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

Berth and quay crane allocation as a scheduling problem

J. Błażewicz, M. MachowiakInstitute of Computing Science, Poznan University of Technology,

C. Oğuz, Department of Industrial Engineering, Koç University, Istanbul

T.C.E.Cheng Hongkong Polytechnic University

Recent survey

Bierwirth, Meisel, EJOR 2009

Berth allocation problem - BAP

gives rise to:

Quay crane assignment problem - QCAP

Quay crane scheduling problem - QCSP

Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28

Planning problems in container terminals


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


Berth and quay relationship


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

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


Classification scheme for BAP formulations


Overview for BAP formulations


Classification scheme for QCSP formulations


Overview for QCSP formulations

Our problem

Cont / stat / QCAP / max(compl)

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.

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 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.

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


From Theorem it follows that the minimum makespan value C*

cont for continuous problem is determined by the intersection point u0 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.


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.

...

T1

T2

Tn

Ccont*

processors

m

r*1

r*2

r*n

time

Turn around time on 1 processor (crane)

2 processors

Concavity justification

processing time

t(1)

processing time

t(2)

berthing time

Documents

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