TOPSIS Approach for Solving Bi-Level Non-Linear Fractional MODM Problems
DOI:
https://doi.org/10.24297/jam.v13i4.6243Keywords:
Multi-objective decision making, Bi-level programming problems, Fractional programming, TOPSIS, Fuzzy goal programmingAbstract
TOPSIS (technique for order preference similarity to ideal solution) is considered one of the known classical multiple criteria decision making (MCDM) methods to solve bi-level non-linear fractional multi-objective decision making (BL-NFMODM) problems, and in which the objective function at each level is considered nonlinear and maximization type fractional functions. The proposed approach presents the basic terminology of TOPSIS approach and the construction of membership function for the upper level decision variable vectors, the membership functions of the distance functions from the positive ideal solution (PIS) and of the distance functions from the negative ideal solution (NIS). Thereafter a fuzzy goal programming model is adopted to obtain compromise optimal solution of BL-NFMODM problems. The proposed approach avoids the decision deadlock situations in decision making process and possibility of rejecting the solution again and again by lower level decision makers. The presented TOPSIS technique for BL-NFMODM problems is a new fuzzy extension form of TOPSIS approach suggested by Baky and Abo-Sinna (2013) (Applied Mathematical Modelling, 37, 1004-1015, 2013) which dealt with bi -level multi-objective decision making (BL-MODM) problems. Also, an algorithm is presented of the new fuzzy TOPSIS approach for solving BL-NFMODM problems. Finally, an illustrative numerical example is given to demonstrate the approach.
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References
[1] C.L. Hwang, K. Yoon., Multiple Attribute Decision Making: Methods and Applications., Springer-Verlag, Heidelberg,(1981).
[2] M.A. Abo-Sinna., Extensions of the TOPSIS for Multi-Objective Dynamic Programming Problems under Fuzziness., Adv. Model. Anal. (AMSE Press, France)., 43 (4), 1-24, (2000).
[3] M. A. Abo-Sinna, A. H. Amer., Extensions of TOPSIS for Multi Objective Large Secale Nonlinear Programming Problems., Appl. Math- Comput.., 162, 243 – 256, (2000).
[4] M.A. Sbo-Sinna, A.H. Amer, A.S. Ibrahim., Extensions of TOPSIS for Large Scale Multi-Objective Non-Linear Programming problems with Block Angular Structure., Appl. Math. Model.., 32, 292-302, (2008).
[5] Y. J. Lai, T.J. Liu, C.L. Hwang., TOPSIS for MODM., Euro. J. Oper. Res.., 76, 486-500, (1994).
[6] M. Zeleny., Multiple Criteria Decision Making., McGraw, Hill Book Company, New York, (1982).
[7] I.A. Baky, M.A. Abo-Sinna., TOPSIS for Bi-Level MODM Problems., Applied Mathematical Modelling., 37, 1004-1015,(2013).
[8] H. Deng, C.H. Yeh, R.J. Willis., Inter-Company Comparison using Modified TOPSIS with Objective Weights., Comput. Oper. Res.., 17, 963-973,(2000).
[9] C.T. Chen., Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment., Fuzzy Sets and Systems., 114, 1-9, (2000).
[10] L.N. Vicent, and P.H. Calamai., Bi-level and Multi-level Programming., a Bibliography Review, J. Global Optim.., 5, 291-306, (1994).
[11] K. Mathur, M.C. Puri., A bi-level Bottleneck Programming Problem., European Journal of Oprational Research., 86, 337-344,(1995).
[12] G. Anandilingam., A Mathematical Programming Model of Decentralized Multi-Level System., Journal of the Operational Research Society., 39 (11), 1021-1033, (1988).
[13] G. Anandilingam, V. Apprey., Multi-Level Programming and Conflicting Resolution., European Journalof Operational Research., 51, 233-247, (1991).
[14] S.R. Arora, R. Gupta., Interactive Fuzzy Goal Programming Approach for Bi-Level Programming Problem., European Journal of Operational Research., 194, 368-376, (2009).
[15] S. Pramanik,T. Kumar Roy., Fuzzy Goal Programming Approach to Multi-Level Programming problems., Euro. J. Oper. Res.., 176, 1151-1166, (2006).
[16] H.S. Shih, Y.J. Lai, E.S. Lee., Fuzzy Approach for Multi-Level Programming Problems., Computers and Operations Research., 23 (1), 773 -791, (1983).
[17] H. S. Shih, E.S,. Lee., Compensatory Fuzzy Multiple Decision Making., Fuzzy Sets and Systems., 14, 71- 87, (2000).
