1. INTRODUCTION

In terms of common sense, people use linguistic terms in nature language for evaluating, reasoning and making decision, rather than crisp numbers. Hence, collection with qualitative information is always got, for example, object-attribute assessment for a car may be “very cheap” in price, “slightly comfortable” in comfort, “somewhat dangerous” in safety. To handle the qualitative information, many linguistic terms models have been proposed based on computing with words (CWW) [1,2], such as fuzzy set theory [3–5], symbolic approaches [6,7] and linguistic truth-valued mode in lattice order [8,9].

Fuzzy set theory was proposed by Zadeh in 1965 using a membership function to model the uncertain information [5]. But it is difficult to find appropriate membership function when we confronted with the circumstances where various concepts are given in various contexts. Type-2 fuzzy set is the extension of ordinary fuzzy set based on the theory of computing with words (CWW) [2]. Unlike a type-1 fuzzy set where the membership grade is a crisp number in [0, 1], the membership grade of Type-2 fuzzy set was extended to an ordinary fuzzy set [2–4]. Type-2 fuzzy set plays a role in wider applications and processes the uncertain information more flexibility [10–12].

Symbolic approaches use linguistic symbols to represent linguistic information directly without the numerical approximation required by fuzzy set based methods, and aggregate or compute on the indexes of these symbols to obtain the final result [13,14]. Symbolic approach has performed well in decision making problems especially semantic interpretation of linguistic terms in nature language [15,16]. But there exists some loss of information when the results come out from the initial expression domain. Francisco Herrera and Luis Martínez proposed a 2-tuple model which are composed by a linguistic term and a numeric value assessed in [−0.5, 0.5) [17,18]. The 2-tuple model is widely used in many real problems for that it can avoid information losing in linguistic information processing and many aggregation operators of 2-tuple model were provided [19–22].

The conventional symbolic approach for linguistic value uses a linear order structure. However, in the natural language, some linguistic values are incomparable, for example, “Exactly True” and “Somewhat False.” Obviously, it is difficult to describe the incomparable values utilizing a linear order structure. Lattice implication algebra (LIA) structure can imitate the uncertain and both comparable and incomparable characteristic [7]. Many researchers have studied in varied direction of LIA [23,24]. Jun Liu et al. have proposed an axiomatizable lattice ordered qualitative linguistic truth-valued logic system, which is a foundation for establishing formal linguistic truth-valued logic [25]. Xingxing He et al. have unified method for finding the structure of k-IESF in linguistic truth-valued lattice-valued propositional logic which provided a theoretical foundations and algorithms for α-resolution automated reasoning [26]. Yang Xu et al. have extended the binary α-resolution to multiary α-resolution in lattice-valued propositional logic LP(X) and lattice-valued first-order logic LF(X), obtained a result that multiary α-resolution principle in LF(X) can be equivalently transformed into that in LP(X) [27]. Yi Liu and Vassilis G. Kaburlasos et al. have introduced a LIA with implication values in a complete lattice of intervals on the real number axis which followed a capacity to optimize [28]. Li Zou proposed a linguistic-valued knowledge representation mode and approximated reasoning approach with linguistic-valued credibility factors [29].

All the above researches provided the feasibility of decision making with linguistic values based on linguistic-valued lattice implication algebra (LV-LIA). Besides object-attribute assessment, the importance of the attributes is often expressed using linguistic terms. However, the linguistic weights are always transformed to numbers or values on a linear order structure in a decision making problem [30]. This paper is aiming at decision making problems not only with the linguistic evaluation set but also the linguistic important degree, i.e., linguistic weight. We will propose a decision making model based on LV-LIA which has a flexible linguistic weighted method.

LV-LIA can express both comparable and incomparable linguistic information, which is consistent with nature language characteristic. In decision making there often exist some incomparable linguistic-valued decision making results. The positive valuation function (PVF) which implies a metric distance as well as an inclusion measure function can mapping a lattice to a real set [31]. We define a linguistic-real valuation function and get a rank with incomparable linguistic values through the linguistic-real metric distance (LRMD) elastic according to the individual preference degree.

Based on the aforementioned academic ideas, we will introduce a decision making approach based on 18-element LV-LIA.

The rest of paper is organized as follows: In section 2, we briefly review the concepts of LIA and LV-LIA. In section 3, we discuss the properties of the operations ⊗ and ⊕ of LV-LIA, and propose a decision making model with linguistic weighted method. In section 4, we introduce a linguistic-real valuation function implied a metric distance to rank the incomparable decision making results. In section 5, some conclusions are summarized.

