Presentation on theme: "Sequencing algorithms for multiple machines"— Presentation transcript:
1 Sequencing algorithms for multiple machines Operations scheduling, Nahmias
2 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 24,833x1010 , possible schedules.
4 Example 8.5 Job Machine A Machine B 1 5 2 6 3 9 7 4 8 10 What is the optimal job sequence ?24351
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 CAi = Processing time of job i on machine A. (Bi ,Ci are defined as similar)min Ai ≥ max Bi or min Ci ≥ max BiDefine Ai‘ = Ai + Bi and define Bi‘ = Bi + Ci
6 Example 8.5 Job Machine A Machine B Machine C 1 4 5 8 2 9 6 10 3 7 11 min Ai = 4max Bi = 6min Ci = 6Check the conditionsmin Ai ≥ max Bi or min Ci ≥ max BiRequired condition is satisfied.JobMachine AMachine B191321516310845What is the optimal job sequence ?14523
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 axisProcessing times for second job on the vertical axisOn 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 Job 1 Job 2 Operation Time A 3 B 4 C 5 6789101112Job 2OperationTimeA3B4C5OperationTimeA3B4C5CBA123456789101112ABC131415A1A2B1B2C1C2