[18] S. Sinha., Fuzzy Mathematical Approach to Multi-Level Programming Problems., Computers and Operations Research., 30, 1259-1268, (2003).
[19] S. Sinha., Fuzzy Programming Approach to Multi-Level Programming Problems., Fuzzy Sets and Systems., 136, 189-202, (2003).
[20] W. Candler, R. Townsley., A Linear Two Level Programming Problem., Comput. Oper Res.., 9 (1), 59-76, (1982).
[21] J. Bracken, J.E. Falk, F.A. Miercort., A Strategic Weapons Exchange Allocation Model., Oper. Res.., 25, 968-976, (1977).
[22] R.G. Cassidy, M.J.L. Kirby, W.H. Raike., Efficient Distribution of Resources through three Levels of Government., Manag. Sci.., 17, 462 -473, (1971).
[23] R.E. Bellmann., Dynamic Programming., Princeton University Press, Princeron, NJ, (1957).
[24] I.A. Baky., Solving Multi-Level Multi-Objective Linear Programming Problems through Fuzzy Goal Programming Approach., Applied Mothematical Modelling., 34,2377-2387, (2010).
[25] M.A. Abo-Sinna, I.A. Baky., Interactive Balance Space Approach for Solving Multi-Level Multi-Objective Programming Problems., Inform. Sci.., 177,3397- 3410, (2007).
[26] I.A. Baky., Fuzzy Goal Programming Algorithm for Solving Decentralized Bi-Level Multi-Objective Programming Problems., Fuzzy Sets and Systems., 160, 2701-2710, (2009).
[27] K. Lachhwani., On Solving Multi-Level Multi-Objective Linear Fractional Programming Problems through Fuzzy Goal Programming Approach., Journal of Operational Research Society of India OPSEARCH.,51 (4), 624-637, (2014).
[28] K. Lachhwani., Modified FGP Approach for Multi-Level Mult- Objective Linear Fractional Programming Problems., Applied Mathematics and Computation., 266, 1038-1049, (2015).
[29] M.A. Abo-Sinna, I.A. Baky., Interactive Balance Space Approach for Solving Bi-Level Multi-Objective Programming Problems., AMSE- Model, Adv. BJ., 40, 43-62. France, (2006).
[30] Dr. Madhuchanda Rakshit, Mr. Suchet Kumar., A Bi-Level Quadratic- Quadratic Fractional Programming through Fuzzy Goal Programming Approach., IJMTT, 38,3,October, (2016).
[31] Deepak Bhati, Pitam Singh and Rubi Arya., A Taxonomy and Review of the Multi-Objective Fractional Programming (MOFP) Problems., Int.J. Appl. Comput. Math.., 21, October, (2016).
[32] F. Chen, G.H. Huang, Y.R. Fan and R. F.Lias., A Non-linear Fractional Programming Approach for Environmental-Economic Power Dispatch., Electrical Power and Energy Systems., 78, 463-469, (2016).
[33] Dr. Savita Mishra., Arun Bihari Verma and Indrani Dey., Quadratic Fractional Multi-Level Programming Problem Based on Fuzzy Goal Programming Approach., V.1 Special Issue, July, (2016).
[34] Nocera Maachou and Mustapha Moulai., Bi-Level Quadratic Fractional/ Quadratic Problem., Springer Int. Publishing Switzerland, (2015).
[35] K.C. Sharma and Jitendra Singh., Quadratic Fractional Optimization through FGP Approach., Int. Journal of Science and Research (IJSR)., V.3, Issues 5, May, (2014).
[36] Savita Mishra and Ajit Ghosh., Interactive Fuzzy Programming Approach to Bi-Level Quadratic Fractional Programming Problems., Ann. Oper. Res.., 143, 251-263, (2006).
[37] A. Charnes, W. Cooper., Programming with Linear Fractional Functions., Naval Research Logistic Quartely., 9(3-4), 181-186, (1962).
[38] S. Mishra. A. Ghosh., Interactive Fuzzy Programming Approach to Bi-Level Quadratic Fractional Programming Problems., Annals of Operations Research., 143, 251-263, (2006).
[39] K. Miettinen., Non-linear Multi-Objective Optimization., Kluwer Academic Publishers, (2002).
[40] P.L. YM, M. Zeleny., the Set of All Non-Dominated Solution in Linear Cases and Multi-Criteria Simplex Method., J. Math. Analy. Appl.., 49, 430-448, (1975).
