The new definitions of intuitionistic and belief-plausibility based local criteria with interval and fuzzy inputs applied to the multiple criteria problem of a raw material supplier selection

The problems of the A – IFS and A – IV IFS theories are analyzed and to solve them, their Dempster-Shafer theory based redefinition is developed and a new concept called Belief-Plausibility number (BPN) is introduced. The BPN can be applied beyond semantics of usual fuzzy sets and the A – IFS. Two main sources of uncertainty generating the A – IV IFS are considered: an uncertainty coursed by the membership and non-membership functions multiplicity and an uncertainty of inputs. For the fuzzy inputs, the definition of the fuzzy-valued BPN is proposed. The developed approach to treat interval-valued intuitionistic fuzzy objects and their Belief-Plausibility re-definitions allowed us to introduce a set of useful definitions. It is important that some of the them were not earlier mentioned in the literature. The corresponding new arithmetical operations with such objects are proposed. The practical usefulness of the introduced approach is illustrated by the solution of the real-world Fuzzy-Valued Belief-Plausibility multiple criteria problem of the raw material suppliers selection on the steel rolling plant. The obtained results allowed us to state that the introduced Belief-Plausibility approach can be successfully used in the analysis and applications independently, without the use of the A – IFS concepts. This approach owns visible advantages in comparison with the A – IFS based one.


I. INTRODUCTION
The classical definitions of the Intuitionistic fuzzy sets theory (A − IF S) and the Interval-valued Intuitionistic fuzzy sets theory (A−IV IF S) were introduced by K. Atanasov [1] and K. Atanassov and G. Gargov [2], respectively, and recognized to be practically useful extensions of the Fuzzy Set theory (F ST ) [3]. In the framework of the A − IF S, besides the degree of membership µ, the non-membership degree ν and the hesitant degree π are considered. It is postulated that 0 ≤ µ, ν, π ≤ 1 and µ + ν + π = 1. The A − IV IF S theory deals with interval extensions of µ, ν and π. Currently, the methods of these theories are actively applied for the solution of multiple criteria decision making problems (M CDM ) [4]- [7] and group M CDM problems [8] in different realworld areas. In this paper, we restrict ourselves to citing only several recently published papers in the field we deal with. We think they more than satisfactory present the nowadays state of art in the field of the MCDM problems solution under intuitionistic uncertainty. But we do not intend to make a comprehensive analysis of the A − IF S and A − IV IF S practical applications. However, there are some methodological problems of the A − IF S and A − IV IF S discussed in the literature. The first of them is the difficulty that an expert experiences when assigning real values to µ and ν. It was noted by many scholars that sometimes experts insist on such values VOLUME X, 2021 that their sum µ + ν occurred to be greater than 1 what is unacceptable in the framework of the intuitionistic theory. To avoid or at least alleviate this problem, some extensions (corrections, modifications) of the A − IF S were developed. The chronologically first of them is the so-called Neutrosophic Set theory (N ST ) [9]- [12]. In terms of A − IF S, the basic assumptions of N ST are presented in the form: 0 ≤ µ + ν + π ≤ 3 for the completely independent components (µ, ν, π ∈ [0, 1]), 0 ≤ µ + ν + π ≤ 1 for the mutually dependent components (µ, ν, π ∈ [0, 1]). It is seen that the N ST conceptually differs from the A − IF S by the accepted assumption that the components µ, ν, π are completely independent. In our opinion, this assumption is not in line with reality and common sense although seems to be correct from the pure mathematical point of view. This assumption in the N ST means that the event µ = 1, ν = 1 and π = 1 is admissible and therefore the first constraint 0 ≤ µ + ν + π ≤ 3 is fulfilled. Let µ, ν and π be the degrees of truth, false and indeterminacy, respectively (in terms of the N ST ). Therefore, in the case of complete truth (µ = 1), according to the formal logic and common sense, the false degree is equal to 0 (ν=0) with no indeterminacy (π=0). Obviously, the similar conclusions can be inferred in the cases of complete false, complete indeterminacy and µ = 0, ν = 0, π = 0. In other words, we can state that the great truth is always accompanied by the small false and indeterminacy. Many of similar statements that will be in compliance with common sense and the formal logic can be claimed. So we can say that the hypothesis of the components µ, ν and π independence seems to be completely invalid in the framework of the problem we deal with and the assumption of mutually dependent components is directly originated from the formal logic and common sense. It is somewhat surprising that the events µ = 1, ν = 1 and π = 1 and µ = 0, ν = 0, π = 0 are qualified in [12] as a paradox and its formal definition is considered as an advantage of the N ST . It is not only our opinion, but in any case, it seems preferable to use theories free of paradoxes, e.g. the A − IF S theory in the considered case. It is also important that described paradoxes occur in the two diametric asymptotic cases µ = 1, ν = 1 and π = 1 and µ = 0, ν = 0, π = 0 achievable in the N ST . Of course, the theory with so bad asymptotic properties is not a good choice if we deal with the real-world applications. Formally, if the hypothesis of mutually dependent components is accepted, then the basic constrain of the N ST 0 ≤ µ + ν + π ≤ 1 is more general than that in the A − IF S (π + µ + ν = 1). This fact was used in the socalled Picture fuzzy sets theory [13]. As a basic of this theory, the so-called refusal degree r such that µ + ν + π + r = 1 was introduced. According to the Oxford Dictionary, the term "refusal" corresponds the rejection from something. Therefore, in considered case, it should be equivalent to the rejection from the analyzing the problem at hand. So, if this degree r is greater than 0, then we can not begin the studies at all and the refusal disappears when we are doing something, e.g. solving the M CDM problem. Therefore, the refusal degree has no sense even on the semantic level of analysis. The operations defined on the mathematical objects of N ST in [14] are shown to have some important drawbacks [15]. As a compromise, in [15] the more simple operations were proposed, but without analysis of their properties. The mathematical operations with the objects of N ST developed in [14], [15] are obtained as simple mechanical extensions of operational laws earlier defined in the A − IF S theory. In our papers [7], [16], [17], we revealed 6 important drawbacks of these operations, which should be directly inherited in operations with objects of N ST . Finally we can say that although the N ST perhaps can be interesting for pure mathematicians, it look as impractical extension of the A − IF S based on the rather artificial hypothesis. The Pythagorean fuzzy set theory (P F SS) was proposed in [18] based on the simple and clear reasoning [19]. Its somewhat modified version is presented here as follows. Let µ = 0.8 be the membership degree and the ν = 0.5 be the corresponding non-membership degree. Since µ + ν > 1, we have no the intuitionistic fuzzy value. Then if in the considered example we will change µ and ν by µ 2 and ν 2 , respectively, we will get µ 2 + ν 2 < 1. So in accordance with the definition introduced in [19], in this example we have the truly intuitionistic fuzzy value. Although such a reasoning seems to be acceptable, two obvious questions arise: Assuming that µ is a membership degree, what is the sense of µ 2 in a natural language? What were logical arguments in favor of the squared degree, while the third, fourth, and so on degrees look to be even more attractive and promising? Because we did not find answers on these questions, we can say that they may be treated as important vital methodological issues of the P F SS. The obvious important practical question can be formulated as follows: does the P F SS always guarantee µ 2 + ν 2 < 1?
