Three-way decisions based on multi-granulation support intuitionistic fuzzy probabilistic rough sets

Abstract

Three-way decisions have become a representative of the models dealing with decision-making problems with uncertainty and fuzziness. However, most of the current models are single granular structures that cannot meet the needs of complex fuzzy environmental decision-making. Multi-granulation rough sets can better deal with fuzzy problems of multiple granularity structures. Therefore, three-way decisions will be a more reasonable decision-making model to address uncertain decision problems in the context of multiple granularity structures. In this paper, firstly we propose the four different conditional probabilities based on support intuitionistic fuzzy sets, which are referred to as support intuitionistic fuzzy probability. Then, a multi-granulation support intuitionistic fuzzy probabilistic approximation space is defined. Secondly, we calculate the thresholds α and β by the Bayesian theory, and construct four different types of multi-granulation support intuitionistic fuzzy probabilistic rough sets models in multi-granulation support intuitionistic fuzzy probabilistic approximation space. Moreover, some properties of lower and upper approximation operators of these models are discussed. Thirdly, by combining these proposed models with three-way decision theory, the corresponding three-way decision models are constructed and three-way decision rules are derived. Finally, an example of person-job fit procedure is given to prove and compare the validity of these proposed models.

Keywords

Support intuitionistic fuzzy sets rough sets support intuitionistic fuzzy probabilistic multi-granulation rough sets three-way decisions

1 Introduction

Due to the complexity and diversity of big data, and it is difficult for people to deal with some uncertain and fuzzy information. Rough set theory [1], was proposed by Pawlak, and can be used to dispose uncertainty and vagueness of data. To date, many extended models have been developed for rough sets, such as rough fuzzy sets [2], fuzzy rough sets [3], decision-theoretic rough sets [4], variable precision rough sets [5] and probabilistic rough sets [6], etc. Nowadays, most scholars still employ rough sets to deal with information uncertainty.

Rough set models are typically characterized by a single granular structure. However, in many cases, it may be described in a multiple granularity structure. Qian et al. [7] extended Pawlak’s rough sets to multi-granulation rough sets (MRSs) by using multiple equivalence relations. Since then, various models of MRSs have been proposed by many scholars. Xu and Wang [8] presented a multiple fuzzy relations-based multi-granulation fuzzy rough set. Qian et al. [9] proposed multi-granulation decision-theoretic rough sets. Zhang et al. [10] pointed out a multi-granulation probabilistic rough sets model. Liu and Pedrycz [11] researched covering approximation space-based multi-granulation fuzzy rough sets, and then discussed multi-granulation decision-theoretic rough sets in a multiple covering approximation space [12]. Mandal and Ranadive [13] designed a preference relation-based multi-granulation decision-theoretic rough set in fuzzy information systems. Kang et al. [14] established a MRS model based on variable precision grey systems. The MRS theory has been employed to research topology analysis [15], feature selection [16], cluster analysis [17], and was widely applied in incomplete information systems [18 –20].

Fuzzy sets [21] have the same characteristics as rough sets, which are used to deal with the uncertainty of data. Subsequently, Zadeh [22] proposed the probability measures based on fuzzy events to address uncertainty problems. Zhao and Hu [23] investigated probability measures-based fuzzy and interval-valued fuzzy decision-theoretic rough sets. Liu et al. [24] combined fuzzy probability measure and Bayesian decisions theory, and studied decision-theoretic rough sets in multi-covering approximation spaces. As a generation of fuzzy sets, intuitionistic fuzzy sets (IFSs), were proposed by Atanassov [25] and can also efficiently address uncertainty and fuzziness problems. Wang and Zhang [26] constructed an intuitionistic fuzzy covering rough set model by combining intuitionistic fuzzy sets and rough sets in covering approximation spaces. Zhang et al. [27] built an attribute reduction method by using tolerance-based intuitionistic fuzzy rough sets. Tan et al. [28] designed a knowledge reduction method based on intuitionistic fuzzy rough sets in the context of granular computing. Moreover, Huang et al. [29] employed intuitionistic fuzzy rough sets to address decision problems at different granularity levels. IFSs have been developed many different models to solve a variety of different uncertainty problems in the framework of MRSs [30 –34]. Combined IFSs with fuzzy sets, the notion of support intuitionistic fuzzy sets (SIFSs) was proposed by Nguyen and Nguyen [35], which is characterized by membership degree, non-membership degree and support degree. It can be considered from three aspects to deal with the uncertainty and fuzziness of data. However, SIFS theory has only been studied on aggregation operators and decision-making theory at present [36]. Therefore, research on SIFSs will deserve our in depth exploration.

Three-way decision theory, proposed by Yao [37], and has been established on the basis of decision-theoretic rough sets. The primary purpose of this model is to research how to divide a whole into three relatively independent parts, that is, three disjoint regions. These three regions are called a positive region, a boundary region and a negative region, respectively. The decision-theoretic rough sets tremendously promoted the development of three-way decisions [38 –40]. In probabilistic approximation space, three-way decision theory with probabilistic rough sets based on stream computing is investigated by Xu [41]. Sun et al. [42] employed probabilistic rough sets-based three-way decisions to settle conflict analysis issues. Three-way decisions have been applied successfully in many ways, such as granular computing [43], cluster analysis [44], attributes reduction [45], cost-sensitive [46] and multi-class decision [47]. Combined with IFSs, three-way decision models with point operators [48], triangular norms and triangular conforms [49] are constructed. In addition, Ye et al. [50] built a three-way decision model by integrating with interval-valued intuitionistic fuzzy sets and decision-theoretic rough sets. Three-way decisions have also achieved considerable results in incomplete information systems [51 –53]. Moreover, many scholars have extended the three-way decision models by considering multiple granular structures. Sun et al. [54] established a three-way group decision model in multi-granulation approximation space. Mandal and Ranadive [55] introduced bipolar-valued fuzzy sets into probabilistic approximation space with multi-granulation rough sets, and proposed corresponding three-way decision models. Xue et al. [56] defined the concept of granularity importance degree in the framework of multiple granularities to address granularity reduction problems, and built an optimal granularity selection method by combining with three-way decisions. In addition, the other multi-granulation three-way decision models have been proposed successively [57 –59].

However, the above mentioned three-way decision models either taken into account IFSs with single-granulation structure to make decisions [48 , 60–62], or merely used a multi-granulation structure without IFS to elaborate their decision-making process [55 , 64], which limits the influence factors of dealing with certain problems from multiple aspects in the context of multi-granulation approximation space. To this end, we develop three-way decision models based on support intuitionistic fuzzy probability (SIFP) from the perspective of multi-granulation, which the risk of decision-making, multiple granularity structures and the three aspects of SIFSs are taken into consideration at the same time. Due to multi-granulation rough sets can better deal with fuzzy problems of multiple granularity structures, and support intuitionistic fuzzy sets can study the uncertainty from the perspective of inherent properties of the object (membership degree and non-membership degree) and external influence on it(support degree), three-way decision theories take into account the uncertainty and cost loss in the decision-making process, which is consistent with the cognitive process and selection habits of human decision-making. Therefore, their combination can more efficient and accurate to make decision-making for uncertainty information. This gives a more reasonable decision-making method for solving complex problems. The main contributions of this paper are as follows:

By combining SIFSs and probability theory, we propose the four different conditional probabilities, which are called SIFP.

A multi-granulation support intuitionistic fuzzy probabilistic approximation space (MSIFPAS) is defined, and the thresholds α and β are calculated by combining the Bayesian theory.

We construct Type-I optimistic multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-I OMSIFPRS), Type-I pessimistic multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-I IMSIFPRS); Type-II optimistic multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-II OMSIFPRS), Type-II pessimistic multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-II IMSIFPRS); Type-III multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-III MSIFPRS) and Type-IV multi-granulation support intuitionistic fuzzy probabilistic rough sets model (Type-IV MSIFPRS).

By combining these proposed models with three-way decision theory, the three-way decision models of four types are established.

The remainder of this paper is organized as follows. Section 2 briefly reviews some fundamental concepts related to SIFSs, rough sets, MRSs, and three-way decisions. In Section 3, the notion of SIFP and the definition of MSIFPAS are introduced, and the calculation method for α and β are defined. In Section 4, we propose four different types of MSIFPRS models based on SIFP, and discuss related properties of these models. Then, we construct some three-way decision models based on above proposed MSIFPRS models. In Section 5, an example is employed to analyze and compare these models. Section 6 concludes the paper.

2 Preliminaries

In this section, we briefly review some basic concepts used the whole paper.

2.1 Support intuitionistic fuzzy sets

Definition 2.1. [35] Let U be a non-empty finite universe, and a SIFS A on U can be expressed as $A = {< x, μ_{A} (x), ν_{A} (x), θ_{A} (x) > | x \in U},$ where μ _A (x): U⟶[0,1], ν _A (x): U⟶[0,1] and θ _A (x): U⟶[0,1] satisfy 0⩽μ _A (x)+ν _A (x)⩽1, 0⩽θ _A (x)⩽1. ∀x∈U, μ _A (x), ν _A (x) and θ _A (x) are referred to as the membership degree, non-membership degree and support degree of x on A, respectively. The support degree, which is the degree of support for element x belonging to A, has similar properties to the membership degree. A support intuitionistic fuzzy number (SIFN) can be simply denoted as A = (μ _A , ν _A , θ _A ). {<x, 1, 0, 1> | x∈U} is expressed as U, and {<x, 0, 1, 0> | x∈U} is represented by ϕ. Given SIFSs A and B, the basic operations are given as follows

A ⊆ B ⇔ μ_A (x) ⩽ μ_B (x) , ν_A (x) ⩾ ν_B (x) , θ_A (x) ⩽ θ_B (x).

A = B ⇔ μ_B (x) = μ_A (x) , ν_A (x) = ν_B (x) , θ_B (x) = θ_A (x).

A ∪ B = {< x, max(μ_A (x) , μ_B (x)) , min(ν_A

(x) , ν_B (x)) , max(θ_A (x) , θ_B (x)) > |x ∈ U}.

