Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP...
Transcript of Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP...
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Solving the Non-permutationFlow Shop Scheduling Problem
Alexander J. Benavides
[email protected]@ucsp.edu.pe
October, 2019
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FSSP Non-permutation Results
Today we’ll see...
How to solve very-difficult combinatorial-optimization problemsby using computers to model problems and produce solutions.
Case Study: Flow Shop Scheduling Problem (FSSP)
Methods: constructive heuristics, local search, meta-heuristics, ...
Thinking out of the box!!! ...
Benavides A.J., & Ritt M., (2016), Two simple and effective heuristics for minimizing themakespan in non-permutation flowshop scheduling problems. Comput. Oper. Res. 60,160–169.
Benavides A.J., & Ritt M., (2018), Fast heuristics for minimizing the makespan innon-permutation flow shops. Comput. Oper. Res. 100, 230–243.
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FSSP Non-permutation Results
OutlineFSSP, Introduction and concepts
FSSP definitionNEH heuristic and Taillard accelerationLocal search heuristics
Non-permutation FSSP, Motivations and proposed heuristicsPermutation FSSP vs. Non-permutation FSSPConstructing non-permutation schedulesConstructing non-permutation schedulesNew permutation representation for non-permutation schedules andnew constructive heuristic NEHBR
Local search heuristics for non-permutation FSSP
Results and RemarksNon-permutation FSSP with makespan (Benavides & Ritt, 2016)Non-permutation FSSP with makespan (Benavides & Ritt, 2018)Concluding Remarks
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)6× 6 instance of the FSSP
Jobs OperationsM1 M2 M3 M4 M5 M6
J1 3 6 3 3 4 3J2 4 3 5 3 5 2J3 6 5 2 2 2 4J4 4 5 2 2 5 5J5 2 2 5 6 3 5J6 2 3 5 5 3 3
A set of jobs J1, . . . , Jn must be processedon a set of machines M1, . . . ,Mm
with given processing times pij for each job Jj onmachine Mi
Objective function:min. Cmax = maxCj (makespan)
There are n! possible solutions
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)
Taillard (1993): 120 instancesn ∈ 20, 50, 100, 200, 500 jobs bym ∈ 5, 10, 20 machines.
Vallada, Ruiz, Framinan (2015)240 small instancesn ∈ 10, 20, 30, 40, 50, 60 jobs bym ∈ 5, 10, 15, 20 machines.
240 large instancesn ∈ 100, 200, 300, 400, 500, 600, 700, 800 jobs bym ∈ 20, 40, 60 machines.
6! 72010! 362880020! 2.43e+1850! 3.04e+64100! 9.33e+157200! 7.88e+374500! 1.22e+1134800! 7.71e+1976Grains of sand on Earth 7.5e+18Stars in the observable universe 2e+204 GHz = 4e+9 op/s = 4 op/s60 s * 60 m * 24 h * 365 d = 31536000operations per year: 1.26144e+17so 2.4e+18/1.2e+17 20 years.
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)
Taillard (1993): 120 instancesn ∈ 20, 50, 100, 200, 500 jobs bym ∈ 5, 10, 20 machines.
Vallada, Ruiz, Framinan (2015)240 small instancesn ∈ 10, 20, 30, 40, 50, 60 jobs bym ∈ 5, 10, 15, 20 machines.
240 large instancesn ∈ 100, 200, 300, 400, 500, 600, 700, 800 jobs bym ∈ 20, 40, 60 machines.
6! 72010! 362880020! 2.43e+1850! 3.04e+64100! 9.33e+157200! 7.88e+374500! 1.22e+1134800! 7.71e+1976Grains of sand on Earth 7.5e+18Stars in the observable universe 2e+204 GHz = 4e+9 op/s = 4 op/s60 s * 60 m * 24 h * 365 d = 31536000operations per year: 1.26144e+17so 2.4e+18/1.2e+17 20 years.
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)
Taillard (1993): 120 instancesn ∈ 20, 50, 100, 200, 500 jobs bym ∈ 5, 10, 20 machines.
Vallada, Ruiz, Framinan (2015)240 small instancesn ∈ 10, 20, 30, 40, 50, 60 jobs bym ∈ 5, 10, 15, 20 machines.
240 large instancesn ∈ 100, 200, 300, 400, 500, 600, 700, 800 jobs bym ∈ 20, 40, 60 machines.
6! 72010! 362880020! 2.43e+1850! 3.04e+64100! 9.33e+157200! 7.88e+374500! 1.22e+1134800! 7.71e+1976Grains of sand on Earth 7.5e+18Stars in the observable universe 2e+204 GHz = 4e+9 op/s = 4 op/s60 s * 60 m * 24 h * 365 d = 31536000operations per year: 1.26144e+17so 2.4e+18/1.2e+17 20 years.
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)
Taillard (1993): 120 instancesn ∈ 20, 50, 100, 200, 500 jobs bym ∈ 5, 10, 20 machines.
Vallada, Ruiz, Framinan (2015)240 small instancesn ∈ 10, 20, 30, 40, 50, 60 jobs bym ∈ 5, 10, 15, 20 machines.
240 large instancesn ∈ 100, 200, 300, 400, 500, 600, 700, 800 jobs bym ∈ 20, 40, 60 machines.
