A useful method for analyzing incomplete and inconsistent information: Paraconsistent soft sets and corresponding decision making methods

Abstract

Soft set theory, proposed by Molodtsov, has been regarded as a generic mathematical tool for dealing with uncertainties. However, classical soft sets are not appropriate to deal with incomplete and inconsistent information. In this paper, we introduce the concept of paraconsistent soft sets combining paraconsistent logic and soft sets. The complement, “And”, restricted intersection, relaxed intersection, restricted cross and relaxed cross operations are defined on paraconsistent soft sets. In order to deal with incomplete and inconsistent information in decision making simultaneously, we also define paraconsistent soft decision system, choice value, decision value, the selected set and the eliminated set, and bring up the decision algorithm. Finally, an investment decision problem with incomplete and inconsistent information is analyzed by paraconsistent soft sets. The result shows that paraconsistent soft sets with more adequate parameterization can solve decision making problems with incomplete and inconsistent information more effectively than classical soft sets.

Keywords

Soft sets paraconsistent logic incomplete information inconsistent information decision making

1 Introduction

Soft set theory [3], as a newly emerging mathematical tool to deal with uncertain problems, is free from the inherent limitations of inadequate parameterization tool in existing methods, such as probability fuzzy set theory [18], vague sets theory [34], interval mathematics theory [19], and rough set theory [43]. Recently, soft set theory has been developed rapidly and lots of relevant concepts, properties, operations and algebraic structures have been brought up [11, 15, 28]. Theories like fuzzy set theory [32], vague set theory [33], interval mathematics theory [36], rough set theory [28], algebras theory [2, 39], description logic [40] are gradually initiated and extended in the frame of soft sets.

With the establishment and development of soft set theory, its applications have been booming and extended to decision making [7 , 37], combining forecasts [44], normal parameter reduction [42], demand analysis [5], data mining [31]. As a powerful tool to deal with uncertainties, soft sets have been widely used in many information analyses, especially information analysis with incomplete and inconsistent information. Zou and Xiao [38] initiated the study on incomplete information analysis approaches based on soft sets. Deng and Wang [30] proposed object–parameter method to predict incomplete information in incomplete fuzzy soft sets. Liu et al. [41] proposed that there are two main limitations in Deng–Wang method, respectively parameter value and concept confusion between objects’ distance and parameters’ distance, and then, they redefined the notion and proposed a new adjustable object-parameter approach. Maji [27] firstly introduced neutrosohic sets into soft set theory to handle problems involving imprecise, indeterminacy and inconsistent data. Karaaslan [15] redefined concepts and operations on neutrosophic soft sets [27], and also gave group decision making method. Later, Karaaslan [9] improved and further extended neutrosophic soft sets. He defined possibility neutrosophic soft sets, each element of it of initial universe has got a possibility degree related to each element of parameter set, and proposed a new decision making method. Peng and Liu [35] proposed three novel algorithms to solve single-valued neutrosophic soft decision making problems by Evaluation based on Distance from Average Solution (EDAS). Deli [13] defined interval valued neutrosophic soft sets (ivn-soft sets) and proposed a decision making method based on ivn-soft sets.

It is noteworthy that the process that soft set theory based on neutrosophic sets deals with inconsistent data suffers from two main limitations: (i) Neutrosophic sets is still based on membership, including “A neutrosophic set is characterized by a truth- membership degree, an indeterminacy-membership degree, and a falsity-membership degree”. However, like fuzzy sets, it is very difficult to determine the membership degree. (ii) It can not process incomplete and inconsistent information simultaneously. However, due to the diversity of information sources, incomplete and inconsistent information often occur at the same time.

In order to overcome these limitations of neutrosophic soft sets, this paper proposes a new type of soft set - paraconsistent soft set theory, which introduces paraconsistent reasoning into soft set theory. Paraconsistent reasoning is one of the most effective means to deal with incomplete and inconsistent information [4 , 25]. The paraconsistent approach allows reasoning in the presence of incomplete and inconsistent information, and incomplete and inconsistent information can be derived or introduced without trivialization. Therefore, based on paraconsistent reasoning, we can extend the expressive ability of parameters in soft sets by using the Belnap’s four-valued structure [24]. Paraconsistent soft sets include not only parameters of classical soft sets, but also three other parameters called approximate opposite, incomplete information and inconsistent information, for designating three other kinds of uncertain information. In other words, we enrich the parameterization of soft sets to deal with incomplete and inconsistent information simultaneously. From what has been discussed above, we may draw the conclusion that (i) Paraconsistent soft sets are based on paraconsistent reasoning [24] rather than membership. (ii) Paraconsistent soft sets can describe and analyze incomplete and inconsistent information simultaneously.

In this paper, some related operations on paraconsistent soft sets are defined, such as complement, “And”, restricted intersection, relaxed intersection, restricted cross and relaxed cross operations. In order to propose the decision algorithm based on paraconsistent soft sets, several concepts are introduced, including paraconsistent soft decision system, choice value, decision value, the selected set and the eliminated set. An investment decision problem with incomplete and inconsistent information is analyzed by paraconsistent soft sets.

The paper is organized as follows. Section 2 introduces the basic definitions of soft sets and paraconsistent logic. Section 3 presents the notion of paraconsistent soft sets and discusses their operations. The paraconsistent soft decision system and decision rules under incomplete and inconsistent information are also described in this section. In Section 4, we apply paraconsistent soft sets into an investment decision problem. Section 5 presents some research conclusions.

2 Brief introduction to soft sets and paraconsistent logic

2.1 Soft sets

Definition 2.1. (See [3]). A pair (F, E) is called a soft set (over U) if and only if F is a mapping of E into the set of all subsets of the set U.

In other words, soft sets are a parameterized family of subsets of the set U. Every set F (ɛ) (ɛ ∈ E), from this family may be considered as the set of ɛ- elements of the soft set (F, E), or as the set of ɛ- approximate elements of the soft set.

To illustrate this idea, let us consider the following example.

Example 2.1. Let universe U = {h₁, h₂, h₃, h₄} be a set of houses, a set of parameters E = {e₁, e₂, e₃, e₄} be a set of status of houses which stand for the parameters “beautiful”, “cheap”, “in green surroundings”, and “in good location”, respectively. Consider the mapping F to be a mapping of E into the set of all subsets of the set U. Now consider a soft set (F, E) that describes the “attractiveness of houses for purchasing”. According to the data collected, the soft set (F, E) is given by $\begin{matrix} \begin{matrix} (F, E) = {(e_{1}, {h_{1}, h_{3}, h_{4}}), (e_{2}, {h_{1}, h_{2}}), \\ (e_{3}, {h_{1}, h_{3}}), (e_{4}, {h_{2}, h_{3}, h_{4}})} \end{matrix} \end{matrix}$ where F (e₁) = {h₁, h₃, h₄} , F (e₂) = {h₁, h₂},

F (e₃) = {h₁, h₃} and F (e₄) = {h₂, h₃, h₄}.

