<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@6869987eb4814ccca4f5a129bb1a3a87" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Slides</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@html+block@5d09194a583c43c68f92e1eee17c8314">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@html+block@5d09194a583c43c68f92e1eee17c8314" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="html">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><i>Slides available <a class="modal-content" href="/assets/courseware/v1/5314d4b66dc387b3a89b5c0eecb16b9b/asset-v1:MITx+15.053x+3T2016+type@asset+block/IP_2.pdf"> here</a></i> </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@74231196b24a4bd8807eddb884108a88" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Intro</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Intro</h3>
<div
id="video_f6f11cee06bd441f9ea8b70af43657c7"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/a5d5c15649cd4172bd18780dcbc3c647.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=sPIaJ%2F%2BnxLPuUSajfCIWb661%2FJY%3D&Expires=1731598181", "streams": "1.00:JSoDxwNNdgA", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001800_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001800/MIT150532016-V001800.m3u8"], "autoplay": false, "duration": 160.53, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="f6f11cee06bd441f9ea8b70af43657c7"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_f6f11cee06bd441f9ea8b70af43657c7">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_f6f11cee06bd441f9ea8b70af43657c7">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@f6f11cee06bd441f9ea8b70af43657c7/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@41616e1f796d4a1690a368e4cb7c6f77" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">More on OR</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">More on OR</h3>
<div
id="video_0e177e65b2974903bebfb40c2ef1ffe6"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/32b7c154a0e54b1b82dcbeef66c58e55.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=7oZz55WdW5Lp6lsMaV4Q05%2BoTJs%3D&Expires=1731598181", "streams": "1.00:zPChdFyWbvw", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002400_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002400/MIT150532016-V002400.m3u8"], "autoplay": false, "duration": 209.43, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="0e177e65b2974903bebfb40c2ef1ffe6"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_0e177e65b2974903bebfb40c2ef1ffe6">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_0e177e65b2974903bebfb40c2ef1ffe6">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0e177e65b2974903bebfb40c2ef1ffe6/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@5e9253ddcfae411f8f48b1d362e3dfa0" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">More on OR Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_5e4bfbda65dc442693c00cf90e5bfefd" class="problems-wrapper" role="group"
aria-labelledby="5e4bfbda65dc442693c00cf90e5bfefd-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="5e4bfbda65dc442693c00cf90e5bfefd-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd-problem-progress" tabindex="-1">
More on OR Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@5e4bfbda65dc442693c00cf90e5bfefd-problem-progress"></div>
<div class="problem">
<div>
<p>Consider three binary variables \(w_1,w_2, w_3\), on
decision variable \(x_1\). Select three constraints that ensures that \(x_1= 13 \text{ or } 39 \text{ or } 88\)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
<fieldset aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_0-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(x_1 = 13w_1 + 39w_2 + 88w_3\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_1-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(w_1 + w_2 + w_3 = 1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_2-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(w_1, w_2, w_3 \in \{0,1\}\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_3-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(x_1 \geq 13w_1 + 39w_2 + 88w_3\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_4-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(w_1 + w_2 + w_3 \geq 1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1[]" id="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="5e4bfbda65dc442693c00cf90e5bfefd_2_1-choice_5-label" for="input_5e4bfbda65dc442693c00cf90e5bfefd_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1">
\(w_1, w_2, w_3 \geq 0\)
</label>
</div>
<span id="answer_5e4bfbda65dc442693c00cf90e5bfefd_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_5e4bfbda65dc442693c00cf90e5bfefd_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_5e4bfbda65dc442693c00cf90e5bfefd_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="More on OR Exercise " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_5e4bfbda65dc442693c00cf90e5bfefd" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_5e4bfbda65dc442693c00cf90e5bfefd">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="5e4bfbda65dc442693c00cf90e5bfefd-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="5e4bfbda65dc442693c00cf90e5bfefd-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="5e4bfbda65dc442693c00cf90e5bfefd-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_583dcd314dd64effbdc402c5aac702e2" class="problems-wrapper" role="group"
aria-labelledby="583dcd314dd64effbdc402c5aac702e2-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="583dcd314dd64effbdc402c5aac702e2-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2-problem-progress" tabindex="-1">
More on OR Exercise 2
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@583dcd314dd64effbdc402c5aac702e2-problem-progress"></div>
<div class="problem">
<div>
<p>Suppose that we want at least two of the following set of constraints to hold: [mathjaxinline]x_1 \geq 20, x_2 \leq 6, x_3 \geq 16, x_4 \leq 18[/mathjaxinline]. Write the model as an integer program. Select all of the contraints below that are constraints of your linear program.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_583dcd314dd64effbdc402c5aac702e2_2_1">