2. LINGUISTIC-VALUED LATTICE IMPLICATION ALGEBRA

We briefly review some concepts of linguistic truth-valued LIA. We refer to the related Ref. [7].

Definition 3.

[7] Let ADn+1=h0,h1,⋯,hn be a set of n hedge operators and h0<h1<⋯<hn, MT=f,t be “false(f)” and “true(t)”, denote f<t and LV((n+1)×2)=ADn+1×MT. Define the mapping g: LV(n+1)×2→Ln+1×L2 as follows.

(2)ghi,mt=di′,b1,mt=f,di,b2,mt=t.

Then g is bijection. Its inverse mapping is g−1. For any x, y ∈LVn+1×2, define

(3)x∨y=g−1gx∨gy,x∧y=g−1gx∧gy,x′=g−1gx′,x→y=g−1gx→gy.

Then LVn+1×2=LVn+1×2,∨,∧,′,→,hn,f,hn,t is called linguistic truth-valued LIA from ADn+1 and MT (Figure 1). g is an isomorphic mapping from LVn+1×2,∨,∧,′,→,hn,f,hn,t to Ln+1×L2.

Figure 1

The Hasse diagram of LVn+1×2.

The operation “→” is

(4)hi,c2→hj,c1=hmax0,i+j−n,c1,hi,c1→hj,c2=hminn,i+j,c2,hi,c2→hj,c2=hminn,n−i+j,c2,hi,c1→hj,c1=hminn,n−j+i,c2.

Note that n should be an even number when we construct the LVn+1×2.

3. THE DECISION MAKING APPROACH WITH LINGUISTIC WEIGHTED BASED ON LV-LIA

In a multi-attributes decision making problem, sometimes attribute importance is expressed with linguistic terms. We explore a linguistic weighted method expressing these linguistic weights with a linguistic lattice structure based on LV-LIA and apply it into decision making with linguistic weight in this section.

3.2. The Decision Making Approach Based on 18LV-LIA

In generally, linguistic terms with linguistic hedges can be seen as linguistic modifiers and prime terms. In the following, we divide the linguistic terms into nine linguistic hedges (modifiers) applying to two prime terms. The linguistic modifiers of LV-LIA are {Slightly (Sl for short), Somewhat (So), Rather (Ra), Almost (Al), Exactly (Ex), Quite (Qu), Very (Ve), Highly (Hi), Absolutely (Ab)} with the semantic ordering relationship Sl < So < Ra < Al< Ex < Qu < Ve < Hi < Ab, i.e., the hedge set is AD9=h0=Sl, h1=So, h2=Ra, h3=Al, h4=Ex, h5=Qu, h6=Ve, h7=Hi, h8=Ab. The prime terms are {Dissatisfied (Ds for short), Satisfied (Sa)} representing a pair opposite meta vague concept with Ds < Sa, i.e., the prime term set MT=c1=Ds, c2=Sa. Applying the linguistic modifiers of AD9 to the prime terms MT, we obtain a partially ordered lattice LV9×2=LV9×2,∨,∧,′,→,h8,c2,h8,c1 called 18-element linguistic-valued lattice implication algebra (18LV-LIA) (Figure 2) [32].

Figure 2

The Hasse diagram of LV9×2.

A decision making model is constructed based on 18LV-LIA. Weights of the attributes are expressed with linguistic terms on LV-LIA. The operation ⊗ can be used to calculate the linguistic weight. The attribute aggregation can be done according to the aggregation rules which can be set flexibly based on the actual cases.

There are some aggregation function about the logical relation “or” and “and”.

For relation “hi,ck and hj,cl” can be aggregated as,

1. (10)Fandhi,ck,hj,cl=hi+j2,cl,if  k=lh8+i⋅sgnk−1.5+j⋅sgnl−1.52,c2, if  k≠l,
   where sgn⋅ is the sign function, the maximum subscript of the hedges in 18LV-LIA is 8, and 1.5 is the median point of the subscript of prime term, in fact, it can be any point in interval (1,2).

2. (11)Fandhi,ck,hj,cl=hi,ck∧hj,cl.

   For relation “hi,ck or hj,cl” can be aggregated as,

3. (12)Forhi,ck,hj,cl=hu,c2,  if k∥l maxhi,ck,hj,cl, others
   where u=maxn,8+i⋅sgnk−1.5+j⋅sgnl−1.5, and the parameters means the same with parameters in Eq. (10).

4. (13)Forhi,ck,hj,cl=hi,ck∨hj,cl.

They all satisfy the associative law and commutative law.

