Intuitionistic fuzzy possibility measure-based three-way decisions for incomplete data

Abstract

Over the last decade, a lot of attentions have been drawn in the problem of multiple attribute decision-making associated with incomplete information tables. The filling missing value method and minimum decision cost are two important challenges for incomplete information tables. But most published studies focus on the optimization of data reckoning without considering the risk appetite of decision makers and decision-making environments. In this paper, considering the above factors, we presented an attribute weight determination method and an effective computing method for dealing with intuitionistic fuzzy incomplete information tables. Then, delay-refused decision in the boundary region and the risk assessment based on cross-entropy were proposed. Finally, a secondary decision strategy based on the probability entropy of delay-accepted decision and the application algorithms were given for improving the quality of decisions.

Keywords

Intuitionistic fuzzy possibility measure incomplete information table three-way decisions secondary decision cross-entropy

1 Introduction

With the development of information technology, many data storage problems have been overcome. However, there are still some irresistible factors, which lead to the data missing. Some scholars have proposed various methods and theories to deal with the problem. It’s beneficial to solve this problem for data analysis, granular computing, and pattern recognition, etc.

Zadeh [1] proposed the concept of fuzzy sets, which attracted great attentions in the academic community [2 –4]. After that, Atanassov further extended fuzzy sets and proposed the intuitionistic fuzzy set theory [5, 6], which has two important metrics: membership degree and non-membership degree. Both of them can describe the fuzziness of things and make the description of the fuzzy and imprecise problems more delicate [7 –9].

On the other hand, Pawlak proposed the rough set theory [10]. As a mathematical method to deal with fuzzy and uncertain information, rough sets have been applied into many fields [11]. Based on the equivalence relation, the rough set theory can divide the universe of discourse into positive domain, negative domain and boundary region. Considering the redundant information in the boundary region, some researchers proposed the 0.5-probability rough set model [12 –14], which enriches and develops the theory of rough sets.

Based on the minimum risk Bayes decision theory, Yao presented a model of decision rough sets [15]. Furthermore, for applying granular computing to rough sets, the theory of three-way decisions was proposed by Yao [16, 17]. Three-way decisions are a superior way to adapt human cognition and habits. Liu [18] researched the dynamic three-way decision-making method considering dynamically changeable characteristics of loss functions. Yu et al. [19] discussed the methods and practices of complex problem solving associated with three-way decisions. On the basis of intuitionistic fuzzy sets, Xue et al. [20] presented a new three-way decision model to solve the descriptive problems of intuitionistic fuzzy data. Zhang et al. [21] studied the F-rough problem for rational decision making. In the process of decision-making, the objectivity of the threshold is positively related to decision-making results. Jia et al. [22] proposed an adaptive learning parameters algorithm in three-way decision rough sets. Zhang et al. [23] studied three-way decision thresholds for the multi-object optimization. But they all disregard the decision cost. Zhu et al. [24] designed a new three-way decision adaptive threshold algorithm. Zhang et al. [25] proposed a three-way decision model based on the cost-sensitivity to solve minimal costs problems.

Recently, the three-way decision theory has been increasingly significant in applications. Zhou [26] and Ayad [27] discussed the three-way decision email filtering problem in detail. The evaluation method of group decision supporting language was proposed by Liang et al. [28]. Zhang et al. [29] applied the multi-granularity cost-sensitive three-way decision method to face recognition. But, there are few studies of applying the three-way decision theory to intuitionistic fuzzy incomplete information tables (IFIIT) [19, 30]. However, three-way decisions have potential to deal with uncertain problems [31]. In the study of IFIIT, we also found that a subtle distinction in data may have a crucial impact on the outcome of decisions. So, how to fill the missing values and minimize the decision cost is a crucial problem. Xu et al. [32] introduced the intuitionistic fuzzy theory to the multi-attribute decision making and got satisfying results. Liu et al. [33] utilized interval numbers to acquire the loss function and applied it to hybrid incomplete information tables. Shi [34] designed a new method, but the method is excessively idealistic. Their results did not consider the cost and cannot reflect the real situation objectively. Yang et al. [35] redefined a similarity class and presented a new method in an incomplete information system. These studies extend the depth and scope of the three-way decisiontheory.

The contribution of this paper can be described as follows:

We make missing values of incomplete information more objective and reasonable under the framework of intuitionistic fuzzy sets.

The decision-making in our model can reflect the appetite of decision makers and minimize decision cost.

We proposed a secondary decision strategy, which is based on the probability entropy of delay-accepted decision and delay-refused decision in the boundary region.

The structure of this paper is organized as follows. Some basic concepts are introduced in Section 2. A method to attribute weight is given in Section 3. The rules of three-way decisions and secondary decisions are given in Section 4. Besides, we introduced the method of decision cost evaluation based on cross-entropy in Section 5 and an algorithm is given in Section 6. Finally, the validity of the new model is verified by an example and the summary of the full text is made in Sections 7 and 8.

2 Preliminaries

2.1 Intuitionistic fuzzy sets and its operations

Definition 2.1. [5, 6] Let U be a finite and nonempty set, then A = {< x, μ _A (x) , ν _A (x) > |x ∈ U} is an intuitionistic fuzzy set in U. μ _A (x) and ν _A (x) are the degrees of membership and non-membership respectively to the set U as follows: $\begin{matrix} μ_{A} (x) : U \to [0, 1], ν_{A} (x) : U \to [0, 1], \\ 0 \leq μ_{A} (x) + ν_{A} (x) \leq 1 . \end{matrix}$

The π _A (x) is called the hesitancy degree, and it can be obtained by membership value and non-membership value: $π_{A} (x) : U \to [0, 1], π_{A} (x) = 1 - (μ_{A} (x) + ν_{A} (x)) .$

All subsets of intuitionistic fuzzy on the domain U are represented by IF (U).

