The operations on interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory

Information Sciences 360 (2016) 256–272

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The operations on interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory Ludmila Dymova, Pavel Sevastjanov∗ Institute of Computer and Information Science, Czestochowa University of Technology, Dabrowskiego 73, 42-200 Czestochowa, Poland

a r t i c l e

i n f o

Article history: Received 5 June 2015 Revised 4 February 2016 Accepted 26 April 2016 Available online 30 April 2016 Keywords: Operations on interval-valued intuitionistic fuzzy values Dempster–Shafer theory of evidence Decision making

a b s t r a c t This work was motivated by the revealed limitations and drawbacks of the classical definition of interval-valued intuitionistic fuzzy set (IVIFS) proposed by Atanassov and Gargov and operations on interval-valued intuitionistic fuzzy values (IVIFV). It is shown that the classical definition of Atanassov’s IVIFS may lead to controversial results. Therefore, a new more constructive definition of IVIFS is proposed. It is shown that this new definition makes it possible to present IVIFVs in the framework of Dempster–Shafer theory (DST) as belief intervals with bounds presented by belief intervals. A new set of operations on IVIFVs in the framework of DST is proposed. It is proved that these operations provide IVIFVs in the sense of new definition of interval-valued intuitionistic fuzzy set. A critical analysis of commonly used operations for comparison of intuitionistic fuzzy values and IVIFVs is made. As a result, a new more justified operation for comparison of IVIFVs in the framework of interval extended Dempster–Shafer theory is proposed. It is proved that introduced set of operations on IVIFVs is free of revealed limitations and drawbacks of commonly used operations based on the classical definition of Atanassov’s IVIFS. The corresponding methods for aggregation of local criteria presented by interval-valued intuitionistic fuzzy values in the framework of interval-extended Dempster–Shafer theory are proposed and analysed. The proposed approach makes it possible to introduce the new interval-valued intuitionistic weighted geometric operators for the cases when the weighs are presented by intuitionistic and IVIFVs. These operators are not defined in the framework of classical approach. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Interval-Valued Intuitionistic Fuzzy Sets abbreviated here as A-IVIFS (the reasons for this are presented in [13]), were introduced in [1,4]. The fundamental characteristic of A-IVIFS is that the values of its membership and non-membership functions are intervals rather than exact numbers. Atanassov [2,3] defined some operations, relations and operators concerning A-IVIFS. Xu and Chen [54] proposed the interval-valued intuitionistic fuzzy weighted averaging operator for aggregation of interval-valued intuitionistic fuzzy values. They presented an application to the multiple criteria decision making (MCDM) with interval-valued intuitionistic fuzzy information.

∗

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

http://dx.doi.org/10.1016/j.ins.2016.04.038 0020-0255/© 2016 Elsevier Inc. All rights reserved.

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

257

In [49,50], the complete set of arithmetical operations on interval-valued intuitionistic fuzzy values (IVIFVs) (including the operations of IVIFVs comparison). Based on them, interval-valued intuitionistic fuzzy weighted averaging and weighted geometric operators were proposed as well. A-IVIFS is one of the possible generalisations of Fuzzy Sets Theory and currently is used mainly for solving MCDM problems and group decision making problems [6–9,22–24,26,28,29,37,42,43,46,47,51,54–58] when the values of local criteria (attributes) of alternatives and/or their weights are interval-valued intuitionistic fuzzy values (IVIFVs). Of course, this list of publications is not complete, but a detailed review of application of A-IVIFS for solving MCDM problems is out of scope of this paper. There are methodological problems in the framework of A-IVIFS theory similar to those in the body of classical (noninterval) Atanassov’s intuitionistic fuzzy set theory (A-IFS). There are many papers devoted to the theoretical problems of A-IFS (see [27] for an overview). There are different links between A-IFS and some other theories modelling imprecision. For example, Deschrijver and Kerre [12] established some interrelations between A-IFS and such theories as Interval Valued Fuzzy Sets, Type 2 Fuzzy Sets and Soft Sets. The semantic aspects of such interrelations were analysed by Grzegorzewski and Mrowka [18]. In our recent paper [14], we showed that there exists also a strong link between A-IFS and the Dempster–Shafer theory of evidence (DST). This link makes it possible to use directly Dempster’s rule of combination to aggregate local criteria presented by intuitionistic fuzzy values (IFVs) in MCDM problems. The usefulness of the developed method was illustrated using the known example of MCDM problem. Another link between A-IFS and DST was shown in [19,20] where the authors developed a theory of mass assignment as a variant of DST linked with A-IFS including inconsistent and contradictory evidence. The most important applications of A-IFS are MCDM problems when the values of local criteria (attributes) of alternatives and/or their weights are IFVs. Therefore it seems quite natural that the resulting alternative’s evaluation should be IFV too. Therefore, appropriate operations on IFVs used for aggregation of local criteria should be properly defined. Obviously, if the final scores of alternatives are IFVs, then appropriate methods for their comparison are needed to select the best alternative. Different definitions of operations on IFVs and their aggregation were proposed in the literature (see, for example [5,14]). Therefore, in our recent paper [15] we analysed their merits and drawbacks and extracted those of them that provide the results of operations on IFVs and aggregation with acceptable properties. As a result, a new set of operations on IFVs in the framework of DST was proposed in [15]. The classical A-IFS is an asymptotic case of A-IVIFS when interval-valued membership and non-membership functions contract to points [4]. Hence, we can expect that undesirable properties of classical operations on IFVs revealed in [5,15] should be the same for operations on IVIFVs presented in [49,50]. Therefore, in the current paper we present a new set of operations on IVIFVs in the framework of interval extension of DST, which are free of revealed in [5,15] undesirable properties. In this paper, we shall use theorems to prove positive properties of operations and convincing critical examples to illustrate undesirable ones. For these reasons the rest of this paper is set out as follows. Section 2 presents the basic definition of A-IVIFS, the commonly used arithmetical operations on IVIFVs, IFVs and the methods for their comparison. The undesirable properties of these operations presented in [5,15] and some new ones revealed recently are analysed. In Section 3, we provide a critical analysis of commonly used basic definition of A-IVIFS proposed in [4] to elicit its disadvantages which may lead to controversial results. In this section, we propose a new more constructive definition of A-IVIFS. We show that this new definition makes it possible to present IVIFVs in the framework of interval-extended DST as belief intervals with bounds presented by belief intervals. A new set of operations on IVIFVs in the framework of interval extension of DST is proposed. It is proved that these operations provide IVIFVs in the sense of new definition of A-IVIFS. In this section, we provide a critical analysis of commonly used operations for comparison of IFVs and IVIFVs. Then we introduce a more justified operation for comparison of IVIFVs in the framework of interval-extended DST. In Section 4, we prove that introduced set of operations on IVIFVs is free of revealed limitations and drawbacks of commonly used operations on IVIFVs based on the classical definition of A-IVIFS proposed in [4]. In Section 5, the advantages of the proposed approach are illustrated by numerical examples of the solution of MCDM problems in IVIF setting. Finally, the concluding section summarises the paper. 2. The basic definitions of interval-valued intuitionistic fuzzy set theory In [4], Atanassov and Gargov defined A-IVIFS as follows. Definition 1. Let X be a finite universal set. Then an interval-valued intuitionistic fuzzy set A˜ in X is an object having the form

A˜ =







x, MA˜ (x ), NA˜ (x ) |x ∈ X ,

(1)

where MA˜ (x ) ⊂ [0, 1] and NA˜ (x ) ⊂ [0, 1] are intervals such that for all x ∈ X

sup MA˜ (x ) + sup NA˜ (x ) ≤ 1.

