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Fuzzy chance-constrained geometric programming: the possibility, necessity and credibility approaches

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Abstract

Geometric programming (GP) is a powerful tool for solving a variety of optimization problems. Most GP problems involve precise parameters. However, the observed values of the parameters in real-life GP problems are often imprecise or vague and, consequently, the optimization process and the related decisions take place in the face of uncertainty. The uncertainty associated with the coefficients of GP problems can be formalized using fuzzy variables. In this paper, we propose chance-constrained GP to deal with the impreciseness and the ambiguity inherent to real-life GP problems. Given a fuzzy GP model, we formulate three variants of chance-constrained GP based on the possibility, necessity and credibility approaches and show how they can be transformed into equivalent deterministic GP problems to be solved via the duality algorithm. We demonstrate the applicability of the proposed models and the efficacy of the introduced procedures with two numerical examples.

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References

  • Banker RD, Maindiratta A (1988) Nonparametric analysis of technical and allocative efficiencies in production. Econometrica 56(6):1315–1332

    Article  Google Scholar 

  • Beightler CS, Phillips DT (1976) Applied geometric programming. Wiley, New York

    Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Boyd S, Patil D, Horowitz M (2005) Digital circuit sizing via geometric programming. Oper Res 53(6):899–932

    Article  Google Scholar 

  • Boyd S, Vandenberghe L, Hossib A (2007) A tutorial on geometric programming. Optim Eng 8(1):67–127

    Article  Google Scholar 

  • Cao BY (1993) Extended fuzzy geometric programming. J Fuzzy Math 1(2):285–293

    Google Scholar 

  • Cao BY (2002) Fuzzy geometric programming. Kluwer, Dordrecht

    Book  Google Scholar 

  • Charnes A, Cooper WW (1959) Chance constrained programming. Manage Sci 6:73–79

    Article  Google Scholar 

  • Danzig GB (1955) Linear programming under uncertainty. Manage Sci 1:197–206

    Article  Google Scholar 

  • Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626

    Article  Google Scholar 

  • Dubois D, Prade H (1980) Fuzzy sets and system: theory and applications. Academic Press, New York

  • Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224

    Article  Google Scholar 

  • Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum, New York

    Book  Google Scholar 

  • Duffin RJ, Peterson EL (1973) Geometric programming with signomials. J Optim Theory Appl 11(1):3–35

    Article  Google Scholar 

  • Duffin RJ, Peterson EL, Zener C (1967) Geometric programming: theory and applications. Wiley, New York

    Google Scholar 

  • Emrouznejad A, Tavana M, Hatami-Marbini A (2014) The state of the art in fuzzy data envelopment analysis. In: Emrouznejad A, Tavana M (eds) Performance measurement with fuzzy data envelopment analysis, studies in fuzziness and soft computing, vol 309. Springer, Berlin, pp 1–45

    Chapter  Google Scholar 

  • Kall P, Mayer J (2005) Stochastic linear programming models, theory, and computation. Springer, New York

    Google Scholar 

  • Khanjani SR, Charles V, Jalalzadeh L (2014) Fuzzy rough DEA model: a possibility and expected value approaches. Expert Syst Appl 41:434–444

    Article  Google Scholar 

  • Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall, New Jersey

    Google Scholar 

  • Kortanek KO, Xu X, Ye Y (1996) An infeasible interior-point algorithm for solving primal and dual geometric programs. Math Program 76:155–181

    Google Scholar 

  • Lai VJ, Hwang CL (1992) Fuzzy mathematical programming. Springer, Berlin

    Book  Google Scholar 

  • Liu B (2002) Theory and practice of uncertain programming. Physica Verlag, New York

    Book  Google Scholar 

  • Liu ST (2006a) A geometric programming approach to profit maximization. Appl Math Comput 182(2):1093–1097

    Google Scholar 

  • Liu ST (2006b) Geometric programming with parametric uncertainty. Eur J Oper Res 168:345–353

    Article  Google Scholar 

  • Liu ST (2007a) Profit maximization with quantity discount: an application of geometric program. Appl Math Comput 190(2):1723–1729

    Google Scholar 

  • Liu ST (2007b) Geometric programming with fuzzy parameters in engineering optimization. Int J Approx Reason 46(3):484–498

