MS&E 263 – HEALTHCARE OPERATIONS MANAGEMENT

 

MS&E 263 – HEALTHCARE OPERATIONS MANAGEMENT

 

Homework 5

QMOS I

30 pts.

 

Engineers and non-engineers complete different assignments.

Non-Engineers:

1. Read Constrained Optimization Methods in Health Services Research
2. Choose an example from Table 2, and describe a potential simple representation of this problem in terms of decision variables, an objective, and constraints
3. Identify the proposed SURF projects for next quarter for which you think optimization may be most relevant, explain why, and describe a potential simple representation of this problem in terms of decision variables, an objective, and constraints

Engineers: Solve the following problems. You may work in a group, but make sure you understand how to do every problem - these kinds of questions will be on the midterm.

1. The Vaden clinic just got once-a-week access to the Stanford Hospital clinic x-ray machines, and they are trying to determine how many clinic resources they should devote to “X-Ray Thursdays.” 

 

The following figure describes the resources that Vaden predicts that they will need for chest x-ray patients and mammography patients (the only two types of patients they are planning to have).  The amount of time, in hours, a resource needs to serve a patient is given below the resource name.  For example, someone who gets a chest x-ray will require 0.25 hours at the front desk checking in, 0.33 hours at the front desk checking out, 0.25 hours of nurse time during each, occupy the exam room for 0.5 hours using 0.5 hours of nurse time and 0.25 hours of doctor time while in that room, etc.  Probabilities are given in underlined bold (for instance, 60% of patients are potential chest x-ray patients, and the other 40% will ask for a mammogram).  After being examined by a doctor, 30% of x-ray patients will be found unsuitable for an x-ray examination and be discharged to the front desk without getting an X-ray (and 10% of potential mammography patients will be found unsuitable). Note that patients may occupy rooms without a staff member present and staff members may work “behind the scenes” analyzing data, so the room and staff resource times do not necessary match up. You may use averages in your analysis, i.e., ignore time of day effects such as the patients who arrive first to the clinic.

 

 

Rooms are available for 24 hours, each X-ray machine can be used for 6 hours, and doctors, nurses, and technicians can all work for 8 hours.  How many of each of the following does Vaden need to commission if they expect 100 patients to come in on X-ray Thursday? (You may use averages and ignore time-of-day specific considerations such as early-morning or end-of-day).

a.       Doctors

 

b.      Nurses

c.       Technicians

d.      Front desk spots

e.       Exam rooms

f.       Prep rooms

g.       X-ray machine

 

 

2. Patients who may need surgery are referred to a surgeon’s clinic for examination. A prominent otolaryngology surgeon has a variable number of patients of varying levels of complexity arrive to her office on a daily bases. She can see 15 patients on her own and has determined that the expected profit (calculated as the probability of the patient needing surgery times the mean profitability of surgery) from seeing each patient is approximately $700. To accommodate each patient beyond 15, she will have to call in a consult from a fellow physician. The cost of scheduling a consult ahead of time is $400 and the cost of calling in a consult on the day of is $800. How many consults should she call in ahead of time in order to maximize profit?

Number of patients	Probability of exactly this many patients arriving
< 20	0
20	0.03
21	0.05
22	0.16
23	0.29
24	0.22
25	0.1
26	0.09
27	0.03
28	0.02
29	0.01
> 30	0

 

 

 

 

 

 

3. An eye care clinic must decide how many surgical procedures to perform in order to maximize profits. The clinic offers the following three procedures:

-          Laser-Assisted in Situ Keratomileusis (LASIK)

-          Photorefractive keratectomy (PRK)

-          Cataract removal

	LASIK	PRK	Cataract
Profit per procedure	$1,000	$750	$500

Resource requirements per procedure and available resources per week are as follows:

	LASIK	PRK	Cataract	Available Resources
Physician time	30 minutes	25 minutes	15 minutes	2,700 minutes
Technician time	75 minutes	60 minutes	30 minutes	4,500 minutes
Operating room time	45 minutes	75 minutes	30 minutes	3,000 minutes

 

Formulate this as a linear programming by defining the variables and constraints. Assume the number of each procedure must be an integer.

 

4. Consider an oncology clinic that cares for immunocompromised patients. Suppose that at any given time approximately 10% of care providers are contagious with COVID-19, that the expected harm associated with a care provider coming to work in the oncology clinic is 1 quality adjusted life year (QUALY), that the cost of keeping someone from coming to work is 0.1 QUALYs whether or not they have COVID-19, and the benefit of a non-contagious provider coming to work is 0.1 QUALYs. Consider 4 criteria for keeping someone from coming to work ( a PCR test is significantly more sensitive, likely to detect COVID-19, than an antigen test).

Cough or fever

Cough and fever

Positive PCR test

Positive antigen test

1. In terms of TPR and FPR, what is the expected utility of a diagnostic test? Hint: the algebra is easier if you multiply everything by 100.
2. In terms of FPR, what must TPR be in order for the expected value of the test to be positive.
3. What are the relative ranks of the true positive rate (TPR) and false positive rate (FPR) of these tests? Do this based only on the definitions of TPR, FPR - you don’t need to look anything up.
4. (Optional) Which test will maximize expected utility for this clinic? You can reason it out or use the internet to find approximate sensitivity and specificity of the tests.

Here is a good reference for more on these definitions and the table from class:

		Status
		Positive	Negative
Result of Test/Classifier	Positive	True Positive (Sensitivity) Utility = TPU Rate = P*TPR Expected utility = TPU*P*TPR	False Positive Utility = FPU Rate = (1-P)*(FPR) Expected utility = FPU*(1-P)*FPR
	Negative	False Negative Utility = FNU Rate = P*(1-TPR) Expected utility = FNU*P*(1-TPR)	True Negative (Specificity) Utility = TNU Rate = (1-P)*(1-FPR) Expected utility = TNU*(1-P)*(1-FPR)

Expected utility = TPU*P*TPR+ FPU*(1-P)*FPR+FNU*P*(1-TPR)+TNU*(1-P)(1-FPR)

P is percentage of population with condition

True positive rate (TPR): # correctly classified positive / total # classified positive

False positive rate (FPR): # incorrectly classified positive / total # classified positive

TPU is the true positive utility, FPU is the false positive utility, FNU is the false negative utility, and TNU is the true negative utility.