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<h2 class="hd hd-2 unit-title">Problem 1</h2>
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PART A
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<p> Consider the NLP formulation below
</p>
<ul>
<li>Decision variables: \(x_1, x_2\) </li>
<li>Objective value: \(z\) </li>
</ul>
<p>Formulation:</p>
<span>
<p class="equation">\[\left.\begin{array}{rrcl} \min &amp; 100(x_2-(\frac{1}{2}+x_1)^2)^2 + (1-x_1)^2 \\
\text {s.t.:} &amp; &amp; &amp; \\
\text{(1)} &amp; x_1 \geq 0\\
\text{(2)} &amp; x_2 \geq 0 \end{array}\right\}\]</p>
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<p>What is the (rounded) optimal solution?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8acad6d8eae742698c95208ce81734ff_2_1">
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\( x_1 = 1, x_2 = 0 \\ z= 506.25
\)
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<input type="checkbox" name="input_8acad6d8eae742698c95208ce81734ff_2_1[]" id="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="8acad6d8eae742698c95208ce81734ff_2_1-choice_1-label" for="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_8acad6d8eae742698c95208ce81734ff_2_1">
\( x_1 = 0, x_2 = 1 \\ z= 57.25\)
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<input type="checkbox" name="input_8acad6d8eae742698c95208ce81734ff_2_1[]" id="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="8acad6d8eae742698c95208ce81734ff_2_1-choice_2-label" for="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_8acad6d8eae742698c95208ce81734ff_2_1">
\( x_1 = 2.25, x_2 = 1\\ z= 4308.203
\)
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<input type="checkbox" name="input_8acad6d8eae742698c95208ce81734ff_2_1[]" id="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="8acad6d8eae742698c95208ce81734ff_2_1-choice_3-label" for="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_8acad6d8eae742698c95208ce81734ff_2_1">
\( x_1 = 1, x_2 = 1\\ z= 156.25\)
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<input type="checkbox" name="input_8acad6d8eae742698c95208ce81734ff_2_1[]" id="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="8acad6d8eae742698c95208ce81734ff_2_1-choice_4-label" for="input_8acad6d8eae742698c95208ce81734ff_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_8acad6d8eae742698c95208ce81734ff_2_1">
\( x_1 = 1, x_2 = 2.25\\ z= 0\)
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PART B
</h3>
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<div>
<p> Consider the NLP formulation below
</p>
<ul>
<li>Decision variables: \(x_1, x_2, x_3 \)</li>
<li>Objective value: \(z\) </li>
</ul>
<p>Formulation:</p>
<span>
<p class="equation">\[\left.\begin{array}{rrcl} \min &amp; (2x_1 + 3x_2 + 4x_3)^2 \\
\text {s.t.:} &amp; &amp; &amp; \\
\text{(1)} &amp; 10x_1 + 11x_2 + 12x_3 \leq 60\\
\text{(2)} &amp; 21x_1 + 22x_2 + 23x_3 \leq 150\\
\text{(3)} &amp; 21x_1 + 22x_2 -23x_3 \geq 110\\
\text{(4)} &amp; 19x_1 + 34x_2 -32x_3 = 180 \\
\text{(5)} &amp; x_1,x_2,x_3 \geq 0
\end{array}\right\}\]</p>
</span>
<p>What is the (rounded) optimal solution? </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_17318a7ccc0c4332932d88bb2916f11f_2_1">
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\( x_1 = 0, x_2 = 5.371, x_3=0.079 \\ z= 269.912\)
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<input type="checkbox" name="input_17318a7ccc0c4332932d88bb2916f11f_2_1[]" id="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="17318a7ccc0c4332932d88bb2916f11f_2_1-choice_1-label" for="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_17318a7ccc0c4332932d88bb2916f11f_2_1">
\( x_1 = 0, x_2 = 0, x_3 = 0 \\ z= 0\)
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<input type="checkbox" name="input_17318a7ccc0c4332932d88bb2916f11f_2_1[]" id="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="17318a7ccc0c4332932d88bb2916f11f_2_1-choice_2-label" for="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_17318a7ccc0c4332932d88bb2916f11f_2_1">
\( x_1 = 5.294, x_2 = 0, x_3 = 0 \\ z= 112.106\)
</label>
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<input type="checkbox" name="input_17318a7ccc0c4332932d88bb2916f11f_2_1[]" id="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="17318a7ccc0c4332932d88bb2916f11f_2_1-choice_3-label" for="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_17318a7ccc0c4332932d88bb2916f11f_2_1">
\( x_1 = 0, x_2 = 5.294, x_3 = 0 \\ z= 252.249\)
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<input type="checkbox" name="input_17318a7ccc0c4332932d88bb2916f11f_2_1[]" id="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="17318a7ccc0c4332932d88bb2916f11f_2_1-choice_4-label" for="input_17318a7ccc0c4332932d88bb2916f11f_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_17318a7ccc0c4332932d88bb2916f11f_2_1">
