Kybernetika 52 no. 4, 531-557, 2016

Linear optimization with bipolar max-parametric hamacher fuzzy relation equation constraints

Samaneh Aliannezhadi, Ali Abbasi Molai and Behnaz HedayatfarDOI: 10.14736/kyb-2016-4-0531

Abstract:

In this paper, the linear programming problem subject to the Bipolar Fuzzy Relation Equation (BFRE) constraints with the max-parametric hamacher composition operators is studied. The structure of its feasible domain is investigated and its feasible solution set determined. Some necessary and sufficient conditions are presented for its solution existence. Then the problem is converted to an equivalent programming problem. Some rules are proposed to reduce the dimensions of problem. Under these rules, some of the optimal variables are found without solving the problem. An algorithm is then designed to find an upper bound for its optimal objective value. With regard to this algorithm, a modified branch and bound method is extended to solve the problem. We combine the rules, the algorithm, and the modified branch and bound method in terms of an algorithm to solve the original problem.

Keywords:

linear optimization, bipolar fuzzy relation equations, bipolar variables, modified branch and bound method, max-parametric hamacher compositions

Classification:

90-xx, 90Cxx, 90C70

paper.pdf

References:

  1. A. Abbasi Molai: Linear optimization with mixed fuzzy relation inequality constraints using the pseudo-t-norms and its application. Soft Computing 19 (2015), 3009-3027.   DOI:10.1007/s00500-014-1464-9
  2. L. Chen and P. P. Wang: Fuzzy relation equations (I): The general and specialized solving algorithms. Soft Computing 6 (2002), 428-435.   DOI:10.1007/s00500-001-0157-3
  3. L. Chen and P. P. Wang: Fuzzy relation equations (II): The branch-poit-solutions and the categorized minimal solutions. Soft Computing 11 (2007), 33-40.   DOI:10.1007/s00500-006-0050-1
  4. B. De Baets: Analytical solution methods for fuzzy relational equations. In: Fundamentals of fuzzy sets, the handbooks of fuzzy sets series (D. Dubois and H. Prade, eds.), Dordrecht, Kluwer 2000, Vol. 1, pp. 291-340.   DOI:10.1007/978-1-4615-4429-6_7
  5. A. Di Nola, W. Pedrycz and S. Sessa: On solution of fuzzy relational equations and their characterization. BUSEFAL 12 (1982), 60-71.   CrossRef
  6. A. Di Nola, W. Pedrycz, S. Sessa and E. Sanchez: Fuzzy relation equations theory as a basis of fuzzy modelling: An overview. Fuzzy Sets and Systems 40 (1991), 415-429.   DOI:10.1016/0165-0114(91)90170-u
  7. A. Di Nola, W. Pedrycz, S. Sessa and P. Z. Wang: Fuzzy relation equations under triangular norms: a survey and new results. Stochastica 8 (1984), 99-145.   CrossRef
  8. A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez: Fuzzy Relation Equations and their Applications to Knowledge Engineering. Kluwer, Dordrecht 1989.   DOI:10.1007/978-94-017-1650-5
  9. S.-C. Fang and G. Li: Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103 (1999), 107-113.   DOI:10.1016/s0165-0114(97)00184-x
  10. S. Feng, Y. Ma and J. Li: A kind of nonlinear and non-convex optimization problems under mixed fuzzy relational equations constraints with max-min and max-average composition. In: Eighth International Conference on Computational Intelligence and Security, Guangzhou 2012.   DOI:10.1109/cis.2012.42
  11. S. Freson, B. De Baets and H. De Meyer: Linear optimization with bipolar max-min constraints. Inform. Sci. 234 (2013), 3-15.   DOI:10.1016/j.ins.2011.06.009
  12. S. Gottwald: Fuzzy Sets and Fuzzy Logic: The Foundations of Application From a Mathematical Point of View. Vieweg, Wiesbaden 1993.   DOI:10.1007/978-3-322-86812-1
  13. S. Gottwald: Generalized Solvability Behaviour for Systems of Fuzzy Equations. In: Discovering the world with fuzzy logic (V. Novák and I. Perfilieva, eds.), Physica-Verlag, Heidelberg 2000, pp. 401-430.   CrossRef
