Preprint 1 (1997) 1{11 1. An improved ant system algorithm for the vehicle. routing problem. Bernd Bullnheimer, Richard F. Hartl and Christine Strauss

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1 Preprint 1 (1997) 1{11 1 An improved ant ytem algorithm for the vehicle routing problem Bernd Bullnheimer, Richard F. Hartl and Chritine Strau Intitute of Management Science, Univerity of Vienna, Bruenner Str. 72, A Vienna, Autria bernd.bullnheimer@univie.ac.at The Ant Sytem i a ditributed metaheuritic that combine an adaptive memory with a local heuritic function to repeatedly contruct olution of hard combinatorial optimization problem. We preent in thi paper an improved ant ytem algorithm for the Vehicle Routing Problem with one central depot and identical vehicle. Computational reult on fourteen benchmark problem from the literature are reported and a comparion with ve other metaheuritic approache to olve vehicle routing problem i made. Keyword: ant ytem, adaptive memory, vehicle routing, metaheuritic 1. Introduction The Ant Sytem, introduced by Colorni, Dorigo and Maniezzo [6], [10], [12] i a new ditributed meta-heuritic for hard combinatorial optimization problem and wa rt ued on the well known Traveling Saleman Problem (TSP). Obervation on real ant earching for food were the inpiration to imitate the behaviour of ant colonie for olving combinatorial optimization problem. Real ant are able to communicate information concerning food ource via an aromatic eence, called pheromone. They mark the path they walk on by laying down pheromone in a uantity that depend on the length of the path and the uality of the dicovered food ource. Other ant can oberve the pheromone trail and are attracted to follow it. Thu, the path will be marked again and will therefore attract more ant. The pheromone trail on path leading to rich food ource cloe to the net will be more freuented and will therefore grow fater. The decribed behaviour of real ant colonie can be ued to olve combina-

2 2 B. Bullnheimer et al. / Ant ytem for vehicle routing torial optimization problem by imulation: articial ant earching the olution pace imulate real ant earching their environment, the objective value correpond to the uality of the food ource and an adaptive memory correpond to the pheromone trail. In addition, the articial ant are euiped with a local heuritic function to guide their earch through the et of feaible olution. The ant ytem ha been applied to the Job Shop Scheduling Problem in [7], to the Graph Colouring Problem in [8], to the Quadratic Aignment Problem in [18] and to the Vehicle Routing Problem in [2]. In thi paper we preent an improved ant ytem algorithm for the Vehicle Routing Problem (VRP) with one central depot and identical vehicle and how that the ant paradigm can be ued to produce competitive reult. The remainder of the paper i organized a follow. Firt we briey decribe the VRP and the two baic ant ytem phae contruction of vehicle route (x2.1) and trail update (x2.2), then we preent our improved ant ytem algorithm (x2.3). In x3 we report on computational reult and a comparion with other metaheuritic for the VRP before we conclude with a dicuion of our nding in x4. 2. Ant Sytem for VRP The Vehicle Routing Problem can be repreented by a complete weighted directed graph G = (V; A; d) where V = fv 0 ; v 1 ; v 2 ; : : : ; v n g i a et of vertice and A = f(v i ; v j ) : i 6= jg i a et of arc. The vertex v 0 denote the depot, the other vertice of V repreent citie or cutomer, and the nonnegative weight d ij, which are aociated with each arc (v i ; v j ), repreent the ditance (or the travel time or the travel cot) between v i and v j. For each cutomer v i a nonnegative demand i and a nonnegative ervice time i i given ( 0 = 0; 0 = 0). The aim i to nd minimum cot vehicle route where every cutomer i viited exactly once by exactly one vehicle all vehicle route begin and end at the depot for every vehicle route the total demand doe not exceed the vehicle capacity Q for every vehicle route the total route length (incl. ervice time) doe not exceed a given bound L. The VRP i a very complicated combinatorial optimization problem that ha been worked on ince the late ftie, becaue of it central meaning in ditribution

3 B. Bullnheimer et al. / Ant ytem for vehicle routing 3 management. Problem pecic method (e.g., [5], [15]) a well a meta-heuritic like tabu earch (e.g., [24]), imulated annealing (e.g., [19]), genetic algorithm (e.g., [16]) and neural network (e.g., [14]) have been propoed to olve it. The VRP and the TSP are cloely related. A oon a the cutomer of the VRP are aigned to vehicle, the VRP i reduced to everal TSP. For that reaon our ant approach i highly inuenced by the TSP ant ytem algorithm by Dorigo et al. [12] Contruction of vehicle route To olve the VRP (or the TSP), the articial ant contruct olution by ucceively chooing citie to viit, until each city ha been viited. Whenever the choice of another city would lead to an infeaible olution for reaon of vehicle capacity or total route length, the depot i choen and a new tour i tarted. For the election of a (not yet viited) city, two apect are taken into account: how good wa the choice of that city, an information that i tored in the pheromone trail ij aociated with each arc (v i ; v j ), and how promiing i the choice of that city. Thi latter meaure of deirability, called viibility and denoted by ij, i the local heuritic function mentioned above. With = fv j 2 V : v j i feaible to be viitedg [ fv 0 g, city v j i elected to be viited after city v i according to a random-proportional rule [11] that can be tated a follow: p ij = 8>< >: P h2 [ ij ] [ ij ] [ ih ] [ ih ] if v j 2 0 otherwie Thi probability ditribution i biaed by the parameter and that determine the relative inuence of the trail and the viibility, repectively. For the TSP Dorigo et al. [12] dene the viibility a the reciprocal of the ditance. The ame i done for the VRP in [2] where the election probability i then further extended by problem pecic information. There, the incluion of aving and capacity utilization both lead to better reult. On the other hand, the latter i relative cotly in term of computation time (a it ha to be calculated in each tep of an iteration) and will therefore not be ued in thi paper. Thu, we introduce the parameter f and g, and ue the following parametrical aving function [20] for the viibility: (1)

