Evaluating evidence reliability based on intuitionistic fuzzy MCDM model

Abstract

In this paper, the combination of unreliable evidence sources is considered based on multiple criteria decision making (MCDM) model in intuitionistic fuzzy environment. In the intuitionistic fuzzy MCDM framework, evidence sources can be evaluated based on the ranking method between intuitionistic fuzzy numbers. A generalized discounting operation on unreliable evidence bodies is firstly proposed to deal with uncertain reliability factors. Then we mainly investigate the evaluation evidence reliability factors without prior knowledge. Our proposed evaluation method is based on the principle of self-assessment. It is implemented by the probabilistic comparison between intuitionistic fuzzy values. Our proposed evaluation method is independent to the dissimilarity measure between basic probability assignments represented by an evidential distance. Numerical examples demonstrate the performance of our proposed reliability evaluation method.

Keywords

Dempster-Shafer theory of evidence intuitionistic fuzzy set multiple criteria decision making (MCDM)reliability evaluation

1 Introduction

Dempster-Shafer evidence theory is an important tool for uncertainty reasoning and decision making [2 , 36]. In evidence theory, when all the sources are considered equally reliable, evidence combinations are implemented by the Dempster’s rule which is commutative and associative. Although Dempster’s rule of combination is well-founded theoretically [5 , 10], its lack of robustness is considered as a roadblock by researchers in this field. This is because counter-intuitive results are obtained in some cases, especially when there is high conflict among bodies of evidence (BOEs) [22, 27]. Such results are harmful for decision-making.

Many works have been done to prevent so-called counter-intuitive combination results. Researchers holding two major viewpoints have debated for decades. The focus of the disputation lies on the cause of counter-intuitive results. The first viewpoint is that the counter-intuitive results are caused by Dempster’s rule of combination, especially by its normalization step. Thus, some researchers have proposed alternative combination rules [9 , 28] that use various strategies to redistribute the conflict to provide a fusion tool that produces results that match expectations.

The second viewpoint is that the counter-intuitive results come from unreliable BOEs to be combined. According to this viewpoint, there are no counter-intuitive behavior results from the use of Dempster’s rule of combination, and the mass functions should be modified before combination [4 , 30]. Evidence discounting and weighted averaging, as a direct evolution of Murphy’s averaging method [3], are two methods for evidence modification. The weighted averaging method essentially depends on the convex combination of all evidence bodies by assigning a weight factor to each evidence body, and all weights sum to one. The basic idea of the discounting method is that if one source of evidence is not fully reliable, it should be discounted by multiplying each basic probability mass by a factor, and assigning the left mass to the full set. In both methods, the reliability of evidence source is applied to evaluate the weight factor or the discounting factor.

Although both two types of viewpoints are rational for certain applications, we prefer the idea that unreliable sources are the cause of the counter-intuitive results. In our view, the problem of Dempster’s combination rule lies on its assumption that all sources of evidence are considered as fully reliable. In fact, sources of information in real systems are always unreliable because different sources have access to different sectors of knowledge and experience. For example, experts differ in their level of expertise; some of them are more reliable than others due to their better knowledge, training, experience, intelligence, etc. To express their opinions, experts may use different background, methodology and knowledge. Hence, it is necessary to consider the expert reliability and consequently their judgments must be appropriately modified. Similarly, information provided by sensors does not have the same degree of reliability. This may be due not only to the same reasons as already mentioned for experts, but also to other factors more specific to sensors. For instance, measurements can differ from one sensor to another in terms of completeness, precision, and certainty. Additionally, the working environment can also affect the sensor reliability, since some of them could be better adapted to the conditions encountered in the considered environment than others.

Therefore, evidence bodies to be combined should be modified according to the reliability of their sources, which reflects the ability of each evidence source to provide a correct assessment of the given problem. The effects of evidence from more reliable sources should be strengthened, and at the same time, the effects of evidence from less reliable sources should be weakened. Thus, evidence reliability must be assessed before combination.

Then, how to determine the reliability of evidence? When the prior information is available, the reliability of evidence can be evaluated by training or optimization. Elouedi et al. [34] have proposed a method of assessing the sensor reliability in a classification problem based on the transferable belief model (TBM), which is an extension of evidence theory. In this method, the sensor reliability is assessed by minimizing the mean square error between the discounted sensor readings and the actual values of data.

The crucial problem is how to access the reliability of each evidence when there is no prior information. As mentioned above, the reliability of evidence is related to the ability of each source to provide correct or creditable information. Without prior information, there is no criterion to quantify the degree of reliability of evidence. Such situations have been regularly studied, and many methods have been proposed for the estimation of the reliability of evidence source based on the “principle of majority”. Following the principle of majority, the dissent or disagreement among different pieces of evidence can be used to generate a measure of reliability as a weight factor or discounting factor. If one source of evidence has great dissimilarity to the other sources, its reliability should be low. The evidential distance measure and evidential conflict measure are usually used to describe the difference or disagreement among different pieces of evidence. Some approaches have also emerged for averaging or discounting the BOEs based on distance of evidence and evidential conflict [17, 33].

Guo et al. [16] extended Elouedi’s work [34] in two aspects. On the one hand, they developed a new evaluation method to improve Elouedi’s method [34] and called it the static evaluation method. On the other hand, they treated the evaluation task as a two-stage training process, namely, supervised (or static) and unsupervised (or dynamic) evaluation, respectively. This leads to a deeper insight into the issue of sensor reliability evaluation. The first one is the static supervised evaluation method. A static discounting factor assigned to a sensor is based on the comparison between its original readings and the actual values of data. Information content contained in the actual values of each target is extracted to determine its influence on the evaluation. The second one is the dynamic evaluation method, which can be used to dynamically evaluate the evidence reliability by adaptive learning and regulation in real-time situations. The dynamic reliability is related to the contexts of sensor acquisitions and sensor dynamic performance.

In fact, the dynamic reliability evaluation is implemented in the situation of lacking prior knowledge, which is more common in practice. By the principle of majority, many other reliability evaluation methods have also been proposed. For example, Schubert [18] proposed a degree of falsity based on which the BOEs could be discounted. Based on Jousselme’s [1] distance measure, Klein and Colot [17] propose the degree of dissent, which is evaluated by comparing a BBA with the average BBA in a set. The distance between a BPA and the average BPA is applied to estimate its reliability. Based on Jousselme’s distance measure and Schubert’s idea, Yang et al. [33] defined a new disagreement measure by borrowing ideas from the design of Schubert’s degree of falsity to estimate the reliability of evidence source. Liu et al. [35] noted that the distance represented the difference between BPAs, whereas the conflict coefficient revealed the divergence degree of the hypotheses that two belief functions strongly support. These two aspects of dissimilarity were complementary in a certain sense, and their combination could be used as the dissimilarity measure. So they presented a new dissimilarity measure by fusing distance and conflict measure based on Hamacher T-conorm fusion rule. In the evaluation of evidence reliability, both its dissimilarity with other sources and their reliability factors were considered.

However, taking a closer examine on these methods, we can find that they all boil down to the definition of dissimilarity measure. In other words, the definition of dissimilarity measure largely determines the evaluation of evidence reliability. Our perspective on evidence combination is that combining evidence from different sources is similar to multiple criteria decision making (MCDM). Under such assumption, we can build MCDM model based on evidence sources to be combined. The main assumption of this study relies on the relation between evidence theory and intuitionistic fuzzy sets (IFSs). By transforming basic probability assignments (BPAs) to IFSs, the problem of evidence combination can be regarded as a multiple criteria decision making model under intuitionistic fuzzy environment.

In the problem of classification based on multisensor data fusion using evidence theory, the readings of each sensor represents certain part of characteristics of the target, i.e., different sensors represent different criteria. Let Ω = {A₁, A₂, ⋯ , A_m} be the discernment frame. The BPA m_i (A_j) with singleton focal elements coming from sensor i can be interpreted as the value of alternative A_j with respect to the criterion i. This discussion of sensor fusion is also potentially applicable to other kind of evidence combination problems. So we can estimate the reliability of evidence source borrowing the idea of evaluating weights of criteria in MCDM model. Based on the reliability factor of each evidence source, the BPA obtained from them can be discounted or weighting averaged. Our focus is on the representation of information about an uncertain variable and on estimating the reliability degrees based on the MCDM model in intuitionistic environment. We do not take the position that the approach suggested in the paper is the best or the only suitable approach to evaluate reliability of evidence sources, or to combine evidence sources; our position is that this may be a reasonable approach that may be of use to some decision makers.

