JIANG Kexin，ZHANG Quan，and YAN Manting

School of Information Science and Engineering，Shenyang University of Technology，Shenyang 110870，China

Abstract: A method is proposed to deal with the uncertain multiple attribute group decision making problems，where 2-dimension uncertain linguistic variables (2DULVs) are used as the reliable way for the experts to express their fuzzy subjective evaluation information. Firstly，in order to measure the 2DULVs more accurately，a new method is proposed to compare two 2DULVs，called a score function，while a new function is defined to measure the distance between two 2DULVs. Secondly，two optimization models are established to determine the weight of experts and attributes based on the new distance formula and a weighted average operator is used to determine the comprehensive evaluation value of each alternative. Then，a score function is used to determine the ranking of the alternatives. Finally，the effectiveness of the proposed method is proved by an illustrated example.

Keywords: 2-dimension uncertain linguistic variables (2DULVs)，multi-attribute group decision making problem，score function，distance formula.

1. Introduction

Multi-attribute group decision making (MAGDM) problem is a process in which decision makers choose the most satisfactory alternative from limited alternatives according to the evaluation information of experts [1-8].To obtain the most satisfactory alternative，experts are invited to give their preference information，which may be expressed in the form of numerical value，such as clear number，interval number，fuzzy number，interval fuzzy number and so on [9-18]. However，some attribute values can only be evaluated qualitatively rather than quantitatively in real scenes，such as automobile comfort. In fuzzy linguistic methods，qualitative attribute values are expressed with fuzzy values through linguistic variables[19-24].

However，sometimes experts are unable to give definite linguistic variables in the course of evaluation due to the lack of expertise and uncertain cognition. For this reason，the concept of uncertain linguistic variables was proposed [25]，however，it failed to describe the reliability of the evaluation information. In response to this problem，the concept of 2-dimension linguistic variables was proposed [26]. The 2-dimension linguistic variables have attracted the attention of scholars because they can accurately describe evaluation information [27-30]. However，experts may hesitate between consecutive linguistic terms. Thus，the concept of 2-dimension uncertain linguistic variables (2DULVs) was proposed [31]. The 2DULVs can clearly and intuitively reflect the expert’s subjective information，which is helpful to improve the accuracy of decision results. They have been widely used for the risk assessment of public private partnership(PPP) waste-to-energy incineration projects，optimal site selection of straw biomass power plant，sustainable supplier selection，energy policy and so on [32-35].

Although in the literature，there is a method comparing two 2DULVs [36]，it compares two 2DULVs according to the product of the median of two dimensional linguistic intervals. Obviously，there is a situation where two 2DULVs are not equal and their medians are equal.Thus，this paper proposes a new method comparing two 2DULVs according to two indicators. Moreover，this paper defines a new distance formula to solve the problem that the existed distance formula is imprecise [37-39].

The remainder of this paper is organized as follows.Section 2 briefly reviews some preliminary concepts related to our research. We propose the new score function and the distance formula of 2DULVs in Section 3.Section 4 gives the application method in the MAGDM problem. Section 5 gives two examples to prove the effectiveness and advantage of the proposed method. The final section summarizes the main work of this paper with a discussion of implications for the future research.

2. Preliminaries

Definition 1[39]Letwhere

3. Score function and distance formula of 2DULVs

We call the function used to measure the size of 2DULVs score function. In the previous studies，the product of the median of two dimensional language intervals is regarded as a scoring function of 2DULVs. However，there may be a situation where two 2DULVs are not equal and their medians are equal. Thus，the existing score function of 2DULVs is not precise. To overcome this shortcoming，this paper adds an index to measure the size of 2DULVs according to the concept of variance of a random variable.

3.1 Scoring function of 2DULVs

3.2 Distance formula for 2DULVs

4. AGDM method for 2DULVs with completely unknown weight information

In the existing MAGDM methods with 2DULVs，experts and attribute weights are mostly known [37-39]. Therefore，this section proposes an MAGDM method with completely unknown weight information under 2DULVs.Firstly，two optimization models are established based on the measure formula to determine the weight of experts and attributes. Then the comprehensive evaluation value of each alternative is determined based on the weighted average operator. Finally，the ranking of alternatives is determined based on the score function of 2DULVs.

4.1 Problem descriptions

4.2 Weight determination model

In this paper，the weighted average operator is used as the aggregation operator to solve the MAGDM problem.Therefore，expert and attribute weights should be determined before expert and attribute information are fused.For the expert weight，the greater the consistency between individual preference and group preference，the greater the weight of the individual should be. Based on above principles，it is suggested to calculate the expert weight model as follows:

To solve this model，we construct the Lagrange function:

where π is the Lagrange multiplier.

Then we compute the partial derivatives ofLas follows:

From (19)，we get a simple and exact formula for determining the experts weight as follows:

As the expert weight determined by (21) and the weighted average operator given by (3)，we can aggregate individual preferences to form group preferences:

Next，we establish the attribute weight model with the attribute value as the 2DULV:

To solve this model，we construct the Lagrange function:

where λ is the Lagrange multiplier.

Then we compute the partial derivatives ofLas follows:

From (24)，we get a simple and exact formula for determining the attributets weights as follows:

4.3 MAGDM method for 2DULVs

The steps for solving the MAGDM of 2DULVs are as follows:

Step 1Establish the distance matrix between experts;

Step 2Calculate the weight of experts;

Step 3Aggregate evaluation information given by experts;

Step 4Establish the distance matrix between alternatives;

Step 5Calculate the weight of attributes;

Step 6Aggregate attribute information;

Step 7Rank each alternative.

5. An illustrated example

Example 2This example is adopted from [39]. A practical use of the proposed approach involves the technological innovation ability evaluation of four enterprises{AS1，AS2，AS3，AS4}，the attributes are shown as follows:the ability of innovative resources input (C1)，the ability of innovation management (C2)，the ability of innovation tendency (C3) and the ability of research and development (C4). Based on the four attributes，three experts{E1，E2，E3}evaluate the technological innovation ability of the four enterprises. λ =(λ1，λ2，λ3)Tis the weight vector of the three experts，which is completely unknown.ω=(ω1，ω2，ω3，ω4)Tis the weight vector of the four attributes，which is completely unknown. The attribute values given by the experts take the form of 2DULVs，which are shown in Tables 1-3. The experts utilize I class linguistic setSI= {S0，S1，S2，S3，S4，S5，S6} and the II class linguistic setRank the four enterprises based on their technological innovation ability.

Step 1Establish the distance matrix between experts.

We establish the distance matrixbetween experts according to (17)，whererepresents the distance between the evaluation information ofEkand the evaluation information ofElwith respect tosij(see Tables 1-3).

Table 1 Attribute values with respect to four enterprises given by expert E1

Table 2 Attribute values with respect to four enterprises given by expert E2

Table 3 Attribute values with respect to four enterprises given by expert E3

6. Conclusions

The score function and distance formula of 2DULVs are two important criteria in the MAGDM problem.However，the existing score function of 2DULVs cannot compare two 2DULVs with the same product of the median of two dimensional linguistic intervals. On the other hand，the existing distance formula of 2DULVs is imprecise in some cases. To overcome these disadvantages，this paper proposes a new scoring function and distance formula of 2DULVs. Comparing with the existing score function and the distance formula of 2DULVs，it is more accurate. On this basis，we propose a method with completely unknown weight information under 2DULVs based on the score function and the distance formula. In further research，it is necessary and meaningful to propose the score function of 2DULVs based on non-normal distribution.