An approach to multiple attribute decision making based on linguistic value soft rough set and VIKOR method

Abstract

Linguistic variable is an effective method of representation the preferences of a decision-maker for inaccuracy available information in decision making under uncertainty. This article investigates a multiple attribute ranking decision making problem with linguistic preference by using linguistic value soft rough set. Firstly, we present the definition of linguistic value fuzzy set by introducing the concept of linguistic variable into the original Zadeh’s fuzzy set. We then define the concept of linguistic value soft set and the pseudo linguistic value soft set over the alternative set and parameter set of discourse. Moreover, we investigate the basic operators and the mathematical properties of the linguistic value soft set. Subsequently, we establish the rough approximation of an uncertainty concept with linguistic value over the object set and parameter set, i.e., the linguistic value soft rough set model. Meanwhile, we discuss several deformations of the linguistic value soft rough lower and upper approximations as well as some fundamental properties of the linguistic value soft approximation operators. With reference on the exploring of the fundamental of linguistic value soft rough set, we construct a new method for handling with the multiple attribute ranking decision making problems with linguistic information by combining the proposed soft rough set and the VIKOR method. Then, we give the detailed decision procedure and steps for the established decision approach. At last, an extensive numerical example is further conducted to illustrate the process of the decision making principle and the results are satisfactory. The main contribution of this paper is twofold. One is to provide a new model of granular computing by infusion the soft set and rough set theory with linguistic valued information. Another is to try making a new way to handle multiple attribute decision making problems based on linguistic value soft rough set and the VIKOR method.

Keywords

Rough set Soft set Linguistic variable Linguistic value soft approximation space VIKOR method

1 Introduction

Generally speaking, the procedure of decision making in socio-economic environment is focused on mankind with their inherent subjectivity, inaccuracy as well as fuzziness in expression of opinions. Some of real life decision making problems are substantially humanistic and therefor subjective essentially, but others are objective, and they are firmly embedded in an imprecise environment. Linguistic variable, introduced by Zadeh [71], has delivered effectiveness approach to deal with these kinds of uncertainty information in practices. Because the information expresses by the mankind could be non-numeric, partial evaluations, preferences and judgments as well as the weights are usually expressed linguistically [5, 66]. Linguistic variable presents a general effective approach to express often not clear-cut human preferences come cross in many practical cases. So far linguistic variable used by decision-makers to represent their individual preference or judgement when comparing decision alternatives in practice of decision making under uncertainty.

To effectively handle of the decision making problems under uncertainty of practice, several uncertainty theories and methodologies including theory of probability [64], fuzzy set [72], Dempster-Shafer theory of evidence [51], other generalized mathematical theories modelling uncertainty available information, and rough set theory which could be regarded as effectively mathematical tools have been established in the past decades years [45]. Moreover, many different new models and methodologies have been presented based on the above mentioned mathematical theories for modeling varieties decision making problems under uncertainty with available information [6 , 60]. Though many uncertainty mathematical theories and approaches have provided useful tool and method to handle of the decision making problems under uncertainty with available information, the existing theories could have the intrinsic insufficient since the insufficiency of the parametrization tool for the considered theories [40]. Consequently, Molodtsov [40] presents a new uncertainty mathematical theory named as soft set considered mathematical methodology for handling uncertainties which is free from the above insufficiencies.

Recently, another theory and methodology named as granular computing is introduced into multiple attribute decision making problems under uncertainty and presents several interesting and valuable models and methods [56]. Granular computing, established by Zadeh [73], as a new perspective and way to handle of the uncertain information has successfully applied to many areas. It presents a kind of theory, technique and tool that using the concept of information granules in the process of problem solving [57]. Rough set [45] is one of effective and important theories and methodologies of granular computing principle and has attracted many researchers and practitioners in the recent years. The original idea and principle of Pawlak rough set is assumed that objects are indiscernible which describing by same characteristics based on the available information data. Then the binary relations is constructed based on the indiscernibility relation on universe of discourse. Meanwhile, the binary classes form a partition over the considered universe and then form the knowledge granules [45]. Then, using the concept of approximate operators for any target set over universe of discourse, the valuable knowledge or information in information systems could be revealed, unrevealed and presented by means of the form of decision rules [59]. Subsequently, several generalized rough set theories [11 , 57] are proposed with different background and requirement in practice according to the granular computing principle. As a matter of fact, from the point of view of dealing with available information, soft set methodology also is an effectively theory and tool of granular computing.

So far there are a larger number of investigations about the original soft set and various improved versions as well as also many successfully applications on the fields of problems in practices under uncertainty in the past recently years, it can be easily concluded that the existing literatures both in the aspects of theoretical and application are based on the concept of Molodtsov soft set [40] and its various generalization of fuzzy forms. In general, many applications like the decision making problems under uncertainty about the available information in management sciences, however, the features of the objects or alternatives could not only be characterized by inaccuracy, imperfection or vague information but also could be expressed by linguistic variable. The linguistic variable provides a means of approximate characterization of phenomena which are too complex or too ill-defined to be amenable to description in conventional quantitative terms. The existing group decision-making approaches prove to be very useful when dealing with some complex, realistic decision-making problems in management science. In general, all decision-makers express their preference evaluations based on the same criteria for the problem considered in many of the group decision-making problems. There are also some limitations, however, when the evaluation criteria of an object are given differently by different decision-makers in the group decision-making process. In other words, earlier approaches could not handle decision-making problems which are characterized by different criteria related to different decision-makers. In order to illustrate the case of linguistic preference information in decision making of reality, we present a problem of decision making in practice: a problem of investment decision making in practice of reality. Suppose the investment enterprise that plans to invest some given funds in the best selection with considering four factors: the risk analysis; the analysis on the growth; the social-political affect analysis and the environmental affect analysis. In reality, the evaluation presented by the invited decision-makers for all candidate objects with respect to the concerned four features may be the linguistic preference information like the “poor”, “good”, “very good”, “extremely good” and etc. This means that the individual preference expression for the considered investment decision making problem expressed by linguistic variable about the concerned features. As aforementioned, there are substantially humanistic and subjective of reality decision making problems and the soft set-based approaches may incapable of handling with the problem of decision making related to linguistic preference information. So, research on another improved soft set theory with linguistic preference information is necessity and valuable.

Based on the above description for multiple attribute decision making problem with linguistic preference information and the basic principle of granular computing, this article attempts to give a new theory and approach to multiple attribute decision making problems by combining the linguistic value soft rough set and the VIKOR method. We first give the definition of linguistic value soft set by combining the linguistic variable with the original soft set. According to the basic framework and principle of rough set, we then establish the concept of linguistic value binary relation over the object set U of discourse and the parameter sets E, which can be seen as a directly generalization of the original binary relation on two different universes [54]. According to the concepts of linguistic value soft set and arbitrary binary relation, we define the notion of the linguistic value soft rough set by integrating the linguistic value soft set and the Pawlak rough set, i.e., the linguistic value soft rough set model. In fact, the linguistic value soft rough set could be regarded as a naturally extension of the existing soft rough set [54] according to linguistic value soft set. Meanwhile, we investigate the fundamental properties of theoretical aspect for the linguistic value soft rough set as well as the interrelationship between the new established model with the existing soft rough set models. Finally, we give an approach to multiple attribute ranking decision making problems based on the linguistic value soft rough set and VIKOR method. Moreover, the detailed process of decision making for the established model and method is presented. Furthermore, the model and method are verified by means of a numerical test case of the evaluation decision making problem of automatics.

The paper is organized as follows. Section 2 presents the literature review of soft sets and linguistic valued variables in decision-making problems. Section 3 presents several basic concepts of soft set theory, Pawlak rough set and linguistic variable as well as the corresponded operations. Section 4 gives the definition of linguistic value soft set and then builds the model of linguistic value soft rough set. Section 5 firstly introduce the basic method and idea of VIKOR, and then presents the model and methodology for multiple attribute ranking problem with linguistic evaluation using the linguistic value soft rough set and the method of VIKOR. At the same time, an illustrate numerical example is applied to test the effectiveness of the established approach in this paper. At last, Section 6 concludes the research and put out further study directions petroleum industry.