(2)

Hereinafter, we will deal with IVIFVs. Therefore we will use the following notation for IVIFV A:





A = [μLA , μUA ], [νAL , νAU ] ,

(3)

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where [μLA , μUA ] and [νAL , νAU ] are interval valued degrees of membership and non-membership to A. The complete set operations on IVIFVs is presented in [49,50] as follows

AB= AB=



   μLA + μLB − μLA μLB , μUA + μUB − μUA μUB , νAL νBL , νAU νBU ,

(4)



   μLA μLB , μUA μUB , νAL + νBL − νAL νBL , νAU + νBU − νAU νBU ,

(5)

    λA = 1 − (1 − μLA )λ , 1 − (1 − μUA )λ , (νAL )λ , (νAU )λ , Aλ =



(6)

   (μLA )λ , (μUA )λ , 1 − (1 − νAL )λ , 1 − (1 − νAU )λ ,

(7)

where λ > 0. Let A1 , A2 , ..., An be IVIFVs representing the values of n local criteria for some alternative and w1 , w2 ,...,wn be real-valued n  weights of local criteria such that wi > 0, i = 1 to n and wi = 1. i=1

Then based on the operations (4)–(7), the following interval-valued intuitionistic weighted arithmetic mean IVIWAM and interval-valued intuitionistic weighted geometric mean IVIWGM operators were obtained [49,50]:



IV IWAM (A1 , A2 , . . . , An ) =

1−

n 

(1 − μ )

L wi , Ai

i=1



IV IW GM (A1 , A2 , . . . , An ) =

n 

1−

n 



(1 − μ )

U wi Ai

i=1

(μLAi )wi ,

i=1

n 



(μUAi )wi , 1 −

i=1

,

n 

(ν )

L wi , Ai

i=1 n  i=1

(1 − νALi )wi , 1 −

n 



(ν )

U wi Ai

,

(8)

i=1 n 

(1 − νAUi )wi

.

(9)

i=1

To compare IVIFVs, the so-called score S(A) and accuracy H(A) functions were introduced in [49] as follows:

S (A ) = H (A ) =

μLA + μUA − νAL − νAU 2

, S (A ) ∈ [−1, 1],

μLA + μUA + νAL + νAU 2

, H (A ) ∈ [0, 1].

(10)

(11)

Then order relations between any pair of IVIFVs A and B were presented in [49] as follows:

I f (S(A ) > S(B )), then B is smal l er than A; I f (S(A ) = S(B )), then

(1 ) I f (H (A ) = H (B )), then A = B; (2 ) I f (H (A ) < H (B )) then A is smal l er than B.

(12)

There are more complicated rules for comparison of IVIFVs proposed in the literature [[38], [44], [45]]. Generally, they are based on the use of more parameters than (12). Nevertheless, at this stage of our analysis the most popular rules (12) seem to be enough justified although their disadvantages are noted in [38]. It was proved in [49,50] that the operations (4)–(7) provide IVIFVs and have the following algebraic properties: Let A and B be IVIFVs. Then

A  B = B  A,

(13)

A  B = B  A,

(14)

λ(A  B ) = λA  λB,

(15)

( A  B )λ = Aλ  Bλ ,

(16)

λ1 A  λ2 A = (λ1 + λ2 )A, λ1 , λ2 > 0,

(17)

Aλ1  Aλ2 = Aλ1 +λ2 ,

λ1 , λ2 > 0. μLA

μUA

(18)

νAL

νAU ,

In the asymptotic case, when = and = IVIFV A reduces to the ordinary IFV a: A = a = μa , νa . Therefore from the above set of operations on IVIFVs (4)–(12) we obtain the set of classical operations on ordinary IFVs:

a  b = μa + μb − μa μb , νa νb ,

(19)

a  b = μa μb , νa + νb − νa νb .

(20)

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

  λa = 1 − (1 − μa )λ , νaλ ,   aλ = μλa , 1 − (1 − νa )λ ,

(21) (22)

where λ > 0.



IWAM = w1 a1  w2 a2  · · ·  wn an = 1 −

n 

( 1 − μa i ) w i ,

i=1

IW GM =

i aw 1



i aw 2

259

n · · ·  aw n =

n 

μ

wi ai , 1

−

i=1

n 

n 

ν

wi ai

.

(23)

i=1

( 1 − νa i ) w i ,

(24)

i=1

S(a ) = μa − νa , S(a ) ∈ [−1, 1],

(25)

H (a ) = μa + νa , H (a ) ∈ [0, 1].

(26)

I f (S(a ) > S(b)), then b is smal l er than a; I f (S(a ) = S(b)), then

(1 ) I f (H (a ) = H (b)), then a = b; (2 ) I f (H (a ) < H (b)) then a is smal l er than b.

(27)

It was proved in [48] that operations (19)–(22) provide IFVs and have good algebraic properties (13)–(18). Nevertheless, in [5,15] it was shown that operation (19)–(24) and (27) have some undesirable properties which may lead to the non-acceptable results in applications. In [5], it was shown that the aggregation operation (23) is not consistent with the aggregation operation on the ordinary fuzzy sets (Ordinary Weighted Arithmetic Mean OWAM, where μ= 1−ν ). In n [5],using corresponding t-norms and t-conorms, the following simple expression was inferred: IWAM = i=1 wi μi , wi νi . It is easy to show that this operator is consistent with the aggregation operator on the ordinary fuzzy sets. Nevertheless, the methodological problem is that this operator cannot be obtained using operations (19) and (23). The undesirable property of ordering (27) revealed in [5] is that it is not preserved under multiplication by a scalar: A < B does not necessarily imply λA < λB, λ > 0. Therefore, in [5], with the use of Lukasiewicz t-norms and t-conorms, the following expression was inferred λA =< λμA , 1 − λ(1 − νA ) > , λ ∈ [0, 1]. It is easy to prove that the use of this operator guarantees that for IFVs A and B such that A < B we always have λA < λB , λ ∈ [0, 1]. Unfortunately, the algebraic properties (15) and (17) with this operator do not hold. Therefore, we can say that the use of Lukasiewicz t-norms and t-conorms is not a reliable way to obtain the set of operations on IVFVs with good properties. In [15], we have shown that the addition (19) is not an addition invariant operation and the aggregation operation (23) is not monotone with respect to the ordering (27). The explanations and numerical examples are presented in [15]. Recently, we have found some undesirable properties of operations (20) and (24): 1. The multiplication (20) is not always monotone with respect to the ordering (27). Consider an example: Example 1. Let a = 0.1, 0.3, b = 0.4, 0.5, c = 0.2, 0.1. Then S(a) = −0.2, S(b) = −0.1 and therefore according to (27) we get b > a. On the other hand, a  c = 0.02, 0.37, b  c = 0.08, 0.55, S(a  c ) = −0.35, S(b  c ) = −0.47. Therefore S(ac) > S(bc) and ac > bc whereas b > a. 2. The aggregation operation (24) is not monotone with respect to the ordering (27). Consider an example: Example 2. Let a = 0.4, 0.5, b = 0.35, 0.448, c = 0.5, 0.5, w1 = 0.5, w2 = 0.5. Since S(a) = −0.1 and S(b) = −0.098 we get b > a. On the other hand, since IWGM(a, c) = 0.4472, 0.5, IWGM(b, c) = 0.41833, 0.474643, S(IWGM(a, c)) = −0.0528 and S(IWGM(b, c)) = −0.0563 we have IWGM(a, c) > IWGM(b, c) whereas b > a. The classical A-IFS is an asymptotic case of A-IVIFS when interval-valued membership and non-membership functions contract to points [4]. Therefore we can expect that undesirable properties of classical operations on IFVs presented above should be the same for operations on IVIFVs from [49,50] presented in this section. In [15], we obtained the complete set of operations on IFVs in the framework of DST, which are free of the drawbacks of classical operation laws of A-IFS. In the following section, we present the set of operations on IVIFVs in the framework of interval-extended DST which is free of the described above drawbacks.

