{"id":737,"date":"2016-02-17T13:01:13","date_gmt":"2016-02-17T12:01:13","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=737"},"modified":"2022-12-03T22:57:31","modified_gmt":"2022-12-03T21:57:31","slug":"analyse-post-optimale-de-sensibilite","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/","title":{"rendered":"Post-optimal sensitivity analysis"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"737\" class=\"elementor elementor-737\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a559033 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a559033\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-0bb3fa8\" data-id=\"0bb3fa8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2d76fce elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2d76fce\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Linear programming<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-253da51\" data-id=\"253da51\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-943d277 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"943d277\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-86d26d5\" data-id=\"86d26d5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-db98ce2 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"db98ce2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Optimisation_lin%C3%A9aire\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-60fea186 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"60fea186\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7c6d0c47\" data-id=\"7c6d0c47\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4d90b5af elementor-widget elementor-widget-text-editor\" data-id=\"4d90b5af\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Analyse-post-optimale-de-sensibilite\" >Post-optimal sensitivity analysis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Couts-marginaux\" >Marginal costs<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Etude-1-variation-dans-la-fonction-objectif\" >Study 1: variation in objective function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Etude-2-variation-dans-le-second-membre\" >Study 2: variation in the second limb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Etude-3-variation-des-variables-hors-base\" >Study 3: variation of non-base variables<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/post-optimal-sensitivity-analysis\/#Etude-4-variation-de-la-production\" >Study 4: variation in production<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Analyse-post-optimale-de-sensibilite\"><\/span>Post-optimal sensitivity analysis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>When the <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/simplex-method-2\/\">basic fix<\/a> optimal of the PL problem is analyzed to answer questions about changes in its formulation, the study is called post-optimal sensitivity analysis.<\/p>\n\n<p>We call post-optimization all the techniques making it possible to obtain the optimum of the PL problem when certain data have undergone modifications.<\/p>\n\n<p>We consider the problem of <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/\">linear programming<\/a> general in its stand art form:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-745 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp-300x95.png\" alt=\"simplex method sensitivity analysis post-optimal primal sensitivity analysis\" width=\"300\" height=\"95\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp-300x95.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp.png 421w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n<p>This study can be motivated by several reasons:<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\">the data of the problem is not known with exactitude, in which case it is important to determine to what extent this affects the proposed solution;<\/li>\n<li style=\"text-align: justify;\">we want to assess the consequences of a policy change that would modify the facts of the problem.<\/li>\n<\/ul>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Couts-marginaux\"><\/span>Marginal costs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The marginal cost of a good is called the minimum increase in expenditure, compared to the optimal solution, which would result from the use of an additional unit of this good, when the problem posed consists of producing goods at the lowest cost.<\/div>\n\n<p>If the problem posed consists in transforming goods to sell a production with a better profit and the maximum increase in income which results from the possibility of having an additional unit of one of the goods, is the marginal value of this good . Very often, the qualifier marginal cost is also used in this case.<\/p>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The marginal costs y * are therefore the net effects associated with the gap variables, since it is these variables that determine the surpluses (or shortages) of goods. <strong>These are the values of the variables in row Z. <\/strong><\/div>\n\n<p>If a difference variable is not zero, in the optimal solution, it means that the corresponding good is already surplus. Therefore, having an additional unit of this good will have no influence on the income. The <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/lp-origin-unrealizable\/\">slack variable<\/a> has zero marginal value.<\/p>\n\n<p>On the other hand, if a variance variable is zero in the optimal solution, it is because the corresponding good is totally used. Subsequently, a variation in availability will generally have an influence on income. This is why this zero deviation variable in the optimal solution has a non-zero marginal value, and this marginal value specifies the variation of the economic function resulting from the use of an additional unit of the associated good.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-2780 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe3.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"379\" height=\"472\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe3.png 379w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe3-241x300.png 241w\" sizes=\"(max-width: 379px) 100vw, 379px\" \/><\/figure>\n<\/div>\n\n<p>with the solution vector x * = (2,6). Attention here the line shows the value of Z and not of -Z (hence the positive values).<\/p>\n\n<p>We can measure the sensitivity of the optimal solution to a change in a line term or a coefficient in the objective.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etude-1-variation-dans-la-fonction-objectif\"><\/span>Study 1: variation in objective function<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">We want to examine how the optimal solution varies when the coefficient of one of the variables in the objective function varies. Edit c<sub>j<\/sub> is equivalent to modifying the slope of the objective function.