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instructions.html
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<!DOCTYPE html>
<HTML lang = "en">
<HEAD>
<meta charset="UTF-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<title>Lake Problem Simulation-Optimization Example</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]},
TeX: { equationNumbers: { autoNumber: "AMS" } }
});
</script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<style>
pre.hljl {
border: 1px solid #ccc;
margin: 5px;
padding: 5px;
overflow-x: auto;
color: rgb(68,68,68); background-color: rgb(251,251,251); }
pre.hljl > span.hljl-t { }
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pre.hljl > span.hljl-kc { color: rgb(59,151,46); font-style: italic; }
pre.hljl > span.hljl-kd { color: rgb(214,102,97); font-style: italic; }
pre.hljl > span.hljl-kn { color: rgb(148,91,176); font-weight: bold; }
pre.hljl > span.hljl-kp { color: rgb(148,91,176); font-weight: bold; }
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pre.hljl > span.hljl-kt { color: rgb(148,91,176); font-weight: bold; }
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pre.hljl > span.hljl-nc { }
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pre.hljl > span.hljl-nd { color: rgb(214,102,97); }
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pre.hljl > span.hljl-nf { color: rgb(66,102,213); }
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pre.hljl > span.hljl-nvm { }
pre.hljl > span.hljl-l { }
pre.hljl > span.hljl-ld { color: rgb(148,91,176); font-style: italic; }
pre.hljl > span.hljl-s { color: rgb(201,61,57); }
pre.hljl > span.hljl-sa { color: rgb(201,61,57); }
pre.hljl > span.hljl-sb { color: rgb(201,61,57); }
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pre.hljl > span.hljl-sdC { color: rgb(201,61,57); }
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pre.hljl > span.hljl-sh { color: rgb(201,61,57); }
pre.hljl > span.hljl-si { }
pre.hljl > span.hljl-so { color: rgb(201,61,57); }
pre.hljl > span.hljl-sr { color: rgb(201,61,57); }
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pre.hljl > span.hljl-nbB { color: rgb(59,151,46); }
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pre.hljl > span.hljl-cpB { color: rgb(153,153,119); font-style: italic; }
pre.hljl > span.hljl-cs { color: rgb(153,153,119); font-style: italic; }
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</style>
<style type="text/css">
@font-face {
font-style: normal;
font-weight: 300;
}
@font-face {
font-style: normal;
font-weight: 400;
}
@font-face {
font-style: normal;
font-weight: 600;
}
html {
font-family: sans-serif; /* 1 */
-ms-text-size-adjust: 100%; /* 2 */
-webkit-text-size-adjust: 100%; /* 2 */
}
body {
margin: 0;
}
article,
aside,
details,
figcaption,
figure,
footer,
header,
hgroup,
main,
menu,
nav,
section,
summary {
display: block;
}
audio,
canvas,
progress,
video {
display: inline-block; /* 1 */
vertical-align: baseline; /* 2 */
}
audio:not([controls]) {
display: none;
height: 0;
}
[hidden],
template {
display: none;
}
a:active,
a:hover {
outline: 0;
}
abbr[title] {
border-bottom: 1px dotted;
}
b,
strong {
font-weight: bold;
}
dfn {
font-style: italic;
}
h1 {
font-size: 2em;
margin: 0.67em 0;
}
mark {
background: #ff0;
color: #000;
}
small {
font-size: 80%;
}
sub,
sup {
font-size: 75%;
line-height: 0;
position: relative;
vertical-align: baseline;
}
sup {
top: -0.5em;
}
sub {
bottom: -0.25em;
}
img {
border: 0;
}
svg:not(:root) {
overflow: hidden;
}
figure {
margin: 1em 40px;
}
hr {
-moz-box-sizing: content-box;
box-sizing: content-box;
