When it is not possible to find a probability distribution for a decision problem, or none exists, that problem is said to be uncertain. In this situation, it is not possible to determine expected values and one must find other factors on which to make a decision. The payoff matrix is still employed for problems under uncertainty, but without any prior probabilities.
There are several methods to deal with this kind of uncertainty. These methods differ mainly in how the decision maker views risk and the state of nature. The different methods could be viewed as being similar to aspects of utility theory in that they try to take into
TABLE 6.8 Payoff Matrix for Snack Production
State of Nature
Alternative Decrease Slightly Steady Marginal Increase Significant Increase
A1 −1500 −400 1100 2150
A2 −450 200 500 500
A3 −850 −75 450 1100
A4 −200 300 300 300
A5 −150 −250 −450 −850
account the decision maker’s preferences and attitudes. These methods are discussed in the context of the following problem.
6.4.1 Snack Production Example
A local food processing plant has developed a low calorie snack that has taken the market by storm. After only being on the market for 3 months sales have outpaced forecasts by more than 60% and the plant cannot meet demand. It is running 24/7 and is lagging behind demand, at the moment, by 25%. To help alleviate this problem, management must decide how to increase production. Marketing was asked to develop new demand forecasts but has not been able to determine whether or not sales will continue to increase, hold steady, or decrease. After several different surveys it was found that sales could hold steady, increase marginally (somewhere between 10% and 30%), increase significantly (somewhere between 30% and 70%) or decrease slightly (somewhere between 1% and 10%), but marketing can- not give a distribution for the possible outcomes. After researching possible solutions the following alternatives have been put forth by the engineering team: (A1) Build an addi- tional plant that would be able to increase the total production of the product by 120%;
(A2) Add an identical additional line to the current plant that would increase production by 30%; (A3) Expand the current plant and replace current line with new technologies that would allow total production to increase by 60%; (A4) Hire a full time operations analyst to increase efficiency with an estimated increase in production of 15%; and (A5) Do Nothing and maintain the status quo. An economic analysis of each alternative has been carried out and the net profit gained from possible increased sales minus implementation costs is used as the payoff for the alternative. For the DN alternative this would simply be lost sales.
Table 6.8 gives the payoff matrix, and the values are in $1000.
6.4.2 Maximin (Minimax) Criterion
This criterion is a conservative approach to managing the unknown risk. The intent is to determine the worst that can happen with each alternative and then pick the alternative that gives the best worst result. When payoffs are profit then the maximin (maximize the minimum value possible) criterion is used and when the payoff is cost then the minimax (minimize the maximum value possible) criterion is used.
Implementation of this criterion is quite simple. For maximin, identify the minimum in each row and then select the alternative with the largest row minimum. This is given in Table 6.9. The best choice under maximin would be alternative A4.
Under this approach, the decision maker is guaranteed they will never lose more than
$200,000 but at the same time they will never make more than $300,000, whereas a different alternative such as A2 would guarantee they would never lose more than $450,000 but could make $500,000 but never any more. By choosing A4, they relinquish the opportunity to make more money if the state of nature is anything other than a decrease in demand.
TABLE 6.9 Maximin for Snack Production
State of Nature
Decrease Marginal Significant
Alternative Slightly Steady Increase Increase Row Minimum
A1 −1500 −400 1100 2150 −1500
A2 −450 200 500 500 −450
A3 −850 −75 450 1,100 −850
A4 −200 300 300 300 −200∗
A5 −150 −250 −450 −850 −850
∗represents best alternative
TABLE 6.10 Maximax for Snack Production
State of Nature
Decrease Marginal Significant
Alternative Slightly Steady Increase Increase Row Maximum
A1 −1500 −400 1100 2150 2150∗
A2 −450 200 500 500 500
A3 −850 −75 450 1100 1100
A4 −200 300 300 300 300
A5 −150 −250 −450 −850 −150
∗represents best alternative
6.4.3 Maximax (Minimin) Criterion
This criterion is the opposite of maximin in that it is very optimistic and risk-seeking. For this approach it is assumed that the best scenario possible will happen. Therefore, for each row the maximum is chosen and then the alternative with the maximum row maximum is selected. Table 6.10 illustrates this.
The alternative selected by this approach is A1, which yields the largest possible pay- off. However, just as with the maximin approach this leaves the decision maker open to significant losses. For A1 two states of nature yield a loss, one of which is quite significant.
6.4.4 Hurwicz Criterion
The two previous criteria were at the extremes, one very pessimistic and the other very optimistic. This criterion is designed to mitigate the extremes and to allow for a range of attitudes of the decision maker. The basis of this approach is an index of optimism given byα, such that 0≤α≤1. The more certain a decision maker is that the better states of nature will occur, the larger the value ofα, the less certain the smaller the value ofα. For a given value ofα, the criterion is implemented by taking for each row (alternative), ((α×row max)−(1−α)×|row min|), and then selecting the alternative with the largest value. For the example, letα= 0.48, which indicates not a strong regard for optimism or pessimism, but with a hint of pessimism. For the alternatives, the computations are
For A1 0.48×(2150)−0.52×(1500) = 44 For A2 0.48×(500)−0.52×(450) = 6 For A3 0.48×(1100)−0.52×(850) = 85 For A4 0.48×(300)−0.52×(200) = 40 For A5 0.48×(−150)−0.52×(850) =−514 The best alternative under this criterion is A3.
TABLE 6.11 Laplace for Snack Production
State of Nature
Decrease Marginal Significant
Alternative Slightly Steady Increase Increase Expected Value
A1 −1500 −400 1100 2150 338∗
A2 −450 200 500 500 188
A3 −850 −75 450 1100 156
A4 −200 300 300 300 175
A5 −150 −250 −450 −850 −425
∗represents best alternative
TABLE 6.12 Minimax Regret for Snack Production
State of Nature
Decrease Marginal Significant
Alternative Slightly Steady Increase Increase Row Maximum
A1 1350 700 0 0 1350
A2 300 100 600 1650 1650
A3 700 375 650 1050 1050∗
A4 50 0 800 1850 1850
A5 0 550 1550 3000 3000
∗represents best alternative
6.4.5 Laplace Criterion or Expected Value
The Laplace criterion is another more optimistic approach to the problem. The basis of this approach is that since the probabilities are not known for the states of nature and there is no reason to think otherwise, each state of nature should be viewed as equally likely. This gives a probability distribution for the states of nature, which then allows the expected value to be determined. The alternative with the best expected value is selected.
For the snack production example, as there are four states of nature, the probability for each state would be 0.25. The expected value for each alternative is determined by taking the sum of the payoff for each state of nature times 0.25. Table 6.11 gives the Laplace (expectation) values and A1 is the best choice under this criterion.
6.4.6 Minimax Regret (Savage Regret)
The last approach is based on the concept of regret discussed previously. The idea is to not look at payoffs, but instead at lost opportunities (regret). The regret is based on each state of nature and is determined by looking at the best outcome of that state against the other possible outcomes. Therefore, regret is computed for each column of the payoff matrix by taking the maximum value in that column and replacing each payoff value in the column with the maximum value minus the payoff value. The regret matrix for the snack production problem is given in Table 6.12. Once the regret is known, the minimax criterion is applied that says to take the maximum value of each row and then choose the minimum of the row maximums. For this criterion the best alternative is A3.
Making decisions under uncertainty is a difficult task. The decision maker’s attitude toward risk will affect the approach taken and the possibilities for large losses or gains as well as regrets. Generally, when the probabilities of the states of nature are truly unknown the decision maker will make more conservative decisions.