tself, also they are allowed to st multple non-hub nodes. There s no upper bound on the number of nodes that a ehcle can st. It s also assumed that larger szed and faster ehcles operate between two hubs. It s not permtted for these ehcles make any addtonal stops along ther trps. One of the man dffcultes for solng hub locaton problems s to deal wth larger numbers of arables and constrants n the formulatons. Therefore, formulatons wth fewer arables and constrants hae been deeloped. Howeer, although the larger formulatons lead to tghter lower bounds, ncreased computatonal tmes are also requred to fnd and exact soluton. For ths reason, we frst explored an nteger lnear programmng model wth O (n 4 ) arables to sole UVRpHCP. The motaton behnd our frst model, whch s based on a classcal four-ndex hub locaton formulaton, s to get tghter lower bounds assocated wth the LP relaxaton. To formulate the problem, we use the dea of the Campbell [1994] study, n whch y s defned as the fracton of flow; as beng 1 f a ehcle s takng mal orgnatng at node and destned for node along the road from node l to node k, wth f beng analogous for the case where both node k and node l are hubs. We note here 1725

3 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem that whle y and f arables are allowed to be fractonal n the formulaton, they are always nteger n the optmal soluton correspondng to the shortest path from node and n the hub network. In ths formulaton, the path followed by ehcles s defned n the model usng the arables x whch take on the alue 1 f ehcle sts node N and node N \ {} n a route, and 0 otherwse. In the suggested formulaton, there s a three ndexed x arable for eery arc-ehcle combnaton. Let h to be a zero-one arable wth h = 1 f node N s a hub and 0 otherwse. Integer arable z {0, 1} to denote f ehcle V s assgned to hub N (z =1) or not (z =0). The mathematcal programmng formulaton of the UVRpHCP wth four-ndexed flow arables s as follows: Model-1 s.t. l k,l k l d y + k ( y lk + f lk ) l mn Z 1, (1) α d f Z 1,, (2) f = f lk,,, k, l k, (3) ( y k + f k ) = 1,,, (4) k l k,l l ( y + f ) = 1,,, k, k =, (5) y x,,, k, l, (6) ( f + f lk ) h k,,, k N, (7) h = p, (8) x x = 0,, (9) z = h n, (10) z x,, (11) x = z + (1 h ), (12) x 1 + h z,, (13) z = 1, (14) z, x, h {0, 1}, N, V (15) 0 f, y 1,, k, l N, (16) The obecte functon (1) mnmzes the maxmum dstance/cost. The frst term n (2) calculates total dstance/cost for a route. The second term calculates the total dstance between two hubs. Constrant (3) ensures that the fracton of flow that traels from node to node a hubs located at k and l s equal to the fracton of flow that follows a route from node to node a hubs l and k. Constrant (4) guarantees that total flow from to s transferred. Constrant (5) s the flow conseraton constrant. Constrant (6) defnes the amount of ehcle flow orgnated at node destned to uses the lnk (, k) from node to node k. Constrant (7) ensures that the total flow s transferred a hub k and hub l. Constrant (8) ensures that exactly p hubs are to be located. Constrant (9) s known as degree constrant. Wth ths constrant t s guaranteed that eery node has as many ehcles arrng as leang. If node s hub, the number of assgned ehcles for ths hub s n(), whch s ensured by constrant (10). If node s hub, the arc between hub node and non-hub node s sered wth exactly one ehcle whch s guaranteed by constrant (11). Constrant (12) states that f node s a hub, the number of arcs that leae from node s equal to the number of assgned ehcles. On the other 1726

4 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem hand, f node s not a hub, then ths constrant guarantees that each non-hub lnk s sted only once by a ehcle. By constrant (13), z takes alue only f node s a hub node. Wth constrant (14) t s guaranteed that each ehcle s only allocated to one hub. Ths model has (n 2 +n +n) bnary arables, (2n 4 ) poste arables and (2n 4 +2n 3 +2n 2 +3n +2n+) constrants. Thus, there are O (n 4 ) arables and O (n 4 ) constrants n total. Secondly, we explored a three ndex ehcle flow formulaton whch uses substantally fewer arables than our O (n 4 ) formulaton. Therefore, we hope to sole much larger problems wth our