Fuzzy Models in Evaluation of Information Uncertainty in Engineering and Technology Applica

Fuzzy Sets and Systems159(2008)3091–3131/locate/fss Low-level interpretability and high-level interpretability:a unified view of data-driven interpretable fuzzy system modellingShang-Ming Zhou a,∗,John Q.Gan ba Centre for Computational Intelligence,School of Computing,De Montfort University,Leicester LE19BH,UKb Department of Computing and Electronic Systems,University of Essex,Colchester CO43SQ,UKReceived28September2007;received in revised form10May2008;accepted26May2008Available online20June2008AbstractThis paper aims at providing an in-depth overview of designing interpretable fuzzy inference models from data within a unified framework.The objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real-world systems whosefirst-principle models are unknown.Because system behaviour can be described naturally as a series of linguistic rules,data-driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose. However,fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability. Model accuracy and interpretability are two conflicting objectives,so interpretation preservation during adaptation in data-driven fuzzy system modelling is a challenging task,which has received much attention in fuzzy system modelling community.In order to clearly discriminate the different roles of fuzzy sets,input variables,and other components in achieving an interpretable fuzzy model, a taxonomy of fuzzy model interpretability isfirst proposed in terms of low-level interpretability and high-level interpretability in this paper.The low-level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level,while the high-level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage,completeness,and consistency of the rules in terms of the criteria on fuzzy rule level. Some criteria for low-level interpretability and high-level interpretability are identified,respectively.Different data-driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low-level interpretability and high-level interpretability.Furthermore,some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested.Crown Copyright©2008Published by Elsevier B.V.All rights reserved.Keywords:Data-driven fuzzy systems;Interpretable;Fuzzy models;Interpretability;Transparency;Criteria;Parsimony;Distinguishability;Low-level interpretability;High-level interpretability.1.IntroductionAs the main objective in system modelling,development of reliable and understandable models is crucial for human being to understand real-world systems or natural phenomena.Generally there are three different strategies for system modelling.Thefirst is white-box modelling,in which the parameters characterizing the systems have clear and interpretable physical meanings(one typical example is the Newton’s universal gravitation law).However,white-box∗Corresponding author.E-mail address:smzhou@(S.-M.Zhou).0165-0114/$-see front matter Crown Copyright©2008Published by Elsevier B.V.All rights reserved.doi:10.1016/j.fss.2008.05.0163092S.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–3131modelling usually becomes impractical when complex systems are considered.The second strategy is black-box mod-elling without using prior knowledge,in which relationships between inputs and outputs are established fully based on observational data.Black-box modelling can simulate a real-world system reliably and precisely,but the model structure and parameters usually give no explicit explanation about the system behaviours.For example,in medical domains, although black-box models,such as bagged decision trees and neural networks,can achieve goodfit performance,the resulting decisions received much suspicion[1,186].The third strategy,grey-box modelling[41,96],can be referred to as an eclecticism between precision and interpretability,in which the prior knowledge about the system is considered and unknown parts of the system are identified by black-box modelling approaches.As one of the most successful tools to develop grey-box models[112],fuzzy modelling describes systems by establishing relationships between input and output variables in terms of a fuzzy logic-based descriptive language[114–116,191,193].Compared to black-box modelling,fuzzy modelling formulates the system knowledge with rules in a transparent way to interpretation and analysis(to a certain degree)so as to gain insights into the system being modelled.This has opened up a brand-new approach to modelling complex systems in system identification and control,fault diagnosis, classification and intelligent data analysis,etc.Fuzzy rules can be generated based on human expert knowledge or heuristics,which offers a good high-level semantic generalization capability.However,for complex systems,the interactions between system behaviours are very difficult to be grasped so that the system model just based on expert knowledge may suffer from a loss of accuracy[68]and combinatorial rule explosion.On the other hand,due to the advanced development of modern information technology,the ever-increasing quantity of data is available from complex industrial and commercial processes(systems).In the past decade or so,data-driven fuzzy rule generation has been widely investigated and shown to be very pared to heuristic fuzzy rules,fuzzy rules generated from data are able to extract more specific knowledge for more complex systems.It is usually assumed that the interpretability of fuzzy models is automatically given just due to using linguistic rules. However,it is not true when adaptive learning techniques are used to optimize the fuzzy inference processes for complex systems.In accuracy-oriented adaptive learning processes,interpretation preservation during adaptation cannot always be guaranteed due to the conflicting objectives of accuracy and interpretation[8,24,28,67,68,86,121,132,160,180],so model interpretability is usually lost,which has been referred to as an incentive for Mamdani,the founder of fuzzy control,to recommend against the use of adaptation of fuzzy inference parameters[115].Particularly,because one of the important incentives of introducing fuzzy sets for modelling complex systems is that they can formulate the knowledge extracted from data or supplied by experts with a more transparent way to gain insights into the complex systems,it is often desirable and sometimes even essential to be able to interpret thefinal model structure.Rule base legibility is an important condition to take full advantage of fuzzy models.Hence,model interpretability improvement is regarded as one of the most important issues in data-driven fuzzy modelling[5,25,46,83,91,94,93,113,156,174,179,190]. However,global model accuracy and interpretability are two conflicting modelling objectives,as improving inter-pretability of fuzzy models generally degrades the global model performance of fuzzy models,and vice versa.Hence one challenging problem is how to construct a fuzzy model with not only good global performance but also good model interpretability,particularly in coping with high-dimensional input space,that is to say,how to achieve a fuzzy model with good performance trade-off between global model accuracy and model interpretability.An interpretable fuzzy model is expected to provide high numeric precision while incurring as little a loss of linguistic descriptive power as possible[5,45,83,139,190,198].It is noteworthy that as we address the interpretability issues in fuzzy modelling,transparency and interpretability issues in traditional statistical system modelling have already received much attention from the aspect of complexity reduction[33,55,73,75,97,164,177,178].The objective of complexity reduction-based statistical system modelling is to provide a trade-off between how well the modelfits the training data and model complexity.Model complexity depends on the number of independent and adjustable parameters,also called degrees of freedom,to be adapted during the learning process.Traditionally,the well-known complexity reduction techniques for achieving transparent and inter-pretable system models include input feature selection[196],orthogonal least square(OLS)[30],basis pursuit[29,31], sparse regression[170],support vector machines(SVM)[175,176],etc.More importantly,these complexity reduc-tion techniques possess strong potentials of improving fuzzy model interpretability when applied to fuzzy modelling.A natural question arises:what is the difference between the interpretability achieved by considering fuzzy systems’own idiosyncrasies and the one obtained by introducing the traditional complexity reduction techniques into fuzzy modelling?Guillaume has provided a good overview of interpretability-oriented fuzzy inference systems designed from data[67].However,the above question was not considered and addressed in[67].The purpose of this paper isS.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–31313093 not only to provide an overview of the state-of-the-art of data-driven interpretable fuzzy modelling techniques,but also to offer an overall view of this domain in a framework by syncretizing the traditional parsimonious statistical system modelling with interpretable fuzzy system modelling,so that one may form a systematic picture of thefield,instead of focusing on specific parts,dealing with particular methods and techniques.To the best of our knowledge,this type of effort has not been reported yet in literature.As a matter of fact,in the traditional statistical modelling techniques,only the influences of input variables on system outputs are taken into account to construct transparent and interpretable system models.While in fuzzy system modelling,due to its own idiosyncrasies,additional influences on system outputs should also be taken into account to achieve a transparent and interpretable fuzzy model,such as the number of membership functions(MFs),coverage and distinguishability of MFs,normality of MFs.In order to clearly discriminate the different roles of fuzzy sets,input variables and other components in achieving a transparent fuzzy model,a framework is suggested in this paper to categorize fuzzy model interpretability into low-level interpretability and high-level interpretability,and the criteria are defined for low-level interpretability and high-level interpretability,respectively.Low-level interpretability of fuzzy models is achieved on fuzzy set level by optimizing MFs in terms of the semantic criteria on MFs,whilst high-level interpretability is