Definition 2 .2. [5, 6] Let A = {< x, μ _A (x) , v _A (x) > |x ∈ U} and B = {< x, μ _B (x) , v _B (x) > |x ∈ U} be two intuitionistic fuzzy sets. The equivalence relation, inclusion relation and union, intersection, product operations are defined as follows:

A = B iff ∀x ∈ U, μ _A (x) = μ _B (x), v _A (x) = v _B (x).

A ⊆ B iff ∀x ∈ U, μ _A (x) ≤ μ _B (x), ν _B (x) ⩾ ν _A (x).

A ∪ B = {< x, μ _A (x) ∨ μ _B (x) , ν _A (x) ∧ ν _B (x) > | ∀ x ∈ U}.

A ∩ B = {< x, μ _A (x) ∧ μ _B (x) , ν _A (x) ∨ ν _B (x) > | ∀ x ∈ U}.

AB = {< x, μ _A (x) μ _B (x) , μ _A (x) + μ _B (x) - ν _A (x) ν _B (x) > | ∀ x ∈ U}.

Definition 2.3. [36] The distance between intuitionistic fuzzy sets A = {x ₁, x ₂, ⋯, x _n} and B = {x ₁, x ₂, ⋯, x _n} is calculated in the following: $\begin{matrix} d (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} \\ \sqrt{| μ_{A} (x_{i}) - μ_{B} (x_{i}) |^{2} + | ν_{A} (x_{i}) - ν_{B} (x_{i}) |^{2} + | π_{A} (x_{i}) - π_{B} (x_{i}) |^{2}} \end{matrix}$

2.2 Cross-entropy

Definition 2.4. [37] Let X = {X ₁, X ₂, ⋯, X _n} be a random variable which has two probability distributions P and Q, where P is the real distribution of X, and Q is the simulated probability distribution of X. We can get the cross-entropy of the simulated probability distribution Q and real distribution P as follows: $H (p, q) = - \sum_{X} P {log}_{2} \frac{P}{Q}$

Obviously, the closer the simulated probability distribution Q gets the real distribution P, the smaller the Cross-entropy is.

2.3 Three-Way decision theory

Definition 2.5. [16, 17] Let U be a finite and nonempty set and R be an equivalence relation on U in rough set. The pair apr = (U, R) is called an approximation space. The equivalence relation R induces a partition of U, denoted as U/R. For a subset A ⊆ U, its lower and upper approximations are defined as follows: $\begin{matrix} \underline{apr} (X) = {x \in U | [x]_{R} \subseteq X} \\ \bar{apr} (X) = {x \in U | [x]_{R} \cap X \neq φ} \end{matrix}$

The equivalence class of object x is denoted as [x] _R. Based on the rough set approximations of X, the universe U can be divided into three disjoint regions: positive region POS (X), boundary region BND (X), and negative region NEG (X): $\begin{matrix} POS (X) = \underline{apr} (X) = {x \in U | [x] \subseteq X} \\ BND (X) = \bar{apr} (X) - \underline{apr} (X) \\ = {POS (X) \cup NEG (X)}^{c} \\ NEG (X) = U - \bar{apr} (X) = {x \in U | [x] \cap X = φ} \end{matrix}$

Definition 2.6. [38] Let S = (U, A, V, f) be a table of information. ∀x ∈ U, X ⊆ U, δ _A is a rough set membership degree: $δ_{A} (x) = P (X | [x]) = \frac{| X \cap [x] |}{| [x] |},$

where P (X| [x]) is the classification of conditional probability.

The objects can be differentiated by a pair of thresholds α and β. Then we will determine which region every object belongs to, positive region, boundary region or negative region.

Let 0 ≤ β < α ≤ 1, then the positive region, boundary region and negative region of (α, β) are defined as follows: $\begin{matrix} {POS}_{(α, β)} (X) = {x \in U | P (X | [x]) ⩾ α} \\ {BND}_{(α, β)} (X) = {x \in U | β < P (X | [x]) < α} \\ {NEG}_{(α, β)} (X) = {x \in U | P (X | [x]) \leq β} \end{matrix}$

2.4 Intuitionistic fuzzy possibility measure

Definition 2.7. [39] Let $R$ be an intuitionistic fuzzy domain which is composed of sets in IF (U), satisfying the following conditions:

$R = {< x, μ_{R} (x), ν_{R} (x) >}$ , $0 \leq μ_{R} (x) + ν_{R} (x) \leq 1$ ;

$\forall A \in R$ , $A^{c} = {< x, ν_{A} (x), μ_{A} (x) > | A \in R}$ ;

For $\forall A_{i} \in R$ , {A _i|i ∈ I} (I = 1, 2, ⋯) has the following properties:

$\cup_{i = 1}^{\infty} μ_{A_{i}} (x) \in R$ , $\cap_{i = 1}^{\infty} ν_{A_{i}} (x) \in R$

We can call $(U, R)$ an intuitionistic fuzzy domain space in the interval [0,1].

