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The Story of An Optimisation Problem

The Story of An Optimisation Problem. Christer Carlsson IAMSR/Åbo Akademi University. In the Beginning There Was an LP-Problem. A Ahlström Oy, systeemianalyysiosasto [1970’es] Tage Carlson, Tor-Erik Holmberg Markku Kallio was there, then he went to Stanford

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The Story of An Optimisation Problem

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  1. The Story of An Optimisation Problem Christer Carlsson IAMSR/Åbo Akademi University Finnish OR Society 13.11 2003

  2. In the Beginning There Was an LP-Problem • A Ahlström Oy, systeemianalyysiosasto [1970’es] • Tage Carlson, Tor-Erik Holmberg • Markku Kallio was there, then he went to Stanford • Bengt Holmström replaced Markku, then he went to Stanford • I did a production planning problem at Varkaus Plywood Mill as part of my licentiate thesis in OR • We had some heated debates on how to do the LP-model; Bengt was not sure that it was smart to do an LP ... Finnish OR Society 13.11 2003

  3. In the Beginning There Was an LP-Problem • The production planning problem • There were around 250 products, too many … • We knew the product margins as we had painstakingly calculated them the previous year • Many of the technical coefficients were not known and we had to get estimates through ad hoc measuring operations in the mill • The foremen got excited and we had to interrupt the operations after only 6 shifts • This became a key problem for the continued use of the model … Finnish OR Society 13.11 2003

  4. In the Beginning There Was an LP-Problem • The production planning problem • The model had about 350+ variables and 60-70 constraints • We found several optimal solutions by changing the RHS vectors a number of times; we could eliminate 25-35 products • Then we run several sensitivity analyses and got a better understanding of the solution • Holmberg: I need integer solutions with the product variables • We did a 10 % branch-and-bound from the Control Data terminal on Lauttasaari; after 1 hour it was still running … Finnish OR Society 13.11 2003

  5. Then It Became a MOLP-Problem • Milan Zeleny: Multiple Criteria Decision Making, McGraw-Hill (1982) • Mr Okaes’ Plywod Production Problem; several objective functions (solvers’ choice) • Milan’s software did not work, no MOLP-solution • Successful solutions done by J.P. Dauer, P. Nijkamp, J.Spronk, J. Wallenius, S. Zionts • Several solutions done by quite a few students in several universities over many years • Some very good ad hoc solution methods used, some of the approaches very smart Finnish OR Society 13.11 2003

  6. Turned Back to a Big LP-Problem • Jaakko Pöyry: Fennia Plywood Mill • Too many products, some of them probably not profitable, the mill should be restructured • A team worked through the mill and collected both profit margins and the technical coefficients (partly on file by Jaakko Pöyry) • A detailed production process was included • The LP model had 1100+ variables 250-350 constraints • The matrix was constructed by hand; in the morning hours of the 5th straight day of working we found inconsistencies in the matrix … Finnish OR Society 13.11 2003

  7. Evolved Into the NPI Case • The LP- and MOLP-cases were interesting and challenging exercises • We always found solutions, sometimes after brute force operations, sometimes after smart improvisations • The mills we worked on got restructured; real decisions were made – thus successful applications • Yet, we were never fully sure that we had managed to find the best possible optimal solution using multiple criteria • The hunt continued … Finnish OR Society 13.11 2003

  8. Evolved Into the NPI Case • The NPI case is a production planning problem in the paper products industry [MCDA Summer School] • Multiple objectives: • Operating result • Productivity • Capacity • Market share • Competitive position • Return on investments • NPI-hard, MILP Finnish OR Society 13.11 2003

  9. Evolved Into the NPI Case • The NPI case is a production planning problem in the paper products industry • Select key objectives: Operating profit, ROI • Select key variables: Production of main qualities in different countries • Discrete decisions, several constraints,lots of input data, a large combinatorial search space • CPLEX – Linear Optimizer Package • Solved as alternative cases • Linux workstation, a few seconds … Finnish OR Society 13.11 2003

  10. OP Inv. Market sh. Prod. costs Shipping Production (MFIM) (MFIM) (%) (MFIM) (MFIM) (tons) npi19,994 19,200 13.6 18,225 2,665 5,590,499 npi29,954 19,200 13.9 18,668 2,668 5,681,437 npi3 6,767 0 11.9 18,102 2,042 4,874,124 npi4 9,589 18,600 14.4 19,974 2,092 5,908,248 npi59,658 20,400 14.9 20,712 2,151 6,098,248 npi611,655 38,400 15.1 19,165 2,874 6,196,000 Evolved Into the NPI Case Finnish OR Society 13.11 2003

  11. Then We Had Some Fuzzy Adventures C. Carlsson-P.Korhonen, A Parametric Approach to Fuzzy Linear Programming, Fuzzy Sets and Systems, Vol. 20, 1986 • Can we prove that it is possible to find optimal solutions to LP-problems with fuzzy parameters? • Can we find compromise solutions in fuzzy MOLP problems with conflicting goals? • Can we do it with interdependent goals? [There is no such thing as interdependence in MCDM – Ron Yager] • Yes, there are a number of good ways with t-norms and possibility theory Finnish OR Society 13.11 2003

  12. Then We Had Some Fuzzy Adventures • Fuzzy optimisation methods turned out to be a fertile field in which a number of old beliefs could be challenged because we had some new and powerful methods • There are classes of interdependent criteria MCDM problems • Complex decision problems are simplified when we work them with fuzzy optimisation methods as we canrelax the requirements on precision • Then we could turn it around and start developing fuzzy number theory to improve the tools we have • Our understanding of complex problems helped us to find good features for fuzzy set theory … Finnish OR Society 13.11 2003

  13. B 1 g A ACB Non-interactive possibility distributions Finnish OR Society 13.11 2003

  14. B 1 g A C The case of rf (A, B) = 1 Finnish OR Society 13.11 2003

  15. B 1 g A D The case of rf (A, B) = –1 Finnish OR Society 13.11 2003

  16. B 1 g A E The case of rf (A, B) = –1/3 Finnish OR Society 13.11 2003

  17. B 1 g A F The case of rf (A, B) = 1/3 Finnish OR Society 13.11 2003

  18. B 1 g A G The case of rf (A, B) = 0 for interactive fuzzy numbers Finnish OR Society 13.11 2003

  19. B 1 g A H The case of rf (A, B) = –1/3 Finnish OR Society 13.11 2003

  20. B 1 g A I The case of rf (A, B) = 1/3 Finnish OR Society 13.11 2003

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