Many optimization problems are formulated from a real scenario involving incomplete information due to uncertainty in reality. The uncertainties can be expressed with appropriate probability distributions or fuzzy numbers with a membership function, if enough information can be accessed for the construction of either the probability density function or the membership of the fuzzy numbers. However, in some cases there may not be enough information for that and grey numbers need to be used. A grey number is an interval number to represent the value of a quantity. Its exact value or the likelihood is not known but the maximum and/or the minimum possible values are. Applications in space exploration, robotics and engineering can be mentioned which involves such a scenario. An optimization problem is called a grey optimization problem if it involves a grey number in the objective function and/or constraint set. Unlike its wide applications, not much research is done in the field. Hence, in this paper, a convex grey optimization problem will be discussed. It will be shown that an optimal solution for a convex grey optimization problem is a grey number where the lower and upper limit are computed by solving the problem in an optimistic and pessimistic way. The optimistic way is when the decision maker counts the grey numbers as decision variables and optimize the objective function for all the decision variables whereas the pessimistic way is solving a minimax or maximin problem over the decision variables and over the grey numbers.
Mots-clés : Grey optimization, interval optimization, convex optimization, uncertainty
@article{RO_2019__53_1_339_0,
author = {Tilahun, Surafel Luleseged},
title = {Convex {Grey} {Optimization}},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {339--349},
publisher = {EDP-Sciences},
volume = {53},
number = {1},
year = {2019},
doi = {10.1051/ro/2018088},
zbl = {1414.90273},
mrnumber = {3912471},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2018088/}
}
TY - JOUR AU - Tilahun, Surafel Luleseged TI - Convex Grey Optimization JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2019 SP - 339 EP - 349 VL - 53 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2018088/ DO - 10.1051/ro/2018088 LA - en ID - RO_2019__53_1_339_0 ER -
Tilahun, Surafel Luleseged. Convex Grey Optimization. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 1, pp. 339-349. doi : 10.1051/ro/2018088. http://www.numdam.org/articles/10.1051/ro/2018088/
[1] and , Lectures On Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics (2001). | DOI | MR | Zbl
[2] , State-of-the-art-surveystochastic programming: computation and applications. INFORMS J. Comput. 9 (1997) 111–133. | DOI | MR | Zbl
[3] , , , and , Treatment parameters optimization to compensate for interfractional anatomy variability and intrafractional tumor motion. Front. Oncol. 5 (2015) 291. | DOI
[4] , and , Grey reinforcement learning for incomplete information processing. In: International Conference on Theory and Applications of Models of Computation. Springer, Berlin Heidelberg (2006). | DOI | MR | Zbl
[5] and , Integration fuzzy preference in genetic algorithm to solve multiobjective optimization problems. Far East Math. Sci. 55 (2011) 165–179. | MR | Zbl
[6] , and , Stochastic optimization problems with incomplete information on distribution functions. SIAM J. Control Optim. 23 (1985) 697–716. | DOI | MR | Zbl
[7] and , Grey theory to predict Ethiopian foreign currency exchange rate. Int. J. Bus. Forecasting Marketing Intell. 2 (2016) 95–116. | DOI
[8] , A survey on Grey Optimization, edited by and . In: Optimization Techniques for Problem Solving in Uncertainty. IGI Global, Hershey (2018).
[9] and , Application of Grey theory and multiobjective programming towards airline network design. Eur. J. Oper. Res. 127 (2000) 44–68. | DOI | Zbl
[10] and , Grey linear programming, its solving approach, and its application. Int. J. Syst. Sci. 24 (1993) 159–172. | DOI | MR | Zbl
[11] , and , A grey linear programming approach for municipal solid waste management planning under uncertainty. Civil Eng. Syst. 9 (1992) 319–335. | DOI
[12] , and , Grey integer programming: an application to waste management planning under uncertainty. Eur. J. Oper. Res. 83 (1995) 594–620. | DOI | Zbl
[13] , and , Application of grey theory to the field of economic forecasting. J. Global Econ. 3 (2014) 1–3.
[14] , and , Application of grey theory to the field of economic forecasting. J. Global Econ. 3 (2015) 1–3.
[15] , , , and , Application of grey system theory to forcast the growth of Larch. Int. J. Inf. Syst. Sci. 5 (2009) 522–527.
[16] and , Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48 (1990) 219–225. | DOI | Zbl
[17] , Control problems of grey systems. Syst. Control Lett. 1 (1982) 288–294. | DOI | MR | Zbl
[18] , Introduction to grey system theory. J. Grey Syst. 1 (1989) 1–24. | Zbl
[19] and , An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming. J. Egypt. Math. Soc. 22 (2014) 292–303. | DOI | MR | Zbl
[20] , Selected Applications of Convex Optimization. Springer, Berlin 103 (2015). | DOI | MR | Zbl
[21] and , Grey programming. In: Grey Information: Theory and Practical Applications. Springer London, London (2006) 367–413.
[22] and , Introduction to grey systems theory. In: Grey Systems: Theory and Applications. Springer Berlin Heidelberg, Berlin, Heidelberg (2011) 1–18.
[23] , , and , Grey System Theory and its Application. Science, Beijing (2004) 150–195.
[24] , , and , General grey numbers and their operations. Grey Syst. Theory App. 2 (2012) 341–349. | DOI
[25] and , On fuzzy stochastic optimization. Fuzzy Sets Syst. 81 (1996) 47–55. | DOI | MR | Zbl
[26] , and , An optimization based heuristic for political districting. Manage. Sci. 44 (1998) 1100–1114. | DOI | Zbl
[27] and , The application of grey system theory in predicting the number of deaths of women by committing suicide – a case study. J. Appl. Quant. Methods 10 (2015) 48–55.
[28] , and , Optimization Methods for Engineering Design. Brigham Young University, Provo, UT 5 (2013).
[29] , Shades of grey: a critical review of grey-number optimization. Eng. Optim. 41 (2009) 573–592. | DOI | MR
[30] and , Dealing with uncertainty in incomplete information system using fuzzy modeling technique. In: Information Sciences Signal Processing and their Applications (ISSPA), 10th International Conference on Information Sciences Signal Processing and their Applications (ISSPA). 10–13 May 2010. IEEE, Kuala Lampur, Malaysia (2010). | DOI
[31] and , Optimization of land use structure applying grey linear programming and analytic hierarchy process. In: 2nd International Conference on Artificial Intelligence, Management Science and Electronic Commerce (AIMSEC), 8–10 August 2011. IEEE, Dengleng, China (2011).
[32] , Mathematical optimization methods: application in project portfolio management. Proc. Soc. Behav. Sci. 210 (2015) 339–347. | DOI
[33] and , Fuzzy optimization theory and methodology survey. Control Theory App. 17 (2000) 159–164.
[34] and , Modeling the expansion of Prosopis juliora and determining its optimum utilization rate to control the invasion in Afar Regional State of Ethiopia. Int. J. Appl. Math. Res. 1 (2012) 726–743. | DOI
[35] and , Optimization techniques for problem solving in uncertainty. IGI Global (2018). DOI: . | DOI
[36] and , Fuzzy preference incorporated evolutionary algorithm for multiobjective optimization. Int. J. Adv. Sci. Eng. Inf. Technol. 1 (2011) 26–30. | DOI
[37] and , Bus timetabling as a fuzzy multiobjective optimization problem using preference-based genetic algorithm. Promet-Traffic Transp. 24 (2012) 183–191. | DOI
[38] and , Fuzzy preference of multiple decision-makers in solving multi-objective optimisation problems using genetic algorithm. Maejo Int. J. Sci. Technol. 6 (2012) 224.
[39] and , On fuzzy preference of decision makers in multiobjective and multilevel decision making. In: Proc. of 2nd International Conference on Management (2012).
[40] and , Vector optimisation using fuzzy preference in evolutionary strategy based firefly algorithm. Int. J. Oper. Res. 16 (2013) 81–95. | DOI | MR | Zbl
[41] and , Prey-predator algorithm: a new metaheuristic algorithm for optimization problems. Int. J. Inf. Technol. Decis. Making 14 (2015) 1331–1352. | DOI
[42] , Interval grey number based project scheduling model and algorithm. Grey Syst. Theory App. 8 (2018) 100–109. | DOI
[43] and , Novel methods on comparing grey numbers. Appl. Math. Model. 34 (2010) 415–423. | DOI | MR | Zbl
[44] , Extended grey numbers and their operations. In: ISIC. IEEE International Conference on Systems, Man and Cybernetics, 7–10 October 2007. IEEE, Montréal, Canada (2007). | DOI
[45] , Fifteen years of grey system theory research: a historical review and bibliometric analysis. Expert Syst. App. 40 (2013) 2767–2775. | DOI
[46] and , On the optimization concept of grey systems. Applied Math. Model. 17 (1993) 388–392. | DOI | Zbl
Cité par Sources :
