By Dr. C. R. Bector, Dr. Suresh Chandra (auth.)
This booklet provides a scientific and centred examine of the applying of fuzzy units to 2 easy parts of selection concept, particularly Mathematical Programming and Matrix online game idea. except offering lots of the simple effects on hand within the literature on those issues, the emphasis is on knowing their traditional dating in a fuzzy setting. The learn of duality concept for fuzzy mathematical programming difficulties performs a key position in knowing this interrelationship. For this, a theoretical framework of duality in fuzzy mathematical programming and conceptualization of the answer of a fuzzy video game is created at the strains in their crisp opposite numbers. many of the theoretical effects and linked algorithms are illustrated via small numerical examples from real applications.
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Extra info for Fuzzy Mathematical Programming and Fuzzy Matrix Games
1 (Addition (+) and subtraction (−)). If x ∈ [a1 , a2 ], y ∈ [b1 , b2 ] then x + y ∈ [a1 + b1 , a2 + b2 ] and x − y ∈ [a1 − b2 , a2 − b1 ]. Therefore the addition of A and B, denoted by A(+)B, is deﬁned as A(+)B = [a1 , a2 ](+)[b1 , b2 ] = [a1 + b1 , a2 + b2 ]. Similarly, the subtraction of A and B, denoted by A(−)B is deﬁned as A(−)B = [a1 , a2 ](−)[b1 , b2 ] = [a1 − b2 , a2 − b1 ]. 2 (Image of an interval). If x ∈ [a1 , a2 ] then its image −x ∈ [−a2 , −a1 ]. Therefore the image of A, denoted by A is deﬁned as A = [a1 , a2 ] = [−a2 , −a1 ].
7 Triangular norms (t-norms) and triangular conorms (t-conorms) 35 (ii) T1 (a, b) = ab, S1 (a, b) = a + b − ab, (iii) T∞ (a, b) = max (a + b − 1, 0), S∞ (a, b) = min (a + b, 1). Here it can be veriﬁed that T0 is not Archimedean but T1 and T∞ are Archimedean. g. Frank’s fundamental family, Yager’s family, Hamacher’s family, Schweizer and Sklar’s family, Sugeno’s family and Dubois and Prade’s family), but these are not discussed here. In fact (T0 , S0 ), (T1 , S1 ) and (T∞ , S∞ ) are particular members of Frank’s fundamental family (Ts , Ss ), 0 ≤ s ≤ ∞.
An (xn ) . 1. (Lin and Lee ). 4)}. Let the function f : X → R be given by y = f (x) = x2 . From this data, one can make the following calculations. 0. 4) . 6 Fuzzy relations Let X and Y be two crisp sets and X × Y be their cross product. A crisp binary relation R is a subset of X × Y. It indicates the presence or absence of certain association between the elements of sets X and Y. If one allows the presence of this association to be of varying degree between 0 and 1 then a binary fuzzy relation is obtained.