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<�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�(� #gekko "display: none; } @media print"#gekko { display: block; } } #gekko2 "display: none; } @media print"#gekko2 "display: block<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�(�(� --></style><script src='https:å]cdnjs.cloudflare.com/aja6libs/mathjax(�W)�.7.5/latest.js?config=TeX-MML-AM_CHTML-full' async></script> <meta name='robots' content='index,follow'ëX> <scrip4�type="text/javascript" src="httpa_/apmonitor.com/pdc/pub/skins/bli6javascript/jquery.min.js">Ã_script><script type="texËzjavascript" src="http:å]apmonitor.coÛwpdupuuskintzblix/javascripËzblix.js"></script><meta name='keywords' content='Optimization, Linear Programming, Python, optimization, engineering optimization'ëX> <meta name='description' content='Linear Programming (LP) has a linear objective function, equality, and inequalit9�constraints. Popular methods to solve LP problems are interior point and simplex methods' /> <script type='texËzjavascript' src='http:å]apmonitor.coÛwpdupuuliupmwiki-utils.js?st³75�0074' � data-config='{"fullname":"Main.LinearProgramming","sortable":1,"highlight":0,"copycode":"","toggle"ES,"localtimes":0,"rediquiet":0,"updrop":false,"pmtoc":{"Enable"ES,"MaxLevel":6,"MinNumber":3,"ParentElement":"","NumberedHeadings":"","EnableBacklinks"ES,"EnableQMarkup"ES,"contents":"Contents","hide":"hide","show":"show"}}' data-fullname='Main.LinearProgramming'></script> </head> <body> <di6�id="container"> <di6�id="header"> <!--pageheaderfmt--> <h1 class='sitetitle'><a class='urllink' href='http:å]apmonitor.coÛwpduindex.php' rel='nofollow'>Dynamics and ControlÃ_a>Ã_È> <div class='sitetag' > </div> <!-q\pageheaderfmt--> Ã_div> <di6�id="navigation"> <!--pagesearchfmt--> <form action='httpa_/apmonitor.com/pdc/index.php' method="get"> <fieldset> <input class="hidden" type='hidden' name='n' value='Main.LinearProgramming' /> <inpu4�class="hidden" type='hidden' name='action' value='search' /> <inpu4�id="searchfield" type='text' name='q' value=''ëX> <input id="searchbutton" type='submit' name="searchbutton" value='Find' /> Ã_fieldset> Ã_form> À--/PageSearchFmt--> À--PageTabsFmt--> <di6�class="navlinks"> <ul><li><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/HomePage'>home</a> Ã_li><li><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/CourseSyllabus'>syllabus</a> Ã_li><li><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/CourseSchedule'>schedule</a> Ã_li></ul> Ã_div> <!-q\pagetabsfmt--> <di6�class="pageactions"> À--PageActionFmt--> <ul><li class='browse' � <a accesskey='' rel='nofollow' class='selflink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming'>ViewÃ_a> Ã_li><li class='edit'> <a accesskey='e' rel='nofollow' class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=edit'>EditÃ_a> </li><li class='diff' � <a accesskey='h' rel='nofollow' class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=diff'>History</a> Ã_li><li class='print' � <a accesskey='' rel='nofollow' class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming?action=print'>Print</a> Ã_li></ul> <!-q\pageactionfmt--> </div> </div> <div id="content-wrapper"> <di6�id="content-container"> <di6�id="content"> À--PageTitleGroupFmt--> <!--<a href='httpa_/apmonitor.com/pdc/index.php/Main' class="pagegroup">Main</a>--> <!-q\pagetitlegroupfmt--> <!--pagetitlefmt--> <È><a href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming'>Linear Programming with Python</a></h1> <!-q\pagetitlefmt--> <!--pagetext--> <di6�id='wikitext'> <di6�class='vspace'></div> <table border='0' ><tr ><th align='left'><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwCourseSchedule'><img height=È0ó05px' src='httpa_/apmonitor.com/pdc/uploads/Maiâwschedule.png' alt=''ëX></a></th><th align='center'><a class='selflink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming'><img height='35px' src='http:å]apmonitor.coÛwpduuploadtzMain/learn2.png' alt='' />Ã_a>Ã_th><th align='center'><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/QuizLinearProgramming'><img height=È0ó05px' src='httpa_/apmonitor.com/pdc/uploads/Maiâwquiz.png' alt=''ëX></a></th><th align='center'><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwProcessControlProject'><img height='35px' src='http:å]apmonitor.coÛwpduuploadtzMain/assignment.png' alt='' />Ã_a>Ã_th><th align='right'><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwArduinoModelin '><img height=È0ó05px' src='httpa_/apmonitor.com/pdc/uploads/Maiâwtclab_icon.png' alt=''ëX></a></th>Ã_tr> Ã_table> <0�class='vspace'>Optimization deals with selecting the bes4�option among a number of possible choices that are feasible or don't violate constraints. Python can be used to optimize parameters in a model to best fit data, increase profitabilit9�of a potential engineering design, or meet some other type of objective that can be described mathematically with variables and equations. </p> <p class='vspace'>Mathematical optimization problems may include equalit9�constraints (e.g. =), inequality constraints (e.g. <, <=, >, >=), objective functions, algebraic equations, differential equations, continuous variables, discrete or integer variables, etc. A general statemen4�of an optimization problem with nonlinear objectives or constraints is given b9�the following: </p> <p class='vspace'>$$\begin{align}\mathrm{minimize} \quad & c\,8�\\ \mathrm{subject\;to}\quad & A \, x=b \\ & A \, x>b \end{align}$$ </p> <p class='vspace'>Two popular numerical methods for solving linear programming problems are the Simple8�method and Interior Poin4�method. Ã_p> <di6�class='vspace'></div> <iframe width="50" height="ª5" src="https:å]www.youtube.coÛwembe(u2dY_tRamSjY" frameborder="."."