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The application of non-linear bi-level programming to the aluminium industry

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Abstract

In this paper, a solution algorithm is presented for the bi-level non-linear programming model developed to represent the complete operations of an aluminium smelter. The model is based on the Portland Aluminium Smelter, in Victoria, Australia and aims at maximising the aluminium production while minimising the main costs and activity associated with the production of this output. The model has two variables, the power input measured in kilo-Amperes (kA) and the setting cycle (of the anode replacement [SC]). The solution algorithm is based on the vertex enumeration approach and uses a specially developed grid search algorithm. An examination of the special nature of the model and how this assists the algorithm to arrive at an optimal unique solution (where there exists one) is undertaken. Additionally, future research into expansion of the model into a multi-period one (i.e., in effect a “staircase” model) allowing the optimisation of the smelter operations over a year (rather than as is currently the case, one month) and the broadening of the solution algorithm to deal with a more general problem, are introduced.

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References

  1. Alexandrov, Natalia. and Dennis, J. EJr. (1994). Multilevel Algorithms for Nonlinear Optimization. Institute for Computer Applications in Science and Engineering, NASA Langley Research Centre, Hampton, Virginia. pp 19.

    Google Scholar 

  2. Anandalingam, G. (1988). A Mathematical Programming Model of Decentralized Multi-Level Systems. J. Opl. Res. Soc, 39. 1021–1033.

    Google Scholar 

  3. Bard, J. F. (1983). An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem. Opns Res, 31. 670–684.

    Google Scholar 

  4. Edmunds, A. and Bard, J. F. (1991) Algorithms for Nonlinear Bilevel Mathematical Programs. IEEE Transactions on Systems Man. and Cybernetics. Vol 21. No. 1. January/February. 83–89.

    Google Scholar 

  5. Hobbs, F. and Nelson, K. (1992). A Nonlinear Bilevel Model for Analysis of Electric Utility Demand-Side Planning Issues. Annals of Operations Research. 34. 225–274.

    Google Scholar 

  6. Jayakumar, M.D. and Ramasesh, R.V. (1994). A Solution Cascading Approach to the Decomposition of Staircase Linear Programming. J. Opl. Res. Soc. 45 No 3. 301–308.

    Google Scholar 

  7. KhayyalAl, A. Horst, R. and Pardalos, M. (1992). Global Optimization of Concave Objective Functions subject to Quadratic Constaints: An Application in Nonlinear Bilevel Programming. Annals of Operations Research. 34. 125–147.

    Google Scholar 

  8. Narula, S. C and Nwosu, A. D. (1985) An Algorithm to Solve a Two-Level Resource Control Pre-emptive Hierarchical Programming Problem. Mathematics of Multiobjectives Optimization, Ed. Serafini, P., Springer, New York. 353–373.

    Google Scholar 

  9. Nicholls, M. G. and Hedditch, D. J. (1993). The Development of an Integrated Mathematical Model of an Aluminium Smelter. J. Opl. Res. Soc, 44, No. 3. 225–235.

    Google Scholar 

  10. Nicholls, M. G. (1993). Production Optimisation and the Resultant Impact on Plant Management-An Australian Study. Proceedings of the Second International Conference of the Decision Sciences Institute, Ed Kang. B-S and Choi. J-U., Seoul, Korea, June 14th–16th. 662–665.

  11. Nicholls, M. G. (1994). Mathematical Modelling-Two Diverse Industrial Applications. Proceedings of the 23rd Annual Meeting of the Western Decision Sciences Institute, Ed Khade, A and Brown, R. Hawaii, March 29th–April 2nd. 689–694.

  12. Nicholls, M. G, (1995). Aluminium Production Modelling-A Non- linear Bi-level Programming Approach. Opns Res. Vol 43, No. 2. 208–218.

    Google Scholar 

  13. Suh, S and Kim, T.J. (1992). Solving Nonlinear Bilvel Programming Models of the Equilibrium Network Design Problem: A Comparative Review. Annals of Operations Research. 34. 203–218.

    Google Scholar 

  14. Wen, U.-P. and Hsu, S.-T. (1991). Linear Bi-Level Programming Problems-A Review. J. Opl. Res. Soc. 42. 125–133.

    Google Scholar 

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Authors and Affiliations

  1. School of Information Systems, Swinburne University of Technology, PO Box 218, 3122, Hawthorn, Victoria, Australia

    Miles G. Nicholls

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  1. Miles G. Nicholls
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Nicholls, M.G. The application of non-linear bi-level programming to the aluminium industry. J Glob Optim 8, 245–261 (1996). https://doi.org/10.1007/BF00121268

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  • DOI: https://doi.org/10.1007/BF00121268

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