© J. Christopher Beck 20081 Lecture 17: Tabu Search.

© J. Christopher Beck 2008 1

Lecture 17: Tabu Search


Outline Review:

Local Search & Crystal Maze Tabu Search Tabu & 1-machine JSP

Example C.5.4 Tabu & JSP

N1 neighborhood


Readings

P Ch C.5


Core Ideas of Local Search

Start with some (random?) assignment of variables

Look in “neighborhood” formed by making a small change to the assignment

Choose best neighbor Repeat


Crystal Maze Place the numbers 1 through 8 in the

nodes such that: Each number appears exactly once

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– No connected nodes have consecutive numbers


Random Initial Solution

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??1

4

3

2 5

6

7

8



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??1

4

3

2 5

6

7

8

“Broken” constraint

Cost = # of broken constraints


What Should We Do Now?

Move: Swap two numbers

Which two numbers? Randomly pick a pair The pair that will lead to the biggest

decrease in cost Cost: number of broken constraints



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??1

4

3

2 5

6

7

8


Cost Difference Table1 2 3 4 5 6 7 8

1 0 0 0 -1 0 -2 -3 -2

2 0 -1 1 -1 -2 -1 -3

3 0 0 0 0 -1 0

4 0 0 0 -1 0

5 0 0 1 -1

6 0 -1 0

7 0 0

8 0



1 0 0 0 -1 0 -2 -3 -2

2 0 -1 1 -1 -2 -1 -3

3 0 0 0 0 -1 0

4 0 0 0 -1 0

5 0 0 1 -1

6 0 -1 0

7 0 0

8 0


Current State

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??1

4

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2 5

6

7

8


Swap 1 & 7: Cost 3

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2 5

6

1

8


New Cost Difference Table1 2 3 4 5 6 7 8

1 0 0 0 0 2 0 3 0

2 0 0 2 0 1 1 1

3 0 0 0 1 1 -1

4 0 0 1 1 1

5 0 1 2 0

6 0 0 0

7 0 1

8 0


Current State

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??7

4

3

2 5

6

1

8


Swap 3 & 8: Cost 2

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??7

4

8

2 5

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1

3


Swap 6 & 7: Cost 1

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??6

4

8

2 5

7

1

3


Moves

Initial State: Cost 6 Swap 1 & 7: Cost 3 Swap 3 & 8: Cost 2 Swap 6 & 7: Cost 1



1 0 1 1 1 2 2 1 1

2 0 1 2 2 1 3 1

3 0 1 1 4 1 2

4 0 2 1 3 1

5 0 2 1 2

6 0 1 1

7 0 1

8 0


Now what?

There are no improving moves to make!

So far, we have been “hill-climbing”

moves

cost


Tabu Search Idea

Local search but: Keep a small list of the moves that

are “tabu”: you can’t (un)do them List has a limited length and the

oldest entries “fall off” when the list is full


Tabu Search

Start with some (maybe random) initial state

Look at the moves in the “neighborhood” and take the best one

Remember the last k moves (“tabu list”) so you don’t undo them


Tabu Search & 1-Machine

Example C.5.4, min wjTj

Tabu List Size = 2

Jobs 1 2 3 4

pj 10 10 13 4

dj 4 2 1 12

wj 14 12 1 12

Jj, pj, dj, wj

J1, 10, 4, 14


TS Step 1: Find Initial Solution

Arbitrarily choose (2, 1, 4, 3)

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

50013612121416128 jjTw

10 20 24 37

Jj, pj, dj, wj


TS Step 2: Evaluate Neighborhood & Select Move

Neighborhood: swap Adjacent pairwise

interchange

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 480

J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12 wjTj = 436

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 652

Choose (1,4)


TS Step 3: Add Moveto Tabu List

J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

wjTj = 436

Tabu: ((1, 4))Best so far:(2, 4, 1, 3): 436



J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 460

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12 wjTj = 500

J3, 13, 1, 1 J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 608

Choose (4,2)

J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

wjTj = 436

Tabu: ((1, 4))


TS Step 3: Add Move to Tabu List

Tabu: ((2,4) (1, 4))

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12wjTj = 460

Best so far:(2, 4, 1, 3): 436



J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 436

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12 wjTj = 440

J3, 13, 1, 1 J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 632

Choose (1,2)

Tabu: ((2,4) (1, 4))

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12wjTj = 460



Tabu: ((1,2), (2,4))

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12wjTj = 440

Best so far:(2, 4, 1, 3): 436



J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 460

J3, 13, 1, 1J1, 10, 4, 14 J2, 10, 2, 12 wjTj = 408

J3, 13, 1, 1J1, 10, 4, 14 J2, 10, 2, 12

J4, 4, 12, 12

wjTj = 586

Choose (1,4)

Tabu: ((1,2), (2,4))

J3, 13, 1, 1J1, 10, 4, 14J2, 10, 2, 12

J4, 4, 12, 12wjTj = 440



Tabu: ((1,4), (1,2))

wjTj = 408

Best so far:(1,4,2,3): 408

J4, 4, 12, 12

J3, 13, 1, 1J1, 10, 4, 14 J2, 10, 2, 12

And so on until a bound on the number of iterations (actually 408 is optimal, but TS has no way of

knowing)


Tabu Search Framework

Find initial solution (something quick)

Until N moves are done Evaluate neighborhood Choose best non-tabu neighbor Update tabu list

Return best solution found


Tabu Search Framework

To completely define a TS algorithm, you must define How the initial solution is found Format and length of tabu list How to generate a neighborhood from

a given state


TS & JSPJ0

J1

J2

J0

J0

J0

J1

J1

J1J2

J2

J2


TS & JSP

How might you find a first solution?

What could your tabu list format be?

Can you define a neighborhood?


The N1 Neighborhood for JSP

J0

J0

J0

J1

J1

J1J2

J2

J2


The N1 Neighborhood for JSP

J0

J0

J0

J1

J1

J1J2

J2

J2

Pick a critical path Neighborhood: all swaps on the

chosen CP


Tabu Example JSP

Run Tabu on JSP from previous lectures

Jobs Processing times

0 J0R0[15] J0R1[50] J0R2[60]

1 J1R1[50] J1R0[50] J1R2[15]

2 J2R0[30] J2R1[15] J2R2[20]

© J. Christopher Beck 20081 Lecture 17: Tabu Search.

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Transcript of © J. Christopher Beck 20081 Lecture 17: Tabu Search.