zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Linear programming with fuzzy parameters: an interactive method resolution. (English) Zbl 1102.90345
Summary: This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.

MSC:
90B50Management decision making, including multiple objectives
90C70Fuzzy programming
WorldCat.org
Full Text: DOI
References:
[1] Bellman, R.; Zadeh, L. A.: Decision making in a fuzzy environment. Management science 17, 141-164 (1970) · Zbl 0224.90032
[2] Bortolan, G.; Degani, R.: A review of some methods for ranking fuzzy subsets. Fuzzy sets and systems 15, 1-19 (1985) · Zbl 0567.90056
[3] Cadenas, J. M.; Verdegay, J. L.: PROBO: an interactive system in fuzzy programming. Fuzzy sets and systems 76, 319-322 (1995) · Zbl 0857.90135
[4] Cadenas, J. M.; Verdegay, J. L.: Using fuzzy numbers in linear programming. IEEE transactions on systems, man and cybernetics --- part B cybernetics 27, No. 6, 1016-1022 (1997)
[5] Delgado, M.; Verdegay, J. L.; Vila, M. A.: A procedure for ranking fuzzy relations. Fuzzy sets and systems 26, 49-62 (1988) · Zbl 0647.94026
[6] Dubois, D.; Prade, H.: Operations on fuzzy numbers. International journal of systems 9, No. 6, 613-626 (1978) · Zbl 0383.94045
[7] Dubois, D.; Kerre, E.; Mesiar, R.; Prade, H.: Fuzzy interval analysis. Fundamentals of fuzzy sets, 483-561 (2000) · Zbl 0988.26020
[8] Heilpern, S.: The expected valued of a fuzzy number. Fuzzy sets and systems 47, 81-86 (1992) · Zbl 0755.60004
[9] Jiménez, M.: Ranking fuzzy numbers through the comparison of its expected intervals. International journal of uncertainty, fuzziness and knowledge-based systems 4, No. 4, 379-388 (1996) · Zbl 1232.03040
[10] Jiménez, M.; Rodrı&acute, M. V.; Guez; Arenas, M.; Bilbao, A.: Solving a possibilistic linear program through compromise programming. Mathware and soft computing 7, No. 2-3, 175-184 (2000) · Zbl 0992.90086
[11] Kaufmann, A.; Aluja, J. Gil: Técnicas de gestión de empresa. (1992)
[12] Lai, Y. -L.; Hwang, C. -L.: Fuzzy multiple objective decision making. (1994) · Zbl 0823.90070
[13] Nakamura, K.: Preference relation on a sets of fuzzy utilities as a basis for decision making. Fuzzy sets and systems 20, 147-162 (1986) · Zbl 0618.90001
[14] Rommelfanger, H.: Fuzzy linear programming and its applications. European journal of operational research 92, 512-527 (1996) · Zbl 0914.90265
[15] Rommelfanger, H.; Slowinski, R.: Fuzzy linear programming with single or multiple objective functions. Fuzzy sets in decision analysis, operation research and statistics (1998)
[16] Sakawa, M.: Fuzzy sets and interactive multiobjective optimization. (1993) · Zbl 0842.90070
[17] Tanaka, H.; Asai, K.: Fuzzy linear programming with fuzzy numbers. Fuzzy sets and systems 13, 1-10 (1984) · Zbl 0546.90062
[18] Tong, R. M.; Bonissone, P. P.: A linguistic approach to decision making with fuzzy sets. IEEE transactions of systems, man and cybernetics 10, 716-723 (1980)
[19] Wang, X.; Kerre, E.: On the classification and the dependencies of the ordering methods. Fuzzy logic foundation and industrial applications, international series in intelligent technologies, 73-90 (1996) · Zbl 0906.04003
[20] Yager, R.R., 1979. Ranking fuzzy subsets over the unit interval. In: Proceedings of 17th IEEE International Conference on Decision and Control, San Diego, CA, pp. 1435 -- 1437. · Zbl 0429.04009
[21] Yuan, Y.: Criteria for evaluating fuzzy ranking methods. Fuzzy sets and systems 44, 139-157 (1991) · Zbl 0747.90003
[22] Zadeh, L. A.: Fuzzy sets. Information and control 8, 338-353 (1965) · Zbl 0139.24606
[23] Zadeh, L.A., 1975. The concept of a linguistic variable and its applications to approximate reasoning. Part I Information Sciences, vol. 8, pp. 199 -- 249; Part II Information Sciences, vol. 8, pp. 301 -- 357; Part III Information Sciences, vol. 9, pp. 43 -- 80. · Zbl 0397.68071
[24] Zadeh, L. A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems 1, 3-28 (1978) · Zbl 0377.04002