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5.5 PROCESSING N JOBS THROUGH THREE MACHINES This sequenecing problem is competely described as follows: (1) only three machines A, B and C are involved, f51 (ii) each job is processed in the prescribed order ABC (first on machine A, then on B and thereafter on C), (iii) no passing of jobs is permitted (i.e., the same order over each machine is maintained), and (iv) the actual or expected processing times A1, A2, ..., Ap B1, B2, ..., B, and C1, C2, ..., C, are known and represented by a table of the type shown below. TABLE 5.10 Machine times for n jobs and three machines Job A B C 1 A B C 2 42 B2 C2 3 Az C : B B3 C. B, The problem, again, is to find the optimum sequence of jobs which minimizes T. No general solution is available at present for such a case. However, the method of section 5.4 can be extended to cover the special cases where either one or both of the following conditions hold good (if neither of the conditions holds good, the method fails and the optimal sequence has to be found by enumerating all the sequences), (1) the minimum time on machine A is 2 maximum time on machine B, and (2) the minimum time on machine C is 2 maximum time on machine B. The method, described here without proof, is to replace the problem by an equivalent prolem involving n jobs and two machines. These two (fictitious) machines are denoted by G and H and their corresponding processing times are given by G = A; + B, H, = B, C, If this new problem with the prescribed order GH is solved by the method of section 5.4, the resulting optimal sequence will also be optimal for the original problem. EXAMPLE 5.5-1 A machine operator has to perform three operations: turning, threading and knurling on a number of different jobs. The time required to perform these operations (in minutes) for each job is known. Determine the order in which the jobs should be processed in order to minimize the total time required to turn out all the jobs. Also find the idle times for the three operations. TABLE 5.11 Job Time for turning (minutes) Time for threading (minutes) Time for knurling (minutes) 1 3 8 13 2 12 6 14 3 5 4 9 2 6 12 5 9 3 8 11 1 13 [G.N.D.U. BBA April, 2008; G.J.U.B.E. (Mech.) 1996]