Berth and quay crane allocation as a scheduling problem J. Błażewicz, M. Machowiak

1. Berth and quay crane allocation as a schedulingproblem 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 simultaneouslyincrease 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 nindependent, 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 pj> 0 of work is associated with each task Tj. • fj(r) 0 is a non-decreasing processing speed function,fj(0) =0. • fj(r)relates processing speed of task Tj to a number of processors allocated. • The criterion assumed is schedule length.

20. The solution of the continuous problem • 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

21. The solution of the continuous problem • From Theorem it follows that the minimum makespan value C*cont for continuous problem is determined by the intersection pointu0 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 Tj uses r*j processors, j = 1,...,n.

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