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A comparative study of two optimization approaches for solving bi-level multi-objective linear fractional programming problem

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Abstract

Despite the important role of bi-level multi-objective linear fractional programming (BL-MOLFP) problem for many hierarchical organizations, a very little success has been achieved to deal with this problem. This paper presents a comparative study between two computational approaches, namely fuzzy TOPSIS (technique for order preference by similarity to ideal solution) approach and Jaya (a Sanskrit word meaning victory) approach, for solving BL-MOLFP problem. The fuzzy TOPSIS (FTOPSIS) approach aims to obtain the satisfactory solution of BL-MOLFP problem by using linearization process as well as formulating the membership functions for the distances of positive ideal solution (PIS) and negative ideal solution (NIS) for each level, respectively. In this sense, the deadlock situations among levels are avoided by establishing the membership functions for the upper level decision variables vector with possible tolerances. On the other hand, Jaya algorithm is proposed for solving BL-MOLFP problem based on nested structure scheme to optimize both levels hierarchically. An illustrative example is presented to describe the proposed approaches. In addition, the performances among the proposed approaches are assessed based on ranking strategy of the alternatives to affirm the superior approach. Based on the examined simulation, Jaya algorithm is preferable than the FTOPSIS approach.

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Abbreviations

BL-MOLFP:

Bi-level multi-objective linear fractional programming

TOPSIS:

Technique for order preference by similarity to ideal solution

FTOPSIS:

Fuzzy TOPSIS

PIS:

Positive ideal solution

NIS:

Negative ideal solution

MLPP:

Multi-level programming problem

BLPP:

Bi-level programming problem

FGP:

Fuzzy goal programming

MCDM:

Multiple criteria decision making

MOLFP:

Multi-objective linear fractional programming

LP:

Linear programming

LFP:

Linear fractional programming

SCCFP:

Standard concave-convex fractional programming

QFP:

Quadratic fractional programming

ULDM:

Upper-level decision maker

LLDM:

Lower- level decision maker

DMs:

Decision makers

MODM:

Multi-objective decision making

\(K\) :

No. of objectives (criteria)

\(G_{1} ,\,G_{2}\) :

The feasible regions of the upper and the lower level decision makers

\(n\) :

Number of control variables

\(q\) :

Number of constraints

\(F(x)\) :

Vector of objective functions

\(F_{{}}^{*}\) :

Reference point or ideal point or PIS

\(F^{ - }\) :

Nadir point or anti-ideal point or NIS

\(\lambda_{j}\) :

Relative significance (weights) of objectives

\(\mu_{1} ,\,\mu_{2}\) :

The membership functions of the two objective functions

\({\mathbf{x}}_{{\mathbf{1}}}\) :

Control vector of the upper-level DM

\({\mathbf{x}}_{{\mathbf{2}}}\) :

Control vector of the lower-level DM

\(\alpha ,\,\delta\) :

The satisfactory levels

\(f({\varvec{x}})\) :

The objective function to be minimized (or maximized)

PS:

Population size

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Acknowledgements

The authors express their gratitude to the editor and the anonymous reviewers for their insightful and constructive suggestions to improve the quality of this paper. First author is very thankful to her beloved wife Mariam Sabry for their endless support to throughout the writing of this paper

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Authors and Affiliations

  1. Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shibīn al-Kawm, Egypt

    Rizk M. Rizk-Allah

  2. Department of Basic Science, Faculty of Engineering, Badr University in Cairo, Badr City, Cairo, Egypt

    Mahmoud A. Abo-Sinna

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  1. Rizk M. Rizk-Allah
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RM designed the algorithm, implemented all experiments, wrote the manuscript, and finalized the manuscript.

MA revised the manuscript.

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Correspondence to Rizk M. Rizk-Allah.

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Rizk-Allah, R.M., Abo-Sinna, M.A. A comparative study of two optimization approaches for solving bi-level multi-objective linear fractional programming problem. OPSEARCH 58, 374–402 (2021). https://doi.org/10.1007/s12597-020-00486-1

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