[41] R.E. Bellmann, L.A. Zadeh., Decision-Making in a Fuzzy Environment, Manag., Sci.., 17, 141-164, (1970).
[42] H.J. Zimmermann., Fuzzy Programming and Linear Programming with Several Objective Functions., Fuzzy Sets Syst.., 1, 45-55, (1978).
[43] H.J. Zimmermann., Fuzzy Sets Decision Making and Expert Systems., Kluwer Academic, Boston, (1987).
[44] Y. J. Lai, C.L. Hwang., A New Approach to Some Possibility Linear Programming Problems., Fuzzy Sets Syst.., 49, 121-134, (1992).
[1] C.L. Hwang, K. Yoon., Multiple Attribute Decision Making: Methods and Applications., Springer-Verlag, Heidelberg,(1981).
[2] M.A. Abo-Sinna., Extensions of the TOPSIS for Multi-Objective Dynamic Programming Problems under Fuzziness., Adv. Model. Anal. (AMSE Press, France)., 43 (4), 1-24, (2000).
[3] M. A. Abo-Sinna, A. H. Amer., Extensions of TOPSIS for Multi Objective Large Secale Nonlinear Programming Problems., Appl. Math- Comput.., 162, 243 – 256, (2000).
[4] M.A. Sbo-Sinna, A.H. Amer, A.S. Ibrahim., Extensions of TOPSIS for Large Scale Multi-Objective Non-Linear Programming problems with Block Angular Structure., Appl. Math. Model.., 32, 292-302, (2008).
[5] Y. J. Lai, T.J. Liu, C.L. Hwang., TOPSIS for MODM., Euro. J. Oper. Res.., 76, 486-500, (1994).
[6] M. Zeleny., Multiple Criteria Decision Making., McGraw, Hill Book Company, New York, (1982).
[7] I.A. Baky, M.A. Abo-Sinna., TOPSIS for Bi-Level MODM Problems., Applied Mathematical Modelling., 37, 1004-1015,(2013).
[8] H. Deng, C.H. Yeh, R.J. Willis., Inter-Company Comparison using Modified TOPSIS with Objective Weights., Comput. Oper. Res.., 17, 963-973,(2000).
[9] C.T. Chen., Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment., Fuzzy Sets and Systems., 114, 1-9, (2000).
[10] L.N. Vicent, and P.H. Calamai., Bi-level and Multi-level Programming., a Bibliography Review, J. Global Optim.., 5, 291-306, (1994).
[11] K. Mathur, M.C. Puri., A bi-level Bottleneck Programming Problem., European Journal of Oprational Research., 86, 337-344,(1995).
[12] G. Anandilingam., A Mathematical Programming Model of Decentralized Multi-Level System., Journal of the Operational Research Society., 39 (11), 1021-1033, (1988).
[13] G. Anandilingam, V. Apprey., Multi-Level Programming and Conflicting Resolution., European Journalof Operational Research., 51, 233-247, (1991).
[14] S.R. Arora, R. Gupta., Interactive Fuzzy Goal Programming Approach for Bi-Level Programming Problem., European Journal of Operational Research., 194, 368-376, (2009).
[15] S. Pramanik,T. Kumar Roy., Fuzzy Goal Programming Approach to Multi-Level Programming problems., Euro. J. Oper. Res.., 176, 1151-1166, (2006).
[16] H.S. Shih, Y.J. Lai, E.S. Lee., Fuzzy Approach for Multi-Level Programming Problems., Computers and Operations Research., 23 (1), 773 -791, (1983).
[17] H. S. Shih, E.S,. Lee., Compensatory Fuzzy Multiple Decision Making., Fuzzy Sets and Systems., 14, 71- 87, (2000).
[18] S. Sinha., Fuzzy Mathematical Approach to Multi-Level Programming Problems., Computers and Operations Research., 30, 1259-1268, (2003).
[19] S. Sinha., Fuzzy Programming Approach to Multi-Level Programming Problems., Fuzzy Sets and Systems., 136, 189-202, (2003).
[20] W. Candler, R. Townsley., A Linear Two Level Programming Problem., Comput. Oper Res.., 9 (1), 59-76, (1982).
[21] J. Bracken, J.E. Falk, F.A. Miercort., A Strategic Weapons Exchange Allocation Model., Oper. Res.., 25, 968-976, (1977).
[22] R.G. Cassidy, M.J.L. Kirby, W.H. Raike., Efficient Distribution of Resources through three Levels of Government., Manag. Sci.., 17, 462 -473, (1971).
[23] R.E. Bellmann., Dynamic Programming., Princeton University Press, Princeron, NJ, (1957).
