Formulate but do not solve the following exercise as a linear programming problem. Kane Manufacturing has a division that produces two models of fireplace grates, x units of model A and y units of model B. To produce each model A requires 3 lb of cast iron and 8 min of labor. To produce each model B grate requires 5 lb of cast iron and 3 min of labor. The profit for each model A grate is $2.00, and the profit for each model B grate is $1.50. If 800 lb of cast iron and 1200 min of labor are available for the production of grates per day, how many grates of each model should the division produce per day to maximize Kane's profits P in dollars?

Maximize P = subject to the constraints

cast iron =

labor =

model A =

y = 0

z - 2*x - 1.5*y = 0 maximize

subject to:

3*x + 5*y ≤ 800

8*x + 3*y ≤ 1200

x, y > 0

Step-by-step explanation:

Formulation:

Kane Manufacturing produce x units of model A (fireplace grates)

and y units of model B

quantity Iron cast lbs labor (min) Profit $

Model A x 3 8 2

Model B y 5 3 1.50

We have 800 lbs of iron cast and 1200 min of labor available

We need to find out how many units x and units y per day to maximiza profit

First constraint Iron cast lbs 800 lbs

3*x + 5*y ≤ 800 3*x + 5*y + s₁ = 800

Second constraint labor 1200 min available

8*x + 3*y ≤ 1200 8*x + 3*y + s₂ = 1200

Objective function

z = 2*x + 1.5*y to maximize z - 2*x - 1.5*y = 0

x > 0 y > 0

The first table is (to apply simplex method)

z x y s₁ s₂ Cte

1 - 2 - 1.5 0 0 0

0 3 5 1 0 800

0 8 3 0 1 1200