<fieldset aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_0-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_1 \geq 20-M(1-w_1)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_1-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_2 \leq 6+M(1-w_2)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_2-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_3 \geq 16-M(1-w_3)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_3-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_4 \leq 18+M(1-w_4)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_4-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]w_1,w_2,w_3,w_4 \in \{0,1\}[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_5-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]w_1+w_2+w_3+w_4 \geq 2[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_6-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_1 \geq 20+M(1-w_1)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_7-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_2 \leq 6-M(1-w_2)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_8-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_3 \geq 16+M(1-w_3)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_9-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_4 \leq 18-M(1-w_4)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_10" class="field-input input-checkbox" value="choice_10"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_10-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_10" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_1 = 20+M(1-w_1)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_11" class="field-input input-checkbox" value="choice_11"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_11-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_11" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_2 = 6-M(1-w_2)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_12" class="field-input input-checkbox" value="choice_12"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_12-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_12" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_3 = 16+M(1-w_3)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_13" class="field-input input-checkbox" value="choice_13"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_13-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_13" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]x_4 = 18-M(1-w_4)[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_14" class="field-input input-checkbox" value="choice_14"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_14-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_14" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]w_1,w_2,w_3,w_4 \geq 0[/mathjaxinline]
</label>
</div>
<div class="field">
<input type="checkbox" name="input_583dcd314dd64effbdc402c5aac702e2_2_1[]" id="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_15" class="field-input input-checkbox" value="choice_15"/><label id="583dcd314dd64effbdc402c5aac702e2_2_1-choice_15-label" for="input_583dcd314dd64effbdc402c5aac702e2_2_1_choice_15" class="response-label field-label label-inline" aria-describedby="status_583dcd314dd64effbdc402c5aac702e2_2_1">
[mathjaxinline]w_1+w_2+w_3+w_4 \leq 2[/mathjaxinline]
</label>
</div>
<span id="answer_583dcd314dd64effbdc402c5aac702e2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_583dcd314dd64effbdc402c5aac702e2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_583dcd314dd64effbdc402c5aac702e2_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="More on OR Exercise 2 " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_583dcd314dd64effbdc402c5aac702e2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_583dcd314dd64effbdc402c5aac702e2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="583dcd314dd64effbdc402c5aac702e2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="583dcd314dd64effbdc402c5aac702e2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="583dcd314dd64effbdc402c5aac702e2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@984aa33801634805b352623200338a36" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Divisibility Constraints</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Divisibility Constraints</h3>
<div
id="video_da983d8ca5874354987885b6d9195414"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/6ee4a871a4964dc7bc7cd5fa2acc92ba.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=mberuMHHUaLl%2BeLIPLmeklmhV0g%3D&Expires=1731598181", "streams": "1.00:z-DPbl3b-JA", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001400_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001400/MIT150532016-V001400.m3u8"], "autoplay": false, "duration": 128.13, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="da983d8ca5874354987885b6d9195414"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_da983d8ca5874354987885b6d9195414">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_da983d8ca5874354987885b6d9195414">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@da983d8ca5874354987885b6d9195414/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@542bc37e5b894d4ba39c3f8a96137fde" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Divisibility Constraints Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_f0479a3b98514b1b95a06e16e15b0356" class="problems-wrapper" role="group"
aria-labelledby="f0479a3b98514b1b95a06e16e15b0356-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="f0479a3b98514b1b95a06e16e15b0356-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356-problem-progress" tabindex="-1">
Divisibility Constraints Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@f0479a3b98514b1b95a06e16e15b0356-problem-progress"></div>
<div class="problem">
<div>
<p>Consider the statement \(x_1\) is NOT divisible by 8. We model the statement starting with the constraint:
\(x_1 = 8y_1 + y_2\) for \(x_1,y_1,y_2 \in \mathbb{Z}_{\geq 0}\). Which two additional constraints are needed?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_f0479a3b98514b1b95a06e16e15b0356_2_1">
<fieldset aria-describedby="status_f0479a3b98514b1b95a06e16e15b0356_2_1">
<div class="field">
<input type="checkbox" name="input_f0479a3b98514b1b95a06e16e15b0356_2_1[]" id="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="f0479a3b98514b1b95a06e16e15b0356_2_1-choice_0-label" for="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_f0479a3b98514b1b95a06e16e15b0356_2_1">
\( y_2 \ge 0.\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f0479a3b98514b1b95a06e16e15b0356_2_1[]" id="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="f0479a3b98514b1b95a06e16e15b0356_2_1-choice_1-label" for="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_f0479a3b98514b1b95a06e16e15b0356_2_1">
\( y_2 \ge 1.\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f0479a3b98514b1b95a06e16e15b0356_2_1[]" id="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="f0479a3b98514b1b95a06e16e15b0356_2_1-choice_2-label" for="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_f0479a3b98514b1b95a06e16e15b0356_2_1">
\( y_2 \le 7.\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_f0479a3b98514b1b95a06e16e15b0356_2_1[]" id="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="f0479a3b98514b1b95a06e16e15b0356_2_1-choice_3-label" for="input_f0479a3b98514b1b95a06e16e15b0356_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_f0479a3b98514b1b95a06e16e15b0356_2_1">
\( y_2 \le 8.\)
</label>
</div>
