Model for evaluating the industrial structure transfer capability with linguistic information

Abstract

The conversion of the industrial structure influences the conversion of the entire economic structure, which involves all aspects of social life inducing consumption, employment and industry and ultimately affects the economic development as a whole. Conversion capacity of the industrial structure is the force to rationalize and sophisticated the industrial structure, the strength of which is directly related to the rationalization and sophistication of industrial structure, the feasibility of upgrading and even the economic development of the region as a whole. In this paper, we investigate the multiple attribute decision making (MADM) problems with fuzzy linguistic information. Motivated by the ideal of generalized Bonferroni mean, we develop the generalized fuzzy linguistic Bonferroni Mean (GFLBM) operator for aggregating the fuzzy linguistic information. For the situations where the input arguments have different importance, we then define the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator, based on which we develop the procedure for multiple attribute decision making under the fuzzy linguistic environments. Finally, a practical example for evaluating the industrial structure transfer capability is proposed to testify the method in this paper.

Keywords

Multiple attribute decision making fuzzy linguistic variables generalized fuzzy linguistic Bonferroni Mean (GFLBM) operator generalized fuzzy linguistic weighted Bonferroni Mean(GFLWBM) operator industrial structure transfer capability

1 Introduction

In the real world, human beings are constantly making decisions under linguistic environment [1 –12]. For example, when evaluating the “comfort” or “design” of a car, linguistic terms like “good”, “fair”, “poor” are usually be used [1]. Sometimes, however, the decision makers are willing or able to provide only triangular fuzzy linguistic information because of time pressure, lack of knowledge, or data, and their limited expertise related to the problem domain [8]. Thus, Xu [8] developed some operators for aggregating triangular fuzzy linguistic variables, such as the fuzzy linguistic averaging (FLA) operator, fuzzy linguistic weighted averaging (FLWA) operator, fuzzy linguistic ordered weighted averaging (FLOWA) operator, and induced FLOWA (IFLOWA) operator, etc. Wei [13] investigated the multiple attribute group decision making problem with triangular fuzzy linguistic information, in which the attribute weights and expert weights take the form of real numbers, and the preference values take the form of triangular fuzzy linguistic variables and proposed some operators for aggregating triangular fuzzy linguistic variables, such as the fuzzy linguistic harmonic mean (FLHM) operator, fuzzy linguistic weighted harmonic mean (FLWHM) operator, fuzzy linguistic ordered weighted harmonic mean (FLOWHM) operator, and fuzzy linguistic hybrid harmonic mean (FLHHM) operator are proposed.

Yager [14] provided an interpretation of BM operator as involving a product of each argument with the average of the other arguments, a combined averaging and “anding” operator. Beliakov et al. [15] presented a composed aggregation technique called the generalized Bonferroni mean (GBM) operator, which models the average of the conjunctive expressions and the average of remaining. In fact, they extended the BM operator by considering the correlations of any three aggregated arguments instead of any two. However, both BM operator and the GBM operator ignore some aggregation information and the weight vector of the aggregated arguments. To overcome this drawback, Xia et al. [16] developed the generalized weighted Bonferroni mean (GWBM) operator as the weighted version of the GBM operator. Based on the GBM operator and geometric mean operator, they also developed the generalized Bonferoni geometric mean (GWBGM) operator. The fundamental characteristic of the GWBM operator is that it focuses on the group opinions, while the GWBGM operator gives more importance to the individual opinions. Because of the usefulness of the aggregation techniques, which reflect the correlations of arguments, most of them have been extended to fuzzy, intuitionistic fuzzy, or hesitant fuzzy environment [17 –24].

Along with the approach of the knowledge-economy era, adjustment and optimization of industrial structure in western region is critical for the western region to become a new economic belt of increase that can stimulate the whole economic development. One of the important part is knowledge of the present situation of industrial structure, the understanding of the industrial structure transfer capability is the most important thing even more, the level of the transfer capability of the industrial structure is related to the economic development, nowadays, this is the one of the main contents of regional economic research, receive the general attention of the regional economists. The western region of china holds plenty of resources, but the economy of it is lagged down apparently-the gross domestic products is the 18.6% of total, but the territory and population take 71.4%, 27.4% respectively. Since the 1980’s, the economic increase in the West has been lower than the other regions, and the gap between the East and the West is wider and wider. Besides the influence of location and policies, the key reason of such difference is that the western region industrial structure doesn’t adapt to its economy particularly. In order to develop the western region well, we must adjust and improve the t the industrial structure transfer capability to meet the economic development and enhance the west performance, and therefore it is necessary to pay more attention to the research on the industrial structure transfer capability in the West. The thesis’s study train of thought is arranged as follows. First, explain the meaning of the industrial structure transfer capability, including industrial structure rationalization and optimization in detail, then narrate the function of the industrial structure transfer capability in regional economic development and understand the importance of the industrial structure transfer capability, thus analyze in depth and advance theory foundation that the industrial structure transfer capability relied on. Then analyze in depth target factor influencing the industrial structure transfer capability, establish the system of evaluation target, and then appraise the industrial structure transfer capability in every western regions. After that probe into the reason that the industrial structure transfer capability of western region is lower than others regions and carry out concrete analysis. At last, advance the suggestion and measure that solved the low the industrial structure transfer capability of western region. In this paper, we investigate the multiple attribute decision making (MADM) problems [25 –39] with fuzzy linguistic information. Motivated by the ideal of generalized Bonferroni mean, we develop the generalized fuzzy linguistic Bonferroni Mean (GFLBM) operator for aggregating the fuzzy linguistic information. For the situations where the input arguments have different importance, we then define the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator, based on which we develop the procedure for multiple attribute decision making under the fuzzy linguistic environments. Finally, a practical example for evaluating the industrial structure transfer capability is proposed to testify the method in this paper.

2 Preliminaries

2.1 Fuzzy Linguistic Variables

LetS ={ s_i |i = 1, 2, ⋯ , t } be a linguistic term set with odd cardinality. Any label,s_i represents a possible value for a linguistic variable, and it should satisfy the following characteristics: ➀ The set is ordered:s_i > s_j, ifi > j; ➁ There is the reciprocal operator:rec (s_i) = s_j such thati = t + 1 - j; ➂ Max operator: max(s_i,s_j) = s_i, ifs_i ≥ s_j; ➃ Min operator: min(s_i,s_j) = s_i, ifs_i ≤ s_j. For example, S can be defined as [5] $\begin{matrix} S & = & {s_{1} = extremely poor, s_{2} = very poor, \\ s_{3} = poor, s_{4} = medium, s_{5} = good, \\ s_{6} = very good, s_{7} = extremely good} \end{matrix}$

To preserve all the given information, we extend the discrete term set S to a continuous term set $\bar{S} = {s_{a} | s_{1} \leq s_{a} \leq s_{q}, a \in [1, q]}$ , whereq is a sufficiently large positive integer. Ifs_a ∈ S, then we calls_a the original linguistic term, otherwise, we calls_a the virtual linguistic term. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation [5].

Definition 1. Let $s_{α}, s_{β} \in \bar{S}$ , then we call $d (s_{α}, s_{β}) = | α - β |$ (1) the distance betweens_α ands_β [8].

In the following we introduce the concept of fuzzy linguistic variable.

Definition 2. Let $\tilde{s} = [s_{α}, s_{β}, s_{γ}] \in \tilde{S}$ , where $s_{α}, s_{β}, s_{γ} \in \bar{S}$ ,s_α,s_β ands_γ are the lower, modal and upper values of $\tilde{s}$ , respectively, then we call $\tilde{s}$ a fuzzy linguistic variable, which is characterized by the following member function [8] $μ_{\tilde{s}} (θ) = {\begin{matrix} 0, & s_{1} \leq s_{θ} \leq s_{α} \\ d (s_{θ}, s_{α}) / - d (s_{β}, s_{α}), & s_{α} \leq s_{θ} \leq s_{β} \\ d (s_{θ}, s_{γ}) / - d (s_{β}, s_{γ}), & s_{β} \leq s_{θ} \leq s_{γ} \\ 0, & s_{γ} \leq s_{θ} \leq s_{q} \end{matrix}$ (2)

Clearly,s_β gives the maximal grade of $μ_{\tilde{s}} (θ) (μ_{\tilde{s}} (θ) = 1)$ ,s_α ands_γ are the lower and upper bounds with limit the field of the possible evaluation. Especially, ifs_α = s_β = s_γ, then $\tilde{s}$ is reduced to a linguistic variable.

Let $\tilde{S}$ be the set of all fuzzy linguistic variables. Consider any three fuzzy linguistic variables $\tilde{s} = (s_{α}, s_{β}, s_{γ})$ , ${\tilde{s}}_{1} = (s_{α_{1}}, s_{β_{1}}, s_{γ_{1}})$ , ${\tilde{s}}_{2} = (s_{α_{2}}, s_{β_{2}}, s_{γ_{2}}) \in \tilde{S}$ , and suppose thatλ ∈ [0, 1], then we define their operational laws as follows:

${\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = (s_{α_{1}}, s_{β_{1}}, s_{γ_{1}}) \otimes (s_{α_{2}}, s_{β_{2}}, s_{γ_{2}}) = (s_{α_{1} α_{2}}, s_{β_{1} β_{2}}, s_{γ_{1} γ_{2}})$ ;

${\tilde{s}}^{λ} = {(s_{α}, s_{β}, s_{γ})}^{λ} = (s_{α^{λ}}, s_{β^{λ}}, s_{γ^{λ}})$ ;

${\tilde{s}}^{- 1} = {(s_{α}, s_{β}, s_{γ})}^{- 1} = (s_{1 / - γ}, s_{1 / - β}, s_{1 / - α})$ .

In the following, we introduce a formula for comparing fuzzy linguistic variables.

Definition 3. Let ${\tilde{s}}_{1} = (s_{α_{1}}, s_{β_{1}}, s_{γ_{1}})$ , ${\tilde{s}}_{2} = (s_{α_{2}}, s_{β_{2}}, s_{γ_{2}}) \in \tilde{S}$ , then the degree of possibility of ${\tilde{s}}_{1} \geq {\tilde{s}}_{2}$ is defined as [8]

$\begin{matrix} p ({\tilde{s}}_{1} \geq {\tilde{s}}_{2}) & = & λ max {1 - max [d (s_{β_{2}}, s_{α_{1}}) / - (d (s_{β_{1}}, s_{α_{1}}) + d (s_{β_{2}}, s_{α_{2}})), 0], 0} \\ + (1 - λ) max {1 - max [d (s_{γ_{2}}, s_{β_{1}}) / - (d (s_{γ_{1}}, s_{β_{1}}) + d (s_{γ_{2}}, s_{β_{2}})), 0], 0} \end{matrix}$ (3) where the valueλ is an index of rating attitude. It reflects the decision maker’s risk-bearing attitude. Ifλ < 0.5, the decision maker is risk lover. Ifλ = 0.5, the decision maker is neutral to risk. Ifλ > 0.5, the decision maker is risk avertor.

From Definition 3, we can easily get the following results easily:

$0 \leq p ({\tilde{s}}_{1} \geq {\tilde{s}}_{2}) \leq 1, 0 \leq p ({\tilde{s}}_{2} \geq {\tilde{s}}_{1}) \leq 1$ ;

$p ({\tilde{s}}_{1} \geq {\tilde{s}}_{2}) + p ({\tilde{s}}_{2} \geq {\tilde{s}}_{1}) = 1$ . Especially, $p ({\tilde{s}}_{1} \geq {\tilde{s}}_{1}) = p ({\tilde{s}}_{2} \geq {\tilde{s}}_{2}) = 0.5$ .

2.2 Bonferroni mean

Bonferroni [40] originally introduced a mean type aggregation operator, called Bonferroni mean, which can provide for aggregation lying between the max, min operators and the logical “or” and “and” operators, which was defined as follows:

Definition 4. [40] Letp,q ≥ 0 anda_i (i = 1, 2, ⋯ ,n) be a collection of non-negative real numbers. Then the aggregaton functions: $\begin{array}{l} B M^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}} \end{array}$ (4) is called the Bonferroni mean (BM) operator.

Beliakov et al. [15] further extended the BM operator by considering the correlations of any three aggregated arguments instead of any two.

Definition 5. Letp,q,r ≥ 0 anda_i (i = 1, 2, ⋯ ,n) be a collection of nonnegative numbers. If $\begin{array}{l} G B M^{p, q, r} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1) (n - 2)} \sum_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} a_{i}^{p} a_{j}^{q} a_{k}^{r})}^{\frac{1}{p + q + r}} \end{array}$ (5) thenGBM^p,q,r is called the generalized Bonferroni mean (GBM) operator.