A linguistic multi-attribute decision making problem consists of a finite and non-empty alternative set A=A1,A2,⋯,Ak, where each alternative is defined by means of a finite set of attribute, C=C1,C2,⋯,Cm which is assessed using linguistic expressions, and the corresponding weight set for attributes is W=ω1,ω2,⋯,ωm.

The decision making approach based on 18LV-LIA consists mainly of the flowing steps.

Step 1. Linguistic assessment matrix and linguistic weight transforming. The linguistic assessment matrix is

E=a11a12⋯a1ma21⋱ a2m  ⋮ ⋱  ⋮aN1aN2⋯akm.

In the linguistic assessment matrix, aij∈LV9×2 is the assessment about the attribute C_j of alternative Ai.

The attribute weights given by the experts are the linguistic terms in LV9×2 as

W=ω1,ω2,⋯,ωm.

In this step, the attribute assessment collection of alternatives is transformed to linguistic terms on LV9×2.

Step 2. Weighted evaluation matrix Construction. Weighted evaluation matrix is constructed from Eq. (8).

M=b11b12⋯b1mb21⋱ b2m  ⋮ ⋱  ⋮bN1bN2⋯bmN=E⊗W,

where bij=aij⊗ωj.

Step 3. Aggregation function setting and comprehensive evaluations aggregation.

The comprehensive evaluation

bi=Fbi1,bi2,⋯,bim,

where F⋅ is the aggregation function of the weighted evaluation.

Step 4. Decision making process. The comprehensive evaluations bi i=1,2,⋯,N are ranked with respect to each alternative according to Fig. 2.

Here are a few notes. The aggregation function is set as

Fbi1,bi2,⋯,bim=Fm−1⋯F2F1bi1,bi2,bi3⋯,bim,

where Fk⋅ is the aggregation operators of bik and bi,k−1 related to their logical relation, bi is the comprehensive evaluation of the ith alternative. The function can be a logical operator, an arithmetical operator or even a hybrid one.

It is also noted that incomparable linguistic results are allowed.

3.3. Examples Illustration

Example 1.

A teaching evaluation system with linguistic information.

To show how the proposed approach works, we give a teaching evaluation system which help decision makers evaluate the teacher's teaching quality. Suppose that the teaching quality depends on four attributes, responsibility, lesson preparation, professional proficiency and teaching efficiency. The attribute set is denoted as C=C1,C2,C3,C4. The alternative set is A=A1,A2,A3,A4 representing four teachers respectively.

Step 1. The assessment collection with respect to the attributes of four teachers about teaching quality is shown in Table 1. The linguistic terms are the linguistic values in LV9×2. We get the initial assessment matrix.

E=h4,c2h5,c2h4,c2h8,c2h4,c2h1,c1h4,c2h8,c1h7,c2h5,c2h7,c2h4,c1h0,c1h3,c1h0,c1h4,c2.

	C1	C2	C3	C4
A1	Ex,Sa	Qu,Sa	Ex,Sa	Ab,Sa
A2	Ex,Sa	So,Ds	Ex,Sa	Ab,Ds
A3	Hi,Sa	Qu,Sa	Hi,Sa	Ex,Ds
A4	Sl,Ds	Al,Ds	Sl,Ds	Ex,Sa

Table 1

The assessment collection about quality of teaching.

W₁, W₂ and W₃ are three weight vectors.

W1=h8,c2,h8,c2,h8,c2,h8,c2,W2=h6,c1,h4,c2,h6,c2,h8,c1,W3=h8,c1,h8,c1,h8,c1,h8,c1.

W1 mans that all of the attribute are equally important. W2 mans that the attributes are with different importance. W3 means that all of the attributes are not important at all, where the case is that unavailable attributes are collected.

Step 2. The weighted matrix are got utilizing the operation ⊗ according to Eq. (8).

M1=h4,c2h5,c2h4,c2h8,c2h4,c2h1,c1h4,c1h8,c1h7,c2h5,c2h7,c2h4,c1h0,c1h3,c1h0,c1h4,c2,

M2=h8,c1h1,c2h2,c2h8,c1h8,c1h5,c1h2,c2h8,c1h7,c1h1,c2h5,c2h8,c1h6,c1h7,c1h2,c1h8,c1,

M3=h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1h8,c1.

Step 3. Suppose there is facts that a teacher who is responsible or prepare lesson fully, and professional proficiency or teaching efficiency should be with high teaching quality. It is a rule, as a teacher,

IF (responsible or prepare lesson fully) and (professional proficiency or teaching efficiency) THEN (with high teaching quality).

