Abstract
Geometric programming is a powerful optimization technique widely used for solving a variety of nonlinear optimization problems and engineering problems. Conventional geometric programming models assume deterministic and precise parameters. However, the values observed for the parameters in real-world geometric programming problems often are imprecise and vague. We use geometric programming within an uncertainty-based framework proposing a chance-constrained geometric programming model whose coefficients are uncertain variables. We assume the uncertain variables to have normal, linear and zigzag uncertainty distributions and show that the corresponding uncertain chance-constrained geometric programming problems can be transformed into conventional geometric programming problems to calculate the objective values. The efficacy of the procedures and algorithms is demonstrated through numerical examples.
Similar content being viewed by others
References
Chu, C., Wong, D.F.: VLSI circuit performance optimization by geometric programming. Ann. Oper. Res. 105, 37–60 (2001)
Hershenson, M.D., Boyd, S.P., Lee, T.H.: Optimal design of a CMOS op-amp via geometric programming. IEEE Trans. Comput. Aid. Design. 20, 1–21 (2001)
Avriel, M., Dembo, R., Passy, U.: Solution of generalized geometric programs. Int. J. Numer. Method Eng. 9, 149–168 (1975)
Beightler, C.S., Philips, D.T.: Applied Geometric Programming. Wiley, New York (1976)
Choi, J.C., Bricker, D.L.: Effectiveness of a geometric programming algorithm for optimization of machining economics models. Comput. Oper. Res. 10, 957–961 (1996)
Scott, C.H., Jefferson, T.R.: Allocation of resources in project management. Int. J. Syst. Sci. 26, 413–420 (1995)
Cheng, T.C.E.: An economic order quantity model with demand-dependent unit production cost and imperfect production process. IIE Trans. 23, 23–28 (1991)
Jung, H., Klein, C.M.: Optimal inventory policies under decreasing cost functions via geometric programming. Eur. J. Oper. Res. 132, 628–642 (2001)
Kim, D., Lee, W.J.: Optimal joint pricing and lot sizing with fixed and variable capacity. Eur. J. Oper. Res. 109, 212–227 (1998)
Lee, W.J.: Determining order quantity and selling price by geometric programming. Optimal solution, bounds, and sensitivity. Decis. Sci. 24, 76–87 (1993)
Roy, T.K., Maiti, M.: A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity. Eur. J. Oper. Res. 99, 425–432 (1997)
Worrall, B.M., Hall, M.A.: The analysis of an inventory control model using posynomial geometric programming. Int. J. Prod. Res. 20, 657–667 (1982)
Duffin, R.J., Peterson, E.L., Zener, C.: Geometric Programming Theory and Applications. Wiley, New York (1967)
Duffin, R.J., Peterson, E.L.: Geometric programming with signomials. J. Optim. Theory Appl. 11, 3–35 (1973)
Fang, S.C., Peterson, E.L., Rajasekera, J.R.: Controlled dual perturbations for posynomial programs. Eur. J. Oper. Res. 35, 111–117 (1988)
Kortanek, K.O., No, H.: A second order affine scaling algorithm for the geometric programming dual with logarithmic barrier. Optimization 23, 303–322 (1992)
Kortanek, K.O., Xu, X., Ye, Y.: An infeasible interior-point algorithm for solving primal and dual geometric programs. Math. Program. 76, 155–181 (1997)
Maranas, C.D., Floudas, C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21, 351–369 (1997)
Rajgopal, J.: An alternative approach to the refined duality theory of geometric programming. J. Math. Anal. Appl. 167, 266–288 (1992)
Rajgopal, J., Bricker, D.L.: Posynomial geometric programming as a special case of semi-infinite linear programming. J. Optim. Theory Appl. 66, 455–475 (1990)
Rajgopal, J., Bricker, D.L.: Solving posynomial geometric programming problems via generalized linear programming. Comput. Optim. Appl. 21, 95–109 (2002)
Yang, H.H., Bricker, D.L.: Investigation of path-following algorithms for signomial geometric programming problems. Eur. J. Oper. Res. 103, 230–241 (1997)
Zhu, J., Kortanek, K.O., Huang, S.: Controlled dual perturbations for central path trajectories in geometric programming. Eur. J. Oper. Res. 73, 524–531 (1992)
Avriel, M., Wilde, D.J.: Engineering design under uncertainty. I&EC Process Des. Dev. 8(1), 127–131 (1969)