<\/div>\n\n<p>The variation of a coefficient in the objective function over a certain interval does not lead to a modification of the optimal solution. Outside this interval, we have a new solution which itself remains optimal over another interval. We can thus highlight a finite number of variation intervals for c<sub>j<\/sub>, with on each of them an invariant solution.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-751 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp2.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"827\" height=\"184\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp2.png 827w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp2-300x67.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp2-768x171.png 768w\" sizes=\"(max-width: 827px) 100vw, 827px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The value of the j-th variable at the optimum x *<sub>j<\/sub> measures the increase in the objective function if the unit cost c is increased by one<sub>j<\/sub>. Logical and trivial behavior since the objective function is composed of the sum of c<sub>j<\/sub>* x<sub>j<\/sub>.<\/div>\n\n<p>Let us change the objective function by max z &#039;= 4 * x<sub>1<\/sub> + 5 * x<sub>2<\/sub>. The value of the objective function will change by x *<sub>1<\/sub> = 2, while the solution vector will not be modified as shown in <a href=\"https:\/\/complex-systems-ai.com\/en\/linear-programming-2\/resolution-graphics\/\">graphics resolution<\/a> :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2819 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe6.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"415\" height=\"371\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe6.png 415w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe6-300x268.png 300w\" sizes=\"(max-width: 415px) 100vw, 415px\" \/><\/figure>\n<\/div>\n\n<p>Likewise if c<sub>1<\/sub> changes from 3 to 2, only the value of the objective function will be modified. To calculate the interval over which the coefficient x *<sub>1<\/sub> is valid, we need to evolve the objective function until it is parallel to the other constraints.<\/p>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">\n<p>That is to say when the slope of the objective function is equal to the slope of the saturated stresses for the solution vector s *:<\/p>\n<ul>\n<li style=\"text-align: justify;\">z = c<sub>1<\/sub>* x<sub>1<\/sub> + 5 * x<sub>2<\/sub> and 2 * x<sub>2<\/sub> = 12 so -c<sub>1<\/sub>\/ 5 = 0, c<sub>1<\/sub> = 0;<\/li>\n<li style=\"text-align: justify;\">z = c<sub>1<\/sub>* x<sub>1<\/sub> + 5 * x<sub>2<\/sub> and 3 * x<sub>1<\/sub> + 2 * x<sub>2<\/sub> = 18 so -c<sub>1<\/sub>\/ 5 = -3\/2, c<sub>1<\/sub> =15\/2.<\/li>\n<\/ul>\n<p>The coefficient x *<sub>1<\/sub> is therefore valid for c<sub>1<\/sub> between 0 and 15\/2.<\/p>\n<\/div>\n\n<p>When the problem is of large dimension, it is possible to calculate the variation of the cost using the simplex by adding a delta on the cost to vary as in the following example:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6835 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp30.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"596\" height=\"233\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp30.png 596w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp30-300x117.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><\/figure>\n<\/div>\n\n<p>The solution remains optimal as long as the line of -Z has negative numbers so if:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6837 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp31.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"396\" height=\"102\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp31.png 396w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp31-300x77.png 300w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><\/figure>\n<\/div>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etude-2-variation-dans-le-second-membre\"><\/span>Study 2: variation in the second limb<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p>When the second member of a constraint varies (within a certain interval), if this constraint was not saturated, then the solution does not change and neither does the optimal value of the objective function. This result is obvious since the optimal solution not satisfying the constraint with equality, one can vary (a little) the second member without \u201ctouching\u201d the optimal solution.<\/p>\n\n<p>On the other hand, if the constraint were checked with equality at the optimum, one has an interval of variation for the second member such as:<\/p>\n\n<ul class=\"wp-block-list\">\n<li style=\"text-align: justify;\">The solution changes but the zero variables remain zero and the non-zero variables remain non-zero: the structure of the solution does not change.<\/li>\n<li style=\"text-align: justify;\">The variation of the second member i causes a variation of the optimal value of the objective function equal to u<sub>i<\/sub>* d<sub>i<\/sub>, therefore proportional to d<sub>i<\/sub>.<\/li>\n<\/ul>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The coefficient of proportionality is called marginal variation or dual cost or marginal profit. The dual cost u<sub>i<\/sub> is equal to the change in the optimal value of the objective function as the second member increases by one.<\/div>\n\n<p>If we leave the interval, we have a new dual cost. We can thus highlight a finite number of variation intervals for the second member with, on each of them, a value for the dual cost. On the different intervals, the sensitivity analysis does not give the optimal solution since the numerical values of the variables depend on the exact value of the second member.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-762 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp3-300x67.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"300\" height=\"67\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp3-300x67.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp3-768x173.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/01\/lp3.png 819w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n<p>Consider in the example that b<sub>1<\/sub> = 4 becomes b &#039;<sub>1<\/sub> = 5. Let&#039;s perform a graphic resolution of the new problem:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2796 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe4.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"458\" height=\"398\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe4.png 458w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe4-300x261.png 300w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">The evolution of this second member did not modify the optimal solution, the value of the objective function does not change. This change was easy to predict because the marginal cost of the gap variable y *<sub>1<\/sub> is zero: z &#039;* - z * = y *<sub>1<\/sub> = 0.