In particular, ifr = 0, then the GBM operator reduces to the BM operator. However, it is noted that both BM operator and the GBM operator do not consider the situation thati = j orj = k ori = k, and the weight vector of the aggregated arguments is

not also considered. To overcome this drawback, Xia et al. [15] defined the weighted version of the GBM operator.

Definition 6. Letp,q,r ≥ 0 anda_i (i = 1, 2, ⋯ ,n) be a collection of nonnegative numbers with the weight vectorw = (w₁,w₂, ⋯ ,w_n) ^T andw_j > 0, $\sum_{j = 1}^{n} w_{j} = 1$ . If $\begin{array}{l} G W B M^{p, q, r} (a_{1}, a_{2}, \dots, a_{n}) \\ = {( \frac{1}{n (n - 1) (n - 2)} \sum_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} w_{i} w_{j} w_{k} a_{i}^{p} a_{j}^{q} a_{k}^{r})}^{\frac{1}{p + q + r}} \end{array}$ (6) thenGWBM^p,q,r is called the generalized weighted Bonferroni mean (GWBM) operator.

3 Generalized fuzzy linguistic weighted Bonferroni mean operator

Based on the well-known generalized Bonferroni Mean operator, in the following, we shall develop the generalized fuzzy linguistic Bonferroni Mean (GFLBM) operator to deal with fuzzy linguistic information.

Definition 7. Let ${\tilde{s}}_{i} = [s_{α_{i}}, s_{β_{i}}, s_{γ_{i}}] (i = 1, 2, \dots, n)$ be a set of fuzzy linguistic variables, and letp,q,r ≥ 0. If $\begin{array}{l} {GFLBM}^{p, q, r} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} ({\tilde{s}}_{i}^{p} \otimes {\tilde{s}}_{j}^{q} \otimes {\tilde{s}}_{k}^{r}))}^{\frac{1}{p + q + r}} \\ = [{(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} ({(s_{α_{i}})}^{p} \otimes {(s_{α_{j}})}^{q} \otimes {(s_{α_{k}})}^{r}))}^{\frac{1}{p + q + r}}, \\ {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} ({(s_{β_{i}})}^{p} \otimes {(s_{β_{j}})}^{q} \otimes {(s_{β_{k}})}^{r}))}^{\frac{1}{p + q + r}}, \\ {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} ({(s_{γ_{i}})}^{p} \otimes {(s_{γ_{j}})}^{q} \otimes {(s_{γ_{k}})}^{r}))}^{\frac{1}{p + q + r}},] \end{array}$ (7) then GFLBM^p,q,r is called the generalized fuzzy linguistic Bonferroni Mean (GFLBM)operator.

Considering that the input arguments may have different importance, here we define the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator.

Definition 8. Let ${\tilde{s}}_{i} = [s_{α_{i}}, s_{β_{i}}, s_{γ_{i}}] (i = 1, 2, \dots, n)$ be a set of fuzzy linguistic variables, and letp,q,r ≥ 0.w = (w₁,w₂, ⋯ ,w_n) ^T is the weight vector of ${\tilde{s}}_{i} = [s_{α_{i}}, s_{β_{i}}, s_{γ_{i}}] (i = 1, 2, \dots, n)$ , wherew_i indicates the importance degree of ${\tilde{s}}_{i} = [s_{α_{i}}, s_{β_{i}}, s_{γ_{i}}]$ , satisfyingw_i > 0 (i = 1, 2, ⋯ ,n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{array}{l} {GFLWBM}^{p, q, r} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (w_{i} w_{j} w_{k} {\tilde{s}}_{i}^{p} \otimes {\tilde{s}}_{j}^{q} \otimes {\tilde{s}}_{k}^{r}))}^{\frac{1}{p + q + r}} \\ = [{(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (w_{i} w_{j} w_{k} {(s_{α_{i}})}^{p} \otimes {(s_{α_{j}})}^{q} \otimes {(s_{α_{k}})}^{r}))}^{\frac{1}{p + q + r}}, \\ {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (w_{i} w_{j} w_{k} {(s_{β_{i}})}^{p} \otimes {(s_{β_{j}})}^{q} \otimes {(s_{β_{k}})}^{r}))}^{\frac{1}{p + q + r}}, \\ {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (w_{i} w_{j} w_{k} {(s_{γ_{i}})}^{p} \otimes {(s_{γ_{j}})}^{q} \otimes {(s_{γ_{k}})}^{r}))}^{\frac{1}{p + q + r}},] \end{array}$ (8) then GFLWBM^p,q,r is called the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator.

