Group Decision Support Using Fuzzy Cognitive Maps for Causal Reasoning

Group Decision and Negotiation 13: 463–480, 2004
C 2004 Kluwer Academic Publishers. Printed in the Netherlands Group Decision Support Using Fuzzy Cognitive Maps for Causal Reasoning M. SHAMIM KHAN School of Information Technology, Murdoch University, Perth, WA 6150, Australia (E-mail: s.khan@murdoch.edu.au) MOHAMMED QUADDUS Graduate School of Business, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia (E-mail: quaddusm@gsb.curtin.edu.au) Abstract Cognitive maps have been used for analysing and aiding decision-making by investigating causal links among relevant domain concepts. A fuzzy cognitive map (FCM) is an extension of a cognitive map with the additional capability of representing feedback through weighted causal links. FCMs can be used as tools for both static as well as dynamic analysis of scenarios evolving with time. An FCM represents an expert’s domain knowledge in a form that lends itself to relatively easy integration into a collective knowledge base for a group involved in a decision process. The resulting group FCM has the potential to serve as a useful tool in a group decision support environment. An appropriate methodology for the development and analysis of group FCMs is required. A framework for such a methodology consisting of the development and application phases is presented. Key words: casual influence, fuzzy cognitive maps, group decision support 1. Introduction A fuzzy cognitive map (FCM) is a representation of the belief system of an expert in a given domain. From a mathematical perspective, it is a directed graph of vertices (or nodes) representing important domain concepts, and edges representing causal links between these vertices. In its graphical form, an FCM is a collection of concept nodes and weighted causal links representing domain knowledge that is relatively easy to visualise and manipulate. An FCM allows feedback among its nodes, enabling its use for modelling domains that evolve with time. It is particularly suited for use in soft knowledge domains with a qualitative, rather than quantitative, emphasis. Decision problems are usually characterised by numerous issues or concepts interrelated in complex ways. They are often dynamic, i.e., they evolve through a sequence of interactions among related concepts. Feedback plays a prominent role in updating the concept states by propagating causal influences through multiple 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. An FCM’s ability to represent unstructured knowledge through causalities expressed in imprecise terms 464 KHAN AND QUADDUS makes it potentially a very useful decision support tool. Although the theory behind FCMs is fairly well developed, to-date, its reported use in the DSS arena has been limited. However, FCMs do have enormous potential for use as a tool for DSS and group decision support system (GDSS). One feature of FCMs is their ability to be merged to create a new FCM that can represent the views of a number of experts in a unified manner. This gives rise to the prospect of using FCMs in a group decision support role. The aim of this paper is to introduce FCMs and propose a framework for using them in a group decision environment. It introduces the FCM as an extension of cognitive mapping, and follows this by a description of the structure, operation and development. The use of FCMs for analysis, carried out with the objective of supporting decision-making, is outlined. Given the importance of combined FCMs in group decision-making, the process of merging FCMs is discussed in some detail. It then presents a framework proposed by us for using FCMs as a GDSS tool. The paper ends with a summary, and an outline of future research work on the application of FCMs in group decision support using the proposed framework. 2. Fuzzy cognitive maps Fuzzy cognitive maps are an extension of the cognitive map (Axelrod 1976; Eden 1990), which is a collection of nodes connected by some causal links or edges. The nodes in a cognitive map represent concepts or variables relevant to a given domain. 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). 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. A cognitive map is static in the sense that it does not allow for influences to be fed back to concept nodes and thus cannot be used to simulate evolution of domain concepts with time. Fuzzy cognitive maps (Caudill 1990; Kosko 1986) were proposed as an extension of the cognitive map, and provide a theoretical basis that overcomes the shortcomings of cognitive maps. An FCM, although based on the cognitive map model, has two additional and significant characteristics: 1. Causal relationships between nodes are fuzzified. Instead of only using signs to indicate positive or negative causality, as shown in Figure 1, a numerical value is associated with the causal link to express varying degrees of causal influence. This allows handling of causal influence levels expressed by domain experts using imprecise or fuzzy linguistic terms. 