A Decision Aid Approach for Optimisation Problems Involving Several Economic Functions

ABSTRACT

Many concrete real life problems ranging from economic and business to industrial and engineering may be cast into a multi-objective optimisation framework. The redundancy of existing methods for solving this kind of problems susceptible to inconsistencies, coupled with the necessity for checking inherent assumptions before using a given method, make it hard for a nonspecialist to choose a method that fits well the situation at hand. Moreover, using blindly a method as proponents of the hammer principle (when you only have a hammer, you want everything in your hand to be a nail) is an awkward approach at best and a caricatural one at worst. This brings challenges to the design of a tool able to help a Decision Maker faced with these kinds of problems. The help should be at two levels. First the tool should be able to choose an appropriate multi-objective programming technique and second it should single out a satisfying solution using the chosen technique. The choice of a method should be made according to the structure of the problem and to the Decision Maker’s judgment value. This paper is an attempt to satisfy that need. We present a Decision Aid Approach that embeds a sample of good multi-objective programming techniques. The system is able to assist the Decision Maker in the above mentioned two tasks.

Many concrete real life problems ranging from economic and business to industrial and engineering may be cast into a multi-objective optimisation framework. The redundancy of existing methods for solving this kind of problems susceptible to inconsistencies, coupled with the necessity for checking inherent assumptions before using a given method, make it hard for a nonspecialist to choose a method that fits well the situation at hand. Moreover, using blindly a method as proponents of the hammer principle (when you only have a hammer, you want everything in your hand to be a nail) is an awkward approach at best and a caricatural one at worst. This brings challenges to the design of a tool able to help a Decision Maker faced with these kinds of problems. The help should be at two levels. First the tool should be able to choose an appropriate multi-objective programming technique and second it should single out a satisfying solution using the chosen technique. The choice of a method should be made according to the structure of the problem and to the Decision Maker’s judgment value. This paper is an attempt to satisfy that need. We present a Decision Aid Approach that embeds a sample of good multi-objective programming techniques. The system is able to assist the Decision Maker in the above mentioned two tasks.

KEYWORDS

Database; Decision Support System; Model-Base; Multi-Objective Program; Pareto Optimality; Software

Database; Decision Support System; Model-Base; Multi-Objective Program; Pareto Optimality; Software

Cite this paper

M. Rangoaga, M. Luhandjula and S. Ruzibiza, "A Decision Aid Approach for Optimisation Problems Involving Several Economic Functions,"*American Journal of Operations Research*, Vol. 2 No. 3, 2012, pp. 331-338. doi: 10.4236/ajor.2012.23040.

M. Rangoaga, M. Luhandjula and S. Ruzibiza, "A Decision Aid Approach for Optimisation Problems Involving Several Economic Functions,"

References

[1] K. Deb, “Multi-Objective Optimization Using Evolutionary Algorithms,” John Wiley and Sons, New York, 2001.

[2] C. L. Hwang and A. S. M. Masud, “Multiple Objective Decision Making: Methods and Applications: A State-of- the-Art Survey,” Lecture notes in economics and mathematical systems, Vol. 164, Springer-Verlag, Berlin, Heidelberg, 1979. doi:10.1007/978-3-642-45511-7

[3] K. M. Miettinen and M. M. Makela, “Interactive Bundle- Based Method for Non-Differentiable Multi-Objective Optimization: NIMBUS,” Optimization, Vol. 34, No. 3, 1995, pp. 231-246. doi:10.1080/02331939508844109

[4] K. C. Kiwiel, “A Descent Method for Non-Smooth Convex Multi-Objective Minimization,” Large Scale Systems, Vol. 8, 1985, pp. 119-129.

[5] B. Render and R.M. Stair, “Quantitative Analysis for Management,” sixth edition, Prentice Hall, New Jersey, 1997.

[6] G. B. Dantzig, “A Complementary Algorithm for an Optimal Capital Path with Invariant Proportions,” International Institute for Applied Systems Analysis, 1973.

[7] D. G. Luenberger, “Introduction to Linear and Nonlinear Programming,” Addison-Wesley Publishing Company, Menlo-Park, 1973.

[8] N. Karmarkar, “A New Polynomial Time Algorithm for Linear Programming,” Combinatorica, Vol. 4, No. 4, 1984, pp. 373-395. doi:10.1007/BF02579150

[9] W. L. Winston, “Operations Research: Applications and Algorithms,” Third Edition, International Thomson Publishing, California, 1994.

[10] M. Avriel “Nonlinear Programming Analysis and Methods,” Prentice-Hall, New Jersey, 1976.

[11] L. Scharage, “Optimization Modeling with LINGO,” Sixth Edition, LINDO Systems Inc, Chicago, 2006.

[12] K. M. Miettinen and M. M. Makela, “Synchronous Approach in Interactive Multi-Objective Optimization,” European Journal of Operational Research, Vol. 170, No. 3, 2006, pp. 909-922. doi:10.1016/j.ejor.2004.07.052

[13] G. W. Evans, “An Overview of Techniques for Solving Multi-Objective Mathematical Programs,” Management Science, Vol. 30, No. 11, 1984, pp. 1268-1282. doi:10.1287/mnsc.30.11.1268

[14] K. M. Miettinen and M. M. Makela, “Optimization System www Nimbus,” Vol. 9, Laboratory of Scientific Computing, Department of Mathematics, University of Jyvaskyla, Finland, 1998.

[15] R. Caballero, M. Luque, J. Molina and F. Ruiz, “Mopen: A Computational Package for Linear Multi-Objective and Goal Programming Problems,” Decision Support Systems, Vol. 41, No. 1, 2005, pp. 160-175. doi:10.1016/j.dss.2004.06.002

[16] M. Ehrgott, “Multicriteria Optimization,” Second Edition, Springer, Auckland, 2005.

[17] K. M. Miettinen, “Nonlinear Multi-Objective Optimization,” First Edition, Kluwer Academic Publishers, Boston, 1999.

[18] K. C. Kiwiel, “Proximity Control in Bundle Methods for Methods for Convex Non-Differentiable Minimization,” Mathematical Programming, Vol. 46, No. 1-3, 1990, pp. 105-122. doi:10.1007/BF01585731

[19] F. Amador and C Romero, “Redundancy in Lexicographic Goal Programming: An Empiricalapproach,” European Journal of Operational Research, Vol. 41, No. 3, 1989, pp. 347-354. doi:10.1016/0377-2217(89)90255-5

[20] R. Chelouah and P. Siarry, “A Hybrid Method Combining Continuous Tabu Search and Nelder-Mead Simplex Algorithms for the Global Optimization of Multiminima Functions,” European Journal of Operational Research, Vol. 161, No. 3, 2005, pp. 636-654. doi:10.1016/j.ejor.2003.08.053

[21] M. Gershon, “The Role of Weights and Scales in the Application of Multi-Objective Decision Making,” European Journal of Operational Research, Vol. 15, No. 2, 1984, pp. 244-250. doi:10.1016/0377-2217(84)90214-5

[22] R. Benayoun, J. de Montgolfier and J. Tergny, “Linear Programming with Multiple Objective Functions: Step Method (Stem),” Mathematical Programming, Vol. 1, No. 1, 1971, pp. 366-375. doi:10.1007/BF01584098

[23] L. R. Gardiner and R. E. Steuer, “Unified Interactive Multiple Objective Programming,” European Journal of Operational Research, Vol. 74, 1984, pp. 371-406.

[24] J. T. Buchanan, “Multiple Objective Mathematical Programming: A Review,” New Zealand Operational Research, Vol. 14, No. 1, 1986, pp. 1-27.

[25] A. M. Geoffrion, “Proper Efficiency and the Theory of Vector Maximization,” Journal of Mathematical Analysis and Applications, Vol. 22, No. 3, 1968, pp. 619-630. doi:10.1016/0022-247X(68)90201-1

[26] M. I. Henig and Z. Ritz, “Multiplicative Decision Rules for Multi-Objective Decision Problems,” European Journal of Operational Research, Vol. 26, No. 1, 1986, pp. 134-141. doi:10.1016/0377-2217(86)90165-7

[27] A. K. Bhunia and J. Majumdar, “Elitist Genetic Algorithm for Assignment Problem with Imprecise Goal,” European Journal of Operational Research, Vol. 177, 2007, pp. 684-692. doi:10.1016/j.ejor.2005.11.034

[28] C. Botha, E. Ferreira, G. Geldenhuys and H. Ittman, “Selected Topics in Operations Research: Quantitative Management,” UNISA, Pretoria, 1998.

[29] C. D. Gelatt, S. Kirkpatrick and M. P. Vecchi, “Optimization by Simulated Annealing,” Science, Vol. 220, 1983, pp. 45-54.

[30] J. W. Barnes, F. W. Glover and M. Laguna, “Tabu Search Methods for a Single Machinescheduling Problem,” Journal of Intelligent Manufacturing, Vol. 2, No. 2, 1991, pp. 63-74. doi:10.1007/BF01471219

[31] M. J. Rangoaga, “A Decision Support System for MultiObjective Programming Problems,” Master’s Thesis, University of South Africa, Pretoria, 2009.

[32] H. S. Solutions, “Textpad,” 1992. http://wapedia.mobi/en/textpad

[33] S. S. Ruzibiza, “Solving Multi-Objective Mathematical Programming Problems with Fixed and Fuzzy Coefficients,” Master’s Thesis, Independent Institute of Lay Aventists of Kigali, Kigali, 2011.

[1] K. Deb, “Multi-Objective Optimization Using Evolutionary Algorithms,” John Wiley and Sons, New York, 2001.

[2] C. L. Hwang and A. S. M. Masud, “Multiple Objective Decision Making: Methods and Applications: A State-of- the-Art Survey,” Lecture notes in economics and mathematical systems, Vol. 164, Springer-Verlag, Berlin, Heidelberg, 1979. doi:10.1007/978-3-642-45511-7

[3] K. M. Miettinen and M. M. Makela, “Interactive Bundle- Based Method for Non-Differentiable Multi-Objective Optimization: NIMBUS,” Optimization, Vol. 34, No. 3, 1995, pp. 231-246. doi:10.1080/02331939508844109

[4] K. C. Kiwiel, “A Descent Method for Non-Smooth Convex Multi-Objective Minimization,” Large Scale Systems, Vol. 8, 1985, pp. 119-129.

[5] B. Render and R.M. Stair, “Quantitative Analysis for Management,” sixth edition, Prentice Hall, New Jersey, 1997.

[6] G. B. Dantzig, “A Complementary Algorithm for an Optimal Capital Path with Invariant Proportions,” International Institute for Applied Systems Analysis, 1973.

[7] D. G. Luenberger, “Introduction to Linear and Nonlinear Programming,” Addison-Wesley Publishing Company, Menlo-Park, 1973.

[8] N. Karmarkar, “A New Polynomial Time Algorithm for Linear Programming,” Combinatorica, Vol. 4, No. 4, 1984, pp. 373-395. doi:10.1007/BF02579150

[9] W. L. Winston, “Operations Research: Applications and Algorithms,” Third Edition, International Thomson Publishing, California, 1994.

[10] M. Avriel “Nonlinear Programming Analysis and Methods,” Prentice-Hall, New Jersey, 1976.

[11] L. Scharage, “Optimization Modeling with LINGO,” Sixth Edition, LINDO Systems Inc, Chicago, 2006.

[12] K. M. Miettinen and M. M. Makela, “Synchronous Approach in Interactive Multi-Objective Optimization,” European Journal of Operational Research, Vol. 170, No. 3, 2006, pp. 909-922. doi:10.1016/j.ejor.2004.07.052

[13] G. W. Evans, “An Overview of Techniques for Solving Multi-Objective Mathematical Programs,” Management Science, Vol. 30, No. 11, 1984, pp. 1268-1282. doi:10.1287/mnsc.30.11.1268

[14] K. M. Miettinen and M. M. Makela, “Optimization System www Nimbus,” Vol. 9, Laboratory of Scientific Computing, Department of Mathematics, University of Jyvaskyla, Finland, 1998.

[15] R. Caballero, M. Luque, J. Molina and F. Ruiz, “Mopen: A Computational Package for Linear Multi-Objective and Goal Programming Problems,” Decision Support Systems, Vol. 41, No. 1, 2005, pp. 160-175. doi:10.1016/j.dss.2004.06.002

[16] M. Ehrgott, “Multicriteria Optimization,” Second Edition, Springer, Auckland, 2005.

[17] K. M. Miettinen, “Nonlinear Multi-Objective Optimization,” First Edition, Kluwer Academic Publishers, Boston, 1999.

[18] K. C. Kiwiel, “Proximity Control in Bundle Methods for Methods for Convex Non-Differentiable Minimization,” Mathematical Programming, Vol. 46, No. 1-3, 1990, pp. 105-122. doi:10.1007/BF01585731

[19] F. Amador and C Romero, “Redundancy in Lexicographic Goal Programming: An Empiricalapproach,” European Journal of Operational Research, Vol. 41, No. 3, 1989, pp. 347-354. doi:10.1016/0377-2217(89)90255-5

[20] R. Chelouah and P. Siarry, “A Hybrid Method Combining Continuous Tabu Search and Nelder-Mead Simplex Algorithms for the Global Optimization of Multiminima Functions,” European Journal of Operational Research, Vol. 161, No. 3, 2005, pp. 636-654. doi:10.1016/j.ejor.2003.08.053

[21] M. Gershon, “The Role of Weights and Scales in the Application of Multi-Objective Decision Making,” European Journal of Operational Research, Vol. 15, No. 2, 1984, pp. 244-250. doi:10.1016/0377-2217(84)90214-5

[22] R. Benayoun, J. de Montgolfier and J. Tergny, “Linear Programming with Multiple Objective Functions: Step Method (Stem),” Mathematical Programming, Vol. 1, No. 1, 1971, pp. 366-375. doi:10.1007/BF01584098

[23] L. R. Gardiner and R. E. Steuer, “Unified Interactive Multiple Objective Programming,” European Journal of Operational Research, Vol. 74, 1984, pp. 371-406.

[24] J. T. Buchanan, “Multiple Objective Mathematical Programming: A Review,” New Zealand Operational Research, Vol. 14, No. 1, 1986, pp. 1-27.

[25] A. M. Geoffrion, “Proper Efficiency and the Theory of Vector Maximization,” Journal of Mathematical Analysis and Applications, Vol. 22, No. 3, 1968, pp. 619-630. doi:10.1016/0022-247X(68)90201-1

[26] M. I. Henig and Z. Ritz, “Multiplicative Decision Rules for Multi-Objective Decision Problems,” European Journal of Operational Research, Vol. 26, No. 1, 1986, pp. 134-141. doi:10.1016/0377-2217(86)90165-7

[27] A. K. Bhunia and J. Majumdar, “Elitist Genetic Algorithm for Assignment Problem with Imprecise Goal,” European Journal of Operational Research, Vol. 177, 2007, pp. 684-692. doi:10.1016/j.ejor.2005.11.034

[28] C. Botha, E. Ferreira, G. Geldenhuys and H. Ittman, “Selected Topics in Operations Research: Quantitative Management,” UNISA, Pretoria, 1998.

[29] C. D. Gelatt, S. Kirkpatrick and M. P. Vecchi, “Optimization by Simulated Annealing,” Science, Vol. 220, 1983, pp. 45-54.

[30] J. W. Barnes, F. W. Glover and M. Laguna, “Tabu Search Methods for a Single Machinescheduling Problem,” Journal of Intelligent Manufacturing, Vol. 2, No. 2, 1991, pp. 63-74. doi:10.1007/BF01471219

[31] M. J. Rangoaga, “A Decision Support System for MultiObjective Programming Problems,” Master’s Thesis, University of South Africa, Pretoria, 2009.

[32] H. S. Solutions, “Textpad,” 1992. http://wapedia.mobi/en/textpad

[33] S. S. Ruzibiza, “Solving Multi-Objective Mathematical Programming Problems with Fixed and Fuzzy Coefficients,” Master’s Thesis, Independent Institute of Lay Aventists of Kigali, Kigali, 2011.