Maximizing X & Y

I want to maximize the value of X & Y in the below equation. The sum of X & Y cannot exceed 260.

55 * X + 100 * Y

[DGM]

That's not an equation. That's an expression. It has to be equal to something in order to be an equation.

You'll have to specify what it's supposed to be equal to, and you'll have to clarify what you mean by "maximize the value of x and y". You can maximize the value of either one at the expense of the other, or you can maximize their sum (or some other combined metric).

If I'm mistaken in my interpretation of the problem, then you'll have to describe it sufficiently that guessing isn't required.

[Suraj Patil]

Sorry. forgot to add

55 * X + 100 * Y >= 12,000

[John D'Errico]

This is a basic linear programming problem, one typically assigned to students. So you might use linprog. Are you asking how to do what appears to be a (your) homework problem? Note that this problem has no maximum as you have posed it so far, since the problem is not sufficiently bounded. So if the sum of x+y cannot exceed 260, then you will maximize the term

55 * X + 100 * Y

by making y approach inf, as x goes to -inf. Since the coefficient of y is the larger, even though the sum will be bounded by 260, the above expression can be made arbitrarily large.

And that suggests your question is not a homework assignment, but merely an improperly formulated problem in mathematics, one with no solution.

If you then counter, oh, I forgot to add one more piece of information, that X and Y must both be non-negative values, then I would point out that then X must be zero, and Y as large as possible, therefore Y = 260. That must maximize the indicated expression.

55*0 + 100*260 = 26000.

So in any case, your problem reduces back to to something that would be assigned to a student to solve, probably using graphical methods or simple linear programming, but MATLAB is not necessary.

[Suraj Patil]

Dear John,

Thank you for your response and the feedback that MATLAB is not necessary and it is a Linear programmaing. But rephrasing my question so far:

I am trying to maximize 55 * X + 100 * Y >= 12,000 with the constraints that

1. sum of X & Y cannot exceed 260
2. X & Y must not be less than zero
3. X & Y must be greater than zero

Answer 1

[DGM]

Okay, well that still leaves the question of what we're actually trying to maximize. Let's look at a graph of what's going on:

The contour plot is x+y, demonstrating the constraint of x+y<=260. The red overlay is 55x+100y>=12E3. All solutions to this optimization problem lie in the illustrated triangular region.

If there are no further constraints on x or y, there is no unique and finite maximizing x,y. If we're trying to maximize x+y, then the maximal sum is 260. There are an infinite number of solutions along the edge of x+y=260 from the point of intersection, extending northwest to [-inf inf].

If instead we're trying to maximize one or the other, then the answer tends to follow the same trajectory. [130 130] is a solution, just as is [1000000 -999740]. You can pick any arbitrarily large value for y and there is an x which satisfies the given constraints. If it's asserted that x >=0, then the largest satisfying value of x or y occurs for x at the point of intersection.

I mentioned the point of intersection. If it's relevant to the actual solution, we can find it.

 syms xx yy aa
e1 = -0.55*xx + aa/100;
e2 = -xx + 260;
ax = solve(e1 == e2,xx)
ay = subs(e1,xx,ax)
double(subs([ax ay],aa,12E3))

ans =
  311.1111  -51.1111

Though I don't know if this is actually helpful.

[Suraj Patil]

Dear DGM,

Thank you for your reply. X & Y must not be less than zero. X & Y must be greater than zero. I am trying to maximize 55 * X + 100 * Y >= 12,000 with the constraints that

1. sum of X & Y cannot exceed 260
2. X & Y must not be less than zero
3. X & Y must be greater than zero

[DGM]

Constraints 2 and 3 are redundant.

• 55 * X + 100 * Y >= 12,000 (what's trying to be maximized)
• X + Y <= 260
• X > 0
• Y > 0

Describes the current degree to which the problem statement has been revealed. If that is the totality of the conditions that apply, then It should be simple.

Since you're trying to maximize f = 55x + 100y instead of trying to maximize "x and y", then simply observe that the gradient of that function is orthogonal to the red boundary line in the plot. Where is the x+y=260 constraint line furthest from the 55x+100y=12000 line?

And of course, since x must be >0, the maximizing coordinates only approach [0 260].

[Suraj Patil]

Dear DGM,

I think I know where you are going with the answer. Let me use a graphical representation to understand my possible set of solutions where X + Y is less than 260 and also 55x+100y >= 12000. X & Y of course reflect two real life variables for a business.

Warm regards

[Walter Roberson]

[X, Y] = ndgrid(linspace(eps, 260, 500));
Z = 55*X + 100*Y;
Z(X+Y > 260) = nan;
Z(Z < 12000) = nan;
surf(X, Y, Z, 'edgecolor', 'none')
xlabel('X'); ylabel('Y'); zlabel('Z')

So the maximum is where X is as close as possible to 0 and Y = 260 - X .

Which makes sense, since Y is weighted nearly twice as highly as X is in the sum, so you want as much Y as you can get, at the expense of X.