Customer Selection and Profit Maximization in Vehicle Routing Problems

Transcription

1 Customer Selecton and Proft Maxmzaton n Vehcle Routng Problems Denz Aksen 1, Necat Aras 2 1 Koç Unversty, College of Admnstratve Scences and Economcs, Rumelfener Yolu, Sarıyer, İstanbul, Turkey 2 Boğazç Unversty, Department of Industral Engneerng, Bebek, İstanbul, Turkey 1 Introducton The capactated vehcle routng problem (CVRP or smply VRP) s one of the most studed combnatoral optmzaton problems n the lterature of operatons research. The man reason for ths much attenton s the abundance of ts real-lfe applcatons n dstrbuton logstcs and transportaton. In ths study we focus on the sngle-depot capactated VRP wth profts and tme deadlnes (VRPP- TD).VRPP-TD s a generalzaton of the VRP where vstng each customer ncurs a fxed revenue, and t s not necessary to vst all customers. The objectve s to fnd the number and routes of vehcles under tme deadlne restrctons so as to maxmze the total proft, whch s equal to the total revenue collected from the vsted customers less the travelng cost. For ths problem we propose an effcent teratve margnal proft analyss method called MPA appled n a two-phase framework. The frst phase nvolves solvng a tme-deadlne constraned vehcle routng problem (VRP-TD) usng smulated annealng gven a set of customers. The second phase s the mplementaton of MPA where each customer s margnal proft value s calculated wth respect to the set of routes found n the frst phase. It s decded upon whch customers to retan and whch ones to dscard. Wth the remanng set of customers determned n phase 2, phase 1 s repeated. A fnal correcton s performed on the fnal soluton n order to make sure that all routes have postve total proft values. In our numercal experments we report the results obtaned wth ths framework and assess the soluton qualty of our approach.

2 2 Aksen and Aras 2 Lterature Revew A recent paper by Fellet et al. (2005) elaborates on the travelng salesman problem wth profts (TSPP) whch s a generalzaton of the travelng salesman problem (TSP) where t s not necessary to vst all vertces of the gven graph. Wth each customer s assocated a proft that s known a pror. TSPP can be formulated as a dscrete bcrtera optmzaton problem where the two goals are maxmzng the proft and mnmzng the travelng cost. It s also possble to defne one of the goals as the objectve functon and the other as a satsfablty constrant. In one verson, whch s known as orenteerng problem (OP), selectve TSP (STSP), or maxmum collecton problem (MCP) n the lterature, the objectve s the maxmzaton of the collected proft such that the total travelng cost (dstance) does not exceed an upper bound. The other verson, named as the prze collectng TSP (PCTSP), s concerned wth determnng a tour wth mnmum total travelng cost where the collected proft s greater than a lower bound. Fellet et al. (2005) provde an excellent survey of the exstng lterature on TSPP. Ther survey lsts varous modelng approaches to TSSP and exact as well as heurstc soluton technques. The extenson of TSPP to multple vehcles s referred to as the VRP wth profts (VRPP). The mult-vehcle verson of the OP s called the team orenteerng problem (TOP) whch s studed by Chao et al. (1996). The authors propose a 5- step metaheurstc based on determnstc annealng for ts soluton. A very recent paper on TOP s due Tang and Mller-Hooks (2005) who develop a tabu search heurstc for the problem. Butt and Cavaler (1994) address the multple tour maxmum collecton problem (MTMCP) n the context of recrutng football players from hgh schools. They propose a greedy tour constructon heurstc to solve ths problem. Later on, Butt and Ryan (1999) develop an exact algorthm for the MTMCP based on branch and prce soluton procedure. Gueguen (1999) also proposes branch and prce soluton procedures for the so-called selectve VRP and for the prze collectng VRP both of whch are further constraned by tme wndows. In ths paper we focus on the VRPP-TD and propose a new heurstc called N = 0,1, 2, K, n MPA for ts soluton. Gven a complete graph G=(N, E), where { } s the set of (n+1) vertces (customers and one depot) and E = {(, j) :, j} s the edge set, the objectve of the VRPP-TD s to fnd the best routes for vehcles whch depart from the depot (vertex 0), vst a set of customers, and then return to the depot so as to maxmze the total proft. Proft equals the total revenue collected from the vsted customers less the travelng cost. We assume that demand and proft of each customer, customer locatons and the locaton of the depot are known wth certanty. In our formulaton we take nto account customer-specfc temporal constrants referred to as tme deadlnes, maxmum route duraton/length constrants and a unform (homogeneous) vehcle capacty. Capacty and tme deadlne constrants n addton to arbtrary customer demands dfferentate our problem from the TOP studed by Chao et al. (1996) and Tang and Mller-Hooks (2005).

