An Effective Multirestart Deterministic Annealing Metaheuristic for the Fleet Size and Mix Vehicle-Routing Problem with Time Windows

The purpose of this paper is to present a new deterministic annealing metaheuristic for the fleet size and mix vehicle routing problem with time windows. The objective is to service, at minimal total cost, a set of customers within their time windows by a heterogeneous capacitated vehicle fleet. First, we motivate and define the problem. We then proceed to give a mathematical formulation of the most studied variant in the literature, in the form of a mixed-integer linear program. We also suggest an industrially relevant, alternative definition that also leads to a linear mixed-integer formulation. The suggested metaheuristic solution method solves both problem variants and comprises three phases. In the Phase 1, high quality initial solutions are generated by means of a savings-based heuristic combining diversification strategies with learning mechanisms. In Phase 2, an attempt is made to reduce the number of routes in the initial solution with a new local search procedure, and in Phase 3 the solution from Phase 2 is further improved by a set of four local search operators that are embedded in a deterministic annealing framework to guide the improvement process. Some new implementation strategies are also suggested for efficient time window feasibility checks. Extensive computational experiments on the 168 benchmark instances of Liu and Shen show that the suggested method outperforms the previously published results and found 167 best-known solutions. Experimental results are also given for the new problem variant.