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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 2

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Machine 3

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Job 1

Job 2

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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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1,2

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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

- Schedule generation using network model

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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

- Global left-shift

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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

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 NN -- 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* = minjSO(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, ...)

- Generating only nondelay schedule

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

- Identify the next bottleneck machine from X'

- Difficulties
- HBT problem is NP-hard
- Nevertheless, there are heuristic procedures, such as Longest Tail (LT) procedure

- HBT problem is NP-hard
- 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

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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 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

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