"Shirts Happen": An application in resource allocation and linear programming
The Shirts Happen Clothing Company manufactures men's shirts and women's blouses. The
production process includes cutting, sewing, and packaging. The table below gives the labor re-
quirements (in minutes) for each type of garment:
Men's shirts
Women's blouses
Minutes per garment
Cutting Sewing Packaging
30
80
12
60
60
4
The maximum number of labor hours available per day at Shirts Happen are 25 hours for
cutting, 35 hours for sewing, and 5 hours for packaging.
1. Let z represent the number of men's shirts produced per day by the Shirts Happen Clothing
Company, and let y represent the number of women's blouses produced per day. Write a
system of five linear inequalities involving aand y that, when solved, give the set of all
ordered pairs (r, y) of number of men's shirts and number of women's blouses that Shirts
Happen can produce per day, given the constraints above. Next to each inequality, write a
brief description or interpretation of its meaning.
2. Carefully graph the system of linear inequalities on a separate sheet of graph paper and then
shade the feasible region. Is it bounded or unbounded? Identify and label on your graph the
coordinates of each of the corner points. Attach the graph along with your work.
3. For each of the following solutions, determine whether or not it is a feasible solution to the
Shirts Happen Clothing Company problem. Show your work and explain your reasoning.

This inequality represents the constraint on the total minutes required for cutting, which should be less than or equal to the available cutting labor hours.

Sewing constraint:

80z + 60y ≤ 35 × 60

This inequality represents the constraint on the total minutes required for sewing, which should be less than or equal to the available sewing labor hours.

Packaging constraint:

12z + 4y ≤ 5 × 60

This inequality represents the constraint on the total minutes required for packaging, which should be less than or equal to the available packaging labor hours.

Non-negativity constraint:

z ≥ 0

y ≥ 0

These inequalities ensure that the number of men's shirts and women's blouses produced cannot be negative.

Integer constraint:

z, y are integers

This inequality ensures that the number of men's shirts and women's blouses produced should be whole numbers.

2. To graph the system of linear inequalities, plot the lines corresponding to each inequality and shade the feasible region where all the inequalities are satisfied.

The feasible region will be bounded since all variables (z and y) are non-negative and there are finite constraints.

To find the corner points, solve the system of equations formed by the intersection of the lines corresponding to the inequalities.

3. To determine whether a given solution is feasible, substitute the values of z and y into the inequalities and check if all the inequalities are satisfied. If all the inequalities are satisfied, the solution is feasible; otherwise, it is not feasible.