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A time variant multi-objective particle swarm optimization algorithm for solving fuzzy number linear programming problems using modified Kerre’s method

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Abstract

Recently, Ghanbari et al. (IEEE Transactions on Fuzzy Systems 27:1286–1294, 2019) have proposed modified Kerre’s method for comparison of LR fuzzy numbers. Here, we make use of the modified Kerre’s method to solve fuzzy linear programming problems with LR coefficients. In an approach to solve a fuzzy linear program with fuzzy LR coefficients, a bi-objective optimization problem is formulated. For the associated bi-objective optimization problem, we present a time variant multi-objective particle swarm optimization (TV-MOPSO) algorithm to compute the Pareto front, a set containing a large number of solutions. Contrary to methods that change the fuzzy optimization problem to a crisp problem by use of a ranking function, using modified Kerre’s method, the fuzzy optimization problem is solved directly, with no need for changing it to a crisp program. A comparative investigation using illustrative examples with triangular fuzzy coefficients show the effectiveness of the proposed algorithm.

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Acknowledgements

The first and second authors thank the Research Council of Ferdowsi University of Mashhad and optimization laboratory of Ferdowsi University of Mashhad and the third author thanks the Research Council of Sharif University of Technology for supporting this work.

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Reza Ghanbari & Khatere Ghorbani-Moghadam

  2. Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Nezam Mahdavi-Amiri

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  1. Reza Ghanbari
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  2. Khatere Ghorbani-Moghadam
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Correspondence to Nezam Mahdavi-Amiri.

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Appendix

Appendix

Here, we show details of the generated test problems and the obtained Pareto fronts (we generated random test problems in Matlab 2015). For all the examples, triangular fuzzy random coefficients were generated uniformly in the interval [1, 100] for the fuzzy linear program posed as (P) in (4) with a given seed value. To generate the fuzzy random coefficients, first we generated three numbers in the interval [1, 100] and then these numbers were sorted in increasing order and used as the tripe for the fuzzy number. All the solutions were obtained by applying Algorithm 1. Fuzzy linear programming problem (P) is solved by methods of Mahavi-Amiri and Nasseri [46], Ganesan and Veeramani [31] and Ebrahimnejad and Nasseri [27] and the results are given in the Tables 1, 3, 5, 7, 9, 11, 13, 15, 17 and 19. In Tables 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20, the first column is the first objective function value of (BOOP) at a point of the Pareto front, the second column shows the second objective function value of (BOOP), the third column shows the comparison of the objective function values corresponding to the point in the Pareto front (ansP) and the one due to Mahavi-Amiri and Nasseri’s method (ansMN) using Theorem 2, the fourth column shows the comparison of the objective function values corresponding to the point in the Pareto front (ansP) and the one due to the Ganesan and Veeramani’s method (ansGV) using Theorem 2 and the fifth column shows the comparison of the objective function values corresponding to the point in the Pareto front (ansP) and the one due to the Ebrahimnejad and Naserri’s method (ansEN) using Theorem 2.

The objective function values for the examples 1–10 are less than the ones due to the answers obtained by Mahavi-Amiri and Nasseri [46], Ganesan and Veeramani [31] and Ebrahimnejad and Nasseri [27], as shown by the Tables 1–20.

Example 1

In Table 2, the results are shown corresponding to a problem with dimension \(2 \times 3\) (2 constraints and 3 variables) and the seed value of 156.

Table 1 The objective function values and penalty of different methods for the test problem 1
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Table 2 The Pareto front of test problem 1 obtained by TV-MOPSO
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Example 2

In Table 4, the results are shown corresponding to a problem with dimension \(3 \times 4\) (3 constraints and 4 variables) and the seed value of 100.

Table 3 The objective function values and penalty of different methods for the test problem 2
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Table 4 The Pareto front of test problem 2 obtained by TV-MOPSO
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Example 3

In Table 6, the results are shown corresponding to a problem with dimension \(5 \times 10\) (5 constraints and 10 variables) and the seed value of 460.

Table 5 The objective function values and penalty of different methods for the test problem 3
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Table 6 The Pareto front of test ptoblem 3 obtained by TV-MOPSO
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Example 4

In Table 8, the results are shown corresponding to a problem with dimension \(10 \times 15\) (10 constraints and 15 variables) and the seed value of 700.

Table 7 The objective function values and penalty of different methods for the test problem 4
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Table 8 The Pareto front of test problem 4 obtained by TV-MOPSO
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Example 5

In Table 10, the results are shown corresponding to a problem with dimension \(10 \times 20\) (10 constraints and 20 variables) and the seed value of 256.

Table 9 The objective function values and penalty of different methods for the test problem 5
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Table 10 The Pareto front of test problem 5 obtained by TV-MOPSO
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Example 6

In Table 12, the results are shown corresponding to a problem with dimension \(25 \times 50\) (25 constraints and 50 variables) and the seed value of 500.

Table 11 The objective function values and penalty of different methods for the test problem 6
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Table 12 The Pareto front of test problem 6 obtained by TV-MOPSO
Full size table

Example 7

In Table 14, the results are shown corresponding to a problem with dimension \(50 \times 100\) (50 constraints and 100 variables) and the seed value of 47.

Table 13 The objective function values and penalty of different methods for the test problem 7
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Table 14 The Pareto front of test problem 7 obtained by TV-MOPSO
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Example 8

In Table 16, the results are shown corresponding to a problem with dimension \(100 \times 200\) (100 constraints and 200 variables) and the seed value of 520.

Table 15 The objective function values and penalty of different methods for the test problem 8
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Table 16 The Pareto front of test problem 8 obtained by TV-MOPSO
Full size table

Example 9

In Table 20, the results are shown corresponding to a problem with dimension \(200 \times 300\) (200 constraints and 300 variables) and the seed value of 520.

Table 17 The objective function values and penalty of different methods for the test problem 9
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Table 18 The Pareto front of test problem 9 obtained by TV-MOPSO
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Example 10

In Table 20, the results are shown corresponding to a problem with dimension \(300 \times 400\) (300 constraints and 400 variables) and the seed value of 420.

Table 19 The objective function values and penalty of different methods for the test problem 10
Full size table
Table 20 The Pareto front of test problem 10 obtained by TV-MOPSO
Full size table

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Ghanbari, R., Ghorbani-Moghadam, K. & Mahdavi-Amiri, N. A time variant multi-objective particle swarm optimization algorithm for solving fuzzy number linear programming problems using modified Kerre’s method. OPSEARCH 58, 403–424 (2021). https://doi.org/10.1007/s12597-020-00482-5

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