The job shop problem
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The Job Shop Problem. Chapter 11 Elements of Sequencing and Scheduling by Kenneth R. Baker Byung-Hyun Ha. Outline. Introduction Types of schedules Disjunctive programming Schedule generation Shifting bottleneck procedure. Introduction. Job shop model

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The Job Shop Problem

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The job shop problem

The Job Shop Problem

Chapter 11

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha


Outline

Outline

  • Introduction

  • Types of schedules

  • Disjunctive programming

  • Schedule generation

  • Shifting bottleneck procedure


Introduction

Introduction

  • Job shop model

    • Each job has operations with precedence constraint

    • Each operation should be done by a specific machine

  • Representation of a job

    • (i, j, k) -- operation j of job i requires machine k

  • Example

    • 4 jobs with 3 operations and 3 machines

Processing time

Machine assignment


Introduction1

Introduction

  • Two views of a feasible schedule

    • 4 jobs with 3 operations and 3 machines (cont’d)

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111

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331

Machine 1

Machine 2

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412

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Machine 3

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14

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Job 1

Job 2

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Job 3

313

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Job 4

412

423

431


Types of schedule

431

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412

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Types of Schedule

  • Semi-active schedules

    • Idle time is not helpful for regular measures

    • Local left-shift

      • Adjusting start time of operations earlier, without altering the sequence, lest superfluous idle time exists


Types of schedule1

Types of Schedule

  • Semi-active schedules (cont’d)

    • Dominant set w.r.t. regular measures

    • Number of semi-active schedules -- not more than (n!)m

    • Network model

      • Precedence constraints

      • Disjunctive arcs

        2 jobs and 2 machines, 3 jobs and 2 machines

Machine assignment

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1,1

1,2

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212

2,1

2,2


Types of schedule2

Types of Schedule

  • Semi-active schedules (cont’d)

    • Schedule generation using network model

      • Resolve disjunctive arcs

      • Schedule schedulable operation, all of whose predecessors are already scheduled

      • Resulting schedules are semi-active, while some are infeasible

      • Makespan is length of longest path

1,1

1,2

1,1

1,2

1,1

1,2

2,1

2,2

2,1

2,2

1,1

1,2

1,1

1,2

2,1

2,2

2,1

2,2

2,1

2,2


Types of schedule3

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Types of Schedule

  • Active schedules

    • Global left-shift

      • Altering sequence and begin some operation earlier, without delaying any other operations

      • Subset of semi-active schedules and dominant w.r.t. regular measures


Types of schedule4

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Types of Schedule

  • Nondelay schedules

    • No machine is kept idle at a time when it could begin processing some operations

    • No guarantee that nondelay schedules contain an optimum


Types of schedule5

Types of Schedule

  • Summary

nondelay

schedules

semi-active

schedules

active

schedules

optimum?


Disjunctive programming

Disjunctive Programming

  • Notation

    • M -- set of machine

    • (i, j) -- operation of job j on machine i

    • pij -- processing time of operation (i, j)

    • N -- set of all the operations

    • A  NN -- set of precedence constrains of the jobs.

    • yij -- starting time of operations (i, j)

  • Objective

    • min. Cmax

  • Constraints

    • yij + pij ykj  ((i, j), (k, j))A

    • yij + pij Cmax (i, j)N

    • yij + pij yilor yil + pil yij (i, j) and (i, l) i M

    • yij 0 (i, j)N


Schedule generation

Schedule Generation

  • Algorithm 1 -- Active Schedule Generation

    1.Let k = 0 and begin with PS(k) as the null partial schedule. Initially, SO(k) includes all operations with no predecessors.

    2.Determine f* = minjSO(k) {fj} and the machine m* on which f* could be realized.

    3.For each operation j  SO(k) that requires machine m* and for whichsj  f*, create a new partial schedule in which operation j is added to PS(k) and started at time sj.

    4.For each new partial schedule PS(k + 1) created in Step 3, update the data set as follows:

    (a) Remove operation j from SO(k).

    (b) Form SO(k + 1) by adding the direct successor of j to SO(k).

    (c) Increment k by one.

    5.Return to Step 2 for each PS(k + 1) created in Step 3, and continue in this manner until all active schedules have been generated.

sj -- the earliest time at which operation j SO(k) could be started

fj -- the earliest time at which operation j SO(k) could be finished


Schedule generation1

Schedule Generation

  • Using Algorithm 1

    • Generating only nondelay schedule

      • Modifying Step 2 and 3

    • Branch and bound

      • Employing lower bound

      • Not much practical (c.f., shifting bottleneck procedure)

    • Dispatching

      • Modifying Step 3 and employing priority rules (e.g. SPT, FCFS, MWKR, ...)


Shifting bottleneck procedure as heuristic

Shifting Bottleneck Procedure as Heuristic

  • Overall procedure

    • X -- set of machines already scheduled, X' -- complement of X

    • Repeat the following steps

      • Identify the next bottleneck machine from X'

        • Solving HBT (head-body-tail) problems for machines in X' using precedence constraints and the confirmed schedule

        • Select the critical machine, i.e., the machine with maximum makespan

      • Schedule all of the operations of the machine by using the result of HBT solution

  • Difficulties

    • HBT problem is NP-hard

      • Nevertheless, there are heuristic procedures, such as Longest Tail (LT) procedure

  • Shifting bottleneck procedure for optimization

    • Most effective optimization algorithm for job shop problems


Shifting bottleneck procedure as heuristic1

Shifting Bottleneck Procedure as Heuristic

  • Example

    • Problem

    • Iteration 1

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Job 1

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Job 2

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Job 3

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Job 4

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Machine 1

Machine 2

Machine 3

Solution 1-2-3-4 (12)

Solution 2-4-3-1 (11)

Solution 3-4-2-1 (12)


Shifting bottleneck procedure as heuristic2

Shifting Bottleneck Procedure as Heuristic

  • Example (cont’d)

    • Problem

    • Iteration 2

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Job 1

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Job 2

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Job 3

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Job 4

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Machine 1

Machine 2

Solution 1-2-3-4 (14)

Solution 2-4-3-1 (13)


Shifting bottleneck procedure as heuristic3

Shifting Bottleneck Procedure as Heuristic

  • Example (cont’d)

    • Problem

    • Iteration 3

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Job 1

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Job 2

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Job 3

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Job 4

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Machine 2

Solution 2-4-3-1 (13)


Shifting bottleneck procedure as heuristic4

Shifting Bottleneck Procedure as Heuristic

  • Example (cont’d)

    • Results

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Job 1

Job 2

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111

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Job 3

313

322

331

Job 4

412

423

431

212

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313

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331

431

Machine 1

412

423

431

Machine 2

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412

322

122

Machine 3

313

423

233

133

14


Summary

Summary

  • Job shop model

  • Mathematical programming

  • Active schedule generation

  • Shifting bottleneck procedure

  • Anyway, optimal?

    • M = 13


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