This article presents a new mathematical formulation for the berth scheduling problem. The objective is to simultaneously minimize the total costs from vessels’ late departure and waiting time, and maximize the total premiums from vessels’ early departures. It is assumed that different vessels have different variable penalty/premium cost functions (PPCFs) that are based on contractual agreements between the liner shipping company and the terminal operator. A genetic algorithms-based heuristic is proposed to solve the resulting problem. A number of computational examples are presented to: (a) assess four different berth scheduling policies that are based on three different contractual agreements and (b) evaluate the effect of non-linear PPCFs. Computational results show that: (a) penalty/premium cost distribution among vessels display distinctive variations between the four berth scheduling policies, and (b) contractual agreements with constant (within a time window), followed by hourly (outside the time window) penalties/premiums should be favored during contractual agreement negotiations between terminal operators and liner shipping companies.

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Keywords berth scheduling, container terminals, genetic algorithms, maritime economics, metaheuristic optimization
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Golias, M.M., & Haralambides, H.. (2011). Berth scheduling with variable cost functions. Maritime Economics & Logistics, 13(2), 174–189. doi:10.1057/mel.2011.4