." allow="accelerometer; autoplay; encrypted-media<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�gyroscope; picture-in-picture" allowfullscreen></iframe> <div class='vspace'>Ã_div><h4>Exercise: Soft Drink Production</h4> <p>A simple production planning problem is given by the use of two ingredients <em>A</em> and <em>BÃ_em �that produce products <em>1</em> and <em(�ÇŒ)�Ã_em>. The available suppl9�is <em>AÃ_em>=30 units and <em>BÃ_em>=44 units. For production i4�requires: Ã_p> <di6�class='vspace'></div><ul><liÜ0ë0È0 units of <em>A</em> and 8 units of <em>B</em> to produce Produc4�<em>1</em> </li><li>5 5 units of <em>AÃ_em �and 2 2 2 units of <em>BÃ_em �to produce Product <em(�ÇŒ)�Ã_em> Ã_li></ul><p class='vspace'>There are at mos4�5 units of Product <em´Ã_em �and 2 2 2 units of Product <em(�ÇŒ)�Ã_em>. Product 1 can be sold for 100 and Product(��N)� can be sold for 125. The objective is to maximize the profi4�for this production problem. </p> <p class='vspace'>For this problem determine: Ã_p> <di6�class='vspace'></div><ol><li>A potential feasible solution </li><li>Identify the constraints on the contour plot </li><li>Mark the set of feasible solutions on the contour plot </li><li>Identify the minimum objective feasible solution </li><li>Identify the maximum objective feasible solution </li><li>Use a solver to find a solution Ã_li></ol><div class='vspace'>Ã_div><div class='img imgonly'><img width='500px' src='http:å]apmonitor.coÛwpduuploadtzMain/lp_contour.png' alt='' />Ã_div> <0�class='vspace'>A contour plo4�can be used to explore the optimal solution. In this case, the black lines indicate the upper and lower bounds on the production of <em>1</em> and <em(�ÇŒ)�Ã_em>. In this case, the production of <em´Ã_em �must be greater than 0 but less than 5. The production of <em>2</em> mus4�be greater than 0 bu4�less than 4. There are a4�mostµ0ó0Á0ü0à00 units of <em>AÃ_em �and 44 units of <em>B</em> ingredients tha4�are available to produce products <em>1</em> and <em(�ÇŒ)�Ã_em>. </p> <div class='vspace'>Ã_div><h4>SolutionÃ_h4> <ul><li><a class='urllink' href='httpsa_/colab.research.google.com/drive/1rAfR-grR6umXs7FSp6HFgcqTEBfnZpMk?usp=sharing' rel='nofollow'>Solve Online with Python (Google Colab)Ã_a> </li><li><a class='urllink' href='httpsa_/apmonitor.com/onlin0uview_pass.php?f=softdrink.apm' rel='nofollow'>Solve Online with APMonitor</a> Ã_li></ul><p class='vspace'><strong>Method 1: Equations and Objective</strong> </p> <p class='vspace'>Using equations and an objective function is good for small problems because i4�is a readable optimization problem and is thereb9�easy to modify. Ã_p> <0�class='vspace'><em>Python (Gekko)</em> </p> <div class='vspace'>Ã_div> <di6�class='sourceblock ' id='sourceblock1'> <div class='sourceblocktext'><di6�class="python"><span class="kw1">from</span> gekko <span class="kw1">import</span> GEKKO<br /> m <span class="sy.".".">=Ã_span �GEKKO<span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><brëx> + <span class="sy.".".">=Ã_span �m.<span class="m">Var</span><span class="br0">(</span>lb<span class="sy0">=</span><span class="nu0">0</span><span class="sy0">,</span> ub<span class="sy.".".">=Ã_span><span class="nu.".".">5Ã_span><span class="br.".".">)Ã_span �<span class="cÝ"># Produc4�1</span><br /> x2 <span class="sy0">=</span> m.<span class="me1">VarÃ_span><span class="br.".".">(Ã_span>lb<span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span �ub<span class="sy0">=</span><span class="nu0">4</span><span class="br0">&#;</span> <span class="co1"># Product(��N)�Ã_span><brëx> m.<span class="m">MaximizeÃ_span><span class="br.".".">(Ã_span><span class="nu."."."´00Ã_span>*++<span class="nu."."."´25Ã_span>*tÎ<span class="br0">&#;</span> <span class="co1"># Profi4�functionÃ_span><brëx> m.<span class="m">EquationÃ_span><span class="br.".".">(Ã_span><span class="nu."."."Ü0ë0È0Ã_span>*++<span class="nu.".".">6Ã_span>*tÎ<span class="sy0"><=</span><span class="nu0">30Ã_span><span class="br.".".">)Ã_span �<span class="cÝ"># Units of AÃ_span><brëx> m.<span class="m">EquationÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">8Ã_span>*++<span class="nu.".".">4Ã_span>*tÎ<span class="sy0"><=</span><span class="nu0">44Ã_span><span class="br.".".">)Ã_span �<span class="cÝ"># Units of BÃ_span><brëx> m.<span class="m">solve</span><span class="br0">(</span>disp<span class="sy0">=</span><span class="k(Ì">False</span><span class="br0">&#;</span><br /> p1 <span class="sy0">=</span> x1.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">;</span> p2 <span class="sy0">=</span> x2.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product 1 (+): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>ß<span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product(��N)� (tÎ): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>ä²<span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Profi4�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   : 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span><span class="nu0">100</span>*p1+<span class="nu0">(�*h)�5</span>*p2<span class="br.".".">)Ã_span><span class="br.".".">)Ã_span></div></div> <div class='sourceblocklink'><a href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming?action=sourceblock&num³' type='text/plain'>[$[Ge4�Code]]Ã_a>Ã_div> </div> <0�class='vspace'><em>MATLAB (Gekko)</em> </p> <div class='vspace'>Ã_div> <di6�class='sourceblock ' id='sourceblock2'> <div class='sourceblocktext'><di6�class="matlab">m = py.<span class="m">gekko</span>.<span class="me1">GEKKOÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span>;<br /> x1 = m.<span class="k(Ì">Var</span><span class="br0">(</span>pyargs<span class="br0">(</span><span class="c˜°">'lb'Ã_span>,<span class="nu0">0</span>,<span class="co2">'ub'</span>,<span class="nu.".".">5Ã_span><span class="br.".".">)Ã_span><span class="br.".".">)Ã_span>; <span class="co1">% Product 1Ã_span><brëx> tÎ = m.<span class="kw2">VarÃ_span><span class="br.".".">(Ã_span>pyargs<span class="br.".".">(Ã_span><span class="co2">'lb'</span>,<span class="nu.".".">0Ã_span>,<span class="c˜°">'ub'Ã_span>,<span class="nu0">4</span><span class="br0">&#;</span><span class="br0">&#;</span><�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�<span class="cÝ">.�.�Produc4�2</span><br /> m.<span class="me1">Maximize</span><span class="br0">(</span><span class="nu0">100</span>*x1+<span class="nu0">(�*h)�5</span>*x2<span class="br.".".">)Ã_span>; <span class="co1">% Profi4�functionÃ_span><brëx> m.<span class="m">EquationÃ_span><span class="br.".".">(Ã_span><span class="nu."."."Ü0ë0È0Ã_span>*++<span class="nu.".".">6Ã_span>*tÎ<=<span class="nu."."."Ü0ë0È00</span><span class="br0">&#;</span><�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�<span class="cÝ">.�.�Units of AÃ_span><brëx> m.<span class="m">EquationÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">8Ã_span>*++<span class="nu.".".">4Ã_span>*tÎ<=<span class="nu.".".">44</span><span class="br0">&#;</span><�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�<span class="cÝ">.�.�Units of BÃ_span><brëx> m.<span class="m">solve</span><span class="br0">(</span>pyargs<span class="br0">(</span><span class="c˜°">'disp'Ã_span>,false<span class="br.".".">)Ã_span><span class="br.".".">)Ã_span>;<br /> p1 = +.<span class="me1">VALUEÃ_span><span class="br.".".">Õ0¡0é0Ã0É0;</span><span class="nu0">1</span><span class="br0">&#(�*h)�5;Ã_span>;<br /> p2 = tÎ.<span class="me1">VALUEÃ_span><span class="br.".".">Õ0¡0é0Ã0É0;</span><span class="nu0">1</span><span class="br0">&#(�*h)�5;Ã_span>;<br /> <span class="kw2">disp</span><span class="br0">(</span><span class="br0">&#;</span><span class="c˜°">'Product 1 (+): 'Ã_span>, <span class="kw2">num2strÃ_span><span class="br.".".">(Ã_span>p1<span class="br.".".">)Ã_span><span class="br.".".">]Ã_span><span class="br.".".">)Ã_span><brëx> <span class="k(Ì">dispÃ_span><span class="br.".".">(Ã_span><span class="br.".".">[Ã_span><span class="co2">'Produc4�2 (x2): '</span>, <span class="k(Ì">nustr</span><span class="br0">(</span>ä²<span class="br0">&#;</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><br /> <span class="kw2">disp</span><span class="br0">(</span><span class="br0">&#;</span><span class="c˜°">'Profi4�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   : 'Ã_span>, <span class="kw2">num2strÃ_span><span class="br.".".">(Ã_span><span class="nu."."."´00Ã_span>*ß+<span class="nu."."."´25Ã_span>*ä²<span class="br0">&#;</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span>Ã_div>Ã_div> � <di6�class='sourceblocklink'><a href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=sourceblock&num=2' type='texËzplain'>[&#Ô0³06;[Get Code]]</a></div> Ã_div> <p class='vspace'><strong>Method(��N)�a: Dense Matrices (Scipy linprog)</strong> </p> <p class='vspace'>For large-scale problems, a matrix forms is best because i4�simplifies the problem description and improves the speed of solution. Scipy.optimize.linprog is one of the available packages to solve Linear programming problems. Another good linear and mixed integer programming Python package is Pul0�with interfaces to dedicate mixed integer linear programming solvers. Ã_p> <di6�class='vspace'></div> <div class='sourceblock ' id='sourcebloc1�9�¹p'> � <di6�class='sourceblocktext'><div class="python"><span class="cÝ"># solve with SciPyÃ_span><brëx> <span class="k)">fromÃ_span �scipy.<span class="m">optimizeÃ_span �<span class="k)">importÃ_span �linprog<brëx> c <span class="sy0">=</span> <span class="br.".".">[Ã_span>-<span class="nu0">100</span><span class="sy0">,</span> -<span class="nu0">(�*h)�5</span><span class="br0">&#Ø0ë0Ä0;</span><br /> A <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="br0">&#;</span><span class="nu0">3</span><span class="sy0">,</span> <span class="nu.".".">6Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span �<span class="br0">&#;</span><span class="nu0">8</span><span class="sy0">,</span> <span class="nu.".".">4Ã_span><span class="br.".".">]Ã_span><span class="br.".".">]Ã_span><brëx> b <span class="sy0">=</span> <span class="br.".".">[Ã_span><span class="nu."."."Ü0ë0È00</span><span class="sy0">,</span> <span class="nu.".".">44</span><span class="br0">&#Ø0ë0Ä0;</span><br /> x0_bounds <span class="sy.".".">=Ã_span �<span class="br0">(</span><span class="nu0">0</span><span class="sy0">,</span> <span class="nu.".".">5Ã_span><span class="br.".".">)Ã_span><brëx> +_bounds <span class="sy0">=</span> <span class="br.".".">(Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">4</span><span class="br0">&#;</span><br /> res <span class="sy.".".">=Ã_span �linprog<span class="br.".".">(Ã_span>c<span class="sy0">,</span> A_ub<span class="sy.".".">=Ã_span>A<span class="sy0">,</span> b_ub<span class="sy.".".">=Ã_span>b<span class="sy0">,</span> \<br />  <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�bounds<span class="sy0">=</span><span class="br0">(</span>x0_bounds<span class="sy.".".">,Ã_span �+_bounds<span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span><brëx>    <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�  options<span class="sy0">=</span><span class="br0">&#(�*h)�3;Ã_span><span class="st.".".">"disp"</span>: <span class="k(Ì">TrueÃ_span><span class="br.".".">}</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span><span class="br.".".">(Ã_span>res<span class="br0">&#;</span>Ã_div>Ã_div> � <di6�class='sourceblocklink'><a href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=sourceblock&num=3' type='texËzplain'>[&#Ô0³06;[Get Code]]</a></div> Ã_div> <p class='vspace'><strong>Method(��N)�b: Dense Matrices (Gekko)</strong> </p> <p class='vspace'>Dense matrix form is also available in Gekko. In this case, two model functions <em>qobjÃ_em �(quadratic objective) and <em>axb</em> (Ax<b) objects are used to create the model. Integer variables for discrete optimization are possible with the APOPT solver (option 1) when the variable is specified with <em>integer=True</em>. See <a class='urllink' href='httpsa_/gekko.readthedocs.iówen/latesËzmodel_methods.html' rel='nofollow'>Model Building Functions</a �in the Gekko documentation. Ã_p> <di6�class='vspace'></div> <div class='sourceblock ' id='sourceblock4'> � <di6�class='sourceblocktext'><div class="python"><span class="k)">fromÃ_span �gekko <span class="k)">importÃ_span �GEKKO<brëx> m <span class="sy0">=</span> GEKKO<span class="br0">(</span>remote<span class="sy0">=</span><span class="k(Ì">False</span><span class="br0">&#;</span><br /> c <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="nu0">100</span><span class="sy0">,</span> <span class="nu."."."´25Ã_span><span class="br.".".">]Ã_span><brëx> A <span class="sy0">=</span> <span class="br.".".">[Ã_span><span class="br.".".">[Ã_span><span class="nu."."."Ü0ë0È0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">6</span><span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">,</span> <span class="br.".".">[Ã_span><span class="nu.".".">8Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">4</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#Ø0ë0Ä0;</span><br /> b <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="nu0">30Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">44Ã_span><span class="br.".".">]Ã_span><brëx> 8�<span class="sy0">=</span> m.<span class="me1">qobj</span><span class="br0">(</span>c<span class="sy.".".">,Ã_span>otype<span class="sy0">=</span><span class="st0">'max'</span><span class="br0">&#;</span><br /> m.<span class="me1">axbÃ_span><span class="br.".".">(Ã_span>A<span class="sy0">,</span>b<span class="sy.".".">,Ã_span>x<span class="sy0">=</span>x<span class="sy.".".">,Ã_span>etype<span class="sy0">=</span><span class="st0">'<'Ã_span><span class="br.".".">)Ã_span><brëx> x<span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span>.<span class="m">lower</span><span class="sy0">=</span><span class="nu0">0</span><span class="sy0">;</span> x<span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span>.