a new approach to the rule-base evidential reasoning in the intuitionistic fuzzy setting

Knowledge-Based Systems xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Knowledge-Based Systems

journal homepage: www.elsevier .com/ locate /knosys

A new approach to the rule-base evidential reasoningin the intuitionistic fuzzy setting

http://dx.doi.org/10.1016/j.knosys.2014.02.0160950-7051/� 2014 Published by Elsevier B.V.

⇑ Corresponding author. Tel./fax: +48 34 3250 589.E-mail address: [email protected] (P. Sevastjanov).

Please cite this article in press as: L. Dymova, P. Sevastjanov, A new approach to the rule-base evidential reasoning in the intuitionistic fuzzyKnowl. Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.02.016

Ludmila Dymova, Pavel Sevastjanov ⇑Institute of Comp. & Information Sci., Czestochowa University of Technology, Dabrowskiego 73, 42-200 Czestochowa, Poland

a r t i c l e i n f o

Article history:Received 22 September 2013Received in revised form 14 January 2014Accepted 24 February 2014Available online xxxx

Keywords:Rule-base evidential reasoningAtannasov’s intuitionistic fuzzy setsDempster-Shafer theoryFuzzy logicOperational lows

a b s t r a c t

A new approach to the rule-base evidential reasoning based on the synthesis of fuzzy logic, Atannasov’sintuitionistic fuzzy sets theory and the Dempster-Shafer theory of evidence is proposed. It is shown thatthe use of intuitionistic fuzzy values and the classical operations on them directly may provide counter-intuitive results. Therefore, an interpretation of intuitionistic fuzzy values in the framework of Dempster-Shafer theory is proposed and used in the evidential reasoning. The merits of the proposed approach areillustrated with the use of developed expert systems for diagnostics of type 2 diabetes. Using the real-world examples, it is shown that such an approach provides reasonable and intuitively obvious resultswhen the classical method of rule-base evidential reasoning cannot produce any reasonable results.

� 2014 Published by Elsevier B.V.

1. Introduction where D00; D01; D10; D11 are fuzzy subsets in

The methods of rule-base evidential reasoning are based on thesynthesis of the tools of Fuzzy Sets theory ðFSTÞ and the Dempster-Shafer theory ðDSTÞ. The integration of FST and DST within sym-bolic, rule-based models primarily was used for solving controland classification problems [5,6,23,36,41]. These models combinethese theories in a synergic way, preserving their strengths whileavoiding disadvantages they present when used as monostrategyapproaches. Generally, such a rule-base evidential reasoningsystem may be presented as in [6]:

IF ððA is LÞ and ðB is MÞÞ THEN C is m0;

IF ððA is HÞ and ðB is LÞÞ THEN C is m1;

where m0 and m1 are two credibility structures with two focalelements and variable C is defined in the universe of discoursewhich usually is a set of classes to deal with in consideredclassification problem.

In the above example adopted from [6], the credibilitystructures were presented as follows:

m0 : D00 ¼l0

00

y0;l0

01

y1

� �; m0ðD00Þ; D01 ¼

l101

y1

� �; m0ðD01Þ;

m1 : D10 ¼l0

11

y1;l0

12

y2

� �; m1ðD10Þ; D11 ¼

l110

y0;l1

11

y1;l1

12

y2

� �; m1ðD11Þ;

Y ¼ ðy0; y1; y2Þ; l000; l0

01; l101; l0

11; l012; l1

10; l111; l1

12 are the cor-responding membership grades, m0ðD00Þ; m0ðD01Þ; m1ðD10Þ;m1ðD11Þ are the basic probability values associated with fuzzy sub-sets D00; D01; D10; D11. The output of the system is obtained in [6]with use of COA method [37]) in the form of defuzzified value �y.

These approaches seem to be justified in the solution of controland classification problems when outputs can be presented by realvalues.

On the other hand, if we deal with decision support systems,system’s outputs can be only the names or labels of correspondingactions or decisions, e.g., the names of medical diagnoses. It is clearthat in such cases, the methods based on conventional fuzzy logic,developed for the controlling cannot be used at least directly. Amore suitable for the building decision support systems seems tobe the so-called RIMER approach proposed in [39,40] based onthe Evidential Reasoning approach [38].

In the belief rule system, each possible consequent of a rule isassociated with a belief degree. Such a rule base is capable tocapture more complicated and continuous causal relationships be-tween different factors than traditional IF-THEN rules. Therefore,the traditional IF-THEN rules may be treated as a special cases ofthe more general belief rule systems [20,25,30]. In the frameworkof rule-base inference methodology, using the evidential reasoning(RIMER) approach [39] a belief IF-THEN rule, e.g., the kth rule Rk, isexpressed as follows:

IF (X1 is Ak1) K (X2 is Ak

2) K . . . K (XTkis Ak

Tk)

THEN (D1, b1k), (D2, b2k), . . ., (DN , bNk), with rule weightshk; k ¼ 1 to L, and attribute weights d1; d2; . . . ; dTk

, where Aki ; i ¼ 1

setting,

http://dx.doi.org/10.1016/j.knosys.2014.02.016

mailto:[email protected]


http://www.sciencedirect.com/science/journal/09507051

http://www.elsevier.com/locate/knosys


2 L. Dymova, P. Sevastjanov / Knowledge-Based Systems xxx (2014) xxx–xxx

to Tk is the referential value of the ith antecedent attribute, Tk isthe number of antecedent attributes used in the kth rule,bik; i ¼ 1 to N, is the belief degree to which Di is believed to bethe consequent of kth antecedent, L is the number of rules in therule-base, K denotes t-norm. If

PNi¼1bik ¼ 1, the kth rule is said to

be complete; otherwise, it is incomplete. The case ofPN

i¼1bik ¼ 0corresponds to the total ignorance about the output given the in-put in the kth rule. This rule is also referred to as a belief rule.

In the framework of RIMER approach, the final outcome ob-tained as the aggregation of belief rules is presented asO ¼ fðDj; bjÞg, where bj; j ¼ 1 to N, is the aggregated degree of be-lief in the decision (hypothesis, action, diagnosis) Dj.

Therefore, the decision characterised by the maximal aggre-gated degree of belief is the best choice. So the RIMER approachcan be used for building decision support systems. Nevertheless,there are two restrictions in the RIMER approach that reduce itsability to deal with uncertainties that decision makers often meetin practice.

The first restriction is that in the framework of RIMER approach,a degree of belief can be assigned only to a particular hypothesis,not to a group of them, whereas the assignment of a belief massto a group of events is a key principle of the DST.

The second restriction is concerned with the observation that inmany real-world decision problems we deal with different sourcesof evidence and the combination of them is needed. The RIMER ap-proach does not provide a technique for the combination of evi-dence from different sources.

It is important that usually the advantages of the approachesbased on the rule-base evidential reasoning were demonstratedusing simple numerical examples and only relatively small numberof examples of solving real-world problems using these approacheswere found in the literature [6,22,29,35]. In [42], a novel updatingalgorithm for RIMER model is proposed based on iterative learningstrategy for delayed coking unit ðDCUÞ. Daily DCU operations underdifferent conditions are modelled by a belief rule-base, which isthen updated using iterative learning methodology, based on a no-vel statistical utility for every belief rule. The paper [33] presents ahybrid evidential reasoning ðERÞ and belief rule-based ðBRBÞmeth-odology for consumer preference prediction and a novel applica-tion to orange juices. In [19], a novel combination of fuzzyinference system and Dempster-Shafer Theory is applied to brainMagnetic Resonance Imaging for the purpose of segmentationwhere the pixel intensity and the spatial information are used asfeatures. The authors of paper [32] proposed a specific algorithm-Evidential Reasoning based Classification algorithm to recognisehuman faces under class noise conditions. The methods used inthese papers are charged with two above mentioned restrictionsof RIMER approach. The method used in [24] is free of the secondrestriction while the first one is retained.

