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Sequencing algorithms for multiple machines

Sequencing algorithms for multiple machines. Operations scheduling , Nahmias. 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 .

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Sequencing algorithms for multiple machines

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  1. Sequencingalgorithmsformultiplemachines Operationsscheduling, Nahmias

  2. Sequencing Algorithms for multiple machines • Assume that njobs are to be processed through mmachines. • 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!)mpossible schedules. • For example, for n=5 and m=5, there are 24,833x1010 , possible schedules.

  3. 2 jobs-2 machines Example Mean Idle Time Mean Flow Time Total Flow Time (or Makespan) Machine 1 I J Machine 2 I J 9 (5+9)/2= 7 (4+4)/2= 4 4 5 9 Machine 1 J I J I Machine 2 6 (5+6)/2=5.5 (1+1)/2=1 1 5 6 Machine 1 I J Machine 2 I J 1 5 6 10 10 (6+10)/2=8 (5+5)/2=5 Machine 1 I J Machine 2 J I 10 (10+9)/2=9.5 (5+5)/2=5 4 5 9 10

  4. Example 8.5 What is the optimal job sequence ? 2 4 3 5 1

  5. 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 • Ai = Processing time of job i on machine A. (Bi ,Ci are defined as similar) • minAi≥ max Bior min Ci≥ max Bi • Define Ai‘ = Ai+ Biand define Bi‘ = Bi+ Ci

  6. Example 8.5 minAi= 4 max Bi = 6 min Ci= 6 Check the conditions minAi≥ max Bior min Ci≥ max Bi Required condition is satisfied. What is the optimal job sequence ? 1 4 5 2 3

  7. 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)

  8. Aker’s 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.

  9. Example 8.7 Job 1 Job 2 10 C 9 8 7 B 6 5 Job 2 4 3 2 A 1 1 2 3 4 5 6 7 8 9 10 11 12 Job 1 A1 A2 A B1 B2 B C C1 C2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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