2. The system is dynamic involving feedback. If the change in a concept node affects one or more other nodes through causal links directed from it to these other nodes, the resulting change in these other nodes can affect the node initiating these changes. The presence of feedback adds a temporal aspect to the operation of the FCM and enables the observation of progressive changes in a scenario as events unfold. 465 FUZZY COGNITIVE MAPS FOR CAUSAL REASONING 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 Figure 1. An example fuzzy cognitive map dealing with public health issues in a city (Hagiwara 1992). 2.1. FCM structure and operation By using Kosko’s conventions, the interconnection strength between two nodes Ci and C j is ei j , with ei j 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 causality. In general, an FCM with n concept nodes is described by an n × n connection matrix, E, whose elements are the causal link strengths (or weights) ei j . The matrix corresponding to the example FCM with seven concepts shown in Figure 1 is:   0 0 0.6 0.9 0 0 0  0.1 0 0 0 0 0 0      0 0.7 0 0 0 0 0    0 0 0 0 0 0 0.9 E =     0 0 0 0 0 −0.9 −0.9    −0.3 0 0 0 0 0 0  0 0 0 0 0 0.8 0 The element in the ith row and jth column of matrix E represents the strength of the causal link directed out of node Ci and into C j . 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. 466 KHAN AND QUADDUS C1(k) e1j Node j Cj (k+1) e2j C2(k) . . . f (Σ Cieij) eij Ci(k) Figure 2. Computation of a concept node’s output. An FCM state vector at any point in time gives a snapshot of concept values or events in the scenario being modelled. In the example FCM shown in Figure 1, node C2 relates to the second component of the state vector, and the state [0 1 0 0 0 0 0] indicates the concept or event “migration into city” is on. This state vector can be used as an input value and its effects on all the concepts can be observed for successive time steps. The new value of any concept is calculated based on the current values of all the concepts, which exert influences on it through causal links. This computation of a node’s output is based on the combination of a summing operation followed by the use of a non-linear transformation function such as thresholding. As shown in Figure 2, the summing operation involves multiplying each input, Ci (causal influence arriving from another concept node), with the weight or strength, ei j , of the corresponding causal link. The summation term,
Ci ei j , is known as the activation of the concept node. The nonlinear function, f, can be a simple thresholding operation with a threshold value T, where the output, C j, is given by  0 if activation j ≤ T Cj = 1 if activation j > T A thresholding function such as the one shown above results in binary concept values. To produce continuous concept values, a continuous-output transformation function may be used. One such popular function is the sigmoid squashing function, giving output Cj = 1 e−gain·activation j 1 + where the term, gain, is a constant which determines how quickly the output reaches the limiting values of 0 and 1. Given the state vector, C, representing all concept node values at any time step k, calculation of the new state vector is performed by multiplying C by the weight matrix E mentioned earlier, 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, and T is the non-linear transformation function. FUZZY COGNITIVE MAPS FOR CAUSAL REASONING 467 With a thresholding transformation function, the FCM reaches either one of two states after a number of passes. It converges to a fixed pattern of node values – the so-called hidden pattern or fixed-point 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. Instead of stabilising, the FCM continues to produce different state vector values for successive cycles. 2.2. Extended FCMs One important consideration in the use of FCMs for dynamic analysis is that of differing lengths of time taken by different concepts to change in response to changes in other causally linked concepts. There may be delays involved in the impact of change in a certain concept becoming evident in other concepts causally linked to it. For example, according to the example FCM of Figure 1, an increase in the number of people living in a city has positive impacts (i.e., leads to an increase) to both modernisation of the city as well as the amount of garbage. Although the effect on garbage generation is immediate, increased modernisation (depending on factors such as public pressure and political will of decision makers) normally would have a time lag relative to population increase. Park (1995) introduces the Fuzzy Time Cognitive Map (FTCM), which allows a time delay before a node xi has an effect on node x j 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 then be translated into numbers of delay units. The size of these units would depend on the problem domain. If the time lag on a causal link ei j is m (m ≥ 1) delay units, then m – 1 dummy nodes are introduced between node i and node j. Another attempt at extending the FCM model is described in Tsadiras and Margaritis (1995b). In this variant of the extended FCM, 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. The effect of causal influence in the prevalent FCM model is represented by a linear function in the form of a simple multiplication of concept values by causal link weights. This approach is not only intuitive, it also maintains the simplicity of the FCM paradigm as one of its major attractions. In some situations though, there may be merit in the argument for replacing the linear causality function by some non-linear functions more reflective of the domain in question (Hagiwara 1992). 2.3. Development of FCMs Compared with other schemes for developing knowledge bases, such as the rule base in an expert system, the process of constructing an FCM is relatively simple. The main steps in this process are as follows. 468 KHAN AND QUADDUS Step 1: Identification of key domain issues or concepts. Step 2: Identification of causal relationships among these concepts. Step 3: Estimation of causal link strengths. The initial development of an FCM is best done manually using pencil and paper. The two-dimensional graphical nature of this knowledge representation scheme helps both the process of development and visual analysis afterwards. Although, as shown in Figure 1, an operational FCM will have a numerical weight value associated with each causal link, the domain expert is expected to express the degrees of causal influences using fuzzy linguistic expressions such as, small positive, moderate negative, strong positive, and so on. These expressions are later mapped to numerical values usually 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 assigned to the causal link leading from A to B. Analytical procedures (e.g. Analytical Hierarchy Process (Saaty 1980)) may be used to find the numerical fuzzy weights. By focussing on one pair of concepts at a time, the expert is relieved of the task of coming up with hidden or indirect cause–effect relationships, which become apparent later through analyses carried out on the FCM once it is completed. In addition, the use of fuzzy expressions for degrees of causality avoids the specification of precise numerical values by the domain expert, who is likely to find such a requirement difficult. It also helps improve the representative accuracy of the FCM by deliberately keeping parameters vague during interaction with domain experts. For example, two experts may both specify a particular causal link qualitatively as strongly positive, but are likely to come up with two different numerical values if asked to express the link strength quantitatively. The outcome of the FCM development exercise is a diagrammatic representation of the FCM, which can be converted into the corresponding connection matrix using computer software and stored for future use. 3. Analysing FCMs for decision support Given an FCM developed to represent a given domain, there can be two distinct methods for analysing this representation with the objective of using it for decision support. An FCM can be used for a static analysis of the domain for establishing (1) the relative importance of concepts, and (2) indirect and total causal effects between concept nodes as mentioned in Axelrod (1976). Dynamic analysis of an FCM is concerned with the evolution over time of a simulated system as a whole and its constituent components. This type of analysis utilises the convergence property of an FCM mentioned earlier when subjected to an input stimulus representing a given situation. Kosko (1986) describes centrality as a measure for determining the importance of nodes in an FCM. Centrality of a concept C j in a cognitive map is given by the sum of the number of concepts causally impinging (directly or indirectly) on C j , and the number of concepts causally impinged on by C j . These numbers are given by the number of edges in the paths leading into and out of C j . In case of an FCM, the number as well as the strengths of these 469 FUZZY COGNITIVE MAPS FOR CAUSAL REASONING C6 Weak C1 Very strong C3 C2 Moderate Strong Weak Moderate C4 Very strong C5 Figure 3. An example FCM. causalities can be taken into account to give a more accurate representation of concept centrality as defined below: concept centrality(Ci ) = IN(Ci ) + OUT(Ci ), where IN(Ci ) is the sum of weights of causal links constituting all paths connecting nodes C j , i = j, to Ci ; OUT(Ci ) the sum of weights of causal links constituting all paths connecting node Ci to all nodes C j , i = j. In summing causal weights, absolute values are