In [16,18,27], a new approach free of both above mentionedrestrictions was developed and used for the solution of real-worldproblems.

It is important that in all above mentioned approaches to therule-base evidential reasoning, the conventional fuzzy logic wasused. For example, the following rule may be used:If x is Low Then D, where Low is some fuzzy class defined by thecorresponding membership function lLowðxÞ; D is a name of deci-sion. Nevertheless, in practice we often deal with the intersectingfuzzy classes, e.g., Low and Middle, and therefore we often havelLowðxÞ > 0 and lMiddleðxÞ > 0. Then if lLowðxÞ > lMiddleðxÞ we statethat x is Low and information of non-zero lMiddleðxÞ is lost, whereasthe difference between lMiddleðxÞ and lLowðxÞ may be very small.

In the current paper, we will show that such a loss of informa-tion may lead to incorrect results in the rule-base evidentialreasoning and a new method for the solution of these problems

Please cite this article in press as: L. Dymova, P. Sevastjanov, A new approacKnowl. Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.02.016

based in the synthesis of Atanassov’s intuitionistic fuzzy sets (A-IFS) [1] and DST will be proposed.

In our recent paper [15], we have shown that there exists also astrong link between A-IFS and DST. This link makes it possible touse directly the Dempster’s rule of combination to aggregate localcriteria presented by IFVs in multiple criteria decision makingproblems ðMCDMÞ. The usefulness of the developed method wasillustrated using the known example of MCDM problem. In [17],we have shown that the classical arithmetical operations on intui-tionistic fuzzy values IFVs have some limitations (drawbacks)which can lead to incorrect results in applications of A-IFS indifferent fields. Therefore, in [17] using interpretation of A-IFS inthe framework of DST, a set of new operations on IFVs treated asbelief intervals was proposed and it was proved that these opera-tions are free of limitations (drawbacks) of the classical operationson IFVs.

To make the presentation of our approach more transparent, weshall use as an illustration the simple enough, but real-worldproblem of diagnostic of type-2 diabetes.

For these reasons, the rest of paper is set out as follows. Sec-tion 2 presents the basic definition of A-IFS and DST, the commonlyused arithmetical operations on IFVs (with their limitations anddrawbacks) and introduced in [17] new operations on IFVs in theframework of DST needed for the subsequent analysis. In Section 3,we present our new approach to the rule-base evidential reasoningbased on the synthesis of A-IFS and DST and perform its advantagesusing examples obtained with the use of expert system for diag-nostics of type 2 diabetes developed on the base of our approach.Finally, the concluding section summarises the paper.

2. Preliminaries

2.1. The basics of A-IFS and problems concerned with operations onIFVs

The concept of A-IFS (the reasons for such a notation are pre-sented in [14]) is based on the simultaneous consideration ofmembership l and non-membership m of an element of a set tothe set itself (see formal definition in [1]). It is postulated that0 6 lþ m 6 1. Following to [1], we call pAðxÞ ¼ 1� lAðxÞ � mAðxÞthe hesitation degree of the element x in the set A. Hereinafter,we shall call an object A ¼ hlAðxÞ; mAðxÞi intuitionistic fuzzy valueðIFVÞ.

The operations of addition � and multiplication � on IFVs weredefined by Atanassov [2] as follows. Let A ¼ hlA; mAi andB ¼ hlB; mBi be IFVs. Then

A� B ¼ hlA þ lB � lAlB; mAmBi; ð1ÞA� B ¼ hlAlB; mA þ mB � mAmBi: ð2Þ

These operations were constructed in such a way that they produceIFVs. Using operations (1) and (2), in [12] the following expressionswere obtained for any integer n ¼ 1;2; . . .:

nA ¼ A� . . .� A ¼ h1� ð1� lAÞn; mn

Ai; An ¼ A� . . .� A

¼ hlnA;1� ð1� mAÞni:

It was proved later that these operations produce IFVs not only forinteger n, but also for all real values k > 0, i.e.

kA ¼ h1� ð1� lAÞk; mk

Ai; ð3ÞAk ¼ hlk

A;1� ð1� mAÞki: ð4Þ

The operations (1)–(4) have good algebraic properties [34]:

Let A ¼ hlA; mAi and B ¼ hlB; mBi be IFVs. Then

h to the rule-base evidential reasoning in the intuitionistic fuzzy setting,


L. Dymova, P. Sevastjanov / Knowledge-Based Systems xxx (2014) xxx–xxx 3

A� B ¼ B� A; ð5ÞA� B ¼ B� A; ð6ÞkðA� BÞ ¼ kA� kB; ð7ÞðA� BÞk ¼ Ak � Bk; ð8Þk1A� k2A ¼ ðk1 þ k2ÞA; k1; k2 > 0; ð9ÞAk1 � Ak2 ¼ Ak1þk2 ; k1; k2 > 0: ð10Þ

The operations (1)–(4) are used to aggregate local criteria for solv-ing MCDM problems in the intuitionistic fuzzy setting.

Let A1; . . . ;An be IFVs representing the values of local criteria andw1; . . . ;wn;

Pni¼1wi ¼ 1, be their weights. Then Intuitionistic

Weighted Arithmetic Mean ðIWAMÞ can be obtained using opera-tions (1) and (3) as follows:

IWAM¼w1A1�w2A2� . . .�wnAn ¼ 1�Yn

i¼1

ð1�lAiÞwi ;Yn

i¼1

mwiAi

* +: ð11Þ

This aggregation operator provides IFVs, is idempotent and cur-rently is the most popular in the solution of MCDM problems inthe intuitionistic fuzzy setting.

An important problem is the comparison of IFVs. Therefore, thespecific methods which are rather of heuristic nature were devel-oped to compare IFVs. For this purpose, Chen and Tan [11] pro-posed to use the so-called score function (or net membership)SðxÞ ¼ lðxÞ � mðxÞ. Let A and B be IFVs. It is intuitively appealingthat if SðAÞ > SðBÞ then A should be greater (better) than B, but ifSðAÞ ¼ SðBÞ this does not always mean that A is equal to B. There-fore, Hong and Choi [21] in addition to the above score functionintroduced the so-called accuracy function HðxÞ ¼ lðxÞ þ mðxÞ andshowed that the relation between functions S and H is similar tothe relation between mean and variance in statistics. Xu [34] usedthe functions S and H to construct order relations between any pairof intuitionistic fuzzy values A and B as follows:

If ðSðAÞ > SðBÞÞ; then B is smaller than A;

If ðSðAÞ ¼ SðBÞÞ; then

ð1Þ If ðHðAÞ ¼ HðBÞÞ; then A ¼ B;

ð2Þ If ðHðAÞ < HðBÞÞ; then A is smaller than B:

ð12Þ

It was shown in [4,17] that operation (1)–(4), (11) and (12) havesome undesirable properties which may lead to the non-acceptableresults in applications:

1. The addition (1) is not an addition invariant operation. LetA; B and C be IFVs. Then A < B (in the sense of (12)) doesnot always lead to ðA� CÞ < ðB� CÞ.