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3. Operations on IVIFVs in the framework of interval-extended DST 3.1. A new definition of IVIFV First of all we will analyse the commonly used definition of A-IVIFS (and IVIFV as well), proposed by Atanassov and Gargov [4] (see Definition 1) and reveal its drawbacks. As a result, a new more constructive definition will be proposed. Let us briefly recall the basics of A-IFS. Let μA (x) be the degree of membership of x in the set A and ν A (x) be the degree of non-membership of x in the set A. Then π A (x) such that μA (x)+ ν A (x)+π A (x) = 1 is the hesitation degree, i.e., degree to which any of considered hypothesis (membership and non-membership) cannot  be definitely   excluded.  It is implicitly assumed in Definition 1 that the upper bound of IVIFV A = μLA , μUA , νAL , νAU , is an ordinary IFV, i.e,

μUA + νAU ≤ 1.

If so, there will be a hesitation degree πAU at the upper bound of A such that μUA + νAU + πAU = 1. It is easy to see that in that case all other possible combinations of μA , ν A , π A in the intervals [μLA , μUA ], [νAL , νAU ], [πAL , πAU ] provide μA + νA + πA < 1. Therefore, there is only one IFV at the upper bound of IVIFV A. This is a first obvious drawback of Definition 1. Let us consider A = [0.6, 0.7], [0.1, 0.25]. Since μUA + νAU = 0.7+0.25=0.95 < 1, then according to the Definition 1, A is a correct IVIFV. On the other hand, on the bounds of A we have the following values of score functions S(A )U = μUA − νAU = 0.45, S(A )L = μLA − νAL = 0.5. Since S(A)L > S(A)U , then according to the rule (27) the lower bound of considered IVIFV A is greater than its upper bound. Obviously, such a result have no reasonable explanations. It is easy to see that the Definition 1 says nothing about the left bound of IVIFV. Toovercome currently the following formal approach is used ( see, for example [43,58]). If A is IVIFV then   this problem,    A = μLA , μUA , νAL , νAU , πAL , πAU , where πAU = 1 − μLA − νAL , πAL = 1 − μUA − νAU . It is easy to show that such approach provides wrong results. Really μUA + νAU + πAU = μUA + νAU + 1 − μLA − νAL = 1 + (μUA − μLA ) + (νAU − νAL ). Since from definition of interval we have (μUA − μLA ) ≥ 0 and (νAU − νAL ) ≥ 0, finally we get μUA + νAU + πAU ≥ 1. Consider an example. Let A = [0.3, 0.6], [0.1, 0.2] be IVIFV. Since μUA + νAU = 0.6+0.2=0.8 < 1, then according to the Definition 1, A is a correct IVIFV. On the other hand, πAU = 1 − μLA − νAL = 1−0.3–0.1=0.6 and πAL = 1 − μUA − νAU = 1− 0.6– 0.2=0.2. Then μUA + νAU + πAU = 1.4. Therefore the upper bound of IVIFV A is not a correct IFV. In our opinion, the above problem is based on faulty treatment of interval arithmetic rules. The expressions for calculation of πAU and πAL are implicitly based on the assumption that [μLA , μUA ]+ [νAL , νAU ]+[πAL , πAU ] = 1. Since in the left hand side of this expression we have an interval and in the right hand side we have a real value, then equality is impossible. Taking into account the above analysis, here we introduce a new definition of IVIFV. In this definition, the upper and lower bounds of IVIFV will be always ordinary IFVs and upper bound will be always greater than a lower one in sense of rule (27). Definition 2. Let X be a finite universal set. Then an interval-valued intuitionistic fuzzy set A˜ in X is an object having the form

A˜ =







x, MA˜ (x ), NA˜ (x ) |x ∈ X ,

(28)

where MA˜ (x ) ⊂ [0, 1] and NA˜ (x ) ⊂ [0, 1] are intervals such that for all x ∈ X

sup MA˜ (x ) + inf NA˜ (x ) ≤ 1, inf MA˜ (x ) + sup NA˜ (x ) ≤ 1.

(29)

The definition for IVIFV is obtained from the Definition 2 as follows. Definition 3. A regular interval-valued intuitionistic fuzzy value A is an object having the form

A= where

μLA ,



   μLA , μUA , νAL , νAU ,

μUA ,

νAL ,

νAU

(30)

∈ [0, 1] and

μLA + νAU ≤ 1, μUA + νAL ≤ 1.

(31)

      It is easy to see that μUA , νAL is the maximal (in sense of rule (27)) IFV attainable in A = μLA , μUA , νAL , νAU . Therefore  U L  L U μA , νA is the upper bound of IVIFV. Similarly, μA , νA is the lower bound of IVIFV A. It is easy to see that upper bound of such IVIFV is always not lesser of its lower bound in sense    of rule  (27) and  these bounds are correct IFVs. It is clear that for any correct IFV μ, ν such that μ ∈ μLA , μUA and ν ∈ νAL , νAU we have μLA , νAU ≤ μ, ν ≤ μUA , νAL . Hence it is implicitly assumed that IVIFV is a set of correct IFVs bounded according to the Definition 3. This is in compliance with a practice of obtaining IVIFVs in many real-world situations and seems to be justified enough from methodological point of view. Really, when we deal with intervals, they should have the left bounds that are lesser than right ones. If we analyse real-valued intervals, then they should consist of real values. If we introduce integer-valued intervals, then they should consist of integer numbers. Similarly, IVIFVs should be intervals consist of only IFVs. The introduced definition  of regular IVIFV may   be illustrated as follows. Let us consider IVIFV A = μLA , μUA , νAL , νAU = {[0.4, 0.8], [0.1, 0.3]}.

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

261

We can see that S(A )U = μUA − νAU = 0.7 and S(A )L = μLA − νAL = 0.1. Hence the upper bound of A is greater than the lover one. We can see also that that μLA + νAU = (0.4 + 0.3 ) < 1, μUA + νAL = (0.8 + 0.1 ) < 1. Therefore, according to the Definition 3, A is a regular IVIFV and, e.g., may be presented by the non-increasing set of IFVs as follows: A = 0.8, 0.1, 0.7, 0.1, 0.7, 0.2, 0.6, 0.3, 0.5, 0.3, 0.4, 0.3. In our recent paper [15], we have presented the definition and operation laws of A-IFS in terms of DST and obtained the operations on IFVs, which are free of the drawbacks (1)–(4) described above. Therefore, here we present the introduced above definitions in terms of interval-extended DST. 3.2. A new definition of IVIFV in the framework of interval-extended DST Here we present a brief description of some basics of the Dempster–Shafer theory of evidence (DST) needed for the subsequent analysis. The DST was developed by Dempster [10,11] and Shafer [34] and its basics may be presented as follows. Assume A is a subset of X. A subset A may be treated also as a question or proposition and X as a set of propositions or mutually exclusive hypotheses or answers [36]. A DST belief structure has the associated mapping m, called basic assignment function (or mass assignment function), from subsets of X into a unit interval, m: 2X → [0, 1] such that  m ( ∅ ) = 0, m(A ) = 1. The subsets of X for which the mapping does not assume a zero value are called focal elements. A⊆X

In the framework of classical DST the null set is never a focal element. In [34], the measures of belief and plausibility associated with DST belief structure were introduced as follows. The measure of belief is a mapping Bel: 2X → [0, 1] such that for any subset B of X



Bel (B ) =

m(A ).

(32)

∅ =A⊆B

The measure of plausibility associated with m is a mapping Pl : 2X → [0, 1] such that for any subset B of X

P l (B ) =



m(A ).

