Multi-objective geometric programming problem with Karush−Kuhn−Tucker condition using ϵ-constraint method

A. K. Ojha; Rashmi Ranjan Ota

RAIRO - Operations Research - Recherche Opérationnelle (2014)

  • Volume: 48, Issue: 4, page 429-453
  • ISSN: 0399-0559

Abstract

top
Optimization is an important tool widely used in formulation of the mathematical model and 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 and multi-choice with certain constraints. There is no such single optimal solution exist which could optimize all the objective functions simultaneously. In this paper, ϵ-constraint method along with Karush−Kuhn−Tucker (KKT) condition has been used to solve multi-objective Geometric programming problems(MOGPP) for searching a compromise solution. To find the suitable compromise solution for multi-objective Geometric programming problems, a brief solution procedure using ϵ-constraint method has been presented. The basic concept and classical principle of multi-objective optimization problems with KKT condition has been discussed. The result obtained by ϵ-constraint method with help of KKT condition has been compared with the result so obtained by Fuzzy programming method. Illustrative examples are presented to demonstrate the correctness of proposed model.

How to cite

top

Ojha, A. K., and Ota, Rashmi Ranjan. "Multi-objective geometric programming problem with Karush−Kuhn−Tucker condition using ϵ-constraint method." RAIRO - Operations Research - Recherche Opérationnelle 48.4 (2014): 429-453. <http://eudml.org/doc/275036>.

@article{Ojha2014,
abstract = {Optimization is an important tool widely used in formulation of the mathematical model and 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 and multi-choice with certain constraints. There is no such single optimal solution exist which could optimize all the objective functions simultaneously. In this paper, ϵ-constraint method along with Karush−Kuhn−Tucker (KKT) condition has been used to solve multi-objective Geometric programming problems(MOGPP) for searching a compromise solution. To find the suitable compromise solution for multi-objective Geometric programming problems, a brief solution procedure using ϵ-constraint method has been presented. The basic concept and classical principle of multi-objective optimization problems with KKT condition has been discussed. The result obtained by ϵ-constraint method with help of KKT condition has been compared with the result so obtained by Fuzzy programming method. Illustrative examples are presented to demonstrate the correctness of proposed model.},
author = {Ojha, A. K., Ota, Rashmi Ranjan},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {geometric programming; Karush-Kuhn-Tucker (KKT) condition; ϵ-constraint method; fuzzy programming; duality theorem; Pareto optimal solution; -constraint method},
language = {eng},
number = {4},
pages = {429-453},
publisher = {EDP-Sciences},
title = {Multi-objective geometric programming problem with Karush−Kuhn−Tucker condition using ϵ-constraint method},
url = {http://eudml.org/doc/275036},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Ojha, A. K.
AU - Ota, Rashmi Ranjan
TI - Multi-objective geometric programming problem with Karush−Kuhn−Tucker condition using ϵ-constraint method
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 429
EP - 453
AB - Optimization is an important tool widely used in formulation of the mathematical model and 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 and multi-choice with certain constraints. There is no such single optimal solution exist which could optimize all the objective functions simultaneously. In this paper, ϵ-constraint method along with Karush−Kuhn−Tucker (KKT) condition has been used to solve multi-objective Geometric programming problems(MOGPP) for searching a compromise solution. To find the suitable compromise solution for multi-objective Geometric programming problems, a brief solution procedure using ϵ-constraint method has been presented. The basic concept and classical principle of multi-objective optimization problems with KKT condition has been discussed. The result obtained by ϵ-constraint method with help of KKT condition has been compared with the result so obtained by Fuzzy programming method. Illustrative examples are presented to demonstrate the correctness of proposed model.
LA - eng
KW - geometric programming; Karush-Kuhn-Tucker (KKT) condition; ϵ-constraint method; fuzzy programming; duality theorem; Pareto optimal solution; -constraint method
UR - http://eudml.org/doc/275036
ER -