height: 0;
}
pre {
overflow: auto;
}
code,
kbd,
pre,
samp {
font-family: monospace, monospace;
font-size: 1em;
}
button,
input,
optgroup,
select,
textarea {
color: inherit; /* 1 */
font: inherit; /* 2 */
margin: 0; /* 3 */
}
button {
overflow: visible;
}
button,
select {
text-transform: none;
}
button,
html input[type="button"], /* 1 */
input[type="reset"],
input[type="submit"] {
-webkit-appearance: button; /* 2 */
cursor: pointer; /* 3 */
}
button[disabled],
html input[disabled] {
cursor: default;
}
button::-moz-focus-inner,
input::-moz-focus-inner {
border: 0;
padding: 0;
}
input {
line-height: normal;
}
input[type="checkbox"],
input[type="radio"] {
box-sizing: border-box; /* 1 */
padding: 0; /* 2 */
}
input[type="number"]::-webkit-inner-spin-button,
input[type="number"]::-webkit-outer-spin-button {
height: auto;
}
input[type="search"] {
-webkit-appearance: textfield; /* 1 */
-moz-box-sizing: content-box;
-webkit-box-sizing: content-box; /* 2 */
box-sizing: content-box;
}
input[type="search"]::-webkit-search-cancel-button,
input[type="search"]::-webkit-search-decoration {
-webkit-appearance: none;
}
fieldset {
border: 1px solid #c0c0c0;
margin: 0 2px;
padding: 0.35em 0.625em 0.75em;
}
legend {
border: 0; /* 1 */
padding: 0; /* 2 */
}
textarea {
overflow: auto;
}
optgroup {
font-weight: bold;
}
table {
font-family: monospace, monospace;
font-size : 0.8em;
border-collapse: collapse;
border-spacing: 0;
}
td,
th {
padding: 0;
}
thead th {
border-bottom: 1px solid black;
background-color: white;
}
tr:nth-child(odd){
background-color: rgb(248,248,248);
}
/*
* Skeleton V2.0.4
* Copyright 2014, Dave Gamache
* www.getskeleton.com
* Free to use under the MIT license.
* http://www.opensource.org/licenses/mit-license.php
* 12/29/2014
*/
.container {
position: relative;
width: 100%;
max-width: 960px;
margin: 0 auto;
padding: 0 20px;
box-sizing: border-box; }
.column,
.columns {
width: 100%;
float: left;
box-sizing: border-box; }
@media (min-width: 400px) {
.container {
width: 85%;
padding: 0; }
}
@media (min-width: 550px) {
.container {
width: 80%; }
.column,
.columns {
margin-left: 4%; }
.column:first-child,
.columns:first-child {
margin-left: 0; }
.one.column,
.one.columns { width: 4.66666666667%; }
.two.columns { width: 13.3333333333%; }
.three.columns { width: 22%; }
.four.columns { width: 30.6666666667%; }
.five.columns { width: 39.3333333333%; }
.six.columns { width: 48%; }
.seven.columns { width: 56.6666666667%; }
.eight.columns { width: 65.3333333333%; }
.nine.columns { width: 74.0%; }
.ten.columns { width: 82.6666666667%; }
.eleven.columns { width: 91.3333333333%; }
.twelve.columns { width: 100%; margin-left: 0; }
.one-third.column { width: 30.6666666667%; }
.two-thirds.column { width: 65.3333333333%; }
.one-half.column { width: 48%; }
/* Offsets */
.offset-by-one.column,
.offset-by-one.columns { margin-left: 8.66666666667%; }
.offset-by-two.column,
.offset-by-two.columns { margin-left: 17.3333333333%; }
.offset-by-three.column,
.offset-by-three.columns { margin-left: 26%; }
.offset-by-four.column,
.offset-by-four.columns { margin-left: 34.6666666667%; }
.offset-by-five.column,
.offset-by-five.columns { margin-left: 43.3333333333%; }
.offset-by-six.column,
.offset-by-six.columns { margin-left: 52%; }
.offset-by-seven.column,
.offset-by-seven.columns { margin-left: 60.6666666667%; }
.offset-by-eight.column,
.offset-by-eight.columns { margin-left: 69.3333333333%; }
.offset-by-nine.column,
.offset-by-nine.columns { margin-left: 78.0%; }
.offset-by-ten.column,
.offset-by-ten.columns { margin-left: 86.6666666667%; }
.offset-by-eleven.column,