(33)

A∩B =∅

It is seen that Bel(B) ≤ Pl(B). An interval [Bel(B),Pl(B)] is called the belief interval (BI). It can also be interpreted as an interval enclosing the “true probability” of B [34]. This information of DST is quite enough to represent A-IFS in terms of DST. In our recent paper [14], we have shown that in the framework of DST the triplet μA (x), ν A (x), π A (x) represents the correct basic assignment function and IFV A(x) = μA (x), ν A (x) may be represented as follows: A(x ) = BIA (x ) = [BelA (x ), P lA (x )] = [μA (x ), 1 − νA (x )] (see [14,15] for more detailed and formal definitions). Obviously, this definition seems to be a redefinition of A-IFS in terms of Interval Valued Fuzzy Sets, but in [14,15] we showed that using DST semantics it is possible to enhance the performance of A-IFS when dealing with the operations on IFVs and MCDM problems. Based on the described above link between A-IFS and DST, the Definition 2 can be rewritten as follows: Definition 4. Let X be a finite universal set. Then an interval-valued intuitionistic fuzzy set A˜ in X is an object having the form

A˜ = where



 

  |x ∈ X ,

x, BIA˜ (x )





(34)



BIA˜ (x ) = BIAL˜ (x ), BIAU˜ (x )

(35)

is the belief interval with bounds presented by the belief intervals:









BIAL˜ (x ) = inf MA˜ (x ), 1 − sup NA˜ (x ) , BIAU˜ (x ) = sup MA˜ (x ), 1 − inf NA˜ (x ) ,

(36)

where MA˜ (x ) ⊂ [0, 1] and NA˜ (x ) ⊂ [0, 1] are intervals such that for all x ∈ X

sup MA˜ (x ) + inf NA˜ (x ) ≤ 1, inf MA˜ (x ) + sup NA˜ (x ) ≤ 1.

(37)

The definition for IVIFV is obtained from Definition 4 as follows. Definition 5. An interval-valued intuitionistic fuzzy value A is an object having the form





A = [BIA ] = BIAL , BIAU ,

(38)

262

where

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272





 μLA , 1 − νAU ,     BIAU = BelAU , P lAU = μUA , 1 − νAL , BIAL = BelAL , P lAL =



(39)

μLA , μUA , νAL , νAU ∈ [0, 1] and

μLA + νAU ≤ 1, μUA + νAL ≤ 1.





(40)

It is important that according to the last definition all intervals [BI] = BIL , BIU with bounds presented by correct belief intervals (0 ≤ Bel ≤ Pl ≤ 1) such that BIU ≥ BIL may be treated as IVIFVs. In the above definitions, we have introduced belief intervals bounded by belief intervals (BIBBI). Therefore, we can say that in our case we deal with the someway interval-extended version of DST. We do not intend to develop here the whole interval-extended DST and restrict ourselves only by consideration of BIBBI. It is worth noting that BIBBI appear naturally in the rule-base evidential reasoning in the interval setting [31]. In the above definitions of IVIFS and IVIFV, we have referred to the rule (12) for IVIFV comparison. Therefore, an appropriate procedure for belief interval comparison is needed as well. In [15], we used directly the rule (27) and its representation in terms of DST. Nevertheless we have revealed some problems rather of methodological nature concerned with the rules (12) and (27). 3.3. The methods for comparison of IFVs, IVIFVs and belief intervals In [16], we analysed the limitations of the known methods for IFVs comparison based on the reasoning (27). There are three important limitations of the method (27) revealed in [16]: (1) This method generally does not provide a technique for estimation of a degree to which an IFV is greater/lesser than another one. But such an information is usually important for a decision maker. (2) The lack of continuity in comparison of IFVs by this method. (3) In the method (27), the implicitly introduced local “net profit” and “risk” criteria are not taken into account simultaneously. Nevertheless, in the decision making practice, a small value of “net profit” criterion may be compensated by the small value of “risk” criterion and so on. It is important that the above mentioned limitations of IFVs comparison based on the method (27) are propagated to the case of IVIFVs comparison. Therefore, to avoid the above mentioned limitations of the known methods, in [16], it was proposed to formulate the problem of IFVs and IVIFVs comparison directly as two-criteria task. It is important that Hong and Choi [21] showed that the relation between functions S and H is similar to the relation between mean and variance in statistics. Strictly speaking, the score function S may be treated as a mean and the hesitation degree or degree of uncertainty π A = 1 − H may be treated as a variance. Based on such “statistical” premise, Xu [48] used the functions S and H to construct order relations (27) between any pair of intuitionistic fuzzy values. Nevertheless, these relations are in contradiction with the basics of conventional statistics. Let A and B be samples of measurements with corresponding uniform or normal probability distributions such that they have a common mean (meanA = meanB ), but different variances (σ A > σ B ). Then using statistical methods it is impossible to prove that B is greater than A or A is greater than B. Therefore we can say that the comparison of IFVs based on the assumption that IFVs having the same values of score functions S are equal ones seems to be more preferable and reasonable. On the other hand, the comparison of intervals, fuzzy values or IFVs is a context dependent problem [16,33]. Nevertheless, for the aims of this paper it is enough to use the simplified, but justified method for IFVs and IVIFVs comparison:

I f (S(A ) > S(B )), then A > B;

(41)

I f (S(A ) = S(B )), then A = B,

where the score function S is calculated using (10) in the case of IVIFVs and with the use of (25) in the case of IFVs. To develop a method for IVIFVs comparison, let us first consider the problem of belief intervals comparison as a part of more general problem of interval comparison. There are many methods for interval comparison proposed in the literature (see reviews in [30,39]). Currently, the most popular are the simple heuristic methods proposed by Facchinetti et al. [[17]], Wang et al. [40,41] and Xu and Da [52]. These methods provide the degree of possibility that an interval is greater/lesser than another one. Xu and Chen [53] proved that these methods are equivalent ones. For intervals B = [bL , bU ], A = [aL , aU ], the possibilities of B ≥ A and A ≥ B are defined in [40,41] as follows:



P (B ≥ A ) =



P (A ≥ B ) =





max 0, bU − aL − max 0, bL − aU aU − aL + bU − bL





max 0, aU − bL − max 0, aL − bU aU − aL + bU − bL



,

(42)

.

(43)



L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

263

The method presented by expressions (42) and (43) have some limitations (e.g., the intersection and inclusion cases should be considered separately [33]) which were analysed in [16]. Therefore, in [16], the simplest method based on the difference of midpoints of compared intervals was proposed. The justification of this method is based on the analysing the operation of interval contraction. The advantages of this method were showed in [16] in comparison with the results obtained using some popular methods including the so-called probabilistic approach to the interval comparison [30]. It is important that for intervals having common midpoint, from (42) and (43) we get P(B ≥ A) = P(A ≥ B) = 0.5, i.e., A = B, and this result does not depend on the widths of compared intervals. Therefore, we can conclude that the equality of intervals with a common centre is an inherent property of interval arithmetic (see [15] for more detail). Hence, we can say that the interval comparison based on the assumption that intervals having a common centre are equal ones seems to be more preferable and reasonable. Nevertheless, generally the interval comparison is a context dependent problem [33]. Based on the above assumptions and the property of belief intervals 0 ≤ Bel ≤ Pl ≤ 1 we propose here the following method for belief intervals comparison. Let A, B be IFVs and BI(A) = [BelA , PlA ]=[μA , 1 − νA ], BI(B) = [BelB , PlB ]=[μB , 1 − νB ] be the corresponding belief intervals. Then

P (A > B ) =

max {0, ( (BelA + P lA ) − (BelB + P lB ) )} , BelA + P lA + BelB + P lB

P (B > A ) =

max {0, ( (BelB + P lB ) − (BelA + P lA ) )} . BelA + P lA + BelB + P lB

(44)

This method has following natural properties: If A = B, i.e., A and B have a common centre we have (BelA + P lA ) = (BelB + P lB ) and P (A > B ) = P (B > A ) = 0, If BelA = P lA = 1 and BelB = P lB = 0, then P (A > B ) = 1, P (B > A ) = 0, If BelB = P lB = 1 and BelA = P lA = 0, then P (A > B ) = 0, P (B > A ) = 1. It is shown in Fig. 1 that proposed method provides reasonable results when one of the compared BI is included into another one, when they intersect and when they have no a common area. The extension of this method to the case if IVIFVs is presented as follows. The centre of IVIFV A may be naturally presented as follows:



C (A ) =



 1 BelAL + PlAL BelAU + PlAU 1 L BIA + BIAU = + . 2 2 2 2

It is easy to see that two IVIFVs A and B are equal ones, i.e., they have a common centre if

BelAL + P lAL + BelAU + P lAU = BelBL + P lBL + BelBU + P lBU .