To facilitate our discussion, in Section 2, we first describe background knowledge related to Dempster-Shafer theory of evidence, intuitionistic fuzzy sets and multiple criteria decision making methods. In Section 3, we propose a dynamic reliability evaluation method based on self-assessment. Numerical examples are given to illustrate the performance of the proposed approaches in Section 4. We come to the conclusion of this paper in Section 5.

2 Preliminaries

The background knowledge presented in this section deals with the following three main points: (1) the interpretation of Dempster-Shafer theory of evidence, (2) axiomatic definitions on intuitionistic fuzzy sets, and (3) the multiple criteria decision making model in intuitionistic fuzzy environment, which will facilitate the reliability estimation of evidence sources.

2.1 Dempster-Shafer theory of evidence

Dempster-Shafer theory of evidence is an approach for uncertainty reasoning. It enables us to combine evidence from different sources and arrive at a degree of belief. It has become an important method for the study of information fusion. To facilitate our discussion, we first briefly review basic concepts of the evidence theory, the combination rule and its counter-intuitive aspect. We then introduce the discounting strategy for unreliable evidence.

2.1.1 Basic concepts

Dempster-Shafer theory of evidence was modeled based on a finite set of mutually exclusive elements, called the frame of discernment denoted by Ω. The power set of Ω, denoted by 2^Ω, contains all possible unions of the sets in Ω, including Ω itself. Singleton sets in a frame of discernment Ω will be called atomic sets because they do not contain nonempty subsets. It is assumed that only one atomic set can be true at any one time. If a set is assumed to be true, then its all supersets are considered true as well.

An observer who believes that one or several sets in the power set of Ω might be true can assign belief masses to these sets. Belief mass on an atomic set A ∈ 2^Ω is interpreted as the belief that the set in question is true. Belief mass on a non-atomic set A ∈ 2^Ω is interpreted as the belief that one of the atomic sets it contains is true, but that the observer is uncertain about which of them is true. The following definitions are significant in the Dempster-Shafer theory.

Definition 2.1. [2] Let Ω = {A₁, ⋯ , A_n} be the frame of discernment. A basic probability assignment (BPA) is a function m : 2^Ω→ [0, 1], satisfying the following conditions: $m (\emptyset) = 0$ (1) $\sum_{A \subseteq Ω} m (A) = 1$ (2) where ∅ denotes empty set, and A is the subset of Ω.

Such a function is also called belief structure (often referred to as a basic belief assignment, BBA, or a basic mass assignment, BMA). For each subset A ⊆ Ω, the value taken by the BPA at A is called the basic probability mass of A, denoted by m (A). The value m (A) can be interpreted as the mass or the proportion of all relevant and available evidence that supports the claim that a particular element of Ω belongs to the set A but to no particular subset of A. The mass of the empty set is 0, and the summation of the masses of all subsets of the power set is 1. The value of 0 indicates total non-belief in a hypothesis, while the value of 1 indicates total belief. The mass of Ω, denoted as m (Ω), indicates the amount of uncertainty, called ignorance.

Definition 2.2. [2] A subset A of Ω is called the focal element of a belief structure if m (A) >0.

Associated with the measure m is a dual measure that plays a significant role in the fusion of information. One part of the dual measure is the belief function, denoted as Bel, which is a mapping from 2^Ω to [0, 1] measuring the total mass that must be distributed, in some way, among the elements of a certain subset of Ω. Thus, we have the definition of belief function as:

Definition 2.3. [3] Given a belief structure m on Ω, the belief function which is in one-to-one correspondence with m can be defined as: $Bel (A) = \sum_{B \subseteq A} m (B)$ (3)

In particular, the belief associated with A can be interpreted as the sum of the proportions of the total belief (masses) in all elements which entail A.

Another important function is the plausibility function, denoted as Pl, which is also a mapping from 2^Ω to [0, 1]. The plausibility function measures the maximal amount of mass that can be distributed among the elements in certain subset of Ω. Thus, the definition of plausibility function is expressed as:

Definition 2.4. [14] Given a belief structure m on Ω, the plausibility function of the element A is: $Pl (A) = \sum_{B \cap A \neq \emptyset} m (B)$ (4)

Bel (A) and Pl (A) are the lower and upper bounds of the belief level of hypothesis A, respectively. The belief value of hypothesis A is regarded as the minimal uncertainty about A, and its plausibility value is regarded as the maximal uncertainty about A. The relation between them is: $Pl (A) = 1 - Bel (\bar{A})$ (5) $Pl (A) \geq Bel (A)$ (6) where $\bar{A}$ is the classical complement of hypothesis A.

[Bel (A) , Pl (A)] is the confidence interval which describes the uncertainty about A, and Pl (A) - Bel (A) represents the level of ignorance about A. If the difference between Bel (A) and Pl (A) increases, then the information for fusion decreases or becomes unreliable. The difference provides a measurement of uncertainty about the level of belief in a decision. The interval [0, Bel (A)] is the range of support for hypothesis A, while the interval [0, Pl (A)] is the range of quasi belief in A, whereas the interval [Pl (A) , 1] is the range of refusal of A.

In evidence theory, we need to transform BPA to probability distribution on the discernment frame for the convenience of decision making. Pignistic transformation proposed by Smets [26] is widely used. The pignistic transformation maps a BPA m to so called pignistic probability function. It is defined as following.

Definition 2.5. [26] The pignistic transformation of BPA m on Ω = {A₁, …, A_n} is given by $BetP (A) = \sum_{B \subseteq Ω} \frac{| A \cap B |}{| B |} \frac{m (B)}{1 - m (\emptyset)}, \forall A \subseteq Ω$ (7) where | · | is the cardinality of a set.

In particular, given m (∅) =0 and A ∈ Ω, i.e., A is a singleton of Ω, we have: $BetP (A) = \sum_{A \in B} \frac{m (B)}{| B |}, A = A_{1}, \dots, A_{n}, B \subseteq Ω$ (8)

2.1.2 Dempster’s rule of combination

Definition 2.6. [2] Given two belief structures m₁ and m₂ on Ω, the belief structure that results from the application of Dempster’s combination rule, denoted as m₁ ⊙ m₂, or m₁₂ for short, is given by:

$\begin{matrix} m_{12} (A) \\ = {\begin{matrix} \frac{\sum_{B \cap C = A} m_{1} (B) m_{2} (C)}{1 - \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)}, \forall A \subseteq Ω, A \neq \emptyset \\ 0, A = \emptyset \end{matrix} \end{matrix}$ (9)

When multiple independent sources of evidence are available, the combined evidence can be obtained as:

$\begin{matrix} m (A) \\ = {\begin{matrix} \frac{\sum_{\cap A_{i} = A} \prod_{i = 1}^{n} m_{i} (A_{i})}{1 - \sum_{\cap A_{i} = \emptyset} \prod_{i = 1}^{n} m_{i} (A_{i})}, \forall A \subseteq Ω, A \neq \emptyset \\ 0, A = \emptyset \end{matrix} \end{matrix}$ (10)

Here, n is the number of evidence pieces in the process of combination, i denotes the ith piece of evidence, m_i (A_i) is the BPA of hypothesis A_i supported by evidence i. The value m (A) reflects the degree of combined support, joint mass, from n mutually independent sources of evidence corresponding to m₁, m₂, ⋯ , m_n, respectively. The quantity k defined in Equation (11) is the amount of conflict among n mutually independent pieces of evidence, which is equal to the mass of the empty set after the conjunctive combination and before the normalization step. It represents contradictory evidence. $k = \sum_{\cap A_{i} = \emptyset} \prod_{i = 1}^{n} m_{i} (A_{i})$ (11)

Dempster’s rule, however, has an inherent problem. When the pieces of evidence are completely contradictory, i.e. k = 1, combination cannot be performed. When they are highly conflicting, i.e. k → 1, the combination results seldom agree with the actual situation, and are counter-intuitive (see the example given by Zadeh [22]).