.offset-by-eleven.columns { margin-left: 95.3333333333%; }
.offset-by-one-third.column,
.offset-by-one-third.columns { margin-left: 34.6666666667%; }
.offset-by-two-thirds.column,
.offset-by-two-thirds.columns { margin-left: 69.3333333333%; }
.offset-by-one-half.column,
.offset-by-one-half.columns { margin-left: 52%; }
}
html {
font-size: 62.5%; }
body {
font-size: 1.5em; /* currently ems cause chrome bug misinterpreting rems on body element */
line-height: 1.6;
font-weight: 400;
font-family: "Raleway", "HelveticaNeue", "Helvetica Neue", Helvetica, Arial, sans-serif;
color: #222; }
h1, h2, h3, h4, h5, h6 {
margin-top: 0;
margin-bottom: 2rem;
font-weight: 300; }
h1 { font-size: 3.6rem; line-height: 1.2; letter-spacing: -.1rem;}
h2 { font-size: 3.4rem; line-height: 1.25; letter-spacing: -.1rem; }
h3 { font-size: 3.2rem; line-height: 1.3; letter-spacing: -.1rem; }
h4 { font-size: 2.8rem; line-height: 1.35; letter-spacing: -.08rem; }
h5 { font-size: 2.4rem; line-height: 1.5; letter-spacing: -.05rem; }
h6 { font-size: 1.5rem; line-height: 1.6; letter-spacing: 0; }
p {
margin-top: 0; }
a {
color: #1EAEDB; }
a:hover {
color: #0FA0CE; }
.button,
button,
input[type="submit"],
input[type="reset"],
input[type="button"] {
display: inline-block;
height: 38px;
padding: 0 30px;
color: #555;
text-align: center;
font-size: 11px;
font-weight: 600;
line-height: 38px;
letter-spacing: .1rem;
text-transform: uppercase;
text-decoration: none;
white-space: nowrap;
background-color: transparent;
border-radius: 4px;
border: 1px solid #bbb;
cursor: pointer;
box-sizing: border-box; }
.button:hover,
button:hover,
input[type="submit"]:hover,
input[type="reset"]:hover,
input[type="button"]:hover,
.button:focus,
button:focus,
input[type="submit"]:focus,
input[type="reset"]:focus,
input[type="button"]:focus {
color: #333;
border-color: #888;
outline: 0; }
.button.button-primary,
button.button-primary,
input[type="submit"].button-primary,
input[type="reset"].button-primary,
input[type="button"].button-primary {
color: #FFF;
background-color: #33C3F0;
border-color: #33C3F0; }
.button.button-primary:hover,
button.button-primary:hover,
input[type="submit"].button-primary:hover,
input[type="reset"].button-primary:hover,
input[type="button"].button-primary:hover,
.button.button-primary:focus,
button.button-primary:focus,
input[type="submit"].button-primary:focus,
input[type="reset"].button-primary:focus,
input[type="button"].button-primary:focus {
color: #FFF;
background-color: #1EAEDB;
border-color: #1EAEDB; }
input[type="email"],
input[type="number"],
input[type="search"],
input[type="text"],
input[type="tel"],
input[type="url"],
input[type="password"],
textarea,
select {
height: 38px;
padding: 6px 10px; /* The 6px vertically centers text on FF, ignored by Webkit */
background-color: #fff;
border: 1px solid #D1D1D1;
border-radius: 4px;
box-shadow: none;
box-sizing: border-box; }
/* Removes awkward default styles on some inputs for iOS */
input[type="email"],
input[type="number"],
input[type="search"],
input[type="text"],
input[type="tel"],
input[type="url"],
input[type="password"],
textarea {
-webkit-appearance: none;
-moz-appearance: none;
appearance: none; }
textarea {
min-height: 65px;
padding-top: 6px;
padding-bottom: 6px; }
input[type="email"]:focus,
input[type="number"]:focus,
input[type="search"]:focus,
input[type="text"]:focus,
input[type="tel"]:focus,
input[type="url"]:focus,
input[type="password"]:focus,
textarea:focus,
select:focus {
border: 1px solid #33C3F0;
outline: 0; }
label,
legend {
display: block;
margin-bottom: .5rem;
font-weight: 600; }
