Operating with Fuzzy Probability Estimates in Decision Making Processes with Risk

Abstract This paper considers different techniques of operating with fuzzy probability estimates of relevant random events in decision making tasks. The recalculation of posterior probabilities of states of nature based on the information provided by indicator events is performed using a fuzzy version of Bayes’ theorem. The choice of an optimal decision is made on the basis of fuzzy expected value maximisation.


I. INTRODUCTION
The theory of decision making under risk is a welldeveloped research and applied area. Effective tools aimed at modelling the initial situations as well as numerous choice criteria are developed that enable determination of optimal decisions for different systems of preferences of decision makers and different attitude to risk. The whole powerful apparatus successfully performs in situations when all factors of the task are set in the deterministic form.
The task of this paper is to represent techniques of decision making under risk in a fuzzy environment when all or some factors of the task are given in the fuzzy form. C , A , and B , respectively. In general form, arithmetic operations on triangular fuzzy numbers are determined as follows [5], [7] (see Fig. 1

II. BASICS OF FUZZY ARITHMETIC
In decision making tasks, these approximate variants of multiplication and division operations on fuzzy numbers are commonly used [11]: if the supports of fuzzy numbers are in the range of positive real numbers (see Fig. 1).

 
,, u l m m l u A C a c a c a c  (6, b) if the support of one fuzzy number is in the range of positive real numbers but the support of the other one is in the range of negative real numbers (see Fig. 1). , In decision making tasks, the necessity to compare fuzzy numbers occurs. A great deal (>60) of relevant detailed methods are developed. These methods can be divided into three large classes.
(1) Methods using estimates of distances from centroids to certain original points. This kind of methods is described in [3] and [13].
(2) Methods using certain specific squares as an evaluation function. A method of this kind is proposed in [9], while a survey of such methods is provided in [6].
(3) Methods using the concept of maximal and minimal sets. An example of this kind of methods is given in [4].
Any of possible methods can be used to choose decisions in a fuzzy environment. But taking into account high computational complexity of most of the methods, the following simplified technique is commonly used. Let us have a look at Fig. 2 In different manuals on fuzzy decision analysis, the value  is calculated differently: . It is apparent that the greater the value  is, the wider possibilities for comparing fuzzy numbers are. In this paper, the value 0.9   will be used.
Since both conditions (9) fulfil for the fuzzy numbers A and B , shown graphically in Fig. 2, an unambiguous conclusion can be made that BA  . pa . Let there be another complete group of random events (let us call them indicator events), whose occurrence probabilities depend on the states of nature. Let us assume that two indicator events exist, and fuzzy conditional probabilities evaluated. In partial case, the values of conditional probabilities can be set in a deterministic way. Unfortunately, this kind of initial information cannot be employed directly in the process of decision making. The matter is that the decision maker is interested in the probabilities of occurrence of the states of nature when one or another indicator event happens. In other words, he is interested in the values of conditional probabilities   The calculation of the posterior conditional probabilities under consideration can be performed using a fuzzy version of Bayes' formula [2]: Symbol S in (10)

IV. CHOOSING DECISIONS IN FUZZY ENVIRONMENT
Let us consider solving that task using an example.   The numbers at the outcomes denote criteria estimates of the outcomes (in conditional monetary units). It is necessary to choose an optimal decision on the basis of the criterion of the expected value maximisation, which is calculated for each alternative decision over the whole set of outcomes.
Let us calculate fuzzy values of outcome probabilities. For that purpose, let us multiply the values of probabilities of random events leading to the given outcome. To make the comparison of the calculated fuzzy expected values, let us represent them graphically (Fig. 4). As can be seen from Fig. 4 Kd , and decision 1 d has to be chosen as optimal. The result obtained is quite correct; however, it might cause mistrust of the decision maker because of the uncertainty of initial estimates and resulting expected values [1], [8], [10]. In this case, the following decision analysis can be made. Let us suppose that the decision maker is a risk-averse person. Then can be fixed that are closer to the right-hand borders of the supports of the corresponding fuzzy numbers. In other words, the question is about greater values of probabilities of unfavourable outcomes. However, the fixation of probability values has to be performed in such a way that those fixed values would be allowable values. In [12], formal algorithms for determining allowable values of probabilities are given for the cases of three and four fuzzy probability estimates. Though, it is possible to operate easier in the above example. Note that the sum of kernels of fuzzy probabilities of the outcomes is equal to 1. Then, to meet the requirement for the fixed values of probabilities, those values have to be such that conditions     If the result of choosing an optimal decision does not coincide with the one made on the basis of the fixed values of outcome probabilities, an additional analysis of decisions can be performed so as to validate one or another result of choice.

V. CONCLUSION
This paper has considered a technique for choosing decisions under risk in a fuzzy environment. Methods of fuzzy arithmetic enable successive execution of all operations related to the choice of an optimal decision. The use of a fuzzy version of Bayes' formula makes it possible to calculate fuzzy posterior probabilities of relevant random events (states of nature) on the basis of fuzzy information provided by indicator events. Fuzzy expected values are calculated over the whole set of consequences for each alternative decision. To compare resulting fuzzy estimates, any method designed for comparing fuzzy numbers can be used. The preference may be given to a simplified method represented in this paper because of its simplicity and visibility of results.
An analysis of the results obtained leads to these generalised conclusions: 1) Fuzzy initial probability estimates of relevant random events in decision making tasks are the consequence of the lack or insufficiency of reliable initial data. 2) Uncertainty of initial data is always translated into the uncertainty of the results. 3) Nowadays, an effective tool is available which helps to successfully solve decision making tasks in a fuzzy environment. The fixation of values of fuzzy probabilities of consequences according to the rules of fuzzy arithmetic makes it possible to successfully perform an additional analysis of decisions, which is in essence a decision sensitivity analysis.