Mixed Integer Models for Naval Structure Optimization. Maud Bay, Yves Crama & Philippe Rigo University of Liège,

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

2. Introduction Application : naval structure design MINLP : problem characteristics Methods (1 & 2) Results Conclusions Plan Mixed Integer Models for Naval Structure Optimization

3. Introduction : naval structure design Corsaire Ship at sea : weight & strength Mixed Integer Models for Naval Structure Optimization

4. Application : naval structure design Corsaire Ship stiffened panels structure Mixed Integer Models for Naval Structure Optimization

5. w t d h δ Δ Application : naval structure model • Panels • Design variables (9/panel) Mixed Integer Models for Naval Structure Optimization

6. 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… MINLP : The mixed integer model is based on a non linear model Mixed Integer Models for Naval Structure Optimization

7. NLP resolution : using LBR5 (Logiciel des Bordages Raidis; ULg) Data initialization xi0, ximin,ximax Objective function computation (COST)f(x0) Structural analysis (CONSTRAINT)ci(x0) Optimisation (OPTI) : local approximationmin f̃0 (x) s.c. c̃i0 (x)≤ cimax Optimal solution of the local approximationxi* Final solution is not a global optimum MINLP : The mixed integer model is based on a non linear model Mixed Integer Models for Naval Structure Optimization

8. Discrete variables (8/panel) : 1 –dimensions (web width, web height and flange width) xi  {xi : xi min ≤ xi ≤ xi max} becomes 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 ≤ 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 ↑ w d h δ Δ MINLP : from a Non Linear Model to a Mixed Integer Model Mixed Integer Models for Naval Structure Optimization

9. Initialization : Problem with all real variables : P0 Black box local optimum with real variables Do : fix some variables to a rounded value or free some rounded variables (groups) new approx : Pi Black box optimum for Pi or unfeasibility until all variables have their values in their sets of admissible values MINLP algorithm :Combination of local search and approximation methods Mixed Integer Models for Naval Structure Optimization

10. MINLP algorithm : INITIALIZATION END END Optimization of P0 LOCAL SEARCH Y generation of approximation Pi N stop ? Black Box Mixed Integer Models for Naval Structure Optimization

11. First Method s1 s2 s5 s3 s4 s6 s7 x5 x5 x1 x1 x1 x1 Mixed Integer Models for Naval Structure Optimization

12. Second Method s1 s5 s2 s7 s6 s3 s4 x5 [x5] x1 [x1] [x1] x1 Mixed Integer Models for Naval Structure Optimization

13. Comments and results Corsaire ship : Structure model of 22 panels discrete variables : 176 (176! = 2x10320 sol) real variables : 22 Local search tips : • space reduction • tree size : 2x10320= 21060 => 28 (groups / binary tree ) • tree depth : 176 => 8 or less (bounds) • order of the groups of variables X5,X1... Mixed Integer Models for Naval Structure Optimization

14. Results * NLP solution Mixed Integer Models for Naval Structure Optimization

15. Method : The local search combined with the approximation method acts as a guided local search with successive rounding steps. Execution time : 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 the same sets of constraints. Conclusions Mixed Integer Models for Naval Structure Optimization

16. Thank you for your attention Mixed Integer Models for Naval Structure Optimization

17. 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 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. Furtherwork Mixed Integer Models for Naval Structure Optimization

18. real-size structure : corsaire 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 0,01 0,04 >200 >23028 84911 85842 90087 1,1% cost 2 1 0,005 0,005 0,04 >200 >22656 84911 85727 90087 1,0% cost 1 2 0,01 0,01 0,04 35 3153 84911 91295 90087 7,5% cost 2 2 0,01 0,01 0,04 33 742 84911 89150 90087 5,0% cost 2 2 0,005 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 0,005 0,04 159 2706 275347 275073 280271 -0,1% weight 2 1 0,005 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 0,005 0,04 25 587 275347 274805 280271 -0,2% Numerical applications Mixed Integer Models for Naval Structure Optimization

19. domaine (variable/panneau indépendant) description #D #P D(x1/ P4,5,6) = {8,9,10,11,12} épaisseur du panneau 5 3 D(x2/P6) = {450,460,470,480,490,500…550} hauteur d’ame des cadres 11 1 D(x3/P6) = {15,16,17,18,19,20,21,22,23,24,25} épaisseur d’ame des cadres 11 1 D(x4/P6) = {250,260,270,280,290,300…350} largeur de semelle des cadres 11 1 D(x5/P6) = {2.7273…3.3333}// N={9,10,11,12} espacement des cadres 4 1 D(x6/P4,5,6) = {80,90,100,110,120} hauteur d’ame des lisses 5 3 D(x7/ P4,5,6) = {8,9,10,11,12} épaisseur d’ame des lisses 5 3 D(x8/ P4,5,6) = {1} largeur de semelle des lisses 1 3 D(x9/P4) = {800,857,923,1000,1091,1200}//N={10…15} esp. des lisses//Nbre 6 1 D(x9/P5) = {900,1125}//N={4,5} espacement des lisses//Nbre 2 1 D(x9/P6) = {875,1167}//N={3,4} espacement des lisses//Nbre 2 1 Nombre de solutions possibles (x1…x9) = 5³.11.11.11.4.5³.5³.1³.6.2.2 = 249.562.500 000  si une évaluation dure une seconde, le temps de calcul est d’environ 7.900 ans. Complexité :calcul du nombre de solutions à analyser pour l’Example935 Mixed Integer Models for Naval Structure Optimization