6! 72010! 362880020! 2.43e+1850! 3.04e+64100! 9.33e+157200! 7.88e+374500! 1.22e+1134800! 7.71e+1976Grains of sand on Earth 7.5e+18Stars in the observable universe 2e+204 GHz = 4e+9 op/s = 4 op/s60 s * 60 m * 24 h * 365 d = 31536000operations per year: 1.26144e+17so 2.4e+18/1.2e+17 20 years.
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FSSP Non-permutation Results
Flow Shop Scheduling Problem (FSSP)
Taillard (1993): 120 instancesn ∈ 20, 50, 100, 200, 500 jobs bym ∈ 5, 10, 20 machines.
Vallada, Ruiz, Framinan (2015)240 small instancesn ∈ 10, 20, 30, 40, 50, 60 jobs bym ∈ 5, 10, 15, 20 machines.
240 large instancesn ∈ 100, 200, 300, 400, 500, 600, 700, 800 jobs bym ∈ 20, 40, 60 machines.
6! 72010! 362880020! 2.43e+1850! 3.04e+64100! 9.33e+157200! 7.88e+374500! 1.22e+1134800! 7.71e+1976Grains of sand on Earth 7.5e+18Stars in the observable universe 2e+204 GHz = 4e+9 op/s = 4 op/s60 s * 60 m * 24 h * 365 d = 31536000operations per year: 1.26144e+17so 2.4e+18/1.2e+17 20 years.
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEH constructive heuristic6× 6 instance of the FSSP
Jobs OperationsM1 M2 M3 M4 M5 M6 Total
J1 3 6 3 3 4 3 22J2 4 3 5 3 5 2 22J3 6 5 2 2 2 4 21J4 4 5 2 2 5 5 23J5 2 2 5 6 3 5 23J6 2 3 5 5 3 3 21
First, determine insertion order:πo = (J4, J5, J1, J2, J3, J6)
The, insert one by oneat the best positionstarting with π = (J4)
1: function NEH Constructive Heuristic( )2: πo := (πo(1), . . . , πo(n)) from large to small3: π := (πo(1))4: for πo(j), j ∈ [2, n] do5: evaluate all the insertion positions of job πo(j) into π6: insert job πo(j) into π at the position which minimizes Cmax
7: end for8: return π9: end function
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEH constructive heuristicπo = (J4, J5, J1, J2, J3, J6) π = (J4) Next job: J5
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1: function NEH Constructive Heuristic( )2: πo := (πo(1), . . . , πo(n)) from large to small3: π := (πo(1))4: for πo(j), j ∈ [2, n] do5: evaluate all the insertion positions of job πo(j) into π6: insert job πo(j) into π at the position which minimizes Cmax
7: end for8: return π9: end function
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEH constructive heuristic
πo = (J4, J5, J1, J2, J3, J6) π = (J5, J4) Next job: J1
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEH constructive heuristic
And so on ... until all jobs are inserted: π = (J5, J4, J6, J2, J1, J3)
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Original NEH has a time complexity of O(n3m)
NEH inserts n jobs, evaluates O(n) insertion positions, (exactly n(n+ 1)/2− 1 evaluations)and each evaluation has a time complexity of O(nm)
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEHT heuristicwith Taillard (1990) acceleration technique for Cmax
Earliest completion times ei,j before insertion position remain unchangedAlso qi,j times after insertion position remain unchanged
ei,j
ll
ll
. . .. . .. . .. . .
. . .. . .. . .. . .
qi,j
Taillard defines:ei,j = max{ei,j−1, ei−1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with e0,j = 0 and ei,0 = 0qi,j = max{qi,j+1, qi+1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with qm+1,j = 0 and qi,k+1 = 0fi,j = max{fi−1,j , ei,j−1}+ pi,l, for i ∈ [m], j ∈ [|π|+ 1], with f0,j = 0
Mj = maxi∈[m]{fi,j + qi,j}, for j ∈ [|π|+ 1]
These calculations evaluate n insertion positions in time O(nm)
This reduces the time complexity of NEHT from O(n3m) to O(n2m)
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEHT heuristicwith Taillard (1990) acceleration technique for Cmax
Earliest completion times ei,j before insertion position remain unchangedAlso qi,j times after insertion position remain unchanged
ei,j
ll
ll
. . .. . .. . .. . .
. . .. . .. . .. . .
qi,j
Taillard defines:ei,j = max{ei,j−1, ei−1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with e0,j = 0 and ei,0 = 0qi,j = max{qi,j+1, qi+1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with qm+1,j = 0 and qi,k+1 = 0fi,j = max{fi−1,j , ei,j−1}+ pi,l, for i ∈ [m], j ∈ [|π|+ 1], with f0,j = 0
Mj = maxi∈[m]{fi,j + qi,j}, for j ∈ [|π|+ 1]
These calculations evaluate n insertion positions in time O(nm)
This reduces the time complexity of NEHT from O(n3m) to O(n2m)
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FSSP Non-permutation Results
Nawaz, Enscore & Ham (1983) NEHT heuristicwith Taillard (1990) acceleration technique for Cmax
Earliest completion times ei,j before insertion position remain unchangedAlso qi,j times after insertion position remain unchanged
ei,j
ll
ll
. . .. . .. . .. . .