2 Literature review

2.1 Soft set

Molodtsov [40] presents a new uncertainty mathematical theory named as soft set considered mathematical methodology for handling uncertainties which is free from the above insufficiencies. There are two fundamental directions about the study of soft set theory: the basic theory and application in management sciences. We briefly review the existing literatures as follows. Maji et al. [35] discuss the operator laws among the soft set and then give an application in the problem of decision making with the established new results. Chen et al. [10] define a new soft set parametrization reduction by analyzing the conclusions given by Maji et al. [35]. The pioneering work of combining the theory of soft set with fuzzy set theory was presented by Maji et al. [36]. Subsequently, a large number of studies about the generalized fuzzy soft set concept were established [37, 62]. The combination of soft set theory and algebraic systems has invoked much attention and then several interesting and valuable results have published in this fields [21]. As far as the aspects of the application for the original soft set principle and idea in decision making problems are concerned, there are also many interesting results were proposed by scholars [12, 70]. Sun and Ma [54] present a general framework and procedure to decision making problems based on soft fuzzy rough set by improving of the insufficient of the existing literature. Cagman and Enginoglu [8, 9] define a new operator of soft matrix and its product for the original soft set theory and then give a kind of improved decision making model and its corresponded method based on the principle of union and intersection operators of any two sets of the universe of discourse. In fact, the key concept of the improved new decision making model is the soft max - min decision making function. Moreover, based on the new defined concept of soft max - min decision making function, the authors also investigate some basic properties and results of the original soft sets [40] and then present the definition of the operation for product of soft sets. Subsequently, an uni - int decision function is defined and then they propose an uni - int decision making method based on the product of soft sets [9]. The established decision making method by Cagman and Enginoglu [8] present an effective approach to find out the optimal alternatives from multiple evaluation attribute sets established by many different decision-makers. Garg [17] proposed a novel TOPSIS method based on correlation coefficient for solving decision-making problems with intuitionistic fuzzy soft set information. Furthermore, there are many investigation of soft rough set approaches and the corresponded applications in the problems of multiple attribute decision making problems under uncertainty by introducing the idea of rough set into soft set [54, 67].

2.2 Linguistic value varibles with decision-making problem

Until now, many theories and methods have been proposed to deal with the decision-making problems under the linguistic environment. The concept of linguistic variable was originally presented by Zadeh [71, 72]. Generally speaking, the linguistic variable and linguistic value are expressed by using some phrases or sentences in a way of natural or artificial language. To portray the qualitative information in a more precise way, researchers have proposed more powerful linguistic computational models. Xu [61] proposed a general definition to express the linguistic variable in group decision-making. Herrera and Martinez [14] proposed 2-tuple linguistic model and use this fuzzy linguistic representation model in a decision-making problem. Zhang [76] proposed the linguistic intuitionistic fuzzy set (LIFS), which is produced by combining IFS with linguistic term set (LTS). Rodriguez et al. [47] present the concept of HFLTS based on the fuzzy linguistic approach. Pang, Wang, and Xu [44] proposed a novel concept called (PLTS) to serve as an extension of the existing tools in multi-attribute group decision making. Garg [15] put forward the linguistic Pythagorean fuzzy sets (LPFSs) in multiattribute decision making process. More and more, many researchers put forward corresponding decision-making methods based on the existing decision-making environment and traditional decision-making methods. Lin [31] set in Multi-attribute group decision-making. Lin [27] proposed a novel linguistic Pythagorean fuzzy TOPSIS (LPF-TOPSIS) method is proposed to solve multiple attribute decisionmaking problems. The traditional TODIM (an acronym in Portuguese for interactive multi-criteria decision making) method is extended to handle the HFLTSs based on the novel comparison function and distance measure [30]. Liu [32] define a new approach to propose a linguistic connection number and a cosine distance measure between the new LCNs to develop the TOPSIS method to the proposed cosine distance measure. A novel probabilistic linguistic ELECTRE II method is put forward to deal with the edge node selection problem [26]. Lin [28] introduced the concept of probabilistic linguistic term sets, probuced a novel probabilistic linguistic best-worst (PLBW) and a novel integrated probabilistic linguistic MCDM model based on TODIM method is proposed to rank IoT platforms. Lin [29] proposed a general form of these two linguistic orthopair fuzzy sets to deal with the multiattribute group decision-making problem. Garg [19] propose the concept of a linguistic interval-valued Atanassov intuitionistic fuzzy and apply to Group Decision Making Problems. Subsequently, Garg [16] present a novel concept of linguistic interval-valued Pythagorean fuzzy set and a multiple attribute group decision-making (MAGDM) algorithm. According to the literatures related to the existing generalized soft set models and methodologies for decision making problems under uncertainty with linguisitic valued information, evaluation information may be incomplete because the fuzziness of human’s thinking and uncertain circumstance. We present a new generalization of original soft set theory by introducing linguistic preference information into the concept soft set under the framework of Pawlak rough set, i.e., the linguistic value soft rough set. furthermore, we establish a new approach to multiple attribute decision making based on the combination of linguistic value soft rough set and VIKOR method.

3 Preliminaries

In this section, we will present several related definitions of the soft set [40], Pawlak rough set [45] and the linguistic variable [71, 72] as well as its basic operations.

3.1 Molodtsov soft set theory

Definition 2.1. [40] Suppose U be a set or universe of all objects or alternatives. E is the parameters set which related to the elements of U . For any A ⊆ E . We call the ordered pairs S = (F, A) a soft set on universe U, where F is defined as: F : A → P (U) . It is easy to know that a soft set S = (F, A) of U is also a parameterized family of subsets of U, which presents a directly characterization for every objects in U . Furthermore, Maji et al. [35] present another explanation for soft set as follows. For a given parameter ε ∈ A, the subset F (ε) ⊆ U can be interpreted as a set of ε-approximate objects about soft set S = (F, A) . Obviously, there may be empty for F (ε) about the given parameter ε ∈ A . Based on this definition, the following basic operations for soft set are clear [35 , 46]. Suppose that (F₁, A) , (F₂, B) be two soft sets on U . Define (1) (F₁, A) ⊆ (F₂, B) , if A ⊆ B and F₁ (ε) = F₂ (ε) for any ε∈ A ; (2) (F₁, A) = (F₂, B) , if (F₁, A) ⊆ (F₂, B) and (F₁, A)⊇ (F₂, B) ; (3) (F₁, A)∩ (F₂, B) = (F₃, C) ; Where C = A ∩ B and F₃ (ε) = F₁ (ε) or F₂ (ε) ∀ε∈ C ; (4) (F₁, A)∪ (F₂, B) = (F₃, C) ; Where C = A ∪ B and ∀ε ∈ C, there has F₃ (ε) = F₁ (ε) when ε ∈ A - B, F₃ (ε) = F₂ (ε) when ε ∈ B - A and F₃ (ε) = F₁ (ε) ∪ F₂ (ε) when ε ∈ A ∩ B .

3.2 Rough set

The concept of equivalence relation over a universe U is the key issue in Pawlak rough set [45]. Here we firstly introduce the definition of binary equivalence relation. Suppose that U be a finite universe. R is any binary relation over universe U . Then R generates a partition over universe U, stands for [x] _R (for short [x]). We use U/R = {[x] |x ∈ U} expression of all binary relation classes of x induced by the binary relation R . We then call (U, R) an approximation space. Then the rough approximation of any target set about the approximation space (U, R) can be given. Suppose that (U, R) be an approximation space. Given arbitrary X ⊆ U, the lower and upper approximations of X about (U, R) are presented as follows: $\underline{R} (X) = {x \in U | [x]_{R} \subseteq X},$

$\bar{R} (X) = {x \in U | [x]_{R} \cap X \neq \emptyset} .$ Based on the definition of the lower and upper approximations of any object X, one can easily present the regions of positive, boundary and negative of X as follows, respectively: $Pos (X) = \underline{R} (X)$

$Bn (X) = \bar{R} (X) - \underline{R} (X)$

$Neg (X) = U - \bar{R} (X)$

3.3 Linguistic value and linguistic variable

The concept of linguistic variable was originally presented by Zadeh [71, 72]. Generally speaking, the linguistic variable and linguistic value are expressed by using some phrases or sentences in a way of natural or artificial language. The way of the expression for individual preference using linguistic variable gives an effective manner to approximate characterization of phenomena that are too complex or too ill-defined to be amenable to characteristic in traditional quantitative terms [69]. For instance, a description of some different cars with the characters of the temperature”, “beauty” or “style”, the evaluation given by individual decision-makers could be “good”, “fair”, “poor”, “very fast”, “fast”, “slow”, etc., but not the quantitative terms. In reality decision making, the linguistic variables are an approximate method for expressing the decision-makers’ individual uncertainty preference information on objects. Xu [61] defines a set L = {l_α|α = - t, ⋯ , -1, 0, 1, ⋯ , t} with odd cardinality as a general definition to express the linguistic variable. Here the symbol l_α stands for a likely value of the linguistic variable. Particularly, l_-t and l_t, respectively, are the minimum and maximum bounds, and t is a given positive integer. In general, a set with 9-terms L could be L = {l_-4 = extremely poor, l_-3 = very poor, l_-2 = poor, l_-1 = slightly poor, l₀ = fair, l₁ = slightly good, l₂ = good, l₃ = very good, l₄ = extremely good} . At the same time, it can be easily proofed that the following results satisfy about the linguistic terms [55, 61]: (1) l_α≥ l_β ⇔ α ≥ β ;

(2) neg(l_a)= l_-α, Particularly, neg(l₀)= l₀ ; (3) Max operator: max (l_α, l_β) = l_α if l_α≥ l_β ;

(4) Min operator: min (l_α, l_β) = l_α if l_α ≤ l_β . In fact, the results of the handling procedure for a considered management decision problem with preference information could not adapt for the given label in L well. Then, for convenient the computing and maintain the original related preference information, an improved version of the original L named as continuous term set is presented as $\bar{L} = {l_{α} | l_{- q} \leq l_{α} \leq l_{q}, α \in [- q, q]}$ (Here we convention that q (q > t) and q is defined as a positive integer) that the continuous term set also satisfies the above properties (1) - (4) [61]. If l_α ∈ L, we call l_α the original term and α the original term index. At the same time, we call l_α the virtual term. Moreover, α also is called the virtual term index. Considered a term set $\bar{L},$ for any $l_{α}, l_{β} \in \bar{L},$ and μ, μ₁, μ₂ ∈ [0, 1] , the following properties are hold [55 , 61]. (1) l_α⊕ l_β = l_α+β ;

(2) l_α⊕ l_β = l_β ⊕ l_α ; (3) μl_α = l_μα ; (4) (μ₁+ μ₂) l_α = μ₁l_α ⊕ μ₂l_α ;

(5) μ (l_α ⊕ l_β) = μl_α ⊕ μl_β .

