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‎notebooks/2/Logical_Modeling_GDP.html‎

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@@ -689,8 +689,8 @@ <h3><span class="section-number">2.1.4.3. </span>Logical Statements and Conjucti
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<p><em>The following excerpts are from Section 15.7 in Biegler, Grossmann, and Westerberg (1997).</em></p>
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<p>The three step procedure below shows how to convert logical statements into conjuctive normal form:</p>
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<p>Step 1: Replace the implication by its equivalent disjunction. Example:</p>
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<div class="amsmath math notranslate nohighlight" id="equation-88bd1afb-bb99-4c79-baf4-7bb2b2736df9">
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<span class="eqno">(2.1)<a class="headerlink" href="#equation-88bd1afb-bb99-4c79-baf4-7bb2b2736df9" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-1492ff8f-640c-45ff-8dd4-fe5cd0dadf32">
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<span class="eqno">(2.1)<a class="headerlink" href="#equation-1492ff8f-640c-45ff-8dd4-fe5cd0dadf32" title="Permalink to this equation">#</a></span>\[\begin{equation}
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P_{1} \Rightarrow P_{2} \Leftrightarrow \neg P_{1} \lor P_{2} \tag{15.18}
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\end{equation}\]</div>
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<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
@@ -735,8 +735,8 @@ <h3><span class="section-number">2.1.4.3. </span>Logical Statements and Conjucti
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</div>
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</div>
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</details><p>Step 2: Distribute the negation by applying DeMorgan’s Theorem. Examples:</p>
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<div class="amsmath math notranslate nohighlight" id="equation-c0a85a6b-a3a0-42f0-b9ee-526bc6ad08ee">
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<span class="eqno">(2.2)<a class="headerlink" href="#equation-c0a85a6b-a3a0-42f0-b9ee-526bc6ad08ee" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-547eb6a6-c578-4e13-89cd-27411b3a6613">
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<span class="eqno">(2.2)<a class="headerlink" href="#equation-547eb6a6-c578-4e13-89cd-27411b3a6613" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\neg (P_{1} \land P_{2}) \Leftrightarrow \neg P_{1} \lor \neg P_{2}\tag{15.19}
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\end{equation}\]</div>
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<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
@@ -781,8 +781,8 @@ <h3><span class="section-number">2.1.4.3. </span>Logical Statements and Conjucti
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</div>
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<p class="sd-card-text">The example holds.</p>
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</div>
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</details><div class="amsmath math notranslate nohighlight" id="equation-7402adea-f735-463b-937c-610a4a2e7bf3">
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<span class="eqno">(2.3)<a class="headerlink" href="#equation-7402adea-f735-463b-937c-610a4a2e7bf3" title="Permalink to this equation">#</a></span>\[\begin{equation}
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</details><div class="amsmath math notranslate nohighlight" id="equation-fd82b6d9-6c04-4099-8eeb-96fa0d8b6f0d">
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<span class="eqno">(2.3)<a class="headerlink" href="#equation-fd82b6d9-6c04-4099-8eeb-96fa0d8b6f0d" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\neg (P_{1} \lor P_{2}) \Leftrightarrow \neg P_{1} \land \neg P_{2}\tag{15.20}
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\end{equation}\]</div>
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<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
@@ -828,8 +828,8 @@ <h3><span class="section-number">2.1.4.3. </span>Logical Statements and Conjucti
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<p class="sd-card-text">The example holds.</p>
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</div>
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</details><p>Step 3: Recursively distribute the <span class="math notranslate nohighlight">\(\textbf{OR}\)</span> over the <span class="math notranslate nohighlight">\(\textbf{AND}\)</span> by using the following equivalence. Example:</p>
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<div class="amsmath math notranslate nohighlight" id="equation-90cd6e9f-02f3-4a88-9518-4af1a53b8ef2">
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<span class="eqno">(2.4)<a class="headerlink" href="#equation-90cd6e9f-02f3-4a88-9518-4af1a53b8ef2" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-c100e8f1-eb73-4f10-8435-90d31ac09e7b">
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<span class="eqno">(2.4)<a class="headerlink" href="#equation-c100e8f1-eb73-4f10-8435-90d31ac09e7b" title="Permalink to this equation">#</a></span>\[\begin{equation}
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(P_{1} \land P_{2}) \lor P_{3} \Leftrightarrow (P_{1} \lor P_{3}) \land (P_{2} \lor P_{3})\tag{15.21}
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\end{equation}\]</div>
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<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
@@ -1007,8 +1007,8 @@ <h2><span class="section-number">2.1.5. </span>Modeling Disjunctions<a class="he
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<p>When modeling disjunctions, we will have to represent logical constraints that involve continuous variables.</p>
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<section id="general-notation">
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<h3><span class="section-number">2.1.5.1. </span>General Notation<a class="headerlink" href="#general-notation" title="Link to this heading">#</a></h3>
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<div class="amsmath math notranslate nohighlight" id="equation-b55a70e5-6725-468b-bae9-e5a7121b54d4">
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<span class="eqno">(2.5)<a class="headerlink" href="#equation-b55a70e5-6725-468b-bae9-e5a7121b54d4" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-a67e1934-4f0e-42a0-9b9a-b944be4466dc">
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<span class="eqno">(2.5)<a class="headerlink" href="#equation-a67e1934-4f0e-42a0-9b9a-b944be4466dc" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\lor_{i \in D_{R}}
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\begin{bmatrix}
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Y_{ik}\\
@@ -1056,16 +1056,16 @@ <h2><span class="section-number">2.1.6. </span>Example: The Reactor Problem<a cl
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<li><p>If <em>reactor 2</em> is selected, then pressure <span class="math notranslate nohighlight">\(P\)</span> must be between <span class="math notranslate nohighlight">\(20\)</span> and <span class="math notranslate nohighlight">\(30\)</span> atmospheres.</p></li>
