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<h2 class="hd hd-2 unit-title">Slides</h2>
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<h2 class="hd hd-2 unit-title">IP Intro</h2>
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<h2 class="hd hd-2 unit-title">Knapsack Problem</h2>
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<h2 class="hd hd-2 unit-title">Knapsack Exercise</h2>
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Knapsack Exercise
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<p>Consider the knapsack problem, in which a subset of items are selected from a set of [mathjaxinline]n[/mathjaxinline] items so as to maximize an objective. Suppose the integrality constraints are relaxed. (That is, the variables are permitted to be fractional.)</p>
<p>Which of the following would be the outcome?</p>
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The optimal objective value increases or remains the same.
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The optimal objective value decreases or remains the same.
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The optimal objective value remains the same.
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The optimal objective value strictly increases.
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The optimal objective value strictly decreases.
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<h2 class="hd hd-2 unit-title">Nonlinear to Integer</h2>
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<h2 class="hd hd-2 unit-title">Nonlinear to Integer Exercise</h2>
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Nonlinear to Integer Exercise
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<p>Consider the nonlinear constraint \(y=x_1x_2, \text{ for } x_1, x_2 \in \{0,1\}\)</p>
<p>Which of the following can together be used to transform the above constraint into linear constraints? You will need to select three of the constraints. (Assume that \(y \in [0,1]\))</p>
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\(y \leq x_1\)
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\(y \geq x_1\)
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\(y \leq x_2\)
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\(y \geq x_2\)
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\(y \geq x_1 + x_2 -1\)
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\(y \leq x_1 + x_2 -1\)
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\(y = x_1 + x_2 -1\)
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<h2 class="hd hd-2 unit-title">Binary Logic</h2>
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<h2 class="hd hd-2 unit-title">Binary Logic Exercise</h2>
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<p>Which of the following is a valid transformation when the objective is to minimize \( z \) subject to \(z = \max \{2x-4, |x-y|\}\)? Select all answers that are correct. </p>
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Max \( \{ z: \, \, z \geq 2x-4, \,\, z \ge x - y, \, \, z \ge y - x \} \).
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Min \( \{ z: \, \, z \geq 2x-4, \,\, z \ge x - y, \, \, z \ge y - x \} \).
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Min \( \{ z: \, \, z \ge 2x-4, \,\, z \ge x - y, \, \, z \le x - y \} \).
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Min \( \{ z: \, \, z \le 2x-4, \,\, z \le x - y, \, \, z \le y - x \} \).
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<h2 class="hd hd-2 unit-title">Big M Exercise</h2>
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Big M Exercise
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<p>Suppose that we are solving a linear program in which &#160; \(0 \leq x_1 \leq 10, &#160;0 \leq x_2 \leq 20, \text{ and } 0 \leq x_3 \leq 30. \)</p>
<p>What is the minimum value of \(M \) under which the following constraint is guaranteed to be redundant?
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<p>\(x_1 -2 x_2 + 3 x_3 \leq M + 5\)</p>
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55
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95
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105
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135
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<h2 class="hd hd-2 unit-title">OR Exercise</h2>
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OR Exercise
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<p>Consider the binary decision variables \( x_1, x_2\). Which of the following is logically equivalent to the statement "\( x_1 = 0 \text{ or } x_2=1\) but not both"?</p>
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\(x_1 = x_2\).
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\(x_1 + x_2 \ge 1\).
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\(x_1 + x_2 \le 1\).
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\(x_1 - x_2 \le 0 \).
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<h2 class="hd hd-2 unit-title">OR2 Exercise</h2>
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&#34;OR2&#34; Exercise
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<p> Consider decision variables \(x_1, x_2\) and a binary variable \(w_1\). </p>
<p>Add two constraints and a binary variable \(w_1\), that ensure the following: \( x_1+x_2 \geq 70\) or \(x_1+x_2 \leq 60\). Select the two constraints below. </p>
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\(x_1 + x_2 \geq 70 - M(1-w_1)\)
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\(x_1 + x_2 \geq 70 + M(1-w_1)\)
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\(x_1 + x_2 \leq 60 + Mw_1\)
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\(x_1 + x_2 \leq 60 - Mw_1\)
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