. . .. . .. . .. . .
qi,jfi,j
Taillard defines:ei,j = max{ei,j−1, ei−1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with e0,j = 0 and ei,0 = 0qi,j = max{qi,j+1, qi+1,j}+ pi,π(j), for i ∈ [m], j ∈ [|π|], with qm+1,j = 0 and qi,k+1 = 0fi,j = max{fi−1,j , ei,j−1}+ pi,l, for i ∈ [m], j ∈ [|π|+ 1], with f0,j = 0
Mj = maxi∈[m]{fi,j + qi,j}, for j ∈ [|π|+ 1]
These calculations evaluate n insertion positions in time O(nm)
This reduces the time complexity of NEHT from O(n3m) to O(n2m)
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FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
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FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
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FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
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FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
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FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
![Page 22: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/22.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
![Page 23: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/23.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
![Page 24: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/24.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Taillard acc. O(n2m)
![Page 25: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/25.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Taillard acc. O(n2m)
![Page 26: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/26.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
part of:
Swapping arbitrary pairs of jobs ((n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Taillard acc. O(n2m)
![Page 27: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/27.jpg)
FSSP Non-permutation Results
Neighborhoods for local searchfor permutation schedules
Swapping adjacent jobs (n− 1 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 part of:Swapping arbitrary pairs of jobs (
(n2
)neighbors)
π = ( )J1, J2, J3, J4, J5, J6
Reinserting a job into another position ( (n− 1)2 neighbors)
π = ( )J1, J2, J3, J4, J5, J6 Taillard acc. O(n2m)
![Page 28: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/28.jpg)
FSSP Non-permutation Results
Iterated Greedy by Ruiz & Stutzle (2007)by Ruiz & Stutzle (2007)
![Page 29: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/29.jpg)
FSSP Non-permutation Results
OutlineFSSP, Introduction and concepts
FSSP definitionNEH heuristic and Taillard accelerationLocal search heuristics
Non-permutation FSSP, Motivations and proposed heuristicsPermutation FSSP vs. Non-permutation FSSPConstructing non-permutation schedulesConstructing non-permutation schedulesNew permutation representation for non-permutation schedules andnew constructive heuristic NEHBR
Local search heuristics for non-permutation FSSP
Results and RemarksNon-permutation FSSP with makespan (Benavides & Ritt, 2016)Non-permutation FSSP with makespan (Benavides & Ritt, 2018)Concluding Remarks
![Page 30: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/30.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Practically are the same problem!
All machines have thesame processing order
Simplified problem:
• Possible solutions: n!disregarding thenumber of machines
• 99% of the literature
Excludes better (optimal)non-permutation schedules
Some machines may havedifferent processing orders
Harder problem
• Possible solutions:n!(m−2) for min. Cmax
n!(m−1) for min. Csum
• 1% of the literature
How to solve this harder problemwith the same effort?
![Page 31: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/31.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Permutation
(J1, J2) Cmax = 11
M1
M2
M3
M4
118 t
(J2, J1) Cmax = 11
M1
M2
M3
M4
118 tFSSP 2× 4 instance.
Jobs OperationsM1 M2 M3 M4
J1 1 3 3 1J2 3 1 1 3
![Page 32: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/32.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Permutation
(J1, J2) Cmax = 11
M1
M2
M3
M4
118 t
(J2, J1) Cmax = 11
M1
M2
M3
M4
118 t
Non-permutation
(J1, J2)
(J2, J1)Cmax = 10
M1
M2
M3
M4
9 10 t
![Page 33: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/33.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Permutation
(J1, J2) Csum = 19M1
M2
M3
9 10 t
(J2, J1) Csum = 19M1
M2
M3
6 13 t
Non-permutation
(J1, J2)
(J2, J1)Csum = 18
M1
M2
M3
7 11 t
![Page 34: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/34.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Practically are the same problem!
All machines have thesame processing order
Simplified problem:
• Possible solutions: n!disregarding thenumber of machines
• 99% of the literature
Excludes better (optimal)non-permutation schedules
Some machines may havedifferent processing orders
Harder problem
• Possible solutions:n!(m−2) for min. Cmax
n!(m−1) for min. Csum
• 1% of the literature
How to solve this harder problemwith the same effort?
![Page 35: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/35.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Practically are the same problem!
All machines have thesame processing order
Simplified problem:
• Possible solutions: n!disregarding thenumber of machines
• 99% of the literature
Excludes better (optimal)non-permutation schedules
Some machines may havedifferent processing orders
Harder problem
• Possible solutions:n!(m−2) for min. Cmax
n!(m−1) for min. Csum
• 1% of the literature
How to solve this harder problemwith the same effort?
![Page 36: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/36.jpg)
FSSP Non-permutation Results
Permutation FSSP vs. Non-permutation FSSP
Practically are the same problem!
All machines have thesame processing order
Simplified problem:
• Possible solutions: n!disregarding thenumber of machines
• 99% of the literature
Excludes better (optimal)non-permutation schedules
Some machines may havedifferent processing orders
Harder problem
• Possible solutions:n!(m−2) for min. Cmax
n!(m−1) for min. Csum
• 1% of the literature
How to solve this harder problemwith the same effort?
![Page 37: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/37.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSP
Optimal schedules have small differences inthe processing order of subsequent machines.