<span id="answer_f0479a3b98514b1b95a06e16e15b0356_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_f0479a3b98514b1b95a06e16e15b0356_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_f0479a3b98514b1b95a06e16e15b0356_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Divisibility Constraints Exercise" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_f0479a3b98514b1b95a06e16e15b0356" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_f0479a3b98514b1b95a06e16e15b0356">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="f0479a3b98514b1b95a06e16e15b0356-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="f0479a3b98514b1b95a06e16e15b0356-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="f0479a3b98514b1b95a06e16e15b0356-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@470570b91ed54bae925c8699048d785c" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Union of Polyhedra</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Union of Polyhedra </h3>
<div
id="video_8f99f7f73ead44cd8b57178feacfbf57"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/0a48460fe0ba49d3afd7a0fd0c3177f8.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=cSlpI6bYuHpeC00x%2BbUOBdGXTvw%3D&Expires=1731598182", "streams": "1.00:B2H84UR5lts", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001600_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001600/MIT150532016-V001600.m3u8"], "autoplay": false, "duration": 244.4, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="8f99f7f73ead44cd8b57178feacfbf57"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_8f99f7f73ead44cd8b57178feacfbf57">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_8f99f7f73ead44cd8b57178feacfbf57">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8f99f7f73ead44cd8b57178feacfbf57/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@b70486573925486784992636e655a3c5" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Fixed Charge Problems</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Fixed Charge Problems</h3>
<div
id="video_a19f57e3bffb45f6a83f45013d2d28b7"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/30a75ce3e5584dc89de23be8ae6af0e1.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=LC1j4SRqVnS5HlBLtQPqqiDPgi8%3D&Expires=1731598183", "streams": "1.00:1M22LD2LpRo", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001500_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001500/MIT150532016-V001500.m3u8"], "autoplay": false, "duration": 251.63, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="a19f57e3bffb45f6a83f45013d2d28b7"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_a19f57e3bffb45f6a83f45013d2d28b7">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_a19f57e3bffb45f6a83f45013d2d28b7">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@a19f57e3bffb45f6a83f45013d2d28b7/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@2da537ed2db14bcdaae24a132d6798d8" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Fixed Charge Problems Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_4789fb43239c453f88c8049711ab789c" class="problems-wrapper" role="group"
aria-labelledby="4789fb43239c453f88c8049711ab789c-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="4789fb43239c453f88c8049711ab789c-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c-problem-progress" tabindex="-1">
Fixed Charge Problems Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@4789fb43239c453f88c8049711ab789c-problem-progress"></div>
<div class="problem">
<div>
<p>Consider the DTC problem in which \(S \) shields and \(K \) slingshot kits are produced, and suppose the fixed costs for making these are 10 and 5, respectively. Recall that the original objective was \(3S+5K\), with demands of 30 and 40 for shields and slingshot kits, respectively. Let \(w_1, w_2 \) be binary variables equal such that </p>
<ul>
<li>\(w_1 = 1\) if there is a setup for shields</li>
<li>\(w_1 = 0\) otherwise no setup for shields</li>
<li>\(w_2 = 1\) if there is a setup for slingshot kits</li>
<li>\(w_2 = 0\) otherwise no setup for slingshot kits</li>
</ul>
<p>Determine the new objective and constraints necessary for modeling the fixed charge</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_4789fb43239c453f88c8049711ab789c_2_1">
<fieldset aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_0-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
MAX \(y_1 + y_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_1-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_1 = -10w_1 + 3S \\ 0 \leq S \leq 30 w_1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_2-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_2 = -5w_2 + 5K \\ 0 \leq K \leq 40 w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_3-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
MIN \(y_1 - y_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_4-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_1 = -10w_1 + 3S \\ 0 \leq S \leq w_1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_5-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_2 = 5w_2 + 5K \\ 0 \leq K \leq w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_6-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
MIN \(y_1 + y_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_7-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_1 = 10w_1 + 3S \\ 0 \leq S \leq w_1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_4789fb43239c453f88c8049711ab789c_2_1[]" id="input_4789fb43239c453f88c8049711ab789c_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="4789fb43239c453f88c8049711ab789c_2_1-choice_8-label" for="input_4789fb43239c453f88c8049711ab789c_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_4789fb43239c453f88c8049711ab789c_2_1">
\(y_2 = 5w_2 + 5K \\ 0 \leq K \leq 100w_2\)
</label>
</div>
<span id="answer_4789fb43239c453f88c8049711ab789c_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_4789fb43239c453f88c8049711ab789c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_4789fb43239c453f88c8049711ab789c_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Fixed Charge Problems Exercise" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_4789fb43239c453f88c8049711ab789c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_4789fb43239c453f88c8049711ab789c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="4789fb43239c453f88c8049711ab789c-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="4789fb43239c453f88c8049711ab789c-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="4789fb43239c453f88c8049711ab789c-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@f8150b94a03947288f64f019c605ab8a" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">The Alchemist's Problem</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Alchemist's Problem</h3>
<div
id="video_8c42582d05ff420498dc81fd8e426d2c"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/a016a3a00cd4419fbdafc70ff45e79a0.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=aWfNrsXnElw1xUJXwpir%2Fx8aQ8A%3D&Expires=1731598183", "streams": "1.00:ADAAlAxHAEs", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002500_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002500/MIT150532016-V002500.m3u8"], "autoplay": false, "duration": 261.5, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="8c42582d05ff420498dc81fd8e426d2c"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_8c42582d05ff420498dc81fd8e426d2c">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_8c42582d05ff420498dc81fd8e426d2c">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@8c42582d05ff420498dc81fd8e426d2c/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@3cfbe60c0d864b69830acd6f90f4c09e" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Piecewise Linear Costs</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Piecewise Linear Costs</h3>
