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I have an exercise to solve, and it is a constrained optimization problem.

Here it is:

"A company makes large championship trophies for youth athletic leagues. At the moment they are planning production for fall sports: football and soccer. Each football trophy has a wood base, an engraved plaque, and a large brass football on top, and returns 12 euro prot. Soccer trophies are similar except that a brass soccer ball is on top, and unit prot is only 9 euro. Since the football has asymmetric shape, its base requires 4 board feet of wood, the soccer base requires only 2 board feet. At the moment there are 1000 brass footballs in stock, 1500 soccer balls, 1750 plaques, and 4800 board feet of wood. Assuming that all that are made can be sold: Formulate a linear programming model to determine an optimal product mix to maximize the profit. Use the decision variables:

- $x_1$: number of football trophies to produce

- $x_2$: number of soccer trophies to produce"

I guess profit will be:

$P = 12x_1 + 9x_2$

Well, the first thing that came to my mind is to set 6 variables (three for $x_1$ and three for $x_2$) defining the three different elements that need each kind of trophy to be produced:

$x_1 = y_1 + z_1 + 4t_1$

$x_2 = y_2 + z_2 + 2t_2$

$y_1:$ brass footballs

$y_2:$ brass soccer balls

$z_1:$ football plaques

$z_2:$ soccer plaques

$t_1:$ football board feet

$t_2:$ soccer board feet

And also, four constraints:

$y_1 ≤ 1000$

$y_2 ≤ 1500$

$z_1 + z_2 ≤ 1750$

$t_1 + t_2 ≤ 4800$

I don't thing all of this is correct, it looks messy, so here I'm asking for help. Thank you!

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1 Answer 1

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You know what each unit of a soccer trophy or football trophy requires. Thus you only need two variables. A soccer trophy needs $4$ board feet of wood and a soccer trophy needs $2$ board feet of wood. And $4800$ board feet of wood are available. Thus the first constraint is

$2x_1+4x_2\leq 4800$

For each unit both require one plaque. And $1750$ plaqes are available. The constraint then is

$x_1+x_2\leq 1750$

A unit football trophy requires 1 brass football ball. For each unit a soccer trophy requires 1 brass soccer ball. There are $1000$ brass footballs and $1500$ soccer balls in stock:

$1\cdot x_1\leq 1000$

$1\cdot x_2\leq 1500$

I agree to your objective function. Finally we have to ensure that the model consider only non negative amounts of soccer an football trophies:

$x_1,x_2\geq 0$

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  • $\begingroup$ Oh, I understand what you've said. It was really easy, I feel so dumb... My mind got stuck thinking about "it's impossible that I can relate everything to $x_1$ and $x_2$!". Thank you! $\endgroup$ Commented May 20, 2016 at 11:59
  • $\begingroup$ At the beginning I often made this kind of substitutions as well. I don´t think it was dumb. It only shows that we do not run things by the rule book. $\endgroup$ Commented May 20, 2016 at 12:04
  • $\begingroup$ Yeah, I guess it's alright. I learned this a week ago. One more thing, is there any way to add the "SOLVED" status or just I add it in the post title? $\endgroup$ Commented May 20, 2016 at 12:09

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