\( x_1 = -\infty, x_2 = 0, x_3 = 0 \\ z= -\infty \)
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PART C
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<div>
<p> Consider the maximization of the same NLP formulation
</p>
<ul>
<li>Decision variables: \(x_1, x_2, x_3\)</li>
<li>Objective value: \(z\) </li>
</ul>
<p>Formulation:</p>
<span>
<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; (2x_1 + 3x_2 + 4x_3)^2 \\
\text {s.t.:} &amp; &amp; &amp; \\
\text{(1)} &amp; 10x_1 + 11x_2 + 12x_3 \leq 60\\
\text{(2)} &amp; 21x_1 + 22x_2 + 23x_3 \leq 150\\
\text{(3)} &amp; 21x_1 + 22x_2 -23x_3 \geq 110\\
\text{(4)} &amp; 19x_1 + 34x_2 -32x_3 = 180 \\
\text{(5)} &amp; x_1,x_2,x_3 \geq 0
\end{array}\right\}\]</p>
</span>
<p>What is the rounded optimal solution?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_4fbe947170e743cea3206a8913eb96bd_2_1">
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<input type="checkbox" name="input_4fbe947170e743cea3206a8913eb96bd_2_1[]" id="input_4fbe947170e743cea3206a8913eb96bd_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="4fbe947170e743cea3206a8913eb96bd_2_1-choice_0-label" for="input_4fbe947170e743cea3206a8913eb96bd_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_4fbe947170e743cea3206a8913eb96bd_2_1">
\( x_1 = 0, x_2 = 5.368, x_3=0.079 \\ z= 269.651\)
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<input type="checkbox" name="input_4fbe947170e743cea3206a8913eb96bd_2_1[]" id="input_4fbe947170e743cea3206a8913eb96bd_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="4fbe947170e743cea3206a8913eb96bd_2_1-choice_1-label" for="input_4fbe947170e743cea3206a8913eb96bd_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4fbe947170e743cea3206a8913eb96bd_2_1">
\( x_1 = 0.079, x_2 = 0, x_3 = 0 \\ z= 0.025\)
</label>
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\( x_1 = 0, x_2 = 5.368, x_3=0 \\ z= 259.339\)
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\( x_1 = 0, x_2 = 5.294, x_3 = -\infty \\ z= \infty \)
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\( x_1 = -\infty, x_2 = 0, x_3 = 0 \\ z= \infty\)
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<h2 class="hd hd-2 unit-title">Problem 2</h2>
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Problem 2
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<p> Consider the Hock-Schittkowski problem, which is defined as follows:
</p>
<ul>
<li>Decision variables: \(x_1, x_2, x_3, x_4 \)</li>
<li>Objective value: \(z\) </li>
</ul>
<p>Formulation:</p>
<span>
<p class="equation">\[\left.\begin{array}{rrcl} \min &amp; z = x_1x_4(x_1 + x_2 + x_3) + x_3 \\
\text {s.t.:} &amp; &amp; &amp; \\
\text{(1)} &amp; x_1x_2x_3x_4 \geq 25\\
\text{(2)} &amp; x^2_1 + x^2_2 +x^2_3 +x^2_4 = 40\\
\text{(3)} &amp; 1 \leq x_1,x_2,x_3,x_4 \leq 5
\end{array}\right\}\]</p>
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<p>What is the (rounded) optimal solution? </p>
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\( x_1 = 1.00, x_2 = 4.74, x_3 = 0, x_4 = 1.00 \\ z= 5.74\)
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\( x_1 = 1.00, x_2 = 0, x_3 = 0, x_4 = 1.38 \\ z= 1.38\)
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\( x_1 = 0, x_2 = 0, x_3 = 3.82, x_4 = 1.38 = 0 \\ z= 3.82
\)
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\( x_1 = 1.00, x_2 = 3.82, x_3 = 4.74, x_4 = 1.38 \\ z= 17.93\)
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\( x_1 = 1.00, x_2 = 4.74, x_3 = 3.82, x_4 = 1.38 \\ z= 17.01\)
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<p id="footnote-1">[1] W. Hock, K. Schittkowski, Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, Springer, No, 187, 1981.</p>
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<h2 class="hd hd-2 unit-title">Problem 3</h2>
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<h3 class="hd hd-3 problem-header" id="1ededb1e70ba4db39bf3913400122a5c-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@1ededb1e70ba4db39bf3913400122a5c-problem-progress" tabindex="-1">
PART A
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<p>In the facility location problem, the goal is to optimally place facilities so as to minimize transportation costs from the facilities to the customers. In practice, one rarely knows the demand of customers with a high degree of accuracy. In addition, the transportation costs themselves may vary with time and are subject to uncertainty. In this version of the facility location problem, we assume that all data is known.</p>
