Abstract
Monomials are widely used. They are basic structural units of geometric programming. In the process of optimization, many objective functions can be denoted by monomials. We can often see them in resource allocation and structure optimization and technology management, etc. Fuzzy relation equations are important elements of fuzzy mathematics, and they have recently been widely applied in fuzzy comprehensive evaluation and cybernetics. In view of the importance of monomial functions and fuzzy relation equations, we present a fuzzy relation geometric programming model with a monomial objective function subject to the fuzzy relation equation constraints, and develop an algorithm to find an optimal solution based on the structure of the solution set of fuzzy relation equations. Two numerical examples are given to verify the developed algorithm. Our numerical results show that the algorithm is feasible and effective.
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Avriel M. (ed). (1980). Advances in geometric programming Volume 21 of mathematical concept and methods in science and engineering. New York, Plenum Press
Beightler C.S., Phillips D.T. (1976). Applied geometric programming. New York, Wiley
Biswal M.P. (1992). Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets and Systems 51(1): 67–71
Cao, B. Y. (1999). Fuzzy geometric programming optimum seeking in power supply radius of transformer substation. 1999 IEEE International Fuzzy Systems Conference Proceedings, Vol. 3, Korea, July 25–29, III-1749–1753.
Cao B.Y. (2001). Fuzzy geometric programming. Boston, Kluwer Academic Publishers
Di Nola A., Sessa S., Pedrycz W., Sanchez E. (1989). Fuzzy relation equations and their applications to knowledge engineering. Dordrecht Boston/London, Kluwer Academic Publishers
Duffin R.J., Peterson E.L., Zener C. (1967). Geometric programming-theory and application. New York, Wiley
Ecker J. (1980). Geometric programming: Methods, computations and applications. SIAM Review 22(3): 338–362
Fang S.C., Li G.Z. (1999). Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103: 107–113
Goldberg D.E. (1989). Genetic algorithms in search, optimization and machine learning. Reading MA, Addison-Wisley
Liu S.T. (2004). Fuzzy geometric programming approach to a fuzzy machining economics model. International Journal of Production Research 42(16): 3253–3269
Lu J.J., Fang S.C. (2001). Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119: 1–20
Peterson E.L. (1976). Geometric programming. SIAM Review 18(1): 1–51
Sanchez E. (1976). Resolution of composite fuzzy relation equations. Information and Control 30: 38–48
Verma R.K. (1990). Fuzzy geometric programming with several objective functions. Fuzzy Sets and Systems 35(1): 115–120
Wang P.Z., Li H.X. (1996). Fuzzy system theory and fuzzy computer. Beijing, Science Press
Wang P.Z., Zhang D.Z., Sanchez E., Lee E.S. (1991). Latticized linear programming and fuzzy relation inequalities. Journal of Mathematical Analysis and Applications 159(1): 72–87
Wang X.P. (2002). Conditions under which a fuzzy relational equation has minimal solutions in a complete Brouwerian lattice. Advances in Mathematics 31(3): 220–228
Yang, J. H., Cao, B. Y. (2005a). Geometric programming with fuzzy relation equation constraints. 2005 IEEE International Fuzzy Systems Conference Proceedings, Reno, Nevada, May 22–25: 557–560
Yang, J. H., & Cao, B. Y. (2005b). Geometric programming with max-product fuzzy relation equation constraints. Proceedings of the 24th North American Fuzzy Information Processing Society, Ann Arbor, Michigan, June 22–25, 650–653.
Zener C. (1971). Engineering design by geometric programming. New York, Wiley
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Yang, J., Cao, B. Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optim Decis Making 6, 337–349 (2007). https://doi.org/10.1007/s10700-007-9017-7
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DOI: https://doi.org/10.1007/s10700-007-9017-7