# MATHEMATICS

MAT 543 Week 6 Homework

Chapter 8: Exercises 8-1 through 8-5 (page 155 of the text) Chapter 9: Exercises 9-1 and 9-2 (page 174 of the text)

8-1 Two types of visits are provided by the Durham Health Clinic, first-time visits and return visits. Table 8-5 provides the processing time for each work station and the available staff hours per week. Determine the production frontiers for this clinic and indicate which station should be expanded to increase the overall capacity of the clinic. Which service station could be reduced?

8-2 Durham Health Clinic has a contribution margin of \$35 per visit. Calculate the break-even point in visits with fixed costs at \$4000, \$6500, and \$8500 per week. Given this analysis, as a manager, what would you recommend and why?

8-3 Durham Health Clinic is considering signing a contract to perform 50 pre-employment physicals per week for a specific corporation. In terms of staff time, a pre-employment physical requires 0.20 hours in Reception/Discharge, 0.45 hours in Nursing and Testing, and 0.20 hours in Medical Examination. By workstation, determine how many work hours per week will be needed to perform these physicals.

8-4 Currently the clinic does 250 visits per week, with 50% of all visits as return visits. Each employee (physician, nurse, and receptionist) is scheduled to work 35 hours per week.

a. How many employees by type does the clinic currently need?

b. How many employees by type will the clinic need if it signs the contract for pre-employment physicals?

c. If return visits shift to 10% of all regular visits, how many employees by type will the clinic need with and without the contract for pre-employment physicals?

d. How will the answers to “b” and “c” change if the number of physicals is modified to 35 pre employment physicals per week?

Throughout these analyses, specify all assumptions, including assumptions concerning worker productivity.

8-5 How would your answers change for problem 8-1 if nursing and testing time was increased to 0.50 hours for both first and repeat visits, and medical exam and treatment time was reduced to 0.30 hours for a first visit and 0.20 hours for a return visit?

9-1 Alpha Walk-in Clinic operates as a single channel single server system. On Tuesdays, its average arrival rate (λ) per hour is 7.0. Analysis indicates that its service rate (μ) is 8.5 patients per hour. Using queuing theory, describe this service system. What is:

a. The probability that the clinic is idle—no patients waiting or being served?

b. The average number of patients in the system?

c. The average time (hours) a patient spends in the system (waiting + service time)?

d. The average number of patients in the queue waiting for service?

e. The average time (hours) a patient spends in the queue waiting?

f. The probability that the patient, upon arrival, must wait?

9-2 The following data have been collected from a hospital pharmacy. This service system operates as a single server, single channel system.

7–3 pm 3–11 pm 11–7 am

Service rate per hour 200 100 50

Arrival rate per hour 60 50  40

The service rate can be increased or decreased in increments of 50 prescriptions per hour. The expense associated with each 50-prescription increment is \$100. In other words, to be able to process 50 additional prescriptions will cost an additional \$100 per hour. If the current rate of processing or service is lowered by 50 prescriptions per hour, the savings are \$100 per hour. Using queuing theory, describe this service system. What is:

a. The probability that the clinic is idle—no patients waiting or being served?

b. The average number of patients in the system?

c. The average time (hours) a patient spends in the system (waiting + service time)?

d. The average number of patients in the queue waiting for service?

e. The average time (hours) a patient spends in the queue waiting?

f. The probability that a patient, upon arrival, must wait?

Given the associated costs, should the service rate be changed? What are the financial implications associated with your recommendations?

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