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The Total Weighted Tardiness of Tasks Minimization on a Single Machine

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Combinatorial Optimization Problems in Planning and Decision Making

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 173))

Abstract

We solve the problem of constructing a schedule for a single machine with various due dates and a fixed start time of the machine that minimizes the sum of weighted tardiness of tasks in relation to their due dates. The problem is NP-hard in the strong sense and is one of the most known intractable combinatorial optimization problems. Unlike other PSC-algorithms in this monograph, in this chapter we present an efficient PSC-algorithm which, in addition to the first and second polynomial components (the first one contains twelve sufficient signs of optimality of a feasible schedule) includes exact subalgorithm for its solving. We have obtained the sufficient conditions that are constructively verified in the process of its execution. If the conditions are true, the exact subalgorithm becomes polynomial. We give statistical studies of the developed algorithm and show the solving of well-known examples of the problem. We present an approximation algorithm (the second polynomial component) based on the exact algorithm. Average statistical estimate of deviation of an approximate solution from the optimum does not exceed 5% of it.

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Notes

  1. 1.

    In this chapter, all schedules obtained during the problem solving are feasible including those with tardy tasks.

  2. 2.

    If we would extend the problem statement to include tasks with negative due dates (when a due date is less than the start time of machine), the change of the algorithm would consist only in adjusting the position of insertion for these tasks. All other procedures would remain unchanged. According to [4], if \( d_{{j_{\left[ g \right]} }} < 0 \), then the position of insertion of the task \( j_{[g]} \) at step 3.2 is defined within the interval \( \overline{p,g - 1} \) as the maximum of three positions: p, q + 1 (defined at step 3.2), and l + 1 if position l is occupied by such a task \( j_{[l]} \) that \( d_{{j_{\left[ l \right]} }} < 0 \), \( l_{{j_{\left[ l \right]} }} < l_{{j_{\left[ g \right]} }} \), \( \upomega_{{j_{\left[ l \right]} }} >\upomega_{{j_{\left[ g \right]} }} \).

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Authors and Affiliations

  1. Kyiv Polytechnic Institute, National Technical University of Ukraine, Kyiv, Ukraine

    Michael Z. Zgurovsky

  2. Faculty of Informatics and Computer Science, National Technical University of Ukraine, Kyiv, Ukraine

    Alexander A. Pavlov

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  1. Michael Z. Zgurovsky
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Correspondence to Michael Z. Zgurovsky .

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Zgurovsky, M.Z., Pavlov, A.A. (2019). The Total Weighted Tardiness of Tasks Minimization 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_4

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  • DOI: https://doi.org/10.1007/978-3-319-98977-8_4

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