Example 2.1. Consider a situation in which we have two belief structures m₁ and m₂ over the same frame of discernment Ω = {A, B, C}. Let these two structures be as follows: $\begin{matrix} m_{1} : m_{1} (A) = 0.9, m_{1} (B) = 0.1, \\ m_{2} : m_{2} (B) = 0.1, m_{2} (C) = 0.9 . \end{matrix}$

Applying Dempster’s rule to these structures yields m (A) = m (C) =0, m (B) =1. We can see that m₁ and m₂ have low support level to hypothesis B, but the resulting structure has complete support to B. On the other hand, m₁ and m₂ have high support level on hypotheses A and C, respectively, but A and C are totally unbelievable in the result. This appears to be counter-intuitive. We can get the conflict degree between m₁ and m₂ is k = 0.99, which is the cause of counter-intuitive result.

Such counter-intuitive behavior indicates that Dempster’s rule cannot handle highly conflicting evidence. This problem can be handled from two main points of view. If the counter-intuitive behavior is believed to be caused by unreliable evidences, then the evidences should be discounted. However, if the counter-intuitive behavior is attributed to the combination rule, improvements of the combination rule, should then be made.

2.1.3 Evidence discounting

When a source of evidence is only partially reliable to a known reliability degree λ ∈ [0, 1], a discounting operation can be defined on the associated BPA [25]. The most common discounting operation was first introduced by Shafer in [14]. The discounting operation is given by ${\begin{matrix} m^{'} (A) = λ m (A), A \subset Ω \\ m^{'} (Ω) = 1 - λ + λ m (Ω) \end{matrix}$ (12) where λ represents the degree of reliability of the evidence. If λ = 1 (i.e. the evidence is completely reliable), then the BPA will remain unchanged. If λ = 0 (i.e. the evidence is completely unreliable), then the BPA will become m (Ω) =1, which means that the evidence provides no supports for decision-making.

2.2 Intuitionistic fuzzy sets

2.2.1 Concepts related to IFSs

Since intuitionistic fuzzy set can be considered as a generation of Zadeh’s fuzzy set, we first give the definition of Zadeh’s fuzzy set, followed by brief description on basic concepts of Intuitionistic fuzzy sets.

Definition 2.7. [21] Let X = {x₁, x₂, ⋯ , x_n} be a universe of discourse, then a fuzzy set A in X is defined as follows: $A = {〈 x, μ_{A} (x) 〉 | x \in X}$ (13) where μ_A (x) : X → [0, 1] is the membership degree.

Definition 2.8. [19, 20] An intuitionistic fuzzy set A in X defined by Atanassov can be written as: $A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ (14) where μ_A (x) : X → [0, 1] and v_A (x) : X → [0, 1] are membership degree and non-membership degree, respectively, with the condition: $0 \leq μ_{A} (x) + v_{A} (x) \leq 1$ (15)π_A (x) determined by the following expression: $π_{A} (x) = 1 - μ_{A} (x) - v_{A} (x)$ (16) is called the hesitancy degree of the element x ∈ X to the set A, and π_A (x) ∈ [0, 1], ∀x ∈ X.

π_A (x) is also called the intuitionistic index of x to A. Greater π_A (x) indicates more vagueness on x. Obviously, when π_A (x) =0, ∀x ∈ X, the IFS degenerates to Zadeh’s fuzzy set.

It is worth noting that besides Definition 2.8 there are other possible representations of IFSs proposed in the literature. Hong and Choi [7] proposed to use an interval representation [μ_A (x) , 1 - v_A (x)] of intuitionistic fuzzy set A in X instead of 〈μ_A (x) , v_A (x)〉. This approach is equivalent to the interval valued fuzzy sets interpretation of IFS, where μ_A (x) and 1 - v_A (x) represent the lower bound and upper bound of membership degree, respectively. Obviously, [μ_A (x) , 1 - v_A (x)] is a valid interval, since μ_A (x) ≤1 - v_A (x) always holds for μ_A (x) + v_A (x) ≤1.

In the sequel, IFSs (X) denotes the set of all IFSs in X. If |X|=1, i.e., there is only one element x in X, then the IFS A in X usually is denoted by A =〈 μ_A, v_A 〉 for short, which is also called intuitionistic fuzzy value (IFV). Denote the hesitancy degree of A as π_A = 1 - μ_A - v_A. Moreover, it is clear that each IFV A =〈 μ_A, v_A 〉 corresponds to an interval-valued value [μ_A, 1 - v_A]. If μ_A = 1 - v_A, the IFV reduces to a real number. The family of all IFVs will be denoted by L^* [3].

Definition 2.9. [20] For A ∈ IFSs (X) and B ∈ IFSs (X), some relations between them are defined as:

(R1) A > B iff ∀x ∈ X μ_A (x) > μ_B (x) , v_A (x) < v_B (x);

(R2) A < B iff ∀x ∈ X μ_A (x) < μ_B (x) , v_A (x) > v_B (x);

(R3) A = B iff ∀x ∈ X μ_A (x) = μ_B (x) , v_A (x) = v_B (x);

(R4) A^C={ 〈 x, v_A (x) , μ_A (x) 〉 |x ∈ X }, where A^C is the complement of A.

Definition 2.10. [19] For two intuitionistic fuzzy values A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉, the ordinary partial ordered relations are defined as: $\begin{matrix} A \geq B \Leftrightarrow μ_{A} \geq μ_{B}, v_{A} \leq v_{B}; \\ A = B \Leftrightarrow μ_{A} = μ_{B}, v_{A} = v_{B} . \end{matrix}$

Following these relations, we can find that the smallest intuitionistic fuzzy value in L^* is 〈0, 1〉 denoted by 0; while the largest element is 〈1, 0〉 denoted by 1.

Definition 2.11. [19] Let A = { 〈 x, μ_A (x) , v_A (x) 〉 | x∈ X }, B ={ 〈 x, μ_B (x) , v_B (x) 〉 |x ∈ X } be two IFSs in X, then the following operations can be defined: $\begin{matrix} A \cap B & = & {〈 x, min (μ_{A} (x), μ_{B} (x)), \\ max (v_{A} (x), v_{B} (x)) 〉 | x \in X}, \\ A \cup B & = & {〈 x, max (μ_{A} (x), μ_{B} (x)), \\ min (v_{A} (x), v_{B} (x)) 〉 | x \in X}, \\ A \oplus B & = & {〈 x, μ_{A} (x) + μ_{B} (x) - μ_{A} (x) \cdot μ_{B} (x), \\ v_{A} (x) \cdot v_{B} (x) 〉 | x \in X}, \\ A \otimes B & = & {〈 x, μ_{A} (x) \cdot μ_{B} (x), v_{A} (x) + v_{B} (x) \\ - v_{A} (x) \cdot v_{B} (x) 〉 | x \in X}, \\ γ \cdot A & = & {〈 x, 1 - {(1 - μ_{A} (x))}^{γ}, {(v_{A} (x))}^{γ} 〉 | x \in X}, \\ A^{γ} & = & {〈 x, {(μ_{A} (x))}^{γ}, 1 - {(1 - v_{A} (x))}^{γ} 〉 | x \in X} . \end{matrix}$

2.2.2 Ranking method of IFVs

In real-life decision situations, comparing or ranking IFVs is an important problem. The ranking method based on the score function and the accuracy function is widely used. It is defined as follows.

Definition 2.12. [7, 29] Let A =〈 μ_A, v_A 〉 be an IFV. Then the score function s and the accuracy function of a can be mathematically expressed as: s (A) = μ_A - v_A and a (A) = μ_A + v_A, respectively.

It is easily derived from Definition 2.12 that s (A) ∈ [-1, 1] and a (A) ∈ [0, 1].

s (A) reflects the difference of the certainty degree to which one element belongs to the IFV A. The larger s (A) the greater A.

a (A) expresses the sum of the certainty degree to which one element does not belong to the IFV A. Larger a (A) indicated that A is more accurate.

It is easy to see that the meanings of the score function and the accuracy function are similar to those of the mean and variance in statistics, respectively. Thus, using the score function and the accuracy function, a ranking method between two IFVs can be established as follows.