To be fair, we should note here that the author of the P F SS said of only limited range of this theory applications: "...we observe that intuitionistic membership grades are all points under the line x + y ≤ 1 and the Pythagorean membership grades are all points with x 2 + y 2 ≤, 1. We see then that the Pythagorean membership grades allow for the representation on a larger body of nonstandard membership grades than intuitionistic membership grades" [19]. In our opinion, we can agree with this statement, but only if the reasonable answers on the above questions will be formulated. Also we can say that because the P F SS pretends to be an extension of the A − IF S, the use of its methods can only enhance the possibility of violating the basic constraint µ + ν ≤ 1. Therefore, the practical usefulness of the P F SS seems to be very doubtable.
The Spherical fuzzy sets theory SF SS [20] is the formal extension of the P F SS since it is based on the introduced constrain µ 2 + ν 2 + π 2 ≤ 1. It is seen that SF SS have all the negative properties of the P F SS additionally complicating the problem (see analysis in [12]). The main problem of the P F SS and SF SS as well as of A − IF S, is the assigning of appropriate values to the components µ, ν and π such that they provide the basic constrain (µ + ν ≤ 1) in the presence of objectively existing, but not explicitly defined (nearly undefined) mutual dependence between the components. Contrary to the N ST , we are insisting that the dependence of components always takes place (may be, implicitly) and is originated from human thinking and logic. It is not so difficult to fancy how important this problem is in practice for an expert who takes part in the solution of, e.g. the M CDM problems. The method for solution of this and some other collateral problems will be presented in this paper below. In this paper, we will present a new method focused on the solution of this and the set of other collateral problems that is based on the redefinition of A − IF S and A − IV IF S in the framework of the Dempster-Shafer theory of evidence (DST ). Based on the provided analysis, we introduce a new mathematical object called Belief-Plausibility number (BP N ), which can be reduced into its particular cases: an ordinary fuzzy number (the value of a membership function) and an intuitionistic fuzzy number. It is important that this object can be applied in an analysis completely separately, without using terms of ordinary fuzzy sets and the A − IF S, as it is typical for more general theories. The above review concerns with the problems of intuitionistic fuzzy numbers. Meanwhile, the second set of important problems we have revealed are typical in the membership µ(x) and nonmembership ν(x) (x ⊆ X) functions building. There are many approaches to the presentation of these functions proposed in the literature. In the papers [21]- [26], the triangular form of these functions was used and in [27] their interval valued triangular versions were proposed. Currently the most used are trapezoidal [28]- [34] and interval valued trapezoidal [35]- [41] µ(x) and ν(x) functions. The natural limitations concerned with the too simple triangular and trapezoidal shapes of µ(x) and ν(x) functions are currently overcoming by the use of piece-wise linear functions consisting of different numbers of pieces. Most known of them are Pentagonal [42]- [46], Hexagonal [47], Heptagonal [48]- [50], Octagonal [51], Nanogonal [52] and Hexadecagonal [53] intuitionistic fuzzy functions. Obviously, with the increasing the number of used pieces, such functions tend to be close to some smooth generally nonlinear functions. The use of non-linear µ(x) and ν(x) functions is presented in [54], [55]. While analyzing the above papers, we have found that their authors do not care of the logically justified properties of these functions in the asymptotic cases when µ(x) = 0 or ν(x) = 0 or µ(x) = 1 or ν(x) = 1. In these papers, we see that according to introduced mathematical presentations of µ(x) and ν(x) and even their graphical illustrations, we can observe in X the whole areas where, e.g. µ(x) = 1, while ν(x) > 0 and even is rising or decreasing in such an area. We insist that this contradicts to common sense. Really, if µ(x) = 1, then we have a complete (100%) certainty that some event has occurred. Therefore there is no place for the opposite event and hesitation, i.e. it should be ν(x) = 0 and π(x) = 0. The similar analyzes can easily be done for other asymptotic cases. Let us consider the more meaningful confirmation of the above statements. Let's assume that we want to establish the truth of the statement that Admiral Nelson was a gentleman based on the judgments of individuals from a sample of 100 people. First consider the case when 20 people called Nelson a gentleman because he bravely fought for his Homeland, 30 people did not call him a gentleman because he was a pirate and the remaining 50 people found it difficult to assess. Then we can introduce the degree of Nelson membership in the "Club of gentleman" as µ=0.2, the non-membership as ν=0.3 and hesitation degree as π=0.5. Then let us suppose that all people in the sample refused to recognize Admiral Nelson as a gentleman. Obviously, we have µ = 0. It is clear that in this case we have ν = 1 and π = 0, since we simply do not have people who could say anything else. Indeed, since everyone completely rejected Nelson's belonging to gentlemen, then there are physically no those who would have hesitated in the assessment, hence π=0. At last, the complete rejection of Nelson's belonging to gentlemen (µ=0) means the complete acceptance of the opposite hypothesis of Nelson not being a gentleman i.e. ν=1. This can be inferred formally as follows : since by definition it always should be µ + ν + π=1 and in our case we have µ = π=0, then ν=1. We have found the only one paper [42], where the authors saying nothing about correct properties of the symmetrical pentagonal µ(x) and ν(x) functions in asymptotic cases, have constructed µ(x) and ν(x) in such a way that the desired properties were fulfilled. Unfortunately, in this paper, the functions µ(x) and ν(x) were defined to be equal to 0.5 in the points of their intersection. It is easy to see that in such a case, for piece-wise linear µ(x) and ν(x) we get µ(x) + ν(x) = 1 for any x ⊆ X. This corresponds to the definition of the ordinary fuzzy set, not an intuitionistic one. It is interesting that there are no definitions of asymptotic properties of the functions µ(x) and ν(x) in the classical Atanassov's works [1], [2], perhaps because they looked to be too obvious. Nevertheless, based on the above analysis, we can state the strong definitions of the discussed properties should be relevant in the basic definitions of the A − IF S and A − IV IF S. In the current paper, they will be included in these definitions. We have not found in the literature clear and