4 Linguistic value soft rough set model and its properties

This section we firstly present the definition of linguistic value soft set and its basic operation as well as some properties. Then we investigate the linguistic value soft rough set model systematically.

4.1 Linguistic value soft set

The motivation for the definition of the linguistic value soft set comes from multiple attribute group decision making under uncertainty. As aforementioned, there are many evaluations for the candidate object given by experts or decision-makers are inherently non-numeric and then using the linguistically expression is a practicable way in reality. The idea of defining the linguistic value soft set is introduced the concept of linguistic variable into Molodtsov soft set theory [56]. In following, we introduce the fundamental definition and the operator laws for the linguistic value soft set. Considering an additive discrete linguistic label set L = {l_α|α = - t, ⋯ , -1, 0, 1, ⋯ , t} . For the sake of convenience, we suppose that linguistic label set L is odd cardinality when there are not confusion arising in the subsequent sections. We then present the definition for the linguistic value fuzzy set of universe. As is well know, fuzzy set [72] is defined as any binary mapping between the universe of discourse and the interval [0, 1] . Clearly, the linguistic value fuzzy set is a generalization of the Zadeh’s fuzzy set with the domain of linguistic variable L, i.e., a linguistic value fuzzy set of U is defined as: A (•) : U ⟶ L, where A (u) denotes the membership of object u (u ∈ U) about the linguistic value fuzzy set A . Here F^L (U) denotes all linguistic value fuzzy subsets on U . Meanwhile, one can easily verified that there are (A ∪ B) (u) = max {A (u) , B (u)} , (A ∩ B) (u) = min {A (u) , B (u)} and A^c (u) = neg (A (u)) for any u ∈ U based on the operation principle of linguistic variable presented in Section 3.3.

Definition 3.1. Suppose U be object set and E be the parameter set that related to the elements of U . Let L be a linguistic label set. We call S_l = (F_l, E) a linguistic value soft set on U, where F is a binary mapping between parameters set E and the linguistic label set L, That is, F_l : E → F^L (U) , where F_l (e) ∈ F^L (U) . Similar to the original soft set [40], the concept of linguistic value soft set also is not the traditional concept of set. Like the concept of original soft set, it is easy to know that linguistic value soft set S_l = (F_l, E) is also a parameterized family of linguistic label subsets of U . Given arbitrary e ∈ E, F_l (e) can be interpreted as the set of e-approximation objects for the linguistic value soft set S_l = (F_l, E) . At the same time, based on the definition F_l (e) (u) ∈ L ∀e ∈ E, u ∈ U, we known that it means the evaluation of linguistic value given by the decision-maker for the decision object u over universe U about the parameter (or attribute) e of parameters set E . Moreover, we easy to known that F_l (e) (u) ^c =neg(F_l (e) (u)) ∈ L (here c stands for the complement of a set with respect to the universe of discourse). To illustrate the above concept, we present the following example.

Example 3.1. Considering an evaluation decision systems of automatics. Suppose U = {u₁, u₂, u₃, u₄, u₅, u₆} be the set of six different cars. E = {e₁, e₂, e₃, e₄, e₅} be the parameter set. Every parameters are presented as a piece of phrase or a short sentence, where e₁ denotes Max-Speed,e₂ denotes Comfort,e₃ denotes Design, e₄ denotes Mileage,e₅ denotes Price. Based on the considered parameters, the individual preference expression for every car about all parameters presented with the decision-makers or invited experts using the linguistic variable. Assume that the used linguistic label set is as follows: L = {l_-4 = extremely poor, l_-3 = very poor, l_-2 = poor, l_-1 = slightly poor, l₀ = fair, l₁ = slightly good, l₂ = good, l₃ = very good, l₄ = extremely good} . So, the soft set S_l = (F_l, E) represents the “attractiveness of the car” which the customers are going to buy. Then the representation for the illustrated example is described as a form of Table 1.

Table 1
A linguistic value soft set

U∖E e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ l ₂ l _-3 l ₀ l ₁ l _-2

u ₂ l ₂ l ₃ l ₁ l ₀ l _-4

u ₃ l _-1 l ₀ l ₂ l ₂ l ₁

u ₄ l ₃ l ₁ l _-1 l ₀ l ₂

u ₅ l _-4 l _-1 l ₂ l ₀ l ₁

u ₆ l ₀ l ₂ l ₃ l ₁ l ₃

U∖E	e ₁	e ₂	e ₃	e ₄	e ₅
u ₁	l ₂	l _-3	l ₀	l ₁	l _-2
u ₂	l ₂	l ₃	l ₁	l ₀	l _-4
u ₃	l _-1	l ₀	l ₂	l ₂	l ₁
u ₄	l ₃	l ₁	l _-1	l ₀	l ₂
u ₅	l _-4	l _-1	l ₂	l ₀	l ₁
u ₆	l ₀	l ₂	l ₃	l ₁	l ₃

By Definition 3.1 (Here the way of representation for linguistic value soft set refers to the fuzzy set theory, the + does not mean the sum), the linguistic value mapping $F_{l} (e_{1}) = \frac{l_{2}}{u_{1}} + \frac{l_{2}}{u_{2}} + \frac{l_{- 1}}{u_{3}} + \frac{l_{3}}{u_{4}} + \frac{l_{- 4}}{u_{5}} + \frac{l_{0}}{u_{6}}$ means that the evaluation for car u₁ and u₂ are “good”, car u₃ is “slightly poor”, car u₄ is “very good”, car u₅ is “extremely poor” and car u₆ is “fair” given by expert or customer (In fact, here the mapping F between the set E and U also could be interpreted as the invited experts or agents). Meanwhile, F_l (e₁) (u₄) = l₃ ∈ S will be interpreted as “very good” of the evaluation for car u₄ given by the expert or agent F according to the index “Max-Speed” e₁ . Furthermore, F_l (e₁) (u₃) = l_-1 ≤ l₃ = F_l (e₁) (u₄) will be interpreted that the evaluation for car u₄ is superior to car u₃ determined by decision-maker F according to the attribute “Max-Speed” e₁ . Other mappings also could make a similar explanation. Clearly, there are F_l (e_i) ⊆ L, (i = 1, 2, 3, 4, 5) hold, Then, the linguistic value soft set S_l = (F_l, E) could be make a similar interpretation as the original soft set concept [40].

Remark 3.1. Generally speaking, if the original linguistic term set L only includes two opposite preferences such as “good” and “not good” and we labeled it by 0 and 1, respectively. Then, we can easily see that the linguistic value soft set will changes into the original soft set [40]. That is, the linguistic value soft set given in Definition 3.1 is a natural generalization of the original soft set concept [40]. According to the interpretation for the concept of linguistic value soft set, we investigate some mathematic properties for the linguistic value soft set in the follows.

Definition 3.2. Suppose U be the set of all objects. E is the parameter set which related to the elements of U . Assume that L be linguistic label set. Given any two linguistic value mappings F and G over E and L . ∀e ∈ E, there are F_l (e) ≤ G_l (e) ⇔ F_l (e) (u) ≤ G_l (e) (u) for any u ∈ U . In what of follows, we establish some basic operation laws for the new defined concept in Definition 3.1.

Definition 3.3. Suppose U be the set of all objects. E is the parameter set which related to the elements of U . Assume that L be linguistic label set. For any A, B ⊆ E, then (F_l, A) and (G_l, B) are the linguistic value soft sets of U, we call (F_l, A) a linguistic value soft subset of (G_l, B) , denote (F_l, A) ⊆ (G_l, A), when the following conditions are satisfied: (i) A ⊆ B, (ii) ∀e ∈ A, u ∈ U, F_l (e) (u) ≤ G_l (e) (u) . Obviously, we can easy to derive the definition of equality for any two linguistic value soft sets based on the former definitions, i.e., given any linguistic value soft sets (F_l, A) and (G_l, B) (A, B ⊆ E), (F_l, A) and (G_l, B) are called equal if and only if (F_l, A) is the linguistic value soft subset of (G_l, B) and (G_l, B) is a linguistic value soft subset of (F_l, A) .

Definition 3.5. Suppose U be the set of all elements. E is the parameter set which related to the objects of U . Let L be linguistic term set. ∀A ⊆ E, then the linguistic value soft complement set of (F_l, A) is assigned as (F_l, A) ^c = (F_l, ¬ A) (where ¬ stands for the not operation of any set) and F_l (e) (u) ^c =neg(F_l (e) (u)) for any e ∈ A, u ∈ U .

