2 questions about fuzzy sets.
Microsoft Word - Assignment2 1.Consider the data given in the attached paper in Table 1, page 43, there are membership grades of two fuzzy sets S and D and the results of their aggregation I. Discuss ways of realizing their aggregation using t- norms and t- conorms, namely I ~ (x)=S(x)tD(x) and I ~ (x)=S(x)sD(x). The t norms and conorms to be used are listed below. minimum product drastic product maximum probabilistic sum drastic sum The suitability of t norm or conorm is evaluated with the use of the following performance index Q Compute the values of this index and identify the best and the worst t norm and t conorm. 2. The three-dimensional fuzzy relation defined on the Cartesian product of X, Y, and Z is given in the following form R (x, y, z) = exp (-(x-1)2 -(y+3)2 -(z-2)2 ) Determine its projections on X, Y, and Z. The reconstruction of R is realized by taking the Cartesian product of these projections by using some t-norm. For which t-norm, the reconstruction returns an original fuzzy relation. PII: 0165-0114(80)90062-7 Fuzzy Sets and Systems 4 (1980) 37-51. © North-Holland Publishing Company L A T E N T C O N N E C T I V E S IN H U M A N D EC ISION M A K I N G H.-J. ZIMMERMANN and P. ZYSNO Lehrstuhi fiir Unternehmens[orschung (Operations Research), RWTH Aachen, 51 Aachen, Federal Republic of Germany Received June 1979 Revised July 1979 The interpretation of a lecision as the intersection of fuzzy sets, computed by applying either the minimum or the pro~,uct operator to the membership functions of the fuzzy sets concerned implies that there is n, compensation between low and high degrees of membership. If, on the other hand, a decision is defined to be the union of fuzzy sets, represented by the maximum or algebraic sum of the degrees of membership, full compensation is assumed. Managerial decisions hardly ever represent any of these extremes. The aggregation of s,bjective categories in the framework of human decisions or evaluations almost always shows some degree of compensation. This indicates that human beings partiahy are using non-verbal aggregation procedures which do not correspond to the verbal and logical connectives 'and' and 'or'. The results of our experiments support the hypothesis that people often use compensatory procedures. Several well-known operators are tested. However, they do not predict our data very well. Therefore a new class of operators is suggested which varies with respect to a parameter of compensation. Our data do confirm this concept. Keywords: Decision, Compensation, Aggregation of fuzzy sets, Class of operators, Empirical results. 1. Introduction A decision in a 'fuzzy environment' has been defined by Bellman and Zadeh [ 1 ] as the intersection of fuzzy sets representing either objectives or constraints. The grade of membership of an object in the intersection of two fuzzy sets, i.e. the fuzzy set 'decision', was determined by use of either the min-operator or the product operator. The fol! ~wing example is an illustration of this: Exmnple 1. The board of directors is trying to find the 'optimal' dividend to be paid to the shareholders. For financial reasons it ought to be attractive and for reasons of wage negotiations it should be modest. The fuzzy set of the objective function 'attractive dividends' could for instance be defined by: 1 1 " x I> 5,8, /Xo(X) = t 2-~0o[-29x3-366x2-877x+540] , 1,0, x~
<~ 1,2,="" tzc(x)="~2-~o0[-29x" 3="" -="" 243x="" 2="" +="" 16x="" +="" 2388],="" lo,="" x="">~6. The fuzzy set 'decision' is then: /xc, = Min(/xo(X),/Xc(X)). The optimal dividend to be paid to the shareholders would be 3.5%, consider- ing the dividend with the highest degree of membership in the fuzzy set 'decision' as the 'most desirable'. Rather than viewing a decision as the intersection of several fuzzy sets, as we did in [6], one could describe it also as the union of all relevant fuzzy sets, using the maximum operator for aggregation: Example 2. An instructor at a university has to decide how to grade written test papers. Let us assume that the problem to be solved in the test was a linear programming problem and that the student was free to solve it either graphically or using the simplex method. The student has done both. The student's perfor- mance is expressednfor graphical solution as well as for the algebraic so lu t ionn as the achieved degree of membership in the fuzzy sets 'good graphical solution' (G) and 'good simplex solution' (S), respectively. Let us assume that he reaches /-tG = 0.9 and /~s = 0.7. If the grade to be awarded by the instructor corresponds to the degree of membership of the fuzzy set 'good solutions of linear programming problems' it 1 . o .6 "3 C . Q ! 1 2 3 zl ~5 6 Fig. 1. A fuzzy decision; x = dividend (%). Latent connectives in human decision making 39 would be quite conceivable that this grade ~u, could be determined by ~ L P - Max(~c, P~s) - Max(0.9, 0.7) = 0.9. The two definitions of decisions--as the intersection or the union of fuzzy setsmimply essentially the following: The interpretation of a decision as the intersection of fuzzy sets implies no positive compensation (trade-off) between the degrees of membership of the fuzzy sets in question, if either the minimum or the product is used as an operator. Each of them yields degrees of membership of the resulting fuzzy set (decision) which are on or below the lowest degree of membership of all intersecting fuzzy sets (see Example 1). The interpretation of a decision as the union of fuzzy sets, using the max- operator, leads to the maximum degree of membership achieved by any of the fuzzy sets representing objectives or constraints. This amounts to a full compensa- tion of lower degrees of membership by the maximum degree of membership (see Example 2). Observing managerial decisions one finds that there are har61y any decisions with no compensation between either different degrees of goal achievement or the degrees to which restrictions are limiting the scope of decisions. The compensa- tion, however, rarely ever seems to be 'complete' such as would be assumed using the max-operator. It may be argued that compensatory tendencies in human aggregation are responsibi:: ~ for the failure of some classical operators (min, product, max) in empirical investigations [4, 6]. Two conclusions can probably be drawn: Neither the non-compensatory 'and' rcpresented by operators which map between zero and the minimum degree of membership (min-operator, product- operator, Hamacher's conjunction operator [3], Yager's operator Cp(x) [7]) nor the fully compensatory 'or ' represented by operators which map between the maximum degree of membership and 1 (maximum, algebraic sum, Hamacher's disjunction operator [3], Yager's Dp(x) [7]) are appropriate to model the aggrega- tion of fuzzy sets representing managerial decisions. New additional operators will have to be de fined which imply some degree of compensation, i.e. which map also between the minimum degree of membership and the maximum degree of membership of the aggregated sets. By contrast to modeilhlg th,. non- compensatory 'and' or the fully-compensatory 'or' they should represent types of aggregation which we shall call 'compensatory and'. Zadeh already pointed out that the type of operator for the aggregation of fuzzy sets in the sense of an intersection might depend on whether the sets arc 'interactive' or 'non-interactive' [8] which sometimes was interpreted as ~depen- dent' or 'independent'. We feel, however, that 'interaction' in the above sense is not a feature of the fuzzy sets aggregated but of the type of aggregation which does not necessarily depend on the sets in question. Before developing more promising models for aggregation we felt it necessary to obtain some more empirical knowledge on the presumption that people do not avoid the area between the minimum and maximum when merging subjective categories. 40 H.-Z Zimmerman, P. Zysno It is possible that human beings use many non-verbal connectives in their thinking and reasoning. One type of these connectives may be called 'merging connectives' which may be represented by the 'compensatory and'. Being forced to verbalize them men possibly map the set of 'merging connectives' into the set of the corresponding language connectives ('and', 'or'). Hence, when talking, they use the verbal connective which they feel closest to their 'real' non-verbal connective. In analogy to the verbal connectives, the logicians defined the connectives 'A ' and ' v ' , assigning certain properties to each of them. By this, compound sentences can be examined for their truth values. In contrary to this constructive process, the empirical researcher has to analyze a given structure. Therefore, in order to induce subjects to use their 'own connectives', we avoided the verbal connectives 'and' and 'or' in our experiment, but tried to ask for combined membership values implicitly presenting a suitable experimental design and instruction, respectively. 2. Experiment As a simple example of a decision or evaluation using such a 'compen'-;atory and' consider the following situation in which the quality of dovetailing of tiles and their solidity determine together the quality of a tile. 2.1. A 'realistic' situation A factory produces fire-resistive tiles for revetting chimneys or heating systems. Let us assume that the quality of the tiles depends on two factors: their dovetailing and their solidity (Fig. 2). \ / X / \ / \ l V--7 \ 1 Fig. 2. Dovetailed tiles. Latent connectives in human decision making 41 \ J a b J J c d Fig