    Article  Google Scholar 

  • Liu ST (2008) Posynomial geometric programming with interval exponents and coefficients. Eur J Oper Res 186(1):17–27

    Article  Google Scholar 

  • Liu ST (2011) Fuzzy measures for profit maximization with fuzzy parameters. J Comput Appl Math 236(6):1333–1342

    Article  Google Scholar 

  • Liu B, Liu YK (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10(4):445–450

    Article  Google Scholar 

  • Luhandjula MK, Gupta MM (1996) On fuzzy stochastic optimization. Fuzzy Sets Syst 81:47–55

    Article  Google Scholar 

  • Mandal NK, Roy TK (2006) A displayed inventory model with L-R fuzzy number. Fuzzy Optim Decis Mak 5(3):227–243

    Article  Google Scholar 

  • Passy U, Wilde DJ (1967) Generalized polynomial optimization. SIAM J Appl Math 15(5):1344–1356

    Article  Google Scholar 

  • Peterson EL (2001) The origins of geometric programming. Ann Oper Res 105:15–19

    Article  Google Scholar 

  • Rommelfanger H (1996) Fuzzy linear programming and applications. Eur J Oper Res 92:512–527

    Article  Google Scholar 

  • Sakawa M (1993) Fuzzy sets and interactive multiobjective optimization. Plenum Press, New York

    Book  Google Scholar 

  • Sakawa M, Katagiri H, Matsui T (2012) Interactive fuzzy stochastic two-level integer programming through fractile criterion optimization. Oper Res Int J 12:209–227

    Article  Google Scholar 

  • Tavana M, Shiraz RKh, Hatami-Marbini A, Agrell PJ, Paryab K (2012) Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC). Expert Syst Appl 39(15):12247–12259

    Article  Google Scholar 

  • Tsai JF, Lin MH, Hu YC (2007) On generalized geometric programming problems with non-positive variables. Eur J Oper Res 178(1):10–19

    Article  Google Scholar 

  • Wang GY, Qiao Z (1993) Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst 57:295–311

    Article  Google Scholar 

  • Wu YK (2008) Optimizing the geometric programming problem with single-term exponents subject to max-min fuzzy relational equation constraints. Math Comput Model 47(3–4):352–362

    Article  Google Scholar 

  • Yang JH, Cao BY (2005) Geometric programming with fuzzy relation equation constraints. In: Proceedings of IEEE international conference on fuzzy systems 2005, pp 557–560

  • Yang JH, Cao BY (2007) Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optim Decis Mak 6(4):337–349

    Article  Google Scholar 

  • Yano H, Sakawa M (2014) Interactive fuzzy programming for multiobjective fuzzy linear programming problems through possibility-based probability maximization. Oper Res Int J 14:51–69

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

  • Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28

    Article  Google Scholar 

  • Zhou X, Ahat R (2011) Geometric programming problem with single term exponents subject to max-product fuzzy relational equations. Math Comput Model 53(1–2):55–62

    Article  Google Scholar 

  • Zimmermann HJ (1996) Fuzzy set theory and its applications, 2nd edn. Kluwer, Dordrecht

    Book  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics Science, University of Tabriz, Tabriz, Iran

    Rashed Khanjani Shiraz

  2. Distinguished Chair of Business Analytics, Business Systems and Analytics Department, La Salle University, Philadelphia, PA, 19141, USA

    Madjid Tavana

  3. Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, 33098, Paderborn, Germany

    Madjid Tavana

  4. Faculty of Commerce, Fukuoka University, Fukuoka, Japan

    Hirofumi Fukuyama

  5. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Debora Di Caprio

  6. Polo Tecnologico IISS G. Galilei, Via Cadorna 14, 39100, Bolzano, Italy

    Debora Di Caprio

Authors
  1. Rashed Khanjani Shiraz
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  2. Madjid Tavana
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  3. Hirofumi Fukuyama
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  4. Debora Di Caprio
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Correspondence to Madjid Tavana.

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Khanjani Shiraz, R., Tavana, M., Fukuyama, H. et al. Fuzzy chance-constrained geometric programming: the possibility, necessity and credibility approaches. Oper Res Int J 17, 67–97 (2017). https://doi.org/10.1007/s12351-015-0216-7

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  • DOI: https://doi.org/10.1007/s12351-015-0216-7

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