1. An Mixed Integer Approach for Optimizing Production Planning Stefan Emet Department of Mathematics University of Turku Finland WSEAS Puerto de la Cruz15-17.12.2008

2. Outline of the talk… Introduction Some notes on Mathematical Programming Chromatographic separation – the process behind the model MINLP model for the separation problem Objective - Maximizing profit under cyclic operation PDA constraints Numerical solution approaches MINLP methods and solvers Solution principles Some advantages and disadvantages Some example problems Solution results - Some different separation sequences Summary Conclusions and some comments on future research issues WSEAS Puerto de la Cruz15-17.12.2008

3. Optimization problems are usually classified as follows; VariablesFunctions continuous: masses, volumes, flowes prices, costs etc. discrete: binary {0, 1} integer {-2,-1,0,1,2} discrete values {0.2, 0.4, 0.6} linearnon-linear non-convex quasi-convex pseudo-convex convex Classification of optimization problems... WSEAS Puerto de la Cruz15-17.12.2008

4. variables functions continuous integermixed linear nonlinear LPILPMILP NLP INLPMINLP On the classification... WSEAS Puerto de la Cruz15-17.12.2008

5. The separationproblem... H2OH2O C1C1 C2C2 C2C2 C1C1 Column 1 A one-column-system: Goal: Maximize the profits during a cycle, i.e. max 1/ T*(incomes-costs) WSEAS Puerto de la Cruz15-17.12.2008

6. A two-column-system with three components: H2OH2O Column 1Column 2 waste H2OH2O C1C1 C2C2 C3C3 C1C1 C2C2 C3C3 C3C3 C2C2 C1C1 Waste (Note 2*3 PDEs) In general C PDEs/Column, i.e. tot. K*C WSEAS Puerto de la Cruz15-17.12.2008

7. Price of products Cycle length Raw-material costs y kij and y ki in are binary decision variables while t i and τ are continuous ones. p j and w are price parameters. K = number of columns, T = number of time intervals, C = number of components to be separated. MINLP model for the SMB process... Objective function: WSEAS Puerto de la Cruz15-17.12.2008

8. MINLP model for the SMB process... PDEs for the SMB process: Logical functions: Boundary and initial conditions: WSEAS Puerto de la Cruz15-17.12.2008

9. MINLP model for the SMB process... Integral constraints for the pure and unpure components; Pure components: Equality constraints: Unpure components: WSEAS Puerto de la Cruz15-17.12.2008

10. MINLP-formulation summary... Linear constraints Non-linear constraints Boundary value problem Objective WSEAS Puerto de la Cruz15-17.12.2008

11. MINLP-methods.. WSEAS Puerto de la Cruz15-17.12.2008

12. NLP-subproblems: + relative fast convergenge if each node can be solved fast. - dependent of the NLPs MINLP-methods (solvers)... Branch&Bound minlpbb, GAMS/SBB Outer Approximation DICOPT ECP Alpha-ECP MILP NLP MILP and NLP-subproblems: + good approach if the NLPs can be solved fast, and the problem is convex. - non-convexities implies severe troubles MILP-subproblems: + good approach if the nonlinear functions are complex, and e.g. if gradients are approximated - might converge slowly if optimum is an interior point of feasible domain. WSEAS Puerto de la Cruz15-17.12.2008

13. SMB example problems... (separation of a fructose/glucose mixture) Problem characteristics: Columns12 3 Variables Continuous 346392 Binary 142771 Constraints Linear 4278114 Non-linear 16 32 48 PDE:s involved2 4 6 WSEAS Puerto de la Cruz15-17.12.2008

14. Feed mixture Collect separated products Purity requirements: 90% of product 1 90% of product 2. Recycle WSEAS Puerto de la Cruz15-17.12.2008

15. Water Mixture Fructose Recycle 1 Glucose 1 14,9 m t=57-124.8 min t=43.5 - 57 min t=57-116 min t= 0- 43.5 min 116-124.8 min 116-124.8 min t=0-43.5 min WSEAS Puerto de la Cruz15-17.12.2008

16. Workload balancing problem... Decision variables: y ikm =1, if component i is in machine k feeder m. z ikm = # of comp. i that is assembled from machine k and feeder m. Feeders: WSEAS Puerto de la Cruz15-17.12.2008

17. Optimize the profits during a period τ: Objective... where τ is the assembly time of the slowest machine: WSEAS Puerto de la Cruz15-17.12.2008

18. constraints... (slot capacity) (component to place) (all components set) WSEAS Puerto de la Cruz15-17.12.2008

19. PCB example problems... Problem characteristics: Machines 3 3336666 Components102040100100140160180 Tot. # comp.404808161640404040565664647272 Variables Binary 901803609001800252028803240 Integer 901803609001800252028803240 Constraints Linear 17233265216123424478454646144 cpu [sec]0.110.033.332.725.476.4411.47121.7 WSEAS Puerto de la Cruz15-17.12.2008

20. Summary... Though the results are encouraging there are issues to be tackled and/or improved in a future research (in order to enable the solving of larger problems in a finite time); - refinement of the models - further development of the numerical methods Some references… Emet S. and Westerlund T. (2007). Solving a dynamic separation problem using MINLP techniques. Applied Numerical Matematics. Emet S. (2004). A Comparative Study of Solving Some Nonconvex MINLP Problems, Ph.D. Thesis, Åbo Akademi University. Westerlund T. and Pörn R. (2002). Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane Techniques. Optimization and Engineering, 3, 253-280. WSEAS Puerto de la Cruz15-17.12.2008