A note on "Berth allocation considering fuel consumption and vessel emissions

Du et al. [Du, Y., Chen, Q., Quan, X., Long, L., Fung, R.Y.K., 2011. Berth allocation considering fuel consumption and vessel emissions. Transportation Research Part E 47, 1021-1037] dealt with a berth allocation problem incorporating ship' fuel consumption minimization. To address the difficulty posed by the power function between fuel consumption rate and sailing speed, they formulated a tractable mixed-integer second-order cone programming model. We propose two quadratic outer approximation approaches that can handle general fuel consumption rate functions more efficiently. In the static quadratic outer approximation approach, the approximation lines are generated a priori. In the dynamic quadratic outer approximation approach, the approximation lines are generated dynamically. Numerical experiments demonstrate the advantages of the two approaches.