<span class="me1">upperÃ_span><span class="sy.".".">=Ã_span><span class="nu.".".">5Ã_span><brëx> x<span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="br.".".">]Ã_span>.<span class="m">lower</span><span class="sy0">=</span><span class="nu0">0</span><span class="sy0">;</span> x<span class="br0">&#;</span><span class="nu0">1</span><span class="br0">&#Ø0ë0Ä0;</span>.<span class="me1">upperÃ_span><span class="sy.".".">=Ã_span><span class="nu.".".">4Ã_span><brëx> m.<span class="m">options</span>.<span class="me1">solver</span> <span class="sy.".".">=Ã_span �<span class="nu0">1</span><br /> m.<span class="me1">solveÃ_span><span class="br.".".">(Ã_span>disp<span class="sy.".".">=Ã_span><span class="kw2">True</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product 1 (+): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>x<span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span>.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product(��N)� (tÎ): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>x<span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="br.".".">]Ã_span>.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Profi4�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   : 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>-m.<span class="me1">optionsÃ_span>.<span class="m">objfcnval</span><span class="br0">&#;</span><span class="br0">&#;</span>Ã_div>Ã_div> � <di6�class='sourceblocklink'><a href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=sourceblock&num=4' type='texËzplain'>[&#Ô0³06;[Get Code]]</a></div> Ã_div> <p class='vspace'><strong>Methodµ0ó0Á0ü0à0: Sparse Matrices (Gekko)</strong> </p> <p class='vspace'>Sparse matrices are faster and use less memory for ver9�large-scale problems with many zeros in <em>A</em>, <em>b</em>, and <em>c</em>. Ã_p> <di6�class='vspace'></div> <div class='sourceblock ' id='sourceblock5'> � <di6�class='sourceblocktext'><div class="python"><span class="cÝ"># solve with GEKKO and sparse matricesÃ_span><brëx> <span class="k)">importÃ_span �nump9�<span class="k)">asÃ_span �np<br /> <span class="kw1">from</span> gekko <span class="kw1">import</span> GEKKO<br /> m <span class="sy.".".">=Ã_span �GEKKO<span class="br.".".">(Ã_span>remote<span class="sy.".".">=Ã_span><span class="kw2">FalseÃ_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># [[ro7�indices],[column indices],[values]]</span><br /> A_sparse <span class="sy0">=</span> <span class="br.".".">[Ã_span><span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span><span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span><span class="br.".".">[Ã_span><span class="nu."."."Ü0ë0È0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">6Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">8Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4Ã_span><span class="br.".".">]Ã_span><span class="br.".".">]Ã_span><brëx> <span class="cÝ"># [[ro7�indices],[values]]Ã_span><brëx> b_sparse <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="br0">&#;</span><span class="nu0">1</span><span class="sy0">,</span><span class="nu0">2</span><span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">30Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">44</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#Ø0ë0Ä0;</span><br /> x <span class="sy.".".">=Ã_span �m.<span class="m">axb</span><span class="br0">(</span>A_sparse<span class="sy0">,</span>b_sparse<span class="sy0">,</span>etype<span class="sy.".".">=Ã_span><span class="st.".".">'<'</span><span class="sy0">,</span>sparse<span class="sy0">=</span><span class="k(Ì">TrueÃ_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># [[ro7�indices],[values]]Ã_span><brëx> c_sparse <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="br0">&#;</span><span class="nu0">1</span><span class="sy0">,</span><span class="nu0">2</span><span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">100</span><span class="sy0">,</span><span class="nu0">(�*h)�5</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#Ø0ë0Ä0;</span><br /> m.<span class="me1">qobj</span><span class="br0">(</span>c_sparse<span class="sy0">,</span>x<span class="sy.".".">=Ã_span>x<span class="sy0">,</span>otype<span class="sy.".".">=Ã_span><span class="st.".".">'max'Ã_span><span class="sy.".".">,Ã_span>sparse<span class="sy.".".">=Ã_span><span class="kw2">True</span><span class="br0">&#;</span><br /> x<span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span>.<span class="me1">lowerÃ_span><span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">;Ã_span �x<span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span>.<span class="m">upper</span><span class="sy0">=</span><span class="nu0">5</span><br /> x<span class="br0">&#;</span><span class="nu0">1</span><span class="br0">&#Ø0ë0Ä0;</span>.<span class="me1">lowerÃ_span><span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">;Ã_span �x<span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="br.".".">]Ã_span>.<span class="m">upper</span><span class="sy0">=</span><span class="nu0">4</span><br /> m.<span class="me1">solveÃ_span><span class="br.".".">(Ã_span>disp<span class="sy.".".">=Ã_span><span class="kw2">True</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product 1 (+): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>x<span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span>.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product(��N)� (tÎ): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>x<span class="br.".".">[Ã_span><span class="nu."."."´Ã_span><span class="br.".".">]Ã_span>.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Profi4�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   : 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>-m.<span class="me1">optionsÃ_span>.<span class="m">objfcnval</span><span class="br0">&#;</span><span class="br0">&#;</span>Ã_div>Ã_div> � <di6�class='sourceblocklink'><a href='http:å]apmonitor.coÛwpduindex.ph:yMain/LinearProgramming?action=sourceblock&num=5' type='texËzplain'>[&#Ô0³06;[Get Code]]</a></div> Ã_div> <p class='vspace'><strong>Contour PlotÃ_strong> Ã_p> <0�class='vspace'>Below are the source files for generating the contour plots in Python. The linear program is solved with the APM model through a web-service while the contour plot is generated with the Python package Matplotlib. Ã_p> <0�class='vspace'><span id='gekko-tog' class='toggle'><inpu4�type='button' class='butto�¬' value='Sho7�GEKKO Python Code' onclick="javascript:toggleObj('gekko')"ëX></span> </p><di6�id='gekko' > <di6�class='vspace'></div> <div class='sourceblock ' id='sourceblock6'> � <di6�class='sourceblocktext'><div class="python"><span class="k)">fromÃ_span �gekko <span class="k)">importÃ_span �GEKKO<brëx> <br /> m <span class="sy.".".">=Ã_span �GEKKO<span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><brëx> <br /> <span class="co1"># variablesÃ_span><brëx> + <span class="sy.".".">=Ã_span �m.<span class="m">Var</span><span class="br0">(</span>value<span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span �<span class="sy0">,</span> lb<span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span �<span class="sy0">,</span> ub<span class="sy.".".">=Ã_span><span class="nu.".".">5Ã_span �<span class="sy0">,</span> name<span class="sy.".".">=Ã_span><span class="st.".".">'+'</span><span class="br0">&#;</span> <span class="co1"># Product 1Ã_span><brëx> tÎ <span class="sy.".".">=Ã_span �m.<span class="m">Var</span><span class="br0">(</span>value<span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span �<span class="sy0">,</span> lb<span class="sy.".".">=Ã_span><span class="nu.".".">0Ã_span �<span class="sy0">,</span> ub<span class="sy.".".">=Ã_span><span class="nu.".".">4Ã_span �<span class="sy0">,</span> name<span class="sy.".".">=Ã_span><span class="st.".".">'tÎ'</span><span class="br0">&#;</span> <span class="co1"># Product(��N)�Ã_span><brëx> profit <span class="sy.".".">=Ã_span �m.<span class="m">Var</span><span class="br0">(</span>value<span class="sy.".".">=Ã_span><span class="nu."."."´Ã_span �<span class="sy0">,</span> name<span class="sy.".".">=Ã_span><span class="st.".".">'profit'</span><span class="br0">&#;</span><br /> <brëx> <span class="cÝ"># profit function</span><br /> m.<span class="me1">ObjÃ_span><span class="br.".".">(Ã_span>-profit<span class="br0">&#;</span><br /> m.<span class="me1">Equation</span><span class="br0">(</span>profit<span class="sy0">==Ã_span><span class="nu."."."´00Ã_span>*++<span class="nu."."."´25Ã_span>*tÎ<span class="br0">&#;</span><br /> m.<span class="me1">Equation</span><span class="br0">(</span><span class="nu0">3</span>*x1+<span class="nu0">6</span>*x2<span class="sy."."."><=Ã_span><span class="nu."."."Ü0ë0È00</span><span class="br0">&#;</span><br /> m.<span class="me1">Equation</span><span class="br0">(</span><span class="nu0">8</span>*x1+<span class="nu0">4</span>*x2<span class="sy."."."><=Ã_span><span class="nu.".".">44</span><span class="br0">&#;</span><br /> <brëx> m.<span class="m">solve</span><span class="br0">(</span><span class="br0">&#;</span><br /> <brëx> <span class="k)">print</span> <span class="br.".".">(Ã_span><span class="st.".".">''</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'--- Results of the Optimization Problem ---'</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product 1 (+): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>+<span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Product(��N)� (tÎ): 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>tÎ<span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <span class="kw1">printÃ_span �<span class="br0">(</span><span class="st0">'Profit: 'Ã_span �+ <span class="k(Ì">str</span><span class="br0">(</span>profit<span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> <brëx> <span class="cÝ">## Generate a contour plotÃ_span><brëx> <span class="cÝ"># Import some other libraries that we'll need</span><br /> <span class="co1"># matplotlib and nump9�packages mus4�also be installed</span><br /> <span class="kw1">import</span> matplotlib<brëx> <span class="k)">importÃ_span �nump9�<span class="k)">asÃ_span �np<br /> <span class="kw1">import</span> matplotlib.<span class="m">pyplotÃ_span �<span class="k)">asÃ_span �plt<brëx> <br /> <span class="co1"># Design variables at mesh pointsÃ_span><brëx> 8�<span class="sy0">=</span> np.<span class="m">arangeÃ_span><span class="br.".".">(Ã_span>-<span class="nu0">1.0</span><span class="sy0">,</span> <span class="nu.".".">8.0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">0.(�åe)�Ã_span><span class="br.".".">)Ã_span><brëx> 9�<span class="sy0">=</span> np.<span class="m">arangeÃ_span><span class="br.".".">(Ã_span>-<span class="nu0">1.0</span><span class="sy0">,</span> <span class="nu.".".">6.0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">0.(�åe)�Ã_span><span class="br.".".">)Ã_span><brëx> +<span class="sy0">,</span> x2 <span class="sy0">=</span> np.<span class="m">meshgridÃ_span><span class="br.".".">(Ã_span>x<span class="sy0">,</span>y<span class="br.".".">)Ã_span><brëx> <br /> <span class="co1"># Equations and ConstraintsÃ_span><brëx> profit <span class="sy.".".">=Ã_span �<span class="nu0">100.0</span> * x1 + <span class="nu0">(�*h)�5.0</span> * x2<brëx> A_usage <span class="sy0">=</span> <span class="nu."."."Ü0ë0È0.0Ã_span �* + + <span class="nu.".".">6.0Ã_span �* tÎ<br /> B_usage <span class="sy.".".">=Ã_span �<span class="nu0">8.0</span> * x1 + <span class="nu0">4.0</span> * x2<brëx> <br /> <span class="co1"># Create a contour plotÃ_span><brëx> plt.<span class="m">figureÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># Weight contours</span><br /> lines <span class="sy.".".">=Ã_span �np.<span class="me1">linspace</span><span class="br0">(</span><span class="nu0">100.0</span><span class="sy0">,</span><span class="nu0">AS0.0</span><span class="sy0">,</span><span class="nu0">8</span><span class="br0">&#;</span><br /> CS <span class="sy0">=</span> plt.<span class="me1">contourÃ_span><span class="br.".".">(Ã_span>x1<span class="sy.".".">,Ã_span>x2<span class="sy.".".">,Ã_span>profit<span class="sy.".".">,Ã_span>lines<span class="sy0">,</span>colors<span class="sy0">=</span><span class="st0">'g'</span><span class="br0">&#;</span><br /> plt.<span class="me1">clabel</span><span class="br0">(</span>CS<span class="sy0">,</span> inline<span class="sy.".".">=Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span �fontsize<span class="sy0">=</span><span class="nu0">10Ã_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># A usage <µ0ó0Á0ü0à00</span><br /> CS <span class="sy0">=</span> plt.<span class="me1">contourÃ_span><span class="br.".".">(Ã_span>x1<span class="sy.".".">,Ã_span>x2<span class="sy.".".">,Ã_span>A_usage<span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">26.0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">28.0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">30.0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span>colors<span class="sy.".".">=Ã_span><span class="st.".".">'r'Ã_span><span class="sy.".".">,Ã_span>linewidths<span class="sy.".".">=Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">0.5Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4.0Ã_span><span class="br.".".">]Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">clabelÃ_span><span class="br.".".">(Ã_span>CS<span class="sy.".".">,Ã_span �inline<span class="sy0">=</span><span class="nu0">1</span><span class="sy0">,</span> fontsize<span class="sy.".".">=Ã_span><span class="nu."."."´0</span><span class="br0">&#;</span><br /> <span class="co1"># B usage <<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�44Ã_span><brëx> CS <span class="sy.".".">=Ã_span �plt.<span class="m">contour</span><span class="br0">(</span>+<span class="sy0">,</span> x2<span class="sy.".".">,Ã_span>B_usage<span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">40.0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">42.0</span><span class="sy0">,</span><span class="nu0">44.0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span>colors<span class="sy.".".">=Ã_span><span class="st.".".">'b'Ã_span><span class="sy.".".">,Ã_span>linewidths<span class="sy.".".">=Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">0.5Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4.0Ã_span><span class="br.".".">]Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">clabelÃ_span><span class="br.".".">(Ã_span>CS<span class="sy.".".">,Ã_span �inline<span class="sy0">=</span><span class="nu0">1</span><span class="sy0">,</span> fontsize<span class="sy.".".">=Ã_span><span class="nu."."."´0</span><span class="br0">&#;</span><br /> <span class="co1"># Container for 0 <= Product 1 <= 500 LÃ_span><brëx> CS <span class="sy.".".">=Ã_span �plt.<span class="m">contour</span><span class="br0">(</span>+<span class="sy0">,</span> x2<span class="sy.".".">,Ã_span>x1 <span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">0.0</span><span class="sy0">,</span> <span class="nu.".".">0.1Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">4.9</span><span class="sy0">,</span> <span class="nu.".".">5.0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span>colors<span class="sy.".".">=Ã_span><span class="st.".".">'k'Ã_span><span class="sy.".".">,Ã_span>linewidths<span class="sy.".".">=Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">4.0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4.0Ã_span><span class="br.".".">]Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">clabelÃ_span><span class="br.".".">(Ã_span>CS<span class="sy.".".">,Ã_span �inline<span class="sy0">=</span><span class="nu0">1</span><span class="sy0">,</span> fontsize<span class="sy.".".">=Ã_span><span class="nu."."."´0</span><span class="br0">&#;</span><br /> <span class="co1"># Container for 0 <= Product(��N)� <= 400 LÃ_span><brëx> CS <span class="sy.".".">=Ã_span �plt.<span class="m">contour</span><span class="br0">(</span>+<span class="sy0">,</span> x2<span class="sy.".".">,Ã_span>x2 <span class="sy0">,</span><span class="br0">&#;</span><span class="nu0">0.0</span><span class="sy0">,</span> <span class="nu.".".">0.1Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">3.9</span><span class="sy0">,</span> <span class="nu.".".">4.0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span>colors<span class="sy.".".">=Ã_span><span class="st.".".">'k'Ã_span><span class="sy.".".">,Ã_span>linewidths<span class="sy.".".">=Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">4.0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´.0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4.0Ã_span><span class="br.".".">]Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">clabelÃ_span><span class="br.".".">(Ã_span>CS<span class="sy.".".">,Ã_span �inline<span class="sy0">=</span><span class="nu0">1</span><span class="sy0">,</span> fontsize<span class="sy.".".">=Ã_span><span class="nu."."."´0</span><span class="br0">&#;</span><br /> <brëx> <span class="cÝ"># Add some labels</span><br /> plt.<span class="me1">titleÃ_span><span class="br.".".">(Ã_span><span class="st.".".">'Soft Drink Production Problem'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">xlabelÃ_span><span class="br.".".">(Ã_span><span class="st.".".">'Produc4�1 (100 L)'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">ylabelÃ_span><span class="br.".".">(Ã_span><span class="st.".".">'Produc4�2 (100 L)'Ã_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># Save the figure as a PNGÃ_span><brëx> plt.<span class="m">savefig</span><span class="br0">(</span><span class="st0">'contour.png'</span><span class="br0">&#;</span><br /> <brëx> <span class="cÝ"># Show the plotsÃ_span><brëx> plt.<span class="m">showÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span></div></div> <div class='sourceblocklink'><a href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming?action=sourceblock&num=6' type='text/plain'>[$[Ge4�Code]]Ã_a>Ã_div> </div> </div> <p class='vspace'><strong>Practice ProblemÃ_strong> Ã_p> <0�class='vspace'>Solve the Linear Programming problem. </p> <p class='vspace'>$$\begin{align}\mathrm{maximize} \quad & x+y \\ \mathrm{subject\;to}\quad &<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�6x+4y\le24 \\ &<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�x+2y\le6 \\ &-x+y\le1 \\ & y\l \\ &<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�x\ge0 \\ & y\ge0 \end{align}$$ </p> <p class='vspace'>A graphical representation of the constraints and objective is shown in the figure below. The contour lines are the objective and the vertices are labeled. Linear programming solutions are found a4�vertices (intersection of constraints). Ã_p> <di6�class='vspace'></div><di6�class='img imgonly'><img width='550px' src='httpa_/apmonitor.com/pdc/uploads/Maiâwlp_example.png' alt=''ëX></div> <p class='vspace'><span id='gekk˜°-tog' class='toggle'><inpu4�type='button' class='butto�¬' value='Sho7�Solution' onclick="javascript:toggleObj('gekko2')"ëX></span> </p><di6�id='gekk˜°' > <p class='vspace'>The optimal solution is shown as an orange dot a4�Ê0Î0,1.5) with a maximized objective function of 4.5. The optimal solution is at the intersection of two constraints. Ã_p> <di6�class='vspace'></div><di6�class='img imgonly'><img width='550px' src='httpa_/apmonitor.com/pdc/uploads/Maiâwlp_solution.png' alt='' />Ã_div> <di6�class='vspace'></div> <div class='sourceblock ' id='sourceblock7'> � <di6�class='sourceblocktext'><div class="python"><span class="k)">importÃ_span �nump9�<span class="k)">asÃ_span �np<br /> <span class="kw1">import</span> matplotlib.