Abstract
In a multi-objective optimization problem there is more than one objective function and there is no single optimal solution which simultaneously optimizes all of the given objective functions. For these unsuitable conditions the decision makers always search for the most ‘‘preferred’’ solution, in contrast to the optimal solution. A number of mathematical programming methods, namely Weighted-sum method, Goal programming, Lexicographic method, Weighted min–max method, Exponential weighted criterion, Weighted product method, Bounded objective function method and Weighted Tchebycheff optimization methods have been applied in the recent past to find the optimal solution. In this paper weighted Tchebycheff optimization methods has been applied to find the optimal solution of a multi-objective optimization problem. A solution procedure of weighted Tchebycheff technique has been discussed to find the optimal solution of the multi-objective optimization problems. For the most part, the parameters of a multi-objective optimization model are thought to be deterministic and settled. Nonetheless, the qualities watched for the parameters in true multi-objective optimization problems are frequently loose and subject to change. In this way, we utilize multi-objective optimization model inside a vulnerability based system and propose a multi-objective optimization model whose coefficients are uncertain in nature. We accept the uncertain variables (UVs) to have linear uncertainty distributions. Finally, a numerical example is solved weighted Tchebycheff technique.
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References
Beightler CS, Philips DT (1979) Foundation of optimization. Prentice-hall, New Jersey
Bishal MP (1992) Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets Syst 51:67–71
Changkong V, Haimes YY (1983) Multi-objective decision making. North-Holland Publishing, New York
Das P, Roy TK (2014) Multi-objective geometric programming and application in gravel box problem. J Glob Res Comput Sci 5(7):6–11
Dey S, Roy TK (2015) Optimum shape design of structural model with imprecise coefficient by parametric geometric programming. Decis Sci Lett 4:407–418
Ding S (2015) The α-maximum flow model with uncertain capacities. Appl Math Model 39(7):2056–2063
Duffin et al (1967) Geometric programming theory and application. Wiley, New York
Ehrgott M (2005) Multi-criteria optimization. Springer, Berlin. ISBN 3-540-21398-8
Ehrogott M (2005) Multi-criterion optimization. Springer, Heidelberg, Berlin
Hang S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inform Sci 265:167–175
Hwang CL, Masud A (1979) Multiple objective decision making methods and applications; a state of art survey. Lecture notes in economics and mathematical systems, vol 164. Springer, Berlin
Islam S, Roy TK (2006) Multi-objective marketing planning inventory model: a geometric programming approach. Appl Math Comput 2005(2008):238–246
Li S, Peng J, Zhang B (2013) The uncertain premium principle based on the distortion function. Insur Math Econ 53:317–324
Lin MH, Tsai JF (2012) Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems. Eur J Oper Res 216:17–25
Liu ST (2006) Posynomial geometric programming with parametric uncertainty. Eur J Oper Res 168:345–353
Liu ST (2007) Profit maximization with quantity discount: an application of geometric program. Appl Math Comput 190(2):1723–1729
Liu ST (2008) Posynomial geometric programming with interval exponents and coefficients. Eur J Oper Res 186(1):17–27
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2009) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin
Liu B (2010) Uncertain risk analysis and uncertain reliability analysis. J Uncertain Syst 4(3):163–170
Liu B (2010) Uncertain set theory and uncertain inference rule with application to uncertain control. J Uncertain Syst 4(2):83–98
Liu B (2015) Uncertainty theory, 4th edn. Springer, Berlin
Liu B, Chen XW (2015) Uncertain multiobjective programming and uncertain goal programming. J Uncertain Anal Appl 3:10
Liu GP, Yang JB, Whidborne JF (2011) Multi-objective optimization control. Research Studies Press, Baldock
Mandal WA (2018) Multi-objective geometric programming problem under uncertainty. Oper Res Decis 27(4):85–109
Maranas CD, Floudas CA (1997) Global optimization in generalized geometric programming. Comput Chem Eng 21:351–369
Miettinen KM (1999) Non-linear multi-objective optimization. Kluwer’s Academic Publishing, New York
Ojha AK, Biswal KK (2014) Multi-objective geometric programming problem with ε-constraint method. Appl Math Model 38(2):747–758
Ojha AK, Das AK (2010) Multi-objective geometric programming problem being cost coefficients as continuous function with mean method. J Comput 2:12
Ojha AK, Ota RR (2014) Multi-objective geometric programming problem with Karush Kuhn Tucker condition using ϵ-constraint method. RAIRO Oper Res 48:429–453
Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18–26
Peterson EL (2001) The fundamental relations between geometric programming duality, parametric programming duality, and ordinary Lagrangian duality. Ann Oper Res 105:109–153
Sadjadi SJ, HamidiHesarsorkh A, Mohammadi M, BonyadiNaeini A (2015) Joint pricing and production management: a geometric programming approach with consideration of cubic production cost function. J Ind Eng Int 11(2):209–223
Samadi F, Mirzazadeh A, Pedram M (2013) Fuzzy pricing, marketing and service planning in a fuzzy inventory model: a geometric programming approach. Appl Math Model 37:6683–6694
Scott CH, Jefferson TR (1995) Allocation of resources in project management. Int J Syst Sci 26:413–420
Shiraz RK, Tavana M, Di Caprio D, Fukuyama H (2016) Solving geometric programming problems with normal, linear and zigzag uncertainty distributions. J Optim Theory Appl 170(1):243–265
Shiraz RK, Tavana M, Fukuyama H, Di Caprio D (2017) Fuzzy chance-constrained geometric programming: the possibility, necessity and credibility approaches. Oper Res Int J 17(1):67–97
Tsai JF (2009) Treating free variables in generalized geometric programming problems. Comput Chem Eng 33:239–243
Tsai JF, Lin MH (2006) An optimization approach for solving signomial discrete programming problems with free variables. Comput Chem Eng 30:1256–1263
Tsai JF, Li HL, Hu NZ (2002) Global optimization for signomial discrete programming problems in engineering design. Eng Optim 34:613–622
Tsai JF, Lin MH, Hu YC (2007) On generalized geometric programming problems with non-positive variables. Eur J Oper Res 178(1):10–19
Wang XS, Gao ZC, Guo HY (2012) Delphi method for estimating uncertainty distributions. Information 15(2):449–460
Wang XS, Gao ZC, Guo HY (2012) Uncertain hypothesis testing for expert’s empirical data. Math Comput Model 55(3–4):1478–1482
Worrall BM, Hall MA (1982) The analysis of an inventory control model using posynomial geometric programming. Int J Prod Res 20:657–667
Yang HH, Bricker DL (1997) Investigation of path-following algorithms for signomial geometric programming problems. Eur J Oper Res 103:230–241
Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547
Zhu J, Kortanek KO, Huang S (1992) Controlled dual perturbations for central path trajectories in geometric programming. Eur J Oper Res 73:524–531
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Mandal, W.A. Weighted Tchebycheff Optimization Technique Under Uncertainty. Ann. Data. Sci. 8, 709–731 (2021). https://doi.org/10.1007/s40745-020-00250-8
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DOI: https://doi.org/10.1007/s40745-020-00250-8