Therefore, for IVIFVs A = [BIA ] = BIAL , BIAU , B = [BIB ] = BIBL , BIBU (see Definition 5) we have

 

P (A > B ) =

max 0,





BelAL + P lAL + BelAU + P lAU + BelBL + P lBL + BelBU + P lBU

 

P (B > A ) =



BelAL + P lAL + BelAU + P lAU − BelBL + P lBL + BelBU + P lBU

max 0,





BelBL + P lBL + BelBU + P lBU − BelAL + P lAL + BelAU + P lAU

BelAL + P lAL + BelAU + P lAU + BelBL + P lBL + BelBU + P lBU

,

 .

(45)

This method has the following natural properties: If A = B, i.e., BelAL + P lAL + BelAU + P lAU = BelBL + P lBL + BelBU + P lBU , then P(A > B) = P (B > A ) = 0, If BelAL + P lAL + BelAU + P lAU = 1 and BelBL + P lBL + BelBU + P lBU = 0, then P (A > B ) = 1, P (B > A ) = 0, If BelBL + P lBL + BelBU + P lBU = 1 and BelAL + P lAL + BelAU + P lAU = 0, then P (A > B ) = 0,P (B > A ) = 1. 3.4. Operations on IVIFVs in the framework of interval-extended DST Here we will use the Definition 5, i.e., we will consider IVIFVs as intervals bounded by belief intervals. The belief interval BIA may be interpreted as an interval enclosing the “true” probability that x ∈ X belongs to the set A ⊆ X [34]. On the other hand, such “probabilistic ” interpretation of belief interval is not unique in the body of DST. Moreover, the basic definitions of DST can be formulated without the term “probability”, as it is known that DST may be treated as a generalisation of the probability theory as well as the possibility theory. Hence the belief interval may be treated, e.g., as an interval enclosing a true power of statement (argument) that x ∈ X belongs to the set A ⊆ X [15]. Obviously, the choice of an appropriate treatment of belief interval is a context dependent problem and different treatments imply different mathematical formalisms when we introduce operations on the belief intervals representing IFVs.

264

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

Fig. 1. The examples of belief intervals relations.

Therefore, in [15], two sets of operations on IFVs based on the interpretation of intuitionistic fuzzy sets in the framework of DST were proposed and analysed. The first set of operations is based on the treatment of belief interval as an interval enclosing a true probability. The second set of operations is based on the treatment of belief interval as an interval enclosing a true power of some statement (argument, hypothesis, ets). It is shown, that the non-probabilistic treatment of belief intervals representing IFVs performs better than the probabilistic one. It is important that operations based on the probabilistic and non-probabilistic treatments of belief intervals representing IFVs perform better than operations on IFVs defined in the framework of conventional A-IFS. Therefore, here we will use the non-probabilistic treatment of belief intervals representing IVIFVs. In [15], we defined the addition of BIs based on the following reasoning. Let X be a universal set. Assume A is a subsets of X. It is important to note that in the framework of DST a subset A may be treated also as a question or proposition and X as a set of propositions or mutually exclusive hypotheses or answers. Therefore, in such a context, a belief interval BIA = [BelA , P lA ] may be treated as an interval enclosing a true power of statement (argument, proposition, hypothesis, etc) that x ∈ X belongs to the set A ⊆ X. Obviously, the value of such a power lies in interval [0,1]. Therefore, a belief interval BIA = [BelA , P lA ] as a whole may be treated as an imprecise (interval-valued) statement (argument, proposition, hypothesis, etc.) that x ∈ X belongs to the set A ⊆ X. Based on this reasoning, we can say that if we pronounce this statement, we can obtain some result, e.g., as a reaction on this statement or as an answer to the question, and if we repeat this statement twice, the result does not change. Such a reasoning implies the following property of addition operator (denoted here by BNP as in [15]): Bel +Bel P l +P l BIA BNP BIA = [ A 2 A , A 2 A ] and BIA = BIA BNP BIA BNP · · · BNP BIA . This is possible only if we define the addition BNP of belief intervals as their averaging. Therefore, if A and B are IFVs, then

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

 A BNP B = BIA BNP BIB =

BelA + BelB P lA + P lB , 2 2

265

 = BIABNP B .

(46)

Therefore, if we have n different statements Ai represented by belief intervals BIAi then their sum BNP can be defined as follows:



A1 BNP A2 BNP · · · BNP An = BIA1 BNP BIA2 BNP · · · BNP BIAn = BIA1 BNP A2 BNP ···BNP An =



n n 1 1 BelAi , P lAi . (47) n n i=1

i=1

The natural extensions of the above definitions to the case of IVIFVs may be presented as follows. Let A = [BIA ]= [BIAL , BIAU ] and B = [BIB ]= [BIBL , BIBU ] be IVIFVs (see Definition 5). Then using interval arithmetic rules and expression (46) we get



A BNP B = [BIA ] BNP [BIB ] = =



 

BIAL BNP BIBL , BIAU BNP BIBU



=







BIAL , BIAU BNP BIBL , BIBU



 

BelAL + BelBL P lAL + P lBL , , 2 2





BelAU + BelBU P lAU + P lBU , 2 2

= BIAL BNP B , BIAUBNP B = BIABNP B Since 0 ≤ BIAL 

BNP B

≤ BIAU







(48)





≤ 1, the resulting BIABNP B is an interval bounded by belief intervals.

BNP B

Therefore, the introduced operation of addition of IVIFVs provides IVIFVs. If we have n different statements Ai represented by intervals [BIAi ] bounded by belief intervals, then



A1 BNP A2 BNP · · · BNP An = BIAL 1 BNP A2 BNP ···BNP An , BIAU1 BNP A2 BNP ···BNP An



1 = n

n 

BelAL i ,



n 

i=1

P lAL i

,

i=1

n 

BelAUi ,

i=1

n 





P lAUi

(49)

i=1

It is easy to see that this operation on IVIFVs provides IVIFVs. The operation of multiplication of IVIFVs A and B presented by intervals [BIA ] and [BIB ] bounded by belief intervals may be defined as interval extension of usual interval multiplication operation [25]. Taking into account that always 0 ≤ BelAL , P lAL , BelAU , P lAU ≤ 1 and 0 ≤ BelBL , P lBL , BelBU , P lBU ≤ 1, i.e., we always deal with positive belief intervals, we get

[BIA ] BNP [BIB ] =







= BIAL BIBL , BIAU BIBU =









BIAL , BIAU BNP BIBL , BIBU





 

BelAL BelBL , P lAL P lBL , BelAU BelBU , P lAU P lBU



= BIAL BNP B , BIAUBNP B .