Definition 3.6. Suppose U be the set of all objects. Let E be the parameter set that related to the elements of U . Assume that L be linguistic lable set and A ⊆ E, if (1) ∀e ∈ A, u ∈ U, F_l (e) (u) = l₀, (F_l, A) is called linguistic value null soft set, denote as (∅ _l, A); (2) ∀e ∈ A, u ∈ U, F_l (e) (u) = l_|q| with $l_{- q}, l_{q} \in \bar{L},$ (F_l, A) is called linguistic value absolute soft set, denote as (I^L, A). In general, the definition of linguistic value null soft set often used to express the attitude of which there is not any preference with respect to the considered attributes related to the invited expert. Meanwhile, the linguistic value absolute soft set often used to express the attitude of which there is a completely opposite attitude with respect to the considered attributes related to the invited expert. Subsequently, we investigate the operation of INTERSECTION and UNION for the linguistic value soft set.

Definition 3.7. Suppose U be the set of all objects. E is the parameter set that related to the objects of U . Assume that L be linguistic term set. ∀A, B ⊆ E, defining that (i) (H_l, C) = (F_l, A) ⋃ (G_l, B) (where C = A ⋃ B). Here we convention F_l (e) (u) = s₀ with ∀e ∈ E, u ∈ U when there is e ∉ A ⊆ E . (ii) We define (H_l, C) = (F_l, A) ⋂ (G_l, B) (where C = A ⋂ B). Clearly, according to the Definition 3.7, the following formulation for the INTERSECTION and UNION can be easily obtained, respectively. H_l (e) (u) = (F_l, A) ∪ (G_l, B) = max(F_l (e) (u) , G_l (e) (u)) , ∀ e ∈ A ∪ B, u ∈ U, H_l (e) (u) = (F_l, A) ∩ (G_l, B) = min(F_l (e) (u) , G_l (e) (u)) , ∀ e ∈ A ∪ B, u ∈ U . Next, we give the linguistic value soft rough set based on the concept established in Definition 3.1.

4.2 Linguistic value soft rough set

As we know, any binary relation of a universe is the key issue in original Pawlak rough set theory and other generalized rough set theory models. So, we should define arbitrary binary relation between universe of objects U and the set of parameters E under the environment of linguistic variable. Then we will present the soft rough approximation operators for any object with linguistic expression based on the defined binary relation between U and E with linguistic variable. From the Definition 3.1, we know that every object owning some specific features over universe U were characterized by selecting the evaluation value with the linguistic variable for given parameter e ∈ E . Like the Example 3.1, the object set $F_{l} (e_{1}) = \frac{l_{2}}{u_{1}} + \frac{l_{2}}{u_{2}} + \frac{l_{- 1}}{u_{3}} + \frac{l_{3}}{u_{4}} + \frac{l_{- 4}}{u_{5}} + \frac{l_{0}}{u_{6}}$ shows that the evaluation established with decision-maker (or expert) F for car u₁ and u₂ are “good”, car u₃ is “slightly poor”, car u₄ is “very good”, car u₅ is “extremely poor” and car u₆ is “fair” with respect to the parameter e₁ (Max-Speed). Conversely, for a concretely object u_i ∈ U, one can also obtain the evaluations about the parameters in E with the linguistic variable. For example, the evaluation of Max-Speed e₁ ∈ E is “good (l₂)” and the evaluation of Comfort e₂ ∈ E is “very poor (l_-3)” for object u₁ ∈ U, and a similar description for object u₁ ∈ U can be given under the other parameters of E . That is, for a specific object u_i ∈ U, decision-maker would like to want what is the characters owned by object u_i ∈ U ? This consideration will give another perspective to explore the linguistic value soft set. Thus, we give the definition of pseudo linguistic value soft set on universe of discourse in following.

Definition 3.8 Suppose U be the set of all objects and E be the parameter set that related to the objects of U . Let L be linguistic term set. We call the ordered pairs $S_{l}^{- 1} = (F_{l}^{- 1}, E)$ a pseudo linguistic value soft set on U if and only if $F_{l}^{- 1} : U ⟶ F^{L} (E),$ i.e., $F_{l}^{- 1}$ is the binary mapping from U to set E with the domain of linguistic term set L . Obviously, we can easy to obtian that a pseudo linguistic value soft set on U is a parameterized family of subsets of universe U . ∀ɛ ∈ E, $F_{l}^{- 1} (ɛ)$ can be regarded as the ɛ-approximation objects of a pseudo linguistic value soft set $S_{l}^{- 1} = (F^{- 1}, E) .$ Such as in Example 3.1, $F_{l}^{- 1} (u_{1}) = \frac{l_{2}}{e_{1}} + \frac{l_{- 3}}{e_{2}} + \frac{l_{0}}{e_{3}} + \frac{l_{1}}{e_{4}} + \frac{l_{- 2}}{e_{5}}$ provides a description of car u₁ under all parameters in E with the linguistic variable given by decision-maker (or expert) F . Furthermore, from the Definition 3.1 and 3.7, there are F_l (e) (u) ∈ F^L (U) and $F_{l}^{- 1} (u) (e) \in F^{L} (E)$ for arbitrary e ∈ E and u ∈ U . Meanwhile, there are e₀ ∈ E and u₀ ∈ U and satisfy $F_{l} (e_{0}) (u_{0}) = F_{l}^{- 1} (u_{0}) (e_{0})$ because both them have the same domain L . However, there is completely different meanings for the decision making process in practice. Therefore, it can easily to see that the linguistic value soft set S_l = (F_l, E) and pseudo linguistic value soft set $S_{l}^{- 1} = (F_{l}^{- 1}, E)$ of universe U give the completely opposite perspective to characterize the relationship between the objects in U and the parameter in E . Actually, according to the definition and its properties, both the linguistic value soft set S_l = (F_l, E) and pseudo linguistic value soft set $S_{l}^{- 1} = (F_{l}^{- 1}, E)$ of universe U may be considered as a kind of linguistic value binary relation between object set U and parameters set E, respectively. So, we give the definition of linguistic value soft approximation space.

Definition 3.9. Let S_l = (F_l, E) be linguistic value soft set on universe U . Denote I = (U, E, S_l) and we call I the linguistic value soft approximation space. Meanwhile, we also call $I^{- 1} = (U, E, S_{l}^{- 1})$ the linguistic value soft approximation space and there are not confusion arising with I = (U, E, S_l) in the context. With reference to the linguistic value soft approximation space, we give the rough approximation of a linguistic value fuzzy set on universe in the following. Let I = (U, E, S_l) be a linguistic value soft approximation space. Given arbitrary linguistic value fuzzy set A ∈ F^L (U) with u ∈ U and e ∈ E, the lower and upper approximations of A about (U, E, S_l) , respectively, are defined as follows. ${\underline{apr}}_{L} (A) (e) = min_{u \in U} max {neg (F_{l} (e) (u)), A (u)},$ ${\bar{apr}}_{L} (A) (e) = max_{u \in U} min {F_{l} (e) (u), A (u)} .$ It is easy to verify that both ${\underline{apr}}_{L} (A)$ and ${\bar{apr}}_{L} (A)$ are the linguistic value fuzzy set of the parameters set E . Then the set-pair $({\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A))$ is called the linguistic value soft rough set and ${\underline{apr}}_{L}$ and ${\bar{apr}}_{L} : F^{L} (U) ⟶ F^{L} (E)$ are regarded as linguistic value lower and upper approximation operators, respectively.

Example 3.2. Such as in Example 3.1, we can obtain F_l (e₁) (u₁) = l₂ ∈ S, suppose $A = \frac{l_{2}}{u_{1}} + \frac{l_{2}}{u_{2}} + \frac{l_{1}}{u_{3}} + \frac{l_{0}}{u_{4}} + \frac{l_{1}}{u_{5}} + \frac{l_{1}}{u_{6}}$ . Then, we have

${\underline{apr}}_{L} (A) (e_{1}) = min_{u \in U} max {neg (F_{l} (e_{1}) (u)), A (u)} = min {max {l_{- 2}, l_{2}}, max {l_{- 2}, l_{2}}, max {l_{1}, l_{1}}, max {l_{- 3}, l_{0}}, max {l_{0}, l_{1}}} = min {l_{2}, l_{2}, l_{1}, l_{0}, l_{1}} = l_{0}$ ${\bar{apr}}_{L} (A) (e_{1}) = max_{u \in U} min {F_{l} (e_{1}) (u), A (u)} = max {min {l_{2}, l_{2}}, max {l_{2}, l_{2}}, max {l_{- 1}, l_{1}}, max {l_{3}, l_{0}}, max {l_{0}, l_{1}}} = max {l_{2}, l_{2}, l_{- 1}, l_{0}, l_{0}} = l_{2}$

