10 DP Sequencing&Routing

10 DP Sequencing&Routing - 1 Sequencing Problems...

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1 Sequencing Problems Consider the problem of sequencing a set of n jobs on a single machine with all of the jobs initially available for processing. The objective is to sequence the jobs on the machine so as to minimize an additive function of the completion times of the jobs. We assume that each job i has a due date d i , a starting time s i that is a function of the sequence, a processing time t i , and a penalty weight of being late of w i . The job completion time c i is s i + t i . The function g measures the penalty of the schedule with respect the job i , where g ( s i , i ) = w i ( s i + t i d i ), if s i + t i d i , 0 otherwise. The problem at hand then is to select the permutation of the jobs {1, " , n } which minimizes the sum of the penalties. Let P be the collection of all permutations of the job set, let p be a particular permutation and define p ( j ) as the j th job in the permutation p . The j th job in the sequence (permutation) we denote s p ( j ) as the starting time, t p ( j ) as the processing time, and d p ( j ) as the job's due date. Then the value of the objective for permutation p P is g ( s p ( j ) , p ( j )) j = 1 n , and the objective is to find the permutation p P which minimizes this function. There are n ! permutations in the permutation set P and for small problems it is relatively easy to compute all of these and select the best one with respect to the objective function measure. For example, the four job problem (sequence the job set {1,2,3,4}) we have only 24 sequences to consider. For a ten-job problem, however, the number of sequences to be explicitly considered is 3,628,800. So exhaustive enumeration quickly becomes infeasible. In his book on sequencing and scheduling, French [1982] gives a comparison between the computational efforts of dynamic programming and exhaustive enumeration for the n -job sequencing problem (see Table 1). To put these numbers in perspective, if it takes one micro-second (10 -6 seconds) to perform one operation (addition, subtraction, or comparison) then to exhaustively solve the 20-job sequencing problem, it would take ten million years. To solve the 20-job sequencing problem by dynamic programming would only take about one minute, but, the 40-job problem would
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2 still take about 4 years. Thus, even though there is a significant improvement in the problem size that can be solved by dynamic programming, realistic problems are still extremely time consuming and research into other methods, heuristics, is warranted. Table 1 Comparison of the number of computations needed to solve the n -job sequencing problem. (From French 1982, p. 97) n enumeration dynamic programming 4 647 237 10 2.286 × 10 8 33,789 20 2.992 × 10 20 6.396 × 10 7 40 1.983 × 10 50 1.352 × 10 14 Rather than exhaustively evaluate all of the possible schedules, we utilize a dynamic programming approach [Held and Karp, 1962]. We consider the set J of all job indices and consider two subsets of J , S and R , where S is the set of jobs we are currently trying to order and R is the set of jobs all of which have been scheduled prior to any jobs in the set S . Since all the jobs in set R
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This note was uploaded on 10/16/2010 for the course ISEN 689 taught by Professor Klutke during the Spring '07 term at Texas A&M.

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10 DP Sequencing&Routing - 1 Sequencing Problems...

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