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<p>Consider three binary variables \(w_1,w_2, w_3\), on
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<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">
\(x_1 = 13w_1 + 39w_2 + 88w_3\)
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<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">
\(w_1 + w_2 + w_3 = 1\)
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<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">
\(w_1, w_2, w_3 \in \{0,1\}\)
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<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>
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<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">
\(w_1 + w_2 + w_3 \geq 1\)
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<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">
\(w_1, w_2, w_3 \geq 0\)
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<span id="answer_5e4bfbda65dc442693c00cf90e5bfefd_2_1"/>
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<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
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<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>
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<h2 class="hd hd-2 unit-title">Divisibility Constraints</h2>
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<h2 class="hd hd-2 unit-title">Divisibility Constraints Exercise</h2>
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Divisibility Constraints Exercise
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<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>
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\( y_2 \ge 0.\)
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\( y_2 \ge 1.\)
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\( y_2 \le 7.\)
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\( y_2 \le 8.\)
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<h2 class="hd hd-2 unit-title">Fixed Charge Problems</h2>
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<h2 class="hd hd-2 unit-title">Fixed Charge Problems Exercise</h2>
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Fixed Charge Problems Exercise
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<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">
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MAX \(y_1 + y_2\)
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\(y_1 = -10w_1 + 3S \\ 0 \leq S \leq 30 w_1\)
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<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">
\(y_2 = -5w_2 + 5K \\ 0 \leq K \leq 40 w_2\)
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MIN \(y_1 - y_2\)
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\(y_1 = -10w_1 + 3S \\ 0 \leq S \leq w_1\)
</label>
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<h2 class="hd hd-2 unit-title">Piecewise Linear Costs Exercise</h2>
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<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>
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<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">
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<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">
\(x=x_1+x_2\)
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<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">
\(y_1=\frac{2x_1}{3}\)
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<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">
\(y_2=2w_2\)
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<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">
\(y=y_1+y_2\)
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<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">
\(3w_2 \leq x_2 \leq 7w_2\)
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<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">
\(w_1, w_2 \in \{0,1 \}\)
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<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">
\(y_2 \geq 2w_2\)
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<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">
\(y \geq y_1+y_2\)
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<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">
\(0 \leq x_2 \leq 7w_2\)
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<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">
\(w_1+w_2 \geq 1 \)
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<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">
\(0 \leq x_1 \leq 7w_1\)
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<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>
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<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">
\(y_2 \geq \frac{2w_1}{3} + 2w_2\)
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<h2 class="hd hd-2 unit-title">Set Covering 1 Exercise</h2>
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Set Covering 1 Exercise
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<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>
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<h2 class="hd hd-2 unit-title">Set Covering 2 Exercise</h2>
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Set Covering 2 Exercise
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<p>Which of the following was <b> not </b> listed in the previous video as an application of the set cover problem?</p>
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Locating hospitals in a city.
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Determining a small set of databases that contains information that you need.
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Figuring out what character strings can be used to identify computer viruses.
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Selecting people for a startup that have the needed set of skills.
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Cutting car doors out of sheet metal.
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<h2 class="hd hd-2 unit-title">Set Packing</h2>
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<h2 class="hd hd-2 unit-title">Set Packing Exercise</h2>
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Set Packing Exercise
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<p>Consider the graph in the diagram below. What is the maximum size of an independent set (packing)?</p>
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3
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4
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5
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<h2 class="hd hd-2 unit-title">Facility Location 2</h2>
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<h2 class="hd hd-2 unit-title">Facility Location 2 Exercise</h2>
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Facility Location 1 Exercise
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<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>
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New constraint: \(\sum_{d \in D} x_{sd} + v_s \leq n_s \text{ for } s \in S \)
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New constraint: \(v_s \geq 0 \text{ for } d \in D\)
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New constraint: \(v_s \geq 0 \text{ for } s \in S\)
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<h2 class="hd hd-2 unit-title">Graph Coloring</h2>
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<h2 class="hd hd-2 unit-title">Graph Coloring Exercise</h2>
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Graph Coloring Exercise
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<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>
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2
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1
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4
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8
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3
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