Decision making in uncertainty (Part 1).

Imagine that a medical practice must decide how much influenza vaccine to order for the next flu season. In this example, the vaccine costs $15 to purchase and is administered for $20, providing the practice with a $5 profit. Any unused vaccine must be discarded resulting in a loss of $15.

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The practice can only buy vaccine lots of 100 doses. The practice has been offered the opportunity to purchase up to 500 doses. The practice is unsure of what the actual demand for the vaccine will be. Once the flu season starts, however, it is impossible to order additional vaccine. How much vaccine should the practice order?

Physician executives and medical practice managers are frequently called upon to make decisions for their organizations without all of the information they may need. Since, this decision making in uncertainty is a daily part of our business experiences, it is no surprise that statisticians and mathematicians have dedicated an entire branch of study to this process. Yet, are there simple techniques we can use to enhance our decision making under these uncertain scenarios?

All decision makers are faced with states of nature and acts or alternatives.

* A state of nature is a situation for which the decision maker has little or no control. One example of a state of nature is the weather.

* An act or alternative is a course of action or strategy available to the decision maker. For example, knowing we cannot control the weather, we may choose the course of action to carry an umbrella.

For each combination of a state of nature and a course of action there is a payoff or outcome. This may be represented in a payoff table, which provides one of the simple and fundamental techniques of decision making under uncertainty.

Table 1 depicts a payoff table constructed for the vaccine example.

As can be seen, if the practice has 100 doses of vaccine available and the demand for vaccine is 100 doses, then the payoff will be $500 (100 doses X $5 profit per dose). If the demand for vaccine exceeds 100 doses, the practice will have only 100 doses to administer despite the increased demand. The payoff, therefore, will be unchanged at $500, despite the increased demand.

If the practice purchases 200 doses of vaccine, but demand is only for 100 doses, then the payoff will be a loss of $1,000. This is because 100 doses will be given at a profit of $500, but 100 doses will be discarded at a loss of $1,500.

If demand is 200 doses, then supply will meet demand and the payoff will be $1,000. The payoff will continue to be $1,000 at demands above 200 doses. In the same manner, the payoffs are calculated for increased supplies of 300, 400, and 500 doses.

Usually, however, the payoff table is not the only information available to the decision maker. Often important historical data is available to aid in the decision making process.

For example, if the practice looks at past year's demands for flu vaccine, they could determine probabilities for demand at each lot level. Let's say that based on previous year's experience, the practice determines there is a 10 percent probability that demand will be for 100 doses, 20 percent for 200 doses, 30 percent for 300 doses, 25 percent for 400 doses and 15 percent for 500 doses.

Given this additional information, a more rational decision can be made. Once the probabilities of various events have been estimated, the expected monetary value, or EMV, of each action or decision can be computed.

The EMV is calculated by multiplying each event's payoff by the probability of its occurrence. For example, the EMV at 200 doses is calculated in Figure 1.

The results for calculating EMV for each dose level are in Table 1. As can be seen, the practice will have the highest expected monetary value by ordering 200 doses of vaccine. Since the EMV for both 200 and 300 doses is close ($800 versus $700), they might decide to order 300 doses to hedge their bet against increased demand.

Finally, it is often of value to determine what perfect information in a complex decision situation is worth. Suppose the practice was given the opportunity to purchase data that would improve its ability to predict its demand for flu vaccine doses. The cost of the data is $500. How could it decide, if the purchase was worth the investment?

The answer is to calculate the expected value of perfect information or EVPI. The EVPI is the difference between the payoff that would result from perfect information and the payoff that would result from uncertainty.

If you assume the practice could predict the demand for flu vaccine with perfect accuracy, then the expected profit under certainty or EPUC would be calculated as shown in Figure 2.

Since the cost of the data ($500) is less than the EVPI, it would be worth purchasing the data to improve the ability to predict demand for the vaccine.

Decision making in uncertainty is always a difficult undertaking. However, by using tools such as the payoff table, executives hopefully can make more rational choices.

Table 1

Probability  Demand  Doses

                     100      200     300     400     500
                     100      200     300     400     500
0.1          100     500    (1000)  (2500)  (4000)  (5500)
0.2          200     500     1000    (500)  (2000)  (3500)
0.3          300     500     1000    1500       0   (1500)
0.25         400     500     1000    1500    2000     500
0.15         500     500     1000    1500    2000    2500
             EMV     500      800     700       0   (1200)

Figure 1

EMV = (0.1)(-$1000) + (0.2)($1000) + (0.3)($1000) + (0.25)($1000) +
      (0.15)($1000)
EMV = $800

Figure 2

EPUC = (0.1)(($500) + (0.2)($1000) + (0.3)($1500) + (0.25)($2000) +
       (0.15)($2500)
EPUC = $1,575
Therefore, EVPI = EPUC - EMV = $1,575 - $800 = $775

By David P. Tarantino, MD, MBA

David P. Tarantino, MD, MBA, is the executive medical director of Shock Trauma Associates, P.A., a 50+ physician, multispecialty practice associated with the University of Maryland School of Medicine. In addition, he is the chief executive officer of The MD Consulting Group, LLC, a health care management consulting firm in Baltimore. He can be reached by phone at 410-328-2036 or by e-mail at [e-mail]