The Job Shop Problem

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# The Job Shop Problem - PowerPoint PPT Presentation

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

Chapter 11

Elements of Sequencing and Schedulingby Kenneth R. Baker

Byung-Hyun Ha

Outline
• Introduction
• Types of schedules
• Disjunctive programming
• Schedule generation
• Shifting bottleneck procedure
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

Introduction
• Two views of a feasible schedule
• 4 jobs with 3 operations and 3 machines (cont’d)

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Machine 1

Machine 2

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Machine 3

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Job 3

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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 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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2,1

2,2

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

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1,2

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1,2

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2,2

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2,2

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1,2

2,1

2,2

2,1

2,2

2,1

2,2

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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

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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 Schedule
• Summary

nondelay

schedules

semi-active

schedules

active

schedules

optimum?

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
• 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 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
• 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 Heuristic
• Example
• Problem
• Iteration 1

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Machine 1

Machine 2

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Solution 1-2-3-4 (12)

Solution 2-4-3-1 (11)

Solution 3-4-2-1 (12)

Shifting Bottleneck Procedure as Heuristic
• Example (cont’d)
• Problem
• Iteration 2

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Machine 1

Machine 2

Solution 1-2-3-4 (14)

Solution 2-4-3-1 (13)

Shifting Bottleneck Procedure as Heuristic
• Example (cont’d)
• Problem
• Iteration 3

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Machine 2

Solution 2-4-3-1 (13)

Shifting Bottleneck Procedure as Heuristic
• Example (cont’d)
• Results

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Machine 1

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Machine 2

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Machine 3

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Summary
• Job shop model
• Mathematical programming
• Active schedule generation
• Shifting bottleneck procedure
• Anyway, optimal?
• M = 13