Fuzzy Cognitive Maps and Intelligent Decision Support – a Review
Fuzzy Cognitive Maps and Intelligent Decision Support
– a Review
M Shamim Khan1, Alex Chong1, and Mohammed Quaddus2
School of Information Technology, Murdoch University, Perth, WA 6150
2
Graduate School of Business, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845
E-mail: s.khan@murdoch.edu.au, alexchong_@hotmail.com, quaddusm@gsb.curtin.edu.au
1
Abstract
Cognitive maps have been utilised in decision analysis in the past. More recently fuzzy cognitive
maps (FCMs) have been proposed as an alternative to knowledge-based expert systems for
representing and analysing complex systems. Increasing interest in adaptive systems, in the form of
artificial neural networks, has also focussed attention on FCMs. This paper presents an overview of
the evolution of this relatively new tool in decision analysis. A brief account of the structure,
operation and use of FCMs are given with examples of some recent work done in the field. The
advantage of FCM-based systems in terms of their relative ease of development, dynamic nature and
adaptive potential are highlighted.
1. Introduction
In generic terms, decision support concerns providing assistance with the process of decision making
(Alter, 1980), (Power, 1997). This involves selecting the best (or optimal) strategy for achieving some
given goals, from among a choice of alternative strategies to achieve the goals. A set of constraints,
in terms of the risks and uncertainties associated with each alternative, influences the decision making
process. In the context of modern information systems, decision support systems (DSS) are software
products aimed at improving the quality of organisational decision making.
The core task of a DSS is decision analysis. Real-life problems are mostly unstructured in nature,
which makes it difficult to apply algorithmic methods based on mathematical models to the process of
decision analysis. Various analytical methods such as decision trees (Render & Stair, 1988) and
influence diagrams (Howard & Matheson, 1984) have been developed as decision analysis tools. The
development of a DSS for a specific problem domain still requires considerable experience with
decision analysis and programming. This has resulted in rather limited use of DSS compared with
other software systems such as database management systems (DBMS).
During the 80s and 90s a new breed of systems known as intelligent DSS has been developed, which
mainly incorporates artificial intelligence (AI) techniques of knowledge representation and rule-based
inferencing. The re-emergence of artificial neural networks during the 80’s has also led to the
application of artificial neural networks in decision support. More recently, fuzzy logic and hybrid
approaches have been tried.
The aim of this paper is to give an overview only of the origin, evolution and application of the fuzzy
cognitive map (FCM) as a new tool for decision support. Section 2 below gives a brief outline of
current intelligent decision support systems. Cognitive maps are introduced in section 3 as an
alternative tool for decision support. Sections 4 and 5 outline the structure, operation and use of FCMs
in decision support applications with examples. This limited review concludes with section 6, which
mentions some future directions of research on FCM-based decision support systems.
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M Shamim Khan, Alex Chong, and Mohammed Quaddus
2 Intelligent Decision Support Systems
Intelligent DSS have resulted from the use of artificial intelligence techniques to improve the
performance of more traditional systems. AI techniques are used in DSS knowledge bases and
inferential procedures. The most prominent among AI techniques used in DSS has been the expert
system (Turban, 1993), which attempts to extract and codify human expertise in a computer program.
An expert system’s ability to draw conclusions as well as explain its reasoning has made it a valuable
asset to DSS studies.
Artificial neural networks (Aleksander, 1989) are systems inspired by biological systems. They
consist of a collection of relatively simple processors, the so-called neurons. During operation, each
neuron receives signals modified by the strength of its input lines. The sum of input signals received
by an artificial neuron is subjected to some non-linear transfer function, which determines the level of
output produced by the neuron. The neurons in an artificial neural network essentially run in parallel
and execution is extremely fast. Learning in artificial neural networks is the process of arriving at the
correct set of interconnection weights needed for solving a particular problem. The most common
learning procedure is called supervised training, in which numerous example input-output pairs are
used repeatedly to modify interconnection weights until the artificial neural network produces the
correct output.
Artificial neural networks represent an alternative modelling technique for use in DSS (Schoken &
Arav, 1994). They can be taught the underlying problem domain model using examples obtained from
historical data. Artificial neural networks are also able to generalise, ie, use the learned model to
respond correctly to previously unseen inputs. The ability of artificial neural networks to learn sets
them apart from expert systems, which rely on rule-bases that can be difficult to build, maintain and
extend.
Fuzzy logic (Zadeh, 1965), (Cox, 1999) allows natural language statements to be expressed and
manipulated using mathematical formalism. It allows us to deal with the inherent imprecision in
everyday life by using the technique of inexact (approximate) reasoning. In decision making
processes, we are often faced with facts and relationships which have varying degrees of truth and
falsehood, and this has to be taken into account in designing systems for decision support. Fuzzy logic
has been applied in rule-based systems, where rules fire with strengths ranging from “none at all” to
“full”. The combined effect of these rules gives rise to assertions having a proportional strength. One
example of fuzzy DSS available commercially is the Fuzzy Decision Maker (McNeill & Thro,1994)
which uses a fuzzy inferencing mechanism (O'Hagan & O'Hagan, 1993). Given a decision making
problem, with its goals, constraints and alternatives, it asks the user to rank the importance of the
goals and constraints using verbal terms ranging from most to least. It then asks for the verbal
specification of how well the various alternatives satisfy each of the goals. The decision result is
given as a graphical display of a relative "value" in a scale of 0 to 100 for each of the alternatives.
Two application examples of the Fuzzy Decision Maker are described in (McNeill & Thro,1994);
One of them is a decision process involving the analysis of an existing transport system with a view to
deciding how it should evolve in the future to become a better system. It uses goals such as "reduction
in air pollution" and "increased use of public transport"; Examples of constraints are " socioeconomic
and land use pattern of area" and "cost of construction and maintenance". Among the alternatives to
choose from are "car pooling", "telecommuting" etc. and combinations of these.
2.1
Drawbacks of rule-based DSS systems
Decision support systems based on expert systems, including those incorporating fuzzy rules and
inferencing, suffer from a number of shortcomings. One of the challenges faced by expert systembased DSS is the difficulty in modeling a human expert's behavior using rules, and maintaining the
resulting rule-base. This is because not all knowledge can be captured in the form of production rules.
Also, as the number of rules grows, so does the system complexity.
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Fuzzy Cognitive Maps and Intelligent Decision Support – a Review
The core component of an expert system is a decision tree with graph search, where the search time
increases with the tree size. The results of merging two trees is not a tree but a cyclic graph
(Taber,1991). This prevents the extension of an expert system’s knowledge base through the
combination of knowledge from multiple expert systems. In addition to this, decision trees are
feedforward structures and are unable to represent environments involving feedback. However, many
of the problem domains in real life do contain feedback loops and are thus unable to be represented
using decision trees.
A desired property of DSS yet to materialise fully is the ability to learn. Although it is possible to
build an expert system capable of learning (e.g. the hybrid approach of combining artificial neural
networks and expert systems), it is generally quite complex to do so. Ideally, a versatile DSS tool
should be capable of modelling environments which evolve dynamically through feedback. The
underlying knowledge base should be expandable by incorporating new knowledge, if possible,
through autonomous learning. One tool promising these attributes is the fuzzy cognitive map (FCM).
The concept of an FCM is derived from that of a cognitive map, which is described next.
3 Cognitive Maps
Cognitive maps (Axelrod, 1976), (Eden, 1990) are a collection of nodes linked by some arcs or edges.
The nodes represent concepts or variables relevant to a given domain. The causal links between these
concepts are represented by the edges. The edges are directed to show the direction of influence.
Apart from the direction, the other attribute of an edge is its sign, which can be positive (a promoting
effect) or negative (an inhibitory effect). Cognitive maps can be pictured as a form of signed directed
graph. Figure 1 shows a cognitive map used to represent a scenario involving some issues in public
health.
C1
Number of
people in the
city
C2
Migration
into city
+
+
C3
Modernisation
+
+
+
C5
Sanitation
-
C4
Garbage
per area
C6
Number of diseases
per 1000 residents
+
-
facilities
+
C7
Bacteria per
area
Fig 1. Cognitive map concerning some public health issues (Margaritis
& Tsadiras, 1997).
The construction of a cognitive map requires the involvement of a knowledge engineer and one or
more experts in a given problem domain. Methods for constructing a cognitive map for a relatively
recent real-world application are discussed in (Margaritis & Tsadiras, 1997).
3
M Shamim Khan, Alex Chong, and Mohammed Quaddus
The main objective of building a cognitive map around a problem is to be able to predict the outcome by letting
the relevant issues interact with one another. These predictions can be used for finding out whether a decision
made by someone is consistent with the whole collection of stated causal assertions. Such use of a cognitive
map is based on the assumption that, a person whose belief system is accurately represented in a cognitive map,
can be expected to make predictions, decisions and explanations that correspond to those generated from the
cognitive map. This leads to the significant question: Is it possible to measure a person’s beliefs accurately
enough to build such a cognitive map? The answer, according to Axelrod and his co-researchers, is a positive
one. Formal methods for analysing cognitive maps have been proposed and different methods for deriving
cognitive maps have been tried in (Axelrod, 1976).
In a cognitive map, the effect of a node A on another node B, linked directly or indirectly to it, is
given by the number of negative edges forming the path between the two nodes. The effect is positive
if the path has an even number of negative edges, and negative otherwise. It is possible for more than
one such paths to exist. If the effects from these paths is a mix of positive and negative influences, the
map is said to have an imbalance and the net effect of node A on node B is indeterminate. This calls
for the assignment of some sort of weight to each inter-node causal link, and a framework for
evaluating combined effects using these numerically weight-ed edges. Fuzzy cognitive maps (FCM)
(Caudill, 1990), (Brubaker, 1996a), (Brubaker, 1996b) were proposed as an extension of cognitive
maps to provide such a framework.
4 Fuzzy Cognitive Maps
The term fuzzy cognitive map (FCM) was coined in (Kosko, 1986) to describe a cognitive map model
with two significant characteristics:
(1) Causal relationships between nodes are fuzzified. Instead of only using signs to indicate positive
or negative causality, a number is associated with the relationship to express the degree of
relationship between two concepts.
(2) The system is dynamic involving feedback, where the effect of change in a concept node affects
other nodes, which in turn can affect the node initiating the change. The presence of feedback
adds a temporal aspect to the operation of the FCM.
The FCM structure can be viewed as a recurrent artificial neural network, where concepts are
represented by neurons and causal relationships by weighted links or edges connecting the neurons.
By using Kosko’s conventions, the interconnection strength between two nodes Ci and Cj is eij, with
eij, taking on any value in the range -1 to 1. Values –1 and 1 represent, respectively, full negative and
full positive causality, zero denotes no causal effects and all other values correspond to different fuzzy
levels of causal effects. In general, an FCM is described by a connection matrix E whose elements are
the connection strengths (or weights) eij. The element in the ith row and jth column of matrix E
represents the connection strength of the link directed out of node Ci and into Cj . If the value of this
link takes on discrete values in the set {-1, 0, 1}, it is called a simple FCM. The concept values of
nodes C1, C2, …, Cn (where n is the number of concepts in the problem domain) together represent the
state vector C.
An FCM state vector at any point in time gives a snapshot of events (concepts) in the scenario being
modelled. In the example FCM shown in figure 2, node C2 relates to the 2nd component of the state
vector and the state [0 1 0 0 0 0 0] indicates the event "migration into city" has happened. To let the
system evolve, the state vector C is passed repeatedly through the FCM connection matrix E. This
involves multiplying C by E, and then transforming the result as follows:
C(k + 1)
=
T[C(k) . E]
where C(k) is the state vector of concepts at some discrete time k, T is the thresholding or nonlinear
transformation function, and E is the FCM connection matrix.
With a thresholding transformation function, the FCM reaches either one of two states after a number
of passes. It settles down to a fixed pattern of node values - the so-called hidden pattern or fixed-point
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Fuzzy Cognitive Maps and Intelligent Decision Support – a Review
attractor. Alternatively, it keeps cycling between a number of fixed states - known as the limit cycle.
With a continuous transformation function, a third possibility known as the chaotic attractor (Elert,
1999) exists, when instead of stabilising, the FCM continues to produce different state vector values
for each cycle.
4.1 Extensions of FCMs
A number of researchers have developed extended versions of the FCM model described above.
Tsadiras et al (1995b) describe the extended FCM, in which concepts are augmented with memory
capabilities and decay mechanisms. The new activation level of a node depends not only on the sum
of the weighted influences of other nodes but also on the current activation of the node itself. A decay
factor in the interval [0,1] causes a fraction of the current activation to be subtracted from itself at
each time step.
Park (1995) introduces the FTCM (Fuzzy Time Cognitive Map), which allows a time delay before a
node xi has an effect on node xj connected to it through a causal link. The time lags can be expressed
in fuzzy relative terms such as “immediate”, “normal” and “long” by a domain expert. These terms
can be assigned numerical values such as 1, 2, 3. If the time lag on a causal link eij is m (1m) delay
units, then m – 1 dummy nodes are introduced between node i and node j.
