{"id":10288,"date":"2024-11-08T16:06:21","date_gmt":"2024-11-08T15:06:21","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=10288"},"modified":"2024-11-08T16:09:05","modified_gmt":"2024-11-08T15:09:05","slug":"probleme-de-surreservation-surbooking","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/08\/probleme-de-surreservation-surbooking\/","title":{"rendered":"Probl\u00e8me de surr\u00e9servation (surbooking)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Salut l&rsquo;ami(e)! Sais-tu ce qu&rsquo;est le surbooking (ou surr\u00e9servation) ? Sais-tu que les compagnies a\u00e9riennes vendent tr\u00e8s souvent plus de billets que ne comporte de places l&rsquo;avion du vol concern\u00e9 ?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons \u00e9tudier cela ensemble si tu veux bien&#8230; Sinon, ciao et bonne route \u00e0 toi dans les m\u00e9andres de la toile \ud83d\ude09<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/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:\/\/www.mathweb.fr\/euclide\/2024\/11\/08\/probleme-de-surreservation-surbooking\/#Surreservation_surbooking_un_exemple_pour_commencer\" >Surr\u00e9servation (surbooking): un exemple pour commencer<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/08\/probleme-de-surreservation-surbooking\/#Mise_en_place_du_probleme\" >Mise en place du probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/08\/probleme-de-surreservation-surbooking\/#Formalisation_du_probleme\" >Formalisation du probl\u00e8me<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/08\/probleme-de-surreservation-surbooking\/#Surreservation_surbooking_optimisation_du_chiffre_daffaire\" >Surr\u00e9servation (surbooking): optimisation du chiffre d&rsquo;affaire<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Surreservation_surbooking_un_exemple_pour_commencer\"><\/span>Surr\u00e9servation (surbooking): un exemple pour commencer<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mise_en_place_du_probleme\"><\/span>Mise en place du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">La compagnie <em>Arnak&rsquo;Air<\/em> vend 105 billets d&rsquo;avion pour un vol qui comporte 100 places.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La place est vendue 100 \u20ac. Si le client arrive \u00e0 l&#8217;embarquement et qu&rsquo;il n&rsquo;y a plus de places, il est d\u00e9dommag\u00e9 \u00e0 hauteur de 250 \u20ac.<\/p>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p> En Europe, le r\u00e8glement CE 261\/2004 impose des <strong>indemnisations fixes<\/strong> pour les passagers victimes de surbooking, bas\u00e9es sur la distance du vol. Pour un vol de moins de 1500 km, elle s&rsquo;\u00e9l\u00e8ve \u00e0 250 \u20ac.<\/p><\/blockquote><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Pour simplifier les choses, nous n&rsquo;allons pas consid\u00e9rer les clients qui se font rembourser leur billet avant le d\u00e9part du vol&#8230; car nous sommes sur <em>Arnak&rsquo;Air<\/em> et que ce n&rsquo;est pas la politique de la maison! \ud83d\ude42<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On estime que la probabilit\u00e9 qu&rsquo;un client ayant achet\u00e9 un billet embarque r\u00e9ellement est \u00e9gale \u00e0 0,95.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formalisation_du_probleme\"><\/span>Formalisation du probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons noter X la variable al\u00e9atoire repr\u00e9sentant le nombre de passagers embarquant r\u00e9ellement sur ce vol. Alors, X suit la loi binomiale \\(\\mathcal{B}(105,0.95)\\). On a alors la loi suivante:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table class=\"has-fixed-layout\" style=\"border-width:1px\"><tbody><tr><td>X<\/td><td class=\"has-text-align-center\" data-align=\"center\">X \u2a7d 100<\/td><td class=\"has-text-align-center\" data-align=\"center\">X = 101<\/td><td class=\"has-text-align-center\" data-align=\"center\">X = 102<\/td><td class=\"has-text-align-center\" data-align=\"center\">X = 103<\/td><td class=\"has-text-align-center\" data-align=\"center\">X = 104<\/td><td class=\"has-text-align-center\" data-align=\"center\">X = 105<\/td><\/tr><tr><td>Probas.