<\/div>\n\n<p>When is the decrease in b<sub>1<\/sub> ? As the explanation on marginal costs explains, decreasing 1 of the second member causes the value of the objective function to decrease by an amount equal to the marginal cost. So decreasing 1 will not result in a change.<\/p>\n\n<p>To know the possibilities of evolution of the stock without changing the value of the optimal solution, it is necessary to add a delta in the second member studied as shown in the following example:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6838 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp32.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"550\" height=\"453\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp32.png 550w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp32-300x247.png 300w\" sizes=\"(max-width: 550px) 100vw, 550px\" \/><\/figure>\n<\/div>\n\n<p>The solution remains optimal as long as the simplex is not degenerated, i.e. the second member is positive:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6839 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp33.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"327\" height=\"97\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp33.png 327w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp33-300x89.png 300w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">The increase and decrease do not change the value of the objective function over a small interval, but if b<sub>1<\/sub> is less than 2, we can see on the graphics resolution that the optimal solution will be changed. The validity interval of y *<sub>1<\/sub> = 0 is therefore for b<sub>1<\/sub> between 2 and infinity.<\/div>\n\n<p>Consider now an increase of the third second member to b &#039;<sub>3<\/sub> = 19. Since the marginal cost is not zero, the optimal solution will be modified as shown by the graphical resolution.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2808 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe5.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"530\" height=\"478\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe5.png 530w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe5-300x271.png 300w\" sizes=\"(max-width: 530px) 100vw, 530px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">We can therefore interpret the marginal cost by: the decrease or the loss of a unit of the third second member will lead to an evolution of y *<sub>3<\/sub> of the value of the objective function over an interval b<sub>3<\/sub> comprised between 12 and 24.<\/div>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etude-3-variation-des-variables-hors-base\"><\/span>Study 3: variation of non-base variables<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The reduced cost of the non-base variable x<sub>j<\/sub>, denoted by<sub>j<\/sub>, measures the increase in the objective function if the value of the non-base variable is increased by one unit. The reduced cost of<sub>j<\/sub> is the opposite of the coefficient of the variable in the objective line Z.<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">The optimal solution will not change until the cost of the non-base variable is better than the optimal value of the objective function (so if the coefficient is between -infinity and Z for a maximization problem).<\/div>\n\n<p>Let&#039;s go back to the previous example with a new constraint and a new variable:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2842 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe7.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"445\" height=\"409\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe7.png 445w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2016\/02\/simplexe7-300x276.png 300w\" sizes=\"(max-width: 445px) 100vw, 445px\" \/><\/figure>\n<\/div>\n\n<p>We have the following reduced cost: d<sub>3<\/sub> = -2. This means that to build a unit of x<sub>3<\/sub>\u00a0 would decrease the value of the objective function by 2 (since it is outside the base x *<sub>3<\/sub> = 0).<\/p>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">Let&#039;s check by calculation: fix x<sub>3<\/sub> to 1. We obtain the following three inequalities: x<sub>1<\/sub> \u2264 3; 2 * x<sub>2<\/sub> \u2264 10; 3 * x<sub>1<\/sub> + 2 * x<sub>2<\/sub> \u2264 15. The vector (5\/3, 5) is solution of the system. We therefore have an evolution of x<sub>1<\/sub> from 5\/3 - 2 = -1\/3 and x<sub>2<\/sub> = 5 -6 = -1 in the objective function (new value - old value). Its cost therefore evolves by -1 \/ 3 * c<sub>1<\/sub> - 1 C<sub>2<\/sub> + 1 * c<sub>3<\/sub> = -2 (1 * c<sub>3<\/sub> because we go from a production of 0 to 1). We find the value of the reduced cost of x<sub>3<\/sub>.<\/div>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">So that x<sub>3<\/sub> to become profitable, its cost must increase at least the opposite of its reduced cost.<\/div>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Etude-4-variation-de-la-production\"><\/span>Study 4: variation in production<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">If the value has<sub>ij<\/sub>\u00a0changes in a saturated constraint, then neither the optimal solution nor the optimal value is kept.<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">If the value has<sub>ij<\/sub>\u00a0changes in an unsaturated constraint and a base variable, then the value can vary between + or - infinity (depending on a min or max) to S<sub>i\u00a0<\/sub>\/ x *<sub>i<\/sub>. with S<sub>i\u00a0<\/sub> the variance variable.<\/div>\n\n<p>Indeed by adding a delta variable in the cost of the variable based on the target constraint then it suffices to inject the optimal vector and solve the equation as in the following example:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6840 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2019\/04\/lp34.png\" alt=\"simplex method sensitivity analysis primal sensitivity analysis\" width=\"271\" height=\"109\" title=\"\"><\/figure>\n<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">If the value has<sub>ij<\/sub>\u00a0changes in any constraint and of a non-base variable, then only a negative variation (in the case of a max) can make the product viable in this constraint. We must then solve the simplex by incorporating the delta and verify its various optimality criteria.<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Linear Programming Wiki Home Page Post-optimal Sensitivity Analysis When the optimal basic solution of the PL problem is analyzed to answer the questions \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":486,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-737","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/737","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=737"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/737\/revisions"}],"predecessor-version":[{"id":17910,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/737\/revisions\/17910"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/486"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}