Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment

ABSTRACT

This paper is comprised of the modeling and optimization of a multi objective linear programming problem in fuzzy environment in which some goals are fractional and some are linear. Here, we present a new approach for its solution by using α-cut of fuzzy numbers. In this proposed method, we first define membership function for goals by introducing non-deviational variables for each of objective functions with effective use of α-cut intervals to deal with uncertain parameters being represented by fuzzy numbers. In the optimization process the under deviational variables are minimized for finding a most satisfactory solution. The developed method has also been implemented on a problem for illustration and comparison.

This paper is comprised of the modeling and optimization of a multi objective linear programming problem in fuzzy environment in which some goals are fractional and some are linear. Here, we present a new approach for its solution by using α-cut of fuzzy numbers. In this proposed method, we first define membership function for goals by introducing non-deviational variables for each of objective functions with effective use of α-cut intervals to deal with uncertain parameters being represented by fuzzy numbers. In the optimization process the under deviational variables are minimized for finding a most satisfactory solution. The developed method has also been implemented on a problem for illustration and comparison.

KEYWORDS

Fuzzy Sets, Trapezoidal Fuzzy Number (TFN), Multi-Objective Linear Programming Problem (MOLPP), Multi-Objective Linear Fractional Programming Problem (MOLFPP)

Fuzzy Sets, Trapezoidal Fuzzy Number (TFN), Multi-Objective Linear Programming Problem (MOLPP), Multi-Objective Linear Fractional Programming Problem (MOLFPP)

Cite this paper

Nishad, A. and Singh, S. (2015) Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment.*Applied Mathematics*, **6**, 2360-2374. doi: 10.4236/am.2015.614208.

Nishad, A. and Singh, S. (2015) Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment.

References

[1] Charnes, A. and Cooper, W.W. (1968) Management Models of Industrial Applications of Linear Program. Vol. 1-2, Wiley, New York.

[2] Lee, S.M. (1972) Goal Programming for Decision Analysis. Auerbach Publishers, Philadelphia.

[3] Ignizio, J.P. (1976) Goal Programming and Extensions. D.C. Heath, Lexington.

[4] Ignizio, J.P. (1982) Notes and Communications on the (Re)Discovery of Fuzzy Goal Programming. Decision Sciences, 13, 331-336.

http://dx.doi.org/10.1111/j.1540-5915.1982.tb00152.x

[5] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[6] Zimmermann, H.J. (1978) Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems, 1, 45-55.

http://dx.doi.org/10.1016/0165-0114(78)90031-3

[7] Narasimhan, R. (1980) Goal Programming in a Fuzzy Environment. Decision Science, 11, 325-336.

http://dx.doi.org/10.1111/j.1540-5915.1980.tb01142.x

[8] Narasimhan, R. (1981) On Fuzzy Goal Programming-Some Comments. Decision Science, 12, 532-538.

http://dx.doi.org/10.1111/j.1540-5915.1981.tb00103.x

[9] Bellman, R.E. and Zadeh, L.A. (1970) Decision Making in a Fuzzy Environment. Management Sciences, 17, 141-164.

http://dx.doi.org/10.1287/mnsc.17.4.B141

[10] Mohanty, B.K. and Vijayaraghawan, T.A.S. (1995) A Multiobjective Programming Problem and Its Equivalent Goal Programming Problem with Approprite Priorities and Aspiration Levels: A Fuzzy Approach. Computers & Operations Research, 22, 771-778.

http://dx.doi.org/10.1016/0305-0548(94)00066-H

[11] Kuwano, H. (1996) On the Fuzzy Multi-Objective Linear Programming Problems: Goal Programming Approach. Fuzzy Sets and Systems, 82, 57-64.

http://dx.doi.org/10.1016/0165-0114(95)00231-6

[12] Chanas, A. and Kuchta, D. (1996) Multiobjective Programming in Optimization of Interval Objective Functions—A Generalized Approach. European Journal of Operational Research, 94, 594-598.

http://dx.doi.org/10.1016/0377-2217(95)00055-0

[13] Chen, L.H. and Tsai, F.C. (2001) Fuzzy Goal Programming with Different Importance and Priorties. European Journal of Operational Research, 133, 548-556.

http://dx.doi.org/10.1016/S0377-2217(00)00201-0

[14] Stanciulescu, C., Fortemps, P., Installe, M. and Wertz, V. (2003) Multi-Objective Fuzzy Linear Programming Problems with Fuzzy Decision Variables. European Journal of Operational Research, 149, 654-675.

http://dx.doi.org/10.1016/S0377-2217(02)00449-6

[15] Lin, C.C. (2004) A Weighted Max-Min Model Fuzzy Goal Programming. Fuzzy Sets and Systems, 142, 407-420.

http://dx.doi.org/10.1016/S0165-0114(03)00092-7

[16] Yaghoobi, M.A. and Tamiz, M. (2007) A Method for Solving Fuzzy Goal Programming Problems Based on MINMAX Approach. European Journal of Operational Research, 177, 1580-1590.

