General Instructions

This homework must be turned in on Brightspace by September 30th 2022, 11:59pm.  It must be your own work, and your own work only—you must not copy anyone’s work, or allow anyone to copy yours.  This extends to writing code.  You may consult with others, but when you write up, you must do so alone. Your homework submission must be a pdf generated from latex/word. No handwritten solutions will be accepted. You should submit:

1. A compiled PDF file named yourNetID solutions.pdf containing your solutions to the prob- lems.

2. An excel file ( .xls or .xlsx) containing the excel worksheet named yourNetID solutions.xlsx or solutions.xls.

Please make sure your answers are clearly structured in the Rmarkdown file:

1. Label each question part(e.g. 3.a).

2. If there is any excel worksheet used to obtain the answer for any question part it should accompany the written answer.

Problem 1 - (15 points)

Evergreen Fertilizer Company produces fertilizer. The company’s fixed monthly cost is $25,000, and its variable cost per pound of fertilizer is $0.15. Evergreen sells the fertilizer for $0.40 per pound.

1.  (3 points) Determine the monthly break-even volume for the company.

2.  (3 points) If the maximum operating capacity is 120,000 pounds of fertilizer per month, determine the break-even volume as a percentage of capacity.

3.  (3 points) If the company changes the price of its fertilizer from $0.40 per pound to $0.60 per pound, what effect will the change have on the break-even volume.

4.  (3 points) If the company changes its production process to add a weed killer to the fertilizer to increase sales, the variable cost per pound will increase from $0.15 to $0.22.  What effect will this change have on the break-even volume computed in Part 3.

5.  (3 points) If the company increases its advertising expenditures by $14,000 per year, what effect will the increase have on the break-even volume computed in Part 4.

Problem 2 - (10 points)

The General Store at State University is an auxiliary bookstore located near the dormitories that sells academic supplies, toiletries, sweatshirts and T-shirts, magazines, packaged food items, and canned soft drinks and fruit drinks. The manager of the store has noticed that several pizza delivery services near campus make frequent deliveries.  The manager is therefore considering selling pizza at the store. She could buy premade frozen pizzas and heat them in an oven. The cost of the oven and freezer would be $27,000. The frozen pizzas cost $3.75 each to buy from a distributor and to prepare (including labor and a box). To be competitive with the local delivery services, the manager believes she should sell the pizzas for $8.95 apiece. The manager needs to write up a proposal for the university’s director of auxiliary services.

1.  (3 points) Determine how many pizzas would have to be sold to break even.

2.  (2 points) If the General Store sells 20 pizzas per day, how many days would it take to break even?

3.  (5 points) The manager of the store anticipates that once the local pizza delivery services start losing business, they will react by cutting prices. If after a month (30 days) the manager has to lower the price of a pizza to $7.95 to keep demand at 20 pizzas per day, as she expects, what will the new break-even point be, and how long will it take the store to break even?

Problem 3 - (15 points)

The Weemow Lawn Service mows its customers’ lawns and provides lawn maintenance starting in the spring, through the summer, and into the early fall.  During the winter, the service doesn’t operate, and Weemow’s owners, Jeff and Julie Weems, find part-time jobs.  They are considering the possibility of doing snow removal during the winter.  A snowblower and a shovel would cost them $700. Because Jeff would do all the work, with occasional help from Julie, their cost per job would be about $3.

1.  (3 points) If they charge $35 to clear a normal-size home driveway, how many jobs would they need to break even?

2.  (4 points) Based on past winters, Jeff and Julie believe they can expect about six major snowfalls in the winter and would be able to work all day for the two days immediately following the snows, when people want their driveways cleared. If they are able to do about

10 snow removal jobs per day (and they believe they will have that much demand because of their existing customer base), how much money can they expect to make?