Some special cases can be obtained as the change of the parameters as follows.

Ifr = 0, then the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator reduces to the fuzzy linguistic weighted Bonferroni Mean (FLWBM) operator.

Ifr = 0, q = 0, the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator reduces to the following: $\begin{array}{l} {GFLWBM}^{p, 0, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} i, j, k = 1 \\ i \neq j \neq k \end{matrix}}^{n} (w_{i} w_{j} w_{k} {\tilde{s}}_{i}^{p} \otimes {\tilde{s}}_{j}^{0} \otimes {\tilde{s}}_{k}^{0}))}^{\frac{1}{p + 0 + 0}} = [{(\frac{1}{n} \oplus_{i = 1}^{n} (w_{i} {(s_{α_{i}})}^{p}))}^{\frac{1}{p}}] \end{array}$ (9)

4 An approach to multiple attribute decision making under fuzzy linguistic environment

In this section, we shall utilize the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator to multiple attribute decision making with fuzzy linguistic information. For a multiple attribute decision making problems with fuzzy linguistic information, letA ={ A₁,A₂, ⋯ ,A_m } be a discrete set of alternatives,G ={ G₁,G₂, ⋯ ,G_n } be the set of attributes, whose weight vector isω = (ω₁,ω₂, ⋯ ,ω_n), withω_j ≥ 0,j = 1, 2, ⋯ ,n,

$\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n}$ is decision matrix, where ${\tilde{r}}_{ij} \in \tilde{S}$ is a preference value, which take the form of fuzzy linguistic variable, given by the decision maker for the alternativeA_i ∈ A with respect to the attributeG_j ∈ G.

Then, we utilize the generalized fuzzy linguistic Bonferroni Mean (GFLWBM) operator to develop an approach to multiple attribute decision making problems with fuzzy linguistic information, which can be described as following:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the GFLWBM operator(in general, we can takep = q = r = 1) $\begin{array}{l} {\tilde{r}}_{i} = [s_{α_{i}}, s_{β_{i}}, s_{γ_{i}}] \\ = {GFLWBM}^{p, q, r} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{i n}) \\ = {( \frac{1}{n (n - 1) (n - 2)} \oplus_{\begin{matrix} m, s, t = 1 \\ m \neq s \neq t \end{matrix}}^{n} (w_{m} w_{s} w_{t} {\tilde{s}}_{i m}^{p} \otimes {\tilde{s}}_{i s}^{q} \otimes {\tilde{s}}_{i t}^{r}) )}^{ \frac{1}{p + q + r}} \\ i = 1, 2, \dots, m . \end{array}$ (10)

to derive the overall fuzzy linguistic preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternativeA_i.

Step 2. To rank these collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ , we first compare each ${\tilde{r}}_{i}$ with all the ${\tilde{r}}_{j} (j = 1, 2, \dots, m)$ by usingEquation (3). For simplicity, we let $p_{ij} = p ({\tilde{r}}_{i} \geq {\tilde{r}}_{j})$ , then we develop a complementary matrix asP = (p_ij) _m×m, wherep_ij ≥ 0,p_ij + p_ji = 1,p_ii = 0.5,i,j = 1, 2, ⋯ ,n.

Summing all the elements in each line of matrixP, we have $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m .$

Then we rank the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ in descending order in accordance with the values ofp_i (i = 1, 2, ⋯ ,m).

Step 3. Rank all the alternativesA_i (i = 1, 2, ⋯ ,m) and select the best one(s) in accordance with the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ .

Step 4. End.

