Question

A factory manufactures two types of gadgets, regular and premium. Each gadget requires the use of two operations, assembly and finishing, and there are at most $120$ hours available for each operation. A regular gadget requires $1$ hour of assembly and $2$ hours of finishing, while a premium gadget needs $3$ hours of assembly and $1$ hour of finishing. Due to other restrictions, the company can make at most $70$ gadgets a week. If a profit of $\$20$ is realized for each regular gadget and $\${}30$ for a premium gadget, how many of each should be manufactured to maximize profit each week? Enter the ordered pair $(x,y).$

Answer 1

To solve this problem we can first set up a system of equations to represent the constraints and objectives of the problem.

Let:

x = number of regular gadgets produced

y = number of premium gadgets produced

Our constraints are as follows:

1) Assembly hours constraint:

1x + 3y ≤ 120

2) Finishing hours constraint:

2x + 1y ≤ 120

3) Production constraint:

x + y ≤ 70

Our objective is to maximize profit which is given by:

Profit = 20x + 30y

Now we can solve this linear programming problem using a graphical method or a mathematical method such as the simplex algorithm.

Using the graphical method we can plot the feasible region determined by the constraints and find the corner points that maximize the profit.

By graphing the constraints we can find the feasible region:

x + 3y ≤ 120

2x + y ≤ 120

x + y ≤ 70

Next we plot the lines and find the region where the shaded area satisfies all the constraints.

After finding the feasible region we evaluate the objective function at each corner point to find the maximum profit.

The corner points of the feasible region are:

(0 0)

(0 40)

(40 0)

(50 20)

(70 0)

Evaluating the profit function at each corner point we find:

Profit(0 0) = 20(0) + 30(0) = $0

Profit(0 40) = 20(0) + 30(40) = $1200

Profit(40 0) = 20(40) + 30(0) = $800

Profit(50 20) = 20(50) + 30(20) = $1600

Profit(70 0) = 20(70) + 30(0) = $1400

Therefore the maximum profit is $1600 when 50 regular gadgets and 20 premium gadgets are produced.

Thus the maximum profit that can be realized is $1600.