Insertion based Ants for Vehicle Routing Problems with Backhauls and Time Windows. Marc Reimann Karl Doerner Richard F. Hartl

Transcription

1 Insertion based Ants for Vehicle Routing Problems with Backhauls and Time Windows Marc Reimann Karl Doerner Richard F. Hartl Report No. 68 June 2002

2 June 2002 SFB Adaptive Information Systems and Modelling in Economics and Management Science Vienna University of Economics and Business Administration Augasse 2 6, 1090 Wien, Austria in cooperation with University of Vienna Vienna University of Technology Papers published in this report series are preliminary versions of journal articles and not for quotations. This paper was accepted for publication in: Dorigo et al. (Eds.): Ant Algorithms, Springer LNCS 2463, Berlin/Heidelberg, pp This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB#010 ( Adaptive Information Systems and Modelling in Economics and Management Science ).

3 Insertion based Ants for Vehicle Routing Problems with Backhauls and Time Windows Marc Reimann, Karl Doerner and Richard F. Hartl University of Vienna, Vienna, Austria fmarc.reimann, karl.doerner, November 7, 2002 Abstract In this paper we present and analyze the application of an Ant System to the Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW). At the core of the algorithm we use an Insertion procedure to construct solutions. We provide results on the learning and runtime behavior of the algorithm as well as a comparison with a custom made heuristic for the problem. 1 Introduction Since their invention in the early 1990s by Colorni et al. (see e.g. [1]), Ant Systems have received increasing attention by researchers, leading to a wide range of applications such as the Graph Coloring Problem ([2]), the Quadratic Assignment Problem (e.g. [3]), the Travelling Salesman Problem (e.g. [4], [5]), the Vehicle Routing Problem ([6], [7]) and the Vehicle Routing Problem with Time Windows ([8]). Recently, a convergence proof for a generalized Ant System has been developed by Gutjahr ([9]). The Ant System approach is based on the behavior of real ants searching for food. Real ants communicate with each other using an aromatic essence called pheromone, which they leave on the paths they traverse. In the absence of pheromone trails ants more or less perform a random walk. However, as soon as they sense a pheromone trail on a path in their vicinity, they are likely to follow that path, thus reinforcing this trail. More specifically, if ants at some point sense more than one pheromone trail, they will choose one of these trails with a probability related to the strenghts of the existing trails. This idea has first been applied to the TSP, where an ant located in a city chooses the next city according to the strength of the artificial trails This leads to a construction process that resembles the Nearest Neighbor heuristic, which makes sense for the TSP. However, most of the applications of Ant Systems to other problems, also used this constructive mechanism. In order to be able to do so, the problem at hand had to be transformed into a TSP first. By doing so, structural characteristics of the problem, may disappear, thus leading to poor solutions. Moreover, for many of the problems solved with Ant Systems so far, problem specific constructive algorithms exist that exploit these structural characteristics. For example, for the classic VRP without side constraints, we have shown in ([7]) that the incorporation of a powerful problem specific heuristic algorithm significantly improves the performance and makes the Ant System competitive to other state-of-the-art methods, such as Tabu Search. Building on these results, in this paper we propose an Ant System, where the constructive heuristic is a sophisticated Insertion algorithm. We apply our approach to the Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW), a problem with high practical relevance. We present some preliminary results, that suggest the potential of our approach. To our best knowledge, we are not aware of existing works that deal with the application of an Ant System to the VRPBTW. The same applies to the incorporation of an Insertion algorithm within Ant Systems. These two points constitute the main contribution of our paper. While revising this paper we became aware of work done by Le Louarn et al. ([10]). In their 1