Definition 2.13. [7] Let A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 be two IFVs. Designate:

If s (A) > s (B), A is bigger than B, denoted by A ≻ B;

If s (A) < s (B), A is smaller than B, denoted by A ≺ B;

When s (A) = s (B), we have:

If a (A) = a (B), A is equal to B, denoted by A = B;

If a (A) < a (B), A is smaller than B, denoted by A ≺ B;

If a (A) > a (B), A is bigger than B, denoted by A ≻ B.

Example 2.2. Assume that there are three IFVs A₁ =〈 0.6, 0.3 〉, A₂ =〈 0.45, 0.2 〉, and A₃ =〈 0.5, 0.25 〉. According to Definition 2.12, scores of the IFSs A₁, A₂ and A₃ are computed as follows: $\begin{matrix} s (A_{1}) & = & 0.6 - 0.3 = 0.3, s (A_{2}) = 0.45 - 0.2 \\ = & 0.25, s (A_{3}) = 0.5 - 0.25 = 0.25 . \end{matrix}$

Hence, A₁ ≻ A₂ = A₃ if the scores of the IFVs A₁, A₂ and A₃ are merely taken into consideration. In order to make a distinction between A₂ and A₃, according to Definition 2.12, accuracy values of the IFSs A₂ and A₃ are obtained as follows: $\begin{matrix} a (A_{2}) & = & 0.45 + 0.2 = 0.65, \\ a (A_{3}) & = & 0.5 + 0.25 = 0.75 . \end{matrix}$

Then, according to Definition 2.13, it is easy to see that A₁ ≻ A₃ ≻ A₂.

2.2.3 MCDM model in IF environment

Suppose that there exists an alternative set A = {A₁, A₂, ⋯ , A_m} which consists of m efficient alternatives from which the best alternative has to be selected. Alternatives are assessed based on n criteria. Denote the set of all criteria by X = {x₁, x₂, ⋯ , x_n}. Usually alternatives are evaluated on quantitative criteria and qualitative criteria, respectively. Generally, alternatives are evaluated on quantitative criteria by using numerical values while they are evaluated on qualitative criteria through IFSs. In the intuitionistic fuzzy environment, numerical values need to be transformed to IFSs for consistency and clarity. So we need not only to acquire IFSs of alternatives on qualitative criteria but also transform numerical values of alternatives on quantitative criteria into IFSs. This is beyond the scope of this paper; hence, it is not presented here. In light of this, we point the reader to some references that may provide the basis for alternative approaches to address this issue [6, 7].

By expressing values of alternatives on qualitative and quantitative criteria as IFSs in a unified way, the vector of IFSs of all n criteria for alternatives A_i ∈ A can be written, respectively, as:

$\begin{matrix} (r_{i 1}, r_{i 2}, \dots, r_{in}) = (〈 μ_{i 1}, v_{i 1} 〉, 〈 μ_{i 2}, v_{i 2} 〉, \dots, \\ 〈 μ_{in}, v_{in} 〉), i = 1, 2, \dots, m \end{matrix}$ (17)

Thus a MCDM problem under intuitionistic fuzzy condition can be concisely expressed in the matrix format as follows: $\begin{matrix} F = {(〈 μ_{ij}, v_{ij} 〉)}_{m \times n} \\ x_{1} x_{2} \dots x_{n} \\ = \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} (\begin{matrix} 〈 μ_{11}, v_{11} 〉 & 〈 μ_{12}, v_{12} 〉 & \dots & 〈 μ_{1 n}, v_{1 n} 〉 \\ 〈 μ_{21}, v_{21} 〉 & 〈 μ_{22}, v_{22} 〉 & \dots & 〈 μ_{2 n}, v_{2 n} 〉 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}, v_{m 1} 〉 & 〈 μ_{m 2}, v_{m 2} 〉 & \dots & 〈 μ_{mn}, v_{mn} 〉 \end{matrix}) \end{matrix}$ (18)

which is referred to as an intuitionistic fuzzy decision matrix used to represent the MCDM model in intuitionistic fuzzy environment.

3 Dynamic reliability evaluation based on MCDM model

A multi-sensor data fusion system is an important component in many fields dealing with pattern recognition, identification, diagnosis, etc. It is used with the hope that the aggregation of several sensors achieves better results. Generally speaking, each sensor provides information of target identification from different aspects. If we suppose that each sensor represents a feature or criterion of the target, the readings of each sensor can be regarded as the degree to which the target matched to each pattern in the discernment frame under such criterion. Apparently, it can be also taken as the evaluation of each alternative (pattern in the discernment frame) on the criterion. So the problem of information fusion is analogous to multiple criteria decision making.

As discussed earlier, in evidence theory, [Bel (A) , Pl (A)] is the confidence interval which describes the uncertainty about A. It can be used to define the lower and upper probability bounds of the imprecise probability of A. Here, Bel (A) is the lower probability, and Pl (A) is the upper probability. Thus, the probability P (A) lies in an interval [Bel (A) , Pl (A)]. We note [Bel (A) , Pl (A)] is a sub-interval the unit interval [0, 1]. In the application of pattern recognition, Bel (A) can be taken as the membership degree to which the object belongs to A, while 1 - Pl (A) is the non-membership degree. Based on such analysis, a belief structure m on the discernment frame Ω = {A₁, A₂, ⋯ A_p} can be transformed to an interval probability distribution on A. Considering the interval representation of IFV, the interval probabilities can be further switched to an intuitionistic fuzzy values 〈μ (A_i) , v (A_i)〉 with μ (A_i) = Bel (A_i) and v (A_i) =1 - Pl (A_i). For the sake of exposition, we call 〈μ (A_i) , v (A_i)〉 as the intuitionistic fuzzy probability of A_i. Therefore, the model of multiple criteria decision making model in the intuitionistic fuzzy environment can be established.

Let Ω = {A₁, A₂, ⋯ , A_p} be the discernment frame. m₁, m₂, ⋯ , m_q are belief structures on Ω. 〈μ_j (A_i) , v_j (A_i)〉, i = 1, 2, ⋯ , p, j = 1, 2, ⋯ , q is the intuitionistic fuzzy probability of A_i generated by m_j, where μ_j (A_i) = Bel_j (A_i), v_j (A_i) =1 - Pl_j (A_i). Then the decision matrix can be written as: $\begin{matrix} \begin{matrix} F = {(〈 μ_{j} (A_{i}), v_{j} (A_{i}) 〉)}_{p \times q} \\ \begin{matrix} \begin{matrix} m_{1} m_{2} \dots m_{q} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{p} \end{matrix} & (\begin{matrix} A_{11} & A_{12} & \dots & A_{1 q} \\ A_{21} & A_{22} & \dots & A_{2 q} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A_{p 1} & A_{p 2} & \dots & A_{pq} \end{matrix}) \end{matrix} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} \begin{matrix} m_{1} m_{2} \dots m_{q} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{p} \end{matrix} & (\begin{matrix} 〈 μ_{1} (A_{1}), v_{1} (A_{1}) 〉 & 〈 μ_{2} (A_{1}), v_{2} (A_{1}) 〉 & \dots & 〈 μ_{q} (A_{1}), v_{q} (A_{1}) 〉 \\ 〈 μ_{1} (A_{2}), v_{1} (A_{2}) 〉 & 〈 μ_{2} (A_{2}), v_{2} (A_{2}) 〉 & \dots & 〈 μ_{q} (A_{2}), v_{q} (A_{2}) 〉 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 〈 μ_{1} (A_{p}), v_{1} (A_{p}) 〉 & 〈 μ_{2} (A_{p}), v_{2} (A_{p}) 〉 & \dots & 〈 μ_{q} (A_{p}), v_{q} (A_{p}) 〉 \end{matrix}) \end{matrix} \end{matrix}$ (19)

Our perspective on evidence combination is that combining evidence from different sources is similar to evaluating different alternatives in multiple criteria decision-making. When the reliability of each evidence source is known, based on the model of multiple criteria decision making in the intuitionistic fuzzy environment, the combination of unreliable evidence sources can be implemented by intuitionistic fuzzy aggregation operators. Thus, final decision can be achieved. Moreover, in the situation where the credibility factor needs to be determined, the reliability of evidence can be evaluated by assessing weights of criteria.

