Mathematics

Total Rate

630 600 708

135

. stated produc- iroduction pro- · pacity could be ipecial x-rays),

Slack (Hoursya

192

492

0

642

90

180

532

0

•ging the mix

Exercises 155

These actions would enhance efficiency and not impact any mix of x-rays · tliat could be produced during the future 3-month period: If imaging hours were reduced by ninety (90), imaging would still have an excess capacity of 102 hours .. Slack would be 102 (i.e., 192 - 90 = 102). If the x-ray department was called upon to only produce special x-rays, reducing imaging by 90 staff hours would leave zero slack. ·

The situation with the developing stage is similar. If only general x-rays were done, reducing staffing by 180 hours over the next 3 months would still leave 312 excess hours (i.e., 492 - 180 = 312). Reducing the developing stage by 180 staff hours would lea\'.e zero excess if only special x-rays ·were produced.

As demonstrated by this example, the capacity analysis model identifies the pro~uction frontier associated with multistep production processes and can be used to develop strategies to increase or reduce resources so that capacity and output are an efficient aspect of the operation of the system or subsystem . .It is important

· to note that different services require different resources. The example classified x-rays into either general or special and states different resource requirements for each at'each stage· in the producticin proc~ss. More often than not, service sta" tions in medical care production ·systems provide more than one type of service. The capacity analysis models are designed to assist when two or more. variables (e.g., special versus general x-rays) are present. These models provide the ability ,to estimate production frontiers when faced with this type of,situation.

EXERCISES

~ Two types of visits are provided by the Durham Health Clinic, first-time visits D 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·?

Table 8-5 Processing Time and Staff Hours Data for Durham Health Clinic (Exercise 14-1)

Work station

. Reception/discharge

Nursing and testing

Medical exam and treatment

Time estimates (hours)

First Visit

0.25

0.40

0.50

Return Visit

0.12 0.38

0.28

I

156 CHAPTER 8: ANALYZiNG CAPACITY AND RESOURCES

@)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.

t:?;;\ Given this analysis, as a manager, what would you recommend and why? \!t.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 work-station, determine how many work hours per week will be needed to perform these physicals.·

@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 /?"S) concerning worker productivity. · · ~ 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? J

i

'

\' ,,, I

174 CHAPTER 9: MANAGING WAITING LINES

The average time a unit spends in the system (Jv):

w _L - fl,,

Whens= 2 _ 9.47 - 6.67 = 1.42 hours or 85.2 minutes

WhenS = 4

I

_ 1.91 - 6.67 = 0.29 hours or 17.4 minutes

The average time a patient spends in the queue waiting for servke (W q):

Lq Wq =-r Whens = 2

= 7.67 12

= 0.64 hours or 38.4 minutes

WhenS = 4

= 0.11 12

= 0.009 hours or 0.54 minute

The proposed expansio'.u to four examination rooms seems justified. It will better

balance this system's service capabilities with patient waiting. ·

EXERCISES (§) Alpha Wilk-in Clinic operates as a single channel single ,erver sys,teni. On Tuesdays, its average arrival rate (µ,) per hour is 7 .0. Analysis indicates that its service rate (1'.) 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 )Vaiting or being served?

b. The average number of patients in the system? I

.c. The averag~ time (hours) a patient spends in the system (waiting +

service time)? '

n lt lS

+

', t_;

Exercises 175

d. The average number of patients in the queue waiting fof"service? e. The average time (hours) a patient spends in the queue waiting? f. The probability that the patient, upon arrival, must wait?

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

Service rate per hour

Arrival ~ate per hour

7-3PM

200

90

3-11 PM

100

50

11-7 AM

, 50

40

The service rate can be increased or decreased in increments of 50 prescrip- tions per hour. The expense associated with each SO-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?

9-3 Consider the data in Table 9-6. This table reports the number of a specific lab test received for processing at an outpatient medical lab over a IO-day period of time. Based on machine capabilities and staffing, the service rate for the day shift (7 AM-3 PM) must be constant even though the arrival rate changes during this time period. The system currently operates as a single channel single server system.

a. What service rate do you recommend for the day shift (7 AM-3 PM)? Remem- ber that the service rate must be greater than the arrival rate. Use queuing parameters to describe this system with different service rates.

b. What are the implications of changing this system from a single server to . a multiple server system in which S = 2. Show the queuing theory param-

etei:s. How would management make this decision?