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<h2 class="hd hd-2 unit-title">Practice Problem 1</h2>
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PART A
</h3>
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<p>Suppose that you are interested in choosing a set of investments, labeled for simplicity \( \{1, \ldots, 7 \}\).
Let \(x_i \) be a binary variable representing whether or not we choose the \(i^{th}\) investment.
</p>
<p>Model the following constraints using integer and linear constraints. (Other nonlinear constraints are not permitted.) In each part of this problem, we will have four choices, and two of the answers will be (alternative) correct answers. Select both of the correct answers. </p>
<p>You cannot invest in all of them. </p>
<p> For every "practice problem" in 15.053x, you may have an unlimited number of attempts and checks. You have the option of revealing the answer any time after the first guess.
</p>
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\(x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 + 2x_6 + 2x_7 \leq 12 \)
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<input type="checkbox" name="input_2813eb219cec4ad0b2959cdc8066f115_2_1[]" id="input_2813eb219cec4ad0b2959cdc8066f115_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="2813eb219cec4ad0b2959cdc8066f115_2_1-choice_1-label" for="input_2813eb219cec4ad0b2959cdc8066f115_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_2813eb219cec4ad0b2959cdc8066f115_2_1">
\(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \leq 7\)
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\(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 \leq 6\)
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\(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \leq 6\)
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PART B
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<p>Investment 1 cannot be chosen if investment 3 is chosen. Select two correct answers from the following four choices.
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<input type="checkbox" name="input_5c246aed277f434792e4b53ba942ddc4_2_1[]" id="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="5c246aed277f434792e4b53ba942ddc4_2_1-choice_0-label" for="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_5c246aed277f434792e4b53ba942ddc4_2_1">
\(x_1 - x_3 \leq 0 \)
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<input type="checkbox" name="input_5c246aed277f434792e4b53ba942ddc4_2_1[]" id="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="5c246aed277f434792e4b53ba942ddc4_2_1-choice_1-label" for="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5c246aed277f434792e4b53ba942ddc4_2_1">
\(x_1 + x_3 \leq 1\)
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<input type="checkbox" name="input_5c246aed277f434792e4b53ba942ddc4_2_1[]" id="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="5c246aed277f434792e4b53ba942ddc4_2_1-choice_2-label" for="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_5c246aed277f434792e4b53ba942ddc4_2_1">
\(3x_1 + 4x_3 \leq 6\)
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<input type="checkbox" name="input_5c246aed277f434792e4b53ba942ddc4_2_1[]" id="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="5c246aed277f434792e4b53ba942ddc4_2_1-choice_3-label" for="input_5c246aed277f434792e4b53ba942ddc4_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_5c246aed277f434792e4b53ba942ddc4_2_1">
\(2x_1 + x_3 \leq 3\)
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PART C
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<p>Investment 4 can be chosen only if investment 2 is also chosen. Select two correct answers from the following four choices.
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_b716c14dbbb944b28682c5e1a0592494_2_1">
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\(-x_2 + x_4 \leq 0 \)
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<input type="checkbox" name="input_b716c14dbbb944b28682c5e1a0592494_2_1[]" id="input_b716c14dbbb944b28682c5e1a0592494_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="b716c14dbbb944b28682c5e1a0592494_2_1-choice_1-label" for="input_b716c14dbbb944b28682c5e1a0592494_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_b716c14dbbb944b28682c5e1a0592494_2_1">
\(x_2 + x_4 \leq 1\)
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\(x_2 + x_4 \geq 1\)
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<input type="checkbox" name="input_b716c14dbbb944b28682c5e1a0592494_2_1[]" id="input_b716c14dbbb944b28682c5e1a0592494_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="b716c14dbbb944b28682c5e1a0592494_2_1-choice_3-label" for="input_b716c14dbbb944b28682c5e1a0592494_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_b716c14dbbb944b28682c5e1a0592494_2_1">
\(x_2 -x_4 \geq 0\)
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<h3 class="hd hd-3 problem-header" id="aa70d3279a3e4d418675865dcf0e5d60-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@aa70d3279a3e4d418675865dcf0e5d60-problem-progress" tabindex="-1">
PART D
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@aa70d3279a3e4d418675865dcf0e5d60-problem-progress"></div>
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<p>You must choose either at least one of the investments 1, 2, 3, or at least two investments from 4, 5, 6, 7 or both. Select two correct answers from the following four choices.
