IOE/MFG 543

1
IOE/MFG 543

• Chapter 14 General purpose procedures for
  scheduling in practice
• Section 14.4 Local search (Simulated annealing
  and tabu search)

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Introduction

• Constructive vs. improvement type algorithms
• Constructive type
• Construct the schedule by, e.g., adding one job
  at the time
• Improvement type
• Start with some complete schedule
• Try to obtain a better schedule by manipulating
  the current schedule

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General local search algorithm

• G(S) is the value of the objective under schedule
  S
• Let k1. Start with a schedule S1 and let the
  best schedule S0S1
• Choose a schedule Sc from the neighborhood of Sk
  N(Sk)
• If Sc is accepted let Sk1 Sc, otherwise let
  Sk1 Sk. If G(Sk1)ltG(S0) let S0Sk1
• Let kk1. Terminate the search if the stopping
  criteria are satisfied. Otherwise return to 2.

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Local search example1SwjTj
Jobs 1 2 3 4
wj 4 5 3 5
pj 12 8 15 9
dj 16 26 25 27
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Local search example (2)Schedule representation

• Let the vector S(j1,,jn) represent the schedule
• jkj if j is the kth job in the sequence
• Use EDD to construct the initial schedule
• S1
• The total weighted tardiness
• Let G(S) be SwjTj under schedule S
• gt G(S1)

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Local search example (3)Neighborhood structure

• Manipulating S1
• Pairwise adjacent interchange
• Try to move a job to a different location in the
  sequence
• Rules 1 and 2 above define two types of
  neighborhoods N1 and N2
• N1(S1)
• N2(S1)

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Local search example (4)Choosing Sc

• Assume we use N1
• Methods for choosing Sc from N1(Sk)
• Randomly
• Move the job forward that has the highest
  contribution to the objective
• Follow rule 2
• Interchange jobs and
• Sc ( , , , )
• G(Sc)

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Local search example (5)Acceptance criteria

• Is G(Sc) lt G(Sk)?
• Should we consider accepting Sc if G(Sc) G(Sk)
  ?
• In this example we only accept if we get an
  improvement in the objective

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Local search example (6)Stopping criteria

• Max number of iterations
• No or little improvement
• We would terminate the search since we did not
  improve the current schedule
• Local optimal solution
• No solution S in N(Sk) satisfies G(S)ltG(Sk)

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Local search example (7)Continuing

• S2S1(1,3,2,4)
• Swap 3 and 2 gt G(1,2,3,4) 115
• S3(1,2,3,4)
• Swap 4 and 3 gt G(1,2,4,3) 67
• S4(1,2,4,3)
• Swap 4 and 2 gt G(1,4,2,3) 72
• S5S4
• Swap 2 and 1 gt G(2,1,4,3) 83
• STOP and return (1,2,4,3) as the solution

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Local search example (8) Total enumeration
j1 j2 j3 j4 SwjTj
1 2 4 3 67
1 4 2 3 72
2 1 4 3 83
4 1 2 3 92
2 4 1 3 109
4 2 1 3 109
1 2 3 4 115
1 4 3 2 123
2 1 3 4 131
4 2 3 1 131
2 4 3 1 133
1 3 2 4 136
j1 j2 j3 j4 SwjTj
2 3 4 1 137
3 2 4 1 137
1 3 4 2 141
3 4 2 1 142
4 3 2 1 142
4 1 3 2 143
2 3 1 4 161
3 2 1 4 161
3 4 1 2 170
4 3 1 2 170
3 1 2 4 174
3 1 4 2 179
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Local searchDesign criteria

1. The representation of the schedule
2. The design of the neighborhood
3. The search process within the neighborhood
4. The acceptance-rejection criteria
5. Stopping criteria

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Simulated Annealing (SA)

• Annealing Heating of a material (metal) to a
  high temperature and then cooling it at a certain
  rate to achieve a desired crystalline structure
• SA Avoids getting stuck at a local minimum by
  accepting a worse schedule Sc with probability

P(Sk,Sc) exp(- G(Sc)-G(Sk) )
P(Sk,Sc) exp(- bk )
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SA Temperature parameter

• bk 0 is the temperature (also called cooling
  parameter)
• Initially the temperature is high making moves to
  a worse schedule more likely
• 50 chance of accepting a slightly worse
  schedule seems to work well
• As the temperature decreases the probability of
  accepting a worse schedule decreases
• Often, bkTak for some .9ltalt1 and Tgt0

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SA algorithm

• Set k1 and select b1.Select S1 and set S0S1.
• Select Sc (randomly) from N(Sk).
• If G(S0)ltG(Sc)ltG(Sk) set Sk1Sc and go to 3
• If G(Sc)ltG(S0) set S0Sk1Sc and go to 3
• If G(Sc)gtG(Sk), generate a uniform random number
  Uk from a Uniform(0,1) distribution (e.g., rand()
  in Excel)
• If UkP(Sk,Sc), set Sk1Sc otherwise set
  Sk1Sk.
• Select bk1 bk.Set kk1.Stop if stopping
  criteria are satisfied otherwise go to 2.

