Health care service delivery

At the outpatient unit of a major hospital in NYC, some patients (with non-critical conditions) experience longer than expected waiting time before seen by a general practitioner. Lately, the hospital has been receiving complaints about long waiting times. When patients must wait more than expected (25 minutes), they complain. Each patient complaint costs the hospital $200. In addition, management team has realized that waiting time are costly for the hospital operations. A minute of wait time per patient cost the hospital $5. The management team believes that the outpatient unit has had sufficient capacity (in terms of the number of doctors) to deal with the demand; however, the problem of waiting and complaints have been prevalent within the unit. The hospital top management team calls the operations managers ask her to improve the service level with respect of the wait time. Upon starting the project, the operations manager collects the following data:
Number of general practitioners in the unit
5
Average time that a doctor spends on each case
9 min/patient
Standard deviation of time a doctor spends on each case
15 min
Average number of patients arriving per hour
30 patient/hour
Acceptable average wait time
25 min
Number of queues
1
She also begins to collect data on the waiting times for individual patients over a month. After collecting data, she observes that the waiting time has an exponential distribution. She also finds out the patients’ arrival rate is quite the same across all days, and patients’ arrivals are independent of each other. Also, data shows patients almost never leave the hospital due to long waiting times.
An apparent solution to her is to increase the service level (in terms of wait time) to reduce to wait time and complaint costs. To do so, she is thinking about hiring additional general practitioners for the unit. But that comes with a cost. The monthly (20 working days per months, 8 hours per day) salary of each doctor is on average $10,000. Answering the following questions should help her to come up with a solution.
a. What is the current service level? How many more general practitioners the unit must hire to achieve 95% service level.
b. What is the current total cost per patient (doctors wage, complaint and wait costs)? What is the total cost per patient with 95% service level?