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Yuri N. Sotskov 1, Omid Gholami 2, Frank Werner 3 1. United Institute of Informatics Problems, Minsk, Belarus, e-mail: sotskov@newman.bas-net.by 2. Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: gholami@iaumah.ac.ir 3. Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany, e-mail: frank.werner@ovgu.de OPTIMA 2012, Costa da Caparica, Portugal September 23-30, 2012 1

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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

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3 Train road map in Belarus

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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

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set of railroad sections (machines) ◦ M ={ M 1, M 2, …, M m } set of trains (jobs) ◦ J ={ J 1, J 2, …, J n } the sequence of the job operations on the corresponding machines is given for any job J i : ◦ O i = (O i 1, O i 2, …, O i ni ) 5

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G = (Q, C, D) -> G= (Q, C D i, Ø) 6 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 000 0 0 9 | 3 4 | 9 3131 0 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 *

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

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The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 8 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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The algorithm considers subsequently the first requests of all jobs, the second requests of all jobs, etc. It compares the operation O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 9 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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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 O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 10 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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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 O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 11 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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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 O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 12 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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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 O i j currently considered with the other operations O k l to be processed on the same machine. Based on the chosen priority rule, a direct arc is created. 13 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 | 3 9 8 9 4 286 8 9 31 2 9 | 4 9 | 2 2 | 2 8 | 2 00 0 00000 0 0 0 0 0 0 9 | 3 4 | 9 3131 0 9 2 9 | 9 8 | 2 1 | 6 8 | 2 0 *

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14 2323 1 2121 4141 6363 5252 4242 32322 1212 5353 43433 1313 6 6 4 2 2 9 8 9 4 286 8 9 31 2 9 2 00 0 00000 0 000 0 0 4 3131 0 9 2 9 8 1 8 Digraph (Q, C D i, Ø) defining a solution of the job-shop problem 0 *

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15 Objective function values of the obtained schedules for the job-shop problems with the criterion ∑T i SRT (Shortest Release Time) SCT (Shortest Completion Time) SDD (Shortest Due-Date)

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16 Required time (Algorithm Ordinal-SCT) to schedule different job-shops: 10 ≤ n = m ≤ 60

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17 Required time (Algorithm Ordinal-SCT) to schedule different job-shops: m = 20 and 10 ≤ n ≤ 110

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18 Required time (Algorithm Ordinal-SCT) to schedule different job-shops: n = 20 and 10 ≤ m ≤ 110

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19 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.

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20 BenchmarkOrdinal-SCTShifting Bottleneck EDDFCFS MT 6 (6×6)59 6365 MT 10 (10×10)1252109412461184 Job-shop 10 (10×10) 829412287 Job-shop 18 (18×5) 1419122012631462 Comparison of different algorithms for the makespan criterion on some benchmark instances

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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: sotskov@newman.bas-net.by 2. Islamic Azad university - Mahmudabad Branch, Mahmudabad, Iran, e-mail: gholami@iaumah.ac.ir 3. Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany, e-mail: frank.werner@ovgu.de 21

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