practical ap-VOLUME X, 2021 proaches to mathematical formulation of the functions µ(x) and ν(x) in such a way that all asymptotic properties and basic constraints µ(x) + ν(x) ≤ 1, µ U (x) + ν U (x) ≤ 1 (µ U (x) and ν U (x) are the upper bounds of corresponding interval-valued membership and non-membership functions) will be fulfilled jointly. In the current paper, such an approach will be proposed. In the literature, we have found only the examples of the use of Atanassov's definition of IV IF S, whereas in [17] we have revealed its limitations and drawbacks and proposed a new correct definition based on representation of the A − IV IF S in the framework of DST . Such a redefinition allowed us to develop new operational laws free of revealed drawbacks. Nevertheless, this definition needs upgrading taking into account the above analysis. Based on our experience, we can state that there are two main sources of uncertainty leading to the need for the use of A − IV IF S. The first of them is an interval uncertainty coursed by the multiplicity of the functions µ(x) and ν(x), e.g. proposed by different experts. The second source is the consequence of the interval uncertainty of input data, e.g. in the M CDM . And if input data we deal with are presented by fuzzy values, we meet the case of fuzzy-valued intuitionistic fuzzy set F V IF S. The described above general approach to treat the interval-valued intuitionistic fuzzy objects and their Belief-Plausibility (based on the DST ) extensions allowed us to introduce a set of new fruitful definitions, some of the them, e.g. the definition of Fuzzy-Valued Belief-Plausibility value (F V BP V ) were not even mentioned in the literature in any form. The corresponding new operational laws with these mathematical objects following from these definitions are proposed. They all and especially the definition of F V BP V , were thoroughly analyzed under general methodological principles and their practical usefulness is illustrated by the real-world case study based on the solution of the DST based uncertain M CDM problem of the raw material supplier selection on the steel rolling plant under triangular fuzzy input data. The obtained results allow us to state that the introduced mathematical objects BP N , IV BP N and F V BP N can be successfully used in the analysis and applications independently, without the use of concepts of the A − IF S and such an approach has obvious advantages in comparison with the A − IF S based approach. The rest of the paper is set out as follows.
In Section 2, we analyze the properties of the DST extension of A − IF S and its practical advantages for a decision maker in assigning certain real values to the parameters of A−IF S. As the result of provided studies, a new mathematical object called Belief-Plausibility number (BP N ) is introduced. It is shown that the BP N has at least two particular cases: usual fuzzy number (the value of a membership function) and intuitionistic fuzzy number. It is emphasized that the BP N can be successfully applied completely separately not using terms of usual fuzzy sets and the A − IF S and this is the typical feature of more general theories. Section 3 is devoted to the justification of new definitions of membership and non-membership functions, their interval extensions inspired by two revealed sources of interval uncertainty. The case of fuzzy-valued input data is considered. It is shown that in such a case we should deal with fuzzy-valued intuitionistic fuzzy set F V IF S. The corresponding definitions in the spirit of BP N , i.e. Belief-Plausibility set (BP S), Belief-Plausibility value (BP V ), Interval-Valued Belief-Plausibility value (IV BP V ) and Fuzzy-Valued Belief-Plausibility value (F V BP V ) are introduced. The new arithmetic operations with local criteria represented by BP V s and dependent on interval and fuzzy arguments are proposed. In Section 4, we present the real-world case study. We have solved the Belief-Plausibility based M CDM problem of the raw material supplier selection on the steel-rolling plant, when Belief-Plausibility local criteria are dependent on fuzzy arguments. The problem formulation based on the F V BP V definition was chosen as the more correct, general and convenient than that in terms of the fuzzy-valued intuitionistic fuzzy set F V IF S. Section 5 concludes the paper with some remarks.

II. THE STUDIES OF THE DST EXTENSION OF THE A − IF S NUMBER. INTRODUCTION OF A NEW MATHEMATICAL OBJECT BELIEF-PLAUSIBILITY NUMBER
The DST extension of A − IF S we have proposed in [56], [57], but here we will analyze it to expand its capabilities to solve the problems described in the previous section. It was found in [57] that the set (µ, ν, π) can represent a mass (or probability) assignment in the framework of the DST [58], [59]. That can be clarified as follows. It can be seen that when analyzing some object O in the framework of the A-IF S, we implicitly deal with the following propositions: Y es: ∈ O cannot be rejected (the case of indeterminacy or hesitation). Therefore µ can be treated as the mass of evidence assigned to O or as the focal element of the basic assignment function: m(Y es)=µ. Similarly we get m(N o)=ν. Since π is usually considered to be a hesitation degree, the evident proposition is m(Y es, N o)=π. Because in the A − IF S theory we have µ + ν + π = 1, we can state that the triplet µ, ν and π represents an ordinary basic assignment function in the DST . Therefore, the DST values of belief and plausibility can be obtained as follows: Bel=m(Y es)=µ and P l=m(Y es)+m(Y es, N o)=µ + π=1 − ν and the belief interval is presented as BI = [Bel, P l]. We can see that as the result of above redefinition, the basic components of the A − IF S (µ, ν, π) transfer to the basic triplet of the DST (m(Y es), m(N o), m(Y es, N o)) or to the belief interval BI. So based on such a redefinition of an intuitioniistic fuzzy number < µ, ν > in the framework of the DST , we obtain a new object, which informal presentation does not include the terms "intuitionistic" and "fuzzy". Therefore, to avoid possible confusions, here we introduce a new notion "Belief-Plausibility Number" (BP N ). Its formal definition formulated also taking into account our remarks concerned to the asymptotic properties (see previous section) is presented as follows: In general, this definition looks as somewhat extended definition of a belief interval in the DST . Nevertheless, the Definition 1 is very important in context of further analysis from the methodological reasons. First, it is remarkable that this definition does not contain any terms of the ordinary and intuitionistic fuzzy sets theories. So it can be used separately, based only on the mathematical tools of the DST . Meanwhile, always, when it is really needed, the BP N can be transformed to the A − IF S presentation of the problem at hand using the expressions Bel = µ and P l = 1 − ν.
