A hybrid metaheuristic algorithm for heterogeneous vehicle routing problem with simultaneous pickup and delivery

Highlights

• The vehicle routing problem with simultaneous pickup and delivery is studied.

• The problem is considered with heterogeneous fleet of vehicles.

• An adaptive local search integrated with tabu search is developed for its solution.

• Proposed approach performs well on the randomly generated problem instances.

Abstract

The Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD) is a variant of the classical Vehicle Routing Problem (VRP) where the vehicles serve a set of customers demanding pickup and delivery services at the same time. The VRPSPD can arise in many transportation systems involving both distribution and collection operations. Originally, the VRPSPD assumes a homogeneous fleet of vehicles to serve the customers. However, in many practical situations, there are different types of vehicles available to perform the pickup and delivery operations. In this study, the original version of the VRPSPD is extended by assuming the fleet of vehicles to be heterogeneous. The Heterogeneous Vehicle Routing Problem with Simultaneous Pickup and Delivery (HVRPSPD) is considered to be an NP-hard problem because it generalizes the classical VRP. For its solution, we develop a hybrid local search algorithm in which a non-monotone threshold adjusting strategy is integrated with tabu search. The threshold function used in the algorithm has an adaptive nature which makes it self-tuning. Additionally, its implementation is very simple as it requires no parameter tuning except for the tabu list length. The proposed algorithm is applied to a set of randomly generated problem instances. The results indicate that the developed approach can produce efficient and effective solutions.

Introduction

The Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD) has attracted research interest due to its applicability in numerous reverse logistic systems involving bi-directional flow of goods. Originally, the VRPSPD assumes a homogeneous fleet of vehicles to serve a set of customers requiring delivery and pickup services simultaneously. However, in many practical situations, companies employ heterogeneous fleet of vehicles to satisfy customer demands. Therefore, this paper extends the original version of the VRPSPD by assuming the fleet of vehicles to be heterogeneous. This version of the problem can arise in many practical applications of reverse logistics, and constructing an effective solution strategy to the problem is one of the most critical issues for operating the transportation system efficiently. In the bottled drinks industry, for example, full bottles are delivered and empty ones which are used for recycling are collected simultaneously from the customers. Other practical applications can be seen in the collection of used materials such as car parts, industrial equipments and computers for remanufacturing or dissembling operations (Zachariadis, Tarantilis, & Kiranoudis, 2009). Moreover, the problem can be seen at grocery stores in which pallets or boxes are collected and reused for transportation (Dethloff, 2001).

The Heterogeneous Vehicle Routing Problem with Simultaneous Pickup and Delivery (HVRPSPD) can be described with graph theory terms as follows: Let G = (N, A) be a complete graph, where N={n₀, n₁,…, n_n} is the set of nodes in which n₀ represents the central depot and the remaining ones the customers. A={(n_i,n_j):n_i, n_j ∈ N,i≠j} is the set of arcs that represent the linkages between the customers and c_ij shows the distance between customers i and j. Each customer {n₁, n₂,…, n_n} requires a nonnegative delivery quantity d_i and a nonnegative pickup quantity p_i. The fleet of vehicles involves M different vehicle types each of which has a vehicle capacity Q_m, a fixed cost F_m and a unit variable cost v_m (m = 1,…, M). Two different cases of the HVRPSPD can occur depending on the availability of the vehicles. In the first case, the number of available vehicles for each type is unlimited and the vehicle fleet mix problem should also be solved, which corresponds to our problem. In the second case, the availability of each type of vehicle is known beforehand. Thus, our objective for the HVRPSPD in our case is to determine the best fleet composition as well as the set of vehicle routes minimizing the total cost and satisfying the following constraints: (i) every route initiates and finishes at the central depot; (ii) each customer must be serviced once by one vehicle; (iii) all delivered goods must be originated from the depot and all pickup goods must be transported back to the depot; (iv) the transported amount of goods cannot exceed the capacity of a vehicle. From the theoretical point of view, the HVRPSPD is an NP-hard combinatorial optimization problem because it is a variant of the standard version of the VRP.

In this study, we develop a simple hybrid local search algorithm based on an adaptive Threshold Accepting (TA) strategy and Tabu Search (TS) for the HVRPSPD with an unlimited number of vehicles. The developed algorithm, HLS, uses a non-monotone threshold function with a tabu list. One of the most important features of the algorithm is that the applied threshold function does not need any parameter tuning which makes the algorithm self-tuning. That is, it self-adaptively adjusts the threshold value in order to provide necessary diversification and intensification in the search process. Moreover, the employed tabu list further improves the diversification of the search process by preventing the algorithm from cycling. The effectiveness of the proposed algorithm is tested on randomly generated instances. The results show that the proposed approach gives good solutions in reasonable computation times.

The rest of this paper is organized as follows: In Section 2, related literature of the problem is given. In Section 3, a mathematical formulation developed for the HVRPSPD is introduced. In Section 4, detailed information about our solution methodology is presented. An illustrative example is given in Section 5. The computational results are presented in Section 6. Finally, concluding remarks and future research are given in Section 7.