Lecture 19:
Timetabling with Workforce
Capacity
© J. Christopher Beck 2008
1
Outline
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Adding Workforce Capacity to
Timetabling
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Examples 9.4.1, 9.4.2
Bin Packing
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Example 9.4.3, 9.4.4
© J. Christopher Beck 2008
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Readings
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P Ch 9.4
Questions:
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Bin packing
Examples in
this lecture
© J. Christopher Beck 2008
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Workforce
Capacity
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n activities
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Processing time of activity j is pj
No pre-emption
An infinite number of resources
Each activity requires Wj workers
You only have W workers
Find a schedule that minimizes
makespan
© J. Christopher Beck 2008
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Example
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W = 10
Find a lower bound on the makespan
Find minimum makespan schedule
activities
1
2
3
4
7
8
9
10
pj
10 3
6
7 11 20 3
1
5
8
Wj
3 10 6
2
4
9
9
© J. Christopher Beck 2008
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1
6
7
3
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Special Case: Exam
Scheduling (Example 9.4.2)
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All exams have the same duration
One exam room with capacity W
Course j has Wj students
All students in course j must write the
exam at the same time
Find a timetable for all n exams in the
minimum amount of time
© J. Christopher Beck 2008
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Special Case = Bin Packing
W5
W1
W6
W3
W2
W4
W8
W7
…
Pack the objects into the bins
to minimize the number of
that are used
Each bin has capacity = W
© J. Christopher Beck 2008
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Special Case = Bin Packing
W5
W8
W3
W1
W6
W2
W7
W4
…
Pack the objects into the bins
to minimize the number of
that are used
Each bin has capacity = W
© J. Christopher Beck 2008
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Example 9.4.3
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Find a lower bound on
the number of bins
Find an upper bound on
the number of bins
Find a solution – is it optimal?
W = 2100
activities
1…6
7…12
13…18
Wj
301
701
1051
© J. Christopher Beck 2008
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Bin Packing Heuristics
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First Fit (FF)
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Order items arbitrarily
Put item into lowest number bin that it will
fit into
First Fit Descending (FFD)
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Order items in descending order
Put item into lowest number bin that it will
fit into
© J. Christopher Beck 2008
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Example 9.4.3
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Find FF solution
Find FFD solution
Is either solution optimal?
W = 2100
activities
1…6
7…12
13…18
Wj
301
701
1051
© J. Christopher Beck 2008
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Example 9.4.4
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Find
Find
Find
Find
LB & UB
FF solution
FFD solution
optimal solution
W = 1000
activities
1…6
7…12
13…18
19…30
Wj
501
252
251
248
© J. Christopher Beck 2008
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