1. Berth and quay crane allocation as a scheduling problem J. Błażewicz, M. Machowiak Institute of Computing Science, Poznan University of Technology, C. Oğuz, Department of Industrial Engineering, Koç University, Istanbul T.C.E.Cheng Hongkong Polytechnic University

7. Recent survey Bierwirth, Meisel, EJOR 2009 Berth allocation problem - BAP gives rise to: Quay crane assignment problem - QCAP Quay crane scheduling problem - QCSP

8. Planning problems in container terminals

9. Space-time representation of a berth plan (a), assignment of cranes to vessels (b)

10. Berth and quay relationship

11. Storage location structure of a vessel (a) and a cross-sectional view of a bay (b)

12. Motivation – Quay crane assignment problem Quay crane assignment problem is among the most important decision problems in a port container terminal since a good allocation of cranes to the incoming ships will enhance ship owners' satisfaction and increase terminal productivity, leading to higher revenues. We model the crane assignment problem as a moldable task scheduling problem by the following transformation quay cranes  processors ships  tasks turn-around time  schedule length

13. Classification scheme for BAP formulations

14. Overview for BAP formulations

15. Classification scheme for QCSP formulations

16. Overview for QCSP formulations

17. Our problem Cont / stat / QCAP / max(compl)

18. We consider the berth allocation and quay crane assignment problems as a moldable task scheduling problem by incorporating the fact that the number of quay cranes allocated to a ship will affect its berthing time. This approach can simultaneously increase the utilization of quay cranes, shorten the turn-around time of ships, and decrease the waiting time of the containers.

19. Moldable Tasks Model We consider a set of m identical processors (quay cranes) using for executing the set of n independent, nonpreemptable moldable tasks (ships).  Each task needs for its execution any number of processors (at least one but less or equal to m).  The total number of processors executing the tasks should not exceed m at any time. An amount p j > 0 of work is associated with each task T j. f j (r)  0 is a non-decreasing processing speed function, f j (0) = 0.  f j (r) relates processing speed of task T j to a number of processors allocated. The criterion assumed is schedule length.

20. To explain the main idea of finding a solution for the continuous problem, we introduce set of feasible resource allocations and set of feasible transformed resource allocations. Denote p = (p 1,…, p n ) Theorem (Weglarz 82) Let n  m, convU be the convex hull of the set U, i.e. the set of all convex combinations of the elements of U, and u = p/C be a straight line in the space of transformed resource allocations given by the parametric equations u j = p j /C, j = 1,…, n. Then the minimum makespan value for continuous problem can be found from The solution of the continuous problem

21. From Theorem it follows that the minimum makespan value C * cont for continuous problem is determined by the intersection point u 0 of the straight line u = p/C, C > 0, and the boundary of the set convU in the n-dimensional space of transformed resource allocations.

22. The solution of the continuous problem The proposed algorithm starts from the continuous version of the problem and transforms the schedule obtained from the continuous version into a feasible schedule for the discrete MT model. We assume that with each task the concave processing speed function is associated. In an optimal schedule for continuous problem all the tasks are processed in the interval [0, C * cont ] and task T j uses r * j processors, j = 1,...,n.

23. Turn around time on 1 processor (crane) 2 processors Concavity justification processing time t(1) processing time t(2) berthing time