Definition 2.2. (See [28]). For two soft sets (F, A) and (G, B) over U, (F, A) is called a soft subset of (G, B) if

A ⊂ B and

∀ɛ ∈ A, F (ɛ) and G (ɛ) are identical approximations.

This relationship is denoted by $(F, A) \tilde{\subset} (G, B)$ .

Similarly, (F, A) is called a soft superset of (G, B), if (G, B) is a soft subset of (F, A). This relationship is denoted by $(F, A) \tilde{\supset} (G, B)$ .

Example 2.2. (See [28]). Let A = { e₁, e₃, e₅ } ⊂ E and B = { e₁, e₂, e₃, e₅ } ⊂ E.

Clearly, A ⊂ B.

Let (F, A) and (G, B) be two soft sets over the same universe U ={ h₁, h₂, h₃, h₄, h₅, h₆ } such that $\begin{matrix} \begin{matrix} G (e_{1}) = {h_{2}, h_{4}}, G (e_{2}) = {h_{1}, h_{3}}, \\ G (e_{3}) = {h_{3}, h_{4}, h_{5}}, G (e_{5}) = {h_{1}}, \end{matrix} \end{matrix}$ $\begin{matrix} F (e_{1}) = {h_{2}, h_{4}}, F (e_{3}) = {h_{3}, h_{4}, h_{5}}, F (e_{5}) = {h_{1}} . \end{matrix}$

Therefore, $(F, A) \tilde{\subset} (G, B)$ .

Definition 2.3. (See [28]). Let E ={ e₁, e₂, e₃, ⋯ e_n } be a set of parameters. The NOT set of E denoted ⌉E by is defined by ⌉E ={ ¬ e₁, ¬ e₂, ¬ e₃, ⋯ ¬ e_n }, where ⌉e_i = not e_i.

Definition 2.4. (See [28]). The complement of a soft set (F, A) is denoted by (F, A) ^c and is defined by (F, A) ^c = (F^c, ⌉ A), where F^c : ⌉ A → P (U) is a mapping given by F^c (α) = U - F (⌉ α), ∀α ∈ ⌉ A. Let us call F^c to be the soft complement function of F.

Definition 2.5. (See [28]). Two soft sets (F, A) and (G, B) over U are called soft equal, if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A).

Definition 2.6. (See [28]). The intersection of two soft sets (F, A) and (G, B) over U is the soft set (H, C), where C = A ∩ B and ∀ɛ ∈ C, H (ɛ) = F (ɛ) or G (ɛ) (as both are the same set). This is denoted by $(F, A) \tilde{\cap} (G, B) = (H, C)$ .

Definition 2.7. (See [28]). The union of two soft sets (F, A) and (G, B) over U is the soft set (H, C) where C = A ∪ B and ∀ɛ ∈ C, $\begin{matrix} H (ɛ) = {\begin{matrix} F (ɛ), & if ɛ \in A - B \\ G (ɛ), & if ɛ \in B - A \\ F (ɛ) \cup G (ɛ), & if ɛ \in A \cap B \end{matrix} \end{matrix}$

This is denoted by $(F, A) \tilde{\cup} (G, B) = (H, C)$ .

Definition 2.8. (See [28]). AND operation on two soft sets. If (F, A) and (G, B) are two soft sets, then (F, A) AND (G, B) denoted by (F, A) ∧ (G, B) is defined by (F, A) ∧ (G, B) = (H, A × B), where H (α, β) = F (α) ∩ G (β) , ∀ (α, β) ∈ A × B .

Example 2.3. (See [28]). Consider the soft set (F, A) which describes the “costs of houses” and the soft set (G, B) which describes the “attractiveness of the houses”.

Suppose that U = {h₁, h₂, h₃, h₄, h₅, h₆, h₇, h₈, h₉, h₁₀},

A = {very costly ; costly ; cheap} and

B = {beautiful ; in the green surroundings ; cheap} .

Let F (very costly) = {h₂, h₄, h₇, h₈}, $\begin{matrix} F (costly) = {h_{1}, h_{3}, h_{5}}, F (cheap) = {h_{6}, h_{9}, h_{10}} \end{matrix}$ and G (beautiful) = {h₂, h₃, h₇} , $\begin{matrix} G (in the green surroundings) = {h_{5}, h_{6}, h_{8}}, \\ G (cheap) = {h_{6}, h_{9}, h_{10}} . \end{matrix}$

Then (F, A) ∧ (G, B) = (H, A × B),

where H (very costly, beautiful) ={ h₂, h₇ }, $\begin{matrix} H (very costly, in the green surroundings) = {h_{8}}, \end{matrix}$ $\begin{matrix} H (very costly, cheap) = φ, \end{matrix}$ $\begin{matrix} H (costly, beautiful) = {h_{3}}, \end{matrix}$ $\begin{matrix} H (costly, in the green surroundings) = {h_{5}}, \end{matrix}$ $\begin{matrix} H (costly, cheap) = φ, H (cheap, beautiful) = φ, \end{matrix}$ $\begin{matrix} H (cheap, in the green surroundings) = {h_{6}}, \end{matrix}$ $\begin{matrix} H (cheap, cheap) = {h_{6}, h_{9}, h_{10}} . \end{matrix}$

2.2 Paraconsistent logic

In paraconsistent reasoning, there exists a broad variety of paraconsistent logic. Among them, four-valued logic [24] is a widely admitted paraconsistent logic and a common basis for various extended and useful paraconsistent logic [12 , 29]. Reasoning with four-valued may be traced back to the 1950 s [1, 14]. Here we use Belnap’s four-valued algebraic structure Four, introduced in [24]. This structure consists four elements. Intuitively, + and – represent the truth values “true” and “false” of classical logic respectively. ⊥ stands for “undefined” or “unknown”. Likewise, T is understood as contradictions. It is necessary to notice that “contradictions” are defined as the inconsistency of the classical truth values.

Four is the simplest nontrivial bilattice, and a double-Hasse diagram of Four is shown in Fig. 1.

Fig.1

Four.

Negation operator ¬ on Four is defined by ¬T = T, ¬+ = - , ¬ - = + , ¬ ⊥ = ¬ ⊥.

AND operator ∧ and OR operator ∨ on Four are defined by Table 1.