<p>In the first part of this problem, the goal is to locate one or more facilities out of five possible sites, which we designated as \(F_1, F_2, F_3, F_4, \) and \(F_5\). The cost of selecting a facility is $40. Coincidentally, there are also five customers that need to be serviced. We refer to them as \(C_1, C_2, C_3, C_4 \text{ and } C_5\). The delivery cost from each possible facility site to each of the five customers is known. The cost of satisfying (all of) the demand of Customer \(C_j\) from site \(F_i \text{ is } d_{ij}\). (Note that the first index is for the site and the second is for the customer.) The delivery cost from all possible sites to all customers are given in Table 1 below.</p>
<div id="tb:1" class="table">
<center>
<table cellspacing="5" class="tabular">
<tr>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="text-align:right">
<p>
Customer \( C_1 \)</p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="text-align:right">
<p>
Customer \( C_2 \) </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="text-align:right">
<p>
Customer \( C_3 \) </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="text-align:right">
<p>
Customer \( C_4 \) </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="text-align:right">
<p>
Customer \( C_5 \) </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
<tr>
<td style="border-top-style:solid; text-align:right; border-top-color:black; border-top-width:1px; border-right:1px solid black">
<p>
Site \( F_1 \) </p>
</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
30 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
15 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
59 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
78 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
27 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
<tr>
<td style="border-top-style:solid; text-align:right; border-top-color:black; border-top-width:1px; border-right:1px solid black">
<p>
Site \( F_2 \) </p>
</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
50 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
42 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
25 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
30 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
53 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
<tr>
<td style="border-top-style:solid; text-align:right; border-top-color:black; border-top-width:1px; border-right:1px solid black">
<p>
Site \( F_3 \) </p>
</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
64 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
14 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
30 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
20 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
62 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
<tr>
<td style="border-top-style:solid; text-align:right; border-top-color:black; border-top-width:1px; border-right:1px solid black">
<p>
Site \( F_4 \) </p>
</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
46 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
19 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
66 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
48 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
11 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
<tr>
<td style="border-top-style:solid; text-align:right; border-top-color:black; border-top-width:1px; border-right:1px solid black">
<p>
Site \( F_5 \) </p>
</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
19 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
40 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
60 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
31 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
<td style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:right">
<p>
27 </p>
</td>
<td style="text-align:right; border-right:1px solid black">&#160;</td>
</tr>
</table>
<div class="caption">
<b>Table 1: Delivery Cost from All Possible Sites to All Customers</b>
<span/>
</div>
<label/>
</center>
</div>
<p>The decision variables for this problem are as follows. We let \(y_j = 1 \)if a facility is located at site \(F_j\). Otherwise, \(y_j = 0\). We let \( 0 \leq x_{ij} \leq 1\) represent the fraction of demand from \(C_j \)satisfied by facility \(F_i \). The cost associated with customer \(C_1\) would be </p>
<p>\(d_{11} x_{11} + d_{21} x_{21} + d_{31} x_{31} + d_{41} x_{41} + d_{51} x_{51}\)</p>
<p>
One would also need to add constraints that ensures that \(x_{ij} = 0, \forall j = 1, \ldots, 5\) whenever \(y_i = 0\), which means we cannot use site \(F_i\) to serve any customer \(C_j\) if we don&#8217;t locate a facility at site i; (2) all the customers&#8217; demand need to be satisfied.