Definition 2.8. [39] Let Ps : x → [0, 1] and Ng : x → [0, 1], Cp : x → [0, 1] and Cn : x → [0, 1] be two pair sets of functions. For ∀A ∈ IF (U), the following conditions are satisfied:

Ps (μ (x)) =1, Ng (ν (x)) =0

${\begin{matrix} Ps (\underset{U_{i} \in U}{\cap} U_{i} (μ (x))) = \inf_{U_{i} \in U} Ps (μ_{U_{i}} (x)) \\ Ng (\underset{U_{i} \in U}{\cup} U_{i} (ν (x))) = \sup_{U_{i} \in U} Ng (ν_{U_{i}} (x)) \end{matrix}$

Then, (Ps, Ng) represents an intuitionistic fuzzy possibility measure (IFPM) about U. Similarly, the intuitionistic fuzzy reliability degree (Cp, Cn) should satisfy the following conditions:

Cp (A) =1, Cn (A) =0

${\begin{matrix} Cp (A) = \frac{1}{2} (0.5 + Ps (μ_{A} (x)) - Ps (μ_{A^{c}} (x))) \\ Cn (A) = \frac{1}{2} (0.5 - Ng (ν_{A} (x)) + Ng (ν_{A^{c}} (x))) \end{matrix}$

Definition 2.9. [39] Let μ (x) and ν (x) be membership degree and non-membership degree of object x. And the threshold value of boundary domain is (μ _(λ), ν _(λ)). We can get the decision rules as follows:

$\begin{matrix} P_{P} (Accept Decision Rules) \\ P_{P} (X) = {< x, μ (x), ν (x) > | \forall x \in U, \\ (μ (x) ⩾ μ_{(λ)}) \land ν (x) < ν_{(λ)}} \end{matrix}$ (1) $\begin{matrix} P_{N} (Negative Decision Rules) \\ P_{N} (X) = {< x, μ (x), ν (x) > | \forall x \in U, \\ (μ (x) < μ_{(λ)}) \land ν (x) ⩾ ν_{(λ)}} \end{matrix}$ (2) $\begin{matrix} P_{B} (Boundary Decision Rules) \\ P_{B} (X) = (P_{P} (X) \cup P_{N} (X))^{c} \end{matrix}$ (3)

3 Some questions in IFIIT

3.1 IFIIT expression form

Definition 3.1. Let the tetrad (U, A, V, ^*) represent an IFIIT. U = {x ₁, x ₂, ⋯, x _n} is a universe of discourse, which is composed of the research objects in IFIIT. A = {a ₁, a ₂, ⋯, a _m} is an attribute set. V is the value of the domain object in the attribute set. * is the loss data of membership degree or non-membership degree. The IFIIT can be expressed as follows:

$D = [\begin{matrix} (μ_{11} (x), ν_{11} (x)) & (μ_{12} (x), ν_{12} (x)) & \dots & (μ_{1 m} (x), ν_{1 m} (x)) \\ (*, ν_{21} (x)) & (μ_{22} (x), ν_{22} (x)) & \dots & (μ_{2 m} (x), ν_{2 m} (x)) \\ (μ_{31} (x), ν_{31} (x)) & (μ_{32} (x), *) & \dots & (*, *) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (*, ν_{n 1} (x)) & (μ_{n 2} (x), ν_{n 2} (x)) & \dots & (μ_{nm} (x), *) \end{matrix}]$ In this matrix, each pair of sequence couple (μ _ij (x) , ν _ij (x)) represents that the membership degree value and non-membership degree value of the object i under the attribute j is μ _ij (x) and ν _ij (x).

We always hold different attitudes towards risks, such as risk pursuit, risk aversion and risk neutrality. So, the corresponding methods to fill missing value should also be different. We can get:

Case 1. Risk loving:

$\begin{matrix} * : μ (x) = \cap Ps {\underset{a_{i} \in A}{\cup} (μ_{ij} (x), μ_{(i + 1, j)} (x))} \\ * : ν (x) = \cup Ng {\underset{a_{i} \in A}{\cap} (ν_{ij} (x), ν_{(i + 1, j)} (x))} \end{matrix}$ (4)

Case 2. Risk aversion:

$\begin{matrix} * : μ (x) = \cup Ps {\underset{a_{i} \in A}{\cap} (μ_{ij} (x), μ_{(i + 1, j)} (x))} \\ * : ν (x) = \cap Ng {\underset{a_{i} \in A}{\cup} (ν_{ij} (x), ν_{(i + 1, j)} (x))} \end{matrix}$ (5)

Case 3.Risk neutrality:

$* : μ (x) = \frac{1}{| \cdot |} \sum_{i = 1}^{n} μ_{ij} (x), * : ν (x) = \frac{1}{| \cdot |} \sum_{i = 1}^{n} ν_{ij} (x)$ (6)

where a ∈ A, μ _ij ∈ U, and | · | is the cardinal number of valid data.

Case 1 is the risk loving type, which has a tendency to seek maximum benefit. Under the same attribute, the smaller values of membership degree and the bigger values of non-membership are chosen through IFPM. Meanwhile, in order to ensure that the cost of decision-making is minimal, the overall membership degree values and non-membership degree values will take the conjunction operation and disjunction operation, respectively. The filling formula’s semantic interpretation of Case 2 is similar to the one of Case 1. Case 3 is the risk neutrality type, which does not pursue the maximization of benefits. Therefore, the treatment of their missing values is to utilize the average value method between the degrees of membership and non-membership.

3.2 The determination of attribute weights in IFIIT

The importance of attribute is positively related to the valuation method, and the weight of attribute has a direct influence on the decision results. So, how to assign weight is a researching focus. The current property assignment methods are the maximizing deviation method, judgment matrix method, PCA, AHP, etc. Inspired by previous researches [19 , 40], in this paper, the attribute weights are obtained by calculating the distance deviation of intuitionistic fuzzy reliability for all objects. It can be described as follows:

Let μ _ij (x) and ν _ij (x) be the membership values and non-membership values of object x _i based on the attribute a _j, i = 1, 2, ⋯, n, j = 1, 2, ⋯, m, a _j ∈ A.

