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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|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|first1 = M. S.|last1 = Pishvaee|first2 = J.|last2 = Razmi|first3 = S. A.|last3 = Torabi}}</ref> in the Journal of 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|journal = Applied Mathematical Modelling|date = 2016-01-01|pages = 407–418|volume = 40|issue = 1|doi = 10.1016/j.apm.2015.04.054|first1 = Mir Saman|last1 = Pishvaee|first2 = Mohamadreza|last2 = Fazli Khalaf|doi-access = free}}</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.
'''Robust fuzzy programming (ROFP)''' ▼
== Definition of robust solution ==
Robust solution is defined
As fuzzy mathematical programming is categorized into ''Possibilistic programming'' and ''Flexible programming'', ROFP also can be classified into:<ref name=":1" />
# Robust possibilistic programming (RPP)
▲Robust solution is defined by Pishvaee and Fazli (2016) 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".
# Mixed possibilistic-flexible robust programming (MPFRP)
▲'''Classification of ROFP methods'''
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" />
▲As fuzzy mathematical programming is categorized into (1) Possibilistic programming and (2) Flexible programming, ROFP also can be classified into (1) Robust possibilistic programming (RPP), (2) Robust flexible programming (RFP) and (3) Mixed possibilistic-flexible robust programming (MPFRP) (see Pishvaee and Fazli, 2016). 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.
# Hard worst case ROFP
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 (1) hard worst case ROFP, (2) soft worst case ROFP and (3) 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).▼
# Soft worst case ROFP
# Realistic ROFP
▲
''' Applications'''▼
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|journal = Computers & Industrial Engineering|date = 2014-08-01|pages = 139–148|volume = 74|doi = 10.1016/j.cie.2014.05.008|first1 = Behzad|last1 = Zahiri|first2 = Reza|last2 = Tavakkoli-Moghaddam|first3 = Mir Saman|last3 = Pishvaee}}</ref> and Mousazadeh et al.<ref>{{Cite journal|title = A robust possibilistic programming approach for pharmaceutical supply chain network design|journal = Computers & Chemical Engineering|date = 2015-11-02|pages = 115–128|volume = 82|doi = 10.1016/j.compchemeng.2015.06.008|first1 = M.|last1 = Mousazadeh|first2 = S. A.|last2 = Torabi|first3 = B.|last3 = Zahiri}}</ref> which consider the planning of an organ transplantation network and a pharmaceutical supply chain, respectively.
* [[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|journal = Industrial & Engineering Chemistry Research|date = 2015-12-22|pages = 237–256|volume = 55|issue = 1|doi = 10.1021/acs.iecr.5b02875|language = EN|first1 = Samira|last1 = Bairamzadeh|first2 = Mir Saman|last2 = Pishvaee|first3 = Mohammad|last3 = Saidi-Mehrabad}}</ref> which uses a multi-objective possibilistic programming model to deal with the design of a bio-ethanol production-distribution network. Also in another research, Zhou et al.<ref>{{Cite journal|title = A robust possibilistic mixed-integer programming method for planning municipal electric power systems|journal = International Journal of Electrical Power & Energy Systems|date = 2015-12-15|pages = 757–772|volume = 73|doi = 10.1016/j.ijepes.2015.06.009|language = EN|first1 = Y.|last1 = Zhou|first2 = Y.P.|last2 = Li|first3 = G.H.|last3 = Huang| bibcode=2015IJEPE..73..757Z }}</ref> developed a robust possibilistic programming model to deal with the planning problem of municipal electric power system.
* [[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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