Furthermore, because the approximation space I = (U, E, S_l) is defined according to the linguistic value soft set S_l = (F_l, E) of universe of discourse U, thus the linguistic value soft rough set $({\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A))$ could be referred as the generalized rough set based on linguistic value soft set S_l = (F_l, E) , or we call $({\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A))$ the generalized rough set on two universes according to linguistic value binary relation S_l = (F_l, E) . Moreover, we have defined another linguistic value soft approximation space $I^{- 1} = (U, E, S_{l}^{- 1})$ based on the pseudo linguistic value soft set $S_{l}^{- 1} = (F_{l}^{- 1}, E)$ over U . Thus, we may define another type of linguistic value soft rough set with respect to linguistic value soft approximation space $I^{- 1} = (U, E, S_{l}^{- 1})$ as the similar way. Let $I^{- 1} = (U, E, S_{l}^{- 1})$ be a linguistic value soft approximation space. Given arbitrary linguistic value fuzzy set A ∈ F^L (E) of parameters set E and u ∈ U, e ∈ U, the lower and upper approximations of A about $(U, E, S_{l}^{- 1}),$ respectively, are defined as follows. ${\underline{apr}}_{L}^{- 1} (A) (u) = min_{e \in E} max {neg (F_{l}^{- 1} (u) (e)), A (e)},$ ${\bar{apr}}_{L}^{- 1} (A) (u) = max_{e \in E} min {F_{l}^{- 1} (u) (e), A (e)} .$ It is easy to verify that both ${\underline{apr}}_{L}^{- 1} (A)$ and ${\bar{apr}}_{L}^{- 1} (A)$ are the linguistic value fuzzy sets of the object set U . Then we call the pair $({\underline{apr}}_{L}^{- 1} (A), {\bar{apr}}_{L}^{- 1} (A))$ the linguistic value soft rough set and ${\underline{apr}}_{L}^{- 1}$ and ${\bar{apr}}_{L}^{- 1} : F^{L} (E) ⟶ F^{L} (U)$ are regarded as linguistic value lower and upper approximation operators, respectively. Because there are clearly definitions and meanings for $({\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A))$ and $({\underline{apr}}_{L}^{- 1} (A), {\bar{apr}}_{L}^{- 1} (A)),$ we then don’t distinguish the name and call both of them linguistic value soft rough set in the following context.

Remark 3.2. In general, because F : E ⟶ F^L (U) and F^-1 : U ⟶ F^L (E) are arbitrary binary mappings (or binary relations) between U and E, then they are not satisfy the reflexive, symmetric and transitive. So, the relationship ${\underline{apr}}_{L} (A) \subseteq {\bar{apr}}_{L} (A), {\underline{apr}}_{L}^{- 1} (A) \subseteq {\bar{apr}}_{L}^{- 1} (A)$ could not hold for any u ∈ U and e ∈ E . Furthermore, from the above definition, the following conclusions could be easily derived.

Remark 3.3. If |L|=2 (where | • | denotes the cardinality of a set), i.e., there are only two elements in term set L and they are opposite each other such as L = {l₀, l₁} (for example l₁ =good and l₀ =not good). Then both the linguistic value soft set S_l = (F_l, E) and pseudo linguistic value soft set $S_{l}^{- 1} = (F_{l}^{- 1}, E)$ of universe U are degenerated into the original soft set [40]. Therefore, the linguistic value soft approximation space I = (U, E, S_l) and $I^{- 1} = (U, E, S_{l}^{- 1})$ are degenerated into soft approximation space (Sun and Ma, 2014). Moreover, the approximated object set A will be a crisp set of the object set U or the parameters set E, i.e., A ∈ P (E) or A ∈ P (U) (where P (U) denotes whole subsets of universe U). So, we have the following results: ${\underline{apr}}_{L} (A) = {u \in U | \exists e \in E, u \in F_{l} (e) \subseteq A},$ A∈ P (U) ; ${\bar{apr}}_{L} (A) = {u \in U | \exists e \in E, u \in F_{l} (e) \cap A \neq \emptyset},$ A∈ P (U) ; and ${\underline{apr}}_{L}^{- 1} (A) = {e \in E | \exists u \in U, e \in F_{l}^{- 1} (u) \subseteq A},$ A∈ P (E) ; ${\bar{apr}}_{L}^{- 1} (A) = {e \in E | \exists u \in U, e \in F_{l}^{- 1} (u) \cap A \neq \emptyset},$ A ∈ P (E) . We can easy to find out that ${\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A) \in P (E)$ and ${\underline{apr}}_{L}^{- 1} (A), {\bar{apr}}_{L}^{- 1} (A) \in P (U),$ i.e., both of them are crisp set on the given universe. In this case, the linguistic value soft rough set $({\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A))$ and $({\underline{apr}}_{L}^{- 1} (A), {\bar{apr}}_{L}^{- 1} (A))$ are degenerated into the soft rough set model or it could be referred as the original model of rough set on two universes (Wong et al. 1993; Yao et al. 1995). According to the above analogizing, we could easily know that linguistic value soft rough set has extended the original soft set [40] well and included the other existing generalized soft set models established by scholars [12, 67], and also has explored a new perspective to integrating the Molodtsov soft set with the existing uncertainty theories and applied to handle of the decision making problems under uncertainty. At the same time, the approximation operations ${\underline{apr}}_{L} (A), {\bar{apr}}_{L} (A)$ and ${\underline{apr}}_{L}^{- 1} (A), {\bar{apr}}_{L}^{- 1} (A)$ are given on two universes (i.e., the object sets U and E), thus the linguistic value soft rough set also can be regarded a new kind of rough set over two universes. So, we obtain another way to discuss the generalized rough set with the framework of two universes. Therefore, we explanation the interrelationship among linguistic value soft rough set model and the other generalized soft set models [10 , 70] as well as the existing generalized rough set model. For the linguistic value approximation operators, the following properties are easily derived.

Theorem 3.1. Suppose I = (U, E, S_l) be linguistic value soft approximation space. Given arbitrary A₁, A₂ ∈ F^L (U) , there are (1) ${\underline{apr}}_{L} (A_{1})^{c} = {\bar{apr}}_{L} (A_{1}^{c}),$

${\bar{apr}}_{L} (A_{1})^{c} = {\underline{apr}}_{L} (A_{1}^{c}),$ (2) ${\underline{apr}}_{L} (A_{1} \cap A_{2}) = {\underline{apr}}_{L} (A_{1}) \cap {\underline{apr}}_{L} (A_{2}),$

${\bar{apr}}_{L} (A_{1} \cup A_{2}) = {\bar{apr}}_{L} (A_{1}) \cup {\underline{apr}}_{L} (A_{2});$ (3) ${\underline{apr}}_{L} (A_{1} \cup A_{2}) \supseteq {\underline{apr}}_{L} (A_{1}) \cup {\underline{apr}}_{L} (A_{2}),$

${\bar{apr}}_{L} (A_{1} \cap A_{2}) \subseteq {\bar{apr}}_{L} (A_{1}) \cap {\underline{apr}}_{L} (A_{2});$ (4) If A₁ ⊆ A₂, then ${\underline{apr}}_{L} (A_{1}) \subseteq {\underline{apr}}_{L} (A_{2}),$

${\bar{apr}}_{L} (A_{1}) \subseteq {\bar{apr}}_{L} (A_{2}) .$ Similar to the results in Theorem 3.1, the following results can be easily given. I = (U, E, S_l) be linguistic value soft approximation space. Given arbitrary A₁, A₂ ∈ F^L (U) , there are (1) ${\underline{apr}}_{L}^{- 1} (A_{1})^{c} = {\bar{apr}}_{L}^{- 1} (A_{1}^{c}),$

${\bar{apr}}_{L}^{- 1} (A_{1})^{c} = {\underline{apr}}_{L}^{- 1} (A_{1}^{c}),$ (2) ${\underline{apr}}_{L}^{- 1} (A_{1} \cap A_{2}) = {\underline{apr}}_{L}^{- 1} (A_{1}) \cap {\underline{apr}}_{L}^{- 1} (A_{2}),$

${\bar{apr}}_{L}^{- 1} (A_{1} \cup A_{2}) = {\bar{apr}}_{L}^{- 1} (A_{1}) \cup {\underline{apr}}_{L}^{- 1} (A_{2});$ (3) ${\underline{apr}}_{L}^{- 1} (A_{1} \cup A_{2}) \supseteq {\underline{apr}}_{L}^{- 1} (A_{1}) \cup {\underline{apr}}_{L}^{- 1} (A_{2}),$

${\bar{apr}}_{L}^{- 1} (A_{1} \cap A_{2}) \subseteq {\bar{apr}}_{L}^{- 1} (A_{1}) \cap {\underline{apr}}_{L}^{- 1} (B_{2});$ (4) If A₁ ⊆ A₂, then ${\underline{apr}}_{L}^{- 1} (A_{1}) \subseteq {\underline{apr}}_{L}^{- 1} (A_{2}),$

${\bar{apr}}_{L}^{- 1} (A_{1}) \subseteq {\bar{apr}}_{L}^{- 1} (A_{2}) .$ The proof of Theorem 3.1 and Theorem 3.1’ can be obtained directly according to the above definitions.

