Chapter 2: Decision Making Under Uncertainty

Introduction

The choice decision between alternatives is influenced by many different criteria such as the manager’s perspective on events, the policies of the business and the optimistic or pessimistic approaches of decision maker.

In decision making under uncertainty, the psychological status of the decision maker and their attitude and approach towards the events affect the decision. Therefore, decision makers may choose different methods. In this case, different decision makers can make different decisions about the same decision problem.

Decision Making Under Uncertainty

In a decision problem, the decision maker is aware of various possible events (states of nature) but has insufficient information to assign any probabilities of occurrence to them. This kind of a decision problem is termed as decision making under uncertainty. There are many unknowns and no possibility of knowing what could occur in the future to change the outcome of a decision in decision making under uncertainty.

• When the decision maker does not know the probabilities of occurrence of events, the decision-making is called “decision making under uncertainty”.

Methods For Decision Making Under Uncertainty

There are some different methods used in decision making under uncertainty. There is no rank or superiority among these methods.

The decision will be changed according to the methods depending on whether the objective of the problem is maximum or minimum. For this reason, in this section, two examples, maximum and minimum, are discussed. Both examples will be analyzed in all five methods.

The decision will be changed according to the methods depending on whether the objective of the problem is maximum or minimum. For this reason, in this section, two examples, maximum and minimum, are discussed. Both examples will be analyzed in all five methods.

By looking at the structure of the problem, the decision maker decides which method will be chosen. Different decision makers can choose different methods.

Equally Likely (Laplace)

This method indicates that all events occur equally likely if the probability of occurrence of events is unknown. Therefore, calculation is made considering that the event’s possibility occur equally. The final decision is made according to the objective (maximum or minimum) of the problem.

When deciding which of the alternatives to choose, the objective function of the problem is looked at. It is decided according to the objective function.

The mathematical representation of the method is as follows:

n: number of events related to the problem

Possibility of occurrence of each event = P (Oj ) = j = 1, 2, 3 …n

If the objective function is maximum, the highest value among the alternatives is chosen. If the objective function is minimum, the lowest value among the alternatives is chosen.

Example 2.1 It is desired to open a health center to provide Physical Therapy and Rehabilitation services. Four different districts were selected for the location of the center. In addition, the main characteristics of the regional population were examined for the selection of the location of the center, and as a result of the feasibility studies, the demand estimates were made for the health center. The table 2.1 provides the possible profit values for the monthly demand levels for each location. The aim of the decision maker is to make the maximum profit. Determine the most appropriate decision for the problem according to the Equally Likely Method.

There are four events in the problem. So, the probability of occurrence of each event is 1/4. District

1 = 10.000*1/4 + 12.500*1/4 + 15.000*1/4 + 16.500*1/4 = 13.500 TL District

2 = 15.000*1/4 + 17.000*1/4 + 18.000*1/4 + 24.500*1/4 = 18.625 TL District

3 = 12.000*1/4 + 13.000*1/4 + 20.000*1/4 + 32.000*1/4 = 19.250 TL District

4 = 21.000*1/4 + 25.000*1/4 + 27.000*1/4 + 40.000*1/4 = 28.250 TL

Since the objective of this decision problem is maximization, decision maker chooses the highest value among the alternatives. The decision maker selects district 4 with 28.250 TL.

Criterion of Optimism (Plunger)

If decision maker has an optimistic approach, this method can be used.

If the problem is the minimization, the best outcome is the lowest value. In this case, the alternative which is the smallest value is chosen. The mathematical representation of the method is as follows (Sonmez, 2016, p.28):

Mini {Min (0i j ) j }

If the problem is the maximization, the best outcome is the highest value. In this case, the alternative which is the highest value is chosen. The mathematical representation of the method is as follows (Sonmez, 2016, p.28):

Maxi {Max (0i j ) j }

When the objective function is minimization, the minimum values for the alternatives are determined. The smallest of these minimum values is selected.

When the objective function is maximization, the maximum values for the alternatives are determined. The highest of these maximum values is selected.

Criterion of Pessimism (Wald)

If decision maker has a pessimistic approach, this method can be used. The decision maker thinks that the worst outcomes for each alternative will be realized. However, when choosing from alternatives, it selects the best alternative for the objective of the problem.

If the problem is minimization, the worst outcome is the highest value. However, the best alternative (minimum value) of these worst outcomes is chosen. The mathematical representation of the method is as follows:

Mini {Max (0i j ) j }

When the objective function is minimization, the maximum values for the alternatives are determined. The smallest of these maximum values is selected.

If the problem is maximization, the worst outcome is the lowest value. However, the best alternative (maximum value) of these worst outcomes is chosen. The mathematical representation of the method is as follows:

Maxi {Min (0i j ) j }

When the objective function is maximization, the minimum values for the alternatives are determined. The highest of these minimum values is selected.

Criterion of Realism (Hurwicz)

The decision maker is a little optimistic, a little pessimistic and the method that involves both cases is called Criterion of Realism.

If the problem is minimization, the best value is the smallest value and this value is multiplied by alpha coefficient. The worst value is the highest value and this value is multiplied by 1-? coefficient. Then, these values are sum up. Finally, the alternative with the best value (minimum value) is selected. The mathematical representation of the method is as follows:

Mini {?*Min (0i j ) j + (1 – ?) *Max (0i j ) j }

When the objective function is minimization, the smallest values of the alternatives are multiplied by alpha and the highest values of the alternatives are multiplied by 1– ?.

If the problem is maximization, the best value is the highest value and this value is multiplied by alpha coefficient. The worst value is the lowest value and this value is multiplied by 1-? coefficient. Then, these values are sum up. Finally, the alternative with the best value (maximum value) is selected. The mathematical representation of the method is as follows:

Maxi {?*Max (0i j ) j + (1 – ?) *Min (0i j ) j }

When the objective function is maximization, the highest values of the alternatives are multiplied by alpha and the smallest values of the alternatives are multiplied by 1– ?.

Minimax Regret (Savage)

This method is based on opportunity loss, also called regret. It is defined as the difference between the optimal and the actual pay-off. Regret is the amount lost when the best alternative is not selected.

The method used for the loss of opportunities when the best alternative is not selected is “Minimax Regret”. If the objective is maximization, the maximum value of each event is determined. All values of the relevant event are subtracted from this maximum value.

If the objective is minimization, the minimum value of each event is determined. The smallest value of each event is subtracted from all values of relevant event. In this way, the regret matrix is obtained.

In this method, regardless of the objective function, the smallest value is always selected from the regret matrix.

Example 2.9 Let’s make a decision with this method using the information in Example 2.9 and Table 2.9. The objective in Example 2.9 is maximization

For the values given in Table 2.9, a regret matrix is created (Table 2.10). Because the objective is maximization, the maximum value of each event is selected. Other values are subtracted from the maximum value of event in relevant column. Thus, a regret matrix is created. Table 2.10 shows the regret matrix of the problem.

According to the results in Table 2.10, maximum regret of each alternatives are as follows:

District 1 = maximum regret = 23.500

District 2 = maximum regret = 15.500

District 3 = maximum regret = 12.000

District 4 = maximum regret = 0

The alternative with the smallest value (District 4) is chosen among the maximum regrets.

*table 2.9 and table 2.10 are on page 26