3. Literature Review of the VRPB and the MT-VRP
3.2. Solution Methods for the VRPB
3.2.3. Metaheuristic Methods
47 clusters are connected with backhaul clusters in order to form mixed routes and any linehaul clusters that are left are connected with the depot. The solution is further improved by using intra-route and inter-route (see Section 2.5.2) neighbourhood moves as refinement routines. This algorithm produced some good quality results compared to the previously published works, and set new benchmark solutions for asymmetric VRPB.
48 algorithm matched the optimal solutions produced by the exact algorithms of Mingozzi et al. (1996).
Brandao (2006) developed a tabu search (TS) algorithm for the VRPB. Two methods called open initial solution and K-tree initial solution are used to generate initial solution. The former involves two steps. In the first step, in order to solve the VRPB, two separate open vehicle routing problems (OVRPs) are solved. This is done because the OVRP is close in structure to the VRPB. Since in OVRP, the vehicles are not required to return to depot at the end of route. The OVRP solution is based on two phases, the initial phase and the improvement phase. In the initial phase, a nearest neighbour (NN) heuristic is used to generate a set of open-ended (i.e., a set of routes consisting of linehaul customers and a set of routes consisting of backhaul customers) routes sequentially. The NN procedure continues until all the customers are routed.
Then in the improvement phase, tabu search is used. Here for the set of linehaul OVRP routes, TS minimises the overall distance travelled by the vehicles. Whereas for the set of backhaul OVRP routes, TS minimises the number of routes as well as distance travelled. In the second step 2, two Hamiltonian solution paths of OVRP for each LH and BH customers are linked together. Note that four different ways of connecting the end points of linehaul and backhaul routes are evaluated and the link returns the least cost is accepted to form a complete VRPB route. This process is repeated until either the backhaul or the linehaul paths are empty. The K-tree initial solution method is based on a lower bound. In this method, linehauls and backhauls are considered as customers only, hence, assuming the VRPB as the VRP. Then the VRP is formulated as a minimum cost K-tree as described in Fisher (1994a) with degree 2K on the depot.
Finally, 10 initial solutions are generated from each of 10 K-trees lower bounds. The solution generated by either (i.e., open initial solution or K-tree initial solution)
49 methods is improved by their TS implementation. The best performance of the TS algorithm is acquired with the K-tree initial solution method.
Ghaziri and Osman (2006) proposed a self-organizing feature maps (SOFM) methodology for the VRP with backhauls which is based on the concept of the Neural Networks. This algorithm is basically an extension of Ghaziri and Osman (2003) algorithm proposed for the Travelling Salesman Problem with backhauls. This algorithm begins by specifying the architecture of the network that comprises of one ring on which artificial neurons are spread spatially. The ring is embedded in the Euclidean space where each neuron is recognized by its position on the ring. Two post- optimisation procedures based on the 2-Opt procedure are used to improve the solution quality. The technique of type one is used at the end of the algorithm; whereas, the type two is used periodically during the search process. Solutions found by their algorithm are of inferior quality compared to the algorithms of Toth and Vigo (1996, 1999) and Osman and Wassan (2002).
Røpke and Pisinger (2006) proposed a unified heuristic for a large class of vehicle routing problems with Backhauls. The unified heuristic uses large neighbourhood search (LNS) meta-heuristics originally developed in Shaw (1998). The LNS shares similarities with the concept of Ruin and Recreate (R&R) which was used in a framework proposed by Schrimpf et al. (2000). Various insertion and removal heuristics are used in this framework, some of them as diversification and others for intensification. Røpke and Pisinger embedded three different configurations and called it a unified heuristic methodology. These configurations (strategies) are named as Standard, 6R-no learning and 6R-normal learning. In the Standard configuration, three removal heuristics are used with a learning mechanism; the 6R-normal learning uses 6
50 different types of removal heuristics without learning mechanism; and the 6R-normal learning employs all 6 removal heuristics with learning mechanism (for full detail of the removal heuristics we refer the reader to their paper). The unified heuristic is tested on various data sets belonging to different backhauling variants including the classical VRPB. The unified heuristic performed very well on all data sets in terms of the solutions quality.
Wassan (2007) studied the VRPB and proposed a hybrid meta-heuristic algorithm that combines the processes of the reactive tabu search and adaptive memory programming (AMP). The RTS and AMP are considered as cutting-edge components of TS. The AMP component is based on long term memory structures and it used a wider framework in which strategies such as intensification and diversification are combined together. Both RTS and AMP approaches are coupled and utilised together in this study intelligently in order to obtain high quality solutions. The savings-insertion and the savings-assignment construction methods developed in Osman and Wassan (2002) are used to construct the initial solution. Solutions are reported for two benchmark VRPB data sets available in the literature. The RTS-AMP algorithm produced better quality solutions (45 new best/optimal) when compared with the best know solutions of two well-known VRPB data sets.
Gajpal and Abad (2009) developed a multi-ant colony system (see Section 2.6.3.1) algorithm called ‘εACS’ for the VRPB. In this study, the authors have used two types of ants, vehicle-ants and route-ants. In order to construct the feasible solution; two types of trail intensities called the vehicle trail intensity and the route trail intensity are used.
After the initial solution constructed by the ants three types of local search procedures are used. These are 2-Opt, customer insertion/interchange multi-route scheme and sub-
51 path exchange multi-route scheme. In order to avoid being trapped in local minima equal importance is given to the elite ants. The MACS algorithm produced competitive results with five new best known solutions compared to the studies published by then.
Moreover it has been reported that the CPU time and solution quality of MACS approach can be controlled by varying the number of ants.
Tutuncu, Carreto and Baker (2009) investigated the classical VRPB and two of its extensions known as the mixed and the restricted VRP with backhauls. A decision support system (DSS), which is based on the GRAMPS (Greedy Randomized Adaptive Memory Programming Search, see Ahmadi and Osman (2005)) algorithm. This is basically a visual approach that is based on the work of Fisher and Jaikumars (1981) proposed for vehicle routing and was later extended by Baker in (1992). Their visual approach which they named as CRUISE2 (Computerised Routing Using Interactive Seeds Entry version 2) consists of three stages. The first stage has two phases called the seed selection and proposition phases respectively. At the seed selection phase, using visual representation of the seeds (customers) on the DSS, users can select customers for each vehicle manually or automatically. Whereas at the proposition phase, GRAMPS meta-heuristic construct routes and also performs a local search with learning process at each iteration. Once the classical VRPB solution is obtained at the first stage, the problem modification stage starts where users are optionally permitted to insert backhaul customers before linehaul customers in order to convert the solution into mixed VRPB or restricting backhaul customers’ positions in order to make it restricted VRPB. Finally in the stage, the solver (GRAMPS) algorithm is called to obtain the final solution for the mixed and restricted VRPB. The visual DSS framework did not find better solutions when compared to the reactive tabu search algorithm of Osman and
52 Wassan (2002), however in terms of computational time and overall solution quality, the proposed framework seems quite competitive.
Zachariadis and Kiranoudis (2012) developed a local search heuristic for the classical VRPB that explores rich solution neighbourhoods (i.e., the neighbourhoods which are composed of variable length customer sequences) and makes use of local search moves stored in Fibonacci Heaps (Fibonacci Heaps are basically special types of priority queue structures that allows a program with capabilities such as fast insertion, deletion and retrieval). Moreover, they propose a parameter-free mechanism called “promises”
which is based on the aspiration criterion mechanism of tabu search to achieve diversification and avoid cycling. The algorithm is tested on a VRPB data set proposed by Goetschalckx and Jacobs-Blecha (1989). The algorithm outperformed other algorithms in the literature in terms of solution quality.
Recently, Cuervo et al. (2013) developed an iterated local search algorithm for the classical VRPB in which an oscillating local search heuristic is used. At each iteration, a broader neighbourhood structure is explored and the information regarding neighbouring solutions is stored in a data structure. At the second stage, a constant transition between feasible and infeasible solution space is achieved by a heuristic while adjusting the transitions by a penalty associated with infeasible solutions dynamically.
The iterated local search algorithm is tested on two VRPB benchmark data sets. The algorithm produced high quality solutions when compared with other state of the art algorithms in the literature.
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