In the other asymptotic case, when Bel = P l, the BP N reduces to the usual fuzzy number. Obviously, based on the above analysis we can conclude that the direct use of the DST formalism seems to be more general approach than those based on the A − IF S or the ordinary fuzzy sets theory. Nevertheless, one can say that as the intuitionistic fuzzy number A=< µ, ν > can be converted to the object A=[µ, 1 − ν], this last object can be naturally treated as a usual interval. But such an interval does not present any useful information with exception that some variable takes values in this interval. Besides, let A=< 0.6, 0.2 > and after conversion we have the usual interval A=[0.6, 0.8]. Then using the corresponding interval arithmetic addition rule, we get A + A=[1.2, 1.6]. This interval is completely out of the unit interval and cannot be reconverted to an intuitionistic fuzzy number. On the other hand, using the idempotent addition operation with belief intervals defined in [16], we have no problems with the reconversion. Therefore, we can say that the interpretation of intuitionistic fuzzy numbers as usual intervals is not useful and provides additional undesirable problems. According to the commonly accepted generic scientific paradigm, the problem, which cannot be solved based on a given theory, may be solved by the use of a more general theory if such a theory exists. Therefore we should to aware that the DST can truly serve as a generalization of the A − IF S. We can see that only few of generic elements of the DST have been applied above for redefinition of all elements of the A − IF S. So the DST looks as a more general theory than the A − IF S and it should be expected that by applying the A−IF S objects in the framework of the DST , we can obtain new results that are not achievable in the A − IF S setting and we can solve problems unsolvable in the A − IF S. This is not especially new idea as the DST is a generic theory for some reputed theories that deal with the uncertainty modeling. It is proved that the probability and possibility theories are the specific asymptotic cases of the DST . In [16], [17], we showed that the DST generalizes the A−IF S and its interval-valued extension. In [60], we proved that the DST can serve as a fruitful generalization of the hesitant and interval-valued hesitant fuzzy sets. In [61], we presented the DST generalization of the rule-base evidential reasoning in the A − IF S environment. This approach we used in [62] for the type 2 diabetes diagnostic. We used the DST extension of the A − IF S to generalize the T OP SIS method in [7]. There is especially close link between the DST and the rough set theory and it is really difficult to point out a more general theory between them [63]. The advantages of the DST were considered in [64], but we are aware that the DST is the more general theory as contrary to the rough set theory it origins the2 set of derived theories. Consider the advantages provided us by the DST redefinition of the A − IF S. The essential practical problem of the classical A − IF S is that decision makers or experts often experience considerable difficulties in assigning of such real values to µ and ν that always the inequality µ + ν ≤ 1 will be fulfilled. When using the DST extension of A − IF S, we are only dealing with the belief interval BI=[Bel, P l], not considering µ and ν at all, but meanwhile we do not forget that initially we had based on the following representation: So an expert or a decision maker is asked to assign to the Bel and P l, the real values laying in the interval [0, 1]. If an expert has assigned a certain value to Bel, then he/she should inevitably assign a greater value to the plausibility P l. This is the natural consequence of two premises: the definitions of belief Bel = µ and plausibility P l = µ + π in context of the A − IF S and the semantics of a natural language reflecting the specificity of human thinking. For instance, imagine we consider the possibility of some future political event E. In this case, Bel(E) is our belief in the appearance of E (somehow guaranteed estimation) and P l(E) is the plausibility of E appearance, which can be naturally treated as the degree to which the event E cannot be excluded or as a greatest admissible degree of belief tolerable in some specific favorable situations, and so on. It is easy to see that in the considered case, an expert should use only such degrees Bel and P l that the constraint Bel ≤ P l should always been held. This also follows from the semantics of a natural language and common sense and does not provide problems for an expert in assigning correct real values to Bel and P l. The constraint Bel ≤ P l is the DST form of µ ≤ 1 − ν and therefore of µ + ν ≤ 1.
We can see that the analyzed practical problem of the classical A − IF S disappears in the framework of the DST extension. It is worthy to note here that using the proposed approach based on the defined above BP N , we do not need to use the somewhat artificial and heuristic Neutrosophic, Pythagorean and Spherical sets theories with VOLUME X, 2021 their derivatives. The other problem of the classical A − IF S is the drawbacks of commonly used mathematical operations with intuitionistic fuzzy numbers (IF N ). In [16], using several persuasive examples we showed that the classical operations with IF N s possess 6 negative properties (two of them were firstly found in [65]), which may lead to unacceptable results of the realworld problems solutions (more information can be found in [16]). This is not so surprising as even the definitions of these operations seem to be controversial, e.g. the multiplication by a scalar looks as completely unacceptable or wrong operation as it is directly defined by the power operation. Therefore in [16], using the dual origin of a belief interval, which is an interval comprising a true power of some statement and at the same time is a usual interval, we inferred in the framework of the DST redefinition of the A − IF S, the set of operations with belief intervals representing corresponding IF N s. These operations are free of above mentioned drawbacks (this was proved formally with the use of relevant theorems). It is important that the introduced DST redefinition of the A − IF S generates new operations, particularly the power operation where both operands are belief intervals, which can be used for the extension of the Weighted Geometric operator, but cannot be defined within the framework of the classical A − IF S. This fact confirms the superiority of the DST based approach over the classical A − IF S, i.e. the DST redefinition is a more general theory in the relation to the A − IF S. The DST extension of the interval-valued A − IF S was introduced in [17].
Here we present ones more important practical argument in favor of the separate use of the BP N (see Definition 1 above) without any terms from A − IF S. In communications with experts, we found that they easily and willingly set the values of the degree of membership. Very often in technological practice, particular criteria are formulated in the form of trapezoidal fuzzy number the top of which is the interval of the best values of the parameter and the bottom is the interval of acceptable values. Experts are used to this and are willing to use it. However, with the degree of non-membership, difficulties arise. Let us say the expert sets µ=0.2. Then he/she automatically sets ν=0.8. Moreover, he/she considers other values of ν as erroneous. It is quite possible that such a probabilistic style of thinking, which is a consequence of a good higher education, prevents the widespread use of methods of A − IF S to solve real-world problems, although there are other reasons of rather psychological nature. But in any case, this is the problem of the A − IF S. At the same time, when the experts were asked to evaluate the BP N , there were no noticeable problems. It is characteristic that sometimes the experts offered their own interpretations of the belief interval BI. Among them the most successful and transparent was the interpretation of the BI = [Bel, P l] as the interval between pessimistic and optimistic assessments of the membership degree.
It is worth noting that our approach is flexible enough. For example, if someone wants to use only the µ and ν formalism of the A − IF S, he/she can first obtain from experts the estimations of BP N s (avoiding the above discussed problems concerned with the estimation of ν ) and then convert BP N s into µ and ν based on the expressions µ=Bel, ν=1 − P l. We should note here that in practice, we can meet the data available with such a structure that the µ and ν appear objectively without expert's judgments. (see examples in [66], [61]). Summarizing we can state that the introduced DST redefinition of the A − IF S is fruitful and justified from both theoretical and practical points of view.