5 The multiple attribute ranking decision making method based on linguistic value soft rough set and VIKOR

Based on the theoretical aspect investigation about the linguistic value soft rough set, this section will establish a new method and model for multiple attribute ranking decision making problem by integrating the linguistic value soft rough set with VIKOR method [41]. As is well known, both the soft set theory and rough set theory are the effectively mathematical theory to deal with decision making problems under uncertainty. Meanwhile, the combination of soft set theory and rough set theory provide a new approach and model to deal with the uncertain decision making problems under the framework of granular computing. In order to ensure the applicability and effectiveness as well as the fusion between the granular computing model and the original multiple attribute decision making models and methodologies [52], we introduce the original VIKOR method [41] into the linguistic value soft rough set model and then establish a new approach to multiple attribute decision making problem with linguistic preference information.

5.1 VIKOR method

The method of Vlsekriterijumska Optimizacija I Kompromisno Resenje (i.e. compromise ranking method, That is, the VIKOR) was original established by Opricovic [41] for deal with the optimization problem of multiple attribute complex systems. It is also a traditional multiple attribute decision making method and is referred as an effectively approach to determine a compromise result or solution from a set of conflicting criteria. The method of VIKOR provides an extraordinary efficient with the case that the decision maker is unable to express his preference with an exactly and quantitative way in reality. In the past few years, the VIKOR method has been extended several improved forms by integrating with other methods such as fuzzy set theory, vague set theory and grey theory as well as interval numbers theory which they are enhance the original model’s performance [42, 43]. Considering its particular advantages, the VIKOR method has been widely used in many complex decision making problems in the recently years [3 , 58].

The main idea of VIKOR is focus on outranking and sorting all considered alternatives against various, or possibly conflicting and non-commensurable criteria related to the given decision making problem supposing which compromising is acceptable to resolve conflicts. Comparing with the existing multiple criteria decision making approaches, VIKOR depends on an aggregating function which describes the closeness to the ideal and deals with the ranking index by using a particular measure of closeness to the ideal solution. Another highlighted character of the VIKOR method is that uses linear normalization to eliminate units of criterion functions [42]. The basic concept of compromise sorting is established according to the l_p-metric depended on an selected aggregating function in a compromise programming [61, 75]: $L_{p, j} = (\sum_{i = 1}^{n} (ω_{i} \frac{f_{i}^{*} - f_{ij}}{f_{i}^{*} - f_{i}^{-}})^{p})^{\frac{1}{p}}, 1 \leq p \leq + \infty, j = 1, 2, \dots, m .$ where w_i (i = 1, 2, ⋯ , n) are the corresponding weights of the criteria. The number n presents criteria or attributes, the number m presents considered alternatives, the number p stands for the weight of the maximal deviation with the ideal solution, $f_{i}^{*} = max_{j} f_{ij}$ and $f_{i}^{-} = min_{j} f_{ij},$ respectively stand for the best values and the worst values. The values L_p,j stand for the distance between the alternative A_i and the ideal solution. Then the VIKOR method uses L_1,j (as S_j) and L_+∞,j (as R_j) to represent the ranking measurement. The result given by $min_{j} {S_{j}}$ is referred as the maximum group utility and the result given by $min_{j} {R_{j}}$ is referred as the minimum individual regret of the opponent. Thus, the main superiority of VIKOR methodology is that it could maximize the group utility and minimize the individual regret of the “opponent”. The compromise alternative F^C is a feasible alternative which is the “closest” to the ideal alternative F^*, and compromise implies an agreement given by mutual concessions.

5.2 Multiple attribute ranking decision making problem statement

In this section, we firstly present the statement of multiple attribute decision making problem with linguistic preference information. Let U = {u₁, u₂, ⋯ , u_n} be a discrete set of alternatives, E = {e₁, e₂, ⋯ , e_m} be the set of attributes, ω = {ω₁, ω₂, ⋯ , ω_m} ^T be the weight vector of attributes where ω_j ≥ 0 and $\sum_{j = 1}^{m} ω_{j} = 1 .$ The values of evaluation for every alternative with respect to all attributes E by the linguistic terms in the set L = {l_α|α = - t, ⋯ , -1, 0, 1, ⋯ , t} . Let F : E ⟶ F^L (U) and F^-1 : U ⟶ F^L (E) be two linguistic value binary mappings (or linguistic value binary relations) between U and E . Then, from the discussion in Section 3, there defines two types linguistic value soft set S_l = (F_l, E) and $S_{l}^{- 1} = (F_{l}^{- 1}, E),$ and also presents two types linguistic value fuzzy set for the considered decision making problem with linguistic variable, i.e., for any u_i ∈ U, $F_{l}^{- 1} (u_{i}) (e_{j}) \in F^{L} (E);$ for any e_j ∈ E, F_l (e_j) (u_i) ∈ F^L (U) , i = 1, 2, ⋯ , n ; j = 1, 2, ⋯ , m . Then the problem for decision maker with respect to this characterization is how to obtain a scientific and reasonable order and then make a decision on what alternatives are the optimal selection.

5.3 The model and methodology

As one of the new approach and tool of mathematics, linguistic value soft rough set theory provides another way to manipulate the problem of decision making under uncertainty. It provides the ability to handle the subjective judgements with their preferences for a given decision making problems of practice. Duo to the subjectivity and uncertainty of the linguistic variable among the process of decision making, this article uses linguistic value soft rough set to combine with VIKOR to deal with individual judgements with preferences and make the ranking for all candidate alternatives. The procedure of the decision making model and method mainly includes two parts and they are described as follows. First part, we modify the evaluation value of every alternative with respect to all attributes. In general, the evaluation value of every alterative with respect to all attributes are obtained by consulting the experts or collecting from data information in reality. Then, in order to eliminate the preferences of experts or the data errors as possible as when begins the process of decision making, we make the modification for the original evaluation value of all alternatives according to the following steps. Step 1. Computing the optimal and the worst evaluations for every alternative with respect to all attributes (denoted as E⁺ and E^-) over universe U, respectively.

$\begin{matrix} E^{+} = \sum \frac{max F_{l}^{- 1} (u_{i}) (e_{j})}{u_{i}}, \forall e_{j} \in E, \\ E^{+} (u_{i}) = max {F_{l}^{- 1} (u_{i}) (e_{j}) | u_{i} \in U} \in F^{L} (U), \end{matrix}$ (1)

$\begin{matrix} E^{-} = \sum \frac{min F_{l}^{- 1} (u_{i}) (e_{j})}{u_{i}}, \forall e_{j} \in E, \\ E^{-} (u_{i}) = min {F_{l}^{- 1} (u_{i}) (e_{j}) | u_{i} \in U} \in F^{L} (U) . \end{matrix}$ (2)

Step 2. Computing the lower approximation ${\underline{apr}}_{L} (E^{+}),$ ${\underline{apr}}_{L} (E^{-})$ and upper approximation ${\bar{apr}}_{L} (E^{+}),$ ${\bar{apr}}_{L} (E^{-}) .$ By the definition in Section 3.2, all of them are linguistic value fuzzy set over the set of attribute E . Then, for any parameter μ ∈ [0, 1] , computing the following two values:

$\begin{matrix} σ_{j}^{+} = μ {\underline{apr}}_{L} (E^{+}) (e_{j}) + (1 - μ) \\ {\bar{apr}}_{L} (E^{+}) (e_{j}), e_{j} \in E, \end{matrix}$ (3)

$\begin{matrix} σ_{j}^{-} = μ {\underline{apr}}_{L} (E^{-}) (e_{j}) + (1 - μ) \\ {\bar{apr}}_{L} (E^{-}) (e_{j}), e_{j} \in E . \end{matrix}$ (4)

Step 3. Updating the value of evaluation for every alternative u_i ∈ U with respect to all attributes e_j ∈ E according to the formula (3) and (4) as follows:

$\begin{matrix} F_{l}^{- 1} (u_{i}^{'}) (e_{j}) = \frac{1}{3} (F_{l}^{- 1} (u_{i}) (e_{j}) \\ + σ_{j}^{+} + σ_{j}^{-}) \in F^{L} (E), u_{i} \in U, e_{j} \in E . \end{matrix}$ (5)

Based on the above three steps, we obtain the updated value of evaluation for every candidate alternative with respect to all attributes. We also use the same symbol $U = {u_{i}^{'} | i = 1, 2, \dots, n}$ represent the set of alternative. Second part, we use the method of VIKOR present the ranking for all candidate alternatives, which is acquired as the way of follows. Step 1. According to the updated data table with the values of evaluation for all alternatives in step 3 of the first part, determine the best value of evaluation $F_{l}^{*} (e_{j})$ and the worst value of evaluation $F_{l}^{-} (e_{j})$ of each attribute e_j in the set of attribute E . As far the benefit attribute that belongs to the larger the better category; as far the cost attribute that belongs to the smaller the better category. That is

$\begin{matrix} F_{l}^{*} (e_{j}) = {{max_{u_{i}} F_{l} (e_{j}) (u_{i}) | e_{j} \in B} or \\ {min_{u_{i}} F_{l} (e_{j}) (u_{i}) | e_{j} \in C}}, e_{j} \in E, u_{i} \in U, \end{matrix}$ (6)

$\begin{matrix} F_{l}^{-} (e_{j}) = {{min_{u_{i}} F_{l} (e_{j}) (u_{i}) | e_{j} \in B} or \\ {max_{u_{i}} F_{l} (e_{j}) (u_{i}) | e_{j} \in C}}, e_{j} \in E, u_{i} \in U, \end{matrix}$ (7) where B is related to the benefit attribute while C is related to the cost attribute. It is easy to know that the value of $F_{l}^{*} (e_{j})$ and $F_{l}^{-} (e_{j})$ (j = 1, 2, ⋯ , m) also are linguistic values. Then, for the sake of facilitation the calculation in following steps, we should make some preconcessions for the linguistic terms, i.e., transforming the linguistic terms into the corresponded term indices. In what of follows, the conversion functions are used to finish this transformation.