4 paper, the authors use their GENI heuristic (proposed earlier in [11]) at the core of an ACO algorithm and show the potential of this approach for the TSP. In the next section we describe the VRPBTW and review the existing literature. Section 3 deals with the Ant System algorithm, and the details of the incorporation of the Insertion algorithm. The numerical analysis in Section 4 focuses on the learning behavior of the Ant System, the effects of backhauls and the general performance of our Ant System when compared with a custom-made heuristic for the problem. Finally, we conclude in Section 5 with a summary of the paper and an outlook on future research. 2 Vehicle Routing Problems with Backhauls and Time Windows Efficient distribution of goods is a main issue in most supply chains. The transportation process between members of the chain can be modeled as a Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW). For example, the distribution of mineral water from a producer to a retailer (linehauls) may be coupled with the distribution of empty recyclable bottles from the retailer to the producer (backhauls). Both linehauls and backhauls may be constrained by possible service times at the producer and the retailers. More formally, the VRPBTW involves the design of a set of pickup and delivery routes, originating and terminating at a depot, which services a set of customers. Each customer must be supplied exactly once by one vehicle route during her service time interval. The total demand of any route must not exceed the vehicle capacity. The total length of any route must not exceed a pre-specified bound. Additionally, it is required that, on each route, all linehauls have to be performed before all backhauls. The intuition for that is, that rearranging goods en route is costly and inefficient. The objective is to minimize the fleet size, and given a fleet size, to minimize operating costs. This problem is a generalization of the VRP, which is known to be NP-hard (cf. [12]), such that exact methods like Branch&Bound work only for relatively small problems in reasonable time. While the VRP has received much attention from researchers in the last four decades (for surveys see [13]), the more constrained variants have only recently attracted scientific attention. The Vehicle Routing Problem with Time Windows (VRPTW) has been studied extensively in the last decade; for a recent overview of metaheuristic approaches see (e.g. [14]). The same applies for the Vehicle Routing Problem with Backhauls (VRPB, see e.g. [15]). The VRPBTW, which combines the issues addressed separately in the works cited above, has received only very little attention. Gelinas et al.([16]) have extended a Branch&Bound algorithm developed for the VRPTW to cope with backhauling. They proposed a set of benchmark sets based on instances proposed earlier for the VRPTW, and solved problems with up to 100 customers to optimality. Their objective was to minimize travel times only. Simple construction and improvement algorithms have been proposed by Thangiah et al. ([17]), while Duhamel et al.([18]) have proposed a Tabu Search algorithm to tackle the problem. While the approach of Thangiah et al. s considered the same objective as we do in this study, namely to minimize fleet sizes first, and then to minimize travel times as a second goal, Duhamel et al. consider the minimization of schedule times (which in addition to travel times include service times and waiting times) as the second goal. 3 Ant System algorithms for the VRPBTW In this section we describe our Ant System algorithm and particularly focus on the constructive heuristic used, as the basic structure of our Ant System algorithm is identical to the one proposed in Bullnheimer et al.([6]) and used in Reimann et al.([7]). The Ant System algorithm mainly consists of the iteration of three steps: ffl Generation of solutions by ants according to private and pheromone information ffl Application of a local search to the ants solutions ffl Update of the pheromone information The implementation of these three steps is described below. 2

