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Sequencing algorithms for multiple machines Operations scheduling, Nahmias

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Sequencing Algorithms for multiple machines Assume that n jobs are to be processed through m machines. The number of possible schedule is staggering, even for moderate values of both n and m. For each machine there are n! different ordering of the jobs. – If the jobs may be processed on the machines in any order, it follows that there are (n!) m possible schedules. – For example, for n=5 and m=5, there are 24,833x10 10, possible schedules.

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2 jobs-2 machines Example Machine 1Machine 2 Job I 41 Job J 14 M achine 1 M achine 2 IJ IJ 459 M achine 1 M achine 2 IJ IJ 1 56 M achine 1 M achine 2 6510 I J IJ 1 M achine 1 M achine 2 IJ IJ 45910 Total Flow Time (or Makespan) Mean Flow Time Mean Idle Time 9 6 10 (5+9)/2= 7 (5+6)/2=5.5 (6+10)/2=8 (10+9)/2=9.5 (4+4)/2= 4 (1+1)/2=1 (5+5)/2=5

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Example 8.5 JobMachine AMachine B 152 216 397 438 5104 What is the optimal job sequence ? 2 2 4 4 3 3 5 5 1 1

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Extension to three machines The problem of scheduling jobs on three machines is considerably more complex. The three machine problem can be reduced to a two machine problem if the following condition is satisfied: – Label the machines A, B and C – A i = Processing time of job i on machine A. (B i,C i are defined as similar) – min A i max B i or min C i max B i – Define A i = A i + B i and define B i = B i + C i

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Example 8.5 JobMachine AMachine BMachine C 1458 29610 3826 4637 55411 What is the optimal job sequence ? 1 1 4 4 5 5 2 2 3 3 min A i = 4 max B i = 6 min C i = 6 Check the conditions min A i max B i or min C i max B i Required condition is satisfied. JobMachine AMachine B 1913 21516 3108 49 5915

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The two-shop Flow shop problem Assume that two jobs are to be processed through m machines. Each job must be processed by the machines in a particular order, but sequences for the two jobs need not to be the same. A graphical procedure for solving this problem is developed by Aker (1954)

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Akers Algorithm Draw a Cartesian coordinate system. – Processing times for first job on the horizontal axis – Processing times for second job on the vertical axis – On each axis, mark off the operation times in the order in which the operations must be performed for that job. Block out areas corresponding to each machine at the intersection of the intervals marked for that machine on the two axis. Determine a path from origin to the end of final block that does not intersect any of the blocks and that minimizes the vertical movement.

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123456789101112 A A B B C C 131415 Job 1 123456789101112 1 2 3 4 5 6 7 8 9 10 Job 2 Example 8.7 OperationTime A3 B4 C5 OperationTime A3 B4 C5 Job 1Job 2 A A B B C C A1A2 B1B2 C1C2

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