# Instructions

Please, follow the following instructions. Failure to do so will result in a 0 in those parts that do not comply with them.

This assignment may be completed by groups of up to 4 students. Only one submission per group is required and expected. This submission will be done through Canvas. Your submission will consist of the following files:

• A computer-typed Word file that contains the solution to the problems in this assignment. For equations, use Microsoft Word’s equation editor. Use figures as needed. If you need to graph any problem, you can use Desmos Graphing Calculator, although you may use any other tool that you see fit.
• For Problem 2, enclose an Excel file with Solver’s solution to the problem. Note that it is imperative that Excel file and Solver are set up so the grader can run the model with Solver and obtain a solution immediately.
• In both files, the names of all the group members must appear.

For general guidelines on how to present your assignments, refer to the course’s syllabus.

For policy on late submissions, refer to the course’s syllabus.

# Problem 1 (40 pt)

Falditas Inc. is a company that is specialized in producing very high-end skirts. Currently, the company produces long and short skirts. Each type of skirt requires a certain amount of processing time distributed as follows:

• Long skirts require 1.2 hours for cutting and 0.7 of hours for sewing.
• Short skirts require 0.8 hours for cutting and 0.6 for sewing.

For producing one long skirt, Falditas needs 6 yards of raw materials. This quantity is smaller (4 yards) when it comes to short skirts. These products have very high quality and their price reflects that: after taking costs into consideration, the marginal profits are \$190 and \$150 for long skirts and short skirts, respectively.

Falditas Inc. have to decide on their production quantities for next week as to maximize their profit. They have 1200 yards of fabric available. They also have 200 hours for cutting and 180 for sewing. However, if needed, they can schedule overtime for these two processes, at a cost of \$15/hour of cutting time and \$10/hour of sewing time. They cannot schedule overtime indefinitely, so they have a limit of 100 hours of overtime.

The company wants to produce a minimum of 100 long skirts and 75 shorts skirts next week. Let L and S be the number of long and short skirts produced, respectively and Oc, Os be the number of hours of overtime or cutting and sewing respectively. The use an LP model to maximize profits, which they solve in Excel. The computer solution is shown below:

Optimal profit: \$ 40,900

 Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \$C\$9 L (units) 100 0 190 35 Inf \$D\$9 S (units) 150 0 150 Inf 23.33 \$E\$9 Oc (hours) 40 0 -15 15 172.5 \$F\$9 Os (hours) 0 -10 -10 10 Inf Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \$B\$15 Cutting LHS 200 15 200 40 60 \$B\$16 Sewing LHS 160 0 180 Inf 20 \$B\$17 Overtime limit LHS 40 0 100 Inf 60 \$B\$18 Material limit LHS 1200 34.5 1200 133.33 200 \$B\$19 Production limit L LHS 100 -35 100 50 100 \$B\$20 Production limit S LHS 150 0 75 75 Inf

1. (14pt) The company is considering raising the price of the long skirt such that the marginal profit is set to \$210. If this change is finally implemented, how will the optimal profit change? (Note: the answer to this question must be made on the basis of the information provided in the table above. You are free to re-run the model if you want, but your answer should be justified just in view of the information given above).
2. (13pt) If the company could order extra material in the last minute with an extra charge of \$8 per yard, would you recommend this? How much more would you be willing to pay for this order? (Note: the answer to this question must be made on the basis of the information provided in the table above. You are free to re-run the model if you want, but your answer should be justified just in view of the information given above).
3. (13pt) If the minimum requirement for long skirts was 75 instead of 100, how would this affect the profit? (Note: the answer to this question must be made on the basis of the information provided in the table above. You are free to re-run the model if you want, but your answer should be justified just in view of the information given above).

# Problem 2 (60 pt)

A medical facility operates between 11 am and 10 pm. Their staff is made of part-time employees who work in shifts of four hours (i.e. an employee that starts at noon will finish at 4 pm). On average, the wage for these employees is \$7.60 per hour. In view of the number of patients that this facility usually has to serve, management considers that they will need the following employees: 8 employees between 11 am and noon; 8 employees between noon and 1 pm; 7 employees between 1 pm and 2 pm; 1 employee between 2 pm and 3 pm; 2 employees between 3 pm and 4 pm; 1 employee between 4 pm and 5 pm; 5 employees between 5 pm and 6 pm; 10 employees between 6 pm and 7 pm; 10 employees between 7 pm and 8 pm; 6 employees between 8 pm and 9 pm; and 6 employees between 9 pm and 10 pm.

Management is asking you to come up with a schedule for part time-employees (i.e. they want to know how many employees should start shifts and 11 am, noon, 1 pm, etc.) such that the total cost in wages is minimized.

1. (25pt) Develop a linear programming model that helps management. As seen in class, follow the following steps:
• Define decision variables.
• Define objective function.
• Define constraints.
• Write down the entire formulation.

1. (25pt) Solve this problem with Microsoft Excel. What is the total payroll per day? What is the optimal schedule?
2. (10pt) Comment on the desirability of scheduling at least some of the employees for three-hour shifts. (Tip: use the surplus variables for this).