III. NEW REPRESENTATIONS OF PEACE-WISE MEMBERSHIP AND NON-MEMBERSHIP FUNCTIONS, THEIR INTERVAL EXTENSIONS BASED ON THE REVEALED SOURCES OF INTERVAL AN FUZZY UNCERTAINTY
In the previous section, we considered only intuitionistic fuzzy numbers and Belief-Plausibility numbers without explicit pointing out their origins. Here we will consider objects characterizing by the functions µ(x) and ν(x), i.e. if A ⊂ X then it is defined by the functions µ A (x) and ν A (x). In the literature [21]- [41] the such type objects as A are called intuitionistic fuzzy numbers. Taking into account that in the previous section we called so (very reasonably) the objects of a different kind, here we will call objects of type A defined by µ A (x) and ν A (x), intuitionistic fuzzy values (IF V ). We are not going to introduce here any new terminology. Here we just use the terms "number" and "value" to indicate the differences between the two analyzed types of objects. Similarly, the objects defined by the Bel(x) and P l(x) will be called Belief-Plausibility values (BP V ). We deliberately do not give strict definitions at the beginning of our analysis as they will appear later when all the necessary prerequisites are formulated.

A. DEFINITION AND BUILDING OF THE REAL-VALUED IF V S AND BP V S
As it was stated in the previous section, one of the problem of the known IF V definitions [21]- [41] are the controversial asymptotic properties. Let us consider a typical case presented in Fig. 1a. We can see that in the interval [x 2 , x 3 ] we have ν(x) = 0, i.e. with the 100% of certainty (π(x) = 0) we should exclude non-membership and therefore we have a case of complete membership (µ(x) = 1). But we have such a result only in one point in the [x 2 , x 3 ]. In the interval [x 4 , x 5 ], we deal with the 100% of non-membership as µ(x) = 0. So it should be ν(x) = 1 in this interval, but we see that 0 < ν(x) ≤ 1. Let us consider this problem from the more general point of view. Really, if µ(x) = 1, then we have a complete (100%) certainty that some event has occurred. Therefore there is no place for the opposite event and hesitation, i.e. it should be ν(x) = 0 and π(x) = 0. The similar analyzes can easily be done for other cases. At first glance, this problem can be solved by the small correction as in Fig. 1)b, but however, this will not save the expert from the need to justify the feasibility of using such uncertain data (in the intervals [x 2 , x 3 ] and [x 4 , x 5 ]) and this is also not always possible.
The next flaw of the known approaches is the lack of clear instructions on how to build the µ(x) and ν(x) functions so that it always executes 0 ≤ µ(x) + ν(x) ≤ 1. Let us consider the simple triangular functions µ A (x) and ν A (x). It is easy to see that if they intersect in some x and x ∈ A and we have no an IF V . If µ A (x) = ν A (x) = 0.5 we obtain an ordinary fuzzy value: x ∈ A, but the problems similar to those presented in Fig. 1a automatically arise. The similar situation we meet in the case of trapezoidal functions µ A (x) and ν A (x). For the sake of simplicity, hereinafter we will consider only triangular (not necessary simple) types of the functions µ A (x) and ν A (x). Although we showed that the simple triangular cannot provide acceptable non-controversial presentation of the functions µ A (x) and ν A (x), the use of the small extension of triangular-the Pentagonal (piece-wise linear functions based on 5 reference points) IF V s [47]- [51] generally can solve this problem. In Fig. 2, we can see the example of such a presentation of the intuitionistic fuzzy value. It is seen that the lesser are the values of µ A (x) and ν A (x) in the point of intersection, the lesser is the uncertainty (hesitation degree) of the IF V . This may help in building the µ A (x) and ν A (x). Also we can see that all reasonable asymptotic properties are fulfilled. Of course, the proposed approach to building the functions µ A (x) and ν A (x), which is focused on their intersection is not unique. Nevertheless, we recommend it as providing great convenience, when it is implemented as the graphical interface: by dragging and dropping with the mouse that pointed out on possible intersections and holding other reference points to be stable, we can easily establish suitable forms of µ A (x) and ν A (x). The above premises allow us to introduce some formal definitions. Theorem 1: If the Pentagonal membership µ A (x) and nonmembership ν A (x) functions such that µ A (x) = 1, when Proof 1: Follows directly from the geometric construction.
. Of course, the same theorems hold for piece-wise linear functions µ(x) and ν(x), e.g., for the Hexagonal, Heptagonal, Octagonal IF V s and so on, as well as for any normalized convex µ(x) and the corresponding concave ν(x).

Definition 2:
The Pentagonal intuitionistic fuzzy value (P IF V ) is the mathematical object defined by the pentagonal membership µ A (x) and non-membership ν A (x) functions intersecting in the points, where µ A (x) = ν A (x) < 0.5 such that µ A (x) = 1, when ν A (x) = 0 and µ A (x) = 0, when ν A (x) = 1. Obviously, in any fixed x 1 ∈ A from P IF V we get the intuitionistic fuzzy number The Pentagonal intuitionistic fuzzy functions µ A (x) and ν A (x) can be presented as follows: Let the interval [x 1 , x 5 ] be the support of A ⊂ X (see Fig.  2). Then based on Fig. 2 we get: Significantly less problems we meet when building the BP V A (x) = [Bel A (x), P l A (x)], as in this case only what we should worried about of is the holding the constraint Bel A (x) ≤ P l A (x) and asymptotic properties seem to be obvious and do not need additional explanations. The formal definition of the BP V A (x) is as follows: Definition 3: The Belief-Plausibility value (BP V ) is the mathematical object defined by the functions Bel A (x) and P l A (x) such that P l A (x) = 0, when Bel A (x) = 0, P l A (x) = 1, when Bel A (x) = 1 and 0 ≤ Bel A (x) ≤ P l A (x) ≤ 1, x ∈ A. For any fixed x 1 ∈ A from BP V we get the Belief- It is remarkable that in comparison with the case of IF V , in this definition there is no need for any simplification, e.g. the use of only Pentagonal functions. The example is shown in Fig. 3.

