Review of Article on Vehicle Routing Problem with LP

Question: Discuss about Review of Article on Vehicle Routing problem with LP ? Answer: Introduction: Linear programing finds vast applications in engineering as well as in several day to day applications. The current discussion is about the vehicle routing problem solution algorithm. The particular characteristic of this problem is the usage of the time windows and forbidden paths as part of the mainstream solutions. The objectives of the problem include determination of the optimal set of routes used by the fleet of vehicles to serve a given set of customers on a predefined path. Objectives: The specific objectives of the problem include minimization of the travelling times and distances. Also it includes minimization of the operational cost as well(Descrochers, Descrosiers, Solomon, 1992). Given data and the constraints: There is provision to use forbidden path constraints as well in the given problem. Basically time windows are used to formulate the problem as well there are procedures included to make vehicle routing indicated by earliest and latest times for the customers to permit the start of the service. Time windows are the constraints in the problems. Customer declared or chosen time windows are the constraints that work in this type of problems and the algorithm generated is flexible to embrace a range of such time windows. Methodology employed: LP relaxation of the set partitioning formulation of the VRPTW (Vehicle routing problem with time windows) is done and subsequently is solved by column generation procedures. Specific observations: The solution provided in the problem is a optimization algorithm for VRPTW, which is found to be capable of optimally solving problems of any size. Typically the problems of size larger than any of the previously seen in the literature can be solved using the algorithm proposed in the research article. The results of the computations and the observations of the graphs and the results are indicated in the solution. Remarks: The new optimization algorithm of column generation model proposed in this solution for a set partitioning formulation for the VRPTW is very versatile and can be applicable to a variety of practical sized benchmark test problems. Generally the size of the problem that can be solved will reduce drastically with the increase in the complexity of the constraints however the column generation approach of solving the problem is immune from such problems, the technique employs complex constraints by incorporating the hardest constraints at key points of the solution procedure employed. Even the approach is flexible enough to yield solutions which are near-optimal in nature by early termination of the algorithms. It is also possible to use multiple time windows for the same customer and this can be modelled by using as many nodes as time windows for each customer. Also the article solution procedure provided is based on the assumption of homogeneous fleet assumptions. The solution consist o f assuming free number of vehicles, which mean that the fleet size is determined simultaneously with the best set of routes and schedules. Demerits: The technique will cease to be competitive if applied in the environments where several customers are visiting the same route. Degeneracy will become a problem in such cases, this follows by increase in the density of the LP. References: Descrochers, M., Descrosiers, J., Solomon, M. (1992). A New optimization Algorithm for the vehicle Routing problem with Time windows. Operations Research, 40(2), 342-354.