AIChE 200 7 Annual Meeting Sa lt Lake , UT November 04 – 09 , 200 7

1. Solving Flexibility Index Problem Using Combined Stochastic Method and Reduced Space Search Algorithms AIChE 2007 Annual Meeting Salt Lake, UT November 04 – 09, 2007 Session #288: Design, Analysis and Operations Under Uncertainty (10A03) Paper 288e Jeonghwa Moon, Kedar Kulkarini, Libin Zhang and Andreas A. Linninger 11/06/2007 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. Introduction • Flexibility index problem (Swaney and Grossman, 1985) • determine the maximum parameter (q) range that a design can tolerate for feasible operation • Previous work • Exhaustive enumeration (Swaney and Grossmann, 1985a) • Implicit vertex enumeration and heuristic vertex search (Swaney and Grossmann, 1985b) • Active constraint formulation (Grossmann and Floudas, 1987) • Convex Hull methods (Ierapetritou et al, 2001) • Deterministic global optimization methods – a-BB algorithm (Floudas et al, 2001) FI : defined as the largest scaled deviation of any of the expected deviations

3. s.t. Feasibility function MINLP Formulation Mathematical formulations Critical point

4. New approach is needed We suggest a New hybrid algorithm (genetic algorithms+ line search for handling constraints) Characteristics of FI problem • Why not deterministic way? (a) (b) Non-differentiable and discontinuous Global optimization problem

5. + penalty Search space Search space Search space Rejection Penalization Repairing Genetic Algorithm for constraint problems • Features of genetic algorithm • Stochastic optimization method which mimics natural selection and principles of genetics . • It is global optimizer • No sensitivity information is required • The ways to treat individuals not in a search space

6. START Cost function, Search space Initial population Repair individuals Cost evaluation Natural Selection No Mating Mutation Is convergent? Yes END Framework Cost (objective) function :infinity norm of Search space: boundary of feasible region

7. Framework START Cost function, Search space Initial population Repair individuals Cost evaluation Natural Selection No Mating Mutation Is convergent? Yes END

8. Framework START Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Special algorithm is needed Mutation Is convergent? Yes END

9. Framework START Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Some of superior individuals are selected Some of inferior individuals are removed Mutation Is convergent? Yes END

10. Framework START Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Mutation Is convergent? Yes END

11. Framework START Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Mutation Some of individuals have new values randomly Is convergent? Yes END

12. Framework START Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Mutation Is convergent? Yes END

13. Framework START Critical point will be found in several iterations Cost function, Search space Initial population Repair procedure Cost evaluation Natural Selection No Mating Mutation This loop is repeated Is convergent? Yes END

14. (+) i=1 (+) i=2 Choose direction (+/-) Θ is feasible i=2 (+) (-) i=1 Θ is infeasible (+) i=2 Move individual until it meets boundary No Select parameter to move Yes End Repair procedure: How to move ? The line search- (without control variable) Step size

15. 1r 1 3r 10r Most individuals converges Example 1-Convex,Nonlinear (4,4) 5.7429,2.2571

16. Example 2:Nonconvex • Non-convex 3D Avg CPU time :230msec MINLP PN=(0.8,0.5,0.8) PN=(0.5,0.5,0.5)

17. Feasibility function when z exists • Movable constraints & actual feasible region θ2 The constraints vary! Z=0.0 Z=0.5 Z=1.0 θ1 z θ2 θ1

18. Select θi and α Select default z value z Zigzag movement θi= θi+ α ∆ θi Repaired individual 1.0 Select zk 0.5 Decide b(+/-) and ∆z Repeat until it reaches boundary zk=zk+bdz 0.0 θ1 :extended  Extended line search • How to get new replacement of individuals when control variables exist ? • Extended line search: • Changes z values to have θ go as far as possible • Zigzag movement in z and θ space

19. z2 z1 θ θ 0.0 0.75 1 z1 0.4 0.0 0.0 z2 0.5 0.5 0.0 Multiple control variables • When many control variables exist • Select one z and do extended line search • Then swap z and do again • Repeat this until every z is involved in search z1=0 z2 z2=0.5 z1 θ θ Select z2 and do line search Select z1 and do line search

20. Case studies • Several studies are done Reactor-cooler (Floudas2001) Heat Exchanger network-1 (Swaney, 1985) Pump and pipe( Grossmann 1987, Floudas 2001)

21. Polymerization Reactor • Average of molecular weight should be kept between 52,569g/mol and 64,361/g/mol • Temperature of reactor must be less than 423K Control variables : QI,Qc

22. Conclusion • It’s new approach for flexibility index problem. • Uses stochastic method with line search algorithms. • Uses geometric characters of uncertain parameters and control variables spaces. • Works regardless of convexity of feasible region. • Easy to formulate. • Proper parameter values of GA are needed. (population size , maximum iteration, selection ratio, mutation ratio) • Our research provides another option for flexibility index problem. • Future work • Genetic algorithm is the best choice for this problem? (PSO, SA) • Another novel repair procedure? • Mating method, termination condition should be studied