</p>
<p>Formulate this problem as a deterministic optimization problem in which the objective is to minimize the expected total cost which is the sum of the facility opening cost and the expected delivery cost. </p>
<p>Choose the correct objection function from below.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_1ededb1e70ba4db39bf3913400122a5c_2_1">
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<input type="radio" name="input_1ededb1e70ba4db39bf3913400122a5c_2_1" id="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="1ededb1e70ba4db39bf3913400122a5c_2_1-choice_0-label" for="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_1ededb1e70ba4db39bf3913400122a5c_2_1">
\(\text{ MIN } \sum_{i=1}^5 40 y_i + \sum_{i=1}^5 \sum_{j=1}^5 x_{ij} \)
</label>
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<input type="radio" name="input_1ededb1e70ba4db39bf3913400122a5c_2_1" id="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="1ededb1e70ba4db39bf3913400122a5c_2_1-choice_1-label" for="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_1ededb1e70ba4db39bf3913400122a5c_2_1">
\(\text{ MAX } \sum_{i=1}^5 40 y_i - \sum_{i=1}^5 \sum_{j=1}^5 x_{ij} \)
</label>
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<input type="radio" name="input_1ededb1e70ba4db39bf3913400122a5c_2_1" id="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="1ededb1e70ba4db39bf3913400122a5c_2_1-choice_2-label" for="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_1ededb1e70ba4db39bf3913400122a5c_2_1">
\(\text{ MAX } \sum_{i=1}^5 40 y_i + \sum_{i=1}^5 \sum_{j=1}^5 d_{ij} x_{ij} \)
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<input type="radio" name="input_1ededb1e70ba4db39bf3913400122a5c_2_1" id="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="1ededb1e70ba4db39bf3913400122a5c_2_1-choice_3-label" for="input_1ededb1e70ba4db39bf3913400122a5c_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_1ededb1e70ba4db39bf3913400122a5c_2_1">
\(\text{ MIN } \sum_{i=1}^5 40 y_i + \sum_{i=1}^5 \sum_{j=1}^5 d_{ij} x_{ij} \)
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<h3 class="hd hd-3 problem-header" id="5d1c7e7f1d1946d8a92fa5c77a031399-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@5d1c7e7f1d1946d8a92fa5c77a031399-problem-progress" tabindex="-1">
PART B
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<p>Choose the correct constraint(s) from below:</p>
<p>
<i>Assume that a facility \(i\) is either used to serve a customer \(j\) or it is not, rather than a fractional amount. </i>
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_5d1c7e7f1d1946d8a92fa5c77a031399_2_1">
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\(\sum_{i=1}^5 y_{i}=1, \forall j=1, \ldots, 5\)
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\(\sum_{i=1}^5 x_{ij}=1, \forall j=1, \ldots, 5\)
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<input type="checkbox" name="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1[]" id="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="5d1c7e7f1d1946d8a92fa5c77a031399_2_1-choice_2-label" for="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_5d1c7e7f1d1946d8a92fa5c77a031399_2_1">
\(\sum_{i=1}^5 \sum_{j=1}^5 x_{ij}=1, \forall i=1, \ldots, 5, \forall j=1, \ldots, 5\)
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<input type="checkbox" name="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1[]" id="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="5d1c7e7f1d1946d8a92fa5c77a031399_2_1-choice_3-label" for="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_5d1c7e7f1d1946d8a92fa5c77a031399_2_1">
\(x_{ij} \leq y_i, \forall i=1,\ldots,5, \forall j=1,\ldots,5\)
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<input type="checkbox" name="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1[]" id="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="5d1c7e7f1d1946d8a92fa5c77a031399_2_1-choice_4-label" for="input_5d1c7e7f1d1946d8a92fa5c77a031399_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_5d1c7e7f1d1946d8a92fa5c77a031399_2_1">
\(x_{ij} \geq y_i, \forall i=1,\ldots,5, \forall j=1,\ldots,5\)
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\( x_{ij} \in \{0,1\}, \forall i=1,\ldots,5, \forall j=1,\ldots,5\)
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\(x_{ij} \geq 1, \forall i=1,\ldots, 5, \forall j=1,\ldots,5\)
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\( y_i \in \{0,1\}, \forall i=1,\ldots,5\)
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\(y_{i} \geq 0, \forall i=1,\ldots,5\)
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PART C
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<p>Solve the deterministic facility location you formulated above using Julia/JuMP. Which facilities are open? Select both of them.</p>
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1
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2
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3
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4
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5
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PART D
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PART E
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<p>In the second part of the problem, we will consider the stochastic version. The delivery cost of satisfying customers from the facilities is uncertain. It depends on which future scenario occurs. There are five possible scenarios. Below is the probability of each scenario: \( p_1, p_2,&#8230;, p_5 \).
