# Math Help - Linear programming problem concerning constraints

1. ## Linear programming problem concerning constraints

I got a LP problem that I'm stuck on.

A furniture factory makes tables and chairs. It takes 2 hours to assemble a table and 30 minutes to assemble a chair. Assembly is carried out by four people on the basis of one eight-hour shift per day. Customers buy at four chairs with each table which means that the factory has to make at most four times as many chairs as tables. The selling price is €135 per table and €50 per chair.

Formulate this as a linear programming problem to determine the daily production of tables and chairs which would maximise the total daily revenue to the factory and solve the problem using the simplex method.

My progress:

X1 = Amount of tables produced
X2 = Amount of chairs produced

Max Z = 135X1+50X2

Subject to:
120X1+30X2 ≤ 1920
4X1 ≥ X2
X1, X2 ≥ 0

Standard form:

Max Z = 135X1+50X2

Subject to:
4X1+X2+S1 = 64
4X1-X2-S2 = 0
X1, X2, S1, S2 ≥ 0

Are my constraints correct?

Thanks for your time.

2. Originally Posted by fobster
I got a LP problem that I'm stuck on.

A furniture factory makes tables and chairs. It takes 2 hours to assemble a table and 30 minutes to assemble a chair. Assembly is carried out by four people on the basis of one eight-hour shift per day. Customers buy at four chairs with each table which means that the factory has to make at most four times as many chairs as tables. The selling price is €135 per table and €50 per chair.

Formulate this as a linear programming problem to determine the daily production of tables and chairs which would maximise the total daily revenue to the factory and solve the problem using the simplex method.

My progress:

X1 = Amount of tables produced
X2 = Amount of chairs produced

Max Z = 135X1+50X2

Subject to:
120X1+30X2 ? 1920
4X1 ? X2
X1, X2 ? 0

Standard form:

Max Z = 135X1+50X2

Subject to:
4X1+X2+S1 = 1920
4X1-X2-S2 = 0
X1, X2, S1, S2 ? 0

Are my constraints correct?

Thanks for your time.
Looks good to me except the first constraint in the standard form has incorrect coefficients for X1 and X2.

3. Hello, fobster!

I simplified the language . . .

A furniture factory makes tables and chairs.
It takes 2 hours to assemble a table and 30 minutes to assemble a chair.
Assembly is carried out by four people on the basis of one eight-hour shift per day.
Customers buy at most four chairs with each table which means that
the factory has to make at most four times as many chairs as tables.
The selling price is €135 per table and €50 per chair.

Formulate this as a linear programming problem to determine the daily production of tables and chairs
which would maximise the total daily revenue to the factory
and solve the problem using the simplex method.

Those subscripts are confusing . . .

Let $T$ = number of tables produced: $T \geq 0$ [1]
Let $C$ = number of chairs produced: $C \geq 0$ [2]

The $T$ tables take a total of $2T$ hours to assemble.
The $C$ chairs take a total of $\frac{C}{2}$ hours to assemble.
So we have: . $2T + \frac{C}{2}\:\leq\:32\quad\Rightarrow\quad 4T + C \:\leq \:64$ [3]

There must be at most four chairs per table: . $C \leq 4T\quad\Rightarrow\quad 4T + C \geq 0$ [4]

Now apply the Simplex Method to the four inequalties
. . and maximize the revenue function: . $R \:=\:135T + 50C$