(50)

It is easy to see that this operation always provides IVIFV. The multiplication λA = λ[BIA ] by scalar λ is defined as follow

       λ[BIA ] = λ BIAL , BIAU = λ BelAL , PlAL , λ BelAU , PlAU       = λBelAL , λP lAL , λBelAU , λP lAU = BIλL A , BIλUA ,

(51)

where λ is a real value such that λ ∈ [0, 1] as for λ > 1 this operation does not always provide a true belief interval. This restriction is justified enough since we define operations on belief intervals to deal with MCDM problems, where λ usually represents the weight of local criterion, which is lesser than 1. Obviously, the operation (51) provides IVIFVs. The multiplication BBNP A, where B = BIB is IFV and A = [BIA ] is IVIFV is defined as follow

B BNP A = BIB BNP [BIA ] = [BelB , P lB ] BNP









 

BelAL , P lAL , BelAU , P lAU



= BelB BelAL , P lAL , P lB BelAU , P lAU =







 

BelB BelAL , BelB P lAL , P lB BelAU , P lB P lAU



.

(52)

It is easy to see that operation (52) provides IVIFVs. Taking into account the properties of belief intervals, the power operation Aλ = [BIA ]λ is defined as follows:



[BIA ]λ = BIAL , BIAU

λ

=



BelAL , P lAL

λ 

, BelAU , P lAU

λ 

=



BelAL

λ 

, P lAL

λ   ,

BelAU

λ 

, P lAU

λ  

=



BIAL

λ 

, BIAU

λ 

, (53)

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L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

where λ ≥ 0. It is easy to see that operation (53) provides IVIFVs. It is easy to prove that introduced operations (48), (50)–(53) have the same properties as (13)–(18). The power operation AB , where A = [BIA ] is IVIFV and B = BIB is IFV, taking into account the properties of belief intervals and the interval arithmetic rules [25], is defined as follows.

[BIA ]

BIB

=



BelAL , P lAL

[BelB ,PlB ] 

, BelAU , P lAU

[BelB ,PlB ] 

=



BelAL

PlB 

, P lAL

BelB   ,

BelAU

PlB 

, P lAU

BelB 

.

(54)

It is easy to see that operation (54) provides IVIFVs. The power operation AB , where A and B are IVIFVs, taking into account the properties of belief intervals and the interval arithmetic rules [25], is defined as follows (the tedious intermediate mathematical manipulations are omitted). BIB ]

[BIA ][

=



BelAL , P lAL

[BIB ] 

, BelAU , P lAU

[BIB ] 

=



BelAL

PlBU 

, P lAL

BelBL   ,

BelAU

PlBU 

, P lAU

BelBL 

.

(55)

It is easy to show that operation (55) provides IVIFVs. n  Let Ai , i = 1 to n be IVIFVs and wi such that wi = 1 be real-valued weights. Then based on the operations (49) and i=1

(51), we get the following interval-valued intuitionistic weighted arithmetic mean:



n 

1 IV IWAM1DST (A1 , A2 , . . . , An ) = n

wi BelAL i ,

n 

i=1



wi P lAL i

n 

,

i=1

wi BelAUi ,

i=1

n 

wi P lAUi

.

(56)

i=1

This aggregation operator is not idempotent. Nevertheless, the small modification of (56) (multiplying by n) provides the idempotent operator:



n 

IV IWAM1DST (A1 , A2 , . . . , An ) =

wi BelAL i ,

i=1

n 



wi P lAL i

n 

,

i=1

wi BelAUi ,

i=1



n 

wi P lAUi

,

(57)

i=1

which provides the results in the form of IVIFVs. It is easy to see that in practice, operators (56) and (57) will produce equivalent orderings of compared alternatives. Let Ai , i = 1 to n be IVIFVs and the weights are presented by IFVs, i.e., wi = BIi = [Beli , Pli ]. Then we get the following interval-valued intuitionistic weighted arithmetic mean:



n 

IV IWAM2DST (A1 , A2 , . . . , An ) =

Beli BelAL i ,

i=1

n 



P li P lAL i

,

i=1

n 

Beli BelAUi ,

i=1

n 



P li P lAUi

.

(58)

i=1

It is important that this operator generally does not provide results in the form of IVIFVs since there is no normalisation of n  [Beli , Pli ], i = 1 to n, e.g., such as wi = 1. The method for normalisation of interval-valued weights is proposed in our paper i=1

[32] and the method for normalisation of Dempster–Shafer structures is proposed in our paper [31], but these problems are out of score of the current paper.     Let Ai , i = 1 to n be IVIFVs and the weights are presented by IVIFVs too. Therefore, wi = [BIi ]= BeliL , P liL , BeliU , P liU . Then we get the following interval-valued intuitionistic weighted arithmetic mean:



n 

IV IWAM3DST (A1 , A2 , . . . , An ) =

BeliL BelAL i ,

i=1

This  operator  generally  does [BIi ]= BeliL , P liL , BeliU , P liU .

not

n 



P liL P lAL i

,

n 

i=1

provide

Let Ai , i = 1 to n be IVIFVs and wi such that

results n  i=1

BeliU BelAUi ,

i=1

in

form

of

n 



P liU P lAUi

.

(59)

i=1

IVIFVs

since

there

is

no

normalisation

of

wi = 1 be real-valued weights. Then based on the operations (50) and

(52), we get the following interval-valued intuitionistic weighted geometric mean operator:

IV IW GM1DST (A1 , A2 , . . . , An ) n  n   w i   w i  w i    U w i  U w i   = BIAi = BelAL i , P lAL i , BelAi , P lAi i=1

i=1



=

n   i=1

n    L wi

BelAi

,

 L wi

P lAi



,

i=1

Obviously, this operator provides IVIFVs.

n   i=1

n    U wi

BelAi

,

i=1

 U wi

P lAi

.

(60)

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

267

In the case, when Ai , i = 1 to n be IVIFVs and the weights are presented by IFVs, i.e., wi = BIi = [Beli , Pli ], omitting the tedious intermediate mathematical manipulations we get the following interval-valued intuitionistic weighted geometric mean operator:



n  

IV IW GM2DST (A1 , A2 , . . . , An ) =

n    L Pli

,

BelAi

i=1

 L Beli



,

P lAi

i=1

n  

n    U Pli

,

BelAi

i=1

 U Beli



.

P lAi

(61)

i=1

It is easy to see that this operator provides IVIFVs.     In the case, when Ai , i = 1 to n be IVIFVs and the weights are presented by IVIFVs, i.e., wi = [BIi ]= BeliL , P liL , BeliU , P liU , omitting the tedious intermediate mathematical manipulations we get the following interval-valued intuitionistic weighted geometric mean operator:



n  

IV IW GM3DST (A1 , A2 , . . . , An ) =



PlU BelAL i i ,

i=1

n  





BeliL P lAL i

,

i=1

n  



PlU BelAUi i ,

i=1

n  





BeliL P lAUi

.