5 3.1 Generation of solutions As stated above, the incorporation of the Insertion algorithm as the solution generation technique within the Ant System is the main contribution of this paper. So far, in most Ant Systems solutions have been built using a Nearest Neighbor heuristic (see e.g. [6]). In Reimann et al. ([7]) we have shown for the VRP the merit of incorporating a powerful problem specific algorithm. However, the Savings algorithm used there does not perform very well for problems with time windows and/or backhauls such that we rather use a different constructive heuristic. The Insertion algorithm used in our current approach for the VRPBTW is derived from the I1 insertion algorithm proposed by Solomon ([19]) for the VRPTW. Solomon tested many different route construction algorithms and found that the I1 heuristic provided the best results. This algorithm works as follows: Routes are constructed one by one. First, a seed customer is selected for the current route, that is, only this customer is served by the route. Sequentially other customers are inserted into this route until no more insertions are feasible with respect to time window, capacity or tour length constraints. At this point, another route is initialized with a seed customer and the insertion procedure is repeated with the remaining unrouted customers. The whole algorithm stops when all customers are assigned to routes. In the above mentioned procedure two types of decisions have to be taken. First, a seed customer has to be determined for each route, second the attractiveness of inserting a customer into the route has to be calculated. These decisions are based on the following criteria, where we will refer to the attractiveness of a customer i as i : Route initialization: i = d 0i 8i 2 N u, where d 0i denotes the distance between the depot and customer i and N u denotes the set of unrouted customers. This route initialization prefers seed customers that are far from the depot. Customer insertion: i = maxf0, max j2rli [ff d 0i fi (d ji + d isj d jsj ) (1 fi) (b i s j b sj )]g 8i 2 N u, where s j is the customer visited immediately after customer j in the current solution, b i s j is the actual arrival time at customer s j,ifiis inserted between customers j and s j, while b sj is the arrival time at customer s j before the insertion of customer i and R li denotes the set of customers assigned to the current tour after which customer i could feasibly be inserted. Thus, this rule not only considers the detour that occurs if customer i is inserted but also the delay in service at the customer s j to be served immediately after i. These two effects are weighted with the parameter fi and compared with customer i 0 s distance to the depot, which is weighted with the parameter ff. A customer being located far from the depot, that causes little detour and little delay is more likely to be chosen than a customer close to the depot and causing detour or delay. Given these values, the procedure for each unrouted customer determines the best feasible insertion position. Afterwards, given these values i, customer i Λ is chosen such that i Λ i 8i 2 N u. Note that, for each customer we have to check the feasibility of inserting it at any position in the current tour. While this is in principle quite tedious, in particular if tours contain a large number of customers, it can be done quite efficiently. Following Solomon we calculate for each position in the current tour the maximum possible delay at that position, that will ensure feasibility of the subour starting at that position. This calculation of the maximum possible delay has to be performed after each change to the current tour. However, by doing so, we then only have to check if the insertion of a customer at a certain position causes less delay than the maximum possible delay, in which case the insertion is feasible. 1 1 This procedure and the two rules for route initialization and customer insertion have been proposed by Solomon ([19]) for the VRPTW. 3

6 To account for the fact that we have to deal with backhauls, we augment these attractiveness values in the following way: i =max f0, max j2rli [ff d 0i fi (d ji + d isj d jsj ) (1 fi) (b i s j b sj )+fl type i ]g 8i 2 N u, where type i is a binary indicator variable denoting whether customer i is a linehaul (type i =0) or a backhaul customer (type i =1). The intuition is that we want to discriminate between linehaul and backhaul customers in some way. In order to use the algorithm described above within the framework of our Ant System we need to adapt it to allow for a probabilistic choice in each decision step. This is done in the following way. First, the attractiveness for inserting each unrouted customer at its best insertion on the current tour is calculated according to the following function: i = max f0, max j2rli [(ff d 0i fi (d ji + d isj d jsj ) (1 fi) (b i s j b sj )+fl type i ) fiji+fiis j ]g 2 fi jsj 8i 2 N u, where fi ji denotes the pheromone concentration on the arc connecting locations (customers or depot) j and i. The pheromone concentration fi ji contains information about how good visiting two customers i and j immediately after each other was in previous iterations. The way we use the pheromone emphasizes the effect of giving up an arc (the arc between customers j and s j in the example above) and adding two other arcs (the arcs between customers j and i and customers i and s j in the example above). In particular, the term fiji+fiis j 2 fi jsj is larger than 1, if the average pheromone value of the arcs to be added exceeds the pheromone value of the arc to be deleted. Note, that the same pheromone utilization is done for route initialization, thus augmenting the attractiveness of initializing a route with an unrouted customer i by the search-historic information. Then, we apply a roulette wheel selection to all unrouted customers with positive attractiveness values i. The decision rule used can be written as P i = 8>< >: i Phj h >0 h if i > 0 0 otherwise. The chosen customer i is then inserted into the current route at its best feasible insertion position. 3.2 Local Search After an ant has constructed its solution, we apply a local search algorithm to improve the solution quality. In particular, we apply Swap and Move operators to the solution. The Swap operator, aims at improving the solution by exchanging a customer i with a customer j. This operator is a special case of the 4-opt operator, where the four arcs deleted are in pairs adjacent. An example of the application of this operator is given in Figure 1, where customers 3 and 4 are exchanged. The Move operator tries to eject a customer i from its current position and insert it at another position. It is a special case of the 3-opt operator, where two of the three arcs deleted are adjacent. This operator is exemplified in Figure 2, where customer 5 is moved from one route to another. Both operators have been proposed by Osman ([20]). We apply these operators until no more improvements are possible. More specifically, we first apply Move and then Swap operators. Note, that we do not accept infeasible solutions. While Osman ([20]) proposed a more general version of these operators, where adjacent customers can be moved or swapped, we restrict our local search to the case where = 1. The reason for this is, that the operators were proposed for the classic VRP without time window and (1) 4