(J1, J2)
(J2, J1)Cmax = 10
M1
M2
M3
M4
9 10 t
(J1, J2)
(J2, J1)Csum = 18
M1
M2
M3
7 11 t
![Page 38: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/38.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPwith anticipation and delay after an intermediate machine
Original NEH inserts jobs only into straight positions. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk
k
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
k′
k′
k′
k′
k′
We also insert jobs with delay after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
k+1k+1
k+1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk+1
k+1
and with anticipation after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kkk−1
k−1k−1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk−1
k−1
![Page 39: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/39.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPwith anticipation and delay after an intermediate machine
Original NEH inserts jobs only into straight positions. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk
k
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
k′
k′
k′
k′
k′
We also insert jobs with delay after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
k+1k+1
k+1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk+1
k+1
and with anticipation after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kkk−1
k−1k−1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk−1
k−1
![Page 40: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/40.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPwith anticipation and delay after an intermediate machine
Original NEH inserts jobs only into straight positions. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk
k
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
k′
k′
k′
k′
k′
We also insert jobs with delay after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
k+1k+1
k+1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk+1
k+1
and with anticipation after an intermediate machine. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kkk−1
k−1k−1
. . .. . .. . .. . .. . .
. . .. . .
. . .. . .
. . .
kk
kk−1
k−1
![Page 41: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/41.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPNEH-like heuristics for non-permutation FSSP
1: function NEH like Constructive Heuristic( )2: πo := (πo(1), . . . , πo(n)) from large to small3: π := (πo(1))4: for πo(j), j ∈ [2, n] do5: for all insertion positions k ∈ [j] do6: evaluate insertion of πo(j) at k with anticipation after Mi with i ∈ [2,m− 2]7: evaluate insertion of πo(j) at k with delay after Mi with i ∈ [2,m− 2]8: evaluate insertion of Jj at k straight9: end for10: Apply the best insertion of job ρo(j) into π which minimizes Cmax
11: end for12: return π13: end function
The number of insertion possibilities goes from O(n) to O(nm)
Inserts n jobs in time O(n3m2) for Csum (cannot use Taillard acceleration)
![Page 42: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/42.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPNEH-like heuristics for non-permutation FSSP
1: function NEH like Constructive Heuristic( )2: πo := (πo(1), . . . , πo(n)) from large to small3: π := (πo(1))4: for πo(j), j ∈ [2, n] do5: for all insertion positions k ∈ [j] do6: evaluate insertion of πo(j) at k with anticipation after Mi with i ∈ [2,m− 2]7: evaluate insertion of πo(j) at k with delay after Mi with i ∈ [2,m− 2]8: evaluate insertion of Jj at k straight9: end for10: Apply the best insertion of job ρo(j) into π which minimizes Cmax
11: end for12: return π13: end function
The number of insertion possibilities goes from O(n) to O(nm)
Inserts n jobs in time O(n3m2) for Csum (cannot use Taillard acceleration)
![Page 43: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/43.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPNon-permutation insertions with Taillard acceleration
Taillard acceleration technique needs adjustments because...
Non-permutation insertions produces invalid ei,j and qi,jwhen used with m-permutation representation, e.g.:π1= (4, 3)π2= (4, 3)π3= (4, 3)π4= (3, 4)π5= (3, 4)
⇒π1= (4, 3, 1)π2= (4, 3, 1)π3= (4, 1, 3)π4= (3, 1, 4)π5= (3, 1, 4)
⇒π1= (4, 3, 2, 1)π2= (4, 3, 2, 1)π3= (4, 1, 3, 2)π4= (3, 1, 4, 2)π5= (3, 1, 4, 2)
Two possible alternative solutions:
Update invalid ei,j and qi,j efficientlyNFS constructive heuristic O(n2m2W ) (Benavides & Ritt, 2016)
Propose a new representation that supports Taillard accelerationNEHBR constructive heuristic O(n2m) (same as NEHT , Benavides & Ritt, 2018)
![Page 44: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/44.jpg)
FSSP Non-permutation Results
Job insertion for non-permutation FSSPNon-permutation insertions with Taillard acceleration
Taillard acceleration technique needs adjustments because...
Non-permutation insertions produces invalid ei,j and qi,jwhen used with m-permutation representation, e.g.:π1= (4, 3)π2= (4, 3)π3= (4, 3)π4= (3, 4)π5= (3, 4)
⇒π1= (4, 3, 1)π2= (4, 3, 1)π3= (4, 1, 3)π4= (3, 1, 4)π5= (3, 1, 4)
⇒π1= (4, 3, 2, 1)π2= (4, 3, 2, 1)π3= (4, 1, 3, 2)π4= (3, 1, 4, 2)π5= (3, 1, 4, 2)
Two possible alternative solutions:
Update invalid ei,j and qi,j efficientlyNFS constructive heuristic O(n2m2W ) (Benavides & Ritt, 2016)
Propose a new representation that supports Taillard accelerationNEHBR constructive heuristic O(n2m) (same as NEHT , Benavides & Ritt, 2018)
![Page 45: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/45.jpg)
FSSP Non-permutation Results
New representation for non-permutation schedules:Permutation of pseudo-jobs
Pseudo-job Jj [i, i′]: operations of job Jj from Mi to Mi′ , others are missing
M1
M2
M3
M4
M5
M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
π = ( J5, J4, J6, J2, J1, J3 )
π′ = ( J5, J6[1, 3], J4, J6[4, 6], J2, J1[1, 4], J3, J1[5, 6] )
Times ei,j and qi,j are valid, but some operations are missing
![Page 46: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/46.jpg)
FSSP Non-permutation Results
Taillard acceleration redefinition: straight insertion
ei,j =
{max{ei,j−1, ei−1,j}+ pi,π(j), if ∃ pi,π(j)ei,j−1, if @ pi,π(j)
for i ∈ [m], j ∈ [|π|],
with e0,j = 0 and ei,0 = 0
qi,j =
{max{qi,j+1, qi+1,j}+ pi,π(j), if ∃ pi,π(j)qi,j+1, if @ pi,π(j)
for i ∈ [m], j ∈ [|π|],
with qm+1,j = 0 and qi,k+1 = 0
fi,j = max{fi−1,j , ei,j−1}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with f0,j = 0
MCj = maxi∈[m]{fi,j + qi,j}, for j ∈ [|π|+ 1]
![Page 47: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/47.jpg)
FSSP Non-permutation Results
Taillard acceleration extension: insertion with anticipationgi,j = max{gi+1,j , qi,j}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with gm+1,j = 0
MC′i,j =
max{fi,j+1 + gi+1,j ,
maxi′∈[i]{gi′,j+1 + ei′,j},
maxi′′∈[i+1,m]
{fi′′,j + qi′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ei,j
lll
l
. . .. . .. . .. . .