obtained on fuzzy rule level by conducting overall complexity reduction in terms of some criteria, such as a moderate number of variables and rules,completeness and consistency of rules.The complexity reduction techniques used in traditional system modelling,if exploited in fuzzy system modelling,can serve as fuzzy rule optimization in nature,which corresponds to aiming at the parsimony of the fuzzy rule base,one of the main high-level interpretability criteria of fuzzy systems.This clarification is helpful,as there are plentiful traditional system modelling methods on complexity reduction with great potentials of being used to induce compact rule base in fuzzy system modelling.This paper is organized as follows.Section2briefly presents the currently used fuzzy rule structures.Various definitions of fuzzy model interpretability are given in Section3,followed by criteria for fuzzy model interpretability in Section4.Constructive techniques for fuzzy model interpretability are analysed and reviewed in Section5.Section 6addresses some open problems and potential new research topics on fuzzy model interpretability,followed by conclusions in Section7.2.Rule structures of fuzzy rule-based systemsBasically,in fuzzy modelling the relationships between input and output variables are established in terms of a descriptive language based on if–then rules:If antecedent proposition then consequent proposition(1) Depending on the consequent part there exist different rule structures with different capabilities of description and approximation for rule-based fuzzy modelling.Linguistic fuzzy model,also known as Mamdani type fuzzy model[114–116,193],is represented by linguistic rules with the following structure:Rule i:If x1is A i,1and···x n is A i,n then y is B i(i=1,...,L)(2) where Rule i denotes the i th rule,L is the number of rules in the rule base,x=(x1,...,x n)T and y are the input and output linguistic variables,respectively,A i,j and B i are the linguistic labels expressed as fuzzy sets with specific semantic meanings regarding the behaviours of the system being modelled.These fuzzy sets are characterized by MFs generated by expert knowledge or fully from data.An outstanding advantage of the linguistic fuzzy model lies in that it may offer a high semantic level with good interpretability and a good generalization capability.In order to enhance the representation power of a fuzzy system,Takagi and Sugeno proposed a fuzzy model(TS fuzzy model)that differs from the linguistic one by using the following different consequent structure[168]: Rule i:If x1is A i,1and···x n is A i,n theny i=a0i+a1i x1+···+a ni x n(i=1,...,L)(3) where x i are input variables and y i are local output variables that determines local linear input–output relations by means of the real-valued coefficients a ji.The output of a fuzzy system with a knowledge base composed of L TS rules3094S.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–3131 is computed as the weighted average of the individual rule outputs y i,i=1,...,L:y=Li=1r i(x)y iLi=1r i(x)(4)with r i(x)=T j i,j(x j)being the matching degree between the antecedent part of the i th rule and the current system inputs x1,...,x n,where T is a t-norm and i,j(·)the MF of fuzzy set A i,j.Compared to a linguistic model,the rule consequent part in the TS model is an affine linear function of input variables. As such,each rule can be considered as a local linear model that is fused together by means of aggregation to produce an overall output y.A special case of the affine function with offset a0i=0(i=1,...,L)results in a homogeneous TS system.Another interesting special case is that the consequent is a constant,i.e.,y i=a0i,which is called0-order TS system.Actually the0-order TS model retains certain linguistic interpretability in the manner of a linguistic model while possessing the attractive properties of the TS system,particularly the automatic determination of model parameters from data[157].Some researchers have made efforts to modify the form of the consequent polynomials of TS models, which allows specific interpretation to the systems being modelled[7,18,54].However,either the Mamdani fuzzy rule in the form of(2)or the TS rule in the form of(3)shares the same fuzzy sets of each input variable with other rules,i.e.,the Mamdani rule(2)and the TS rule(3)are based on the global grid, which offers good linguistic readability[67].The disadvantage is that the fuzzy systems built based on(2)or(3)suffer from the curse of dimensionality in high-dimensional input space:the problem will increase exponentially in volume associated with adding extra dimensions to the input space.The curse of dimensionality is a significant obstacle to solving machine learning problems that involve learning a“state-of-nature’’from afinite number of data samples in a high-dimensional space.One efficient way of evading this problem for fuzzy systems is to generate the input space via scatter-partition(prototype)rather than grid-partition.The Mamdani and TS fuzzy rules generated by scatter-partition have the following forms respectively:Rule i:If x is A i then y is B i(5) andRule i:If x is A i then y i=a0i+a1i x1+···+a ni x n(6) where x=t(x1,x2,...,x n)∈ n is an n-dimensional input vector and A i∈F( n)are the fuzzy sets defined on n with multi-dimensional MFs.Very interestingly,the fuzzy systems consisting of multi-dimensional Mamdani rules(5) are totally equivalent to fuzzy graphs introduced by Zadeh[192,195].A fuzzy graph is build by a collection of fuzzy points corresponding to fuzzy rules,which represents a functional dependency f∗of Y on X:f∗=A1×B1+···+A n×B n(7) Some researchers have made efforts to construct fuzzy graphs from data[6,13].More details about how to generate the prototype-based fuzzy systems and their pros and cons will be reviewed in Section5.2.4)of this paper. Interestingly,different from the above rule structures Gegov proposed to use Boolean matrices and binary relations to express fuzzy models[62].For example,given the following fuzzy rule base:Rule1:If x1is Small and x2is Small then y is NegativeRule2:If x1is Small and x2is Big then y is PositiveRule3:If x1is Big and x2is Small then y is NegativeRule4:If x1is Big and x2is Big then y is ZeroIf Small=1,Big=2,Negative=1,Zero=2,Positive=3,then a Boolean relation and a binary relation that equivalently express the above fuzzy system individually are indicated in Table1and(8).{[and(11,1)]or[and(12,3)]or[and(21,1)]or[and(22,2)]}(8) Moreover,for the sake of inducing compact rule base a formal approach to manipulation of rule bases is suggested by performing operations on Boolean matrices and binary relations[62].Because Boolean transformations of rule baseS.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–31313095 Table1Boolean relation for a fuzzy systemInputs Outputs123 11100 12001 21100 22010can lead to more compact representations,Klose and Nurnberger presented an approach to building more expressive rules by performing Boolean transforming during and after learning from data[103].3.Definitions of fuzzy model interpretabilityAlthough interpretability issues in fuzzy system modelling and statistical system modelling have received much attention in recent years,there is no well-established definition about model interpretability.In this section,we review the existing efforts made by the researchers on addressing the connotations of model interpretability and propose a new framework for fuzzy model interpretability.3.1.Transparency and interpretabilityModel transparency is defined as a property that enables us to understand and analyse the influence of each system parameter on the system output[22,74,156].In connection with transparency in fuzzy modelling,another symbiotic term is interpretability mentioned by Jang et al.[86],in which some basic ideas were discussed about how to constrain the optimization of the adaptive-network-based fuzzy inference system(ANFIS)[87]to preserve the interpretability. In Riid’s opinion[142],transparency and interpretability do not match when used to characterize fuzzy systems. Interpretability is a property of fuzzy systems which exists by default,being established with linguistic rules and fuzzy sets associated with these rules.Transparency,on the other hand,is not a default property of fuzzy systems,but a measure of how valid or how reliable the linguistic interpretation of the system is.However,the meanings of these two concepts possess so much synonymity in describing system modelling.Particularly,in fuzzy modelling,it may not be reasonable to distinguish them,because the two terms had been used in parallel in traditional statistical system modelling[30,33,55,73,164,170],long before they were used in fuzzy system modelling[3,11,19,20,26].Transparency and interpretability share the same connotations according to the two definitions in practice.In this paper,the two terms are used in parallel with the same meaning in fuzzy modelling,unless otherwise stated.3.2.Formalized definitionsSome researchers have tried to give formalized definitions for interpretability or transparency of fuzzy models.Definition1(Bodenhofer and Bauer[19,20]).Consider a linguistic variable V=(N,T,X,G,S)and an index set I, where N is the name of the linguistic variable V,G is a grammar,T is the so-called term set,i.e.,the set of linguistic expressions resulting from G,X is the universe of discourse,and S is a T→F(X)mapping which defines the semantics, i.e.,a fuzzy set on X,of each linguistic expression in T.Let RE=(RE i)i∈I be a family of relations on the set of verbal values T,where each relation RE i has afinite entry a i.Assume that,for every relation RE i,there exists a relation Q i on the fuzzy power set F(X)with the same entry.1Let Q represent the family(Q i)i∈I.Then the linguistic variable V is called R-Q-interpretable if and only if the following holds for all i∈I and all x1,...,x a i∈T: RE i(x1,...,x a i)⇒Q i(s(x1),...,s(x a i))(9) where⇒represent an implication relation.1Qi is associated with the“semantic counterpart’’of RE i,i.e.,the relation that models RE i on the semantic level.3096S.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–3131Riid and Rustern gave a formalized definition for transparency of linguistic fuzzy model as follows.Definition2(Riid[142],Riid et al.