7 Figure 1: An example of the application of the Swap operator Figure 2: An example of the application of the Move operator backhauling constraints. Given these additional constraints, most possible operations with >1 lead to infeasible solutions. Thus, the additional computation effort to perform these more complex operations, will in general not be justified. 3.3 Pheromone update After all ants have constructed their solutions, the pheromone trails are updated on the basis of the solutions found by the ants. According to the rank based scheme proposed in ([5]) and ([6]), the pheromone update is as follows fi ij := ρfi ij + px μ=1 fi μ ij + ff fi Λ ij (2) where 0» ρ» 1 is the trail persistance and ff = p +1is the number of elitists. Using this scheme two kinds of trails are laid. First, the best solution found during the process is updated as if ff ants had traversed it. The amount of pheromone laid by these elitists is fi Λ ij =1=LΛ, where L Λ is the objective value of the best solution found so far. Second, the p best ants of the iteration are allowed to lay pheromone on the arcs they traversed. The quantity laid by these ants depends on their rank μ as well as their solution quality L μ, such that the μ-th best ant lays fi μ ij =(p μ +1)=Lμ. Arcs belonging to neither of those solutions just lose pheromone at the rate (1 ρ), which constitutes the trail evaporation. 4 Numerical analysis In this section we turn to the numerical analysis of our proposed approach. First we will describe the benchmark problem instances. After providing details about the parameter settings, we evaluate the influence of the pheromone information. Finally, we compare the results of our Ant System with those of Thangiah et al. s heuristic algorithms. All our comparisons will be on the basis of the objective to first minimize fleet sizes and then minimize travel times as a second goal. This objective was established by minimization of the following objective function: L = FS + TT; (3) 5

