Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:06:59.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-objective geometric programming problem withKarush−Kuhn−Tucker condition using ϵ-constraintmethod

Published online by Cambridge University Press:  10 June 2014

A.K. Ojha  and
Rashmi Ranjan Ota
A.K. Ojha
Affiliation:
School of Basic Sciences, Indian Institute of Technology, Bhubaneswar, 751013 Bhubaneswar, Odisha, India.. akojha57@yahoo.com; akojha@iitbbs.ac.in
Rashmi Ranjan Ota
Affiliation:
Department of Mathematics, ITER, SOA University, 751030 Bhubaneswar, Odisha, India.; otamath@yahoo.co.in
Get access

Abstract

Optimization is an important tool widely used in formulation of the mathematical modeland design of various decision making problems related to the science and engineering.Generally, the real world problems are occurring in the form of multi-criteria andmulti-choice with certain constraints. There is no such single optimal solution existwhich could optimize all the objective functions simultaneously. In this paper,ϵ-constraint method along with Karush−Kuhn−Tucker (KKT) condition has been used tosolve multi-objective Geometric programming problems(MOGPP) for searching a compromisesolution. To find the suitable compromise solution for multi-objective Geometricprogramming problems, a brief solution procedure using ϵ-constraint method hasbeen presented. The basic concept and classical principle of multi-objective optimizationproblems with KKT condition has been discussed. The result obtained by ϵ-constraint method withhelp of KKT condition has been compared with the result so obtained by Fuzzy programmingmethod. Illustrative examples are presented to demonstrate the correctness of proposedmodel.

Keywords

Geometric ProgrammingKarush−Kuhn−Tucker (KKT) conditionϵ-constraint methodfuzzy programmingduality theoremPareto optimal solution
Type
Research Article
Information
RAIRO - Operations Research , Volume 48 , Issue 4 , October 2014 , pp. 429 - 453
Copyright
© EDP Sciences, ROADEF, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

