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Interactive group decision-making: Modeling and application
~
Socio-Econ. Plann. Sci. Vol. 32, No. 2, pp. 113-121, 1998
Pergamon
PII: S0038-0121(97)00008-6
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-0121/98 $19.00+ 0.00
Interactive Group Decision-making: Modeling and Application KAUNG-HWA CHEN Department of Business Management, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C.
HSIN-HUI LIN* Department of Information Management, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C. Abstract--Past research in group decision-making has focused mainly on aggregating individual preferences among group members and rating the alternatives in order to obtain the best one with the highest evaluation score. Recently, the concept of unanimous agreement (or consensus) has been developed based on the notion of a fuzzy relation and its aggregation operations. However, the study of correlation or interaction among group members is rarely mentioned. Consequently, the main purpose of this research is to propose an interactive group decision model for describing the influences and changes of individual preference within a decision group such as in the Delphi decision process. The revised fuzzy correlation is adopted to construct an interaction function that represents the interaction between two individuals in the decision group. Also, three types of interaction--positive interaction, negative interaction, and irrelevance, are classified and defined in the proposed function. A two-dimensional interaction function is derived to describe the interactive trend if both runs and alternatives are considered. Finally, a numerical example is presented. © 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION Traditionally, decision theory deals with uncertainty in utility values, subjective preferences, and judgements through the use of probabilistic models [14]. However, many authors have stated that a stochastic decision model such as statistical decision analysis cannot measure the vagueness in human behavior [18-21]. Because the evaluation of alternatives or criteria is often accompanied by fuzziness, especially when subjective factors are involved, decision making should take into account human subjectivity, rather than employing a purely objective probability measure. Thus, fuzzy set theory is a subtle way for modeling uncertainty arising from mental phenomena, which are neither random nor stochastic [1-3]. In group decision-making, the complexity of process is perhaps the most recognizable since agreement among interactive group members is associated with many factors, such as the self-interest of individuals, aggregation of individual evaluations, and information exchange. In a word, group dynamics, organizational setting, social context, and behavioral aspects might very well impact the outcomes of group decision-making [13]. Individual decision-making in a group environment was considered by Hare [5]. While not addressing the motions of rational or optimal decisions in his model, Hare explored the process of reaching decisions within a group. In particular, he gave groups a rank-ordering task and defined 'consensus' as the difference between the average correlation of all rank orders before any discussion, and average correlation after discussion. Goodwin and Restle [4] discussed group members' choices among four alternative solutions to a problem. Group members were assumed to prefer only one of the alternatives and consensus was
Author for correspondence: Tel: 5252000 ext 4712; Fax: 886-7-5316988, E-mail: hhlin(amis.nsysu.edu.tw. 113
114
Kaung-Hwa Chen and Hsin-Hui Lin
said to be reached only when all group members preferred the same alternative. Kline [9] gave groups six alternative choices. The deviation from consensus was determined at the conclusion of the discussion by measuring the total number of positions members were away from the most agreed-upon policy. Hill [6] measured consensus by asking individual group members to rate their own level of agreement with the final group decision. In general, past research has focused mainly on the rating of alternatives or consensus (or degree of agreement) as measured by individual preferences across group members. The degree of consensus is used to overcome the rigidity of the conventional concept of consensus in which (full) consensus occurs only when all individuals agree on all issues. However, this may often be counterintuitive and not consistent with real human perception [7, 16]. Kacprzyk et al. [8] derived the measure 'degree of majority consensus' under fuzzy preferences and a fuzzy majority. The concept of 'consensus' in previous studies is limited in definition to a positive interaction, meaning the degree of agreement, which can be aggregated by defining a positive preference weighting. However, situations involving hostility and irrelevance among group members are generally not considered. In fact, taking a microscopic view, it is obvious that consensus, hostility, and irrelevance are mixed together among group members [17]. In this regard, the aggregation of individual preferences in a decision group, as performed in previous studies, will realize some loss of information. This is the case since the interactions among individuals will change over time as information exchanges occur among group members. Silver [15] recognizes the dual or competing motives of members who act as both individuals and group members, and proposes some requisite conditions for individual members in the process of information exchange. In doing so, Silver aims to minimize the probability of receiving negative evaluations weighted by the sender's status. In fact, he attempts to describe the interaction among different personalities. For our purpose, we view fuzzy correlation as the interrelation or closeness between two fuzzy subsets. It can represent the interaction between two fuzzy subsets when its values are placed on [ 1,1]. In this research, we propose a model that systematically captures the process of interaction among group members throughout a multiple-stage decision process based on the concept of fuzzy correlation. Under a Delphi process, due to the communication and exchange of information in each decision 'run', the group decision process is dynamic. For generality, we construct a dynamic model as well.
I N T E R A C T I V E G R O U P DECISION M O D E L In this section, three types of interaction among group members, namely, positive interaction, negative interaction, and irrelevance, are presented. Let X = {x~,x2..... x,} be the set of all alternatives of the decision problem and D = { 1,2 ..... m} be the set of all decision makers within the decision group. Here, two kinds of group decision problems, namely simple evaluation and two-dimensional evaluation under a Delphi process, are considered [12]:
I. Simple evaluation of the alternatives. At first, let f ( x ) be the membership function of decision maker i associated with fuzzy subset Ai. It represents the degree of 'easeness' that x could be rated as an optimal alternative by decision maker i, for all x~X, when an individual judgement is expressed by a fuzzy number. Therefore, a function that captures a degree of interrelation and direction of interaction between two decision makers must be defined. In this article, interrelation means the 'closeness' of two fuzzy opinions with no direction implied, while interaction represents the positive, negative, or independent relationship between two fuzzy opinions. So, a measure of interrelation falls in [0,1], while a measure of interaction falls in [ - 1,1]. Let F be the set of all membership functions. An interaction index, C, is a function mapping from F x F to [ - 1,1]; that is,
C:F x F - * [ - 1,1].
Interactive group decision-making
115
In this proposed function, a common domain ~ of membership f u n c t i o n s f and f, must satisfy the following conditions: 1. 2. 3. 4.
fl is a finite set in R; f:fl~[0,1]f,:fl~[0,1]; f ( f l ) = [0,1], £ ( ~ ) = [0,1]; and if xE~", then f(x) and f ( x ) are zero or undefined.
To guarantee that these two functions have an overlapping area, let (~.0 = {S:S satisfies the above four conditions} and fl must satisfy the following form: ~2=
0S.
~%;,!;
Further, let £2 be the minimum domain o f f and ~. Function C ~ f ) represents the degree and direction of interaction between decision makers i and j. Based on the nature and characteristics of the interaction among group members, C ( f ~ ) must satisfy the following properties: pl. If, for higher values o f f ( x ) f ( x ) takes higher values, and if the converse is also true, that is, the direction of values that f~f2 take are identical, and they have a high degree of consensus, then C ( f ~ ) > 0 where the value must be very high, VxEX,idED. p2. If, for higher values o f f ( x ) f ( x ) takes lower values, and if the converse is true, that is, the direction of values that f~f: take are opposite, and they have a high degree of hostility, then C ( f f , ) < 0 where the value must be very low, VxEX, idED. p3. Whether or not f(x) takes either higher or lower values, and the values o f f ( x ) are more or less the same, namely, they are irrelevant, then C ( f , f ) must be very near to zero, VxEX, idED. p4. (i) C ( f f ) = C ( f f ) , VZfEF. (ii) C ( f f ) = 1, VfEF. (iii) C(f,1 - f ) = - 1, VfEF. The main properties that C ( f , f ) must satisfy are pl, p2, and p3, as mentioned above. They represent the degree and direction of interaction between two individuals in a decision group, where p4 is a common property. Murthy et al. [10] defined a fuzzy correlation coefficient as follows:
Cs,.,,_= 1
X, + )(2 (j] -fz)Zdx'
where
X, = fQ(2f~-- 1)2dx and X2= fa(2f2-1)2dx. The above form has the following properties: Proposition 1. (i) - I _< c~_ 1:, -< 1
(ii) Cr,j2 = - Cs,r,.t_,,Vfl~ (iii) @,,r, = 1, and ci,.~-s, = - l,Vf~ (iv) G,r_, = %,f~, and c~-I,.n-s, = cs,4_,,s,4_," In addition, if a is finite, the form of G,.I2 is given as follows:
¢r,Ji = 1
4
X, + )(2 ~,(J~(x) --A(x)) 2
= lifX,+X,=O SEPS 32/2
B
.v¢:~
116
Kaung-Hwa Chen and Hsin-Hui Lin
where X, = ~ ( 2 f t ( x ) - 1)2, and X_, = ~ ( 2 f 2 ( x ) - 1)2. .',-El?
.,,El2
Obviously, the above formula coincides with the direction of interaction within the proposed function. In the finite case, Murthy's correlation coefficient will lose information and likely cause distortion. The distortion would occur at those points where the two fuzzy subsets are equivalent, with the exception that the values are equal to 0.5. Therefore, a modified correlation coefficient is defined below by removing equivalent elements of the two fuzzy subsets except that the values of ones are equal to 0.5. Thus, let D = { x f ( x ) = f ( x ) , VxEf~}, E = { x f ( x ) = ~ ( x ) = 0.5, VxEf~}, and let f~' = D - E. Then, a revised form of Murthy's fuzzy correlation is given by:
+ °,__J ,°-[ , , o ,
4 XI +
q
x2 y' ~(x)-£(x))2J ,.El? -- ~'
= 1, ~fX, + )(2 = 0
(2.1)
where X,=
}-" [ 2 f ( x ) - 1]z, and )(2= vEf~ -
O'
~ 3.E~ -
[ 2 f j ( x ) - 112 f2'
and I*l is the number of elements in set*. Clearly, this formula satisfies the properties stated earlier.
H. Two-dimensional evaluation of the alternatives If the decision process is to be carried out according to the Delphi method, then it becomes a dynamic process since the exchange of information and discussion among group members may very well influence and change an individual's decision within each 'run'. In general, it is easy to see that the interaction between two individuals involve three possible situations: cooperation, deviation, and independence. Under the Delphi decision process, preferences, which are characterized by membership functions, are dynamic. We now seek to explore how individuals' preferences varied with the decision runs. In this regard, the interaction among individuals during the dynamic decision process was captured in two dimensions: runs and alternatives. Let N = {run I, run2 ..... funk} be the ordered runs within the Delphi decision process. The membership function of decision maker i is thus defined by: Definition 1. /~i:N x X~[O,1]. It can be presented by a relation matrix Ri:
I~ll(run 1,x,) Ihl(run2,xO
• ..
~l.(run
l,x.)
-
p2,(run2,x,)
R i =
pk,(runk,xO
I~k,(runk,x,)
The formula for the above interaction between decision makers i and j is derived as follows: Definition 2. Let #R, and pR, be such that ]2,%(X,yl) =
#R,(x,y2),Vy,,y2EX, VxEX,
and
#R,(x~,y) = #~,(xz,y),Vx,,x2EX,VyEX. Then #R, and/~R, are non-interactive.
Interactive group decision-making
117
Definition 3. Let, Ai = {x:xEX,laR,(x,y,) = #R,(x,y2)Vy,,y2EX}, Aj = {x:xEX,laR,(x,y,) = laR,(x,y2)Vy~,y2EX}, B, = {y:yEX,I~R,(x~,y) = #e,(x2,y)Vx,,x2EX}, Bj = {y:yEX,l~R,(x,,y) = I~a,(X2,y)Vx,,x2EX}, E' = {(x,y):#R,(x,y) = I~R,(x,y) = 0 . 5 , V x , y E X } , E = {(x,y):I~R,(x,y) = I~R,(x,y),Vx,yEX}, D=E-E',
R3 + (Ai x B~)U(Aj x B,),R4 = D - R3,R5 = R~UD, and
X, =
Y' [2/~R,(x)- 112,X2= ~ xER 2 - R 5
[2UR,(x)- 112.
x~R- -- R 5
Then, the revised structure of Murthy's et al. [10, 1 1] formulation is given as follows:
riyal
+ IR2 - R,I 1-1
c(.u.~,,,u, 0 = t IR~I
4
v..
#R,)Q (2.2)
IR,[ . IR 2 - R 5 1
I-k5 +
........
ik f
where I*1 is again the number of elements in set*. By (2.2), the correlations between decision makers i and j for all i,jED can be obtained. These values represent the degree of overlapping and direction between two individuals across the decision runs. The two interaction functions, (2.1) and (2.2), stated above generally have different formulas where the type II formula may not be equal to 1 if [R31 > 0.
Definition 4. Suppose f ( x ) f j ( x ) are membership functions of decision makers i and j, and are associated with fuzzy subsets Ai and Aj, respectively. Given this, 1. If C ( f f ) > 0, then decision makers i and j have a positive interaction (PI), i.e. the trend is convergent. 2. If C ( f f ) = 1, then decision makers i and j have a strictly positive interaction (SPI), i.e. they are completely in the same direction. 3. If C ( f f ) < 0, then decision makers i and j have a negative interaction (NI), i.e. the trend is divergent. 4. If C ( f , f ) = - 1, then decision makers i and j have a strictly negative interaction (SNI), i.e. they have completely different decisions. 5. If C ( f f ) = 0, then decision makers i and j are irrelevant (II), i.e. their decisions are independent of each other. Now, let M be the correlation matrix, and )'C(f~), /f i < j, } ( M ) , = [ 0, otherwise. ' that is,
Ii c~) c~)... 1 C(f2~) M =
''"
"0
l'''
"'"
118
Kaung-Hwa Chen and Hsin-Hui Lin
Next, we define Sp, = { ( i f f ) l C ( f f ) > O,VidED},SN, =
{(id)lC(ff)
< 0,VialED}
and S,, = {(id')lC(f~) = O,VijED}. For simplicity, let S = {Se~, SN~, S,} represent the classified set of group members. Hence, the system IG = < X , C , S , M > denotes the Interactive Group Decision Making (IGDM)model. This model contains alternatives X, interaction index C, classified set S, and correlation matrix M. So, after several runs of a Delphi decision process, the final conclusions of a decision group can be classified as follows: 1. If consensus is reached, then the limit values of elements in M are positive and close to 1. 2. If limit values of the elements are negative and close to - 1, then members, except for the diagonal elements of the decision group, are hostile. 3. If limit values of the elements are close to zero, then members of the decision group are viewed as independent or uncorrelated. 4. If the elements are irregular, that is, their limit values in M have positive, negative, and zero values, then the elements of classified set: S = {Sp~,SN~,S,} represent the consensus, hostile, and independent subgroups, respectively. A N U M E R I C A L EXAMPLE OF T H E PROPOSED I G D M M O D E L To illustrate our proposed model, a numerical example is given as follows: Suppose that the cement industry is a type of oligopoly and it comprises four firms, Dj,D2,D3, and D4 [13]. By this model, the firms' policies toward each other would be consensus (C(f, f j ) > 0), hostility (C(f,,fj) < 0), and irrelevance (C(f,f/) = 0), which would be reflected in the price. The classified set S can reveal this equilibrium sufficiently. Namely, we can conclude that: 1. If the element Sel of S is full and others vanish, then collusive agreement is attained; that is, firms form a cartel. Additionally, the equilibrium price can be suggested by final correlation matrix M and membership functions of the firms. 2. If the element SN/of S is full and others vanish, then firms are competed and contained completely. 3. If the element Sn of S is full and others vanish, then firms are 'non-interactive'. 4. If none of the Se~, SNI, and S~ vanishes, then the firms construct three blocks that are competed and contained with each other. For purposes of illustrating the proposed model, a numerical example is given as follows: Let the pricing domain be given by X = {600, 760, 900, 1100, 1200, 1250}, and let f ( x ) be the degree of 'easeness' to obtain 'competitive advantage' with respect to Di associated with fuzzy subset A :eX. Now, assume that
AI
f~(x/) 0.2 0.7 1 0.9 /=./'2' X, = 600 + ~ + 9-~ + ~ 6
A2 =
L
f~(x3
i= ~ XJ
=
0.8 0.9 0.5 0.2 600 + 7-6-6 + 9-66 + ~
A3 = ~ f3(x/) 0.1 0.1 0.5 0.7 j=, XJ = 600 + ~ + ~ + ~ A4= ~ f4(x./) 0.1 0.5 1 ,=, x/ = 60----0+ ~ + ~
0.2 + ~ 0.2
+ 1-~
0.8 + ~
0.9 + ~
0.3 + 125------0' 0.3
+ 125-----6' 0.7 + 1-~-0'and
0.5 + ~
0.3 + 125----0
Interactive group decision-making
119
are the m e m b e r s h i p functions of D,, D2, D3 and D4, respectively, associated with X. By (2.1), the interactive index between A~ and A2 is calculated as follows: F r o m m e m b e r s h i p functions AI and A2, D = {1200,1250} and E--- ~b are obtained. Hence, X, = (0.2*2 -- 1) 2 + (0.7*2 -- 1) 2 + (2 -- 1) 2 + (0.9*2 -- 1) 2 = 2.16, X2 = (0.8*2 -- 1) 2 + (0.9*2 -- 1) 2 + (0.5*2 -- 1) 2 + (0.8*2 -- 1) 2 = 1.36, ~ ( ] ] ( x ) - f f f x ) ) 2 = 1.14
SO,
24[
COqlf2) = ~ + ~
4"1.14 2 . 1 6 + 1.36
1
1
=0.136.
Repeating the same procedure, we can obtain C0qJf3) = - 0.01,C(J]f4) = 0.816,C(f2f3) = - 0.968 C(f2f4) = - 0.239, and C(f3f4) = 0.252. So, Sp~ = {(1,2),(1,4),(3,4)},SN, = {(1,3,(2,3),(2,4)}, and S , = ¢, li also, M -
0.136 1 0 0
-0.01 -0.968 1
0.816-0.239 0.252
0
1
F r o m the set S = {Se~,SN1,Su} and matrix M, we find that firms 1 and 2, firms 1 and 4, and firms 3 and 4, have positive interactive: in particular, firms 1 and 4 reveal a high positive interaction. Similarly, firms 2 and 3 have a high opposite trend, with no independent relationships a m o n g the four firms. Next, the two dimensional model under a Delphi decision process is considered. T h a t is, interactions a m o n g decision makers and changes t h r o u g h o u t runs of the decision process are considered. Suppose that R~ ..... R4 are given by
100 Ol O109 09 05] RI=
0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.0
1.0 1.0 0.9
0.8 0.9 0.8
0.5 0.7 0.9
0.1 0.3 0.0
i05 06 Ol Ol Ol oo ,
R2=
1.0 1.0 0.1 0.0
R3=
0.1 0.0 0.1 0.1
0.2 0.3 0.4 0.7
0.7 0.9 1.0 1.0
0.8 0.8 0.7 0.7
0.5 0.4 0.3 0.2
0.8 0.9 1.0
0.3 0.4 0.2
1.0 0.9 0.1
EOl0.20.20.50.90.01 0.1 0.2 0.5 0.6
0.7 0.8 0.9
,
0.1 0.2 0.1 0.0
0.0 0.0 0.0 0.0
0.1 0.1 0.2
0.1
lot 001090906j R4=
0.0 0.0 0.1 0.0
0.1 0.5 0.7 0.6
0.1 0.3 0.4 0.0
0.8 1.0 1.0 0.9
1.0 0.9 0.9 1.0
0.5 0.3 0.3 0.1
.
are the degrees of agreement of six prices within five runs and are denoted by D~, D2, D3, and D4, respectively. By (2.2), the interaction between D~ and D2 is c o m p u t e d as follows: At first, E = {(1,3),(2,6)}, E ' = q~, and, hence, D = E - E ' = E. So, the interaction index within five runs o f D~ and D2 is given as:
CS(/a"/h) =
28[
+ 3-0 1
4"14.03 1 2 0 . 8 + 16.6 = - 0 . 4 .
120
Kaung-Hwa Chen and Hsin-Hui Lin
Similarly, C5(~Ul,~U3) =
0.58,C5(]¢~1,1.~4)
C5(#2,/~4) = -
=
0.44,C5(p2,#3) =
0.49,CS(p3,p4) =
-
0.46,
0.64.
Thus, S~, = {(1,3),(1,4),3,4)},S~, = {(1,2),(2,3),(2,4)},S~1 = ~b, and
m 5
1 0
-0.4 1
I0°°0
0.58 -0.46
0.44 1 - 0 . 4 9 [!
01 0164J
From matrix M 5, firms 1 and 3, firms 1 and 4, and firms 2 and firm 4 all have the same trends after some runs (or a given period of time). In particular, firms 3 and 4 have considerable consensus, while firms 2 and 1, firms 2 and 3, and firms 2 and 4 all have divergent trends. CONCLUSIONS A N D D I R E C T I O N S FOR F U R T H E R RESEARCH An interactive group decision-making model is proposed based on the concept of fuzzy correlation between two fuzzy sets. Three formulas of fuzzy correlation are stated and their characteristics pointed out. Moreover, three types of interaction, namely, positive interaction, negative interaction, and irrelevance, are proposed to represent decision behaviors. Successively, we use interaction among members of a decision group to describe behaviors of firms or industries in an oligopolistic market. We suggest that this approach differs from that of game theory. In the future, the prediction capability of the proposed model under a Delphi decision process should be explored. The results, in terms of convergent and divergent trends, can be obtained after running the model several times, and analyzed via mathematical or fuzzy reasoning. Additionally, the aggregation function for evaluating each subgroup must be defined. Implementation of the model through software development should also be a priority of further research. Acknowledgements--We are grateful to the Editor-in-Chief and the referees for their valuable suggestions and efforts which led to an improvement of the paper. This work was supported in part by the National Science Council of Taiwan.
REFERENCES 1. Bellman, R. E. and Zadeh, Z. A., Decision-making in a fuzzy environment. Management science, 1970, 17, 141-164. 2. Buckley, J. J., The multiple judge, multiple criteria raking problem: A fuzzy set approach. Fuzz), Sets and Systems, 1984, 13, 25-38. 3. Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Application. Academic Press, 1980. 4. Goodwin, W. F. and Restle, W., The road to agreement: Subgroup pressures in small group consensus processes. Journal of Personality Social Psychology, 1974, 3, 500. 5. Hare, A. P., A Study of Interaction and Consensus in Different Sized Groups, American Sociological Review, 1952, 262-267. 6. Hill, T. A., An experimental study of the relationship between opinionated leadership and small group consensus. Comm. Mono., 1976, 246-257. 7. Kacprzyk, J., Fedrizzi, M., (eds), Multiperson Decision-Making Using Fuzzy Sets and Possibility Theory, Kluwer, Netherlands, 1990, 231-241. 8. Kacprzyk, J., Fedrizzi, M. and Nurmi, H., Group decision-making consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems, 1992, 49, 21-31. 9. Kline, J. A., Orientation and group consensus. Centr. States Speech J., 1972, 44. 10. Murthy, C. A., Pal, S. K. and Majumder, D. D., Correlation between two fuzzy membership functions. Fuzz), Sets and Systems, 1985, 17, 23-38. 11. Murthy, C. A. and Pal, S. K., Bounds for membership functions: A correlation-based approach. Information Science, 1992, 65, 143-171. 12. Ovchinnikov, S. V. and Ozernoy, V. M., Using fuzzy binary relations for identifying noninferior decision alternatives. Fuzzy Sets and Systems, 1982, 17, 21-32. 13. Parkin, M., Economics, Addison Wesley, 1990. 14. Raiffa and Keeney, Decision-Making Under Multiple Objectives. Wiley, New York, 1976.
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15. Silver, S. D., A dual-motive heuristic for member information initiation in group decision making: Managing risk and commitment. Decision Support Systems, 1995, 15, 83-97. 16. Terano, T., Asai, K. and Sugeno, M., Fuzzy Systems Theory and Its Application. Academic Press, 1992. 17. Yager, R. R., Fuzzy decision making including unequal objectives. Fuzz), Sets and Systems, 1978, 1, 87-95. 18. Yager, R. R., On ordered weighted averaging aggregation operators in multicriteria decision-making. IEEE Trans. on Systems, Man, Cybernetic, 1988, lg, 183-190. 19. Zadeh, L. A., Fuzzy sets. Information and Control, 1965, 8,' 338-353. 20. Zadeh, L. A., Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man. Cybern., 1973, 3, 28-44. 21. Zimmermann, H. J. and Zysno, P., Decision and evaluations by hierarchical aggregation of information. Fuzzy Sets and Systems, 1983, 10, 243-266.
Socio-Econ. Plann. Sci. Vol. 32, No. 2, pp. 113-121, 1998
Pergamon
PII: S0038-0121(97)00008-6
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-0121/98 $19.00+ 0.00
Interactive Group Decision-making: Modeling and Application KAUNG-HWA CHEN Department of Business Management, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C.
HSIN-HUI LIN* Department of Information Management, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C. Abstract--Past research in group decision-making has focused mainly on aggregating individual preferences among group members and rating the alternatives in order to obtain the best one with the highest evaluation score. Recently, the concept of unanimous agreement (or consensus) has been developed based on the notion of a fuzzy relation and its aggregation operations. However, the study of correlation or interaction among group members is rarely mentioned. Consequently, the main purpose of this research is to propose an interactive group decision model for describing the influences and changes of individual preference within a decision group such as in the Delphi decision process. The revised fuzzy correlation is adopted to construct an interaction function that represents the interaction between two individuals in the decision group. Also, three types of interaction--positive interaction, negative interaction, and irrelevance, are classified and defined in the proposed function. A two-dimensional interaction function is derived to describe the interactive trend if both runs and alternatives are considered. Finally, a numerical example is presented. © 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION Traditionally, decision theory deals with uncertainty in utility values, subjective preferences, and judgements through the use of probabilistic models [14]. However, many authors have stated that a stochastic decision model such as statistical decision analysis cannot measure the vagueness in human behavior [18-21]. Because the evaluation of alternatives or criteria is often accompanied by fuzziness, especially when subjective factors are involved, decision making should take into account human subjectivity, rather than employing a purely objective probability measure. Thus, fuzzy set theory is a subtle way for modeling uncertainty arising from mental phenomena, which are neither random nor stochastic [1-3]. In group decision-making, the complexity of process is perhaps the most recognizable since agreement among interactive group members is associated with many factors, such as the self-interest of individuals, aggregation of individual evaluations, and information exchange. In a word, group dynamics, organizational setting, social context, and behavioral aspects might very well impact the outcomes of group decision-making [13]. Individual decision-making in a group environment was considered by Hare [5]. While not addressing the motions of rational or optimal decisions in his model, Hare explored the process of reaching decisions within a group. In particular, he gave groups a rank-ordering task and defined 'consensus' as the difference between the average correlation of all rank orders before any discussion, and average correlation after discussion. Goodwin and Restle [4] discussed group members' choices among four alternative solutions to a problem. Group members were assumed to prefer only one of the alternatives and consensus was
Author for correspondence: Tel: 5252000 ext 4712; Fax: 886-7-5316988, E-mail: hhlin(amis.nsysu.edu.tw. 113
114
Kaung-Hwa Chen and Hsin-Hui Lin
said to be reached only when all group members preferred the same alternative. Kline [9] gave groups six alternative choices. The deviation from consensus was determined at the conclusion of the discussion by measuring the total number of positions members were away from the most agreed-upon policy. Hill [6] measured consensus by asking individual group members to rate their own level of agreement with the final group decision. In general, past research has focused mainly on the rating of alternatives or consensus (or degree of agreement) as measured by individual preferences across group members. The degree of consensus is used to overcome the rigidity of the conventional concept of consensus in which (full) consensus occurs only when all individuals agree on all issues. However, this may often be counterintuitive and not consistent with real human perception [7, 16]. Kacprzyk et al. [8] derived the measure 'degree of majority consensus' under fuzzy preferences and a fuzzy majority. The concept of 'consensus' in previous studies is limited in definition to a positive interaction, meaning the degree of agreement, which can be aggregated by defining a positive preference weighting. However, situations involving hostility and irrelevance among group members are generally not considered. In fact, taking a microscopic view, it is obvious that consensus, hostility, and irrelevance are mixed together among group members [17]. In this regard, the aggregation of individual preferences in a decision group, as performed in previous studies, will realize some loss of information. This is the case since the interactions among individuals will change over time as information exchanges occur among group members. Silver [15] recognizes the dual or competing motives of members who act as both individuals and group members, and proposes some requisite conditions for individual members in the process of information exchange. In doing so, Silver aims to minimize the probability of receiving negative evaluations weighted by the sender's status. In fact, he attempts to describe the interaction among different personalities. For our purpose, we view fuzzy correlation as the interrelation or closeness between two fuzzy subsets. It can represent the interaction between two fuzzy subsets when its values are placed on [ 1,1]. In this research, we propose a model that systematically captures the process of interaction among group members throughout a multiple-stage decision process based on the concept of fuzzy correlation. Under a Delphi process, due to the communication and exchange of information in each decision 'run', the group decision process is dynamic. For generality, we construct a dynamic model as well.
I N T E R A C T I V E G R O U P DECISION M O D E L In this section, three types of interaction among group members, namely, positive interaction, negative interaction, and irrelevance, are presented. Let X = {x~,x2..... x,} be the set of all alternatives of the decision problem and D = { 1,2 ..... m} be the set of all decision makers within the decision group. Here, two kinds of group decision problems, namely simple evaluation and two-dimensional evaluation under a Delphi process, are considered [12]:
I. Simple evaluation of the alternatives. At first, let f ( x ) be the membership function of decision maker i associated with fuzzy subset Ai. It represents the degree of 'easeness' that x could be rated as an optimal alternative by decision maker i, for all x~X, when an individual judgement is expressed by a fuzzy number. Therefore, a function that captures a degree of interrelation and direction of interaction between two decision makers must be defined. In this article, interrelation means the 'closeness' of two fuzzy opinions with no direction implied, while interaction represents the positive, negative, or independent relationship between two fuzzy opinions. So, a measure of interrelation falls in [0,1], while a measure of interaction falls in [ - 1,1]. Let F be the set of all membership functions. An interaction index, C, is a function mapping from F x F to [ - 1,1]; that is,
C:F x F - * [ - 1,1].
Interactive group decision-making
115
In this proposed function, a common domain ~ of membership f u n c t i o n s f and f, must satisfy the following conditions: 1. 2. 3. 4.
fl is a finite set in R; f:fl~[0,1]f,:fl~[0,1]; f ( f l ) = [0,1], £ ( ~ ) = [0,1]; and if xE~", then f(x) and f ( x ) are zero or undefined.
To guarantee that these two functions have an overlapping area, let (~.0 = {S:S satisfies the above four conditions} and fl must satisfy the following form: ~2=
0S.
~%;,!;
Further, let £2 be the minimum domain o f f and ~. Function C ~ f ) represents the degree and direction of interaction between decision makers i and j. Based on the nature and characteristics of the interaction among group members, C ( f ~ ) must satisfy the following properties: pl. If, for higher values o f f ( x ) f ( x ) takes higher values, and if the converse is also true, that is, the direction of values that f~f2 take are identical, and they have a high degree of consensus, then C ( f ~ ) > 0 where the value must be very high, VxEX,idED. p2. If, for higher values o f f ( x ) f ( x ) takes lower values, and if the converse is true, that is, the direction of values that f~f: take are opposite, and they have a high degree of hostility, then C ( f f , ) < 0 where the value must be very low, VxEX, idED. p3. Whether or not f(x) takes either higher or lower values, and the values o f f ( x ) are more or less the same, namely, they are irrelevant, then C ( f , f ) must be very near to zero, VxEX, idED. p4. (i) C ( f f ) = C ( f f ) , VZfEF. (ii) C ( f f ) = 1, VfEF. (iii) C(f,1 - f ) = - 1, VfEF. The main properties that C ( f , f ) must satisfy are pl, p2, and p3, as mentioned above. They represent the degree and direction of interaction between two individuals in a decision group, where p4 is a common property. Murthy et al. [10] defined a fuzzy correlation coefficient as follows:
Cs,.,,_= 1
X, + )(2 (j] -fz)Zdx'
where
X, = fQ(2f~-- 1)2dx and X2= fa(2f2-1)2dx. The above form has the following properties: Proposition 1. (i) - I _< c~_ 1:, -< 1
(ii) Cr,j2 = - Cs,r,.t_,,Vfl~ (iii) @,,r, = 1, and ci,.~-s, = - l,Vf~ (iv) G,r_, = %,f~, and c~-I,.n-s, = cs,4_,,s,4_," In addition, if a is finite, the form of G,.I2 is given as follows:
¢r,Ji = 1
4
X, + )(2 ~,(J~(x) --A(x)) 2
= lifX,+X,=O SEPS 32/2
B
.v¢:~
116
Kaung-Hwa Chen and Hsin-Hui Lin
where X, = ~ ( 2 f t ( x ) - 1)2, and X_, = ~ ( 2 f 2 ( x ) - 1)2. .',-El?
.,,El2
Obviously, the above formula coincides with the direction of interaction within the proposed function. In the finite case, Murthy's correlation coefficient will lose information and likely cause distortion. The distortion would occur at those points where the two fuzzy subsets are equivalent, with the exception that the values are equal to 0.5. Therefore, a modified correlation coefficient is defined below by removing equivalent elements of the two fuzzy subsets except that the values of ones are equal to 0.5. Thus, let D = { x f ( x ) = f ( x ) , VxEf~}, E = { x f ( x ) = ~ ( x ) = 0.5, VxEf~}, and let f~' = D - E. Then, a revised form of Murthy's fuzzy correlation is given by:
+ °,__J ,°-[ , , o ,
4 XI +
q
x2 y' ~(x)-£(x))2J ,.El? -- ~'
= 1, ~fX, + )(2 = 0
(2.1)
where X,=
}-" [ 2 f ( x ) - 1]z, and )(2= vEf~ -
O'
~ 3.E~ -
[ 2 f j ( x ) - 112 f2'
and I*l is the number of elements in set*. Clearly, this formula satisfies the properties stated earlier.
H. Two-dimensional evaluation of the alternatives If the decision process is to be carried out according to the Delphi method, then it becomes a dynamic process since the exchange of information and discussion among group members may very well influence and change an individual's decision within each 'run'. In general, it is easy to see that the interaction between two individuals involve three possible situations: cooperation, deviation, and independence. Under the Delphi decision process, preferences, which are characterized by membership functions, are dynamic. We now seek to explore how individuals' preferences varied with the decision runs. In this regard, the interaction among individuals during the dynamic decision process was captured in two dimensions: runs and alternatives. Let N = {run I, run2 ..... funk} be the ordered runs within the Delphi decision process. The membership function of decision maker i is thus defined by: Definition 1. /~i:N x X~[O,1]. It can be presented by a relation matrix Ri:
I~ll(run 1,x,) Ihl(run2,xO
• ..
~l.(run
l,x.)
-
p2,(run2,x,)
R i =
pk,(runk,xO
I~k,(runk,x,)
The formula for the above interaction between decision makers i and j is derived as follows: Definition 2. Let #R, and pR, be such that ]2,%(X,yl) =
#R,(x,y2),Vy,,y2EX, VxEX,
and
#R,(x~,y) = #~,(xz,y),Vx,,x2EX,VyEX. Then #R, and/~R, are non-interactive.
Interactive group decision-making
117
Definition 3. Let, Ai = {x:xEX,laR,(x,y,) = #R,(x,y2)Vy,,y2EX}, Aj = {x:xEX,laR,(x,y,) = laR,(x,y2)Vy~,y2EX}, B, = {y:yEX,I~R,(x~,y) = #e,(x2,y)Vx,,x2EX}, Bj = {y:yEX,l~R,(x,,y) = I~a,(X2,y)Vx,,x2EX}, E' = {(x,y):#R,(x,y) = I~R,(x,y) = 0 . 5 , V x , y E X } , E = {(x,y):I~R,(x,y) = I~R,(x,y),Vx,yEX}, D=E-E',
R3 + (Ai x B~)U(Aj x B,),R4 = D - R3,R5 = R~UD, and
X, =
Y' [2/~R,(x)- 112,X2= ~ xER 2 - R 5
[2UR,(x)- 112.
x~R- -- R 5
Then, the revised structure of Murthy's et al. [10, 1 1] formulation is given as follows:
riyal
+ IR2 - R,I 1-1
c(.u.~,,,u, 0 = t IR~I
4
v..
#R,)Q (2.2)
IR,[ . IR 2 - R 5 1
I-k5 +
........
ik f
where I*1 is again the number of elements in set*. By (2.2), the correlations between decision makers i and j for all i,jED can be obtained. These values represent the degree of overlapping and direction between two individuals across the decision runs. The two interaction functions, (2.1) and (2.2), stated above generally have different formulas where the type II formula may not be equal to 1 if [R31 > 0.
Definition 4. Suppose f ( x ) f j ( x ) are membership functions of decision makers i and j, and are associated with fuzzy subsets Ai and Aj, respectively. Given this, 1. If C ( f f ) > 0, then decision makers i and j have a positive interaction (PI), i.e. the trend is convergent. 2. If C ( f f ) = 1, then decision makers i and j have a strictly positive interaction (SPI), i.e. they are completely in the same direction. 3. If C ( f f ) < 0, then decision makers i and j have a negative interaction (NI), i.e. the trend is divergent. 4. If C ( f , f ) = - 1, then decision makers i and j have a strictly negative interaction (SNI), i.e. they have completely different decisions. 5. If C ( f f ) = 0, then decision makers i and j are irrelevant (II), i.e. their decisions are independent of each other. Now, let M be the correlation matrix, and )'C(f~), /f i < j, } ( M ) , = [ 0, otherwise. ' that is,
Ii c~) c~)... 1 C(f2~) M =
''"
"0
l'''
"'"
118
Kaung-Hwa Chen and Hsin-Hui Lin
Next, we define Sp, = { ( i f f ) l C ( f f ) > O,VidED},SN, =
{(id)lC(ff)
< 0,VialED}
and S,, = {(id')lC(f~) = O,VijED}. For simplicity, let S = {Se~, SN~, S,} represent the classified set of group members. Hence, the system IG = < X , C , S , M > denotes the Interactive Group Decision Making (IGDM)model. This model contains alternatives X, interaction index C, classified set S, and correlation matrix M. So, after several runs of a Delphi decision process, the final conclusions of a decision group can be classified as follows: 1. If consensus is reached, then the limit values of elements in M are positive and close to 1. 2. If limit values of the elements are negative and close to - 1, then members, except for the diagonal elements of the decision group, are hostile. 3. If limit values of the elements are close to zero, then members of the decision group are viewed as independent or uncorrelated. 4. If the elements are irregular, that is, their limit values in M have positive, negative, and zero values, then the elements of classified set: S = {Sp~,SN~,S,} represent the consensus, hostile, and independent subgroups, respectively. A N U M E R I C A L EXAMPLE OF T H E PROPOSED I G D M M O D E L To illustrate our proposed model, a numerical example is given as follows: Suppose that the cement industry is a type of oligopoly and it comprises four firms, Dj,D2,D3, and D4 [13]. By this model, the firms' policies toward each other would be consensus (C(f, f j ) > 0), hostility (C(f,,fj) < 0), and irrelevance (C(f,f/) = 0), which would be reflected in the price. The classified set S can reveal this equilibrium sufficiently. Namely, we can conclude that: 1. If the element Sel of S is full and others vanish, then collusive agreement is attained; that is, firms form a cartel. Additionally, the equilibrium price can be suggested by final correlation matrix M and membership functions of the firms. 2. If the element SN/of S is full and others vanish, then firms are competed and contained completely. 3. If the element Sn of S is full and others vanish, then firms are 'non-interactive'. 4. If none of the Se~, SNI, and S~ vanishes, then the firms construct three blocks that are competed and contained with each other. For purposes of illustrating the proposed model, a numerical example is given as follows: Let the pricing domain be given by X = {600, 760, 900, 1100, 1200, 1250}, and let f ( x ) be the degree of 'easeness' to obtain 'competitive advantage' with respect to Di associated with fuzzy subset A :eX. Now, assume that
AI
f~(x/) 0.2 0.7 1 0.9 /=./'2' X, = 600 + ~ + 9-~ + ~ 6
A2 =
L
f~(x3
i= ~ XJ
=
0.8 0.9 0.5 0.2 600 + 7-6-6 + 9-66 + ~
A3 = ~ f3(x/) 0.1 0.1 0.5 0.7 j=, XJ = 600 + ~ + ~ + ~ A4= ~ f4(x./) 0.1 0.5 1 ,=, x/ = 60----0+ ~ + ~
0.2 + ~ 0.2
+ 1-~
0.8 + ~
0.9 + ~
0.3 + 125------0' 0.3
+ 125-----6' 0.7 + 1-~-0'and
0.5 + ~
0.3 + 125----0
Interactive group decision-making
119
are the m e m b e r s h i p functions of D,, D2, D3 and D4, respectively, associated with X. By (2.1), the interactive index between A~ and A2 is calculated as follows: F r o m m e m b e r s h i p functions AI and A2, D = {1200,1250} and E--- ~b are obtained. Hence, X, = (0.2*2 -- 1) 2 + (0.7*2 -- 1) 2 + (2 -- 1) 2 + (0.9*2 -- 1) 2 = 2.16, X2 = (0.8*2 -- 1) 2 + (0.9*2 -- 1) 2 + (0.5*2 -- 1) 2 + (0.8*2 -- 1) 2 = 1.36, ~ ( ] ] ( x ) - f f f x ) ) 2 = 1.14
SO,
24[
COqlf2) = ~ + ~
4"1.14 2 . 1 6 + 1.36
1
1
=0.136.
Repeating the same procedure, we can obtain C0qJf3) = - 0.01,C(J]f4) = 0.816,C(f2f3) = - 0.968 C(f2f4) = - 0.239, and C(f3f4) = 0.252. So, Sp~ = {(1,2),(1,4),(3,4)},SN, = {(1,3,(2,3),(2,4)}, and S , = ¢, li also, M -
0.136 1 0 0
-0.01 -0.968 1
0.816-0.239 0.252
0
1
F r o m the set S = {Se~,SN1,Su} and matrix M, we find that firms 1 and 2, firms 1 and 4, and firms 3 and 4, have positive interactive: in particular, firms 1 and 4 reveal a high positive interaction. Similarly, firms 2 and 3 have a high opposite trend, with no independent relationships a m o n g the four firms. Next, the two dimensional model under a Delphi decision process is considered. T h a t is, interactions a m o n g decision makers and changes t h r o u g h o u t runs of the decision process are considered. Suppose that R~ ..... R4 are given by
100 Ol O109 09 05] RI=
0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.0
1.0 1.0 0.9
0.8 0.9 0.8
0.5 0.7 0.9
0.1 0.3 0.0
i05 06 Ol Ol Ol oo ,
R2=
1.0 1.0 0.1 0.0
R3=
0.1 0.0 0.1 0.1
0.2 0.3 0.4 0.7
0.7 0.9 1.0 1.0
0.8 0.8 0.7 0.7
0.5 0.4 0.3 0.2
0.8 0.9 1.0
0.3 0.4 0.2
1.0 0.9 0.1
EOl0.20.20.50.90.01 0.1 0.2 0.5 0.6
0.7 0.8 0.9
,
0.1 0.2 0.1 0.0
0.0 0.0 0.0 0.0
0.1 0.1 0.2
0.1
lot 001090906j R4=
0.0 0.0 0.1 0.0
0.1 0.5 0.7 0.6
0.1 0.3 0.4 0.0
0.8 1.0 1.0 0.9
1.0 0.9 0.9 1.0
0.5 0.3 0.3 0.1
.
are the degrees of agreement of six prices within five runs and are denoted by D~, D2, D3, and D4, respectively. By (2.2), the interaction between D~ and D2 is c o m p u t e d as follows: At first, E = {(1,3),(2,6)}, E ' = q~, and, hence, D = E - E ' = E. So, the interaction index within five runs o f D~ and D2 is given as:
CS(/a"/h) =
28[
+ 3-0 1
4"14.03 1 2 0 . 8 + 16.6 = - 0 . 4 .
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Kaung-Hwa Chen and Hsin-Hui Lin
Similarly, C5(~Ul,~U3) =
0.58,C5(]¢~1,1.~4)
C5(#2,/~4) = -
=
0.44,C5(p2,#3) =
0.49,CS(p3,p4) =
-
0.46,
0.64.
Thus, S~, = {(1,3),(1,4),3,4)},S~, = {(1,2),(2,3),(2,4)},S~1 = ~b, and
m 5
1 0
-0.4 1
I0°°0
0.58 -0.46
0.44 1 - 0 . 4 9 [!
01 0164J
From matrix M 5, firms 1 and 3, firms 1 and 4, and firms 2 and firm 4 all have the same trends after some runs (or a given period of time). In particular, firms 3 and 4 have considerable consensus, while firms 2 and 1, firms 2 and 3, and firms 2 and 4 all have divergent trends. CONCLUSIONS A N D D I R E C T I O N S FOR F U R T H E R RESEARCH An interactive group decision-making model is proposed based on the concept of fuzzy correlation between two fuzzy sets. Three formulas of fuzzy correlation are stated and their characteristics pointed out. Moreover, three types of interaction, namely, positive interaction, negative interaction, and irrelevance, are proposed to represent decision behaviors. Successively, we use interaction among members of a decision group to describe behaviors of firms or industries in an oligopolistic market. We suggest that this approach differs from that of game theory. In the future, the prediction capability of the proposed model under a Delphi decision process should be explored. The results, in terms of convergent and divergent trends, can be obtained after running the model several times, and analyzed via mathematical or fuzzy reasoning. Additionally, the aggregation function for evaluating each subgroup must be defined. Implementation of the model through software development should also be a priority of further research. Acknowledgements--We are grateful to the Editor-in-Chief and the referees for their valuable suggestions and efforts which led to an improvement of the paper. This work was supported in part by the National Science Council of Taiwan.
REFERENCES 1. Bellman, R. E. and Zadeh, Z. A., Decision-making in a fuzzy environment. Management science, 1970, 17, 141-164. 2. Buckley, J. J., The multiple judge, multiple criteria raking problem: A fuzzy set approach. Fuzz), Sets and Systems, 1984, 13, 25-38. 3. Dubois, D., Prade, H., Fuzzy Sets and Systems: Theory and Application. Academic Press, 1980. 4. Goodwin, W. F. and Restle, W., The road to agreement: Subgroup pressures in small group consensus processes. Journal of Personality Social Psychology, 1974, 3, 500. 5. Hare, A. P., A Study of Interaction and Consensus in Different Sized Groups, American Sociological Review, 1952, 262-267. 6. Hill, T. A., An experimental study of the relationship between opinionated leadership and small group consensus. Comm. Mono., 1976, 246-257. 7. Kacprzyk, J., Fedrizzi, M., (eds), Multiperson Decision-Making Using Fuzzy Sets and Possibility Theory, Kluwer, Netherlands, 1990, 231-241. 8. Kacprzyk, J., Fedrizzi, M. and Nurmi, H., Group decision-making consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems, 1992, 49, 21-31. 9. Kline, J. A., Orientation and group consensus. Centr. States Speech J., 1972, 44. 10. Murthy, C. A., Pal, S. K. and Majumder, D. D., Correlation between two fuzzy membership functions. Fuzz), Sets and Systems, 1985, 17, 23-38. 11. Murthy, C. A. and Pal, S. K., Bounds for membership functions: A correlation-based approach. Information Science, 1992, 65, 143-171. 12. Ovchinnikov, S. V. and Ozernoy, V. M., Using fuzzy binary relations for identifying noninferior decision alternatives. Fuzzy Sets and Systems, 1982, 17, 21-32. 13. Parkin, M., Economics, Addison Wesley, 1990. 14. Raiffa and Keeney, Decision-Making Under Multiple Objectives. Wiley, New York, 1976.
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15. Silver, S. D., A dual-motive heuristic for member information initiation in group decision making: Managing risk and commitment. Decision Support Systems, 1995, 15, 83-97. 16. Terano, T., Asai, K. and Sugeno, M., Fuzzy Systems Theory and Its Application. Academic Press, 1992. 17. Yager, R. R., Fuzzy decision making including unequal objectives. Fuzz), Sets and Systems, 1978, 1, 87-95. 18. Yager, R. R., On ordered weighted averaging aggregation operators in multicriteria decision-making. IEEE Trans. on Systems, Man, Cybernetic, 1988, lg, 183-190. 19. Zadeh, L. A., Fuzzy sets. Information and Control, 1965, 8,' 338-353. 20. Zadeh, L. A., Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Syst. Man. Cybern., 1973, 3, 28-44. 21. Zimmermann, H. J. and Zysno, P., Decision and evaluations by hierarchical aggregation of information. Fuzzy Sets and Systems, 1983, 10, 243-266.