5 FCM-based decision support
Decision makers often find it difficult to cope with significant real-world systems. These systems are
usually characterised by a number of concepts or facts interrelated in complex ways. They are often
dynamic ie, they evolve through a series of interactions among related concepts. Feedback plays a
C1
Number of
people in the
city
+0.1
+0.6
C3
Modernisation
C2
Migration
into city
+0.7
+0.9
+0.9
-0.3
C4
Garbage
per area
C6
Number of diseases
per 1000 residents
-0.9
C5
Sanitation
facilities
-0.9
+0.8
+0.9
C7
Bacteria per
area
Fig 2. Fuzzified version of the cognitive map shown in figure 1.
prominent role among them by propagating causal influences in complicated pathways. Formulating a
quantitative mathematical model for such a system may be difficult or impossible due to lack of
numerical data, its unstructured nature, and dependence on imprecise verbal expressions. FCMs
provide a formal tool for representing and analysing such systems with the goal of aiding decision
making.
Given an FCM's edge matrix and an input stimulus in the form of a state vector, each of the three
possible outcomes mentioned above can provide an answer to a causal “what if” question. The
5
M Shamim Khan, Alex Chong, and Mohammed Quaddus
inference mechanism of FCMs works as follows. The node activation values representing different
concepts in a problem domain are set based on the current state. The FCM nodes are then allowed to
interact (implemented through the repeated matrix multiplication mentioned above). This interaction
continues until:
(1) The FCM stabilises to a fixed state (the fixed-point attractor), in which some of the concepts are
'on' and others are not.
(2) A limit cycle is reached.
(3) The FCM moves into a chaotic attractor state instead of stabilising as in (1) and (2) above.
The usefulness of the three different types of outcomes depends on the user’s objectives. A fixedpoint attractor can provide straightforward answers to causal “what if” questions. The equilibrium
state can be used to predict the future state of the system being modelled by the FCM for a particular
initial state. As an example based on figure 2, the state vector [0 1 0 0 0 0 0], provided as a stimulus to
the FCM, may cause it to equilibrate to the fixed-point attractor at [0 0 0 1 0 0 0]. Such an
equilibrium state would indicate that an increase in “migration into city” eventually leads to the
increase of “garbage per area”.
A limit cycle provides the user with a deterministic behaviour of the real-life situation being
modelled. It allows the prediction of a cycle of events that the system will find itself in, given an
initial state and a causal link (edge) matrix. For FCMs with continuous transformation function and
concept values, a resulting chaotic attractor can assist in simulation by feeding the simulation
environment with endless sets of events so that a realistic effect can be obtained.
5.1 Development of FCMs for decision support
FCMs can be based on textual descriptions given by an expert on a problem scenario or on interviews
with the expert. The steps followed are:
Step 1: Identification of key concepts/issues/factors influencing the problem.
Step 2: Identification of causal relationships among these concepts/issues/factors.
Experts give qualitative estimates of the strengths associated with edges linking nodes. These
estimates are translated into numeric values in the range –1 to 1. For example, if an increase in the
value of concept A causes concept B to increase significantly (a strong positive influence), a value of
0.8 may be associated with the causal link leading from A to B. Experts themselves may be asked to
assign these numerical values. The outcome of this exercise is a diagrammatic representation of the
FCM, which is converted into the corresponding edge matrix.
5.1.2 Learning in FCMs
FCM learning involves updating the strengths of causal links. Combining multiple FCMs is the
simplest form of learning. An alternative learning strategy is to improve the FCM by fine-tuning its
initial causal link or edge strengths through training similar to that in artificial neural networks. Both
these approaches are outlined below.
Multiple FCMs constructed by different experts can be combined to form a new FCM. FCM
combination can provide the following advantages:
1. It allows the expansion of an FCM by incorporating new knowledge embodied in other FCMs.
2. It facilitates the construction of a relatively bias-free FCM by merging different FCMs
representing belief systems of a number of experts in the same problem domain.
The procedures for combining FCM are outlined in (Kosko, 1988). Generally, combination of FCMs
involves summing the matrices that represent the different FCMs. The matrices are augmented to
ensure conformity in addition. Each FCM drawn by different experts may be assigned a credibility
weight. The combined FCM is given by
6
Fuzzy Cognitive Maps and Intelligent Decision Support – a Review
E=
k=N
 Wk Ek
k=1
Where E is the edge matrix of the new combined FCM, Ek is the edge matrix of FCM k, Wk is the
credibility weight assigned to FCM k, and N is the number of FCMs to be combined. Siegel and
Taber (1987) outlines procedures for credibility weights assignment in FCMs.