(62)

i=1

It is seen that this operator provides IVIFVs. 4. The properties of operations on IVIFVs in the framework of DST In this section, we show that operations on IVIFVs obtained in the framework of interval-extended DST are free of undesirable properties 1–4 (see Section 2) of classical operation on IVIFVs. Opposite to the case of classical addition operations (4) and (19), the addition operation (48) is an addition invariant operator. Theorem 1. The operation (48) is an addition invariant operator. Proof. Let A = [BIA ], B = [BIB ] and C = [BIC ] be IVIFVs such that [BIA ] > [BIB ] in sense of (45). Then according to (48)

   1  L BelA + BelCL , P lAL + P lCL , BelAU + BelCU , P lAU + P lCU , 2    1  L BelB + BelCL , P lBL + P lCL , BelBU + BelCU , P lBU + P lCU . [BIC ] = 2

[BIA ] BNP [BIC ] = [BIB ] BNP

To compare [BIA ]BNP [BIC ] and [BIB ]BNP [BIC ] according to (45) the following real values should be compared: V (A, C ) = BelAL + BelCL + P lAL + P lCL + BelAU + BelCU + P lAU + P lCU , V (B, C ) = BelBL + BelCL + P lBL + P lCL + BelBU + BelCU + P lBU + P lCU . It is easy to see that V (A, C ) = V (A ) + V (C ), where V (A ) = BelAL + P lAL + BelAU + P lAU , V (C ) = BelCL + P lCL + BelCU + P lCU and V (B, C ) = V (B ) + V (C ), where V (B ) = BelBL + P lBL + BelBU + P lBU . Since A > B, then according to (45) V(A) > V(B) and therefore V(A, C) > V(B, C). Therefore, finally we get [BIA ]BNP [BIC ] > [BIB ]BNP [BIC ].  Opposite to the classical multiplication operations (6) and (21), the operation (51) is preserved under multiplication by a scalar. Theorem 2. The operation (51) is preserved under multiplication by a scalar. Proof. Let A = [BIA ] and B = [BIB ] be IVIFVs such that [BIA ] > [BIB ] in sense of (45) and λ ∈ [0, 1] be a real value. Since [BIA ] > [BIB ], then according to (45) V(A) > V(B), where V (A ) = BelAL + P lAL + BelAU + P lAU , V (B ) = BelBL + P lBL + BelBU + P lBU . Obviously λV(A) > λV(B) and therefore λ[BIA ] > λ[BIB ].  Opposite to the classical aggregation operations IVIWAM (8), the aggregation operator IVIWAM1DST (57) is consistent with the aggregation operator on the ordinary (when μ = 1 − ν ) interval-valued fuzzy sets. Theorem 3. The aggregation operator IVIWAM1DST (57) is consistent with the aggregation operator on the ordinary (when μ = 1 − ν ) interval-valued fuzzy sets. Proof. Since in the considered case BelAL = μLA , P lAL = 1 − νAL = μLA and BelAU = μUA , P lAU = 1 − νAU = μUA , then from (57) we i

obtain

i

i



IV IWAM1DST (A1 , A2 , . . . , An ) =

n  i=1

wi μ

L Ai ,

i

n  i=1

i

i



wi μ

L Ai

,

n  i=1

wi μ

U Ai ,

n  i=1

i

i

i

wi μ

U Ai

=

n  i=1

wi μ

i

L Ai ,

n 

wi μ

U Ai

.

i=1



268

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272

Opposite to the classical aggregation operations IVIWAM (8), the aggregation operator IVIWAM1DST (57) is monotone with respect to the ordering (45). Theorem 4. The aggregation operator IVIWAM1DST (57) is monotone with respect to the ordering (45). Proof. Let A = [BIA ], B = [BIB ] and C = [BIC ] be IVIFVs such that [BIA ] > [BIB ] in sense of (45) and w1 , w2 be the weights such that w1 +w2 = 1. Then from (57) we get IVIWAM1DST (A, C)

= [[w1 BelAL + w2 BelCL , w1 P lAL + w2 P lCL ], [w1 BelAU + w2 BelCU , w1 P lAU + w2 P lCU ]], IVIWAM1DST (B, C)

= [[w1 BelBL + w2 BelCL , w1 P lBL + w2 P lCL ], [w1 BelBU + w2 BelCU , w1 P lBU + w2 P lCU ]]. To compare IVIWAM1DST (A, C) and IVIWAM1DST (B, C) according to (45) the following real values should be compared

V (A, C ) = w1 BelAL + w2 BelCL + w1 P lAL + w2 P lCL + w1 BelAU + w2 BelCU + w1 P lAU + w2 P lCU , V (B, C ) = w1 BelBL + w2 BelCL + w1 P lBL + w2 P lCL + w1 BelBU + w2 BelCU + w1 P lBU + w2 P lCU . It is easy to see that

V (A, C ) = V (A ) + V (C ) where

V (A ) = w1 BelAL + w1 P lAL + w1 BelAU + w1 P lAU , V (C ) = w2 BelCL + w2 P lCL + w2 BelCU + w2 P lCU . V (B, C ) = V (B ) + V (C ), where

V (B ) = w1 BelBL + w1 P lBL + w1 BelBU + w1 P lBU .

Since [BIA ] > [BIB ], then according to (45) V(A) > V(B). Therefore V(A, C) > V(B, C) and finally we get IVIWAM1DST (A, C) > IVIWAM1DST (B, C).  An important property of operation of multiplication of IVIFVs is that it is not always monotone with respect to the ordering (45), although we can present many examples where this operation is monotone with respect to (45). Let us first consider the multiplication of IFVs. Example 3. Let A, B and C be IFVs presented by corresponding belief intervals BIA = [0.1,0.7], BIB = [0.4,0.5] and BIC = [0.2.0.4]. Then using any rule for interval comparison (see, for example [30,39]) we get BIB > BIA . Since by definition a belief interval is positive one: BI = [Bel , P l ], 0 ≤ Bel ≤ Pl ≤ 1, then using classical rule for interval multiplication [25] we get BIA BIC = [BelA BelC , PlA PlC ]=[0.02,0.28] and BIB BIC = [BelB BelC , PlB PlC ]=[0.08,0.2]. Then using the rule for interval comparison we obtain BIA BIC > BIB BIC (opposite to BIB > BIA ). At first glance, this property seems to be a negative one. On the other hand, it can be considered as one of the specific properties of interval arithmetic (such that for intervals A, B and C we have AB + AC = A(B + C ) and only A(B + C ) ⊂ AB + AC, A/A = 1, A − A = 0 and so on). Since the operator of multiplication of IFVs is an asymptotic case of multiplication of IVIFVs, we can conclude that multiplication of IVIFVs is not always monotone with respect to the ordering. The interval-valued intuitionistic weighted geometric mean operator (60) is not always monotone with respect to the ordering. Example 4. Let us first consider an asymptotic case of (60) where IVIFVs A1 , A2 , ...,An contract to the IFVs and the weights of local criteria contract to real values. In such a case, the aggregation (60) transforms to the operator  n wi n wi IW GMDST (A1 , A2 , . . . , An ) = introduced in [15]. Let us consider the case when IFVs A = BIA = [0.4,0.5], i=1 BelA , i=1 P lA i

i

B = BIB = [0.35,0.56], C = BIC = [0.1,0.1] and w1 = 0.3, w2 = 0.7. Using any method for interval comparison [30,39] we get B > A Based on these data we get IWGMDST (A, C) = [0.15157,0.16207] and IWGMDST (B, C) = [0.14562,0.16767]. Using any method for interval comparison [30,39] we get IWGM1DST (A, C) > IWGM1DST (B, C), whereas BIB > BIA .