new formulaton whch has O (n 2 ) arables. Usng the same notaton as for the four ndex hub locaton formulaton, we defne addtonal decson arables and parameters; a : 1, f node s the successor node of non-hub node ; 0 otherwse. b w : 1, f ehcle w traels from node to node ; 0 otherwse. ab : 1, f ehcle traels from node to node, and node s not a hub; 0 otherwse. transfer w : the dstance between two hub nodes. collect : the longest collecton dstance from any locaton sered by ehcle to the hub. dstrb w : the longest dstrbuton dstance to any locaton sered by ehcle w. L : the maxmum number of nodes a ehcle may st. u : auxlary arable - defnes dstance to the hub for subtour elmnaton. The second mathematcal formulaton that we deeloped for UVRpHCP uses O (n 2 ) arables. Note that snce s typcally ery small, ths formulaton s not sgnfcantly larger than the thrd model wth O (n 2 ) arables. Model-2 mn Z 2, (17) subect to: (8) (15) a = 1, (18) a 1 h,, (19) a x,, (20) a x h,,, (21) u u + L a L 1,, (22) b w = 1, (23) w b w x w h,,, w (24) (b w d ) = dstrb w, w (25) ab = 1, (26) ab x h,,, (27) (ab d ) = collect, (28) d z w max{d } (1 z ) transfer w,,, w (29) d x d x M (1 x ) collect + dstrb, k, l, (30) collect + dstrb w + α transfer w Z 2, w (31) a, b, ab {0, 1}, N, V (32) collect, dstrb w, transfer w, max 0,, N,, w V (33) 1727

5 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem The obecte functon (17) mnmzes the maxmum dstance/cost. Ths cost ncludes the total dstance between hubs, plus the dstance begnnng from the frst non-hub node n a ehcle route to the hub (collecton) and lastly the dstance/cost from the last non-hub node n a ehcle route to the hub (dstrbuton). Snce the sngle allocaton scheme s used, each non-hub node s sted only once whch s guaranteed by constrant (18). Constrant (19) ensures that f node s a hub node, a takes on alue 0. On the contrary, f node s not a hub node, then there should be a lnk between node and node. If node s not a hub node and node s a hub node, then constrant (20) lmts the number of enterng arcs nto the hub node to the number of aalable ehcles. By constrant (21) a s one f s a non-hub node and arc s sted by exactly one ehcle. Constrants (22) are sub-tour elmnaton constrants. Constrants (23) (25) together calculate the maxmum dstrbuton cost/dstance amongst all ehcles. Smlarly, constrants (26) (28) together calculate the maxmum collecton cost/dstance amongst all ehcles. Constrant (29) calculates the maxmum cost/dstance amongst all hub pars. When constrant (30) s acte the maxmum dstance s determned by two successe nodes on the route of ehcle, that s we are calculatng the dstance from l to the hub and then from the hub to node k where arc (k, l) s on the route of. Lastly, constrant (31) ensures that the obecte s no less than the trael dstance/cost between any par of nodes allocated to the ehcle w and, snce collect and dstrb w ge the longest dstances of ehcle and w for dfferent hubs, respectely. Model-2 for UVRpHCP s a mxed nteger program that has (3n 2 +n 2 +n+n) bnary arables, (n+ 2 +2) poste arables and (n n 2 + 3n 2 + n n + 5n + 3) constrants. Thus, there are O (n 2 ) arables and O (n 2 2 ) constrants n total. Fnally, we descrbe a thrd formulaton whch s based on a two-ndex ehcle flow formulaton that uses O (n 2 ) bnary arables, that keeps eery arc; for the determnaton path of the ehcle routes. In ths formulaton, the path followed by ehcles are defned n the model by means of x arables whch takes on the alue 1 f node N precedes node N \ {} and 0 otherwse. Integer arables s {0, 1} s used to represent f node N s serced by ehcle V (s = 1) or not (s = 0). Followng the dea of p-hub center formulaton n Ernst et al. [2009], whch s based on the concept of a hub radus, the other arables are r 1 the total dstance from the frst non-hub node to hub node n a ehcle route ; smlarly, r 2 the total dstance from the hub node to last node n a ehcle route. And lastly, n order to calculate the dstances of two dfferent ehcles for the same hub, we defne r 1 to be the total dstance from the frst non-hub node to hub node n a ehcle route ; r 2 denotes the the total dstance from the hub node to last node n a ehcle route. The thrd formulaton whch has O (n 2 ) bnary arables wth two ndex ehcle flow formulaton s then gen below: Model-3 mn Z 3, (34) subect to: (8), (14) x x = 0, (35) x = 1 + (n 1) h, (36) s = 1 + (n 1) h, (37) s 2, (38) z h,, (39) z s,, (40) z h + s 1,, (41) s s + x 1 h,,, (42) s s + x 1 h,,, (43) r 1 r 1 + d M (1 x ) M h M (1 s ),,, (44) r 2 r 2 + d M (1 x ) M h M (1 s ),,, (45) r 1 r 1 + d M (1 x ) M h M (1 s ),,, (46) 1728