Table 1

AND and OR operators on Four

∧				∨
	_T	+	–	⊥	_T	+	–	⊥
_T	_T	_T	–	–	_T	+	_T	+
+	_T	+	–	⊥	+	+	+	+
–	–	–	–	–	_T	+	–	⊥
⊥	–	⊥	–	⊥	+	+	⊥	⊥

3 Paraconsistent soft sets

3.1 Concept of paraconsistent soft sets and operations

Definition 3.1. Let (F, P) be a soft set over a common universe U, where F is a mapping F : P → P (U) and P is a family of the parameters set. We say that (F, P) is a paraconsistent soft set, if (F, P) such that $\begin{matrix} (i) ɛ = {ɛ^{+}, ɛ^{-}, ɛ^{⊥}, ɛ^{T}}, ɛ \in P . \end{matrix}$

We say ɛ^* is a cell parameter. ɛ⁺ and ɛ^- represent “approximate belonging to ɛ” and “approximate not belonging to ɛ ” respectively. ɛ^⊥ represents “lack of information about parameter ɛ”, and ɛ^T stands for “inconsistency of ɛ”. $\begin{matrix} (ii) \cup_{ɛ^{*} \in ɛ \in P} F (ɛ) = U . \end{matrix}$

(iii) For any two cell parameters ɛⁱ, ɛ^j ∈ ɛ, ɛⁱ ≠ ɛ^j, F (ɛⁱ)∩ F (ɛ^j) = φ.

Example 3.1. Let U be the set of real estate companies. Suppose that there are six companies in the universe U given by U ={ u₁, u₂, u₃, u₄, u₅, u₆ }.

P is a family of the parameter set, and each parameter is a word or sentence.

P ={ e₁, e₂, e₃ } where e₁ stands for operational capacity; e₂ stands for probability; e₃ stands for development.

In this case, we define a paraconsistent soft set (F, P) to describe the different comprehensive abilities of real estate companies, the mapping of (F, P) is given as below: $\begin{matrix} \begin{matrix} F (e_{1}^{+}) = {u_{1}, u_{3}}, F (e_{1}^{-}) = {u_{2}, u_{4}, u_{6}}, F (e_{1}^{⊥}) = {u_{5}}, \\ F (e_{1}^{T}) = φ; \\ F (e_{2}^{+}) = {u_{1}}, F (e_{2}^{-}) = {u_{2}, u_{4}, u_{6}}, F (e_{2}^{⊥}) = {u_{5}}, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} F (e_{2}^{T}) = {u_{3}}; \\ F (e_{3}^{+}) = {u_{1}, u_{3}}, F (e_{3}^{-}) = {u_{2}, u_{4}, u_{6}}, F (e_{3}^{⊥}) = {u_{5}}, \\ F (e_{3}^{T}) = {u_{3}} . \end{matrix} \end{matrix}$

From Definition 3.1 $(F, {e_{1}, e_{2}})$ is a paraconsistent soft set, while $(F, {e_{1}, e_{3}})$ and $(F, {e_{2}, e_{3}})$ are not paraconsistent soft sets. Further, $F (e_{2}^{+}) = {u_{1}}$ can be viewed that the company u₁ owns profitability approximately, while $F (e_{2}^{-}) = {u_{2}, u_{4}, u_{6}}$ can be thought that companies u₂, u₄, u₆ have no profitability approximately. $F (e_{2}^{⊥}) = {u_{5}}$ represents the company u₅ lack of exact information about profitability, and $F (e_{2}^{T}) = {u_{3}}$ represents that the probability of company u₃ is inconsistent, which could be due to the different evaluation institutes or periodic estimations.

We can represent the paraconsistent soft set $(F, {e_{1}, e_{2}})$ in the form of Table 2.

Table 2
Tabular representation of the paraconsistent soft set (F, {e₁, e₂})

e ₁ e ₂

u ₁ + +

u ₂ – –

u ₃ + _T

u ₄ – –

u ₅ ⊥ ⊥

u ₆ – –

	e ₁	e ₂
u ₁	+	+
u ₂	–	–
u ₃	+	_T
u ₄	–	–
u ₅	⊥	⊥
u ₆	–	–

Definition 3.2. Let (F, P) and (G, Q) be two paraconsistent soft sets over U, we say that (H, P × Q) is a N2- paraconsistent soft set where H (αⁱ, β^j) = F (αⁱ) ∩ G (β^j), $\begin{matrix} \forall (α^{i}, β^{j}) \in (α, β) \in P \times Q, (i, j \in {+, -, ⊥, T}) . \end{matrix}$

We say (αⁱ, β^j) is N2- cell parameter.

Example 3.2. Suppose that (F, P) and (G, Q) are two paraconsistent soft sets over the same universe U = {u₁, u₂, u₃, u₄, u₅, u₆}, and P = {e₁, e₂} , Q = {e₁, e₃}.

Let $F (e_{1}^{+}) = {u_{1}, u_{3}}, F (e_{1}^{-}) = {u_{2}, u_{4}, u_{6}}$ , $F (e_{1}^{⊥}) = {u_{5}}$ , $F (e_{1}^{T}) = φ$ ; $F (e_{2}^{+}) = {u_{1}}$ , $F (e_{2}^{-}) = {u_{2}, u_{4}, u_{6}}$ , $F (e_{2}^{⊥}) = {u_{5}}, F (e_{2}^{T}) = {u_{3}}$ ; and $G (e_{1}^{+}) = {u_{2}}, G (e_{1}^{-}) = {u_{1}, u_{3}, u_{4}, u_{6}}$ , $G (e_{1}^{⊥})$ = {u₅}, $G (e_{1}^{T}) = φ$ , $G (e_{3}^{+}) = {u_{3}}$ , $G (e_{3}^{-}) = {u_{2}, u_{4}$ , u₆}, $G (e_{3}^{⊥}) = {u_{5}}$ , $G (e_{3}^{T}) = {u_{1}}$ .

Then N2 – paraconsistent soft set of (F, P) and (G, Q) is (H, P × Q), where $\begin{matrix} H (e_{1}^{+}, e_{1}^{+}) = φ, H (e_{1}^{+}, e_{1}^{-}) = {u_{1}, u_{3}}, H (e_{1}^{+}, e_{1}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{1}^{+}, e_{1}^{T}) = φ; H (e_{1}^{-}, e_{1}^{+}) = {u_{2}}, H (e_{1}^{-}, e_{1}^{-}) = {u_{4}, u_{6}}, \end{matrix}$ $\begin{matrix} H (e_{1}^{-}, e_{1}^{⊥}) = φ, H (e_{1}^{-}, e_{1}^{T}) = φ, H (e_{1}^{⊥}, e_{1}^{+}) = φ, \end{matrix}$ $\begin{matrix} H (e_{1}^{⊥}, e_{1}^{-}) = φ, H (e_{1}^{⊥}, e_{1}^{⊥}) = {u_{5}}, H (e_{1}^{⊥}, e_{1}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{1}^{T}, e_{1}^{+}) = φ, H (e_{1}^{T}, e_{1}^{-}) = φ, H (e_{1}^{T}, e_{1}^{⊥}) = φ, \\ H (e_{1}^{T}, e_{1}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{1}^{+}, e_{3}^{+}) = {u_{3}}, H (e_{1}^{+}, e_{3}^{-}) = φ, H (e_{1}^{+}, e_{3}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{1}^{+}, e_{3}^{T}) = {u_{1}}; H (e_{1}^{-}, e_{3}^{+}) = φ, \\ H (e_{1}^{-}, e_{3}^{-}) = {u_{2}, u_{4}, u_{6}}, \end{matrix}$ $\begin{matrix} H (e_{1}^{-}, e_{3}^{⊥}) = φ, H (e_{1}^{-}, e_{3}^{T}) = φ; H (e_{1}^{⊥}, e_{3}^{+}) = φ, \end{matrix}$ $\begin{matrix} H (e_{1}^{⊥}, e_{3}^{-}) = φ, H (e_{1}^{⊥}, e_{3}^{⊥}) = {u_{5}}, H (e_{1}^{⊥}, e_{3}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{1}^{T}, e_{3}^{+}) = φ, H (e_{1}^{T}, e_{3}^{-}) = φ, H (e_{1}^{T}, e_{3}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{1}^{T}, e_{3}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{2}^{+}, e_{1}^{+}) = φ, H (e_{2}^{+}, e_{1}^{-}) = {u_{1}}, H (e_{2}^{+}, e_{1}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{+}, e_{1}^{T}) = φ; H (e_{2}^{-}, e_{1}^{+}) = {u_{2}}, H (e_{2}^{-}, e_{1}^{-}) = {u_{4}, u_{6}}, \end{matrix}$ $\begin{matrix} H (e_{2}^{-}, e_{1}^{⊥}) = φ, H (e_{2}^{-}, e_{1}^{T}) = φ; H (e_{2}^{⊥}, e_{1}^{+}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{⊥}, e_{1}^{-}) = φ, H (e_{2}^{⊥}, e_{1}^{⊥}) = {u_{5}}, H (e_{2}^{⊥}, e_{1}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{2}^{T}, e_{1}^{+}) = φ, H (e_{2}^{T}, e_{1}^{-}) = {u_{3}}, H (e_{2}^{T}, e_{1}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{T}, e_{1}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{2}^{+}, e_{3}^{+}) = {u_{1}, u_{3}}, H (e_{2}^{+}, e_{3}^{-}) = φ, H (e_{2}^{+}, e_{3}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{+}, e_{3}^{T}) = φ; H (e_{2}^{-}, e_{3}^{+}) = φ, H (e_{2}^{-}, e_{3}^{-}) = {u_{2}, u_{4}, u_{6}}, \end{matrix}$ $\begin{matrix} H (e_{2}^{-}, e_{3}^{⊥}) = φ, H (e_{2}^{-}, e_{3}^{T}) = φ; H (e_{2}^{⊥}, e_{3}^{+}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{⊥}, e_{3}^{-}) = φ, H (e_{2}^{⊥}, e_{3}^{⊥}) = {u_{5}}, H (e_{2}^{⊥}, e_{3}^{T}) = φ; \end{matrix}$ $\begin{matrix} H (e_{2}^{T}, e_{3}^{+}) = φ, H (e_{2}^{T}, e_{3}^{-}) = φ, H (e_{2}^{T}, e_{3}^{⊥}) = φ, \end{matrix}$ $\begin{matrix} H (e_{2}^{T}, e_{3}^{T}) = φ; \end{matrix}$

By Definition 3.1 and 3.2, N2- paraconsistent soft set (H, P × Q) satisfy the following propositions. $\begin{matrix} (i) & U_{(α^{i}, β^{j}) \in (α, β) \in P \times Q} H (α^{i}, β^{j}) = \\ U (i, j \in {+, -, ⊥, T}) . \end{matrix}$

(ii) For any two different N2 – cell parameter $\begin{matrix} (α^{i}, β^{j}), (α^{m}, β^{n}) \in (α, β), \end{matrix}$ $\begin{matrix} (α^{i}, β^{j}) \neq (α^{m}, β^{n}), H (α^{i}, β^{j}) \cap H (α^{m}, β^{n}) = φ, \end{matrix}$ $\begin{matrix} (i, j \in {+, -, ⊥, T}) . \end{matrix}$

Proof. From Definition 3.2, $\begin{matrix} H (α^{i}, β^{j}) = F (α^{i}) \cap G (β^{j}), \end{matrix}$ $\begin{matrix} \forall (α^{i}, β^{j}) \in (α, β) \in P \times Q (i, j \in {+, -, ⊥, T}) . \end{matrix}$

Suppose e ∈ P × Q is a parameter of (H, Y).

Therefore, H (e) = F (αⁱ) ∩ G (β^j), $\begin{matrix} \begin{matrix} \cup_{e \in Y} H (e) = \cup_{α^{i} \in α \in P} \cup_{β^{j} \in β \in Q} F (α^{i}) \cap G (β^{j}) \\ = \cup_{α^{i} \in α \in P} F (α^{i}) \cap (\cup_{β^{j} \in β \in Q} G (β^{j})) \\ = \cup_{α^{i} \in α \in P} F (α^{i}) \cap U \\ = U \end{matrix} \end{matrix}$

Suppose that e_i, e_j ∈ Y, e_i ≠ e_j . e_i is the Cartesian product of α (α ∈ P) and β (β ∈ Q) , e_j is the Cartesian product of δ (δ ∈ P) and γ (γ ∈ Q) .

Then $\begin{matrix} \begin{matrix} H (e_{i}) \cap H (e_{j}) \\ = (F (α) \cap G (β)) \cap (F (δ) \cap G (γ)) = φ . \end{matrix} \end{matrix}$

Similarly, N3-paraconsistent soft set, N4- paraconsistent soft set, . . . , Nn- paraconsistent soft set can be defined.□

Definition 3.3. For two paraconsistent soft sets (F, P) and (G, Q) over a common universe U, we say that (F, P) is a paraconsistent soft subset of (G, Q), if P ⊆ Q and ɛ^* ∈ ɛ ∈ P, F (ɛ^*) and G (ɛ^*) are identical approximations. We write $(F, P) \tilde{\subseteq} (G, Q)$ .

Similarly, (F, P) is said to be a paraconsistent soft superset of (G, Q), if (G, Q) is a paraconsistent soft subset of (F, P) .We denote it by $(F, P) \tilde{\supseteq} (G, Q)$ .

Definition 3.4. Two paraconsistent soft sets (F, P) and (G, Q) over a common universe U are said to be paraconsistent soft equal, if (F, P) is a paraconsistent soft superset of (G, Q), and (G, Q) is a paraconsistent soft superset of (F, P).

Example 3.3. Let (F, P) and (G, Q) be two paraconsistent soft sets over the same universe U ={ u₁, u₂, u₃, u₄, u₅, u₆ }, and P ={ e₁, e₂ },Q ={ e₁, e₂, e₃ }.