By transforming evidence combination to MCDM model in intuitionistic fuzzy environment, the dynamic reliability of evidence source is corresponding to the weight of criterion. So we can evaluate the dynamic reliability of evidence sources using criteria weighting approaches. Although many researchers [24] have been dedicated to developing methods of determining criteria weights in MCDM, most of them are based on the ordered weighting averaging (OWA) operation and non-linear optimization. Moreover, the so-called ideal solutions are needed for some approaches to minimize the instantaneous errors. We notice that these methods may be not applicable in the evaluation of evidence reliability, which concerns more about timeliness without enough a priori knowledge. So we will propose a new reliability evaluation approach based on probabilistic comparison between IFVs.

3.1 Comparison between IFVs

In Definition 2.10, the ordinary partial ordered relations between two IFVs A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 have been defined as A ≥ B ⇔ μ_A ≥ μ_B, v_A ≤ v_B. But for the situation of μ_A ≥ μ_B, v_A ≥ v_B, we cannot give their order. In addition, the ranking methods in Definition 2.10 can provide the qualitative order between IFVs, which is less applicable than quantitative comparison between IFVs. Considering the equivalence between IFVs and interval values, we can probabilistically compare IFVs based on the comparison between interval values.

Definition 3.1. [31, 32] Let α = [a^L, a^U] and β = [b^L, b^U] be two interval values such that a^L< a^U and b^L< b^U. The possible degree of the interval value α preceding the interval value β, denoted as P_(α≥β), is defined as:

$\begin{matrix} P_{(α \geq β)} \\ = min {1, max {\frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})}, 0}} \end{matrix}$ (20) It is worth mentioning that if α and β both reduce to precise values, i.e., a^L= a^U= a, b^L= b^U= b, Equation (20) is undefined owing to the (a^U- a^L) + (b^U- b^L) =0. In this special case, we take P_(α≥β) = 1 for a > b. Particularly, we assume P_(α≥β) = 0.5 for a = b.

Theorem 3.1. [13, 32] For two interval valuesα = [a^L, a^U] andβ = [b^L, b^U], t he possible degreeP_(α≥β)satisfies the following properties:

0 ≤ P_(α≥β) ≤ 1;

P_(α≥α) = 0.5;

P_(α≥β) = 1 ⇔ a^L≥ b^U;

P_(α≥β) + P_(β≥α) = 1.

Proof. (1) and (2) are both straightforward from the definition of P_(α≥β).

(3) $\begin{matrix} P_{(α \geq β)} = 1 \Rightarrow max {\frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})}, 0} \geq 1 \\ \Rightarrow \frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} \geq 1 \end{matrix}$

Since the denominator (a^U- a^L) + (b^U- b^L) is positive, we can get a^U- b^L> 0. Then it follows that: a^U- b^L≥ (a^U- a^L) + (b^U- b^L) = a^U- b^L- (a^L- b^U), which indicates that a^L- b^U≥ 0, i.e., a^L≥ b^U.

The inverse proof is trivial.

(4) Three cases listed in Fig. 1 need to be considered.

(i) For the separation case, we get a^L≥ b^U, P_(α≥β) = 1 and P_(β≥α) = 0. Thus P_(α≥β) + P_(β≥α) = 1.

(ii) The overlapping case indicates that b^L≤ a^L≤ b^U≤ a^U. Then we have: $\begin{matrix} P_{(α \geq β)} = \frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})}, \\ P_{(β \geq α)} = \frac{b^{U} - a^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} . \end{matrix}$

Thus, $\begin{matrix} P_{(α \geq β)} + P_{(β \geq α)} = \frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} \\ + \frac{b^{U} - a^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} = 1 . \end{matrix}$

(iii) In the inclusion case, we notice that b^L≥ a^L, b^U≤ a^U. Then we get: $\begin{matrix} P_{(α \geq β)} = \frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})}, \\ P_{(β \geq α)} = \frac{b^{U} - a^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} . \end{matrix}$

Therefore, $\begin{matrix} P_{(α \geq β)} + P_{(β \geq α)} = \frac{a^{U} - b^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} \\ + \frac{b^{U} - a^{L}}{(a^{U} - a^{L}) + (b^{U} - b^{L})} = 1 . \end{matrix}$

Since IFVs A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 can be expressed by interval values A = [μ_A, 1 - v_A] and B = [μ_B, 1 - v_B], respectively, we can also define the possibility of A ≥ B form Definition 3.1.

Definition 3.2. Let A =〈 μ_A, v_A 〉 and B =〈 μ_B, v_B 〉 be two IFVs with π_A · π_B ≠ 0. The possible degree of A ≥ B can defined as: $P_{(A \geq B)} = min {1, max {\frac{1 - v_{A} - μ_{B}}{π_{A} + π_{B}}, 0}}$ (21)

In the case of π_A · π_B = 0, A and B are both associated with precise real numbers. So we have P_(A≥B) = 1 if μ_A > μ_B and P_(A≥B) = 0.5 if μ_A = μ_B.

Theorem 3.2.Given two IFVsA =〈 μ_A, v_A 〉 andB = [μ_B, 1 - v_B], we have:

0 ≤ P_(A≥B) ≤ 1;

P_(A≥A) = 0.5;

P_(A≥B) = 1 ⇔ μ_A ≥ 1 - v_B;

P_(A≥B) + P_(B≥A) = 1.

Proof. They are all straightforward from Theorem 3.1 and Definition 3.2. □

It is worth noticing that P_(A≥B) = 1 ⇒1 - v_A ≥ μ_A ≥ 1 - v_B ≥ μ_B ⇒ μ_A ≥ μ_B, v_A ≤ v_B can be obtained from the third property in Theorem 3.2, but not vice versa. This is different from the ordinary partial ordered relation defined in Definition 2.10. So the relation A ≥ B defined in Definition 3.2 is much stricter than ordinary partial ordered relation in Definition 2.10.

We now suppose that there are n input arguments IFVs A₁, A₂, ⋯ , A_n with the forms of A_i =〈 μ_{A
_i}, v_{A
_i} 〉, i = 1, 2, ⋯ , n. To rank these arguments, we first compare each argument A_i with all arguments A_j, i, j = 1, 2, ⋯ , n. By Definition 3.2, we have:

$\begin{matrix} P_{(A_{i} \geq A_{j})} = min {1, max {\frac{1 - v_{A_{i}} - μ_{A_{j}}}{π_{A_{i}} + π_{A_{j}}}, 0}}, \\ i, j = 1, 2, \dots, n \end{matrix}$ (22)

For simplicity, we let P_ij = P_{(A_i≥A_j)}. Then we can construct a complementary comparison matrix as follows: $P = [\begin{matrix} P_{11} & P_{12} & \dots & P_{1 n} \\ P_{21} & P_{22} & \dots & P_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ P_{n 1} & P_{n 2} & \dots & P_{nn} \end{matrix}]$ (23)

Obviously, ∀i, j ∈ {1, 2, ⋯ , n}, we have 0 ≤ P_ij ≤ 1, P_ij + P_ji = 1, and P_ii = 0.5.

Summing all elements in each row of matrix P , we have: $P_{i} = \sum_{j = 1}^{n} P_{ij}$ (24)

Then we can rank the arguments A₁, A₂, ⋯ , A_n in accordance with the value of P_i, i ∈ {1, 2, ⋯ , n}.

To make this ranking method more transparent and comparable with ranking methods proposed in Definition 2.13, we reconsider Example 2.2.

By Definition 3.2, we get the complementary matrix: $P = [\begin{matrix} 0.50 & 0.56 & 0.57 \\ 0.44 & 0.50 & 0.50 \\ 0.43 & 0.50 & 0.50 \end{matrix}] .$

Then, $\begin{matrix} P_{1} = 0.50 + 0.56 + 0.57 = 1.63, \\ P_{2} = 0.44 + 0.50 + 0.50 = 1.44, \\ P_{3} = 0.43 + 0.50 + 0.50 = 1.43 . \end{matrix}$

So the ranking order is A₁ ≻ A₂ ≻ A₃.