Definition 4.1 Let $\bar{L} = {l_{α} | α \in [- q, q]}$ be a set of extended continuous linguistic terms of discrete term set L, where q (q > t) is a sufficiently large positive integer, $l_{α} \in \bar{L}$ is a linguistic term, then the corresponded term index α can be obtained according to the function I as follows: $I : \bar{L} ⟶ [- q, q], I (l_{α}) = α, l_{α} \in \bar{L} .$ Step 2. Calculating the values S_i and R_i (i = 1, 2, ⋯ , n) , respectively.

$\begin{matrix} S_{i} = \sum ω_{j} (\frac{I (F_{l}^{*} (e_{j})) - I (F_{l} (e_{j}) (u_{i}))}{I (F_{l}^{*} (e_{j})) - I (F_{l}^{-} (e_{j}))}), \\ e_{j} \in E, u_{i} \in U, j = 1, 2, \dots, m, \end{matrix}$ (8)

$\begin{matrix} R_{i} = max_{e_{j}} {ω_{j} (\frac{I (F_{l}^{*} (e_{j})) - I (F_{l} (e_{j}) (u_{i}))}{I (F_{l}^{*} (e_{j})) - I (F_{l}^{-} (e_{j}))})}, \\ e_{j} \in E, u_{i} \in U, j = 1, 2, \dots, m, \end{matrix}$ (9)

Step 3. Calculating the values Q_i .

$\begin{matrix} Q_{i} = ν \frac{S_{i} - S^{-}}{S^{*} - S^{-}} + (1 - ν) \\ \frac{R_{i} - R^{-}}{R^{*} - R^{-}}, i = 1, 2, \dots, n \end{matrix}$ (10) where $S^{*} = max_{i} {S_{i}}, S^{-} = min_{i} {S_{i}}, R^{*} = max_{i} {R_{i}} R^{-} = min_{i} {R_{i}},$ Q_i is an aggregating index, ν is defined as the weight of the strategy for the majority of attribute, and the 1 - ν is assigned as the weight of the individual regret and ν ∈ [0, 1] . Generally speaking, suppose that the value is set to 0.5 . Step 4. Ranking the alternatives according to the values S_i, R_i, and Q_i in an increasing order. Then there are three ranking list results. Step 5. Making a compromise solution for all alternative (u⁽¹⁾) , which is the best ranked according to the measure Q, if and only if it satisfies the following two conditions:

Condition 1. Acceptable advantage:

$Q (u^{(2)}) - Q (u^{(1)}) \geq \frac{1}{n - 1},$ (11) where u⁽²⁾ is the alternative with second position among of the ranking list with Q ; n is the number of the whole alternatives.

Condition 2. Acceptable stability in decision making: the alternative (u⁽¹⁾) must be the best ranked by S or/and R, too. This compromise solution is stable within a decision making procedure. If ν > 0.5, then it may be the tactics of maximum group utility, or “with veto” when ν < 0.5, or “by the consensus” while ν = 0.5 .

If one of the conditions is not hold, then go to the next step and then there can obtain the compromise solution: (a) If Condition 2 is not hold, then both u⁽¹⁾ and u⁽²⁾ are compromise solutions. (b) If Condition 1 is not hold, then exploring the maximum value of k in u⁽ⁱ⁾ which is determined by the following relationship:

$Q (u^{(k)}) - Q (u^{(1)}) < \frac{1}{k - 1}$ (12)

By using the above two related phases, the methodology for the multiple attribute decision making according to the linguistic value soft rough set and VIKOR is established well.

5.4 An illustrate example

This subsection considers a multiple attribute decision making problem of an evaluation decision making of automatics (Example 3.1) given in Section 3 to show the principal procedure one-by-one for the decision method proposed in Section 4.3. Let U = {u₁, u₂, u₃, u₄, u₅, u₆} be the considered set of six different cars. E = {e₁, e₂, e₃, e₄, e₅} is the parameter set (i.e., the attribute of evaluation for the cars). ω = (0.15, 0.19, 0.25, 0.12, 0.29) ^T be the weight vector of attributes. The linguistic term in set L = {l_α|α = -4, ⋯ , -1, 0, 1, ⋯ , 4} . Where l_-4=extremely poor, l_-3=very poor, l_-2=poor, l_-1=slightly poor, l₀=fair, l₁=slightly good, l₂=good, l₃=very good, l₄=extremely good.

Then, we can calculate the optimal and worst evaluations for every car with respect to all attributes (i.e., E⁺ and E^-) over U based on the formula (1) and (2) . Subsequently, we calculate the lower and upper approximations of E⁺ and E^- and also can obtain the $σ_{j}^{+}$ and $σ_{j}^{-}$ by the formula (3) and (4) , respectively. In general, the parameter μ in formula (3) and (4) characterizes the attitude for risk of a decision-maker in practice and it selected by the decision-maker according to his/her preference about the risk. Here we select μ = 0.5 . All results are presented in Table 2.

Table 2
The linguistic preference evaluation for cars

U∖E e ₁ e ₂ e ₃ e ₄ e ₅ E ⁺ E ^-

u ₁ l ₂ l _-3 l ₀ l ₁ l _-2 l ₂ l _-3

u ₂ l ₂ l ₃ l ₁ l ₀ l _-4 l ₃ l _-4

u ₃ l _-1 l ₀ l ₂ l ₂ l ₁ l ₂ l _-1

u ₄ l ₃ l ₁ l _-1 l ₀ l ₂ l ₃ l _-1

u ₅ l _-4 l _-1 l ₂ l ₀ l ₁ l ₂ l _-4

u ₆ l ₀ l ₂ l ₃ l ₁ l ₃ l ₃ l ₀

$σ_{j}^{+}$ l _2.5 l _2.5 l _2.5 l ₂ l _2.5 - -

$σ_{j}^{-}$ l _-1 l _-0.5 l _-1 l _-0.5 l _-0.5 - -

U∖E	e ₁	e ₂	e ₃	e ₄	e ₅	E ⁺	E ^-
u ₁	l ₂	l _-3	l ₀	l ₁	l _-2	l ₂	l _-3
u ₂	l ₂	l ₃	l ₁	l ₀	l _-4	l ₃	l _-4
u ₃	l _-1	l ₀	l ₂	l ₂	l ₁	l ₂	l _-1
u ₄	l ₃	l ₁	l _-1	l ₀	l ₂	l ₃	l _-1
u ₅	l _-4	l _-1	l ₂	l ₀	l ₁	l ₂	l _-4
u ₆	l ₀	l ₂	l ₃	l ₁	l ₃	l ₃	l ₀
$σ_{j}^{+}$	l _2.5	l _2.5	l _2.5	l ₂	l _2.5	-	-
$σ_{j}^{-}$	l _-1	l _-0.5	l _-1	l _-0.5	l _-0.5	-	-

The updated values of evaluation for all cars according to formula (5) are represented as Table 3.

Table 3

The updated values of evaluation for cars

U∖E	e ₁	e ₂	e ₃	e ₄	e ₅
$u_{1}^{'}$	l _1.167	l _-0.333	l _0.500	l _0.833	l _0.000
$u_{2}^{'}$	l _1.333	l _1.667	l _0.833	l _0.500	l _-0.667
$u_{3}^{'}$	l _0.167	l _0.667	l _1.167	l _1.167	l _1.000
$u_{4}^{'}$	l _1.500	l _1.333	l _0.167	l _0.500	l _1.333
$u_{5}^{'}$	l _-0.833	l _0.333	l _1.167	l _0.500	l _1.000
$u_{6}^{'}$	l _0.500	l _1.333	l _1.500	l _0.833	l _1.667

Then the the best value of evaluation $F_{l}^{*} (e_{j})$ and the worst value of evaluation $F_{l}^{-} (e_{j})$ of each attribute e_j ∈ E according to the formula (6) and (7) are represented as Table 4.

Table 4

The best and the worst value of evaluation

	e ₁	e ₂	e ₃	e ₄	e ₅
$F_{l}^{*} (e_{j})$	l _1.333	l _1.667	l _1.500	l _1.167	l _1.667
$F_{l}^{-} (e_{j})$	l _-0.883	l _-0.333	l _0.167	l _0.500	l _-0.667

The values of S, R and Q are calculated by the formula (8) , (9) and (10) and the results are presented as Table 5.

Table 5

The values of S, R and Q

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
S _i	0.6560	0.3692	0.3194	0.4319	0.4187	0.1482
R _i	0.2071	0.1251	0.0950	0.2500	0.1466	0.0601
Q _i	0.8871	0.3888	0.2605	0.7794	0.3096	0.0000

So, the The ranking of six cars in increasing order by S, R and Q are as follows in Table 6.

Table 6

The Ranking by S, R and Q for all cars

	1	2	3	4	5	6
By S	u ₆	u ₃	u ₂	u ₅	u ₄	u ₁
By R	u ₆	u ₃	u ₂	u ₅	u ₁	u ₄
By Q	u ₆	u ₃	u ₅	u ₂	u ₄	u ₁

It is easy to verify that both the Condition 1 and Condition 2 given in the Step 5 of the second part are satisfied well, then the car u₆ has the best ranking and then it selected as the optimal selection.

5.5 Comparison and discussion

The results of MAGDM are often sensitive to different aggregation operators. Accordingly, we conduct a comparative analysis to ascertain whether the results obtained by the proposed method are comparable in the context of other MAGDM method. The TOPSIS is also employed to obtain the ranking of alternatives for this example. The result is u₆ ≽ u₅ ≽ u₃ ≽ u₂ ≽ u₄ ≽ u₁, and thus, the best alternatives is u₆. It can be seen that the proposed method and the TOPSIS obtain the same best alternative. However, the ranking of alternative is different. The proposed method can not only deal with individual judgements with preferences, but also we apply non-aggregate operators to obtain the fuse information which make the method given in this paper is reasonable and effective.

Comparing to the literatures related to the existing generalized soft set models and methodologies for decision making problems under uncertainty [2 , 66], this paper provides the following main contributions: On the one hand, we present a new generalization of original soft set theory [40] by introducing linguistic preference information into the concept soft set under the framework of Pawlak rough set [45], i.e., the linguistic value soft rough set. The linguistic value soft rough set also is a new model of granular computing for dealing with decision making problem under uncertainty. Though there are many literatures about the generalized soft set models and methods with the requirement of decision making problems under uncertainty [1 , 67], less effort has been made to study the combination of classical soft set and linguistic variable based on the original rough set theory. So, the generalization of this paper enriches the fundamental theory as well as the application fields of the soft set theory for handling with the decision making problems under uncertainty. On the other hand, we establish a new approach to multiple attribute decision making based on the combination of linguistic value soft rough set and VIKOR method. The existing approaches to decision making under uncertainty based on generalized soft set theories for multiple attribute decision making problems are based on the concept of sore function and its various improved version [12 , 70]. This paper gives a new model and method for multiple attribute decision making problems with linguistic information by using the linguistic value soft rough set and VIKOR method. Meanwhile, from the point of view of granular computing, the established method provides a new perspective and effective way of granular computing to deal with decision making problems under uncertainty. As is well know, it could not good differentiate among of all alternatives according to the score function-based soft set approach to decision making problems under uncertainty [12 , 70], then the final optimal solutions or results may be not stable. As a direct result, the final optimal solutions or results could not be correct. Therefore, it limits the ability of the application of the model and method of decision making based on generalized soft set theory. Therefore, searching improved model and methodology with a robustness solutions based on original soft set theory and its generalized forms are an interesting and valuable topic and direction. This paper focuses on this issue and presents a new model for multiple attribute decision making problems under uncertainty, i.e., we establish an improved soft set approach to multiple attribute decision making problems under uncertainty by combining linguistic value soft set and rough set (i.e., linguistic value soft rough set). This is another contribution of our proposed decision making model and method. Granular computing, established by Zadeh [71], as a new perspective and way to handle of the uncertain information has successfully applied to many areas. Granular computing, as a cognitive calculating process, provides a new perspective and methodology to deal with the decision making problem under uncertainty. It presents a kind of theory, technique and tool that using the concept of information granules in the process of problem solving [57]. As is well known, there are large number of literatures about fuzzy or fuzzy-extended soft computing theories that are also capable to deal with linguistic vagueness [7 , 74]. The existing literatures provide many useful and valuable models and methodologies to handle of multiple criteria/attribute decision making problems under uncertainty. As aforementioned, soft set methodology also is an effectively granular computing tool and model as similar as the rough set theory. The distinct advantage of rough set theory is that does not require any prior knowledge, and through the effective analysis and reasoning of the current data, and then discovers the implicit knowledge and finally reveals the underlying law of the data. Comparing to the existing literatures for multiple attribute decision making problems under uncertainty [7 , 74?], the established model and method based on linguistic value soft rough set in this paper are based on the framework and principle of granular computing theory and then gives a new granular computing approach to multiple attribute decision making problems under uncertainty by combining the rough set and soft set. Meanwhile, the given approach to multiple attribute decision making problem in this paper also provides a new perspective and valuable attempt to combining the traditional VIKOR method and the granular computing methodology. Therefore, the proposed model and method enrich and perfect both the granular computing theory and the decision making approaches under uncertainty.

6 Conclusions

To deal with the inaccuracy and unstructured problems of decision making with complex and uncertainty environment in practice, a large number of scholars and practitioners have emphasized the problems of uncertainty decision making fields and developed several effectively generalized and extended methodologies to improve efficiency and flexibility of decision making under uncertainty. This article proposes an new approach to handle with the multiple attribute decision making problem with linguistic preference information by integrating soft rough set theory and VIKOR methodology. This paper constructs a new granular computing model and method named as linguistic value soft rough set by introducing the linguistic preference information into the original soft set theory under the framework of Pawlak rough set theory. A series of basic theories are investigated for the linguistic value soft rough set by means of the constructed method. Subsequently, the proposed linguistic value soft rough set is applied to integrate with the VIKOR method to handle with a kind of multiple attribute decision making problem under uncertainty. Both the fundamental principle as well as the decision making steps of the proposed method are established. At last, the proposed model and method is applied to a numerical case study with assist multiple attribute ranking. The results are more reasonable and the vagueness and inaccuracy also are quantified and handled properly. This paper focuses on the theoretical aspect of linguistic value soft rough set theory as well as the feasibility and effectiveness of the combination of granular computing methodology with traditional VIKOR method. In fact, There are several improved version of VIKOR method [3–5 , 68] to deal with multiple criteria/attribute decision making problems under uncertainty. This paper makes an tentative exploring to the infusion between the traditional decision making methodology and granular computing principle. Moreover, this paper emphasize on the steps and procedure of the established decision making method and then use a numerical example test the results. In fact, the application of the problems of decision making with real data and background in reality is more useful. Different evaluation methods can be constructed based on the above two theories in future research. Improving the evaluation model by combining other theories and methodologies can also be a worthwhile direction for study [16, 19]. Therefore, we will explore the improved version by infusion the extended VIKOR and the granular computing methodologies as well as the application of problems of decision making of reality in future.

Footnotes

Acknowledgment

The work was partly supported by the National Natural Science Foundation of China (No. 72071152 and 71571090), the Xi’an Science and Technology Projects (No. XA2020-RKXYJ-0086), the Youth Innovation Team of Shaanxi Universities, the China Postdoctoral Science Foundation (No. 2020M670046ZX), the Science and Technology Plan Project of Yulin (No.19-50), the Zhejiang Provincial Natural Science Foundation (No. LY20G010007), the Joint Fund of Zhejiang Provincial Natural Science Foundation (LHY2 0F030001).

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An approach to multiple attribute decision making based on linguistic value soft rough set and VIKOR method

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Soft set

2.2 Linguistic value varibles with decision-making problem

3 Preliminaries

3.1 Molodtsov soft set theory

3.2 Rough set

3.3 Linguistic value and linguistic variable

4 Linguistic value soft rough set model and its properties

4.1 Linguistic value soft set

Table 1 A linguistic value soft set U∖E e 1 e 2 e 3 e 4 e 5 u 1 l 2 l -3 l 0 l 1 l -2 u 2 l 2 l 3 l 1 l 0 l -4 u 3 l -1 l 0 l 2 l 2 l 1 u 4 l 3 l 1 l -1 l 0 l 2 u 5 l -4 l -1 l 2 l 0 l 1 u 6 l 0 l 2 l 3 l 1 l 3

5 The multiple attribute ranking decision making method based on linguistic value soft rough set and VIKOR

5.1 VIKOR method

5.2 Multiple attribute ranking decision making problem statement

5.3 The model and methodology

6 Conclusions

Footnotes

Acknowledgment

References

Table 1
A linguistic value soft set

U∖E e ₁ e ₂ e ₃ e ₄ e ₅

u ₁ l ₂ l _-3 l ₀ l ₁ l _-2

u ₂ l ₂ l ₃ l ₁ l ₀ l _-4

u ₃ l _-1 l ₀ l ₂ l ₂ l ₁

u ₄ l ₃ l ₁ l _-1 l ₀ l ₂

u ₅ l _-4 l _-1 l ₂ l ₀ l ₁

u ₆ l ₀ l ₂ l ₃ l ₁ l ₃