# Decision making under uncertainty: Part 2.

The physician executive overseeing an outpatient surgical center is planning for increased volume with the addition of a new surgical group. Given the anticipated increased surgical caseload, she must decide whether to hire one or two additional anesthesiologists.

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She can hire two now, or can hire one with the option of hiring an additional person later, if the volume warrants it. (Assume there are anesthesiologists readily available to hire). She is not sure what the actual increase in volume will be. How does she decide what to do?

Physician executives and medical practice managers are frequently asked to make decisions for their organizations without all of the information they need. In my previous column (Jan/Feb 2005, The Physician Executive), I discussed the use of the payoff table as a tool to assist in making rational choices when faced with this uncertainty.

Another useful tool for making decisions under these uncertain conditions is tree diagram analysis. Tree diagrams are used to summarize complex situations so their essential elements can be identified.

* A box is used to represent a decision node from which one of several alternatives may be selected.

* A circle is used to represent a state-of-nature node.

* 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 in case it rains.

* For each combination of a state of nature and a course of action there is a payoff or outcome. While it is useful for visualizing the problem, by itself the tree diagram does not provide a solution to the problem. (Figure 1).

Decisions like the one in our example involve the possibility of more or less profit for the organization. So, the next step in our analysis is to determine the cash value for each branch of our tree diagram, since the final cash values will represent the payoffs for these branches. In addition, there are costs associated with branches that must be identified. Finally, probabilities of the occurrence of the state-of-nature nodes must be assigned.

Assume that the following additional information is known to our physician executive:

* The cost of hiring an additional anesthesiologist is \$200,000.

* The additional revenue from a high caseload from the new group will be \$1 million.

* The probability of the higher caseload is 70 percent, the probability of the lower caseload is 30 percent.

Figure 1 shows the tree diagram with the costs, payoffs, and probabilities.

The final step is to find the best decision branch. The best decision branch is found by finding the alternative with the highest monetary value determining EMV.

The EMV is calculated by multiplying each event's payoff by the probability of its occurrence. The EMV is determined by working backward along the tree diagram, starting at the right side of the tree and moving to the left until the first decision point is reached.

First, we compute the EMV for the "hire 2 anesthesiologists" branch of the diagram.

EMVHire 2 = (revenue from high caseload)(probability of high caseload) + (revenue from low caseload)(probability of low caseload) - cost of hiring 2 anesthesiologists

EMVHire 2 = (\$1,000,000)(0.7) + (\$500,000)(0.3) - \$400,000 = \$450,000

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Next, we calculate the EMV for the "hire 1 anesthesiologist" branch of the diagram. Here, however, we have the option of hiring or not hiring a second anesthesiologist depending on caseload.

Again, we move from right to left in the diagram. Payoff for the "high caseload, hire" branch is \$1 million. However, the cost to hire an anesthesiologist is \$200,000. So the EMV for this branch is \$800,000. This value is higher than the "high caseload, don't hire" branch of \$500,000. Therefore, we can eliminate the "high caseload, do not hire" branch of the tree.

This is designated in Figure 1 by using a red line. Looking at the "low caseload, do hire" branch reveals a payoff of \$500,000. The cost is \$200,000. Therefore, the EMV for this branch is \$300,000. Since this value is less than the \$500,000 for the "low caseload, don't hire" branch, we can eliminate the "low caseload, do hire" branch with a red line, as well.

With these calculations, we can then work backward from right to left and calculate the EMV for the "hire 1 anesthesiologist" branch.

EMVHire 1 = (revenue from high caseload)(probability of high caseload) + (revenue from low caseload)(probability of low caseload) - cost of hiring 1

EMVHire 1 = (\$800,000)(0.7) + (\$500,000)(0.3) - \$200,000 = \$510,000

Based on this analysis, the physician executive determines it is best to hire one anesthesiologist, with the option of hiring a second, if the increased demand warrants it, since the payoff for this option is higher.

Unfortunately, in the real world, decisions like this are not so easy. For example, we assumed the physician executive would have no problem hiring a second anesthesiologist, if demand increased. The truth is that she may have great difficulty recruiting a second anesthesiologist given the current market for anesthesiologists. This could adversely affect her ability to meet the demand if the caseload is high.

However, like the payoff table, the tree diagram provides a useful tool for analyzing complex decisions under uncertain conditions.

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 mdcg@verizon.net
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