[24] I.A. Baky., Solving Multi-Level Multi-Objective Linear Programming Problems through Fuzzy Goal Programming Approach., Applied Mothematical Modelling., 34,2377-2387, (2010).
[25] M.A. Abo-Sinna, I.A. Baky., Interactive Balance Space Approach for Solving Multi-Level Multi-Objective Programming Problems., Inform. Sci.., 177,3397- 3410, (2007).
[26] I.A. Baky., Fuzzy Goal Programming Algorithm for Solving Decentralized Bi-Level Multi-Objective Programming Problems., Fuzzy Sets and Systems., 160, 2701-2710, (2009).
[27] K. Lachhwani., On Solving Multi-Level Multi-Objective Linear Fractional Programming Problems through Fuzzy Goal Programming Approach., Journal of Operational Research Society of India OPSEARCH.,51 (4), 624-637, (2014).
[28] K. Lachhwani., Modified FGP Approach for Multi-Level Mult- Objective Linear Fractional Programming Problems., Applied Mathematics and Computation., 266, 1038-1049, (2015).
[29] M.A. Abo-Sinna, I.A. Baky., Interactive Balance Space Approach for Solving Bi-Level Multi-Objective Programming Problems., AMSE- Model, Adv. BJ., 40, 43-62. France, (2006).
[30] Dr. Madhuchanda Rakshit, Mr. Suchet Kumar., A Bi-Level Quadratic- Quadratic Fractional Programming through Fuzzy Goal Programming Approach., IJMTT, 38,3,October, (2016).
[31] Deepak Bhati, Pitam Singh and Rubi Arya., A Taxonomy and Review of the Multi-Objective Fractional Programming (MOFP) Problems., Int.J. Appl. Comput. Math.., 21, October, (2016).
[32] F. Chen, G.H. Huang, Y.R. Fan and R. F.Lias., A Non-linear Fractional Programming Approach for Environmental-Economic Power Dispatch., Electrical Power and Energy Systems., 78, 463-469, (2016).
[33] Dr. Savita Mishra., Arun Bihari Verma and Indrani Dey., Quadratic Fractional Multi-Level Programming Problem Based on Fuzzy Goal Programming Approach., V.1 Special Issue, July, (2016).
[34] Nocera Maachou and Mustapha Moulai., Bi-Level Quadratic Fractional/ Quadratic Problem., Springer Int. Publishing Switzerland, (2015).
[35] K.C. Sharma and Jitendra Singh., Quadratic Fractional Optimization through FGP Approach., Int. Journal of Science and Research (IJSR)., V.3, Issues 5, May, (2014).
[36] Savita Mishra and Ajit Ghosh., Interactive Fuzzy Programming Approach to Bi-Level Quadratic Fractional Programming Problems., Ann. Oper. Res.., 143, 251-263, (2006).
[37] A. Charnes, W. Cooper., Programming with Linear Fractional Functions., Naval Research Logistic Quartely., 9(3-4), 181-186, (1962).
[38] S. Mishra. A. Ghosh., Interactive Fuzzy Programming Approach to Bi-Level Quadratic Fractional Programming Problems., Annals of Operations Research., 143, 251-263, (2006).
[39] K. Miettinen., Non-linear Multi-Objective Optimization., Kluwer Academic Publishers, (2002).
[40] P.L. YM, M. Zeleny., the Set of All Non-Dominated Solution in Linear Cases and Multi-Criteria Simplex Method., J. Math. Analy. Appl.., 49, 430-448, (1975).
[41] R.E. Bellmann, L.A. Zadeh., Decision-Making in a Fuzzy Environment, Manag., Sci.., 17, 141-164, (1970).
[42] H.J. Zimmermann., Fuzzy Programming and Linear Programming with Several Objective Functions., Fuzzy Sets Syst.., 1, 45-55, (1978).
[43] H.J. Zimmermann., Fuzzy Sets Decision Making and Expert Systems., Kluwer Academic, Boston, (1987).
[44] Y. J. Lai, C.L. Hwang., A New Approach to Some Possibility Linear Programming Problems., Fuzzy Sets Syst.., 49, 121-134, (1992).
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Published
2018-02-09
How to Cite
ABO-SINNA, M. A., & AMER, A. H. (2018). TOPSIS Approach for Solving Bi-Level Non-Linear Fractional MODM Problems. JOURNAL OF ADVANCES IN MATHEMATICS, 13(4), 7353–7370. https://doi.org/10.24297/jam.v13i4.6243
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