Berth and quay crane allocation as a scheduling problem J. Błażewicz, M. Machowiak
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Transcript of 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
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
Space-time representation of a berth plan (a), assignment of cranes to vessels (b)
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
Berth and quay relationship
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
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
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
Classification scheme for BAP formulations
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
Overview for BAP formulations
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
Classification scheme for QCSP formulations
Machowiak, Błażewicz, Oguz - PMS 2006, Poznań, April 26-28
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
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 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.
...
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