Yuri N. Sotskov 1, Omid Gholami 2, Frank Werner 3 1. United Institute of Informatics Problems,...
21
HEURISTIC ALGORITHMS FOR A JOB-SHOP PROBLEM WITH MINIMIZING TOTAL JOB TARDINESS Yuri N. Sotskov 1 , Omid Gholami 2 , Frank Werner 3 1. United Institute of Informatics Problems, Minsk, Belarus, e-mail: [email protected] 2. Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: [email protected] 3. Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany , e-mail: [email protected] OPTIMA 2012, Costa da Caparica, Portugal September 23-30, 2012 1
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Transcript of Yuri N. Sotskov 1, Omid Gholami 2, Frank Werner 3 1. United Institute of Informatics Problems,...
HEURISTIC ALGORITHMS FOR A JOB-SHOP PROBLEM
WITH MINIMIZING TOTAL JOB TARDINESS
Yuri N. Sotskov1, Omid Gholami2, Frank Werner3
1.United Institute of Informatics Problems, Minsk, Belarus, e-mail: [email protected]
2.Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: [email protected]
3.Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany , e-mail: [email protected]
OPTIMA 2012, Costa da Caparica, PortugalSeptember 23-30, 2012
1
Introduction
Literature Review for Single-Track Railway Systems
Problem Setting in Terms of a Job-Shop
Mixed (Disjunctive) Graph Formulation of a Job-Shop Scheduling Problem
Heuristic Algorithms
Computational Results
2
OUTLINE OF THE TALK
3
INTRODUCTION
Train road mapin Belarus
Szpigel (1973): B&B algorithm, results for 5 sections and 10 trains
Cai and Goh (1994): greedy algorithm
Carey and Lockwood (1995): binary mixed integer programming model
Mladenovic and Cangalovic (2007): constraint programming approach
Zhou and Zhong (2007): B&B algorithm, resource-constrained project scheduling problem
Liu and Kozan (2011): no-wait condition for prioritized trains, recursive procedure
Sotskov and Gholami (2012): shifting bottleneck procedure
4
LITERATURE REVIEW FOR SINGLE-TRACK RAILWAY PROBLEMS
set of railroad sections (machines) ◦ M ={ M1, M2, …, Mm }
set of trains (jobs) ◦ J ={ J1, J2, …, Jn }
the sequence of the job operations on the corresponding machines is given for any job Ji :
◦ Oi = (Oi1 , Oi
2 , … , Oini )
5
PROBLEM SETTING IN TERMS OF A JOB-SHOP
G = (Q, C, D) -> G= (Q, C Di, Ø)
6
MIXED (DISJUNCTIVE) GRAPH FORMULATIONOF A JOB-SHOP SCHEDULING PROBLEM
23
11
21
41
63
52
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22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
Mixed graph G=(Q, C, D) for a job-shop problem with three jobs (trains) and seven machines (railroad sections)
0 *
Algorithms:
Ordinal-algorithm MaxPT-algorithm MinPT-algorithm
Priority rules for comparing conflict jobs:
Release time Completion time Due date
7
HEURISTIC ALGORITHMS
The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation Oi
j currently considered with the other operations Okl to be processed on the same machine. Based on the chosen priority rule, a
direct arc is created.
8
Ordinal-algorithm
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21
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22
12
53
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6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0
*
The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation Oi
j currently considered with the other operations Okl to be processed on the same machine. Based on the chosen priority rule, a
direct arc is created.
9
Ordinal-algorithm
23
11
21
41
63
52
42
32
22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0 *
Sort the jobs in non-increasing order of their total processing times and consider all operations of a job subsequently. Then it compares the operation Oi
j currently considered with the other operations Okl to be processed on the same
machine. Based on the chosen priority rule, a direct arc is created.
10
MaxPT-algorithm
23
11
21
41
63
52
42
32
22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
29 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0 *
Sort the jobs in non-increasing order of their total processing times and consider all operations of a job subsequently. Then it compares the operation Oi
j currently considered with the other operations Okl to be processed on the same machine. Based
on the chosen priority rule, a direct arc is created.