6 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem r 2 r 2 + d M (1 x ) M h M (1 s ),,, (47) Z 3 r 1 + α d + r 2 w M (1 h ) M (1 h ),,, w (48) Z 3 r 1 + r 2 M (1 x ),, (49) x, s, h {0, 1},, (50) r 1, r 2, r 1, r 2 0,,,, (51) The obecte functon (34) mnmzes maxmum dstance between hubs, plus the dstance begnnng from the frst non-hub node n a ehcle route to the hub (collecton) and lastly the dstance/cost from the last nonhub node n a ehcle route to the hub (dstrbuton). Constrant (35) s known as the degree constrant. By constrants (36) t s guaranteed that f node s a hub, the number of arcs that leae from node s equal to the number of assgned ehcles. On the other hand, f node s not a hub, then these constrants guarantee that each non-hub lnk s sted only once. Constrant (37) guarantees that each non-hub lnk s sted by only one ehcle unless node s not a hub node. Constrant (38) guarantees that for any ehcle the number of sted nodes ncludng hub node tself s greater than or equal to two. Wth constrant (39) t s guaranteed that each ehcle s only allocated to one hub. By constrant (40) no ehcle s assgned to a node unless a hub s opened at that ste. Constrant (41) allows assgnng a ehcle to a hub node. Constrants (42) and (43) together guarantee that f there s a lnk between node and node, both of these nodes are sted by the same ehcle. Constrant (44) keeps the longest dstance for a ehcle from the frst non-hub node to hub node n the route (collecton) where M s a large number. Smlarly, constrant (45) keeps the longest dstance/cost for a ehcle from the hub to the last node of the route (dstrbuton). Constrant (46) calculates the longest collecton dstance/cost between two nodes n a ehcle route. Constrant (47) calculates the longest dstrbuton dstance/cost between two nodes n a ehcle route. If the collecton plus dstrbuton cost/dstance for two ehcles of one hub s greater than the two dfferent ehcles of dfferent hubs plus the cost/dstance between the hubs, constrant (49) allows that the obecte functon alue must be greater than ths alue. Notce also here that due to constrants (44), (45), (46) and (47) sub-tours are elmnated. For solng Model-1, M s taken as the longest dstance tmes number of nodes n the data set. Ths mathematcal model has (n 2 + 4n + n) bnary arables, (4n) poste arables and (n n 2 + 3n + 3n + 2) constrants. Thus, there are O (n 2 ) arables and constrants n total. 3 COMPUTATIONAL RESULTS In ths secton we present the results wth regard to the three formulatons that are descrbed n ths paper. We soled these usng GAMS 23.8 wth the GUROBI 4.6 soler on a 64 bt Intel 5 (3.30 Ghz) machne wth 8 GB RAM. The effecteness of our mathematcal models were tested on the Cl Aeronautcs Board (CAB) data set. Ths data set, ntroduced n O Kelly [1987], s dered from arlne passenger data between 25 US ctes that ncludes demands and dstances. In order to prode results, we took n = 10 and n = 15 nodes problem nstances wth dfferent numbers of ehcles and hubs. Whle solng the mathematcal models, we lmted the CPU tme to two hours on GAMS. We report the results from GUROBI performance measures n Table 1. For each nstance n Table 1, the frst column ges the characterstcs of the test problem n.p. where n s the number of nodes, p s the number of hubs and s the number of ehcles that are assgned for each hub. In each nstance we set α = 1. The second column ges the optmal soluton. In the nstances when any of the mathematcal models could not fnd the optmal soluton, reported wth an astersk (*) n the second column, we ge the best nteger soluton found wth one of the mathematcal models. The rest of the table has three man parts. We ge the same performance measures reported by GUROBI for O (n 4 ) arable formulaton, O (n 2 ) arable wth three ndex ehcle flow