{"id":10937,"date":"2020-11-11T23:16:24","date_gmt":"2020-11-11T22:16:24","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=10937"},"modified":"2022-12-03T23:05:39","modified_gmt":"2022-12-03T22:05:39","slug":"corrected-exercises-assignment-problem","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/planning-problem\/corrected-exercises-assignment-problem\/","title":{"rendered":"Corrected Exercises: Assignment Problem"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"10937\" class=\"elementor elementor-10937\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a35a90c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a35a90c\" 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-1ff64ce\" data-id=\"1ff64ce\" 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-9218fd8 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"9218fd8\" 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\/planning-problem\/\">\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\">Planning problem<\/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-58c25fc\" data-id=\"58c25fc\" 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-3704630 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"3704630\" 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-91193b9\" data-id=\"91193b9\" 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-64b91eb elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"64b91eb\" 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:\/\/en.wikipedia.org\/wiki\/Automated_planning_and_scheduling\" 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-8df6b1e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8df6b1e\" 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-03192d7\" data-id=\"03192d7\" 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-85bd61d elementor-widget elementor-widget-heading\" data-id=\"85bd61d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<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\/planning-problem\/corrected-exercises-assignment-problem\/#Corrected-exercises-about-assignment-problems\" >Corrected exercises about assignment problems<\/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\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-1\" >Exercise 1<\/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\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-2\" >Exercise 2<\/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\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-3\" >Exercise 3<\/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\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-4\" >Exercise 4<\/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\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-5\" >Exercise 5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-6\" >Exercise 6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-7\" >Exercise 7<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/corrected-exercises-assignment-problem\/#Exercise-8\" >Exercise 8<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Corrected-exercises-about-assignment-problems\"><\/span>Corrected exercises about assignment problems<span class=\"ez-toc-section-end\"><\/span><\/h2>\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-cea304b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cea304b\" 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-72860f6\" data-id=\"72860f6\" 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-08c09b5 elementor-widget elementor-widget-text-editor\" data-id=\"08c09b5\" 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<p>The page presents several corrected exercises about automated planning and scheduling problems, especially about assignment problems.<\/p>\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-4e6d1ff elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4e6d1ff\" 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-f377523\" data-id=\"f377523\" 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-a15dc44 elementor-widget elementor-widget-text-editor\" data-id=\"a15dc44\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-1\"><\/span><strong><b>Exercise 1<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>The Atlantic Coast Grid deserves four cities. The office wants to assign four plants. The price to send energy from a plant to each city is described below:<\/p><table><tbody><tr><td width=\"122\"><p>\u00a0<\/p><\/td><td width=\"122\"><p>Raleigh<\/p><\/td><td width=\"122\"><p>Atlanta<\/p><\/td><td width=\"122\"><p>Durham<\/p><\/td><td width=\"122\"><p>Clemson<\/p><\/td><\/tr><tr><td width=\"122\"><p>Plant A<\/p><\/td><td width=\"122\"><p>210<\/p><\/td><td width=\"122\"><p>90<\/p><\/td><td width=\"122\"><p>180<\/p><\/td><td width=\"122\"><p>160<\/p><\/td><\/tr><tr><td width=\"122\"><p>Plant B<\/p><\/td><td width=\"122\"><p>100<\/p><\/td><td width=\"122\"><p>70<\/p><\/td><td width=\"122\"><p>130<\/p><\/td><td width=\"122\"><p>200<\/p><\/td><\/tr><tr><td width=\"122\"><p>Plant C<\/p><\/td><td width=\"122\"><p>175<\/p><\/td><td width=\"122\"><p>105<\/p><\/td><td width=\"122\"><p>140<\/p><\/td><td width=\"122\"><p>170<\/p><\/td><\/tr><tr><td width=\"122\"><p>Plant D<\/p><\/td><td width=\"122\"><p>80<\/p><\/td><td width=\"122\"><p>65<\/p><\/td><td width=\"122\"><p>105<\/p><\/td><td width=\"122\"><p>120<\/p><\/td><\/tr><\/tbody><\/table><p>A plant can deserve only one city, and a city can take energies only from one plant. Find the best assignment at lowest cost.