Improved ant colony optimization algorithms for continuous ... - CoDE

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60 Appendix Table 6.1: The mathematical formulation of the welded beam design problem case A. g1 g2 g3 g4 g5 g6 g7 g8 g9 min f(x) = 1.10471 x 2 1 x2 + 0.04811 x3x4 (14 + x2) where τ(x) = τ(x) − τmax ≤ 0 σ(x) − σmax ≤ 0 x1 − x4 ≤ 0 0.10471 x2 1 + 0.04811 x3x4 (14 + x2) − 5 ≤ 0 0.125 − x1 ≤ 0 δ(x) − δmax ≤ 0 P − Pc(x) ≤ 0 0.1 ≤ x1, x4 ≤ 2.0 0.1 ≤ x2, x3 ≤ 10.0 (τ ′ ) 2 + 2τ ′ τ ′′ x2 2R + (τ ′′ ) 2 τ ′ = P √ , τ 2x1x2 ′′ = MR X2 J , M = P (L + 2 ) x2 2 x1+x3 R = 4 + ( 2 ) 2 √ x2 2 J = 2 2x1x2 12 + x1+x3 2 2 6P L σ(x) = x4x2 4P L3 , δ(x) = 3 Ex3 3x4 Pc(x) = 4.013E x2x 3 6 4 36 L2 1 − x3 E 2L 4G P = 6000lb, L = 14in., E = 30 × 106psi, G = 12 × 106psi τmax = 1360psi, σmax = 30000psi, δ = 0.25in. Table 6.2: Material properties for the welded beam design problem case B Methods x5 S(10 3 psi) E(10 6 psi) G(10 6 psi) c1 c2 Steel 30 30 12 0.1047 0.0481 Cast iron 8 14 6 0.0489 0.0224 Aluminum 5 10 4 0.5235 0.2405 Brass 8 16 6 0.5584 0.2566 min f(x) = (1 + c1) x 2 1 x2 + c2 x3x4 (14 + x2) S − τmax √ ≤ 0 x2 2 J = 2 2x1x2 12 + x1+x3 2 2 , if x6 : two side √ x2 2 J = 2 2x1x2 12 + x1+x3 2 2 , if x6 : four side τmax = 0.577 · S (6.1)
6.1 Mathematical formulation of engineering problems 61 Figure 6.2: Schema of the pressure vessel to be designed. Table 6.3: The mathematical formulation of the cases (A, B, C and D) of the pressure vessel design problem. No Case A Case B Case C Case D min f = 0.6224 TsRL + 1.7781 ThR 2 + 3.1611 T 2 s L + 19.84 T 2 g1 s R −Ts + 0.0193 R ≤ 0 g2 −Th + 0.00954 R ≤ 0 g3 −π R 2 L − 4 3 π R3 g4 + 750 · 1728 ≤ 0 L − 240 ≤ 0 g5 1.1 ≤ Ts ≤ 12.5 1.125 ≤ Ts ≤ 12.5 1 ≤ Ts ≤ 12.5 0 ≤ Ts ≤ 100 g6 0.6 ≤ Th ≤ 12.5 0.625 ≤ Th ≤ 12.5 0 ≤ Th ≤ 100 g7 0.0 ≤ R ≤ 240 10 ≤ R ≤ 200 g8 0.0 ≤ L ≤ 240 10 ≤ L ≤ 200 6.1.3 Pressure Vessel Design Problems The pressure vessel design problems requires designing a pressure vessel consisting of a cylindrical body and two hemispherical heads such that the cost of its manufacturing is minimized subject to certain constraints. The schematic picture of the vessel is presented in Figure 6.2. There are four variables where values must be chosen: the thickness of the main cylinder Ts, the thickness of the heads Th, the inner radius of the main cylinder R, and the length of the main cylinder L. While variables R and L are continuous, the thickness for variables Ts and Th may be chosen only from a set of allowed values, these being the integer multiples of 0.0625 inch. The mathematical formulation for the cases (A, B, C and D) is given in Table 6.3. 6.1.4 Coil Spring Design Problem The problem consists in designing a helical compression spring that will hold an axial and constant load. The objective is to minimize the volume of the spring wire used to manufacture the spring. A schematic of the coil spring to be designed is shown in Figure 6.3. The decision variables are the number of spring coils N, the outside diameter of the spring D, and the spring wire diameter d. The number of coils N is an integer variable, the outside diameter of the spring D is a continuous variable, and finally, the spring wire diameter is a discrete variable, whose possible values are given
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Université Libre de Bruxelles Facu
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Acknowledgements I thank Prof. Marc
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Contents Abstract iii Acknowledgeme
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Chapter 1 Introduction Metaheuristi
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them in the smallest function evalu
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6 Ant Colony Optimization Algorithm
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8 Ant Colony Optimization Algorithm
Page 20 and 21: 10 Ant Colony Optimization of Sep-A
Page 22 and 23: 12 Ant Colony Optimization Average
Page 25 and 26: Chapter 3 An Incremental Ant Colony
Page 27 and 28: 3.2 IACOR with Local Search 17 3.2
Page 29 and 30: 3.3 Experimental Study 19 Algorithm
Page 31 and 32: 3.3 Experimental Study 21 Table 3.2
Page 33 and 34: 3.4 Conclusions 23 3.4 Conclusions
Page 35 and 36: 3.4 Conclusions 25 G−CMA−ES (5
Page 37: 3.4 Conclusions 27 Table 3.4: The m
Page 40 and 41: 30 Ant Colony Optimization for Mixe
Page 66 and 67: 56 Conclusions and Future Work of t
Page 69: Chapter 6 Appendix 6.1 Mathematical
Page 73 and 74: 6.1 Mathematical formulation of eng
Page 75 and 76: Bibliography [1] K. Abhishek, S. Le
Page 77 and 78: BIBLIOGRAPHY 67 [20] D. Datta and J
Page 79 and 80: BIBLIOGRAPHY 69 [44] Q. He and L. W
Page 81 and 82: BIBLIOGRAPHY 71 [65] S. Lucidi, V.
Page 83 and 84: BIBLIOGRAPHY 73 [87] Thomas Stützl
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