. . .. . .. . .. . .
e+ g
f + q
f+ g
gi,j qi,jfi,j
![Page 48: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/48.jpg)
FSSP Non-permutation Results
Taillard acceleration extension: insertion with anticipationgi,j = max{gi+1,j , qi,j}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with gm+1,j = 0
MC′i,j =
max{fi,j+1 + gi+1,j ,
maxi′∈[i]{gi′,j+1 + ei′,j},
maxi′′∈[i+1,m]
{fi′′,j + qi′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ei,j
lll
l
. . .. . .. . .. . .
. . .. . .. . .. . .
e+ g
f + q
f+ g
gi,j qi,jfi,j
![Page 49: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/49.jpg)
FSSP Non-permutation Results
Taillard acceleration extension: insertion with anticipationgi,j = max{gi+1,j , qi,j}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with gm+1,j = 0
MC′i,j =
max{fi,j+1 + gi+1,j ,
maxi′∈[i]{gi′,j+1 + ei′,j},
maxi′′∈[i+1,m]
{fi′′,j + qi′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ei,j
lll
l
. . .. . .. . .. . .
. . .. . .. . .. . .
e+ g
f + q
f+ g
gi,j qi,jfi,j
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FSSP Non-permutation Results
Taillard acceleration extension: insertion with anticipationgi,j = max{gi+1,j , qi,j}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with gm+1,j = 0
MC′i,j =
max{fi,j+1 + gi+1,j ,
maxi′∈[i]{gi′,j+1 + ei′,j},
maxi′′∈[i+1,m]
{fi′′,j + qi′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ei,j
lll
l
. . .. . .. . .. . .
. . .. . .. . .. . .
e+ g
f + q
f+ g
gi,j qi,jfi,j
![Page 51: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/51.jpg)
FSSP Non-permutation Results
Taillard acceleration extension: insertion with anticipationgi,j = max{gi+1,j , qi,j}+ pi,πo(l), for i ∈ [m], j ∈ [|π|+ 1] with gm+1,j = 0
MC′i,j =
max{fi,j+1 + gi+1,j ,
maxi′∈[i]{gi′,j+1 + ei′,j},
maxi′′∈[i+1,m]
{fi′′,j + qi′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ei,j
lll
l
. . .. . .. . .. . .
. . .. . .. . .. . .
e+ g
f + q
f+ g
gi,j qi,jfi,j
![Page 52: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/52.jpg)
FSSP Non-permutation Results
Taillard acceleration extension: insertion with delaye′i,j =
{max{e′i−1,j , fi,j}+ pi,π(j), if ∃ pi,π(j)fi,j , if @ pi,π(j)
for i ∈ [m], j ∈ [|π|] with e′0,j = 0
q′i,j =
{max{q′i+1,j , gi,j+1}+ pi,π(j), if ∃ pi,π(j)gi,j+1, if @ pi,π(j)
for i ∈ [m], j ∈ [|π|] with q′m+1,j = 0
MC′′i,j =
max{e′i,j + q′i+1,j ,
maxi′∈[i]
{fi′,j + qi′,j},
maxi′′∈[i+1,m]
{gi′′,j+1 + ei′′,j}}, if ∃pi,π(j) ∧ ∃pi+1,π(j)
∞, if @pi,π(j) ∨ @pi+1,π(j)
for i ∈ [2,m− 2], j ∈ [|π|]
ll
ll
. . .. . .. . .. . .
. . .. . .. . .. . .