References

top
  1. [1] C.S. Beightler and D.T. Phillips. Appl. Geometric Programming, John Wiley and Sons, New York (1976). MR680967
  2. [2] M.P. Biswal, Fuzzy Programming technique to solve multi-objective Geometric Programming Problems. Fuzzy Sets Syst.51 (1992) 67–71. Zbl0786.90086MR1187372
  3. [3] S.J. Chen and C.L. Hwang, Fuzzy multiple Attribute decision making methods and Applications, Springer, Berline (1992). Zbl0768.90042MR1223777
  4. [4] R.J. Duffin, E.L. Peterson and C.M. Zener, Geometric Programming Theory and Application, John Wiley and Sons, New York (1967). Zbl0171.17601MR214374
  5. [5] A.K. Bit, Multi-objective Geometric Programming Problem: Fuzzy programming with hyperbolic membership function. J. Fuzzy Math.6 (1998) 27–32. Zbl0905.90181MR1609923
  6. [6] B.Y. Cao, Fuzzy Geometric Programming (i). Fuzzy Sets Syst.53 (1993) 135–153. Zbl0804.90135MR1202983
  7. [7] B.Y. Cao, Solution and theory of question for a kind of Fuzzy Geometric Program, proc, 2nd IFSA Congress, Tokyo 1 (1987) 205–208. 
  8. [8] S.T. Liu, Posynomial Geometric Programming with parametric uncertainty. Eur. J. Oper. Res.168 (2006) 345–353. Zbl1083.90040MR2170242
  9. [9] Y. Wang, Global optimization of Generalized Geometric Programming. Comput. Math. Appl.48 (2004) 1505–1516. Zbl1066.90096MR2107107
  10. [10] H.R. Maleki and M. Maschinchi, Fuzzy number Multi-objective Geometric Progrmming. In 10th IFSA World congress, IFSA (2003), Istanbul, Turkey 536–538. 
  11. [11] S. Islam and T.K. Ray, A new Fuzzy Multi-objective Programming: Entropy based Geometric Programming and its Applications of transportation problems. Eur. J. Oper. Res.173 (2006) 387–404. Zbl1113.90021MR2230181
  12. [12] G. Mavrotas, Effective implementation of the ϵ-constraint method in Multi-objective mathematical programming problems. Appl. Math. Comput. 213 (2009) 455–465. Zbl1168.65029MR2536670
  13. [13] V. Chankong and Y.Y. Haimes, Multiobjective decision making: Theory and Methodology, North-Holland, New York (1983). Zbl0622.90002MR780745
  14. [14] A.P. Wierzbicki, On the Completeness and constructiveness of parametric characterization to vector optimization problems, OR Spectrum8 (1986) 73-87. Zbl0592.90084MR848533
  15. [15] D. Diakoulaki, G. Mavrotas and L. Papayannakis, Determining objective weights in multiple criteria problems: The critical method. Comput. Oper. Res.22 (1995) 763–770. Zbl0830.90079
  16. [16] H. Asham and A.B. Khan, A simplex type algorithm for general transportation problems. An alternative to steping stone. J. Oper. Res. Soc. 40 (1989) 581–590. Zbl0674.90066
  17. [17] S.P. Evans, Derivation and Analysis of some models for combining trip distribution and assignment. Transp. Res.10 (1976) 37–57. 
  18. [18] F.L. Hitchcack, The distribution of a product from several sources to numerous localities. J. Math. Phys.20 (1941) 224–236. Zbl0026.33904MR4469JFM67.0528.04
  19. [19] S. Islam, Multi-objective marketing planning inventory model: A Geometric programming approach. Appl. Math. Comput.205 (2008) 238–246. Zbl1151.90313MR2466627
  20. [20] L.V. Kantorovich, Mathematical Methods of organizing and planning production in Russia. Manag. Sci.6 (1960) 366–422. Zbl0995.90532MR129016
  21. [21] A.G. Wilson, Entropy in urban and regional modeling, Pion, London (1970). 
  22. [22] S.J. Boyd, D. Patil and M. Horowitz, Digital circuit sizing via Geometric Programming. Oper. Res.53 (2005) 899–932. Zbl1165.90655MR2193868
