MATH 1324: Mathematics for Business & Social Sciences - Signature Assignment

Mathematics Core Learning Outcome assessed: Empirical and Quantitative Reasoning

Course Learning Outcome assessed: Formulate and solve linear programming problems by graphical methods

Directions: Please read and complete the following on a separate sheet of paper.

Clearly explain your answers, using your own words

"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 requirements (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 a 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 a and y that, when solved, give the set of all ordered pairs (2,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. (a). 5 men's shirts per day, 20 women's blouses per day (b). 25 men's shirts per day, 25 women's blouses per day (c). 0 men's shirts per day, 0 women's blouses per day (d). -10 men's shirts per day, -20 women's blouses per day (e). 10 men's shirts per day, 10 women's blouses per day (f). 18 men's shirts per day, 14 women's blouses per day

4. 4. Suppose that each men's shirt produced and sold yields a $25 profit and each women's blouse produced and sold yields a $50 profit.

   (a). An isoprofit line is a line of solution points (2,y) within the feasible region that each yield the same profit. Find two feasible solutions to the Shirts Happen Clothing Company problem that produce a total profit of $200, and then find two feasible solutions that produce a total profit of $500. Explain your answers, and then draw these two isoprofit lines on your graph - one for a total profit of $200, and the other for a total profit of $500. (b). Finding optimal solution (s): How many men's shirts and women's blouses should the Shirts Happen Clothing Company produce each day so as to maximize its daily profit? What is the company's maximum daily profit as a result? Hint: There are actually multiple optimal solutions. List and explain two of them. Would there be a reason for the Shirts Happen Clothing Company to favor one optimal solution over another? Clearly explain your reasoning along with any additional assumptions that need to be made in determining an optimal solution to this problem.

MATH 1324: Mathematics for Business \& Social Sciences - Signature Assignment Mathematics Core Learning Oufcome assessed: Empirical and Quantitative Ressoning Course Learning Outcome assessed: Formulate and solve linear programming problems by graphical methods Directions: Please read and complete the following on a separate sheet of paper. Clearly explain your answers, using your own words. "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, seaing, and packiging. The table below gives the labor requirements (in minutes) for each type of garment: The maximum number of labor hours available per day at Slairts Happen are 25 hours for eutting, 35 hours for sewing, and 5 hours for packaging. 1. Let x 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 x and y that, when solved, give the set of all ordered pairs (x,y) of number of men's shirts and number of woenen's blouses that Shirts Happen can produce per day, given the constraints ahove. Next to each inequality, write a brief description or interpretation of its meaning. 2. Carefully graph the system of linear inequalities on a sepurate 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, deternine whether or not it is a foasible solution to the Shirts Happen Clothing Coanpany problens. Show your work and explain your rensoning. (a). 5 men's shirts per day, 20 women's blouses per day (b). 25 men's shirts per duy, 25 women's blouses per day (c). 0 men's shirts per dny, 0 women's blouses per day (d). 10 men's shirts per day, 20 woneen's blouses per day (e). 10 men's shirts per day, 10 wotnen's blouses per day (f). 18 men's shirts per day, 14 women's blouses per day 4. Suppose that each men's shirt produced and sold yields a $25 profit and cach women's blouse produced and sold yields a 850 protit. (a). An isoprofit line is a line of solution points (x,y) within the feasible region that each yield the same profit. Find two feasible solutions to the Shirts Happen Clothing Cocnpany problem that produce a total profit of $200, and then find two fensible solutions that produce a total profit of 8500 . Explain your answers, and then draw these two isoptofit lines on your graph - one for a total profit of $200, and the other for a total profit of 8500 . (b). Finding optimal solution(s): How many men's shirts and women's blouses should the Shirts Happen Clothing Conapany produce each day so as to marimise its daily peofit? What is the company's maximam daily proft as a result? Hint: There are actually multiple optimal solutions. List and explain two of them. Woeld there be a resson for the Shirts Happen Clothing Company to favor one optimal solution over another? Clearly explain your reasoning along with any additional assumptions that need to be made in deternining an optimal solution to this problem. MATH 1324: Mathematics for Business \& Social Sciences - Signature Assignment Mathematics Core Learning Oufcome assessed: Empirical and Quantitative Ressoning Course Learning Outcome assessed: Formulate and solve linear programming problems by graphical methods Directions: Please read and complete the following on a separate sheet of paper. Clearly explain your answers, using your own words. "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, seaing, and packiging. The table below gives the labor requirements (in minutes) for each type of garment: The maximum number of labor hours available per day at Slairts Happen are 25 hours for eutting, 35 hours for sewing, and 5 hours for packaging. 1. Let x 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 x and y that, when solved, give the set of all ordered pairs (x,y) of number of men's shirts and number of woenen's blouses that Shirts Happen can produce per day, given the constraints ahove. Next to each inequality, write a brief description or interpretation of its meaning. 2. Carefully graph the system of linear inequalities on a sepurate 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, deternine whether or not it is a foasible solution to the Shirts Happen Clothing Coanpany problens. Show your work and explain your rensoning. (a). 5 men's shirts per day, 20 women's blouses per day (b). 25 men's shirts per duy, 25 women's blouses per day (c). 0 men's shirts per dny, 0 women's blouses per day (d). 10 men's shirts per day, 20 woneen's blouses per day (e). 10 men's shirts per day, 10 wotnen's blouses per day (f). 18 men's shirts per day, 14 women's blouses per day 4. Suppose that each men's shirt produced and sold yields a $25 profit and cach women's blouse produced and sold yields a 850 protit. (a). An isoprofit line is a line of solution points (x,y) within the feasible region that each yield the same profit. Find two feasible solutions to the Shirts Happen Clothing Cocnpany problem that produce a total profit of $200, and then find two fensible solutions that produce a total profit of 8500 . Explain your answers, and then draw these two isoptofit lines on your graph - one for a total profit of $200, and the other for a total profit of 8500 . (b). Finding optimal solution(s): How many men's shirts and women's blouses should the Shirts Happen Clothing Conapany produce each day so as to marimise its daily peofit? What is the company's maximam daily proft as a result? Hint: There are actually multiple optimal solutions. List and explain two of them. Woeld there be a resson for the Shirts Happen Clothing Company to favor one optimal solution over another? Clearly explain your reasoning along with any additional assumptions that need to be made in deternining an optimal solution to this