[143]).The i th rule of the linguistic fuzzy system(2)is transparent if its activation degree satisfiesr i(x)=T j i,j(x j)=1(10) and results in system output as y=y i,where y i is the centre of the output MF B i(y)associated with the activated rule. In such a way,to define the transparency of1-order TS fuzzy system(3)and(4)is much more complicated[142–144]. Riid et al.suggested that thefirst order TS fuzzy system is transparent if the global output y can be derived directly on the basis of its local output y i,and they proposed a measure for transparency evaluation,but this measure results in further problems[143],i.e.,the minimum transparency error is obtained for systems that are non-fuzzy,and there is no trivial way for preserving this transparency for thefirst order TS fuzzy system.Unlike the above two formalized definitions,Nauck suggested an index to measure the interpretability of fuzzy rule bases for classification problems in terms of complexity,the degree of coverage of fuzzy partition over the domain,and a partition index that penalizes partitions with a high granularity[124].However,this definition only considers special circumstances of model construction for classification instead of prediction.More importantly,it ignores many other aspects of interpretable fuzzy models,such as distinguishability,rule base simplicity,etc.In interpretable fuzzy modelling,it might be not feasible to give a formalized definition for transparency of fuzzy models in practice[28],because the interpretability of fuzzy models heavily depends on human’s prior knowledge [58,59,198].To decide whether a specific fuzzy model is interpretable or not is a highly subjective task and it is contro-versial sometimes[38,121].Some researchers have become aware of this matter and proposed some constraint criteria that fuzzy models should meet to assure good interpretability during model adaptation or optimization[45,67,135–139].3.3.A framework for fuzzy model interpretabilityIn order to clearly discriminate the different roles of fuzzy sets,input variables and other components in achieving an interpretable fuzzy model,we propose in this paper a taxonomy of fuzzy model interpretability:low-level interpretability and high-level interpretability.First let us review fuzzy models from the perspective of statistical system modelling.3.3.1.Review of fuzzy models from the perspective of traditional statistical system modellingAtfirst sight,fuzzy modelling algorithms for the purpose of improving interpretability may seem rather strange and hardly related to the existing methods of traditional statistical parsimonious system modelling techniques.Once a fuzzy model is cast into a more standard mathematical notation,we will observe its connections to the traditional statistical system modelling methods.In the following,we consider the TS fuzzy models.However,depending on the defuzzification method,the Mamdani models can also be expressed in the similar manner,i.e.,as linear-in-parameters models for regression problems.Eq.(4)can be rewritten in a slightly different form,y=Li=1p i(x)(a0i+a1i x1+···+a ni x n)(11)wherep i(x)=r i(x)Li=1r i(x)(12)is the normalizedfiring strength of the i th rule.The great advantage of the fuzzy model is its representative power,that is,to describe a highly nonlinear system by using simple local models(rules).On the other hand,Eq.(11)can also be viewed as a linear-in-parameters model,y(x, )=Li=1w T i(x) i(13)S.-M.Zhou,J.Q.Gan/Fuzzy Sets and Systems159(2008)3091–31313097 where w i(x)=[p i(x),p i(x)x T]T∈R n+1and i=[a0i,a1i,...,a ni]T∈R n+1.If basis functions w i(x)arefixed, parameterization of(13)becomes linear with respect to parameters i.Given N input–output data sequences{x(k),y(k)}, k=1,...,N,(13)can be expressed in the matrix form,Y=W + (14) where Y=[y(1),y(2),...,y(N)]T∈R N,W=[w(1),w(2),...,w(L)]∈R N×(n+1)L,with w(i)=[w i(x(1)), w i(x(2)),...,w i(x(N))]T∈R N×(n+1),which is calledfiring strength matrix with each column corresponding to one of the fuzzy rules, =[ 1, 2,..., L]T∈R(n+1)L,and represents model error.Thus,the linear-in-parameters model can be addressed further from the angle of statistical regression analysis.According to the taxonomy developed in[33]to categorize methods for estimating continuous-valued functions from noisy samples,(13)is a dictionary representation in nature,comparing to kernel representation in which a continuous-valued function estimator is expressed as a distance-weighted combinations of observed output values.The set of w i(x)is called a dictionary.The goal of a predictive learning system that employs(13)for function regression is to adjust the degrees of freedom in(13)based on training samples,in such a way,the approximation function provides minimum prediction risk.In case the transparency of the fuzzy model with(13)and(14)is considered,the problem is reduced to the automatic selection of the dictionary from observed data accounting for the interpretability of the rule base.Thereupon,many statistical regression techniques,such as regularization and sparse regression,can be used or extended to attack this problem.However,one fundamental problem is that as the number of predictors in the regression function increases the curse of dimensionality applies.On the other hand,when viewing regression model(13)as a function estimation problem,one attractive approach for ameliorating the curse of dimensionality is to model the regression function as an additive function of the predictors [164,177].This approach has been popularized by Hastie and Tibshirani[75],who emphasized the use of back-fitting together with a one-dimensional smoother tofit the additive models to data as follows:f(x1,...,x n)=f0+ni=1f i(x i)(15)where f i are“smooth’’functions obtained by some smoothing processes,including the use of cubic smoothing splines. More general models for f,which allow the explicit modelling and visualization of possible interactions between variables,are expressed via functional analysis of variance decompositions,that is,the output can be represented by the ANalysis Of V Ariance(ANOV A)decomposition[177,178].An often used special ANOV A decomposition is expressed as follows:f(x)=f0+ni=1f i(x i)+ni=1nj=i+1f i,j(x i,x j)+···+f1,2,...,n(x)(16)where f0is a constant,f i are the x i main effects,f i,j are the effects from interactions between x i and x j,and so on.The regression function is thus represented as additive components by the subset of the terms from this expansion,which shows the attraction of decomposing the model into simpler and more transparent pieces that can be easily interpreted. Another statistical system modelling technique that can produce a parsimonious model structure is the method of orthogonal-least squares(OLS)[30]when applied to a linear-in-parameters model(14).The OLS method transforms the columns of the coefficient matrix W into a set of orthogonal basis vectors.The Gram–Schmidt orthogonalization procedure is used to perform this decomposition,i.e.,W=UA,where U is an orthogonal matrix such that U T U=I(I is the unity matrix)and A is an upper-triangular matrix with unity diagonal elements.Substituting W=UA into(14), we haveY=U A + =U z+ (17) where z=A .Let u i be the i th column of U,so the OLS solution of the system isˆz i=u T i Yu i u i(i=1,...,L)(18)。

第18卷 第6期 太赫兹科学与电子信息学报Vo1.18，No.6 2020年12月 Journal of Terahertz Science and Electronic Information Technology Dec.，2020 文章编号：2095-4980(2020)06-0984-08卫星S频段下行链路频谱占用建模与预测刘稳1，洪涛1，王忠2，张更新1(1.南京邮电大学通信与信息工程学院，江苏南京 210003；2.中国人民解放军 31006部队，北京 100840)摘 要：目前提出的频谱占用模型能够在时域上描述和重现基本的统计特征，如传统的地面移动通信的频谱占用/空闲周期长度可以用经典的广义帕累托(GP)分布、指数分布等分布来拟合。

然而在某些复杂的如卫星链路频谱占用场景中，传统的参数估计分布无法给出良好的拟合。

为此提出了用核密度估计(KDE)的方法来进行概率密度分布的拟合，在此基础上，分别采用差分整合移动平均自回归模型(ARIMA)和模糊神经网络对频谱占用模型的时间序列进行预测并进行对比。

结论表明，核密度估计的使用可以更加准确地描述并再现卫星下行链路所使用S频段的占用时间序列的统计特征，而模糊神经网络的预测比ARIMA模型预测更加精确。

关键词：频谱占用模型；概率密度分布；核密度估计；差分整合移动平均自回归模型预测；模糊神经网络预测中图分类号：TN129文献标志码：A doi：10.11805/TKYDA2019270Modeling and prediction of time series for S-band spectrum use in satellite downlinkLIU Wen1，HONG Tao1，WANG Zhong2，ZHANG Gengxin1(1.Institute of Communication and Information Engineering，Nanjing University of Posts and Telecommunications，Nanjing Jiangsu 210003，China；2.Unit 31006 of PLA，Beijing 100840，China)Abstract：The development of the Cognitive Radio(CR) technology has benefited from the availability of realistic and accurate spectrum occupancy models. The spectrum occupancy models proposed in theliteratures so far are able to capture and reproduce the statistical characteristics of occupied time series.For example, the busy/idle-period lengths of terrestrial wireless network can be fitted by GeneralizedPareto(GP) distribution, exponential distribution, etc. However, the traditional parameter estimationdistribution cannot give a good fit in satellite link spectrum occupancy. In this context, a method of KernelDensity Estimation(KDE) is proposed to fit the probability density distribution. On this basis, the AutoRegressive Integrated Moving Average Model(ARIMA) and fuzzy neural network are adopted to predict andcompare the time series of the spectrum occupancy models. The conclusion shows that the proposedmethod can describe and reproduce the statistical characteristics of the occupied time series of the S-bandused in the satellite downlink more accurately; while the prediction of the fuzzy neural network is moreaccurate than that of the ARIMA model.Keywords：spectrum occupancy model；probability density distribution；kernel density estimation；Auto Regressive Integrated Moving Average Model(ARIMA) forecast；fuzzy neural network prediction近年来，随着移动通信技术的高速发展，第5代移动通信系统也将进入商用阶段。

Generalized Aggregation Operators for Intuitionistic Fuzzy SetsHua Zhao,1,2Zeshui Xu,2∗Mingfang Ni,1Shousheng Liu21Institute of Communications Engineering,PLA University of Science and Technology,Nanjing210007,People’s Republic of China2Institute of Sciences,PLA University of Science and Technology,Nanjing 210007,People’s Republic of ChinaThe generalized ordered weighted averaging(GOW A)operators are a new class of operators, which were introduced by Yager(Fuzzy Optim Decision Making2004;3:93–107).However,it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information.In this paper,wefirst develop some new general-ized aggregation operators,such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator,generalized intuitionistic fuzzy hybrid averaging operator,generalized interval-valued intuitionistic fuzzy weighted averag-ing operator,generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval-valued intuitionistic fuzzy hybrid average operator,which extend the GOW A operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function,and interval-valued intuitionistic fuzzy sets,whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers,and study their properties.Then,we apply them to multiple attribute decision making with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information.C 2009Wiley Periodicals,Inc.1.INTRODUCTIONSince its appearance in1965,1the fuzzy set theory has been widely used in manyfields in our modern society.Atanassov2generalized fuzzy set to intuitionistic fuzzy set(IFS),which is a powerful tool to deal with vagueness.3A prominent characteristic of IFS is that it assigns to each element a membership degree and a nonmembership degree.Gau and Buehrer4introduced the concept of vague set.Chen and Tan5and Hong and Choi6presented some techniques for handling multicriteria fuzzy decision-making problems based on vague sets.But Bustince and Burillo7 showed that vague sets are intuitionistic fuzzy sets.In the past two decades,many authors have paid attention to the IFS theory.For example,Atanassov and Gargov8∗Author to whom all correspondence should be addressed:e-mail:xu zeshui@ INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS,VOL.25,1–30(2010)C 2009Wiley Periodicals,Inc.Published online in Wiley InterScience().•DOI10.1002/int.203862ZHAO ET AL.defined the notion of interval-valued intuitionistic fuzzy set,which is characterized by a membership function and a nonmembership function whose values are inter-vals rather than exact numbers.Interval-valued intuitionistic fuzzy set(IVIFS)is a generalization of the notion of IFS in the spirit of interval-valued fuzzy set.9They also showed that IFSs are equivalent to interval-valued fuzzy sets.Up to now,the research on IFSs and IVIFSs is only about their basic theory and applications,but there is little research on the aggregation method of intuitionistic fuzzy or interval-valued intuitionistic fuzzy information.10,11So it is necessary to investigate such questions.The research on aggregation operators is an interesting topic,which has received increasing attention in a number of research papers12–14and also in other papers.Yager15extended the ordered weighted averaging(OWA)operator,16which has been used in many applications to provide a new class of operators called the generalized OWA(GOWA)operators.These operators add to the OWA operator an additional parameter controlling the power to which the argument values are raised. At the same time,he proved that the GOWA operators are mean operators.Yet it is worthy of pointing out that the GOWA operators have not been extended to ac-commodate intuitionistic fuzzy or interval-valued intuitionistic fuzzy environment. However,in many real-life problems,we need to aggregate the given intuitionistic fuzzy or interval-valued intuitionistic fuzzy arguments into a single one.In such cases,information fusion techniques are necessary.In this paper,based on the GOWA operators we will introduce some new in-tuitionistic fuzzy aggregation operators,such as generalized intuitionistic fuzzy weighted averaging(GIFWA)operator,generalized intuitionistic fuzzy ordered weighted average(GIFOWA)operator,and generalized intuitionistic fuzzy hybrid average(GIFHA)operator.Then,we will extend them to the interval-valued intu-itionistic fuzzy environments and study the properties of all the above-mentioned aggregation operators.Furthermore,we apply them to multiple attribute decision making with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information.2.PRELIMINARIESAtanassov2generalized the concept of fuzzy set1and defined the concept of IFS as follows:Let a set X befixed.An IFS A in X is an object having the form:A={ x,μA(x),νA(x) |x∈X}(1)where the functionsμA:X→[0,1]and v A:X→[0,1]define the degree of membership and the degree of nonmembership of the element x∈X to A,respec-tively,and for every x∈X:0≤μA(x)+νA(x)≤1(2)For each IFS A in X,ifπA(x)=1−μA(x)−νA(x),for all x∈X(3)thenπA(x)is called the degree of indeterminacy of x to A.International Journal of Intelligent Systems DOI10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS3For computational convenience,Xu 10called α=(μα,να)an intuitionistic fuzzy value (IFV).The IFV α=(μα,να)has a physical interpretation,for ex-ample,if (μα,να)=(0.3,0.2),then it can be interpreted as “the vote for resolution is 3in favor,2against,and 5abstentions”.4D EFINITION 1.10,11Let α=(μα,να),α1=(μα1,να1),and α2=(μα2,να2)be three IFVs,then the following operational laws are valid:1.α1⊕α2=(μα1+μα2−μα1μα2,να1να2);2.α1⊗α2=(μα1μα2,να1+να2−να1να2);3.λα=(1−(1−μα)λ,νλα),λ>0;4.αλ=(μλα,1−(1−να)λ),λ>0.T HEOREM 1.10,11Let α=(μα,να),α1=(μα1,να1),and α2=(μα2,να2)be threeIFVs,and let ˙α1=α1⊕α2,˙α2=α1⊗α2,˙α3=λα,˙α4=αλ,λ>0,then all ˙αi (i =1,2,3,4)are IFVs.By the operational laws in Definition 1,we haveT HEOREM 2.10,11Let α=(μα,να),α1=(μα1,να1),α2=(μα2,να2),and α3=(μα3,να3)be four IFVs,λ,λ1,λ2>0then1.λ1α⊕λ2α=(λ1+λ2)α;2.(α1⊕α2)⊕α3=α1⊕(α2⊕α3);3.((α)λ1)λ2=(α)λ1λ2.Based on the IFVs,Chen and Tan 5introduced a score function s to evaluate the degree of suitability that an alternative satisfies a decision maker’s requirement.Let α=(μα,να)be an IFV ,whereμα∈[0,1],να∈[0,1],μα+να≤1(4)The score of αcan be evaluated by the score function s shown ass (α)=μα−να(5)where s (α)∈[−1,1].The function s is used to measure the score of an IFV .From (5),we know that the score of αis directly related to the deviation between μαand να,i.e.,the higher the degree of deviation between μαand να,the bigger the score of α,thus,the larger the IFV α.On the basis of the minimum and maximum operations,Chen and Tan 5utilized the score function to develop a technique for handling multiple attribute decision-making problems in which the characteristics of alternatives are represented by ter,Hong and Choi 6defined an accuracy function to evaluate the degree ofInternational Journal of Intelligent SystemsDOI 10.1002/int4ZHAO ET AL.accuracy of the IFV α=(μα,να)ash (α)=μα+να(6)where h (α)∈[0,1].The larger the value of h (α),the higher the degree of accuracy of the degree of membership of the IFV α.Then,Hong and Choi 6utilized the score function,the accuracy function,and the minimum and maximum operations 5to develop another technique for handling multiple attribute decision-making problems based on intuitionistic fuzzy information.As presented above,the score function s and the accuracy function h are,respectively,defined as the difference and the sum of the membership function μand the nonmembership function ν.Hong and Choi 6showed that the relation between the score function s and the accuracy function h is similar to the relation between mean and variance in statistics.On the basis of the score function s and the accuracy function h ,in the following,Xu and Yager 11gave an order relation between two IFVs,which is defined as follows:D EFINITION 2.11Let α=(μα,να)and β=(μβ,νβ)be two IFVs,s (α)=μα−ναand s (β)=μβ−νβbe the scores of αand β,respectively,and h (α)=μα+ναand h (β)=μβ+νβbe the accuracy degrees of αand β,respectively,then1.If s (α)<s (β),then αis smaller than β,denoted by α<β;2.If s (α)=s (β),theni.If h (α)=h (β),then αand βrepresent the same information,denoted by α=β;ii.If h (α)<h (β),then αis smaller than β,denoted by α<β.On the basis of the GOWA operator and Definitions 1and 2,we have introduced some new aggregation operators.3.THE GIFWA,GIFOWA,AND GIFHA OPERATORS3.1.The GWA and GOWA OperatorsD EFINITION 3.A generalized weighted averaging (GWA)operator of dimension n is a mapping GWA:(R +)n →R +,which has the following form:GWA (a 1,a 2,...,a n )=⎛⎝n j =1w j a λj⎞⎠1/λ(7)where λ>0,w =(w 1,w 2,...,w n )T is the weight vector of the arguments a j (j =1,2,...,n ),with w j ≥0,j =1,2,...,n ,and n j =1w j =1,and R +is the set of all nonnegative real numbers.Another aggregation operator called the GOWA operators is the generalization of the OWA operator.International Journal of Intelligent SystemsDOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS5D EFINITION 4.15A generalized ordered weighted averaging (GOWA)operator of dimension n is a mapping GOWA:I n →I ,which has the following form:GOWA (a 1,a 2,...,a n )=⎛⎝n j =1w j b λj⎞⎠1/λ(8)where λ>0,w =(w 1,w 2,...,w n )T is the weight vector of (a 1,a 2,...,a n ),w j ≥0,j =1,2,...,n ,and n j =1w j =1,b j is the jth largest of a i ,I =[0,1].When we use different choices of the parameters λand w ,we will get some special cases.Some of the special operators have been used in situations where the input arguments are IFVs,such as the weighted averaging (WA)operator,the ordered weighted averaging (OWA)operator,10the weighted geometric (WG)operator,the ordered weighted geometric (OWG)operator.11But there are still a large number of special operators that have not been extended to the situations where the input arguments are intuitionistic fuzzy or interval-valued intuitionistic fuzzy ones.In the following section,we will extend the GWA and GOWA operators to accommodate the situations where the input arguments are intuitionistic fuzzy or interval-valued intuitionistic fuzzy ones.3.2.The GIFWA OperatorFor convenience,let V be the set of all IFVs.D EFINITION 5.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs and GIFWA:V n →V ,ifGIFWA w (α1,α2,...,αn )=w 1αλ1⊕w 2αλ2⊕···⊕w n αλn1/λ(9)then the function GIFWA is called a GIFWA operator,where λ>0,w =(w 1,w 2,...,w n )T is a weight vector associated with the GIFWA operator,with w j ≥0,j =1,2,...,n ,and nj =1w j =1.T HEOREM 3.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,then their aggregated value by using the GIFWA operator is also an IFV ,andGIFWA w (α1,α2,...,αn )=⎛⎜⎝⎛⎝1−n j =11−μλαjw j⎞⎠1/λ,1−⎛⎝1−n j =11−(1−ναj )λ w j ⎞⎠1/λ⎞⎟⎠(10)International Journal of Intelligent Systems DOI 10.1002/int6ZHAO ET AL.where λ>0,w =(w 1,w 2,...,w n )T is a weight vector associated with the GIFWA operator,with w j ≥0,j =1,2,...,n ,and n j =1w j =1.Proof.The first result follows quickly from Definition 3and Theorem 1.In the following,we first provew 1αλ1⊕w 2αλ2⊕···⊕w n αλn=⎛⎝1−n j =11−μλαjw j,n j =11−(1−ναj )λ wj ⎞⎠(11)by using mathematical induction on n :1.For n =2:sinceαλ1= μλα1,1−(1−να1)λ ,αλ2= μλα2,1−(1−να2)λ thenw 1αλ1⊕w 2αλ2=⎛⎝1−2 j =11−μλαjw j ,2 j =11−(1−ναj )λ w j⎞⎠2.If Equation 11holds for n =k ,that isw 1αλ1⊕w 2αλ2⊕···⊕w k αλk =⎛⎝1−k j =11−μλαjw j ,k j =11−(1−ναj )λ w j⎞⎠then,when n =k +1,by the operational laws (1),(2),and (4)in Definition 1,we havew 1αλ1⊕w 2αλ2⊕···⊕w k +1αλk +1=⎛⎝1−k j =11−μλαjw j ,k j =11−(1−ναj )λ w j ⎞⎠⊕ 1− 1−μλαk +1w k +1, 1−(1−ναk +1)λ w k +1 =⎛⎝1−k +1 j =11−μλαjw j ,k +1 j =11−(1−ναj )λ wj ⎞⎠International Journal of Intelligent Systems DOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS 7i.e.Equation 11holds for n =k +1.Thus,Equation 11holds for all n .ThenGIFW A w (α1,α2,...,αn )=⎛⎝1−n j =11−μλαjw j ,n j =11−(1−ναj )λ w j⎞⎠1/λ=⎛⎝⎛⎝1−n j =11−μλαjw j ⎞⎠1/λ,1−⎛⎝1−nj =11−(1−ναj )λ w j ⎞⎠1/λ⎞⎠.Example 1.Let α1=(0.1,0.7),α2=(0.4,0.3),α3=(0.6,0.1),and α4=(0.2,0.5)be four IFVs,w =(0.2,0.3,0.1,0.4)T be the weight vector of αj (j =1,2,3,4),and be λ=2,thenμα1=0.1,μα2=0.4,μα3=0.6,μα4=0.2να1=0.7,να2=0.3,να3=0.1,να4=0.5ThusGIFWA w (α1,α2,α3,α4)=(0.3381,0.3717).On the basis of Theorem 2,we have the following properties of the GIFWA operators:T HEOREM 4.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,λ>0,and w =(w 1,w 2,...,w n )T be the weight vector associated with the GIFWA op-erator,with w j ≥0,j =1,2,...,n ,and n j =1w j =1.If all αj (j =1,2,...,n )are equal,i.e.αj =α,for all j ,thenGIFWA w (α1,α2,...,αn )=α.Proof.By Theorem 2,we haveGIFWA w (α1,α2,...,αn )=(w 1αλ1⊕w 2αλ2⊕···⊕w n αλn )1/λ=(w 1αλ⊕w 2αλ⊕···⊕w n αλ)1/λ=((w 1+w 2+···+w n )αλ)1/λ=(αλ)1/λ=α.T HEOREM 5.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,λ>0,and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFWA operator,International Journal of Intelligent SystemsDOI 10.1002/int8ZHAO ET AL.with w j ≥0,j =1,2,...,n,n j =1w j =1,and letα−= min j(μαj ),max j(ναj ) ,α+= max j(μαj ),min j(ναj )Thenα−≤GIFWA w (α1,α2,...,αn )≤α+.(12)Proof.Since min j(μαj )≤μαj ≤max j(μαj )and min j(ναj )≤ναj ≤max j(ναj ),for all j ,thennj =1(1−μλαj )w j≥n j =11−(max j(μαj ))λw j=1−(max j(μαj ))λand then⎛⎝1−n j =11−μλαjw j⎞⎠1/λ≤max j(μαj )(13)Similarly,we have⎛⎝1−n j =1 1−μλαjw j⎞⎠1/λ≥min j(μαj )(14)n j =11−(1−ναj )λ wj ≤n j =11− 1−max j(ναj )λ w j =1− 1−max j(ναj ) λ1−n j =11−(1−ναj )λ wj ≥ 1−max j(ναj )λ⎛⎝1−n j =11−(1−ναj )λ wj ⎞⎠1/λ≥1−max(jναj )1−⎛⎝1−n j =11−(1−ναj )λ wj ⎞⎠1/λ≤max j(ναj ).(15)International Journal of Intelligent Systems DOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS9Similarly,we have1−⎛⎝1−n j =11−(1−ναj )λ wj ⎞⎠1/λ≥min j(ναj ).(16)Let GIFWA w (α1,α2,...,αn )=α=(μα,να),thens (α)=μα−να≤max j(μαj )−min j(ναj )=s (α+)s (α)=μα−να≥min j(μαj )−max j(ναj )=s (α−).If s (α)<s (α+)and s (α)>s (α−),then by Definition 2,we haveα−<GIFWA w (α1,α2,...,αn )<α+.(17)If s (α)=s (α+),i.e.μα−να=max j(μαj )−min j(ναj ),then by (13)and (16),we haveμα=max j(μαj ),να=min j(ναj ).Soh (α)=μα+να=max j(μαj )+min j(ναj )=h (α+).Then by Definition 2,we haveGIFWA w (α1,α2,...,αn )=α+.(18)If s (α)=s (α−),i.e.μα−να=min j(μαj )−max j(ναj ),then by (14)and (15),we haveμα=min j(μαj ),να=max j(ναj ).Soh (α)=μα+να=min j(μαj )+max j(ναj )=h (α−).Thus,from Definition 2,it follows thatGIFWA w (α1,α2,...,αn )=α−(19)and then from Equations 17–19,we know that Equation.12always holds.International Journal of Intelligent SystemsDOI 10.1002/int10ZHAO ET AL.T HEOREM 6.Let αj =(μαj ,ναj )(j =1,2,...,n )and α∗j =(μα∗j ,να∗j )(j =1,2,...,n )be two collections of IFVs and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFWA operator,where w j ≥0,j =1,2,...,n ,and nj =1w j =1,λ>0,if μαj ≤μα∗j and ναj ≥να∗j ,for all j ,thenGIFWA w (α1,α2,...,αn )≤GIFWA w (α∗1,α∗2,...,α∗n ).(20)Proof.Since μαj ≤μα∗j and ναj ≥να∗j ,for all j ,thenn j =11−μλαj w j ≥n j =11−μλα∗j w j1−n j =11−μλαj w j≤1−n j =11−μλα∗jw j ⎛⎝1−n j =11−μλαjw j⎞⎠1/λ≤⎛⎝1−n j =11−μλα∗jw j⎞⎠1/λn j =11−(1−ναj )λ wj ≥n j =11−(1−να∗j )λ w j 1−n j =11−(1−ναj )λ w j ≤1−n j =11−(1−να∗j )λ w j ⎛⎝1−n j =11−(1−ναj )λ w j ⎞⎠1/λ≤⎛⎝1−n j =11−(1−να∗j )λ wj ⎞⎠1/λ1−⎛⎝1−n j =11−(1−ναj )λ wj ⎞⎠1/λ≥1−⎛⎝1−n j =11−(1−να∗j )λ wj ⎞⎠1/λInternational Journal of Intelligent Systems DOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS11⎛⎝1−nj=11−μλαjwj⎞⎠1/λ−⎛⎜⎝1−⎛⎝1−nj=11−(1−ναj)λwj⎞⎠1/λ⎞⎟⎠≤⎛⎝1−nj=11−μλα∗jwj⎞⎠1/λ−⎛⎜⎝1−⎛⎝1−nj=11−(1−να∗j)λwj⎞⎠1/λ⎞⎟⎠.(21) Letα=GIFWA w(α1,α2,...,αn),α∗=GIFWA w(α∗1,α∗2,...,α∗n),then byEquation21,we haves(α)≤s(α∗).If s(α)<s(α∗),then by Definition2,we haveGIFWA w(α1,α2,...,αn)<GIFWA w(α∗1,α∗2,...,α∗n)(22) If s(α)=s(α∗),then⎛⎝1−nj=11−μλαjwj⎞⎠1/λ−⎛⎜⎝1−⎛⎝1−nj=11−(1−ναj)λwj⎞⎠1/λ⎞⎟⎠=⎛⎝1−nj=11−μλα∗jwj⎞⎠1/λ−⎛⎜⎝1−⎛⎝1−nj=11−(1−να∗j)λwj⎞⎠1/λ⎞⎟⎠.Sinceμαj ≤μα∗jandναj≥να∗j,for all j,then we have⎛⎝1−nj=11−μλαjwj⎞⎠1/λ=⎛⎝1−nj=11−μλα∗jwj⎞⎠1/λ1−⎛⎝1−nj=11−(1−ναj)λwj⎞⎠1/λ=1−⎛⎝1−nj=11−(1−να∗j)λwj⎞⎠1/λ.Henceh(α)=⎛⎝1−nj=11−μλαjwj⎞⎠1/λ+⎛⎜⎝1−⎛⎝1−nj=11−(1−ναj)λwj⎞⎠1/λ⎞⎟⎠International Journal of Intelligent Systems DOI10.1002/int12ZHAO ET AL.=⎛⎝1−n j =11−μλα∗jw j⎞⎠1/λ+⎛⎜⎝1−⎛⎝1−n j =11−(1−να∗j )λ w j ⎞⎠1/λ⎞⎟⎠=h (α∗)thus,by Definition 2,we haveGIFWA w (α1,α2,...,αn )=GIFWA w (α∗1,α∗2,...,α∗n ).(23)From Equations 22and 23,we know that Equation 20always holds.We now look at some special cases obtained by using different choices of theparameters w and λ.T HEOREM 7.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,λ>0,and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFWA operator with w j ≥0(j =1,2,...,n ),and n j =1w j =1,then1.If λ=1,then the GIFWA operator (9)is reduced to the following :IFW A w (α1,α2,...,αn )=w 1α1⊕w 2α2⊕···⊕w n αn ,which is called an intuitionistic fuzzy weighted average operator .102.If λ→0,then the GIFWA operator (9)is reduced to the following :IFWG w (α1,α2,...,αn )=αw 11⊗αw 22⊗···⊗αw nn ,which is called an intuitionistic fuzzy weighted geometric operator .113.If λ→+∞,then the GIFWA operator (9)is reduced to the following :IFMAX w (α1,α2,...,αn )=max j(αj ),which is called an intuitionistic fuzzy maximum operator .54.If w =(1/n,1/n,...,1/n )T and λ=1,then the GIFWA operator (9)is reduced to the following :IFA w (α1,α2,...,αn )=1n(α1⊕α2⊕···⊕αn ),which is called an intuitionistic fuzzy average operator .105.If w =(1/n,1/n,...,1/n )T and λ→0,then the GIFWA operator (9)is reduced to the following :IFG w (α1,α2,...,αn )=(α1⊗α2⊗···⊗αn )1/n ,which is called an intuitionistic fuzzy geometric operator .11International Journal of Intelligent SystemsDOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS133.3.The GIFOWA OperatorD EFINITION 6.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,and let GIFOWA:V n →V ,ifGIFOWA w (α1,α2,...,αn )= w 1αλσ(1)⊕w 2αλσ(2)⊕···⊕w n αλσ(n ) 1/λ(24)where λ>0,w =(w 1,w 2,···,w n )T is an associated weight vector such that w j ≥0,j =1,2,...,n ,and n j =1w j =1,ασ(j )is the j th largest of αj ,then the function GIFOWA is called a GIFOWA operator.The GIFOWA operator has some properties similar to those of the GIFWA operator.T HEOREM 8.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,then their aggregated value by using the GIFOWA operator is also an IFV ,and GIFOWA w (α1,α2,...,αn )=⎛⎜⎝⎛⎝1−n j =11−μλασ(j ) w j ⎞⎠1/λ,1−⎛⎝1−n j =11−(1−νασ(j ))λ w j ⎞⎠1/λ⎞⎟⎠(25)where λ>0,w =(w 1,w 2,···,w n )T is an weight vector related to the GIFOWA operator such that w j ≥0,j =1,2,...,n ,and n j =1w j =1,ασ(j )is the j th largest of αj .Example 2.Let α1=(0.3,0.6),α2=(0.4,0.5),α3=(0.6,0.3),α4=(0.7,0.1),and α5=(0.1,0.6)be five IFVs,and w =(0.1117,0.2365,0.3036,0.2365,0.1117)T be the weight vector of αj (j =1,2,3,4,5).Let λ=2,thenμα1=0.3,μα2=0.4,μα3=0.6,μα4=0.7,μα5=0.1να1=0.6,να2=0.5,να3=0.3,να4=0.1,να5=0.6.Let us calculate the scores of αj (j =1,2,3,4,5):s (α1)=0.3−0.6=−0.3,s (α2)=0.4−0.5=−0.1,s (α3)=0.6−0.3=0.3s (α4)=0.7−0.1=0.6,s (α5)=0.1−0.6=−0.5.Sinces (α4)>s (α3)>s (α2)>s (α1)>s (α5)International Journal of Intelligent SystemsDOI 10.1002/int14ZHAO ET AL.thenασ(1)=(0.7,0.1),ασ(2)=(0.6,0.3),ασ(3)=(0.4,0.5)ασ(4)=(0.3,0.6)ασ(5)=(0.1,0.6)and thus,by Equation 25,we haveGIFOWA w (α1,α2,α3,α4,α5)=(0.4762,0.3762).T HEOREM 9.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,andw =(w 1,w 2,...,w n )T be the weight vector related to the GIFOWA operator,with w j ≥0,j =1,2,...,n ,andn j =1w j =1.If all αj (j =1,2,...,n )are equal,i.e.αj =α,for all j ,thenGIFOWA w (α1,α2...,αn )=α.T HEOREM 10.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,andw =(w 1,w 2,...,w n )T be the weight vector related to the GIFOWA operator,with w j ≥0,j =1,2,...,n ,andn j =1w j =1,and letα−=(min j(μαj ),max j(ναj )),α+=(max j(μαj ),min j(ναj ))thenα−≤GIFOWA w (α1,α2...,αn )≤α+.T HEOREM 11.Let αj =(μαj ,ναj )(j =1,2,...,n )and α∗j =(μα∗j ,να∗j )(j =1,2,...,n )be two collections of IFVs and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFOWA operator,with w j ≥0,j =1,2,...,n ,and nj =1w j =1.If μαj ≤μα∗j and ναj ≥να∗j ,for all j ,thenGIFOWA w (α1,α2...,αn )≤GIFOWA w (α∗1,α∗2...,α∗n ).T HEOREM 12.Let αj =(μαj ,ναj )(j =1,2,...,n )and αj =(μα j ,να j )(j =1,2,...,n )be two collections of IFVs,λ>0,and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFOWA operator,with w j ≥0,j =1,2,...,n ,and nj =1w j =1,thenGIFOWA w (α1,α2...,αn )=GIFOWA w (α 1,α 2...,α n )(26)where (α 1,α 2...,α n )Tis any permutation of (α1,α2...,αn )T .International Journal of Intelligent Systems DOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS15Proof.LetGIFOWA w (α1,α2...,αn )= w 1αλσ(1)⊕w 2αλσ(2)⊕···⊕w n αλσ(n )1/λGIFOWA w (α 1,α 2...,α n )= w 1(α σ(1))λ⊕w 2(α σ(2))λ⊕···⊕w n (α σ(n ))λ 1/λ.Since (α 1,α 2...,α n )T is any permutation of (α1,α2...,αn )T ,thenασ(j )=ασ(j ),j =1,2,...,nthen Equation 26holds.From (26),we know that the GIFOWA operator has commutativity propertythat we desire.It is worth noting that the GIFWA operator does not have this property.We now look at some special cases obtained by using different choices of the parameters w and λ.T HEOREM 13.Let αj =(μαj ,ναj )(j =1,2,...,n )be a collection of IFVs,λ>0and w =(w 1,w 2,...,w n )T be the weight vector related to the GIFOWA operator,with w j ≥0,j =1,2,...,n,n j =1w j =1,then1.If λ=1,then the GIFOWA operator (24)is reduced to the following :IFOW A w (α1,α2,...,αn )=w 1ασ(1)⊕w 2ασ(2)⊕···⊕w n ασ(n ),which is called an intuitionistic fuzzy ordered weighted average operator .102.If λ→0,then the GIFOWA operator (24)is reduced to the following :IFOWG w (α1,α2,...,αn )=αw 1σ(1)⊗αw 2σ(2)⊗···⊗αw nσ(n ),which is called an intuitionistic fuzzy ordered weighted geometric operator .113.If λ→+∞,then the GIFOWA operator (24)is reduced to the following :IFMAX w (α1,α2,...,αn )=max j(αj ),which is called an intuitionistic fuzzy maximum operator .54.If w =(1/n,1/n,...,1/n )T and λ=1,then the GIFOWA operator (24)is reduced to the following :IFA w (α1,α2,...,αn )=1n(α1⊕α2⊕···⊕αn ),which is called an intuitionistic fuzzy average operator .10International Journal of Intelligent SystemsDOI 10.1002/int16ZHAO ET AL.5.If w =(1/n,1/n,...,1/n )T and λ→0,then the GIFOWA operator (24)is reduced to the following :IFG w (α1,α2,...,αn )=(α1⊗α2⊗···⊗αn )1/n ,which is called an intuitionistic fuzzy geometric operator .116.If w =(1,0,...,0)T ,then the GIFOWA operator (24)is reduced to the following :IFMAX w (α1,α2,...,αn )=max j(αj ),which is called an intuitionistic fuzzy maximum operator .57.If w =(0,0,...,1)T ,then the GIFOWA operator (24)is reduced to the following :IFMIN w (α1,α2,...,αn )=min j(αj ),which is called an intuitionistic fuzzy minimum operator .53.4.The GIFHA OperatorConsider that the GIFWA operator weighs only the IFVs,whereas the GIFOWA operator weighs only the ordered positions of the IFVs instead of weighing the IFVs themselves.To overcome this limitation,motivated by the idea of combining the WA and OWA operators,14,17in what follows,we developed a generalized intuitionistic fuzzy hybrid aggregation (GIFHA)operator,which weighs both the given IFV and its ordered position.D EFINITION 7.A GIFHA operator of dimension n is a mapping GIFHA:V n →V ,which has an associated vector w =(w 1,w 2,···,w n )T ,with w j ≥0,j =1,2,...,n ,and nj =1w j =1,such thatGIFHA w,ω(α1,α2...,αn )=(w 1(˙ασ(1))λ⊕w 2(˙ασ(2))λ⊕···⊕w n (˙ασ(n ))λ)1/λ(27)where λ>0,˙ασ(j )is the j th largest of the weighted IFVs ˙αj (˙αj =nωj αj ,j =1,2,...,n ),ω=(ω1,ω2,...,ωn )Tis the weight vector of αj (j =1,2,...,n )with ωj ≥0,and n j =1ωj =1,and n is the balancing coefficient,which plays a role of balance (in such a case,if the vector (ω1,ω2,...,ωn )T approaches (1/n,1/n,...,1/n )T ,then the vector (nω1α1,nω2α2,...,nωn αn )T approaches (α1,α2,...,αn )T ).Let ˙ασ(j )=(μ˙ασ(j ),ν˙ασ(j )),then,similar to Theorem 3,we have GIFHA w,ω(α1,α2,...,αn )=⎛⎜⎝⎛⎝1−n j =11−μλ˙ασ(j )w j ⎞⎠1/λ,1−⎛⎝1−n j =11−(1−ν˙ασ(j ))λ w j ⎞⎠1/λ⎞⎟⎠,(28)International Journal of Intelligent Systems DOI 10.1002/intAGGREGATION OPERATORS FOR INTUITIONISTIC FUZZY SETS17 and the aggregated value derived by using the GIFHA operator is also an IFV. Especially,ifλ=1,then(28)is reduced to the following form:IFHA w,ω(α1,α2,...,αn)=⎛⎝1−nj=1(1−μ˙ασ(j))w j,nj=1νw j˙ασ(j)⎞⎠,which is called an intuitionistic fuzzy hybrid-averaging(IFHA)operator.10T HEOREM14.The GIFOWA operator is a special case of the GIFHA operator. Proof.Letω=(1/n,1/n,...,1/n)T,then˙αj=αj(j=1,2,...,n),so we have GIFHA w,ω(α1,α2,...,αn)=(w1(˙ασ(1))λ⊕w2(˙ασ(2))λ⊕···⊕w n(˙ασ(n))λ)1/λ=(w1αλσ(1)⊕w2αλσ(2)⊕···⊕w nαλσ(n))1/λ=GIFOWA w(α1,α2,...,αn).This completes the proof of Theorem14.Example3.Letα1=(0.2,0.5),α2=(0.4,0.2),α3=(0.5,0.4),α4=(0.3,0.3), andα5=(0.7,0.1)befive IFVs,and letω=(0.25,0.20,0.15,0.18,0.22)T be the weight vector ofαj(j=1,2,3,4,5),then by the operational law(3)in Definition1, we get the weighted intuitionistic fuzzy values as˙α1=(0.234,0.42),˙α2=(0.4,0.2),˙α3=(0.405,0.503),˙α4=(0.275,0.338),˙α5=(0.734,0.079).By Equation5,we calculate the scores of˙αj(j=1,2,3,4,5):s(˙α1)=−0.177,s(˙α2)=0.2,s(˙α3)=−0.098,s(˙α4)=−0.063,s(˙α5)=0.655.Sinces(˙α5)>s(˙α2)>s(˙α4)>s(˙α3)>s(˙α1)then˙ασ(1)=(0.734,0.079),˙ασ(2)=(0.4,0.2),˙ασ(3)=(0.275,0.338)˙ασ(4)=(0.405,0.503),˙ασ(5)=(0.234,0.42).International Journal of Intelligent Systems DOI10.1002/int。

A fuzzy approach to construction project risk assessmentA.Nieto-Morote a,*,F.Ruz-Vila b,1a Project Engineering Department,Polytechnic University of Cartagena,c/Dr.Fleming,s/n,30202Cartagena,Spain bElectric Engineering Department,Polytechnic University of Cartagena,c/Dr.Fleming,s/n,30202Cartagena,SpainReceived 15July 2009;received in revised form 28January 2010;accepted 2February 2010AbstractThe increasing complexity and dynamism of construction projects have imposed substantialuncertainties and subjectivities in the risk analysis process.Most of the real-world risk analysis problems contain a mixture of quantitative and qualitative data;therefore quan-titative risk assessment techniques are inadequate for prioritizing risks.This article presents a risk assessment methodology based on the Fuzzy Sets Theory,which is an effective tool to deal with subjective judgement,and on the Analytic Hierarchy Process (AHP),which is used to structure a large number of risks.The proposed methodology incorporates knowledge and experience acquired from many experts,since they carry out the risks identification and their structuring,and also the subjective judgements of the parameters which are considered to assess the overall risk factor:risk impact,risk probability and risk discrimination.All of these factors are expressed by qualitative scales which are defined by trapezoidal fuzzy numbers to capture the vagueness in the linguistic variables.The most nota-ble differences with other fuzzy risk assessment methods are the use of an algorithm to handle the inconsistencies in the fuzzy preference relation when pair-wise comparison judgements are necessary,and the use of trapezoidal fuzzy numbers until the defuzzification step.An illustrative example on risk assessment of a rehabilitation project of a building is used to demonstrate the proposed methodology.Ó2010Elsevier Ltd and IPMA.All rights reserved.Keywords:Risk assessment;Linguistics variables;Trapezoidal fuzzy numbers;Risk factor1.IntroductionAccording to Mark et al.(2004),risk is simply the potential for complications and problems with respect to the completion of a project task and the achievement of a project goal.Risk is inherent in all project undertakings,as such it can never be fully eliminated,although can be effectively managed to mitigate the impacts to the achieve-ment of project’s goals.Other definitions of risk are available in the literature such as “the exposure the possibility of economic or financial loss or gain,physical damage or injury,or delay,as a consequence of the uncertainty associated with pursu-ing a particular course of action ”(Perry and Hayes,1985;Chapman and Ward,1997),“the probability of losses in aproject ”(Jaafari,2001;Kartam and Kartam,2001),“the likelihood of a detrimental event occurring to the project ”(Baloi and Price,2003);or “a barrier to success ”(Hertz and Thomas,1994).Although risk has been defined in var-ious ways,some common characteristics can be found (Chia,2006):A risk is a future event that may or may not occur. A risk must also be an uncertain event or condition that,if it occurs,has an effect on,at least,one of the project objectives,such as scope,schedule,cost or quality. The probability of the future event occurring must be greater than 0%but less than 100%.Future events that have a zero or 100%chance of occurrence are not risks. The impact or consequence of the future event must be unexpected or unplanned for.There are many different risk sources in the construction projects and some approaches have been suggested in the literature for classifying them.Some classifications are0263-7863/$36.00Ó2010Elsevier Ltd and IPMA.All rights reserved.doi:10.1016/j.ijproman.2010.02.002*Corresponding author.Tel.:+34968326551.E-mail addresses:Ana.nieto@upct.es (A.Nieto-Morote),Paco.ruz@upct.es (F.Ruz-Vila).1Tel.:+34968325351.International Journal of Project Management 29(2011)220–231focused on the risks nature and their magnitude(Cooper and Champan,1987)or on the risks origin(Edwards and Bowen,1998;Zhou et al.,2008).Other use a hierarchical structure of risks(Tah et al.,1993;Wirba et al.,1996)to classify risks according to their origin and to the location of the risk impact in the project.The increasing size and complexity of the construction projects have added risks to their execution.With the need for improved performance in construction project and increasing contractual obligations,the requirement of an effective risk management approach has never been more necessary.On the subject of risk management process,there have recently been a large number of researchers which have proposed different processes.Some of the most important approaches are:PRAM(Chapman,1997),RAMP(Institu-tion of Civil Engineering,2002),PMBOK(Project Man-agement Institute,2008),RMS(Institute of Risk Management,2002).Almost all of these approaches have a similar framework with differences in the established steps in order to get the risks control.Effective risk management involves a four-phase process:1.Risks identification:The process of determining whichrisks may affect the project and documenting their characteristics.2.Risk assessment:The process of prioritizing risks for fur-ther analysis by assessing and combining,generally, their probability of occurrence and impact.3.Risk response:The process of developing options andactions to enhance opportunities and to reduce threats to the project objectives.4.Risk monitoring and reviewing:The process of imple-a risk response plan,tracking identified risks,monitoring residual risks,identifying new risks,and evaluating the risk process effectiveness throughout the project.Risk project management is beneficial if it is imple-mented in a systematic manner from planning stage through the project completion.The unsystematic and arbitrary risk management can endanger the success of the project since most of the risks are very dynamic throughout the project lifetime.2.Fuzzy risk assessment procedureThe nature of construction project has imposed,in the risk analysis process,substantial uncertainties and subjec-tivities,which have hampered the applicability of many risk assessment methods,that are used widely in construc-tion projects and require high quality data,such as Fault Tree Analysis(FTA),Event Tree Analysis(ETA),Proba-bility and impact grids,Sensitivity Analysis,Estimation of System Reliability,Failure Mode and Effect Analysis (Ahmed et al.,2007).Recently,many risk assessment approaches have been based on using linguistic assessments instead of numerical ing Fuzzy Sets Theory(Zadeh,1965),data may be defined on vague,linguistic terms such as lowity,serious impact,or high risk.These terms cannot be defined meaningfully with a precise single value,but Fuzzy Sets Theory provides the means by which these terms may be formally defined in mathematical logic.Several research studies on the risk assessment of con-struction projects using fuzzy approaches have been per-formed.Some fuzzy proposals have been inspired in the classical risk assessment methods,such as,ETA and FTA:Fujino(1994)demonstrates the applicability of the proposed fuzzy FTA methodology to some cases of con-struction site accidents in Japan;Huang et al.(2001)pro-poses a fuzzy formal procedure in order to integrate both human-error and hardware failure events into a ETA meth-odology;Cho et al.(2002)proposes a fuzzy ETA method-ology characterized by the use of new forms of fuzzy membership curves.However,the research studies have not only been focused on using fuzzy concepts into conventional risk assessment frameworks,but rather new methods have been proposed.Carr and Tah(2001)define a formal model based on a hierarchical risk breakdown structure.The risks descriptions and their consequences are defined using lin-guistic variables and the relationship between the likeli-hood of occurrence(L),the severity(V)and the effect of a risk factor(E)is represented by rules such as“If L and V then E”.Zeng et al.