2. The operation (3) is not preserved under multiplication bya real-valued k > 0, i.e., inequality A < B (in the sense of(12)) does not necessarily imply kA < kB.In [4], with the use of Lukasiewicz t-conorm and t-norm,the following expression was inferred:

PleaseKnowl

kA ¼ hklA;1� kð1� mAÞi; k 2 ½0;1�: ð13Þ

It is easy to prove that the use of (13) guarantees that forIFVs A and B the inequality A < B always imply kA < kB(k 2 ½0;1�), but unfortunately, the properties (7) and (9) withoperation (13) do not hold.

3. An important problem with the aggregation operation (11)is that it is not consistent with the aggregation operationon the ordinary fuzzy sets (when l ¼ 1� m). This can beeasily seen from the following example.

cite this article in press as: L. Dymova, P. Sevastjanov, A new approac. Based Syst. (2014), http://dx.doi.org/10.1016/j.knosys.2014.02.016

Example 1. Let A ¼ h0:1;0:9i; B ¼ h0:9; 0:1i and w1 ¼ w2 ¼ 0:5. Itis easy to see that A and B are IF representations of ordinary fuzzynumbers. Then in the framework of ordinary fuzzy sets, we getOrdinary Weighted Arithmetic Mean OWAM ¼ w1lA þw2lB ¼0:5 � 0:1þ 0:5 � 0:9 ¼ 0:5 and in the framework of A-IFS from (11),we obtain IWAM ¼ h0:7;0:3i. We can see that the resulting valueof l obtained using IWAM is considerably greater than thatobtained from OWAM.

4. Another problem with the aggregation operation (11) isthat it is not monotone with respect to the ordering (12).Consider the following example:

Example 4. Let A ¼ h0;1i; B ¼ h0:5;0:4i and C ¼ h0:3;0:2i. SinceSðAÞ ¼ �1; SðBÞ ¼ 0:1; SðCÞ ¼ 0:1 and HðAÞ ¼ 1; HðBÞ ¼ 0:9;HðCÞ ¼ 0:5, from (12) we obtain B > C > A. Suppose w1 ¼w2 ¼ 0:5. Then from (11) we get IWAMðA;CÞ ¼ h0:1634;0:4472i; IWAMðA;BÞ ¼ h0:2928;0:6423i. The score functions ofthese results are as follows: SðIWAMðA;CÞÞ ¼ �0:2838; SðIWAMðA;BÞÞ ¼ �0:3396.

We can see that SðIWAMðA;CÞÞ > SðIWAMðA;BÞÞ.Summarising, we can say that the undesirable properties of

addition (1) and multiplication (3) operations lead to the undesir-able properties of intuitionistic weighted arithmetic mean operator(11). Moreover, this operator seems rather like some modificationof intuitionistic fuzzy weighted geometric operator since in (11)the multiplication by weights wi is changed by the power operator,i.e., the usual weighted averaging structure is violated.

Therefore, in our paper [17] using interpretation of A-IFS in theframework of DST, a set of new operations on IFVs treated as beliefintervals was proposed and it was proved that these operations arefree of limitations (drawbacks) of the classical operations on IFVs.We will recall briefly these results in the following subsection.

2.2. The operations on IFVs in the framework of DST

Firstly we recall briefly the basics of DST needed for the subse-quent analysis.

The origins of the Dempster-Shafer theory ðDSTÞ go back to thework by A.P. Dempster [13] who developed a system of upper andlower probabilities. Following this work his student G. Shafer [28]provided a more thorough explanation of belief functions.

Assume A are subsets of X. It is important to note that a subset Amay be treated also as a question or proposition and X as a set ofpropositions or mutually exclusive hypotheses or answers. A DSbelief structure has associated with it a mapping m, called basicprobability assignment (bpa), from subsets of X into a unit interval,m : 2X ! ½0;1� such that mð;Þ ¼ 0;

PA # XmðAÞ ¼ 1. The subsets of X

for which the mapping does not assume a zero value are called fo-cal elements.

Shafer [28] introduced a number of measures associated withDS belief structure.

The measure of belief is a mapping Bel : 2X ! ½0;1� such that forany subset B of X it can be presented as

BelðBÞ ¼X;–A # B

mðAÞ: ð14Þ

A second measure introduced by Shafer [28] is a measure of plausi-bility. The measure of plausibility associated with m is a mappingPl : 2X ! ½0;1� such that for any subset B of X it can be presented as

PlðBÞ ¼X

A\B–;mðAÞ: ð15Þ

It is easy to see that BelðBÞ 6 PlðBÞ. DS provides an explicit measureof ignorance about an event B and its complementary B as a length




of an interval ½BelðBÞ; PlðBÞ� called the belief interval ðBIÞ. It can alsobe interpreted as imprecision of the ‘‘true probability’’ of B [28].

The core of the evidence theory is the Dempster’s rule of com-bination of evidence from different sources. The rule assumes thatinformation sources are independent. With two belief structuresm1;m2, the Dempster’s rule of combination is defined as follows:

m12ðAÞ ¼

XB\C¼A

m1ðBÞm2ðCÞ

1� K;A – ;;m12ð;Þ ¼ 0; ð16Þ

where K ¼P

B\C¼;m1ðBÞm2ðCÞ is called the degree of conflict whichmeasures the conflict between the pieces of evidence. Zadeh [43]underlined that this rule involves counter-intuitive behaviours inthe case of considerable conflict.

It is important to note that the Dempster’s rule is commutativeand associative, but not idempotent operator. Nevertheless, inspite of the lack of idempotency, the Dempster’s rule is successfullyused in different real-world applications.

The above information of DST is quite enough for the presenta-tion of A-IFS in terms of DST.

It was shown in [15] that convenient for practical applicationsmethods for MCDM can be developed using DST semantics for A-IFS.

It was shown in [15] that in the framework of DST the tripletlAðxÞ; mAðxÞ; pAðxÞ represents the basic assignment function.Really, when analysing any situation in context of A-IFS, we implic-itly deal with the following three hypotheses: x 2 A; x R A and thesituation when both the hypotheses x 2 A; x R A cannot be re-jected (the case of hesitation). In the spirit of DST, we can denotethese hypotheses as Yes (x 2 A), No (x R A) and ðYes;NoÞ (the caseof hesitation when both the hypotheses x 2 A and x R A cannotbe rejected).

In this context, lAðxÞ may be treated as the probability or evi-dence of x 2 A, i.e., as the focal element of the basic assignment func-tion: mðYesÞ ¼ lAðxÞ. Similarly, we can assume that mðNoÞ ¼ mAðxÞ.Since pAðxÞ is usually treated as a hesitation degree, a naturalassumption is mðYes;NoÞ ¼ pAðxÞ. Taking into account thatlAðxÞ þ mAðxÞ þ pAðxÞ ¼ 1 we come to the conclusion that the tripletlAðxÞ; mAðxÞ; pAðxÞ represents a correct basic assignment function.According to the DST formalism we get BelAðxÞ ¼ mðYesÞ ¼ lAðxÞand PlAðxÞ ¼ mðYesÞ þmðYes;NoÞ ¼ lAðxÞ þ pAðxÞ ¼ 1� mAðxÞ.

