1. Topic 15
Job Shop Scheduling

2. Job Shop Scheduling Have m machines and n jobs
Each job visits some or all of the machines
Customer order of small batches
Wafer fabrication in semiconductor industry
Hospital
Very difficult to solve
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3. Job Shop Example Constraints Job follows a specific route
One job at a time on each machine
Machine 1
(1,1)
(1,2)
(1,3)
Machine 2
(2,3)
(2,1)
(2,2)
Machine 3
(3,1)
(3,3)
Machine 4
(4,3)
(4,2)
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4. Graph Representation
Each job follows a specific route through the job shop ...
(1,1)
(2,1)
(3,1)
Sink
Source
(1,2)
(2,2)
(4,2)
(2,3)
(1,3)
(4,3)
(3,3)
(Conjuctive arcs)
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5. Graph Representation Machine constraints must also be satisfied ...
(1,1)
(2,1)
(3,1)
(1,2)
(2,2)
(4,2)
(2,3)
(1,3)
(4,3)
(3,3)
Sink
Source
(Disjunctive arcs)
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6. Solving the Problem Select one of each pair of disjunctive arcs
The longest path in this graph G(D) determines the makespan
(1,1)
(2,1)
(3,1)
Sink
Source
(1,2)
(2,2)
(4,2)
(2,3)
(1,3)
(4,3)
(3,3)
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7. Feasibility of the Schedule
Are all selections feasible?
(1,1)
(2,1)
(3,1)
Sink
Source
(1,2)
(2,2)
(4,2)
(2,3)
(1,3)
(4,3)
(3,3)
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8. Disjunctive Programming
Minimize
Subject to
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9. Solution Methods Exact solution Dispatching rules (16+)
Branch and Bound
20 machines and 20 jobs
Dispatching rules (16+)
Shifting Bottleneck
Search heuristics
Tabu, SA, GA, etc.
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10. Topic 16
Types of Schedules

11. Definitions
A schedule is nondelay if no machine is idled when there is an operation available
A schedule is called active if no operation can be completed earlier by altering the sequence on machines and not delaying other operations
For “regular” objectives the optimal schedule is always active but not necessarily nondelay
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12. Schedule Space Semiactive Nondelay Active All Schedules Optimum
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13. Nonactive Schedule Machine 1 (1,1) (2,1) Machine 2 (2,3) (2,2) (2,1)
(3,2)
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14. Active Schedule, not Nondelay
Machine 1
(1,1)
Delay (Operation (2,1) does not fit)
Machine 2
(2,3)
(2,2)
(2,1)
Machine 3
(3,2)
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15. Nondelay schedule Machine 1 (1,1) Machine 2 (2,3) (2,1) (2,2)
(3,2)
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16. Branch & Bound for Job Shops
Topic 17
Branch & Bound for
Job Shops

17. Branch and Bound Operation (i,j) with duration pij Minimize makespan
Branch by generating all active schedules
Notation
Let W denote operations whose predecessors have been scheduled
Let rij be the earliest possible starting time of (i,j) in W.
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18. Generating Active Schedules
Step 1. (Initialize)
Let W contain the first operation of each job
Let rij = 0 for all (i,j)W.
Step 2. (Machine selection)
Compute the current partial schedule
and let i* denote the machine where minimum achieved
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19. Generating Active Schedules
Step 3. (Branching)
Let W’ denote all operations on machine i such that
For each operation in W’ consider a partial schedule with that operation next on i*
For each partial schedule, delete operation from W and include immediate follower in W. Go back to Step 2.
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20. Branching Tree
Selection of (i*,j)
Selection of (i*,l)
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21. Example (1,1) 10 (2,1) 8 (3,1) 4 (2,2) 8 (1,2) 3 (4,2) 5 (3,2) 6 3
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3
(1,3) 4
(2,3) 7
(4,3)
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22. Partitioning Tree
Level 1
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23. Level 1: select (1,1) (1,1) 10 (2,1) 8 (3,1) 4 10 10 (2,2) 8 (1,2) 310
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3
(1,3) 4
(2,3) 7
(4,3)
Disjunctive Arcs
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24. Level 1: Select (1,3) (1,1) 10 (2,1) 8 (3,1) 4 4 (2,2) 8 (1,2) 3 (4,2)4
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 4 3
(1,3) 4
(2,3) 7
(4,3)
Disjunctive Arcs
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25. Branching Tree No disjunctive arcs (1,1) scheduled first
on machine 1
LB = 24
(1,3) scheduled first
on machine 1
LB = 26
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26. Branching Tree
Level 2:
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27. Level 2: Select (2,2) (1,1) 10 (2,1) 8 (3,1) 4 10 10 (2,2) 8 (1,2) 310
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3
(1,3) 4
(2,3) 7
(4,3)
Disjunctive Arcs
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28. Branching Tree No disjunctive arcs (1,1) scheduled first on machine 1
LB = 24
(1,3) scheduled first
on machine 1
LB = 26
(1,1) first on M1 and
(2,2) first on M2
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29. Lower Bounds Lower bounds The length of the critical path in G(D’)
Quick but not very tight
Linear programming relaxation
A maximum lateness problem (see book)
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30. The Shifting Bottleneck Heuristic
Topic 18
The Shifting Bottleneck Heuristic

31. Shifting Bottleneck Minimize makespan in a job shop
Let M denote the set of machines
Let M0  M be machines for which disjunctive arcs have been selected
Basic idea:
Select a machine in M - M0 to be included in M0
Sequence the operations on this machine
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32. Example
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33. Iteration 1 (1,1) 10 (2,1) 8 (3,1) 4 (2,2) 8 (1,2) 3 (4,2) 5 (3,2) 6 3
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3
(1,3) 4
(2,3) 7
(4,3)
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34. Selecting a Machine
Set up a nonpreemptive single machine maximum lateness
problem for Machine 1:
Optimum sequence is 1,2,3 with Lmax(1)=5
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35. Selecting a Machine
Set up a nonpreemptive single machine maximum lateness
problem for Machine 2:
Optimum sequence is 2,3,1 with Lmax(2)=5
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36. Selecting a Machine Similarly,
 Either Machine 1 or Machine 2 is the bottleneck
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37. Iteration 2 (1,1) 10 (2,1) 8 (3,1) 4 10 (2,2) 8 (1,2) 3 (4,2) 5 (3,2)10
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3 3
(1,3) 4
(2,3) 7
(4,3)
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38. Selecting a Machine
Set up a nonpreemptive single machine maximum lateness
problem for Machine 2:
Optimum sequence is 2,1,3 with Lmax(2)=1
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39. Selecting a Machine Similarly,
 Either Machine 2 or Machine 3 is the bottleneck
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40. Iteration 3 (1,1) 10 (2,1) 8 (3,1) 4 10 8 8 (2,2) 8 (1,2) 3 (4,2) 58 8
Source
(2,2) 8
(1,2) 3
(4,2) 5
(3,2) 6
Sink 3 3
(1,3) 4
(2,3) 7
(4,3)
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41. Discussion
Procedure continues until all the disjunctive arcs have been added
Extremely effective
Fast
Good solutions
‘Just a heuristic’
No guarantee of optimum
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42. Solving the Maximum Lateness Problem
(•,•,•)
(1,•,•)
(2,•,•)
(3,•,•)
(1,2,3)
(1,3,2)
(3,1,2)
(3,2,1)
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43. Discussion
The solution is actually a little bit more complicated than before
Precedence constraints because of other (already scheduled) machines
Delay precedence constraints
(see example in book)
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44. Discussion Shifting bottleneck can be applied generally Basic idea
Solve problem “one variable at a time”
Determine the “most important” variable
Find the best value of that variable
Move on to the “second most important” ….
Here we treat each machine as a variable
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45. Shifted Bottleneck for Total Weighted Tardiness Objective
Topic 19
Shifted Bottleneck for Total Weighted Tardiness Objective

46. Total Weighted Tardiness
We now apply a shifted bottleneck procedure to a job shop with total weighted tardiness objective
Need n sinks in disjunctive graph
Machines scheduled one at a time
Given current graph calculate completion time of each job
Some n due dates for each operation
Piecewise linear penalty function
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47. Cost Function for Operation (i,j)
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48. Machine Selection Machine criticality Solve a single machine problem
Piecewise linear cost function
May have delayed precedence constraints
Generalizes single-machine with n jobs, precedence constraints, and total weighted tardiness objective
ATC rule
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49. Generalized ATC Rule Earliest time machine can be used Ranks jobs
 good schedule
Scaling constant
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50. Criticality of Machines
Criticality = subproblem objective function
Simple
More effective ways, e.g.
Add disjunctive arcs for each machine
Calculate new completion times and
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51. Example (1,1) 5 (2,1) 10 (3,1) 4 5 (3,2) 4 (1,2) 5 (2,2) 6 (3,3) 5
Sink 5
Source
(3,2) 4
(1,2) 5
(2,2) 6
Sink
(3,3) 5
(2,3) 3
(1,3) 7
Sink
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52. Subproblem: Machine 1
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53. Subproblem: Machine 2
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54. Subproblem: Machine 3
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55. Subproblem Solutions Solve using dispatching rule
Use K=0.1
Have t = 4,
For machine 1 this results in
Schedule
first
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56. Solution to Subproblems
Schedule
first
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57. First Iteration (1,1) 5 (2,1) 10 (3,1) 4 5 5 (3,2) 4 (1,2) 5 (2,2) 6 5
Sink 5 5
Source
(3,2) 4
(1,2) 5
(2,2) 6
Sink 5
(3,3) 5
(2,3) 3
(1,3) 7
Sink
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58. Subproblem: Machine 2
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59. Subproblem: Machine 3
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60. Solution to Subproblems
Schedule
first
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61. Second Iteration (1,1) 5 (2,1) 10 (3,1) 4 5 5 10 (3,2) 4 (1,2) 5 (2,2)
Sink 5 5 10
Source
(3,2) 4
(1,2) 5
(2,2) 6
Sink 5 3
(3,3) 5
(2,3) 3
(1,3) 7
Sink
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62. Subproblem: Machine 3
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63. Third Iteration (1,1) 5 (2,1) 10 (3,1) 4 5 5 10 4 (3,2) 4 (1,2) 5
Sink 5 5 10 4
Source
(3,2) 4
(1,2) 5
(2,2) 6
Sink 5 5 3
(3,3) 5
(2,3) 3
(1,3) 7
Sink
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64. Final Schedule Machine 1 1,1 1,2 1,3 Machine 2 2,3 2,1 2,2 Machine 3
3,3
3,2
3,1
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65. Performance Objective function
Try finding a better schedule using LEKIN
(optimal value = 18)
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66. Random Search for Job Shop Scheduling
Topic 20
Random Search for Job Shop Scheduling

67. Random Search Methods
Popular to use genetic algorithms, simulated annealing, tabu search, etc.
Do not work too well
Problems defining the neighborhood
Do not exploit special structure
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68. Defining the Neighborhood
(1,1) 5
(2,1) 10
(3,1) 4
Sink 5 5 10 4
Source
(3,2) 4
(1,2) 5
(2,2) 6
Sink 5 5 3
(3,3) 5
(2,3) 3
(1,3) 7
Sink
Approximately nm neighbors!
Simply too inefficient
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69. Job Shop with Makespan Random search methods can be applied
Use ‘critical path’ neighborhood
Can eliminate many neighbors immediately
Specialized methods usually better
Random search = ‘giving up’ !
Traveling Salesman Problem (TSP)
Very well studied
Lin-Kernighan type heuristics (1970)
Order of 1000 times faster than random search
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70. Comparison of Methods Percentage over known optimum: Winner
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71. The Nested Partitions Method
Partitioning
by scheduling the bottleneck machine first
Random sampling
using randomized dispatching rules
Calculating the promising index
incorporating local improvement heuristic
Can incorporate any special structure!
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72. Special Case: Flow Shops
Topic 21
Special Case:
Flow Shops

73. A Flexible Flow Shop with Setups
Stage 1 Stage Stage Stage 4
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74. Applications Very common in applications: Paper mills Steel lines
Bottling lines
Food processing lines
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75. Classical Literature Exact solutions
Simple flow shop with makespan criterion
Two machine case (Johnson’s rule)
Realistic problems require heuristic approaches
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76. Objectives Multiple objectives usual Meet due dates
Maximize throughput
Minimize work-in-process (WIP)
Setting for job
j on Machine i
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77. Generating Schedules Identify bottlenecks
Compute time windows at bottleneck stage
Compute machine capacity at bottleneck
Schedule bottleneck stage
Schedule non-bottlenecks
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78. Identifying Bottlenecks
In practice usually known
Schedule downstream bottleneck first
Determining the bottleneck
loading
number of shifts
downtime due to setups
Bottleneck assumed fixed
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79. Identifying Time Window
Due date
Shipping day
Multiply remaining processing times with a safety factor
Release date
Status sj of job j
Release date if sj = l
Decreasing
function -
determined
empirically
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80. Computing Capacity Capacity of each machine at bottleneck Two cases:
Speed
Number of shifts
Setups
Two cases:
Identical machines
Non-identical machines
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81. Scheduling Bottleneck
Jobs selected one at a time
Setup time
Due date
Capacity
For example ATCS rule
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82. Schedule Remaining Jobs
Determined by sequence at bottleneck stage
Minor adjustments
Adjacent pairwise interchanges to reduce setup
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83. Summary: Job Shops Representation: graph w/disjoint arcs
Solution methods
Branch-and-bound
Shifted bottleneck heuristic
Beam search
Can incorporate bottleneck idea & dispatching rules
Random search methods
Special case: flow shops
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