B. INTERVAL-VALUED EXTENSIONS OF THE IF V S AND BP V S RESULTING FROM THE MULTIPLICITY OF MEMBERSHIP AND NON-MEMBERSHIP FUNCTIONS PRESENTATION
This is the case when, e.g., differed experts are asked to present own versions of functions µ A (x) and ν A (x). The correct interval-valued Pentagonal Intuitionistic fuzzy value IV P IF V defined by the interval valued functions [µ A (x)] and [ν A (x)] in presented in Fig. 4. It is easy to see that to get a correct IV P IF V it is enough to take care that at the points of intersection of the upper bounds of the interval functions [µ A (x)] and [ν A (x)] it is always performed µ AU (x) = ν AU (x) ≤ 0.5. VOLUME X, 2021 BelA(x)   Bel AU (x) ≤ P l AU (x). This constraint can be redefined in the framework of the A − IF S (taking into account that Bel AU (x) = µ AU (x) and P l AU (x) = 1 − ν AL (x)) as follows: µ AU (x) + ν AL (x) ≤ 1. This important constraint is absent in the Atanassov's definition [2] and was first introduced in our paper [17]. But here we will call it "soft constraint" since it does not prevent the cases when Bel AU (x) ≥ P l AL (x) as in the example in Fig. 5 (left side), which can be treated as controversial situations. Therefore, the "strong constraint" Bel AU (x) ≤ P l AL (x) that prevents the controversial situations like that in Fig. 5 (right side) reduces to its A − IF S form µ AU (x) + ν AU (x) ≤ 1, which is the same as in [2]. It is easy to see that if the strong constraint is held, then the soft constraint is fulfilled as well, but not vice verse. Besides, as we deal with interval extensions of belief and plausibility functions, the natural thing is the extension of the basic inequality Bel ≤ P L to its interval extension , where inequality is treated in the interval form.
There are many methods for interval comparison proposed in the literature. The most popular ones were studied in [67], where it was found that the simplest method based of intervals midpoints comparison provides results not the worst than even the complex methods based on the probabilistic approach [68]. Then the above interval inequality can be presented as Bel AL (x)+Bel AU (x) ≤ P l AL (x)+P l AU (x). Of course, having three possible constraints, we should present thee different definitions of IV BP V A (x), but since the last of them seems to be the some compromise between two first considered constraints, we propose here the following definition: For any x 1 ∈ A from IV BP V we obtain the Interval-Valued Belief-Interval number, which can be defined as follows: Definition 6: The Interval-Valued Belief-Plausibility number under uncertainty coursed by multiplicity of belief and plausibility functions is presented by the belief interval bounded belief interval: The introduced IV BP N is the belief interval bounded by two intersecting or not belief intervals. We can state that the intervals [Bel AL (x 1 ), Bel AU (x 1 )] and [P l AL (x 1 ), P l AU (x 1 )] can be formally treated as the belief intervals since in the Exp.
(2) we have 0 ≤ Bel AL (x 1 ) ≤ Bel AU (x 1 ) ≤ 1 and 0 ≤ P l AL (x 1 ) ≤ P l AU (x 1 ) ≤ 1. In the Definition 6, the presentation of IV BP N differs from that in [17], which in the current notation can be presented as follows: IV BP N = [[Bel AL , P l AL ], [Bel AU , P l AU ]]. We can say that a new Definition 6 may be treated as the upgraded version of that introduced in [17] as it is the direct interval extension of the real-valued BP N when the actual interval uncertainty is originated from multiplicity of belief and plausibility functions, whereas the definition from [17] is formulated without specifying the source of the interval uncertainty.

C. INTERVAL-VALUED EXTENSIONS OF THE IF V S AND BP V S AS THE CONSEQUENCE OF THE INTERVAL NATURE OF INPUT DATA
First we should say that introducing interval and fuzzy input data in the intuitionistic framework is not a purely mathematical exercise, but is justified by the specificity of real-world problems (see Section 4). Suppose x be some parameter defining values of the corresponding local criterion represented by the IV IF V (x) or IV BP V (x), where values of x belong to the interval of unstable measurements: x ∈ [x 1 , x 2 ] as it is shown in Fig. 6. Generally for calculating the value of IV BP N , the νA(x) following definition can be used: Definition 7: The Interval-Valued Belief-Plausibility number under uncertainty coursed by interval nature of input data is the belief interval bounded interval number defined by the belief and plausibility function as follows: It is important that in this case, we obtain again the belief interval bounded by belief intervals, since in the Exp. (3) we have always 0 ≤ inf Bel ≤ supBel ≤ 1 and 0 ≤ inf P l ≤ supP l ≤ 1. So the Definition 7 is the upgraded version of that in [17] as it directly specifies the source of interval uncertainty. In the considered case (see Fig. 6), the general expression (3) can be simplified to In this example, there are no intersections between the resulting intervals.
In the example presented in Fig. 7, the functions Bel A (x) and P l A (x) have their maximum equal to 1 and the function ν A (x) has its minimum equal to 0 in [x 1 , x 2 ].
As the result, we obtain: It is seen that in this case, all the resulting intervals are intersecting. It is shown in Fig. 7 and in Fig. 8 that all possible mutual locations of resulting intervals can be obtained. It is important that we often PlA(x)

meet the intersecting intervals [Bel
with the relatively wide area of intersection as in Fig. 7. Of course, we can complete eliminate this area redefining the results of the IV BP N calculation as follows: It is easy to see (Fig. 6) that in this case, we get two relatively small intervals [Bel A (x 1 ), P l A (x 1 )] and [Bel A (x 2 ), P l A (x 2 )] with the great gap between them. So instead of obtained in the direct calculations information we have the gap, which the width can serve as the measure of information lost. So such an approach seems to be at least unproductive with great information losses. Therefore, we will use here the described above approach based on the Exp.(3) and its simplification. It should be stressed that there are no any confrontational or debatable assumptions and methods in the Definition 7. So we should not reject from any information obtained with its use, but only properly treat and apply it in the analysis. For example, we have found that the width of the Bel and P l intervals intersection is gradually decreasing along with lowering an interval uncertainty, i.e. the width of input interval. So this intersection area should be used in analysis. Also, it can be considered as an indeterminacy area, where nor the belief estimation and no the plausibility one cannot be excluded and this is entirely in the spirit of the Dempster-Shafer theory formalism. It is important that although we have introduced here two different definitions 6 and 7 for the IV BP N and they somewhat differ from that in [17], the common thing in these definitions is the introduced concept of belief interval bounded by belief intervals. On the other hand, the Definitions 6 and 7 are the specified and upgraded versions of that in [17] as they originated from the different sources of interval uncertainty, while in the definition from [17] there are no any indications of uncertainty sources.