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</ul>
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<p><strong>Linear Disjunction Form:</strong></p>
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<div class="amsmath math notranslate nohighlight" id="equation-307c7665-77e0-4295-b2e4-3c5b221a0e16">
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<span class="eqno">(2.6)<a class="headerlink" href="#equation-307c7665-77e0-4295-b2e4-3c5b221a0e16" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-793d63d3-84fe-4b27-978a-aae1c166b9aa">
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<span class="eqno">(2.6)<a class="headerlink" href="#equation-793d63d3-84fe-4b27-978a-aae1c166b9aa" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\lor_{i \in D}
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\begin{bmatrix}
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A_{i}x \leq b_{i}(x)
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\end{bmatrix}
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\end{equation}\]</div>
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<p><strong>Applied to the Reactor Problem:</strong></p>
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<div class="amsmath math notranslate nohighlight" id="equation-263545db-509f-4c88-b8b9-e31b6fbb900e">
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<span class="eqno">(2.7)<a class="headerlink" href="#equation-263545db-509f-4c88-b8b9-e31b6fbb900e" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-b824f9d7-d358-4c7a-b4ba-7020fca75b1e">
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<span class="eqno">(2.7)<a class="headerlink" href="#equation-b824f9d7-d358-4c7a-b4ba-7020fca75b1e" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\begin{bmatrix}
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y_1\\
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P \leq 10\\
@@ -1149,21 +1149,21 @@ <h3><span class="section-number">2.1.6.2. </span>Transform and Solve with Big-M
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<section id="big-m-relaxation-approach">
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<h3><span class="section-number">2.1.6.3. </span>Big-M Relaxation Approach<a class="headerlink" href="#big-m-relaxation-approach" title="Link to this heading">#</a></h3>
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<p><strong>General Notation:</strong></p>
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<div class="amsmath math notranslate nohighlight" id="equation-2a686b36-3e88-45e6-8461-9f17efa58e52">
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<span class="eqno">(2.8)<a class="headerlink" href="#equation-2a686b36-3e88-45e6-8461-9f17efa58e52" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-8df56953-df18-4e56-b06e-389f3f44a43f">
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<span class="eqno">(2.8)<a class="headerlink" href="#equation-8df56953-df18-4e56-b06e-389f3f44a43f" title="Permalink to this equation">#</a></span>\[\begin{equation}
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A_{i}(x) \leq b_{i} + M_{i}(1-y_{i}) \ , \ \forall i \in D
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\end{equation}\]</div>
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<div class="amsmath math notranslate nohighlight" id="equation-6b8cfee9-3405-45a8-9b51-6d04588e0c8c">
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<span class="eqno">(2.9)<a class="headerlink" href="#equation-6b8cfee9-3405-45a8-9b51-6d04588e0c8c" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-cc47056c-8c50-4253-a54c-ad9830193a5c">
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<span class="eqno">(2.9)<a class="headerlink" href="#equation-cc47056c-8c50-4253-a54c-ad9830193a5c" title="Permalink to this equation">#</a></span>\[\begin{equation}
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\sum_{i \in D} y_{i} = 1
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\end{equation}\]</div>
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<div class="amsmath math notranslate nohighlight" id="equation-cc931efd-1327-4f7b-ae4e-6e0fb01c1a72">
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<span class="eqno">(2.10)<a class="headerlink" href="#equation-cc931efd-1327-4f7b-ae4e-6e0fb01c1a72" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-f0d20539-82e5-4c03-bd9e-f6e8fbb1a24a">
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<span class="eqno">(2.10)<a class="headerlink" href="#equation-f0d20539-82e5-4c03-bd9e-f6e8fbb1a24a" title="Permalink to this equation">#</a></span>\[\begin{equation}
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y_{i} \in \{0,1\} \ , \ \forall i \in D
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\end{equation}\]</div>
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<p><strong>Applied to the Reactor Problem:</strong></p>
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<div class="amsmath math notranslate nohighlight" id="equation-1b821df3-e053-4630-bec3-10ad124586ed">
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<span class="eqno">(2.11)<a class="headerlink" href="#equation-1b821df3-e053-4630-bec3-10ad124586ed" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-18261702-834a-412c-9800-8d9d13823e03">
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<span class="eqno">(2.11)<a class="headerlink" href="#equation-18261702-834a-412c-9800-8d9d13823e03" title="Permalink to this equation">#</a></span>\[\begin{equation}
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P \leq 10 + M_{1}(1-y_{1})\\
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-P \leq -5 + M_{1}(1-y_{1})\\
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P \leq 30 + M_{2}(1-y_{2})\\
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\]</div>
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<p><span class="math notranslate nohighlight">\(z_{i}\)</span>: continuous variables separated into as many new variables as there are terms for the disjunctions.</p>
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<p><strong>Applied to the Reactor Problem:</strong> <br></p>
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<div class="amsmath math notranslate nohighlight" id="equation-cd43e01d-07fc-4b06-b1a2-f062055a2d81">
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<span class="eqno">(2.12)<a class="headerlink" href="#equation-cd43e01d-07fc-4b06-b1a2-f062055a2d81" title="Permalink to this equation">#</a></span>\[\begin{equation}
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<div class="amsmath math notranslate nohighlight" id="equation-b3ac98c1-da03-419a-bda6-0158796afdc1">
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<span class="eqno">(2.12)<a class="headerlink" href="#equation-b3ac98c1-da03-419a-bda6-0158796afdc1" title="Permalink to this equation">#</a></span>\[\begin{equation}
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P = P_{1} + P_{2} \\
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P_{1} \leq 10y_{1}\\
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P_{2} \leq 30y_{2} \\

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