Strict limits on the maximum sulphur content in fuel used by ships have recently been imposed in some Emission Control Areas (ECAs). In order to comply with these regulations many ship operators will switch to more expensive low-sulphur fuel when sailing inside ECAs. Since they are concerned about minimizing their costs, it is likely that speed and routing decisions will change because of this. In this paper, we develop an optimization model to be applied by ship operators for determining sailing paths and speeds that minimize operating costs for a ship along a given sequence of ports. We perform a computational study on a number of realistic shipping routes in order to evaluate possible impacts on sailing paths and speeds, and hence fuel consumption and costs, from the ECA regulations. Moreover, the aim is to examine the implications for the society with regards to environmental effects. Comparisons of cases show that a likely effect of the regulations is that ship operators will often choose to sail longer distances to avoid sailing time within ECAs. Another effect is that they will sail at lower speeds within and higher speeds outside the ECAs in order to use less of the more expensive fuel. On some shipping routes, this might give a considerable increase in the total amount of fuel consumed and the CO 2 emissions.

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The inventory-routing problem (IRP) integrates two well-studied problems, namely, inventory management and vehicle routing. Given a set of customers to service over a multi-period horizon, the IRP consists of determining when to visit each customer, which quantity to deliver in each visit, and how to combine the visits in each period into feasible routes such that the total routing and inventory costs are minimized. In this paper, we propose an innovative mathematical formulation for the IRP and develop a state-of-the-art branch-price-and-cut algorithm for solving it. This algorithm incorporates known and new families of valid inequalities, including an adaptation of the well-known capacity inequalities, as well as an ad hoc labeling algorithm for solving the column generation subproblems. Through extensive computational experiments on a widely used set of 640 benchmark instances involving between two and five vehicles, we show that our branch-price-and-cut algorithm clearly outperforms a state-of-the-art branch-and-cut algorithm on the instances with four and five vehicles. In this instance set, 238 were still open before this work and we proved optimality for 49 of them.

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