The distance deviation of intuitionistic fuzzy reliability degree between objects can be calculated by: $\begin{matrix} d ((u_{ij} (x), ν_{ij} (x)), (Cp, Cn)) = \\ \frac{1}{2} \sum_{i = 1}^{n} \sqrt{| μ_{ij} (x) - Cp |^{2} + | ν_{ij} (x) - Cn |^{2} + | π_{ij} (x) - (1 - Cp - Cn) |^{2}} \end{matrix}$ (7)

If an attribute is more important, it will correspond to the bigger of total deviations. Also, if an attribute has little effect on all scenarios, the total deviation will be smaller. For the convenience of calculation, the results of weight are normalized.

Definition 3.2. Let Ω = (U, A, V, *) be an IFIIT. U = {x ₁, x ₂, ⋯, x _n} is the universe of discourse. The attribute set is A = {a ₁, a ₂, ⋯, a _m}, a _j ∈ A, and the distance measure between each objects and its intuitionistic fuzzy reliability degree are as follows: $\begin{matrix} η_{j} = \sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i})) \cdot w_{j} \\ i = 1, 2, \dots, n; j = 1, 2, \dots, m; k = 1, 2, \dots, n . \end{matrix}$ (8)

According to the deviation, we can get the following weight optimization problems: ${\begin{matrix} η_{j} = \sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i})) \cdot w_{j} \\ w_{j} ⩾ 0; i = 1, 2, \dots, n; j = 1, 2, \dots, n; k = 1, 2, \dots, n \\ s . t . \sum_{j = 1}^{m} w_{j}^{2} = 1 \end{matrix}$

By Lagrange’s operator, we can get the optimal solution: $w_{j} = \frac{\sum_{k = 1}^{n} \sum_{j = 1}^{n} d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i}))}{\sqrt{\sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} [d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i}))]^{2}}}$

Normalizing result: $w_{j}^{\circ} = \frac{w_{j}}{\sum_{j = 1}^{m} w_{j}} = \frac{\sum_{k = 1}^{n} \sum_{j = 1}^{n} d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i}))}{\sum_{j = 1}^{m} \sum_{i = 1}^{n} \sum_{k = 1}^{n} d ((u_{kj} (x), ν_{kj} (x)), ({Cp}_{i}, {Cn}_{i}))}$ (9)

4 Secondary decision

In IFIIT, according to the different risk preferences of filling incomplete data, it’s easy to conduct the three-way decision division by utilizing the comprehensive value and threshold to make decisions. And the partition method has been demonstrated in Section 2.3.

Definition 4.1. Let Ω = (U, A, V, *) be an IFIIT. U = {x ₁, x ₂, ⋯, x _n} is the universe of discourse. The attribute set is A = {a ₁, a ₂, ⋯, a _m}. W = {w ₁, w ₂, ⋯, w _m} is the attribute weight. The comprehensive value of object’s x _i is as follows: $x_{i} = \sum_{j = 1}^{m} (μ_{i} (x) \cdot w_{j}, ν_{i} (x) \cdot w_{j}), i = 1, 2, \dots, n$ (10)

According to the relationship between comprehensive value and threshold value, the three-way decision division can separate the acceptance decision and the rejection decision. However, this treatment is quite stringent for delaying decision-making. For example, if the boundary threshold is (0.6, 0.15), then, x ₁ (0.6, 0.15) will be designated as the acceptance domain and x ₂ (0.59, 0.01) will be divided into the delay decision domain. In the decision making, the object x ₁ must be superior to object x ₂? It is necessary to ensure the secondary decisions within the objects in the boundary domain.

The decision maker to choose the possibility of delayed decision making H _B should consist of two parts: delay-accepted decision H _BP and delay-refused decision H _BN. Delay-accepted decision means that policymakers choose to delay the decision, but the actual decision action should accept the decision. Similarly, delay-refused decision is that although policymakers choose to delay the decision, the actual decision action should reject the decision.

Definition 4.2. Let Δ _μ = μ _(λ) (x) - μ _(λ) and Δ _ν = ν _(λ) (x) - ν _(λ) represent the differences between the membership degree, the non-membership degree and the threshold value of the objects in the boundary domain, respectively. There are nine kinds of situations as follows: $\begin{matrix} I) {\begin{matrix} Δ_{μ} > 0 \\ Δ_{ν} > 0 \end{matrix} & II) {\begin{matrix} Δ_{μ} > 0 \\ Δ_{ν} < 0 \end{matrix} & III) {\begin{matrix} Δ_{μ} > 0 \\ Δ_{ν} = 0 \end{matrix} \\ IV) {\begin{matrix} Δ_{μ} < 0 \\ Δ_{ν} > 0 \end{matrix} & V) {\begin{matrix} Δ_{μ} < 0 \\ Δ_{ν} < 0 \end{matrix} & XI) {\begin{matrix} Δ_{μ} < 0 \\ Δ_{ν} = 0 \end{matrix} \\ XII) {\begin{matrix} Δ_{μ} = 0 \\ Δ_{ν} > 0 \end{matrix} & XIII) {\begin{matrix} Δ_{μ} = 0 \\ Δ_{ν} < 0 \end{matrix} & XIV) {\begin{matrix} Δ_{μ} = 0 \\ Δ_{ν} = 0 \end{matrix} \end{matrix}$

Among them, according to decision rules Equations (1–s3)

Cases II and XIII belong to the positive region;

Cases IV and XI belong to the negative region;

Other cases belongs to the boundary region.

As mentioned in the preceding paragraphs, the greater value of Δ _μ and smaller value of Δ _ν, are more likely inclined to be the delay-accepted decision. In order to calculate H _BP and H _BN easily, the probability of delay-accepted decision and delay-refused decision are put forward: $P_{BP} = \frac{Δ_{μ}}{u (x_{i})}, P_{BN} = \frac{Δ_{ν}}{ν (x_{i})}$ (11)

In order to guarantee that the probability is a positive value, its absolute value is used as the final result. If H _BN is not smaller than H _BP, the delay decision will be transformed into the refusal decision. We can get, $H_{BP} ⩾ H_{BN}$ (12)

On the contrary, the delay decision will be transformed into the acceptance decision. We can get, $H_{BP} < H_{BN}$ (13)

H _BP = - P _BP log ₂ P _BP, H _BN = - P _BN log ₂ P _BN, represent the amount of residual information, which is carried by Δ _μ and Δ _ν. Furthermore, if Δ _μ and Δ _ν are the same amount of residual information, i.e. H _BP = H _BN, we can compare the decision cost of objects between them. Hence, through the secondary decision strategy, we can classify the redundant information in the boundary domain directly.

5 Decision cost evaluation based on cross-entropy

In general, the cost of three-way decisions is calculated according to the cost functions, which are shown in Table 1.

Table 1
Classical cost function of three-way decisions

Decision action Cost function

X X ^C

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Decision action	Cost function
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

Normally, according to the logic relationship of the decision cost we can get the rank, λ _PP ≤ λ _BP < λ _NP, λ _NN ≤ λ _BN < λ _PN. Then, the domain experts might obtain specific values based on their experience, with a great subjectivity.

In order to make the risk function more objective, we have the following definitions:

Definition 5.1. Given the domain U = {x ₁, x ₂, ⋯, x _n} and the threshold (μ _(λ), ν _(λ)), the comprehensive value of object x _i is (μ _i (x) , ν _i (x)), which is short for $\sum_{j = 1}^{m} (μ_{i} (x) \cdot w_{j}, ν_{i} (x) \cdot w_{j})$ . The cost of three-way decisions can be expressed as: $C_{1} = \frac{1}{n} \sum_{i = 1}^{n} (μ_{i} (x) {log}_{2} \frac{μ_{i} (x)}{μ_{(λ)}} - ν_{i} (x) {log}_{2} \frac{ν_{i} (x)}{ν_{(λ)}})$ (14)

where μ _i (x) ≠ μ _(λ) or ν _i (x) ≠ ν _(λ).

When μ _i (x) = μ _(λ) and ν _i (x) = ν _(λ), we can calculate the cost in another form: $C_{2} = \frac{1}{n} \sum_{i = 1}^{n} (μ_{i} (x) {log}_{2} μ_{(λ)} - ν_{i} (x) {log}_{2} ν_{(λ)})$ (15)

If C < 0, it means that membership degree cost is smaller than non-membership degree cost. We can get result that the smaller the value C, the lower the cost. So, the cost of C < 0 is smaller than C ⩾ 0.

On the contrary, if C ⩾ 0, the bigger the value C, the higher the cost.

Proposition 5.1. The larger the deviation of membership degree in the domain is, the smaller the overall decision cost is.

Proof. For an object, the larger the membership degree value becomes, the farther it gets away from the threshold.

Under the constraints of 0 ≤ μ _i (x) + ν _i (x) ≤1 and μ _i (x) ≤1 - ν _i (x), when the membership degree μ _i (x) of object x _i increases, the non-membership degree becomes smaller. That is, the degree of deviation from the threshold ν _(λ) becomes larger, and the cost of non-membership degree becomes smaller.

We use the positive value expression to standardize the non-membership degree costs. The deviation value changes larger, the cost changes smaller.

Proposition 5.2. The decision cost of positive domain is less than the decision cost of negative domain.

Proof. i) Positive region P _P

When the objects are in the positive region P _P, μ _i (x) > μ _(λ), ν _i (x) < ν _(λ).

Membership degree cost: $μ_{i} (x) \log_{2} \frac{μ_{i} (x)}{μ_{(λ)}} > 0,$

Non-membership degree cost: $ν_{i} (x) \log_{2} \frac{ν_{i} (x)}{ν_{(λ)}} < 0 .$

Therefore, the value of decision cost is positive.

ii) Negative region P _N

When the objects are in the negative region P _N, μ _i (x) < μ _(λ), ν _i (x) > ν _(λ).

Membership degree cost: $μ_{i} (x) \log_{2} \frac{μ_{i} (x)}{μ_{(λ)}} < 0,$

Non-membership degree cost: $ν_{i} (x) \log_{2} \frac{ν_{i} (x)}{ν_{(λ)}} > 0 .$

Therefore, the value of decision cost is negative.

In conclusion, the decision cost of positive domain is less than the decision cost of negative domain.

Proposition 5.3. When μ _i (x) =1, ν _i (x) =0, the decision cost is minimum.

Proof. Given the domain U = {x ₁, x ₂, ⋯, x _n} and that the threshold be (μ _(λ), ν _(λ)). $\begin{matrix} μ_{1} (x) = 1 > μ_{2} (x) > \dots > μ_{n} (x) ⩾ 0, \\ ν_{1} (x) = 0 < ν_{2} (x) < \dots < ν_{n} (x), 0 \leq μ_{i} (x) + ν_{i} (x) \leq 1 . \end{matrix}$

We can get, $\begin{matrix} μ_{1} (x) \log_{2} \frac{μ_{1} (x)}{μ_{(λ)}} > \dots > μ_{n} (x) \log_{2} \frac{μ_{n} (x)}{μ_{(λ)}} \\ ν_{1} (x) \log_{2} \frac{ν_{1} (x)}{ν_{(λ)}} < \dots < ν_{n} (x) \log_{2} \frac{ν_{n} (x)}{ν_{(λ)}} \end{matrix}$

So, $\begin{matrix} μ_{1} (x) \log_{2} \frac{μ_{1} (x)}{μ_{(λ)}} - ν_{1} (x) \log_{2} \frac{ν_{1} (x)}{ν_{(λ)}} > \dots \\ > μ_{n} (x) \log_{2} \frac{μ_{n} (x)}{μ_{(λ)}} - ν_{n} (x) \log_{2} \frac{ν_{n} (x)}{ν_{(λ)}} \end{matrix}$

Thus, when μ _i (x) =1, ν _i (x) =0, the decision of the domain has the minimum cost.

