Modelling path flows for a combined ship routingand inventory management problem

We consider a combined time constrained ship routing and inventory managementproblem. A fleet of ships transports a single product between production and consumptionharbours. The transporter has the responsibility for keeping the stock level within its limitsat all actual harbours, and there should be no need to stop the production at any harboursdue to missing transportation possibilities. The number of arrivals to each harbour and thequantities loaded and discharged at each arrival are determined by the continuous productionrates at the harbours, the stock limits and the actual ships visiting the harbours. We use apath flow formulation for this planning problem, and generate paths for each ship includinginformation about the geographical route, the load quantity and start time at each harbourarrival. In addition, we generate paths for each harbour including information about thenumber of arrivals to the harbour, the load quantity and start time at each harbour arrival.We emphasise the formulation of the path generation problems which are subproblems inthe total planning problem. The generated paths appear as columns in a path flow problemwhich corresponds to a master problem. We use a column generation approach to solve thecontinuous problem. The solution is made integer optimal by branch-and-bound. Computationalresults indicate that a path flow formulation and an optimisation based solutionapproach work for real instances of the planning problem. Copyright Kluwer Academic Publishers 1998