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SA example1SwjTj
Jobs 1 2 3 4
wj 4 5 3 5
pj 12 8 15 9
dj 16 26 25 27
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SA exampleIteration 1

• Step 1 S0S1(1,3,2,4). G(S1)136. Let T10 and
  a.9 gt b19
• Step 2. Select randomly which jobs to swap,
  suppose a Uniform(0,1) random number is V1 .24
  gt swap first two jobs
• Sc(3,1,2,4), G(Sc)174, P(Sk,Sc)1.5
• U1.91 gt Reject Sc
• Step 3 Let k2

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SA exampleIteration 2

• Step 2. Select randomly which jobs to swap,
  suppose a Uniform(0,1) random number is V2 .46
  gt swap 2nd and 3rd jobs
• Sc(1,2,3,4), G(Sc)115
• gt S3S0Sc
• Step 3 Let k3

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SA exampleIteration 3

• Step 2. V3 .88 gt swap jobs in 3rd and 4th
  position
• Sc(1,2,4,3), G(Sc)67
• gt S4S0Sc
• Step 3 Let k4

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SA exampleIterations 4 and 5

• Step 2 V4 .49 gt swap jobs in 2nd and 3rd
  position
• Sc(1,4,2,3), G(Sc)72, b410(.9)46.6
• P(Sk,Sc)47, U4.90 gt S5S4
• Step 3 Let k5
• Step 2 V5 .11 gt swap 1st and 2nd jobs
• Sc(2,1,4,3), G(Sc)83, b510(.9)55.9
• P(Sk,Sc)7, U5.61 gt S6S5
• Step 3 Let k6
• Are you bored yet?

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Tabu (taboo?) search

• Tabu search tries to model human memory processes
• A tabu-list is maintained throughout the search
• Moves according to the items on the list are
  forbidden

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Tabu search algorithm

• Set k1. Select S1 and set S0S1.
• Select Sc from N(Sk).
• If the move Sk?Sc is on the tabu list set Sk1Sk
  and go to 3
• If Sk?Sc is not on the tabu list set Sk1Sc.Add
  the reverse move to the top of the tabu list and
  delete the entry on the bottom.If G(Sc)ltG(S0),
  set S0Sc.
• Set kk1.Stop if stopping criteria are
  satisfied otherwise go to 2.

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Tabu search example1SwjTj
Jobs 1 2 3 4
wj 4 5 3 5
pj 12 8 15 9
dj 16 26 25 27

• Determine Sc by the best schedule in the
  neighborhood that is not tabu
• Use tabu-list length 2
• The tabu list is denoted by L

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Tabu search exampleIteration 1

• Step 1 S0S1(1,3,2,4). G(S1)136. Set L.
• Step 2. N(S1) (3,1,2,4), (1,2,3,4), (1,3,4,2)
  with respective cost 174, 115, 141 gt
  ScS0S2(1,2,3,4). Set L(3,2), i.e.,
  swapping 3 and 2 is not allowed
• Step 3 Let k2

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Tabu search exampleIteration 2

• Step 2.
• N(S2) (2,1,3,4), (1,3,2,4), (1,2,4,3)
• with respective costs 131, - , 67
• gt ScS3(1,2,4,3)
• Set S0Sc
• Set L(3,4),(3,2)
• Step 3 Let k3

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Tabu search exampleIteration 3

• Step 2
• N(S3) (2,1,4,3), (1,4,2,3), (1,2,3,4)
• with respective costs 83, 72, -
• gt ScS4(1,4,2,3)
• Set L(2,4),(3,4)
• Step 3 Let k4

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Tabu search exampleIteration 4

• Step 2
• N(S4) (4,1,2,3), (1,2,4,3), (1,4,3,2)
• with respective costs 92, -, 123
• gt ScS5(4,1,2,3)
• Set L(1,4),(2,4)
• Step 3 Let k5

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Tabu search exampleIteration 5

• Step 2
• N(S5) (1,4,2,3), (4,2,1,3), (4,1,3,2)
• with respective costs -, 109, 143
• gt ScS6(4,2,1,3)
• Set L(2,1),(4,1)
• Step 3 Let k6