[17] Cheng, H.F., Huang, W.L., Zhon, Q. and Cai, J. (2013) Solving Fuzzy Multiobjective Linear Programming Problems Using Deviation Degree Measures and Max-Min Method. Applied Mathematical Modeling, 73, 6855-6869.

http://dx.doi.org/10.1016/j.apm.2013.01.048

[18] Soliman, O., Sarker Rahul, A. and Zahir, S. (2008) Fuzzy Goal Programming Model with Parametric Analysis for Regional Sustainability Development under Climate Change: A Case of Agriculture Sector. ASOR Bulletin, 27, 2-15.

[19] Mishra, B. and Singh, S.R. (2013) Linear Fractional Programming Procedure for Multi Objective Linear Programming Problem in Agriculture System. International Journal of Computer Applications, 61, 45-52.

[20] Bharati, S.K., Nishad, A.K. and Singh, S.R. (2014) Solution of Multi-Objective Linear Programming under Intuitionistic Fuzzy Environment. In: Babu, B.V., Nagar, A., Deep, K., Pant, M., Bansal, J.C., Ray, K. and Gupta, U., Eds., Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, Advances in Intelligent Systems and Computing, Vol. 236, Springer India, New Delhi, 161-171.

http://dx.doi.org/10.1007/978-81-322-1602-5_18

[21] Sen, S. and Pal, B.B. (2013) Interval Goal Programming Approach to Multiobjective Fuzzy Goal Programming Problems with Interval Weights. Procedia Technology, 10, 587-595.

http://dx.doi.org/10.1016/j.protcy.2013.12.399

[22] Seyed, H. and Razavi, H. (2013) Multi-Objective Linear Programming with Interval Coefficients: A Fuzzy Set Based Approach. Emerald, 42, 482-496.

[23] Azzabi, L., Dorra, A., Kaddour, B. and Abdessamad, K. (2014) Fuzzy Goal Programming to Optimization the Multi-Objective Problem. Science Journal of Applied Mathematics and Statistics, 2, 14-19.

http://dx.doi.org/10.11648/j.sjams.20140201.12

[24] Marbini, A.H. and Tavana, M. (2011) An Extension of the Linear Programming Method with Fuzzy Parameters. International Journal of Mathematics in Operational Research, 3, 44-55.

[25] Luhandjula, M.K. (1984) Fuzzy Approaches for Multiple Objective Linear Fractional Optimization. Fuzzy Sets and Systems, 13, 11-23.

http://dx.doi.org/10.1016/0165-0114(84)90023-X

[26] Chakraborty, M. and Gupta, S. (2002) Fuzzy Mathematical Programming for Multi Objective Linear Fractional Programming Problem. Fuzzy Sets and Systems, 125, 335-342.

http://dx.doi.org/10.1016/S0165-0114(01)00060-4

[27] Pal, B.B., Moitra, B.N. and Maulik, U. (2003) A Goal Programming Procedure for Fuzzy Multiobjective Linear Fractional Programming Problem. Fuzzy Sets and Systems, 139, 395-405.

http://dx.doi.org/10.1016/S0165-0114(02)00374-3

[28] Pop, B., Minasian, I. and Stancu, M. (2008) A Method of Solving Fully Fuzzified Linear Fractional Programming Problems. Journal of Applied Mathematics and Computing, 27, 227-242.

http://dx.doi.org/10.1007/s12190-008-0052-5

[29] Cui, Y.H., Qu, J., Peng, Y., Wang, L. and Li, B. (2010) The Study of the Solution on Multi-Objective Linear Programming Problem under Fuzzy. 2010 Asia-Pacific Conference on Wearable Computing Systems (APWCS), Shenzhen, 17-18 April 2010, 286-290.

http://dx.doi.org/10.1109/APWCS.2010.79

[30] Mehrjerdi, Y.Z. (2011) Solving Fractional Programming Problem through Fuzzy Goal Setting and Approximation. Applied Soft Computing, 11, 1735-1742.

[31] Pitam, S., Kumar, S.D. and Singh, R.K. (2011) Fuzzy Multiobjective Linear plus Linear Fractional Programming Problem: Approximation and Goal Programming Approach. International Journal of Mathematics and Computers in Simulation, 5, 395-404.

[32] Animesh, B. and Dewan, S. (2012) Priority Based Fuzzy Goal Programming Technique to Fractional Fuzzy Goals Using Dynamic Programming. Fuzzy Information and Engineering, 4, 165-180.