8 where L denotes the total costs of a solution, FS denotes the fleet size found, and TT corresponds to the total travel time (or distance). The parameter was chosen to ensure that a solution that saves a vehicle always outperforms a solution with a higher fleet size. 4.1 The benchmark problem instances The benchmark problem instances we used for our numerical tests were developed by Gelinas et al. ([16]). They used the first five problems of the r1 instances, namely r101 to r105, originally proposed by Solomon ([19]) for the vehicle routing problem with time windows (VRPTW). Each of these problems consists of 100 customers to be serviced from a central depot. The customers are located randomly around the depot. Service has to take place within a short time horizon (230 time units), and vehicle capacities are fairly loose when compared with the time window requirements at the customers. These time window requirements are varying in the data sets. The average time window length in the five instances are given in Table 1. Instance r101 r102 r103 r104 r105 Avg. Length of TW Table 1: Average length of the time windows Given these data sets Gelinas et al. randomly chose 10%, 30% and 50% of the customers to be backhaul customers with unchanged quantities, thus creating 15 different 100 customer instances. 4.2 Parameter settings Our aim is to use standard approaches and standard parameter settings as much as possible in order to demonstrate the benefit of intelligent combination of these approaches. However, we also performed parameter tests in order to find good combinations of these parameters for our problem. It turned out, that most of the parameters should be in the range documented in the literature. More specifically, we analyzed the three parameters of the insertion algorithm in the ranges ff 2 f0; 0:1; :::; 2g, fi 2 f0; 0:1; :::; 1g and fl 2 f0; 1; :::; 20g on the instances described above. From this analysis we obtained the following values: ff =1, fi =1and fl =13. Note, that the parameter fl =13leads to a discrimination between linehaul and backhaul customers in favor of the backhaul customers. While this seems counterintuitive, it can be explained in the following way. Given the parameters ff and fi, the attractiveness values can become negative. However, negative attractiveness values prevent insertion. Thus, feasible solutions may be prohibited and this will generally lead to larger fleet sizes. This effect is reduced by the parameter fl, leading to tighter packed vehicles and smaller fleet sizes at the cost of increased travel times. To balance this trade-off, we chose to make the backhaul customers more attractive as they represent the minority of customers. 2 Let n be the problem size, i.e. the number of customers to be served, then the Ant System parameters were: m = dn=2e ants, ρ = 0:95 and p = 4 elitists. These parameters were not extensively tested, as our experience suggests that the rank based Ant System is quite robust. However, the number of ants was reduced to m = dn=2e to be able to run more iterations. For each instance we performed 10 runs of 2.5 minutes each. All runs were performed on a Pentium 3 with 900MHz. The code was implemented in C. 4.3 Evaluation of pheromone influence In this section we will analyze whether our approach features pheromone learning or not. As we diverge from the standard constructive approach used in Ant Systems for Vehicle Routing, we have to check whether the utilization of the pheromone trails helps or hinders the constructive process of the ants. To do this, we compare the proposed Ant System with a stochastic implementation of the underlying algorithm, where no pheromone is used. The results are shown in Table 2. We provide averaged results for the different backhaul densities after runtimes of 15, 30, 75 and 150 seconds. The table confirms that the use of the pheromone trails in decision 2 However, we plan and already started to investigate other methods to solve this problem. One idea is to adjust the attractiveness values corresponding to feasible insertions, to ensure that a certain percentage of these values is positive. 6

9 Time 10% BH 30% BH 50% BH in ASinsert StochInsert ASinsert StochInsert ASinsert StochInsert sec. FS TT FS TT FS TT FS TT FS TT FS TT 15 17, ,0 17, ,3 18, ,2 18,4 1629,2 18, ,2 18, , , ,4 17, ,0 18, ,3 18,3 1628,9 18, ,7 18, , , ,1 17, ,9 18, ,0 18, ,3 18, ,9 18, , ,3 1499,2 17,3 1546,7 18, ,8 18, ,9 18, ,6 18, ,0 FS...fleet size TT...travel times Table 2: Influence of the pheromone on the solution quality making significantly improves solution quality. The Ant System (referred to as ASinsert) does find better solutions than the stochastic implementation of the Insertion algorithm (referred to as StochInsert). This can be seen from the last row of the table. Furthermore, the table shows that the Ant System finds good solutions faster than the stochastic Insertion algorithm. After 15 seconds the solutions of the Ant System are already superior to those of the stochastic Insertion algorithm, albeit the difference is small. As more and more pheromone information is built and this matrix better reflects the differences between good and bad solutions the Ant System clearly outperforms the stochastic algorithm without pheromone information. This fact is also shown in Figure 3. We chose the case with 10% backhauls, as in this case the evolution of the fleet sizes, which are the primary goal, is similar in both approaches. So we can compare just travel times, and the figure shows that the Ant System at any point in time finds better solutions than the stochastic Insertion algorithm and moreover the difference gets larger as the number of iterations increases. travel times sec. 30 sec. 75 sec. 150 sec. runtime in seconds Figure 3: Ant System(light columns) and stochastic Insertion algorithm performance on problems with 10% backhauls 4.4 Comparison of our Ant System with existing results Let us now turn to the analysis of the solutions found with respect to absolute solution quality. As stated above, there exists one paper that studies the same objective as we do. In this paper, Thangiah et al. ([17]), propose a constructive algorithm and a number of local search descent algorithms to tackle the problem. In Table 3 we show in detail, that is for each instance, the results of this approach together with the results of our Ant System and the stochastic Insertion algorithm. Note, that Thangiah et al. propose five different algorithms and the results in the first column of Table 3 are the best ones for each instance regardless of the version that found the solution. For our approaches we present average results over ten runs. In the rightmost two columns we report fleet sizes and travel times of the best solutions we found regardless of the algorithm that found this solution. The last row of the table reports the aggregate numbers for the approaches. Our Ant System outperforms the Thangiah s simple heuristics by approximately 2% with respect to fleet sizes, and by 2.7% with respect to travel times. However, we also see that the simple heuristics outperform our approach in three instances, namely r102 with 30% and 50% backhauls and r103 with 10% backhauls. In these instances, we did not find the same fleetsize as Thangiah s algorithms. As these instances are spread over all possible backhaul densities and our approach on average outperforms Thangiah s algorithm for each density of backhaul customers, the effect has to stem from the characteristics of the backhaul customers. 7