We can see that this result is not identical to that from the ranking method in Definition 2.13. If we take $P_{i} = \sum_{j = 1}^{n} P_{ij}$ as a special score function, it is much more convenient for ranking IFVs than the score function s (A_i) and the accuracy function a (A_i). Moreover, since s (A_i) and a (A_i) are used separately, we cannot get a comprehensive evaluation on IFVs. So P_i is more applicable in ranking IFVs quantitatively.

3.2 Evaluate criteria weights based on self-assessment

In the application of MCDM, we can get the evaluation of an alternative for each criterion. If we make an assumption of economic person, each alternative can be treated as an economic person. Rationality, self-benefit and pursuit the biggest interest are the major features of an economic person. Based on this assumption, when an economic person (alternative) is evaluated by different criteria, it would assign the greatest weight to the criterion providing the maximum support for it. In light of this observation, a perfect competition can be achieved naturally, where the largest weight is assigned as 1, while others are 0. However, this assignment may cause a tremendous loss of information, which is unfavorable to rank all alternatives. So we need to evaluate criteria weights in an imperfect competition environment to make the best use of the superiority information as well as other information. This leaves us with the question of how to determine the weighs in an imperfect competition environment. It is natural and reasonable to use the probabilistic comparison between IFVs, which may be beneficial to construct an imperfect competition environment.

Let {A₁, A₂, ⋯ , A_p} be an alternative set to be assessed across a set of criteria {C₁, C₂, ⋯ , C_q}. The evaluation of the alternative A_i over criterion C_j by IFVs is expressed as: $\begin{matrix} A_{ij} & = & 〈 μ_{j} (A_{i}), v_{j} (A_{i}) 〉, \\ i & = & 1, 2, \dots, p, j = 1, 2, \dots, q . \end{matrix}$

Then the criteria weights can be assessed by alternative A_i, i = 1, 2, ⋯ , p, through the following three steps.

Step 1. By Definition 3.2 and Equation (23), construct a complementary matrix P ⁽ⁱ⁾= [P_mn] _q×q based on the evaluation results of an alternative A_i over all criteria.

Step 2. Sum all elements in each row of matrix P ⁽ⁱ⁾ by Equation (24): $P_{m}^{(i)} = \sum_{n = 1}^{q} P_{mn}, m = 1, 2, \dots, q$ (25)

Step 3. Calculate the weight of each criterion. The weight of C_j is determined by: $w_{j}^{(i)} = \frac{P_{j}^{(i)}}{\sum_{m = 1}^{q} P_{m}^{(i)}}$ (26)

The weighting vector of all criteria based on alternative A_i can be obtained as: $w^{(i)} = {(w_{1}^{(i)}, w_{2}^{(i)}, \dots, w_{q}^{(i)})}^{T}$ (27)

Since the weights of all criteria are evaluated by the assessment of each alternative, we call this method as self-assessment. We can note that the weighting vector obtained by different alternatives may be different. So we need to determine the final weighting vector. This is similar to the problem of group decision making, where a group of experts provide their

preferences over criteria with w ⁽ⁱ⁾. Here we assume that the ideal weighting vector should have the least distance to all w ⁽ⁱ⁾, i = 1, 2, ⋯ , p. If we let W = [ w ⁽¹⁾, w ⁽²⁾, ⋯ , w ^(p)], The Perron-Frobenius vector (the eigenvector associated to the maximal positive eigen value) of WW ^T can be used as the ideal weighting vector w , i.e., WW ^T· w = λ_max w . Usually, w need to be normalized by:

$w^{'} = w / \sum_{j = 1}^{q} w_{j}$ (28)

In the light of the correspondence between dynamic reliability of evidence sources and criteria weights, we can evaluate the dynamic reliability of evidence bodies by self-assignment-based method mentioned above.

Now we give an example to illustrate the given dynamic reliability evaluation for evidence bodies.

Example 3.1. Consider a situation in which we have five belief structures m₁, m₂, ⋯ , m₅ over the same frame of discernment Ω = {A₁, A₂, A₃, A₄}. Let these five belief structures be as follows: $\begin{matrix} m_{1} : m_{1} (A_{1}) = 0.2, m_{1} (A_{2}) = 0.3, m_{1} (A_{3}) = 0.15, \\ m_{1} (A_{4}) = 0.15, m_{1} (A_{1} A_{2}) = 0.1, m_{1} (A_{2} A_{3} A_{4}) = 0.1; \\ m_{2} : m_{2} (A_{1}) = 0.15, m_{2} (A_{2}) = 0.25, m_{2} (A_{3}) = 0.15, \\ m_{2} (A_{4}) = 0.1, m_{2} (A_{1} A_{3} A_{4}) = 0.15, m_{2} (A_{2} A_{4}) = 0.2; \\ m_{3} : m_{3} (A_{1}) = 0.3, m_{3} (A_{2}) = 0.35, m_{3} (A_{3}) = 0.15, \\ m_{3} (A_{4}) = 0.1, m_{3} (A_{1} A_{2} A_{3} A_{4}) = 0.1; \\ m_{4} : m_{4} (A_{1}) = 0.15, m_{4} (A_{2}) = 0.3, m_{4} (A_{3}) = 0.2, \\ m_{4} (A_{1} A_{2} A_{3}) = 0.15, m_{4} (A_{3} A_{4}) = 0.15; \\ m_{5} : m_{5} (A_{1}) = 0.3, m_{5} (A_{2}) = 0.2, m_{5} (A_{3}) = 0.1, \\ m_{5} (A_{4}) = 0.1, m_{5} (A_{2} A_{3}) = 0.2, m_{5} (A_{1} A_{3} A_{4}) = 0.1 . \end{matrix}$ We can get the intuitionistic fuzzy MCDM model as: $F = \begin{matrix} \begin{matrix} m_{1} m_{2} m_{3} m_{4} m_{5} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix} & (\begin{matrix} < 0.2, 0.7 > & < 0.15, 0.7 > & < 0.3, 0.6 > & < 0.15, 0.7 > & < 0.3, 0.6 > \\ < 0.3, 0.5 > & < 0.25, 0.55 > & < 0.35, 0.55 > & < 0.3, 0.55 > & < 0.2, 0.6 > \\ < 0.15, 0.75 > & < 0.15, 0.8 > & < 0.15, 0.75 > & < 0.2, 0.5 > & < 0.1, 0.6 > \\ < 0.15, 0.75 > & < 0.1, 0.65 > & < 0.1, 0.8 > & < 0, 0.85 > & < 0.1, 0.8 > \end{matrix}) \end{matrix}$

By Equations (22) and (23), we can get comparison matrices as following: $\begin{matrix} P^{(1)} = [\begin{matrix} 0.5 & 0.6 & 0 & 0.6 & 0 \\ 0.4 & 0.5 & 0 & 0.5 & 0 \\ 1 & 1 & 0.5 & 1 & 0.5 \\ 0.4 & 0.5 & 0 & 0.5 & 0 \\ 1 & 1 & 0.5 & 1 & 0.5 \end{matrix}], \end{matrix}$ $\begin{matrix} P^{(2)} = [\begin{matrix} 0.5 & 0.625 & 0.5 & 0.571 & 0.75 \\ 0.375 & 0.5 & 0.333 & 0.429 & 0.625 \\ 0.5 & 0.667 & 0.5 & 0.6 & 0.833 \\ 0.429 & 0.571 & 0.4 & 0.5 & 0.714 \\ 0.25 & 0.375 & 0.167 & 0.286 & 0.5 \end{matrix}], \\ P^{(3)} = [\begin{matrix} 0.5 & 0.667 & 0.5 & 0.125 & 0.375 \\ 0.333 & 0.5 & 0.333 & 0 & 0.286 \\ 0.5 & 0.667 & 0.5 & 0.125 & 0.375 \\ 0.875 & 1 & 0.875 & 0.5 & 0.667 \\ 0.625 & 0.714 & 0.625 & 0.333 & 0.5 \end{matrix}], \\ P^{(4)} = [\begin{matrix} 0.5 & 0.429 & 0.75 & 1 & 0.75 \\ 0.571 & 0.5 & 0.714 & 0.875 & 0.714 \\ 0.25 & 0.286 & 0.5 & 0.8 & 0.833 \\ 0 & 0.125 & 0.2 & 0.5 & 0.2 \\ 0.25 & 0.286 & 0.5 & 0.8 & 0.5 \end{matrix}] . \end{matrix}$

