*(English)*Zbl 1179.90068

Summary: This paper proposes a formulation of the vehicle routing problem with simultaneous pickup and delivery (VRPSPD) and a particle swarm optimization (PSO) algorithm for solving it. The formulation is a generalization of three existing VRPSPD formulations. The main PSO algorithm is developed based on GLNPSO, a PSO algorithm with multiple social structures. A random key-based solution representation and decoding method is proposed for implementing PSO for VRPSPD. The solution representation for VRPSPD with $n$ customers and $m$ vehicles is a (n+2m)-dimensional particle. The decoding method starts by transforming the particle to a priority list of customers to enter the route and a priority matrix of vehicles to serve each customer. The vehicle routes are constructed based on the customer priority list and vehicle priority matrix. The proposed algorithm is tested using three benchmark data sets available from the literature. The computational result shows that the proposed method is competitive with other published results for solving VRPSPD. Some new best known solutions of the benchmark problem are also found by the proposed method.

Scope and Purpose

This paper applies a real-value version of particle swarm optimization (PSO) algorithm for solving the vehicle routing problem with simultaneous pickup and delivery (VRPSPD). The VRPSPD formulation is reformulated and generalized from three existing formulations in the literature. The purposes of this paper are to explain the mechanism of the PSO for solving VRPSPD and to demonstrate the effectiveness of the proposed method.

##### MSC:

90B20 | Traffic problems |

##### Keywords:

vehicle routing problem; simultaneously pickup and delivery; particle swarm optimization; random key representation; metaheuristics##### References:

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