<span class="m">pyplotÃ_span �<span class="k)">asÃ_span �plt<brëx> <span class="k)">fromÃ_span �gekko <span class="k)">importÃ_span �GEKKO<brëx> <br /> <span class="co1"># solve LP</span><br /> m <span class="sy.".".">=Ã_span �GEKKO<span class="br.".".">(Ã_span>remote<span class="sy.".".">=Ã_span><span class="kw2">FalseÃ_span><span class="br.".".">)Ã_span><brëx> x<span class="sy.".".">,Ã_span>y <span class="sy.".".">=Ã_span �m.<span class="m">Array</span><span class="br0">(</span>m.<span class="m">Var</span><span class="sy0">,</span><span class="nu0">2</span><span class="sy0">,</span>lb<span class="sy0">=</span><span class="nu0">0</span><span class="br0">&#;</span><br /> m.<span class="me1">EquationsÃ_span><span class="br.".".">(Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">6Ã_span>*x+<span class="nu0">4</span>*y<span class="sy0"><=</span><span class="nu0">24Ã_span><span class="sy.".".">,Ã_span>x+<span class="nu."."."(�ÇŒ)�Ã_span>*y<span class="sy."."."><=Ã_span><span class="nu.".".">6Ã_span><span class="sy.".".">,Ã_span>-x+y<span class="sy."."."><=Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span>y<span class="sy0"><=</span><span class="nu0">2</span><span class="br0">&#Ø0ë0Ä0;</span><span class="br0">&#;</span><br /> m.<span class="me1">Maximize</span><span class="br0">(</span>x+y<span class="br.".".">)Ã_span><brëx> m.<span class="m">solve</span><span class="br0">(</span>disp<span class="sy0">=</span><span class="k(Ì">False</span><span class="br0">&#;</span><br /> xop4�<span class="sy0">=</span> x.<span class="me1">valueÃ_span><span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">;Ã_span �yopt <span class="sy.".".">=Ã_span �y.<span class="m">value</span><span class="br0">&#;</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><br /> <brëx> <span class="cÝ"># visualize solutionÃ_span><brëx> g <span class="sy0">=</span> np.<span class="m">linspaceÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">5Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�00Ã_span><span class="br.".".">)Ã_span><brëx> x<span class="sy.".".">,Ã_span>y <span class="sy.".".">=Ã_span �np.<span class="me1">meshgrid</span><span class="br0">(</span>g<span class="sy.".".">,Ã_span>g<span class="br0">&#;</span><br /> obj <span class="sy.".".">=Ã_span �x+y<brëx> plt.<span class="m">imshowÃ_span><span class="br.".".">(Ã_span><span class="br.".".">(Ã_span><span class="br.".".">(Ã_span><span class="nu.".".">6Ã_span>*x+<span class="nu0">4</span>*y<span class="sy0"><=</span><span class="nu0">24Ã_span><span class="br.".".">)Ã_span>&<span class="br0">(</span>x+<span class="nu0">2</span>*y<span class="sy0"><=</span><span class="nu0">6</span><span class="br0">&#;</span>&<span class="br.".".">(Ã_span>-x+y<span class="sy."."."><=Ã_span><span class="nu."."."´Ã_span><span class="br.".".">)Ã_span>&<span class="br0">(</span>y<span class="sy."."."><=Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="br.".".">)Ã_span>&<span class="br0">(</span>x<span class="sy.".".">>=Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">)Ã_span>&<span class="br0">(</span>y<span class="sy.".".">>=Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">)Ã_span><span class="br.".".">)Ã_span>.<span class="m">astypeÃ_span><span class="br.".".">(Ã_span><span class="kw2">intÃ_span><span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span �<br />  <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�  extent<span class="sy.".".">=Ã_span><span class="br.".".">(Ã_span>x.<span class="kw2">minÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span>x.<span class="kw2">maxÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span>y.<span class="kw2">minÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span>y.<span class="kw2">maxÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">,Ã_span>origin<span class="sy.".".">=Ã_span><span class="st.".".">'lower'Ã_span><span class="sy.".".">,Ã_span>cmap<span class="sy.".".">=Ã_span><span class="st.".".">'Greys'Ã_span><span class="sy.".".">,Ã_span>alpha<span class="sy0">=</span><span class="nu0">0.3</span><span class="br0">&#;</span><span class="sy0">;</span><br /> <span class="co1"># plo4�constraints</span><br /> x0 <span class="sy0">=</span> np.<span class="m">linspaceÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span �<span class="nu0">5</span><span class="sy0">,</span> <span class="nu."."."(�ÇŒ)�000</span><span class="br0">&#;</span><br /> y0 <span class="sy0">=</span> <span class="nu.".".">6Ã_span>-<span class="nu0">1.5</span>*x0   <span class="co1"># 6*x+4*y<(�O)�4</span><br /> y1 <span class="sy0">=</span> <span class="nu."."."Ü0ë0È0Ã_span>-<span class="nu0">0.5</span>*x0   <span class="co1"># x(�kp)�*y<=6Ã_span><brëx> ÀÐ <span class="sy.".".">=Ã_span �<span class="nu0">1</span>+x0    <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�  <span class="co1"># -x+y<=1</span><br /> y2 2 <span class="sy0">=</span> <span class="br.".".">(Ã_span>x0*<span class="nu0">0</span><span class="br0">&#;</span> + <span class="nu."."."(�ÇŒ)�Ã_span �<span class="cÝ"># 9�<= 2</span><br /> y2 2 2 <span class="sy0">=</span> x0*<span class="nu0">0</span>  <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�   <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�<span class="cÝ"># 8�>= 0</span><br /> plt.<span class="me1">plot</span><span class="br0">(</span>x0<span class="sy0">,</span> y0<span class="sy.".".">,Ã_span �label<span class="sy.".".">=Ã_span>r<span class="st0">'$6x+4y<span class="es.".".">\l</span>eq24$'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">plotÃ_span><span class="br.".".">(Ã_span>x0<span class="sy.".".">,Ã_span �,<span class="sy0">,</span> label<span class="sy0">=</span>r<span class="st.".".">'$x(�kp)�y<span class="es.".".">\l</span>eq6$'</span><span class="br0">&#;</span><br /> plt.<span class="me1">plot</span><span class="br0">(</span>x0<span class="sy0">,</span> y2<span class="sy.".".">,Ã_span �label<span class="sy.".".">=Ã_span>r<span class="st0">'$-x+y<span class="es0">\lÃ_span>e$'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">plotÃ_span><span class="br.".".">(Ã_span>x0<span class="sy.".".">,Ã_span �<span class="nu0">2</span>*np.<span class="m">ones_like</span><span class="br0">(</span>x0<span class="br0">&#;</span><span class="sy0">,</span> label<span class="sy0">=</span>r<span class="st.".".">'$y<span class="es0">\lÃ_span>e|·$'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">plotÃ_span><span class="br.".".">(Ã_span>x0<span class="sy.".".">,Ã_span �y4<span class="sy0">,</span> label<span class="sy0">=</span>r<span class="st.".".">'$x<span class="es0">\gÃ_span>eq0$'Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">plotÃ_span><span class="br.".".">(Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span><span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."