Optimization of a Nonlinear Workload Balancing Problem

Transcription

1 Optimization of a Nonlinear Workload Balancing Problem Stefan Emet Department of Mathematics and Statistics University of Turku Finland

2 Outline of the talk Introduction Some notes on Mathematical Programming MINLP methods and solvers Solution principles Some advantages and disadvantages MINLP model for the PCB-problem Objective - Maximizing profit under cyclic operation Some example problems Solution results Summary Conclusions and some comments on future research issues

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

4 linear nonlinear On the classification... NLP INLP MINLP LP ILP MILP continuous integer mixed variables

5 MINLP-methods.. Branch and Bound Methods Dakin R. J. (1965). Computer Journal, 8, Gupta O. K. and Ravindran A. (1985). Management Science, 31, Leyffer S. (2001). Computational Optimization and Applications, 18, Cutting Plane Methods Westerlund T. and Pettersson F. (1995). An Extended Cutting Plane Method for Solving Convex MINLP Problems. Computers Chem. Engng. Sup., 19, Westerlund T., Skrifvars H., Harjunkoski I. and Porn R. (1998). An Extended Cutting Plane Method for Solving a Class of Non-Convex MINLP Problems. Computers Chem. Engng., 22, Westerlund T. and Pörn R. (2002). Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane Techniques. Optimization and Engineering, 3, Decomposition Methods Generalized Benders Decomposition Geoffrion A. M. (1972). Journal of Optimization Theory and Appl., 10, Outer Approximation Duran M. A. and Grossmann I. E. (1986). Mathematical Programming, 36, Viswanathan J. and Grossmann I. E. (1990). Computers Chem. Engng, 14, Generalized Outer Approximation Yuan X., Piboulenau L. and Domenech S. (1989). Chem. Eng. Process, 25, Linear Outer Approximation Fletcher R. and Leyffer S. (1994). Mathematical Programming, 66,

6 MINLP-methods (solvers)... Branch&Bound minlpbb, GAMS/SBB Outer Approximation DICOPT ECP Alpha-ECP NLP NLP NLP MILP MILP NLP NLP NLP NLP-subproblems: + relative fast convergenge if each node can be solved fast. - dependent of the NLPs 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.

7 Workload balancing problem... Feeders: Placementhead PCB 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.

8 Production balancing problem... Production lines: Line 1: Line 2: Line n:

9 Objective (one production line).. Optimize the profits during a period τ: K c Y k k k 1 max where τ is the assembly time of the slowest machine: s. t. M I m1 i1 tik zikm, k 1,..., K

10 constraints... (slot capacity) M sik yikm m1 S km (all components set) K M zikm k1 m1 d i (component to place) z ikm d i y ikm 0

11 PCB example problems... Problem characteristics: Machines Components Tot. # comp Variables Binary Integer Constraints Linear cpu [sec]

12 Objective (multiline system)... min max l1,, L I i1 z il t il where τ is the assembly time of the slowest machine: Decision variables: z il = # of products i that are assembled on line l t il = production time of product i on line

13 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 - implementation of convexification strategies Some references Emet S. et al. (2010). Workload balancing in printed circuit board assembly line. Int. Journal of Adv Manufacturing Technology, 50, Porn R. et al. (2008). Global Solution of Optimization Problems with Signomial Parts. Discrete Optimization, 5, Emet S. (2004). A Comparative Study of Solving Some Nonconvex MINLP Problems, Ph.D. Thesis, Åbo Akademi University.