Minimizing the weighted number of tardy jobs on parallel processors

doi:10.1016/j.ejor.2003.06.027

European Journal of Operational Research

Volume 160, Issue 2, 16 January 2005, Pages 471-484

https://doi.org/10.1016/j.ejor.2003.06.027 Get rights and content

Abstract

We present a branch-and-bound algorithm to minimize the weighted number of tardy jobs on either identical or non-identical processors. Bounds come from a surrogate relaxation resulting in a multiple-choice knapsack. Extensive computational experiments indicate problems with 400 jobs and several machines can be solved quickly. The results also indicate what parameters affect solution difficulty for this algorithmic approach.

Introduction

We examine both identical and unrelated processor scheduling to minimize both the weighted and unweighted number of tardy jobs. Using the three-field notation of Graham et al. [4], the problems are denoted as P∥∑U_j (identical machines, unweighted jobs), P∥∑w_jU_j (identical machines, weighted jobs), R∥∑U_j (unrelated machines, unweighted jobs), and R∥∑w_jU_j (unrelated machines, weighted jobs). Proportional processors (Q∥∑U_j and Q∥∑w_jU_j) are not considered, although the algorithms presented can solve them as a special case of R∥∑w_jU_j. All of these problems are known to be NP-hard. Several heuristic and exact algorithms have been proposed for the problem, but little testing has been done. Existing exact algorithms require one-third to one hour for problems with 100 jobs and 10 machines. We develop an exact algorithm capable of solving problems with up to 400 jobs.

Section 2 gives a brief overview of previous work on the problem. Section 3 contains a mathematical programming formulation used to develop a surrogate bound based on the multiple-choice knapsack problem, followed by a formal statement of the algorithm. Section 4 describes and analyzes computational experiments. Section 5 summarizes the paper.

Section snippets

Literature review

P∥∑U_j is NP-hard even when m=2 [3], or the due dates are common [7]. Few heuristics exist and they focus on identical processor, unweighted problems [5], [11]. They are straightforward extensions of Moore's algorithm [9] for the single machine case and are time consuming, requiring 44 seconds to solve a 400 job 20 machine problem on an IBM PS/2 Model 70.

P∥∑w_jU_j is very hard to solve; there are only two exact algorithms. Van den Akker et al. [12] solve the identical parallel machine sum of

Algorithm

We follow the approach of M'Hallah and Bulfin [8] for the single machine problem. We formulate the problem mathematically and use surrogate multipliers to compute bounds. The formulation is different, resulting in a multiple-choice knapsack bounding problem rather than a standard knapsack problem.

We present a mathematical model for the most general problem, R∥∑w_jU_j; other versions follow easily. Also, for ease of discussion, we will maximize the weighted number of on-time jobs, an equivalent

Computational experiments

The preceding algorithm was extensively tested on a battery of randomly generated problems. The purpose of this experiment is to examine the performance of the algorithm, to investigate the possible influence of certain problem parameters on the difficulty of the problems, and to evaluate the difficulty of the four variations of the problem: P∥∑U_i, R∥∑U_i, P∥∑w_iU_i, and R∥∑w_iU_i. We follow the experimental procedure of Villarreal and Bulfin [13].

Summary

This paper contains results on scheduling parallel machines to minimize the weighted number of tardy jobs. An exact algorithm based on a surrogate constraint is proposed. Extensive experimentation indicates the exact method solves these problems optimally very quickly. Furthermore, these results provide insight into how the problem parameters influence solution difficulty and indicate the algorithm is robust with respect to problem type and factor levels.

Acknowledgements

This research was partially supported under grants NSF-INT-0116655: Tunisia Cooperative Research and NSF-DMI-0200409: Algorithms for Scheduling with Due Dates. We are grateful for their support. We would also like to thank the referees for their valuable comments.

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Discrete OptimizationMinimizing the weighted number of tardy jobs on parallel processors

Abstract

Introduction

Section snippets

Literature review

Algorithm

Computational experiments

Summary

Acknowledgements

Annals of Discrete Mathematics

European Journal of Operational Research

Discrete Applied Mathematics

European Journal of Operational Research

Computers and Industrial Engineering

Solving parallel machine scheduling problems by column generation

INFORMS Journal on Computing

Discrete Optimization
Minimizing the weighted number of tardy jobs on parallel processors