Ü0ë0È0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">,Ã_span �label<span class="sy.".".">=Ã_span>r<span class="st0">'$y<span class="es.".".">\g</span>eq0$'</span><span class="br0">&#;</span><br /> x6�<span class="sy0">=</span> <span class="br.".".">[Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."(�ÇŒ)�Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."Ü0ë0È0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">4Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">0Ã_span><span class="br.".".">]Ã_span><span class="sy.".".">;Ã_span �yv <span class="sy.".".">=Ã_span �<span class="br0">&#;</span><span class="nu0">0</span><span class="sy0">,</span><span class="nu0">1</span><span class="sy0">,</span><span class="nu0">2</span><span class="sy0">,</span><span class="nu0">2</span><span class="sy0">,</span><span class="nu0">1.5</span><span class="sy0">,</span><span class="nu0">0</span><span class="sy0">,</span><span class="nu0">0</span><span class="br0">&#Ø0ë0Ä0;</span><br /> plt.<span class="me1">plot</span><span class="br0">(</span>xv<span class="sy0">,</span>yv<span class="sy0">,</span><span class="st0">'ka--'Ã_span><span class="sy.".".">,Ã_span>markersize<span class="sy.".".">=Ã_span><span class="nu.".".">7Ã_span><span class="sy.".".">,Ã_span>linewidth<span class="sy0">=</span><span class="nu0">2</span><span class="br0">&#;</span><br /> <span class="kw1">forÃ_span �i <span class="k)">inÃ_span �<span class="k(Ì">range</span><span class="br0">(</span><span class="k(Ì">len</span><span class="br0">(</span>xv<span class="br0">&#;</span><span class="br0">&#;</span>:<brëx>    <�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�s�i�g�n�"�>�; <�/�s�p�a�n�>�<�s�p�a�n� �c�l�a�s�s�=�"�n�a�k�e�d�_�a�u�r�a�l�"�>�(�èlŠ)�<�/�s�p�a�n�>�plt.<span class="m">textÃ_span><span class="br.".".">(Ã_span>xv<span class="br.".".">[Ã_span>i<span class="br0">&#Ø0ë0Ä0;</span>+<span class="nu.".".">0.1Ã_span><span class="sy.".".">,Ã_span>yv<span class="br.".".">[Ã_span>i<span class="br0">&#Ø0ë0Ä0;</span>+<span class="nu.".".">0.1Ã_span><span class="sy.".".">,Ã_span>f<span class="st0">'({xv[i]V�{yv[i]})'</span><span class="br0">&#;</span><br /> <span class="co1"># objective contours</span><br /> CS <span class="sy0">=</span> plt.<span class="me1">contourÃ_span><span class="br.".".">(Ã_span>x<span class="sy0">,</span>y<span class="sy.".".">,Ã_span>obj<span class="sy0">,</span>np.<span class="me1">arange</span><span class="br0">(</span><span class="nu0">1</span><span class="sy0">,</span><span class="nu0">7</span><span class="br0">&#;</span><span class="br0">&#;</span><br /> plt.<span class="me1">clabel</span><span class="br0">(</span>CS<span class="sy0">,</span> inline<span class="sy.".".">=Ã_span><span class="nu."."."´Ã_span><span class="sy.".".">,Ã_span �fontsize<span class="sy0">=</span><span class="nu0">10Ã_span><span class="br.".".">)Ã_span><brëx> <span class="cÝ"># optimal point</span><br /> plt.<span class="me1">plot</span><span class="br0">(</span><span class="br0">&#;</span>xopt<span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">,</span><span class="br0">&#;</span>yopt<span class="br0">&#Ø0ë0Ä0;</span><span class="sy0">,</span>marker<span class="sy0">=</span><span class="st0">'o'</span><span class="sy0">,</span>color<span class="sy.".".">=Ã_span><span class="st.".".">'orange'</span><span class="sy0">,</span>markersize<span class="sy0">=</span><span class="nu0">10Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">xlimÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu.".".">5Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">;Ã_span �plt.<span class="m">ylimÃ_span><span class="br.".".">(Ã_span><span class="nu.".".">0Ã_span><span class="sy.".".">,Ã_span><span class="nu."."."Ü0ë0È0Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">;Ã_span �plt.<span class="m">gridÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><span class="sy.".".">;Ã_span �plt.<span class="m">tight_layoutÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">legendÃ_span><span class="br.".".">(Ã_span>loc<span class="sy0">=</span><span class="nu0">1</span><span class="br0">&#;</span><span class="sy0">;</span> plt.<span class="me1">xlabel</span><span class="br0">(</span><span class="st0">'x'</span><span class="br0">&#;</span><span class="sy0">;</span> plt.<span class="me1">ylabel</span><span class="br0">(</span><span class="st0">'y'</span><span class="br0">&#;</span><br /> plt.<span class="me1">savefigÃ_span><span class="br.".".">(Ã_span><span class="st.".".">'plot.png'</span><span class="sy0">,</span>dpi<span class="sy.".".">=Ã_span><span class="nu."."."Ü0ë0È000Ã_span><span class="br.".".">)Ã_span><brëx> plt.<span class="m">showÃ_span><span class="br.".".">(Ã_span><span class="br.".".">)Ã_span></div></div> <div class='sourceblocklink'><a href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming?action=sourceblock&num=7' type='text/plain'>[$[Ge4�Code]]Ã_a>Ã_div> </div> </div> <p class='vspace'><strong>Additional ResourcesÃ_strong> Ã_p> <di6�class='vspace'></div><ul><li><a class='urllink' href='https:å]apmonitor.coÛwwiki/index.php/MaiâwIntegerProgramming' rel='nofollow'>Integer Linear Programming (ILP)</a> Ã_li><li><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/NonlinearProgramming'>Nonlinear Programming (NLP)Ã_a> </li><li><a class='urllink' href='httpsa_/apmonitor.com/wik}vindex.ph:yMain/IntegerBinaryVariables' rel='nofollow'>Mixed-Integer Nonlinear Programming (MINLP)</a> Ã_li><li><a class='urllink' href='https:å]apmonitor.coÛwme57ô^index.ph:yMain/DiscreteOptimization' rel='nofollow'>Discrete Optimization</a> Ã_li></ul> </div> Ã_div> </div> </div> <div id="subcontent"> <div id='subcontent_bg'>À-- -->Ã_div> <!--pagerightfmt--> <0�class='sidehead' �Course Information </p><ul><li><a class='wikilink' href='http:å]apmonitor.coÛwpduindex.ph:yMain/HomePage'>Course OverviewÃ_a> </li><li><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwCourseSyllabus'>SyllabusÃ_a> </li><li><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwCourseSchedule'>BYU ScheduleÃ_a> </li><li><a class='wikilink' href='httpa_/apmonitor.com/pdc/index.php/MaiâwProcessControlSchedule'>PDC ScheduleÃ_a> </li><li><a class='urllink' href='httpsa_/github.coÛwAPMonito¾ypdc' rel='nofollow'>Course on GitHubÃ_a> </li><li><a 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