  14. S.-M. Guu and Y.-K. Wu: Minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 1 (2002), 347-360.   DOI:10.1023/a:1020955112523
  15. S.-M. Guu and Y.-K. Wu: Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint. Fuzzy Sets and Systems 161 (2010), 285-297.   DOI:10.1016/j.fss.2009.03.007
  16. G. Klir and B. Yuan: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Upper Saddle River, Prentice Hall, NJ 1995.   DOI:10.1021/ci950144a
  17. J. Li, S. Feng and H. Mi: A Kind of nonlinear programming problem based on mixed fuzzy relation equations constraints. Physics Procedia 33 (2012), 1717-1724.   DOI:10.1016/j.phpro.2012.05.276
  18. P. Li and S.-C. Fang: A survey on fuzzy relational equations, Part I: Classification and solvability. Fuzzy Optimization and Decision Making 8 (2009), 179-229.   DOI:10.1007/s10700-009-9059-0
  19. P. Li and Q. Jin: Fuzzy relational equations with min-biimplication composition. Fuzzy Optimization and Decision Making 11 (2012), 227-240.   DOI:10.1007/s10700-012-9122-0
  20. P. Li and Q. Jin: On the resolution of bipolar max-min equations. In: IEEE Trans. Fuzzy Systems, second review, 2013.   CrossRef
  21. P. Li and Y. Liu: Linear optimization with bipolar fuzzy relational equation constraints using the Lukasiewicz triangular norm. Soft Computing 18 (2014), 1399-1404.   DOI:10.1007/s00500-013-1152-1
  22. J. Loetamonphong and S.-C. Fang: Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118 (2001), 509-517.   DOI:10.1016/s0165-0114(98)00417-5
  23. A. Markovskii: Solution of fuzzy equations with max-product composition in inverse control and decision making problems. Automation and Remote Control 65 (2004), 1486-1495.   DOI:10.1023/b:aurc.0000041426.51975.50
  24. A. Markovskii: On the relation between equations with max-product composition and the covering problem. Fuzzy Sets and Systems 153 (2005), 261-273.   DOI:10.1016/j.fss.2005.02.010
  25. M. Miyakoshi and M. Shimbo: Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems 16 (1985), 53-63.   DOI:10.1016/s0165-0114(85)80005-1
  26. W. Pedrycz: Fuzzy Control and Fuzzy Systems. Research Studies Press/Wiely, New York 1989.   CrossRef
  27. W. Pedrycz: Processing in relational structures: Fuzzy relational equations. Fuzzy Sets and Systems 40 (1991), 77-106.   DOI:10.1016/0165-0114(91)90047-t
  28. K. Peeva and Y. Kyosev: Fuzzy Relational Calculus: Theory, Applications and Software. World Scientific, New Jersey 2004.   CrossRef
  29. E. Sanchez: Resolution of composite fuzzy relation equation. Inform. Control 30 (1976), 38-48.   DOI:10.1016/s0019-9958(76)90446-0
  30. E. Sanchez: Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Fuzzy Automata and Decision Processes (M. M. Gupta, G. N. Saridis, B. R. Gaines, eds.), North-Holland, Amsterdam 1977, pp. 221-234.   CrossRef
  31. B.-S. Shieh: Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint. Inform. Sci. 181 (2011), 832-841.   DOI:10.1016/j.ins.2010.10.024
  32. A. Thapar, D. Pandey and S. K. Gaur: Optimization of linear objective function with max-t fuzzy relation equations. Applied Soft Computing 9 (2009), 1097-1101.   DOI:10.1016/j.asoc.2009.02.004
  33. Y.-K. Wu and S.-M. Guu: A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optimization and Decision Making 3 (2004), 271-278.   DOI:10.1023/b:fodm.0000036862.45420.ea
  34. Y.-K. Wu and S.-M. Guu: Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets and Systems 150 (2005), 147-162.   DOI:10.1016/j.fss.2004.09.010
  35. Y.-K. Wu, S.-M. Guu and J. Y.-C. Liu: An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Trans. Fuzzy Systems 10 (2002), 552-558.   DOI:10.1109/tfuzz.2002.800657
  36. S.-J. Yang: An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition. Fuzzy Sets and Systems 255 (2014), 41-51.   DOI:10.1016/j.fss.2014.04.007