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

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

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 Cruz

variables functions continuous integermixed linear nonlinear LPILPMILP NLP INLPMINLP On the classification... WSEAS Puerto de la Cruz

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 Cruz

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 Cruz

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 Cruz

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

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

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

MINLP-methods.. WSEAS Puerto de la Cruz

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 Cruz

SMB example problems... (separation of a fructose/glucose mixture) Problem characteristics: Columns12 3 Variables Continuous Binary Constraints Linear Non-linear PDE:s involved2 4 6 WSEAS Puerto de la Cruz

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

Water Mixture Fructose Recycle 1 Glucose 1 14,9 m t= min t= min t= min t= min min min t= min WSEAS Puerto de la Cruz

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 Cruz

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

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

PCB example problems... Problem characteristics: Machines Components Tot. # comp Variables Binary Integer Constraints Linear cpu [sec] WSEAS Puerto de la Cruz

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, WSEAS Puerto de la Cruz