Journal of the Operational Research Society (2011) 62, 1189--1197  2011 Operational Research Society Ltd. All rights reserved. 0160-5682/11 www.palgrave-journals.com/jors/ Berth and quay crane allocation: a moldable task scheduling model J Blazewicz 1 , TCE Cheng 2 , M Machowiak 1 and C Oguz 3 ∗ 1 Institute of Computing Science, Poznan University of Technology, Poland; 2 The Hong Kong Polytechnic University, Hong Kong SAR, People’s Republic of China; and 3 Koc ¸ University, Istanbul, Turkey We study the problem of allocating berths to incoming ships and assigning the necessary quay cranes to the ships at a port container terminal. We formulate the problem as the moldable task scheduling problem by considering the tasks as ships and processors as quay cranes assigned to the ships based on the observation that the berthing duration of a ship depends on the number of quay cranes allocated to it. In the model, the processing speed of a task is considered to be a non-linear function of the number of processors allocated to it. We present a suboptimal algorithm that obtains a feasible solution to the discrete version of the problem from the continuous version, that is, where the tasks may require fractional quantities of the resources. We conducted computational experiments to evaluate the performance of the algorithm. The computational results show that the average behaviour of the algorithm is very good. Journal of the Operational Research Society (2011) 62, 1189 – 1197. doi:10.1057/jors.2010.54 Published online 16 June 2010 Keywords: berth allocation; quay crane allocation; moldable task; resource allocation; parallel computing Introduction A port container terminal is a site comprising a quayside and a storage yard. The quayside is divided into a number of berths, at which incoming ships moor and have their containers loaded/unloaded by the quay cranes. In this setting, ships’ arrival and loading/unloading of containers are consid- ered as two parts of the operational activities that take place at the quayside. For detailed descriptions of the activities in a port container terminal, the reader is referred to Vis and de Koster (2003) and Steenken et al (2004). The first part of the quayside operational activities, that is, the arrival of ships, is commonly modelled as the berth allocation problem, which seeks to determine the berthing time and the position of each ship along the quayside. The second part, that is, the loading/unloading of containers to/from ships, includes quay crane scheduling to determine the exact movements of the quay cranes during the loading/unloading operation of a berthed ship. These two decisions are dependent on each other. Murty et al (2005) stated that the management of terminals will allocate berths and quay cranes to incoming ships at two levels: First, the allocation decision is made based on the projected workload of the ships a few weeks before the actual arrival of the ships. Here the workload of a ship is determined by the number of containers to be handled (loaded/unloaded) at the terminal. Second, the actual ∗ Correspondence: C Oguz, Department of Industrial Engineering, Koc ¸ University, Istanbul 34450, Turkey. E-mail: coguz@ku.edu.tr berth and quay crane allocations will be made based on the information received about the ships one or a few days before their actual arrival. It is clear that the terminal operator can utilize the resources (ie, berths and quay cranes) efficiently by taking both berth allocation and quay crane allocation decisions simultaneously. As a matter of fact, these are among the most important decision problems in a port container terminal since efficient allocation of berths and quay cranes to incoming ships will enhance ship owners’ satisfaction (by reducing their waiting times) and increase terminal productivity (by utilizing the resources more efficiently), leading to higher revenues. In this paper we consider the berth and quay crane allo- cation problems jointly without considering quay crane scheduling. The crane scheduling problem concerns with the detailed tasks to be performed by each crane in handling the containers on the ships and it will be out of the scope of our study. We aim to provide a helpful tool for terminal managers who will make berth and quay crane allocation decisions a few weeks before the actual arrival of the ships so that these resources are utilized efficiently. Since the number of quay cranes allocated to a ship will define the duration that the ship spends at the berth, it is reasonable to consider the handling time for each ship as a function of the number of quay cranes assigned to it. This is important because quay cranes should be positioned at the berths that are allocated to incoming ships according to the requirements of the ships since physical constraints, such as non-crossing and neighbourhood constraints, limit the flexibility of the