[33] Ohta, H. and Yamaguchi, T. (1996) Linear Fractional Goal Programming in Consideration of Fuzzy Solution. European Journal of Operation Research, 92, 157-165.

http://dx.doi.org/10.1016/0377-2217(95)00052-6

[34] Pal, B.B. and Sen, S. (2008) A Goal Programming Procedure for Solving Interval Valued Multi-Objective Fractional Programming Problems. 16th International Conference on Advanced Computing and Communications, ADCOM 2008, Chennai, 14-17 December 2008, 297-302.

[35] Sohrab, E. and Morteza, P. (2012) Solving the Interval Valued Linear Programming Problem. American Journal of Computational Mathematics, 2, 51-55.

http://dx.doi.org/10.4236/ajcm.2012.21006

[36] Kornbluth, J.S.H. and Steuer, R.E. (1981) Multiple Objective Linear Fractional Programming. Management Science, 27, 1024-1039.

http://dx.doi.org/10.1287/mnsc.27.9.1024

[1] Charnes, A. and Cooper, W.W. (1968) Management Models of Industrial Applications of Linear Program. Vol. 1-2, Wiley, New York.

[2] Lee, S.M. (1972) Goal Programming for Decision Analysis. Auerbach Publishers, Philadelphia.

[3] Ignizio, J.P. (1976) Goal Programming and Extensions. D.C. Heath, Lexington.

[4] Ignizio, J.P. (1982) Notes and Communications on the (Re)Discovery of Fuzzy Goal Programming. Decision Sciences, 13, 331-336.

http://dx.doi.org/10.1111/j.1540-5915.1982.tb00152.x

[5] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[6] Zimmermann, H.J. (1978) Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems, 1, 45-55.

http://dx.doi.org/10.1016/0165-0114(78)90031-3

[7] Narasimhan, R. (1980) Goal Programming in a Fuzzy Environment. Decision Science, 11, 325-336.

http://dx.doi.org/10.1111/j.1540-5915.1980.tb01142.x

[8] Narasimhan, R. (1981) On Fuzzy Goal Programming-Some Comments. Decision Science, 12, 532-538.

http://dx.doi.org/10.1111/j.1540-5915.1981.tb00103.x

[9] Bellman, R.E. and Zadeh, L.A. (1970) Decision Making in a Fuzzy Environment. Management Sciences, 17, 141-164.

http://dx.doi.org/10.1287/mnsc.17.4.B141

[10] Mohanty, B.K. and Vijayaraghawan, T.A.S. (1995) A Multiobjective Programming Problem and Its Equivalent Goal Programming Problem with Approprite Priorities and Aspiration Levels: A Fuzzy Approach. Computers & Operations Research, 22, 771-778.

http://dx.doi.org/10.1016/0305-0548(94)00066-H

[11] Kuwano, H. (1996) On the Fuzzy Multi-Objective Linear Programming Problems: Goal Programming Approach. Fuzzy Sets and Systems, 82, 57-64.

http://dx.doi.org/10.1016/0165-0114(95)00231-6

[12] Chanas, A. and Kuchta, D. (1996) Multiobjective Programming in Optimization of Interval Objective Functions—A Generalized Approach. European Journal of Operational Research, 94, 594-598.

http://dx.doi.org/10.1016/0377-2217(95)00055-0

[13] Chen, L.H. and Tsai, F.C. (2001) Fuzzy Goal Programming with Different Importance and Priorties. European Journal of Operational Research, 133, 548-556.

http://dx.doi.org/10.1016/S0377-2217(00)00201-0

[14] Stanciulescu, C., Fortemps, P., Installe, M. and Wertz, V. (2003) Multi-Objective Fuzzy Linear Programming Problems with Fuzzy Decision Variables. European Journal of Operational Research, 149, 654-675.

http://dx.doi.org/10.1016/S0377-2217(02)00449-6

[15] Lin, C.C. (2004) A Weighted Max-Min Model Fuzzy Goal Programming. Fuzzy Sets and Systems, 142, 407-420.

http://dx.doi.org/10.1016/S0165-0114(03)00092-7

[16] Yaghoobi, M.A. and Tamiz, M. (2007) A Method for Solving Fuzzy Goal Programming Problems Based on MINMAX Approach. European Journal of Operational Research, 177, 1580-1590.

[17] Cheng, H.F., Huang, W.L., Zhon, Q. and Cai, J. (2013) Solving Fuzzy Multiobjective Linear Programming Problems Using Deviation Degree Measures and Max-Min Method. Applied Mathematical Modeling, 73, 6855-6869.

http://dx.doi.org/10.1016/j.apm.2013.01.048

[18] Soliman, O., Sarker Rahul, A. and Zahir, S. (2008) Fuzzy Goal Programming Model with Parametric Analysis for Regional Sustainability Development under Climate Change: A Case of Agriculture Sector. ASOR Bulletin, 27, 2-15.

