Engineering Optimization: Theory and Practice, Fourth Edition

14.10 Multiobjective Optimization 763 60 40 20 (^012345678) f 1 = (x−3)^4 f 2 = (x−6)^2 P Q x f Figure 14.9 Pareto optimal solut ...

764 Practical Aspects of Optimization wherewiis a scalar weighting factor associated with theith objective function. This method ...

14.10 Multiobjective Optimization 765 14.10.5 Lexicographic Method In the lexicographic method, the objectives are ranked in ord ...

766 Practical Aspects of Optimization subject to gj( X)≤ 0 , j= 1 , 2 ,... , m fj(X)+dj+−dj−=bj, j= 1 , 2 ,... , k dj+≥ 0 , j= 1 ...

14.11 Solution of Multiobjective Problems Using MATLAB 767 with the weights satisfying the normalization condition ∑k i= 1 wi= 1 ...

768 Practical Aspects of Optimization 4- x(2); ... x(2)- 4; ... x(2)+4*x(1)- 4; ... 1- x(1); ... x(1)- 2- x(2)] ceq = []; Step ...

References and Bibliography 769 14.3 R. L. Fox and H. Miura, An approximate analysis technique for design calculations, AIAA Jou ...

770 Practical Aspects of Optimization 14.23 L. A. Schmit and C. Fleury, Structural synthesis by combining approximation concepts ...

Review Questions 771 14.44 W. Stadler, Ed.,Multicriteria Optimization in Engineering and in the Sciences, Plenum Press, New York ...

772 Practical Aspects of Optimization (e)Bounded objective function method (f)Lexicographic method Problems 14.1 Consider the mi ...

Problems 773 (a)Exact displacement solutionU 0 atX 0 (b)Exact displacement solution(U 0 +U)at the perturbed design,(X 0 +X) (c ...

774 Practical Aspects of Optimization 14.10 The eigenvalue problem for the stepped bar shown in Fig. 14.11 can be expressed as [ ...

Problems 775 Y 1 Y 2 k 1 k 2 k 3 m 1 m 2 Figure 14.13 Two-degree-of-freedom spring–mass system. whereEis Young’s modulus,Ithe ar ...

776 Practical Aspects of Optimization Bar 1 (area = A 1 ) Bar 2(area = A 2 ) P = 1000 N 9 m 1 m Figure 14.14 Two-bar truss. indu ...

Problems 777 (^12) 3 P^45 ° X Y x x h Figure 14.15 Two-bar truss. subject to g 1 (X)= P ( 1 +x 1 ) √ 1 +x 12 2 √ 2 x 1 x 2 −σ 0 ...

778 Practical Aspects of Optimization f 1 (X)= − 25 (x 1 − 2 )^2 −(x 2 − 2 )^2 −(x 3 − 1 )^2 −(x 4 − 4 )^2 −(x 5 − 1 )^2 f 2 (X) ...

A Convex and Concave Functions Convex Function. A functionf (X)is said to be convex if for any pair of points X 1 =      ...

780 Convex and Concave Functions Figure A.1 Functions of one variable:(a)convex function in one variable;(b)concave function in ...

Convex and Concave Functions 781 Theorem A.1A functionf (X)is convex if for any two pointsX 1 andX 2 , we have f(X 2 )≥f(X 1 ) + ...

782 Convex and Concave Functions The following theorem establishes a very important relation, namely, that any local minimum is ...