This paper develops a method to solve multi-level multi-objective linear fractional programming problem (ML-MOLFPP) with interval parameters as the coefficients of decision variables and the constants involved in both the objectives and constraints. The objectives at each level are transformed into interval-valued fractional functions and approximated by intervals of linear functions using variable transformation and Taylor series expansion. Interval analysis and weighting sum method with analytic hierarchy process (AHP), are used to determine the non-dominated solutions at each level from which the aspiration values of the controlled decision variables are ascertained and linear fuzzy membership functions are constructed for all the objectives. Two multi-objective linear problems are equivalently formulated for the ML-MOLFPP with interval parameters and fuzzy goal programming is used to compute the optimal lower and upper bounds of all the objective values. A numerical example is solved to demonstrate the proposed solution approach.
Accepté le :
DOI : 10.1051/ro/2018063
Mots-clés : Multi-level multi-objective optimization, linear fractional programming, interval parameters, fuzzy goal programming
@article{RO_2019__53_5_1601_0, author = {Nayak, Suvasis and Ojha, Akshay}, title = {On multi-level multi-objective linear fractional programming problem with interval parameters}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {1601--1616}, publisher = {EDP-Sciences}, volume = {53}, number = {5}, year = {2019}, doi = {10.1051/ro/2018063}, mrnumber = {4016087}, zbl = {1431.90155}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ro/2018063/} }
TY - JOUR AU - Nayak, Suvasis AU - Ojha, Akshay TI - On multi-level multi-objective linear fractional programming problem with interval parameters JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2019 SP - 1601 EP - 1616 VL - 53 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2018063/ DO - 10.1051/ro/2018063 LA - en ID - RO_2019__53_5_1601_0 ER -
%0 Journal Article %A Nayak, Suvasis %A Ojha, Akshay %T On multi-level multi-objective linear fractional programming problem with interval parameters %J RAIRO - Operations Research - Recherche Opérationnelle %D 2019 %P 1601-1616 %V 53 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2018063/ %R 10.1051/ro/2018063 %G en %F RO_2019__53_5_1601_0
Nayak, Suvasis; Ojha, Akshay. On multi-level multi-objective linear fractional programming problem with interval parameters. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1601-1616. doi : 10.1051/ro/2018063. http://www.numdam.org/articles/10.1051/ro/2018063/
[1] Interactive balance space approach for solving multi-level multi-objective programming problems. Inform. Sci. 177 (2007) 3397–3410. | DOI | Zbl
and ,[2] Fuzzy goal programming procedure to bilevel multiobjective linear fractional programming problems. Int. J. Math. Math. Sci. 2010 (2010) 148975. | MR | Zbl
and ,[3] Interactive fuzzy programming for decentralized two-level linear fractional programming (dtllfp) problems. Omega 35 (2007) 432–450. | DOI
and ,[4] Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems. Fuzzy Sets Syst. 160 (2009) 2701–2713. | DOI | MR | Zbl
,[5] Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Appl. Math. Modell. 34 (2010) 2377–2387. | DOI | MR | Zbl
,[6] Interactive topsis algorithms for solving multi-level non-linear multi-objective decision-making problems. Appl. Math. Model. 38 (2014) 1417–1433. | DOI | MR | Zbl
,[7] Decision-making in a fuzzy environment. Manag. Sci. 17 (1970) B–141. | DOI | MR | Zbl
and ,[8] Linear programming with a fractional objective function. Oper. Res. 21 (1973) 22–29. | DOI | MR | Zbl
and ,[9] Management models and industrial applications of linear programming. Manag. Sci. 4 (1957) 38–91. | DOI | MR | Zbl
and ,[10] Solving the linear fractional programming problem in a fuzzy environment: Numerical approach. Appl. Math. Model. 40 (2016) 6148–6164. | DOI | MR | Zbl
, ,[11] Fractional Programming. Heldermann Verlag, Berlin (1988). | MR | Zbl
,[12] A note on the analysis of subjective judgment matrices. J. Math. Psycho. 29 (1985) 387–405. | DOI | Zbl
and ,[13] On nonlinear fractional programming. Manag. Sci. 13 (1967) 492–498. | DOI | MR | Zbl
,[14] The Analytic Hierarchy Process. Springier-Verlag, New York (1989). | DOI
, and ,[15] Attrition games, Naval Res. Logist. Quart. 3 (1956) 71–94. | DOI | MR | Zbl
and ,[16] An alternative scaling method for priorities in hierarchical structures. J. Math. Psycho. 28 (1984) 317–332. | DOI
,[17] On solving multi-level multi objective linear programming problems through fuzzy goal programming approach. Opsearch 51 (2014) 624–637. | DOI | MR | Zbl
,[18] Modified fgp approach for multi-level multi objective linear fractional programming problems. Appl. Math. Comput. 266 (2015) 1038–1049. | DOI | MR | Zbl
,[19] Geometric programming with fuzzy parameters in engineering optimization. Int. J. Approx. Reason. 46 (2007) 484–498. | DOI | Zbl
,[20] Nonlinear Multiobjective Optimization. Springer Science & Business Media. Vol. 12 (2012). | Zbl
,[21] Weighting method for bi-level linear fractional programming problems. Eur. J. Oper. Res. 183 (2007) 296–302. | DOI | Zbl
,[22] The relationship between goal programming and fuzzy programming. Fuzzy Sets Syst. 89 (1997) 215–222. | DOI | MR
,[23] Interval Analysis. Prince-Hall, Englewood Cliffs, NJ (1966). | MR | Zbl
,[24] A multi-level non-linear multi-objective decision-making under fuzziness. Appl. Math. Comput. 153 (2004) 239–252. | DOI | MR | Zbl
, , and ,[25] Fuzzy goal programming approach to multilevel programming problems. Euro. J. Oper. Res. 176 (2007) 1151–1166. | DOI | Zbl
and ,[26] Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. Fuzzy Sets Syst. 115 (2000) 93–103. | DOI | MR | Zbl
, and ,[27] Fuzzy approach for multi-level programming problems. Comp. & Oper. Res. 23 (1996) 73–91. | DOI | MR | Zbl
, and ,[28] Fractional Programming: Theory, Methods and Applications. Kluwer Academic Publishers, South Holland (1997). | DOI | MR | Zbl
,[29] Taylor series approach to fuzzy multiobjective linear fractional programming. Inform. Sci. 178 (2008) 1189–1204. | DOI | MR | Zbl
,[30] Taylor series approach for bi-level linear fractional programming problem. Selcuk University Research Center of Applied Mathematics, Konya (2010). | Zbl
,[31] Interactive fuzzy goal programming based on jacobian matrix to solve decentralized bi-level multi-objective fractional programming problems. Int. J. Fuzzy Syst. 17 (2015) 499–508. | DOI | MR
and ,[32] On interval-valued nonlinear programming problems. J. Math. Anal. Appl. 338 (2008) 299–316. | DOI | MR | Zbl
,[33] Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1 (1978) 45–55. | DOI | MR | Zbl
,[34] Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Boston (1985). | DOI | MR | Zbl
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