<\/p>\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-d2575a8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d2575a8\" 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-102711f\" data-id=\"102711f\" 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-00254ae elementor-widget elementor-widget-toggle\" data-id=\"00254ae\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1521\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1521\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1521\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1521\"><p>After first reduction:<\/p><table><tbody><tr><td width=\"122\"><p>\u00a0<\/p><\/td><td width=\"122\"><p>R<\/p><\/td><td width=\"122\"><p>TO<\/p><\/td><td width=\"122\"><p>D<\/p><\/td><td width=\"122\"><p>VS<\/p><\/td><\/tr><tr><td width=\"122\"><p>TO<\/p><\/td><td width=\"122\"><p>105<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>55<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><\/tr><tr><td width=\"122\"><p>B<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>25<\/p><\/td><td width=\"122\"><p>75<\/p><\/td><\/tr><tr><td width=\"122\"><p>VS<\/p><\/td><td width=\"122\"><p>35<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>10<\/p><\/td><\/tr><tr><td width=\"122\"><p>D<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>5<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><\/tr><\/tbody><\/table><p>After second reduction:<\/p><table><tbody><tr><td width=\"122\"><p>\u00a0<\/p><\/td><td width=\"122\"><p>R<\/p><\/td><td width=\"122\"><p>TO<\/p><\/td><td width=\"122\"><p>D<\/p><\/td><td width=\"122\"><p>VS<\/p><\/td><\/tr><tr><td width=\"122\"><p>TO<\/p><\/td><td width=\"122\"><p>90<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>40<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><\/tr><tr><td width=\"122\"><p>B<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>10<\/p><\/td><td width=\"122\"><p>60<\/p><\/td><\/tr><tr><td width=\"122\"><p>VS<\/p><\/td><td width=\"122\"><p>55<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>10<\/p><\/td><\/tr><tr><td width=\"122\"><p>D<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><td width=\"122\"><p>5<\/p><\/td><td width=\"122\"><p>0<\/p><\/td><\/tr><\/tbody><\/table><p>Solution: 90 + 100 + 140 + 120 = 450.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-418d8da elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"418d8da\" 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-44d2a4a\" data-id=\"44d2a4a\" 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-b859a9f elementor-widget elementor-widget-text-editor\" data-id=\"b859a9f\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-2\"><\/span><strong><b>Exercise 2<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Permute the columns of a square matrix so as to minimize the sum of elements on the main diagonal.<\/p><table><tbody><tr><td width=\"122\"><p>8<\/p><\/td><td width=\"122\"><p>16<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><td width=\"122\"><p>91<\/p><\/td><td width=\"122\"><p>64<\/p><\/td><\/tr><tr><td width=\"122\"><p>83<\/p><\/td><td width=\"122\"><p>42<\/p><\/td><td width=\"122\"><p>93<\/p><\/td><td width=\"122\"><p>75<\/p><\/td><td width=\"122\"><p>27<\/p><\/td><\/tr><tr><td width=\"122\"><p>76<\/p><\/td><td width=\"122\"><p>95<\/p><\/td><td width=\"122\"><p>75<\/p><\/td><td width=\"122\"><p>81<\/p><\/td><td width=\"122\"><p>50<\/p><\/td><\/tr><tr><td width=\"122\"><p>20<\/p><\/td><td width=\"122\"><p>42<\/p><\/td><td width=\"122\"><p>96<\/p><\/td><td width=\"122\"><p>90<\/p><\/td><td width=\"122\"><p>24<\/p><\/td><\/tr><tr><td width=\"122\"><p>38<\/p><\/td><td width=\"122\"><p>28<\/p><\/td><td width=\"122\"><p>2<\/p><\/td><td width=\"122\"><p>15<\/p><\/td><td width=\"122\"><p>81<\/p><\/td><\/tr><\/tbody><\/table>\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-c12b6f9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c12b6f9\" 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-04a2fdb\" data-id=\"04a2fdb\" 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-8a89634 elementor-widget elementor-widget-toggle\" data-id=\"8a89634\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1451\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1451\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1451\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1451\"><p>Find an assignment solution to this matrix. Reorganize the columns such as the solution forms a diagonal.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-6770d28 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6770d28\" 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-7aaf931\" data-id=\"7aaf931\" 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-d8386ed elementor-widget elementor-widget-text-editor\" data-id=\"d8386ed\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-3\"><\/span><strong><b>Exercise 3<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Three robots {a, b, c} need to finish three tasks {t1, t2, t3} in the following grid. It take one day for a robot to move from one cell to one of its neighbors.<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-10943 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image30-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"205\" height=\"202\" title=\"\"><\/p><p>In the following table, we list the days that each robot can finish each task alone. The tasks need to be finished as soon as possible.<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-10944 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image31-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"151\" height=\"96\" title=\"\"><\/p>\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-9c74c3c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9c74c3c\" 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-8a494d2\" data-id=\"8a494d2\" 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-d5a3e15 elementor-widget elementor-widget-toggle\" data-id=\"d5a3e15\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2241\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2241\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2241\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2241\"><p>Add Manhattan distance in days to the last table. Solve the assignment problem.