f + q
e′+q′
e+ g
![Page 53: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/53.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
has time complexity of O(n2m)
Besides calculating
MC: makespan for O(n) straight insertions (like NEH)
Calculations are triplicated to obtain:
MC ′: makespan for O(nm) insertions with anticipation
MC ′′: makespan for O(nm) insertions with delay
Calculations have time complexity of O(|π|m), n ≤ |π| ≤ 2n
NEHBR evaluates O(nm) insertion possibilities in time O(nm)
NEHBR has time complexity of O(n2m)Same time complexity but three times more expensive
than NEHT for permutation FSSP
![Page 54: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/54.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J4, J2, J1) Next job: J3
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
4
4
4
4
4
4
5
5
5
5
5
5
j MCj
1 442 413 404 405 39straight
j MC ′2,j MC ′
3,j MC ′4,j
1 46 43 412 41 40 403 43 40 404 40 39 38
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 46 46 462 42 41 413 42 42 424 44 44 44
delay
![Page 55: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/55.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J4, J2, J1[1, 4], J3, J1[5, 6]) with anticipation
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
4
4
4
4
4
4
5
5
5
5
5
5
j MCj
1 442 413 404 405 39straight
j MC ′2,j MC ′
3,j MC ′4,j
1 46 43 412 41 40 403 43 40 404 40 39 38
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 46 46 462 42 41 413 42 42 424 44 44 44
delay
![Page 56: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/56.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J4, J2, J1[1, 4], J3, J1[5, 6]) with anticipation
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
4
4
4
4
4
4
5
5
5
5
5
5
3
3
3
3
3
3
j MCj
1 442 413 404 405 39straight
j MC ′2,j MC ′
3,j MC ′4,j
1 46 43 412 41 40 403 43 40 404 40 39 38
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 46 46 462 42 41 413 42 42 424 44 44 44
delay
![Page 57: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/57.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J4, J2, J1[1, 4], J3, J1[5, 6]) Next job: J6
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
4
4
4
4
4
4
5
5
5
5
5
5
3
3
3
3
3
3
j MCj
1 432 423 414 435 466 487 44straight
j MC ′2,j MC ′
3,j MC ′4,j
1 45 47 452 46 46 463 44 46 464 47 47 465 51 51 516 48 48 48
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 45 48 462 42 40 443 46 46 454 49 48 455 50 47 476 44 44 48
delay
![Page 58: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/58.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J6[1, 3], J4, J6[4, 6], J2, J1[1, 4], J3, J1[5, 6]) with delay
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
j MCj
1 432 423 414 435 466 487 44straight
j MC ′2,j MC ′
3,j MC ′4,j
1 45 47 452 46 46 463 44 46 464 47 47 465 51 51 516 48 48 48
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 45 48 462 42 40 443 46 46 454 49 48 455 50 47 476 44 44 48
delay
![Page 59: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/59.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J6[1, 3], J4, J6[4, 6], J2, J1[1, 4], J3, J1[5, 6]) with delay
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
j MCj
1 432 423 414 435 466 487 44straight
j MC ′2,j MC ′
3,j MC ′4,j
1 45 47 452 46 46 463 44 46 464 47 47 465 51 51 516 48 48 48
anticipation
j MC ′′2,j MC ′′
3,j MC ′′4,j
1 45 48 462 42 40 443 46 46 454 49 48 455 50 47 476 44 44 48
delay
![Page 60: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/60.jpg)
FSSP Non-permutation Results
Constructive heuristic NEHBR
π = (J5, J6[1, 3], J4, J6[4, 6], J2, J1[1, 4], J3, J1[5, 6])
M1M2M3M4M5M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
M1
M2
M3
M4
M5
M6
0 5 10 15 20 25 30 35 40 t
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
NEH producesπ′ = (J5, J4, J6, J2, J1, J3)
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FSSP Non-permutation Results
Local search heuristics for non-permutation FSSP
M1
M2
M3
M4
0 5 10 15 17 t
Extended Neighbourhood of Nowicki & Smutnicki (1996)Used in (Benavides & Ritt, 2016)
Interchange the first two (or the last two) operations in a critical blockEvaluate the interchange only on critical machine Mi
Evaluate the interchange on machines M1, . . . ,M′i for all i′ ≥ i
Evaluate the interchange on machines M ′′i , . . . ,Mm for all i′′ ≤ iEvaluates O(nm) neighbours in time O(n2m2)
proposed before pseudo-jobs permutation representation
![Page 62: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/62.jpg)
FSSP Non-permutation Results
Local search heuristics for non-permutation FSSP
M1
M2
M3
M4
0 5 10 15 17 t
Extended Neighbourhood of Nowicki & Smutnicki (1996)Used in (Benavides & Ritt, 2016)
Interchange the first two (or the last two) operations in a critical blockEvaluate the interchange only on critical machine Mi
Evaluate the interchange on machines M1, . . . ,M′i for all i′ ≥ i
Evaluate the interchange on machines M ′′i , . . . ,Mm for all i′′ ≤ iEvaluates O(nm) neighbours in time O(n2m2)
proposed before pseudo-jobs permutation representation
![Page 63: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/63.jpg)
FSSP Non-permutation Results
Local search heuristics for non-permutation FSSPwith pseudo-jobs and acceleration
π = (. . . , Ja[1, 2], Jb, Ja[3, 4], . . . ). . .. . .. . .. . .
. . .. . .. . .. . .
e+ gf+ g
f + q
Non-permutation insertion local search π = ( )J1, J2, J3, J4, J5, J6
evaluates (n− 1)2(2m− 5) non-permutation neighbours in time O(n2m) same as the
insertion local search for (n− 1)2 permutation neighbours
New BRN local search π = ( )J1, J2, J3, J4, J5, J6
based on swapping adjacent jobs completely or partiallyevaluates (n− 1)(2m− 5) non-permutation neighbours in time O(nm)
![Page 64: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/64.jpg)
FSSP Non-permutation Results
Local search heuristics for non-permutation FSSPwith pseudo-jobs and acceleration
π = (. . . , Ja[1, 2], Jb, Ja[3, 4], . . . ). . .. . .. . .. . .