10 Note that we do not compare computation times for the approaches. We believe that a comparison of runtimes is meaningless in this case. First, the machines are very different, in particular ours are much faster. Second, a metaheuristic can never be expected to be as fast as a custom-made approach. Thangiah s heuristics find their solutions within less than a minute, while our approach runs for 2.5 minutes, we nevertheless believe that the results obtained by our Ant System justify the application and show the potential savings that can be achieved through the use of a metaheuristic approach. Moreover, the computation time reported for our approach refers to the total execution time of the algorithm. Of course we find good, and even better solutions than Thangiah, earlier in the search. Thangiah StochInsert ASinsert Best solutions best avg. avg. identified by 5 versions 10 runs 10 runs our algorithms Instance FS TT FS TT FS TT FS TT r101 10% BH , , , ,68 r101 30% BH ,6 23,3 2000, , ,16 r101 50% BH ,6 24,1 1981, , ,29 r102 10% BH ,1 19,5 1695,13 19,8 1636, ,62 r102 30% BH , , , ,43 r102 50% BH , , , ,21 r103 10% BH , , , ,41 r103 30% BH , , , ,88 r103 50% BH , , , ,66 r104 10% BH , ,73 11,9 1142, ,78 r104 30% BH , , , ,30 r104 50% BH ,6 12,6 1282, , ,46 r105 10% BH , ,16 16,8 1523, ,81 r105 30% BH ,7 17,1 1664,66 16,1 1596, ,23 r105 50% BH , ,84 17,2 1644, ,01 Sum ,9 270, ,13 268, , ,92 FS...fleet size TT...travel times Table 3: Comparison of solution quality between our Ant System and other approaches Finally, we show in Figure 4 the effects of the density of backhaul customers on both fleet sizes and travel times for our Ant System. Clearly, both fleet sizes and travel times increase with the density of backhaul customers. Note however, that increasing the percentage of backhaul customers further beyond 50% will not further increase the travel times and fleet sizes. On the contrary, fleet sizes and travel times will fall again. In the extreme case of 100% backhauls we will have the same solution as in the other extreme case of 100% linehauls. Generally, with a mix of 50% linehauls and 50% backhauls, there is the smallest degree of freedom for the optimization, such that in these cases the solution quality should be worst. (a) (b) Figure 4: Effect of backhaul customer density on fleet sizes (a) and travel times (b) 8