5 Illustrative example

The industrial structure is the core of economic structure, the industrial structure’s comprehensive quality has reflected regional economic development level. The regional disparity, if said is the total quantity difference, would rather said is the structure nature condition difference. Looking from the long economy developing process that, the economic growth and the industrial structure change is a cause and effect relationship, the quicker economic growth, the higher rate of conversion of the industrial structure, the industrial structure changes along with the economic growth, and also has a significant contribution to economic growth. Regional industrial structure transformation whether is timely and reasonable, has a relationship to economic sustainability, stability and healthy development, and relates to the economy development efficiency too. Especially when a regional economic development conditions changed significantly, we need to study on the transformation about industrial structure. Let us suppose there is an investment company, which wants to invest a sum of money in the best option. There is a panel with five possible provinces to invest the money. The investment company must take a decision according to the following four attributes: ➀ G₁ is the technical innovation; ➁ G₂ is the supply capacity; ➂ G₃ is the the development of foreign trade; ➃ G₄ is the demand. The five possible provincesA_i (i = 1, 2, ⋯ , 5) are to be evaluated using the linguistic term setS by the decision makers under the above four attributes, and construct, respectively, the decision matrices as follows $\tilde{R} = {({\tilde{r}}_{ij})}_{5 \times 4}$ : $\tilde{R} = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} (\begin{matrix} (s_{1}, s_{2}, s_{3}) & (s_{1}, s_{4}, s_{5}) & (s_{1}, s_{2}, s_{3}) & (s_{3}, s_{5}, s_{6}) \\ (s_{1}, s_{3}, s_{4}) & (s_{2}, s_{4}, s_{5}) & (s_{2}, s_{3}, s_{4}) & (s_{2}, s_{3}, s_{5}) \\ (s_{2}, s_{4}, s_{6}) & (s_{1}, s_{2}, s_{3}) & (s_{1}, s_{3}, s_{5}) & (s_{2}, s_{3}, s_{4}) \\ (s_{2}, s_{3}, s_{5}) & (s_{3}, s_{4}, s_{5}) & (s_{3}, s_{5}, s_{6}) & (s_{1}, s_{2}, s_{3}) \\ (s_{2}, s_{4}, s_{5}) & (s_{4}, s_{5}, s_{6}) & (s_{1}, s_{3}, s_{4}) & (s_{3}, s_{4}, s_{5}) \end{matrix})$

Then, we utilize the approach developed to get the most desirable province(s).

Step 2. Utilize the GFLWBM operator (letω = (0.3, 0.1, 0.2, 0.4) ^T), we get $\begin{matrix} {\tilde{r}}_{1} & = & (s_{1.45}, s_{2.53}, s_{3.87}) \\ {\tilde{r}}_{2} & = & (s_{1.72}, s_{2.56}, s_{4.22}) \\ {\tilde{r}}_{3} & = & (s_{2.62}, s_{4.12}, s_{5.09}) \\ {\tilde{r}}_{4} & = & (s_{2.34}, s_{3.23}, s_{4.67}) \\ {\tilde{r}}_{5} & = & (s_{2.55}, s_{4.23}, s_{5.09}) \end{matrix}$

Step 3. Suppose that the decision maker’s attitude neutral to the risk, i.e.,ρ = 0.5, then by usingEquation (3), and develop a complementary matrix: $P = [\begin{matrix} 0.500 & 0.373 & 0.005 & 0.163 & 0.025 \\ 0.627 & 0.500 & 0.077 & 0.279 & 0.103 \\ 0.995 & 0.923 & 0.500 & 0.714 & 0.545 \\ 0.837 & 0.721 & 0.286 & 0.500 & 0.323 \\ 0.975 & 0.897 & 0.455 & 0.677 & 0.500 \end{matrix}]$

Summing all the elements in each line of matrixP, we have $\begin{matrix} p_{1} & = & 1.045, p_{2} = 1.575 \\ p_{3} & = & 3.609, p_{4} = 2.633 p_{5} = 3.954 . \end{matrix}$

Then we rank the overall preference values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ in descending order in accordance with the values ofp_i (i = 1, 2, ⋯ , 5): ${\tilde{r}}_{3} ≻ {\tilde{r}}_{5} ≻ {\tilde{r}}_{4} ≻ {\tilde{r}}_{2} ≻ {\tilde{r}}_{1}$ .

Step 4. Rank all the provincesA_i (i = 1, 2, ⋯ , 5) in accordance with the overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ :A₃ ≻ A₅ ≻ A₄ ≻ A₂ ≻ A₁, and thus the most desirable province isA₃.

6 Conclusion

In this paper, we investigate the multiple attribute decision making (MADM) problems with fuzzy linguistic information. Motivated by the ideal of generalized Bonferroni mean, we develop the generalized fuzzy linguistic Bonferroni Mean (GFLBM) operator for aggregating the fuzzy linguistic information. For the situations where the input arguments have different importance, we then define the generalized fuzzy linguistic weighted Bonferroni Mean (GFLWBM) operator, based on which we develop the procedure for multiple attribute decision making under the fuzzy linguistic environments. Finally, a practical example for evaluating the industrial structure transfer capability is proposed to testify the method in this paper. In the future, we shall continue working in the developed models to other application domains and other environments [41 –55].

Footnotes

Acknowledgments

The paper is supported the National Natural Science Foundation of China (Grant No. 71373059).

References

Delgado

and Herrera

, A communication model based on the 2-tuple fuzzy linguistic representation for a distributed intelligent agent system on internet, Soft Computing 6 (2002), 320–328.

Herrera

and Martínez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000), 746–752.

Z.S.

,Uncertain Multiple Attribute Decision Making: Methods and Applications.Beijing:Tsinghua University Press,2004.(in Chinese).

Z.S.

,Group decision making with triangular fuzzy linguistic variables, Yin

et al. (Eds.):IDEAL 2007, LNCS 4881, pp.17–26.

Z.S.

, On the method of multi-attribute group decision making under pure linguistic information, Control and Decision 19(7) (2004), 778–782.

Herrera

and Herrera-Viedma

, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems 115(10) (2000), 67–82.

Wei

G.W.

, Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Expert Systems with Applications 38(5) (2011), 4824–4828.

Z.S.

, Harmonic mean operators for aggregating linguistic information, Proceedings of the 4th International Conference on Natural Computation,Shandong,2008, pp.204–208.

Z.S.

, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences 166(1) (2004), 19–30.

10.

Z.S.

, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Science 168 (2004), 171–184.

11.

Zhou

L.Y.

, Lin

, Zhao

X.F.

and Wei

G.W.

, Uncertain linguistic prioritized aggregation operators and their application to multiple attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(4) (2013), 603–627.

12.

Wei

G.W.

, Uncertain linguistic hybrid geometric mean operator and its Application to group decision making under uncertain linguistic environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17(2) (2009), 251–267.

13.

Wei

G.W.

and Yi

W.D.

, Fuzzy linguistic hybrid harmonic mean operator and its application to software selection, Journal of Software 4(9) (2009), 1037–1042.

14.

Yager

R.R.

, On generalized Bonferroni mean operators for multi-criteria aggregation, International Journal of Approximate Reasoning 50(8) (2009), 1279–1286.

15.

Beliakov

, James

, Mordelova

, Ŕuckschlossova

and Yager

R.R.

, Generalized Bonferroni mean operators in multicriteria aggregation, Fuzzy Sets and Systems 161(17) (2010), 2227–2242.

16.

Xia

, Xu

and Zhu

, Generalized intuitionistic fuzzy Bonferroni means, International Journal of Intelligent Systems 27(1) (2012), 23–47.

17.

Tan

and Chen

, Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert Systems with Applications 37(1) (2010), 149–157.

18.

, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences 180(5) (2010), 726–736.

19.

and Yager

R.R.

, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems, Man, and Cybernetics B 41(2) (2011), 568–578.

20.

, Wu

and Zhou

, Generalized hesitant fuzzy bonferroni mean and its application in multi-criteria group decision making, Journal of Information and Computational Science 9(2) (2012), 267–274.

21.

, Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators, Knowledge-Based Systems 24(6) (2011), 749–760.

22.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27 (2014), 2153–2162.

23.

Wei

G.W.

, Zhao

X.F.

, Lin

and Wang

H.J.

, Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Applied Mathematical Modelling 37 (2013), 5277–5285.

24.

, Wei

G.W.

, Alsaadi

Fuad E.

, Hayat

Tasawar

, Alsaedi

Ahmed

, Bipolar 2-tuple linguistic aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1197–1207.

25.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure, International Journal of Fuzzy Systems 19(3) (2017), 607–614.

26.

Wei

G.W.

, Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems 46 (2013), 43–53.

27.

Wei

G.W.

and Wang

J.M.

, A comparative study of robust efficiency analysis and data envelopment analysis with imprecise data, Expert Systems with Applications 81 (2017), 28–38.

28.