23. Reference • Perkins, J.D. & S.P.K. Walsh, 1994, Optimization as a Tool for Design/Control Integration. Proceedings of the IFAC workshop on Integration of Process Design and Control CE. 7_afiriou Ed.), 1. • I. E Grossmann and M. Morari, Operability, resiliency and flexibility : process design objectives for a changing world proc 2nd Int. Conf Foundations Computer Aided Process Design (Weterberg and Chien Eds). CACHE 937 (1984) • Halemane K, Grossmann IE. Optimal Process Design under Uncertainty. AIChE J. 1983; 29(3):425-432. • Swaney R, Grossmann IE. An Index for Flexibility in Chemical Process Design Part I. AIChE J. 1985; 31(4): 621-630. • Grossmann IE, Floudas CA. Active Constraint Strategy for Flexible Analysis in Chemical processes. Comp. Chem. Eng. 1986;11(6):675-693. • Floudas CA, Gumus ZH, Ierapetritou MG. Global Optimization in Design under Uncertainty: Feasibility Test and Flexibility Index Problems. Ind. Eng Chem Res. 2001; 40: 4267-4282. • Ierapetritou MG. New Approach for Quantifying Process Feasibility: Convex and 1-D Quasi-Convex Regions • Holland JH. Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press, 1975. • Goldberg DE. Genetic Algorithms in Search, Optimization, and Machine Learning. Reading ,MA: Addison-Wesley, 1989. • Michalewicz Z. Genetic Algorithms + Data Structures = Evolution Programs, New York, NY: Springer-Verlag, Second Edition, 1994. • Radcliff, N.J. 1991. Forma analysis and random respectful recombination. In proc 4th Int. Conf. on genetic algorithms, San Mateo, CA: Morgan Kauffman. • Maner BR, Doyle FJ, Ogunnaike BA, Pearson RK. Nonlinear model predictive control of a simulated multivariable polymerization reactor using second order Volterra models. Automatica, 32(9):1285-1301, 1996.

24. THANK YOU!

25. Four infeasible areas are found simultaneously Future work-feasibility test • Feasibility test problem • Practical and robust • less expensive than flexibility index problem • Multiple infeasible areas can be found! • Using niche technic Finding multiple solutions in feasibility test problem using fitness sharing

26. θ2 θ1 Z=0.0->0.5 θ2 θ2 θ2 z Zigzag movement 1.0 θ1 θ1 θ1 0.5 Z=0.5 Z=0.5->1.0 Z=1.0 0.0 θ1 Example of extended line search θ2 θ1 Z=0.0

27. θ2 θ1 Z=0.0->0.5 θ2 θ2 θ2 z Zigzag movement 1.0 θ1 θ1 θ1 0.5 Z=0.5 Z=0.5->1.0 Z=1.0 0.0 θ1 Example of extended line search θ2 θ1 Z=0.0

28. z2 z1 θ z2=0.5 z1 θ z1=0 z2 θ Multiple control variables m=2 k=1 Line search with (θi,zk) Local termination? Yes k=1 k=1? No No Yes Any movement of θi is done? k=2 Yes No k=1 k=k+1 k=1 k>m Yes (global termination) k=2 End k=3 : globally terminated m : no. of control variables

29. Reference • Grossmann,I.;Halemane,K.;Swaney, R., “Optimization Strategies For Flexible Chemical Processes” ,Computers and Chemical Eng.,Vol.7(4),439-462,1983. • Halemane,K.; Grossmann,I., “Optimal Process Design under Uncertainty” , AIChE Journal,Vol.29(3),425-432,1983. • Grossmann,I.; Swaney, R., “An Index for Flexibility in Chemical Process Design Part I” ,AIChE Journal,Vol.31(4),621-630,1985. • Grossmann,I.; Swaney, R., “An Index for Flexibility in Chemical Process Design Part II” ,AIChE Journal,Vol.31(4),631-641,1985. • Grossmann,I.; Floudas,C., “Active Constraint Strategy for Flexible Analysis in Chemical Processes” , Computers and Chemical Eng.,Vol.11(6),675-693,1986.

30. g2 g1 Zi θ g2 Zi g1 θ g2 Zi g1 θ Checking moving direction of Z • Procedure • G+=Max(gi(θ, zi+h)) and G-=Max(gi(θ, zi-h)) • When h <0 • Terminate it. G+<0 G- >0 G+<0, G- <0 Zi g1 θ G+>0, G- >0 h gets less than 0

31. z2 g2 znew Zi z1 g1 Moving direction :+ θ Movement of z & θ • Move control variable (z) • Find z2 moving z from z1 with direction until it meets constraint. • Change z value • znew=(z1+z2)/2 • Move uncertain parameter • Go until it meets constraint. • θ(i+1)= θ(i)+∆h • Two types of moving • From feasible region • From infeasible region g2 θ θnew θnew θ Zi g1 Moving direction :+ θ

32. Contents • Introduction • Flexibility analysis • Genetic algorithm • Part I – FI without control variables • Cost function definition • Repairing procedure • Case studies • Part II – FI with control variables • Characteristic of geometry with control variables • Framework for flexibility index problem with control variable • Repair procedure • Case study: • Heat Exchanger network[5][6] • Pump and pipe[3][5][6] • Reactor-cooler system • conclusion • Future work

33. θ2 ? ? Move θ θ1 ? Move z ? Move z Move θ Move θ Individual movement using repairing algorithm Repair procedure • How to get new replacement of individuals ? • Extended line search: • Changes z values to have θ go as far as it can • Zigzag movement in z and θ space • Local optimization method Select θi and α Select default z value θi= θi+ α h Select zk Decide b(+/-) and ∆z Repeat until it reaches boundary zk=zk+bdz :extended 