formulaton, and lastly O (n 2 ) arable wth two ndex ehcle flow formulaton, referred to as Model-1, Model-2 and Model-3, respectely. In the three columns for three models, we frst prode the number of nodes n the branch and bound tree, the CPU tme requrement n seconds and gap.e., ub lb ub 100 as reported by GUROBI, where lb s the fnal lower bound and ub s the fnal upper bound. Obsere from Table 1 that, Model-2 and Model-3 soled all the nstances wth 10 nodes optmally n less than one hour of CPU tme. Model-2 also soled to optmalty wthn two hours. We remark here that, the lowest obecte alues amongst three mathematcal models s found by Model-2. We also obsere that Model-1 has the tghtest LP relaxaton oer all the problem nstances, although LP relaxaton alues are stll far from the optmal upper bound.the reason s that the O(n 4 ) arable formulaton s the strongest formulaton of the three models. Howeer, ths tghtness comes at the cost of greatly ncreased computatonal tme. Usng ths formulaton, no optmal soluton found wthn two hours. We note here that 1729

7 Z.Kartal and A.Ernst, Integer Programmng Formulatons for the Uncapactated Vehcle Routng p-hub Center Problem Problem Optmal Table 1. Results for Models on CAB data sets Model-1 Model-2 Model-3 B&B nodes CPU % Gap B&B nodes CPU % Gap B&B nodes CPU % Gap , , , , , , , , , , , , , ,071, * NA 7200 NA 39, , * NA 7200 NA 85, , * NA 7200 NA 5, , * , , * , , the LP relaxaton yelded by Model-1 s always the longest dstance n the data set we consdered. Howeer, the LP relaxaton alue of O (n 2 ) arable models are always zero, except n the frst problem nstance for Model-2. Lastly, we remark here that the O (n 2 ) model wth two ndex ehcle flow formulaton has the largest branch and bound tree to be explored compared to other models. We also note here that although Model-3 s based on the most successful formulaton n terms of the soluton tme n the lterature for the standard sngle allocaton p-hub center problem, the performance of t s weak on the uncapactated ehcle routng p-hub center problem. Clearly, the O (n 2 ) model wth three ndex ehcle flow formulaton s superor to Model-1 and Model-3 n terms of the soluton qualty acheed n reasonable amounts of CPU tme. 4 CONCLUSIONS In ths paper we studed a new hub locaton problem UVRpHCP. To the best of the authors knowledge, ours s the frst study n the lterature that combnes ehcle routng and the p-hub center problem. To sole ths problem we deeloped three nteger programmng models rangng n sze form O(n 2 ) to O(n 4 ) arables. We tested the performances of the formulatons on the well-known CAB data set. The numercal results showed the formulaton Model-2 wth O (n 2 ) arables (where n s the number of nodes and the number of ehcles) s superor to the other formulatons n terms of computatonal tme and soluton qualty. Real world problems are, unfortunately, sgnfcantly larger than any soled exactly n ths paper. Therefore, future work must focus on solng larger problems. Ths could be acheed by deelopng new mathematcal models that reduce symmetry. For example n all of our formulatons renumberng the ehcles leads to equalent (symmetrc) solutons. Reducng such symmetry s lkely to hae a poste mpact but more sgnfcant algorthmc mproements are expected to be necessary to obtan exact solutons to medum and large szed nstances. REFERENCES Alumur, S. and B. Kara (2008). Network hub locaton problems: The state of the art. European Journal of Operatonal Research 190(1), Campbell, J. (1994). Integer programmng formulatons of dscrete hub locaton problems. European Journal of Operatonal Research 72(2), Campbell, J. F. and M. O Kelly (2012). Twenty-fe years of hub locaton research. Transportaton Scence 46(2), Ernst, A. T., H. Hamacher, H. Jang, M. Krshnamoorthy, and G. Woegnger (2009). Uncapactated sngle and multple allocaton p-hub center problems. Computers & Operatons Research 36(7), O Kelly, M. (1987). A quadratc nteger program for the locaton of nteractng hub facltes. European Journal of Operatonal Research 32(3), Rodrguez-Martn, I., J.-J. Salazar-Gonzalez, and H. Yaman (2014). A branch-and-cut algorthm for the hub locaton and routng problem. Computers & Operatons Research 50(0),