A new saving-based ant algorithm for the Vehicle Routing Problem with Simultaneous Pickup and Delivery
Introduction
The classical Vehicle Routing Problem (VRP) involves a set of delivery customers to be serviced by a homogeneous fleet of vehicles housed at a central depot. The item to be delivered is a single commodity. The objective of the problem is to develop a set of vehicle routes originating and terminating at the depot such that all customers are serviced, the demands of the customers assigned to each route do not violate the capacity of the vehicle that services the route, and the total distance traveled by all vehicles is minimized. The Vehicle Routing Problem with Pickup and Delivery (VRPPD) is a variant of the VRP where the vehicles are not only required to deliver goods to customers but also to pick up some goods from the customers. Customers receiving goods are called linehauls and customers sending goods are called backhauls. VRPPD may be classified into three categories (Nagy & Salhi, 2005):
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Delivery First, Pickup Second: the vehicles pick up goods only after they have delivered their goods (VRPB).
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Mixed Pickup and Delivery: the vehicles deliver and pick up goods in any sequence along their routes (MVRPB).
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Simultaneous Pickup and Delivery: the vehicles simultaneously deliver and pick up goods (VRPSPD).
VRP with delivery first, pickup second is the first VRPPD problem introduced in the literature and is known as the VRP with backhauls (VRPB). However, several variants of this problem have been proposed over time relaxing the restriction of servicing backhaul customers after the linehauls as well as introducing multiple-depot cases. In this paper, we consider the VRP with Simultaneous Pickup and Delivery (VRPSPD). In the VRPSPD the objective and constraints are the same as in the VRPB except the servicing order of the customers, which makes the problem more complicated because of the fluctuating loads on the vehicle along the route. In the VRPB, the loads of linehaul customers and backhaul customers can be checked separately during the delivery route and pickup route, respectively, to ensure that the vehicle capacity is not exceeded. In VRPSPD, however, the net change (decrease or increase) on the vehicle load at each customer must be monitored for capacity feasibility.
In this paper, we propose an Ant Colony Optimization (ACO) approach for the VRPSPD. The remainder of the paper is organized as follows: In Section 2, the notation and description of the problem is introduced. In Section 3, we review the related literature. Section 4 is devoted to the discussion of the proposed ACO approach. Section 5 discusses computational experiments and numerical results. Finally, concluding remarks are given in Section 6.
Section snippets
Problem description
The VRPSPD deals with a single depot distribution/collection system servicing a set of customers by means of a homogeneous fleet of vehicles, i.e., all vehicles have the same capacity. Each customer requires two types of service: a delivery and a pickup. Products to be delivered are loaded at the depot and products picked up are transported back to the depot. The objective is to find the set of routes servicing all the customers with the minimum total distance. The critical feature of the
Literature review
VRPB has been intensively studied in the literature. However, research on VRPSPD has recently gained momentum. Since VRPB is beyond the scope of this paper we omit the discussion and refer the interested reader to Goetschalckx and Jacobs-Blecha, 1989, Halse, 1992, Toth and Vigo, 1997, Brandão, 2006 for details and references.
VRPSPD was first introduced by Min (1989) as a book distribution and collection problem between a central library and 22 remote libraries in Ohio using two vehicles. Min
Overview of Ant Colony Optimization
ACO is a population-based metaheuristic that can be used to find approximate solutions to difficult optimization problems (Dorigo, 2008). It was first introduced for solving the TSP (Colorni, Dorigo, & Maniezzo, 1991). Since then many implementations of ACO have been proposed for a variety of combinatorial optimization problems such as quadratic assignment problem (Stützle & Dorigo, 1999), scheduling problems (Colorni, Dorigo, Maniezzo, & Trubian, 1994), sequential ordering problem (Gambardella
Description of the saving-based ant algorithm
In this section, we present the proposed saving-based ant algorithm, namely SavAnt, describing the mechanisms of ACO we employed in more detail.
Experimental study
The proposed algorithm is coded using C++ and executed on an Intel Pentium D 2.8 GHz processor with 2 Gb RAM. We fixed α = 1 to reduce the number of user controlled parameters and varied parameter β to control the relative weight of visibility versus pheromone intensity. We performed an extensive experimental design considering different combinations of parameter values: q0 = (0.5, 0.6, 0.7, 0.8, 0.9); β = (0.5, 1, 1.5, 2, 3, 4, 5); ρ = (0.05, 0.1, 0.15, 0.2, 0.25, 0.3). The following setting is observed to provide
Conclusion
In this paper, we address the VRPSPD which has a growing practical relevance in the reverse logistics literature. The computational complexity of the problem necessitates good heuristic solution procedures. To solve this problem efficiently, we develop an ant colony algorithm equipped with a new saving-based visibility function and pheromone update procedure. To the best of our knowledge, this is the first ACO approach proposed for this problem. The experimental analysis reveals good results
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