A Fuzzy Random Minimum Cost Network Flow Programming Problem

Document Type: Research Paper


1 Department of Industrial Engineering, University of Tabriz, Iran

2 Department of Systems Engineering, Sharif University of Technology, Tehran, Iran


In this paper, a fuzzy random minimum cost flow problem is presented. In this problem, cost parameters and decision variables are fuzzy random variables and fuzzy numbers respectively. The object of the problem is to find optimal flows of a capacitated network. Then, two algorithms are developed to solve the problem based on Er-expected value of fuzzy random variables and chance-constrained programming. Furthermore, the results of two algorithms will be compared. An illustrative example is also provided to clarify the concept.


Main Subjects

[1] Ahuja R. K., Magnanti T. L., Orlin J. B. (1993), Network Flows; Prentice-Hall, EnglewoodCliffs, NJ.
[2] Bellman R., Zadeh L. A. (1970), Decision-making in a fuzzy environment; Management Science 17;
[3] Buckley J.J., Feuring T. (2000), Evolutionary algorithm solution to fuzzy problems: Fuzzy linear
programming; Fuzzy Sets and Systems 109; 35-53.
[4] Charnes A., Cooper w. w. (1959), Chance-constrained programming; Management science 6; 73-79.
[5] Diamond P., Korner R. (1997), Extended fuzzy linear models and least squares estimates; Computers
and Mathematics with Applications 33; 15–32.
[6] Dubois D., Prade H. (1979), Fuzzy real algebra: Some results; Fuzzy Sets and System 2; 327-348.
[7] Dubois D., Prade H. (1980), Fuzzy Sets and Systems: Theory and Application; Academic Press; New
[8] Dubois D., Prade H. (1988), Fuzzy numbers: An overview. In Analysis of fuzzy information; CRC
Press 2; pp 3-29.
[9] Eshghi K., Nematian J. (2008), Special classes of mathematical programming models with fuzzy
random variables; Journal of Intelligent & Fuzzy Systems 19; 131-140.
[10] Hukuhara M. (1967), Integration des applications measurable dont la valeur est un compact convexe;
Funkcialaj Ekvacioj 10; 205–223.
[11] Katagiri H., Ishii H. (2000), Linear programming problem with fuzzy random constraint;
Mathematica Japonica 52; 123-129.
[12] Kwakernaak H. (1978), Fuzzy random variable-1: Definitions and theorems; Information Sciences 15;

[13] Lin C. J., Wen U. P. (2004), A labeling algorothm for the fuzzy assignment problem; Fuzzy Sets and
Systems 142; 373-391.
[14] Liu Y-K., Liu B. (2003), Fuzzy random variables: A scalar expected value operator; Fuzzy
Optimization and Decision Making 2; 143-160.
[15] Puri M. L., Ralescu D. A. (1986), Fuzzy random variables; Journal of Mathematical Analysis and
Application 114; 409-422.
[16] Okada S., Soper T. (2000), A shortest path problem on a network with fuzzy arc lengths; Fuzzy Sets
and Systems 109; 129-140.
[17] Shih H. S., Stanley Lee E. (1999), Fuzzy multi-level minimum cost flow problems; Fuzzy Sets and
Systems 107; 159-176.
[18] Wang G.-Y., Zang Yue (1992), The theory of fuzzy stochastic processes; Fuzzy Sets and Systems 51;
[19] Wang G-Y., Zhong Q. (1993), Linear programming with fuzzy random variable coefficients; Fuzzy
sets and Systems 57; 295-311.
[20] Wang J. R. (1999), A fuzzy set approach to activity scheduling for product development; Journal of
the Operational Research Society 50;1217–1228.
[21] Williams H. P. (1999), Model building in mathematical programming, Fourth edition;John Wiley &
[22] Zadeh L. A. (1973), The concept of linguistic variable and its application to approximate reasoning;
Memorandum ERL- M 411; Berkeley.
[23] Zimmermann, H.J. (1978), Fuzzy programming and linear programming with several objective
function; Fuzzy Sets and Systems 1; 45-55.