How to find maximum ConditionalExpression among all solutions

Question

For example I want to find solution with maximum value among solutions (and possibly plot it):

answers = Piecewise[List @@@ Last @@@ N @Solve[{
       z == 40 x + 50 y,
       6 x + 10 y <= 672,
       0.25 x + 0.15 y <= 24,
       1.5 y <= 42,
       0 <= x,
       0 <= y
       }, {z}, Integers]]

In this case it is : {4230.,x\[LongEqual]87.\[And]y\[LongEqual]15.}

Answer 1

I'm not sure why you are using piecewice to hold the answers, but you could just use sort and get the last element:

 answers = 
  List @@@ Last @@@ 
  N@Solve[{z == 40 x + 50 y, 6 x + 10 y <= 672, 
  0.25 x + 0.15 y <= 24, 1.5 y <= 42, 0 <= x, 0 <= y}, {z}, 
  Integers]

 Sort[answers][[-1]]

 (*=> *) {4230., x == 87. && y == 15.}

As for plotting the solution points, you could turn the conditional expressions into individual points and plot those using ListPlot3D:

 zxy = answers //. {{a_, Or[b_, c_]} :> Sequence[{a, b}, {a, c}] , 
 And -> Sequence, Equal[_, b_] :> b};

 ListPlot3D[zxy[[1 ;;, {2, 3, 1}]], AxesLabel -> {"x", "y", "z"} ]

This doesn't work for arbitrary answer lists, but works in your case. If you have results with different logic constructs you can modify the replacement rules accordingly.

Answer 2

Could use Maximize.

Maximize[{z, z == 40*x + 50*y,
       6 x + 10 y <= 672,
       x/4 + 3*y/20 <= 24,
       3*y/2 <= 42,
       0 <= x,
       0 <= y}, {x,y,z}, Integers]

(* {4230, {x -> 87, y -> 15, z -> 4230}} *)

Answer 3

 answers[[1, -1]]
 (* {4230., x == 87. && y == 15.}   *)

EDIT: An alternative series of replacements to get the data for plotting:

pltdata =  (List @@@ Last @@@ N@
     Solve[{z == 40 x + 50 y, 6 x + 10 y <= 672, 
       0.25 x + 0.15 y <= 24, 1.5 y <= 42, 0 <= x, 0 <= y}, {z}, Integers] /.
  {And -> List, Or -> List, Equal[_, a_] :> a} //
  If[Depth[#] == 3, Reverse[#], Sequence @@ Reverse /@ Thread[#, List, 2]] & /@ # &) /.
  {{a_, b_}, c_} :> {a, b, c};

ListPointPlot3D[pltdata, ColorFunction -> (Hue[#3] &), BoxRatios -> 1]