(2007)propose a risk assessment model based on fuzzy reasoning and AHP approach.A modified analytical hierarchy process is used to structure and prioritize risks considering three fundamental risk parameters:risk likelihood(RL),risk severity(RS)and factor index(FI),defined all of them in terms of linguistic variables which are transformed into trapezoidal fuzzy numbers.The relations between input parameters FI,RL, RS and output named Risk magnitude(RM)are presented in form of“if...then”rules.Dikmen et al.(2007)propose a methodology for risk rating of international construction projects.Once the risks have been identified and modelled using Influence Diagrams,they are assessed by linguistic terms.The relationships between risks and influencing fac-tors are captured from the knowledge of experts by using ‘‘aggregation rules’’,where the risk knowledge is explained in form of“if...then”rules.The aggregation of fuzzy rules into a fuzzy cost overrun risk rating is carried out by fuzzy operations.Wang and Elhag(2007)proposes a risk assessment methodology which allows experts to eval-uate risk factors,in terms of likelihood and consequences, using linguistic terms.Also it is provided two alternative algorithms to aggregate the assessments of multiple risk factors,one of which offers a rapid assessment and the other one leads to an exact assessment.Zhang and Zou (2007)propose a methodology based on a hierarchical structure of risks associated with a construction project. Based on expert judgment,the weight coefficients of riskA.Nieto-Morote,F.Ruz-Vila/International Journal of Project Management29(2011)220–231221groups and risk factors are acquired with the aid of the AHP techniques and the fuzzy evaluation matrixes of risk factors.Then the aggregation of weight coefficients and fuzzy evaluation matrices produces the appraisal vector of risky conditions of the construction project.It can be affirmed that all the proposed fuzzy risk assess-ment methods have a common procedure(Lyons and Skit-more,2004):1.Definition and measurement of parameters:The funda-mental parameters,by which the risks associated witha project are assessed,are risk probability and riskseverity,although other parameters can be defined.The measurement of these parameters frequently is dif-ficult due to the great uncertainty involved.In these cases,the measurement of each parameter is made in vague data or linguistic terms and converted into its cor-responding fuzzy number.2.Definition of fuzzy inference:The relations between inputparameters and output parameters can be defined in form of“if-then”rules or in form of mathematical function defined by an appropriated fuzzy arithmetic operator. 3.Defuzzification:As the result of a fuzzy inference phaseis a fuzzy number,this step is used to convert the fuzzy result into a exact numerical value that can adequately represent it.In some risk assessment methodologies,some judge-ments are done by means of pair-wise comparisons(Zeng et al.,2007;Wang and Elhag,2007;Zhang and Zou, 2007).Preference information of alternatives generally pre-sents inconsistency problems.Although there are a large number of studies about the inconsistency of fuzzy prefer-ence relations(Dong et al.,2008;Ghazanfari and Nojavan, 2004);(Herrera-Viedma et al.,2004;Ma et al.,2006;Wang and Chen,2008),none of the proposed fuzzy risk assess-ment methodologies take into account the inconsistency of the judgements.This paper presents a fuzzy risk assess-ment model which most significant difference with other fuzzy risk assessment methods is the use of an algorithm to handle the inconsistencies in the fuzzy preference rela-tion when pair-wise comparison judgements are necessary.3.Fuzzy Sets TheoryThe Fuzzy Set Theory introduced by Zadeh(1965)is suitable for dealing with imprecision and uncertainty asso-ciated with data in risk assessment problems.In a universal set of discourse X,a fuzzy subset A of X is defined by a membership function l A(x),which maps each element x in X to a real number in the interval[0,1].The function value of l A(x)signifies the grade of membership of x in A.When l A(x)is large,its grade of membership of x in A is strong(Kaufmann and Gupta,1991).Among the various types of fuzzy sets of special signif-icance are fuzzy numbers(Dubois and Prade1978)defined as A={x,l A(x)}where x takes its number on the real line R and membership function l A:R!½0;1 ,which have the following properties:A continuous mapping from R to the closed interval[0,1].Constant on(À1,a]:l A(x)=0"x(À1,a].Strictly increasing on[a,b].Constant on[b,c]:l A(x)=1"x[b,c].Strictly decreasing on[c,d].Constant on[d,1):l A(x)=0"x[d,1).where a,b,c and d are real numbers and eventually a=À1,or b=c,or a=b,or c=d or d=1.For convenience,l LAis named as left membership func-tion of a fuzzy number A,defining l LAðxÞ¼l AðxÞ,for allx½a;b ;l RAis named as right membership function of a fuzzynumber A,defining l RAðxÞ¼l AðxÞ,for all x[c,d].A trapezoidal fuzzy number A is a fuzzy number denoted as A=(a,b,c,d)which membership function is defined as:l AðxÞ¼0for x<al LAðxÞ¼xÀa for a6x6b1for b6x6cl RAðxÞ¼xÀd for c6x6d0for x>d26666664ð1Þwhere a,b,c and d are real numbers and a<b<c<d.If b=c,it is defined a triangular fuzzy number.By the extension principle(Zadeh,1965),the fuzzy arithmetic operations of any two trapezoidal fuzzy num-bers follow these operational laws:Fuzzy addition:A1ÈA2¼ða1þa2;b1þb2;c1þc2;d1þd2Þð2ÞFuzzy subtraction:A1H A2¼ða1Àd2;b1Àc2;c1Àb2;d1Àa2Þð3ÞFuzzy multiplication:A1 A2%ða1Âa2;b1Âb2;c1Âc2;d1Âd2Þð4ÞFuzzy division:A1;A2%ða1=d2;b1=c2;c1=b2;d1=a2Þð5ÞThe fuzzy addition or the fuzzy subtraction of any two fuz-zy trapezoidal numbers is also a trapezoidal fuzzy number. But the fuzzy multiplication or the fuzzy division is only approximate a trapezoidal fuzzy number.The scalar multiplication of a trapezoidal fuzzy number is also a trapezoidal fuzzy number defined as:kÂA¼ðkÂa;kÂb;kÂc;kÂdÞif k>0kÂA¼ðkÂd;kÂc;kÂb;kÂaÞif k<0ð6Þ3.1.The algebraic operations of fuzzy numbers based on a-cut conceptThe a-cuts of a fuzzy number A with membership func-tion l A(x)are defined as the crisp set that contains all the222 A.Nieto-Morote,F.Ruz-Vila/International Journal of Project Management29(2011)220–231elements of R whose membership grades in A are greater or equal to the specified value of a:A a¼f x j l AðxÞP a;0P a P1gð7Þand denoted it by A al ;A arÂÃ,i.e.,A a¼½A al ;A ar.The multiplication and division operations on closedintervals,A a¼A al ;A arÂÃand B a¼B al ;B arÂÃ,are defined as (Klir and Yuan,1995):Multiplication:ðAÂBÞa¼½A al ÂB al;A arÂB arð8ÞDivision:ðAÄBÞa¼½A al =B ar;A ar=B alð9ÞAccording the extension principle(Zadeh,1965),an arbi-trary fuzzy set A can fully and uniquely be represented as: A¼[a2½0;1a A aðxÞð10Þwhere U denotes the standard fuzzy union and a aA denotesthe special fuzzy set which membership function is defined as:la A a ¼a for x2A a0for x R A a&ð11ÞTherefore,the multiplication and division operations of4.Proposed risk assessment modelA risk assessment model,based on fuzzy reasoning,is proposed as shown in Fig.1.The model consists of three steps:preliminary step,definition of risk factor function and measurement of variables step and fuzzy inference step.The details are described in the following sections.4.1.Preliminary step4.1.1.Establish a risk assessment groupThe members in a risk assessment group must be care-fully selected.The selected experts will have a high degree of knowledge and previous experience in similar construc-tion projects.The risk assessment team must include the following experts:project managers,project team mem-bers,customers,subject matter experts from outside the project team,end users,stakeholders and risk management group.The members in the risk assessment group will under-take the risk identification,even though all project person-nel should be encouraged to identify risks.Also,the risk assessment group will undertake the measure of risk func-tion parameters.A.Nieto-Morote,F.Ruz-Vila/International Journal of Project Management29(2011)220–231223The risks identification is an iterative process because the risks may evolve or new risks become known as the project progresses through its life cycle.The iteration fre-quency and who participate in each cycle will depend on the characteristics of the project.The experts in a risk assessment group have intuitive methods of recognizing a risk situation.Anyway,there are some risks identification tools as:Checklist,Influence Diagrams,Cause and Effect Diagrams,Failure Mode and Effect Analysis,Hazard and Operability Study,Fault Trees and Event Tree(Ahmed et al.,2007).4.1.3.Construct hierarchical structure of risksThe members in the risk assessment group are required to identify and classify the risks associated with the con-struction project.To decompose the risks into adequate details in which they can be efficiently assessed,a hierarchi-cal structure of risks is generated.The risks are sorted into n groups on the basis of the types of risks,as shown in Fig.2.More levels of decomposition can be incorporated into the hierarchical structure whenever the elements of a given level are mutually independent,but comparable to the elements of the same level.4.2.Definition of risk factor function and measurement of variables step4.2.1.Define risk factor functionThe risk factor(RF)can usually be assessed by consid-ering two fundamental risk parameters:risk impact(RI) and risk probability(RP).The risk impact parameter inves-tigates the potential effect of the risk on a project objective such as schedule,cost,quality or performance.The risk probability parameter investigates the likelihood that each specific risk will occur.These parameters do not take into consideration the impact of the risk to the overall frame-work of the project.In order to assess the risks efficiently and effectively,a parameter,named risk discrimination (RD),is proposed by Cervone(2006).The risk discrimina-tion parameter provides an additional perspective because it gauges the impact of the risk to the overall framework of the project,rather than looking at each risk as an inde-pendent variable within the project.With each risk evalu-ated in the context of the three dimensions,a value can be assigned to each risk using Eq.(14):Overall risk factor¼risk impactÂrisk probabilityrisk discriminationð14Þ4.2.2.Define linguistic scales and associated fuzzy numbersFrequently,it may be extremely difficult to assess the risk associated with a project due to the great uncertainty involved.The imprecision comes from a variety of sources such as,unquantifiable information,incomplete informa-tion or non-obtainable information(Chen and Hwang, 1992).When the members in a risk assessment group have inexact information about risks associated with a project, the assessments cannot be exact but approximate.In these circumstances,the judgements of the members in a risk assessment group are expressed by means of linguistic term instead of real numbers.An important aspect to handle this kind of problems is to define the linguistic terms that will be used.The possible linguistic terms that can be used depends on the nature of the problem.The linguistic terms which generally can be used to assess the parameters of a risk assessment problem, risk impact,risk probability and risk discrimination are the following ones:For evaluating thefirst dimension,RI,afive-point scale is defined:Critical(C),Serious(S),Moderate(Mo), Minor(Mi)and Negligible(N).For the second dimension,RP,a three-point scale is proposed:High(H),Medium(M)and Low probability (L).The third parameter,RD,will be obtained by pair-wise comparison between risks.The selected comparison scale is:Much less(Ml),Less(L),Same(S),More(M) and Much more(Mm).The linguistics terms must be transformed into a fuzzy numbers by using appropriate conversion scale.One of the key points in fuzzy modelling is the definition of fuzzy numbers which represent vague concepts and imprecise terms expressed in a natural language.The representation does not only depend on the concept but also on the context in which it is used.Even for similar contexts,fuzzy numbers representing the same concept may vary considerably.When operating with fuzzy numbers,the result of our calculations strongly depend on the shape of the member-ship functions of these numbers.Less regularmembership Fig.2.Generic hierarchical structure of risks.functions lead to more complicated calculations.More-over,fuzzy numbers with simpler shape of membership functions often have more intuitive and more natural interpretation.All these reasons cause a natural need of simple approx-imations of fuzzy numbers which are easy to handle and have a natural interpretation.For the sake of simplicity the trapezoidal or triangular fuzzy numbers are most com-mon in current applications.As noted in a research study (Mayor and Trillas,1986),the precision in the shape of the membership functions often is not important because of the quantitative nature of the problems with vague pred-icates;thus they are generally written as linearly as possi-ble.Normally it is sufficient to use a trapezoidal representation,as it makes it possible to define them with no more than four parameters.Based on some researches,a numerical approximation system is proposed to systematically convert linguistic terms into their corresponding fuzzy numbers(Chen and Hwang, 1992).The used linguistic terms,their meaning and their associated membership function are shown in Table1. 4.2.3.Define of RI and RP parameters4.2.3.1.Measure RI and RP.The RI and RP parameters of each risk at the bottom level of the hierarchy must be mea-sured by each member in the risk assessment group using the defined linguistic scales.The linguistic measures for RI and RP assigned by each member in the group are con-verted into their corresponding fuzzy numbers according to Table1.The fuzzy numbers obtained for RI and RPparameters are RI mi and RP miwhere i is the number of risksat the bottom level of the hierarchy and m is the number of members in risk assessment group.4.2.3.2.Aggregate individual fuzzy numbers into a groupfuzzy number.The individual fuzzy numbers RI mi and RP mi,corresponding to each one of the m expert in the risk assessment group,are aggregated into group fuzzy number by using the fuzzy arithmetic average,which is defined as:RI i¼1mÂX mn¼1RI ni¼1mÂðRI1iÈRI2iÈÁÁÁÈRI miÞð15ÞRP i¼1mÂX mn¼1RP ni¼1mÂðRP1iÈRP2iÈÁÁÁÈRP miÞð16Þwhere i is each one of the risks at the bottom level of the hierarchy,m is the number of members in risk assessment group,Âis the scalar multiplication defined in Eq.(6) andÈis the fuzzy addition defined in Eq.(2).4.2.4.Measure RD parameterpare risks pair-wise.The members in the risk assessment group are required to provide their comparative judgement on the impact on overall framework of the pro-ject for each risk pair-wise of the same level and group in the hierarchical structure.These linguistic measures are converted into their corre-sponding fuzzy members according to Table1.For each member in risk assessment group it is obtained the follow-ing comparison matrix for the group g and the level l in the hierarchy:A mgl¼r1r2ÁÁÁr nr1r2ÁÁÁr n–ðRDCÞm12ÁÁÁRDC1n mðRDCÞm21–ÁÁÁRDC m2nÁÁÁÁÁÁÁÁÁÁÁÁðRDCÞn1m RDC n2mÁÁÁ–2666666437777775ð17Þwhere n is the number of risks of the group g and the level l in the hierarchy and m is the number of members in risk assessment group.4.2.4.2.Aggregate individual fuzzy numbers into a group fuzzy number.The comparative fuzzy numbers of each one of the members in the risk assessment group are aggregated into a group fuzzy number by using the fuzzy arithmetic average which is defined as:Table1Descriptions of RI,RP and RD comparison.Description of RI General interpretation Fuzzy number Critical(C)Involved very highly impact(0.8,0.9,1,1) Serious(S)Involved highly impact(0.6,0.75,0.75,0.9) Moderate(Mo)Involved moderate impact(0.3,0.5,0.5,0.7) Minor(Mi)Involved only small impact(0.1,0.25,0.25,0.4) Negligible(N)Involved no substantive impact(0,0,0.1,0.2) Description of RP General interpretationHigh(H)Very likely to occur(0.7,0.9,1,1) Medium(M)Likely to occur(0.2,0.5,0.5,0.8) Low(L)Occurrence is unlikely(0,0,0.1,0.2) Description of RDC General interpretationMuch more Much more impact on overall framework of project than(0,0,0,0.3)More More impact on overall framework of project than(0,0.25,0.25,0.5) Same Same impact on overall framework of project than(0.3,0.5,0.5,0.7) Less Less impact on overall framework of project than(0.5,0.75,0.75,1) Much less Much less impact on overall framework of project than(0.7,1,1,1)A.Nieto-Morote,F.Ruz-Vila/International Journal of Project Management29(2011)220–231225RDC ij¼1mÂX mn¼1RDC nij¼1mÂðRDC1ijÈRDC2ijÈÁÁÁÈRDC mijÞð18Þwhere i and j are the risks of the group g and the level l in the hierarchy and m is the number of members in risk assessment group,Âis the scalar multiplication defined in Eq.(6)andÈis the fuzzy addition defined in Eq.(2).The matrix for the group g and the level l in the hierar-chy of comparative fuzzy numbers is defined as:A gl¼r1r2ÁÁÁr nr1r2ÁÁÁr n–ðRDCÞ12ÁÁÁRDC1nðRDCÞ21–ÁÁÁRDC2nÁÁÁÁÁÁÁÁÁÁÁÁðRDCÞn1RDC n2ÁÁÁ–2666666437777775ð19Þwhere n is the number of risks of the group g and the level l in the hierarchy.4.2.4.3.Estimate RD local(RD*).This problem is solved by using fuzzy classical methods of criteria weight calcula-tion adapted to operate with trapezoidal fuzzy numbers.In the fuzzy classical methods,the decision maker(DM) provides its fuzzy preference relations on criteria pair-wise, W ij.The ratio W ij denotes the preference degree of the cri-teria c i over c j and it is usually defined as a fuzzy singleton, which is a fuzzy set which contains only one element.If values of the fuzzy preference relations on criteriapair-wise were consistent,W0ij ,by the reciprocal propertyof fuzzy preference relation,i.e.,W0ij +W0ij=1,therewould be an explicit functional relation between W0ij andthe fuzzy values w i and w j,which would reflect the ranking values of the criteria c i and c j(Ma et al.,2006):W0ij¼0:5½1þwðw iÞÀwðw jÞ ð20Þwhere w(w i)can be any non-decreasing function andP w i=1.By the reciprocal property of preference of the valuesW0ij ,the Eq.(20)satisfies the additive transitivity property,i.e.,W0ik +W0ik+W0ki=1.5In order to keep the simplicity of the method,if w (w i)=w i then Eq.(20)is transformed into a new equation defined as:W0ij ¼w iþð1Àw jÞð21ÞGenerally the pair-wise comparison information given bythe DM has inconsistency,that is W0ik <0.5for W ij P0.5and W jk P0.5.Transitivity is the property that is usually accepted to deal with problems of fuzzy preference relations consistency(Wang and Chen,2008;Dong et al.,2008).Due to the fuzziness of the opinions and the weak tran-sitivity restriction considered,that is,W ik P0.5for W ij P 0.5and W jk P0.5(Herrera-Viedma et al.,2004),an accu-rate solution for this problem could not be found.The w i and w j values are calculated by difference minimization method of the value W ij,obtained directly from the experts, and the value W0ij,defined as a ideal fuzzy preference rela-tions which are consistent:minX ni¼1X nj¼1i¼1ðW0ijÀW ijÞ2264375ð22ÞIn our case,the values of fuzzy preference relations on risks obtained directly from the experts,RDC ij,and the values of the ideal fuzzy preference relations on risks,which areconsistent,RDC0ij,are trapezoidal fuzzy numbers.The extension of this classical method to fuzzy method can be expressed as:minX ni¼1X nj¼1i¼1RDC0ijH RDC ij2264375ð23Þwhere RDC0ijis defined in terms of the fuzzy values of RDÃi and RDÃj,which reflect the ranking values of the risks r i and r j,as:RDC0ij¼RDÃiÈð1H RDÃjÞð24Þwhere i and j are risks of the group g and the level l in the hierarchy andÈand H represents the fuzzy addition and subtraction using Eqs.(2)and(3).The main implication of this method is that the sum ofRDÃishould be now a trapezoidal fuzzy number“around one”that must be defined correctly to get a solution.The matrix for the group g and the level l in the hierar-chy of the RD*fuzzy values are defined as:B gl¼r1r2ÁÁÁr nRDÃ1RDÃ2ÁÁÁRDÁÁÁn2666437775ð25Þwhere n is the number of risks of the group g and the level l in the hierarchy.4.2.4.4.Aggregate RD*in hierarchy.Assume the risk r i has t upper groups at different level in the risk structure hierar-chy and RDÃðjÞgroupis the value RD*of the jth upper group which contain the risk r i in the hierarchy.Thefinal value of RD for each risk r i can be calculated by:RD¼iRDÃiY tj¼1ðRDÃÞðjÞgroupð26Þwhere i is each one of the risks at the bottom level of the hierarchy and represent the fuzzy multiplication using arithmetic operations on their a-cuts in Eq.(12).4.3.Fuzzy inference stepIn the fuzzy inference step,risk analysts convert the aggregated fuzzy number of RI,RP and RD into a fuzzy226 A.Nieto-Morote,F.Ruz-Vila/International Journal of Project Management29(2011)220–231。