By Equation (25), we can get: $\begin{matrix} P_{1}^{(1)} = 1.7, P_{2}^{(1)} = 1.4, P_{3}^{(1)} = 4, \\ P_{4}^{(1)} = 1.4, P_{5}^{(1)} = 4; \\ P_{1}^{(2)} = 2.946, P_{2}^{(2)} = 2.262, P_{3}^{(2)} = 3.1, \\ P_{4}^{(2)} = 2.614, P_{5}^{(2)} = 1.577; \\ P_{1}^{(3)} = 2.167, P_{2}^{(3)} = 1.452, P_{3}^{(3)} = 2.167, \\ P_{4}^{(3)} = 3.917, P_{5}^{(3)} = 2.798; \\ P_{1}^{(4)} = 3.429, P_{2}^{(4)} = 3.375, P_{3}^{(4)} = 2.336, \\ P_{4}^{(4)} = 1.025, P_{5}^{(4)} = 2.336 . \end{matrix}$

By Equations (26) and (27), the weighting vectors evaluated by different “alternatives” A₁, A₂, A₃, A₄ can be given as: $\begin{matrix} w^{(1)} = {(0.136, 0.112, 0.320, 0.112, 0.320)}^{T}, \\ w^{(2)} = {(0.236, 0.181, 0.248, 0.209, 0.126)}^{T}, \\ w^{(3)} = {(0.173, 0.116, 0.173, 0.314, 0.224)}^{T}, \\ w^{(4)} = {(0.274, 0.270, 0.187, 0.082, 0.187)}^{T} . \end{matrix}$ So we have: $\begin{matrix} W = [w^{(1)}, w^{(2)}, w^{(3)}, w^{(4)}] \\ = [\begin{matrix} 0.136 & 0.236 & 0.173 & 0.274 \\ 0.112 & 0.181 & 0.116 & 0.270 \\ 0.320 & 0.248 & 0.173 & 0.187 \\ 0.112 & 0.209 & 0.313 & 0.082 \\ 0.320 & 0.126 & 0.223 & 0.187 \end{matrix}], \\ S = {WW}^{T} = [\begin{matrix} 0.179 & 0.152 & 0.183 & 0.141 & 0.163 \\ 0.152 & 0.132 & 0.151 & 0.109 & 0.135 \\ 0.183 & 0.151 & 0.229 & 0.157 & 0.207 \\ 0.141 & 0.109 & 0.157 & 0.161 & 0.148 \\ 0.163 & 0.135 & 0.207 & 0.148 & 0.203 \end{matrix}] . \end{matrix}$

The maximum eigenvalue of S is 0.811, and its corresponding vector is: $w = {(0.453, 0.375, 0.518, 0.396, 0.478)}^{T} .$

The final weighting vector for all belief structures is the normalized form of w , calculated as: $w^{'} = {(0.204, 0.169, 0.233, 0.178, 0.215)}^{T} .$

4 Numerical examples

We have proposed an alternative to evaluate dynamic reliability of evidence bodies based on self-assessment. When no a priori knowledge is available, we can evaluate evidence reliability without defining distance/similarity measure between BPAs. It can be well adapted for the fusion of highly conflicting sources of information for decision-making support. Two simple illustrative examples will be presented in this section to show the interest of our new reliability evaluation approach.

The context of these examples could correspond to an automatic target identification system using some independent sensors where the signals arising from these sensors are supposed to have been processed into BPAs by some given methods. The construction of BPAs is application dependent and is out of the scope of this paper. Here, we assume no prior knowledge about reliability, nor importance about the sources of evidence.

Firstly, we consider the example in [35] to show how the proposed method works for the decision making from Bayesian BPAs.

Example 4.1. Consider three simple Bayesian BPAs over the frame Ω = {A₁, A₂, A₃} as in Table 1. It is assumed that sources of evidence No. 3 can provide possibly two similar BPAs denoted m_3a and m_3b.

For Bayesian BPAs, the intuitionistic fuzzy MCDM model reduces to a fuzzy MCDM model. In this example, two decision matrices correspond to all BPAs can be obtained as: $\begin{matrix} F_{1} = \begin{matrix} \begin{matrix} m_{1} m_{2} m_{3 a} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix} & (\begin{matrix} < 0, 1 > & < 0.6, 0.4 > & < 0.75, 0.25 > \\ < 0.9, 0.1 > & < 0.25, 0.75 > & < 0.15, 0.85 > \\ < 0.1, 0.9 > & < 015, 0.85 > & < 0.1, 0.9 > \end{matrix}) \end{matrix}, \\ F_{2} = \begin{matrix} \begin{matrix} m_{1} m_{2} m_{3 b} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix} & (\begin{matrix} < 0, 1 > & < 0.6, 0.4 > & < 0.7, 0.3 > \\ < 0.9, 0.1 > & < 0.25, 0.75 > & < 0.2, 0.8 > \\ < 0.1, 0.9 > & < 015, 0.85 > & < 0.1, 0.9 > \end{matrix}) \end{matrix} . \end{matrix}$

Considering BPAs m₁, m₂ and m_3a, we can get their dynamic reliability factors using our proposed evaluation approach. The weighting vector is w _a = (0.294, 0.411, 0.295)^T. If m₁, m₂ and m_3b are combined, based on F ₂ the dynamic reliability can be evaluated as w _b = (0.294, 0.411, 0.295)^T. We can notice that these two dynamic reliability vectors are identical. That is because the probabilistic ranking result stays unchanged when sustaining m_3b for m_3a.

In this example, we compare original Dempster’s rule and Dempster’s rule with different discounting factors obtained by our proposed methods. To make our methods more transparent and comparable, the method proposed in [35] is also taken to make comparison.

The results obtained for this example are shown in Table 2. In the first row of Table 2, $m_{1}^{i}$ corresponds to the (sequential) fusion of sources m₁, m₂, ⋯ , m_i. The first column of Table 2 describes the method used for combining the sources of evidence. In the second method, we discount evidence bodies by the dynamic reliability factors, and combine them using Dempster’s rule.

Dempster’s rule provides the unreasonable result that A₁ is impossible to happen, which is illogical since there are two sources among three that consider A₁ as being most possibly true. Once the discounting approach by our proposed dynamic reliability is applied before the fusion of Dempster’s rule, we get the largest mass of belief to A₁ as expected.

We can see that m₂ and m_3a or m_3b commit most belief on A₁, and m_3a is very close to m_3b, but m₁, which distributes the largest belief to A₂, highly conflicts with m₂ and m_3a/m_3b. Thus, in Liu’s method, m₁ is not considered so reliable or important as the other ones according to the principle of majority. However, in our proposed self-assessment-based method, the dynamic reliability of m₂ is highest, while the reliability factor of m₁ is similar to that of m_3a/m_3b. For decision-making purpose, the fusion results presented in Tables 2 and 3 show that all methods are sensitive to the change on the third evidence body. We can also note that the discounted & Dempster’s rue can work as well as Liu’s method thanks to the dynamic reliability evaluation based on self-assessment.

This example illustrate that self-assessment-based reliability evaluation method can work as well as those based on the principle of majority. Moreover, less information loss will be caused by our proposed evaluation method. Due to the difference on the principle of majority and self-assessment, different combination results can be obtained. The truth may depend on the principle we prefer to apply in the given application under consideration.

To substantiate our arguments, we use another numerical example in [35] to demonstrate the performance of the proposed approaches when combining more than two pieces of evidence.

Example 4.2. Let’s consider a set of five normalized BPAs with imprecise focal elements over the frame of discernment Ω = {A₁, A₂, A₃} as given in Table 4. In this example m₃ is a Bayesian BPA, whereas all other BPAs are non-Bayesian.

