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Omid Gholami 1 , Yuri N. Sotskov 2 , Frank Werner 3

JOB-SHOP PROBLEMS WITH OBJECTIVES APPROPRIATE FOR TRAIN SCHEDULING IN A SINGLE-TRACK RAILWAY. Omid Gholami 1 , Yuri N. Sotskov 2 , Frank Werner 3 Islamic Azad university - Mahmudabad Branch, Mahmudabad , Iran, e-mail: gholami@iaumah.ac.ir

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Omid Gholami 1 , Yuri N. Sotskov 2 , Frank Werner 3

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  1. JOB-SHOP PROBLEMS WITH OBJECTIVES APPROPRIATE FOR TRAIN SCHEDULING IN A SINGLE-TRACK RAILWAY Omid Gholami1, Yuri N. Sotskov2, Frank Werner3 Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: gholami@iaumah.ac.ir United Institute of Informatics Problems, Minsk, Belarus, e-mail: sotskov@newman.bas-net.by Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany e-mail: frank.werner@ovgu.de SIMULTECH 2012; JULY 28 - 31, 2012; ROME / ITALY

  2. 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 Outline of the Talk

  3. Train road map in Belarus Introduction

  4. 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 LITERATURE REVIEW FOR SINGLE-TRACK RAILWAY PROBLEMS

  5. 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 , Oi2 , … , Oini ) PROBLEM SETTING IN TERMS OF A JOB-SHOP

  6. 2 2 9 11 21 41 31 0 0 0 0 • G = (Q, C, D) -> G= (Q, C  Di, Ø) 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 13 0 23 0 33 43 0 53 63 0 0 Mixed graph G=(Q, C, D) for a job-shop problem with three jobs (trains) and seven machines (railroad sections) MIXED (DISJUNCTIVE) GRAPH FORMULATIONOF A JOB-SHOP SCHEDULING PROBLEM

  7. Algorithms: • Ordinal-algorithm • MaxPT-algorithm • MinPT-algorithm Priority rules for comparing conflict jobs: • Release time • Completion time • Due date HEURISTIC ALGORITHMS

  8. 2 2 9 21 41 31 0 0 0 • The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 Ordinal-algorithm

  9. 2 2 9 21 41 31 0 0 0 • The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 Ordinal-algorithm

  10. 2 2 9 21 41 31 0 0 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 Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 MaxPT-algorithm

  11. 2 2 9 21 41 31 0 0 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 Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 MaxPT-algorithm

  12. 2 2 9 21 41 31 0 0 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 Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 MinPT-algorithm

  13. 2 2 9 21 41 31 0 0 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 Oij 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 0 8 6 2 | 3 8 | 2 4 | 9 8 | 2 9 | 2 2 | 2 4 6 9 2 8 6 12 22 32 42 52 0 0 0 0 0 0 * 9 | 3 9 | 4 1 | 6 2 9 | 9 4 8 | 2 9 8 3 1 9 0 33 43 0 53 63 0 0 0 13 23 0 MinPT-algorithm

  14. 2 2 9 11 21 41 31 0 0 0 0 8 6 2 8 9 4 2 4 6 9 2 8 6 * 12 22 32 42 52 0 0 0 0 0 0 1 2 9 8 4 9 8 3 1 9 0 13 0 23 0 33 43 0 53 63 0 0 Digraph (Q, C  Di, Ø) defining a solution of the job-shop problem A complete schedule for the instance

  15. SRT (Shortest Release Time) SCT (Shortest Completion Time) SDD (Shortest Due-Date) Objective function values of the obtained schedules for the job-shop problems with the criterion Cmax Computational Results

  16. SRT (Shortest Release Time) SCT (Shortest Completion Time) SDD (Shortest Due-Date) Objective function values of the obtained schedules for the job-shop problems with the criterion ∑Ci Computational Results

  17. SRT (Shortest Release Time) SCT (Shortest Completion Time) SDD (Shortest Due-Date) Objective function values of the obtained schedules for the job-shop problems with the criterion ∑Ti Computational Results

  18. 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 Intel Core 2 Due CPU, 2.00 GHz, Ram 2 GB, Windows 7 Ultimate, Borland Delphi programming language. Best algorithm for the train scheduling problem is the Ordinal-SCT (Shortest Completion Time) algorithm. Computational Results

  19. Thanks JOB-SHOP PROBLEMS WITH OBJECTIVES APPROPRIATE FOR TRAIN SCHEDULING IN A SINGLE-TRACK RAILWAY Omid Gholami1, Yuri N. Sotskov2, Frank Werner3 Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: gholami@iaumah.ac.ir United Institute of Informatics Problems, Minsk, Belarus, e-mail: sotskov@newman.bas-net.by Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany, email: frank.werner@ovgu.de

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