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스케줄 이론 Ch 4 Heuristic Method PowerPoint PPT Presentation


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스케줄 이론 Ch 4 Heuristic Method. 1. Introduction. Heuristic Procedure Capable of obtaining good solutions with limited computational effort. We need a measure for performances of heuristic procedures. Procedures based on Local Search: Simulated Annealing Tabu-search Genetic Algorithm

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스케줄 이론 Ch 4 Heuristic Method

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Ch 4 heuristic method

스케줄 이론

Ch 4 Heuristic Method


1 introduction

1. Introduction

  • Heuristic Procedure

    • Capable of obtaining good solutions with limited computational effort.

    • We need a measure for performances of heuristic procedures.

  • Procedures based on Local Search:

    • Simulated Annealing

    • Tabu-search

    • Genetic Algorithm

    • Neighborhood search (Adjacent Pair-wise Interchange, One-step look-back, Multi-step look-back, One-step look-ahead, Multi-step look-ahead)


2 dispatching and construction procedures

2. Dispatching and Construction Procedures

  • Sorting

    • The use of a ranking scheme where the relative ranking of two jobs does not change with time.  Static priorities

  • Dispatching

    • A procedure that uses a decision rule to select the next job when the machine becomes free.

    • Include dynamic sorting rules

      • Example: At time t

t=0

t=3


2 dispatching and construction procedures1

2. Dispatching and Construction Procedures

  • Construction

    • A procedure where a schedule is built from scratch, normally adding a job(s) to the schedule one at a time. (but not necessarily adding jobs in order from earliest to latest)

    • Greedy procedure: T-problem

      • O(n2)

    • Insertion procedure

      • {1, 2} 에서 1-2, 2-1 중  (2, 1) 선택

      • 3을 삽입 3-2-1, 2-3-1, 2-1-3 중  (2, 3, 1) 선택

      • O(n3)

  • Table1. Comparison of heuristic algorithm(p. 4.5)


3 neighborhood search

3. Neighborhood Search

  • Neighborhood

    • ⓐ Initial seed (Initial trial solution)

    • ⓑ Neighborhood generating mechanism

      • Adjacent pair-wise interchange       n-1

      • Pair-wise interchange

      • Last job inserted to previous jobs    n-1

    • ⓒ New seed selection rule (Only one, or search all possible improvements)

  • Neighborhood Search Procedure:

    • ① A trial solution

    • ② Generate and compare neighborhood solutions and find new trial solution

    • ③ Go to ② until no new trial solution is possible

  • Example (p. 4.7)

Example: p 4.5


3 neighborhood search1

3. Neighborhood Search

  • Neighborhood Search  Descent technique

  • Fig. Improvement of the objective function in Neighborhood Search

  • 단점: Trapped at local optima

Objective

Function

Seed Number


4 tabu search

4. Tabu-Search

  • A modified form of neighborhood search

    • Do not stop even when a local optimum is encountered

  • Move: Change from one schedule (solution, seed) to another

    • Neighborhood definition

    • Search the candidate

  • Tabu move: A move back to the previous seed (schedule, solution)

  • Tabu-list: A list of tabu moves

    • Fixed number of entries (5~9)

    • To avoid cycling

  • Mutations:

    • Every time a move is made thru a mutation in the current schedule, the reverse mutation is entered at the top of the Tabu-list.

  • Basic difference: Mechanism used for approving candidate moves

    • Simulated Annealing: Probabilistic

    • Tabu Search: Deterministic by a Tabu-List


4 tabu search1

4. Tabu-Search

  • Algorithm(Tabu-Search)

    • Step 1:

    • Step 2:

    • Step 3:

      • Increment k by 1

      • If k=K, then stop; otherwise, go to Step 2.


4 tabu search2

4. Tabu-Search

  • (Example)

    • Single machine sum of the weighted tardiness problem( )

    • Neighborhood: Adjacent Pair-wise Interchange

    • Tabu-list: a list of pairs of jobs (j, k) that swapped within the last two moves and cannot be swapped again.

    • Initial Tabu-list; empty


4 tabu search3

4. Tabu-Search

  • Iteration 1:

    • Neighborhood: 3 schedules

      • (1,2,4,3): 480

      • (2,4,1,3): 436

      • (2,1,3,4): 652

    • Selection of the best non-Tabu sequence

    • Tabu-list: {(1,4)}

  • Iteration 2:

    • Neighborhood: 3 schedules

      • (4,2,1,3): 460

      • (2,1,4,3): 500  Tabu

      • (2,4,3,1): 608

    • The First move results in a schedule worse than the best schedule so far. Nevertheless

    • Tabu-list: {(2,4), (1,4)}


4 tabu search4

4. Tabu-Search

  • Iteration 3:

    • Neighborhood: 3 schedules

      • (2,4,1,3): 436  Tabu

      • (4,1,2,3): 440

      • (4,2,3,1): 632

    • Tabu-list:{(1,2),(2,4)} ( max. length of Tabu-list=2 in this case)

  • Iteration 4:

    • Neighborhood: 3 schedules

      • (1,4,2,3): 408

      • (4,2,1,3): 460  Tabu

      • (4,1,3,2): 586

    • Tabu-list:{(1,4),(1,2)}

  • Actually, S5 is a global optimum, but tabu-search, being unaware of this fact, continues.

