Abstract
The paper proposed a method to study and obtain a set of Pareto optimal solutions or a set of representative solutions to a quadratically constrained multi-level multiobjective quadratic fractional programming problem. This problem involves several objectives to be fulfilled at multi levels under a common set of quadratic constraints. Initially, we used parametric approach to convert the fractional programming model to an equivalent non-fractional programming model by allocating a parametric vector to each fractional objective. Then, \(\varepsilon\)-constraint method is used to convert this multiobjective programming model into an equivalent model with single objective. The solution of every previous level is followed by the next level in succession to find a solution which is suitable to each level decision maker. An algorithm and numerical example are also presented at the end of the paper to validate the proposed methodology for the Model.
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References
Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13(7), 492–498 (1967)
Valiaho, H.: A unified approach to one-parametric general quadratic programming. Math. Program. 33(3), 318–338 (1985)
Wolf, H.: Solving special nonlinear fractional programming problems via parametric linear programming. Eur. J. Oper. Res. 23(3), 396–400 (1986)
Emam, O.E.: Interactive approach to bi-level integer multi-objective fractional programming problem. Appl. Math. Comput. 223, 17–24 (2013)
Emam, O.E.: Interactive bi-level multi-objective integer non-linear programming problem. Appl. Math. Sci. 5(65), 3221–3232 (2011)
Heesterman, A.R.G.: Parametric methods in quadratic programming. In: Matrices and simplex algorithms. pp. 516–555, Springer, Dordrecht (1983). https://doi.org/10.1007/978-94-009-7941-3_17
Osman, M.S., Emam, O.E., Sayed, M.A.E.: On parametric multi-level multi-objective fractional programming problems with fuzziness in the constraints. J. Adv. Math. Comput. Sci. 18(5), 1–19 (2016)
Borza, M., Rambely, A.S., Saraj, M.: Parametric approach for an absolute value linear fractional programming with interval coefficients in the objective function. AIP Conf. Proc. 1602(1), 415–421 (2014)
Nayak, S., Ojha, A.K.: Generating pareto optimal solutions of multi-objective lfpp with interval coefficients using \(\varepsilon\)-constraint method. Math. Model. Anal. 20(3), 329–345 (2015)
Nayak, S., Ojha, A.K.: Solution approach to multi-objective linear fractional programming problem using parametric functions. OPSEARCH 56(1), 174–190 (2019)
Almogy, Y., Levin, O.: A class of fractional programming problems. Oper. Res. 19(1), 57–67 (1971)
Ehrgott, M., Ruzika, S.: Improved \(\varepsilon\)-constraint method for multiobjective programming. J. Optim. Theory Appl. 138(3), 375–396 (2008)
Chircop, K., Zammit-Mangion, D.: On-constraint based methods for the generation of pareto frontiers. J. Mech. Eng. Autom. 3(5), 279–289 (2013)
Emmerich, M.T.M., Deutz, A.H.: A tutorial on multiobjective optimization: fundamentals and evolutionary methods. Nat. Comput. 17(3), 585–609 (2018)
Valipour, E., Yaghoobi, M.A., Mashinchi, M.: An approximation to the nondominated set of a multiobjective linear fractional programming problem. Optimization 65(8), 1539–1552 (2016)
Stanojević, B., Stanojević, M.: A computationally efficient algorithm to approximate the pareto front of multi-objective linear fractional programming problem. RAIRO Oper. Res. 53(4), 1229–1244 (2019)
Nikas, A., Fountoulakis, A., Forouli, A., Doukas, H.: A robust augmented \(\varepsilon\)-constraint method (augmecon-r) for finding exact solutions of multi-objective linear programming problems. Oper. Res. 1–42 (2020)
Zhong, Z., You, F.: Parametric solution algorithms for large-scale mixed-integer fractional programming problems and applications in process systems engineering. Comput. Aided Chem. Eng. 33, 259–264 (2014)
Baky, I.A.: Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Appl. Math. Model. 34(9), 57–67 (2010)
Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142(3), 444–462 (2002)
Stein, O.: Bi-level strategies in semi-infinite programming. In: Noncovex optimization and its applications, vol. 71, pp. 1–202. Springer Science & Business Media, New York (2013)
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Goyal, V., Rani, N. & Gupta, D. Parametric approach to quadratically constrained multi-level multi-objective quadratic fractional programming. OPSEARCH 58, 557–574 (2021). https://doi.org/10.1007/s12597-020-00497-y
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DOI: https://doi.org/10.1007/s12597-020-00497-y