An Effective Search Framework Combining Meta-Heuristics to Solve the Vehicle Routing Problems with Time Windows

Many delivery problems in real-world applications such as the newspaper delivery and courier services can be formulated as capacitated vehicle routing problems (VRPs) [10], which we want to route a number of vehicles with limited capacity in order to satisfy customer requests with the minimal operational cost. This is usually measured by the number of vehicles used multiplied by the total distance travelled. Often, a customer may specify a time-window with the earliest and latest time for the delivery, which gives rise to the VRP with time windows (VRP-TWs). In other words, a vehicle must arrive at a customer within the interval as specified by that customer in a VRP-TWs. The arrival of a vehicle before the earliest time as specified by a customer in a VRP-TWs will result in idle time. On the other hand, a vehicle is not allowed to reach a customer after the specified latest time. Moreover, a service time is commonly associated with servicing each customer. A real-life example of the VRP-TWs is furniture delivery in which each customer often requests the furniture to be delivery within a given period of a day. Unfortunately, the VRP-TWs are shown to be NP-complete, implying an exponential growth in the time complexity for a general algorithm to solve any of these delivery problems in the worst case. In practice, there are many instances of VRP-TWs involving 100 customers [1, 10] or more [9] which are difficult to solve optimally [1, 3, 6]. In the past two decades, VRP-TWs, due to their challenging nature and practical value, have continuously attracted many interesting proposals for useful heuristics and search algorithms [1, 3] to effectively solve problems in the areas of Artificial Intelligence [3], Constraint Programming [1] and Operations Research [3]. Among the heuristics [1, 2] proposed to solve the vehicle routing problems, there are some interesting proposals to initialize the search while others are targeted at advancing the search from a given initial state in an effective manner. As far as search initialization is concerned, there are two useful heuristics, namely the push-forward insertion heuristic (PFIH) [6] and the virtual vehicle heuristic (VVH) [2] which was proposed to generate more feasible initial states that may lead to better results. The PFIH is a simple-yet-efficient method to compute every route by comparing the cost of inserting a new customer into the O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m