  23. [23] S.J. Boyd, L. Vandenberghe and A. Hossib, A tutorial on Geometric Programming. Optim. Eng.8 (2007) 67–127. Zbl1178.90270MR2330467
  24. [24] M. Chiang and S.J. Boyd, Geometric Programing duals of channel capacity and rate distortion. IEEE Trans. Inf Theory50 (2004) 245–258. Zbl1288.94032MR2044071
  25. [25] 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. 
  26. [26] K. Seong, R. Narasimhan and J.M. Cioffi, Queue proportional Scheduling via Geometric programming in fading broadcast channels, IEEE J. Select. Areas Commun.24 (2006) 1593–1602. 
  27. [27] B.Y. Cao, The further study of posynomial GP with Fuzzy co-efficient. Math. Appl.5 (1992) 119–120. 
  28. [28] B.Y. Cao, Extended Fuzzy GP. J. Fuzzy Math.1 (1993) 285–293. Zbl0800.90763MR1230319
  29. [29] 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). Zbl0397.90001MR567888
  30. [30] 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. Zbl0224.93016MR300411
  31. [31] S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge (2004). Zbl1058.90049MR2061575
  32. [32] T. Soorpanth, Multi-objective Analog Design via Geometric programming. ECTI Conference2 (2008) 729–732. 
  33. [33] F. Waiel and El. Wahed, A Multi-objective transportation problem under fuzzyness. Fuzzy Sets Syst. 117 (2001) 27–33. Zbl0965.90001MR1802773
  34. [34] T.K. Ray, S. Kar and M. Maiti, Multi-objective inventory model of deteriorating items with space constraint in a Fuzzy environment. Tamsui Oxford J. Math. Sci.24 (2008) 37–60. Zbl1141.90005MR2428957
  35. [35] H.S. Hall and S.R. Knight, Higher Algebra, Macmillan, New York (1940). JFM21.0076.01
  36. [36] S. Islam, Multi-objective marketing planning inventory model. A Geometric programming approach. Appl. Math. Comput. 205 (2008) 238–246. Zbl1151.90313MR2466627
  37. [37] K.M. Miettinen, Non-linear Multi-objective optimization, Kluwer Academic Publishers, Boston, Massachusetts (1999). 
  38. [38] E.L. Peterson, The fundamental relations between Geometric programming duality, Parametric programming duality and Ordinary Lagrangian duality. Annal. Oper. Res. 105 (2001) 109–153. Zbl0996.90060MR1879422
  39. [39] J. Rajgopal and D.L. Bricker, Solving posynomial Geometric programming problems via Generalized linear programming. Comput. Optim. Appl.21 (2002) 95–109. Zbl0988.90021MR1883091
  40. [40] R.E. Bellman and L.A. Zadeh, Decision making in Fuzzy environment. Mang. Sci. 17B (1970) 141–164. Zbl0224.90032MR301613
  41. [41] H.J. Zimmermann, Fuzzy set theory and its Applications, 2nd ed. Kluwer Academic Publishers, Dordrecht-Boston (1990). Zbl0719.04002MR1174743
  42. [42] Surabhi Sinha and S.B. Sinha, KKT transportation approach for Multi-objective multi-level linear programming problem. Eur. J. Oper. Res.143 (2002) 19–31. Zbl1073.90552MR1922619
  43. [43] Jean-Francois Berube, M.Gendreau and J. Potvin, 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. Zbl1179.90274MR2469192
  44. [44] M. Laumanns, L. Thiele and E. Zitzler, An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur. J. Oper. Res.169 (2006) 932–942. Zbl1079.90122MR2174001
  45. [45] M. Luptacik, Kuhn−Tucker Condition. Math. Optim. Economic Anal. 36 (2010) 25–58. 
  46. [46] L. Pascual and A. Ben-Israel, Vector-Valued Criteria in Geometric Programming. Oper. Res. 19 98–104. Zbl0232.90058MR274041

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.