  • Stop only when K reaches.


5 simulated annealing

5. Simulated Annealing

  • (Definition)

    • First developed as a simulation model for describing the physical annealing process for condensed matter.


5 simulated annealing1

5. Simulated Annealing

  • Algorithm(Simulated Annealing)

    • Step 1:

    • Step 2:

    • Step 3:

    • Effectiveness depends on

      • The design of the neighborhood

      • Search policy: Randomly or in an Organized way.


6 genetic algorithms

6. Genetic algorithms

  • More general and abstract

    • Individuals (Chromosomes): Sequences or schedules

    • Population: Constant size

    • Fitness: f (Obj. fn.)

    • Generation: Iteration

      • Consists of surviving individuals of the previous generation and new solutions(Children).

      • Children generation  Reproduction and mutation of individuals that are  a part of the previous generation(Parents).

    • Chromosome consists of sub-chromosomes containing the information regarding job sequence.

    • Mutation in a parent chromosome  Adjacent pair-wise interchange

    • Fittest individual reproduces and the least fit dies.


6 genetic algorithms1

6. Genetic algorithms

  • Algorithm(Genetic Algorithms)

    • Step 1:

    • Step 2:

    • Step 3:

      • Increment k by 1

      • If k=K, then stop; otherwise, go to Step 2.


6 genetic algorithms2

6. Genetic algorithms

  • (Example) The same data with previous Tabu-Search example

    • 1st Generation

      • 3 individuals are selected at random

      • 4,1,3,2 reproduces 4,3,1,2 with random swap and 3,4,1,2 dies.

    • 2nd Generation

      • 4,1,3,2 reproduces 1,4,3,2 with random swap and 4,3,1,2 dies.

    • 3rd Generation

      • 1,4,3,2 reproduces 1,4,2,3 with random swap and 2,1,3,4 dies.

    • 4th Generation


7 filtered beam search

7. Filtered Beam Search

  • Adaptation of B&B

  • Only the most promising nodes at level k are selected as nodes to branch from.

    • Beam width of the search: # of nodes retained

    • Filter width: # of nodes selected for a thorough search

    • Crude prediction  Thorough evaluation

  • (Crude prediction)

    • Contribution of the partial schedule to the objective

    • Due-date tightness or other statistic of the jobs remaining to be scheduled

    •   The nodes are compared and overall assessment are made.

  • (Thorough evaluation)

    • All the jobs not yet scheduled are scheduled using a composite dispatching rule

    • Indication of promise of the node

      •   Nodes may be filtered out

      •   Retained nodes may be analyzed more thoroughly  by having all its remaining jobs scheduled with the composite dispatching rule

    • The value of this schedule's objective - upper bound on the best schedule among the offspring of that node.


7 filtered beam search1

7. Filtered Beam Search

  • (Example) The same data with previous Tabu-Search example

    • Beam width =2

    • No filtering mechanism

    • Prediction: ATC(Apparent tardiness cost: Combination of WSPT and MS) rule

      • Highest ranking index first

      • Due-date tightness factor

      • Due-date range factor

      • Look-ahead parameter: k =5


7 filtered beam search2

7. Filtered Beam Search

  • (Level 1)

    • (1, *, *, *)  1,4,2,3 : 408 by ATC rule - Retained

    • (2, *, *, *)  2,4,1,3 : 436 by ATC rule - Retained

    • (3, *, *, *)  3,4,1,2 : 814 by ATC rule

    • (4, *, *, *)  4,1,2,3 : 440 by ATC rule

  • (Level 2)

    • From (1, *, *, *)

      • (1, 2, *, *)

      • (1, 3, *, *)

      • (1, 4, *, *) - Retained

    • From (2, *, *, *)

      • (2, 1, *, *)

      • (2, 3, *, *)

      • (2, 4, *, *) - Retained

  • (Level 3)

    • (1, 4, 2, 3) - Best( Optimal )

    • (1, 4, 3, 2)

    • (2, 4, 1, 3)

    • (2, 4, 3, 1)


8 random sampling

8. Random Sampling

  • K samples 일 경우, 총 가능해의 수 =

  • (Terminology) Biased Random Sampling

    • At each stage, each job has unequal chance to be selected.

    • (e.g.) higher selection probability for first job in EDD sequence or SPT sequence.⇒ 모두 combinatorial problem이기 때문에 마지막 수단으로 등장한 것임.


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