By Jiuping Xu, Xiaoyang Zhou
Decision makers frequently face a number of, conflicting goals and the advanced fuzzy-like environments within the actual global. What are the fuzzy-like environments? How will we version the a number of aim choice making difficulties below fuzzy-like environments? How do you take care of those types? with a view to solution those questions, this e-book offers an updated technique approach for fuzzy-like a number of goal selection making, inclusive of modelling approach, version research process, set of rules procedure and alertness method in constitution optimization challenge, choice challenge, buying challenge, stock challenge, logistics challenge etc. Researchers, practitioners and scholars in administration technological know-how, operations learn, info technological know-how, process technology and engineering technology will locate this paintings an invaluable reference.
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Additional resources for Fuzzy-Like Multiple Objective Decision Making (Studies in Fuzziness and Soft Computing)
Triangular and trapezoidal fuzzy number Let x, l, m, n ∈ R. 21) (n − x)/(n − m), m < x ≤ n ⎪ ⎪ ⎩ 0, x ≥ n. 12, M = (l, m, n) with l and n being the lower and upper bounds of fuzzy number M. 4 Fuzzy Arithmetic 17 Fig. 12 Triangular and trapezoidal fuzzy numbers. When there are multiple peaks, fuzzy number M is represented by M = (a, b, c, d), with the [b, c] interval being the most likely values for M and any value below a and above d being totally impossible. The membership value decrease gradually (or linearly) from b to a and from c to d.
7 to explain. Fig. 7 α -cut of triangular fuzzy number Although characteristic function can be assigned by any number, a normalized value between 0 and 1 is always preferred. Thus let us introduce the normality as follows. 5). A fuzzy set is subnormal if it is not normal. A non-empty subnormal fuzzy set can be normalized by dividing each μA (x) by the factor sup μA (x). (A fuzzy x set is empty if and only if μA (x) = 0, ∀ x ∈ U). It is noted that a characteristic function is always a normalized function through this study.
M(∗)N (Z) = α α · (Mα ∗ Nα ) = ( α · Mα ) ∗ ( α · Nα ) α α = μM(Z) (∗)μN(Z) . 1.  (M(∗)N)α = Mα ∗ Nα . This implies fuzzy number operations by interval-valued operations. Thus, if * is commutative, (*) is commutative, and if * is associative, (*) is associative too. Addition of fuzzy numbers The addition of two fuzzy numbers M and N may be done in two different ways. 24) and Nα = [n1 , n2 ]. 25) The addition of M and N may be rewritten as: Mα (+)Nα = [m1 + n1 , n1 + n2 ]. 26) This is equivalent to adding two intervals of confidence level by Kaufmann and Gupta.