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)
A labeling algorithm for the fuzzy assignment problem. (English) Zbl 1044.90097
Summary: This paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones. Here, the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions, whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function. By the max--min criterion suggested by Bellman and Zadeh, the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem. Here, we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case. The algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found. The computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem.

MSC:
90C70Fuzzy programming
90B80Discrete location and assignment
WorldCat.org
Full Text: DOI
References:
[1] Anzai, Y.: On integer fractional programming. J. oper. Res. soc. Jpn. 17, 49-66 (1974) · Zbl 0278.90052
[2] Balinski, M. L.; Gomory, R. E.: A primal method for the assignment and transportation problems. Management sci. 10, 578-593 (1964)
[3] Bazaraa, M. S.; Jarvis, J. J.; Sherali, H. D.: Linear programming and network flow. (1990) · Zbl 0722.90042
[4] Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M.: Nonlinear programming--theory and algorithms. (1993) · Zbl 0774.90075
[5] Bellman, R. E.; Zadeh, L. A.: Decision-making in a fuzzy environment. Management sci. 17B, 141-164 (1970) · Zbl 0224.90032
[6] Chanas, S.; Kolodziejczyk, W.; Machaj, A.: A fuzzy approach to the transportation problem. Fuzzy sets and systems 13, 211-221 (1984) · Zbl 0549.90065
[7] Chanas, S.; Kuchta, D.: A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy sets and systems 82, 299-305 (1996)
[8] Chanas, S.; Kuchta, D.: Fuzzy integer transportation problem. Fuzzy sets and systems 98, 291-298 (1998) · Zbl 0931.90066
[9] Charnes, A.; Cooper, W. W.: Programming with linear fractionals. Naval res. Logist. quart. 9, 181-186 (1962) · Zbl 0127.36901
[10] Chen, M. S.: On a fuzzy assignment problem. Tamkang J. 22, 407-411 (1985) · Zbl 0656.90083
[11] Dubois, D.; Fortemps, P.: Computing improved optimal solutions to MAX--MIN flexible constraint satisfaction problems. European J. Oper. res. 118, 95-126 (1999) · Zbl 0945.90087
[12] Gen, M.; Ida, K.; Li, Y.: Bicriteria transportation problem by hybrid genetic algorithm. Comput. indust. Eng. 35, 363-366 (1998)
[13] Gillett, B. E.: Introduction to operations research--A computer-oriented algorithm approach. (1976) · Zbl 0359.90001
[14] Haken, H.; Schanz, M.; Starke, J.: Treatment of combinatorial optimization problems using selection equations with cost terms. Part I. Two-dimensional assignment problems. Physica D 134, 227-241 (1999) · Zbl 0931.90039
[15] Kuhn, H. W.: The hungarian method for the assignment problem. Naval res. Logist. quart. 2, 253-258 (1956)
[16] Li, L.; Lai, K. K.: A fuzzy approach to the multiobjective transportation problem. Comput. oper. Res. 27, 43-57 (2000) · Zbl 0973.90010
[17] Malhotra, R.; Bhatia, H. L.; Puri, M. C.: Bi-criteria assignment problem. Opsearch 19, 84-96 (1982) · Zbl 0487.90058
[18] Éigeartaigh, M. Óh: A fuzzy transportation algorithm. Fuzzy sets and systems 8, 235-243 (1982)
[19] Sakawa, M.; Nishizaki, I.; Uemura, Y.: Interactive fuzzy programming for two-level linear and linear fractional production and assignment problemsa case study. European J. Oper. res. 135, 142-157 (2001) · Zbl 1077.90564
[20] Schaible, S.; Ibaraki, T.: Fractional programming. European J. Oper. res. 12, 325-338 (1983) · Zbl 0529.90088
[21] Shigeno, M.; Saruwatari, Y.; Matsui, T.: An algorithm for fractional assignment problems. Discrete appl. Math. 56, 333-343 (1995) · Zbl 0820.90121
[22] Tada, M.; Ishii, H.: An integer fuzzy transportation problem. Comput. math. Appl. 31, 71-87 (1996) · Zbl 0853.90123
[23] Taha, H. A.: An algorithm for zero-one fractional programming. AIIE trans. 7, 29-34 (1975)
[24] Wang, X.: Fuzzy optimal assignment problem. Fuzzy math. 3, 101-108 (1987) · Zbl 0645.90041
[25] Werners, B.: Interactive multiple objective programming subject to flexible constraints. European J. Oper. res. 31, 342-349 (1987) · Zbl 0636.90085