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Solving a job-shop scheduling problem by an adaptive algorithm based on learning 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 IFAC MIM '2013 Conference, June 19-21, St Petersburg. 1

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Outline of the Talk 2 Introduction Literature review The job-shop problem An adaptive scheduling algorithm Priority dispatching rules The learning stage The examination stage Procedures for monitoring variable parameters Three strategies for considering a set of conflict edges Computational results

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Introduction 3 The job-shop problem is a scheduling problem arising in real world, see e.g.: Muth and Thompson (1963); Panwalkar and Iskander (1977); Haupt (1989); Tanaev et al. (1994). No priority dispatching rule performs globally better than other ones tested for a wide class of scheduling problems which are NP-hard, see e.g.: Shakhlevich et al. (1996); Geiger et al. (2006); Mouelhi-Chibani and Pierreval (2010); Gholami et al. (2012). A particular priority dispatching rule may provide a good solution for a concrete scheduling problem but applied to another NP-hard problem, it may provide a bad solution. Several researchers developed tools to discover effective priority dispatching rules automatically: Li and Shi (1994); Dorndorf and Pesch (1995); Shakhlevich et al. (1996); Geiger et al. (2006); Gabel and Riedmiller (2007); Abdolzadeh and Rashidi (2009).

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Literature review Shifting bottleneck procedure: Adams et al. (1988), Dauzere-Peres and Laserre (1993) Tabu search: Glover (1989), Nowicki and Smutnicki (1996, 2005) Simulated annealing: van Laarhoven et al. (1992) Genetic algorithms: Della Croce et al. (1995), Bierwirth (1995), Park et al. (2003) Genetic programming: Geiger et al. (2006) Multi-agent reinforcement learning: Gabel and Riedmiller (2007) Cellular learning automata: Abdolzadeh and Rashidi (2009) Neural network approach: Mouelhi-Chibani and Pierreval (2010) 4

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The job-shop problem 5 There are n jobs J={J 1, J 2,..., J n }, which need to be processed on m different machines M={M 1, M 2,..., M m }. The machine (technological) route O i =(O i1, O i2,..., O ini ) of each job J i through the machines is fixed. The machine routes may be different for different jobs. The time for processing each operation of a job on the corresponding machine is known. The objective of the problem under consideration is to minimize the makespan.

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Mixed (Disjunctive) graph formulation of a job-shop scheduling problem 6 G = (O, C, D) => G= (O, C D i, Ø) 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=(O, C, D) for job-shop problem with three jobs and seven machines 0 *

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Figure 1: Scheme of the adaptive scheduler An adaptive scheduling algorithm An adaptive scheduling algorithm

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Priority dispatching rules There are a lot of priority dispatching rules which are used in a variety of heuristic algorithms for scheduling the jobs in a job-shop, see e.g.: Haupt (1989), Muth and Thompson (1963), Panwalkar and Iskander (1977). First come first served Shortest release time Shortest completion time Shortest due date Shortest processing time Shortest remaining processing time Smallest number of remaining operations …. 8

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The learning stage Conflict resolution for the edge 9

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The learning stage... 10 Conflict edge X1X1 X2X2 …XnXn Ω class [O ij 1, O uv 1 ]g11g11 g21g21 …gr1gr1 Ω1Ω1 [O ij 2, O uv 2 ]g12g12 g22g22 …gr2gr2 Ω2Ω2 … ………… [O ij w, O uv w ]g1wg1w g2wg2w …grwgrw ΩwΩw [O ij, O uv ]… ? Table 1: Conflict resolutions in optimal schedules

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The examination stage Let the characteristic vector (here the index is fixed) be the closest one to the vector among all vectors (here the index varies) presented in Table1. To resolve a conflict edge in the mixed graph, an adaptive scheduler uses the same decision as in the class stored in Table 1.

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Procedures for monitoring variable parameters 12

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Ordinal-algorithm 13 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 u v to be processed on the same machine (based e.g. on release time, completion time or due date). 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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Ordinal-algorithm 14 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 u v to be processed on the same machine (based e.g. on release time, completion time or due date). 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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MaxPT-algorithm 15 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 u v to be processed on the same machine (based e.g. on release time, completion time or due date). 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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MaxPT-algorithm 16 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 u v to be processed on the same machine (based e.g. on release time, completion time or due date). 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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Three strategies for considering a set of conflict edges 17 Max-PT (Maximum Processing Time first) Min-PT (Minimum Processing Time first) Ordinal

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Computational Results 18 Benchmark problems Ordinal-ECT algorithm Shifting bottleneck EDD- rule FCFS- rule LPT- rule SPT- rule Adaptive algorithm MT-6 (6×6) 59 6365677358 Job-shop-10 (10×10) 8294122878611886 MT-10 (10×10) 1252109412461184116813381167 Job-shop-18 (18×5) 1419122012631462139314511370

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Computational Results 19 Figure 3: Average CPU-times used by the adaptive algorithm for solving a set of problems with a fixed number of jobs

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Computational Results 20 Figure 4: Average CPU-time used by the adaptive algorithm for solving a set of problems with a fixed number of machines

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Thanks Solving a job-shop scheduling problem by an adaptive algorithm based on learning 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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