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Quay crane allocation as a moldable task scheduling problem

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Motivation - Crane scheduling problem ... quay cranes processors. ships tasks. turn-around time schedule length ... The crane allocation as moldable tasks ... – PowerPoint PPT presentation

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Title: Quay crane allocation as a moldable task scheduling problem


1
  • Quay crane allocation as a moldable task
    scheduling problem
  • J. Blazewicz, M. Machowiak
  • Institute of Computing Science, Poznan University
    of Technology,
  • C. Oguz,
  • Department of Industrial Engineering, KoƧ
    University, Istanbul

2
Motivation - Crane scheduling problem
  • Berth allocation problem is among the most
    important decision problems in a port container
    terminal since a good allocation of berths to the
    incoming ships will enhance ship owners'
    satisfaction and increase terminal productivity,
    leading to higher revenues.
  • We model the berth allocation problem as a
    moldable task scheduling problem by considering
    its relation to the crane scheduling problem.
  • quay cranes ?? processors
  • ships ?? tasks
  • turn-around time ?? schedule length

3
  • We consider the berth allocation problem 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.

4
Literature
  • Despite the strong dependence of the berth
    allocation decisions on the number of quay cranes
    allocated to a ship, the majority of research in
    this area has not taken this relation into
    account when analyzing the berth allocation
    problem (see, for example, Daganzo (1989),
    Peterkofsky and Daganzo (1990), Imai et al.
    (2001, 2003), Li et al. (1998)).
  • To the best of our knowledge, only Park and Kim
    (2003) incorporated the crane scheduling
    decisions into the berth allocation problem.

5
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 tj gt 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.
  • The problem is NP-hard (Du Leung, 1989), thus,
    suboptimal algorithms were looked for.

6
The solution of continuous problem
  • To explain 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 (p1, , pn)
  • 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 uj pj /C, j
    1, , n.
  • Then the minimum makespan value for continuous
    problem can be found from

7
The solution of continuous problem
  • From Theorem it follows that the minimum makespan
    value Ccont for continuous problem is determined
    by the intersection point u0 of the straight line
    u p/C, C gt 0, and the boundary of the set convU
    in the n-dimensional space of transformed
    resource allocations.

8
The solution of continuous problem
  • In an optimal schedule for continuous problem all
    the tasks are processed in the interval 0,
    Ccont and task Tj uses rj processors, j
    1,...,n.
  • 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.

9
Concavity justification
departure time
  • Turn around time on 1 processor (crane)
  • 2 processors

unloading time
t(1)
berthing time
unloading time
t(2)
10
Rounding scheme
  • if rj ? 1 then rj 1
  • When the number of processors assigned in the
    continuous solution is between 0 and 1 for each
    task, the speed function is linear, and then the
    surface (defined as a product of a task duration
    and a number of processors allocated to it) of a
    task does not change.
  • else if rj gt 2 or rj lt 1.5 then rj ?
    rj ?
  • In this case, the completion time of a task grows
    but no more than 50 to the continuous solution.
    Moreover, the sum of the processors allocated to
    these tasks is lower than m.
  • else (1.5 ? rj ? 2 ) rj 2.
  • Here a processing time of a task is smaller than
    in continuous solution but, using the concavity
    assumption, its surface increases. When we have
    many of such tasks, the sum of the discrete
    processors allocated to these tasks is greater
    than m and the Algorithm has to perform many
    steps of rounding off.

11
Algorithm
  • calculate the Ccont and the optimal continuous
    processor allocations rj.
  • round the continuous allocation of processors to
    the integer values.
  • calculate the new processing times of the tasks
    and sort them in non-increasing order of
    processing times.
  • m ? rj (i1,...,n) (discrete number of
    processors)
  • C max(Cj)
  • if m m then GoToEnd
  • if m lt m then allocate an excess of
    processors to the biggest task GoToEnd
  • if m gt m then assign the remaining tasks
    after the tasks already scheduled till the
    max(C, ?tj(1) /m)
  • if m m then GoToEnd
  • else take the minimum value of the schedule
    length constructed by A or B
  • A while m gt m then
  • reduce a number of processors assigned to the
    task (group of tasks) with the biggest number of
    processors alloted and assign the tasks on the
    freed processor and on the other processors till
    the finishing time of the longest task (C).
  • B the remaining tasks schedule after the tasks
    already scheduled treating them as a new
    instance.
  • End Cmax C

12
Example
  • Let us consider the following data
  • n6, m4, T T1, T2, , T6,
  • t(1) 32, 34, 14, 92, 18, 55,
  • f(r) ? r, the same for each task.
  • Ccont 64.46,
  • r 0.49 0.53 0.22 1.63 0.28 0.85.
  • From this solution we proceed according to the
    rounding algorithm.

Ccont64,46
t3(1)14
t5(1)18
t1(1)32
4
t2(1)34
t4(2)54,7
processors
t6(1)55
time
13

Ccont64,46
t3(1)14
t5(1)18
t1(1)32
4
t2(1)34
t4(2)54,7
processors
t6(1)55
time
14
Ccont64,46
t3(1)14
t1(1)32
4
t2(1)34
t5(1)18
t4(2)54,7
processors
t6(1)55
time
15
Ccont64,46
t1(1)32
t3(1)14
4
t2(1)34
t5(1)18
t4(2)54,7
processors
t6(1)55
time
16
Step A
Ccont64,46
t1(1)32
t3(1)14
4
t2(1)34
t5(1)18
processors
t4(1)92
t6(1)55
time
17
Step A
Ccont64,46
4
t1(1)32
t3(1)14
t2(1)34
t5(1)18
processors
t4(1)92
t6(1)55
time
18
Step B
Ccont64,46
t1(1)32
t3(1)14
4
t2(1)34
t5(1)18
t4(2)54,7
processors
t6(1)55
time
19
Step B
Tasks of new instance
t1(1)32
Ccont64,46
t3(1)14
4
t2(1)34
t5(1)18
t4(2)54,7
processors
t6(1)55
time
20
Step B
Tasks of new instance
Ccont64,46
4
t2(1)34
t5(1)18
t1(3)14,04
t4(2)54,7
processors
t6(1)55
t3(1)14
time
21
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22
Computational experiments
  • To evaluate the mean behavior of Algorithm we
    use the following measure
  • Relative error Cmax / Ccont
  • where
  • Cmax schedule length obtained by Algorithm
  • Ccont optimal schedule length of the
    continuous solution
  • Task processing times ti(1) have been generated
    from a uniform distribution in interval 1, 100.
  • Processing speed functions have been chosen as
  • fi(r) ra, 0 lt a ? 1
  • Each entry in the table is a mean value for 25
    instances randomly generated.

23
Average behavior of the Algorithm for different
shapes of processing speed functions and varying
number of processors
24
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25
Conclusion
  • The crane allocation as moldable tasks scheduling
    problem has been considered.
  • Starting from the continuous version of the
    problem (i.e. where the tasks may require a
    fractional part of the resources),
  • we proposed an approximation algorithm with a
    very good average behavior.
  • Further investigations in this area will take
    into account a construction of the algorithm with
    a good average and worst case performance
    guarantee for arbitrary non-decreasing processing
    speed function.
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