Dupačová, J.: Stochastic geometric programming with an application. Kybernetika 46(3), 374–386 (2010)
Liu, S.T.: A geometric programming approach to profit maximization. Appl. Math. Comput. 182(2), 1093–1097 (2006)
Liu, S.T.: Posynomial geometric programming with parametric uncertainty. Eur. J. Oper. Res. 168, 345–353 (2006b)
Liu, S.T.: Geometric programming with fuzzy parameters in engineering optimization. Int. J. Approx. Reason. 46(3), 484–498 (2007)
Liu, S.T.: Posynomial geometric programming with interval exponents and coefficients. Eur. J. Oper. Res. 186(1), 17–27 (2008)
Liu, S.T.: Fuzzy measures for profit maximization with fuzzy parameters. J. Comput. Appl. Math. 236(6), 1333–1342 (2011)
Tsai, J.F., Li, H.L., Hu, N.Z.: Global optimization for signomial discrete programming problems in engineering design. Eng. Optim. 34, 613–622 (2002)
Li, H.L., Tsai, J.F.: Treating free variables in generalized geometric global optimization programs. J. Glob. Optim. 33, 1–13 (2005)
Tsai, J.F., Lin, M.H., Hu, Y.C.: On generalized geometric programming problems with non-positive variables. Eur. J. Oper. Res. 178(1), 10–19 (2007)
Tsai, J.F.: Treating free variables in generalized geometric programming problems. Comput. Chem. Eng. 33, 239–243 (2009)
Tsai, J.F., Lin, M.H.: An optimization approach for solving signomial discrete programming problems with free variables. Comput. Chem. Eng. 30, 1256–1263 (2006)
Lin, M.H., Tsai, J.F.: Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems. Eur. J. Oper. Res. 216, 17–25 (2012)
Liu, B.: Uncertainty Theory, 4th edn. Springer, Berlin (2015)
Liu, B.: Some research problems in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009)
Peng, J., Yao, K.: A new option pricing model for stocks in uncertainty markets. Int. J. Oper. Res. 8(2), 18–26 (2011)
Liu, B.: Uncertain risk analysis and uncertain reliability analysis. J. Uncertain Syst. 4(3), 163–170 (2010)
Wang, X.S., Gao, Z.C., Guo, H.Y.: Delphi method for estimating uncertainty distributions. Information 15(2), 449–460 (2012)
Wang, X.S., Gao, Z.C., Guo, H.Y.: Uncertain hypothesis testing for expert’s empirical data. Math. Comput. Model. 55(3–4), 1478–1482 (2012)
Liu, B.: Uncertain set theory and uncertain inference rule with application to uncertain control. J. Uncertain Syst. 4(2), 83–98 (2010)
Zhu, Y.: Uncertain optimal control with application to a portfolio selection model. Cybern. Syst. 41(7), 535–547 (2010)
Li, S., Peng, J., Zhang, B.: The uncertain premium principle based on the distortion function. Insur. Math. Econ. 53, 317–324 (2013)
Han, S., Peng, Z., Wang, S.: The maximum flow problem of uncertain network. Inf. Sci. 265, 167–175 (2014)
Ding, S.: The \(\alpha \)-maximum flow model with uncertain capacities. Appl. Math. Model. 39(7), 2056–2063 (2015)
Peterson, E.L.: The fundamental relations between geometric programming duality, parametric programming duality, and ordinary Lagrangian duality. Ann. Oper. Res. 105, 109–153 (2001)
Liu, B.: Extreme value theorems of uncertain process with application to insurance risk model. Soft Comput. 17(4), 549–556 (2013)
Liu, S.T.: Profit maximization with quantity discount: an application of geometric program. Appl. Math. Comput. 190(2), 1723–1729 (2007a)
Hamidi, Sadjadi S.J., Hesarsorkh, A., Mohammadi, M., Bonyadi Naeini, A.: Joint pricing and production management: a geometric programming approach with consideration of cubic production cost function. J. Ind. Eng. Int. 11(2), 209–223 (2015)
Samadi, F., Mirzazadeh, A., Pedram, M.: Fuzzy pricing, marketing and service planning in a fuzzy inventory model: a geometric programming approach. Appl. Math. Model. 37, 6683–6694 (2013)
Acknowledgments
The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khanjani Shiraz, R., Tavana, M., Di Caprio, D. et al. Solving Geometric Programming Problems with Normal, Linear and Zigzag Uncertainty Distributions. J Optim Theory Appl 170, 243–265 (2016). https://doi.org/10.1007/s10957-015-0857-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0857-y
Keywords
- Uncertainty theory
- Uncertain variable
- Linear uncertainty distribution
- Normal uncertainty distribution
- Zigzag uncertainty distribution