„Decision making in fuzzy environment
- Ways for getting practical Decison models“
Heinrich J. Rommelfanger[*]
Abstract. In real decision situations we are often confronted with the problem that the very demanding conditions of classical decision models are often not fulfilled or the costs for getting this information seem too high. Subsequently, the decision maker usually abstains from constructing a decision model; he fears that this model is not a real image of his real problem.
The fuzzy set theory offers the possibility to construct decision models with vague data. Consequently, a lot of decision models with fuzzy components are proposed in literature. In my opinion, only fuzzy utilities (fuzzy results) and fuzzy probabilities are important for practical applications. Therefore, the focus of this paper is concentrated on these subjects.
At first, fuzzy intervals of the e-l-type are introduced. These special fuzzy sets offer a practical way for modeling vague data. Moreover, the arithmetic operations can be calculated with little effort. Afterwards, different preference orderings on fuzzy intervals are discussed.
Based on these definitions the principle of Bernoulli can easily be extended to decision models with fuzzy consequences. The use of additional information for improving the a priori probabilities is also possible. Moreover, fuzzy probabilities can be used combined with crisp or with fuzzy utilities. Here, we introduce several new algorithms for calculating the fuzzy expected values.
A disadvantageous consequence of the use of vague data is the fact that an absolutely best alternative is not identified in all applications. But normally it is possible to reject the majority of the alternatives as inferior ones. For getting the optimal alternative additional information on the results of the remaining alternatives can be used; but this should be done under consideration of cost-benefit-relations.
Apart from the fact that fuzzy models offer a more realistic modeling of decision situations the proposed interactive solution process leads to a reduction of information costs. That circumstance is caused by the fact that additional information is gathered in correspondence to the requirements and under consideration of cost-benefit-relations.
Keywords: decision theory, fuzzy utilities, fuzzy probabilities, information costs,
1 Introduction
Looking at modern theories in management sciences and business administration one recognizes that the majority of these concepts are based on decision theory in the sense of Neumann and Morgenstern [1953]. However, empirical surveys, see e.g. [Lilien 1987], [Tingley 1987], [Meyer zu Selhausen 1989], reveal that statistical decision models are hardly used in practice to solve real-life problems. This neglect of recognized theoretical concepts may be caused by the fact that the very demanding conditions of classical decision models are often not fulfilled in real decision situations or the costs for getting this information seem too high.
For modeling a decision problem by a classical decision system, the decision maker (DM) must be able to specify the following elements:
1. A set A of alternative courses of action (acts), ,
2. A set S of possible events associated with each course of action, ,
3. A value (result, gain) to be associated with each act-event combination,
. G is the set of possible values .
4. The degree of knowledge with regard to the chance of each of events occurring. Usually only partial knowledge is assumed in form of a probability distribution .
5. A criterion by which a course of action is selected:
In literature, the Bernoulli-criterion is recommended for rational behavior, i.e. the expected utility should be maximized:
.
6. A posteriori probability distribution:
The only chance for improving the solution of a classical decision model is to use additional information of a test market . Knowing the Likelihoods, the a priori probability distribution can be substituted by the a posteriori probability distribution
= Bayes’s formula .
Since the paper "Fuzzy Sets" of Lofti A. Zadeh was published in 1965, the fuzzy sets theory has been considered as a new way for modeling more realistic decision models. Especially between 1975 and 1985 several decision models with various fuzzy components were introduced. Without any claim on completeness, the following fuzzy elements have been proposed for use in decision models:
1. Fuzzy acts , Tanaka, Oukuda; Asai 1976;
2. Fuzzy events , Tanaka, Oukuda; Asai 1976.
3. Fuzzy probabilities , Watson; Weiss; Donell 1979; Dubois; Prade 1982; Whalen 1984.
4. Fuzzy utility values , where U is the set of given crisp utilities associated with each act-event combination, Jain 1976; Watson; Weiss; Donell 1979; Yager 1979; Rommelfanger 1984; Whalen 1984.
5. Fuzzy information , Tanaka, Oukuda; Asai 1976; Sommer 1980.
6. Moreover, some authors propose to substitute the probability distribution by a possibility distribution , see e.g. Yager 1979 and Whalen 1984. They assume that utilities can be measured on an ordinal scale only and therefore expected values do not exist.
These new ideas, however were not applied to practice, either because they did not become known to the public or because they are of little use for real decision problems.
In my opinion the latter statement is correct as far as the points 1, 2, 5 and 6 are concerned:
· DMs need workable but not fuzzy acts.
· In real problems, the events and the information are usually described in a fuzzy way. In these cases one is able to assign directly probabilities to those elements; that means that we have probability distributions and and values directly associated with the combinations . Therefore, we can use the classical procedure as we replace by and by . But for simplifying the presentation, we will use crisp notations in this paper.
· In my opinion persons have no idea how to interpret possibility degrees in contrast to the interpretation of probability degrees. Moreover, possibility measures allow no addition or multiplication but only the comparison of possibility values by using the min- or max-operator. Therefore I prefer to use probabilities, even though we have only subjective ones.
I think the best chance for increasing the acceptance of decision models in practice is to use fuzzy utilities and fuzzy probabilities. Therefore, I will concentrate on these two extensions. At first, we will discuss the use of fuzzy utilities or fuzzy values associated with each act-event combination. In this case the well known Bernoulli-principle can be extended to the fuzzy model. Moreover, if it is possible to get additional information of a test market, we can improve the solution by using a posteriori probability distributions. The concept of „value of additional information“ can also be extended to fuzzy models by using fuzzy values of information.
Crucial topics of decision models with fuzzy utilities are:
a) The modeling of fuzzy utilities associated with each act-event combination,
b) the definition of expected utility values,
c) the preference orderings of expected utility values.
In addition to the fuzzy utilities fuzzy probabilities will be used in the second part of the paper. There the main problem is the calculation of expected utility values.
2 Modeling fuzzy values associated with each act-event combination
One of the most difficult problems in classical decision theory is the transformation of the values in utility values . Working with fuzzy values we have the same difficulties. In this contribution I do not want to discuss the question, how to get (fuzzy) utility functions. Therefore, we assume that the DM knows his utility function u = u(gij); then the fuzzy results are mapped in the fuzzy utilities . Alternatively we can suppose that the DM is able to specify directly utility values , where U is the possible set of crisp utility values. Obviously in the case of risk neutrality, we can use instead of .
In literature values or are usually modeled in form of triangular fuzzy numbers. In my opinion this shape with a mean value is too special, the application of fuzzy intervals is more realistic. On the other side a DM has often more information, so that he can characterize the fuzzy interval in even more detail.
As an efficient way of getting suitable membership functions we propose the following procedure, which is in a similar form used in the program FULPAL for solving (multiobjective) fuzzy linear programming problems, see [Rommelfanger 1994].
At first the DM specifies some prominent membership values and relates them to special meanings. This step can be clarified by using three levels which appear to be sufficient for practical applications.
a = 1 : = 1 means that u has the highest chance of realization,
a = l : means that the decision maker is willing to accept u as an available value for the time being. A value y with has a good chance of belonging to the set of utility values associated with the act-event combination . Corresponding values of u are relevant for the decision.
a = e : < e means that u has only a very little chance of belonging to the set of utility values associated with the act-event combination . The decision maker is willing to neglect the values u with < e.
Accordingly the DM should specify numbers , so that
,
and .
The lower the information of the DM, the larger are the intervals , . The special case = can also be imagined, but in my opinion it is rarely realistic to assume that all coefficients are fuzzy numbers.
Consequently the polygon line from (, e) over (, l), (, 1), (, 1), (, l) to (,e) is a suitable approach to the membership function of on .
Figure 1: =
We characterize a fuzzy interval with this kind of membership function by . For simplification we call this special fuzzy set a fuzzy interval of l-e-type. If required the DM can specify additional membership levels and additional points of the polygon line.
An advantage of fuzzy intervals of e-l-type that the arithmetic operations based on Zadeh's extension principle can be calculated extremly simple. Moreover the approximation of the product is very much better compared with the terms for fuzzy intervals of L-R-type and it can be improved by using additional levels.
=
=
Ä =
3 Fuzzy expected values and preference relations
As each real number a can be modeled as a fuzzy number
with mA(x) = ,
we assume the general case that each act-event combination is valued by a fuzzy interval =.
If the DM is able to specify a priori probabilities , we can calculate the fuzzy expected value of each act ai:
= =,
where
Example
A manufacturer is confronted with the problem of determining the output of a product. Based on his pattern of production he has the choice between five alternatives which are ordered according their size: a1< a2 < a3 < a4 < a5.
The profit earned with a specific output depends on the demand, which is not known with absolute certainty. Due to his amount of information the manufacturer considers either a „high“ (state of nature s1) or an „average“ (state of nature s2) or a „low“ (state of nature s3) demand. He assigns the following a-priori-probabilities to the states of nature:
p(s1) = 0,5, p(s2) = 0,3, p(s3) = 0,2 .
The succeeding matrix of profits displays which profits measured in 1.000 $ correspond to the alternative constellations of output and demand. In order to elude the problem of obtaining utility values we assume risk neutrality. Then we can simply use the maximization of expected profit criterion for the decision and are sure that the selected acts are those that are consistent with the true preferences based on expected utilities. As in the case of risk neutrality it does not influence the decision whether we employ expected profits or expected utilities we are going to apply either of them in accordance with the content.
s1 / s2 / s3a1 / (170; 180; 200; 220; 225; 230) / (70; 83; 90; 100; 110; 120) / (-110; -97; -90; -77; -60, -50)
a2 / (140; 155; 165; 175; 180; 190) / (85; 93; 100; 110; 115; 125) / (-85; -80; -70; -58; -50; -40)
a3 / (120; 135; 145; 150; 160; 170) / (115; 130; 135; 140; 145; 150) / (-30; -20; -10; 0; 5; 10)
a4 / (85; 90; 100; 110; 115; 125) / (85; 93; 100; 105; 108; 115) / (-15; -10; -5; 5; 10; 15)
a5 / (45; 48; 50; 53; 58; 60) / (40; 45; 50;50; 53; 55) / (35; 40; 45; 50; 55; 60)
Table 1: A priori profit matrix
expected profita1
a2
a3
a4
a5 / (84; 95.5; 109; 124.6; 133.5; 141)
(78.5; 89.4; 98.5; 108.9; 114.5; 124.5)
(87; 102.5; 111; 117; 124.5; 132)
(65; 70.9; 79; 87.5; 91.5; 100)
(41.5; 45.5; 49; 51.5; 55.9; 58.5)
Table 2: Expected profit matrix
Figure 2: Membership functions of the expected profits
Comparing the membership functions of the expected profits in Figure 2, it becomes evident that the alternatives a4 and a5, eventually even alternative a2 come off a lot worse than the alternatives a1 and a3. Yet the decision whether a1 or a3 should be selected is not as trivial as in the classical model because of the fact that fuzzy sets are not well ordered.
In the literature various concepts are proposed for comparing fuzzy sets and for constructing preference orderings, see e.g. [Dubois, Prade 1983], [Bortolan, Degani 1985], [Rommelfanger 1986], [Chen, Hwang 1992]. Essential preference criteria are the r-Preference and the e-Preference.
Definition: r-Preference:
A set is preferred to a set on the r-level, r Î [0, 1], written as , if r is the least real number, so that