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Multicriteria scheduling optimization using an elitist multiobjective population heuristic: the h-NSDE algorithm

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Abstract

In today’s manufacturing industry more than one performance criteria are considered for optimization to various degrees simultaneously. To deal with such hard competitive environments it is essential to develop appropriate multicriteria scheduling approaches. In this paper consideration is given to the problem of scheduling n independent jobs on a single machine with due dates and objective to simultaneously minimize three performance criteria namely, total weighted tardiness (TWT), maximum tardiness and maximum earliness. In the single machine scheduling literature no previous studies have been performed on test problems examining these criteria simultaneously. After positioning the problem within the relevant research field, we present a new heuristic algorithm for its solution. The developed algorithm termed the hybrid non-dominated sorting differential evolution (h-NSDE) is an extension of the author’s previous algorithm for the single-machine mono-criterion TWT problem. h-NSDE is devoted to the search for Pareto-optimal solutions. To enable the decision maker for evaluating a greater number of alternative non-dominated solutions, three multiobjective optimization approaches have been implemented and tested within the context of h-NSDE: including a weighted-sum based approach, a fuzzy-measures based approach which takes into account the interaction among the criteria as well as a Pareto-based approach. Experiments conducted on existing data set benchmarks problems show the effect of these approaches on the performance of the h-NSDE algorithm. Moreover, comparative results between h-NSDE and some of the most popular multiobjective metaheuristics including SPEA2 and NSGA-II show clear superiority for h-NSDE in terms of both solution quality and solution diversity.

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  1. Available on the public OR-Library via http://people.brunel.ac.uk/~mastjjb/jeb/info.html.

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  1. Department of Business Administration, University of Patras, 26 500, Rio, Patras, Greece

    Andreas C. Nearchou

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Appendix

Appendix

1.1 The sub-range encoding method

Floating-point representation is the representation used by several well-known evolutionary algorithms (EAs) including evolution strategies, particle swarm optimization and differential evolution. These types of EAs were introduced for global optimization over continuous spaces and therefore their application to integer problems with discrete decision variables is not straightforward. In this appendix we present the sub-range encoding method, a special genotype–phenotype mapping method for encoding integer problems by using floating-point vectors.

Assuming a n-job single-machine scheduling problem, each individual solution in the genotypic level is encoded as a random n-dimensional floating-point vector. Sub-range encoding builds a schedule solution represented by permutation on the set {1, 2, …, n} of jobs’ indices by mapping the genotype’s components to unique integers. The general logic of sub-range encoding is as follows: first, the range [1…n] is divided into n equal sub-ranges and the upper bound of each sub-range is saved in an array of floating-point numbers called SR (stands for Sub-Range array). Hence, \( SR = \left[ {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{2 \mathord{\left/ {\vphantom {2 n}} \right. \kern-0pt} n},{3 \mathord{\left/ {\vphantom {3 n}} \right. \kern-0pt} n}, \ldots , \, {n \mathord{\left/ {\vphantom {n n}} \right. \kern-0pt} n}} \right]^{T} \). Then, a phenotype (a job sequence) for a particular genotype is constructed by examining the sub-range in which each allele (genotype’s component value) belongs to. The derived phenotype solution is finally checked for feasibility in order to present a valid job sequence solution. Below we describe this method for a 5-job single-machine scheduling problem. This means that, SR has the form:

$$ SR = [1 / 5 , { 2/5, 3/5, 4/5, 5/5}]^{T} = [ 0. 2 , { 0} . 4 , { 0} . 6 , { 0} . 8 , 1]^{T} . $$

Let us assume the genotype \( {\mathbf{x}} = \left( {0.81, \, 0.34, \, 0,12, \, 0.05, \, 0.66} \right) \). The phenotype corresponding to x is therefore build as in the following.

  1. 1.

    The 1st allele (= 0.81) lies in the 5th sub-range since 0.8 < 0.81 ≤ 1, therefore l = 5 and the generated proto-phenotype is (5 _ _ _ _).

  2. 2.

    The 2nd allele (= 0.34) lies in the 2nd sub-range (since 0.2 < 0.34 ≤ 0.4), i.e. the phenotype becomes (5 2 _ _ _).

  3. 3.

    The 3rd allele (= 0.12) lies in the 1st sub-range (0.0 < 0.12 ≤ 0.2), i.e. the phenotype becomes (5 2 1 _ _).

  4. 4.

    The 4th allele (= 0.05) lies in the 1st sub-range (0.0 < 0.05 ≤ 0.2), i.e. the phenotype becomes (5 2 1 1 _).

  5. 5.

    The 5th allele (= 0.66) lies in the 4th sub-range (0.6 < 0.66 ≤ 0.8), i.e. the phenotype becomes (5 2 1 1 4).

Obviously, the generated phenotype is illegal since it contains duplicated genes. To finally produce a valid version of the phenotype vector the following simple two-steps repairing procedure is applied on the proto-phenotype:

  1. (a)

    Delete the duplicate genes: (5 2 1 _ 4)

  2. (b)

    Fill the empty locations in the proto-phenotype with the remaining (unused) SR indices (follow an ascending order of the indices). Hence, the final legal schedule is (5, 2, 1, 3, 4). Meaning that the order in which the jobs are to be executed on the machine is, job 5, followed by job 2, followed by job 1 etc.

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Nearchou, A.C. Multicriteria scheduling optimization using an elitist multiobjective population heuristic: the h-NSDE algorithm. J Heuristics 24, 817–851 (2018). https://doi.org/10.1007/s10732-018-9378-9

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