Singh

, Correlation coefficients for picture fuzzy sets, Journal of Intelligent and Fuzzy Systems 28(2) (2015), 591–604.

29.

Zhao

X.F.

and Wei

G.W.

, Some intuitionistic fuzzy einstein hybrid aggregation operators and their application to multiple attribute decision making, Knowledge-Based Systems 37 (2013), 472–479.

30.

Wei

G.W.

and Zhao

X.F.

, Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making, Expert Systems with Applications 39 (2012), 5881–5886.

31.

Merigó

J.M.

and Gil-Lafuente

A.M.

, Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making, Information Sciences 236(1) (2013), 1–16.

32.

Wei

G.W.

and Zhao

X.F.

, Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making, Expert Systems with Applications 39(2) (2012), 2026–2034.

33.

Merigo

J.M.

and Gil-Lafuente

A.M.

, Fuzzy induced generalized aggregation operators and its application in multi-person decision making, Expert Systems with Applications 38(8) (2011), 9761–9772.

34.

Wei

G.W.

, Picture fuzzy aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 713–724.

35.

Wei

G.W.

, Hesitant Fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems 31 (2012), 176–182.

36.

Robinson

J.P.

and Henry Amirtharaj

E.C.

, MAGDM problems with correlation coefficient of triangular fuzzy IFS, International Journal of Fuzzy System Applications 4(1) (2015), 1–32.

37.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning and Cybernetics 7(6) (2016), 1093–1114.

38.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17(4) (2016), 491–502.

39.

Liao

, Decision method of optimal investment enterprise selection under uncertain information environment, International Journal of Fuzzy System Applications 4(1) (2015), 33–42.

40.

Bonferroni

, Sulle medie multiple di potenze, Bolletino Matematica Italiana 5 (1950), 267–270.

41.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

42.

Ran

L.G.

and Wei

G.W.

, Uncertain prioritized operators and their application to multiple attribute group decision making, Technological and Economic Development of Economy 21(1) (2015), 118–139.

43.

Mohammadian

, Intelligent decision making and risk analysis of B2c E-commerce customer satisfaction, International Journal of Fuzzy System Applications 4(1) (2015), 60–76.

44.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems 17(3) (2015), 484–489.

45.

Ojokoh

B.A.

, Omisore

O.M.O.

, Samuel

O.W.

and Otunniyi

, A fuzzy-based recommender system for electronic products selection using users’ requirements and other users’ opinion, International Journal of Fuzzy System Applications 4(1) (2015), 77–87.

46.

Wei

G.W.

, Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making, Expert Systems with Applications 38(9) (2011), 11671–11677.

47.

Wei

G.W.

, Chen

and Wang

J.M.

, Stochastic efficiency analysis with a reliability consideration, Omega 48 (2014), 1–9.

48.

Kharola

, Design of a hybrid adaptive neuro fuzzy inference system (ANFIS) controller for position and angle control of inverted pendulum (IP) systems, International Journal of Fuzzy System Applications 5(1) (2016), 27–42.

49.

Lin

, Zhao

X.F.

and Wei

G.W.

, Models for selecting an ERP system with hesitant fuzzy linguistic information, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2155–2165.

50.

Nan

, Wang

and An

, Intuitionistic fuzzy distance based TOPSIS method and application to MADM, International Journal of Fuzzy System Applications 5(1) (2016), 43–56.

51.

Wang

H.J.

, Zhao

X.F.

and Wei

G.W.

, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 26(5) (2014), 2281–2290.

52.

Subramanian

and Savarimuthu

, Cloud service evaluation and selection using fuzzy hybrid MCDM approach in marketplace, International Journal of Fuzzy System Applications 5(2) (2016), 120–154.

53.

Q.X.

, Zhao

X.F.

and Wei

G.W.

, Model for software quality evaluation with hesitant fuzzy uncertain linguistic information, Journal of Intelligent and Fuzzy Systems 26(6) (2014), 2639–2647.

54.

, Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment, European Journal of Operational Research 205 (2010), 202–204.

55.

, Wei

G.W.

, Alsaadi

Fuad E.

, Hayat

Tasawar

, Alsaedi

Ahmed

, Hesitant Pythagorean Fuzzy Hamacher Aggregation Operators and Their Application to Multiple Attribute Decision Making, Journal of Intelligent and Fuzzy Systems 33(2) (2017), 1105–1117.