6 Algorithm

In this section, we give the IFPM and its three-way decision model by an application algorithm for IFIIT. The basic idea in the algorithm is as follows.

Let a non-empty finite set U consist of research objects in the universe of discourse. First of all, according to the subjective risk attitude of decision makers, the filling method of missing value will be selected in IFIIT. Then, the distance deviation d between the intuitionistic fuzzy reliability degree (Cp, Cn) and attribute values a ∈ A of each objects x ∈ U is computed. In addition, according to the relationship between the total of deviations η and the boundary threshold value (μ _(λ), ν _(λ)), we obtained the attribute weights w and the comprehensive values. Moreover, three-way decisions are divided by the relationship between the comprehensive values and the threshold value. Finally, the objects x ∈ U in the boundary domain are re-divided into the positive domains POS (X) and the negative domains NEG (X) by the secondary decision strategy. And cross-entropy C is used to select the smallest decision cost scheme in the positive domain.

The pseudo-code of Algorithm 1 is described as follow.

7 Example analysis

In this section, we will illuminate the effectiveness of our proposed model by giving an example. Problem description:

Initially we have five motorcycle types, x ₁, x ₂, x ₃, x ₄, x ₅, x ₆ and four criteria, “consumption“, “maximum speed”, “price”, “after-sales service”, which are represented by c ₁, c ₂, c ₃ and c ₄, respectively. According the field expertise, the boundary threshold value is obtained with (0.48, 0.2). The evaluation value of each scheme is an intuitionistic fuzzy number, due to some objective reasons, data is not complete. And incomplete intuitionistic fuzzy matrix from the data is shown in Table 2.

Table 2
The incomplete information of different motorcycle types

c ₁ c ₂ c ₃ c ₄

x ₁ (0.6,0.2) (0.7,0.2) (0.3,0.4) (,0.1)

x ₂ (0.4, ) (0.5,0.2) (0.6,0.2) (0.8,0.1)

x ₃ (0.7,0.3) (,0.1) (0.5,0.5) (0.3,0.2)

x ₄ (0.3,0.4) (0.7,0.1) (, ) (0.4,0.3)

x ₅ (,0.1) (0.7,0.2) (0.6,0.3) (0.7,0.2)

x ₆ (0.3,0.4) (0.2,0.3) (0.2,0.5) (0.3,0.3)

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.6,0.2)	(0.7,0.2)	(0.3,0.4)	(*,0.1)
x ₂	(0.4, *)	(0.5,0.2)	(0.6,0.2)	(0.8,0.1)
x ₃	(0.7,0.3)	(*,0.1)	(0.5,0.5)	(0.3,0.2)
x ₄	(0.3,0.4)	(0.7,0.1)	(, )	(0.4,0.3)
x ₅	(*,0.1)	(0.7,0.2)	(0.6,0.3)	(0.7,0.2)
x ₆	(0.3,0.4)	(0.2,0.3)	(0.2,0.5)	(0.3,0.3)

In this case, it is necessary to determine which kind of motorcycle types should be more appropriate. Hypothesizing that the decision maker is a risk neutrality type, the detail steps to solve the problem are shown as follows:

Step 1. According to the decision maker’s risk preference, the complete data are shown in Table 3.

Table 3

The completion information of Table 2

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.6,0.2)	(0.7,0.2)	(0.3,0.4)	(0.5,0.1)
x ₂	(0.4,0.28)	(0.5,0.2)	(0.6,0.2)	(0.8,0.1)
x ₃	(0.7,0.3)	(0.56,0.1)	(0.5,0.5)	(0.3,0.2)
x ₄	(0.3,0.4)	(0.7,0.1)	(0.44,0.38)	(0.4,0.3)
x ₅	(0.46,0.1)	(0.7,0.2)	(0.6,0.3)	(0.7,0.2)
x ₆	(0.3,0.4)	(0.2,0.3)	(0.2,0.5)	(0.3,0.3)

Step 2. Calculate the IFPM and intuitionistic fuzzy reliability degree of each object.

IFPM: $\begin{matrix} Ps (x_{1}) = 0.3, Ng (x_{1}) = 0.4; & Ps (x_{2}) = 0.4, Ng (x_{2}) = 0.28; \\ Ps (x_{3}) = 0.3, Ng (x_{3}) = 0.5; & Ps (x_{4}) = 0.3, Ng (x_{4}) = 0.4; \\ Ps (x_{5}) = 0.46, Ng (x_{5}) = 0.3; & Ps (x_{6}) = 0.2, Ng (x_{6}) = 0.5 . \end{matrix}$

Intuitionistic fuzzy reliability degree: $Cp (U) = 0.2, Cn (U) = 0.2 .$

Then, the weight of attribute set a ∈ A is calculated by Equaiton (9): $w_{j}^{\circ} = (0.236, 0.262, 0.278, 0.224)^{T}, j = 1, 2, \dots, m .$

Step 3. We can get partition result by three-way decisions.