<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.6076<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.1680<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.1252<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0693<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0253<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0046<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><em>Loi de probabilit\u00e9s de X<\/em><\/figcaption><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure data-wp-context=\"{&quot;imageId&quot;:&quot;6a3a11926f092&quot;}\" data-wp-interactive=\"core\/image\" data-wp-key=\"6a3a11926f092\" class=\"aligncenter size-medium wp-lightbox-container\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"219\" data-wp-class--hide=\"state.isContentHidden\" data-wp-class--show=\"state.isContentVisible\" data-wp-init=\"callbacks.setButtonStyles\" data-wp-on--click=\"actions.showLightbox\" data-wp-on--load=\"callbacks.setButtonStyles\" data-wp-on--pointerdown=\"actions.preloadImage\" data-wp-on--pointerenter=\"actions.preloadImageWithDelay\" data-wp-on--pointerleave=\"actions.cancelPreload\" data-wp-on-window--resize=\"callbacks.setButtonStyles\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-300x219.png\" alt=\"probl\u00e8me surbooking surr\u00e9servation\" class=\"wp-image-10291\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-300x219.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image.png 560w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><button\n\t\t\tclass=\"lightbox-trigger\"\n\t\t\ttype=\"button\"\n\t\t\taria-haspopup=\"dialog\"\n\t\t\tdata-wp-bind--aria-label=\"state.thisImage.triggerButtonAriaLabel\"\n\t\t\tdata-wp-init=\"callbacks.initTriggerButton\"\n\t\t\tdata-wp-on--click=\"actions.showLightbox\"\n\t\t\tdata-wp-style--right=\"state.thisImage.buttonRight\"\n\t\t\tdata-wp-style--top=\"state.thisImage.buttonTop\"\n\t\t>\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12\" height=\"12\" fill=\"none\" viewBox=\"0 0 12 12\">\n\t\t\t\t<path fill=\"#fff\" d=\"M2 0a2 2 0 0 0-2 2v2h1.5V2a.5.5 0 0 1 .5-.5h2V0H2Zm2 10.5H2a.5.5 0 0 1-.5-.5V8H0v2a2 2 0 0 0 2 2h2v-1.5ZM8 12v-1.5h2a.5.5 0 0 0 .5-.5V8H12v2a2 2 0 0 1-2 2H8Zm2-12a2 2 0 0 1 2 2v2h-1.5V2a.5.5 0 0 0-.5-.5H8V0h2Z\" \/>\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Exhaustion de cas sur le chiffre d&rsquo;affaire (C.A.) de ce vol:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Si au plus 100 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 100 \u00d7 100 = 10 000 \u20ac;<\/li>\n\n\n\n<li>Si 101 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 10 000 \u20ac &#8211; 1 \u00d7 250 \u20ac (d\u00e9dommagement) = 9 750 \u20ac;<\/li>\n\n\n\n<li>Si 102 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 10 000 \u20ac &#8211; 2 \u00d7 250 \u20ac (d\u00e9dommagement) = 9 500 \u20ac;<\/li>\n\n\n\n<li>Si 103 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 10 000 \u20ac &#8211; 3 \u00d7 250 \u20ac (d\u00e9dommagement) = 9 250 \u20ac;<\/li>\n\n\n\n<li>Si 104 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 10 000 \u20ac &#8211; 4 \u00d7 250 \u20ac (d\u00e9dommagement) = 9 000 \u20ac;<\/li>\n\n\n\n<li>Si 105 passagers se pr\u00e9sentent, le C.A. est \u00e9gal \u00e0 10 000 \u20ac &#8211; 5 \u00d7 250 \u20ac (d\u00e9dommagement) = 8 750 \u20ac.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Si on note C la variable al\u00e9atoire correspondant au CA du vol, l&rsquo;esp\u00e9rance math\u00e9matique de C est alors:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">E(C) = 10 000\u00d70.6076 + 9 750\u00d70.1680 + 9 500\u00d70.1252 + &#8230; + 8 750\u00d70.0046 = 9812.375<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ainsi, la compagnie a\u00e9rienne peut esp\u00e9rer avoir un chiffre d&rsquo;affaire de 9812.375 \u20ac.