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An Mixed Integer Approach for Optimizing Production Planning

An Mixed Integer Approach for Optimizing Production Planning. Stefan Emet. Department of Mathematics University of Turku Finland. WSEAS Puerto de la Cruz 15-17.12.2008. Outline of the talk…. Introduction Some notes on Mathematical Programming

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An Mixed Integer Approach for Optimizing Production Planning

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  1. An Mixed Integer Approach for Optimizing Production Planning Stefan Emet Department of Mathematics University of Turku Finland WSEAS Puerto de la Cruz 15-17.12.2008

  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 Cruz 15-17.12.2008

  3. Classification of optimization problems... Optimization problems are usually classified as follows; Variables Functions • discrete: • binary {0, 1} • integer {-2,-1,0,1,2} • discrete values {0.2, 0.4, 0.6} linear • non-linear • non-convex • quasi-convex • pseudo-convex • convex • continuous: • masses, volumes, flowes • prices, costs etc. WSEAS Puerto de la Cruz 15-17.12.2008

  4. INLP MINLP NLP nonlinear functions LP ILP MILP linear integer mixed continuous variables On the classification... WSEAS Puerto de la Cruz 15-17.12.2008

  5. C2 H2O C1 Column 1 C2 C1 The separationproblem... A one-column-system: Goal: Maximize the profits during a cycle, i.e. max 1/T*(incomes-costs) WSEAS Puerto de la Cruz 15-17.12.2008

  6. C2 C3 H2O H2O C1 Column 1 Column 2 Waste waste C1 C2 C3 C1 C2 C3 A two-column-system with three components: (Note 2*3 PDEs) In general C PDEs/Column, i.e. tot. K*C WSEAS Puerto de la Cruz 15-17.12.2008

  7. MINLP model for the SMB process... Objective function: Raw-material costs Price of products Cycle length ykij and ykiinare binary decision variables while ti and τ are continuous ones. pj and w are price parameters. K = number of columns, T = number of time intervals, C = number of components to be separated. WSEAS Puerto de la Cruz 15-17.12.2008

  8. MINLP model for the SMB process... PDEs for the SMB process: Boundary and initial conditions: Logical functions: WSEAS Puerto de la Cruz 15-17.12.2008

  9. Pure components: Unpure components: MINLP model for the SMB process... Integral constraints for the pure and unpure components; Equality constraints: WSEAS Puerto de la Cruz 15-17.12.2008

  10. MINLP-formulation summary... Objective Linear constraints Boundary value problem Non-linear constraints WSEAS Puerto de la Cruz 15-17.12.2008

  11. MINLP-methods.. WSEAS Puerto de la Cruz 15-17.12.2008

  12. NLP NLP NLP NLP NLP MINLP-methods (solvers)... Outer Approximation DICOPT ECP Alpha-ECP Branch&Bound minlpbb, GAMS/SBB MILP MILP NLP 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. MILP and NLP-subproblems: + good approach if the NLPs can be solved fast, and the problem is convex. - non-convexities implies severe troubles NLP-subproblems: + relative fast convergenge if each node can be solved fast. - dependent of the NLPs WSEAS Puerto de la Cruz 15-17.12.2008

  13. SMB example problems... (separation of a fructose/glucose mixture) Problem characteristics: Columns 1 2 3 Variables Continuous 34 63 92 Binary 14 27 71 Constraints Linear 42 78 114 Non-linear 16 32 48 PDE:s involved 2 4 6 WSEAS Puerto de la Cruz 15-17.12.2008

  14. Purity requirements: 90% of product 1 90% of product 2. Recycle Collect separated products Feed mixture Recycle WSEAS Puerto de la Cruz 15-17.12.2008

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

  16. Workload balancing problem... Feeders: Decision variables: yikm=1, if component i is in machine k feeder m. zikm= # of comp. i that is assembled from machine k and feeder m. WSEAS Puerto de la Cruz 15-17.12.2008

  17. Objective... Optimize the profits during a period τ: where τ is the assembly time of the slowest machine: WSEAS Puerto de la Cruz 15-17.12.2008

  18. constraints... (slot capacity) (all components set) (component to place) WSEAS Puerto de la Cruz 15-17.12.2008

  19. PCB example problems... Problem characteristics: Machines 3 3 3 3 6 6 6 6 Components 10 20 40 100 100 140 160 180 Tot. # comp. 404 808 1616 4040 4040 5656 6464 7272 Variables Binary 90 180 360 900 1800 2520 2880 3240 Integer 90 180 360 900 1800 2520 2880 3240 Constraints Linear 172 332 652 1612 3424 4784 5464 6144 cpu [sec] 0.11 0.03 3.33 2.72 5.47 6.44 11.47 121.7 WSEAS Puerto de la Cruz 15-17.12.2008

  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 Cruz 15-17.12.2008

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