In using the optimistic criterion, the best (maximum) payoff for each alternative is considered, and the alternative with the best (maximum) of these is selected. Hence, the optimistic criterion is sometimes called the maximax criterion. In Table 3.2, we see that Thompson’s optimistic choice is the first alternative, “construct a large plant.” By using this criterion, the highest of all possible payoffs ($200,000 in this example) may be achieved, while if any other alternative were selected, it would be impossible to achieve a payoff this high.

	STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MAXIMUM IN A ROW ($)
Construct a large plant	200,000	$-$ 180,000
Construct a small plant	100,000	$-$ 20,000	100,000
Do nothing	0	0	0

In using the optimistic criterion for minimization problems in which lower payoffs (e.g., cost) are better, you would look at the best (minimum) payoff for each alternative and choose the alternative with the best (minimum) of these.

In using the pessimistic criterion, the worst (minimum) payoff for each alternative is considered, and the alternative with the best (maximum) of these is selected. Hence, the pessimistic criterion is sometimes called the maximin criterion. This criterion guarantees the payoff will be at least the maximin value (the best of the worst values). Choosing any other alternative may allow a worse (lower) payoff to occur.

Thompson’s maximin choice, “do nothing,” is shown in Table 3.3. This decision is associated with the maximum of the minimum number within each row or alternative.

In using the pessimistic criterion for minimization problems in which lower payoffs (e.g., cost) are better, you would look at the worst (maximum) payoff for each alternative and choose the alternative with the best (minimum) of these.

	STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MINIMUM IN A ROW ($)
Construct a large plant	200,000	$-$ 180,000	$-$ 180,000
Construct a small plant	100,000	$-$ 20,000	$-$ 20,000
Do nothing	0	0

Both the maximax and maximin criteria consider only one extreme payoff for each alternative, while all other payoffs are ignored. The next criterion considers both of these extremes.

Often called the weighted average, the criterion of realism (the Hurwicz criterion) is a compromise between an optimistic and a pessimistic decision. To begin, a coefficient of realism, $a,$ is selected. This measures the degree of optimism of the decision maker and is between 0 and 1. When $\alpha$ is 1, the decision maker is 100% optimistic about the future. When $\alpha$ is 0, the decision maker is 100% pessimistic about the future. The advantage of this approach is that it allows the decision maker to build in personal feelings about relative optimism and pessimism. The weighted average is computed as follows:

For a maximization problem, the best payoff for an alternative is the highest value, and the worst payoff is the lowest value. Note that when $\alpha =1,$ this is the same as the optimistic criterion, and when $\alpha =0$, this is the same as the pessimistic criterion. This value is computed for each alternative, and the alternative with the highest weighted average is then chosen.

If we assume that John Thompson sets his coefficient of realism, $\alpha ,$ to be 0.80, the best decision would be to construct a large plant. As seen in Table 3.4, this alternative has the highest weighted average: $\$124,000=\left(0.80\right)\left(\$200,000\right)+\left(0.20\right)\left(\mathrm{-}\$180,000\right).$

	STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	CRITERION OF REALISM OR WEIGHTED AVERAGE (α = 0.8) ($)
Construct a large plant	200,000	$-$ 180,000
Construct a small plant	100,000	$-$ 20,000	76,000
Do nothing	0	0	0

In using the criterion of realism for minimization problems, the best payoff for an alternative would be the lowest payoff in the row, and the worst would be the highest payoff in the row. The alternative with the lowest weighted average is then chosen.

Because there are only two states of nature in the Thompson Lumber example, only two payoffs for each alternative are present and both are considered. However, if there are more than two states of nature, this criterion will ignore all payoffs except the best and the worst. The next criterion will consider all possible payoffs for each decision.

One criterion that uses all the payoffs for each alternative is the equally likely, also called Laplace, decision criterion. This involves finding the average payoff for each alternative and selecting the alternative with the best or highest average. The equally likely approach assumes that all probabilities of occurrence for the states of nature are equal, and thus each state of nature is equally likely.

The equally likely choice for Thompson Lumber is the second alternative, “construct a small plant.” This strategy, shown in Table 3.5, is the one with the maximum average payoff.

In using the equally likely criterion for minimization problems, the calculations are exactly the same, but the best alternative is the one with the lowest average payoff.

The next decision criterion that we discuss is based on opportunity loss or regret. Opportunity loss refers to the difference between the optimal profit or payoff for a given state of nature and the actual payoff received for a particular decision for that state of nature. In other words, it’s the amount lost by not picking the best alternative in a given state of nature.

	STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	ROW AVERAGE ($)
Construct a large plant	200,000	$-$ 180,000	10,000
Construct a small plant	100,000	$-$ 20,000
Do nothing	0	0	0

The first step is to create the opportunity loss table by determining the opportunity loss for not choosing the best alternative for each state of nature. Opportunity loss for any state of nature, or any column, is calculated by subtracting each payoff in the column from the best payoff in the same column. For a favorable market, the best payoff is $200,000 as a result of the first alternative, “construct a large plant.” The opportunity loss is 0, meaning that it is impossible to achieve a higher payoff in this state of nature. If the second alternative is selected, a profit of $100,000 would be realized in a favorable market, and this is compared to the best payoff of $200,000. Thus, the opportunity loss is $200,000-100,000=100,000.$ Similarly, if “do nothing” is selected, the opportunity loss would be $200,000-0=200,000.$

For an unfavorable market, the best payoff is $0 as a result of the third alternative, “do nothing,” so this has 0 opportunity loss. The opportunity losses for the other alternatives are found by subtracting the payoffs from this best payoff ($0) in this state of nature, as shown in Table 3.6. Thompson’s opportunity loss table is shown as Table 3.7.

Using the opportunity loss (regret) table, the minimax regret criterion first considers the maximum (worst) opportunity loss for each alternative. Next, looking at these maximum values, pick the alternative with the minimum (or best) number. By doing this, the opportunity loss actually realized is guaranteed to be no more than this minimax value. In Table 3.8, we can see that the minimax regret choice is the second alternative, “construct a small plant.” When this alternative is selected, we know the maximum opportunity loss cannot be more than 100,000 (the minimum of the maximum regrets).

In calculating the opportunity loss for minimization problems such as those involving costs, the best (lowest) payoff or cost in a column is subtracted from each payoff in that column. Once the opportunity loss table has been constructed, the minimax regret criterion is applied in exactly the same way as just described. The maximum opportunity loss for each alternative is found, and the alternative with the minimum of these maximums is selected. As with maximization problems, the opportunity loss can never be negative.

We have considered several decision-making criteria to be used when probabilities of the states of nature are not known and cannot be estimated. Now we will see what to do if the probabilities are available.

	STATE OF NATURE
ALTERNATIVE	FAVORABLE MARKET ($)	UNFAVORABLE MARKET ($)	MAXIMUM IN A ROW ($)
Construct a large plant	0	180,000	180,000
Construct a small plant	100,000	20,000
Do nothing	200,000	0	200,000