Slack Based Lower Bound And Heuristic for Permutation Flowshop Models

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Slack Based Lower Bound And Heuristic for Permutation Flowshop Models
Anantaram Balakrishnan
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If the tour contains arcs (n+lj) and (j'. N+l) then jobs j and j' are the first and last jobs in the sequence. and if it contains arc (p, q), for p, q < n, then job q follows job p. Since the makespan of an\ job sequence (for NIS/NW systems) is the interval time on machine 1 plus the processing time for the last job on machines 2 through n, the length of the corresponding traveling salesman tour plus the total processing time TPj for all jobs on machine I equals the makespan. Minimizing the length of the traveling salesman tour, therefore, solves the NIS/NW/M model optimally. Instead of solving the equivalent TSP optimally, we apply the following assignment- patching (AF) heuristic. We can formulate the TSP as an integer program using the binar\ sequencing variables Y^: for all i, j = 1, ... , n+1, i *]. Y- is 1 if job i precedes job j. And otherwise. The formulation has assignment constraints (to assign each node to one position in the sequence, and one node to each of the n positions) and subtour breaking constraints (Dantzig, Fulkerson, and Johnson [1954]).

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