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Scenario 1</p>
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Scenario 2</p>
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Scenario 3</p>
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Scenario 4</p>
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Scenario 5</p>
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Probability</p>
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0.15 </p>
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0.15 </p>
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0.25</p>
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0.2</p>
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0.25</p>
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<p>We will model this as a 2-stage stochastic optimization problem. In the first stage, the decision is which of the five facilities to select. Similarly, we let \(y_j = 1 \) if a facility is located at site \( F_i \). Otherwise, \(
y_j= 0. \) The cost of opening up a facility is $40.</p>
<p>After the facilities are selected, the decision maker learns which of the five scenarios occurs. We let the scenarios be designated as \( S_1, S_2, S_3, S_4 \text{ and } S_5\). We let \(d_{ij}^k \) denote the cost of satisfying demand of customer \(C_j \) using facility \(F_i \) under scenario \(S_k\). Note that the first subscript corresponds to the facility and the second subscript corresponds to the customer. The data of \(d^k_{ij}\) is stored in <a class="modal-content" href="/assets/courseware/v1/7df1df661b2d8f5b6bd12b9d3906616f/asset-v1:MITx+15.053x+3T2016+type@asset+block/pset6_p1_data.xlsx"> pset6_p1_data.xlsx</a></p>
<p>We let \(x_{ij}^k=1\) if customer \(C_j \)is served by facility \(F_i \) under scenario \(S_k\). Otherwise, it is 0. </p>
<p>Model the problem of minimizing the following sum: the cost of opening facilities plus the expected cost of satisfying demand for customers from the open facilities</p>
<p>Choose the correct objection function from below.</p>
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\(\text{ MIN } \sum_{i=1}^5 40 y_i + \sum_{k=1}^5 p_k \sum_{i=1}^5 \sum_{j=1}^5 d_{ij}^k x_{ij}^k \)
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PART F
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<p>Choose all necessary constraint(s) from below.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
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\(\sum_{i=1}^5 x_{ij}^k=1, \forall j=1,\ldots,5, \forall k=1,\ldots,5\)
</label>
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<input type="checkbox" name="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1[]" id="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="48c86eb530fd4e7d9e63f97a16eb499e_2_1-choice_1-label" for="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
\(\sum_{j=1}^5 x_{ij}^k=1, \forall i=1,\ldots,5, \forall k=1,\ldots,5\)
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<input type="checkbox" name="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1[]" id="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="48c86eb530fd4e7d9e63f97a16eb499e_2_1-choice_2-label" for="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
\(x_{ij}^k \leq y_i, \forall i=1,\ldots,5, \forall j=1,\ldots,5, \forall k=1,\ldots,5\)
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<input type="checkbox" name="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1[]" id="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="48c86eb530fd4e7d9e63f97a16eb499e_2_1-choice_3-label" for="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
\(x_{ij}^k \geq y_i, \forall i=1,\ldots,5, \forall j=1,\ldots,5, \forall k=1,\ldots,5\)
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<input type="checkbox" name="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1[]" id="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="48c86eb530fd4e7d9e63f97a16eb499e_2_1-choice_4-label" for="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
\( x^k_{ij} \in \{0,1\}, \forall i=1,\ldots,5, \forall j=1,\ldots,5\)
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<input type="checkbox" name="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1[]" id="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="48c86eb530fd4e7d9e63f97a16eb499e_2_1-choice_5-label" for="input_48c86eb530fd4e7d9e63f97a16eb499e_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_48c86eb530fd4e7d9e63f97a16eb499e_2_1">
\(x^k_{ij} \geq 1, \forall i=1,\ldots,5, \forall j=1,\ldots,5\)
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\( y_j \in \{0,1\}, \forall j=1,\ldots,5\)
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data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@031a78a01d3c44c0b7d5c2d073798bb9" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@031a78a01d3c44c0b7d5c2d073798bb9/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="031a78a01d3c44c0b7d5c2d073798bb9-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@031a78a01d3c44c0b7d5c2d073798bb9-problem-progress" tabindex="-1">
PART G
</h3>
