Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> R.E. Bellman and L.A. Zadeh's approach <s> It is pointed out that there exists a vast amount of fuzzy information in both objective and constraint functions of optimum design of structures (ODS) Thus, relatively reasonable optimum designs can only be obtained by fuzzy programming methods. While all fuzzy constraints enclose a fuzzy feasible region in decision-making space, the fuzzy optimum solution will be a sequence of points in a small fuzzy optimum subregion. In the presented procedure, optimum structural design with fuzzy constraints is transformed into a set of ordinary optimum problems by a level cuts approach which results in a sequence of optimum design schemes with different design levels. The concept of optimum design level corresponding to the most suitable scheme among the obtained sequence is also advanced in this paper. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> R.E. Bellman and L.A. Zadeh's approach <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> BIB002
In BIB002 approach, the problem in fuzzy environment can be stated as, Find X which minimizes f(X) (17) subject to gj(X) ∈ Gj j=1,2,…,n where ordinary subset Gj denotes the allowable interval for the constraint function gj, Gj =[gj(l) , gj(u)] the bold face symbols indicate that the operations or variables contain fuzzy information. The constraint gj(X) ∈ Gj means that gj is a member of a fuzzy subset Gj in the sense of µGj(gj) > 0. The fuzzy feasible region is defined by considering all the constraints as And the membership degree of any design vector X to fuzzy feasible region S is given by (20) i.e., the minimum degree of satisfaction of the design vector X to all of the constraints. A design of vector X is considered feasible provided µS(X) > 0 and the differences in the membership degrees of two design vectors X1 and X2 imply nothing but variations in the minimum degrees of satisfaction of X1 and X2 to the constraints. Thus the optimum solution will be a fuzzy domain D in S with f(X). The fuzzy domain D is defined by (21) that is www.intechopen.com  If the membership function of D is unimodal and has a unique maximum, then the maximum solution X* is one for which the membership function is maximum: 4.2 Verdegay's approach: α-cuts method  considered that if the membership function of the fuzzy constraints has the following form: Simultaneously, the membership functions of µgi(X), ∀I, are continuous and monotonic functions, and trade-off between those fuzzy constraints are allowed; then problem is equivalent to the following formulation: Subject to X ε Xα where Xα= {x ¡ µg i (X) ≥α, ∀i X ≥ 0}, for each αε [0, 1]. This is the fundamental concepts of α-level cuts method of fuzzy mathematical programming. The membership function indicates that if g i (X) ε(b i ,b i + p i ); then the memberships functions are monotonically decreasing. That also can means, the more resource consumed, the less satisfaction the decision maker thinks. One can then obtain the following formulation: where XL ≤X ≤XU and αε[0, 1]. Thus, the problem is equivalent to a crisp parametric programming formulation while α= 1-θ. For each α, one will have an optimal solution; therefore, the solution with αgrade of membership function is fuzzy. This model was applied by BIB001 and  in structural design problems.  proposed the objective function should be fuzzy due to the fuzziness existing in fuzzy inequality constraints. For solving equations, one needs to define f max and f min as follows: