3. Literature Review of the VRPB and the MT-VRP
3.4. Solution methods for the MT-VRP
3.4.3. Metaheuristic Methods
Taillard et al. (1996) were the first researchers to study the MT-VRP. They developed a three phase tabu search heuristic to solve the MT-VRP which is based on the tabu search adaptive memory algorithm of Taillard (1993). In the first phase, a large set of vehicle routes of the classical VRP is produced using the algorithm of Taillard (1993) and routes forming the VRP solution are stored in the list (data structure). Secondly, an enumerative algorithm is used to select a subset of routes generated in the first phase.
Finally, a Bin Packing Problem is solved for each VRP solution stored in the list and then the best solution is selected from all the packed solutions. The tabu search algorithm is tested on a number of MT-VRP instances which they generated from the VRP data instances of Christofides et al. (1979) and Fisher (1994). Their tabu search algorithm successfully found feasible solutions for most of the instances within reasonable times. Moreover, given the way MT-VRP instances were generated, the authors state that the results show that the feasible solutions (i.e., solutions found without overtime) are on average within 5% to 10% of the best known VRP solutions.
Brandao and Mercer (1997) studied a practical MT-VRP for the British company Burton’s Biscuits δtd. and termed it as the multi-trip vehicle routing and scheduling problem (MTVRSP). Many time related scheduling constraints close to practical world constraints are taken into account in their study. Moreover, as this problem deals with solving the real distribution problems with practical constraints and actual costs, real
59 distances are used. To solve the problem they developed a tabu search algorithm which consists of three phases. In the first phase, the tabu algorithm generates the initial solution by using nearest neighbour and insertion heuristics. At this initial stage the routes are created in a sequential constructive manner and all the routes are feasible in terms of routing constraints. There may be a possibility that the routes being constructed are infeasible in terms of scheduling constraints but this constraint is not considered at this stage. In the second phase, two objectives, to make the solution feasible (i.e., solution where no overtime is used) in terms of maximum driving time and time windows while decreasing the cost of the solution as much as possible are taken into account simultaneously. In the third phase, a set of swap and insert moves are performed to reduce the solution cost while maintaining feasibility. It has been reported that this algorithm improved over the manual solutions obtained by the company by approximately 20% on average.
In (1998) Brandao and Mercer provided a simplification of the above tabu search algorithm for the MT-VRP. This algorithm has no additional constraints as compared to the above real-world application algorithm. This tabu search algorithm generates the initial solution by using a nearest neighbour insertion heuristic and utilises insertion and swap moves in its search process. Moreover this algorithm takes into account the variable-size tabu list and aspiration criteria. Furthermore infeasible solutions (solutions found with overtime) are also allowed with respect to the maximum overtime permitted.
The algorithm is tested on the data set proposed by Taillard et al. (1996). The solutions obtained through the proposed tabu search algorithm are compared with the solutions obtained by Taillard et al. (1996).
60 Salhi and Petch (2007) developed a genetic algorithm (GA) based heuristic to solve the MT-VRP (see Section 2.6.2.1). They claim this is the first GA based approach proposed for MT-VRP in the literature. The power of GA lies in that of new solutions can be generated simultaneously. Moreover, classical GAs normally involves binary based chromosome representation. But in practice it is difficult to convert a solution into binary representation. So in this study, the authors have developed a flexible non-binary chromosome structure that is established upon the circle partition concept of Thangiah and Salhi (2001) to address the above hurdle. The initial population of chromosome is obtained by the circle partition scheme that facilitates in providing a base for clustering and finally route generation. In order to maintain the solution quality and population diversity two mechanisms called Injection and Cloning are used. To generate new chromosome or offspring, the “extraction” and “mutation” operators are used. A savings heuristic is used to solve small VRP sub-problems whereas a bin packing heuristic is used to obtain the final set of vehicle trips. In order to further optimise the trips some post optimisation refinement modules proposed in Petch and Salhi (2004) are used. The algorithm is tested on MT-VRP data set proposed by Taillard et al. (1996) in the literature. According to the solutions quality, it appears that the proposed GA approach does not produce better results when compared with other algorithms proposed in the literature for the MT-VRP. However GA found solutions of reasonable quality in short time when compared to the other algorithms.
Olivera and Viera (2007) developed an adaptive memory programming (AMP) approach based on the AMP principle of Rochat and Taillard (1995) to solve the MT- VRP. The authors have also presented the mathematical programming formulation of the problem which is based on the set covering formulation of the VRPTW. The sweep algorithm is used to generate the initial solution by selecting customers randomly each
61 time. Initial solutions are then improved by using tabu search (TS) algorithm before storing in the memory M (data structure), hence storing the top quality solutions. In addition, the data structure in which routes are stored is sorted is ascending order only.
This is performed according to the lexicographic criteria to ensure that good routes reside in the first positions of the memory. After that, a new solution s is selected from the memory M and a bin packing approach is utilised to pack the routes into vehicles while using some local search refinements based on reducing the driver overtime. The memory M is updated with new routes while poor solutions are discarded. The AMP algorithm is tested on the 104 benchmark instances proposed by Taillard et al. (1996).
The AMP algorithm found 98 feasible solutions out of 104 when compared with the algorithms of Taillard et al. (1996), Brandao and Mercer (1998) and Petch and Salhi (2004).
Alonso et al. (2008) developed a tabu search algorithm for the periodic vehicle routing problem with multiple vehicle trips and accessibility restrictions. The authors call this problem the site-dependent multi-trip periodic vehicle routing problem (SDMTPVRP).
This problem combines some of the characteristics of the VRP, PVRP (for periodic VRP, see Chao et al. (1995) and Cordeau et al. (1997), SDVRP (for site-dependant VRP, see Nag et al. (1988), Chao et al. (1999) and Cordeau and Laporte (2001)) and MT-VRP. The tabu search approach used to solve the SDMTPVRP and its particular cases is a modification of the tabu search algorithm developed in Cordeau et al. (1997) for the periodic VRP; hence the authors call this algorithm TS-ABB. However this approach differs in many ways that is; the definition of solution attributes, the construction of the neighbourhood, the evaluation of the objective function and finally the construction of the initial solution. According to the authors, the SDMTPVRP is the first problem of its kind so some new data instances are created to test the TS-ABB
62 algorithm. Moreover the PVRP and the SDVRP test problems are also solved through this approach. The algorithm is tested on the MT-VRP problems proposed by Taillard et al. (1996). The computational results obtained show that the TS-ABB algorithm found feasible solutions for most of the MT-VRP problems when compared to those obtained by Taillard et al. (1996) while taking approximately the same time.
Cattaruzza et al. (2014a) proposed a hybrid genetic algorithm for the MT-VRP that uses some adaptations from the literature. A new local search operator called the combined local search (CLS) is introduced that combines the standard VRP moves and performs the reassignments of trips to vehicles by using a swapping procedure to obtain a better solution. This algorithm produced good quality results. Cattaruzza et al. (2014b) extended the model to include time windows aspect and developed an iterated local search methodology to solve the problem.