The demand weighted vehicle routing problem

We consider the problem of constructing optimal bus routes for carrying students from residences to campus and then back home. In typical routing problems the objective is to minimize distance traveled subject to vehicle capacity constraints or meeting time window constraints. However, in this application the vehicles carry people and each person travels the entire length of the route. Therefore it is critical to consider how long the people are riding on the bus. Bus shuttles from airports to hotel locations are another example where riders often traverse the entire route, first from the airport to the hotel, and then from the hotel to the airport. This leads to the objective of minimizing the distance traveled weighted by the number of riders on the route. The resulting model is a non-convex integer program that is extremely difficult to optimize. We show how to reformulate this problem as a linear integer program with auxiliary variables. The resulting formulation provides very tight lower bounds on the optimal solution value and little enumeration is required. However, the reformulated model is very large and requires the use of column generation to solve. Using COIN-OR software, we provide computational results based on actual student demand data.