Fuzzy Optimization via Multi-Objective Evolutionary Computation for Chocolate Manufacturing

  • Fernando Jiménez
  • Gracia Sánchez
  • Pandian Vasant
  • José Luis Verdegay
Part of the Springer Optimization and Its Applications book series (SOIA, volume 16)


This chapter outlines, first, a real-world industrial problem for product mix selection involving 8 variables and 21 constraints with fuzzy coefficients and, second, a multi-objective optimization approach to solve the problem. This problem occurs in production planning in which a decision maker plays a pivotal role in making decisions under a fuzzy environment. Decision maker should be aware of his/her level-of-satisfaction as well as degree of fuzziness while making the product mix decision. Thus, the authors have analyzed using a modified S-curve membership function for the fuzziness patterns and fuzzy sensitivity of the solution found from the multi-objective optimization methodology. An ad hoc Pareto-based multi-objective evolutionary algorithm is proposed to capture multiple nondominated solutions in a single run of the algorithm. Results obtained have been compared with the well-known multi-objective evolutionary algorithm NSGA-II.

Key words

Multi-objective optimization evolutionary algorithm NSGA-II 


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  1. Ali, F.M., 1998, A differencial equarion approach to fuzzy non-linear programming problems, Fuzzy Sets and Systems, 93(1): 57-61.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bellman, R.E., Zadeh, L.A., 1970, Decision Making in a fuzzy environment, Management Science, 17: 141-164.CrossRefMathSciNetGoogle Scholar
  3. Coello, C.A., Veldhuizen, D.V., Lamont, G.V., 2002, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic/Plenum publishers, New York.zbMATHGoogle Scholar
  4. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T., 2000, A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGAII, In: Proceedings of the Parallel Problem Solving from Nature VI (PPSN-VI), pp. 849-858.Google Scholar
  5. Deb, K., 2001, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley and Sons, New York.zbMATHGoogle Scholar
  6. Ekel, P., Pedrycz, W., Schinzinger, R., 1998, A general approach to solving a wide class of fuzzy optimization problems, Fuzzy Sets and Systems, 97(1): 49-66.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Jiménez, F., Gómez-Skarmeta, A.F., Sánchez, G., Deb, 2002, K., An evolutionary algorithm for constrained multi-objective optimization, Proceedings IEEE World Congress on Evolutionary Computation.Google Scholar
  8. Jiménez, F., Gómez-Skarmeta, A.F, Sánchez, G., 2004, A multi-objective evolutionary approach for nonlinear constrained optimization with fuzzy costs, IEEE International Conference on Systems, Man & Cybernetics (SMC’04) The Hague, Netherlands.Google Scholar
  9. Jiménez, F., Gómez-Skarmeta, A.F, Sánchez, G., 2004, Nonlinear optimization with fuzzy constraints by multi-objective evolutionary algorithms, Advances in Soft Computing. Computational Intelligence, Theory and Applications, pp. 713-722Google Scholar
  10. Jiménez, F., Cadenas, J.M., Sánchez, G., Gómez-Skarmeta, A.F., Verdegay, J.L., 2006, Multi-objective evolutionary computation and fuzzy optimization, International Journal of Approximate Reasoning, 43: 59-75.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Laumanns, M., Zitzler, E., and Thiele, L., 2001, On the effects of archiving, elitism, and density based selection in evolutionary multi-objective optimization, Proceedings of the First International Conference on Evolutionary Multi-Criterion Optimization (EMO 2001), Zitzler, E., et al. (eds.), pp. 181-196.Google Scholar
  12. Purshouse, R.C., Fleming, P.J., 2002, Why use elitism and sharing in a multiobjective genetic algorithm, Proceedings of the Genetic and Evolutionary Computation Conference, pp. 520-527.Google Scholar
  13. Ramik, J., Vlach, M., 2002, Fuzzy mathematical programming: a unified approach based on fuzzy relations, Fuzzy Optimization and Decision Making, 1: 335-346.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Tabucanon, T.T., 1996, Multi objective programming for industrial engineers, Mathematical Programming For Industrial Engineers, pp. 487-542, Marcel Dekker, Inc., New York. Google Scholar
  15. Tanaka, H., Okuda, T., Asai, K., 1974, On fuzzy mathematical programming, Journal of Cybernetics, 3: 37-46.CrossRefMathSciNetGoogle Scholar
  16. Vasant, P., 2003, Application of fuzzy linear programming in production planning, Fuzzy Optimization and Decision Making, 2(3): 229-241.CrossRefMathSciNetGoogle Scholar
  17. Vasant, P., 2004, Industrial production planning using interactive fuzzy linear programming, International Journal of Computational Intelligence and Applications, 4(1): 13-26.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Vasant, P., 2006, Fuzzy production planning and its application to decision making, Journal of Intelligent Manufacturing, 17(1): 5-12.CrossRefGoogle Scholar
  19. Veldhuizen, D.V., Lamont, G.B., 1999, Multiobjective evolutionary algorithms: classifications, analyses, and new innovations, Ph.D. thesis, Air Force Institute of Technology. Technical Report No. AFIT/DS/ENG/99í01, Dayton, Ohio:Google Scholar
  20. Zimmermann, H.J., 1976, Description and optimization of fuzzy systems, International Journal of General Systems, 2: 209-215.CrossRefGoogle Scholar
  21. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V., 2003, Performance assessment of multiobjective optimizers: an analysis and review, IEEE Transactions on Evolutionary Computation, 7(2): 117-132.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2008

Authors and Affiliations

  • Fernando Jiménez
    • 1
  • Gracia Sánchez
    • 1
  • Pandian Vasant
    • 2
  • José Luis Verdegay
    • 3
  1. 1.Department of Ingeniería de la Informatión y las ComunicacionesUniversity of MurciaSpain
  2. 2.Universiti Teknologi PetronasMalaysia
  3. 3.Department of Ciencias de la Computaci’on e Inteligencia ArtificialUniversity of GranadaSpain

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