The probabilistic uncapacitated open vehicle routing location problem

Suppose that m$$ m $$ mobile service units are located at a base station (depot) in a transportation network with n$$ n $$ nodes. On any day, the nodes of the network may generate calls for service independently with known probabilities. The calls are centrally allocated to the service units who then visit the allocated customers on shortest open tours, that is, for each service unit, the way back to the depot from the last served customer is not counted towards the length of the tour. It is required to find an optimal location for the depot to minimize the expected travel distance. We obtain bounds on the approximation ratios for two simple and fast heuristics for the problem on a general network. For the problem on a tree, we present an O(nm+1)$$ O\left({n}^{m+1}\right) $$ exact algorithm.