used to give positive and negative causalities equal importance. Concepts with high centrality values deserve special attention in any analysis for decision support. Kosko (1986) describes a formal method for obtaining both the indirect and total effects between two nodes in an FCM using fuzzy causal algebra as follows. Let there be m causal paths from Ci to C j , each of which may be expressed as (i, kl1 , kl2 , . . . , kln , j), for 1 ≤ l ≤ m. Let I l (Ci , C j ) denote the indirect effect of concept Ci on concept C j through the lth causal path, and T(Ci , C j ) denote the total effect of Ci on C j over all m causal paths. Then Il (Ci , C j ) = min {e(C p , C p+1 ) : (p, p + 1)ε(i, kl1 , kl2 , . . . , kln , j)}, T(Ci , C j ) = max(Il (Ci , C j ), where p and p + 1 are contiguous left to right path indices. In other words, the indirect effect of concept Ci on concept C j is given by the weight of the weakest causal link in the path leading from Ci to C j . The total effect of concept Ci on concept C j over all the paths leading from Ci to C j amounts to the strongest of the indirect effects of Ci on C j . For example, suppose for a particular application, the fuzzy causal link strengths are specified as: weak, moderate, strong, and very strong, and the corresponding FCM is as shown in Figure 3. Then, indirect effect I(C1, C6) = weak, total effectI(C1, C3) = strong. 470 KHAN AND QUADDUS The dynamic analysis of an FCM can be carried out to observe and explore the impact of changes in the decision domain with time. Given an FCM’s connection matrix and an input stimulus in the form of a state vector, the resulting state (or states in case of a limit cycle or chaotic attractor) can provide useful insights into the likely impacts of any changes made to the system modelled by the FCM. These changes may be made for example by turning a particular concept “on” to provide an answer to a “what-if” question. The 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 value of each node is determined by available data on the concept it represents. While some of the concepts may be quantitative in nature, e.g., “population of a city”, or “unemployment rate”, others will be qualitative, e.g., “user awareness of available technology”, or “level of modernisation”. Qualitative concepts will involve subjective judgement in assigning their values. Concept values need to be normalised in order to prevent particular concepts assuming disproportionate significance in the causal chains. The FCM nodes are then allowed to interact (implemented in computer software through the repeated matrix multiplication mentioned in section on Applications of FCMS in Decision Support). This interaction continues until: 1. The FCM stabilises to a fixed state (the fixed-point attractor), in which some of the concepts are ‘on’ (or, in case of a continuous transformation function, have relatively large values), and others are not. 2. The FCM keeps cycling through the same set of output states (the limit cycle). 3. The FCM exhibits unstable behaviour (the chaotic attractor) and keeps changing state instead of stabilising as in (1) and (2) above. This third possibility exists only when a continuous valued transformation function is used for calculating concept node outputs. The usefulness of the three different types of outcomes depends on the user’s objectives. A fixed-point 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. For example, with the FCM shown in Figure 1, the state vector [0 1 0 0 0 0 0], provided as a stimulus to the FCM, may cause it to stabilise 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 decision domain being modelled. It allows the prediction of a cycle of events that the domain will find itself in, given its cause–effect relationships as represented in its FCM, and a specific initial state. For FCMs implemented with a continuous transformation function, a resulting chaotic attractor can provide a realistic and informative effect in simulation by feeding the simulation environment with an endless sequence of state vectors. FCMs can also be used as a convenient tool for performing sensitivity analysis. Complex relationships between concepts can be explored by holding some concepts values while allowing others to change. FUZZY COGNITIVE MAPS FOR CAUSAL REASONING 471 3.1. Applications of FCMs in decision support Success in today’s highly competitive and often rapidly changing business environment depends on fast and reliable decision making. Various decision support tools attempt to aid the process of decision making by utilising knowledge extracted from domain experts and applying various inference methods. Being able to make predictions into the future, given past and present knowledge about the behaviour of a problem domain, is a major concern for managers and policy makers. As the predecessor of FCMs, cognitive maps have been applied with different objectives in decision support environments since the 1970s (Eden and Ackermann 1998; Tsadiras and 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 planning. Strategic issues are complicated, unstructured and not readily quantifiable. A number of investigations into the use of FCMs in this area have been reported. Tsadiras and Margaritis (1994) describe an example application in strategic planning in the automobile industry. An FCM consisting of the concepts: “High Profits”, “Customer Satisfaction”, “High Sales”, “Union Raises”, “Safer Vehicles”, “Foreign Competition” and “Lower Prices” is used. Initialising these concepts with values for each concept best representing the prevailing situation according to expert opinion results in the FCM converging to a limit cycle with the value of each node varying periodically within a range. Lowering of prices is a common strategy for minimising the impact of foreign competition. This is reflected in the FCM by a weight of +0.5 assigned to the causal link leading to “Lower Prices” from “Foreign Competition”. Assuming that discounts may become difficult to make, this causal link is weakened to a value of +0.1 to observe the consequences of an increase in foreign competition. The modified FCM is observed to converge to a fixed-point equilibrium where although safer car design and customer satisfaction remain high, there are significant reductions in sales and profits. Lee et al. (1998) describe the use of an FCM for strategic planning simulation. Causal knowledge stored in an FCM is combined with a differential game-based simulation mechanism, which enables time-variant competition to be incorporated in the simulation process. The FCM helps decision makers understand the complex dynamics between a certain strategic goal and related environmental factors. In strategic information systems planning (SISP), planners develop scenarios and assess alternative ways of applying information technology in order to improve organisational performance. Kardaras and Karakostas (1999) describe the use of FCMs as an alternative modelling approach for simulating the SISP process. The proposed FCM considers both organisational and information technology (IT) related concepts and their causal relationships. It consists of 165 concepts (referred to as variables) and 210 causal links, which were derived from theoretical frameworks, case studies and relevant practical experience. The FCM can be used to assist planners to identify specific IT projects and assess their impact on an organisation. It is shown that the FCM-based model is a powerful representation 472 KHAN AND QUADDUS technique, which is more flexible than other SISP models and allows accommodation of changes in business and IT environments. 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, the FCM modelling tool has been used to deal with a relatively complex domain by following the strategy of decomposing the problem into a number of smaller sub-problems, each of which is more manageable in terms of expressing the relevant concepts and their causal links. This approach creates a number of FCMs representing different levels – the market level, industry level and company level, which are combined to form a hierarchical knowledge base. This involves the construction of three intra-level FCM matrices, such as the one involving concepts at the market level and their cause–effect relationships, as well as two inter-level FCM matrices; namely, those relating market–industry and industry–company relationships. Bi-directional (downward or upward) inferences can be performed by starting at a particular level and propagating the inference reached at that level to the next higher or lower level through the inter-level matrices. An inference propagated from another level to a particular level can be combined with its own inference to infer a composite effect. For example, the effect of a strong yen on a particular company can be inferred by combining the inference of the company level FCM with that propagated downward from the market level and industry level. Electronic data interchange (EDI) control design is another ill-structured problem domain requiring consideration of the complex causal relationships among various components of control. It is difficult for EDI experts to predict the causal effects of one control on another, requiring the application of statistical estimation techniques on opinions expressed by different experts. Lee and Han (2000) describe the EDIFCM in which the interrelationships among seven components are modelled using structural equations. The latent variables in the causal paths represent factors. Modelling with linear structural relationships is used to estimate the standardised causal relation. These estimates are mapped into values ranging from –1 to 1. The overall fit of the model is assessed by generating fitness indices among the chi-square statistics. FCMs have also been used for decision support in industrial production processes. An on-line fault-diagnosis system has been developed successfully (Lee and Sakawa 1996). Quantitative approaches to fault-diagnosis systems are based on the accurate measurement of model parameters. However, for some processes, accurate model parameters, and accurate and direct measurements of some process variables may not be available. This limits the applicability of quantitative diagnostic approaches. The FCM-based approach is a qualitative alternative requiring much less process knowledge. In this particular application, an FCM generates a sequence of patterns, which is compared with an observed sequence for identifying the origin of fault. 4. Group FCM Matrix representation of FCMs makes it a relatively simple procedure to merge multiple FCMs for creating an aggregate representation of knowledge extracted from a number of experts. In the context of group decision support, we refer to such an FCM as a group FCM or simply GFCM. The ability to merge FCMs offers the following advantages. FUZZY COGNITIVE MAPS FOR CAUSAL REASONING 473 1. It provides a methodology for an adaptive FCM, which can accumulate knowledge progressively as new expertise and ideas become available. 2. It can increase the reliability of an FCM by incorporating the opinions of more than one domain experts, who may be ranked according to their credibility. 3. It has the potential to be a useful tool for group decision-making environments, allowing the rapid development of a collective knowledge base for group decision analysis and support. The procedures for combining FCMs are described in Kosko (1988). Generally, combination of FCMs involves summing the matrices that represent the different FCMs. In all probability, different versions of an FCM specific to the same domain will consist of an unequal number of concepts. This results in these FCM matrices having different sizes. This requires augmentation of some matrices to ensure conformity in addition. Matrices with fewer concepts are augmented by including any missing concept(s) through addition of extra rows and columns of all zeros. If the total number of distinct concepts over all FCMs is n, then each connection matrix is augmented to become an n × n matrix. For k experts, the combined FCM connection matrix, E, is given by E = 1/k(E1 + E2 + · · · + Ek ) FCMs obtained from different experts may be assigned a credibility weight. If each expert is assigned a credibility weight wi in the range 0 to 1, the combined FCM connection matrix is given by Ew = 1/k(w1 E1 + w2 E2 + · · · + wk Ek ) Taber and Siegel (1987) discuss procedures for credibility weights assignment in FCMs. 5. A framework for group decision support using fuzzy cognitive maps A group fuzzy cognitive map (GFCM) can be used as an effective tool in a group decision support environment, provided an appropriate methodology is utilised for GFCM development and analysis. We propose the following framework for the utilisation of a GFCM as a tool for aiding group decision. It is described below in terms of its two phases concerned with the development and application of a GFCM. The development phase is illustrated with a hypothetical group decision example following on from the city planning FCM shown in Figure 1. Members of a group participating in a decision process may start the development phase individually or in sub-groups. For example, the sales, marketing and manufacturing departments of an industrial organisation can be represented by sub-groups within the group involved in deciding on the launch of a new product range. The following description, however, assumes the decision group to consist of individuals rather than sub-groups, e.g., each member of the group may represent one particular department of an organisation. 474 KHAN AND QUADDUS Development phase 1. Each participant of the group decision process identifies concepts he or she considers to be important and relevant to the decision domain. This reflects their perception of the domain and any bias. Concepts are drawn as nodes of the FCM to be developed. 2. Each decision group member next specifies causal links between the identified concepts. Attention is given to so-called policy nodes, which are not affected by any other nodes (no causal links lead into them). 3. Strengths of the inter-concept causal links are then expressed by each group member using fuzzy linguistic terms, and assigned as numerical weights to these links. Each decision group member thus completes one FCM on paper. 