Journal of Global Optimization

, Volume 41, Issue 2, pp 283–298

# The optimization problem over a distributive lattice

Article

## Abstract

In this paper we give a necessary and sufficient condition for existence of minimal solution(s) of the linear system A * X ≥ b where A, b are fixed matrices and X is an unknown matrix over a lattice. Next, an algorithm which finds these minimal solutions over a distributive lattice is given. Finally, we find an optimal solution for the optimization problem min {Z = C * X | A * X ≥ b} where C is the given matrix of coefficients of objective function Z.

## Keywords

Distributive lattice Linear programming Fuzzy linear systems Fuzzy relational equation Optimization

## Mathematics Subject Classification (2000)

06Dxx 34A30 20M14 16Y60

## References

1. 1.
Birkhoff, G.: Lattice Theory. American Mathematical Society, Third Edition, Second Printing (1973)Google Scholar
2. 2.
Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M. and Scott D.S. (2003). Continuous Lattices and Domains. Cambridge University Press, Cambridge Google Scholar
3. 3.
Grätzer G. (1978). General Lattice Theory. Academic Press, New York Google Scholar
4. 4.
Higashi M. and Klir G.J. (1984). Resolution of finite fuzzy relation equations. Fuzzy Sets Syst. 13(1): 65–82
5. 5.
Hosseinyazdi M., Hassankhani A. and Mashinchi M. (2005). Optimization over pseudo-Boolean latices. WSEAS Trans. Syst. 4(7): 970–973 Google Scholar
6. 6.
Hosseinyazdi, M., Hassankhani, A., Mashinchi, M.: Linear systems and optimization over lattices. Int. Rev. Fuzzy Math. 2(1) (2007) To appearGoogle Scholar
7. 7.
Miyakoshi M. and Shimbo M. (1987). Sets of solution-set-invariant matrices of simple fuzzy relation equations. Fuzzy Sets Syst. 21: 59–83
8. 8.
Peeva K. (1992). Fuzzy linear systems. Fuzzy Set Syst. 49: 339–355
9. 9.
Peeva K. and Kyosev Y. (2004). Fuzzy Relational Calculus, Advances in Fuzzy Systems-Applications and Theory, vol. 22. World Scientific Publishing Co. Pte. Ltd, Singapore Google Scholar
10. 10.
Peeva K. (2006). Universal algorithm for solving fuzzy relational equations. Ital. J. Pure Appl. Math. 19: 9–20 Google Scholar
11. 11.
Sanchez E. (1976). Resolution of composite fuzzy relation equation. Inform. Control 30: 38–48
12. 12.
Sessa S. (1989). Finite fuzzy relation equations with unique solution in complete Brouwerian lattices. Fuzzy sets Syst. 29: 103–113
13. 13.
Xiong, Q., Wang, X.: Some properties of sup-product fuzzy relational equation. In: Proceeding of IFSA 2005, Bejing, pp. 19–24 (2005)Google Scholar
14. 14.
Zimmermann U. (1981). Linear and Combinatorial Optimization in Ordered Algebraic Structures. North-Holland Publishing Company, Amesterdam Google Scholar

## Authors and Affiliations

1. 1.Shiraz Payam-e-Noor UniversityShirazIran