MAT/543 MAT543 MAT 543 WEEK 7 Homework

MAT 543 WEEK 7 Homework

Week 7 Homework

• Homework
• Chapter 10: Exercise 10-2 (page 208 of the text)

Table 10-2 Expected Payoffs Umbrella Example

Expected payoff [carry]

=   (0.8 × –$1) + (0.2 × –$1) = –$1

Expected payoff [not carry]

=   (0.8 × –$50) + (0.2 × $0) = –$40

Break Even Analysis

Thinking about the preceding example, one question that may come to mind would be, How low does the probability of rain have to be before we leave our umbrella at home? In essence we are asking at what probabilities for future states of the world we are indifferent to our decision choices. This is a break even question, presented in Chapter 8. Expressed as a break even question, with the payoffs we have estimated, what probability value will give an expected payoff if we do not carry an umbrella equal to the expected payoff if we do?

In this case, we would set the expected payoff of not carrying an umbrella be equal to –$1, which is the value of the expected payoff for carrying an umbrella. We can state this algebraically:

–$1 = (p × –$50) + [(1 – p) × 0)]

Here p represents the probability of rain. Remembering that all probabilities add up to 1, we use p to represent the probability of rain and 1 – p to represent the remaining probability of no rain.

Solving this equation, we find:

    –1 = p × –50

and thus

p = 1/50

              = 0.02 or 2%

Given the parameters used in this example, when the chance of rain equals 2%, we are indifferent as to whether we carry an umbrella or not. If the chance of rain is less than 2% we choose not to carry our umbrella.

Again it is important to realize that our payoff amounts for losing an umbrella or the value of our clothes are often estimates. At times we may need to alter those estimates and reassess our analysis. If, for example, we estimate the value of our clothes to be equal to $20 rather than $50 when our clothes got wet, then the break even probability for carrying an umbrella changes to:

   –1 = p × –20

and thus

p = 1/20

              = 0.05 or 5%

Changing the value of our expected loss from ruined clothes from $50 to $20, changes the break even point for our decision alternatives—in such an example the probability of rain must be 5% or less for us not to carry an umbrella.

LEARNING OBJECTIVE 2: TO BE ABLE TO CALCULATE EXPECTED PAYOFF AND SELECTING A DECISION BASED UPON MAXIMUM PAYOFF USING SPECIFIC EXAMPLES

Health services managers make decisions under conditions of uncertainty all the time. In some instances, uncertainty may be relatively low. For example, purchasing a new imaging device to meet existing volume demands more efficiently involves a low level of uncertainty. Conversely, purchasing a new imaging device to meet an unknown demand, or to compete with another similar service provider, might represent a situation characterized by a high level of uncertainty. Consider the following examples.

Clinic Renovation

An ambulatory care clinic administrator is trying to decide whether to renovate to accommodate possible increased demand. The manager could plan a major renovation costing $700,000 that would allow 50 patients per day to be served, or a minor renovation costing $225,000 that would allow 35 patients a day to be served. The final alternative is to do nothing, thus keeping the status quo by not renovating. This continues the existing capacity of accommodating the current 20 patients per day, but no more. Presently the clinic earns $75 per patient served. Assume that the clinic is open 300 days per year and that management wants to cover the costs of the renovation from first-year earnings.

To begin quantitatively analyzing our decision options, we first go back to the three decision steps listed previously. The first step is to state the alternatives. These are to do nothing, undergo a minor renovation, or undergo a major renovation. The second step is to determine the future states of the world. These are the unknowns in our decision. Here they are the estimates of future demand. Because our decisions limit future capacity, we will use these limits as estimates of future demand. Thus, let us describe the potential for 20 patients per day, 35 patients per day, or 50 patients per day to be served, defined by current capacity, capacity given a minor renovation, and capacity given a major renovation.

There are three alternatives and three possible states of the world. This means that there are nine possible outcomes. These are listed in Table 10-3. The third step is to determine the payoffs for each of the potential outcomes.

Earnings are based upon patients served; therefore, part of the payoff involves earnings. For each state of the world of future demand, there are different potential maximum patients who can be seen. Each of these brings in revenue of $75. This amount is then multiplied by the 300 days the clinic is open yearly to calculate the total revenue per year. The maximum revenue generated in each state of the world can be seen in Table 10-4.