. . .. . .. . .. . .
e+ gf+ g
f + q
Non-permutation insertion local search π = ( )J1, J2, J3, J4, J5, J6
evaluates (n− 1)2(2m− 5) non-permutation neighbours in time O(n2m) same as the
insertion local search for (n− 1)2 permutation neighbours
New BRN local search π = ( )J1, J2, J3, J4, J5, J6
based on swapping adjacent jobs completely or partiallyevaluates (n− 1)(2m− 5) non-permutation neighbours in time O(nm)
![Page 65: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/65.jpg)
FSSP Non-permutation Results
BRN local search
First calculates ei,j and qi,j in a time of complexity O(nm)
Best-improvement
chooses the best in the adjacent job swap neighbourhood
Reduced-neighbourhood
(π(j), π(j + 1)) ∈ R ⇐⇒ ei,j + qi+1,j = Cmax(π) ∨ ei,j+1 + qi+1,j+1 = Cmax(π)
Either π(j) or π(j + 1) has critical operations on consecutive machines
Like Nowicki & Smutnicki but considering all the critical paths
Non-permutation
Calculates the makespan of swapping two consecutive jobs π(j), π(j + 1)MC: swap completelyMC ′: swap on the first machines (like insertion with anticipation)MC ′′: swap on the last machines (like insertion with delay)
with a time complexity of O(m) for each (π(j), π(j + 1)) ∈ R, with |R|<|π|
![Page 66: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/66.jpg)
FSSP Non-permutation Results
BRN local search
First calculates ei,j and qi,j in a time of complexity O(nm)
Best-improvement
chooses the best in the adjacent job swap neighbourhood
Reduced-neighbourhood
(π(j), π(j + 1)) ∈ R ⇐⇒ ei,j + qi+1,j = Cmax(π) ∨ ei,j+1 + qi+1,j+1 = Cmax(π)
Either π(j) or π(j + 1) has critical operations on consecutive machines
Like Nowicki & Smutnicki but considering all the critical paths
Non-permutation
Calculates the makespan of swapping two consecutive jobs π(j), π(j + 1)MC: swap completelyMC ′: swap on the first machines (like insertion with anticipation)MC ′′: swap on the last machines (like insertion with delay)
with a time complexity of O(m) for each (π(j), π(j + 1)) ∈ R, with |R|<|π|
![Page 67: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/67.jpg)
FSSP Non-permutation Results
BRN local search
First calculates ei,j and qi,j in a time of complexity O(nm)
Best-improvement
chooses the best in the adjacent job swap neighbourhood
Reduced-neighbourhood
(π(j), π(j + 1)) ∈ R ⇐⇒ ei,j + qi+1,j = Cmax(π) ∨ ei,j+1 + qi+1,j+1 = Cmax(π)
Either π(j) or π(j + 1) has critical operations on consecutive machines
Like Nowicki & Smutnicki but considering all the critical paths
Non-permutation
Calculates the makespan of swapping two consecutive jobs π(j), π(j + 1)MC: swap completelyMC ′: swap on the first machines (like insertion with anticipation)MC ′′: swap on the last machines (like insertion with delay)
with a time complexity of O(m) for each (π(j), π(j + 1)) ∈ R, with |R|<|π|
![Page 68: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/68.jpg)
FSSP Non-permutation Results
BRN local search
First calculates ei,j and qi,j in a time of complexity O(nm)
Best-improvement
chooses the best in the adjacent job swap neighbourhood
Reduced-neighbourhood
(π(j), π(j + 1)) ∈ R ⇐⇒ ei,j + qi+1,j = Cmax(π) ∨ ei,j+1 + qi+1,j+1 = Cmax(π)
Either π(j) or π(j + 1) has critical operations on consecutive machines
Like Nowicki & Smutnicki but considering all the critical paths
Non-permutation
Calculates the makespan of swapping two consecutive jobs π(j), π(j + 1)MC: swap completelyMC ′: swap on the first machines (like insertion with anticipation)MC ′′: swap on the last machines (like insertion with delay)
with a time complexity of O(m) for each (π(j), π(j + 1)) ∈ R, with |R|<|π|
![Page 69: Solving the Non-permutation Flow Shop Scheduling Problem · 2019. 12. 1. · Non-permutation FSSP with makespan (Benavides & Ritt, 2016) Non-permutation FSSP with makespan (Benavides](https://reader033.fdocuments.in/reader033/viewer/2022051903/5ff42b3cfb8a685d8f4e543d/html5/thumbnails/69.jpg)
FSSP Non-permutation Results
OutlineFSSP, Introduction and concepts
FSSP definitionNEH heuristic and Taillard accelerationLocal search heuristics
Non-permutation FSSP, Motivations and proposed heuristicsPermutation FSSP vs. Non-permutation FSSPConstructing non-permutation schedulesConstructing non-permutation schedulesNew permutation representation for non-permutation schedules andnew constructive heuristic NEHBR
Local search heuristics for non-permutation FSSP
Results and RemarksNon-permutation FSSP with makespan (Benavides & Ritt, 2016)Non-permutation FSSP with makespan (Benavides & Ritt, 2018)Concluding Remarks
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax (2016)
Benavides, A. J.; Ritt, M. (2016). (first attempt)
Two simple and effective heuristics for minimizing the makespan in non-permutationflow shops.Computers & Operations Research, Elsevier, v. 66, p. 160–169.CAPES WebQualis A1; Impact Factor 1.861; 5-Year Impact Factor 2.454
Iterated greedy algorithm for non-permutation FSSP with Cmax
Greedy Reconstruction Perturbation scheme:Based on NFS, O(nm2W ) per insertion
Local search scheme:Extended Neighbourhood of Nowicki & Smutnicki
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax (2016)
Demirkol Lin & Rossi & Our IGA
instances Ying Lanzetta min avg max
Averages 0.00 7.99 -1.98 -1.57 -1.13