3 Proft Maxmzaton n Vehcle Routng Problems 3 3 Model Formulaton and Soluton Methodology In the mxed-nteger lnear program (MIP) below, p s the revenue collected by vstng customer ; q s the demand of customer ; d j s dstance between customers and j; β denotes the unt travelng cost, and Q s the vehcle capacty. Decson varable y s 1 f customer s vsted by some vehcle, 0 otherwse; x j s 1 f customer j s vsted after customer by some vehcle, 0 otherwse; u s a weght assocated wth each customer bounded between the demand of the customer and the vehcle capacty. maxmze s.t. py j j N\0 N j j 0 N\0 N\0 j j j j { } x = y N \ 0 j \ 0 u u + Qx + ( Q q q ) x Q q, \ 0 j j j j j j y, x 0,1, β d x (1) x = x N Q x q y q u Q N j In ths formulaton, the frst constrant s a degree balance constrant for all vertces n N. The second constrant ensures that a customer has no ncomng and outgong arcs unless t s vsted. The thrd constrant mposes a mnmum number of vehcle routes accordng to vsts to customers. The next two constrants are lfted Mller-Tucker-Zemln subtour elmnaton constrants for the VRP frst proposed by Desrochers and Laporte (1991), corrected later by Kara et al. (2005). Fnally, the last constrant pertans to the ntegralty of y and x j s. Note that Q must be chosen larger than the maxmum customer demand such that every customer s elgble to be vsted by a vehcle. The MIP formulaton n (1)-(7) needs to be supplemented by tme deadlne constrants and consequently arrval tme varables for each customer node f we want to solve a VRPP-TD nstance wth t. However, after the ncorporaton of those extra constrants and varables, a commercal solver such as CPLEX cannot solve problems wth more than 20 customers. Therefore, we need effcent heurstcs to tackle large sze nstances of VRPP-TD. To ths end, we propose a new heurstc referred to as teratve margnal proft analyss (MPA) whch s based on the dea of the margnal proft of a customer. We employ MPA n the followng soluton framework. 1. Solve the gven problem nstance as a VRP-TD wth the current set of vsted customers usng smulated annealng (SA). (2) (3) (4) (5) (6) (7)

4 4 Aksen and Aras 2. For the current set of routes apply MPA untl the margnal proft of each and every remanng customer s postve. 3. If MPA does not modfy the set of vsted customers (.e., no customers are dropped from the current routes), then go to step 4. Otherwse (.e., some customers are dropped), go to step Check the proft of all routes. If they are postve, then stop. Otherwse, for each route wth a nonpostve total proft drop the customer wth the lowest margnal proft untl the total proft of the route becomes postve and the removal of ths customer does not mprove the total proft of the route. Step 4 of the above soluton framework s necessary snce t s possble that the total proft assocated wth a route can be negatve even f the margnal profts of customers on that route are all postve. Therefore, we make a fnal check to detect such routes, and try to modfy them by removng some customers so as to make the profts of these routes postve. It s clear that at the very begnnng of step 1 the ntal set of vsted customers s taken as the set of all customers N \{0}. In step 1 of the above procedure we solve a VRP-TD usng SA. Gven a combnatoral optmzaton problem wth a fnte set of solutons and an objectve functon, the SA algorthm s characterzed by a rule to randomly generate a new feasble soluton n the neghborhood of the current soluton. The new soluton s accepted f there s an mprovement n the objectve value. In order to escape local mnma, new solutons wth worse objectve values are also accepted wth a certan probablty that depends both on the magntude of the deteroraton and the annealng temperature T. The acceptance probablty s taken as e / T. A certan number of teratons (Lk) are performed at a fxed temperature (T k ), then the temperature s reduced every L k teratons by the coolng rate α. The most mportant characterstc of an SA-based heurstc s the defnton of the neghborhood structure that determnes how new solutons are generated from the current soluton. We use three dfferent neghborhood structures, whch are 1-0 move, 1-1 exchange, and 2-Opt. In 1-0 move, a customer selected arbtrarly s removed from ts current poston and nserted n another poston on the same or a dfferent route. 1-1 exchange swaps the postons of two customers that are ether on the same route or on two dfferent routes. Fnally, 2-Opt removes two arcs, whch are ether n the same route or n two dfferent routes, and replaces them wth two new arcs. For a detaled explanaton and pctoral descrpton of these local mprovement heurstcs we refer the reader to Tarantls et al. (2005). In step 1 of the proposed procedure where VRP-TD s solved by SA wth the current set of vsted customers, SA has to be provded wth an ntal soluton. For ths purpose we use the parallel savngs heurstc of Clarke and Wrght (1964). At each teraton of the SA heurstc we randomly choose one of the local mprovement heurstcs to generate a new feasble soluton from the current soluton. Note that t s possble that one move of the selected local mprovement heurstc may result n an nfeasble soluton to the VRP-TD because ths soluton may volate vehcle capacty or tme deadlne constrants. If ths happens, we try all possble moves that can be performed by the local mprovement heurstc untl a feasble soluton s found. Ths mples that one of the feasble solutons n the neghbor-

