Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Xu's approach: Bound search method <s> By decision-making in a fuzzy environment is meant a decision process in which the goals and/or the constraints, but not necessarily the system under control, are fuzzy in nature. This means that the goals and/or the constraints constitute classes of alternatives whose boundaries are not sharply defined. ::: ::: An example of a fuzzy constraint is: “The cost of A should not be substantially higher than α,” where α is a specified constant. Similarly, an example of a fuzzy goal is: “x should be in the vicinity of x0,” where x0 is a constant. The italicized words are the sources of fuzziness in these examples. ::: ::: Fuzzy goals and fuzzy constraints can be defined precisely as fuzzy sets in the space of alternatives. A fuzzy decision, then, may be viewed as an intersection of the given goals and constraints. A maximizing decision is defined as a point in the space of alternatives at which the membership function of a fuzzy decision attains its maximum value. ::: ::: The use of these concepts is illustrated by examples involving multistage decision processes in which the system under control is either deterministic or stochastic. By using dynamic programming, the determination of a maximizing decision is reduced to the solution of a system of functional equations. A reverse-flow technique is described for the solution of a functional equation arising in connection with a decision process in which the termination time is defined implicitly by the condition that the process stops when the system under control enters a specified set of states in its state space. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Xu's approach: Bound search method <s> Abstract This work presents the two-phase method for fuzzy optimization of structures. In the first phase the fuzzy solution is obtained by using the level cuts method and in the second phase the crisp solution, which maximizes the membership function of fuzzy decision making, is found by using the bound search method. Illustrative numerical examples involving skeletal structures and reinforced concrete slabs are solved. <s> BIB002
Suppose there are a fuzzy goal function f and a fuzzy constraint C in a decision space X, which are characterized by their membership functions µf(X) and µC(X), respectively. The combined effect of those two can be represented by the intersection of the membership functions and the following formulation. Then BIB001 proposed that a maximum decision could be defined as: If µD(X) has a unique maximum at XM, then the maximizing decision is a uniquely defined crisp decision. From equations and following the procedure given, one can obtain the particular optimum level α* corresponding to the optimum point XM such that: where Cα* is the fuzzy constraint set C of α*-level cut. BIB002 used a goal membership function of f(X) as following: (36) www.intechopen.com where fmin has been defined as before. It is apparent that the upper and lower bound of this goal membership function is between 1 and fmin /fmax. As a result, the optimum α* can be achieved through an iteration computation. This method has been called the 2nd phase of α-cuts method in his paper BIB002 ).