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-e3b9ab9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e3b9ab9\" 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-69cc512\" data-id=\"69cc512\" 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-d77be63 elementor-widget elementor-widget-text-editor\" data-id=\"d77be63\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-4\"><\/span><strong><b>Exercise 4<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Braneast Airlines must staff the faily flights between New York and Chicago shown in the table below. Each of Braneast&#039;s crews lives in either New York or Chicago. Each day a crew must fly one NY-Chicago and one Chicago-NY flight with at least one hour of downtime between flights.<\/p><p>Braneast wants to schedule the crews to minimize the total downtime. Set up an assignment problem that can be used to accomplish this goal. Of course, some assignments are not possible. Find the flight assignments that minimize the total downtime. How many crews should be based in each city? Assume that at the end of the day, each crew must be in tis home city.<\/p><p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter wp-image-10945 size-large\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1-1024x547.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"1024\" height=\"547\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1-1024x547.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1-300x160.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1-768x410.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1-1000x534.png 1000w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image32-1.png 1098w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\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-6784ef3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6784ef3\" 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-3ba1f3d\" data-id=\"3ba1f3d\" 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-32deaae elementor-widget elementor-widget-toggle\" data-id=\"32deaae\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5331\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5331\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-5331\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5331\"><p>The assignment table is constructed as follows: for the first flight from Chicago, considering the time it arrives at NY, how much time the crew has to wait at the airport. For example for the first flight from Chicago, it arrives at 10 AM:<\/p><table><tbody><tr><td width=\"83\"><p>Flight<\/p><\/td><td width=\"76\"><p>1<\/p><\/td><td width=\"76\"><p>2<\/p><\/td><td width=\"76\"><p>3<\/p><\/td><td width=\"76\"><p>4<\/p><\/td><td width=\"76\"><p>5<\/p><\/td><td width=\"76\"><p>6<\/p><\/td><td width=\"75\"><p>7<\/p><\/td><\/tr><tr><td width=\"83\"><p>Leave New York<\/p><\/td><td width=\"76\"><p>7<\/p><\/td><td width=\"76\"><p>8<\/p><\/td><td width=\"76\"><p>10<\/p><\/td><td width=\"76\"><p>12<\/p><\/td><td width=\"76\"><p>14<\/p><\/td><td width=\"76\"><p>16<\/p><\/td><td width=\"75\"><p>18<\/p><\/td><\/tr><tr><td width=\"83\"><p>Waiting time<\/p><\/td><td width=\"76\"><p>impossible<\/p><\/td><td width=\"76\"><p>impossible<\/p><\/td><td width=\"76\"><p>impossible<\/p><\/td><td width=\"76\"><p>2<\/p><\/td><td width=\"76\"><p>4<\/p><\/td><td width=\"76\"><p>6<\/p><\/td><td width=\"75\"><p>8<\/p><\/td><\/tr><\/tbody><\/table><p>We note that flight 4, 5, 6, 7 from Chicago can&#039;t be starting point, so we compute waiting time from NY flight. Note that flight 7 from NY can&#039;t be a starting point, so we compute waiting time from Chicago flight. From each elements of the table, we keep the lowest value computed (for flight from Chicago and NY).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10946 size-large\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1-1024x316.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"1024\" height=\"316\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1-1024x316.png 1024w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1-300x93.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1-768x237.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1-1000x309.png 1000w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image33-1.png 1120w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p><p>Solution:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10947 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image34-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"989\" height=\"333\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image34-1.png 989w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image34-1-300x101.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image34-1-768x259.png 768w\" sizes=\"(max-width: 989px) 100vw, 989px\" \/><\/p><p>Starts in NY: (1.3), (2.4), (3.5), (5.6), (6.7).<\/p><p>Starts in Chicago: (4.1), (7.2).<\/p><p>Total downtime = 25h.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10948 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image35-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"927\" height=\"578\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image35-1.png 927w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image35-1-300x187.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image35-1-768x479.png 768w\" sizes=\"(max-width: 927px) 100vw, 927px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-bd7dcfc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bd7dcfc\" 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-8209f67\" data-id=\"8209f67\" 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-ebd7f12 elementor-widget elementor-widget-text-editor\" data-id=\"ebd7f12\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-5\"><\/span><strong><b>Exercise 5<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Consider the data of table below. If a crew based in Mumbai arrives at Delhi on a given flight, it must return to Mumbai on a later flight. Assume that for any given pairing, the crew cill be based in the city that results in the smaller layover.<\/p><p>The problem is to find the pairings so as to minimize the time on ground away from home, subject to a minimum interval of one hour between arrival and departure. Given the pairs of flights, where should the crews be based?<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10949 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image36-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"430\" height=\"297\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image36-1.png 430w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image36-1-300x207.png 300w\" sizes=\"(max-width: 430px) 100vw, 430px\" \/><\/p>\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-1f36aef elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1f36aef\" 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-92b4cd3\" data-id=\"92b4cd3\" 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-eb0859b elementor-widget elementor-widget-toggle\" data-id=\"eb0859b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2461\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2461\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2461\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2461\"><p>As the previous exercise, firstly compute the layover time matrices, one for layover in Mumbai and the other for Delhi.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10950 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image37-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"510\" height=\"195\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image37-1.png 510w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image37-1-300x115.png 300w\" sizes=\"(max-width: 510px) 100vw, 510px\" \/><\/p><p>We now compute the minimum of the values for the 36 pairs and construct the table below. For example (7,2) is the minimum between (7,2) in first matrix and (2,7) of the second one.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10951 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image38-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"307\" height=\"163\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image38-1.png 307w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image38-1-300x159.png 300w\" sizes=\"(max-width: 307px) 100vw, 307px\" \/><\/p><p>The solution of this problem is pairs (7,3) from Mumbai, (8,4) from Mumbai, (9,2) from Delhi, (10,5) from Mumbai, (11,6) from Delhi, (12, 1) from Delhi with a total layoff time = 18h.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-07cec61 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"07cec61\" 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-d50daea\" data-id=\"d50daea\" 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-d6fd32f elementor-widget elementor-widget-text-editor\" data-id=\"d6fd32f\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-6\"><\/span><strong><b>Exercise 6<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Resolve the following problem as an assignment problem.<\/p><p>Minimize 4X<sub>11<\/sub>+ 6X<sub>12<\/sub>+ 5X<sub>13<\/sub>+ 5X<sub>14<\/sub>+ 7X<sub>21<\/sub>+ 4X<sub>22<\/sub>+ 5X<sub>23<\/sub>+ 6X<sub>24<\/sub>\u00a0\u00a0+ 4X<sub>31<\/sub>+ 7X<sub>32<\/sub>+ 6X<sub>33<\/sub>+ 4X<sub>34<\/sub>\u00a0\u00a0+ 5X<sub>41<\/sub>+ 3X<sub>42<\/sub>+ 4X<sub>43<\/sub>+ 7X<sub>44 <\/sub><\/p><p>St. X<sub>11<\/sub>+ X<sub>12<\/sub>+ X<sub>13<\/sub>+ X<sub>14<\/sub>=1<\/p><p>X<sub>21<\/sub>+ X<sub>22<\/sub>+ X<sub>24<\/sub>+ X<sub>24<\/sub>=1<\/p><p>X<sub>31<\/sub>+ X<sub>32<\/sub>+ X<sub>33<\/sub>+ X<sub>34<\/sub>=1<\/p><p>X<sub>41<\/sub>+ X<sub>42<\/sub>+ X<sub>43<\/sub>+ X<sub>44<\/sub>=1<\/p><p>X<sub>11<\/sub>+ X<sub>21<\/sub>+ X<sub>31<\/sub>+ X<sub>41<\/sub>=1<\/p><p>X<sub>12<\/sub>+ X<sub>22<\/sub>+ X<sub>32<\/sub>+ X<sub>42<\/sub>=1<\/p><p>X<sub>13<\/sub>+ X<sub>23<\/sub>+ X<sub>33<\/sub>+ X<sub>43<\/sub>=1<\/p><p>X<sub>14<\/sub>+ X<sub>24<\/sub>+ X<sub>34<\/sub>+ X<sub>44<\/sub>=1<\/p>\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-595a28f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"595a28f\" 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-00a6b2d\" data-id=\"00a6b2d\" 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-30c04eb elementor-widget elementor-widget-toggle\" data-id=\"30c04eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-5111\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5111\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-5111\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5111\"><p>Here is the table to resolve:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10952 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image39-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"171\" height=\"146\" title=\"\"><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-2412235 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2412235\" 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-496533f\" data-id=\"496533f\" 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-bfd8372 elementor-widget elementor-widget-text-editor\" data-id=\"bfd8372\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-7\"><\/span><strong><b>Exercise 7<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>Resolve the following problem as an assignment problem.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10953 size-medium\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image40-1-300x186.