Therefore, in [15] we introduced the following definition:

Definition 1. Let X ¼ fx1; x2; . . . ; xng be a finite universal set and xj

is an object in X presented by the functions lAðxjÞ; mAðxjÞ whichrepresent the degree of membership and degree of non-member-ship of xj 2 X to the set A # X such that lA : X ! ½0;1�; xj 2 X !lAðxjÞ 2 ½0;1� and mA : X ! ½0;1�; xj 2 X ! mAðxjÞ 2 ½0;1� and forevery xj 2 X; 0 6 lAðxjÞ þ mAðxjÞ 6 1. An intuitionistic fuzzy set Ain X is an object having the following form: A ¼ f< xj;BIAðxjÞ >jxj 2 Xg, where BIAðxjÞ ¼ ½BelAðxjÞ; PlAðxjÞ� is the belief interval,BelAðxjÞ ¼ lAðxjÞ and PlAðxjÞ ¼ 1� mAðxjÞ are the measures of beliefand plausibility that xj 2 X belongs to the set A # X.

At first glance, the Definition 1 seems as a simple redefinition ofA-IFS in terms of Interval Valued Fuzzy Sets, but this interpretationmakes it possible to represent mathematical operations on IFVs asoperations on belief intervals. The use of the semantics of DSTmakes it possible to enhance the performance of A-IFS when deal-ing with the operations on IFVs.

In [17], the two sets of operations on IFVs based on the interpre-tation of intuitionistic fuzzy sets in the framework of DST were pro-posed and analysed. The first set of operations is based on thetreatment of belief interval as an interval enclosing a true probabil-ity. The second set of operations is based on the treatment of beliefinterval as an interval enclosing a true power of some statement


(argument, hypothesis, etc.). It was shown in [17] that the non-probabilistic treatment of belief intervals representing IFVsperforms better than the probabilistic one and operations basedon the probabilistic and non-probabilistic treatments of beliefintervals representing IFVs perform better than operations on IFVsdefined in the framework of conventional A-IFS.

Therefore, here we shall use only the treatment of belief intervalas an interval enclosing a true power of some statement.

Let X ¼ fx1; x2; . . . ; xng be a finite universal set. Assume A is asubset of X. It is important to note that in the framework of DSTa subset A may be treated also as a question or proposition and Xas a set of propositions or mutually exclusive hypotheses or an-swers. In such a context, a belief interval BIðAÞ ¼ ½BelðAÞ; PlðAÞ�may be treated as an interval enclosing a true power of statement(argument, proposition, hypothesis, etc.) that xj 2 X belongs to thesubset A # X. Obviously, the value of such a power lies in interval[0, 1].

Therefore, a belief interval BIðAÞ ¼ ½BelðAÞ; PlðAÞ� as a whole maybe treated as an imprecise (interval-valued) statement (argument,proposition, hypothesis, etc.) that xj 2 X belongs to the set A # X.

Based on this reasoning, we can say that if we pronouncethis statement, we can obtain some result, e.g., as a reaction on thisstatement or as an answer to some question, and if we repeat thisstatement twice, the result does not change.

Such a reasoning implies the following property of additionoperator:

BIðAÞ ¼ BIðAÞ�BBIðAÞ�B . . .�BBIðAÞ:

This is possible only if we define the addition �B of belief intervals

as follows: BIðAÞ�BBIðAÞ ¼ BelðAÞþBelðAÞ2 ; PlðAÞþPlðAÞ

2

h i. So the addition of

belief intervals is represented by their averaging.Therefore, if we have n different statements represented by be-

lief intervals BIðAiÞ then their sum �B can be defined as follows:

BIðA1Þ�BBIðA2Þ�B . . .�BBIðAnÞ ¼1n

Xn

i¼1

BelðAiÞ;1n

Xn

i¼1

PlðAiÞ" #

: ð17Þ

The other operations on belief intervals are presented in [17] asfollows:

BIðAÞ�BBIðBÞ ¼ ½BelðAÞBelðBÞ; PlðAÞPlðBÞ�; ð18ÞkBIðAÞ ¼ ½kBelðAÞ; kPlðAÞ�; ð19Þ

where k is a real value in the interval ½0;1� as for k > 1 this operationdoes not always provide a true belief interval. This restriction is jus-tified enough since we define operations on belief intervals to dealwith decision making problems, where k usually represents theweight of local criterion, which is lesser than 1.

BIðAÞk ¼ ½BelðAÞk; PlðAÞk�; ð20ÞBIðAÞBIðBÞ ¼ ½BelðAÞPlðBÞ

; PlðAÞBelðBÞ�: ð21Þ

These operations on belief intervals produce belief intervals, too.The defined set of operations have good algebraic properties

(the same as in the case of conventional A-IFS, see (5)–(10)) whichfollow immediately from the definitions (17)–(21):

BIðAÞ�BBIðBÞ¼BIðBÞ�BBIðAÞ; BIðAÞ�BBIðBÞ¼BIðBÞ�BBIðAÞ;ðBIðAÞ�BBIðBÞÞk¼BIðAÞk�BBIðBÞk; BIðAÞk1�BBIðAÞk1 ¼BIðAÞk1þk2 ;

kBIðAÞ�BkBIðBÞ¼ kðBIðAÞ�BBIðBÞÞ; k1BIðAÞ�Bk2BIðAÞ¼ ðk1þk2ÞBIðAÞ:

Using expressions (17) and (19) we obtain the following Intuition-istic Weighted Arithmetic Mean:

IWAMDSTNPðA1;A2; . . . ;AnÞ ¼1n

Xn

i¼1

wiBelAi;1n

Xn

i¼1

wiPlAi

" #: ð22Þ




This aggregation operator is not idempotent. For example, considerBel1 ¼ Bel2 ¼ 0:4; Pl1 ¼ Pl2 ¼ 0:6 and w1 ¼ w2 ¼ 0:5. Then from (22)we obtain IWAMDSTNP ¼ ½0:2; 0:3�, not the IWAMDSTNP ¼ ½0:4; 0:6�.

Nevertheless, the small modification of (22) (multiplying by n)provides the idempotent operator

IWAMIDSTNPðA1;A2; . . . ;AnÞ ¼Xn

i¼1

wiBelAi;Xn

i¼1

wiPlAi

" #: ð23Þ

It is easy to see that operators (22) and (23) in practice will produceequivalent orderings of compared alternatives.

Since BIs are regular intervals, the problem of their comparison isthe problem of interval comparison. A lot of methods for intervalcomparison is proposed in the literature. To compare intervals, usu-ally the quantitative indices are used (see reviews in [31,26]). Themost of them are simple enough, but some methods seem to be com-plicated ones. For example, in [26] it was shown that the result ofinterval comparison is belief interval. In [9], the authors introducedthe concept of an admissible order as a total order that extends theusual partial order between intervals and proposed a method tobuild these admissible orders in terms of two aggregation functions.In [10], the admissible order is used to study the construction ofinterval-valued ordered weighted aggregation operators.

Nevertheless, all proposed approaches to the interval compari-son provide the same qualitative intuitively clear results.

Therefore, here we will use the simple approach to the intervalcomparison based on the interval subtraction. It is justified in [17]that based on this approach, to compare belief intervals it is en-ough to compare their centres.

With the use of corresponding theorem, it was proved in [17]that introduced operations on belief intervals are free of undesir-able properties (1)–(4) of conventional operations on IFVs.

3. The synthesis of fuzzy logic and A� IFS in the rule-basedevidential reasoning

Strictly speaking, the synthesis of fuzzy logic and A� IFS is not anew idea, since in [8] a method for inference in approximate rea-soning based on the normal A� IFS was proposed. The authors ofthis paper wrote that they used a very specific method of inferencein one dimensional case and for normal A� IFS.

Here we present an another approach to the solution this prob-lem based on the synthesis of fuzzy logic, A� IFS and DST.

Of course, in practice the rule-based evidential reasoning sys-tems may involve hundreds of rules [16]. But the analysis of meth-odological aspects on the base of such great systems is usually verydifficult. Therefore, to present our approach in a more transparentform, in this section we shall use a relatively simple example ofbuilding the expert system for diagnosing type 2 diabetes whichmakes it possible to show the features of the proposed approachand avoid here the use of complicated general expressions.

It is important also that in spite of simplicity this system is thereal-world one.

The following tests are recommended by the World HealthOrganization (WHO) for diagnosing of the type 2 diabetes:

Test 1. A fasting plasma glucose test measures blood glucose ina person who has not eaten anything for at least 8 h.Test 2. An oral glucose tolerance test measures blood glucoseafter a person fasts at least 8 h and 2 h after the person drinksa glucose- containing beverage.

These tests are used to detect diabetes and pre-diabetes.Although, WHO proposes crisp intervals for blood glucose whichcorrespond to the health (H), Pre-diabetes (H,D) and diabetes (D),in practice doctors use for diagnostics, e.g., such fuzzy concepts


as Low, Medium and Big blood glucose, which can be presentedby corresponding membership functions as in Fig. 1, wherelL;lM;lB correspond to the lLow; lMedium; lBig , respectively. Herewe shall treat the diagnosis Pre-diabetes as an intermediate onewhen a doctor hesitates in choice between the Health and Diabetesdiagnoses. Therefore, the diagnosis Pre-diabetes in the spirit of DSTcan be treated as the compound hypothesis (H,D). Based on theknown approaches to the rule-base evidential reasoning [16,18]we get the following rules:

IF x1 is Low Then m�1ðHÞ ¼ lLðx1Þ;IF x1 is Medium Then m�1ðH;DÞ ¼ lMðx1Þ;IF x1 is Big Then m�1ðDÞ ¼ lBðx1Þ;IF x2 is Low Then m�2ðHÞ ¼ lLðx2Þ;IF x2 is Medium Then m�2ðH;DÞ ¼ lMðx2Þ;IF x2 is Big Then m�2ðDÞ ¼ lBðx2Þ;

ð24Þ

where bpas m�1ðHÞ; m�1ðH;DÞ; m�1ðDÞ and m�2ðHÞ; m�2ðH;DÞ; m�2ðDÞshould be additionally normalised:

m1ðHÞ ¼ m�1ðHÞ=S1; m1ðH;DÞ ¼ m�1ðH;DÞ=S1; m1ðDÞ ¼ m�1ðDÞ=S1;

m2ðHÞ ¼ m�2ðHÞ=S2; m2ðH;DÞ ¼ m�2ðH;DÞ=S2; m2ðDÞ ¼ m�2ðDÞ=S2;

S1 ¼ m�1ðHÞ þm�1ðH;DÞ þm�1ðDÞ; S2 ¼ m�2ðHÞ þm�2ðH;DÞ þm�2ðDÞ:

Then based on the principles of classical fuzzy logic, for x1 ¼ x�1 andx2 ¼ x�2 (see Fig. 1) using the above rules (24) from the first test weget the diagnosis H (with m�1ðHÞ ¼ l1

L ðx�1Þ) and from the second one– (H,D) (with m�2ðH;DÞ ¼ l1

Mðx�2Þ).Since these two tests are different sources of evidence, to obtain

the final diagnosis they should be combined using an appropriatecombination rule.

It is important that the classical fuzzy logic may lead to thecounterintuitive results. That may be explained as follows. In thetest 1, we take into account only diagnosis H, whereas the diagno-sis (H,D) is possible as well with a non-zero value of membershipfunction l1

M x�1� �

. Similarly, in the test 2 we are not taking into ac-count the possible diagnosis D.

Let us consider a hypothetical situation when in the Test 2 wehave two parameters x2 and x3 used to detect diabetes. Then thecorresponding rules may be presented, e.g., as follows:

IF ((x2 is Low) and (x3 is High)) Then m�21ðH;DÞ,IF ((x2 is Low) and (x3 is Medium)) Then m�22ðH;DÞ,IF ((x2 is Low) (x3 is Low)) Then m�23ðHÞ,IF ((x2 is Medium) (x3 is Low)) Then m�24ðH;DÞ,IF ((x2 is Medium) (x3 is Medium)) Then m�25ðH;DÞ,IF ((x2 is Medium) (x3 is High)) Then m�26ðDÞ,IF ((x2 is High) (x3 is Low)) Then m�27ðH;DÞ,IF ((x2 is High) (x3 is Medium)) Then m�28ðH;DÞ,IF ((x2 is High) (x3 is High)) Then m�29ðDÞ,

where the consequent attributes m�21;m�22; . . . ;m�29 may be calcu-

lated using different kinds of aggregation in the antecedent partof rules, e.g., to calculate m�21 we can apply the following expres-sions m�23ðHÞ ¼minðlLðx2Þ;lLðx3ÞÞ; m�23ðHÞ ¼ lLðx2ÞlLðx3Þ andm�23ðHÞ ¼ 1

2 ðlLðx2Þ þ lLðx3ÞÞ the same operators may be used forall other antecedents.

Thus, a natural question arises: what type of aggregation is thebest one? The problem of choosing an appropriate aggregation isout of frameworks of fuzzy logic and DST. Generally, this is contextdependent problem [44].

The final bpa for the Test 2 may be calculated as follows:

m�2ðHÞ¼m�23ðHÞ;m�2ðH;DÞ¼m�21ðH;DÞþm�22ðH;DÞþm�24ðH;DÞþm�25ðH;DÞþm�27ðH;DÞþm�28ðH;DÞ;m�2ðDÞ¼m�26ðDÞþm�29ðDÞ:



Fig. 1. Test 1 (blood glucose x1) and Test 2 (blood glucose x2).


It is easy to see that the part of components of above sums may beequal to 0 in the concrete situations. Of course, the obtained bpashould be normalised.

Another problem of classical approach is that it is possible inthe considered case to use two different rules:

IF ((x2 is Low) (x3 is Medium)) Then m�22ðH;DÞ,IF ((x2 is Low) (x3 is Medium)) Then m�22ðDÞ.

Since we cannot use these rules simultaneously, we need anadditional information to choose a correct rule and this informa-tion usually may be found only out of the framework of classicalfuzzy logic.

Taking into account all above mentioned problem of classicalfuzzy logic we prefer in such situations as we considered aboveto treat x3 as the separate source of evidence. In such a case, allwhat we need is to add to the set (24) three new rules:

IF x3 is Low Then m�3ðHÞ ¼ lLðx3Þ,IF x3 is Medium Then m�3ðH;DÞ ¼ lMðx3Þ,IF x3 is High Then m�3ðDÞ ¼ lBðx3Þ.

Then the final combination of rules based on x1; x2 and x3 maybe obtained using the Dempster’s rule (16) (firstly we combine therules based on x1; x2 and the obtained new bpa is combined withthe rules based on x3).

Summarising, we can say that the known methods of rule-baseevidential seasoning have some inherent limitations and draw-backs and may lead to the loss of important information whichmay affect the final results.

To avoid this problem, we propose to use the tools of intuition-istic fuzzy set theory. To clarify the basics of proposed approach,consider an example which can be treated as an extension of rea-soning used by Atannasov [3] to explain the essence of A� IFS.

Let us consider a general presidential election where 30% of eli-gible population votes for the first candidate (X) and 50% – for thesecond one (Y). The rest of the votes, 20%, are for none or lost forsome other reason. For the first candidate X one can state thatthe membership lðXÞ of the eligible population to those who sup-port the first candidate is equal to 0.3; the membership lðYÞ tothose who does not support the first candidate because they preferthe second candidate is equal to 0.5; Therefore, we can say thatmðXÞ ¼ lðYÞ is the non-membership to those who support the can-didate X. Similarly, for the second candidate we have mðYÞ ¼ lðXÞand the uncertain part also known as hesitation degree pðX;YÞ isequal to 0.2.

This reasoning may be used for the presentation of intersectingmembership functions. Let us consider the Fig. 1.

It is seen that x�1 belongs to the fuzzy class Low with the mem-bership equal to l1

L x�1� �

, but at the same time it belongs to the com-peting fuzzy class Medium with the membership equal to l1

M x�1� �

.


Therefore, based on the above reasoning, we can say that thenon-membership m1

L x�1� �

of x�1 to the fuzzy class Low is equal tol1

Mðx�1Þ as with this degree of membership it belongs to thecompeting class Medium. Using the same reasoning for the classMedium we can present the final result by twoIFVs : hl1

L x�1� �

; m1L x�1� �i; hl1

M x�1� �

; m1M x�1� �i, where l1

L x�1� �

¼ m1M x�1� �

and l1M x�1� �

¼ m1L x�1� �

. Similarly, for the test 2 (see Fig. 1) we getthe following result: hl2

M x�2� �

; m2M x�2� �i; hl2

B x�2� �

; m2B x�2� �i, where

l2M x�2� �

¼ m2B x�2� �

and l2B x�2� �

¼ m2M x�2� �

.It is easy to see that the example presented in Fig. 1 is con-

structed in such a way that always l1L ðx1Þ þ m1

L ðx1Þ 6 1;l1

Mðx1Þ þ m1Mðx1Þ 6 1; l1

Bðx1Þ þ m1Bðx1Þ 6 1 and l2

L ðx2Þ þ m2L ðx2Þ 6 1;

l2Mðx2Þ þ m2

Mðx2Þ 6 1; l2Bðx2Þ þ m2

Bðx2Þ 6 1. Therefore, in the frame-work of proposed approach, the hesitation degreespLðx1Þ; pMðx1Þ; pBðx1Þ and pLðx2Þ; pMðx2Þ; pBðx2Þ can be analysedas well.

An important question arises: does the fundamental property ofA� IFS ðlðxÞ þ mðxÞÞ 6 1 hold for all cases when we deal with theintersecting fuzzy classes? Below we shall show that this propertyholds only if membership functions of competing fuzzy classes sat-isfy jointly some reasonable and justified conditions which are notso important in the framework of traditional approach.

Let us consider some illustrative examples. In Fig. 1, we can seethat competing membership functions intersect in the pointswhere the values of membership functions are less or equal to0.5, and therefore the fundamental property of A� IFS ðlðxÞþmðxÞÞ 6 1 holds. In Fig. 2, we can see that if competing membershipfunctions intersect in the points where the values of membershipfunctions are equal to 0.5 we have ðlLðxÞ þ mLðxÞÞ ¼ 1; ðlMðxÞþmMðxÞÞ ¼ 1 and ðlBðxÞ þ mBðxÞÞ ¼ 1. Obviously there are no any hes-itation degrees in this case.

On the other hand, if the membership functions intersect in thepoint where their values are greater than 0.5 (see Fig. 2), the fun-damental property of A� IFS may be violated. It is easy to see thatin the interval ½x1; x3� we have ðlBðxÞ þ mBðxÞÞ > 1. In our opinion,this non-acceptable result is obtained as the membership func-tions lMðxÞ and lBðxÞ were built improperly. Really, all x 2 ½x1; x2�belong completely to the fuzzy class Medium and therefore theycannot belong to the another class, whereas we have lBðxÞP 0in this interval. Maybe, this type of reasoning seems to be toorestrictive one, but it reflects well the specificity of the decisionmaking based on the intersecting membership functions repre-senting competing fuzzy classes, such as Low; Medium;Big; ets.To avoid the above-mentioned problem, the membership func-tions of competing fuzzy classes should be constructed in such away that if one of them is equal to 1 then the other one shouldbe equal to 0.

Using the notation of A� IFS and DST (the basic probabilityassignment ðbpaÞ; m) we can represent the rules for the examplepresented in Fig. 1 as follows:



Fig. 2. The example of an inappropriate building of membership functions.


IF x1 ¼ x�1� �

Then m1ðHÞ ¼ l1L ðx�1Þ; m1

L ðx�1Þ� �

;

m1ðH;DÞ ¼ l1M x�1� �

; m1M x�1� ��

;m1ðDÞ ¼ l1B x�1� �

; m1B x�1� ��

;

IF x2 ¼ x�2� �

Then m2ðHÞ ¼ l2L x�2� �

; m2L x�2� ��

;

m2ðH;DÞ ¼ l2M x�2� �

; m2M x�2� ��

;m2ðDÞ ¼ l2B x�2� �

; m2B x�2� ��

:

ð25Þ

Of course, in this case we have: m1ðDÞ ¼ l1B x�1� �

; m1B x�1� ��

¼h0;1i; m2ðHÞ ¼ l2

L x�2� �

; m2L x�2� ��

¼ h0;1i.The next step is the combination of obtained bpas. To do that we

propose to use the direct intuitionistic fuzzy extension of theDempster’s rule (16). If bpas are usual real values, the Dempster’srule produces a resulting bpa m such that

PA # XmðAÞ ¼ 1. It is

important to note that in this case the normalisation (division by1-K) is obligatory. Obviously, if bpas are presented by IFVs there isno way to provide such a normalisation. On the other hand, all clas-sical operations on IFVs produce IFVs as well. Hence, using only thenumerator of direct intuitionistic fuzzy extension of (16) we obtain

m12ðAÞ ¼ �B\C¼Aðm1ðBÞ �m2ðCÞÞ ð26Þ

from which, in our example we get

m12ðHÞ ¼ m1ðHÞ �m2ðHÞ �m1ðHÞ �m2ðH;DÞ �m2ðHÞ �m1ðH;DÞ;m12ðDÞ ¼ m1ðDÞ �m2ðDÞ �m1ðDÞ �m2ðH;DÞ �m2ðDÞ �m1ðH;DÞ;m12ðH;DÞ ¼ m1ðH;DÞ �m2ðH;DÞ:

ð27Þ

Since the sum of m12ðHÞ �m12ðDÞ �m12ðH;DÞ is IFV, we can say thatcombination (27) is normalised in the sense of A� IFS.

In the case when belief intervals are used for representation ofIFVs, we can represent the rules for the example presented in Fig. 1as follows:

IF x1 ¼ x�1� �

Then m1ðHÞ ¼ BI1L x�1� �

; m1ðH;DÞ ¼ BI1M x�1� �

;

m1ðDÞ ¼ BI1B x�1� �

;

IF x2 ¼ x�2� �

Then m2ðHÞ ¼ BI2L x�2� �

; m2ðH;DÞ ¼ BI2M x�2� �

;

m2ðDÞ ¼ BI2B x�2� �

;

ð28Þ

where BI1L x�1� �

¼ l1L x�1� �

;1� m1L x�1� ��

;BI1M x�1� �

¼ l1M x�1� �

;1� m1M x�1� ��

;

BI1B x�1� �¼ l1

B x�1� �

;1�m1B x�1� ��

; BI2L x�2� �¼ l2

L x�2� �

;1�m2L x�2� ��

; BI2M x�2� �¼

l2M x�2� �

;1�m2M x�2� ��

;BI2B x�2� �¼ l2

B x�2� �

;1�m2B x�2� ��

.Since the operation �B and �B on belief intervals provide belief

intervals too, as in the case of representation of focal elements ofbpa directly by IFVs, we can formulate the combination rule, basedon the expression (26), for our example as follows:

m12ðHÞ ¼ m1ðHÞ�Bm2ðHÞ�Bm1ðHÞ�Bm2ðH;DÞ�Bm2ðHÞ�Bm1ðH;DÞ;m12ðDÞ ¼ m1ðDÞ�Bm2ðDÞ�Bm1ðDÞ�Bm2ðH;DÞ�Bm2ðDÞ�Bm1ðH;DÞ;m12ðH;DÞ ¼ m1ðH;DÞ�Bm2ðH;DÞ:

ð29Þ

In the considered case, we deal with the fuzzy rules with interval-valued consequent attributes. This can be treated as a considerable


complication of the problem. On the other hand, in [7] the solutionof much more complicated problem of approximate reasoningbased on the interval-valued fuzzy sets is presented.

Since m12ðHÞ�Bm12ðDÞ�Bm12ðH;DÞ is belief interval, we can saythat combination (29) is normalised in the sense of DST when be-lief intervals are treated as intervals enclosing a true power ofstatement (argument, proposition, hypothesis, etc.) that xj 2 X be-longs to the subset A # X (a new approach to the normalisation ofDempster’s combination rule in the case when the focal elementsare usual intervals, is presented in [27]). To represent the advanta-ges of proposed approaches, consider the critical example pre-sented in Fig. 3. Using conventional fuzzy logic, from (6) we getbpas m�1ðHÞ ¼ 0:55 and m�2ðDÞ ¼ 0:6. In this case, we can say onlythat we deal with a high conflict between the pieces of evidenceand cannot obtain a reasonable solution (diagnosis) in the consid-ered real-world example. On the other hand, according to the doc-tor’s opinion, in our case the diagnosis Pre-diabetes seems to bemore justified than Diabetes and Diabetes is more preferable thanHealth. The Pre-diabetes is intuitively obvious for the doctor in theconsidered example. Moreover, in his informal, but based on com-mon sense analysis, the doctor considered the values m1ðHÞ ¼0:55; m1ðH;DÞ ¼ 0:45; m1ðDÞ ¼ 0 and m2ðHÞ ¼ 0; m2ðH;DÞ ¼ 0:4;m2ðDÞ ¼ 0:6 as the arguments in favour of corresponding diagno-ses. It is easy to see that the sum of arguments in favour of Pre-diabetes m1ðH;DÞ þm2ðH;DÞ is grater than the sum of argumentsin favour of Diabetes m1ðDÞ þm2ðDÞ which is greater thanm1ðHÞ þm2ðHÞ.

Nevertheless, from (25) and (27) we get the following intuition-istic fuzzy-valued result:

m12ðHÞ ¼ h0:22;0:19i; m12ðH;DÞ ¼ h0:18;0:82i; m12ðDÞ ¼ h0:27;0:175i. To compare obtained IFVs, the rules (12) was used. The fol-lowing values of score function have been obtained:

Sðm12ðHÞÞ ¼ 0:027; Sðm12ðH;DÞÞ ¼ �0:64; Sðm12ðDÞÞ ¼ 0:095:

Since Sðm12ðDÞÞ > Sðm12ðHÞÞ > Sðm12ðH;DÞÞ then from (12) we ob-tain the counter-intuitive diagnosis-Diabetes.

Then using in (28) and (29) we get the following resulting beliefintervals m12ðHÞ¼ ½0:073;0:073�; m12ðH;DÞ¼ ½0:18;0:18�; m12ðDÞ¼½0:09;0:09� (factually, we have obtained real-valued results, butthis a specificity of our example and usually we obtain interval-valued results).

Since in our case, m12ðH;DÞ > m12ðDÞ > m12ðHÞ we obtain theintuitively obvious diagnosis – Pre-diabetes.

Let us consider the another critical example presented in Fig. 4.From (25) and (27) we get m12ðHÞ ¼ h0:6303;0:3696i;

m12ðH;DÞ ¼ h0;16;0:84i; m12ðDÞ ¼ h0;1i.In this case we have SðHÞ¼ 0:2607; SðH;DÞ¼�0:68; SðHÞ¼�1.

Therefore according to (12), the diagnosis Health is more prefera-ble than Pre-diabetes, but Pre-diabetes is not excluded.

The similar results we get from (28) and (29):

m12ðHÞ ¼ ½0:28;0:28�; m12ðH;DÞ ¼ ½0:16;0:16�; m12ðHÞ ¼ ½0;0�:



Fig. 3. Critical Example 1.

Fig. 4. Critical Example 2: x�1 ¼ 107:5; x�2 ¼ 153.


We can see that, as in the previous case, the diagnosis Health is onlymore preferable than Pre-diabetes, since argument in favour of Pre-diabetes seems to be considerable enough.

But using conventional fuzzy logic, from (24) we getbpas m1ðHÞ ¼ 1; m1ðH;DÞ ¼ 0; m1ðDÞ ¼ 0. We can see that in thiscase the diagnosis Pre-diabetes is completely excluded. This resultis in contradiction with common sense since in both test (seeFig. 4) we have non-zero and even considerable arguments infavour of Pre-diabetes.

Therefore, the use of conventional fuzzy logic may lead to theloss of information, which may be very important, e.g., if we extendthe task considering the third source of evidence where the diag-nosis Pre-diabetes is dominating one.

Summarising, we can say that the interpretation of A� IFS inthe framework of DST makes it possible to use more informationin the evidential reasoning, and as a consequence to obtain reason-able results when the synthesis of classical fuzzy logic and DST isfailed.

The counter-intuitive results obtained in the first critical exam-ple with the use of (25) and (27) when operations the � and � areused may be caused by bad properties of these operations (seesubSection 2.1).

4. Conclusion

In this paper, a new method for the rule-base evidential reason-ing based on the synthesis of A� IFS, fuzzy logic and the DST is pro-posed and analysed using the critical real-world example of type 2diabetes diagnostics. It is shown that the direct use of intuitionisticfuzzy values and classical operations on them may lead to thecounter-intuitive results. This may be a consequence of bad prop-erties of classical operation on intuitionistic fuzzy values. It isshown that the interpretation of A� IFS in the framework of DSTmakes in possible to use more information in the evidential


reasoning and as a consequence to obtain reasonable results whenthe synthesis of classical fuzzy logic and DST is failed.

References

[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.[2] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy

Sets Syst. 61 (1994) 137–142.[3] K. Atanassov, Intuitionistic Fuzzy Sets, Springer Physica-Verlag, Berlin, 1999.[4] G. Beliakov, H. Bustince, D.P. Goswami, U.K. Mukherjee, N. R Pal, On averaging

operators for Atanassov’s intuitionistic fuzzy sets, Inform. Sci. 182 (2011)1116–1124.

[5] E. Binaghi, P. Madella, Fuzzy Dempster-Shafer reasoning for rule-basedclassifiers, Intell. Syst. 14 (1999) 559–583.

[6] E. Binaghi, I. Gallo, P. Madella, A neural model for fuzzy Dempster-Shaferclassifiers, Int. J. Approx. Reason. 25 (2000) 89–121.

