Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Classification based on the number of objective functions <s> 1 Introduction.- 1.1 Objectives of This Study.- 1.2 (Fuzzy) Multiple Objective Decision Making.- 1.3 Classification of (Fuzzy) Multiple Objective Decision Making.- 1.4 Applications of (Fuzzy) Multiple Objective Decision Making.- 1.5 Literature Survey.- 1.6 Fuzzy Sets.- 2 Multiple Objective Decision Making.- 2.1 Introduction.- 2.2 Goal Programming.- 2.2a A Portfolio Selection Problem.- 2.2b An Audit Sampling Problem.- 2.3 Fuzzy Programming.- 2.3.1 Max-Min Approach.- 2.3.1a A Trade Balance Problem.- 2.3.1b A Media Selection Problem.- 2.3.2 Augmented Max-Min Approach.- Example.- 2.3.2a A Trade Balance Problem.- 2.3.2b A Logistics Planning Model.- 2.3.3 Parametric Approach.- Example.- 2.4 Global Criterion Approach.- 2.4.1 Global Criterion Approach.- 2.4.1a A Nutrition Problem.- 2.4.2 TOPSIS for MODM.- 2. .2a A Water Quality Management Problem.- 2.5 Interactive Multiple Objective Decision Making.- 2.5.1 Optimal System Design.- 2.5.1a A Production Planning Problem.- 2.5.2 KSU-STEM.- 2.5.2a A Nutrition Problem.- 2.5.2b A Project Scheduling Problem.- 2.5.3 ISGP-II.- 2.5.3a A Nutrition Problem.- 2.5.3b A Bank Balance Sheet Management Problem.- 2.5.4 Augmented Min-Max Approach.- 2.5.4a A Water Pollution Control Problem.- 2.6 Multiple Objective Linear Fractional Programming.- 2.6.1 Luhandjula's Approach.- Example.- 2.6.2 Lee and Tcha's Approach.- 2.6.2a A Financial Structure Optimization Problem.- 2.7 Multiple Objective Geometric Programming.- Example.- 2.7a A Postal Regulation Problem.- 3 Fuzzy Multiple Objective Decision Making.- 3.1 Fuzzy Goal Programming.- 3.1.1 Fuzzy Goal Programming.- 3.1.1a A Production-Marketing Problem.- 3.1.1b An Optimal Control Problem.- 3.1.1c A Facility Location Problem.- 3.1.2 Preemptive Fuzzy Goal Programming.- Example: The Production-Marketing Problem.- 3.1.3 Interpolated Membership Function.- 3.1.3.1 Hannan's Method.- Example: The Production-Marketing Problem.- 3.1.3.2 Inuiguchi, Ichihashi and Kume's Method.- Example: The Trade Balance Problem.- 3.1.3.3 Yang, Ignizio and Kim's Method.- Example.- 3.1.4 Weighted Additive Model.- 3.1.4.1 Crisp Weights.- 3.1.4.1a Maximin Approach.- Example: The Production-Marketing Problem.- 3.1.4.1b Augmented Maximin Approach.- 3.1.4.1c Supertransitive Approximation.- Example: The Production-Marketing Problem.- 3.1.4.2 Fuzzy Weights.- Example: The Production-Marketing Problem.- 3.1.5 A Preference Structure on Aspiration Levels.- Example: The Production-Marketing Problem.- 3.1.6 Nested Priority.- 3.1.6a A Personnel Selection Problem.- 3.2 Fuzzy Global Criterion.- Example.- 3.3 Interactive Fuzzy Multiple Objective Decision Making.- 3.3.1 Werners's Method.- Example: The Trade Balance Problem.- 3.3.1a An Aggregate Production Planning Problem.- 3.3.2 Lai and Hwang's Method.- 3.3.3 Leung's Method.- Example.- 3.3.4 Fabian, Ciobanu and Stoica's Method.- Example.- 3.3.5 Sasaki, Nakahara, Gen and Ida's Method.- Example.- 3.3.6 Baptistella and Ollero's Method.- 3.3.6a An Optimal Scheduling Problem.- 4 Possibilistic Multiple Objective Decision Making.- 4.1 Introduction.- 4.1.1 Resolution of Imprecise Objective Functions.- 4.1.2 Resolution of Imprecise Constraints.- 4.2 Possibilistic Multiple Objective Decision Making.- 4.2.1 Tanaka and His Col1eragues' Methods.- Example.- 4.2.1.1 Possibilistic Regression.- Example 1.- Example 2.- 4.2.1.2 Possibilistic Group Method of Data Handling.- Example 28.- 4.2.2 Lai and Hwang's Method.- 4.2.3 Negi's Method.- Example.- 4.2.4 Luhandjula's Method.- Example.- 4.2.5 Li and Lee's Method.- Example.- 4.2.6 Wierzchon's Method.- 4.3 Interactive Methods for PMODM.- 4.3.1 Sakawa and Yano's Method.- Example.- 4.3.2 Slowinski's Method.- 4.3.2a A Long-Term Development Planning Problem of a Water Supply System.- 4.3.2b A Land-Use Planning Problem.- 4.3.2c A Farm Structure Optimization Problem.- 4.3.3 Rommelranger's Method.- Example.- 4.4 Hybrid Problems.- 4.4.1 Tanaka, Ichihashi and Asai's Method.- Example.- 4.4.2 Inuiguchi and Ichihashi's Method.- Example.- 4.5 Possibilistic Multiple Objective Linear Fractional Programming.- 4.6 Interactive Possibilistic Regression.- 4.6.1 Crisp Output and Crisp Input.- Example.- 4.6.2 Imprecise Output and Crisp Input.- Example.- 4.6.3 Imprecise Output and Imprecise Input.- Example.- 5 Concluding Remarks.- 5.1 Future Research.- 5.2 Fuzzy Mathematical Programming.- 5.3 Multiple Attribute Decision Making.- 5.4 Fuzzy Multiple Attribute Decision Making.- 5.5 Group Decision Making under Multiple Criteria.- Books, Monographs and Conference Proceedings.- Journal Articles, Technical Reports and Theses.- Appendix: Stochastic Programming.- A.1 Stochastic Programming with a Single Objective Function.- A.1.1 Distribution Problems.- A.1.2 Two-Stage Programming.- A.1.3 Chance-Constrained Programming.- A.2 Stochastic Programming with Multiple Objective Functions.- A.2.1 Distribution Problem.- A.2.2 Goal Programming Problem.- A.2.3 Utility Function Problem.- A.2.4 Interactive Problem.- References. <s> BIB001 </s> Survey on Design of Truss Structures by Using Fuzzy Optimization Methods <s> Classification based on the number of objective functions <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