<div
id="video_2a677dd9d54f4692aa8f1850b5663619"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/aa466ea7f8de486a84d8ad3040742ffa.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=RBTxuQO8aczpr7gDDVPUVqt5f0s%3D&Expires=1731598183", "streams": "1.00:5JBEpsaZ6KQ", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001900_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001900/MIT150532016-V001900.m3u8"], "autoplay": false, "duration": 412.93, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="2a677dd9d54f4692aa8f1850b5663619"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_2a677dd9d54f4692aa8f1850b5663619">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_2a677dd9d54f4692aa8f1850b5663619">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@2a677dd9d54f4692aa8f1850b5663619/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@812ef01f76b9419f9a4d7a819149f7d2" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Piecewise Linear Costs Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_61400c47798d4dc289b82841f29dfd4e" class="problems-wrapper" role="group"
aria-labelledby="61400c47798d4dc289b82841f29dfd4e-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="61400c47798d4dc289b82841f29dfd4e-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e-problem-progress" tabindex="-1">
Piecewise Linear Costs Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@61400c47798d4dc289b82841f29dfd4e-problem-progress"></div>
<div class="problem">
<div>
<p>Suppose we have the following piecewise linear function of \(x\)</p>
<ul>
<li>\(y = \frac{2x}{3} \text{ if } 0 \leq x \leq 3\)</li>
<li>\(y = 2 \text{ if } 3 \leq x \leq 7\)</li>
</ul>
<p>Select the necessary constraints for the formulation</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_61400c47798d4dc289b82841f29dfd4e_2_1">
<fieldset aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_0-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(w_1+w_2 = 1 \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_1-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(0 \leq x_1 \leq 3w_1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_2-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(x=x_1+x_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_3-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y_1=\frac{2x_1}{3}\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_4-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y_2=2w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_5-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y=y_1+y_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_6-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(3w_2 \leq x_2 \leq 7w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_7-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(w_1, w_2 \in \{0,1 \}\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_8-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y_2 \geq 2w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_9" class="field-input input-checkbox" value="choice_9"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_9-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_9" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y \geq y_1+y_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_10" class="field-input input-checkbox" value="choice_10"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_10-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_10" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(0 \leq x_2 \leq 7w_2\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_11" class="field-input input-checkbox" value="choice_11"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_11-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_11" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(w_1+w_2 \geq 1 \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_12" class="field-input input-checkbox" value="choice_12"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_12-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_12" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(0 \leq x_1 \leq 7w_1\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_13" class="field-input input-checkbox" value="choice_13"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_13-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_13" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y \geq \frac{2x_1}{3}\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_61400c47798d4dc289b82841f29dfd4e_2_1[]" id="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_14" class="field-input input-checkbox" value="choice_14"/><label id="61400c47798d4dc289b82841f29dfd4e_2_1-choice_14-label" for="input_61400c47798d4dc289b82841f29dfd4e_2_1_choice_14" class="response-label field-label label-inline" aria-describedby="status_61400c47798d4dc289b82841f29dfd4e_2_1">
\(y_2 \geq \frac{2w_1}{3} + 2w_2\)
</label>
</div>
<span id="answer_61400c47798d4dc289b82841f29dfd4e_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_61400c47798d4dc289b82841f29dfd4e_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_61400c47798d4dc289b82841f29dfd4e_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Piecewise Linear Costs Exercise" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_61400c47798d4dc289b82841f29dfd4e" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_61400c47798d4dc289b82841f29dfd4e">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="61400c47798d4dc289b82841f29dfd4e-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="61400c47798d4dc289b82841f29dfd4e-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="61400c47798d4dc289b82841f29dfd4e-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@ab8b2b4d32bd42d0ba02803bfa746afd" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Overview of Combinatorial Problems</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Overview of Combinatorial Problems</h3>
<div
id="video_3416f8b9799e437c93b854643087eb37"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/d92172087ce54bd280758cd2aff4c283.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=zHJu1QFXo0njLaQW2pOiJRT21H8%3D&Expires=1731598183", "streams": "1.00:Yt-XIrSVNd8", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V001700_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V001700/MIT150532016-V001700.m3u8"], "autoplay": false, "duration": 76.27, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="3416f8b9799e437c93b854643087eb37"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_3416f8b9799e437c93b854643087eb37">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_3416f8b9799e437c93b854643087eb37">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@3416f8b9799e437c93b854643087eb37/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@ea73987711e943a8b2e8ccfd03517bfa" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Covering 1</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Set Covering 1</h3>
<div
id="video_7fb452bab709465bb4a6ad4beba0381e"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/f60061cfb6314a26b3496fcc9d6e1b68.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=z%2FmPzGEIk2av1STeuPiuMa98S1c%3D&Expires=1731598183", "streams": "1.00:L6t4LUs2sgc", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002300_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002300/MIT150532016-V002300.m3u8"], "autoplay": false, "duration": 342.03, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="7fb452bab709465bb4a6ad4beba0381e"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_7fb452bab709465bb4a6ad4beba0381e">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_7fb452bab709465bb4a6ad4beba0381e">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fb452bab709465bb4a6ad4beba0381e/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@c0b8151fbc38466789be4f87123f33f9" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Covering 1 Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_107d37dadb604c24ac4884330017bf16" class="problems-wrapper" role="group"
aria-labelledby="107d37dadb604c24ac4884330017bf16-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="107d37dadb604c24ac4884330017bf16-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16-problem-progress" tabindex="-1">
Set Covering 1 Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@107d37dadb604c24ac4884330017bf16-problem-progress"></div>
<div class="problem">
<div>
<p>
<img width="400" src="/assets/courseware/v1/c547bc9c6b1adbc22c7809799a3adb71/asset-v1:MITx+15.053x+3T2016+type@asset+block/map.png"/>
</p>
<p>There are two different solutions to the 053 Chocolate problem with three stores. Find one of these solutions. </p>
<p>What is the sum of the district numbers of the three stores?&#8221; (Your answer will be correct regardless of which of the two solutions you found.)</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_107d37dadb604c24ac4884330017bf16_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_107d37dadb604c24ac4884330017bf16_2_1" id="input_107d37dadb604c24ac4884330017bf16_2_1" data-input-id="107d37dadb604c24ac4884330017bf16_2_1" value="" aria-describedby="status_107d37dadb604c24ac4884330017bf16_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_107d37dadb604c24ac4884330017bf16_2_1"/>
<span class="status unanswered" id="status_107d37dadb604c24ac4884330017bf16_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_107d37dadb604c24ac4884330017bf16_2_1" class="answer"/>
<div id="input_107d37dadb604c24ac4884330017bf16_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_107d37dadb604c24ac4884330017bf16_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Set Covering 1 Exercise " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_107d37dadb604c24ac4884330017bf16" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_107d37dadb604c24ac4884330017bf16">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="107d37dadb604c24ac4884330017bf16-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="107d37dadb604c24ac4884330017bf16-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="107d37dadb604c24ac4884330017bf16-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@9aa1a132f75f4ac2af156da060a46fd3" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Covering 2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Set Covering 2</h3>
<div
id="video_75b71c7ff284433194ced1d600e8909e"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/b8d7181099124236891004165eb447c5.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=NEctzyyYh1ijatj7eITHECrozb8%3D&Expires=1731598183", "streams": "1.00:unG5gLf9eH0", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002200_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002200/MIT150532016-V002200.m3u8"], "autoplay": false, "duration": 232.03, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="75b71c7ff284433194ced1d600e8909e"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_75b71c7ff284433194ced1d600e8909e">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_75b71c7ff284433194ced1d600e8909e">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@75b71c7ff284433194ced1d600e8909e/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@0f8ee2684f734dff904312c23362cabe" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Covering 2 Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_7149d699fcc644ac9cc5cca9ad70065b" class="problems-wrapper" role="group"
aria-labelledby="7149d699fcc644ac9cc5cca9ad70065b-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="7149d699fcc644ac9cc5cca9ad70065b-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b-problem-progress" tabindex="-1">
Set Covering 2 Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@7149d699fcc644ac9cc5cca9ad70065b-problem-progress"></div>
<div class="problem">
<div>
<p>Which of the following was <b> not </b> listed in the previous video as an application of the set cover problem?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_7149d699fcc644ac9cc5cca9ad70065b_2_1">
<fieldset aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
<div class="field">
<input type="radio" name="input_7149d699fcc644ac9cc5cca9ad70065b_2_1" id="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="7149d699fcc644ac9cc5cca9ad70065b_2_1-choice_1-label" for="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
Locating hospitals in a city.
</label>
</div>
<div class="field">
<input type="radio" name="input_7149d699fcc644ac9cc5cca9ad70065b_2_1" id="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="7149d699fcc644ac9cc5cca9ad70065b_2_1-choice_4-label" for="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
Determining a small set of databases that contains information that you need.
</label>
</div>
<div class="field">
<input type="radio" name="input_7149d699fcc644ac9cc5cca9ad70065b_2_1" id="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="7149d699fcc644ac9cc5cca9ad70065b_2_1-choice_2-label" for="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
Figuring out what character strings can be used to identify computer viruses.
</label>
</div>
<div class="field">
<input type="radio" name="input_7149d699fcc644ac9cc5cca9ad70065b_2_1" id="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="7149d699fcc644ac9cc5cca9ad70065b_2_1-choice_3-label" for="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
Selecting people for a startup that have the needed set of skills.
</label>
</div>
<div class="field">
<input type="radio" name="input_7149d699fcc644ac9cc5cca9ad70065b_2_1" id="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="7149d699fcc644ac9cc5cca9ad70065b_2_1-choice_0-label" for="input_7149d699fcc644ac9cc5cca9ad70065b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_7149d699fcc644ac9cc5cca9ad70065b_2_1">
Cutting car doors out of sheet metal.
</label>
</div>
<span id="answer_7149d699fcc644ac9cc5cca9ad70065b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_7149d699fcc644ac9cc5cca9ad70065b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_7149d699fcc644ac9cc5cca9ad70065b_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Set Covering 2 Exercise " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_7149d699fcc644ac9cc5cca9ad70065b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_7149d699fcc644ac9cc5cca9ad70065b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7149d699fcc644ac9cc5cca9ad70065b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7149d699fcc644ac9cc5cca9ad70065b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="7149d699fcc644ac9cc5cca9ad70065b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@20d17075d02440cd8dfbcc6d7b3164c1" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Packing</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Set Packing</h3>
<div
id="video_7fc2ff8bc7da47e9a57bd7d7ccf861ef"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/9f41f2b41bd04610898a6dc9e1d06e6a.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=CB6T901HPHfPqd5g%2ByJkk3ubNYY%3D&Expires=1731598183", "streams": "1.00:XC-mhYC8WvY", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002000_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002000/MIT150532016-V002000.m3u8"], "autoplay": false, "duration": 390.43, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="7fc2ff8bc7da47e9a57bd7d7ccf861ef"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_7fc2ff8bc7da47e9a57bd7d7ccf861ef">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_7fc2ff8bc7da47e9a57bd7d7ccf861ef">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@7fc2ff8bc7da47e9a57bd7d7ccf861ef/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@337562ae0e6e4d4997c739dddafaa255" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Set Packing Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_38aa8ded009e4f7b929e73b53557fde1" class="problems-wrapper" role="group"
aria-labelledby="38aa8ded009e4f7b929e73b53557fde1-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="38aa8ded009e4f7b929e73b53557fde1-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1-problem-progress" tabindex="-1">
Set Packing Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@38aa8ded009e4f7b929e73b53557fde1-problem-progress"></div>
<div class="problem">
<div>
<p>Consider the graph in the diagram below. What is the maximum size of an independent set (packing)?</p>
<img width="275" src="/assets/courseware/v1/564a5df2967976f3338bf45a6a31b583/asset-v1:MITx+15.053x+3T2016+type@asset+block/packing.png"/>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_38aa8ded009e4f7b929e73b53557fde1_2_1">
<fieldset aria-describedby="status_38aa8ded009e4f7b929e73b53557fde1_2_1">
<div class="field">
<input type="radio" name="input_38aa8ded009e4f7b929e73b53557fde1_2_1" id="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="38aa8ded009e4f7b929e73b53557fde1_2_1-choice_0-label" for="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_38aa8ded009e4f7b929e73b53557fde1_2_1">
2
</label>
</div>
<div class="field">
<input type="radio" name="input_38aa8ded009e4f7b929e73b53557fde1_2_1" id="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="38aa8ded009e4f7b929e73b53557fde1_2_1-choice_1-label" for="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_38aa8ded009e4f7b929e73b53557fde1_2_1">
3
</label>
</div>
<div class="field">
<input type="radio" name="input_38aa8ded009e4f7b929e73b53557fde1_2_1" id="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="38aa8ded009e4f7b929e73b53557fde1_2_1-choice_2-label" for="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_38aa8ded009e4f7b929e73b53557fde1_2_1">
4
</label>
</div>
<div class="field">
<input type="radio" name="input_38aa8ded009e4f7b929e73b53557fde1_2_1" id="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="38aa8ded009e4f7b929e73b53557fde1_2_1-choice_3-label" for="input_38aa8ded009e4f7b929e73b53557fde1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_38aa8ded009e4f7b929e73b53557fde1_2_1">
5
</label>
</div>
<span id="answer_38aa8ded009e4f7b929e73b53557fde1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_38aa8ded009e4f7b929e73b53557fde1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_38aa8ded009e4f7b929e73b53557fde1_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Set Packing Exercise" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_38aa8ded009e4f7b929e73b53557fde1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_38aa8ded009e4f7b929e73b53557fde1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="38aa8ded009e4f7b929e73b53557fde1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="38aa8ded009e4f7b929e73b53557fde1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="38aa8ded009e4f7b929e73b53557fde1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@1a8358bb15504c899a1466dbd8f486dc" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Facility Location 1</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Facility Location 1</h3>
<div
id="video_49b98a021c04464b911abd45c283165c"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/15012021e03e47df80368a3476137bce.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=f3s4HET%2F%2F34wWMVBVCp0ctrmIXQ%3D&Expires=1731598184", "streams": "1.00:6CPVfG8b0xY", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002700_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002700/MIT150532016-V002700.m3u8"], "autoplay": false, "duration": 356.63, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="49b98a021c04464b911abd45c283165c"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_49b98a021c04464b911abd45c283165c">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_49b98a021c04464b911abd45c283165c">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@49b98a021c04464b911abd45c283165c/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@faa64239a7ab4f41b6a9bfb0f9e073b9" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Facility Location 2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Facility Location 2</h3>
<div
id="video_172b9ffd0813472b9b272876e0ad391b"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/110b7512840c45f6915a725103d34629.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=kwv5TMhUiobzWxaBl%2FIi4UVPCy8%3D&Expires=1731598184", "streams": "1.00:9-Lwlz2MWbY", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002100_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002100/MIT150532016-V002100.m3u8"], "autoplay": false, "duration": 291.1, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="172b9ffd0813472b9b272876e0ad391b"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_172b9ffd0813472b9b272876e0ad391b">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_172b9ffd0813472b9b272876e0ad391b">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@172b9ffd0813472b9b272876e0ad391b/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@db7b7c97e2404ba991714439993736d1" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Facility Location 2 Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_a4b23a382cfe492f9846cbf6e796632a" class="problems-wrapper" role="group"
aria-labelledby="a4b23a382cfe492f9846cbf6e796632a-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="a4b23a382cfe492f9846cbf6e796632a-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a-problem-progress" tabindex="-1">
Facility Location 1 Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@a4b23a382cfe492f9846cbf6e796632a-problem-progress"></div>
<div class="problem">
<div>
<p>Suppose that students are not required to dine, and let \(v_s\) be the number of such students. If a student at location s chooses not to dine, then the disutility for that student is \(g_s\). How would the model change if this possibility were permitted? Select one objective and two constraints.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a4b23a382cfe492f9846cbf6e796632a_2_1">
<fieldset aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_0-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New objective: \(\sum_{d \in D} f_d y_d + \sum_{s \in S} \sum_{d \in D} c_{sd} x_{sd} + \sum_{s \in S} g_s v_s \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_1-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New objective: \(\sum_{d \in D} f_d y_d + \sum_{s \in S} \sum_{d \in D} c_{sd} x_{sd} - \sum_{s \in S} g_s v_s \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_2-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New constraint: \(\sum_{d \in D} x_{sd} + v_s = n_s \text{ for } s \in S \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_3-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New constraint: \(\sum_{d \in D} x_{sd} + v_s \leq n_s \text{ for } s \in S \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_4-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New constraint: \(\sum_{d \in D} x_{sd} + v_s = n_s \text{ for } d \in D \)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_5-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New constraint: \(v_s \geq 0 \text{ for } d \in D\)
</label>
</div>
<div class="field">
<input type="checkbox" name="input_a4b23a382cfe492f9846cbf6e796632a_2_1[]" id="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="a4b23a382cfe492f9846cbf6e796632a_2_1-choice_6-label" for="input_a4b23a382cfe492f9846cbf6e796632a_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_a4b23a382cfe492f9846cbf6e796632a_2_1">
New constraint: \(v_s \geq 0 \text{ for } s \in S\)
</label>
</div>
<span id="answer_a4b23a382cfe492f9846cbf6e796632a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_a4b23a382cfe492f9846cbf6e796632a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_a4b23a382cfe492f9846cbf6e796632a_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Facility Location 1 Exercise " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_a4b23a382cfe492f9846cbf6e796632a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_a4b23a382cfe492f9846cbf6e796632a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="a4b23a382cfe492f9846cbf6e796632a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="a4b23a382cfe492f9846cbf6e796632a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="a4b23a382cfe492f9846cbf6e796632a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@feb596b113cf4ebcaa604c4c9fa6dfe4" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Graph Coloring</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="video">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Graph Coloring</h3>
<div
id="video_0219a8c930864d9ba8ff51025c57c82c"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "saveStateEnabled": false, "autoAdvance": false, "savedVideoPosition": 0.0, "showCaptions": "true", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/264fd540db494109a5ea04226250f2ae.jpeg?x-amz-security-token=IQoJb3JpZ2luX2VjEGcaCXVzLWVhc3QtMSJHMEUCIQDUCJVhzGUgs%2B%2BbLazCptI1ejNDQMreGm%2BgMMK%2Bw1CUXwIgQyDcGqcfCQF4cw40lP4xpBW5kH%2BO0mMzEGVBNvNEeB8qxQUI8P%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDKL1qETrivS5ecEmNiqZBaL8wdgu0yBvoUvjyaSiHWLc11BAPrJsNzLpH2V6t%2F54bdN4zF%2BmeaS1cM9sAU2CZjNOSrYBA2432BdPVmUG8rf6FJs85LhsMNm1jRjZN87G54VHKJekSStsQ9ZhdlgQZ9xhSCYY4c2TWUGYr9t4bBbhugaMgsTzpMqMfNobd13E5VRGZyT1yNMy2YxDEjyYSJyHaBRZ2%2Bu4bNoVkyt2ncoyVP5vBZHvRW80WOQ8GRIKgdY61y863cnBNyiv2D4Qx8W%2Ba9161t%2FeS5kk7IR2LfLCLGYgSaw5Zk6tEOIJndMNfQQwUGdubQwVd6k51JjFUkTGXN%2BnOva%2BKe5q80yYiAzUM%2B3hNbOsdeeWll3hfbd3IWKgO1oPPmUe7BwzV9DYn4b11oP%2FoMtu6yiKJ377A3iwS1aLfra%2FKBlGfkuRRg2ryVH8PCWn7lqiSEApfY6Gk%2BEf7xJFyzrplOBEeLQulBM8ENWhEj1E9QM2CaUkX42U%2BrGptEXp7iwEpr8iN4YPxBqpNn7Ybg1RKGejXRPmUG5ARXxnHjeCbAqsLloN5Mds43JjZ7skcjNAyUc7NtcI6OgmoedYecV4aQvZtscsrIQzjIzUaGLNvXTTVgsDDTmjAwItX2iDY%2BDpMsPqNijF%2B3PCIhZzB2k8lfOc9yPtdEWoGqEJKXgEGyr43%2FXKz8lSkwVNLjMyRGnkTvkudl5gnWbaYCHkMRJ%2FHGEt3hM3bGUBR1N6DpXB1Ml%2FxdXWtoJgZ0VaNuIQ%2FcNrOPgD8UsZnVz9hojH4mGlr%2Byrir0xQTwTz2B5CCKTCNnmuWhAILozZqbzjIkhXRTu6XTcy0P2JU3IW10Uh5g%2BDyxG0VGLBfWYKoKRr3Jnn%2BU3aGkKBMx5xka43WkkxtpTMJGb2LkGOrEBB9JgSSl%2B579iwExP3duY2tF2JjnIoqiigp72RgFnUPMgcSU21V90no19c1%2BqS6PcC%2BfEFCCbmY3iIbHTTqzgvRpeTWoDqVYUlpjEo5nzUMsEqvtAS5Bc893xHrvbeULo14P1dZUAEf8CX3IcyasMhNh%2BukCYjGqECQne8F8o%2F3Z2AwQ9yFvTwIPGCRajakOyXiYMoF7P12JTmW4Flmc0FtS2B%2FKOW73Bw4eZrvy5IcLv&AWSAccessKeyId=ASIA4QDFFSURVFALOVB5&Signature=0sawuaF%2FjUHaeo0x2YJ0JyQdEPY%3D&Expires=1731598184", "streams": "1.00:r28eS4hmoQI", "speed": null, "transcriptTranslationUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "saveStateUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/transcript/available_translations", "ytTestTimeout": 1500, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT15053/MIT150532016-V002600_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT150532016-V002600/MIT150532016-V002600.m3u8"], "autoplay": false, "duration": 447.93, "autohideHtml5": false, "publishCompletionUrl": "/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/publish_completion", "transcriptLanguages": {"en": "English"}, "start": 0.0, "generalSpeed": 1.0, "completionEnabled": false, "captionDataDir": null, "prioritizeHls": false, "completionPercentage": 0.95, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="0219a8c930864d9ba8ff51025c57c82c"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_0219a8c930864d9ba8ff51025c57c82c">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_0219a8c930864d9ba8ff51025c57c82c">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@video+block@0219a8c930864d9ba8ff51025c57c82c/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@vertical+block@132ae095ab814d81b6ec89d029cd8b1a" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="vertical">
<h2 class="hd hd-2 unit-title">Graph Coloring Exercise</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-graded="True" data-request-token="91741596a29c11ef8fa716ffea5863df" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-usage-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+15.053x+3T2016" data-block-type="problem">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_cbf3cedaf77c43bf9353b7a3cc76db9d" class="problems-wrapper" role="group"
aria-labelledby="cbf3cedaf77c43bf9353b7a3cc76db9d-problem-title"
data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="cbf3cedaf77c43bf9353b7a3cc76db9d-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d-problem-progress" tabindex="-1">
Graph Coloring Exercise
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@cbf3cedaf77c43bf9353b7a3cc76db9d-problem-progress"></div>
<div class="problem">
<div>
<p>Consider the following graph. What is the minimum number of colors needed to color the vertices (nodes) so that adjacent vertices receive different colors?</p>
<img width="100" src="/assets/courseware/v1/6e548820d953fe419797e333c02a8a4f/asset-v1:MITx+15.053x+3T2016+type@asset+block/coloring.png"/>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
<fieldset aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
<div class="field">
<input type="radio" name="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" id="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1af#D" class="field-input input-radio" value="choice_1af#D"/><label id="cbf3cedaf77c43bf9353b7a3cc76db9d_2_1-choice_1af#D-label" for="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1af#D" class="response-label field-label label-inline" aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
2
</label>
</div>
<div class="field">
<input type="radio" name="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" id="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="field-input input-radio" value="choice_1f#D"/><label id="cbf3cedaf77c43bf9353b7a3cc76db9d_2_1-choice_1f#D-label" for="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="response-label field-label label-inline" aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
1
</label>
</div>
<div class="field">
<input type="radio" name="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" id="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="field-input input-radio" value="choice_1f#D"/><label id="cbf3cedaf77c43bf9353b7a3cc76db9d_2_1-choice_1f#D-label" for="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="response-label field-label label-inline" aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
4
</label>
</div>
<div class="field">
<input type="radio" name="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" id="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="field-input input-radio" value="choice_1f#D"/><label id="cbf3cedaf77c43bf9353b7a3cc76db9d_2_1-choice_1f#D-label" for="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="response-label field-label label-inline" aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
8
</label>
</div>
<div class="field">
<input type="radio" name="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" id="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="field-input input-radio" value="choice_1f#D"/><label id="cbf3cedaf77c43bf9353b7a3cc76db9d_2_1-choice_1f#D-label" for="input_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1_choice_1f#D" class="response-label field-label label-inline" aria-describedby="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1">
3
</label>
</div>
<span id="answer_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_cbf3cedaf77c43bf9353b7a3cc76db9d_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_cbf3cedaf77c43bf9353b7a3cc76db9d_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Graph Coloring Exercise " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_cbf3cedaf77c43bf9353b7a3cc76db9d" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_cbf3cedaf77c43bf9353b7a3cc76db9d">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="cbf3cedaf77c43bf9353b7a3cc76db9d-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="cbf3cedaf77c43bf9353b7a3cc76db9d-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="cbf3cedaf77c43bf9353b7a3cc76db9d-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved