- Email: [email protected]
Circuits, networks, and power systems
Math/ Compu~. Modelling, Vol.14,pp.336-339,1990
0895-7177/90 %3.00+0.00 Pergamon Pressplc
Printed inGreatBritain
Circuits, Networks, and Power Systems A FUZZY NFXWORK MODEL
Gerry M. Klein Department of Industrial Engineering, University of Missouri-Columbia, Columbia, Missouri 65211 Jose A. Ventura Department of Industrial and Management Systems Engineering, 207 Hammond Building, The Pennsylvania State University, University Park, PA 16802 Abstract. Many problems can be modeled by networks. However, due to the imprecise nature of much of the information decision makers have available it is sometimes difficult to determine a "best" approach to the problem. To help alleviate this exigency, a network model that combines both precise and nonprecise information is presented. Key Words:
Networks, Decision Theory, Dynamic Programming, Fuzzy Sets If the uncertainty is of some kind other than randomness then probability theory cannot be properly applied. Decision classic which making is a area in randomness does not explain or model many situations. This is due to the human factor in decision making and the vagueness it introduces in terms of perceptions, subjectivities, attitudes of goals, conceptions, and emotions Rosenfeld (1975). In decision making, the person making the decision is actually what is being modelled and a modelling language is needed that can capture all aspects of the model. To this end, fuzzy set theory was developed. For a complete discussion on fuzzy set theory and its applications to decision making, the reader is referred to Jain (1976, 1977). (1975) and Zimmerman Yager and Basson (1987).
INTRODUCTION imprecise information in Dealing with decision making has always been a problem. In classical decision theory, imprecision has been dealt with from a statistical or utility based approach. However, some feel this approach is to cumbersome and does not reflect the nature of the truly information. To help model imprecise information, Zadeh (1965) introduced the concept of a fuzzy A fuzzy set is defined by a set. membership function and allows the modeling of vagueness. Since their introduction, fuzzy sets have received a great deal of attention. The evolution of fuzzy set theory seems to be paralleling classical mathematics since the "fuzzy" counterpart of many fundamental Fuzzy sets concepts have been studied. have also been employed for a wide variety of applications. They have been applied in such diverse areas as pattern recognition, approximate reasoning, artificial social sciences, intelligence, control, and medical sciences, psychology. operations research.
Zadeh defined a fuzzy set A in E by a membership function p (x) which associates a real number in the Aterval [O.ll to each element x in E. The value of u,(x) is the grade of membership of xeE in the fuzzy set A, where E represents the universal set. A fuzzy set is designed not to have sharp boundaries and is characterized by its membership function for each element of the set. The membership function clA(x) takes on values between and including 0 and 1. If p (x) = 0 then the element is not in A, if pA(x) = 1 then the element is in A, and if pA(x) = .7 then the element is more in A As u (x) approaches the thanAnot in A. value of 1. the more x b elongs to A.
One area, though, that has received a great deal of attention as a realm of application is decision theory. The number of articles and books dealing with fuzzy sets in decision making is almost overwhelming. However, very few of these articles deal with methods of using fuzzy sets that are directly assessable by a decision maker. It is the purpose of this paper to present that a not work based decision model combines fuzzy and nonfuzzy information into a setting and allows easy solution procedures and analysis.
It is easy to see that fuzzy set theory is based on the membership function pA,(x) that describes the grade of membership of the Therefore, element x in the fuzzy set A. the determination of the membership function is of importance in terms of applications as well as theoretical developments. The determination of membership functions is not easy and is a self-contained reThe search area within fuzzy set theory. interested reader is referred to Chandhuri and Majumder (1982). Giles (1976. 1988) and Hisdal (1988) for detailed information on membership function determination.
BACKGROUND Fuzzy sets were first introduced by Zadeh The concept of fuzzy sets was (1965). that because it was felt introduced probability theory did not adequately model all aspects of uncertainty or vagueness. 336
337
Proc. 7th Int. Conf. on Mathematical and Computer Modelling Assuming that a membership function exists, it is possible to develop many different operations and properties for fuzzy sets. Several simple operations that are used in the paper are presented below. Let A 5 E. B 5 E, be fuzzy sets. Then the union of A and B and the intersection of A and B are fuzzy sets and are defined by their membership function; uAUB(x) = max (u,(x). uB(x)),
(1)
uADB(x) = min (uA(x), u,(x)).
(2)
The o - cut of a fuzzy set A is denoted by A, and is given by Ao-
{xeE;
u(x) A
>a).
(3)
Note that the set A, is not a fuzzy set. One of the most basic ideas of fuzzy set theory is the extension principle. This principle allows one to take nonfuzzy concepts and extend them into the fuzzy That is, a nonfuzzy mathematical domain. concept can be extended to handle fuzzy quantities. The extension principle takes the fuzzy sets Ai, i=l,..,n and induces the fuzzy set B through some function f(x). The principle is stated in terms of the membership function for B. uB(Y) =
min (uA1,(x),..,uAn(xn)) (4) sup X1'X2'..Xn
y=f(Xl...GQ
.
Any binary operation in R, the reals, can be extended to fuzzy sets by the extension principle. One operator in particular that will be used is the sum. The extended sum, i, is given by uA i
B
(2) = sup
min (uA(x). u,(y)). (5)
z=x+y FUZZY NETWORK MODELS
Most network models involve concerns that can be modeled as fuzzy sets. It is possible to include them in the problem structure in several ways. The concerns could be introduced into a network model in standard fuzzy methods by letting an edge or vertex of the network be a fuzzy set. An example of this is to let the length of an edge be a fuzzy set representing safety and then finding the path with the highest safety (i.e. the longest path). However, in the use of fuzzy edge lengths a new problem is introduced. That problem is the interpretation of the final result. In order to obtain the fuzzy longest path, the extension principle is used in place of the normal addition and in place of nonfuzzy minimization or maximization. This leads to the possibility of obtaining a result that does not actually correspond to an existing path. To circumvent this problem the network models to be presented do not view edges as fuzzy sets but as elements of
fuzzy sets that depict the needed information about an edge (i.e. travel time, safety, etc.). This allows one to always find an existing path that has the appropriate fuzzy qualifications. These models are presented next. The first model, denoted Hl, deals with finding the shortest route. Each road segment can be viewed as having several attributes in addition to its length. Each of these attributes will be modeled by a Each road segment will then fuzzy set. have a grade of membership in that fuzzy could be set. These side attributes considered safety of transversal, traffic flow, environmental conditions, population within the area, etc. One objective of model Hl would be to find the shortest route from a waste producer to a waste site that maintains certain levels of the fuzzy sets. That is, find the shortest route that is in a given a-cut of the fuzzy sets representing the side attributes. A second A second objective would be to find the "best" route in the sense of a domination criteria. Domination would be dependent on the given criteria, i.e. more is better for safety, less is better for traffic flow or population. Note that the attributes themselves actually represent objectives of the decision constraints.
maker
as
well
as
possible
The second model, denoted H2, is the same as the first model, except that one wishes The objective of to determine locations. model H2 is to find the anti-absolutecenter for a given road structure and it's attributes. PATH MRMBRRSHIP In order to formulate and solve these issue must be one main problems, considered. This issue is related to fuzzy membership. The attributes that are modeled by the fuzzy sets are for each The overall individual edge or vertex. solution to each model given above will and combination of edges require a given the In other words, vertices. membership grade of each edge in a network, how does one determine the member ship grade of a path? If p. is a path, then P, then p. is a set of Ages e that connect some vartices s to t.The length of p. is then defined as in classical (nonfuizy) network theory, d(pi) =
C d(e)
.
eaPi Let u (p.) be the membership grade of the pathsSaskociated with the fuzzy set S. The relationship of the membership of the path to the memberships of the individual arcs are flexibility. There much allows of possible obviously a large number relationships. We will consider two: u,(p) = v us(e), eep
(6)
Proc. 7th Int. Conf on Mathematical and Computer Modelling
338
‘I,
(P) = g ps(e), eeP
(7)
dam (pij,
pkm,...,pxy)
= {pip
is
a
nondominated N-tuple). (p) represents the;;yrli;;i membersh?p grade of p. membership grade denoted by n (p) is not interval ' [O,ll. A restricted to the normalized membership function can easily be computed and given by where
q
P,(p) =
E us(e) (8)
p is nondominated if there does not exist an N-tuple p' such that each element of p' dominates the corresponding element in p. The concept of domination is dependent on the decision maker and can be mixed within the N-tuple. That is, for the first element of p, more may be better, but for the third element, less may be better.
-""g-* A modified version of equation (91, which is a multi-criteria DP approach, is given below.
SOLUTION TECHNIQUES Based on these path membership functions several different solution techniques can be developed. However, the methods present ed here are based on a dynamic programming approach that incorporates the fuzzy aspects with multi-criteria concepts. For model Hl, the major concern is to find "shortest" given the the best path auxiliary attributes of the edges. There are a variety of ways to find the shortest We will use the path for a network. dynamic programming formulation for the shortest path problem in a directed acyclic network which is: f(N) = 0 f(i) = min (dij + f(j)l(i,j) e El i
(9)
where d.. is the length of arc (i,j) and f(i) isit!he length of the shortest path or paths from vertex i to vertex N. The objective for model Hl presents a Although it is still a slight problem. shortest path problem, the inclusion of the restriction on the membership grade of the path prevents the standard dynamic programming formulation from being used. Therefore, a hybrid dynamic programming approach that can be used with any membership function will be used. Let d .eR+ be the length or distance from vertei'i to vertex j given that edge (i,j) e E. If (i.j) C E then let d..= + -. PO?2 each (s,t) path in G(V,E) ddfine an Ntuple, pst, such that P st = (Dst, PS2(s.t),PS3(s,t),...,oSN(s,t)), where 52, s3,..., SN are N-l fuzzy sets associated with each edge, D n&be:shii o~idatahn~s~t!~i?thteheu~~~de Of set Si, i=Z,...N. For the N-tuples pst define the operator $ as follows. Pim'P.. (P),u (P) ikm'& nathS3fo&h"bv u,..(P?4, pfbs (i,)) and (k,m). If the paths (i,jj and (k,m) do not have a linking edge (j,k) then they do not form a path and p is given an associated value of +-. Lais"tly, define the domination operator, dom, as
Let f(i) be an N-tuple representing the "shortest" nondominated path distance and path membership from vertex i to an identified sink vertex t. f(i) can be found by DP with the following functional equation. f(t) = (0,0,0,...,0) f(i) = dom (pij @ f(j)) . (10) (i,j)eE The shortest nondominated path from the source s to the sink t would be determined by finding f(s) and then backtracking to find the corresponding path. If there are restrictions on path membership grade in some fuzzy set, then only those paths satisfying the restriction are considered. can be used Likewise, the restriction during the recursion to eliminate potential paths. CONCLUSIONS In this paper two network models involving presented. attributes have been fuzzy These models are designed to take into account any aspects of a problem. These aspects or concerns are incorporated into the model through the use of fuzzy sets. In addition to the models, a hybrid multicriteria dynamic programming approach based on domination concepts was presented. The method was based on the shortest path algorithm and on any one of a variety of path membership functions. The combination of fuzzy and nonfuzzy attributes for each edge as well as the construction of the path membership function helps these models circumvent interpretation problems that can occur in strictly fuzzy graphs.
~roc.
7th Int.
Conf.
on Mathematical
RRFERENCES
Chaudhuri, B. 8. and Majumder, D. D., (1982). CmMembership Evaluation in Fuzzy Sets, in M.M. Gupta and E. Sanchez, eds. &proximate Reasoning in Decision Analysis, Amsterdam, North Holland, pp. 3-11. Giles, R., (1976) A Logic for Subjective Belief, Foundations of Probability Theory. Statistical Inference and Statistical Theories of Science, Vol. I, Dordrecht, Reidel. Giles, R., (1988) The Concept of Grade of Membership, Fuzzy Sets and Systems, 25, 297-323. Hisdal, E., (1988) Are Grades of Membership Probabilities?, Pussy Sets and Systems. 25. 325-348. Jain, R., (1976) Decision-Making in the Presence of Fuzzy Variables, IIE Transactions Systems Man CybeGt. 6, 698-703. Jain, R., (1977) A Procedure for MultipleAspect Decision-Making Using Fuzzy Numbers. International Journal Systems Science. 8, l-7. Rosenfeld, A., (1975) Fussy Graphs.in L.A. Zadeh. K.S. Fu. K. Tanaka, and M. Shimura, eds. Fuzzy Sets and Their Applications to Cognitive and Decision Processes, New York, Academic Press, pp 75-95 Yager, R. R. and Basson, D.. (1975) Decision Making with Fuzzy Sets, Decision Sciences, 6, 590-600. Zadeh, L. A., (1965) Fuzzy Sets, Informa tion and Control, 8, 338-353. Zimmerman, H.J., (1987) Puszv Sets, Decision Making, and Expert Systems, Nowell, MA, Kluwer Academic Publishers.
and Computer
Modelling
339
0895-7177/90 %3.00+0.00 Pergamon Pressplc
Printed inGreatBritain
Circuits, Networks, and Power Systems A FUZZY NFXWORK MODEL
Gerry M. Klein Department of Industrial Engineering, University of Missouri-Columbia, Columbia, Missouri 65211 Jose A. Ventura Department of Industrial and Management Systems Engineering, 207 Hammond Building, The Pennsylvania State University, University Park, PA 16802 Abstract. Many problems can be modeled by networks. However, due to the imprecise nature of much of the information decision makers have available it is sometimes difficult to determine a "best" approach to the problem. To help alleviate this exigency, a network model that combines both precise and nonprecise information is presented. Key Words:
Networks, Decision Theory, Dynamic Programming, Fuzzy Sets If the uncertainty is of some kind other than randomness then probability theory cannot be properly applied. Decision classic which making is a area in randomness does not explain or model many situations. This is due to the human factor in decision making and the vagueness it introduces in terms of perceptions, subjectivities, attitudes of goals, conceptions, and emotions Rosenfeld (1975). In decision making, the person making the decision is actually what is being modelled and a modelling language is needed that can capture all aspects of the model. To this end, fuzzy set theory was developed. For a complete discussion on fuzzy set theory and its applications to decision making, the reader is referred to Jain (1976, 1977). (1975) and Zimmerman Yager and Basson (1987).
INTRODUCTION imprecise information in Dealing with decision making has always been a problem. In classical decision theory, imprecision has been dealt with from a statistical or utility based approach. However, some feel this approach is to cumbersome and does not reflect the nature of the truly information. To help model imprecise information, Zadeh (1965) introduced the concept of a fuzzy A fuzzy set is defined by a set. membership function and allows the modeling of vagueness. Since their introduction, fuzzy sets have received a great deal of attention. The evolution of fuzzy set theory seems to be paralleling classical mathematics since the "fuzzy" counterpart of many fundamental Fuzzy sets concepts have been studied. have also been employed for a wide variety of applications. They have been applied in such diverse areas as pattern recognition, approximate reasoning, artificial social sciences, intelligence, control, and medical sciences, psychology. operations research.
Zadeh defined a fuzzy set A in E by a membership function p (x) which associates a real number in the Aterval [O.ll to each element x in E. The value of u,(x) is the grade of membership of xeE in the fuzzy set A, where E represents the universal set. A fuzzy set is designed not to have sharp boundaries and is characterized by its membership function for each element of the set. The membership function clA(x) takes on values between and including 0 and 1. If p (x) = 0 then the element is not in A, if pA(x) = 1 then the element is in A, and if pA(x) = .7 then the element is more in A As u (x) approaches the thanAnot in A. value of 1. the more x b elongs to A.
One area, though, that has received a great deal of attention as a realm of application is decision theory. The number of articles and books dealing with fuzzy sets in decision making is almost overwhelming. However, very few of these articles deal with methods of using fuzzy sets that are directly assessable by a decision maker. It is the purpose of this paper to present that a not work based decision model combines fuzzy and nonfuzzy information into a setting and allows easy solution procedures and analysis.
It is easy to see that fuzzy set theory is based on the membership function pA,(x) that describes the grade of membership of the Therefore, element x in the fuzzy set A. the determination of the membership function is of importance in terms of applications as well as theoretical developments. The determination of membership functions is not easy and is a self-contained reThe search area within fuzzy set theory. interested reader is referred to Chandhuri and Majumder (1982). Giles (1976. 1988) and Hisdal (1988) for detailed information on membership function determination.
BACKGROUND Fuzzy sets were first introduced by Zadeh The concept of fuzzy sets was (1965). that because it was felt introduced probability theory did not adequately model all aspects of uncertainty or vagueness. 336
337
Proc. 7th Int. Conf. on Mathematical and Computer Modelling Assuming that a membership function exists, it is possible to develop many different operations and properties for fuzzy sets. Several simple operations that are used in the paper are presented below. Let A 5 E. B 5 E, be fuzzy sets. Then the union of A and B and the intersection of A and B are fuzzy sets and are defined by their membership function; uAUB(x) = max (u,(x). uB(x)),
(1)
uADB(x) = min (uA(x), u,(x)).
(2)
The o - cut of a fuzzy set A is denoted by A, and is given by Ao-
{xeE;
u(x) A
>a).
(3)
Note that the set A, is not a fuzzy set. One of the most basic ideas of fuzzy set theory is the extension principle. This principle allows one to take nonfuzzy concepts and extend them into the fuzzy That is, a nonfuzzy mathematical domain. concept can be extended to handle fuzzy quantities. The extension principle takes the fuzzy sets Ai, i=l,..,n and induces the fuzzy set B through some function f(x). The principle is stated in terms of the membership function for B. uB(Y) =
min (uA1,(x),..,uAn(xn)) (4) sup X1'X2'..Xn
y=f(Xl...GQ
.
Any binary operation in R, the reals, can be extended to fuzzy sets by the extension principle. One operator in particular that will be used is the sum. The extended sum, i, is given by uA i
B
(2) = sup
min (uA(x). u,(y)). (5)
z=x+y FUZZY NETWORK MODELS
Most network models involve concerns that can be modeled as fuzzy sets. It is possible to include them in the problem structure in several ways. The concerns could be introduced into a network model in standard fuzzy methods by letting an edge or vertex of the network be a fuzzy set. An example of this is to let the length of an edge be a fuzzy set representing safety and then finding the path with the highest safety (i.e. the longest path). However, in the use of fuzzy edge lengths a new problem is introduced. That problem is the interpretation of the final result. In order to obtain the fuzzy longest path, the extension principle is used in place of the normal addition and in place of nonfuzzy minimization or maximization. This leads to the possibility of obtaining a result that does not actually correspond to an existing path. To circumvent this problem the network models to be presented do not view edges as fuzzy sets but as elements of
fuzzy sets that depict the needed information about an edge (i.e. travel time, safety, etc.). This allows one to always find an existing path that has the appropriate fuzzy qualifications. These models are presented next. The first model, denoted Hl, deals with finding the shortest route. Each road segment can be viewed as having several attributes in addition to its length. Each of these attributes will be modeled by a Each road segment will then fuzzy set. have a grade of membership in that fuzzy could be set. These side attributes considered safety of transversal, traffic flow, environmental conditions, population within the area, etc. One objective of model Hl would be to find the shortest route from a waste producer to a waste site that maintains certain levels of the fuzzy sets. That is, find the shortest route that is in a given a-cut of the fuzzy sets representing the side attributes. A second A second objective would be to find the "best" route in the sense of a domination criteria. Domination would be dependent on the given criteria, i.e. more is better for safety, less is better for traffic flow or population. Note that the attributes themselves actually represent objectives of the decision constraints.
maker
as
well
as
possible
The second model, denoted H2, is the same as the first model, except that one wishes The objective of to determine locations. model H2 is to find the anti-absolutecenter for a given road structure and it's attributes. PATH MRMBRRSHIP In order to formulate and solve these issue must be one main problems, considered. This issue is related to fuzzy membership. The attributes that are modeled by the fuzzy sets are for each The overall individual edge or vertex. solution to each model given above will and combination of edges require a given the In other words, vertices. membership grade of each edge in a network, how does one determine the member ship grade of a path? If p. is a path, then P, then p. is a set of Ages e that connect some vartices s to t.The length of p. is then defined as in classical (nonfuizy) network theory, d(pi) =
C d(e)
.
eaPi Let u (p.) be the membership grade of the pathsSaskociated with the fuzzy set S. The relationship of the membership of the path to the memberships of the individual arcs are flexibility. There much allows of possible obviously a large number relationships. We will consider two: u,(p) = v us(e), eep
(6)
Proc. 7th Int. Conf on Mathematical and Computer Modelling
338
‘I,
(P) = g ps(e), eeP
(7)
dam (pij,
pkm,...,pxy)
= {pip
is
a
nondominated N-tuple). (p) represents the;;yrli;;i membersh?p grade of p. membership grade denoted by n (p) is not interval ' [O,ll. A restricted to the normalized membership function can easily be computed and given by where
q
P,(p) =
E us(e) (8)
p is nondominated if there does not exist an N-tuple p' such that each element of p' dominates the corresponding element in p. The concept of domination is dependent on the decision maker and can be mixed within the N-tuple. That is, for the first element of p, more may be better, but for the third element, less may be better.
-""g-* A modified version of equation (91, which is a multi-criteria DP approach, is given below.
SOLUTION TECHNIQUES Based on these path membership functions several different solution techniques can be developed. However, the methods present ed here are based on a dynamic programming approach that incorporates the fuzzy aspects with multi-criteria concepts. For model Hl, the major concern is to find "shortest" given the the best path auxiliary attributes of the edges. There are a variety of ways to find the shortest We will use the path for a network. dynamic programming formulation for the shortest path problem in a directed acyclic network which is: f(N) = 0 f(i) = min (dij + f(j)l(i,j) e El i
(9)
where d.. is the length of arc (i,j) and f(i) isit!he length of the shortest path or paths from vertex i to vertex N. The objective for model Hl presents a Although it is still a slight problem. shortest path problem, the inclusion of the restriction on the membership grade of the path prevents the standard dynamic programming formulation from being used. Therefore, a hybrid dynamic programming approach that can be used with any membership function will be used. Let d .eR+ be the length or distance from vertei'i to vertex j given that edge (i,j) e E. If (i.j) C E then let d..= + -. PO?2 each (s,t) path in G(V,E) ddfine an Ntuple, pst, such that P st = (Dst, PS2(s.t),PS3(s,t),...,oSN(s,t)), where 52, s3,..., SN are N-l fuzzy sets associated with each edge, D n&be:shii o~idatahn~s~t!~i?thteheu~~~de Of set Si, i=Z,...N. For the N-tuples pst define the operator $ as follows. Pim'P.. (P),u (P) ikm'& nathS3fo&h"bv u,..(P?4, pfbs (i,)) and (k,m). If the paths (i,jj and (k,m) do not have a linking edge (j,k) then they do not form a path and p is given an associated value of +-. Lais"tly, define the domination operator, dom, as
Let f(i) be an N-tuple representing the "shortest" nondominated path distance and path membership from vertex i to an identified sink vertex t. f(i) can be found by DP with the following functional equation. f(t) = (0,0,0,...,0) f(i) = dom (pij @ f(j)) . (10) (i,j)eE The shortest nondominated path from the source s to the sink t would be determined by finding f(s) and then backtracking to find the corresponding path. If there are restrictions on path membership grade in some fuzzy set, then only those paths satisfying the restriction are considered. can be used Likewise, the restriction during the recursion to eliminate potential paths. CONCLUSIONS In this paper two network models involving presented. attributes have been fuzzy These models are designed to take into account any aspects of a problem. These aspects or concerns are incorporated into the model through the use of fuzzy sets. In addition to the models, a hybrid multicriteria dynamic programming approach based on domination concepts was presented. The method was based on the shortest path algorithm and on any one of a variety of path membership functions. The combination of fuzzy and nonfuzzy attributes for each edge as well as the construction of the path membership function helps these models circumvent interpretation problems that can occur in strictly fuzzy graphs.
~roc.
7th Int.
Conf.
on Mathematical
RRFERENCES
Chaudhuri, B. 8. and Majumder, D. D., (1982). CmMembership Evaluation in Fuzzy Sets, in M.M. Gupta and E. Sanchez, eds. &proximate Reasoning in Decision Analysis, Amsterdam, North Holland, pp. 3-11. Giles, R., (1976) A Logic for Subjective Belief, Foundations of Probability Theory. Statistical Inference and Statistical Theories of Science, Vol. I, Dordrecht, Reidel. Giles, R., (1988) The Concept of Grade of Membership, Fuzzy Sets and Systems, 25, 297-323. Hisdal, E., (1988) Are Grades of Membership Probabilities?, Pussy Sets and Systems. 25. 325-348. Jain, R., (1976) Decision-Making in the Presence of Fuzzy Variables, IIE Transactions Systems Man CybeGt. 6, 698-703. Jain, R., (1977) A Procedure for MultipleAspect Decision-Making Using Fuzzy Numbers. International Journal Systems Science. 8, l-7. Rosenfeld, A., (1975) Fussy Graphs.in L.A. Zadeh. K.S. Fu. K. Tanaka, and M. Shimura, eds. Fuzzy Sets and Their Applications to Cognitive and Decision Processes, New York, Academic Press, pp 75-95 Yager, R. R. and Basson, D.. (1975) Decision Making with Fuzzy Sets, Decision Sciences, 6, 590-600. Zadeh, L. A., (1965) Fuzzy Sets, Informa tion and Control, 8, 338-353. Zimmerman, H.J., (1987) Puszv Sets, Decision Making, and Expert Systems, Nowell, MA, Kluwer Academic Publishers.
and Computer
Modelling
339