11
MaxPT-algorithm
23
11
21
41
63
52
42
32
22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0 *
Sort the jobs in non-decreasing order of their total processing times and consider all operations of a job subsequently. Then it compares the operation Oi
j currently considered with the other operations Okl to be processed on the same
machine. Based on the chosen priority rule, a direct arc is created.
12
MinPT-algorithm
23
11
21
41
63
52
42
32
22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0 *
Sort the jobs in non-decreasing order of their total processing times and consider all operations of a job subsequently. Then it compares the operation Oi
j currently considered with the other operations Okl to be processed on the same
machine. Based on the chosen priority rule, a direct arc is created.
13
MinPT-algorithm
23
11
21
41
63
52
42
32
22
12
53
43
33
13
6
6
4
2
2 | 3
9
8
9 4 2 8 6
8 9 3 1
2
9 | 4
9 | 22 | 2
8 | 2
0 0 0
0 0 0 0 0
00 0 0 0 0
9 | 3
4 | 9
310
9
2
9 | 9
8 | 2
1 | 6
8 | 2
0 *
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A complete schedule for the instance
23
11
21
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12
53
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6
6
4
2
2
9
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9 4 2 8 6
8 9 3 1
2
9 2
0 0 0
0 0 0 0 0
00 0 0 0 0
4
310
9
2
9 8 1
8
Digraph (Q, C Di, Ø) defining a solution of the job-shop problem
0 *
15
COMPUTATIONAL RESULTS
Objective function values of the obtained schedules for the job-shop problems with the criterion ∑Ti
SRT (Shortest Release Time)SCT (Shortest Completion Time)SDD (Shortest Due-Date)
16
COMPUTATIONAL RESULTS
Required time (Algorithm Ordinal-SCT) to schedule different job-shops: 10 ≤ n = m ≤ 60
10*10 20*20 30*30 40*40 50*50 60*600
20
40
60
80
100
120
140
1 26
20
64
116
f(x) = 0.541809411887023 x^2.74653717927989
Times in seconds
Job-shop Power (Job-shop)
17
COMPUTATIONAL RESULTS
Required time (Algorithm Ordinal-SCT) to schedule different job-shops: m = 20 and 10 ≤ n ≤ 110
10*20 30*20 50*20 70*20 90*20 110*200
50
100
150
200
250
1 318
42
77
218
f(x) = 0.673148376055515 x^3.01397386033606
Times in seconds
Job-shop Power (Job-shop)
18
COMPUTATIONAL RESULTS
Required time (Algorithm Ordinal-SCT) to schedule different job-shops: n = 20 and 10 ≤ m ≤ 110
10*20 30*20 50*20 70*20 90*20 110*200
50
100
150
200
250
1 318
42
77
218
f(x) = 0.673148376055515 x^3.01397386033606
Times in seconds
Job-shop Power (Job-shop)
19
COMPUTATIONAL RESULTS
Best constructive algorithm for the train scheduling problem among the tested ones is the Ordinal-SCT (Shortest Completion Time) algorithm.
Intel Core 2 Due CPU, 2.00 GHz, Ram 2 GB, Windows 7 Ultimate, Borland Delphi programming language.
Ranking for ∑Ti
---------------------
1.Ordinal-SCT
2.Ordinal-SRT
3.Min-PTRT
4.Min-PTCT
5.Max-PTCT
6.Max-PTRT
7.Ordinal-SDD
8.Min-PTDD
9.Max-PTDD
20
Some computational results for an alternative criterion: MAKESPAN
Benchmark
Ordinal-SCT
Shifting Bottleneck
EDD FCFS
MT 6 (6×6) 59 59 63 65
MT 10 (10×10) 1252 1094 1246 1184
Job-shop 10 (10×10)
82 94 122 87
Job-shop 18 (18×5)
1419 1220 1263 1462
Comparison of different algorithms for the makespan criterion on some benchmark instances
ThanksHeuristic Algorithms for a Job-Shop Problem with Minimizing Total Job Tardiness
Yuri N. Sotskov1, Omid Gholami2, Frank Werner3
1.United Institute of Informatics Problems, Minsk, Belarus, e-mail: [email protected] Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: [email protected] of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany , e-mail: [email protected]
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