Answer to Question #236166 in Operations Research for Bebe

Question

The Colonial pottery company produces two products daily – bowls and mugs. The Company has limited amounts of resources used in the production of these products – clay and labor. For the bowl, it would require 1 hr of labor and 4 lb. of clay to produce while the mug requires 4 lb of clay and 2 hrs of labor to produce. The items are sold at $4/unit for the bowl while $5/unit for the mug. There are 40 hrs of labor and 120 lbs. of clay available each day for production. What should be the optimal number of products to have maximum profit?

Answer 1

From the question, we will form a linear program.

Let $x$ and $y$ represent the number of bowls and mugs produced

The cost of production which is to be maximized is given as:

$$z=4x+5y$$

We have two constraints which are clay and labor for production.

Constraint on labor

$$x+2y\leq40$$

Constraint on clay

$$4x+4y \leq 120$$

We have the linear program as below:

$$\text{Max: } z=4x+5y\\ \text{Subject to: } x+2y \leq 40\\ 4x+4y \leq 120$$

Using graphical method to solve, we have the graph as below:

We have four points as our feasible solution. Substitute into the objective function to get the optimal solution.

$A(0,20), z=4(0)+5(20)=100\\ B(0,0), ~~z=4(0)+5(0)=0\\ C(30,0),z=4(30)+5(0)=120\\ D(20,10), z=4(20)+5(10)=130$

We have the maximum at $D(20,10)$ with value 130.

Hence, the optimal number of products to be produced are 20 bowls and 10 mugs to have a profit of 130.