When we use our proposed methods to determine the weights of evidence sources, decision matrices are needed. For convenience, we first give the intuitionistic fuzzy MCDM model F as shown in Equation (20) based on all BPAs. We should extract different columns to yield decision matrix according to evidence bodies to be combined. $F = \begin{matrix} \begin{matrix} m_{1} m_{2} m_{3} m_{4} m_{5} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix} & (\begin{matrix} < 0.8, 0.1 > & < 0.4, 0.3 > & < 0, 1 > & < 0.3, 0.45 > & < 0.45, 0.25 > \\ < 0.1, 0.8 > & < 0.2, 0.5 > & < 0.95, 0.05 > & < 0.2, 0.55 > & < 0.1, 0.45 > \\ < 0 0.9 > & < 0.1, 0.9 > & < 0.05, 0.95 > & < 0.25, 0.7 > & < 0, 0.55 > \end{matrix}) \end{matrix}$

Table 5 shows the combination of all the sources. We can see that when the first two evidence sources are combined, all methods can derive the result the hypothesis A₁ is very unlikely to happen. But when m₃ is considered, m (A₁) decreases while m (A₂) increases for all methods. Moreover, Dempster’s rule presents opposite conclusion. That is because BPAs m₁, m₂, m₄ and m₅ assign most of their belief to A₁, but m₃ oppositely commits its largest mass of belief to A₂. By our proposed reliability evaluation method, we can get w = [0 . 295, 0 . 411, 0 . 294]^T. The reliability factor of m₃ is close to that of m₁. So the discounting operation is competent enough to reduce its affect. With the participation of other evidence sources supporting A₁, the probability assigned to A₂ decreases, which is better to decision making. It is worth noticing that the classical Dempster’s rule is easy to fall into “veto-by-one-vote”.

In this example, we can note that the dynamic reliability factor estimated based on self-assessment is sensitive to the change of evidence sources. This reliability factor can be applied to discount evidence to reduce the influence of unreliable evidence bodies. The interest of the new reliability estimation method lies in the principle of self-assessment, where each criterion is regarded as an estimation agent. The debatable distance/similarity measure between

evidence bodies is unnecessary in the proposed evaluation method.

5 Conclusion

In this paper we have mainly investigated the evaluation of evidence reliability in the frame of multiple criteria decision making model under intuitionistic fuzzy environment. By transforming the problem of evidence combination to MCDM model in intuitionistic fuzzy environment, we propose a novel reliability evaluation method for unreliable evidence sources. Unlike many existing reliability evaluation method, our proposed method is based on the principle of self-assessment and the probabilistic comparison between intuitionistic fuzzy values. Generally, when combining unreliable evidence sources, we can evaluate their reliability factors based on our proposed evaluation method. Then the evidence sources can be discounted by the reliability factors. Finally the discounted evidence sources are combined by Dempster’s rule. Numerical examples demonstrate that the proposed method is capable of reducing the influence of unreliable evidence sources.

Footnotes

Acknowledgments

This research is supported by the Natural Science Foundation of China under Grants No. 71572082 and No. 61503407.

References

Jousselme

A.-L.

, Grenier

and Bosse

, A new distance between two bodies of evidence, Information Fusion2(2) (2001), 91–101.

Dempster

A.P.

, Upper and lower probabilities induced by a multiple valued mapping, The Annals of Mathematical Statistics38 (1967), 325–339.

and He

, Intuitionistic fuzzy PRI-AND and PRI-OR aggregation operators, Information Fusion14 (2013), 450–459.

Murphy

C.K.

, Combining belief functions when evidence conflicts, Decision Support Systems29(1) (2000), 1–9.

Dubois

and Prade

, On the unicity of Dempster rule of combination, International Journal of Intelligent Systems1 (1986), 133–142.

D.-F.

, The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets, Mathematical and Computer Modelling53 (2011), 1182–1196.

Hong

D.H.

and Choi

C.H.

, Multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems114 (2000), 103–113.

Klawonn

and Schwecke

, On the axiomatic justification of Dempster’s rule of combination, International Journal of Intelligent Systems7(5) (1992), 469–478.

Lefevre

, Colot

and Vannoorenberghe

, Belief functions combination and conflict management, Information Fusion3(2) (2002), 149–162.

10.

Pichon

and Denoeux

, The unnormalized Dempster’s rule of combination: A new justification from the Least Commitment Principle and some extensions, Journal of Automated Reasoning45(1) (2010), 61–87.

11.

Smarandache

and Dezert

, Applications and Advances of DSmT for Information Fusion, vol. 3, American Research Press, Rehoboth, 2009, pp. 4–32.

12.

Beliakov

, Bustince

, Goswami

D.P.

, Mukherjee

U.K.

and Pal

N.R.

, On averaging operators for Atanassov’s intuitionistic fuzzy sets, Information Sciences181 (2011), 1116–1124.

13.

Facchinetti

, Ricci

and Muzzioli

, Note on ranking fuzzy triangular numbers, International Journal of Intelligent Systems13 (1998), 613–622.

14.

Shafer

, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976.

15.

Shafer

, Perspectives on the theory and practice of belief functions, International Journal of Approximate Reasoning4(5–6) (1990), 323–362.

16.

Guo

, Shi

and Deng

, Evaluating sensor reliability in classification problems based on evidence theory, IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics36(5) (2006), 970–981.

17.

Klein

and Colot

, Automatic discounting rate computation using a dissent criterion, in: Proceedings of the Workshop on the Theory of Belief Functions, Brest, France, 2010, pp. 1–6.

18.

Schubert

, Conflict management in Dempster–Shafer theory using the degree of falsity, International Journal of Approximate Reasoning52(3) (2011), 449–460.

19.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(1) (1986), 87–96.

20.

Atanassov

K.T.

, On intuitionistic fuzzy sets theory, Berlin: Springer-Verlag; 2012.

21.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–353.

22.

Zadeh

L.A.

, A simple view of the Dempster–Shafer theory of evidence and its implication for the rule of combination, AI Magazine2 (1986), 85–90.

23.

Dymova

and Sevastjanov

, An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect, Knowledge-Based Systems23 (2010), 772–782.

24.

Chen

L.-H.

, Hung

C.-C.

and Tu

C.-C.

, Considering the decision maker’s attitudinal character to solve multi-criteria decision-making problems in an intuitionistic fuzzy environment, Knowledge-Based Systems36 (2012), 129–138.

25.

Florea

M.C.

, Jousselme

A.-L.

and Bosse

, Robust combination rules for evidence theory, Information Fusion10(2) (2009), 183–197.

26.

Smets

, Data fusion in the transferable belief model, in: Proceedings of the 3rd International Conference on Information Fusion, Paris, France, 2000, pp. PS21–PS33.

27.

Haenni

, Shedding new light on Zadeh’s criticism of Dempster’s rule of combination, in: Proceedings of the Eighth International Conference on Information Fusion, Philadelphia, USA, IEEE,Piscataway, NJ, 2005, pp. 879–884.

28.

Yager

R.R.

, On the Dempster–Shafer framework and new combination rules, Information Science41(2) (1987), 93–137.

29.

Chen

S.M.

and Tan

J.M.

, Handling multicriteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems67 (1994), 163–172.

30.

Deng

, Shi

W.K.

, Zhu

Z.F.

and Liu

, Combining belief functions based on distance of evidence, Decision Support Systems38(3) (2004), 489–493.

31.

Nakahara

, User oriented ranking criteria and its application to fuzzy mathematical programming problems, Fuzzy Sets and Systems94 (1998), 275–276.

32.

Nakahara

, Sasaki

and Gen

, On the linear programming problems with interval coefficients, Computers & Industrial Engineering23 (1992), 301–304.

33.

Yang

, Han

and Han

, Discounted combination of unreliable evidence using degree of disagreement, International Journal of Approximate Reasoning54 (2013), 1197–1216.

34.

Elouedi

, Mellouli

and Smets

, Assessing sensor reliability for multisensor data fusion within the transferable belief model, IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cyetics34(4) (2004), 782–787.

35.

Liu

Z.-g.

, Dezert

, Pan

and Mercier

, Combination of sources of evidence with different discounting factors based on a new dissimilarity measure, Decision Support Systems52 (2011), 133–141.

36.

Liu

Z.-g.

, Pan

, Dezert

and Martin

, Adaptive imputation of missing values for incomplete pattern recognition, Pattern Recognition52 (2016), 85–95.