Positive region: {x ₂}

Boundary region: {x ₁, x ₃, x ₅}

Negative region: {x ₄, x ₆}

Step 4. Based on the Equations (12–15), we can get final division results of boundary domain objects by secondary decisions. $\begin{matrix} Δ_{μ} (x_{1}) = - 0.04, & Δ_{ν} (x_{1}) = - 0.033; & Δ_{μ} (x_{3}) = - 0.038 \\ Δ_{ν} (x_{3}) = - 0.081; & Δ_{μ} (x_{5}) = - 0.136, & Δ_{ν} (x_{5}) = - 0.004 . \end{matrix}$

Probability of delay-accepted decision and delay-refused decision: $\begin{matrix} P_{BP} (x_{1}) = 0.078, & P_{BN} (x_{1}) = 0.142; \\ P_{BP} (x_{3}) = 0.074, & P_{BN} (x_{3}) = 0.288; \\ P_{BP} (x_{5}) = 0.220, & P_{BN} (x_{5}) = 0.021; \end{matrix}$

Residual information: $\begin{matrix} H_{BP} (x_{1}) = 0.286, & H_{BN} (x_{1}) = 0.4; \\ H_{BP} (x_{3}) = 0.277, & H_{BN} (x_{3}) = 0.517; \\ H_{BP} (x_{5}) = 0.481, & H_{BN} (x_{5}) = 0.115; \end{matrix}$

So, we can get

$\begin{matrix} H_{BP} (x_{1}) < H_{BN} (x_{1}), H_{BP} (x_{3}) < H_{BN} (x_{3}), \\ H_{BP} (x_{5}) > H_{BN} (x_{5}) . \end{matrix}$

In other words, x ₁ and x ₃ are re-divided from the negative domain, and x ₅ is re-divided from the positive domain. The final three-way decisions partition as follows:

Positive region: {x ₁, x ₂, x ₃}

Negative region: {x ₄, x ₅, x ₆}

The scheme in the positive domain can be eventually chosen, and the scheme in the negative domain can be discarded directly.

Step 5. In the positive domain, we can choose the optimal scheme by calculating the decision cost.

Calculation results as follows:

C (x ₁) =0.009, C (x ₂) =0.023, C (x ₃) =0.215

In summary, C (x ₁) < C (x ₂) < C (x ₅), it is more appropriate to choose the scheme x ₁ in consideration of the cost to decision making.

8 Conclusion

This paper studies the application of incomplete data filling and decision making in IFIIT based IFPM and its three-way decisions. Firstly, according to the decision maker’s risk preference, the incomplete data is processed and the weights w are determined by the maximum distance measure deviation η. Then, according to the relationship between the comprehensive value and the threshold value (μ _(λ), ν _(λ)), the three-way decisions are divided. Finally, the decision of objects within the boundary domain is made for the secondary decisions. The method has the following advantages:

Objective values. In the IFIIT, the filling of missing values and the determination of attribute weights are based on the consideration of the cost, which can be reduced by the subjective factors.

Boundary factor. No longer ignore the role of the boundary region. Quantitative analysis was also carried out, which can be comprehensive considered the final decision.

There are also some shortcomings in the simulation, such as the subjectivity of the threshold, the dynamic change of the elements in the domain and how to establish of knowledge map for decision results, etc. These factors will certainly have an impact on decision-making and we will focus on the changes in the future.

Footnotes

Acknowledgments

This work is supported by the national natural science foundation of China under Grant Nos. 61772176, 61402153, and the scientific and technological project of Henan Province of China under Grant Nos. 182102210078, 182102210362, and the Plan for Scientific Innovation of Henan Province of China under Grant No. 18410051003, and the key scientific and technological project of Xinxiang City of China under Grant No. CXGG17002.

References

Zadeh

L.A.

, Fuzzy sets, Information & Control 8(65) (1965), 338–353.

Abdullah

and Najib

, A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process, Expert Systems with Applications 41(7) (2014), 3297–3305.

Mesiar

and Stupnanova

, Open problems from the 12th international conference on fuzzy set theory and its applications, Fuzzy Sets and Systems 261 (2015), 112–123.

Wang

W.N.

and Liu

X.D.

, Fuzzy forecasting based on automatic clustering and axiomatic fuzzy set classification, Information Sciences 294 (2015), 78–94. (In Chinese).

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Atanassov

K.T.

, Intuitionistic fuzzy sets. New York: Physica-Verlag, Heidelberg, 1999.

Deli

and Cagman

, Intuitionistic fuzzy parameterized soft set theory and its decision making, Applied Soft Computing 28 (2015), 109–113.

De Miguel

, Bustince

, Fernandez

, et al., Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making, Information Fusion 27 (2016), 189–197.

D.J.

and Shi

S.S.

, Researching the development of Atanassov intuitionistic fuzzy set: Using a citation network analysis, Applied Soft Computing 32 (2015), 189–198. (In Chinese).

10.

Pawlak

, Rough sets, International Journal of Computer & Information Sciences 11(5) (1982), 341–356.

11.

Miao

D.Q.

and Li

G.D.

, The algorithms and applications of Rough set theory. Beijing: Tsinghua University Press, 2008, pp. 1–23. (In Chinese)

12.

Sle

and Ziarko

, The investigation of the Bayesian rough set model, International Journal of Approximate Reasoning 40(1) (2005), 81–91.

13.

Ziarko

, Variable precision rough set model, Journal of Computer and System Sciences 46(1) (1993), 39–59.

14.

Yao

Y.Y.

, Probabilistic rough set approximations, International Journal of Approximate Reasoning 49(2) (2008), 255–271.

15.

Yao

Y.Y.

and Wong

S.K.M.

, A decision-theoretic rough set model, In: Proceedings of the 5th International Symposium on Methodologies for Intelligent System, Knoxville, Tennessee, 1990.

16.

Yao

Y.Y.

, Three-Way Decision: An interpretation of rules in rough set theory, In: International Conference on Rough Sets and Knowledge Technology, Springer-Verlag, 2009, pp. 642–649.