11 5 Conclusions and future research In this paper we have proposed a promising approach for the Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW). This Ant System approach is based on the well known Insertion algorithm proposed for the VRPTW by Solomon ([19]). We have proposed an approach to use pheromone information in the context of such an Insertion algorithm and shown that our algorithm benefits from this pheromone information and outperforms a custom-made heuristic proposed for the VRPBTW by Thangiah et al. ([17].) Future research will deal with a more detailed analysis of the approach, and its application to other combinatorial problems. For the VRPBTW we will analyze the approach on larger instances. We also plan to incorporate swarm-like features by equipping each ant with its own set of parameters (of the heuristic) and adjusting these parameters in an evolutionary way during the optimization process. Furthermore, we will embed this approach in our multi-colony Ant System proposed already in Doerner et al. ([21]). This approach should help us deal better with the multiple objectives that have to be tackled in the problem. Acknowledgments We are grateful to two anonymous referees who provided valuable comments that improved the presentation of the paper. This work was supported by the Oesterreichische Nationalbank (OeNB) under grant #8630. References [1] Colorni, A., Dorigo, M. and Maniezzo, V.: Distributed Optimization by Ant Colonies. In: Varela, F. and Bourgine, P. (Eds.): Proc. Europ. Conf. Artificial Life. Elsevier, Amsterdam (1991) [2] Costa, D. and Hertz, A.: Ants can colour graphs. Journal of the Operational Research Society 48(3) (1997) [3] Stützle, T. and Dorigo, M.: ACO Algorithms for the Quadratic Assignment Problem. In: Corne, D. et al. (Eds.): New Ideas in Optimization. Mc Graw-Hill, London (1999) [4] Dorigo, M. and Gambardella, L. M.: Ant Colony System: A cooperative learning approach to the Travelling Salesman Problem. IEEE Transactions on Evolutionary Computation 1(1) (1997) [5] Bullnheimer, B., Hartl, R. F. and Strauss, Ch.: A new rank based version of the ant system: a computational study. Central European Journal of Operations Research 7(1) (1999) [6] Bullnheimer, B., Hartl, R. F. and Strauss, Ch.: An improved ant system algorithm for the vehicle routing problem. Annals of Operations Research 89 (1999) [7] Reimann, M., Stummer, M. and Doerner, K.: A Savings based Ant System for the Vehicle Routing Problem. to appear in: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2002), Morgan Kaufmann, San Francisco (2002) [8] Gambardella, L. M., Taillard, E. and Agazzi, G.: MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows. In: Corne, D. et al. (Eds.): New Ideas in Optimization. McGraw-Hill, London (1999) [9] Gutjahr, W. J.: ACO algorithms with guaranteed convergence to the optimal solution. Information Processing Letters. 82 (2002) [10] Le Louarn, F. X., Gendreau, M. and Potvin, J. Y.: GENI Ants for the Travelling Salesman Problem. CRT Research Report, Montreal (2001) [11] Gendreau, M., Hertz, A. and Laporte, G.: New Insertion and Postoptimization Procedures for the Travelling Salesman Problem. Operations Research. 40 (1992) [12] Garey, M. R. and Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP Completeness. W. H. Freeman & Co., New York (1979) [13] Toth, P. and Vigo, D. (Eds.): The Vehicle Routing Problem. Siam Monographs on Discrete Mathematics and Applications, Philadelphia (2002) 9

12 [14] Bräysy, O. and Gendreau, M.: Metaheuristics for the Vehicle Routing Problem with Time Windows. Sintef Technical Report STF42 A01025 (2001) [15] Toth, P. and Vigo, D.: VRP with Backhauls. In Toth, P. and Vigo, D. (Eds.): The Vehicle Routing Problem. Siam Monographs on Discrete Mathematics and Applications, Philadelphia (2002) [16] Gelinas, S., Desrochers, M., Desrosiers, J. and Solomon, M. M.: A new branching strategy for time constrained routing problems with application to backhauling. Annals of Operations Research. 61 (1995) [17] Thangiah, S. R., Potvin, J. Y. and Sun, T.: Heuristic approaches to Vehicle Routing with Backhauls and Time Windows. Computers and Operations Research. 23 (1996) [18] Duhamel, C., Potvin, J. Y. and Rousseau, J. M.: A Tabu Search Heuristic for the Vehicle Routing Problem with Backhauls and Time Windows. Transportation Science. 31 (1997) [19] Solomon, M. M.: Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints. Operations Research. 35 (1987) [20] Osman, I. H.: Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Annals of Operations Research. 41 (1993) [21] Doerner, K. F., Hartl, R.F., and Reimann, M.: Are CompetANTS competent for problem solving - the case of a transportation problem. POM Working Paper 01/2001 (2001) 10