Depending on the number of objective functions to be minimized, optimization problem can be classified as single and multi-objective programming problem. Multiobjective Programming Problem: Multiobjective optimization in last two decades has been acknowledged as an advanced design technique in structural optimization (Eschenawer et.al., 1990 ). The reason is that most of the real-world problems are multidisciplinary and complex, as there is always more than one important objective in each problem. To accommodate many conflicting design goals, one needs to formulate the optimization problem with multiple objectives. One important reason for the success of the multiobjective optimization approach is its natural property of allowing the designer to participate in the design selection process even after the formulation of the mathematical optimization model. The main task in structural optimization is determining the choice of the design variables, objectives, and constraints. Sometimes only one dominating criterion may be a sufficient objective for minimization, especially if the other requirements can be presented by equality and inequality constraints. But generally the choice of the constraint limits may be a difficult task in a practical design problem. These allowable values can be rather fuzzy, even for common quantities such as displacements, stresses, and natural frequencies. If the limit cannot be determined, it seems reasonable to treat that quantity as an objective. In addition, usually several competing objectives appear in a real-life application, and thus the designer is faced with a decision-making problem in which the task is to find the best compromise solution between the conflicting requirements. A multiobjective optimization problem can be formulated as follow: subject to hj(x) = 0j = 1,2,...,p< n (8) where x is n-dimensional design variable vector, fi(x) is objective function. A variety of techniques and applications of multiobjective optimization have been developed over the past few years. The progress in the field of multicriteria optimization was summarized by BIB001 and later by  . Stadler inferred from his survey that if one has decided that an optimal design is to be based on the consideration of several criteria, then the multicriteria theory (Pareto theory) provides the necessary framework. In addition, if the minimization or maximization is the objective for each criterion, then an optimal solution should be a member of the corresponding Pareto set. Only then does any further improvement in one criterion require a clear tradeoff with at least one other criterion. Radfors, et al (1985) in their study has explored the role of Pareto optimization in computer-aided design. They used the weighting method, noninferior set estimation (NISE) method, and constraint method for generating the Pareto optimal. The authors discussed the control and derivation of meaning from the Pareto sets. Pareto optimality serves as the basic multicriteria optimization concept in virtually all of the previous literature BIB002 . A general multiobjective optimization problem is to find the vector of design variables X = ( x 1 , x 2 , …, x n )T that minimize a vector objective function F(X) over the feasible design space X. It is the determination of a set of nondominated solutions (Pareto optimum solutions or noninferior solutions) that achieves a compromise among several different, usually conflicting, objective functions. The Pareto optimal is stated in simple words as follows: A vector X* is Pareto optimal if there exists no feasible vector X which would increase some objective function without causing a simultaneous decrease in at least one objective function. This definition can be explained graphically. An arbitrary collection of feasible solutions for a two-objective maximization problem is shown in Figure 2 . The area inside of the shape and its boundaries are feasible. The axes of this graph are the objectives F1 and F2. It can be seen from the graph that the noninferior solutions are found in the portion of the boundary between points A and B. Thus, here arises the decision-making problem from which a partial or complete ordering of the set of nondominated objectives is accomplished by considering the preferences of the decision maker. Most of the multiobjective optimization techniques are based on how to elicit the preferences and determine the best compromise solution.