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"300\" height=\"186\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image40-1-300x186.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image40-1.png 330w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\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-17c9e20 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"17c9e20\" 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-de0650e\" data-id=\"de0650e\" 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-bcc3e1f elementor-widget elementor-widget-toggle\" data-id=\"bcc3e1f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1971\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1971\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1971\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1971\"><p>The assignment problem is described in the table below:<\/p><table><tbody><tr><td width=\"76\"><p>\u00a0<\/p><\/td><td width=\"76\"><p>1<\/p><\/td><td width=\"76\"><p>2<\/p><\/td><td width=\"76\"><p>3<\/p><\/td><\/tr><tr><td width=\"76\"><p>1<\/p><\/td><td width=\"76\"><p>5<\/p><\/td><td width=\"76\"><p>9<\/p><\/td><td width=\"76\"><p>inf<\/p><\/td><\/tr><tr><td width=\"76\"><p>2<\/p><\/td><td width=\"76\"><p>inf<\/p><\/td><td width=\"76\"><p>2<\/p><\/td><td width=\"76\"><p>inf<\/p><\/td><\/tr><tr><td width=\"76\"><p>3<\/p><\/td><td width=\"76\"><p>inf<\/p><\/td><td width=\"76\"><p>1<\/p><\/td><td width=\"76\"><p>1<\/p><\/td><\/tr><\/tbody><\/table><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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-d6a5381 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d6a5381\" 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-6a89913\" data-id=\"6a89913\" 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-9b33793 elementor-widget elementor-widget-text-editor\" data-id=\"9b33793\" 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<h2><span class=\"ez-toc-section\" id=\"Exercise-8\"><\/span><strong><b>Exercise 8<\/b><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2><p>The Art History Department wishes to offer six courses in a semester. There are seven professors in the department, each of which can teach only certain courses, as shown in the table. Is it possible to assign the six courses to the professors so that no professor teaches more than one course?<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10954 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image41-1.png\" alt=\"corrected exercises automated planning and scheduling problems assignment problems\" width=\"354\" height=\"174\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image41-1.png 354w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/11\/Image41-1-300x147.png 300w\" sizes=\"(max-width: 354px) 100vw, 354px\" \/><\/p>\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-3eb9da6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3eb9da6\" 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-007a27a\" data-id=\"007a27a\" 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-94d37f4 elementor-widget elementor-widget-toggle\" data-id=\"94d37f4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1561\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1561\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1561\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1561\"><p>The assignment problem is described in the table below:<\/p><table><tbody><tr><td width=\"101\"><p>\u00a0<\/p><\/td><td width=\"73\"><p>Ant<\/p><\/td><td width=\"73\"><p>Bat<\/p><\/td><td width=\"73\"><p>Cat<\/p><\/td><td width=\"74\"><p>Dodo<\/p><\/td><td width=\"74\"><p>Frog<\/p><\/td><td width=\"74\"><p>Gnat<\/p><\/td><td width=\"72\"><p>Hog<\/p><\/td><\/tr><tr><td width=\"101\"><p>Antique<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"72\"><p>inf<\/p><\/td><\/tr><tr><td width=\"101\"><p>Renaissance<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"72\"><p>inf<\/p><\/td><\/tr><tr><td width=\"101\"><p>Baroque<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"72\"><p>inf<\/p><\/td><\/tr><tr><td width=\"101\"><p>Impressionism<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"72\"><p>inf<\/p><\/td><\/tr><tr><td width=\"101\"><p>Modern<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"72\"><p>1<\/p><\/td><\/tr><tr><td width=\"101\"><p>Contemporary<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"73\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>inf<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"72\"><p>inf<\/p><\/td><\/tr><tr><td width=\"101\"><p>dummy<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"73\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"74\"><p>1<\/p><\/td><td width=\"72\"><p>1<\/p><\/td><\/tr><\/tbody><\/table><p>In order to find an assignment, it&#039;s better to use <a href=\"https:\/\/complex-systems-ai.com\/en\/planning-problem\/stepping-stone\/\">stepping stone<\/a> method instead of Hungarian method.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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>The page presents several corrected exercises about automated planning and scheduling problems, especially about\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":868,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"footnotes":""},"class_list":["post-10937","page","type-page","status-publish","hentry"],"amp_enabled":false,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10937","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=10937"}],"version-history":[{"count":6,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10937\/revisions"}],"predecessor-version":[{"id":19088,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10937\/revisions\/19088"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/868"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=10937"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}