[19] Mishra, B. and Singh, S.R. (2013) Linear Fractional Programming Procedure for Multi Objective Linear Programming Problem in Agriculture System. International Journal of Computer Applications, 61, 45-52.

[20] Bharati, S.K., Nishad, A.K. and Singh, S.R. (2014) Solution of Multi-Objective Linear Programming under Intuitionistic Fuzzy Environment. In: Babu, B.V., Nagar, A., Deep, K., Pant, M., Bansal, J.C., Ray, K. and Gupta, U., Eds., Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), December 28-30, 2012, Advances in Intelligent Systems and Computing, Vol. 236, Springer India, New Delhi, 161-171.

http://dx.doi.org/10.1007/978-81-322-1602-5_18

[21] Sen, S. and Pal, B.B. (2013) Interval Goal Programming Approach to Multiobjective Fuzzy Goal Programming Problems with Interval Weights. Procedia Technology, 10, 587-595.

http://dx.doi.org/10.1016/j.protcy.2013.12.399

[22] Seyed, H. and Razavi, H. (2013) Multi-Objective Linear Programming with Interval Coefficients: A Fuzzy Set Based Approach. Emerald, 42, 482-496.

[23] Azzabi, L., Dorra, A., Kaddour, B. and Abdessamad, K. (2014) Fuzzy Goal Programming to Optimization the Multi-Objective Problem. Science Journal of Applied Mathematics and Statistics, 2, 14-19.

http://dx.doi.org/10.11648/j.sjams.20140201.12

[24] Marbini, A.H. and Tavana, M. (2011) An Extension of the Linear Programming Method with Fuzzy Parameters. International Journal of Mathematics in Operational Research, 3, 44-55.

[25] Luhandjula, M.K. (1984) Fuzzy Approaches for Multiple Objective Linear Fractional Optimization. Fuzzy Sets and Systems, 13, 11-23.

http://dx.doi.org/10.1016/0165-0114(84)90023-X

[26] Chakraborty, M. and Gupta, S. (2002) Fuzzy Mathematical Programming for Multi Objective Linear Fractional Programming Problem. Fuzzy Sets and Systems, 125, 335-342.

http://dx.doi.org/10.1016/S0165-0114(01)00060-4

[27] Pal, B.B., Moitra, B.N. and Maulik, U. (2003) A Goal Programming Procedure for Fuzzy Multiobjective Linear Fractional Programming Problem. Fuzzy Sets and Systems, 139, 395-405.

http://dx.doi.org/10.1016/S0165-0114(02)00374-3

[28] Pop, B., Minasian, I. and Stancu, M. (2008) A Method of Solving Fully Fuzzified Linear Fractional Programming Problems. Journal of Applied Mathematics and Computing, 27, 227-242.

http://dx.doi.org/10.1007/s12190-008-0052-5

[29] Cui, Y.H., Qu, J., Peng, Y., Wang, L. and Li, B. (2010) The Study of the Solution on Multi-Objective Linear Programming Problem under Fuzzy. 2010 Asia-Pacific Conference on Wearable Computing Systems (APWCS), Shenzhen, 17-18 April 2010, 286-290.

http://dx.doi.org/10.1109/APWCS.2010.79

[30] Mehrjerdi, Y.Z. (2011) Solving Fractional Programming Problem through Fuzzy Goal Setting and Approximation. Applied Soft Computing, 11, 1735-1742.

[31] Pitam, S., Kumar, S.D. and Singh, R.K. (2011) Fuzzy Multiobjective Linear plus Linear Fractional Programming Problem: Approximation and Goal Programming Approach. International Journal of Mathematics and Computers in Simulation, 5, 395-404.

[32] Animesh, B. and Dewan, S. (2012) Priority Based Fuzzy Goal Programming Technique to Fractional Fuzzy Goals Using Dynamic Programming. Fuzzy Information and Engineering, 4, 165-180.

[33] Ohta, H. and Yamaguchi, T. (1996) Linear Fractional Goal Programming in Consideration of Fuzzy Solution. European Journal of Operation Research, 92, 157-165.

http://dx.doi.org/10.1016/0377-2217(95)00052-6

[34] Pal, B.B. and Sen, S. (2008) A Goal Programming Procedure for Solving Interval Valued Multi-Objective Fractional Programming Problems. 16th International Conference on Advanced Computing and Communications, ADCOM 2008, Chennai, 14-17 December 2008, 297-302.

[35] Sohrab, E. and Morteza, P. (2012) Solving the Interval Valued Linear Programming Problem. American Journal of Computational Mathematics, 2, 51-55.

http://dx.doi.org/10.4236/ajcm.2012.21006

[36] Kornbluth, J.S.H. and Steuer, R.E. (1981) Multiple Objective Linear Fractional Programming. Management Science, 27, 1024-1039.

http://dx.doi.org/10.1287/mnsc.27.9.1024