[7] H. Bustince, Indicator of inclusion grade for interval-valid fuzzy sets:application to approximate reasoning based on interval-valid fuzzy sets, Int.J. Approx. Reason. 23 (2000) 137–209.

[8] H. Bustince, P. Burillo, V. Mohedano, A method for inference in approximatereasoning based on normal intuitionistic fuzzy sets, Notes Intuition. Fuzzy Sets1 (1995) 51–55.

[9] H. Bustince, J. Fernandez, A. Kolesárová, R. Mesiar, Generation of linear ordersfor intervals by means of aggregation functions, Fuzzy Sets Syst. 220 (2013)69–77.

[10] H. Bustince, M. Galar, B. Bedregal, A. Kolesárová, R. Mesiar, A new approach tointerval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy sets applications, IEEE Trans. Fuzzy Syst. 21 (2013) 1150–1162.

[11] S.M. Chen, J.M. Tan, Handling multicriteria fuzzy decision-making problemsbased on vague set theory, Fuzzy Sets Syst. 67 (1994) 163–172.

[12] S.K. Dey, R. Biswas, A.R. Roy, Some operations on intuitionistic fuzzy sets,Fuzzy Sets Syst. 114 (2000) 477–484.

[13] A.P. Dempster, Upper and lower probabilities induced by a muilti-valuedmapping, Ann. Math. Stat. 38 (1967) 325–339.

[14] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk, H. Prade, Terminologicaldifficulties in fuzzy set theory – the case of ‘‘Intuitionistic Fuzzy Sets’’, FuzzySets Syst. 156 (2005) 485–491.

[15] L. Dymova, P. Sevastjanov, An interpretation of intuitionistic fuzzy sets interms of evidence theory: decision making aspect, Knowl.-Based Syst. 23(2010) 772–782.

[16] L. Dymova, P. Sevastianov, P. Bartosiewicz, A new approach to the rule-baseevidential reasoning: stock trading expert system application, Expert Syst.Appl. 37 (2010) 5564–5576.

[17] L. Dymova, P. Sevastjanov, The operations on intuitionistic fuzzy values in theframework of Dempster-Shafer theory, Knowl.-Based Syst. 35 (2012) 132–143.


http://refhub.elsevier.com/S0950-7051(14)00071-9/h0005











































[18] L. Dymova, P. Sevastjanov, K. Kaczmarek, A stock trading expert system basedon the rule-base evidential reasoning using Level 2 Quotes, Expert Syst. Appl.39 (2012) 7150–7157.

[19] J. Ghasemi, R. Ghaderi, M.R. Karami Mollaei, S.A. Hojjatoleslami, A novel fuzzyDempster-Shafer inference system for brain MRI segmentation, Inform. Sci.223 (2013) 205–220.

[20] J. Hodges, S. Bridges, C. Sparrow, B. Wooley, B. Tang, C. Jun, The development ofan expert system for the characterization of containers of contaminated waste,Expert Syst. Appl. 17 (1999) 167–181.

[21] D.H. Hong, C.-H. Choi, Multicriteria fuzzy decision-making problems based onvague set theory, Fuzzy Sets Syst. 114 (2000) 103–113.

[22] R. Huber, Scene classification of SAR images acquired from antiparallel tracksusing evidential and rule-based fusion, Image Vis. Comput. 19 (2001) 1001–1010.

[23] M. Ishizuka, K.S. Fu, J.T.P. Yao, Inference procedure and uncertainty for theproblem reduction method, Inform. Sci. 28 (1982) 179–206.

[24] V. Khatibi, G.A. Montazer, A fuzzy-evidential hybrid inference engine forcoronary heart disease risk assessment, Expert Syst. Appl. 37 (2010) 8536–8542.

[25] S. Parson, Current approaches to handling imperfect information in data andknowledge bases, IEEE Trans. Knowl. Date Eng. 8 (1996) 353–372.

[26] P. Sevastjanov, Numerical methods for interval and fuzzy number comparisonbased on the probabilistic approach and Dempster-Shafer theory, Inform. Sci.177 (2007) 4645–4661.

[27] P. Sevastianov, L. Dymova, P. Bartosiewicz, A framework for rule-baseevidential reasoning in the interval setting applied to diagnosing type 2diabetes, Expert Syst. Appl. 39 (2012) 4190–4200.

[28] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press,Princeton, 1976.

[29] E. Straszecka, Combining uncertainty and imprecision in models of medicaldiagnosis, Inform. Sci. 176 (2006) 3026–3059.

[30] R. Sun, Robust reasoning: integrating rule-based and similarity basedreasoning, Artif. Intell. 75 (1995) 241–295.

[31] X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities(I) (II), Fuzzy Sets Syst. 112 (2001) 387–405.


[32] X. Wang, F. Liu, L.C. Jiao, Z. Zhou, J. Yu, B. Li, J. Chen, J. Wu, F. Shang, Anevidential reasoning based classification algorithm and its application for facerecognition with class noise, Pattern Recogn. 45 (2012) 4117–4128.

[33] Y.- M. Wang, J.- B. Yang, D.- L. Xu, K.- S. Chin, Consumer preference predictionby using a hybrid evidential reasoning and belief rule-based methodology,Expert Syst. Appl. 36 (2009) 8421–8430.

[34] Z. Xu, Intuitionistic preference relations and their application in groupdecision making, Inform. Sci. 177 (2007) 2363–2379.

[35] D.-L. Xu, J. Liu, J.-B. Yang, G.-P. Liu, J. Wang, I. Jenkinson, J. Ren, Inference andlearning methodology of belief-rule-based expert system for pipeline leakdetection, Expert Syst. Appl. 32 (2007) 103–113.

[36] R.R. Yager, Generalized probabilities of fuzzy events from belief structures,Inform. Sci. 28 (1982) 45–62.

[37] R.R. Yager, D.P. Filev, Including probabilistic uncertainty in fuzzy logiccontroller modeling using Dempster-Shafer theory, IEEE Trans. Syst. ManCybernet. 25 (1995) 1221–1230.

[38] J.B. Yang, Rule and utility based evidential reasoning approach for multi-attribute decision analysis under uncertainties, Eur. J. Oper. Res. 131 (2001)31–61.

[39] J.B. Yang, J. Liu, J. Wang, H.S. Sii, H. Wang, Belief rule-base inferencemethodology using the evidential reasoning approach – RIMER, IEEE Trans.Syst. Man Cybernet. Part A – Syst. Humans 36 (2006) 266–285.

[40] J.B. Yang, J. Liu, D.L. Xu, J. Wang, H. Wang, Optimization models for trainingbelief-rule-based systems, IEEE Trans. Syst. Man Cybernet. Part A – Syst.Humans 37 (2007) 569–585.

[41] J. Yen, Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Trans. Syst.Man Cybernet. 20 (1990) 559–570.

[42] X. Yu, D. Huang, Y. Jiang, Y. Jin, Iterative learning belief rule-base inferencemethodology using evidential reasoning for delayed coking unit, Control Eng.Pract. 20 (2012) 1005–1015.

[43] L. Zadeh, A simple view of the Dempster-Shafer theory of evidence and itsapplication for the rule of combination, AI Mag. 7 (1986) 85–90.

[44] H.J. Zimmermann, P. Zysno, Latest connectives in human decision making,Fuzzy Sets Syst. 4 (1980) 37–51.









































































a new approach to the rule-base evidential reasoning in the intuitionistic fuzzy setting

Documents