A ∩ B = {< x, min(μ_A (x) , μ_B (x)) , max(ν_A

(x) , ν_B (x)) , min(θ_A (x) , θ_B (x)) > |x ∈ U}.

∼A = {< x, ν_A (x) , μ_A (x) , 1-θ_A (x) > |x ∈ U}.

A⊕ B = {< x, μ_A (x) + μ_B (x) - μ_A (x) μ_B (x) , ν_A (x) ν_B (x) , θ_A (x) + θ_B (x)-θ_A (x) θ_B (x) >

|x ∈ U}.

A ⊗ B = {< x, μ_A (x) μ_B (x) , ν_A (x) + ν_B (x)-ν_A (x) ν_B (x) , θ_A (x) θ_B (x) > |x ∈ U}.

λA = {< x, 1-(1-μ_A (x)) ^λ, (ν_A (x)) ^λ, 1-(1-θ_A (x)) ^λ > |x ∈ U} , λ > 0.

A^λ = {< x, (μ_A (x)) ^λ, 1-(1-ν_A (x)) ^λ, (θ_A (x)) ^λ > |x ∈ U} , λ > 0.

Definition 2.2. [31] Let U and V be two non-empty finite sets, and a support intuitionistic fuzzy subset R of U×V is called a support intuitionistic fuzzy relation (SIFR) from U to V. Thus, we can define the SIFR R as follows $\begin{matrix} R = & {< (x, y), μ_{R} (x, y), ν_{R} (x, y), θ_{R} (x, y) \\ > | (x, y) \in U \times V}, \end{matrix}$ where μ _R (x, y): U×V⟶[0,1], ν _R (x, y): U×V⟶[0,1], θ _R (x, y): U×V⟶[0,1], and satisfy the conditions 0⩽μ _R (x, y)+ν _R (x, y)⩽1, 0⩽θ _R (x, y)⩽1, ∀(x, y)∈U×V. The SIFR R from U to V can be expressed by the relational matrix as $M_{R} = [< μ_{R} (x, y), ν_{R} (x, y), θ_{R} (x, y) >]_{n \times n} .$

The family of all SIFRs from U to V is denoted by SIFR(U×V). Particularly, if U = V, then we say that R is an SIFR on U.

2.2 Probability theory on fuzzy sets

Definition 2.3. [22] Let (U, ζ, P) be a probability space, where U is a sample space of events, ζ is an σ-algebra of events in U, namely, the measurable subsets of U and P is a probability measure over ζ. A fuzzy event on U is deemed as a fuzzy set $\tilde{A}$ on U. $\forall \tilde{A} \in \tilde{E} (U)$ , $\tilde{A} \in \tilde{G} (ζ) = {\tilde{A} \in \tilde{E} (U) : {\tilde{A}}_{\tilde{ϑ}} \in ζ, \tilde{ϑ} \in [0, 1]}$ . Then the probability of fuzzy event $\tilde{A}$ is defined as $P (\tilde{A}) = \int_{U} \tilde{A} (x) dP = \int_{U} μ_{\tilde{A}} (x) dP = P (μ_{A})$ .

Zadeh [22] proposed the notions of the independence and the conditional probability of two fuzzy events. We can describe in detail as follows.

Definition 2.4. [22] Suppose (U, ζ, P) to be a probability space, $\tilde{A}$ and $\tilde{B}$ be two fuzzy events. If P( $\tilde{A} \tilde{B}$ )=P( $\tilde{A}$ )P( $\tilde{B}$ ), then $\tilde{A}$ and $\tilde{B}$ will be independence. The conditional probability of $\tilde{A}$ given $\tilde{B}$ is defined as $P (\tilde{A} | \tilde{B}) = \frac{P (\tilde{A} \tilde{B})}{P (\tilde{B})}$ , where P( $\tilde{B}$ )≠0.

Provided U = {x₁, x₂, ... , x_n}, and p_i = p({x_i}) (i = 1, 2, ... , n). $\forall \tilde{A}, \tilde{B} \in \tilde{E} (U)$ , we have $P (\tilde{A}) = \sum_{i = 1}^{n} \tilde{A} (x_{i}) p_{i}$ .

Thus we can deduce the conditional probability of $\tilde{A}$ given $\tilde{B}$ as follows $P (\tilde{A} | \tilde{B}) = \frac{P (\tilde{A} \tilde{B})}{P (\tilde{B})} = \frac{\sum_{i = 1}^{n} \tilde{A} (x_{i}) \tilde{B} (x_{i}) p_{i}}{\sum_{i = 1}^{n} \tilde{B} (x_{i}) p_{i}} .$

Note that we employ the algebraic product $\tilde{A} \tilde{B}$ rather than the intersection $\tilde{A} \cap \tilde{B}$ in defining the independence of two fuzzy events. Moreover, if $\tilde{A}$ and $\tilde{B}$ are independent, then P( $\tilde{A} | \tilde{B}$ ) = P( $\tilde{A}$ ) if and only if in the case of non-fuzzy independence events.

Proposition 2.1. [23] Let $\tilde{A}$ , $\tilde{B}$ and $\tilde{C}$ be three fuzzy events on a probability space (U, ζ, P). If P( $\tilde{A}$ )≠0, then the following conclusions hold.

$P (ϕ | \tilde{A}) = 0, P (U | \tilde{A}) = 1$ .

If $\tilde{B} \subseteq \tilde{C}$ , then $P (\tilde{B}) ⩽ P (\tilde{C})$ and $P (\tilde{B} | \tilde{A}) ⩽$ $P (\tilde{C} | \tilde{A})$ .

$P (\tilde{B} | \tilde{A}) + P (\sim \tilde{B} | \tilde{A}) = 1$ .

2.3 Rough sets

Definition 2.5. [1] Assume U be a non-empty and finite universe and R be an equivalence relation on U×U. [x] _R is an equivalence class of x induced by R, and U / R = {[x] _R | x∈U} represent the family of all equivalence classes. ∀X ⊆ U, the lower and upper approximations of X w.r.t. R are defined as $\underline{R} (X) = {x | [x]_{R} \subseteq X, x \in U}$ (1) $\bar{R} (X) = {x | [x]_{R} \cap X \neq ϕ, x \in U}$ (2)

If $\underline{R} (X) \neq \bar{R} (X)$ , the pair $(\underline{R} (X), \bar{R} (X))$ is called rough sets of X w.r.t. R. Otherwise, X is called definable. According to the lower and upper approximations, we can obtain the positive, negative, and boundary regions of X, as $\begin{matrix} {POS}_{R} (X) = \underline{R} (X), \\ {NEG}_{R} (X) = U - \bar{R} (X), \\ {BND}_{R} (X) = \bar{R} (X) - \underline{R} (X) . \end{matrix}$

2.4 Multi-granulation rough sets

Definition 2.6. [7] Let U be a non-empty finite set and AT is a non-empty attribute set on an information system IS = (U, AT). A family of attribute subsets on AT is denote by {A₁, A₂, ... , A_m}. [x] _{A
_i} is an equivalence class of x under the attribute subsets A_i. ∀X⊆U, the optimistic MRS lower and upper approximations of A_i are defined, respectively, by

$\begin{matrix} {\underline{\sum_{i = 1}^{m} A_{i}}}^{O} (X) & = {x \in U | [x]_{A_{1}} \subseteq X \lor [x]_{A_{2}} \\ \subseteq X \lor \dots \lor [x]_{A_{m}} \subseteq X} \end{matrix}$ (3) ${\bar{\sum_{i = 1}^{m} A_{i}}}^{O} (X) = \sim ({\underline{\sum_{i = 1}^{m} A_{i}}}^{O} (\sim X))$ (4)

Definition 2.7. [7] Suppose that IS = (U, AT) be an information system, {A₁, A₂, ... , A_m} is a family of attribute subsets on AT. [x] _{A
_i} is an equivalence class of x w.r.t. A_i. ∀X⊆U, the pessimistic MRS lower and upper approximations of X w.r.t. A_i is defined, respectively, as

$\begin{matrix} {\underline{\sum_{i = 1}^{m} A_{i}}}^{I} (X) & = {x \in U | [x]_{A_{1}} \subseteq X \land [x]_{A_{2}} \\ \subseteq X \land \dots \land [x]_{A_{m}} \subseteq X} \end{matrix}$ (5) ${\bar{\sum_{i = 1}^{m} A_{i}}}^{I} (X) = \sim ({\underline{\sum_{i = 1}^{m} A_{i}}}^{I} (\sim X))$ (6)

2.5 Three-way decision theory

Definition 2.8. [37] Let U = {x₁, x₂, ... , x_n} be a finite universe, the set of states ω= {C, ∼C} indicates an object x is in state C and not in state C, respectively. The set of actions is denoted by ξ= {a_P, a_B, a_N}, where a_P, a_B and a_N means that the acceptance decision, deferment decision and rejection decision in classifying x into a positive region, a boundary region and a negative region, respectively. Different actions lead to different costs, λ _PP , λ _BP and λ _NP represent the cost caused by taking actions a_P, a_B and a_N, respectively, when x belongs to C. Similarly, λ _PN , λ _BN and λ _NN denote the cost of taking the same actions, when x does not belong to C. The cost matrix of decision actions are shown by Table 1.

Table 1
The cost matrix of decision actions

Decision Actions Loss Functions

C ∼C

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Decision Actions	Loss Functions
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

According to Bayesian decision theory, we can obtain three-way decision rules as follows

Acceptance decision rule: If P (C| [x]) ⩾ α′, then x ∈ POS (C);

Deferment decision rule: If β′ < P (C| [x]) < α′, then x ∈ BND (C);

Rejection decision rule: If P (C| [x]) ⩽ β′, then x ∈ NEG (C).

where α′, β′ and γ′ can be computed by the following formula: $α^{'} = \frac{λ_{PN} - λ_{BN}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})},$ $β^{'} = \frac{λ_{BN} - λ_{NN}}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})},$ $γ^{'} = \frac{λ_{PN} - λ_{NN}}{(λ_{PN} - λ_{NN}) + (λ_{NP} - λ_{PP})} .$

3 The basic ingredients of three-way decisions based on MSIFPRS

In this section, we primarily introduce the important components in the construction of MSIFPRS models.

3.1 Probability theory on support intuitionistic fuzzy sets

Definition 3.1. Let (U, ζ, P) be a probability space and A is a SIFS, where U is a sample space of events, ζ denotes an σ-algebra of events, i.e., the measurable subsets of U; and P is defined as a probability measure on ζ. ∀A ∈ E (U), A ∈ G (ζ) = {A ∈ E (U) : A_ϑ ∈ ζ, ϑ ∈ [0, 1]}, then A is called a support intuitionistic fuzzy (SIF) event on U. The probability of SIF event A is defined as $\begin{matrix} P (A) & = \int_{U} A (x) dP \\ = (\int_{U} μ_{A} (x) dP, \int_{U} ν_{A} (x) dP, \int_{U} θ_{A} (x) dP) \end{matrix}$ (7) $= (P (μ_{A}), P (ν_{A}), P (θ_{A}))$

Assume that U = {x₁, x₂, ... , x_n}, and p_i = p({x_i}) (i = 1, 2, ... , n). ∀A∈E(U), based on Definition 2.4,we have

$\begin{matrix} P (A) & = \sum_{i = 1}^{n} A (x_{i}) p_{i} \\ = (\sum_{i = 1}^{n} μ_{A} (x_{i}) p_{i}, \sum_{i = 1}^{n} ν_{A} (x_{i}) p_{i}, \sum_{i = 1}^{n} θ_{A} (x_{i}) p_{i}) \end{matrix}$ (8)

Definition 3.2. Provided (U, ζ, P) be a probability space, A = (μ _A , ν _A , θ _A ) and B = (μ _B , ν _B , θ _B ) be two SIF events on U. If P(B)≠0, then the conditional probability of A given B is denoted by $P (A | B) = \frac{P (AB)}{P (B)}$ (9)

Note that, ∀A, B∈E(U) and ∀x ∈ U, we can obtain the product of two SIFSs based on Definition 2.1 as follows: AB = {< x, μ_A (x) μ_B (x) , ν_A (x) ν_B (x) , θ_A (x) θ_B (x) > |x ∈ U}.

Proposition 3.1. Let (U, ζ, P) be a probability space, and A = (μ_A, ν_A, θ_A), B = (μ_B, ν_B, θ_B) be two SIF events on U. If P(B)≠0, then we have following conclusion, P (A|B) = (P (μ_A|μ_B) , P (ν_A|ν_B) , P (θ_A|θ_B)).

Proof. This can be proved easily based on Definition 3.2, as follows $\begin{matrix} P (A | B) = \frac{P (AB)}{P (B)} = \frac{P ((μ_{A}, ν_{A}, θ_{A}) (μ_{B}, ν_{B}, θ_{B}))}{P (μ_{B}, ν_{B}, θ_{B})} \\ = \frac{P (μ_{A} μ_{B}, ν_{A} ν_{B}, θ_{A} θ_{B})}{P (μ_{B}, ν_{B}, θ_{B})} \\ = \frac{(P (μ_{A} μ_{B}), P (ν_{A} ν_{B}), P (θ_{A} θ_{B}))}{(P (μ_{B}), P (ν_{B}), P (θ_{B}))} \\ = (\frac{P (μ_{A} μ_{B})}{P (μ_{B})}, \frac{P (ν_{A} ν_{B})}{P (ν_{B})}, \frac{P (θ_{A} θ_{B})}{P (θ_{B})}) \\ = (P (μ_{A} | μ_{B}), P (ν_{A} | ν_{B}), P (θ_{A} | θ_{B})) \end{matrix}$

Therefore, P (A|B) = (P (μ_A|μ_B) , P (ν_A|ν_B) , P (θ_A|θ_B)) be proved. □

Let U = {x₁, x₂, ... , x_n}, and p_i = p(x_i) (i = 1, 2, ... , n). ∀A, B∈E(U), according to Definition 3.2 and Proposition 3.1,we can obtain

$\begin{matrix} P (A | B) & = \frac{P (AB)}{P (B)} \\ = (P (μ_{A} | μ_{B}), P (ν_{A} | ν_{B}), P (θ_{A} | θ_{B})) \end{matrix}$ (10) where $P (μ_{A} | μ_{B}) = \frac{\sum_{i = 1}^{n} μ_{A} (x_{i}) μ_{B} (x_{i}) p_{i}}{\sum_{i = 1}^{n} μ_{B} (x_{i}) p_{i}},$ $P (ν_{A} | ν_{B}) = \frac{\sum_{i = 1}^{n} ν_{A} (x_{i}) ν_{B} (x_{i}) p_{i}}{\sum_{i = 1}^{n} ν_{B} (x_{i}) p_{i}},$ $P (θ_{A} | θ_{B}) = \frac{\sum_{i = 1}^{n} θ_{A} (x_{i}) θ_{B} (x_{i}) p_{i}}{\sum_{i = 1}^{n} θ_{B} (x_{i}) p_{i}} .$

Proposition 3.2. Let (U, ζ, P) be a probability space, A = (μ_A, ν_A, θ_A), B = (μ_B, ν_B, θ_B) and C = (μ_C, ν_C, θ_C) be three SIF events on U. If P(A)≠ 0, then the following conclusions hold.

P (ϕ|A) =0, P (U|A) =1.

If B ⊆ C, then P (B|A) ⩽ P (C|A).

3.2 Multi-granulation support intuitionistic fuzzy probabilistic approximation space

Definition 3.3. Let U and V be two non-empty finite sets, and R_k (1⩽k⩽m) be m SIFRs on U × V; P is defined as a probability measure on σ-algebra formed by the image of element x ∈U, i.e. the subset of the universe V. Then, (U, V, R_k, P) (1⩽k⩽m) be referred to as a MSIFPAS.

Definition 3.4. Suppose (U, V, R_k, P) (1⩽k⩽m) to be a MSIFPAS, for each x ∈ U, ∀y ∈ V, $[x]_{\cap_{k = 1}^{m} R_{k}}$ and $[x]_{\cup_{k = 1}^{m} R_{k}}$ are expressed by $\begin{matrix} [x]_{\cap_{k = 1}^{m} R_{k}} (y) = \land_{k = 1}^{m} R_{k} (x, y) \\ = (\land_{k = 1}^{m} μ_{R_{k}} (x, y), \lor_{k = 1}^{m} ν_{R_{k}} (x, y), \land_{k = 1}^{m} θ_{R_{k}} (x, y)) \end{matrix}$ (11) $\begin{matrix} [x]_{\cup_{k = 1}^{m} R_{k}} (y) = \lor_{k = 1}^{m} R_{k} (x, y) \\ = (\lor_{k = 1}^{m} μ_{R_{k}} (x, y), \land_{k = 1}^{m} ν_{R_{k}} (x, y), \lor_{k = 1}^{m} θ_{R_{k}} (x, y)) \end{matrix}$ (12)

3.3 The determination of thresholds

On the basis of Definition 2.8, the loss functions can be expressed by the cost matrix shown in Table 2, where the loss function satisfy the following constraints: λ_PP ⩽ λ_BP ⩽ λ_NP, λ_NN⩽λ_BN ⩽ λ_PN.

Table 2
The SIF cost matrix of decision actions

C ∼C

a_P λ_PP =< μ_{λ
_PP}, ν_{λ
_PP}, θ_{λ
_PP} > λ_PN =< μ_{λ
_PN}, ν_{λ
_PN}, θ_{λ
_PN} >

a_B λ_BP =< μ_{λ
_BP}, ν_{λ
_BP}, θ_{λ
_BP} > λ_BN =< μ_{λ
_BN}, ν_{λ
_BN}, θ_{λ
_BN} >

a_N λ_NP =< μ_{λ
_NP}, ν_{λ
_NP}, θ_{λ
_NP} > λ_NN =< μ_{λ
_NN}, ν_{λ
_NN}, θ_{λ
_NN} >

	C	∼C
a_P	λ_PP =< μ_{λ _PP}, ν_{λ _PP}, θ_{λ _PP} >	λ_PN =< μ_{λ _PN}, ν_{λ _PN}, θ_{λ _PN} >
a_B	λ_BP =< μ_{λ _BP}, ν_{λ _BP}, θ_{λ _BP} >	λ_BN =< μ_{λ _BN}, ν_{λ _BN}, θ_{λ _BN} >
a_N	λ_NP =< μ_{λ _NP}, ν_{λ _NP}, θ_{λ _NP} >	λ_NN =< μ_{λ _NN}, ν_{λ _NN}, θ_{λ _NN} >

Next, we define a new method for calculating thresholds. The thresholds α, β and γ are denoted by α= (μ_α, ν_α, θ_α), β= (μ_β, ν_β, θ_β) and γ= (μ_γ, ν_γ, θ_γ), where $\begin{matrix} μ_{α} = \frac{μ_{λ_{PN}} - μ_{λ_{BN}}}{(μ_{λ_{PN}} - μ_{λ_{BN}}) + (μ_{λ_{BP}} - μ_{λ_{PP}})}, \\ ν_{α} = \frac{ν_{λ_{PN}} - ν_{λ_{BN}}}{(ν_{λ_{PN}} - ν_{λ_{BN}}) + (ν_{λ_{BP}} - ν_{λ_{PP}})}, \\ θ_{α} = \frac{θ_{λ_{PN}} - θ_{λ_{BN}}}{(θ_{λ_{PN}} - μ_{λ_{BN}}) + (θ_{λ_{BP}} - θ_{λ_{PP}})}; \end{matrix}$ (13) $\begin{matrix} μ_{β} = \frac{μ_{λ_{BN}} - μ_{λ_{NN}}}{(μ_{λ_{BN}} - μ_{λ_{NN}}) + (μ_{λ_{NP}} - μ_{λ_{BP}})}, \\ ν_{β} = \frac{ν_{λ_{BN}} - ν_{λ_{NN}}}{(ν_{λ_{BN}} - ν_{λ_{NN}}) + (ν_{λ_{NP}} - ν_{λ_{BP}})}, \\ θ_{β} = \frac{θ_{λ_{BN}} - θ_{λ_{NN}}}{(θ_{λ_{BN}} - θ_{λ_{NN}}) + (θ_{λ_{NP}} - θ_{λ_{BP}})}; \end{matrix}$ (14) $\begin{matrix} μ_{γ} = \frac{μ_{λ_{PN}} - μ_{λ_{NN}}}{(μ_{λ_{PN}} - μ_{λ_{NN}}) + (μ_{λ_{NP}} - μ_{λ_{PP}})}, \\ ν_{γ} = \frac{ν_{λ_{PN}} - ν_{λ_{NN}}}{(ν_{λ_{PN}} - ν_{λ_{NN}}) + (ν_{λ_{NP}} - ν_{λ_{PP}})}, \\ θ_{γ} = \frac{θ_{λ_{PN}} - θ_{λ_{NN}}}{(θ_{λ_{PN}} - θ_{λ_{NN}}) + (θ_{λ_{NP}} - θ_{λ_{PP}})} . \end{matrix}$ (15)

Provided the loss function satisfy the following conditions: $(R_{λ_{PN}} - R_{λ_{BN}}) (R_{λ_{NP}} - R_{λ_{BP}}) > (R_{λ_{BN}} - R_{λ_{NN}})$ $(R_{λ_{BP}} - R_{λ_{PP}})$ ( $R$ =μ, ν, θ), then we can obtain α> γ> β.

4 The four MSIFPRS models and their corresponding three-way decisions

In this section, we propose the four kinds of MSIFPRS models based on the MSIFPAS. The first model is called Type-I MSIFPRS, which is a class based on combination of multiple binary relationships first and then establishment of approximations. The second model, called Type-II MSIFPRS, is constructed by creating another class in the reverse order. The other models are improved on the basis of Type-II MSIFPRS. Then, we combine these models with three-way decision theory to create novel models, respectively.

4.1 Three-way decisions of Type-I MSIFPRS based on SIFP

Definition 4.1. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, and a pair of parameters α= (μ_α, ν_α, θ_α), β= (μ_β, ν_β, θ_β) satisfy 0 ⩽ β < P (A) < α ⩽ 1. ∀A∈ SIFS, the Type-I OMSIFPRS lower approximation ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A)$ and upper approximation ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A)$ w.r.t. A are defined, respectively, as follows ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) = {x | P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) ⩾ α, x \in U}$ (16) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) = {x | P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) > β, x \in U}$ (17) where assume that V = {y₁, y₂, ... , y_n}, and p(y_i) = p_i (i = 1, 2, ... , n); ∀x ∈ U,

$\begin{matrix} P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) = & (P (μ_{A} | [x]_{\cup_{k = 1}^{m} μ_{R_{k}}}), P (ν_{A} | [x]_{\cup_{k = 1}^{m} ν_{R_{k}}}), \\ P (θ_{A} | [x]_{\cup_{k = 1}^{m} θ_{R_{k}}})), \end{matrix}$

$P (μ_{A} | [x]_{\cup_{k = 1}^{m} μ_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} μ_{R_{k}} (x, y_{i})) μ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} μ_{R_{k}} (x, y_{i})) p_{i}},$ $P (ν_{A} | [x]_{\cup_{k = 1}^{m} ν_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\land_{k = 1}^{m} ν_{R_{k}} (x, y_{i})) ν_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\land_{k = 1}^{m} ν_{R_{k}} (x, y_{i})) p_{i}},$ $P (θ_{A} | [x]_{\cup_{k = 1}^{m} θ_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} θ_{R_{k}} (x, y_{i})) θ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} θ_{R_{k}} (x, y_{i})) p_{i}} .$

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A)$ , the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A))$ is referred to as Type-I OMSIFPRS of A. Otherwise, A is said to be definable. By the lower approximation ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A)$ and upper approximation ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A)$ , the positive, negative and boundary regions of three-way decisions based on Type-I OMSIFPRS are defined as $\begin{matrix} {POS}^{I (O)} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A), \\ {NEG}^{I (O)} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A), \\ {BND}^{I (O)} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) . \end{matrix}$

Definition 4.2. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, where V = {y₁, y₂, ... , y_n} and p(y_i) = p_i (i = 1, 2, ... , n). A is a SIFS, and α= (μ_α, ν_α, θ_α), β= (μ_β, ν_β, θ_β) satisfy 0 ⩽ β < P (A) < α ⩽ 1, the Type-I IMSIFPRS lower and upper approximations of A are denoted by ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A)$ and ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A)$ , respectively, are defined as follows ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) = {x | P (A | [x]_{\cap_{k = 1}^{m} R_{k}}) ⩾ α, x \in U}$ (18) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A) = {x | P (A | [x]_{\cap_{k = 1}^{m} R_{k}}) > β, x \in U}$ (19) where

$\begin{matrix} P (A | [x]_{\cap_{k = 1}^{m} R_{k}}) = & (P (μ_{A} | [x]_{\cap_{k = 1}^{m} μ_{R_{k}}}), P (ν_{A} | [x]_{\cap_{k = 1}^{m} ν_{R_{k}}}), \\ P (θ_{A} | [x]_{\cap_{k = 1}^{m} θ_{R_{k}}})), \end{matrix}$

$P (μ_{A} | [x]_{\cap_{k = 1}^{m} μ_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\land_{k = 1}^{m} μ_{R_{k}} (x, y_{i})) μ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\land_{k = 1}^{m} μ_{R_{k}} (x, y_{i})) p_{i}},$ $P (ν_{A} | [x]_{\cap_{k = 1}^{m} ν_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} ν_{R_{k}} (x, y_{i})) ν_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\lor_{k = 1}^{m} ν_{R_{k}} (x, y_{i})) p_{i}},$ $P (θ_{A} | [x]_{\cap_{k = 1}^{m} θ_{R_{k}}}) = \frac{\sum_{i = 1}^{n} (\land_{k = 1}^{m} θ_{R_{k}} (x, y_{i})) θ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} (\land_{k = 1}^{m} θ_{R_{k}} (x, y_{i})) p_{i}} .$

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A)$ , the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A))$ is called a Type-I IMSIFPRS model w.r.t. A. Otherwise, A is said to be definable. Then the positive, negative and boundary regions of three-way decisions based on Type-I IMSIFPRS are defined as follows ${POS}^{I (I)} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A),$ ${NEG}^{I (I)} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A),$ ${BND}^{I (I)} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) .$

Theorem 4.1. Suppose (U, V, R_k, P) (1⩽k⩽m) to be a MSIFPAS. A, B be two SIFSs, and 0 ⩽ β < P (A) < α ⩽ 1, then we have following properties:

(1) ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A)$ .

(2) ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (ϕ) = ϕ$ ,

${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (ϕ) = ϕ$ .

(3) ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (U) = U$ ,

${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (U) = U$ .

(4) If A ⊆ B, then ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (B)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (B)$ .

(5) If 0 ⩽ β₁ < β₂ < α₁ < α₂ ⩽ 1, then, ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{2} (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{1} (O)} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{2} (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{1} (O)} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{2} (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{1} (I)} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{2} (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{1} (I)} (A)$ .

Proof. It is easy to prove (1)-(3) based on Definitions 4.1, 4.2 and Proposition 3.2.

(4) According to the Proposition 3.2, we can obtain $P (μ_{A} | [x]_{\cup_{k = 1}^{m} μ_{R_{k}}}) ⩽ P (μ_{B} | [x]_{\cup_{k = 1}^{m} μ_{R_{k}}})$ , $P (ν_{A} | [x]_{\cup_{k = 1}^{m} ν_{R_{k}}}) ⩽$ $P (ν_{B} | [x]_{\cup_{k = 1}^{m} ν_{R_{k}}})$ and $P (θ_{A} | [x]_{\cup_{k = 1}^{m} θ_{R_{k}}}) ⩽ P (θ_{A} | [x]_{\cup_{k = 1}^{m} θ_{R_{k}}})$ , Thus, $P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) ⩽ P (B | [x]_{\cup_{k = 1}^{m} R_{k}})$ can be easily obtained, it is obvious that $P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) > α \subseteq$ $P (B | [x]_{\cup_{k = 1}^{m} R_{k}}) > α$ . Therefore, ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (B)$ be proved. Similarly, we can easily obtain ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (B)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (I)} (B)$ and ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (I)} (B)$ .

(5) If 0 ⩽ α₁ < α₂ ⩽ 1, then we have $P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) ⩾ α_{2} \subseteq P (A | [x]_{\cup_{k = 1}^{m} R_{k}}) ⩾ α_{1}$ , thus, ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{2} (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{1} (O)} (A)$ can be directly obtained. Similarly, we can easily prove ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{2} (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{1} (O)} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{2} (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α_{1} (I)} (A)$ and ${\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{2} (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β_{1} (I)} (A)$ . □

4.2 Three-way decisions of Type-II MSIFPRS based on SIFP

Definition 4.3. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, where V = {y₁, y₂, ... , y_n} and p(y_i) = p_i (i = 1, 2, ... , n). The parameters α and β satisfy 0 ⩽ β < P (A) < α ⩽ 1, ∀A ∈ SIFS (V), the lower approximation ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A)$ and upper approximation ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A)$ of Type-II OMSIFPRS w.r.t. A are defined, respectively, by the following formulas: ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A) = {x | \cup_{k = 1}^{m} P (A | [x]_{R_{k}}) ⩾ α, x \in U}$ (20) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A) = {x | \cup_{k = 1}^{m} P (A | [x]_{R_{k}}) > β, x \in U}$ (21) where

$\begin{matrix} \cup_{k = 1}^{m} P (A | [x]_{R_{k}}) = & (\cup_{k = 1}^{m} P (μ_{A} | [x]_{μ_{R_{k}}}), \cup_{k = 1}^{m} P (ν_{A} | [x]_{ν_{R_{k}}}), \\ \cup_{k = 1}^{m} P (θ_{A} | [x]_{θ_{R_{k}}})), \end{matrix}$

$\cup_{k = 1}^{m} P (μ_{A} | [x]_{μ_{R_{k}}}) = \lor_{k = 1}^{m} \frac{\sum_{i = 1}^{n} μ_{R_{k}} (x, y_{i}) μ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} μ_{R_{k}} (x, y_{i}) p_{i}},$ $\cup_{k = 1}^{m} P (ν_{A} | [x]_{ν_{R_{k}}}) = \land_{k = 1}^{m} \frac{\sum_{i = 1}^{n} ν_{R_{k}} (x, y_{i}) ν_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} ν_{R_{k}} (x, y_{i}) p_{i}},$ $\cup_{k = 1}^{m} P (θ_{A} | [x]_{θ_{R_{k}}}) = \lor_{k = 1}^{m} \frac{\sum_{i = 1}^{n} θ_{R_{k}} (x, y_{i}) θ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} θ_{R_{k}} (x, y_{i}) p_{i}} .$

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A)$ , the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A))$ is referred to as a Type-II OMSIFPRS model w.r.t. A. Otherwise, A is said to be definable. In light of the formulas (20) and (21), the positive, negative and boundary regions of three-way decisions based on Type-II OMSIFPRS are described as follows ${POS}^{II (O)} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A),$ ${NEG}^{II (O)} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A),$ ${BND}^{II (O)} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A) .$

Definition 4.4. Suppose (U, V, R_k, P) (1⩽k⩽m) to be a MSIFPAS, where V = y₁, y₂, ... , y_n and p(y_i) = p_i (i = 1, 2, ... , n). ∀A ∈ SIFS (V), if 0 ⩽ β < P (A) < α ⩽ 1, then the lower and upper approximations of Type-II IMSIFPRS w.r.t. A are denoted by ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A)$ and ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A)$ , respectively, in which ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A) = {x | \cap_{k = 1}^{m} P (A | [x]_{R_{k}}) ⩾ α, x \in U}$ (22) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A) = {x | \cap_{k = 1}^{m} P (A | [x]_{R_{k}}) > β, x \in U}$ (23) where

$\begin{matrix} \cap_{k = 1}^{m} P (A | [x]_{R_{k}}) = & (\cap_{k = 1}^{m} P (μ_{A} | [x]_{μ_{R_{k}}}), \cap_{k = 1}^{m} P (ν_{A} | [x]_{ν_{R_{k}}}), \\ \cap_{k = 1}^{m} P (θ_{A} | [x]_{θ_{R_{k}}})), \end{matrix}$

$\cap_{k = 1}^{m} P (μ_{A} | [x]_{μ_{R_{k}}}) = \land_{k = 1}^{m} \frac{\sum_{i = 1}^{n} μ_{R_{k}} (x, y_{i}) μ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} μ_{R_{k}} (x, y_{i}) p_{i}},$ $\cap_{k = 1}^{m} P (ν_{A} | [x]_{ν_{R_{k}}}) = \lor_{k = 1}^{m} \frac{\sum_{i = 1}^{n} ν_{R_{k}} (x, y_{i}) ν_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} ν_{R_{k}} (x, y_{i}) p_{i}},$ $\cap_{k = 1}^{m} P (θ_{A} | [x]_{θ_{R_{k}}}) = \land_{k = 1}^{m} \frac{\sum_{i = 1}^{n} θ_{R_{k}} (x, y_{i}) θ_{A} (y_{i}) p_{i}}{\sum_{i = 1}^{n} θ_{R_{k}} (x, y_{i}) p_{i}} .$

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A)$ , the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A))$ is called a Type-II IMSIFPRS model of A. Otherwise, A is said to be definable. By the lower and upper approximations, the positive, negative and boundary regions of three-way decisions based on Type-II IMSIFPRS are expressed as follows $\begin{matrix} {POS}^{II (I)} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A), \\ {NEG}^{II (I)} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A), \\ {BND}^{II (I)} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A) . \end{matrix}$

Theorem 4.2. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS. A, B be two SIFSs, and if 0 ⩽ β < P (A) < α ⩽ 1, then we have following properties:

${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A)$ .

${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (ϕ) = ϕ$ ,

${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (ϕ) = ϕ$ .

${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (U) = U$ ,

${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (U) = U$ .

(4) If A ⊆ B, then ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (O)} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (O)} (B)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α (I)} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β (I)} (B)$ .

(5) If 0 ⩽ β₁ < β₂ < α₁ < α₂ ⩽ 1, then ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α_{2} (O)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α_{1} (O)} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β_{2} (O)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β_{1} (O)} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{II α_{2} (I)} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{II α_{1} (I)} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{II β_{2} (I)} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{II β_{1} (I)} (A)$ .

Proof. The proof is similar to Theorem 4.1, and this can be easily proved from Definitions 4.3, 4.4 and Proposition 3.2. □

On the basis of Type-II MSIFPRS models, we come up with the two other MSIFPRS models. The two kinds of MSIFPRS models without optimism and pessimism are described in detail as follows.

4.3 Three-way decisions of other two MSIFPRS based on SIFP

Definition 4.5. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, where V = {y₁, y₂, ... , y_n} and p(y_i) = p_i (i = 1, 2, ... , n). ∀A ∈ SIFS (V),0 ⩽ α ⩽ 1, the lower and upper approximations of Type-III MSIFPRS w.r.t. A are calculated by the following formulas, respectively, ${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) = {x | \cap_{k = 1}^{m} P (A | [x]_{R_{k}}) ⩾ α, x \in U}$ (24) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) = {x | \cup_{k = 1}^{m} P (A | [x]_{R_{k}}) ⩾ α, x \in U}$ (25)

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A)$ , we can call the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A))$ Type-III MSIFPRS w.r.t. A. Otherwise, A is said to be definable. The positive, negative and boundary regions of three-way decisions based on Type-III MSIFPRS are defined as follows $\begin{matrix} {POS}^{III} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A), \\ {NEG}^{III} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A), \\ {BND}^{III} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) . \end{matrix}$

Definition 4.6. Assume (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, where V = {y₁, y₂, ... , y_n} and p(y_i) = p_i (i = 1, 2, ... , n). ∀A ∈ SIFS (V),0 ⩽ β ⩽ 1, the lower and upper approximations of Type-IV MSIFPRS w.r.t. A are denoted by ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A)$ and ${\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A)$ , respectively, are define as follows ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) = {x | \cap_{k = 1}^{m} P (A | [x]_{R_{k}}) > β, x \in U}$ (26) ${\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) = {x | \cup_{k = 1}^{m} P (A | [x]_{R_{k}}) > β, x \in U}$ (27)

If ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) \neq {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A)$ , the pair $({\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A), {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A))$ is known as a Type-IV MSIFPRS model w.r.t. A. Otherwise, A is said to be definable. According to the lower and upper approximations, the positive, negative and boundary regions of three-way decisions based on Type-IV MSIFPRS are defined as follows $\begin{matrix} {POS}^{IV} (A) = {\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A), \\ {NEG}^{IV} (A) = U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A), \\ {BND}^{IV} (A) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) . \end{matrix}$

Theorem 4.3. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS. Given two SIFSs A and B, if 0 ⩽ α ⩽ 1 and 0 ⩽ β ⩽ 1, then we can obtain the following properties:

${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A)$

${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (ϕ) = ϕ$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (ϕ) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (ϕ) = ϕ$ .

${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (U) = U$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (U) = {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (U) = U$ .

(4) If A ⊆ B, then ${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{III α} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{III α} (B)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β} (B)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β} (B)$ .

(5) If 0 ⩽ α₁ < α₂ ⩽ 1 and 0 ⩽ β₁ < β₂ ⩽ 1, then ${\underline{\sum_{k = 1}^{m} R_{k}}}^{III α_{2}} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{III α_{1}} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{III β_{2}} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{III β_{1}} (A)$ , ${\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β_{2}} (A) \subseteq {\underline{\sum_{k = 1}^{m} R_{k}}}^{IV β_{1}} (A)$ , ${\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β_{2}} (A) \subseteq {\bar{\sum_{k = 1}^{m} R_{k}}}^{IV β_{1}} (A)$ .

Proof. It is similar to the proof of Theorem 4.1. □

4.4 Three-way decisions algorithm based on MSIFPRSs

In this subsection, the three-way decision algorithms based on MSIFPRSs are constructed. The basic idea of the algorithm is as follows:

According to the creation of different classes, the corresponding SIFPs are calculated. Then, by calculating α and β, the SIFPs satisfying the conditions are found, the lower and upper approximation of MSIFPRSs is obtained, and the corresponding positive region, negative region and boundary region are derived. That is, three-way decision algorithms based on MSIFPRSs are generated. In what follows, we take a Type-I OMSIFPRS model as an example, and the details can be found in Algorithm 1.

Algorithm 1: Three-way decision rules extraction algorithm

based on Type-I OMSIFPRS. Input: MSIFPAS (U, V, R_k, P) (1⩽k⩽m), SIFS A.

Output:POS^I(O) (A), NEG^I(O) (A), BND^I(O) (A).

1: begin

2: initialize POS^I(O) (A) ← ϕ,NEG^I(O) (A) ← ϕ,BND^I(O) (A) ← ϕ;

3: Calculate the thresholds α and β based on loss functions;

4: for each x ∈ Udo

5: Calculate

[x]_{\cup_{k = 1}^{m} R_{k}}

;

6: Compute

P (A | [x]_{\cup_{k = 1}^{m} R_{k}})

of A given

[x]_{\cup_{k = 1}^{m} R_{k}}

;

7: end for

8: fori = 1: n

9: if

P (A | [x_{i}]_{\cup_{k = 1}^{m} R_{k}}) ⩾ α

, then

10:

{\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A) \leftarrow x_{i}

;

11: if

P (A | [x_{i}]_{\cup_{k = 1}^{m} R_{k}}) > β

, then

12:

{\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) \leftarrow x_{i}

;

13: end if

14: end if

15: end for

16:

{POS}^{I (O)} (A) \leftarrow {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A)

;

17:

{NEG}^{I (O)} (A) \leftarrow U - {\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A)

;

18:

{BND}^{I (O)} (A) \leftarrow ({\bar{\sum_{k = 1}^{m} R_{k}}}^{I β (O)} (A) - {\underline{\sum_{k = 1}^{m} R_{k}}}^{I α (O)} (A))

;

19: returnPOS^I(O) (A), NEG^I(O) (A), BND^I(O) (A).

20: end

5 Example analysis

The MSIFPRS models can provide an effective method to address the decision problems with uncertainty and vagueness. In this section, we study a person-job fit procedure to verify the validity of the proposed models.

Example 1. Suppose that there is six job seekers who intends to know the most suitable job position in an enterprise, and they invite three experts to instruct them to conduct a vocational ability evaluation. Let (U, V, R_k, P) (1⩽k⩽m) be a MSIFPAS, and suppose U = {x₁, x₂, x₃, x₄, x₅, x₆} be six different job seekers. V = {y₁, y₂, y₃, y₄, y₅, y₆} be six professional abilities in an enterprise, where y₁ = “organization and management ability”, y₂ = “mathematical ability”, y₃ = “writing ability”, y₄ = “software application capability”, y₅ = “social skills”, y₆ = “market development and analysis ability”. It is known the probability distribution on V is p_i = {0.15, 0.12, 0.25, 0.08, 0.28, 0.12}. Moreover, the three experts constructs the relationships between six job seekers and six kinds of abilities, the relationships are denoted by M_{R
_k}, as follows. The semantics of <μ_{R
₁} (x₁, y₂) , ν_{R
₁} (x₁, y₂), θ_{R
₁} (x₁, y₂)>= <0.6, 0.2, 0.4 > can be interpreted as 60% of experts believe that x₁ has good mathematical ability, while 20% of experts think that x₁ has poor mathematical ability. On the other hand, x₁ has only 40% confidence in his/her mathematical ability. Let a SIFS on V be denoted by A = {<y₁, 0.7, 0.2, 0.6>, <y₂, 0.6, 0.2, 0.4>, <y₃, 0.7, 0.2, 0.3>, <y₄, 0.4, 0.5, 0.3>, <y₅, 0.3, 0.6, 0.7>, <y₆, 0.5, 0.4, 0.5> }, and means that job seekers conducts a self-evaluation.

$M_{R_{1}} = [\begin{matrix} < 1, 0, 1 > & < 0.6, 0.2, 0.4 > & < 0.3, 0.7, 0.6 > & < 0.2, 0.7, 0.5 > & < 0.4, 0.6, 0.3 > & < 0.5, 0.5, 0.7 > \\ < 0.6, 0.2, 0.4 > & < 1, 0, 1 > & < 0.3, 0.6, 0.8 > & < 0.2, 0.7, 0.6 > & < 0.5, 0.4, 0.3 > & < 0.6, 0.3, 0.4 > \\ < 0.3, 0.7, 0.6 > & < 0.2, 0.7, 0.6 > & < 1, 0, 1 > & < 0.6, 0.3, 0.2 > & < 0.3, 0.6, 0.7 > & < 0.7, 0.2, 0.4 > \\ < 0.2, 0.7, 0.5 > & < 0.3, 0.7, 0.1 > & < 0.6, 0.3, 0.2 > & < 1, 0, 1 > & < 0.6, 0.4, 0.5 > & < 0.2, 0.8, 0.6 > \\ < 0.4, 0.6, 0.3 > & < 0.5, 0.4, 0.3 > & < 0.3, 0.6, 0.7 > & < 0.6, 0.4, 0.5 > & < 1, 0, 1 > & < 0.6, 0.4, 0.6 > \\ < 0.5, 0.5, 0.7 > & < 0.6, 0.3, 0.4 > & < 0.7, 0.2, 0.4 > & < 0.2, 0.8, 0.6 > & < 0.6, 0.4, 0.6 > & < 1, 0, 1 > \end{matrix}]$ $M_{R_{2}} = [\begin{matrix} < 1, 0, 1 > & < 0.4, 0.5, 0.6 > & < 0.7, 0.2, 0.6 > & < 0.3, 0.6, 0.4 > & < 0.3, 0.7, 0.2 > & < 0.8, 0.2, 0.5 > \\ < 0.4, 0.5, 0.6 > & < 1, 0, 1 > & < 0.5, 0.4, 0.3 > & < 0.2, 0.8, 0.7 > & < 0.3, 0.7, 0.4 > & < 0.6, 0.3, 0.6 > \\ < 0.7, 0.2, 0.6 > & < 0.5, 0.4, 0.3 > & < 1, 0, 1 > & < 0.5, 0.4, 0.7 > & < 0.2, 0.7, 0.5 > & < 0.3, 0.6, 0.4 > \\ < 0.3, 0.6, 0.4 > & < 0.2, 0.8, 0.7 > & < 0.5, 0.4, 0.7 > & < 1, 0, 1 > & < 0.7, 0.3, 0.8 > & < 0.4, 0.5, 0.3 > \\ < 0.3, 0.7, 0.2 > & < 0.3, 0.7, 0.4 > & < 0.2, 0.7, 0.5 > & < 0.7, 0.3, 0.8 > & < 1, 0, 1 > & < 0.5, 0.4, 0.6 > \\ < 0.8, 0.2, 0.5 > & < 0.6, 0.3, 0.6 > & < 0.3, 0.6, 0.4 > & < 0.4, 0.5, 0.3 > & < 0.5, 0.4, 0.6 > & < 1, 0, 1 > \end{matrix}]$

$M_{R_{3}} = [\begin{matrix} < 1, 0, 1 > & < 0.8, 0.2, 0.6 > & < 0.6, 0.4, 0.3 > & < 0.2, 0.8, 0.4 > & < 0.2, 0.7, 0.7 > & < 0.4, 0.5, 0.5 > \\ < 0.8, 0.2, 0.6 > & < 1, 0, 1 > & < 0.7, 0.2, 0.7 > & < 0.3, 0.6, 0.6 > & < 0.4, 0.6, 0.5 > & < 0.6, 0.4, 0.2 > \\ < 0.6, 0.4, 0.3 > & < 0.7, 0.2, 0.7 > & < 1, 0, 1 > & < 0.3, 0.6, 0.5 > & < 0.4, 0.6, 0.2 > & < 0.3, 0.6, 0.4 > \\ < 0.2, 0.8, 0.4 > & < 0.3, 0.6, 0.6 > & < 0.3, 0.6, 0.5 > & < 1, 0, 1 > & < 0.7, 0.2, 0.3 > & < 0.7, 0.2, 0.6 > \\ < 0.2, 0.7, 0.7 > & < 0.4, 0.6, 0.5 > & < 0.4, 0.6, 0.2 > & < 0.7, 0.2, 0.3 > & < 1, 0, 1 > & < 0.5, 0.4, 0.5 > \\ < 0.4, 0.5, 0.5 > & < 0.6, 0.4, 0.2 > & < 0.3, 0.6, 0.4 > & < 0.7, 0.2, 0.6 > & < 0.5, 0.4, 0.5 > & < 1, 0, 1 > \end{matrix}]$

First, assume that the six risk coefficients are given, and as shown in Table 3.

Table 3
The SIF cost function values of different actions

C ∼C

a_P λ_PP = <0.20, 0.60, 0.30> λ_PN = <0.60, 0.30, 0.80>

a_B λ_BP = <0.30, 0.50, 0.40> λ_BN = <0.50, 0.50, 0.60>

a_N λ_NP = <0.60, 0.30, 0.70> λ_NN = <0.25, 0.75, 0.48>

	C	∼C
a_P	λ_PP = <0.20, 0.60, 0.30>	λ_PN = <0.60, 0.30, 0.80>
a_B	λ_BP = <0.30, 0.50, 0.40>	λ_BN = <0.50, 0.50, 0.60>
a_N	λ_NP = <0.60, 0.30, 0.70>	λ_NN = <0.25, 0.75, 0.48>

In light of the formulas (13)-(15), the thresholds α and β can be calculated as follows $\begin{matrix} α = (μ_{α}, ν_{α}, θ_{α}) \\ = (\frac{μ_{λ_{PN}} - μ_{λ_{BN}}}{(μ_{λ_{PN}} - μ_{λ_{BN}}) + (μ_{λ_{BP}} - μ_{λ_{PP}})}, \\ \frac{ν_{λ_{PN}} - ν_{λ_{BN}}}{(ν_{λ_{PN}} - ν_{λ_{BN}}) + (ν_{λ_{BP}} - ν_{λ_{PP}})}, \\ \frac{θ_{λ_{PN}} - θ_{λ_{BN}}}{(θ_{λ_{PN}} - μ_{λ_{BN}}) + (θ_{λ_{BP}} - θ_{λ_{PP}})}) \\ = (0.5, 0.4, 0.5) \end{matrix}$ $\begin{matrix} β = (μ_{β}, ν_{β}, θ_{β}) \\ = (\frac{μ_{λ_{BN}} - μ_{λ_{NN}}}{(μ_{λ_{BN}} - μ_{λ_{NN}}) + (μ_{λ_{NP}} - μ_{λ_{BP}})}, \\ \frac{ν_{λ_{BN}} - ν_{λ_{NN}}}{(ν_{λ_{BN}} - ν_{λ_{NN}}) + (ν_{λ_{NP}} - ν_{λ_{BP}})}, \\ \frac{θ_{λ_{BN}} - θ_{λ_{NN}}}{(θ_{λ_{BN}} - θ_{λ_{NN}}) + (θ_{λ_{NP}} - θ_{λ_{BP}})}) \\ = (0.45, 0.43, 0.29) \end{matrix}$

Then, the four kinds of MSIFPRS models based on SIFP can be obtained, respectively.

(1) Three-way decisions of Type-I MSIFPRS based on SIFP

a) Three-way decision model of Type-I OMSIFPRS

Step 1: The calculation of $\lor_{k = 1}^{3} R_{k} (x_{i}, y_{j})$ is shown in the matrix below.

$\begin{matrix} \lor_{k = 1}^{3} R_{k} (x_{i}, y_{j}) \\ = [\begin{matrix} < 1, 0, 1 > & < 0.8, 0.2, 0.6 > & < 0.7, 0.2, 0.6 > & < 0.3, 0.6, 0.5 > & < 0.4, 0.6, 0.7 > & < 0.8, 0.2, 0.7 > \\ < 0.8, 0.2, 0.6 > & < 1, 0, 1 > & < 0.7, 0.2, 0.8 > & < 0.3, 0.6, 0.7 > & < 0.5, 0.4, 0.5 > & < 0.6, 0.3, 0.6 > \\ < 0.7, 0.2, 0.6 > & < 0.7, 0.2, 0.8 > & < 1, 0, 1 > & < 0.6, 0.3, 0.7 > & < 0.4, 0.6, 0.7 > & < 0.7, 0.2, 0.4 > \\ < 0.3, 0.6, 0.5 > & < 0.3, 0.6, 0.7 > & < 0.6, 0.3, 0.7 > & < 1, 0, 1 > & < 0.7, 0.2, 0.8 > & < 0.7, 0.2, 0.6 > \\ < 0.4, 0.6, 0.7 > & < 0.5, 0.4, 0.5 > & < 0.4, 0.6, 0.7 > & < 0.7, 0.2, 0.8 > & < 1, 0, 1 > & < 0.6, 0.4, 0.6 > \\ < 0.8, 0.2, 0.7 > & < 0.6, 0.3, 0.6 > & < 0.7, 0.2, 0.4 > & < 0.7, 0.2, 0.6 > & < 0.6, 0.4, 0.6 > & < 1, 0, 1 > \end{matrix}] \end{matrix}$

Based on Definitions 3.4, ∀x ∈ U, the class $[x_{i}]_{\cup_{k = 1}^{3} R_{k}} = \lor_{k = 1}^{3} R_{k} (x_{i}, y_{j})$ can be easily obtained.

Step 2: The SIFP $P (A | [x_{i}]_{\cup_{k = 1}^{3} R_{k}})$ of A given $[x_{i}]_{\cup_{k = 1}^{3} R_{k}}$ can be computed as follows $\begin{matrix} P (A | [x_{1}]_{\cup_{k = 1}^{3} R_{k}}) = (0.5763, 0.4752, 0.5130), \\ P (A | [x_{2}]_{\cup_{k = 1}^{3} R_{k}}) = (0.5624, 0.4406, 0.4614), \\ P (A | [x_{3}]_{\cup_{k = 1}^{3} R_{k}}) = (0.5764, 0.4933, 0.4498), \\ P (A | [x_{4}]_{\cup_{k = 1}^{3} R_{k}}) = (0.4922, 0.2858, 0.4900), \\ P (A | [x_{5}]_{\cup_{k = 1}^{3} R_{k}}) = (0.4624, 0.2409, 0.5168), \\ P (A | [x_{6}]_{\cup_{k = 1}^{3} R_{k}}) = (0.5380, 0.4033, 0.5119) . \end{matrix}$

Step 3: According to Definitions 4.1 and the thresholds α= (0.5, 0.4, 0.5), β= (0.45, 0.43, 0.29), the lower approximation ${\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (O)} (A)$ and upper approximation ${\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (O)} (A)$ of Type-I OMSIFPRS model can be calculated that $\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (O)} (A) & = {x | P (A | [x]_{\cup_{k = 1}^{3} R_{k}}) \\ ⩾ α, x \in U} = ϕ, \end{matrix}$ $\begin{matrix} {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (O)} (A) & = {x | P (A | [x]_{\cup_{k = 1}^{3} R_{k}}) > β, x \in U} \\ = {x_{4}, x_{5}, x_{6}} . \end{matrix}$

Step 4: The positive region POS^I(O) (A), negative region NEG^I(O) (A) and boundary region BND^I(O) (A) can be obtained by the lower and upper approximations of Type-I OMSIFPRS, as follows $\begin{matrix} {POS}^{I (O)} (A) & = {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (O)} (A) = ϕ, \\ {NEG}^{I (O)} (A) & = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (O)} (A) = {x_{1}, x_{2}, x_{3}}, \\ {BND}^{I (O)} (A) & = {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (O)} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (O)} (A) \\ = {x_{4}, x_{5}, x_{6}} . \end{matrix}$

b) Three-way decision model of Type-I IMSIFPRS

Step 1: The $\land_{k = 1}^{3} R_{k} (x_{i}, y_{j})$ can be calculated as follows

$\begin{matrix} \land_{k = 1}^{3} R_{k} (x_{i}, y_{j}) \\ = [\begin{matrix} < 1, 0, 1 > & < 0.4, 0.5, 0.4 > & < 0.3, 0.7, 0.3 > & < 0.2, 0.8, 0.4 > & < 0.2, 0.7, 0.2 > & < 0.4, 0.5, 0.5 > \\ < 0.4, 0.5, 0.4 > & < 1, 0, 1 > & < 0.3, 0.6, 0.3 > & < 0.2, 0.8, 0.6 > & < 0.3, 0.7, 0.3 > & < 0.6, 0.4, 0.2 > \\ < 0.3, 0.7, 0.3 > & < 0.3, 0.6, 0.3 > & < 1, 0, 1 > & < 0.3, 0.6, 0.2 > & < 0.2, 0.7, 0.2 > & < 0.3, 0.6, 0.4 > \\ < 0.2, 0.8, 0.4 > & < 0.2, 0.8, 0.6 > & < 0.3, 0.6, 0.2 > & < 1, 0, 1 > & < 0.6, 0.4, 0.3 > & < 0.2, 0.8, 0.3 > \\ < 0.2, 0.7, 0.2 > & < 0.3, 0.7, 0.3 > & < 0.2, 0.7, 0.2 > & < 0.6, 0.4, 0.3 > & < 1, 0, 1 > & < 0.5, 0.4, 0.5 > \\ < 0.4, 0.5, 0.5 > & < 0.6, 0.4, 0.2 > & < 0.3, 0.6, 0.4 > & < 0.2, 0.8, 0.3 > & < 0.5, 0.4, 0.5 > & < 1, 0, 1 > \end{matrix}] \end{matrix}$

∀x ∈ U, the class $[x_{i}]_{\cap_{k = 1}^{3} R_{k}} = \land_{k = 1}^{3} R_{k} (x_{i}, y_{j})$ can be easily obtained by Definitions 3.4.

Step 2: Then, the SIFP $P (A | [x_{i}]_{\cap_{k = 1}^{3} R_{k}})$ of A given $[x_{i}]_{\cap_{k = 1}^{3} R_{k}}$ can be calculated as follows $\begin{matrix} P (A | [x_{1}]_{\cap_{k = 1}^{3} R_{k}}) = (0.5941, 0.3975, 0.5000), \\ P (A | [x_{2}]_{\cap_{k = 1}^{3} R_{k}}) = (0.5482, 0.4011, 0.4664), \\ P (A | [x_{3}]_{\cap_{k = 1}^{3} R_{k}}) = (0.6096, 0.4174, 0.4089), \\ P (A | [x_{4}]_{\cap_{k = 1}^{3} R_{k}}) = (0.4546, 0.3115, 0.4728), \\ P (A | [x_{5}]_{\cap_{k = 1}^{3} R_{k}}) = (0.4183, 0.2432, 0.5846), \\ P (A | [x_{6}]_{\cap_{k = 1}^{3} R_{k}}) = (0.5059, 0.3425, 0.5172) . \end{matrix}$

Step 3: According to Definitions 4.2, the lower and upper approximations of Type-I IMSIFPRS model can be calculated as follows $\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (I)} (A) & = {x | P (A | [x]_{\cap_{k = 1}^{m} R_{k}}) ⩾ α, x \in U} \\ = {x_{1}, x_{6}}, \\ {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (I)} (A) & = {x | P (A | [x]_{\cap_{k = 1}^{m} R_{k}}) > β, x \in U} \\ = {x_{1}, x_{2}, x_{3}, x_{4}, x_{6}} . \end{matrix}$

Step 4: By the lower and upper approximations of Type-I IMSIFPRS, the positive, negative and boundary regions can be obtained as follows $\begin{matrix} {POS}^{I (I)} (A) & = {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (I)} (A) = {x_{1}, x_{6}}, \\ {NEG}^{I (I)} (A) & = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (I)} (A) = {x_{5}}, \\ {BND}^{I (I)} (A) & = {\bar{\sum_{k = 1}^{3} R_{k}}}^{I β (I)} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{I α (I)} (A) \\ = {x_{2}, x_{3}, x_{4}} . \end{matrix}$

(2) Three-way decisions of Type-II MSIFPRS based on SIFP

a) Three-way decision model of Type-II OMSIFPRS

Step 1: We can calculate the SIFP P (A| [x_i] _{R
_k}) (i = 1, 2, 3) of A given [x_i] _{R
_k} for each relationship R_k based on formula (10).

Step 2: ∀x ∈ U, the $\cup_{k = 1}^{3} P (A | [x_{i}]_{R_{k}})$ can be easily obtained as follows $\begin{matrix} \cup_{k = 1}^{3} P (A | [x_{1}]_{R_{k}}) = (0.6101, 0.3988, 0.5438), \\ \cup_{k = 1}^{3} P (A | [x_{2}]_{R_{k}}) = (0.5642, 0.3792, 0.4870), \\ \cup_{k = 1}^{3} P (A | [x_{3}]_{R_{k}}) = (0.6130, 0.4015, 0.4790), \\ \cup_{k = 1}^{3} P (A | [x_{4}]_{R_{k}}) = (0.4933, 0.2645, 0.5047), \\ \cup_{k = 1}^{3} P (A | [x_{5}]_{R_{k}}) = (0.4533, 0.2368, 0.5789), \\ \cup_{k = 1}^{3} P (A | [x_{6}]_{R_{k}}) = (0.5351, 0.3237, 0.5165) . \end{matrix}$

Step 3: According to Definitions 4.3, the lower and upper approximations of Type-II OMSIFPRS model can be calculated as $\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (O)} (A) & = {x | \cup_{k = 1}^{3} P (A | [x]_{R_{k}}) \\ ⩾ α, x \in U} = {x_{1}, x_{6}}, \\ {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (O)} (A) & = {x | \cup_{k = 1}^{3} P (A | [x]_{R_{k}}) > β, x \in U} \\ = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} = U . \end{matrix}$

Step 4: The positive, negative and boundary regions of three-way decisions based on Type-II OMSIFPRS model are deduced by the lower and upper approximations, as follows $\begin{matrix} {POS}^{II (O)} (A) & = {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (O)} (A) = {x_{1}, x_{6}}, \\ {NEG}^{II (O)} (A) & = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (O)} (A) = ϕ, \\ {BND}^{II (O)} (A) & = {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (O)} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (O)} (A) \\ = {x_{2}, x_{3}, x_{4}, x_{5}} . \end{matrix}$

b) Three-way decision model of Type-II IMSIFPRS

Step 1: The SIFP P (A| [x_i] _{R
_k}) (i = 1, 2, 3) of A given [x_i] _{R
_k} for each relationship R_k can be calculated.

Step 2: ∀x ∈ U, the $\cap_{k = 1}^{3} P (A | [x_{i}]_{R_{k}})$ can be easily obtained as follows $\begin{matrix} \cap_{k = 1}^{3} P (A | [x_{1}]_{R_{k}}) = (0.5581, 0.4582, 0.4665), \\ \cap_{k = 1}^{3} P (A | [x_{2}]_{R_{k}}) = (0.5300, 0.4651, 0.4307), \\ \cap_{k = 1}^{3} P (A | [x_{3}]_{R_{k}}) = (0.5750, 0.4709, 0.4031), \\ \cap_{k = 1}^{3} P (A | [x_{4}]_{R_{k}}) = (0.4549, 0.3356, 0.4485), \\ \cap_{k = 1}^{3} P (A | [x_{5}]_{R_{k}}) = (0.4260, 0.2522, 0.5215), \\ \cap_{k = 1}^{3} P (A | [x_{6}]_{R_{k}}) = (0.5011, 0.3899, 0.5069) . \end{matrix}$

Step 3: According to Definitions 4.4, the lower and upper approximations of Type-II IMSIFPRS model can be calculated as $\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (I)} (A) & = {x | \cap_{k = 1}^{3} P (A | [x]_{R_{k}}) \\ ⩾ α, x \in U} = {x_{6}}, \\ {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (I)} (A) & = {x | \cap_{k = 1}^{3} P (A | [x]_{R_{k}}) > β, x \in U} \\ = {x_{4}, x_{6}} . \end{matrix}$

Step 4: In light of the lower and upper approximations of Type-II IMSIFPRS, the positive, negative and boundary regions can be expressed as follows

$\begin{matrix} {POS}^{II (I)} (A) = {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (I)} (A) = {x_{6}}, \\ {NEG}^{II (I)} (A) = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (I)} (A) = {x_{1}, x_{2}, x_{3}, x_{5}}, \\ {BND}^{II (I)} (A) = {\bar{\sum_{k = 1}^{3} R_{k}}}^{II β (I)} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{II α (I)} (A) = {x_{4}} . \end{matrix}$

(3) Three-way decisions of Type-III MSIFPRS based on SIFP

On the basis of Type-II MSIFPRS models, the lower and upper approximations of Type-III MSIFPRS model can be computed as $\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) & = {x | \cap_{k = 1}^{3} P (A | [x]_{R_{k}}) \\ ⩾ α, x \in U} = {x_{6}}, \\ {\bar{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) & = {x | \cup_{k = 1}^{3} P (A | [x]_{R_{k}}) \\ ⩾ α, x \in U} = {x_{1}, x_{6}} . \end{matrix}$

Then, the positive, negative and boundary regions of three-way decisions based on Type-III MSIFPRS are derived as $\begin{matrix} {POS}^{III} (A) = {\underline{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) = {x_{6}}, \\ {NEG}^{III} (A) = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) = {x_{2}, x_{3}, x_{4}, x_{5}}, \\ {BND}^{III} (A) = {\bar{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{III α} (A) = {x_{1}} . \end{matrix}$

(4) Three-way decisions of Type-IV MSIFPRS based on SIFP

In light of the result of Type-II MSIFPRS models, the lower and upper approximations of Type-IV MSIFPRS can be directly obtained, as follows

$\begin{matrix} {\underline{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) & = {x | \cap_{k = 1}^{3} P (A | [x]_{R_{k}}) > β, x \in U} \\ = {x_{4}, x_{6}}, \\ {\bar{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) & = {x | \cup_{k = 1}^{3} P (A | [x]_{R_{k}}) > β, x \in U} = U . \end{matrix}$

Then, according to the lower and upper approximations, the positive, negative and boundary regions of three-way decisions based on Type-IV MSIFPRS are derived as follows $\begin{matrix} {POS}^{IV} (A) & = {\underline{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) = {x_{4}, x_{6}}, \\ {NEG}^{IV} (A) & = U - {\bar{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) = ϕ, \\ {BND}^{IV} (A) & = {\bar{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) - {\underline{\sum_{k = 1}^{3} R_{k}}}^{IV β} (A) \\ = {x_{1}, x_{2}, x_{3}, x_{5}} . \end{matrix}$

Through the above calculation, we can obtain three regions of three-way decisions for four kinds of MSIFPRSs. In what follows, we compare the four kinds of models, as shown in Table 4 in detail.

Table 4

Comparison with four kinds of MSIFPRS models

Models		Positive region	Negative region	Boundary region
Type-I MSIFPRS	Type-I OMSIFPRS	ϕ	{x₁, x₂, x₃}	{x₄, x₅, x₆}
	Type-I IMSIFPRS	{x₁, x₆}	{x₅}	{x₂, x₃, x₄}
Type-II MSIFPRS	Type-II OMSIFPRS	{x₁, x₆}	ϕ	{x₂, x₃, x₄, x₅}
	Type-II IMSIFPRS	{x₆}	{x₁, x₂, x₃, x₅}	{x₄}
Type-III MSIFPRS		{x₆}	{x₂, x₃, x₄, x₅}	{x₁}
Type-IV MSIFPRS		{x₄, x₆}	ϕ	{x₁, x₂, x₃, x₅}

On the one hand, as seen from Table 4, different models have different decision results. We observed that in Type-I OMSIFPRS, the job seekers x₁, x₂ and x₃ are not suitable for the job position in this enterprise, while x₄, x₅ and x₆ are not sure whether to work in the job position. In Type-I IMSIFPRS model, the job seekers x₁ and x₆ can work in the job position, but x₅ not suitable for this job. However, x₂, x₃ and x₄ can consider whether or not to work in this job position. The results of Type-II OMSIFPRS model reveal that the job seekers x₁ and x₆ can be competent for this job, but x₂, x₃, x₄ and x₅ are not sure whether to work in this job position, which requires further consideration before making a decision. Regarding the Type-II IMSIFPRS model, the job seeker x₆ is good for the job, but x₁, x₂, x₃ and x₅ are not to work in the job position. For x₄, it is also worth considering whether to work or not. In Type-III MSIFPRS model, we can see that x₆ can work in this job position, but x₂, x₃, x₄ and x₅ are not suitable for working, and the job seeker x₁ is hesitant to work in the job position. In Type-IV MSIFPRS model, it is obvious that the job seekers x₄ and x₆ are qualified for this work, while for x₁, x₂, x₃ and x₅, job seekers need to improve their abilities in various aspects before deciding whether they are suitable for this job.

On the other hand, by comparing the decision results of the four models of MSIFPRSs, we can observe that the boundary regions of Type-II IMSIFPRS and Type-III MSIFPRS is the smallest, while that of Type-II OMSIFPRS and Type-IV MSIFPRS is the largest. The job seeker x₆ is hesitant about the job position in Type-I OMSIFPRS model, while x₆ thinks this job position is most suitable in other models. In addition, x₄ attitude toward the job position is uncertain in Type-I MSIFPRS and Type-II MSIFPRS models. In Type-III MSIFPRS model, x₄ considers that this is not the most suitable job position, while it is considers that this is the most suitable job position in Type-IV MSIFPRS model.

In light of the above analysis, it is shows that different models have their unique advantages. In real-world applications, we can choose an appropriate model to provide a decision-making method for ourselves. The three-way decision models of these MSIFPRS are constructed based on SIFP by combining multi-granulation rough sets with three-way decision theory, and it is much more appropriate for decision-making problems in the SIF context. Consequently, we can use these proposed models to address the decision problems and reduce the uncertainty of decision-making.

6 Conclusion

In this paper, we propose four different SIFPs by combining SIFSs with probability theory, and a MSIFPAS is further defined. Subsequently, we construct four different types of MSIFPRS models by using different SIFPs in the context of MSIFPAS, and discuss their related properties. Moreover, the decision rules and uncertainty measures of three-way decisions under diverse MSIFPRS models are further derived. Meanwhile, an algorithm is designed and an example of person-job fit procedure is used to verify the validity of these proposed models. In the future, it is worth considering that the proposed MSIFPRS models with three-way decisions are applied to multi-scale information systems and the optimal scale selection.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Footnotes

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Grant Nos. 61772176, 61402153, 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.

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Three-way decisions based on multi-granulation support intuitionistic fuzzy probabilistic rough sets

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Support intuitionistic fuzzy sets

2.2 Probability theory on fuzzy sets

2.3 Rough sets

Table 1 The cost matrix of decision actions Decision Actions Loss Functions C ∼C a P λ PP λ PN a B λ BP λ BN a N λ NP λ NN

3.1 Probability theory on support intuitionistic fuzzy sets

4.1 Three-way decisions of Type-I MSIFPRS based on SIFP

5 Example analysis

Table 3 The SIF cost function values of different actions C ∼C a P λ PP = <0.20, 0.60, 0.30> λ PN = <0.60, 0.30, 0.80> a B λ BP = <0.30, 0.50, 0.40> λ BN = <0.50, 0.50, 0.60> a N λ NP = <0.60, 0.30, 0.70> λ NN = <0.25, 0.75, 0.48>

Conflicts of interest

Footnotes

Acknowledgments

References

Table 1
The cost matrix of decision actions

Decision Actions Loss Functions

C ∼C

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Table 3
The SIF cost function values of different actions

C ∼C

a_P λ_PP = <0.20, 0.60, 0.30> λ_PN = <0.60, 0.30, 0.80>

a_B λ_BP = <0.30, 0.50, 0.40> λ_BN = <0.50, 0.50, 0.60>

a_N λ_NP = <0.60, 0.30, 0.70> λ_NN = <0.25, 0.75, 0.48>