Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fig. 2. Graphical Interpretation of Pareto Optimum <s> This is part 1 of a survey of recent developments in goal programming and multiple objective optimizations. In this part, attention is directed to goal programming with emphasis on the authors' own work (with others) in a variety of applications. This includes goal and goal interval programming along with definitions and examples of proper goal functionals. Characterizations are also supplied for alternate representations and explicit solutions from special structural properties involving piecewise linear functions. Possibilities for various other kinds of goal functionals are explored and delineated. One class of examples is developed in detail and an algorithm is supplied which utilizes sequences of ordinary linear programming problems to solve certain nonlinear and non-convex problems involving maxima of ratios of linear forms. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Fig. 2. Graphical Interpretation of Pareto Optimum <s> This paper presents a multiobjective optimization algorithm based on generalized compound scaling techniques. The algorithm handles any number of objective functions, similar to handling behavior constraints. This technique generates a partial Pareto set while solving the optimization problem. A reliability-ba sed decision criterion is used for selecting the best compromise design. The example cases considered in this work include various disciplines in airframe structures, such as stress, displacement, and frequency with hundreds of design variables and constraints. This paper also discusses the concept of Pareto-optima l solutions in the context of a multiobjective structural optimization problem and the commonly used methods of generating Pareto-optimal solutions. <s> BIB002
Nearly all of the solution schemes used in multiobjective optimization involve some sort of scalarization of the vector optimization problem. The vector problem is replaced by some equivalent scalar minimization problem. Because the Pareto set is generally infinite, an additional use of scalarization is the selection of a unique member of the Pareto set as the optimum for the vector optimization problem. Usually, a problem is scalarized either by defining an additional supercriterion function or by considering the criteria sequentially. There are various techniques for generating noninferior solutions  Radford et.al., 1985; BIB002 . Weighting Method: This technique is based on the preference techniques of the weights' prior assessment for each objective function. It transforms the multicriteria function to a single criterion function through a parameterization of the relative weighting of the criteria. With the variation of the weights, the entire Pareto set can be generated. Because the results of solving an optimization problem can vary significantly as the weighting coefficients change, and very little is usually known about how to choose these coefficients, a necessary approach is to solve the same problem for many different values of weighting factors. However, because the shape and distribution characteristics of the Pareto set are unknown, it is difficult to determine beforehand the nature of the variations requi re d in t he we ight s s o as to pr o duc e a new solution at each pass. The second important disadvantage of the method is that it will not identify the Pareto solutions in a nonconvex part of the set. The idea of this technique consists in adding all the objective functions together using different coefficients for each. It means that we change our multicriteria optimization problem to a scalar optimization problem by creating one function of the form (9) where wi ≥ 0 are the weighting coefficients representing the relative importance of the criteria. It is usually assumed that Since the results of solving an optimization model using Eq. (9) can vary significantly as the weighting coefficients change and since very little is usually known about how to choose these coefficients, a necessary approach is to solve the same problem for many different values of w i . Note that the weighting coefficients do not reflect proportionally the relative importance of the objectives but are only factors which when varied locate points in the domain. For the numerical methods of seeking the minimum of Eq. (10) this location depends not only on values of w i but also on units in which the functions are expressed. The best results are usually obtained if objective functions are normalized. In this case the vector function is normalized to the following form where Here, fio is generally the maximum value of ith objective function. A condition fio≠0 is assumed and if it is not satisfied which rarely happens; another value of normalizing function must be chosen by the decision maker. Game Theory: Game theory deals with decision situations in which two intelligent opponents with conflicting objectives are trying to outdo one another. It is a mathematical theory that deals with the general features of competitive situations like these in a formal, abstract way. It places particular emphasis on the decision-making processes of the adversaries. Typical examples include launching advertising campaigns for competing products and planning strategies for warring armies. In a game conflict, two opponents, known as players, will each have a (finite or infinite) number of alternatives or strategies. Associated with each pair of strategies is a payoff that one player receives from the other. Such games are known as two-person zero-sum games because a gain by one player signifies an equal loss to the other. It suffices, then, to summarize the game in terms of the payoff to one player. Because games are rooted in conflict of interest, the optimal solution selects one or more strategies for each player such that any change in the chosen strategies does not improve the payoff to either player. These solutions can be in the form of a single pure strategy or several strategies mixed according to specific probabilities (Frederick & Gerald, 2001 ). Goal Programming: Goal programming was proposed by  for a linear model. It has been further developed by others  BIB001 . This method requires the decision maker (DM) to set goals for each objective that he wishes to attain. A preferred solution is then defined as the one, which minimizes the deviations from the set goals. Thus a simple GP formulation of the multiobjective optimization problem is given by Min Subject to: where bj's are the goals set by the DM for the objectives, and are respectively the underachievement and over-achievement of the jth goal. The value of p is based on the utility function of the DM. Other than p = 1 results in a nonlinear goal programming problem. The most common form of GP requires that the DM, in addition to setting the goals for objectives, also be able to give an ordinal ranking of the objectives. This may result in a nonlinear goal-programming problem if objectives or constraints are nonlinear. Goal Attainment Method, Global Criterion Method and Utility Function Method are also used to solve multiobjective optimization problems.