Mixed Integer Models for Naval Structure Optimization Maud

Mixed Integer Models for Naval Structure Optimization. Maud Bay, Yves Crama & Philippe Rigo University of Liège, Belgium. • • • Maud Bay Yves Crama HEC-Management School of the University of Liège 24 -05 -2006 Mixed Integer Models for Naval Philippe Rigo ANAST, University of Liège Structure Optimization

Plan • Introduction • Naval structure • Introduction • MINLP • Application : naval structure design • Method 1 • MINLP : problem characteristics • Method 2 • Results • Methods (1 & 2) • Conclusion • Results • Conclusions OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 2

Introduction : naval structure design • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results Corsaire Ship at sea : • Conclusion weight & strength OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 3

Application : naval structure design • Introduction • Naval structure • MINLP • Method 1 • Method 2 Corsaire Ship stiffened panels structure • Results • Conclusion OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 4

Application : naval structure model • Panels • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion • Design variables (9/panel) w t d h δ Δ OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 5

MINLP : The mixed integer model is based on a non linear model • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion • non linear objective function F = γ. L. B. Σ{ δ + [(h. d + w. t)x / Δx ] + [(h. d + w. t)y / Δy ] } • explicit non linear constraints (bounds & geometry) • implicit constraints (sets of behavior models - systems of differential equations) • Real variables : panel thickness, number of members… OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 6

MINLP : The mixed integer model is based on a non linear model • Introduction • Naval structure • MINLP resolution : using LBR 5 (Logiciel des Bordages Raidis; ULg) • Method 1 • Method 2 • Results • Conclusion Data initialization xi 0, ximin, ximax Objective function computation (COST) f (x 0) Structural analysis (CONSTRAINT) ci(x 0) Optimisation (OPTI) : local approximation min f 0 (x) s. c. c i 0 (x) ≤ cimax Optimal solution of the local approximation xi* Þ Final solution is not a global optimum OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 7

MINLP : from a Non Linear Model to a Mixed Integer Model Discrete variables (8/panel) : • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion 1 –dimensions (web width, web height and flange width) w h d δ xi {xi : xi min ≤ xi max} becomes Δ yi D with D={yimin, yimin+step, yimin+2*step… yimin+n*step = yimax} avec yimin = ximin/step * step ; yimax = ximax/step * step 2 – number of stiffeners ( frame spacing) xi {xi : xi min ≤ xi max} becomes yi D with D={yimin, … yimax } avec yimin= w /nmax, , …, yimax= w /nmin et nmin = w / ximax ; nmax = w / ximin variables ↑ weight ↑ strength ↑ OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 8

MINLP algorithm : Combination of local search and approximation methods • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion OPT & ENG LLN - 24 -05 -2006 Initialization : • Problem with all real variables : • Black box Ø local optimum with real variables P 0 Do : • fix some variables to a rounded value or free some rounded variables (groups) new approx : Pi • Black box Ø optimum for Pi or unfeasibility until all variables have their values in their sets of admissible values Mixed Integer Models for Naval Structure Optimization 9

MINLP algorithm : INITIALIZATION • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results END Optimization of P 0 LOCAL SEARCH Y N stop ? generation of approximation Pi • Conclusion Black Box OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 10

First Method • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion s 1 x 5 x 5 s 2 x 1 s 3 s 4 OPT & ENG LLN - 24 -05 -2006 s 5 x 1 s 6 s 7 Mixed Integer Models for Naval Structure Optimization 11

Second Method • Introduction • Naval structure s 1 • MINLP • Method 1 • Method 2 • Results • Conclusion x 5 s 5 x 1 [x 1] s 7 s 6 OPT & ENG LLN - 24 -05 -2006 [x 5] s 2 [x 1] x 1 s 3 s 4 Mixed Integer Models for Naval Structure Optimization 12

Comments and results • Introduction • Naval structure • MINLP • Method 1 Corsaire ship : Structure model of 22 panels discrete variables : 176 (176! = 2 x 10320 sol) real variables : 22 • Method 2 • Results • Conclusion Local search tips : • space reduction – tree size : 2 x 10320 = 21060 => 28 (groups / binary tree ) – tree depth : 176 => 8 or less (bounds) • order of the groups of variables X 5, X 1. . . OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 13

Results Method # itera time # itera until best sol Best solution Δ solutions 0 (Bbox runs) • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion Black-box 1 30 1 275 323* Method 1 59 1976 17 274 986 0. 12% Method 2 14 338 12 274 979 0. 13% * NLP solution OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 14