4 4 B. Bullnheimer et al. / Ant ytem for vehicle routing ij = d i0 + d 0j? g d ij + f jd i0? d 0j j 2.2. Trail update After an articial ant ha contructed a feaible olution, the pheromone trail are laid depending on the objective value of the olution. In early ant ytem approache all ant contributed to the trail update (ee [2], [12]). In more recent paper on the TSP, better reult were obtained for update rule where only the bet ant contribute to the pheromone trail (ee [11], [23]). In another paper Bullnheimer et al. [1] ugget to rank the ant according to olution uality and to ue only the bet ranked ant a well a o-called elitit ant to update the pheromone trail. For the VRP thi update rule i a follow new ij X?1 = ij old + ij + ij (2) =1 where i the trail peritence (with 0 1), thu the trail evaporation i given by (1? ). Only if an arc (v i ; v j ) wa ued by the -th bet ant, the pheromone trail i increaed by a uantity ij which i then eual to (?)=L, and zero otherwie (cf. econd term in (2)). In addition to that, all arc belonging to the o far bet olution (objective value L ) are emphaized a if elitit ant had ued them. Thu, each elitit ant increae the trail intenity by an amount ij that i eual to 1=L if arc (v i ; v j ) belong to the o far bet olution, and zero otherwie (cf. third term in (2)) Ant ytem algorithm After initializating the ant ytem, the two baic tep contruction of vehicle route and trail update, are repeated for a given number of iteration. Concerning the initial placement of the articial ant it wa found in [2] that the number of ant hould be eual to the number of cutomer and that one ant hould be placed at each cutomer at the beginning of an iteration. To improve the performance of a metaheuritic it i common practice to include a local earch procedure and ue o-called candidate lit 1. Bullnheimer et al. [2] how that uing the 2-opt-heuritic for the TSP [9] to horten the vehicle 1 The idea of candidate lit i to concentrate the earch on promiing candidate thu aving

5 B. Bullnheimer et al. / Ant ytem for vehicle routing 5 route generated by the articial ant, coniderably improve the olution uality 2. In addition to thi traight forward local earch we alo introduce candidate lit for the election of cutomer which are determined in the initialization phae of the algorithm. For each location v i we ort V n fv i g according to increaing ditance d ij to obtain the candidate lit. The propoed ant ytem for the VRP can be decribed by the chematic algorithm given in Figure 1. I II Initialize For I max iteration do: (a) For all ant generate a new olution uing Formula (1) and the candidate lit (b) Improve all vehicle route uing the 2-opt-heuritic (c) Update the pheromone trail uing Formula (2) Figure 1. Ant Sytem Algorithm for VRP 3. Computational experience In thi ection we dicu the parameter etting for the propoed ant ytem algorithm and preent computational reult Benchmark problem The ant ytem for VRP wa teted on fourteen benchmark problem decribed in [4]. Thee problem contain between 50 and 199 cutomer in addition to the depot. The cutomer in problem 1-10 are randomly ditributed in the plane, while they are clutered in problem Problem 1-5 and 6-10 are identical, except that for the latter the total length of each vehicle route i bounded, wherea for the former there i no route length retriction. The ame i true for the clutered problem: problem are the counterpart of problem with additional route length contraint. For the problem with bounded route length all cutomer reuire the ame ervice time = 1 = : : : = n. run time which can be better ued for further iteration. Candidate lit were rt ued for an ant ytem approach in [11]. 2 The ue of a more ophiticated heuritic uch a Lin-Kernighan [17] could be reaonable, but only for problem where the number of cutomer per tour i large.