C.S. Beightler and D.T. Phillips. Appl. Geometric Programming, John Wiley and Sons, New York (1976).
Biswal, M.P., Fuzzy Programming technique to solve multi-objective Geometric Programming Problems. Fuzzy Sets Syst. 51 (1992) 67–71. Google Scholar
S.J. Chen and C.L. Hwang, Fuzzy multiple Attribute decision making methods and Applications, Springer, Berline (1992).
R.J. Duffin, E.L. Peterson and C.M. Zener, Geometric Programming Theory and Application, John Wiley and Sons, New York (1967).
Bit, A.K., Multi-objective Geometric Programming Problem: Fuzzy programming with hyperbolic membership function. J. Fuzzy Math. 6 (1998) 27–32. Google Scholar
Cao, B.Y., Fuzzy Geometric Programming (i). Fuzzy Sets Syst. 53 (1993) 135–153. Google Scholar
B.Y. Cao, Solution and theory of question for a kind of Fuzzy Geometric Program, proc, 2nd IFSA Congress, Tokyo 1 (1987) 205–208.
Liu, S.T., Posynomial Geometric Programming with parametric uncertainty. Eur. J. Oper. Res. 168 (2006) 345–353. Google Scholar
Wang, Y., Global optimization of Generalized Geometric Programming. Comput. Math. Appl. 48 (2004) 1505–1516. Google Scholar
H.R. Maleki and M. Maschinchi, Fuzzy number Multi-objective Geometric Progrmming. In 10th IFSA World congress, IFSA (2003), Istanbul, Turkey 536–538.
Islam, S. and Ray, T.K., A new Fuzzy Multi-objective Programming: Entropy based Geometric Programming and its Applications of transportation problems. Eur. J. Oper. Res. 173 (2006) 387–404. Google Scholar
Mavrotas, G., Effective implementation of the ϵ-constraint method in Multi-objective mathematical programming problems. Appl. Math. Comput. 213 (2009) 455–465. Google Scholar
V. Chankong and Y.Y. Haimes, Multiobjective decision making: Theory and Methodology, North-Holland, New York (1983).
Wierzbicki, A.P., On the Completeness and constructiveness of parametric characterization to vector optimization problems, OR Spectrum 8 (1986) 73-87. Google Scholar
Diakoulaki, D., Mavrotas, G. and Papayannakis, L., Determining objective weights in multiple criteria problems: The critical method. Comput. Oper. Res. 22 (1995) 763–770. Google Scholar
Asham, H. and Khan, A.B., A simplex type algorithm for general transportation problems. An alternative to steping stone. J. Oper. Res. Soc. 40 (1989) 581–590. Google Scholar
Evans, S.P., Derivation and Analysis of some models for combining trip distribution and assignment. Transp. Res. 10 (1976) 37–57. Google Scholar
Hitchcack, F.L., The distribution of a product from several sources to numerous localities. J. Math. Phys. 20 (1941) 224–236. Google Scholar
Islam, S., Multi-objective marketing planning inventory model: A Geometric programming approach. Appl. Math. Comput. 205 (2008) 238–246. Google Scholar
Kantorovich, L.V., Mathematical Methods of organizing and planning production in Russia. Manag. Sci. 6 (1960) 366–422. Google Scholar
A.G. Wilson, Entropy in urban and regional modeling, Pion, London (1970).
Boyd, S.J., Patil, D. and Horowitz, M., Digital circuit sizing via Geometric Programming. Oper. Res. 53 (2005) 899–932. Google Scholar
Boyd, S.J., Vandenberghe, L. and Hossib, A., A tutorial on Geometric Programming. Optim. Eng. 8 (2007) 67–127. Google Scholar
Chiang, M. and Boyd, S.J., Geometric Programing duals of channel capacity and rate distortion. IEEE Trans. Inf Theory 50 (2004) 245–258. Google Scholar
K.L. Hsiung, S.J. Kim and S.J. Boyd, Power control in lognormal fading wireless channels with optimal probability specifications via robust Geometric programming. In Proceeding IEEE, American Control Conference, Portland, OR 6 (2005) 3955–3959.
Seong, K., Narasimhan, R. and Cioffi, J.M., Queue proportional Scheduling via Geometric programming in fading broadcast channels, IEEE J. Select. Areas Commun. 24 (2006) 1593–1602. Google Scholar
Cao, B.Y., The further study of posynomial GP with Fuzzy co-efficient. Math. Appl. 5 (1992) 119–120. Google Scholar
Cao, B.Y., Extended Fuzzy GP. J. Fuzzy Math. 1 (1993) 285–293. Google Scholar
C.L. Hwang and A. Masud, Multiple objective decision making methods and applications, A state of art survey series Lect. Notes Econ. Math. Syst., Springer-Varlag, Berlin vol. 164 (1979).
Y.Y. Haimes, L.S. Lasdon and D.A. Wismer, On a Bicriterion formulation of problems integrated System identification and System optimization. IEEE Trans. Syst. Man Cybern. (1971) 296–297.
S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge (2004).
Soorpanth, T., Multi-objective Analog Design via Geometric programming. ECTI Conference 2 (2008) 729–732. Google Scholar
Waiel, F. and Wahed, El., A Multi-objective transportation problem under fuzzyness. Fuzzy Sets Syst. 117 (2001) 27–33. Google Scholar
Ray, T.K., Kar, S. and Maiti, M., Multi-objective inventory model of deteriorating items with space constraint in a Fuzzy environment. Tamsui Oxford J. Math. Sci. 24 (2008) 37–60. Google Scholar
H.S. Hall and S.R. Knight, Higher Algebra, Macmillan, New York (1940).
Islam, S., Multi-objective marketing planning inventory model. A Geometric programming approach. Appl. Math. Comput. 205 (2008) 238–246. Google Scholar
K.M. Miettinen, Non-linear Multi-objective optimization, Kluwer Academic Publishers, Boston, Massachusetts (1999).
Peterson, E.L., The fundamental relations between Geometric programming duality, Parametric programming duality and Ordinary Lagrangian duality. Annal. Oper. Res. 105 (2001) 109–153. Google Scholar
Rajgopal, J. and Bricker, D.L., Solving posynomial Geometric programming problems via Generalized linear programming. Comput. Optim. Appl. 21 (2002) 95–109. Google Scholar
Bellman, R.E. and Zadeh, L.A., Decision making in Fuzzy environment. Mang. Sci. 17B (1970) 141–164. Google Scholar
H.J. Zimmermann, Fuzzy set theory and its Applications, 2nd ed. Kluwer Academic Publishers, Dordrecht-Boston (1990).
Sinha, Surabhi and Sinha, S.B., KKT transportation approach for Multi-objective multi-level linear programming problem. Eur. J. Oper. Res. 143 (2002) 19–31. Google Scholar
Berube, Jean-Francois, Gendreau, M. and Potvin, J., An exact [epsilon]-constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits. Eur. J. Oper. Res. 194 (2009) 39–50. Google Scholar
Laumanns, M., Thiele, L. and Zitzler, E., An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur. J. Oper. Res. 169 (2006) 932–942. Google Scholar
Luptacik, M., Kuhn−Tucker Condition. Math. Optim. Economic Anal. 36 (2010) 25–58. Google Scholar
Pascual, L. and Ben-Israel, A., Vector-Valued Criteria in Geometric Programming. Oper. Res. 19 98–104.