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<p>Solve the stochastic facility location problem you formulated above using Julia/JuMP. Complete <a class="modal-content" href="/assets/courseware/v1/7cd3614805df7ac68ec46abdfc4c291d/asset-v1:MITx+15.053x+3T2016+type@asset+block/PS6_p1_partfg.jl"> PS6_p1_partfg.jl</a> to solve the problem. We already provide data in it.</p>
<p>Which facilities will be opened under the optimal solution?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
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<input type="checkbox" name="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1[]" id="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="031a78a01d3c44c0b7d5c2d073798bb9_2_1-choice_0-label" for="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
1
</label>
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<input type="checkbox" name="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1[]" id="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="031a78a01d3c44c0b7d5c2d073798bb9_2_1-choice_1-label" for="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
2
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<div class="field">
<input type="checkbox" name="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1[]" id="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="031a78a01d3c44c0b7d5c2d073798bb9_2_1-choice_2-label" for="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
3
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<input type="checkbox" name="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1[]" id="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="031a78a01d3c44c0b7d5c2d073798bb9_2_1-choice_3-label" for="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
4
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<input type="checkbox" name="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1[]" id="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="031a78a01d3c44c0b7d5c2d073798bb9_2_1-choice_4-label" for="input_031a78a01d3c44c0b7d5c2d073798bb9_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_031a78a01d3c44c0b7d5c2d073798bb9_2_1">
5
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data-problem-id="block-v1:MITx+15.053x+3T2016+type@problem+block@5090de5e3e9a4c7fa9e582ea7693eb75" data-url="/courses/course-v1:MITx+15.053x+3T2016/xblock/block-v1:MITx+15.053x+3T2016+type@problem+block@5090de5e3e9a4c7fa9e582ea7693eb75/handler/xmodule_handler"
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PART H
</h3>
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<h2 class="hd hd-2 unit-title">Problem 4</h2>
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PART A
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<div><p> For each part, determine whether the function is convex or not over its domain.</p><p><a class="modal-content" href="/assets/courseware/v1/fef336272ea5044ee0ed873fafc9f940/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex8.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/fef336272ea5044ee0ed873fafc9f940/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex8.png"/></a></p>
\(f(x) = \frac{1}{x} \text{ for } x \gt 0\)
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Not Convex
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Convex
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PART B
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<div><p><a class="modal-content" href="/assets/courseware/v1/c16da83227701cbaf57c9ed1a43372db/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex9.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/c16da83227701cbaf57c9ed1a43372db/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex9.png"/></a></p>
\(f(x) = max \{x-3, -3x+5\}\)
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Convex
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PART C
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<div><p><a class="modal-content" href="/assets/courseware/v1/e57c682c27b0b4e17fccd65dff078e66/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex10.png"><img width="600" height="439" alt="Plot1" src="/assets/courseware/v1/e57c682c27b0b4e17fccd65dff078e66/asset-v1:MITx+15.053x+3T2016+type@asset+block/convex10.png"/></a></p>
\(f(x) = min \{x-3, -3x+5\}\)
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Not Convex
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Convex
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PART D
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\(f(x) = \lfloor x \rfloor, \text{ which is the largest integer not greater than x } \)
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Not Convex
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Convex
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PART E
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\(f(x) = sin(x)\)
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Not Convex
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Convex
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PART F
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\( f(x) = sin(x), \pi \le x \le 2 \pi\)
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Not Convex
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Convex
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