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<title>Robust fuzzy programming</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Robust fuzzy programming</span></span>

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<p><b>Robust fuzzy programming (ROFP)</b> is a powerful <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a> approach to deal with optimization problems under <a href="Uncertainty" title="Uncertainty">uncertainty</a>. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in the Journal of Fuzzy Sets and Systems. ROFP enables the decision makers to be benefited from the capabilities of both <a href="Fuzzy_set" title="Fuzzy set">fuzzy</a> mathematical programming and <a href="Robust_optimization" title="Robust optimization">robust optimization</a> approaches. At 2016 Pishvaee and Fazli<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> put a significant step forward by extending the ROFP approach to handle flexibility of constraints and goals. ROFP is able to achieve a <i>robust solution</i> for an optimization problem under uncertainty.

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<div class="mw-heading mw-heading2"><h2 id="Definition_of_robust_solution">Definition of robust solution</h2></div>

<p>Robust solution is defined as a solution which has "both <i>feasibility robustness</i> and <i>optimality robustness</i>; 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".<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Classification_of_ROFP_methods">Classification of ROFP methods</h2></div>

<p>As fuzzy mathematical programming is categorized into <i>Possibilistic programming</i> and <i>Flexible programming</i>, ROFP also can be classified into:<sup id="cite_ref-:1_2-2" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<ol><li>Robust possibilistic programming (RPP)</li>

<li>Robust flexible programming (RFP)</li>

<li>Mixed possibilistic-flexible robust programming (MPFRP)</li></ol>

<p>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.

</p><p>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:<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<ol><li>Hard worst case ROFP</li>

<li>Soft worst case ROFP</li>

<li>Realistic ROFP</li></ol>

<p>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).

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>ROFP is successfully implemented in different practical application areas such as the following ones.

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<ul><li><a href="Supply_chain_management" title="Supply chain management">Supply chain management</a> such as the work by Pishvaee et al.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> which addresses the design of a social responsible supply chain network under epistemic uncertainty.</li>

<li>Healthcare management such as the works by Zahiri et al.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and Mousazadeh et al.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> which consider the planning of an organ transplantation network and a pharmaceutical supply chain, respectively.</li>

<li><a href="Energy_planning" title="Energy planning">Energy planning</a> such as Bairamzadeh et al.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> developed a robust possibilistic programming model to deal with the planning problem of municipal electric power system.</li>

<li><a href="Sustainability" title="Sustainability">Sustainability</a> such as Xu and Huang<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> which employ ROFP to cope with an air quality management problem.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFPishvaeeRazmiTorabi2012" class="citation journal cs1">Pishvaee, M. S.; Razmi, J.; Torabi, S. A. (2012-11-01). "Robust possibilistic programming for socially responsible supply chain network design: A new approach". <i>Fuzzy Sets and Systems</i>. Theme : Operational Research. <b>206</b>: <span class="nowrap">1–</span>20. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.fss.2012.04.010">10.1016/j.fss.2012.04.010</a>.</cite></span>

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<li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFPishvaeeFazli_Khalaf2016" class="citation journal cs1">Pishvaee, Mir Saman; Fazli Khalaf, Mohamadreza (2016-01-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apm.2015.04.054">"Novel robust fuzzy mathematical programming methods"</a>. <i>Applied Mathematical Modelling</i>. <b>40</b> (1): <span class="nowrap">407–</span>418. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apm.2015.04.054">10.1016/j.apm.2015.04.054</a></span>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFZahiriTavakkoli-MoghaddamPishvaee2014" class="citation journal cs1">Zahiri, Behzad; Tavakkoli-Moghaddam, Reza; Pishvaee, Mir Saman (2014-08-01). "A robust possibilistic programming approach to multi-period location–allocation of organ transplant centers under uncertainty". <i>Computers & Industrial Engineering</i>. <b>74</b>: <span class="nowrap">139–</span>148. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cie.2014.05.008">10.1016/j.cie.2014.05.008</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMousazadehTorabiZahiri2015" class="citation journal cs1">Mousazadeh, M.; Torabi, S. A.; Zahiri, B. (2015-11-02). "A robust possibilistic programming approach for pharmaceutical supply chain network design". <i>Computers & Chemical Engineering</i>. <b>82</b>: <span class="nowrap">115–</span>128. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.compchemeng.2015.06.008">10.1016/j.compchemeng.2015.06.008</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBairamzadehPishvaeeSaidi-Mehrabad2015" class="citation journal cs1">Bairamzadeh, Samira; Pishvaee, Mir Saman; Saidi-Mehrabad, Mohammad (2015-12-22). "Multiobjective Robust Possibilistic Programming Approach to Sustainable Bioethanol Supply Chain Design under Multiple Uncertainties". <i>Industrial & Engineering Chemistry Research</i>. <b>55</b> (1): <span class="nowrap">237–</span>256. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.iecr.5b02875">10.1021/acs.iecr.5b02875</a>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFZhouLiHuang2015" class="citation journal cs1">Zhou, Y.; Li, Y.P.; Huang, G.H. (2015-12-15). "A robust possibilistic mixed-integer programming method for planning municipal electric power systems". <i>International Journal of Electrical Power & Energy Systems</i>. <b>73</b>: <span class="nowrap">757–</span>772. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015IJEPE..73..757Z">2015IJEPE..73..757Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijepes.2015.06.009">10.1016/j.ijepes.2015.06.009</a>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFXuHuang2015" class="citation journal cs1">Xu, Ye; Huang, Guohe (2015-10-15). "Development of an Improved Fuzzy Robust Chance-Constrained Programming Model for Air Quality Management". <i>Environmental Modeling & Assessment</i>. <b>20</b> (5): <span class="nowrap">535–</span>548. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015EMdAs..20..535X">2015EMdAs..20..535X</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10666-014-9441-3">10.1007/s10666-014-9441-3</a>.</cite></span>

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