Hybrid Insertion Heuristics for Vehicle Routing Problem with Soft Time Windows

Industrial growth and expanding employment opportunities have led to the urban-oriented economic development in many countries. Demand of transportation, both in terms of passengers as well as for freight is also increasing in and around these big urban conurbations. A high proportion of total goods movement occurs within cities 1) , and most of this movement is based on road transport. Traffic congestion, noise, vibrations, generation of NOx, SPM, CO2 and other environmental problems, crashes and loading and unloading on the street side are typical problems caused by the roadbased freight transport in urban areas. Such freight movement related problems has magnified the need for research in the field of city logistics. The Vehicle Routing and scheduling Problem with Time Windows (VRPTW) can be used as a tool for evaluating many city logistics schemes. For example, the VRPTW could be used in the analysis of cooperative delivery systems and ideal location of logistics terminals, which belong to infrastructure planning and management problems in city logistics. Depending on the nature of the time windows, the VRPTW is further expanded to the Vehicle Routing and scheduling Problem with Hard Time Windows (VRPHTW) and the Vehicle Routing and scheduling Problem with Soft Time Windows (VRPSTW). Most of the exact optimization research has been directed towards the hard time windows variant, though it lacks the practicality found in real life problems. Furthermore, these exact techniques allow waiting at no penalty cost when a vehicle arrives earlier than the start time of service. This results in more waiting time as compared to the case when waiting is penalized 2) . On the other hand, soft time windows are often encountered in practical freight transport, yet mostly heuristics (approximate) solutions are used for the VRPSTW in city logistics related research. Recently column generation techniques have been used efficiently for the VRPHTW, successfully solving large size problems as well as significantly reducing the required computational efforts. This could be desirable to develop exact solution approaches for the soft time windows variant as well. However, complex soft time windows constraints and time dependent costs have been the greatest barriers in the way of these efforts. This paper presents a hybrid solution technique for the VRPSTW embedding a heuristics solution technique (Insertion heuristics) within the exact solution techniques (Column Generation) in order to improve the solution quality and to reduce computational times. The basic idea is to replace the NP-hard subproblem of column generation scheme i.e. the Elementary Shortest Path Problem with Resource Constraint (ESPPRC) by an insertion heuristics, which can efficiently handle complex soft time windows constraints. The shadow prices generated by column generation method are utilized to find the reduced costs, which are then used in the insertion heuristics to find negative reduced cost routes (columns) for the master problem.