5 Proft Maxmzaton n Vehcle Routng Problems 5 ( ) k l kl hood of the current soluton s found wth certanty f there exsts one. In case there are no feasble solutons wth respect to the selected local mprovement heurstc, then we randomly choose another one. If we cannot generate a feasble soluton, the procedure s stopped, and we report the best feasble soluton found so far. In step 2 of our soluton framework, we apply MPA for the current set of routes found n step 1. Gven a customer, ts margnal proft s defned as π = p β d + d d where k and l are those nodes that, respectvely, precede and succeed customer. If π 0, then customer s not worth vstng. It can be dropped from ts route. Otherwse, t s proftable to vst customer ; thus t s kept between nodes k and l. The formal defnton of MPA s gven below. 1. For the current set of routes compute each customer s margnal proft π. Sort π 0 values n nondecreasng order and obtan a sorted stack π[]., \ N 2. If π [1] of customer [1] (.e, customer wth the lowest margnal proft) s postve, then ext MPA. Otherwse, go to step Let succ [1] and pred [1] be the successor and predecessor nodes, respectvely, of customer [1] on ts current route r. Delete [1] from the route. Update the π values of succ [1] and pred [1] f they are customer nodes. Cancel route r f [1] was the only customer on t. 4. Set π [1] to nfnty such that t s put at the end of the stack. Restore the nondecreasng order of π values n the stack and go to step 2. 4 Computatonal Results In order to test the proposed heurstc we generate random VRPP-TD nstances wth up to 20 customers. These nstances are solved by the proposed heurstc nvolvng MPA, and the objectve values are compared to those found optmally by the solver CPLEX 8.1 operatng under GAMS. The heurstc s coded and compled n Vsual C and run on a 3.20GHz Pentum 4/HT PC wth 1 GB RAM. In the mplementaton of SA, we adopt the followng choces. The number of teratons, L k, whch have to be performed at temperature T k s set to 250n 2 where n s the number of customers. The parameter α whch determnes the coolng rate s assgned a value of SA algorthm s termnated when the mprovement n the objectve value of the best soluton durng the last fve temperature updates s less than 1%. The routes of the best soluton found by the SA are an nput to our heurstc nvolvng MPA. Results are shown n Table 1 where we report the percent gaps from the best objectve values found by CPLEX. In some nstances, as can be seen n the table, the maxmum allowable CPU tme (3 hours) s not suffcent for CPLEX to arrve at a proven optmal soluton. One possble mprovement to the method proposed n ths paper s n step 2 where MPA s carred out on a gven soluton wth a set of vsted customers. On

6 6 Aksen and Aras the bass of margnal profts, MPA dentfes whch customers should be dropped from ths set. It does not take nto account the possblty of addng any currently unvsted customer to ths set. We are currently workng on the extenson of our method that wll account also for the repatraton of dscarded customers. Table 4.1. Accuracy of the results obtaned by MPA No. of Best Proven Gap Prob. No. Customers. Value Optmal Heurstc % yes yes yes yes yes no yes no yes no no no References Butt SE and Cavaler T (1994) A heurstc for the multple tour maxmum collecton problem. Computers & Operatons Research 21: Butt SE and Ryan DM (1999) An optmal soluton procedure for the multple tour maxmum collecton problem usng column generaton. Computers & Operatons Research 26: Chao I, Golden B and Wasl E (1996) The team orenteerng problem. European Journal of Operatonal Research 88: Clarke G, Wrght JW (1964) Schedulng of vehcles from a central depot to a number of delvery ponts. Operatons Research 12: Desrochers M and Laporte G (1991) Improvements and extensons to the Mller-Tucker- Zemln subtour elmnaton constrants. Operaton Research Letters 10: Fellet D, Dejax P and Gendreau M (2005) Travelng salesman problems wth profts. Transportaton Scence 39: Gueguen C (1999) Méthodes de résoluton exacte pour les problèmes de tournées de véhcules. Ph.D. thess, Laboratore Productque Logstque, Ecole Centrale Pars. Kara İ, Laporte G and Bektaş T (2004) A note on the lfted Mller Tucker Zemln subtour elmnaton constrants for the capactated vehcle routng problem. European Journal of Operatonal Research 158: Tang H and Mller-Hooks E (2005) A tabu search heurstc for the team orenteerng problem. Computers & Operatons Research 32: Tarantls CD, Ioannou G, Kranouds CT and Prastacos GP (2005) Solvng the open vehcle routeng problem va a sngle parameter metaheurstc algorthm. Journal of the Operatonal Research Socety 56: