Applied Soft Computing - Hashemite University

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Applied Soft Computing 48 (2016) 91–110

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

novel semi-quantitative Fuzzy Cognitive Map model for complexystems for addressing challenging participatory real life problems

amoon Obiedat a, Sandhya Samarasinghe b,∗

Department of Computer Information Systems, The Hashemite University, JordanIntegrated Systems Modelling Group, Lincoln University, Christchurch, New Zealand

r t i c l e i n f o

rticle history:eceived 27 July 2015eceived in revised form 29 May 2016ccepted 1 June 2016vailable online 23 June 2016

eywords:omplex real-life problemsocio-ecological systemsuzzy Cognitive Map-Tuple fuzzy linguistic representationentrality measuresredibility weightsolicy scenario simulations

a b s t r a c t

Fuzzy Cognitive Maps (FCM) are a promising approach for socio-ecological systems modelling. FCMsrepresent problem knowledge extracted from different stakeholders in the form of connected fac-tors/variables with imprecise cause-effect relationships and many feedback loops. These typically largemaps are condensed and aggregated to obtain a summary view of the system. However, representa-tion, condensation and aggregation of previous FCM models are qualitative due to lack of appropriatequantitative methods. This study tackles these drawbacks by developing a semi-quantitative FCM modelconsisting of robust methods for adequately and accurately representing and manipulating imprecise datadescribing a complex problem involving stakeholders for pragmatic decision making. The model startswith collecting qualitative imprecise data from relevant stakeholders. These data are then transformedinto stakeholder perceptions/FCMs with different causal relationship formats (linguistic or numeric)which the proposed model then represents in a unified format using a 2-tuple fuzzy linguistic represen-tation model which allows combining imprecise linguistic and numeric values with different granularityand/or semantic without loss of information. The proposed model then condenses large FCMs using asemi-quantitative method that allows multi-level condensation. In each level of condensation, groups ofsimilar variables are subjectively condensed and the corresponding imprecise connections are computa-tionally condensed using robust calculations involving credibility weights assigned to variables (variables’importance). The model then uses a quantitative fuzzy method to aggregate perceptions/FCMs into astakeholder group or social perception/FCM based on the 2-tuple model and credibility weights assigned
to FCMs (stakeholders’ importance). Thereafter, the structure of produced FCMs is analysed using graphtheory indices to examine differences in perceptions between stakeholders or groups. Finally, the modelapplies various what-if policy scenario simulations on group FCMs using a dynamical systems approachwith neural networks and analyses scenario outcomes to provide appropriate recommendations to deci-sion makers. An example application illustrates method’s effectiveness and usefulness.
© 2016 Elsevier B.V. All rights reserved.

. Introduction

Soft computing approach, coined by Zadeh in (1994) [62], haseen proposed to deal with reasoning and approximation andxploit the tolerance for imprecision and uncertainty of complexroblems. Zadeh (1994) also pointed to the key techniques of soft
omputing: fuzzy logic, neural network and probabilistic reasoning,ith emphasis on fuzzy logic technique for approximate decisionaking. The last three decades have witnessed the growth of sev-
∗ Corresponding author.E-mail addresses: [email protected] (M. Obiedat),

[email protected] (S. Samarasinghe).

ttp://dx.doi.org/10.1016/j.asoc.2016.06.001568-4946/© 2016 Elsevier B.V. All rights reserved.

eral soft computing approaches such as intelligent systems andexpert systems proposed to address real life ecological problems.For example, Yamada et al. [58] used expert systems for elicit-ing expert knowledge for wildlife habitat modelling. This studyprovided promising results for wildlife habitat management. How-ever, the authors found out that the elicitation of knowledge basewas complicated and very time consuming. Another example ofaddressing ecological problems using intelligent systems was inChen and Liu [9]. In this study, artificial neural network and mul-tilinear regression models were developed for modelling water
quality in reservoirs. It provided valuable information that couldhelp decision-makers in managing reservoir water quality. How-ever, this study requires many complex calculations and largeamount of data for the water quality variables needed for model
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imulation. Moreover, it lacks an effective method for the determi-ation of sufficient variables needed to deal with the complexitynd dynamicity of the problem.

Many real-life problems such as ecosystems management areualitative, complex and dynamic such that their knowledge cannly be represented in the form of relevant factors linked to eachther through imprecise and uncertain cause-effect relationshipsnd many feedback loops. Fuzzy Cognitive Map (FCM) presentedy Kosko [31] is one such directed graph that represents domainnowledge on a map by way of nodes and imprecise directed con-ections between them to represent the factors and relationships,espectively. FCM incorporates fuzzy logic to deal with uncertainonnections and allows feedback loops to appropriately representystem dynamics. Compared to various soft computing approachesuch as intelligent systems and expert systems, FCM model is sim-ler and quicker to extract knowledge from diverse sources andifferent levels of stakeholder experience at the lowest cost. This isecause FCM model can represent and deal with the knowledge ofarious complex problems in an intuitive, appropriate and flexibleay that other intelligent systems cannot. It represents knowledge

s factors and relationships, rather than using complicated IF/THENules as expert systems. It has the ability to deal with ill-definedariables and relationships. It is also useful if detailed scientificata or knowledge are lacking or insufficient but local knowledgef people is available [43] thus making them highly valuable foruch domains in a way that neural networks are not. Further, FCModel can combine diverse knowledge sources (individual FCMs)

nto a Group map that represents overall knowledge. Additionally,t allows simplification of a large complex map into a simpler mapo that the important processes of the domain become easily under-tandable, traceable and predictable. Finally, FCM model can bereated as a dynamical system that helps making inferences fromystem outcomes using simple operations and scenario simulationsnstead of using complex and costly calculations as in most otherpplicable soft computing and quantitative methods.

Fig. 1 shows an example FCM from a water scarcity problemhowing a number of factors (water resources, demand, wastagend economic, legal, technological factors etc.) affecting the watercarcity situation and their relationships (positive, negative andeedback). Feedback loops in an FCM mean that the effect of changen one node on other nodes can, in turn, affect the node initiatinghe change. FCM with feedback therefore is a dynamical systemnd it can be used not only to understand systems properties andehaviour but also to conduct what-if policy scenarios simula-ions that can be the basis for recommending effective policies formproved management of the system concerned.

FCM model is a useful approach to model and reflect the real-ty of complex ecological or environmental problems involvingumans. In Papageorgiou et al. [44,45], the researchers developedCM models using ‘expert’ knowledge to capture the effective fac-ors and causal relationships in agriculture applications. These

odels could assist decision makers in agricultural yield man-gement. However, the drawback is that these FCM models onlyapture experts’ knowledge which could narrowly represent thenowledge of complex domains. For example, addressing complexcological problems could involve many relevant stakeholders witharying levels of knowledge and perceptions that often exist in annvironment of conflict. Perceptions (point of view) of each ‘expert’takeholder could be represented by an individual FCM as in Fig. 1ut the development of FCM models based solely on experts can-ot be completely representative of the entire domain knowledge.

n such cases, FCM model can benefit from local people’s knowl-
dge to extend or complement experts’ knowledge in knowledgextraction [43]. Several studies have addressed the limitation ofhe dependency on experts’ knowledge alone in developing FCM
odels in ecological problems. Kafetzis et al. [19] combined the

oft Computing 48 (2016) 91–110

perceptions of farmers, local residents and government officialsto develop a social FCM model to analyse conflicts of stakeholderviews on water problems [19]. Further, owing to the diversity ofopinions of the stakeholders, the individual and aggregated percep-tions/FCMs are often large and complex. Such large FCM needs tobe simplified (condensed) before or after aggregation in order to beunderstood, analysed and later simulated in search of effective pol-icy options that could alleviate the ecological problem concerned.Ozesmi and Ozesmi [43] suggested a novel multi-step FCM modelfor ecological problems. This model can extract different types ofknowledge from experts as well as from lay experts, combines theperceptions of different stakeholder groups and allows simplifica-tion/condensation of large FCMs. The approach has proven to beuseful and gained good results in addressing a number of environ-mental problems ([61,54]).

Despite its capabilities in modelling ecosystems management,currently FCM model has a number of limitations and building arobust and comprehensive model is still a challenge. Most impor-tant challenges can be summarised as follows. The participants,especially humans, would like to express their knowledge in a waythat is natural to them rather than in ways that are artificial orenforced by the researcher (system developer). FCM model lackssatisfactory mathematical methods to represent and deal with dif-ferent imprecise numeric values and/or linguistic expressions ofvarious perceptions that are more likely to arise due to differ-ent levels of stakeholder experiences. When an FCM model dealswith different perceptions of various participants and stakehold-ers, there is a need to combine these perceptions in order to obtaina group or social perception [33,42,43]. FCM model lacks a fuzzypersuasive method to aggregate conflicting and shared perceptionsand also suffers from the inability to assign acceptable credibil-ity weights to these perceptions during the aggregation processin order to successfully attain a more realistic group consensus.After aggregating FCMs into a group or social FCM, the result of theaggregation process is often a complex map and it is very difficultto gain insights and understanding from it. Even individual stake-holder FCMs may be large and complex in a large problem domain.In such cases, there is a need to simplify a complex map into asimpler one. A qualitative condensation method has been previ-ously used to combine, subjectively, the variables into a higher levelof abstraction [43]. This qualitative condensation lacks a suitableapproach to multi-level condensation as well as adequate repre-sentation and calculation of the connections resulting from thecondensation process. Finally, FCM model lacks sound objectivecriteria for evaluating the simulated maps for choosing the mosteffective policies.

The above discussion makes the case that FCM representationand calculations involved in the condensation and aggregation pro-cesses as well as the evaluation of its outcomes should be carriedout using appropriate methods that can deal with imprecise orfuzzy values, avoid loss of information, reflect process validity, andlead to accurate outcomes. Therefore, improvements are neededin approaches to all aspects of FCM ranging from representation,development, synthesis, condensation, simulation and evaluation. Themain goal of this article is to develop an advanced FCM modelthat improves FCM representation, development, synthesis, con-densation and evaluation in a systemic way to provide a robustsemi-quantitative FCM model for addressing and modelling com-plex real-life problems. The main contributions of the article canbe summarised as follows. It uses an appropriate fuzzy repre-sentation method to deal with imprecise numeric values as wellas linguistic expressions, individually or together. It also benefits
from various intelligent systems, especially graph theory and fuzzylogic approaches, in handling imprecise data during calculations.In terms of FCM aggregation process, this article enhances this pro-cess by, firstly, using an appropriate fuzzy aggregation method, and

M. Obiedat, S. Samarasinghe / Applied Soft Computing 48 (2016) 91–110 93

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ig. 1. An example FCM depicting the perception of a stakeholder on a Water Scarciashed arrows indicate negative and positive influences, respectively.

econdly, proposing a new method for calculating the credibilityeights for variables and FCMs and then using credibility weights

f FCMs in the aggregation process. In terms of FCM condensa-ion, this article replaces the qualitative or subjective condensation

ethod with a semi-quantitative or subjective-objective conden-ation method and uses more than one level of condensation. Theubjective part of the proposed condensation method is identi-cation of the condensed variables in the higher levels and thebjective part is in using suitable calculation methods for calculat-

ng new connection weights and credibility weights for variablest each level of condensation. Furthermore, the study proposesn insightful, objective and effective multiple criteria evaluationpproach for assessing FCM simulation outcomes to select the mostppropriate policies. These improvements are expected to makehe whole FCM development process robust leading to effectivetrategy/policy development from map simulations.

This article is structured as follows: Section 1 provides anntroduction to socio-ecological systems, challenges to their mod-lling and the important role models play in our understandingnd managing these systems. It summarises current modellingpproaches and their limitations, introduces FCM as an alterna-ive and highlights the attractive properties of FCM for modellingocio-ecological systems relative to other soft computing and otherpproaches. Section 2 gives a brief introduction and backgroundo FCM highlighting specific limitations and issues in current FCM

odels and formulates the research problem. Section 3 providesn overview of the methods and concepts used to develop theroposed advanced FCM model based on a fuzzy 2-tuple method,raph theory and social network theory to address the current lim-tations and deliver the aforementioned contributions. Section 4resents the whole model. Finally, Section 5 presents an applicationf the developed model to a real life problem involving ‘Mitigat-

ng Groundwater Degradation and Depletion in Jordan’ before weonclude the article and suggest some future work.

. Fuzzy Cognitive Maps (FCM)

FCM is a hybrid soft computing approach incorporating proper-
ies of fuzzy logic, nonlinear systems dynamics models and neuraletworks techniques [31]. As stated before, the structure of FCM
s a directed graph with feedback loops. This structure gives FCMpproach significant capability to represent domain knowledge

blem. For clarity, the weights of connections are not shown on the graph; solid and

of qualitative complex problems, capture their nonlinear systemsdynamics, and help generate potential solutions to domain prob-lems. Human perceptions are rift with uncertainty, imprecision andincompleteness that necessitate a cognitive model to deal withthese data. FCM with fuzzy weights has the ability to deal withuncertain and imprecise data [33,35], depict the knowledge fromhuman perceptions and represent this knowledge in a symbolicmanner. The significance of the FCM model lies in representingthe factors and relationships of the problem in a fuzzy (imprecise)way. In other words, FCM consists of nodes, also known as vari-ables or concepts, and imprecise causal connections, also known ascause-effect relationships among nodes (see Fig. 1). The nodes rep-resent the related causal factors that describe the states, behavioursor characteristics of the domain knowledge. These factors cor-respond to variables, concepts, events, input, output, trends andother characteristics of the domain knowledge being modelled [12].The imprecise connections among nodes represent their cause andeffect relationships. A connection between two nodes has a sign,magnitude and arrow to express type, strength and direction of theconnection, respectively. The experts/designers draw their FCMs torepresent a variety of domain problems. This process is called FCMdevelopment or domain knowledge representation.

Due to the flexibility and ability of FCM to represent humanknowledge, FCM has been used for modelling numerous differentill-structured domains such as: political decisions and devel-opments [53,56], socio-ecological systems [14,17,21,26,42,43,54],socio-economical systems [10], agriculture [27,44], medicaldecision-making [1,23,52], catastrophic natural events, e.g., earth-quakes [50,51], and water and drought management [17–19].

FCM model can represent the knowledge of various complexproblems and provide pragmatic solutions in a way that other intel-ligent systems cannot. It can also play a significant role in helpingdecision-makers devise appropriate strategies through simulatingcomplex dynamics of the system, particularly if the system con-sists of different participants or stakeholders. Importantly, FCMmodel can combine perceptions of different stakeholders into aGroup stakeholder map that represents overall Group perception.This process is called FCM combination or aggregation. The aggre-gation process, usually, produces a large complex FCM with a large
number of nodes and connections. FCM model allows simplifica-tion of the large complex FCM into a simpler one with a smallnumber of nodes and connections [43]. This process is called FCM

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ondensation. Furthermore, FCM model can utilize graph theoryndices and/or statistical methods to analyse the structure of theCM [42,43]. This process is called statistical analysis of the struc-ure of the FCM. FCM model can reveal the effects of nodes on eachther and the whole system by simulating the FCM as a dynami-al system based on known initial state of nodes in the system tond their final state. This process is called FCM inference. Finally,

t can perform different scenario (policy) simulations using “whatf” questions to see what state the model will reach if the initialtate of some specific nodes are altered [33] in order to find moreesirable outcomes for the system concerned to help in decisionaking. Considering the versatility and flexibility of FCM, there is a

reat scope for and advantages in advancing FCMs in all its aspectsescribed above. The specific aspects advanced in this research areddressed in the next sub-sections.

.1. Complete representation of the forms and range oferceptions: enhancing the versatility of FCMs

FCM is a network of concepts and their relations (weights). InCMs for complex real-life participatory problems, elicited rela-ions between concepts can be a mix of positive or negativeinguistic statements and numeric values depending on the par-icipants’ level of understanding and skills. Therefore, representingll such relations in a uniform format greatly aids and simplifies anyubsequent FCM processing, be it condensation, aggregation or sim-lation. Currently, methods to unify weights presented in a mix of

inguistic statements and numeric values in FCMs are lacking. Fuzzy-tuple model has been proposed to represent symbolic or/andumeric values by a pair of values called 2-tuple—a fuzzy set and aymbolic translation [25], which can be useful here (the details ofhis model are presented in Section 3). However, the current 2-tupleormat deals only with positive linguistic information in a linguisticerm set. Therefore, one of the main objectives of this article is toropose enhancements to this model to accept negative symbolicr numeric information while keeping model functions operatings they are. These enhancements do not change model’s purpose oronflict with its definitions, propositions and rules. Importantly, weake use of the proposed enhanced 2-tuple representation to con-

uct all FCM processing – from map representation to aggregation in a unified and robust manner.

.2. Finding key players in the system: important nodes in anCM that control information flow

Not all nodes in an FCM have equal value; there can be highlynfluential nodes that control information flow and therefore,efine the final state reached by the system. Understanding theseodes cannot only be useful in understanding the structure of anCM but also in affecting changes to move the system to a moreesirable state, such as water abundance in the case of watercarcity. In the past, only a local centrality measure, degree central-ty, has been used to identify important/central nodes [31,43]. This

easure is based on the connectivity of a node to other nodes inhe system and highly connected nodes are assumed to be central.owever, high connectivity (local measure) alone may not accu-

ately identify important/central nodes. Global measures such asow quickly a node reaches or is reached by other nodes (Close-ess measure) and how often information flows through a nodeBetweenness measure) are likely to play an influential role here.
his article provides a novel method to derive a new centrality mea-ure, called “Consensus Centrality Measure” (CCM) [40] to define aode’s centrality in its FCM based on the local and two global cen-rality measures mentioned above. Additionally, this article defines

a CCM for each FCM (in a group of FCMs) that helps identify theinfluence of FCMs (stakeholders) on system outcomes.

2.3. Representing whole community perception: FCM aggregation

Large applied problems, such as socio-ecological problems,constitute different perceptions from various relevant stakehold-ers/participants. Thus, it is necessary to integrate varied types ofknowledge (i.e., from lay experts to experts) and different levelsof perception (i.e., from local people to high level managers) fromdifferent stakeholders [43]. Such participatory knowledge wouldorganize the complexity of a problem [21], and then lead to theencapsulation of a successful, flexible and adaptable system [42].FCM is an effective and useful approach for handling such prob-lems. FCM allows any number of participants to take part and eachparticipant can build a different FCM with different variables andconnections. Each participant can easily identify the relevant vari-ables and connections among them, but identifying the relevantstrengths of the connections, e.g., accurate magnitude of strength,is a challenge. This is compounded by the fact that the percep-tions of individual stakeholders are generally dominated by theirknowledge and preferences.

That notwithstanding, a large sample size of FCMs or percep-tions leads to finding out the relevant variables and connectionsamong variables with more certainty [37], reducing additional newvariables, stable connection strength values as well as high relia-bility in the combined knowledge [33], and a better representationof the problem [13]. A collection of different perceptions is partic-ularly useful for decision making as it encapsulates a wide rangeof skills, interests and knowledge representation [55], and it leadsto useful information [39]. Therefore, to improve the accuracy ofoutcomes and obtain a reliable and acceptable consensus repre-sentation of knowledge from several individual stakeholders, theindividual perceptions (FCMs) are aggregated into a group or com-bined FCMs [32,34].

Fig. 2 shows a simple example of graphical representation ofFCM aggregation. It shows two FCMs in relation to a water scarcityproblem, each described by a member of the stakeholder groupsof experts and local people, respectively, with imprecise numericand linguistic values. They reveal how the two stakeholders viewwater scarcity situation as being affected by factors such as waterresources, technology, water demand and wastage of water etc.with only some variables overlapping between the two. Fig. 2 alsoshows that the group FCM resulting from the aggregation of thetwo maps includes all distinct variables in the individual FCMs andall possible connections among them. Different methods have beenused to combine individual FCMs into a group FCM and define thevalues of the new connections.

The most common method that has been used for aggregatingindividual FCMs into a combined one is additive superimpositionmethod [12,20,33,43,50]. In this method, each adjacency matrixWk, of individual FCM connection strengths or weights is aug-mented by including all distinct variables from all individual FCMs.Every new variable in the augmented matrix of an FCM is encodedby zeros in its column and row. Then all augmented matricesare added together to form a combined or group FCM connectionmatrix as:

W =M∑k=1

Wk (1)

where, W is the group FCM adjacency matrix (or FCM matrix), M isthe number of augmented FCMs, and Wk is the augmented matrixof kth FCM.


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ig. 2. A simple example of graphical representation of aggregation of two differeositive influences, respectively.

The group FCM could be normalized to present its connectioneights wij in the interval [−1, 1]. This could be done by averaging

he group FCM to produce a group mean FCM matrix [35] as

= 1M

M∑k=1

Wk (2)

here, W is the group mean FCM matrix.This method is easy and requires simple mathematical calcula-

ions. It also allows embedding any new FCM easily by adding itsugmented matrix to the collection of FCMs. However, the mainrawback of this method is that such combination of the causalonnections from multiple individuals or experts may incorrectlyeflect either agreement or disagreement of the individual FCMs.his is called conflicting causal connection. For example, when con-ection between two variables has opposite signs (disagreement)

n two different maps, this will lead to weakening the combinedonnection or cancelling each other which is not a true repre-entation of the situation. On the other hand, the agreement ofonnection between two variables will lead to reinforcing the com-ined connections.

An appropriate way to combine conflicting knowledge of grouparticipants is to involve their credibility weights in the combina-ion [32]. Each participant is assigned a non-negative credibilityeight bi in the range [0, 1] [33,53] and a weighted Group FCMatrix W is defined by

=M∑k=1

bk × Wk (3)

ividual FCMs into one group FCM. Solid and dashed arrows indicate negative and

where bk is the credibility weight of kth participant (the developerof kth FCM), and b1 + b2 + . . . + bM = 1.

Engaging credibility weights in the combination process givesa better representation of different levels of experience of partic-ipants in extracting knowledge. It also produces a weighted FCMthat can better reflect the conflicts in the knowledge of the par-ticipants [33]. However, a suitable approach to obtain realisticcredibility weights is lacking. In this article, we present an objectivemethod to determine FCM credibility weights and then use theseweights in FCM aggregation to obtain weighted Group FCM. Pro-posed credibility weights are derived from the Consensus CentralityMeasure (CCM) described previously.

2.4. Simplifying knowledge network: FCM condensation

When an FCM includes a large number of nodes, it becomes com-plex making it harder to understand its structure and gain insightfrom it. The ideal number of nodes in an FCM is around 12 [6],and then on, the more the number of nodes, the more complexFCM is. A large number of nodes might be identified in an individ-ual FCM. After combining the individual FCMs into a group FCM,even larger number of nodes and interconnections result in thegroup FCM. Therefore, there is a need to simplify the complex FCMinto a smaller FCM with fewer nodes and connections. This processis called condensation. The condensed FCM is more understand-able, traceable and predictable than the larger one, and thereby
leads to improved outcomes. There are two types of condensation:qualitative and quantitative condensation.
In the qualitative condensation approach, as has been used inRefs. [39,43,50,51,59], the experts or developers of the system con-

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erned use their knowledge and judgement to group, in a subjectiveay, similar or related variables into high level categories (groups)

23,49]. Thereafter, the connections among the variables are alsoondensed and then transferred to new connections among theroups. Ozesmi and Ozesmi [43] suggested the cognitive interpre-ation diagram (CID) to identify connections among the groups andheir signs and magnitudes. Another suggestion for identifying theeights of connections among groups was presented in Young et al.

59]. In the latter article, the researchers converted the connectioneights between variables in two groups into linguistic (fuzzy)

ariables and summed them into an overall linguistic value, andhen defuzzified this linguistic value into a numerical value in thenterval [−1, 1] that became the new connection weight betweenhe two groups of variables.

In the quantitative aggregation, similar nodes are objectivelyggregated into a single unit [23]. Here, the map which is a digraphs divided into sub graphs. Each sub graph encompasses a group oftrongly connected variables. Then each sub graph is replaced by

single unit and the connections between sub graphs are main-ained. Another quantitative approach used by Samarasinghe andtrickert [47] to aggregate FCM concepts is Self-Organising MapSOM) that assembles nodes with similar connection weight vec-ors based on a distance metric. Both of the above approachesequire a further step to condense original weights between vari-bles into new weights between the discovered groups. In allrevious condensation methods, there are limitations; especially,

n the way new weights between the groups are determined. Theres information loss in the transfer of weights from the node to groupevel. In this article, we propose a robust semi-quantitative FCMondensation method. This method can use more than one level ofondensation and at each level group of similar variables are sub-ectively condensed. In particular, their corresponding connectionsre condensed using robust calculations in such a way that infor-ation at the lower (node) level is preserved in the higher (group)

evel. In the next section, methods used to accomplish the proposeddvancements are presented.

. Materials and methods

The main aim of this article is to propose a robust semi-uantitative FCM model. This section reviews the effective andobust methods and models that contribute to the development ofhis proposed model. It, first, discusses the concept of the 2-tupleuzzy linguistic representation model and its efficiency in represen-ation and computation of imprecise numeric and fuzzy values in anified format. Then, it gives an overview of some social networkeasures and discusses how these measures can be exploited to

evelop a new significant measure, Consensus Centrality MeasureCCM), to be used in various aspects of FCM processing includinghe determination of credibility weight for concepts as well as FCMso allow realistic incorporation of diverse perceptions.

.1. 2-Tuple fuzzy linguistic representation method

One of the important objectives of this article is to extend a-tuple fuzzy linguistic approach [25] for representing linguisticnd numeric imprecise connections uniformly in a novel fuzzy wayhat can be effectively used in various subsequent computationalrocesses of FCMs. The proposed model can represent and dealith imprecise values without any loss of information, and it keeps

he consistency of these values throughout any subsequent com-utational processes. Specifically, it represents each fuzzy valuelinguistic or numeric) by a pair of symbolic values called a 2-tuple.t can combine linguistic and numeric imprecise or fuzzy values


with different granularity and/or semantic and retain their fuzzyrepresentation during the combination process.

The 2-tuple fuzzy linguistic representation model is based on theconcept of a symbolic translation [25]. It takes a symbolic linguisticmodel proposed in Delgado et al. [11] as the basis for representingthe symbolic translation. The symbolic model represents linguisticinformation by an ordered linguistic term set, S = {s0, . . ., sg}, whereg + 1 is the number of linguistic terms (or fuzzy sets) in S (cardinal-ity). It uses the ordered structure of the set (labelled index i = 0,. . .g)to perform the calculations.

For example, a linguistic term set S with seven terms couldbe given as: S = {s0 = ‘extremely low’, s1 = ‘very low’, s2 = ‘low’,s3 = ‘medium’, s4 = ‘high’, s5 = ‘very high’, s6 = ‘extremely high’}. Thefollowing definitions apply to linguistic term set S.

Definition 1. Let S = {s0, . . ., sg} be a linguistic term set and � [0, g]the numerical result of a linguistic symbolic aggregation of labelledindices of linguistic terms i in the linguistic term set S, i.e., the resultof a symbolic aggregation operator (see Eqs. (8) and (9) for moreclarity). Let i = round (ˇ), where round (.) is the mathematical round-ing operation, and = − i be two values, such that i � [0, g] and

� [−0.5, 0.5); then is called a symbolic translation [25].

From Definition 1, value represents the difference between value resulting from a symbolic aggregation operation and the

index value i of the linguistic term si in S closest to ˇ. Then, a pairof symbolic values (si, �), where si C-- S and C-- [−0.5, 0.5), whichrepresents the means of the 2-tuple linguistic representation modelis defined.

Definition 2. Let S = {s0, . . ., sg} be a linguistic term set and � [0,g] represents the result of linguistic symbolic aggregation, then the2-tuple (si, �) equivalent of denoted as �(ˇ) is obtained with thefollowing function [25]:

�(ˇ) ={si, i = round(ˇ)

= − i, ∈ [−0.5, 0.5)(4)

where si has the closest index label to and is the value of thesymbolic translation.

Definition 3. Let S = {s0, . . ., sg} be a linguistic term set and (si, ˛)be a 2-tuple, then the equivalent numerical value (ˇ) to 2-tuple (si,�) denoted as �−1(si, ˛) is obtained with the following function:

�−1(si, ˛) = i + = (5)

Definition 4. Let si ∈ S be a linguistic term, then its equivalent 2-tuple representation is obtained by adding a value 0 as a symbolictranslation [25]:

�(si) = (si, 0) (6)

Definition 5. Negation Operator of a 2-tuple: Let S = {s0, . . ., sg} bea linguistic term set and g + 1 is the cardinality of S, the negationoperator over 2-tuples is defined as follows:

Neg(si, ˛) = �(g − (�−1(si, ˛))) (7)

As stated before, the 2-tuple deals with only positive linguisticinformation in a linguistic term set and there is a need to extendits range to incorporate negative linguistic and numeric imprecisevalues for FCM connection weights. This article therefore proposesenhancements to fuzzy 2-tuple model that do not change its pur-pose or conflict with its definitions, propositions and rules; andthey are:

1. The linguistic information is represented by S = {s−g , . . ., s0, . . .,sg} linguistic term set, where 2 × g + 1 is the number of linguisticterms in S (cardinality) and � [−g, g]

2. Neg (si, �) = �(−(�−1(si, �)))


F to rep

.cvdbbr2betabsiuti

�

bf

ˇ

2paea‘p‘

su

ig. 3. A graphical representation of the 13 triangular membership functions used

In this research, 2-tuples based on a linguistic term set S = {s-g , . ., s0, . . ., sg}are used to represent linguistic and numeric impre-ise connection weights in the FCM and their equivalent numericalues are used to perform calculations (i.e., aggregation and con-ensation calculations). Each linguistic term si in S is representedy real valued triangular parameters (ai, bi, ci) (i.e., triangular mem-ership function). Now, we discuss how this model can be used toepresent linguistic and numeric weights in the unified format of-tuple and �. The 2-tuple value of a linguistic weight representedy a linguistic term si in S, is directly obtained using Eq. (6), and itsquivalent value is calculated using Eq. (5). It is important to men-ion here that different linguistic sets (e.g., if different linguistic setsre used for different FCMs) are normalized into one standard setefore any calculations. In case of a linguistic weight being repre-ented by a numeric imprecise value in the range [−1, 1], its � values calculated first and then its equivalent 2-tuple value is obtainedsing Eq. (4). To find value of a numeric imprecise value (n), firsthe membership function of value n associated with si (i.e., �si(n))s calculated using the following equation:

si (n) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

n − aibi − ai

, a ≤ n ≤ b

ci − n

ci − bi, b < n ≤ c

0, otherwise

(8)

Then, value is the result of a symbolic aggregation of mem-ership functions over labels i assessed in S and obtained using theollowing equation:

=

g∑i=−g

i × �si

g∑i=−g

�si

(9)

Now we demonstrate the application of above definitions in the-tuple representation to the two FCMs from the ‘expert’ and ‘localeople’ stakeholder groups described in Fig. 2. Suppose numericnd linguistic imprecise connection weights in these FCMs arexpressed using [−1, 1] numeric interval and {‘very very highly neg-tive’, ‘very highly negative’, ’ highly negative’, ‘medium negative’,

low negative’, ‘very low negative’, ‘none’, ‘very low positive’, ‘lowositive’, ‘medium positive’, ‘highly positive’, ‘very highly positive’,

very very highly positive’} linguistic expressions, respectively.Let the linguistic set S = {s−6, s−5, s−4, s−3, s−2, s−1, s0, s1, s2, s3,

4, s5, s6} is used to represent these weights in 2-tuple and � valuessing 2-tuple model. Let linguistic terms in the linguistic set S are

resent the 13 terms in the linguistic term set S over [−1, 1] universe of discourse.

represented by triangular membership functions (fuzzy subsets) asshown in Fig. 3.

The connection weights in the ‘local people’ FCM are linguisticvalues, and hence, their 2-tuple and � values are calculated usingEqs. (6) and (5), respectively. For example, on this FCM, the influ-ence of Water Resources on Water (scarcity) Situation is very high(VH). Therefore, from Fig. 3, si = s5 and from Eq. (6), i = 5 and = 0;therefore, 2-tuple is (s5, 0) and from Eq. (5), = i + � = 5 (refer tovertical line on the right hand side of Fig. 3). On the other hand, theconnection weights on ‘expert’ FCM are numeric values, and hence,their � values are calculated first using Eqs. (8) and (9), and thenequivalent 2-tuple values of these values are calculated using Eq.(4). For example, on this FCM, Technology has a negative influence(−0.3) on Water Demand. This value belongs to two membershipfunctions (−L and −VL) in Fig. 3 (refer to the vertical line placedat −0.3 in the Universe of Discourse). The corresponding indicesof the fuzzy sets are i = −2 and −1 and membership function val-ues calculated from Eq. (8) are approximately 0.8135 and 0.1865,respectively. The resulting value from Eq. (9) is −1.813 and thecorresponding 2-tuple from Eq. (4) is (s−2, 0.187). Table 1 andFig. 4 show the resulting representation of the connection weightsof both FCMs in � and 2-tuple values, respectively, using the 2-tuple representation model. As stated before, 2-tuple is used forrepresenting both linguistic and numeric connections in a uniformformat so that they can be easily handled together and all subse-quent computations are carried out using their equivalent numericvalues of as will be explained in detail in the next segment of thearticle.

3.2. Centrality measures—inspirations from social networks

A common and suitable way to represent and analyse complexinterconnected systems with many connections and feedbacks ingeneral is by using network models. Social networks are networkmodels which have grown rapidly in the last decades in the SocialSciences [29] and as such advances in social network studies canbe brought to bear on FCMs. Typically, socio-ecological systems arerepresented by coherent social networks that can represent inter-actions in both social and ecological systems as well as interactionsbetween them. The key aspects of social networks are networkanalysis and connections that determine the interactions amongnodes [48,57]. Social networks are modelled by graphs of nodesand connections between these nodes [4,7], which can then be eas-
ily analysed using graph theory methods [60]. A graph could bedirected or undirected or weighted or unweighted. The attentionof this article is a directed weighted graph as FCM belongs to thiscategory.

98 M. Obiedat, S. Samarasinghe / Applied Soft Computing 48 (2016) 91–110

F 2-tun

cicihoua[tapi

denuptapoaianTmAd

TR

ig. 4. Representation of connection weights on ‘expert’ and ‘local people’ FCMs inegative and positive influences, respectively.

FCM is a digraph consisting of a network of nodes and weightedonnections that can represent any social structure [40]. Kosko [31]ntroduced the concept of centrality to depict and understand theontribution of a node in an FCM. He used the degree central-ty measure to reflect the importance of a node; i.e., a node withigher degree value is of higher importance to the causal flowf information in the FCM. Beyond Kosko, few researchers havesed the degree centrality to identify important nodes in FCMsnd used this information for various purposes. Ozesmi and Ozesmi42,43] utilized the degree centrality to characterize the most cen-ral/important nodes of their social FCMs in order to understandnd analyse the structure of these FCMs. Ozesmi and Ozesmi [43]ointed out that the important nodes play a large role in an FCM by

nfluencing and/or being influenced by other nodes.Another attempt to use the most important nodes based on their

egree centrality in social group FCMs was presented in Strickertt al. [50]. In this article, the authors utilized the importance ofodes to define decision nodes that could be useful in scenario sim-lations, i.e., the altered state of these decision nodes represents aarticular policy scenario and these nodes are fixed (or clamped) atheir respective altered state throughout the simulation process toscertain the final state of the other nodes in the system under thisolicy scenario. In the feedback cycle of FCM, the effect of a node inne cycle may influence other nodes through a chain of connectionsmong them. This means that the influence of a node is not only onts adjacent nodes, but it may extend far beyond in complex waysffecting and/or transferring the accumulated effects of upstreamodes to all other nodes existing in the path of the feedback chain.
his issue poses a challenge that lies in two questions: what are theost important nodes in an FCM that can make influential changes?
nd how can such nodes be defined? This issue inherent in theynamics of an FCM should be considered when identifying the

able 1epresentation of connection weights of ‘expert’ and ‘local people’ FCMs in values using

Connection weights on Expert FCM represented in valuesWater situation Water resources

Water situation 0 0

Water resources 5.412 0

Water demand −6 0

Technology 0 3.588

Wastage of water −2.412 0

Connection weights on Local People FCM represented in valuesWater situation Water resources

Water situation 0 0

Water resources 5 0

Water demand −6 0

Water projects 0 4

Economic situation 0 0

ple values using 2-tuple representation model. Solid and dashed arrows indicate

most important nodes in the FCM. The previous studies show thatthe only measure that has been used to define the importance of anode in an FCM is the degree centrality measure.

The degree centrality measure is considered a straightforwardand efficient local measure to identify the centrality of a node ina social network (and an FCM). It articulates the direct connec-tions of a node in an FCM to other nodes [43]. It measures thecontribution or activity of a node in an FCM and does give a strongindication of the effective nodes that can be useful for understand-ing and analysing the structure and other characteristics such assystem dynamics of the FCM. However, a node connected with asmall number of highly effective nodes may be more powerful andinfluential than a node connected with a large number of lowlyeffective nodes. In this case, local measures, such as the degree cen-trality measure, are less relevant and therefore, global measures,such as closeness and betweenness, can be more suitable for iden-tifying a node’s centrality and can give better results as has beenreported in Social Science literature [8]. Specifically, a node mayhave a strategic location, i.e., if a node is located in between and/orclose to other nodes, it may have a considerable influence on theflow of information (communication control) through that node.This feature can be taken into account through the considerationof the concept of the shortest paths rather than direct connectionsbetween nodes. Closeness and betweenness centrality measurestake into account the indirect connections of a node by emphasiz-ing the shortest paths between this node and the other nodes toshow how a node could be globally quite central in the whole net-work rather than just locally [22]. As such, these latter measures
reflecting global centrality of a node can be more effective than thedegree centrality measure which only emphasizes the direct con-nections of a node and shows how much the node is central in alocal neighbourhood.
2-tuple representation model.

Water demand Technology Wastage of water

0 0 00 0 00 0 0

−1.813 0 −4.1881.188 0 0

Water demand Water projects Economic situation

0 0 30 0 00 0 00 0 −5

−3 4 0

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M. Obiedat, S. Samarasinghe / App

In fact, the selection of a suitable centrality measure or a com-ination of the above three measures depends on the context ofhe application concerned [16]. Therefore, this research employs aovel method to derive a new centrality measure, called “Consensusentrality Measure” (CCM) that we introduced in Obiedat et al. [40],o define a node’s centrality in its FCM. It combines the degree cen-rality measure with the two significant centrality measures justescribed—closeness and betweenness. CCM combines the threeeasures using the weighted average operator to take into account

he relative importance of a certain measure over other measures.hese ideas are presented in the next sections.

.2.1. Degree centrality measureThe degree centrality measure is the simplest measure used in

ocial network analysis. In a directed graph, two components, in-egree and out-degree centrality measures, are used to find theegree centrality of a node [5,23,30]. In a signed weighted digraph

ike FCM, the in-degree centrality of a node equals the sum of itsbsolute incoming connection weights from its neighbours:

d(ci) =N∑j=1

|wji| (10)

here id (ci) is the in-degree centrality of node ci, N is the num-er of nodes connected to node ci in FCM, and wji is the weight ofhe connection entering node ci from node cj . In contrast, the out-egree centrality of a node in an FCM equals the sum of its absoluteutgoing connection weights to its neighbours:

d(ci) =N∑j=1

|wij| (11)

here od (ci) is the out-degree centrality of node ci. The overallegree centrality of a node in FCM is the summation of its in-degreend out-degree [5,31]:

enD(ci) = id(ci) + od(ci) (12)

here CenD(ci) is the degree centrality of the node ci.

.2.2. Closeness centrality measureThe closeness centrality is considered a global centrality mea-

ure because it is based on the shortest paths (geodesic distances)oncept [3,46]. It finds the sum of geodesic distances between aiven node and the remaining ones. In this sense, the lower theum is, the higher the centrality. To reflect this idea, the inverse ofhe sum is taken to express the closeness centrality. As with degreeentrality, this research considers two aspects of closeness central-ty measure, in-closeness and out-closeness, to obtain the closenessentrality of a node in an FCM. In addition, the distances betweenodes are to be represented by the absolute value of the connec-ion weights. Thus, the in-closeness centrality of a node in an FCM

easures how much other nodes are close to this node, and it isxpressed as follows:

c(ci) = 1N∑t=1

dG(t, ci)

(13)

here t /= ci, ic(ci) is the in-closeness centrality of the node ci, Ns the number of nodes in FCM, and dG(t, ci) is the shortest pathrom node t to node ci. The out-closeness centrality of a node in an

oft Computing 48 (2016) 91–110 99

FCM measures how much this node is close to other nodes, and itis expressed as follows:

oc(ci) = 1N∑t=1

dG(ci, t)

(14)

where t /= ci and oc(ci) is the out-closeness centrality of the node.The closeness centrality of a node in FCM is the summation of itsin-closeness and out-closeness, and it is defined as follows:

CenC (ci) = ic(ci) + oc(ci) (15)

where CenC (ci) is the closeness centrality of the node ci.

3.2.3. Betweenness centrality measureLike closeness, the betweenness is a global centrality measure

based on the concept of shortest paths. The betweenness centralityof a node is determined by summing the proportion of shortestpaths between all node pairs that go through a particular node[2,15]. To find the betweenness centrality of a node in an FCM, theabsolute value of the connection weights between nodes representsthe distances between these nodes, and hence, the betweennesscentrality of a node is defined as:

CenB(ci) =N∑

s,t=1

�st(ci)�st

(16)

where s /= t /= ci, CenB(ci) is the betweenness centrality of the nodeci, �st is the number of shortest paths from node s to node t, and �st(ci) is the number of shortest paths from node s to node t that passthrough node ci.

3.2.4. Consensus centrality measure (CCM) of a node and FCMWe proposed a novel CCM measure for a node and FCM in

Obiedat et al. [40] and Obiedat and Samarasinghe [41], respec-tively. They were obtained from the above degree, closeness andbetweenness measures. CCM of a node is obtained as [40]:

CenCons(ci) = bD × CenD(ci) + bC × CenC (ci) + bB × CenB(ci) (17)

where CenCons(ci) is the CCM of node ci, i = 1 to the number of nodesin FCM, CenD(ci), CenC (ci) and CenB(ci) are the degree, closeness andbetweenness centrality values of node ci, respectively, and bD, bCand bB are the prioritization weights for the degree, closeness andbetweenness measures, respectively, where bD + bC + bB = 1.

Similar to the centrality of a node in an FCM, centrality of anFCM in a group of FCMs can indicate the influence of that FCM onthe overall group system behaviour. To obtain CCM for an FCM(CenCons(FCM)), the degree (CenD(FCM)), closeness (CenC (FCM))and betweenness (CenB(FCM)) centrality measures of FCM need tobe obtained. Each of these measures is calculated based on the cor-responding node centrality measures in respective FCMs. Let Cen∗

D,Cen∗

C and Cen∗B are the maximum degree, closeness and between-

ness centrality values of nodes in an FCM respectively. Then, thedegree, closeness and betweenness for the FCM are calculated fromthe following Eqs. (18)–(20), respectively [40]:

CenD(FCM) =

N∑i=1

(Cen∗D − CenD(ci))

N − 1(18)

CenC (FCM) =

N∑i=1

(Cen∗C − CenC (ci))

(N − 1)(N − 2)/(N − 3)(19)


Table 2Adjacency matrix of the FCM in Fig. 1.

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13

c1 0 0 0 0.37 0 0 0 0.68 0 0 0 −0.71 −0.15c2 0.76 0 0 0 0 0 0 0.27 0 0 0 −0.31 −0.26c3 −0.71 −0.13 0 0 0 0 0 −0.36 0 0 0 0.26 0c4 0 0.04 0 0 0 0.7 0.61 0.11 0.62 0 0 0 0c5 −0.63 0 0 −0.34 0 0 0 0 0 0 0 0 0c6 0.58 0.3 −0.33 −0.48 0 0 0 0 0 0 0 0 0c7 0.48 0.29 −0.17 0 −0.62 0.62 0 0 0.55 0.34 0.37 −0.18 −0.58c8 −0.1 0 0.55 0.48 0 0 0 0 0 0 0 0 0c9 0.48 0.27 −0.27 −0.47 −0.22 0.25 0.5 0.14 0 0 0.35 0 0c10 0 0 0 0 − 0.09 0.32 0.2 0 0 0 0.41 0 0c11 0 0 −0.47 0 −0.55 0.26 0.58 0 0.43 0.24 0 0 −0.46c12 0 −0.21 0 0 0 0 0 −0.51 0 0 0 0 0.26c13 −0.17 −0.63 0.63 −0.34 0.31 0 0 0 0 0 −0.17 0.42 0

Table 3Values for Degree, Closeness, Betweenness, and CCM centrality measures of FCM nodes considered in Example 1.

Node Degree centrality Closeness centrality Betweenness centrality CCM centrality

c1 5.66 4.12 1.27 3.69c2 2.52 5.45 2.81 3.59c3 3.06 0.71 0.38 1.38c4 3.95 3.11 2.56 3.21c5 1.56 0.75 0.5 0.94c6 2.99 0 0 1c7 6 1.74 0.5 2.75c8 2.14 3.6 1.13 2.29c9 3.93 3.17 0 2.37c10 0 2.28 1.27 1.18

3.56 3.640.25 1.436 5.24

C

C

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amrciRcdFt

Table 4Values of Degree, Closeness, Betweenness and CCM centrality measures of the FCMin Example 1.

Degreecentrality

Closenesscentrality

Betweennesscentrality

CCM centrality

c11 3.58 3.78

c12 1.68 2.37

c13 3.71 6

enB(FCM) =

N∑i=1

(Cen∗B − CenB(ci))

N − 1(20)

Then, the CCM of the FCM is obtained as follows [40]:

enCons(FCM) = bD × CenD(FCM) + bC × CenC (FCM)

+bB × CenB(FCM) (21)

CCM is a weighted average of the three centrality measures witheights indicating their priority. Prioritisation weights are sub-

ectively assigned by the developer of the system. The developerssigns a non-negative weight for each measure according to theeasure’s effectiveness in determining node or FCM centrality. This

ives the FCM developer flexibility, depending on his/her in-depthnderstanding of the context of FCM, to prioritize between theseeasures by assigning a non-negative weight to each in the interval

0, 1]. This means that the CCM can endow a measure with a pref-rence over others if the weight of this measure takes on a highalue, for example, close to 1, or even ignore one or two measuresrom calculations if they are given zero or very low values.

To illustrate the generation of centrality measures, considers an example FCM nodes in Fig. 1 and its adjacency (weight)atrix presented in Table 2 below. As FCM weight values are rep-

esented in numeric values in the interval [0, 1], this requires firstonverting them to values based on the degree of membershipn relevant fuzzy sets (Eqs. (8) and (9)) as described previously.esulting � values of the weights are used in calculations of the
entrality measures using Eqs. (10)–(16). Tables 3 and 4 show theegree, closeness, betweenness, and CCM centrality measures ofCM nodes (Eqs. (10)–(17)) and FCM itself (Eqs. (18)–(21)), respec-ively, represented in values. In this example, we consider that
3.10 3.09 4.81 3.67

degree, closeness and betweenness measures are equally impor-tant and so have the same weight (i.e., 0.33).

3.2.5. Credibility weight of a node and FCMThe credibility weights (CW) are assigned to nodes and FCMs

based on the calculated CCM of nodes and FCMs, respectively, rep-resented in values. This proposition has been developed fromthe idea that if a node/FCM is more central (important) than othernodes/FCMs, this means that it occupies a higher or more powerfulposition with greater ability to influence and change the outcomesof the domain of concern compared to other nodes/FCMs. Conse-quently, this node/FCM should take a higher credibility value thanother nodes/FCMs, and so on. The CCM is more expressive about thecentrality than other measures because it combines the three cen-trality measures incorporating local and global influence of nodes.Therefore, this measure is utilized to assign credibility weights inthe interval [0, 1] to both nodes and FCMs. First, the CCM of nodesin FCM and FCMs in the system are normalized in the interval [0,1]. For this calculation, we use the already described 2-tuple modelas it has the advantage of being able to convert � values into [0, 1]or [−1, 1]. The transformation process is presented in Herrera andMartinez [25]. It uses the function ı to compute two 2-tuples fromCCM value based on the degree of membership in linguistic terms
that supports the same counting of information as follows:
ı(ˇ) ={

(sh, 1 − �), (sh+1, �)}

(22)

M. Obiedat, S. Samarasinghe / Applied S

Table 5Consensus centrality measure (CCM) in values, its representation in [0, 1] and thecredibility weight values of nodes of the example FCM.

Node CCM in values CCM in [0, 1] Credibility weight

c1 3.69 0.616 0.113c2 3.59 0.601 0.11c3 1.38 0.231 0.042c4 3.21 0.535 0.098c5 0.94 0.159 0.029c6 1 0.169 0.031c7 2.75 0.457 0.084c8 2.29 0.380 0.07c9 2.37 0.392 0.072c10 1.18 0.199 0.036c11 3.64 0.609 0.112c12 1.43 0.239 0.044c 5.24 0.87 0.159

wS

n

�

wo

T(id

otae4is

4

piccaapdSmfisiwuFtrfss

Fourth step initiates data manipulation and calculations that

13

Total 1

here h = trunc(ˇ), trunc is the usual trunc operation, � = − h, andh and Sh+1 are linguistic terms in S, in which has membership.

Then it uses the function � to convert these two 2-tuples into aumerical value assessed in the interval [0, 1] or [−1, 1] as follows:

((sh, 1 − �), (sh+1, �)) = CV(sh) × (1 − �) + CV(sh+1) × (�) (23)

here CV(.) is a function providing a characteristic value like onef the defuzzification techniques.

For example, considering CCM (� values) of nodes in FCM inable 3 (last column and repeated in Table 5 column 2), Table 5column 3) shows these values represented in [0, 1] after apply-ng the above transformation using Mean of Maximum (MoM) asefuzzification method.

Once the CCM values in the interval [0, 1] are obtained from theriginal CCM values, the credibility weight of each node (cwn) inhe FCM is calculated by dividing its CCM by the sum of CCMs ofll nodes in the FCM. The credibility weight values of nodes in thexample FCM, resulted from this calculation, are shown in column

of Table 5. Similarly, the credibility weight of each FCM(cwFCM)n the system is calculated by dividing its normalised CCM by theum of CCMs of all FCMs in the system.

. Proposed robust semi-quantitative FCM model

This article enhances the existing qualitative FCM approaches toroduce a robust semi-quantitative approach. The proposed model

ncludes several coherent and consistent steps of reasoning andalculations based on robust methods and models making it moreontextually realistic and reasonable and computationally practicalnd manageable. Accordingly, the proposed model is suitable for,nd effective in, addressing a range of currently challenging com-lex dynamical problems dominated by uncertainty and impreciseata such as participatory real-life or environmental problems.pecifically, the model includes eight advanced processes thatake it coherent and robust. Fig. 5 shows in sequence the con-

guration of the 8 steps involved in the proposed FCM model. Ittarts from the phase of the data collection and FCM developmentn the form of ambiguous perceptions based on in-depth interviews

ith relevant stakeholders (steps 1–3), and then FCM data manip-lation without loss of information throughout different stages ofCM processing (FCM representation, condensation and aggrega-ion) using the novel computational approaches proposed in thisesearch (steps 4–6), to the phase of knowledge extraction in the
orm of map structure and properties (step 7) as well as generatingolutions or recommendations through simulations (step 8). Theseteps are described in the next sections.
oft Computing 48 (2016) 91–110 101

4.1. Interview process (step 1)

In problem solving involving participation of people (participa-tory systems), one of the most common methods to collect usefuldata about the problem is to conduct qualitative interviews withrelevant respondents [43]. Qualitative interviews are attempts tounderstand the problem from people’s point of view and to uncoverthe meaning of their experiences [36]. The first step of the proposedmodel is to collect significant qualitative data about the problemfrom relevant stakeholders using in-depth semi-structured qualita-tive interviews based on open-ended questions. Depending on thecomplexity and domain of the problem, the relevant stakeholderscan be divided into groups based on their attributes (e.g., experts,non-experts etc.). An interview may take a long time and generatea lot of data, and hence it is highly recommended that the interviewis recorded and useful notes are taken. This helps in the subsequentprocesses of Development (step 2) and Review (step 3) of FCM.

4.2. FCM development process (step 2)

After each interview with a stakeholder, the raw data collectedfrom it are reported. The second step of the proposed model is todepict the collected data in the form of FCM of the stakeholder.The data are converted into causal factors/variables and relation-ships/causalities between them. These variables may representactions, events, inputs, outputs and other states that could affecteach other either negatively or positively, and this in turn describesthe behaviour of the domain problem. The variables are transferredinto nodes and the causalities into connections to draw the FCM ina suitable media such as a white board. The structure of the FCMshows the influence of domain variables on each other.

4.3. FCM review process (step 3)

Typically, an FCM developed immediately from an interviewmay not be comprehensive or reflective of all the important datathat have been made available in the interview. One of the rea-sons for this situation is the generally limited time available forthe interview and FCM development processes considering theenormity of the domain problem tackled. Eliciting knowledge fromparticipants usually takes a long time, particularly if the problemconcerned is substantial. An interview contains many open-endedquestions to investigate comprehensively most of the influentialfactors and relationships between them in relation to the prob-lem, which requires much reflection and probing. Accordingly, thedevelopment of FCM may also take a long time if many factors andconnections are defined. Another reason is the difficulty in depict-ing all the identified factors as nodes in FCM and then tracing theconnections between them. To handle this issue, the third processof the proposed model is to hold an FCM review by the systemdeveloper (person who conducted the interviews). The review pro-cess includes reviewing each developed FCM, recorded interviewand written notes to determine if there are any missing impor-tant factors and connections mentioned in the interview but notincluded in the developed FCM. Accordingly, the system devel-oper makes any necessary updates to the FCM so that it includesall important factors and connections, and thus the FCM becomesreflective/representative of all that has been revealed in the inter-view.

4.4. FCM fuzzy representation process (step 4)

continue through to step 8. The purpose of step 4 of the proposedmodel is to represent variant (numeric and linguistic) imprecisedata in a uniform way to enhance the quality of data representation


proce

irsepbicF

Fig. 5. Configuration of the eight

n FCM. For this purpose, this article uses the extended 2-tuple fuzzyepresentation model stated earlier. The 2-tuple model can repre-ent and deal with imprecise numerical and vague linguistic dataither separately or together. It represents these data in the form ofairs of symbolic values. It can also transform these pairs of sym-olic data into numeric � values and vice versa for easier handling

n further computations. The option of selecting between numeri-
al values and linguistic words for connection strengths gives theCM designer/participants the freedom in the expression of FCM
sses of the proposed FCM model.

connection weights, especially when it is difficult to give a straightforward or explicit value.

The first step of the FCM fuzzy representation method is todetermine a linguistic term set, S, consisting of an expressivenumber of linguistic terms (membership functions) for all FCMconnection weights. In the case of FCM weights being expressedin linguistic values, Eq. (6) is used to represent FCM weights in 2-
tuple values (pairs of (si, ˛)) and then find equivalent values tothese 2-tuple values using Eq. (5). In the case of FCM weights beingexpressed in numeric values in the interval [−1, 1], first, they are


Table 6The values representing the adjacency matrix of connection weights in the example FCM in Fig. 1.

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13

c1 0 0 0 2.21 0 0 0 4.09 0 0 0 −4.27 −0.87c2 4.54 0 0 0 0 0 0 1.60 0 0 0 −1.89 −1.59c3 −4.28 −0.77 0 0 0 0 0 −2.18 0 0 0 1.57 0c4 0 0.24 0 0 0 4.21 3.65 0.64 3.69 0 0 0 0c5 −3.76 0 0 −2.07 0 0 0 0 0 0 0 0 0c6 3.44 1.78 −2.01 −2.89 0 0 0 0 0 0 0 0 0c7 2.85 1.78 −0.98 0 − 3.69 3.72 0 0 3.27 2.07 2.22 −1.04 −3.47c8 −0.61 0 3.27 2.88 0 0 0 0 0 0 0 0 0c9 2.89 1.63 −1.63 −2.80 −1.32 1.47 3.02 0.80 0 0 2.10 0 0c10 0 0 0 0 −0.56 1.95 1.16 0 0 0 2.48 0 0c11 0 0 −2.84 0 −3.30 1.49 3.45 0 2.60 1.43 0 0 −2.76c12 0 −1.22 0 0 0 0 0 −3.03 0 0 0 0 1.55c13 −1.01 −3.78 3.76 −2.06 1.90 0 0 0 0 0 −0.98 2.51 0

Table 7The 2-tuple values that are equivalent to values of weights in Table 6.

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13

c1 (s0, 0) (s0, 0) (s0, 0) (s2, 0.21) (s0, 0) (s0, 0) (s0, 0) (s4, 0.09) (s0, 0) (s0, 0) (s0, 0) (s−4, −0.27) (s−1, 0.13)c2 (s5,−0.46) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s2, −0.4) (s0, 0) (s0, 0) (s0, 0) (s−2, 0.11) (s−2, 0.41)c3 (s-4, −0.28) (s−1, 0.23) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s−2, −0.18) (s0, 0) (s0, 0) (s0, 0) (s2, −0.43) (s0, 0)c4 (s0, 0) (s0, 0.23) (s0, 0) (s0, 0) (s0, 0) (s4, 0.2) (s4, −0.35) (s1, −0.36) (s4, −0.3) (s0, 0) (s0, 0) (s0, 0) (s0, 0)c5 (s−4, 0.24) (s0, 0) (s0, 0) (s−2, −0.07) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0)c6 (s3, 0.44) (s2, −0.22) (s−2, −0.01) (s−3, 0.11) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0)c7 (s3, −0.15) (s2, −0.22) (s−1, 0.02) (s0, 0) (s−4, 0.31) (s4, −0.28) (s0, 0) (s0, 0) (s3, 0.27) (s2, 0.07) (s2, 0.22) (s−1, −0.04) (s−2, −0.47)c8 (s−1, 0.39) (s0, 0) (s3, 0.27) (s3, −0.12) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s0, 0)c9 (s3, −0.11) (s2, −0.37) (s2, −0.37) (s−3, 0.2) (s−1, −0.32) (s1, 0.47) (s3, 0.02) (s1, −0.2) (s0, 0) (s0, 0) (s2, 0.1) (s0, 0) (s0, 0)c10 (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s-1, 0.44) (s2, −0.05) (s1, 0.16) (s0, 0) (s0, 0) (s0, 0) (s2, 0.48) (s0, 0) (s0, 0)

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c11 (s0, 0) (s0, 0) (s-3, 0.16) (s0, 0) (s−3, −0.3) (s1, 0.49)

c12 (s0, 0) (s−1, 0.22) (s0, 0) (s0, 0) (s0, 0) (s0, 0)

c13 (s−1, −0.01) (s−4, 0.22) (s4, −0.24) (s−2, −0.06) (s2, −0.1) (s0, 0)

onverted into fuzzy subsets (membership functions) in S by cal-ulating the membership function values for each weight using Eq.8). Then, the membership values are transformed into numeric ˇalues using Eq. (9). Finally, values are represented in 2-tupleuzzy linguistic values (si, ˛) using Eq. (4).

xample 1. Consider the FCM in Fig. 1 and its numeric connectioneights described by adjacency matrix shown in Table 2. Assume

hat the expressive linguistic term set S consists of 13 linguis-ic symbolic terms si which express the numerical values in thenterval [−1, 1] with following specific linguistic values “Very Veryighly Negative (−VVH), Very Highly Negative (−VH), Highly Neg-tive (−H), Medium Negative (−M), Low Negative (−L), Very Lowegative (−VL), Zero (Null), Very Low Positive (VL), Low Positive

L), Medium Positive (M), Highly Positive (H), Very Highly PositiveVH), Very Very Highly Positive (VVH)” as follows: S = {s−6, s−5, s−4,−3, s−2, s−1, s0, s1, s2, s3, s4, s5, s6}. Assume also that these symbolicerms are represented by 13 triangular membership functions on−1, 1] universe of discourse using the graphical representation inig. 3.

As the FCM connection weights are numerical values, the nexttep is to convert FCM connection weights into fuzzy subsets in. For each FCM connection weight (wjk), its membership functionalue associated with every si in S, (�si (wjk)), where, for this exam-le i = −6. . . 6, is calculated using Eq. (8). Thus, all membershipalues of (wjk) in the functions associated with the linguistic termet S, (�(wjk)) are as follows:

(wjk) = {(s−6, b−6), (s−5, b−5), . . ., (s5, b5), (s6, b6)} (24)

here �(wjk) are the membership values of (wjk) in the functionsssociated with the linguistic term set S; for example, b−6 � [0, 1]s the membership function value of (wjk) associated with the lin-uistic term s−6, and so on.

.45) (s0, 0) (s3, −0.4) (s1, 0.43) (s0, 0) (s0, 0) (s−3, 0.24)) (s−3, −0.03) (s0, 0) (s0, 0) (s0, 0) (s0, 0) (s2, −0.45)) (s0, 0) (s0, 0) (s0, 0) (s−1, 0.18) (s3, −0.49) (s0, 0)

Based on the membership values �(wjk), we obtain a (numeri-cal) value for (wjk) using Eq. (9) [25] which for this case is:

ˇ(wjk) =

i=6∑i=−6

i × bi

i=6∑i=−6

bi

(25)

where ˇ(wjk), is the numerical value of the FCM connectionweight (wjk). Then, we apply the previous step to all FCM con-nection weights in the example to obtain the � values for all FCMconnection weights and these values for weights are shown inTable 6. Finally, from the values, we obtain their equivalent 2-tuple (si, ˛) values using Eq. (4). The 2-tuple (si, ˛) values of theweights are shown in Table 7.

4.5. Fuzzy condensation of FCM (step 5)

In large complex participatory problems, especially if the prob-lem includes a range of stakeholder groups, many factors andrelationships among these factors are likely to be defined. As aresult, the FCMs designed by the relevant stakeholders could bevery complex and include a large number of nodes and connec-tions. In addition, these FCMs may need to be aggregated to obtainstakeholder group FCMs or social FCM (sum total of all FCMs). Ingroup or social FCM, the number of nodes could reach hundredsand the number of connections could be up to thousands. With suchFCMs, it is a big challenge to understand, analyse or gain insightsfrom them. To handle this challenge, the fifth step of the proposed
model is the FCM condensation process. It is a semi-quantitativemethod and simplifies a large FCM with a large number of nodes(variables) and connections into a smaller FCM with a small numberof higher level categories (groups) of variables and connections.

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The proposed condensation method is based on multi-levelondensation determined by the system developer. The systemeveloper should take into account several issues when choos-

ng the number of condensation levels, such as the size of FCM,he strength of interdependence between nodes, the smoothnessnd easiness of the transition process from the lower level of con-ensation to a higher level, and how to categorize the nodes andonnections at the lower level into new nodes (groups) at the higherevel without loss or distortion of information in the FCM. Theondensation method proposed in this article uses a number ofobust calculations and utilizes the credibility weights of nodes athe lower level in the process of transferring the nodes and theironnections to a higher level. Moreover, the values used in thisethod are represented by fuzzy numeric � values in all com-

utational processes to avoid loss of information. Consequently,his novel method can be considered as a semi-quantitative fuzzy

ethod proposed to overcome the shortage of previous qualitativeondensation methods.

The proposed method consists of two major phases in eachevel of condensation. The first phase is subjective (qualitative);t involves identification of the groups of nodes at the higher level.he system developer uses their knowledge and experience in theroblem domain to identify similar nodes in the lower level to beondensed into groups at the higher level of FCM condensation anddentifies the names of these groups. The second phase is objectivequantitative); it includes steps and calculations to ensure valid andccurate transition of weights from the lower level to the identi-ed higher level. It uses the credibility weights of nodes at the lower

evel, which are assigned based on the CCM of these nodes, to cal-ulate the connection values at the higher level. Typically, the FCMay have nodes and connections at the lower level, which are con-

ensed to the same group at the higher level. However, some nodesn the same group may, through each other, have indirect connec-ions to nodes in other groups; and some of these nodes in turnhrough other nodes in the same group may already have indirectinks to nodes in other groups. Rather than eliminating or disregard-ng such indirect connections, the proposed method uses a proper

ay to refine these connection weights before proceeding with theondensation process. The refinement process redistributes thesendirect links to direct links between appropriate nodes in differ-nt groups. This helps avoiding some nodes to have connections tohemselves (self-reference connections due to connections within

group) in the resulting condensed FCM and avoiding the loss ofndirect interactions between nodes prevailing at the lower levelefore condensation.

The refinement process is performed for each node ci in a groupf nodes that require redistribution of indirect connections at the

ower level to nodes in other groups. It starts after the group ofodes at the higher level and their nodes at the lower level areefined and its steps are described in Algorithm 1.

lgorithm 1. Steps in the refinement process of the indirect con-ections of node ci.

For each node cj in the same group that node ci belongs to do thefollowing:

If there is a connection between ci and cj (wij), then

If there is a connection between cj and any node ck (wjk) outsidethis group in the FCM, then

assign a new connection between ci and ck as follows:

ew wik = New cwi × wij + New cwj × wjk (26)


where New wik is the new connection between ci and ck, andNew cwi and New cwj are the new credibility weights for ciand cj nodes, respectively, where New cwi = cwi/(cwi + cwj) andNew cwj = cwj/(cwj + cwk).

b If the absolute value of New wik is greater than the absolute valuewik, then wik = New wik

ii Remove the connection wij

2 Repeat step 1 for other nodes in the group

The advantage of the proposed refinement of every connectionbetween any two nodes located in the same group over its cancel-lation is that the cancellation may adversely affect the behaviourof the FCM, for example, when a node influences another nodein the same group and the latter in turn influences other nodesin other groups. In this novel method described in Algorithm 1,the indirect influences of connections between nodes in differ-ent groups are converted to direct influences between them. Themethod calculates the weights for the newly created direct influ-ences by considering the connection weight and credibility weightsof nodes that established the indirect influences. Eq. (26) expressesthis calculation. Finally, if the nodes which formed the newly cre-ated direct influences already have connection weights betweenthem, then the highest of the absolute value of the newly createddirect influence and the existing weight is assigned as the connec-tion weight between these nodes.

After the refinement process, FCM condensation process foreach level of condensation resumes with the Algorithm 2 shownbelow. The process involves computing new weights betweengroups at the higher level. The first step in this process is to initial-ize to zero the connection weights between groups at the higherlevel. Then, for the connection between two groups, we identifyat the lower level only the non-zero connections between nodesin these groups. For the identified nodes in each group, we utilizetheir calculated credibility weights at the lower level to calculate anew credibility weight for them using Eq. (27). Then, Eq. (28) usesthese new credibility weights of nodes and their connections at thelower level to calculate a condensed connection between the twogroups at the higher level. These steps are repeated until all con-nections between all groups at the higher level are calculated. Inthis way, the adjacency matrix of the condensed FCM is created. Itis easy from this matrix to draw the condensed FCM and show theconnections between its nodes (groups).

Algorithm 2. Steps in the fuzzy condensation process for FCM ateach level of condensation.

1 Initialise an adjacency matrix (G) for the groups of condensednodes at the higher level and fill its elements (connection weightvalues between groups) with zero values

2 For each connection weight gij in the matrix G between group giand group gj do the following:

A Obtain the nodes at the lower level that belong to gi and gj groupsat the higher level

B Initialise two dimensional matrix (Grpij), i = 1 to the number ofnodes in gi and j = 1 to the number of nodes in gj

C Store in Grpij the connection values at the lower level from nodes
in gi to nodes in gj
D Select from Grpij the nodes in both gi and gj groups that have atleast one non-zero connection value and reassign new credibilityweights to each node using the following equation:

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ew cwni =cwni

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here New cwni is the new credibility weight of a selected node nn gi, n = 1 . . . N, N is the total number of selected nodes in gi, cwni ishe lower level credibility weight of the selected node n in gi, andum(cwNi) is the sum of credibility weights at the lower level of allelected nodes in group gi

Hint: the above equation calculates the new credibility weightf a selected node in group gi, taking into account in this case that

is the total number of selected nodes in gi

Assign new weight value to the connection gij between gi and gjgroups at the higher level as follows:

ij = sum(New cwni × New cwnj × wij) (28)

here sum(New cwni× New cwnj × wij) is the sum of all connec-ion weights wij at the lower level between nodes in gi and nodesn gj after multiplying each connection weight between two nodesy their new credibility weights assigned using Eq. (27)

Repeat step 2 for all connection weights in matrix G at the higherlevel

Construct from the matrix G the condensed FCM at the higherlevel

Once the condensed FCM is constructed at this level, the CCMnd credibility weights values for its nodes can be easily obtainedsing calculations stated before. These values are required for theext level of FCM condensation process. After the completion of all

evels of the FCM condensation process, a simpler FCM with fewodes and connections is constructed. In this case, it is easy to gain

nsights from FCM and analyse its structure as well as apply it forimulating what-if scenarios. It also helps produce a simpler groupCM that would result from combining condensed individual FCMs.ext section presents the FCM aggregation process that combinesCMs in a group (such as a stakeholder group) into the correspond-ng group FCM as well as combine multiple group FCMs (such as

ultiple stakeholder groups) into a social FCM.

.6. Fuzzy aggregation process of FCM (step 6)

According to literature, obtaining a consensus FCM from a groupf different FCMs pertaining to stakeholders with diverse percep-ions is currently a challenge. The FCMs are typically developedor complex domains characterised by uncertain and imprecisenowledge. As such, the issue of conflicting perceptions betweenhe stakeholders (designers of FCMs) is natural to arise. In real-liferoblems, incorporation of the human dimension through humanerceptions/opinions has become a necessity to fully characterisend study such problems. However, people vary in the level ofnowledge and experience and their opinions should be taken

nto account accordingly when their FCMs are combined into anverall expression for a group. Hence, the FCMs that depict theseerceptions should be weighted according to an objective cred-

bility measure. This issue constitutes a barrier for developing aalid and accurate group FCM that describes a consensus percep-ion. Another issue that has been stressed throughout this articles that different people may need different measures to expressheir knowledge in the form of nodes and connection betweenodes. Some could prefer to use linguistic terms while others pre-

er numeric values; or they may use different scales of linguistic
nd numeric measures. Finally, as complex problems addressed byCM are typically characterised by ambiguity and uncertainty, themprecise values describing the connections between nodes in FCMhould be represented by a suitable model that retains the accuracy
oft Computing 48 (2016) 91–110 105

when combining weights expressed in different formats in differ-ent FCMs. To overcome the stated challenges, this article utilizesthe FCM aggregation method we proposed in Obiedat et al. [40] tointroduce the sixth process of the proposed model − robust fuzzyaggregation of FCM.

The introduced FCM aggregation method can be applied on theFCMs before or after condensation process, or even at any level of it.This method uses the 2-tuple model, described previously, to rep-resent the imprecise connection values between nodes in FCMs infuzzy values. It also takes into consideration the importance ofthese connection values by weighting them according to the cred-ibility weight of their FCMs. Credibility weights of FCMs addressthe importance of different levels of knowledge of the stakehold-ers and are used to prioritize the importance of the connectionsbefore they are aggregated with the connection values in otherFCMs. Finally, the proposed fuzzy aggregation method can deal withmultiple linguistic and numeric scales used to describe impreciseconnection values in different FCMs. In other words, the methodallows describing imprecise connections by different fuzzy sets asappropriate for the stakeholders. Fuzzy sets could include differentnumber of linguistic terms represented by different membershipfunctions (fuzzy subsets) to deal with multiple linguistic and/ornumeric scales. The different linguistic sets are to be converted intoone uniform set called Base Linguistic Term Set (BLTS) [24,25].

The BLTS is chosen such that it contains a number of LinguisticTerms appropriate to represent the overall group of FCMs beingassembled. This process is called the normalization of the linguisticsets; and for each FCM, before combining with other FCMs, its fuzzysubsets are converted into fuzzy subsets in the BLTS. The objectiveof this process is to manage the imprecise connection values ofFCMs in a way that prevents any loss of information during theaggregation process by normalizing the fuzzy subsets representingthese imprecise values into a standard set, BLTS [24]. To do so, weconvert the connection weight values of FCMs into the interval[−1, 1]; the procedure for this step was explained previously. Then,we transform the linguistic set of FCM into the BLTS. Let S = (s−p,. . ., sp) be the linguistic set of an FCM and BLTS = (t−g , . . ., sg) be thelinguistic set of the BLTS, such that p ≤ g; then the function TS−BLTSdefines each fuzzy subset of S in the BLTS as follows [24]:

TS−BLTS(si) = {(tk, ˛ik), k ∈ {0, . . ., g}}, ∀si ∈ S (29)

˛ik = maxy

(min{�si (y), �tk (y)}) (30)

where max and min are the usual maximum and minimum oper-ations, respectively, y is a connection weight value on [−1, 1]universe of discourse, and �si(y) and �tk(y) are the membershipfunctions of the fuzzy subsets associated with the terms si and tk,respectively.

As the next step in the process, using the transformed uniformlinguistic set of FCM, a conversion of connection values in the inter-val [−1, 1] back to values is carried out. Once the above steps areperformed on all FCMs in the system, the next step of the FCMaggregation is to initialize an adjacency matrix (Soc) for the groupFCM to include all possible nodes and connection weights betweenthem that could result from combining the FCMs into the group andfill its elements by zero values as follows:

Socij ={

0 i, j = 1. . .NC} (31)

where Socij represents the connection weight between node ci andnode cj in the group or social FCM and NC is the number of distinctnodes in all FCMs to be aggregated in the system.

This step is followed by an FCM augmentation process; here,the adjacency matrix of each FCM (i.e., FCMk, where k = 1 to thenumber of FCMs in the system) is augmented to include all nodesin all FCMs. The column and row of each new node added to the


Table 8Connection weights (in values) of the group FCM resulting from the fuzzy aggregation of the ‘expert’ and ‘local people’ FCMs presented in Fig. 4.

Water situation Water resources Water demand Technology Wastage of water Water projects Economic situation

Water situation 0 0 0 0 0 0 1.200Water resources 5.247 0 0 0 0 0 0Water demand −6.000 0 0 0 0 0 0Technology 0 2.153 −1.088 0 −2.513 0 0

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Wastage of water −1.447 0 0.713

Water projects 0 1.600 0

Economic situation 0 0 −1.200

atrix are filled with zero values. The augmented matrix is thenultiplied by the FCM credibility weight as follows:

CMwk = FCMk × cwk (32)

here FCMwk is the resulting weighted matrix of FCM obtainedrom multiplying the FCM connection weight values FCMk by itsredibility weight value cwk.

Finally, the weighted augmented matrix is added to the adja-ency matrix of group FCM as follows:

oc = Soc + FCMwk (33)

The above steps of the FCM aggregation process are repeatedntil the last FCM in the group has been added to the group FCM.

To aggregate the example ‘expert’ and ‘local people’ FCMs pre-ented in Fig. 4 into a group FCM, consider their values in Table 1.uppose the results from calculating the credibility weights (seeection 3.2.5) of the ‘expert’ and ‘local people’ FCMs are 0.6 and 0.4,espectively, and suppose the linguistic term set S used to repre-ent their connection weights is BLTS itself. Therefore, there is noeed for the normalization process. Applying the stated steps ofhe fuzzy aggregation (Eqs. (31)–(33)), the weight values of theesulting group FCM are presented in Table 8. This table shows thathe group FCM includes all distinct nodes and all possible connec-ions between these nodes that existed in both FCMs. Fig. 6 showsexpert’ and ‘local people’ and resulting group FCMs. It shows theonnection weights of these FCMs represented by 2-tuple valuesequivalent of obtained from Eq. (4)).

In addition to individual (original) FCMs, the condensation andggregation processes create a number of diverse FCMs (groupCMs and a social group FCM) to be analysed and simulated. Thisllows us to reach towards a comprehensive knowledge and under-tanding of the problem and allows access to a wide range ofndividual and group perceptions such that analysing and simu-ating these perceptions can lead to many reliable suggestions orotential solutions to the problem.

.7. FCM analysis process (step 7)

After the completion of the various FCM processes describedn the previous sections, the developer of the system becomesamiliar with the details/issues of the problem and has the under-tanding and experience to initiate FCM analysis and simulationrocesses. Analysis and reflection on the original, condensed andggregated perceptions/FCMs lead to a greater understanding theroblem domain and examination of different perceptions of thetakeholders and their interests. Therefore, the seventh process ofhe proposed FCM model is to analyse the structures of differentCMs such as those for individual stakeholders, stakeholder groupCMs and the social FCM and then make comparisons among them.he proposed FCM model can benefit from graph theory indices tonalyse the structures of the perceptions/FCMs and compare them
ccordingly. It can also use different statistical measures to findimilarities and dissimilarities between nodes in individual FCMs,nd between FCMs in FCM groups. In addition, the CCM index iden-ifies the importance of FCMs and nodes in FCMs, and based on this,
0 0 0 00 0 0 −2.0000 0 1.600 0

the most important (central) nodes are determined. These nodesmake great contributions to their FCMs and could be used as influ-ential instruments to change the behaviour and outcome of theFCMs (system) when these FCMs are simulated.

4.8. FCM simulation process (step 8)

Because systems representing complex and dynamic real-lifeproblems include many feedback loops, system simulation andpolicy scenario simulations are required to explore the systembehaviour and reach better outcomes. The last process of the pro-posed FCM model is the FCM simulation. Different FCMs can besimulated and used as simulation models to conduct differentwhat-if policy scenarios. The first step of simulating any FCM isto simulate the FCM from the current (initial states of nodes) untilsteady state is reached. This steady state is the status quo outcomeof the FCM. For this purpose, the FCM is treated as a neural networkwith feedback (a recurrent network) where each neuron representsa concept [47]. Initial state (or initial input) of each neuron is thestate of a variable (or a group of variables) in the range of [0, 1]specified by stakeholders or determined from existing informa-tion. Each neuron or node computes its output as the weightedsum of inputs coming from the concepts connected to it processedthrough a sigmoid function. Weights are the fuzzy connection val-ues assigned as discussed previously. As there are feedback loops,the simulation will run many cycles before the states of all nodesreach equilibrium, which represents the steady state of the system.

After the status quo outcome is obtained, different policy sce-narios (what-if questions) can be applied to test whether the newscenario leads to a desirable outcome. These what-if scenarios con-stitute proportional changes or perturbations to one or several mostcentral variables in an FCM identified previously. Typically, thealtered states of the chosen variables are clamped (fixed) through-out the iterations until equilibrium (new steady state) is reached.Then, a comparison between this new state and the status quosteady state of the FCM reveals the influence of the new policy onthe system outcomes. With a good understanding of the domain,what-if scenarios can be designed such that the simulated scenar-ios are practically reasonable and effective in producing desirablepolicy outcomes. Finally, the analysis and comparison betweenthe results from simulating multiple policy scenarios would helpexplore and rank potential solutions to the problem. These solu-tions are introduced to decision makers as recommendations thatcould help them in making effective decisions (for example, actionsto be taken in the form of changes to be made to the influentialfactors (variables) in order to alleviate an acute water scarcity prob-lem). A brief summary of a case study using the proposed method is
given below to demonstrate its application and usefulness. A largerand more comprehensive study for alleviating water scarcity sit-uation in a country using the proposed method is planned for aforthcoming article.


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ig. 6. A graphical representation of aggregation of ‘expert’ and ‘local people’ FCMs pnd dashed arrows indicate negative and positive influences, respectively.

. An example from reality: mitigating groundwateregradation and depletion in Jordan

Water is critical to survival and a significant matter for the safetynd wellbeing of any society. Jordan faces severe water crises andhere are signs of increasing water problems in Jordan. Govern-

ent in Jordan is paying a great deal of attention to address theater issues [38,28]. Here, we present the application of the pro-

osed model to this real-life problem—‘Groundwater Degradationnd Depletion in Jordan’. We address challenges of Jordan ground-ater problem and suggest some meaningful recommendations

hat could help decision makers in managing and improving theroundwater situation in Jordan to mitigate its degradation andepletion. This study was carried out according to the steps of theroposed model as summarised below.

Firstly, we collected knowledge and perceptions on the ground-ater problem in Jordan from the perspective of different relevant

takeholder groups using in-depth semi-structured interviews. Thedea was to get as much information as possible on the problemncluding factors affecting ground water degradation/depletion,trength of their interactions and any other helpful informationith a view to improving the situation through the proposedethod. 15 interviews were conducted with 3 stakeholder groups:

ocal people, farmers and managers. Then, 15 stakeholder FCMsere developed. 9 stakeholders preferred to draw the connection
eights between variables using linguistic values, while the rest
sed the interval [0, 1]. These FCMs were then thoroughly reviewedo ensure that all variables and connections that were mentioned in

ed in Fig. 4. Connection weights of all FCMs are represented by 2-tuple values. Solid

the recorded interviews are present in the FCMs. The stakeholdersdefined 67 different original variables in total.

The connection weights were then represented in a unified for-mat using the proposed 2-tuple fuzzy linguistic model. We used alinguistic term set, S, consisting of 11 linguistic terms (membershipfunctions) as follows: S = {s−5, s−4, s−3, s−2, s−1, s0, s1, s2, s3, s4, s5}.Then we calculated and 2-tuple fuzzy linguistic values (si, ˛) forthese weights. The next step was calculating Consensus Central-ity Measure (CCM) and credibility weights for variables and FCMs.We present here the CCM and credibility weights only for the threecondensed group FCMs and the social group FCM in Table 9 whichcould be clearer if read in conjunction with the following discussionon FCM condensation.

The next stage was using one level of FCM condensation tosimplify the 67 variables into following 12 subjectively definedconcepts at a higher level of abstraction: groundwater improve-ment, causes of degradation, modern technology, law enforcement,groundwater basins, groundwater uses, stakeholder awareness, finan-cial support, other water resources, highland water extraction, privatesector, and pollution. The indirect connections between variableswere manipulated using the proposed refinement process dur-ing the FCM condensation. The individual condensed FCMs werecombined into 3 stakeholder group FCMs using the obtained cred-ibility weights of these FCMs and the proposed FCM aggregationprocess. Then, we combined the group FCMs into one social FCM
using the above process and credibility weights of these FCMs areshown in Table 9. We analysed the condensed individual, groupand social FCMs using CCM of concepts, most mentioned conceptsand map density measures. Table 9 (bottom row) shows the calcu-


Table 9Some graph theory indices of individual, group and social condensed FCMs at the level of the 12 concepts: frequency of concept (i.e., how many of the 15 individual condensedFCMs contain the concept), the CCM and credibility weight values of concepts in groups and social FCMs, and the map density, CCM and credibility weight values of groupand social FCMs.

Concept name No. of condensedindividual FCMscontaining concept

CCM and credibility weight (CW) values of concepts in

Local people group FCM Farmer group FCM Manager group FCM Social FCM

CCM CW CCM CW CCM CW CCM CW

Groundwater improvement 15 0.479 0.168840324 0.484 0.182228916 0.551 0.131755141 0.524 0.169689119Causes of degradation 11 0.202 0.071201974 0.124 0.046686747 0.356 0.085126734 0.217 0.070272021Modern technology 13 0.442 0.155798379 0.537 0.202183735 0.547 0.130798661 0.621 0.201101036Law enforcement 12 0.302 0.106450476 0.237 0.089231928 0.647 0.154710665 0.532 0.172279793Groundwater basins 11 0.099 0.034896017 0.107 0.040286145 0.135 0.032281205 0.087 0.028173575Groundwater uses 10 0.187 0.065914699 0.22 0.082831325 0.201 0.048063128 0.128 0.041450777Stakeholder awareness 11 0.207 0.072964399 0.098 0.03689759 0.372 0.088952654 0.221 0.071567358Financial support 12 0.245 0.08635883 0.342 0.12876506 0.318 0.076040172 0.207 0.067033679Other water resources 10 0.078 0.027493832 0.113 0.042545181 0.149 0.035628886 0.096 0.031088083Highland water extraction 10 0.253 0.08917871 0.156 0.05873494 0.401 0.095887135 0.205 0.06638601Private sector 10 0.286 0.100810716 0.213 0.080195783 0.304 0.072692492 0.205 0.06638601Pollution 9 0.057 0.020091646 0.025 0.009412651 0.201 0.048063128 0.045 0.014572539

FCM density 0.318 0.379 0.394 0. 530FCM CCM 0.592 0.654 0.712 0.605FCM credibility weight (CW) 0.302 0.334 0.364 1

Table 10Outcomes of the applied policy scenario simulations in group and social FCM systems by fixing the influential concepts at high values to show their impacts on GroundwaterImprovement concept. The impact is shown in %.

Influential concept The impact of the influential concepts on groundwater improvement concept in

Local people group FCM % Farmer group FCM % Manager group FCM % Social FCM % Average impact %

Modern technology 30% 28% 24% 25% 27%Law enforcement 33% 25% 23% 26% 27%

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wc

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aifiGctto

Highland water extraction 15% 13%

Stakeholder awareness 11% 12%

Financial support 11% 22%

ated values of these measures. The map density (D) (extent of FCMonnectivity) is calculated as follows:

= C

(N × (N − 1))(34)

here C is the number of connections and N is the number of con-epts in FCM. Density of a fully connected map is 1.0.

The analysis process revealed that the stakeholders in differ-nt groups defined many similar concepts and their perceptionsere approximately similar to each other with regard to the groundater situation/improvement. For example, Table 9 shows that the

ensity of group FCMs is very close to each other. Also, the den-ity is relatively high meaning that the stakeholders defined a largeumber of connections among the concepts. According to the CCMalues, Modern Technology and Law Enforcement concepts have highalues in all groups and social FCMs. Highland Water Extraction andtakeholder Awareness concepts have high CCM values in managerroup FCM, and Financial Support concept has high CCM value inarmer and manager FCMs. Therefore, these concepts are the mostmportant (influential) concepts in FCMs and could be considereds the most appropriate and effective concepts for policy scenarioimulations.

Several policy scenarios were simulated in the condensed groupnd social FCMs to provide useful recommendations for “Mitigat-ng Groundwater Degradation and Depletion”. The objective is tond policies (concepts) that most strongly influence the conceptroundwater Improvement. As an interconnected systems, all con-
epts that significantly impact Groundwater Improvement will findheir steady states impacted in the simulation. Initially, the sys-ems consisting of group FCMs and the social FCM at the levelf concepts were simulated to find the steady state of concepts
26% 23% 19%17% 12% 13%10% 14% 14%

under the present conditions. These outcomes indicate the even-tual state of systems if the present conditions persist, according tothe perceptions of the groups of stakeholders. Then, several policyscenarios were applied by fixing the above mentioned influentialconcepts at high values, for example 1, throughout the simula-tion process to see if the system reaches desirable states towardsmitigating the problem. Specifically, these policy scenario simu-lations were implemented to test the effectiveness of each of theselected influential concepts. Table 10 summarises the outcomes ofthe applied policy scenario simulations in group and social FCM sys-tems. It shows the contributions of the selected influential conceptsto Groundwater Improvement. Finally, we analysed the outcomes ofthese scenario simulations in search of appropriate recommenda-tions based on how much the scenarios contribute to GroundwaterImprovement.

As a result, we proposed the following three useful recommen-dations to the decision makers of water management in Jordan:(a) use modern technologies to recharge groundwater basins inJordan; (b) prevent the overuse of highland water that lead toover-exploitation of groundwater; and (c) reinforce and activatedeterrent and effective laws to reduce theft and exploitation of unli-censed wells and protect groundwater resources from attacks andillegal use.

6. Conclusions and future work

This article proposes a robust semi-quantitative FCM modelthat can address and model various complex contemporary real-life problems. The advantages of this model over other previousFCM models lie in its ability to make the FCM model more com-

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utationally robust, consistent, functional and efficient. This ischieved by the enhancement of the existing processes of the FCMpproach or adding new ones. The proposed FCM model consists ofight coherent and consistent processes. These processes model theroblem from the phase of data collection in the form of ambigu-us perceptions to the phase of knowledge extraction in the formf solutions or recommendations. The model collects comprehen-ive data (FCMs/perceptions) using qualitative interviews with aufficient number of relevant stakeholders. It allows, representsnd deals with multiple types (symbolic and numeric) of impre-ise human perceptions throughout all computational processesithout information loss or inconsistency using a 2-tuple fuzzy

epresentation method. It introduces a novel Consensus Central-ty Measure and credibility weight for nodes and FCMs to identifymportant nodes and FCMs as well as to properly incorporateifferences in nodes and FCMs during FCM condensation and aggre-ation. It includes a novel semi-quantitative condensation methodor FCM which consists of multi-level condensation to simplify aomplex and large FCM into a simpler and smaller one for ease ofnderstanding, traceability and predictability in a way that infor-ation is not lost between levels of condensation. It provides a

uzzy weighted aggregation method to obtain a reasonable con-ensus FCM from many relevant FCMs/stakeholders. It analyseshe structure of the FCMs to probe into the reasonableness of rep-esentation and understanding of the problem. Finally, it allowsimulation of FCM systems to study their steady states (status quo)nd examines them for several policy scenarios based on identifiednfluential variables to select scenarios that could shift the sys-em behaviour towards a more desirable outcome than the statusuo. Accordingly, appropriate recommendations or solutions coulde defined and introduced to decision makers for addressing theroblem concerned. To test the applicability of the proposed FCModel, we applied it in a real life problem—Mitigating Groundwateregradation and Depletion in Jordan. Accordingly, useful outcomesere acquired in the form of recommendations that could help

ecision makers toward mitigating the problem. In a future pub-ication, the authors would report on the detailed results from auccessful application of this model to a more comprehensive prob-em and one of the acutely stressful real-life problems, namelyMitigating the Water Scarcity Problem in Jordan”. Finally, we haveoticed that there could be room for improvement in the processf obtaining credibility weights for variables and FCMs using otherethods besides CCM measure. In addition, the qualitative process

f the selection of group variables at the higher level of condensa-ion should be replaced by quantitative methods such as intelligentomputing methods. These limitations are the focus of our futureork.

cknowledgments

Research in this article was supported by a scholarship offeredy Lincoln University, New Zealand.

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Mamoon Obiedat was born in Jordan in 1970. He receivedBSc. in Computer Science and MSc. in Computer Informa-tion Systems form Yarmouk University, Jordan in 1992,2005, respectively. He was a Lecturer at Al-balqa AppliedUniversity in Jordan from 1998 until he received PhDdegree in Computer Science from Lincoln University, NewZealand in 2014. He is currently an Assistant Professorat the Department of Computer Information Systems inAl-Hashemite University.

Sandhya Samarasinghe is a Professor of Integrated Sys-tems Modelling and Head of the Integrated SystemsModelling Group at Lincoln University, New Zealand. She




































































































































































































































































































































































































































































































































































Applied Soft Computing - Hashemite University

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