Duality results in fuzzy linear programming problems based on the concept of goal programming

References

• Amiri, N. M., & Nasseri, S. H. (2006). Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied Mathematics and Computation, 180, 206–216. doi: 10.1016/j.amc.2005.11.161
   Web of Science ®Google Scholar
• Bazaraa, M. S, Sherali, H. D., & Shetty, C. M. (2004). Nonlinear programming. Noida: Wiley.
   Google Scholar
• Bector, C. R., & Chandra, S. (2002). On duality in linear programming under fuzzy environment. Fuzzy Sets and Systems, 125, 317–325. doi: 10.1016/S0165-0114(00)00122-6
   Web of Science ®Google Scholar
• Bector, C. R., Chandra, S., & Vijay, V. (2004a). Duality in linear programming with fuzzy parameters and matrix games with fuzzy payoffs. Fuzzy Sets and Systems, 146, 253–269. doi: 10.1016/S0165-0114(03)00260-4
   Web of Science ®Google Scholar
• Bector, C. R., Chandra, S., & Vijay, V. (2004b). Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optimization and Decision Making, 3, 255–269. doi: 10.1023/B:FODM.0000036866.18909.f1
   Google Scholar
• Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17(4), 141–164. doi: 10.1287/mnsc.17.4.B141
   Web of Science ®Google Scholar
• Chakraborty, A., Tiwari, S. P., Chattopadhyay, A., & Chatterjee, K. (2014). Duality in fuzzy multi objective linear programming problem with multi constraint. International Journal of Mathematics in Operational Research, 6, 297–315. doi: 10.1504/IJMOR.2014.060850
   Google Scholar
• Gupta, S. K., & Dangar, D. (2010). Duality in fuzzy quadratic programming with exponential membership functions. Fuzzy Information and Engineering, 4, 337–346. doi: 10.1007/s12543-010-0054-5
   Google Scholar
• Gupta, P., & Mehlawat, M. K. (2009). Bector-Chandra type duality in fuzzy linear programming with exponential membership functions. Fuzzy Sets and Systems, 160, 3290–3308. doi: 10.1016/j.fss.2009.04.012
   Web of Science ®Google Scholar
• Hosaeni, S., & Cheshmavar, J. (2016). Some results of duality in fuzzy linear programming. International Journal of Pure and Applied Mathematics, 108, 859–864. doi: 10.12732/ijpam.v108i4.11
   Google Scholar
• Mohamed, R. H. (1997). The relationship between goal programming and fuzzy programming. Fuzzy Sets and Systems, 89, 215–222. doi: 10.1016/S0165-0114(96)00100-5
   Web of Science ®Google Scholar
• Pandey, D., & Kumar, S. (2011). Fuzzy optimization of primal-dual pair using piecewise linear membership functions. Yugoslav Journal of Operations Research, 21(2), 97–106.
   Google Scholar
• Ramík, J., & Vlach, M. (2016). Intuitionistic fuzzy linear programming and duality: A level sets approach. Fuzzy Optimization and Decision Making, 15, 457–489. doi: 10.1007/s10700-016-9233-0
   Web of Science ®Google Scholar
• Rodder, W., & Zimmermann, H. -J. (1980). Duality in fuzzy linear programming. In A.V. Fiacco and K.O. Kortanek (Eds.) External methods and system analysis. (pp. 415–427). Heidelberg: Springer-Verleg.
   Google Scholar
• Saxena, P., & Jain, R. (2017). Duality for a nonlinear fractional programming under fuzzy environment with parabolic concave membership functions. International Journal of Advanced and Applied Sciences, 4, 131–136. doi: 10.21833/ijaas.2017.01.019
   Web of Science ®Google Scholar
• Schryen, G., & Hristova, D. (2015). Duality in fuzzy linear programming: A survey. OR Spectrum, 37(1), 1–48. doi: 10.1007/s00291-013-0355-2
   Web of Science ®Google Scholar
• Verdegay, J. L. (1984). A dual approach to solve the fuzzy linear programming problem. Fuzzy Sets and Systems, 14, 131–141. doi: 10.1016/0165-0114(84)90096-4
   Web of Science ®Google Scholar
• Vijay, V., Chandra, S., & Bector, C. R. (2005). Matrix games with fuzzy goals and fuzzy payoffs. Omega, 33, 425–429. doi: 10.1016/j.omega.2004.07.007
   Web of Science ®Google Scholar
• Wu, H. C. (2003a). Duality theorems in fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems, 139, 363–377. doi: 10.1016/S0165-0114(02)00575-4
   Web of Science ®Google Scholar
• Wu, H. C. (2003b). Duality theory in fuzzy linear programming problems with fuzzy coefficients. Fuzzy Optimization and Decision Making, 2, 61–73. doi: 10.1023/A:1022852314914
   Google Scholar
• Zhang, C., Yuan, X. H., & Lee, E. S. (2005). Duality theory in fuzzy mathematical programming problems with fuzzy coefficients. Computers and Mathematics with Applications, 49, 1709–1730. doi: 10.1016/j.camwa.2004.12.012
   Web of Science ®Google Scholar
• Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several obective functions. Fuzzy Sets and Systems, 1, 45–55. doi: 10.1016/0165-0114(78)90031-3
   Google Scholar
• Zou, T. (2015). Extended duality in fuzzy optimization problems. Mathematical Problems in Engineering, 2015, 11.
   Web of Science ®Google Scholar