fieldset {
padding: 0;
border-width: 0; }
input[type="checkbox"],
input[type="radio"] {
display: inline; }
label > .label-body {
display: inline-block;
margin-left: .5rem;
font-weight: normal; }
ul {
list-style: circle; }
ol {
list-style: decimal; }
ul ul,
ul ol,
ol ol,
ol ul {
margin: 1.5rem 0 1.5rem 3rem;
font-size: 90%; }
li > p {margin : 0;}
th,
td {
padding: 12px 15px;
text-align: left;
border-bottom: 1px solid #E1E1E1; }
th:first-child,
td:first-child {
padding-left: 0; }
th:last-child,
td:last-child {
padding-right: 0; }
button,
.button {
margin-bottom: 1rem; }
input,
textarea,
select,
fieldset {
margin-bottom: 1.5rem; }
pre,
blockquote,
dl,
figure,
table,
p,
ul,
ol,
form {
margin-bottom: 1.0rem; }
.u-full-width {
width: 100%;
box-sizing: border-box; }
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<h1 class="title">Lake Problem Simulation-Optimization Example</h1>
<h5>BEE 4750/5750</h5>
<h5>Fall 2022</h5>
</div>
<h1>Background</h1>
<p>Due to ongoing economic activity, a town emits phosphorous into a shallow lake (with a concentration of <span class="math">$a_t$</span>), which also receives non-point source runoff (concentration <span class="math">$y_t$</span>) from the surrounding area. The concentration of the lake at time <span class="math">$t+1$</span> is given by</p>
<p class="math">\[
X_{t+1} = X_t + a_t + y_t + \frac{X_t^q}{1+X_t^q} - bX_t,
\]</p>
<p>where:</p>
<table><tr><th>Parameter</th><th>Value</th></tr><tr><td><span class="math">$a_t$</span></td><td>point-source phosphorous concentration from the town</td></tr><tr><td><span class="math">$y_t$</span></td><td>non-point-source phosphorous concentration</td></tr><tr><td><span class="math">$q$</span></td><td>rate at which phosphorous is recycled from sediment</td></tr><tr><td><span class="math">$b$</span></td><td>rate at which phosphorous leaves the lake</td></tr></table>
<p>and <span class="math">$X_0 = 0$</span>, <span class="math">$y_t \sim LogNormal(\log(0.02), 0.2)$</span>, <span class="math">$q=2.5$</span>, and <span class="math">$b=0.4$</span>.</p>
<h1>Questions (we might not get to address all of these)</h1>
<ol>
<li><p>Try to find the best solution you can (maximizing the objective <span class="math">$1/T \sum_{t=1}^T a_t$</span>) which does not allow the lake to eutrophy (exceeding its critical threshold value) with more than 20% probability.</p>
<p>Some things to experiment with:</p>
<ul>
<li><p>How many Monte Carlo samples are you using?</p>
</li>
<li><p>What bounds are you searching over for the releases?</p>
</li>
<li><p>How many function evaluations are you using for the search?</p>
</li>
<li><p>What is the impact of different <a href="https://docs.juliahub.com/Metaheuristics/aJ70z/3.2.12/algorithms/"><code>Metaheuristics.jl</code> algorithms</a>?</p>
</li>
</ul>
</li>
<li><p>What are the tradeoffs that you notice between these choices, the performance of any found feasible solutions, and speed? What might you recommend?</p>
</li>
<li><p>How well does your solution perform against slightly different of <span class="math">$q$</span>, <span class="math">$b$</span>, and/or the variance and mean of <span class="math">$y$</span>? Is the solution reasonably "robust" (in other words, can it satisfy the constraints if your estimate of those parameters is wrong)? How might you decide this?</p>
</li>
</ol>
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Published from <a href="instructions.jmd">instructions.jmd</a>
using <a href="http://github.com/JunoLab/Weave.jl">Weave.jl</a> v0.10.9 on 2022-11-01.
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