McNeill and Thro (1994) discuss the training of FCMs for prediction. A list of state vectors is
supplied as historical data. An initial FCM is constructed with arbitrary weight values. It is then
trained to make predictions of future average value in a stock market using historical stock data. The
FCM runs through the historical data set one state at a time. For each input state, the ‘error’ is
determined by comparing the FCM's output with the expected output provided in the historical data.
Weights are adjusted when error is identified. The data set is cycled until the error has been reduced
sufficiently for no more changes in weights to occur.
If a correlated change between two concepts is observed, then a causal relation between the two is
likely and the strength of this relationship should depend on the rate of the correlated change. This
proposition forms the basis of the Differential Hebbian Learning (DHL). Kosko (1992) discusses the
use of DHL as a form of unsupervised learning for FCMs. DHL can simplify the construction of
FCMs by allowing the expert to enter approximate values (or even just the signs) for causal link
strengths. DHL can then be used to encode some training data to improve the FCM’s representation of
the problem domain and consequently its performance.
5.2
Current applications of FCMs in decision support
As the predecessor of FCMs, cognitive maps have been applied with different objectives in decision
support environments since the 70’s (Tsadiras & Margaritis, 1995a). FCMs can exhibit enhanced
usefulness through their ability to handle imprecise information, to evolve dynamically through
recurrent feedback and the facility to combine knowledge bases through the union of a number of
FCMs.
One application domain known for its complexity is that of strategic information systems planning
(SISP). SISP requires planners to develop scenarios and assess alternative ways of applying IT in
order to improve organisational performance. Strategic issues are complicated, unstructured and not
readily quantifiable. At least four investigations into the use of FCMs for strategic planning have been
reported (Tsadiras et al 1994), (Tsadiras & Margaritis, 1995b), (Lee, Lee, Kwon, Han, & Yu, 1998),
(Kardaras, & Karakostas, 1999).
In the field of financial application, the use of FCMs for stock investment analysis has been reported
in (Lee and Kim, 1997). In this case, FCMs have been used to extract causal knowledge, construct a
hierarchical knowledge base and perform bi-directional inference.
FCMs have also been used for decision support in industrial production processes. An on-line faultdiagnosis system has been developed successfully (Lee & Sakawa, 1996). In this application, an FCM
generates a sequence of patterns, which is compared with an observed sequence for identifying the
origin of fault.
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M Shamim Khan, Alex Chong, and Mohammed Quaddus
6 Conclusion and Future directions
Fuzzy systems in general have already proven themselves capable of dealing with the inherent
imprecision encountered in real-world problem domains. This is borne out particularly by the success
of fuzzy commercial products. Fuzzy cognitive maps combine this advantage with the wellestablished utility of cognitive maps as a decision support tool, and the dynamic and adaptive nature
of artificial neural networks.
This relatively short review has dealt mainly with the origin and underlying principles of the
development and operation of fuzzy cognitive maps in the intelligent decision support context. Only
brief mentions of FCM application examples found in the literature have been made. The main
objective has been to introduce the FCM as a new paradigm for decision support systems research and
development. The use of FCMs in decision support is still in its infancy and there is ample scope for
further investigation and development.
One promising feature of fuzzy cognitive maps is their adaptive nature. Although differential Hebbian
learning (DHL) has been proposed by Kosko (1992) as a learning scheme for encoding a sequence of
states into the FCM, no concrete procedures exist for applying DHL in FCMs. A formal methodology
suitable for practical application needs to be developed. One such methodology has been proposed in
(Chong, Khan, & Gedeon, 1999). It describes the development of an adaptive FCM by starting
initially with a simple FCM with crisp causal link strengths (+1, -1 or 0). Differential Hebbian
learning is then used to fuzzify these values and produce the desired FCM. Initial performance results
of the FCM developed using this methodology are encouraging. Further investigations involving
industrial problems are needed to establish its validity.
As is evident from the publications years of the application examples cited in this survey, fuzzy
cognitive map-based decision support systems have only recently started appearing on the scene.
Development of a suitable framework for making FCMs effective and user-friendly tools in DSS
environments is required. One step in this direction is the development of DSS generators (Bhargava,
Herrick & Sridhar, 1999) for FCM-based decision support systems. One such experimental system
has been developed recently and is described in (Chong, 1999b). Further work on FCM-based DSS
systems are planned for future.
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