# Thread: Formulate LP to maximize profit

1. ## Formulate LP to maximize profit

Question:
Vicky's manufactures chairs and tables. Each chair or table must go through the painting shop and assembly shop. If the painting shop were completely devoted to painting chairs, 900 per day could be painted, whereas if it were completely devoted to painting tables, 500 per day could be painted. If the assemly shop were completely devoted to assemble the chairs, 1500 can be assembled, whereas if it were completely devoted to assemble the tables, 1100 per day can be assembled. Each chair brings a profit of 110 and each table brings a profit of 160. Formulate an LP to maximize vicky's profit. *Note, don't have to solve the LP, just define the decision variables, state the objective function and the constraints.*

i can't figure out if you only need 2 decision variables (X1 = number of chairs, X2 = number of tables)
or 4 variables (X1= number of chairs painted , X2 = number of tables painted ,X3 = number of chairs assembled, X2 = number of tables assembled.)

and other than the obvious constraints X1 <= 900, X2 <= 500 , X3 <= 1500, X4 <= 1100 i can't see how they help .

Any help would be greatly appreciated.

2. ## Re: Formulate LP to maximize profit

i think its safe to assume you are going to be painting everything you assemble and vice-versa, so:
X1=X3, X2=X4.

also, you cant have negative numbers: $X_i \geq 0$

There are 2 more constraints i can think of:
what can you say about the relationship between X1 and X2? (hint: inequality)
what can you say about the relationship between X3 and X4? (hint: inequality)

3. ## Re: Formulate LP to maximize profit

would it be
X1 + (9/5)X2 <= 900

X3 + (15/11)X4 <= 1500

4. ## Re: Formulate LP to maximize profit

Let x denote the number of chairs produced.
Let y denote the number of tables produced.

Objective function: P(x, y) = 110x + 160y

Constraints:

Painting constraint:
The boundary of this constraint would be the line connecting (900, 0) with (0, 500). The slope of this line is 500/(-900) = -5/9. Since (0, 500) is the y-intercept of this line, its equation is

y = (-5/9)x + 500

Assembly constraint:
The boundary of this constraint would be the line connecting (1500, 0) with (0, 1100). The slope of this line is (1100 - 0)/(0 - 1500) = -11/15. Since (0, 1100) is the y-intercept of this line, its equation is

y = (-11/15)x + 1100

Physical reality constraints:
x ≥ 0
y ≥ 0

From a graph of these inequalities it appears the assembly constraint is irrelevant.

This is what I came up with