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Consider

$1\rightarrow{}x+2y\leq{500}$,

$2\rightarrow{}2x +y \leq{520}$,

$3\rightarrow{}2x+5y \leq{}1200$,

$4\rightarrow{}x \geq{}0$,

$5\rightarrow{}y \geq{}0$,

The above is the set of inequalities that generates a linear programming problem whose objective function is

$f(x,y)=9x +12y$,

I need to maximize the objective function, but the graphics are unclear when it comes to the points,

a) The problem to be solved is to produce a plot with enough detail with the coordinates of the solution polygon, drawn it by hand, with geogebra, symbolab and it is not possible to appreciate it well

b) I have never solved a linear programming problem with Mathematica and I would like to know how it can be done using mathematical syntax.

If you can help me, I would appreciate it very much

Notes

I do not know the symplex method.
This is not homework or a work assignment, it is for personal knowledge.

Update

error corrected

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  • 1
    $\begingroup$ Look at the condition: x,y <=0 and you want to maximize the objective function . The largest value the objective function can have under this assumption is zero. Therefore, I think there is something wrong with your question. $\endgroup$ Nov 7 '20 at 21:03
  • $\begingroup$ @Daniel Hube see correction above $\endgroup$
    – rpujadas
    Nov 7 '20 at 21:27
  • $\begingroup$ There is a function LinearProgamming" in MMA for minimization. In your case for maximization you would define: c= -{..},m=-{{},{},..}; b=- {..}. Note the minus signs. $\endgroup$ Nov 8 '20 at 8:38
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obj = 9 x + 12 y;
c1 = x + 2 y <= 500;
c2 = 2 x + y <= 520;
c3 = 2 x + 5 y <= 1200;
c4 = x >= 0;
c5 = y >= 0;

{max, argmax} = {#[[1]], {x, y} /. #[[2]]} & @
   NMaximize[{obj, And[c1, c2, c3, c4, c5]}, {x, y}]
 {3540., {180., 160.}}
Show[RegionPlot[And[c1, c2, c3, c4, c5], {x, 0, 300}, {y, 0, 300}, 
  ColorFunction -> (ColorData[{"TemperatureMap", {0, max}}][9 # + 12 #2] &), 
  ColorFunctionScaling -> False], 
 Normal[ContourPlot[obj, {x, 0, 300}, {y, 0, 300}, 
    ContourShading -> None, Contours -> Subdivide[0, max, 5]]] /. 
  Tooltip[l : {__, Line[x_, ___]}, t_] :> {l, Text[t, Mean[x]]}, 
 ListPlot[{Callout[argmax, argmax]}, PlotStyle -> Directive[Red, PointSize[Large]]]]

enter image description here

A variation: You can add a legend showing objective function value over the feasible region and use custom arrowheads to place and orient the contour labels along the contour lines:

Show[RegionPlot[And[c1, c2, c3, c4, c5], {x, 0, 300}, {y, 0, 300}, 
  ColorFunction -> (ColorData[{"TemperatureMap", {0, max}}][ 
      9 # + 12 #2] &), ColorFunctionScaling -> False,
   PlotLegends ->
    BarLegend[{ColorData[{"TemperatureMap", {0, max}}], {0, max}}, 
     LegendMarkerSize -> 400, LegendFunction -> "Frame", 
     LegendLabel -> "obj"]], 
 Normal[ContourPlot[obj, {x, 0, 300}, {y, 0, 300}, 
    ContourShading -> None, Contours -> Subdivide[0, max, 5]]] /. 
  Tooltip[{dir__, Line[x_, ___]}, t_] :> {dir, 
    Arrowheads[{{Automatic, .3, 
       Graphics @ Text[Framed[Style[t, 16, Bold, Opacity[1], Black], 
          Background -> White, FrameStyle -> None]]}}], 
    Arrow[Line @ SortBy[First] @ x]}, 
 ListPlot[{Callout[argmax, Style[argmax, 16, Bold]]}, 
  PlotStyle -> Directive[Red, PointSize[Large]]], ImageSize -> 500]

enter image description here

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  • $\begingroup$ kglr__wow your first code works very well the second and third send me these errors, the graph appears but not with the color variation) "Part::partd: Part specification nmax[[1]]] is longer than depth of object. ColorData::notent: {TemperatureMap,{0,nmax[[1]]}} is not a known entity, class, or tag for ColorData. Use ColorData[] for a list of entities. (use MMA 12.1).What does the value 3540 mean in the first result? $\endgroup$
    – rpujadas
    Nov 7 '20 at 21:53
  • 1
    $\begingroup$ @rpujadas, sorry, nmax[[1]] should be max . Fixed now. $\endgroup$
    – kglr
    Nov 7 '20 at 21:57
  • $\begingroup$ @kglr_What does the value 3540 mean in the first result? $\endgroup$
    – rpujadas
    Nov 7 '20 at 22:19
  • 2
    $\begingroup$ @rpujadas - The 3540 is the maximum value of the objective function subject to the given constraints. The system can be solved exactly {max, argmax} = {#[[1]], {x, y} /. #[[2]]} &@Maximize[{obj, c1, c2, c3, c4, c5}, {x, y}] $\endgroup$
    – Bob Hanlon
    Nov 7 '20 at 22:34
  • $\begingroup$ @Bob Hanlon ,I had not understood the result format of the code. Thank you $\endgroup$
    – rpujadas
    Nov 7 '20 at 22:36

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