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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<h3 class="hd hd-2">Introduction</h3>
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<h2 class="hd hd-2 unit-title">Introduction Exercise</h2>
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Applications of Optimization
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<legend id="fb073d7b8a184fbbb4324c1c5fd255b2_2_1-legend" class="response-fieldset-legend field-group-hd">Which of the following are applications of optimization?</legend>
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Manufacturing
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Marketing
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Finance
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Transportation
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Machine Learning
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<h2 class="hd hd-2 unit-title">Differences 1</h2>
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<h2 class="hd hd-2 unit-title">Differences 1 Exercise</h2>
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Differences 1 Exercise
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<p>Suppose that you want to partition the following set of numbers so as to minimize the difference of the sum of the two parts \(\{1, 2, 4, 8, 16, 32\}\). What is the difference of the best partition? </p>
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\(0\)
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\(1\)
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\(2\)
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\(3\)
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Larger than \(3\)
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<h2 class="hd hd-2 unit-title">Differences 2</h2>
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<h2 class="hd hd-2 unit-title">Differences 2 Exercise</h2>
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Sum of Two Partitions (15 numbers)
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<p>Now we repeat the exercise with 15 integers</p>
<p>Suppose we have \((I_1, I_2, \ldots, I_{15}) \in [200000, 1000000]\)</p>
<p>Using <a class="modal-content" href="/assets/courseware/v1/776f0c34524e596d47315110cceaf90a/asset-v1:MITx+15.053x+3T2016+type@asset+block/Differences.xlsx"> Differences.xlsx</a> (<i>Google sheets version available <a class="modal-content" href="/assets/courseware/v1/38d5bad32bdb5d585fb38c31c6963209/asset-v1:MITx+15.053x+3T2016+type@asset+block/Differences_google.xlsx"> here</a></i>), we wish to minimize the difference in sums between the two partitions. In Differences.xlsx, the optimum solution for this problem is given on the tab/sheet titled "Opt 15 Variables". Based on this sheet, answer the following question. </p>
<p>
Which of the following is the sum of the 15 integers?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
<fieldset aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
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<input type="radio" name="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1" id="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="4db7c33a188e4ffa959d7d70a0ea66ba_2_1-choice_1a#B-label" for="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
\(907,670\)
</label>
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<input type="radio" name="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1" id="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#E" class="field-input input-radio" value="choice_1a#E"/><label id="4db7c33a188e4ffa959d7d70a0ea66ba_2_1-choice_1a#E-label" for="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#E" class="response-label field-label label-inline" aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
\(679,097\)
</label>
</div>
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<input type="radio" name="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1" id="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#C" class="field-input input-radio" value="choice_1a#C"/><label id="4db7c33a188e4ffa959d7d70a0ea66ba_2_1-choice_1a#C-label" for="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#C" class="response-label field-label label-inline" aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
\(90,097,670\)
</label>
</div>
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<input type="radio" name="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1" id="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#D" class="field-input input-radio" value="choice_1a#D"/><label id="4db7c33a188e4ffa959d7d70a0ea66ba_2_1-choice_1a#D-label" for="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#D" class="response-label field-label label-inline" aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
\(6,709,097\)
</label>
</div>
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<input type="radio" name="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1" id="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="4db7c33a188e4ffa959d7d70a0ea66ba_2_1-choice_1a#A-label" for="input_4db7c33a188e4ffa959d7d70a0ea66ba_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_4db7c33a188e4ffa959d7d70a0ea66ba_2_1">
\(9,097,670\)
</label>
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<span id="answer_4db7c33a188e4ffa959d7d70a0ea66ba_2_1"/>
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Minimizing the Difference (15 numbers)
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<p>
<strong>Finding the Minimum Difference of Sums</strong>
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<p>
Suppose the optimal parts of the partition are denoted as \(S_1, S_2\).
</p>
<p>
<legend>What is \(| \sum_{j \in S_1} I_j - \sum_{j \in S_2} I_j |\)?</legend>
</p>
<p>
Note that the partition already in the sheet is not necessaily the optimal one. To find the correct answer, you will need to use Excel Solver or preferably OpenSolver. (You can skip this exercise for now if neither is installed.) If you use Solver, then select solver, click on "options" in the dialogue box, and set the value of "Integer Optimality(%)" to .0001. The default is 1, which permits the solutions provided by Solver for integer programs to be 1% from optimal.
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Sum of Two Partitions (25 numbers)
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<p>Now we repeat the exercise with 25 integers. As one would expect, this makes the problem more difficult. You will observe in the next part that solvers take more time to output the optimal partition.</p>
<p>Suppose we have \((I_1, I_2, \ldots, I_{15}) \in [200000, 1000000]\)</p>
<p>Using "Opt 25 Var" in <a class="modal-content" href="/assets/courseware/v1/776f0c34524e596d47315110cceaf90a/asset-v1:MITx+15.053x+3T2016+type@asset+block/Differences.xlsx"> Differences.xlsx</a> (<i>Google sheets version available <a class="modal-content" href="/assets/courseware/v1/38d5bad32bdb5d585fb38c31c6963209/asset-v1:MITx+15.053x+3T2016+type@asset+block/Differences_google.xlsx" target="_blank"> here</a></i>), we wish to minimize the difference in sums between the two partitions</p>
<p>Which of the following is the sum of the 25 integers?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_8a39b44a044a4f76bf905f7885e0793b_2_1">
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<input type="radio" name="input_8a39b44a044a4f76bf905f7885e0793b_2_1" id="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#B" class="field-input input-radio" value="choice_1a#B"/><label id="8a39b44a044a4f76bf905f7885e0793b_2_1-choice_1a#B-label" for="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#B" class="response-label field-label label-inline" aria-describedby="status_8a39b44a044a4f76bf905f7885e0793b_2_1">
\(6,257,235\)
</label>
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<input type="radio" name="input_8a39b44a044a4f76bf905f7885e0793b_2_1" id="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#E" class="field-input input-radio" value="choice_1a#E"/><label id="8a39b44a044a4f76bf905f7885e0793b_2_1-choice_1a#E-label" for="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#E" class="response-label field-label label-inline" aria-describedby="status_8a39b44a044a4f76bf905f7885e0793b_2_1">
\(257,235\)
</label>
</div>
<div class="field">
<input type="radio" name="input_8a39b44a044a4f76bf905f7885e0793b_2_1" id="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#C" class="field-input input-radio" value="choice_1a#C"/><label id="8a39b44a044a4f76bf905f7885e0793b_2_1-choice_1a#C-label" for="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#C" class="response-label field-label label-inline" aria-describedby="status_8a39b44a044a4f76bf905f7885e0793b_2_1">
\(9,097,670\)
</label>
</div>
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<input type="radio" name="input_8a39b44a044a4f76bf905f7885e0793b_2_1" id="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#D" class="field-input input-radio" value="choice_1a#D"/><label id="8a39b44a044a4f76bf905f7885e0793b_2_1-choice_1a#D-label" for="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#D" class="response-label field-label label-inline" aria-describedby="status_8a39b44a044a4f76bf905f7885e0793b_2_1">
\(160,257,235\)
</label>
</div>
<div class="field">
<input type="radio" name="input_8a39b44a044a4f76bf905f7885e0793b_2_1" id="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#A" class="field-input input-radio" value="choice_1a#A"/><label id="8a39b44a044a4f76bf905f7885e0793b_2_1-choice_1a#A-label" for="input_8a39b44a044a4f76bf905f7885e0793b_2_1_choice_1a#A" class="response-label field-label label-inline" aria-describedby="status_8a39b44a044a4f76bf905f7885e0793b_2_1">
\(16,257,235\)
</label>
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Minimizing the Difference (25 numbers)
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<strong>Finding the Minimum Difference of Sums</strong>
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<p>
Suppose the optimal parts of the partition are denoted as \(S_1, S_2\).
</p>
<p>
<legend>What is \(| \sum_{j \in S_1} I_j - \sum_{j \in S_2} I_j |\)?</legend>
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Note that the partition already in the sheet is not necessaily the optimal one. To find the correct answer, you will need to use Excel Solver or preferably OpenSolver. (You can skip this exercise if neither is installed.)
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<h2 class="hd hd-2 unit-title">Tom and Pie Eating Exercise 1</h2>
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Time Constraint
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<p>Suppose Tom has to clean up after himself, and must do so within the same time period.</p>
<p>It takes Tom one minute to clean for each torte plate and two minutes to clean after each pie plate.</p>
<p>What is the new constraint on time?</p>
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<h2 class="hd hd-2 unit-title">Tom and Pie Eating Exercise 2</h2>
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<p>In a linear program, there can be more than one optimal solution.</p>
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<p>In a linear program, the optimum value must be bounded.</p>
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<h2 class="hd hd-2 unit-title">Linear Programs Exercise</h2>
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Linear Programs
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<p>Which of the following is a linear program? (select all that apply)</p>
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<td class="equation">\[\left.\begin{array}{rrcl} \max &amp; y \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; y \ge 10 \\ \end{array}\right\}\]</td>
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<td class="equation">\[\left.\begin{array}{rrcl} \max &amp; 2 + 2y^2 \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; xy=5 \\ \end{array}\right\}\]</td>
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<h2 class="hd hd-2 unit-title">Optimization Paradigm</h2>
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<h2 class="hd hd-2 unit-title">Optimization Paradigm Exercise</h2>
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Nonlinear Optimization
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<p>Which of the following is a non-linear program? (select all that apply)</p>
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<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; y \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; y \ge 10 \\ \end{array}\right\}\]</p>
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<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; 2 + 2y^2 \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; xy=5 \\ \end{array}\right\}\]</p>
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<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; 3+sin(2y) \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; y \leq \pi \\ \end{array}\right\}\]</p>
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<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; log(5x)+y \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; \vert y \vert \leq 10 \\ \end{array}\right\}\]</p>
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<p class="equation">\[\left.\begin{array}{rrcl} \max &amp; 5x \\ \text {s.t.:} &amp; &amp; &amp; \\ &amp; x \leq 10 \\ &amp; x \geq 10 \end{array}\right\}\]</p>
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<h2 class="hd hd-2 unit-title">DTC Problem I</h2>
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<h2 class="hd hd-2 unit-title">DTC Problem I Exercise</h2>
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DTC Problem I Exercise
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<p>Suppose the profit for stone shields included a cost of manufacturing. Suppose further that the cost of manufacturing stone shields decreased by 1 shekel. What is the (revised) objective function?</p>
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MAX \(3K + 6S\)
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MAX \( 6K + 3S\)
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MAX \( 6K + 6S\)
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MAX \( 3K + 3S\)
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<h2 class="hd hd-2 unit-title">DTC Problem II</h2>
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<h2 class="hd hd-2 unit-title">DTC Problem II Exercise</h2>
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Minimum Resources
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<p>Suppose we have a constraint that creates a lower bound \(q_i\) on the amount of resource \(i\) used. The other notation is given next.</p>
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<li>\(S\): Set of products</li>
<li>\(R\): Set of resources</li>
<li>\(r_i\): Amount of resource \(i\) available</li>
<li>\(x_j\): Amount of product \(j\) for \(j \in S\)</li>
<li>\(a_{ij}\): Amount of resource \(i\) used to make 1 unit of product \(j\)</li>
<li>\(p_j\): Profit of one unit of \(j\)</li>
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<p>Which of the following represents the constraint?</p>
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\(\sum_{j \in S} a_{ij}x_j \geq r_i\)
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\(\sum_{j \in R} a_{ij}x_j \geq q_i\)
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\(\sum_{j \in S} a_{ij}x_j \leq q_i\)
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\(\sum_{j \in S} a_{ij}x_j \leq r_i\)
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\(\sum_{j \in S} a_{ij}x_j \geq q_i\)
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<h2 class="hd hd-2 unit-title">Diet Problem Exercise</h2>
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Cost of Eating
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<p>Suppose we wish to minimize the amount of calories consumed in one week.</p>
<p>Suppose further that we wish to spend <b>no more than</b> \(D\) dollars on eating in one week.</p>
<p>
<ul>
<li>\(n\): number of different foods</li>
<li>\(z\): Total number of calories consumed in one week</li>
<li>\(A\): Number of dollars spent in one week</li>
<li>\(c_j\): Number of calories in one unit of food \(j\)</li>
<li>\(x_j\): Number of units of food \(j\) consumed in the week</li>
<li>\(p_{j}\): Cost of one unit of food \(j\) in dollars</li>
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<p>Which of the following represents the constraint on cost?</p>
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\( \sum_{j=1} ^n p_j x_j \geq D\)
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\( \sum_{j=1} ^n x_j \leq p_j\)
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\( \sum_{j=1} ^n p_j x_j = D\)
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\( \sum_{j=1} ^n x_j \geq p_j\)
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\( \sum_{j=1} ^n p_j x_j \leq D\)
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Calories as an Objective
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<p>Which of the following represents the objective on calories?</p>
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MAX \(z = \sum_{j=1} ^n c_j x_j\)
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MIN \( z = \sum_{j=1} ^n c_j p_j\)
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MAX \(z = \sum_{j=1} ^n p_j x_j\)
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<h2 class="hd hd-2 unit-title">Meet the Characters</h2>
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<p><i>Slides available <a class="modal-content" href="/assets/courseware/v1/5c47c9c99311b404a8c2c45f995cdef2/asset-v1:MITx+15.053x+3T2016+type@asset+block/1K_Meet_the_characters.pdf"> here</a></i> </p>
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<h2 class="hd hd-2 unit-title">LP Reformulation Tricks</h2>
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