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Surreservation_surbooking_optimisation_du_chiffre_daffaire\"><\/span>Surr\u00e9servation (surbooking): optimisation du chiffre d&rsquo;affaire<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On cherche maintenant le nombre de billets pour optimiser le chiffre d&rsquo;affaire, c&rsquo;est-\u00e0-dire le rendre le plus grand possible. Pour cela, nous allons noter <em>n<\/em> le nombre de billets \u00e0 vendre, <em>n<\/em> &gt; 100.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Dans ce cas,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Le CA g\u00e9n\u00e9r\u00e9 par la vente des <em>n<\/em> billets est \u00e9gal \u00e0 100<em>n<\/em>;<\/li>\n\n\n\n<li>Si <em>n<\/em> = 100 + <em>k<\/em>, alors le d\u00e9dommagement s&rsquo;\u00e9l\u00e8ve \u00e0 250<em>k<\/em>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 partir de cela, on peut exprimer l&rsquo;esp\u00e9rance math\u00e9matique du chiffre d&rsquo;affaire comme suit:$$E(C)=100n\\times P(X\\leqslant100) + \\sum_{k=1}^{n-100} (100n-250k)\\times P(X=k).$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nulle question d&rsquo;e tenter d&rsquo;exprimer alg\u00e9briquement d&rsquo;aller plus loin du point de vue alg\u00e9brique ici. Cela serait ridicule. En effet, nous allons nous servir de Python  pour d\u00e9terminer la valeur de <em>n<\/em> pour laquelle E(C) est optimale.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from probastat import binom\n\n\ndef E(n):\n    X = binom(n,0.95)\n    s = 100*n + X.proba_cdf(100)\n    \n    for k in range(1,n-99):\n        s += (100*n-250*k)*X.proba(100+k)\n        \n    return s\n\nn = 101\nM = 10000\n\nwhile True:\n    if E(n) > M:\n        M = E(n)\n        n += 1\n        print(n,M)\n    else:\n        print(\"Valeur maximale atteinte pour n =\", n)\n        break\n<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>102 10156.395727676516\n103 10538.21115720697\n104 11361.376663985022\n105 12708.638659358936\n106 14433.538106350177\n107 16236.015988536621\n108 17825.23156864628\n109 19036.042126823977\n110 19845.99773814133\n111 20325.129641238662\n112 20573.85340106389\n113 20681.87354363916\n114 20712.818941198555\nValeur maximale atteinte pour n = 114<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Je pr\u00e9cise ici que j&rsquo;ai fait appel \u00e0 mon propre <a href=\"https:\/\/www.mathweb.fr\/euclide\/2021\/04\/09\/probabilites-et-python-au-lycee-loi-binomiale-et-variables-aleatoires\/\" target=\"_blank\" rel=\"noreferrer noopener\">module<\/a> (car les autres sont, \u00e0 mon go\u00fbt, trop chiants \u00e0 manipuler d\u00e8s lors qu&rsquo;il s&rsquo;agit de calculer une probabilit\u00e9).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La compagnie a\u00e9rienne a dont int\u00e9r\u00eat \u00e0 vendre 114 billets pour optimiser son chiffre d&rsquo;affaire.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Salut l&rsquo;ami(e)! Sais-tu ce qu&rsquo;est le surbooking (ou surr\u00e9servation) ? Sais-tu que les compagnies a\u00e9riennes vendent tr\u00e8s souvent plus de billets que ne comporte de places l&rsquo;avion du vol concern\u00e9 ? Nous allons \u00e9tudier cela ensemble si tu veux bien&#8230; Sinon, ciao et bonne route \u00e0 toi dans les m\u00e9andres [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":10293,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[427,169],"class_list":["post-10288","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python","tag-binomiale","tag-probabilites"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Probl\u00e8me de surr\u00e9servation (surbooking) - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Salut l&#039;ami! Sais-tu ce qu&#039;est le surbooking (ou surr\u00e9servation) ? 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