The aggregation function for relation “or” is set as the logical operation Eq. (13). The aggregation function for relation “and” is set as the arithmetic operations Eq. (10). The weighted evaluations under W2 are listed in the Table 2.

	A1	A2	A3	A4
C1	h8,c1	h1,c2	h2,c2	h8,c1
C2	h8,c1	h5,c1	h2,c2	h8,c1
C3	h7,c1	h1,c2	h5,c2	h8,c1
C4	h6,c1	h7,c1	h2,c1	h8,c1

Table 2

The weighted evaluations values under W₂.

And by the aggregation rule, we get the assessments of four teachers with different attribute weights (Table 3).

	A1	A2	A3	A4
W1	h6,c2	h5,c2	h7,c2	h8,c2
W2	h1,c2	h3,c2	h3,c2	h4,c1
W3	h8,c1	h8,c1	h8,c1	h8,c1
Without weights	h6,c2	h5,c2	h7,c2	h8,c2

Table 3

Comprehensive evaluations.

Step 4: The results in Table 3 are ranked (Table 4).

W1	W2	W3
A4>A3>A1>A2,	A2=A3>A1,	A1=A2=A3=A4.
	A1∥A4,
	A2∥A4,
	A3∥A4.

Table 4

The final results with different weights.

From Table 2, we can see that the four teachers get the same assessments in attribute C1 and C3 but they get different weighted evaluations because the weights of attribute C1 and C3 are different. In fact, the two weights has the same linguistic hedges but the different prime terms, that can't expressed utilizing a linear structure.

From Table 3, we can see that when all attributes are absolutely important, the results equal to the results without any weight. Therefore, the present method not considering weights is a special case of the proposed approach.

When the weights of attribute are different, the results will be changed. For example, Teacher A2 get higher score under the weight W2 than it under the weight W1.

From the results, we can see that when we consider all attribute absolutely unimportant, i.e., the weight vector is W₃, the four teachers get the same and lowest evaluation. It means that we collected unavailable data.

From Table 4, there are incomparable values in the results. For example, there exists the relation for W₂ as follows (Figure 3),

Figure 3

Relations of teachers on LV9×2 under W₂.

It is reasonable for that the incomparable cases exist in reality because of the characteristic of the linguistic value in nature language.

Remark:

In step 3, taking the teacher A1 when the weight set is W2 as an example, we show the aggregation process.

F=FandForh8,c1,h1,c2,Forh2,c2,h8,c1 =Fandh8,c1∨h1,c2,h2,c2∨h8,c1 =h1+22,c2 =h1,c2.

Example 2.

To illustrate the effectiveness of the method, an example of car evaluation in Ref. [30] is considered.

Suppose that there are three kinds of cars: BMW x1, Hyundai x2 and Passat x3 are under evaluation according to four attributes: safety f1, price f2, comfort f3 and fuel economy f4. The linguistic judgments for evaluating the cars on LV9×2 is as Table 5. The weights associated with the four criteria are supposed to be

W1=h8,c2,h6,c2,h3,c2,h5,c2,

or

W2=h8,c1,h6,c2,h3,c1,h5,c2.

	f1	f2	f3	f4
x1	h6,c2	h4,c1	h7,c2	h1,c2
x2	h1,c1	h5,c2	h1,c2	h4,c2
x3	h6,c2	h1,c2	h5,c2	h2,c1

Table 5

Linguistic assessment about cars on LV9×2.

The comprehensive evaluations is as Table 6.

Weight	W1			W2
Method	x1	x2	x3	x1	x2	x3
Our Method	h6,c2	h7,c2	h6,c2	h4,c2	h3,c2	h3,c2
Chen's method	h6,c2	h7,c2	h6,c2	–	–	–

Table 6

The comprehensive evaluations from two methods.

From Table 6, we can see that when the weights are positive prime terms, the two methods get the same results. But when the weights consist of negative prime terms, the method in Ref. [30] is hard to handle it.

In Ref. [30], Shuwei Chen et al. used the linguistic multi-criteria decision making approach based on logical reasoning. Our method is more widely in weight setting and more flexible in comprehensive way.

In Ref. [30], weights are from the positive prime terms only. But in this paper, weights can be positive prime terms or negative prime terms. And the aggregation function can be logical operators or arithmetic operators or hybrid operators in this paper. Our method can handle more widely cases. It is more flexible facing the complex problems in reality.

From the results in Table 4, incomparable linguistic values are existed rationally because of the inherent characteristic of linguistic value. However, a certain decision is demanded sometimes. Then some further works should be done in this case.