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I am following a course on linear optimization; and unfortunately my brain is already stuck on the first toy example. The example is as follows:

Your company produces Duplo chairs and tables. Profit per chair is 15, profit per table is 20 dollars. We only have 8 small bricks available and 6 big bricks. Each chair needs 1 big brick and 2 small bricks; each table needs 2 big bricks and 2 small bricks. How many chairs and tables should you produce to maximixe profit?

Then, they move on to developing a mathematical model. We introduce two decision variables

  • $X_1$: The number of chairs to produce
  • $X_2$: the number of tables to produce

To solve this problem, we need to determine the optimal solution of the following linear programming model:

Maximize $15 x_{1}+20 x_{2}$ (Total profit)

Subject to:

$x_{1}+2 x_{2} \leq 6$ (only 6 big bricks)

$2 x_{1}+2 x_{2} \leq 8$ (Only 8 small bricks)

$x_{1} \geq 0, x_{2} \geq 0$ (Produce non-negative quantities)

I get that $15 x_{1}+20 x_{2}$ defines the total profit.

What I don't understand is the following: Why do we express the first two constraints as such? What I don't get is why $x_{1}+2 x_{2} \leq 6$ and $2 x_{1}+2 x_{2} \leq 8$ must hold. For me this reads as the "ingredients" of a chair, and the "ingredients" of a table. Since one chair needs 1 small brick and 2 big bricks, and 1 table needs 2 small bricks and 2 big bricks.

How I see it now, according to these 2 equalities, $x_{1}$ is the number of big bricks and $x_{2}$ is the number of small bricks. $x_{1}$ must be smaller then 6, as we only have 6 big bricks, and $x_{2}$ must be smaller then 8, as we only have 8 small bricks. But that is not how the constraints are specified, and if we specify them in that way (e.g. $x_{1} \leq 6$ and $x_{2} \leq 8$), we lose the power of specifying how many ingredients go into one table or one chair. I don't understand why x1 is the number of chairs, and x2 is the number of tables. How do these constraints work?

I hope I explained my doubts clear enough for someone to help me.

amWhy
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Psychotechnopath
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  • It might help to use letters corresponding to what they represent. For example $C$ for the number of chairs produced, $T$ for the number of tables produced, $s$ for the number of small bricks used, and $b$ for the number of big bricks used. The total number of big bricks used is $C+2T=b$ and since we are limited in the number of big bricks used we have $b\leq 6$. Together, this implies $C+2T\leq 6$. Similarly the next reads $s=2C+2T\leq 8$ and so on... Seeing it written this way it should hopefully be clearer. Using $x_1,x_2$ you can easily forget what it represents. – JMoravitz Feb 11 '20 at 13:23
  • This explanation clears everything up. I knew i was overlooking something simple! x1 and x2 in the first inequality represent the big bricks used, and x1 and x2 in the second represent the small bricks used. The only unfortunate thing in this example is that the formula is the same for the chair and the number of big bricks used, and the table and the small bricks used. If you craft it into an answer I will make sure to accept it! – Psychotechnopath Feb 11 '20 at 15:03

1 Answers1

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Let me rephrase the problem as the numbers make it hard to understand. Lets say you build another set of chairs and tables. Now Each chair needs 1 big brick and 2 small bricks; each table needs 3 big bricks and 4 small bricks.

You would still maximize $15x_1+20x_2$, as $x_1$ and $x_2$ describe the numbers of chairs and tables you build.

Now the first constraint is as follows:

$$1x_1+3x_2\leq 6$$

because for each chair you need 1 big brick and for each table you need 3. With a total of 6 big bricks available, you can not use more than you have.

Your misunderstanding comes from the fact that the number of small bricks needed per chair is the same as the number of big bricks needed per table. Thus, when changing the numbers to be different, hopefully the idea is clearer.

Nurator
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