6 6 B. Bullnheimer et al. / Ant ytem for vehicle routing 3.2. Parameter etting Following our nding in [2] we ued n articial ant, initially placed at the cutomer v 1 ; : : :; v n and et = = 5 and = 0:75. For the other parameter we found f = g = 2 a a good etting. For all problem I max = 2n iteration were imulated with = 6 elitit ant. Thu, the ve bet ant of an iteration further contributed to the pheromone trail update, which i alo propoed in [1]. The candidate lit ize wa et to bn=4c, i.e. only one fourth of the location, namely the cloet one, were conidered 3. Figure 2 depict the 50-cutomer problem (C1) and the correponding candidate et (each of ize 12) for two elected cutomer. Cutomer v 11 and it candidate are marked by a bold circle, cutomer v 40 and it candidate by a uare, and the depot i marked by a tar. D? Figure Cutomer Problem (C1) with two Candidate Set The coneuence of the ue of the candidate lit were twofold: run time went down and olution uality improved. The former i obviou a time conuming calculation of p ij are aved, the latter i due to the fact that election probabilitie are no longer watered down by 'very unlikely' cutomer. Suppoe an articial ant i located at cutomer v 11 in Figure 2. Then calculating the 3 If all candidate have been viited already the ant return to the depot.

7 B. Bullnheimer et al. / Ant ytem for vehicle routing 7 probability for electing for example cutomer v 40 to be viited next i obviouly neither reaonable from a olution point of view nor from a run time point of view, epecially a the correponding pheromone trail value will be very low after a few iteration Ant ytem reult Table 1 give the computational reult for the fourteen tet problem obtained by our ant ytem. For each problem the column give the problem ize n, the vehicle capacity Q, the maximal route length L and the ervice time. In the lat three column the bet olution obtained with our improved ant ytem (denoted by AS) are compared to previou ant ytem reult (denoted by AS old ) from [2] and the bet publihed olution. Bold character indicate that the bet known olution wa found by the ant ytem Comparion with other metaheuritic In Table 2 we compare deviation from the bet known olution, average deviation (denoted by ) a well a computation time of our ant ytem algorithm (denoted by AS) and ve other metaheuritic approache, where computation time were reported 4. Thee are the parallel tabu earch algorithm by Rego and Roucairol [21] (denoted by RR-PTS), TABUROUTE, a tabu earch algorithm by Gendreau, Hertz and Laporte [13] (denoted by GHL-TS), the rt-bet-admiible variant of Oman' [19] tabu earch (denoted by Om-TS), hi imulated annealing algorithm (denoted by Om-SA) and a neural network implementation by Ghaziri [14] (denoted by Gha-NN). The tabu earch approache eem to be uperior in term of olution uality with an average deviation of 0:5%? 1%. Thi tatement relie not only on thi comparion but alo on the fact that the bet known olution where optimality ha not been hown were found uing tabu earch [22], [24]. Oman' imulated annealing i on average wore than the ant ytem (2.03% v. 1.51%) but that i mainly due to it very bad performance on problem C11. Other than that, the performance of the two algorithm i fairly comparable. Only the neural network approach with an average deviation of 5:30%, four problem with deviation > 8% and only 12 problem olved, can not compete with the other metaheuritic. 4 A there are dierence in the imulation etup of the variou method a comparion on bai of execution time i hardly meaningful.

8 8 B. Bullnheimer et al. / Ant ytem for vehicle routing Table 1 Problem Characteritic and Solution Value for Ant Sytem Random Problem Prob. n Q L bet publ. AS AS old C a C a C a C a C b C a C a C a C a C b Clutered Problem Prob. n Q L bet publ. AS AS old C a C a C a C a a Taillard (1993) b Rochat and Taillard (1995) In the next ection we briey ummarize and dicu our tudy and give an outlook on apect that hould be ubject of further reearch. 4. Dicuion and concluion In thi paper we how the application and the improvement of an ant ytem algorithm to the VRP. The computational reult conrm the poitive experience made with the ant ytem by applying it to the TSP [1], [11], [23]. Although ome very good olution for the VRP intance were obtained, the bet known olution for the fourteen tet problem could not be improved. For practical purpoe deviation up to 5% are more than acceptable a uncertainty about travel cot, demand, ervice time etc. make perfect planning impoible. Therefore the preented ant ytem approach i a valid alternative to tackle VRP. A detailed invetigation of parameter value (by extenive teting and/or

9 B. Bullnheimer et al. / Ant ytem for vehicle routing 9 Table 2 Relative Percentage Deviation and Run Time for everal Metaheuritic Approache RR-PTS GHL-TS Om-TS Om-SA Gha-NN AS Prob. [%] [min] [%] [min] [%] [min] [%] [min] [%] [min] [%] [min] C C C C C C C C C C C C C C % 0.86% 1.03% 2.09% 5.30% 1.51% Sun Sparc 4 Silicon 4D/35 VAX 8600 VAX 8600 VAX 8600 Pentium 100 thorough theoretical analyi) could yield better calibration and hould allow further improvement. On top of that, applying a more ophiticated local earch procedure uch a Lin-Kernighan [17] to the generated vehicle route, or even a VRP-pecic local earch that work on entire olution, i another promiing direction for future work. Finally, the algorithm eem to be well uited for parallel implementation [3]. Again, thee improvement hould lead to ant ytem algorithm that are able to nd better olution needing le run time. Beide thee methodological conideration, additional modication of the algorithm to extenion of the VRP, e.g. multiple depot or problem with time window are of interet. Acknowledgement The author would like to thank Marco Dorigo and Herbert Dawid a well a three anonymou referee for valuable comment that helped to improve the uality of thi paper.

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