Conclusions Method : • • Introduction The local search combined with the approximation method acts as a guided local search with successive rounding steps. • Naval structure Execution time : • MINLP • • Method 1 • Method 2 • Results • Conclusion The method is fast: a good solution is found in less than 20 runs of the black-box Quality : • The best solution of the MINLP is of the same magnitude than the best solution of the NL problem (with all real variables). Advantage : • The method provides several solutions of comparable quality ► quite different structures with comparable weights and convenient for the same sets of constraints. OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 15

• Introduction • Naval structure • MINLP • Method 1 • Method 2 Thank you for your attention • Results • Conclusion OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 16

Further work • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion • As the objective function is increasing with the variables • as an increase of any variables of a feasible solution leads to a feasible solution • as a decrease of any variables of an unfeasible solution leads to an unfeasible solution Ø each solution analysed provides information reducing the solution space to be explored. we would like to analyse a minimum number of solutions to explore the entire solution space. OPT & ENG LLN - 24 -05 -2006 Mixed Integer Models for Naval Structure Optimization 17

Numerical applications • real-size structure : corsaire • Introduction • Naval structure • MINLP • Method 1 • Method 2 • Results • Conclusion OPT & ENG LLN - 24 -05 -2006 relative loss in the objective value : delta = (D*-Z*)/ Z* influence of TBRA, TBRO, depending on the objective function results (NB : violation % not considered) Obj TBRO TBRA tolarr toldis tolcontr nodes time Z* D* initial UB delta cost 2 1 0, 01 0, 04 >200 >23028 84911 85842 90087 1, 1% cost 2 1 0, 005 0, 04 >200 >22656 84911 85727 90087 1, 0% cost 1 2 0, 01 0, 04 35 3153 84911 91295 90087 7, 5% cost 2 2 0, 01 0, 04 33 742 84911 89150 90087 5, 0% cost 2 2 0, 005 0, 04 37 811 84911 90000 90087 6, 0% weight 1 1 0, 100 0, 050 0, 04 119 1976 275347 274986 279966 -0, 1% weight 1 1 0, 005 0, 04 159 2706 275347 275073 280271 -0, 1% weight 2 1 0, 005 0, 04 147 2559 275347 275253 280271 0, 0% weight 1 2 0, 100 0, 050 0, 04 37 654 275347 275540 279966 0, 1% weight 2 2 0, 100 0, 050 0, 04 14 338 275347 274979 279966 -0, 1% weight 2 2 0, 005 0, 04 25 587 275347 274805 280271 -0, 2% Mixed Integer Models for Naval Structure Optimization 18

Complexité : calcul du nombre de solutions à analyser pour l’Example 935 • • domaine (variable/panneau indépendant) description #D #P Introduction D(x 1/ P 4, 5, 6) = {8, 9, 10, 11, 12} épaisseur du panneau 5 3 Naval structure D(x 2/P 6) = {450, 460, 470, 480, 490, 500… 550} hauteur d’ame des cadres 11 MINLP 1 D(x 3/P 6) = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} épaisseur d’ame des cadres 11 Method 1 1 Method 2 D(x 4/P 6) = {250, 260, 270, 280, 290, 300… 350} largeur de semelle des cadres 11 Results 1 D(x 5/P 6) = {2. 7273… 3. 3333}// N={9, 10, 11, 12} espacement des cadres 4 Conclusion 1 D(x 6/P 4, 5, 6) = {80, 90, 100, 110, 120} hauteur d’ame des lisses 5 3 D(x 7/ P 4, 5, 6) = {8, 9, 10, 11, 12} épaisseur d’ame des lisses 5 3 D(x 8/ P 4, 5, 6) = {1} largeur de semelle des lisses 1 3 D(x 9/P 4) = {800, 857, 923, 1000, 1091, 1200}//N={10… 15} esp. des lisses//Nbre 6 1 D(x 9/P 5) = {900, 1125}//N={4, 5} espacement des lisses//Nbre 2 1 D(x 9/P 6) = {875, 1167}//N={3, 4} espacement des lisses//Nbre 2 1 OPT & ENG Mixed Integer Models for Naval Structure Optimization 19 LLN - 24 -05 -2006 Nombre de solutions possibles (x 1…x 9) = 5³. 11. 11. 4. 5³. 1³. 6. 2. 2 = 249. 562. 500 000