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Linear Programming with PythonÃ_title>
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<È><a href='httpa_/apmonitor.com/pdc/index.php/MaiâwLinearProgramming'>Linear Programming with Python</a></h1>
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<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.coÛwpd uuploadtzMain/learn2.png' alt='' />Ã_a>Ã_th><th align='center'><a class='wikilink' href='http:å]apmonitor.coÛwpd uindex.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.coÛwpd uuploadtzMain/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>
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<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.coÛwpd uuploadtzMain/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.coÛwpd uindex.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.coÛwpd uindex.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.coÛwpd uindex.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.coÛwpd uindex.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>
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