D. TRIANGULAR FUZZY EXTENSION OF THE IF V S AND BP V S ORIGINATED FROM THE TRIANGULAR FUZZY NATURE OF INPUT DATA
is the simple triangular fuzzy value representing the range of measured values of the input parameter x, where [x 1 , x 3 ] is the interval of all measurements and x 2 is the point, around which the measurements are observed most frequently, so that in the point x=x 2 , we have the top of A, i.e. the value of its membership function is equal to 1. Let BP V A (x) be the Belief-Plausibility value representing some criterion in the M CDM task by the corresponding belief and plausibility functions Bel A (x) and P l A (x). Then the value of this criterion for the fuzzy argument A = (x 1 , x 2 , x 3 ) can be obtained as the Triangular Fuzzy-Valued Belief-Plausibility Number T F V BP N A = BP V A (A), which in our case of simple triangular A, can be presented by the mapping of A on the Bel A (x) and P l A (x) as it is shown in Fig. 9. We can see that the resulting T F V BP N A is the belief In this case (see Fig. 9), the values of these bounding triangles can be calculated as follows: It is seen that Bel A (A) and P l A (A) can intersect. In such cases, the small interval in the intersecting area, similarly to the considered in the above section interval extension, can serve as the measure of uncertainty coursed by the uncertainty of the input triangular fuzzy value A = (x 1 , x 2 , x 3 ). This is an interesting and useful feature of the developed approach, but now it is out of scope of this paper. Then the corresponding operational laws with these numbers should be defined. In [16], [17], we showed that the classical arithmetical operations with the IF N s and IV IF N s are suffered from a set of important drawback that may lead to unacceptable, illogical results. Therefore, in [16], [17]  So based on the first above treatment of a belief interval, the addition operation with belief intervals was inferred in such a way that it is idempotent one and provides a belief interval as the result. These demands can be provided only by the averaging operation. The other operations used in M CDM were defined as those in the classical interval arithmetic. So for the real-valued BP N 1 =[Bel 1 , P l 1 ], BP N 2 =[Bel 2 , P l 2 ], ..., BP N n =[Bel n , P l n ] in [16], it was obtained: where α is a real value. This operation is correct only for α ≤ 1 since for α > 1 this operation cannot not always provide a correct belief interval. This restriction is admissible as α is usually used in the operations with belief intervals to solve M CDM problems, where α is less of 1 as it serves being the local criterion weight.
It is worth noting that the last operation cannot not be defined in the body of classical A − IF S. With the use of corresponding theorems, we have proved in [16] that these operations have good algebraic properties and free of revealed drawbacks of the conventional operations defined in the framework of the A − IF S.
Here we present the more justified than in [17], upgraded set of arithmetical operations with IV BP N s obtained with the use of Definition 7 for the case of interval uncertainty of input data, as based on these operations the operation laws with T F V BP N s used in the case study in Section 4 were inferred. Let These operations have good algebraic properties and are free of the drawbacks of corresponding operations defined in the A−IV IF S. This is easily proved using the same theorems as in [17] replacing the Bel L in them with Bel 11 , the P l L with Bel 12 , the Bel U with P l 11 and P l U with P l 12 in the first operand and by making similar replacements in the second one. Let T F V BP N 1 =((Bel 11 , Bel 12 , Bel 13 ), (P l 11 , P l 12 , P l 13 )), T F V BP N 1 =((Bel 21 , Bel 22 , Bel 23 ), (P l 21 , P l 22 , P l 23 )). Then T F V BP N 1 ⊗ f uzzy T F V BP N 2 = ((Bel 11 Bel 21 , Bel 12 Bel 22 , Bel 13 Bel 23 ), (P l 11 P l 21 , P l 12 P l 22 , P L 13 P l 23 )).
The introduced operations with T F V BP N s have good algebraic properties and are free of the drawbacks of those defined in the A − IF S for IN IF V s. The corresponding theorems are obtained from those proved in [17] by simply replacing the notation and so they are not represented here.

IV. CASE STUDY: THE SOLUTION OF TRIANGULAR FUZZY BELIEF-PLAUSIBILITY M CDM PROBLEM OF THE RAW MATERIAL SUPPLIER SELECTION ON THE STEEL-ROLLING PLANT
In this section, we present the example of the practical application of the developed methods for the solution of the triangular fuzzy Belief-Plausibility M CDM problem of the raw material supplier selection for the Belorussian Mini Metallurgical Plant BM M P based on the quality estimation of the ingot heating process before the hot rolling at the rolling mill 850. The plant was erected and commissioned on a "turn-key" basis in 1984 by "Voest-Alpine". The basic raw material used in the BM M P is the metal scrap delivered by different suppliers and therefore characterizing by different chemical compositions affected the product quality. For example, if one scrap consist mainly of cutted ships and another one contains many pressed cars then to use them often different technologies should be applied. In [69], we presented the solutions of two fuzzy multiple criteria optimization problems on the BM M P , but their applicability was restricted by the need to solve these problems for everyone new supplier. Therefore, based on the compromise technology providing not the best, but at least acceptable product quality for the considerable group of suppliers, it was proposed to make the multiple-criteria selection of them. The compromise technology was proposed by the Voest-Alpine, but not taking into account the specificity of the Belorussian suppliers. The ingot heating process was chosen as a the basic one as it is the process immediately preceding the rolling and in a great extent determining its quality. Therefore, the problem was formulated as follows: to make the selection of the metal scrap suppliers based on their ability to ensure high multiple criteria estimations of the ingot heating process quality. In this process, the ingots are heated in methodical gas furnaces. The methodical furnace with walking beams of the mill 850 is designed for heating continuously cast blooms with a cross section of 250x300 and 300x400 mm with a length of 2.5 to 5.5 m with a layout step of 150 and 200 mm, respectively. The total length of the furnace is 23 m, according to the location of the burners, it is divided into seven sections, which can be divided into methodical, two the so-called welding and homogenizing zones. Three parameters of the process quality were considered: the scale thickness δ to be minimized as it directly represent the metal losses, the accuracy of heating T to be maximized, and the maximum temperature difference in the ingot ∆ T at the time of unloading to be minimized because it causes thermal stresses that can lead to the destruction of the ingot during rolling. The corresponding membership functions µ δ , µ T and µ ∆T (or Bel δ , Bel T and Bel ∆T ) representing the local criteria of the process quality are presented in Fig. 10. The membership functions of the criteria µ δ , µ T and µ ∆T were proposed by the technologists based on the carried earlier industrial ex- periments, their experience and intuition. The criteria µ δ and µ T were nearly directly based on the existing technological instructions. More details are presented in [69]. The crucial for the considered problem thing is