3.  (4 points) Another possibility for Weemow is to remove snow from business parking lots. Weemow would need a small tractor with a snow plow, which costs $1,800, and would have to hire someone on an hourly basis to help, which with gas would cost about $28 per job. Jeff and Julie estimate that they could do four of these large jobs per day. If they charged $150 per job, would this be a better alternative than clearing individuals’ driveways?

4.  (4 points) If Weemow wanted to do both (b) and (c), Jeff and Julie would need to hire one more person to do the driveways, while Jeff worked with the other person on the parking lots. This would add $15 in cost per driveway job. Should they do this?

Problem 4 - (15 points)

A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four.  A necklace earns $300 in profit and a bracelet, $400.  The store wants to determine the number of necklaces and bracelets to make to maximize profit.

1.  (2 points) Formulate a linear programming model for this problem.

2.  (3 points) Solve this model by using graphical analysis.

3.  (2 points) Explain the effect on the optimal solution of increasing the profit on a bracelet from $400 to $600.

4.  (3 points) What will be the effect of changing the platinum requirement for a necklace from

2 ounces to 3 ounces?

5.  (3 points) The maximum demand for bracelets is 4. If the store produces the optimal number of bracelets and necklaces, will the maximum demand for bracelets be met?  If not, by how much will it be missed?

6.  (2 points) What profit for a necklace would result in no bracelets being produced, and what would be the optimal solution for this profit?

Problem 5 - (15 points)

Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are permanent and some of whom are temporary.  A permanent operator can process 16 claims per day, whereas a temporary operator can process 12 per day, and on average the company processes at least 450 claims each day.  The company has 40 computer workstations.  A permanent operator generates about 0.5 claim with errors each day, whereas a temporary operator averages about 1.4 defective claims per day. The company wants to limit claims with errors to 25 per day. A permanent operator is paid $64 per day, and a temporary operator is paid $42 per day. The company wants to determine the number of permanent and temporary operators to hire to minimize costs.

1.  (3 points) Formulate a linear programming model for this problem.

2.  (3 points) Solve this model by using graphical analysis.

3.  (4 points) Explain the effect on the optimal solution of changing the daily pay for a permanent claims processor from $64 to $54. Explain the effect of changing the daily pay for a temporary claims processor from $42 to $36.

4.  (2 points) What would be the effect on the optimal solution if Universal Claims Processors decided not to try to limit the number of defective claims each day?

5.  (3 points) Explain the effect on the optimal solution if the minimum number of claims the firm processes each day increased from 450 to at least 650.

Problem 6 - (15 points)

Mega-Mart, a discount store chain, is to build a new store in Rock Springs. The parcel of land the company has purchased is large enough to accommodate a store with 140,000 square feet of floor space. Based on marketing and demographic surveys of the area and historical data from its other stores, Mega-Mart estimates its annual profit per square foot for each of the store’s departments to be as shown in Table 1.  Each department must have at least 15,000 square feet of floor space, and no department can have more than 20% of the total retail floor space.  Men’s, women’s, and children’s clothing plus housewares keep all their stock on the retail floor; however, toys, electronics, and auto supplies keep some items (such as bicycles, televisions, and tires) in inventory. Thus, 10% of the total retail floor space devoted to these three departments must be set aside outside the retail area for stocking inventory.  Mega-Mart wants to know the floor space that should be devoted to each department to maximize profit.

Table 1: Estimated annual profit per square foot of Mega-Mart in Problem 6

1.  (3 points) Formulate a linear programming model for this problem.

2.  (4 points) Solve this model by using the computer.

3.  (4 points) Mega-Mart is considering purchasing a parcel of land adjacent to the current site on which it plans to build its store.  The cost of the parcel is $190,000, and it would enable Mega-Mart to increase the size of its store to 160,000 square feet. Discuss whether Mega-Mart should purchase the land and increase the planned size of the store.

4.  (4 points) Suppose that the profit per square foot will decline in all departments by 20% if the store size increases to 160,000 square feet.  (If the stock does not turn over as fast, increasing inventory costs will reduce profit.) How might this affect Mega-Mart’s decision in Part 3?