4. Individual FCMs are converted into computer representation using available FCM software, which provides a user-interface for interactive construction of an FCM. 5. Individual FCMs are merged by software to create the preliminary version of the group FCM. 6. The decision group then reviews the preliminary group FCM to identify groups of two or more concepts that are too similar in their meaning but have different concept names given to them by different experts. For each of these groups all but one of the concepts is removed from the GFCM. This is aimed at improving clarity and operational efficiency of the GFCM by eliminating redundant concepts. The final GFCM is thus created. The steps in the development phase are shown in Figure 4. Although the process of merging FCMs is relatively straightforward, and as such can be performed automatically once the individual FCMs are input, reviewing and modifying the resulting preliminary GFCM requires human intervention. The decision group has to carefully consider the meaning and significance of each node to determine any redundancy that may have been introduced by different experts viewing the problem domain with a different perspective and expressing perhaps the same concepts using different verbal expressions. Once two or more concepts are identified as too similar in meaning to one another, all except one of them are removed from the merged preliminary GFCM. But the decision to FCM1 FCM2 .. . FCMn Figure 4. Development of a group FCM. FCM merging GFCM review Preliminary GFCM Final GFCM 475 FUZZY COGNITIVE MAPS FOR CAUSAL REASONING C1 City population +0.1 +0.4 C2 Migration into city C3 Modernisation +0.4 +0.9 +0.8 C5 Sanitation facilities -0.3 C4 Amount of garbage produced -0.6 C6 Number of diseases per 1000 residents +0.8 +0.8 C1 City population +0.1 +0.5 +0.9 C2 Migration into city C3 Infrastructure improvement +0.5 +0.7 +0.8 C4 Garbage per area C5 Sanitation facilities +0.3 C5 Healthcare facilities -0.4 -0.9 -0.9 +0.9 C6 Number of diseases per 1000 residents +0.8 C7 Bacteria per area Figure 5. Two different versions of the city health issues FCM from Figure 1. 476 KHAN AND QUADDUS Table 1. FCM Matrix of the Preliminary GFCM Consisting of 10 Concept Nodes. Concept (id) City population (C1 ) Migration into city (C2 ) Modernisation (C3 ) Garbage per area (C4 ) Sanitation facilities (C5 ) Number of diseases per 1000 (C6 ) Bacteria per area (C7 ) Garbage produced (C8 ) Infrastructure improvement (C9 ) Healthcare facilities (C10 ) C1 0 0.1 0 0 0 −0.3 0 0 0 0.3 Degree of causal influence, ei j of Ci on C j (i, j = 1–10) C2 C3 C4 C5 C6 C7 C8 C9 0 0.5 0.9 0 0 0 0.9 0.5 0 0 0 0 0 0 0 0 0.55 0 0 0.85 0 0 0 0 0 0 0 0 0 0.9 0 0 0 0 0 0 −0.8 −0.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.8 0 0 0 0 0 0 0 0.8 0 0 0 0.5 0 0 0.7 0 0 0 0 0 0 0 0 0 0 0 0 C10 0 0 0 0 0 0 0 0 0.8 0 Table 2. Similar Concepts Found in the Preliminary GFCM. Concept Garbage produced Modernisation Total causal influence 0.8 1.4 Similar concepts Garbage per area Infrastructure improvement Total causal influence 0.9 2.0 select one of these for retention should be made after considering each of them in terms of the impact of its removal on the GFCM. For instance, the removal of a concept node also removes all causal links passing through it, leaving some other nodes with fewer outgoing and incoming links. Concepts, which have fewer outgoing links representing their influence on other nodes, should be considered as more suitable for removal, as their absence is likely to affect relatively few other concepts. With equal number of outgoing links, the concept with the least total causal influence on other concepts in the GFCM is considered first for removal. The causal influence of a node is measured by summing the weights associated with all its outgoing links with their signs ignored. The FCM matrix corresponding to the merged example FCMs shown in Figure 5 is given in Table 1. It consists of two pairs of similar concepts as listed in Table 2. Garbage per area is a linear function of Garbage produced and both represent the role of garbage generated in city life. The decision in step 6 above to select Garbage produced for removal is based on the fact that its outgoing causal link has a slightly lower weight of 0.8 associated with it compared with that of Garbage per area equaling 0.9, as shown in Table 2. Similarly, concept Modernisation is regarded as carrying the same meaning as concept Infrastructure improvement. The former is selected for removal because of its lower total causal influence compared with the latter. As pointed out earlier, the aftermath of deleting redundant concept nodes is the disappearance of associated causal links. The impact of this, particularly for outgoing links of a deleted node, needs to be analysed in order to decide whether extra links should be added to the retained equivalent concept node. In the group FCM example shown in Figure 6, the removal of the node Garbage produced also removes the causal link from it to the concept Number of diseases per 1000 residents. This removal is not compensated for with the 477 FUZZY COGNITIVE MAPS FOR CAUSAL REASONING C3 Modernisation +0.5 C1 City population +0.1 +0.5 +0.9 C3 Infrastructure improvement +0.5 +0.7 +0.8 C5 Healthcare facilities -0.3 C6 Garbage produced -0.8 +0.8 +0.9 +085 C5 Sanitation facilities +0.3 +0.9 +0.55 C2 Migration into city -0.9 C6 Number of diseases per 1000 residents +0.8 C7 Bacteria per area +0.9 C4 Garbage per area Figure 6. Preliminary GFCM created by merging the three FCMs from Figures 1 and 5. introduction of any new link from the equivalent node Garbage per area, due to the fact that Garbage per area already exerts positive causal influence on Number of diseases per 1000 residents through the concept Bacteria per area. In contrast, the positive influence of the deleted concept Infrastructure improvement on concept Healthcare facilities is not reflected by any pathway from its equivalent concept Modernisation. The eliminated causal link is therefore maintained by replacing its old deleted source node by the source node Modernisation. In other words, the expert opinion – “Increased improvement in infrastructure leads to more healthcare facilities” continues to be part of the GFCM in the equivalent form – “Increased modernisation leads to more healthcare facilities”. The resulting GFCM after the deletion of redundant concepts and associated adjustments in the preliminary GFCM from Figure 6 is shown in Figure 7. 5.1. Application phase This phase consists of both static and dynamic analyses. It must be carried out in the group environment for immediate feedback and discussion. 478 KHAN AND QUADDUS C3 Modernisation +0.5 C1 City population +0.1 +0.5 +0.9 C3 Infrastructure improvement +0.5 +0.7 +0.8 +085 C5 Sanitation facilities +0.3 C5 Healthcare facilities -0.3 +0.55 C2 Migration into city -0.8 -0.9 C6 Number of diseases per 1000 residents +0.9 +0.8 C7 Bacteria per area +0.9 C4 Garbage per area Figure 7. GFCM derived from preliminary GFCM shown in Figure 6. 5.1.1. Static analysis Full range of static analyses can be carried out as discussed in Eden and Ackermann (1998). At least the following must be performed. 1. Concept centrality (identification of the relative significance of a concept in the domain). 2. Causal link analysis (estimation of the degree of influence between pairs of concepts). 5.1.2. Dynamic analysis Dynamic analysis of FCM can be carried out to study the behaviour of the simulated system over time. As mentioned before, the system might stabilise to a fixed state, enter into a limit cycle, or a chaotic attractor. All of these possible behaviours can provide very important information for decision support. The dynamic analyses can be structured as follows. 1. An initial dynamic analysis to study the behaviour of the simulated system. 2. A range of what-if analyses by subjecting the GFCM to a range of initial state vector values of interest. FUZZY COGNITIVE MAPS FOR CAUSAL REASONING 479 3. Sensitivity analyses for specific concepts. Complex relationships between concepts can be explored by holding some concepts values constant while allowing others to change. 4. Summarising implications. 5.2. Summary and future directions Cognitive maps have been extensively used for policy analyses and decision support (Eden 1990; Eden and Ackermann 1998). Fuzzy cognitive maps add a new dimension to cognitive maps by adding the capability of dynamic feedback analysis. Although the theory behind FCMs is well developed (Kosko 1986, 1987, 1997), its application in the decision support arena has been limited. This paper provided an overview of the FCM and its structure and operation with some examples of FCM applications in decision support. The development of group FCMs and a framework for group decision support using them has been presented. The proposed framework includes procedures for the development and application of FCMs in a group decision environment. The framework presented remains to be tried and tested. Our immediate research plan is to implement it in a practical group decision application to further refine and validate the methodology contained in it. Future research involving group FCMs will deal with (i) developing a computer aided decision support tool based on the proposed framework, (ii) testing the GFCM-based decision tool in various field environments, and (iii) comparing and contrasting the group FCM approach with other dynamic feedback system analysis approaches such as system dynamics (Forrester 1968). Note 1. One such software, the Virtual World Mapper (School of Information Technology 2003), was used to develop the example FCMs shown in Figure 5. References Axelrod, R. (1976). Structure of Decision. Princeton, USA: Princeton University Press. Caudill, M. (1990). “Using Neural Nets: Fuzzy Cognitive Maps”, AI Expert, 49–53. Eden, C. (1999). Strategic Thinking with Computers, Long Range Planning, Vol. 23, No. 6, 35–43. Eden, C. and F. Ackerman. (1998). Making Strategy: The Journey of Strategic Management. London, UK: SAGE Publications. Elert, G. (1999). 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