Our IGA is better in the same adjusted time
Our IGA finds new BKV for the 40 instances
28 Taillard Yagmahan & Rossi & Lanzetta Our IGA
instances Yenisey min avg min avg max
Averages 6.86 5.02 5.98 -0.69 -0.51 -0.25
Our IGA is better in the less than their adjusted time
Our IGA finds new BKV for 13 of those 28 instances and 32 of all 120
Better results for 30nm2 ms in both cases
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax (2016)
Demirkol Lin & Rossi & Our IGA
instances Ying Lanzetta min avg max
Averages 0.00 7.99 -1.98 -1.57 -1.13
Our IGA is better in the same adjusted time
Our IGA finds new BKV for the 40 instances
28 Taillard Yagmahan & Rossi & Lanzetta Our IGA
instances Yenisey min avg min avg max
Averages 6.86 5.02 5.98 -0.69 -0.51 -0.25
Our IGA is better in the less than their adjusted time
Our IGA finds new BKV for 13 of those 28 instances and 32 of all 120
Better results for 30nm2 ms in both cases
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax (2016)compared to permutation FSSP
Taillard Permutation Our IGA 30nm ms Our IGA 30nm2 ms
instances RS FF min avg max min avg max
Averages 0.44 0.38 -0.22 -0.03 0.21 -0.41 -0.32 -0.20
Fernandez-Viagas & Framinan (2014) is 0.06% better thanRuiz & Stutzle (2007)
Our IGA is 0.4% better than Fernandez-Viagas & Framinan (2014)in the same time, and is 0.7% better in 30nm2 ms
Small job reordering
Buffer sizes Non-permutation schedules require slightly smaller buffers
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax using pseudo-jobs
Benavides, A. J.; Ritt, M. (2018).
Novel pseudo-jobs permutation representationfor non-permutation flow shop schedules
Extended acceleration tech. with the same time complexity
NEHT, NEHBR: O(n2m) (Permutation and non-permutation)
BRN local search: O(nm) per neighbourhood (non-permutation)
Insertion local search: O(n2m) per neighbourhood (Permutation and non-permutation)
FRBBR based on Farahmand Rad, Ruiz & Boroojerdian (2009)– produces better results than NEHBR, more complex and expensive– different initial solutions slightly affect IGb
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax using pseudo-jobs
Benavides, A. J.; Ritt, M. (2018).
Novel pseudo-jobs permutation representationfor non-permutation flow shop schedules
Extended acceleration tech. with the same time complexity
NEHT, NEHBR: O(n2m) (Permutation and non-permutation)
BRN local search: O(nm) per neighbourhood (non-permutation)
Insertion local search: O(n2m) per neighbourhood (Permutation and non-permutation)
FRBBR based on Farahmand Rad, Ruiz & Boroojerdian (2009)– produces better results than NEHBR, more complex and expensive– different initial solutions slightly affect IGb
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax using pseudo-jobs
Benavides, A. J.; Ritt, M. (2018).Makespan in non-permutation flow shop scheduling problemby the price of permutation.
Iterated greedy algorithmsIGA Reconstr. Local search
IGb NEHBR BRNIGi NEHBR InsertionIGbi NEHBR BRN, InsertionIGp NEH Permutation insertion
NEHT, NEHBR: O(n2m) (Permutation and non-permutation)
BRN local search: O(nm) per neighbourhood (non-permutation)
Insertion local search: O(n2m) per neighbourhood (Permutation and non-permutation)
IGb is the best combination for non-permutation FSSP
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FSSP Non-permutation Results
Non-permutation FSSP with Cmax using pseudo-jobs
Benavides, A. J.; Ritt, M. (2018).
Novel pseudo-jobs permutation representationfor non-permutation flow shop schedules
Extended acceleration tech. with the same time complexity
NEHT, NEHBR: O(n2m) (Permutation and non-permutation)
BRN local search: O(nm) per neighbourhood (non-permutation)
Insertion local search: O(n2m) per neighbourhood (Permutation and non-permutation)
FRBBR based on Farahmand Rad, Ruiz & Boroojerdian (2009)– produces better results than NEHBR, more complex and expensive– different initial solutions slightly affect IGb
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FSSP Non-permutation Results
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FSSP Non-permutation Results
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FSSP Non-permutation Results
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FSSP Non-permutation Results
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FSSP Non-permutation Results
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FSSP Non-permutation Results
Non-permutation FSSPConcluding Remarks
Non-permutation schedules can be represented as a permutation of pseudo-jobs, andthis allows the use of an extended taillard acceleration and a BRN local search.
Strategic operation reordering leads to non-permutation schedules with better qualitythan the best possible permutation schedules.
Non-permutation schedules can be found using the same computational effort than theused for permutation scheduleswith the makespan and the total completion time criteria.
Non-permutation schedules can be implemented in practice without strongtechnological differences.
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Solving the Non-permutationFlow Shop Scheduling Problem
Alexander J. Benavides
[email protected]@ucsp.edu.pe
Thank you!Questions?