Models and exact solutions for a class of stochastic location-routing problems

This paper describes a family of stochastic location-routing problems which consist of simultaneously locating a depot among a set of potential sites, of determining the vehicle fleet size and of designing collection routes through a set of customers having random supplies. In a first stage, decisions regarding depot location, fleet size and planned routes have to be made without knowing the actual supplies, so that the total supply of a route may turn out to exceed the vehicle capacity (i.e. failures may occur). In such an event, a corrective recourse action is taken at a second stage: the vehicle returns to the depot and empties its load before resuming its journey. Such an action generates a penalty. In this paper, two variants are studies: (P1) minimize first stage costs so that the probability of route failure does not exceed a preset threshold; (P2) minimize first stage costs so that the expected penalty of any route does not exceed a fraction of its planned cost. The problems are modelled as integer linear programs and solved to optimality.