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_aa70d3279a3e4d418675865dcf0e5d60_2_1">
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<input type="checkbox" name="input_aa70d3279a3e4d418675865dcf0e5d60_2_1[]" id="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="aa70d3279a3e4d418675865dcf0e5d60_2_1-choice_0-label" for="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_aa70d3279a3e4d418675865dcf0e5d60_2_1">
\(x_1 + x_2 + x_3 + 2x_4 + 2x_5 + 2x_6 + 2x_7 \geq 2 \)
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\(2x_1 + 2x_2 + 2x_3 + x_4 + x_5 + x_6 + x_7 \geq 2\)
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<input type="checkbox" name="input_aa70d3279a3e4d418675865dcf0e5d60_2_1[]" id="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="aa70d3279a3e4d418675865dcf0e5d60_2_1-choice_2-label" for="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_aa70d3279a3e4d418675865dcf0e5d60_2_1">
\(2x_1 + 2x_2 + 2x_3 + 3x_4 + 3x_5 + 3x_6 + 3x_7 \geq 3\)
</label>
</div>
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<input type="checkbox" name="input_aa70d3279a3e4d418675865dcf0e5d60_2_1[]" id="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="aa70d3279a3e4d418675865dcf0e5d60_2_1-choice_3-label" for="input_aa70d3279a3e4d418675865dcf0e5d60_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_aa70d3279a3e4d418675865dcf0e5d60_2_1">
\(3x_1 + 3x_2 + 3x_3 + 2x_4 + 2x_5 + 2x_6 + 2x_7 \geq 3\)
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PART E
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<p>You must choose both of investments 1 and 2, or choose investment 4, or choose all three of these investments. Select two correct answers from the following four choices.
</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_4607a6aed80d49b090f8c234f0a6c70a_2_1">
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<input type="checkbox" name="input_4607a6aed80d49b090f8c234f0a6c70a_2_1[]" id="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="4607a6aed80d49b090f8c234f0a6c70a_2_1-choice_0-label" for="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_4607a6aed80d49b090f8c234f0a6c70a_2_1">
\( x_1 + x_2 + 2x_4 \geq 2 \)
</label>
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<input type="checkbox" name="input_4607a6aed80d49b090f8c234f0a6c70a_2_1[]" id="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="4607a6aed80d49b090f8c234f0a6c70a_2_1-choice_1-label" for="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4607a6aed80d49b090f8c234f0a6c70a_2_1">
\(x_1x_2 + x_4 \geq 1\)
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\(x_1 +x_2 + x_4 \geq 2\)
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<input type="checkbox" name="input_4607a6aed80d49b090f8c234f0a6c70a_2_1[]" id="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="4607a6aed80d49b090f8c234f0a6c70a_2_1-choice_3-label" for="input_4607a6aed80d49b090f8c234f0a6c70a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_4607a6aed80d49b090f8c234f0a6c70a_2_1">
\(3x_1 + 4x_2 + 7x_4 \geq 7\)
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<h2 class="hd hd-2 unit-title">Practice Problem 2</h2>
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PART A
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<p>Consider two non-negative integral variables \( x_1 \text{ and }x_2\). Suppose that \(x_1 \leq 60 \text{ and }x_2 \leq 50\). Model the following constraint using an additional binary variable \(w\): </p>
<p> \(x_1 \lt 30 \text{ or }x_2 \geq 20\).</p>
<p>Note that the first inequality is strict. Select the correct way of modeling this &#8220;OR&#8221; constraint using linear and integer constraints. Select the (one) correct answer.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_3acda8e70b1342139f8f278d9d868d60_2_1">
<fieldset aria-describedby="status_3acda8e70b1342139f8f278d9d868d60_2_1">
<div class="field">
<input type="radio" name="input_3acda8e70b1342139f8f278d9d868d60_2_1" id="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#D" class="field-input input-radio" value="choice_1a#D"/><label id="3acda8e70b1342139f8f278d9d868d60_2_1-choice_1a#D-label" for="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#D" class="response-label field-label label-inline" aria-describedby="status_3acda8e70b1342139f8f278d9d868d60_2_1">
\( x_1 \leq 29 - M(1-w)\\
x_2 \geq 20 +Mw \)
</label>
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<input type="radio" name="input_3acda8e70b1342139f8f278d9d868d60_2_1" id="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="3acda8e70b1342139f8f278d9d868d60_2_1-choice_1a#B-label" for="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_3acda8e70b1342139f8f278d9d868d60_2_1">
\( x_1 \leq 29 + M(1-w)\\
x_2 \geq 20 -Mw \)
</label>
</div>
<div class="field">
<input type="radio" name="input_3acda8e70b1342139f8f278d9d868d60_2_1" id="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#C" class="field-input input-radio" value="choice_1a#C"/><label id="3acda8e70b1342139f8f278d9d868d60_2_1-choice_1a#C-label" for="input_3acda8e70b1342139f8f278d9d868d60_2_1_choice_1a#C" class="response-label field-label label-inline" aria-describedby="status_3acda8e70b1342139f8f278d9d868d60_2_1">
\( x_1 \leq 30 - M(1-w) \\
x_2 \geq 20 +Mw \)
</label>
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\( x_1 \leq 30 + M(1-w) \\
x_2 \geq 20 -Mw \)
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PART B
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<p>How small can M be in the correct answer for PART A?</p>
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\(\)
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<h2 class="hd hd-2 unit-title">Practice Problem 3</h2>
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PART A
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<p>A basketball coach is trying to come up with a team for the following match. He has a total of \( 9\) players from which he must select \( 5\) . Assume these \( 5\) players will play the whole game. Consider there are three positions players can play: <i>guard</i>, <i> forward</i>, and <i> center</i>. So the team of \( 5\) players will consist of \( 2\) guards, \( 2\) forwards, and \( 1\) center. Players \( 1,2,3,4,8,9\) can play guard, \( 3,4,5,6,7,8,9\) can play forward, and \( 8,9\) can play center. </p>
<p>Consider the decision variables as \( x_i\) for \( i\in\{1,\dots,9\}\) so that \( x_i=1\) if the coach selects player \( i\) , and \( x_i=0\) otherwise and examine the IP below. Note that since there is no objective, we state the linear objective as 0. </p>
<p>The IP formulated below requires that we select five players, and that at least two of these five can play guard; at least two can play forward; at least one can play center. If we solve this integer program, and obtain a feasible integer solution, is the solution guaranteed to produce a feasible five person team? </p>
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<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation">\[\left.\begin{array}{rrcl} \min &amp; 0 \\
\text {s.t.:} &amp; &amp; &amp; \\
&amp; x_{1} + x_{2} + x_{3} + x_{4}+x_8+x_9 \geq 2 \text{ (2 Guards)}\\
&amp; x_{3} + x_{4} + x_{5} + x_{6}+x_{7}+x_{8}+x_9 \ge 2 \text{ (2 Forwards)} \\
&amp; x_{8} + x_{9} \ge 1 \text{ (1 Center)} \\
&amp; \sum_{i=1}^{9}x_{i} = 5 \text{ (5 players)} \\
&amp; x_{i} \in \{0,1\} \quad \forall i\in\{1,\dots,9\} \text{ (Binary)} \\
\end{array}\right\}\]</td>
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