17.

Yao

Y.Y.

, Three-way decisions with probabilistic rough sets, Information Sciences 180(3) (2010), 341–353.

18.

Liu

, Li

T.R.

and Liang

D.C.

, Three-way decisions in dynamic decision-theoretic rough sets, In: International Conference on Rough Sets and Knowledge Technology, Springer Berlin Heidelberg, 2013, pp. 291–301.

19.

, Wang

G.Y.

, Li

T.Y.

, et al., Three-way decisions: Methods and practices for complex problem solving. Beijing: Science Press, 2015, pp. 269–285. (In Chinese).

20.

Xue

Z.A.

, Zhu

T.L.

, Xue

T.Y.

, et al., Three-way decisions model based on intuitionistic fuzzy sets, Computer Science 43(6) (2016), 285–288,297. (In Chinese).

21.

Zhang

, Deng

D.Y.

and Pei

M.H.

, A model of three-way decision based on F-rough sets, Journal of NanJing University: Natural Sciences 49(5) (2013), 582–587. (In Chinese).

22.

Jia

X.Y.

, Li

W.W.

, Shang

, et al., An adaptive learning parameters algorithm in Three-Way Decision theoretic rough set model, Acta Elfctronica Sinica 39(6) (2013), 2520–2525. (In Chinese).

23.

Zhang

Y.J.

, Miao

D.Q.

, Xu

J.F.

, et al., Determining thresholds in Three-way decisions: A multi-object optimization view, International Joint Conference on Rough Sets, Springer, Cham, 2017, pp. 326–339.

24.

Zhu

Y.H.

, Tian

H.L.

, Zhang

Y.P.

, et al., Adaptive algorithm for learning optimal threshold in three-way decisions based on two factors, Journal of Chinese Computer Systems 37(6) (2016), 1303–1307. (In Chinese).

25.

Zhang

Y.P.

, Zou

H.J.

and Zhao

, Cost sensitive three-way decisions model based on CCA, Journal of NanJing University: Natural Sciences 51(2) (2015), 447–452. (In Chinese).

26.

Zhou

, Yao

Y.Y.

and Luo

J.G.

, A Three-way decision approach to email spam filtering, In: Proceedings of the Advances in Artificial Intelligence, Canadian Conference on Artificial Intelligence, Canadian, Canada: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG., 2010, pp. 28–39.

27.

Ayad

R.A.

and Liu

, Supporting E-learning system with modified Bayesian rough set model, In: Proceedings of the 6th International Symposium on Neural Networks 2009, Lecture Notes in Computer Science 5552, Berlin: Springer, 2009, pp. 192–200.

28.

Liang

D.C.

, Witold

, Liu

, et al., Three-way decisions based on decision-theoretic rough sets under linguistic assessment with the aid of group decision making, Applied Soft Computing 29 (2015), 256–269. (In Chinese).

29.

Zhang

L.B.

, Li

H.X.

, Zhou

X.Z.

, et al., Multi-granularity cost-sensitive three-way decision for face recognition, Journal of Shandon University: Natural Science 49(8) (2014), 48–57. (In Chinese).

30.

Zhao

Y.P.

, An algorithm for attribute reduction of incomplete information system, In: Intelligent Computing and Intelligent Systems 2010 IEEE International Conference on IEEE, 2, 2010, pp. 635–638.

31.

Yao

Y.Y.

, Three-way decisions and cognitive computing, Cognitive Computation 8(4) (2016), 543–554.

32.

Z.S.

and Hui

, Models for intuitionistic fuzzy multiple attribute decision making, International Journal of Information Technology & Decision Making 9(02) (2010), 267–280.

33.

Liu

, Liang

D.C.

and Wang

C.C.

, A novel three-way decision model based on incomplete information system, Know-ledge-Based Systems 91 (2016), 32–45.

34.

Shi

Z.H.

, Zhang

D.H.

and Liu

, A TOPSIS method based on incomplete intuitionistic fuzzy information, Mathematics Practice and Theory 42(19) (2012), 234–240. (In Chinese).

35.

Yang

X.P.

and An

H.T.

, Three-way decisions based on intuitionistic fuzzy sets, In: International Joint Conference on Rough Sets, Springer, Cham, 2017, pp. 290–299.

36.

Szmidt

and Kacprzyk

, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114(3) (2000), 505–518.

37.

Kullback

and Leibler

R.A.

, On Information and sufficiency, Annals of Mathematical Statistics 22(22) (1951), 79–86.

38.

Liu

, Li

T.R.

, Miao

D.Q.

, et al., Three-way decision and granular computing. Beijing: Science Press, 2013, pp. 14–41. (In Chinese).

39.

Xue

Z.A.

, Xin

X.W.

, Yuan

Y.L.

, et al., Research on the Three-way Decisions model based on intuitionistic fuzzy possibility measures, Journal of Nanjing University: Natural Sciences 52(6) (2016), 1065–1074. (In Chinese).

40.

Wang

Y.M.

, Using the method of maximizing deviation to make decision for multiindices, Systems Engineering and Electronics 8(3) (1997), 21–26.

Decision action	Cost function
	X	X ^C
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

Intuitionistic fuzzy possibility measure-based three-way decisions for incomplete data

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Intuitionistic fuzzy sets and its operations

2.2 Cross-entropy

2.3 Three-Way decision theory

2.4 Intuitionistic fuzzy possibility measure

3.1 IFIIT expression form

Table 1 Classical cost function of three-way decisions Decision action Cost function X X C a P λ PP λ PN a B λ BP λ BN a N λ NP λ NN

7 Example analysis

Footnotes

Acknowledgments

References

Table 1
Classical cost function of three-way decisions

Decision action Cost function

X X ^C

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN