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Scheduling identical parallel machines with machine eligibility restrictions to minimize total weighted flowtime in automobile gear manufacturing

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Abstract

In this paper, we address a scheduling problem for minimizing total weighted flowtime, observed in automobile gear manufacturing. Specifically, the bottleneck operation of the pre-heat treatment stage of gear manufacturing process has been dealt with in scheduling. Many real-life scenarios like unequal release times, sequence dependent setup times, and machine eligibility restrictions have been considered. A mathematical model taking into account dynamic starting conditions has been proposed. The problem is derived to be NP-hard. To approach the problem, a few heuristic algorithms have been proposed. Based on planned computational experiments, the performance of the proposed heuristic algorithms is evaluated: (a) in comparison with optimal solution for small-size problem instances and (b) in comparison with the estimated optimal solution for large-size problem instances. Extensive computational analyses reveal that the proposed heuristic algorithms are capable of consistently yielding near-statistically estimated optimal solutions in a reasonable computational time.

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Acknowledgment

The authors gratefully acknowledge the valuable comments and remarks of the anonymous referees that improved the quality of the paper.

Author information

Authors and Affiliations

  1. Indian Institute of Management Indore, Indore, 453331, India

    Ravindra Gokhale

  2. Department of Management Studies, Indian Institute of Science, Bangalore, 560 012, India

    M. Mathirajan

Authors
  1. Ravindra Gokhale
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  2. M. Mathirajan
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Correspondence to M. Mathirajan.

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Appendix

Appendix

A procedure for EOS

The idea of statistical estimation techniques for optimal values based on Weibull distribution is explained here as per Rardin and Uzsoy [30].

Begin by letting Z(1), Z(2), . . ., Z(N) be the n independent group minima, and sort them so that: Z(1) ≤ Z(2) ≤. . . Z(n).

Then, the location parameter or optimal value a can be estimated as follows:

$$ a = \left\{ {Z(1) \times Z(n) - {{\left[ {Z(2)} \right]}^2}} \right\}/\left\{ {{ }Z(1) + Z(n) - 2 \times Z(2)} \right\} $$

This value of a is the EOS that is used for evaluation.

A reliable estimate of the true optimum would be valuable in itself, but Golden and Alt [11] introduce more certainty by computing a lower confidence limit Z l which should be less than or equal to the optimal value Z * with high probability. Specifically, they estimate Weibull scale parameter b from a as

$$ b = Z\left[ {0.63\, \times \,n + 1} \right]--a $$

and then compute

$$ {Z_l} = a--b $$

In theory, this interval should cover the optimal value Z * with probability approximately (1−e −n).

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Gokhale, R., Mathirajan, M. Scheduling identical parallel machines with machine eligibility restrictions to minimize total weighted flowtime in automobile gear manufacturing. Int J Adv Manuf Technol 60, 1099–1110 (2012). https://doi.org/10.1007/s00170-011-3653-3

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