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Table 1 Decision matrix presented by belief intervals. Ai /C i

C1

C2

C3

A1 A2 A3 A4

[0.3, 0.4], [0.3, 0.5] [0.3, 0.3], [0.6, 0.7] [0.2, 0.4], [0.3, 0.6] [0.1, 0.2], [0.7, 0.8]

[0.2, 0.4], [0.4, 0.6] [0.2, 0.3], [0.6, 0.7] [0.3, 0.4], [0.5, 0.6] [0.1, 0.3], [0.6, 0.7]

[0.1, [0.1, [0.1, [0.1,

0.3], 0.2], 0.3], 0.2],

[0.4, [0.4, [0.5, [0.3,

0.5] 0.7] 0.6] 0.4]

Since the operator IWGMDST is an asymptotic case of operators IVIWGM1DST , IVIWGM2DST and IVIWGM3DST , we can say that these operators are not always monotone with respect to the ordering. Therefore, the interval-valued intuitionistic weighted geometric mean operators should be used in practice with caution. 5. Application to MCDM When comparing methods used for MCDM, the methods which make it possible to take into account more available information or based on more correct operation laws are usually treated as the better ones. In this paper, we have revealed the serious shortcomings of the classical definition of IVIFS and the operation laws defined on IVIFVs. Therefore we proposed a new correct definition of IVIFS and the operation laws based on the representation of IVIFVs by belief intervals with the bounds presented by belief intervals. These new operation laws are free of drawbacks of classical operation laws. Therefore we can say that the use of new operation laws on IVIFVs in the solution of MCDM in the interval-valued intuitionistic fuzzy setting may provide more correct results than those obtained using classical operation laws on IVIFVs. Nevertheless, often the proposed new methods for the solution of MCDM problems based on more correct operation laws and more information, in practice do not provide results that are qualitatively different from those obtained with the use of known methods. For example, a new method provides the same ranking of alternatives as the known ones. Obviously, the use of new more correct methods of MCDM in such case is justified enough if they are more understandable, more simple in calculations or make it possible to provide new operators of aggregation with more information, which are not introduced within the framework of known approaches. Let us consider interval-valued intuitionistic weighted arithmetic mean IVIWAM in the interval-valued intuitionistic fuzzy setting. Suppose we have four alternative A1 , A2 , A3 , A4 , three local criteria C1 , C2 , C3 and rij be the IVIFV rating of Ai with respect to Cj . Let w1 ,w2 ,w3 be the interval-valued intuitionistic fuzzy weighs of local criteria. Then in the framework of classical approach we get:

IV IWAMC (Ai ) = w1  ri1  w2  ri2  w3  ri3

(63)

and using our approach we obtain

IV IWAMDST (Ai ) = w1 BNP ri1 BNP w2 BNP ri2 BNP w3 BNP ri3 .

(64)

Let us consider the following example. Example 5. For our case of four alternatives A1 , A2 , A3 , A4 and three local criteria C1 , C2 , C3 the decision matrix presented by belief intervals bounded by belief intervals is presented in Table 1. According to the Definition 5 all entries rij of this table are DST representations of IVIFVs in sense of Definition 3. The weights of local criteria are presented by belief intervals bounded by belief intervals: w1 = [[0.05, 0.14], [0.19, 0.29]], w2 = [[0.14, 0.19], [0.19, 0.33]], w3 =[[0.19, 0.24], [0.29, 0.38]]. They are normalised in such a way that the sum of the right bounds is equal to 1 (0.29+0.33+0.38=1). Then from (49), (50), (64) and (45) we get the final ranking of alternatives:

A2 > A3 > A4 > A1 .

(65)

With the use of Definition 5, we transform the above weights and decision matrix presented by belief intervals into intervalvalued intuitionistic fuzzy form as follows. Let us consider the weight w1 = [[0.05, 0.14], [0.19, 0.29]] presented by the bounded by belief intervals.   belief interval  Then according to the Definition 5 it can be presented as w1 = μL , 1 − ν U , μU , 1 − ν L , where μL = 0.05, μU = 0.19, ν L = 1 − 0.29 = 0.71, ν U = 1 − 0, 14 = 0.86. Therefore intuitionistic fuzzy weight w1 as follows:  we get   the interval-valued  w1 = μL , μU , ν L , ν U = [0.05, 0.19], [0.71, 0.86]. In the same way, we obtain w2 = [0.14, 0.19], [0.67, 0.81], w3 = [0.19, 0.29], [0.62, 0.76] and interval-valued intuitionistic fuzzy ratings presented in Table 2.

270

L. Dymova, P. Sevastjanov / Information Sciences 360 (2016) 256–272 Table 2 Interval-valued intuitionistic fuzzy decision matrix. Ai /C i

C1

C2

C3

A1 A2 A3 A4

[0.3, 0.3], [0.5, 0.6] [0.3, 0.6], [0.3, 0.7] [0.2, 0.3], [0.4, 0.6] [0.1, 0.7], [0.2, 0.8]

[0.2, 0.4], [0.4, 0.6] [0.2, 0.6], [0.3, 0.7] [0.3, 0.5], [0.4, 0.6] [0.1, 0.6], [0.3, 0.7]

[0.1, [0.1, [0.1, [0.1,

0.4], 0.4], 0.5], 0.3],

[0.5, [0.3, [0.4, [0.6,

0.7] 0.8] 0.7] 0.8]

Then from (8), (63) and (41) we get the final ranking of alternatives which is qualitatively the same as the rating (65). We have made more than hundred tests using different decision matrices and in all cases we have obtained qualitatively equivalent final ratings of alternatives. The use of weighted sums is not the best approach to the aggregation of local criteria in many real-world situations, since the small values of some criteria can be counterbalanced by large values of other ones which in practice may be less important for the decision maker. It is worth noting that in some fields the weighted sums aggregation is not used at all [35]. Therefore, let as consider the interval-valued intuitionistic weighted geometric mean operators (9) and (60). Then for w1 = 0.1, w2 = 0.5, w3 = 0.4 using operator (60) from Table 1 we obtain A2 > A3 > A1 > A4 and with the use of operator (9) from Table 2 we get the same result. We have made many tests using different decision matrices and in all cases we have obtained qualitatively equivalent final ratings of alternatives. These results can be explained by the fact that the negative properties of the classical operations on IFVs (see Section 2) do not considerable effect on the operators of aggregation. On the other hand, we cannot guarantee that this is always the case. Summarising, we can say that the main practical advantages of our approach in the decision making in comparison with the classical one is that it is more understandable and simple in calculations. It is important also that our approach allows us to introduce the new interval-valued intuitionistic weighted geometric operators (61) and (62) for the cases when the weighs are presented by IFVs and IVIFVs. These operators are not defined in the framework of classical approach. Therefore, we can say that our approach is efficient in the solution of the MCDM problems. 6. Conclusion It is shown that DST may serve as a good methodological base for interpretation of A-IVIFS. A critical analysis of conventional operations on IVIFVs and their applicability to the solution of multiple criteria decision making problems in the interval-valued intuitionistic fuzzy setting is provided. It is shown that the classical definition of Atanassov’s A-IVIFS has some important drawbacks which may lead to controversial results. Therefore, in this paper, a new more constructive definition of A-IVIFS is proposed. It is shown that this new definition makes it possible to present IVIFVs in the framework of interval-extended Dempster–Shafer theory of evidence as belief intervals with bounds presented by belief intervals. A new set of operations on IVIFVs in the framework of interval extension of DST is proposed. These operations are based on the treatment of belief interval as an interval enclosing a true power of some statement (argument, hypothesis, ets). It is proved that these operations provide IVIFVs in the sense of new definition of A-IVIFS. A critical analysis of commonly used operations for comparison of IFVs and IVIFVs is made and a new more justified operation for comparison of IVIFVs in the framework of interval-extended DST is proposed. It is proved that introduced set of operations on IVIFVs is free of revealed limitations and drawbacks of commonly used operations on IVIFVs based on the classical definition of A-IVIFS proposed by Atanassov and Gargov. The corresponding methods for aggregation of local criteria presented by IVIFVs in the framework of interval-extended DST are proposed and analysed. The proposed approach allows us to solve MCDM problems without intermediate defuzzification when not only criteria, but their weights are IVIFVs. It is important also that our approach allows us to introduce the new interval-valued intuitionistic weighted geometric operators for the cases when the weighs are presented by IFVs and IVIFVs. These operators are not defined in the framework of classical approach. Acknowledgements The research has been supported by the grant financed by National Science Centre (Poland) on the basis of decision number DEC-2013/11/B/ST6/00960. References [1] [2] [3] [4] [5]

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