that after defining the membership functions, the technologists refused to present the non-membership ones. They argued such a rejection insisting that presenting the criteria by membership functions they had used exhaustively all the information available and therefore there is no place for any nonmembership functions, which in their opinion seem to be something artificial senseless mathematical constructions. Perhaps this was coursed by the psychological problem to give a negative assessment (non-membership), when a positive one (membership) is already established. Therefore, we were forced to abandon the use of the A−IF S methods to solve the problem, at least explicitly. Meanwhile, technologists experts were willing to answer the question: What would be your maximum permissible (optimistic) estimate of membership (plausibility, P L) if the Bel = µ is a guaranteed minimum estimate? The obtained estimates of the plausibility degrees are approximately presented in Fig. 10 by the piece-wise linear functions. The weights (relative importance) of the local criteria were calculated using the matrix of the linguistic pairwise comparisons of the criteria provided in the verbal form by experts. The method developed in [70] was used (see more details in [69]). The following weights were obtained: w δ =0.076, w T = 0.477 and w ∆T = 0.447 for the local criteria µ δ , µ T and µ ∆T , respectively. For all considered parameters of the process quality, their measurements were carried out for the sets of ingots produced from the scrap delivered by competing suppliers. Since the measurements of δ and ∆T required the use of very expensive methods based on the partial destruction of ingots it was not possible to obtain a sufficient number of measurements to apply rigorous statistical methods. Therefore, the results of measurement for each considered ith supplier were presented by triangular fuzzy numbers as follows: 1i , a 2i , a 3i ),  for the four considered suppliers are presented in Table 1. The values of the obtained T F BP N s were calculated as follows: Bel T rδi = (Bel(a 1i ), Bel(a 2i ), Bel(a 3i )), P l T rδi = (P l(a 1i ), P l(a 2i ), P l(a 3i )), Bel T rT i = (Bel(b 1i ), Bel(b 2i ), Bel(b 3i )), P l T rT i = (P l(b 1i ), P l(b 2i ), P l(b 3i )), Bel T r∆T i = (Bel(c 1i ), Bel(c 2i ), Bel(c 3i )), P l T r∆T i = VOLUME X, 2021  (Bel(c 1i ), Bel(c 2i ), Bel(c 3i )). Then using the operations (15) and (17), we inferred the generalized criterion GC i presentation for the considered problem as follow: GC i = (Bel GCi , P l GCi ) = ((w δ Bel T rδi + w T Bel T rT i + w ∆T Bel T r∆T i ), (w δ P l T rδi + w T P l T rT i + w ∆T P l T r∆T i )).
The results are presented in Fig. 14 and Table 2.
Analyzing the results presented in Fig. 14 and Table 2, with- out additional calculations we can conclude that Bel GC1 < Bel GC2 < Bel GC3 < Bel GC4 and P l GC1 < P l GC2 < P l GC3 < P l GC4 . Therefore the final rating of the suppliers can be presented according to the rating of the generalized criterion of the heating process quality, which was obtained using the delivered by the competing suppliers raw materials as follows: GC 1 < GC 2 < GC 3 < GC 4 . Of course, in general, the comparison of T F BP N s is not so easy problem as above and the corresponding method for its solution should be developed. But this is out of scope of this paper and now it is in the final stage of elaboration. The considered case study illustrates the ability of the proposed approach, based on the Belief-Plausibility formalism, to solve the real-world problems independently, without use the A − IF S concepts. It is shown that such an approach owns obvious advantages in comparison with the A − IF S based one.

V. CONCLUSION
The several important internal theoretical and practical problems of the A − IF S and A − IV IF S theories are analyzed and approaches to their solutions based on the redefinition of them in the framework of the DST are developed. When analyzing the ability of the novel Neutrosophic, Picture, Pythagorean and Spherical sets theories to solve the problems of the classical A − IF S theory, some their own limitations and drawbacks are revealed. It is shown that these drawbacks prevent the solution of the A − IF S problems, whereas the proposed approach based on the DST is free of them and performs better than these novel theories and the A − IF S. The properties of the DST extension of the A − IF S and its practical advantages for a decision maker in assigning certain real values to the parameters of A−IF S are analyzed. As the result of provided studies, a new mathematical object called Belief-Plausibility number (BP N ) is introduced. It is shown that the BP N has at least two particular cases: an usual fuzzy number (the value of a membership function) and an intuitionistic fuzzy number. It is emphasized that the BP N can be successfully applied completely separately not using terms of usual fuzzy sets and the A − IF S and this is the typical feature of more general theories. It were not found in the literature clear and practical approaches to mathematical formulation of the membership µ(x) and non-membership ν(x) functions in such a way that all asymptotic properties and basic constraints µ(x)+ν(x) ≤ 1, µ U (x) + ν U (x) ≤ 1 (µ U (x) and ν U (x) are the upper bounds of corresponding interval-valued membership and non-membership functions) will be fulfilled jointly. Therefore, in the current paper, such an approach is proposed. It was found that there are two main sources of uncertainty leading to the need for the use of A − IV IF S, The first of them is an interval uncertainty coursed by the multiplicity of the functions µ(x) and ν(x), e.g. proposed by different experts. The second is the consequence of interval uncertainty of input data, e.g. in the M CDM . And if input data we deal with are presented by fuzzy values, we meet the case of fuzzy-valued intuitionistic fuzzy set F V IF S. The described above general approach to treatment of interval-valued intuitionistic fuzzy objects and their Belief-Plausibility (based on the DST ) extensions allowed us to introduce a set of new fruitful definitions, some of the them, e.g. the definition of Fuzzy-Valued Belief-Plausibility value (F V BP V ) were not even mentioned in the literature in any form. The corresponding new arithmetical operations with such mathematical objects, following from these definitions, are proposed. They all, and especially the definition of F V BP V , were thoroughly analyzed under general methodological principles and their practical usefulness is illustrated by the real-world case study based on the solution of the DST based uncertain M CDM problem of the raw material supplier selection on the steel rolling plant under triangular fuzzy input data. The obtained results allowed us to state that the introduced mathematical objects BP N , IV BP N and F V BP N can be successfully used in the analysis and applications independently, without the use of the A − IF S concepts and such an approach possesses obvious advantages in comparison with the A − IF S based approach. The main directions of our future researches will be an extension of the proposed general approach to the cases when an uncertainty is originated jointly from the multiplicity of the Bel and P l functions formulations and uncertainty of input data (functions arguments), and the development of methods for the comparison of the Belief-Plausibility based objects characterising by different uncertainty levels.