Abstract
We consider the problem of constructing a schedule for a single machine that minimizes the total weighted completion time of tasks when the restrictions on their processing order are given by an arbitrary oriented acyclic graph. The problem is NP-hard in the strong sense. Efficient polynomial algorithms for its solving are known only for cases when the oriented acyclic graph is a tree or a series-parallel graph. We give a new efficient PSC-algorithm of its solving. It is based on our earlier theoretical and practical results and solves the problem with precedence relations specified by an oriented acyclic graph of the general form. The first polynomial component of the PSC-algorithm contains sixteen sufficient signs of optimality. One of them will be statistically significantly satisfied at each iteration of the algorithm when solving randomly generated problem instances. In case when the sufficient signs of optimality fail, the PSC-algorithm is an efficient approximation algorithm. If the sufficient signs of optimality are satisfied at each iteration then the algorithm becomes exact. We present the empirical properties of the PSC-algorithm on the basis of statistical studies.
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Notes
- 1.
A schedule is feasible if it does not violate the precedence relations.
- 2.
All fifteen SSOs are the signs of optimality for feasible subsequences obtained at the current iterations. We give more detailed justification for the implementation of insertion procedures for separate tasks or their constructions in the PSC-algorithm description (Sect. 7.3.4). We base the justification on an analysis of the priority and precedence relations. The presented logic of justifying the rules for constructing a p-ordered schedule in the PSC-algorithm causes the necessity of a more detailed investigation of these relations.
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Zgurovsky, M.Z., Pavlov, A.A. (2019). The Total Weighted Completion Time of Tasks Minimization with Precedence Relations on a Single Machine. In: Combinatorial Optimization Problems in Planning and Decision Making. Studies in Systems, Decision and Control, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-319-98977-8_7
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