Clearly, P ⊆ Q. $\begin{matrix} \begin{matrix} F (e_{1}^{+}) = {u_{1}, u_{3}}, F (e_{1}^{-}) = {u_{2}, u_{4}, u_{6}}, \\ F (e_{1}^{⊥}) = {u_{5}}, F (e_{1}^{T}) = φ; \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} F (e_{2}^{+}) = {u_{1}}, F (e_{2}^{-}) = {u_{2}, u_{4}, u_{6}}, \\ F (e_{2}^{⊥}) = {u_{5}}, F (e_{2}^{T}) = {u_{3}}; \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G (e_{1}^{+}) = {u_{1}, u_{3}}, G (e_{1}^{-}) = {u_{2}, u_{4}, u_{6}}, \\ G (e_{1}^{⊥}) = {u_{5}}, G (e_{1}^{T}) = φ; \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G (e_{2}^{+}) = {u_{1}}, G (e_{2}^{-}) = {u_{2}, u_{4}, u_{6}}, \\ G (e_{2}^{⊥}) = {u_{5}}, G (e_{2}^{T}) = {u_{3}}; \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G (e_{3}^{+}) = {u_{3}}, G (e_{3}^{-}) = {u_{2}, u_{4}, u_{6}}, \\ G (e_{3}^{⊥}) = {u_{5}}, G (e_{3}^{T}) = {u_{1}}; \end{matrix} \end{matrix}$

Therefore, $(F, P) \tilde{\subseteq} (G, Q)$ .

Definition 3.5. Let $\begin{matrix} \begin{matrix} P = {{e_{1}^{+}, e_{1}^{-}, e_{1}^{⊥}, e_{1}^{T}}, {e_{2}^{+}, e_{2}^{-}, e_{2}^{⊥}, e_{2}^{T}}, \\ {e_{3}^{+}, e_{3}^{-}, e_{3}^{⊥}, e_{3}^{T}}, \dots, {e_{n}^{+}, e_{n}^{-}, e_{n}^{⊥}, e_{n}^{T}}} \end{matrix} \end{matrix}$ be a family of parameters sets. The NOT set of P denoted by ⌉Pis defined by $\begin{matrix} \begin{matrix} ⌉ P = {{\neg e_{1}^{+}, \neg e_{1}^{-}, \neg e_{1}^{⊥}, \neg e_{1}^{T}}, {\neg e_{2}^{+}, \neg e_{2}^{-}, \neg e_{2}^{⊥}, \neg e_{2}^{T}}, \\ {\neg e_{3}^{+}, \neg e_{3}^{-}, \neg e_{3}^{⊥}, \neg e_{3}^{T}}, \dots, {\neg e_{n}^{+}, \neg e_{n}^{-}, \neg e_{n}^{⊥}, \neg e_{n}^{T}}}, \end{matrix} \end{matrix}$ where ¬e⁺ = e^-, ¬ e^- = e⁺, ¬ e^⊥ = e^⊥ and ¬e^T = e^T.

Definition 3.6. The complement of a paraconsistent soft set (F, P) is denoted by (F, P) ^c and is defined by (F, P) ^c = (F^c, ⌉ P), where F^c : ⌉ P → P (U) is a mapping given by F^c (ɛ^*) = F (¬ ɛ^*) , ∀ ɛ^* ∈ ɛ ∈ P.

We say F^c is the paraconsistent soft complement function of F. Clearly, (F^c) ^c is the same as F and ((F, P) ^c) ^c = (F, P).

Example 3.4. For Example 3.1, the complement of a paraconsistent soft set (F, { e₁, e₂ }) is given by $\begin{matrix} \begin{matrix} (F, {e_{1}, e_{2}})^{c} = \\ {(e_{1}^{+} = {u_{2}, u_{4}, u_{6}}, e_{1}^{-} = {u_{1}, u_{3}}, e_{1}^{⊥} = {u_{5}}, e_{1}^{T} = φ), \\ (e_{2}^{+} = {u_{2}, u_{4}, u_{6}}, e_{2}^{-} = {u_{1}}, e_{2}^{⊥} = {u_{5}}, e_{2}^{T} = {u_{3}})} \end{matrix} \end{matrix}$

Definition 3.7. If (F, P) and (G, Q) are two paraconsistent soft sets over U, (F, P) AND (G, Q) denoted by (F, P) ∧ (G, Q) is defined by (F, P) ∧ (G, Q) = (H, P × Q),

where H (αⁱ, β^j) = F (αⁱ) ∩ G (β^j) $\begin{matrix} \forall (α^{i}, β^{j}) \in (α, β) \in P \times Q, (i, j \in {+, -, ⊥, T}) . \end{matrix}$

From Definition 3.2, (F, P) ∧ (G, Q) = (H, P × Q) is also a N2- paraconsistent soft set.

Definition 3.8. (Restricted intersection operation on two paraconsistent soft sets) Let (F, P) and (G, Q) be two paraconsistent soft sets over a common universe U. The restricted intersection of (F, P) AND (G, Q) is denoted by and is defined as ,

where Y = P ∩ Q, and ∀ɛ^* ∈ ɛ ∈ Y, H (ɛ^*) is given by H (ɛ⁺) = F (ɛ⁺) ∩ G (ɛ⁺), $\begin{matrix} \begin{matrix} H (ɛ^{-}) = (F (ɛ^{-}) \cap G (ɛ^{-})) \cup (F (ɛ^{+}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{+})) \cup (F (ɛ^{⊥}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{⊥})) \cup (F (ɛ^{T}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{T})) \cup (F (ɛ^{⊥}) \cap G (ɛ^{T})) \cup \\ (F (ɛ^{T}) \cap G (ɛ^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} H (ɛ^{⊥}) = (F (ɛ^{+}) \cap G (ɛ^{⊥})) \cup (F (ɛ^{⊥}) \cap G (ɛ^{+})) \\ \cup (F (ɛ^{⊥}) \cap G (ɛ^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} H (ɛ^{T}) = (F (ɛ^{+}) \cap G (ɛ^{T})) \cup (F (ɛ^{T}) \cap G (ɛ^{+})) \\ \cup (F (ɛ^{T}) \cap G (ɛ^{T})) . \end{matrix} \end{matrix}$

In the above Example 3.2,

where $H (e_{1}^{+}) = φ, H (e_{1}^{-}) = {u_{1}, u_{2}, u_{3}, u_{4}, u_{6}}$ , $\begin{matrix} H (e_{1}^{⊥}) = {u_{5}}, H (e_{1}^{T}) = φ . \end{matrix}$

Definition 3.9. (Relaxed intersection operation on two paraconsistent soft sets) Let (F, P) and (G, Q) be two paraconsistent soft sets over a common universe U. The relaxed intersection of (F, P) and (G, Q) is denoted by and is defined as where Y = P ∩ Q, and ∀ɛ^* ∈ ɛ ∈ Y, H (ɛ^*) is given by $\begin{matrix} \begin{matrix} H (ɛ^{+}) = (F (ɛ^{+}) \cap G (ɛ^{+})) \cup (F (ɛ^{+}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{+})) \cup (F (ɛ^{+}) \cap G (ɛ^{⊥})) \cup \\ (F (ɛ^{⊥}) \cap G (ɛ^{+})) \cup (F (ɛ^{+}) \cap G (ɛ^{T})) \cup \\ (F (ɛ^{T}) \cap G (ɛ^{+})) \cup (F (ɛ^{⊥}) \cap G (ɛ^{T})) \cup \\ (F (ɛ^{T}) \cap G (ɛ^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} H (ɛ^{-}) = (F (ɛ^{-}) \cap G (ɛ^{-})), \end{matrix}$ $\begin{matrix} \begin{matrix} H (ɛ^{⊥}) = (F (ɛ^{⊥}) \cap G (ɛ^{⊥})) \cup (F (ɛ^{⊥}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} H (ɛ^{T}) = (F (ɛ^{T}) \cap G (ɛ^{T})) \cup (F (ɛ^{T}) \cap G (ɛ^{-})) \cup \\ (F (ɛ^{-}) \cap G (ɛ^{T})) . \end{matrix} \end{matrix}$

In the above Example 3.2, where $H (e_{1}^{+}) = {u_{1}, u_{2}, u_{3}}, H (e_{1}^{-}) = {u_{4}, u_{6}}$ , $H (e_{1}^{⊥}) = {u_{5}}$ , $H (e_{1}^{T}) = φ$ .

