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{{short description|Mathematical optimization approach to deal with optimization problems under uncertainty}}
'''Robust fuzzy programming (ROFP)''' is a powerful [[mathematical optimization]] approach to deal with optimization problems under [[uncertainty]]. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi<ref name=":0">{{Cite journal|title = Robust possibilistic programming for socially responsible supply chain network design: A new approach
▲'''Robust fuzzy programming (ROFP)''' is a powerful [[mathematical optimization]] approach to deal with optimization problems under [[uncertainty]]. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi<ref name=":0">{{Cite journal|title = Robust possibilistic programming for socially responsible supply chain network design: A new approach|url = http://www.sciencedirect.com/science/article/pii/S0165011412001819|journal = Fuzzy Sets and Systems|date = 2012-11-01|pages = 1–20|volume = 206|series = Theme : Operational Research|doi = 10.1016/j.fss.2012.04.010|first = M. S.|last = Pishvaee|first2 = J.|last2 = Razmi|first3 = S. A.|last3 = Torabi}}</ref> in the Journal of [http://www.journals.elsevier.com/fuzzy-sets-and-systems/ Fuzzy Sets and Systems]. ROFP enables the decision makers to be benefited from the capabilities of both [[fuzzy set|fuzzy]] mathematical programming and [[robust optimization]] approaches. At 2016 Pishvaee and Fazli<ref name=":1">{{Cite journal|title = Novel robust fuzzy mathematical programming methods|url = http://www.sciencedirect.com/science/article/pii/S0307904X15003686|journal = Applied Mathematical Modelling|date = 2016-01-01|pages = 407–418|volume = 40|issue = 1|doi = 10.1016/j.apm.2015.04.054|first = Mir Saman|last = Pishvaee|first2 = Mohamadreza|last2 = Fazli Khalaf}}</ref> put a significant step forward by extending the ROFP approach to handle flexibility of constraints and goals.ROFP is able to achieve a ''robust solution'' for an optimization problem under uncertainty.
== Definition of Robust solution == ▼
Robust solution is defined as a solution which has "both ''feasibility robustness'' and ''optimality robustness''; Feasibility robustness means that the solution should remain feasible for (almost) all possible values of uncertain parameters and flexibility degrees of constraints and optimality robustness means that the value of objective function for the solution should remain close to optimal value or have minimum (undesirable) deviation from the optimal value for (almost) all possible values of uncertain parameters and flexibility degrees on target value of goals".<ref name=":1" />
== Classification of ROFP methods ==
As fuzzy mathematical programming is categorized into ''Possibilistic programming'' and ''Flexible programming'', ROFP also can be classified into:<ref name=":1" />
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# Mixed possibilistic-flexible robust programming (MPFRP)
The first category is used to deal with imprecise input parameters in optimization problems while the second one is employed to cope with flexible constraints and goals. Also, the last category is capable to handle both uncertain parameters and flexibility in goals and constraints.
From another point of view, it can be said that different ROFP models developed in the literature can be classified in three categories according to degree of conservatism against uncertainty. These categories include:<ref name=":0" />
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# Realistic ROFP
Hard worst case ROFP has the most conservative nature among ROFP methods since it provides maximum safety or immunity against uncertainty. Ignoring the chance of infeasibility, this method immunizes the solution for being infeasible for all possible values of uncertain parameters. Regarding the optimality robustness, this method minimizes the worst possible value of objective function (min-max logic). On the other hand, Soft worst case ROFP method behaves similar to hard worst case method regarding optimality robustness, however does not satisfy the constraints in their extreme worst case. Lastly, realistic method establishes a reasonable trade-off between the robustness, the cost of robustness and other objectives such as improving the average system performance (cost-benefit logic).
== Applications ==
ROFP is successfully implemented in different practical application areas such as the following ones.
▲ROFP is successfully implemented in different practical application areas such as the following ones.
* [[Supply chain management]] such as the work by Pishvaee et al.<ref name=":0" /> which addresses the design of a social responsible supply chain network under epistemic uncertainty.
* Healthcare management such as the works by Zahiri et al.<ref>{{Cite journal|title = A robust possibilistic programming approach to multi-period ___location–allocation of organ transplant centers under uncertainty
* [[Energy planning]] such as Bairamzadeh et al.<ref>{{Cite journal|title = Multiobjective Robust Possibilistic Programming Approach to Sustainable Bioethanol Supply Chain Design under Multiple Uncertainties
* [[Sustainability]] such as Xu and Huang<ref>{{Cite journal|title = Development of an Improved Fuzzy Robust Chance-Constrained Programming Model for Air Quality Management|journal = Environmental Modeling & Assessment|date = 2015-10-15|pages = 535–548|volume = 20|issue = 5|doi = 10.1007/s10666-014-9441-3|language = EN|first1 = Ye|last1 = Xu|first2 = Guohe|last2 = Huang| bibcode=2015EMdAs..20..535X }}</ref> which employ ROFP to cope with an air quality management problem.
== References ==
{{Reflist}}
[[Category:Optimization algorithms and methods]]
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