Strategic Planning of Transportation Assets

Logistics service providers use transportation assets to offer services to their customers. To cope with demand variability that cannot be accommodated by their existing assets, they may acquire additional assets on a one-off  (spot) basis. The planner's problem is to determine the optimal level of assets acquired upfront such that their cost and the expected cost of spot assets is jointly minimized, for a given planning horizon. We study this problem using a multi-period, two-stage stochastic programming formulation, having spot assets acquired in each period as recourse actions. Our formulation captures news-vendor trade-offs with a non-trivial complication: while ordering quantities are pertinent to asset acquisition, customer demand is in the form of service requests. Not only has each such request a stochastic duration, but also the total number of requests per customer is uncertain. Furthermore, even the deterministic version of determining the number of assets required to cover such customer requests is combinatorial. Using structural properties of our model, we decompose it to a series of period- and scenario-specific subproblems, each solved by a column generation reformulation, whose solutions are used at a reduced formulation that can be optimized in sub-linear time. Our method finds optimal solutions to instances intractable by commercial solvers, within orders of magnitude less CPU time. Finally, we investigate demand variability by means of a factorial experiment. We find that, while variability in the number of requests leads to higher costs, variability in each request's duration reduces costs. Our results provide a practical approach to the planning of transportation assets and offer insights regarding the differing impact of demand uncertainty on the planning cost.