Definition 3.10. (Restricted cross operation on two paraconsistent soft sets) The restricted cross of two paraconsistent soft sets (F, P) and (G, Q) over a common universe U is the soft set (C, R), where $\begin{matrix} \begin{matrix} R = \cup_{ɛ_{i} \in P, ɛ_{j} \in Q, ɛ_{m} \in P \cap Q} {e_{mj}, e_{im} : e_{mj} = {ɛ_{m}, ɛ_{j}} \\ e_{im} = {ɛ_{i}, ɛ_{m}}; m \neq i; m \neq j}, \end{matrix} \end{matrix}$ and $\forall e_{ij}^{*} \in e_{ij} \in R, C (e_{ij}^{*})$ is given by $\begin{matrix} C (e_{ij}^{+}) = F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{+}), \end{matrix}$ $\begin{matrix} \begin{matrix} C (e_{ij}^{-}) = (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{-})) \cup (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{+})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{⊥})) \cup (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{T})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{T})) \cup \\ (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} C (e_{ij}^{⊥}) = (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{⊥})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{+})) \cup \\ (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} C (e_{ij}^{T}) = (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{T})) \cup (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{+})) \cup \\ (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{T})) . \end{matrix} \end{matrix}$

We write

In the above Example 3.2, where $C (e_{13}^{+}) = {u_{3}}, C (e_{13}^{-}) = {u_{2}, u_{4}, u_{6}}, C (e_{13}^{⊥}) = {u_{5}}, C (e_{13}^{T}) = {u_{1}};$ $C (e_{21}^{+}) = φ, C (e_{21}^{-}) = {u_{1}, u_{2}, u_{3}, u_{4}, u_{6}}, C (e_{21}^{⊥}) = {u_{5}}, C (e_{21}^{T}) = φ$ .

Definition 3.11. (Relaxed cross operation on two paraconsistent soft sets) The relaxed cross of two paraconsistent soft sets (F, P) and (G, Q) over a common universe U is the soft set (C, R), where $\begin{matrix} \begin{matrix} R = \cup_{ɛ_{i} \in P, ɛ_{j} \in Q, ɛ_{m} \in P \cap Q} {e_{mj}, e_{im} : e_{mj} = {ɛ_{m}, ɛ_{j}} \\ e_{im} = {ɛ_{i}, ɛ_{m}}; m \neq i; m \neq j}, \end{matrix} \end{matrix}$

and $\forall e_{ij}^{*} \in e_{ij} \in R, C (e_{ij}^{*})$ is given by $\begin{matrix} \begin{matrix} C (e_{ij}^{+}) = (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{+})) \cup (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{+})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{+})) \cup \\ (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{⊥})) \cup (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{+})) \cup \\ (F (ɛ_{i}^{+}) \cap G (ɛ_{j}^{T})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{T})) \cup \\ (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} C (e_{ij}^{-}) = F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{-}), \end{matrix}$ $\begin{matrix} \begin{matrix} C (e_{ij}^{⊥}) = (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{⊥})) \cup (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{⊥}) \cap G (ɛ_{j}^{⊥})), \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} C (e_{ij}^{T}) = (F (ɛ_{i}^{-}) \cap G (ɛ_{j}^{T})) \cup (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{-})) \cup \\ (F (ɛ_{i}^{T}) \cap G (ɛ_{j}^{T})) . \end{matrix} \end{matrix}$

We write

In the above Example 3.2, where $C (e_{13}^{+}) = {u_{1}, u_{3}}$ , $\begin{matrix} C (e_{13}^{-}) = {u_{2}, u_{4}, u_{6}}, C (e_{13}^{⊥}) = {u_{5}}, C (e_{13}^{T}) = φ; \end{matrix}$ $\begin{matrix} C (e_{21}^{+}) = {u_{1}, u_{2}}, C (e_{21}^{-}) = {u_{4}, u_{6}}, \\ C (e_{21}^{⊥}) = {u_{5}}, C (e_{21}^{T}) = {u_{3}} . \end{matrix}$

3.2 Paraconsistent soft decision system

Definition 3.12. (Paraconsistent soft decision system of imperfect information) Let (H, Y) be the intersection of two paraconsistent soft sets (F, P) and (G, Q), and (C, R) be the cross of (F, P) and (G, Q). Suppose n₊ and n_- stand for the number of objects u ∈ U that belong to $C (e_{ij}^{+})$ and $C (e_{ij}^{-})$ , respectively. The paraconsistent soft decision system is defined as (H_d, Y), and ∀ɛ^* ∈ ɛ ∈ Y, decision rule of imperfect information is given by

If u ∈ H (ɛ⁺), then u∈ H_d (ɛ⁺) ;

If u ∈ H (ɛ^-), then u∈ H_d (ɛ^-) ;

If n₊ > n_-, u ∈ H (ɛ^⊥) ∪ H (ɛ^T), then u ∈ H_d (ɛ⁺) ;

If n_- > n₊, u ∈ H (ɛ^⊥) ∪ H (ɛ^T), then u ∈ H_d (ɛ^-) ;

If n₊ = n_-, u ∈ H (ɛ^⊥), then u∈ H_d (ɛ^⊥) ;

If n₊ = n_-, u ∈ H (ɛ^T), then u ∈ H_d (ɛ^T).

When (H, Y) is the restricted intersection and (C, R) is restricted cross of two paraconsistent soft sets (F, P) and (G, Q), we call (H_d, Y) is the restricted paraconsistent soft decision system, denoted by

When (H, Y) is the relaxed intersection and (C, R) is relaxed cross of two paraconsistent soft sets (F, P) and (G, Q), we call (H_d, Y) is the relaxed paraconsistent soft decision system, denoted by

By applying the decision rule as above, we can reduce the quantity of unknown information and inconsistent information in decision system.

Definition 3.13. (Choice value on paraconsistent soft decision system) (H_d, Y) is the paraconsistent soft decision system, and (C, R) be the cross of (F, P) and (G, Q). Suppose n₊, n_-, n_⊥ and n_T stand for the number of objects u_p ∈ U that belong to $C (e_{ij}^{+}), C (e_{ij}^{-}), C (e_{ij}^{⊥})$ and $C (e_{ij}^{T})$ , respectively. The choice value of an object u_p ∈ U and ɛ_q ∈ P ∩ Q is given by $\begin{matrix} c_{pq} = {\begin{matrix} \frac{n_{+}}{n_{+} + n_{-} + n_{⊥} + n_{T}} \begin{matrix} , & u_{p} \in H_{d} (ɛ_{q}^{+}) \end{matrix} \\ \frac{n_{-}}{n_{+} + n_{-} + n_{⊥} + n_{T}} \begin{matrix} \times (- 1) & , & u_{p} \in H_{d} (ɛ_{q}^{-}) \end{matrix} \\ 0 \begin{matrix} , & u_{p} \in H_{d} (ɛ_{q}^{⊥}) \cup H_{d} (ɛ_{q}^{T}) \end{matrix} \end{matrix} \end{matrix}$

Definition 3.14. (Decision value on paraconsistent soft decision system) The decision value of an object u_p ∈ U is given by

d_p = ∑_qc_pq, where ɛ_q = P ∩ Q.

According to the positive and negative of d_p, the objects can be decided. If d_p > 0, the objects are considered to be good and should be chosen. The selected set consisted of corresponding qualifying objects is denoted by . On the contrary, if d_p < 0, the objects are considered to be bad and should be deleted. The eliminated set consisted of corresponding qualifying objects is denoted by .

Therefore, the algorithm of decision making problems with incomplete and inconsistent information based on paraconsistent soft sets is as follows:

Algorithm.

Step 1: Choose feasible subsets of the set of parameters.

Step 2: Construct paraconsistent soft sets for each set of parameters.

Step 3: Compute restricted and relaxed intersection by Definition 3.8 and 3.9, the restricted and relaxed cross by Definition 3.10 and 3.11.

Step 4: Construct restricted and relaxed paraconsistent soft decision system respectively by Definition 3.12.

Step 5: Calculate choice value and decision value of both restricted and relaxed paraconsistent soft decision system by Definition 3.13 and 3.14 respectively.

Step 6: Find the selected set of restricted and relaxed paraconsistent soft decision system, and , and the eliminated set of restricted and relaxed paraconsistent soft decision system, and .

Note that if it necessary to make a subdivision of objects to support more complicated decision-making, further we can compute , and further.

denotes the optimal objects.

denotes the suboptimal objects.

denotes the removable objects.

4 Application in an investment decision problem

There is an investment company that wants to invest in real estate, and 6 real estate companies are as candidates. And the investment company needs to evaluate comprehensive ability of these candidates. The investment company collects relevant information from two sources. One is real estate companies themselves, and the other is a third party. Two data sets include incomplete and inconsistent information because of data missing, different time and different means of collect. Some factors affecting comprehensive ability should be considered. These factors are “short-term liquidity”, “long-term solvency”, “operational capacity”, “profitability”, “low risk level”, “development”, “excellent skill of leader”, which are denoted by e₁ to e₇, respectively. Assume that the set of candidates is U = {u₁, u₂, u₃, u₄, u₅, u₆}, and we can construct paraconsistent soft sets (F, P) and (G, Q) over U.

Step 1: Real estate companies and the third party focus on different factors. Real estate companies consider set of parameters $P = {e_{1}, e_{2}, e_{3}, e_{4}}$ and the third party considers set of parameters Q = {e₃, e₄, e₅, e₆, e₇}.

Step 2: Real estate companies and the third party evaluate comprehensive ability for the real estate companies based on data and decision criteria owned by themselves. Then they construct the following two paraconsistent soft sets and in Table 3 over according to their parameters, respectively.

Table 3
Tabular representation the paraconsistent soft sets (F, P) and (G, Q)

(F, P) (G, Q)

e ₁ e ₂ e ₃ e ₄ e ₃ e ₄ e ₅ e ₆ e ₇

u ₁ – + + – + – + + –

u ₂ _T + _T + + + + – +

u ₃ + – – ⊥ – _T + – –

u ₄ + + + + + + + + _T

u ₅ ⊥ _T + _T ⊥ + – + –

u ₆ + + ⊥ + – + ⊥ – –

(F, P)	(G, Q)
u ₁	–	+	+	–	+	–	+	+	–
u ₂	_T	+	_T	+	+	+	+	–	+
u ₃	+	–	–	⊥	–	_T	+	–	–
u ₄	+	+	+	+	+	+	+	+	_T
u ₅	⊥	_T	+	_T	⊥	+	–	+	–
u ₆	+	+	⊥	+	–	+	⊥	–	–

Table 4

The table representation $(F, P) \underset{˜}{\cap} (G, Q)$


	e ₃	e ₄	e ₃	e ₄
u ₁	+	–	+	–
u ₂	_T	+	+	+
u ₃	–	–	–	+
u ₄	+	+	+	+
u ₅	⊥	_T	+	+
u ₆	–	+	⊥	+

Table 5

The table representation

	e ₃₄	e ₃₅	e ₃₆	e ₃₇	e ₁₃	e ₂₃	e ₄₃	e ₄₃	e ₄₅	e ₄₆	e ₄₇	e ₁₄	e ₂₄	e ₃₄
u ₁	–	+	+	–	–	+	–	–	–	–	–	–	–	–
u ₂	_T	_T	–	_T	_T	+	+	+	+	–	+	_T	+	_T
u ₃	–	+	–	–	–	–	–	–	–	–	⊥	_T	–	–
u ₄	+	+	+	_T	+	+	+	+	+	+	–	+	+	+
u ₅	+	–	+	–	⊥	–	–	–	–	_T	–	⊥	_T	–
u ₆	⊥	⊥	–	–	–	–	–	–	⊥	–	–	+	–	⊥

Table 6

The table representation

	e ₃₄	e ₃₅	e ₃₆	e ₃₇	e ₁₃	e ₂₃	e ₄₃	e ₄₃	e ₄₅	e ₄₆	e ₄₇	e ₁₄	e ₂₄	e ₃₄
u ₁	+	+	+	+	+	+	+	+	+	+	–	–	+	+
u ₂	+	+	_T	+	+	+	+	+	+	+	+	+	+	+
u ₃	_T	+	–	–	+	–	⊥	⊥	+	⊥	–	+	_T	_T
u ₄	+	+	+	+	+	+	+	+	+	+	+	+	+	+
u ₅	+	+	+	+	⊥	+	+	+	T	+	_T	+	+	+
u ₆	+	⊥	⊥	⊥	+	+	+	+	+	+	+	+	+	+

Step 3: In order to construct restricted and relaxed paraconsistent soft decision systems, firstly, we compute the restricted intersection and the relaxed intersection given in Table 4. Similarly, we compute restricted cross and relaxed cross given in Tables 5 and 6, respectively.

Step 4: Now, we can construct restricted paraconsistent soft decision system and relaxed paraconsistent soft decision system as Table 7 and Table 8.

Table 7

The restricted paraconsistent soft decision system

		$e_{3}^{d}$		$e_{4}^{d}$
u ₁	$u_{1} \in H (e_{3}^{+})$	+	$u_{1} \in H (e_{4}^{-})$	–
u ₂	n₊> n_- · u₂ ∈	+	$u_{2} \in H (e_{4}^{+})$	+
	$H (e_{3}^{⊥}) \cup H (e_{3}^{T})$
u ₃	$u_{3} \in H (e_{3}^{-})$	–	$u_{3} \in H (e_{4}^{-})$	–
u ₄	$u_{4} \in H (e_{3}^{+})$	+	$u_{4} \in H (e_{4}^{+})$	+
u ₅	n_-> n₊ · u₅ ∈	–	n_-> n₊ · u₅ ∈	–
	$H (e_{3}^{⊥}) \cup H (e_{3}^{T})$		$H (e_{4}^{⊥}) \cup H (e_{4}^{T})$
u ₆	$u_{6} \in H (e_{3}^{-})$	–	$u_{6} \in H (e_{4}^{+})$	+

Table 8

The relaxed paraconsistent soft decision system

		$e_{3}^{d}$		$e_{4}^{d}$
u ₁	$u_{1} \in H (e_{3}^{+})$	+	$u_{1} \in H (e_{4}^{-})$	–
u ₂	$u_{2} \in H (e_{3}^{+})$	+	$u_{2} \in H (e_{4}^{+})$	+
u ₃	$u_{3} \in H (e_{3}^{-})$	–	$u_{3} \in H (e_{3}^{+})$	+
u ₄	$u_{4} \in H (e_{3}^{+})$	+	$u_{4} \in H (e_{4}^{+})$	+
u ₅	$u_{5} \in H (e_{3}^{+})$	+	$u_{5} \in H (e_{3}^{+})$	+
u ₆	n_-> n₊ · u₆ ∈	+	$u_{6} \in H (e_{4}^{+})$	+
	$H (e_{3}^{⊥}) \cup H (e_{3}^{T})$

Step 5: Calculate choice value c_p3 and c_p4 and decision value d_p of and respectively. The results are shown in Table 9.

Table 9

Choice value and decision value in and


	c _p3	c _p4	d _p	c _p3	c _p4	d _p
u ₁	3/7	–1	–4/7	1	–2/7	5/7
u ₂	2/7	4/7	6/7	6/7	1	13/7
u ₃	–6/7	–5/7	–11/7	–3/7	2/7	–1/7
u ₄	6/7	6/7	12/7	1	1	14/7
u ₅	–4/7	–4/7	–8/7	6/7	5/7	11/7
u ₆	–5/7	1/7	–4/7	4/7	1	11/7

Step 6: From Table 9, we can get

Hence, when the market is bad, real estate companies should qualify higher and stricter standard. The investment company will invest in u₂, u₄ instead of u₁, u₃, u₅, u₆. Conversely, when the market is good, real estate companies may not be required so severely. The investment company will invest in u₁, u₂, u₄, u₅, u₆, instead of u₃.

If the investment company requires more complex strategies on the real estate companies, we can get:

According to the decision sets, it is clear that the real estate companies u₂, u₄ are given priority to be invested, and the real estate companies u₁, u₅, u₆ are the suboptimum to be invested, and the real estate company u₃ can be removed.

5 Conclusions

Soft set theory has been regarded as an effective mathematical tool to deal with uncertain information, and is widely applied in decision making. However, it is difficult to cope with problems with incomplete and inconsistent information simultaneously. In this paper, paraconsistent soft sets combining paraconsistent logic and soft sets was proposed. Moreover, N2- paraconsistent soft set, paraconsistent soft subsets, complement, “And”, restricted intersection, relaxed intersection, restricted cross and relaxed cross operations are defined on paraconsistent soft sets. In order to deal with incomplete and inconsistent information in decision making, we also defined paraconsistent soft decision system, choice value, decision value, the selected set, the eliminated set, and brought up the decision algorithm. Finally, we applied paraconsistent soft sets into the investment decision making. According to the example, we can conclude that the parameterization of paraconsistent soft sets is more adequate than that of classical soft sets. Without ignoring the incomplete and inconsistent information, parameterization decision algorithm can make a decision without losing important information. Therefore, paraconsistent soft sets can handle effectively decision making problems with incomplete and inconsistent information simultaneously.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant No. 71701116), Ministry of Education in China Project of Humanities and Social Sciences (Grant No. 15YJC630016) and the Subject of Philosophy and Social Sciences in Shanxi “Research on the construction of credit system in Shanxi Province under the background of Internet finance: credit collection model and assessment support”.

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∧				∨
	_T	+	–	⊥	_T	+	–	⊥
_T	_T	_T	–	–	_T	+	_T	+
+	_T	+	–	⊥	+	+	+	+
–	–	–	–	–	_T	+	–	⊥
⊥	–	⊥	–	⊥	+	+	⊥	⊥

∧				∨
	_T	+	–	⊥	_T	+	–	⊥
_T	_T	_T	–	–	_T	+	_T	+
+	_T	+	–	⊥	+	+	+	+
–	–	–	–	–	_T	+	–	⊥
⊥	–	⊥	–	⊥	+	+	⊥	⊥

A useful method for analyzing incomplete and inconsistent information: Paraconsistent soft sets and corresponding decision making methods

Abstract

Keywords

1 Introduction

2 Brief introduction to soft sets and paraconsistent logic

2.1 Soft sets

2.2 Paraconsistent logic

3.1 Concept of paraconsistent soft sets and operations

Table 2 Tabular representation of the paraconsistent soft set (F, {e1, e2}) e 1 e 2 u 1 + + u 2 – – u 3 + T u 4 – – u 5 ⊥ ⊥ u 6 – –

4 Application in an investment decision problem

Table 3 Tabular representation the paraconsistent soft sets (F, P) and (G, Q) (F, P) (G, Q) e 1 e 2 e 3 e 4 e 3 e 4 e 5 e 6 e 7 u 1 – + + – + – + + – u 2 T + T + + + + – + u 3 + – – ⊥ – T + – – u 4 + + + + + + + + T u 5 ⊥ T + T ⊥ + – + – u 6 + + ⊥ + – + ⊥ – –

Footnotes

Acknowledgments

References

Table 2
Tabular representation of the paraconsistent soft set (F, {e₁, e₂})

e ₁ e ₂

u ₁ + +

u ₂ – –

u ₃ + _T

u ₄ – –

u ₅ ⊥ ⊥

u ₆ – –

∧				∨
	_T	+	–	⊥	_T	+	–	⊥
_T	_T	_T	–	–	_T	+	_T	+
+	_T	+	–	⊥	+	+	+	+
–	–	–	–	–	_T	+	–	⊥
⊥	–	⊥	–	⊥	+	+	⊥	⊥