Research on the management performance appraisal for the transnational corporation with 2-tuple linguistic information

Abstract

In this paper, we investigate the multiple attribute decision making problems for evaluating the management performance for the transnational corporation with 2-tuple linguistic information. Motivated by the ideal of generalized weighted Bonferroni mean and dual generalized weighted Bonferroni mean, we develop the 2-tuple linguistic generalized Bonferroni mean (2TLGBM) operator and the 2-tuple linguistic dual generalized Bonferroni mean (2TLDGBM) operator for aggregating the 2-tuple linguistic information. For the situations where the input arguments have different importance, we then define the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator and the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM), based on which we develop the procedure for multiple attribute decision making under the 2-tuple linguistic environments. At last, a numerical example for evaluating the management performance for the transnational corporation is provided to illustrate the proposed method. The result shows the approach is simple, effective and easy to calculate.

Keywords

Multiple attribute decision making 2-tuple linguistic variables generalized Bonferroni mean generalized geometric Bonferroni mean generalized weighted Bonferroni mean management performance transnational corporation

1 Introduction

Multiple attribute decision making is a usual task in human activities. It consists of finding the most preferred alternative from a given alternative set. The increasing complexity of the socio-economic environment makes it less and less possible for a single decision maker to consider all relevant aspects of a problem. As a result, many decision making processes take place in group settings in the real life situation. However, under many conditions, for the real multiple attribute decision making problems, the decision information about alternatives is usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the vagueness of inherent subjective nature of human think, thus, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in linguistic terms [1 –30].

Ever since 1990s, Corporate Social Responsibility (CSR) has been an issue that attracts the attention from enterprises, academia, governments and non-governmental organizations. Multinational Corporations (MNCs) as enterprises operated across cultures and regions, their CSR actions will affect in even a wider scope, thus cause more attention. Before finally propose the new CSR Standards System for MNCs, the author first categorize the current research on CSR for MNCs home and abroad. Generally speaking, currently the number of researches on the CSR issues of MNCs has been increasing. Internationally, researchers mainly focus on using institutional structure difference and stakeholder theories to explain the difference in MNCs’ CSR practices in the home countries and host countries, some empirical studies have been carried out as well. While as the FDI’s increase in both scale and scope in China, there are also more researches on MNCs’ CSR in China as well, but generally speaking the number is not so big, some empirical studies have been done, but mainly based on single cases or focused on descriptive study. Nowadays, the researches in China on MNCs’ CSR are still mainly about the introduction of the CSR standards, especially the international standards, it is lack of the theoretical explanation on the CSR practices of MNCs in China, for the phenomenon of ‘double standards’ adopted by MNCs and the ‘inconsistency’ of MNCs’ CSR practice in home countries and host countries, almost no explanation is provided, let alone presenting a new standards system can be used to direct MNCs’ CSR practice in China.

In this paper, we investigate the multiple attribute decision making problems with 2-tuple linguistic information. Motivated by the ideal of Bonferroni mean [31] and geometric Bonferroni mean [32], we investigate the multiple attribute decision making problems for evaluating the management performance for the transnational corporation with 2-tuple linguistic information. Motivated by the ideal of generalized weighted Bonferroni mean and dual generalized weighted Bonferroni mean, we develop the 2-tuple linguistic generalized Bonferroni mean (2TLGBM) operator and the 2-tuple linguistic dual generalized Bonferroni mean (2TLDGBM) operator for aggregating the 2-tuple linguistic information. For the situations where the input arguments have different importance, we then define the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator and the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM), based on which we develop the procedure for multiple attribute decision making under the 2-tuple linguistic environments. At last, a numerical example for evaluating the management performance for the transnational corporation is provided to illustrate the proposed method. The result shows the approach is simple, effective and easy to calculate.

2 Preliminaries

Herrera [1, 2] first introduced the 2-tuple fuzzy linguistic approach for overcoming the drawback of the classical computational models, which include the semantic model and symbolic model. The 2-tuple linguistic model is a kind of new information processing method. It takes 2-tuple to represent linguistic assessment information and carry out operation. The basic concept of linguistic 2-tuple is symbolic translation. The 2-tuple linguistic representation and computational model has received more and more attention since its appearance.

In the following, we shall introduce the definition of the 2-tuple linguistic representation and computational model.

Let S ={ 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 [1, 2]:

(1) The set is ordered:s_i > s_j, if i > j; (2) Max operator:max(s_i, s_j) = s_i, if s_i ≥ s_j; (3) Min operator: min(s_i, s_j) = s_i, if s_i ≤ s_j. For example, S can be defined as

$\begin{matrix} S = {s_{1} = extremely poor (EP), s_{2} = very poor (VP), \\ s_{3} = poor (P), s_{4} = medium (M), s_{5} = good (G), \\ s_{6} = very good (VG), s_{7} = extremely good (EG)} \end{matrix}$

Herrera and Martinez [1, 2] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple (s_i, α_i), where s_i is a linguistic label from predefined linguistic term set S and α_i is the value of symbolic translation, and α_i ∈ [- 0.5, 0.5) .

Definition 1. [1, 2] Let β be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, β ∈ [1, t], being t the cardinality of S. Let i = round (β) and α = β - i be two values, such that, i ∈ [1, t] and α ∈ [- 0.5, 0.5) then α is called a symbolic translation.

Definition 2. [1, 2] Let S ={ s₁, s₂, ⋯ , s_t } be a linguistic term set and β ∈ [1, t] is a number value representing the aggregation result of linguistic symbolic. Then the function Δ used to obtain the 2-tuple linguistic information equivalent to βis defined as: $Δ : [1, t] \to S \times [- 0.5, 0.5)$ (1) $Δ (β) = {\begin{matrix} s_{i}, & i = round (β) \\ α = β - i, & α \in [- 0.5, 0.5) \end{matrix}$ (2) where round(.) is the usual round operation, s_i has the closest index label to β and α is the value of the symbolic translation.

Definition 3. [1, 2] Let S ={ s₁, s₂, ⋯ , s_t } be a linguistic term set and (s_i, α_i) be a 2-tuple. There is always a function Δ^-1 can be defined, such that, from a 2-tuple (s_i, α_i) it return its equivalent numerical value β ∈ [1, t] ⊂ R, which is. $Δ^{- 1} : S \times [- 0.5, 0.5) \to [1, t]$ (3) $Δ^{- 1} (s_{i}, α) = i + α = β$ (4)

From Definitions 1 and 2, we can conclude that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation: $Δ (s_{i}) = (s_{i}, 0)$ (5)

Definition 4. [1, 2] Let (s_k, a_k) and (s_l, a_l) be two 2-tuple, they should have the following properties:

If k < l then (s_k, a_k) is smaller than (s_l, a_l);

If k = l then

if a_k = a_l, then (s_k, a_k), (s_l, a_l) represents the same information;

if a_k < a_l then (s_k, a_k) is smaller than (s_l, a_l); if a_k > a_l then (s_k, a_k) is bigger than (s_l, a_l).

3 Some generalized Bonferroni mean aggregating operators with 2-tuple linguistic information

Beliakov et al. [33] further extended the BM operator by considering the correlations of any three aggregated arguments instead of any two.

Definition 5. Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If

$\begin{matrix} {GBM}^{p, q, r} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1) (n - 2)} \underset{\begin{array}{l} i, j, k = 1 \\ i \neq j \neq k \end{array}}{\sum^{n}} a_{i}^{p} a_{j}^{q} a_{k}^{r})}^{\frac{1}{p + q + r}} \end{matrix}$ (6) then GBM^p,q,r is called the generalized Bonferroni mean (GBM) operator.

In particular, if r = 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 that i = j or j = k or i = k, and the weight vector of the aggregated arguments is not also considered. To overcome this drawback, Xia et al. [34] defined the weighted version of the GBM operator.

Definition 6. Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T and w_j > 0, $\sum_{j = 1}^{n} w_{j} = 1$ . If $\begin{matrix} {GWBM}^{p, q, r} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\sum_{i, j, k = 1}^{n} w_{i} w_{j} w_{k} a_{i}^{p} a_{j}^{q} a_{k}^{r})}^{\frac{1}{p + q + r}} \end{matrix}$ (7) then GWBM^p,q,r is called the generalized weighted Bonferroni mean (GWBM) operator.

In the following, we shall develop 2-tuple linguistic generalized Bonferroni mean (2TLGBM) operator and 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator.

Definition 7. Let p, q, r ≥ 0 and { (r₁, a₁) , (r₂, a₂) , … , (r_n, a_n) } be a set of 2-tuple linguistic variables, If $\begin{matrix} 2 {TLGBM}^{p, q, r} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\frac{1}{n (n - 1) (n - 2)} \underset{\begin{array}{l} i, j, k = 1 \\ i \neq j \neq k \end{array}}{\sum^{n}} {(Δ^{- 1} (r_{i}, a_{i}))}^{p} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{q} {(Δ^{- 1} (r_{k}, a_{k}))}^{r})}^{\frac{1}{p + q + r}} \end{matrix}$ (8) then 2TLGBM^p,q,r is called the 2-tuple linguistic generalized Bonferroni mean (2TLGBM) operator.

If r = 0, then the 2TLGBM operator reduces to the 2TLBM operator. $\begin{matrix} 2 {TLGBM}^{p, q, 0} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = 2 {TLBM}^{p, q} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\frac{1}{n (n - 1)} \underset{\begin{array}{l} i, j \\ i \neq j \end{array}}{\sum^{n}} {(Δ^{- 1} (r_{i}, a_{i}))}^{p} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (9)

However, it is noted that both 2TLBM operator and the 2TLGBM operator do not consider the situation that i = j or j = k or i = k, and the weight vector of the aggregated arguments is not also considered. To overcome this drawback, we shall propos the weighted version of the 2TLGBM operator.

Definition 8. Let { (r₁, a₁) , (r₂, a₂) , … , (r_n, a_n) } be a set of 2-tuple linguistic variables, and let p, q, r ≥ 0.w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of { (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } (i = 1, 2, ⋯ , n), where w_i indicates the importance degree of (r_i, a_i), satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} 2 TLG W {BM}_{w}^{p, q, r} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\sum_{i, j, k = 1}^{n} {(w_{i} w_{j} w_{k} Δ^{- 1} (r_{i}, a_{i}))}^{p} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{q} {(Δ^{- 1} (r_{k}, a_{k}))}^{r})}^{\frac{1}{p + q + r}} \end{matrix}$ (10) then $2 TLG W {BM}_{w}^{p, q, r}$ is called the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator.

It can be easily proved that the 2TLGWBM operator has the following properties.

Theorem 1. (Idempotency)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } be a set of 2-tuple linguistic variables. If all (r_j, a_j) (j = 1, 2, ⋯ , n) are equal, i.e. (r_j, a_j) = (r, a) for all j, then $\begin{matrix} 2 TLG W {BM}_{w}^{p, q, r} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = (r, a) \end{matrix}$ (11)

Theorem 2. (Boundedness)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } be a set of 2-tuple linguistic variables, and let $(r^{-}, a^{-}) = min_{j} (r_{j}, a_{j})$ , $(r^{+}, a^{+}) = max_{j} (r_{j}, a_{j})$

Then $(r^{-}, a^{-}) \leq 2 T L D G W B M_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \leq (r^{+}, a^{+})$ (12)

Theorem 3. (Monotonicity)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } and $x^{'} = ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'}))$ be two set of 2-tuple linguistic variables, if $(r_{j}, a_{j}) \leq (r_{j}^{'}, a_{j}^{'})$ , for all j, then $\begin{matrix} 2 TLG W {BM}_{w}^{p, q, r} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \leq \\ 2 TLG W {BM}_{w}^{p, q, r} ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'})) \end{matrix}$ (13)

Theorem 4. (Commutativity)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } and $x^{'} = ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'}))$ be two set of 2-tuple, where $((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'}))$ is any permutation of { (r₁, a₁) , (r₂, a₂) , … , (r_n, a_n) }, then $\begin{matrix} 2 TLG W {BM}_{w}^{p, q, r} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) = \\ 2 TLG W {BM}_{w}^{p, q, r} ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'})) \end{matrix}$ (14)

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

If r = 0, then the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator reduces to the 2-tuple linguistic weighted Bonferroni mean (2TLGWBM) operator. $\begin{matrix} {2 TLGWBM}_{w}^{p, q, 0} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = {2 TLWBM}_{w}^{p, q, 0} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\sum_{i, j = 1}^{n} {(w_{i} w_{j} Δ^{- 1} (r_{i}, a_{i}))}^{p} {(Δ^{- 1} (r_{j}, a_{j}))}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (15)

If r = 0, q = 0, the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator reduces to the following:

$\begin{matrix} 2 TLG W {BM}_{w}^{p, 0, 0} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\sum_{i, j, k = 1}^{n} {(w_{i} w_{j} w_{k} Δ^{- 1} (r_{i}, a_{i}))}^{p} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{0} {(Δ^{- 1} (r_{k}, a_{k}))}^{0})}^{\frac{1}{p}} \\ = Δ {(\sum_{i = 1}^{n} {(w_{i} Δ^{- 1} (r_{i}, a_{i}))}^{p})}^{\frac{1}{p}} \end{matrix}$ (16) which is the 2-tuple linguistic generalized weighted averaging (2TLGWA) operator. Furthermore, in this case, let us look at the 2TLGWBM operator for some special cases of p.

If p = 1, the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator reduces to 2-tuple linguistic weighted averaging (2TLWA) operator.

If p → 0, the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator reduces to 2-tuple linguistic weighted geometric (2TLWG) operator.

If p→ ∞, the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator reduces to 2-tuple linguistic max operator.

4 Some Dual generalized Bonferroni mean aggregating operators with 2-tuple linguistic information

Beliakov et al. [33] further extended the BM operator by considering the correlations of any number of aggregated arguments instead of any two.

Definition 5. Let t_j ≥ 0 (i = 1, 2, ⋯ , k) and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If $\begin{matrix} {DGBM}^{t_{j}} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{\prod_{x = 0}^{k - 1} (n - x)} \underset{\begin{array}{l} i_{1}, i_{2}, \dots, i_{k} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{k} \end{array}}{\sum^{n}} (\prod_{j = 1}^{k} a_{i_{j}}^{t_{j}}))}^{1 / \sum_{j = 1}^{k} t_{j}} \end{matrix}$ (17) then DGBM^{r
_j} is called the Dual generalized Bonferroni mean (DGBM) operator.

Definition 6. Let t_j ≥ 0 (j = 1, 2, ⋯ , k) and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T and w_j > 0, $\sum_{j = 1}^{n} w_{j} = 1$ . If $\begin{matrix} {DGWBM}_{w}^{t_{j}} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\sum_{i_{1}, i_{2}, \dots, i_{k} = 1}^{n} (\prod_{j = 1}^{k} ω_{i_{j}} a_{i_{j}}^{t_{j}}))}^{1 / - \sum_{j = 1}^{k} t_{j}} \end{matrix}$ (18) then DGWBM^{r
_j} is called the Dual generalized weighted Bonferroni mean (DGWBM) operator.

In the following, we shall develop 2-tuple linguistic Dual generalized Bonferroni mean (2TLDGBM) operator and 2-tuple linguistic Dual generalized weighted Bonferroni mean (2TLDGWBM) operator.

Definition 7. Let t_j ≥ 0 (j = 1, 2, ⋯ , k) and { (r₁, a₁) , (r₂, a₂) , … , (r_n, a_n) } be a set of 2-tuple linguistic variables, If $\begin{matrix} 2 {TLDGBM}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\frac{1}{\prod_{x = 0}^{k - 1} (n - x)} \underset{\begin{array}{l} i_{1}, i_{2}, \dots, i_{k} = 1 \\ i_{1} \neq i_{2} \neq \dots \neq i_{k} \end{array}}{\sum^{n}} (\prod_{j = 1}^{k} {(Δ^{- 1} (r_{i}, a_{i}))}^{t_{j}}))}^{1 / - \sum_{j = 1}^{k} t_{j}} \end{matrix}$ (19)

then 2TLDGBM^{t
_j} is called the 2-tuple linguistic Dual generalized Bonferroni mean (2TLDGBM) operator.

If t_j = 0 (j = 4, 5, ⋯ , k), then the 2TLDGBM operator reduces to the 2TLGBM operator. $\begin{matrix} 2 {TLDGBM}^{t_{1}, t_{2}, t_{3}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = 2 {TLGBM}^{t_{1}, t_{2}, t_{3}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\frac{1}{n (n - 1) (n - 2)} \underset{\begin{array}{l} i, j, k = 1 \\ i \neq j \neq k \end{array}}{\sum^{n}} {(Δ^{- 1} (r_{i}, a_{i}))}^{t_{1}} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{t_{2}} {(Δ^{- 1} (r_{k}, a_{k}))}^{t_{3}})}^{\frac{1}{t_{1} + t_{2} + t_{3}}} \end{matrix}$ (20)

If t_j = 0 (j = 3, 4, ⋯ , k), then the 2TLDGBM operator reduces to the 2TLBM operator. $\begin{matrix} 2 {TLDGBM}^{t_{1}, t_{2}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = 2 {TLGBM}^{t_{1}, t_{2}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\frac{1}{n (n - 1)} \underset{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}{\sum^{n}} {(Δ^{- 1} (r_{i}, a_{i}))}^{t_{1}} {(Δ^{- 1} (r_{j}, a_{j}))}^{t_{2}})}^{\frac{1}{t_{1} + t_{2}}} \end{matrix}$ (21)

However, it is noted that the 2TLDGBM operator do not consider the weight vector of the aggregated arguments. To overcome this drawback, we shall propos the weighted version of the 2TLDGBM operator.

Definition 8. Let {(r₁, a₁), (r₂, a₂), …, (r_n, a_n)} be a set of 2-tuple linguistic variables, and let t_j ≥ 0 (j = 1, 2, ⋯, k). w = (w₁, w₂, ⋯, w_n) ^T is the weight vector of {(r₁, a₁), (r₂, a₂), …, (r_n, a_n)} (i = 1, 2, ⋯, n), where w_i indicates the importance degree of (r_i, a_i), satisfying w_i > 0 (i = 1, 2, ⋯, n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} 2 {TLDGWBM}_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\sum_{i_{1}, i_{2}, \dots, i_{k} = 1}^{n} (\prod_{j = 1}^{k} ω_{i_{j}} {(Δ^{- 1} (r_{i}, a_{i}))}^{t_{j}}))}^{1 / - \sum_{j = 1}^{k} t_{j}} \end{matrix}$ (22) then $2 TLDG W {BM}_{w}^{t_{j}}$ is called the 2-tuple linguistic Dual generalized weighted Bonferroni mean (2TLDGWBM) operator.

It can be easily proved that the 2TLDGWBM operator has the following properties.

Theorem 1. (Idempotency)

Let x = {(r₁, a₁), (r₂, a₂), …, (r_n, a_n)} be a set of 2-tuple linguistic variables. If all (r_j, a_j) (j = 1, 2, ⋯, n) are equal, i.e. (r_j, a_j) = (r, a) for all j, then $\begin{matrix} 2 TLDG W {BM}_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = (r, a) \end{matrix}$ (23)

Theorem 2. (Boundedness)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } be a set of 2-tuple linguistic variables, and let $(r^{-}, a^{-}) = min_{j} (r_{j}, a_{j})$ , $(r^{+}, a^{+}) = max_{j} (r_{j}, a_{j})$

Then $\begin{matrix} (r^{-}, a^{-}) \leq 2 TLDG W {BM}_{w}^{t_{j}} \\ {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \leq (r^{+}, a^{+}) \end{matrix}$ (24)

Theorem 3. (Monotonicity)

$\begin{matrix} 2 TLDG W {BM}_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \leq \\ 2 TLDG W {BM}_{w}^{t_{j}} {(r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'})} \end{matrix}$ (25)

Theorem 4. (Commutativity)

Let x ={ (r₁, a₁) , (r₂, a₂) , …, (r_n, a_n) } and $x^{'} = ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'}))$ be two set of 2-tuple, where $((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'}))$ is any permutation of { (r₁, a₁) , (r₂, a₂) , … , (r_n, a_n) }, then $\begin{matrix} 2 TLDG W {BM}_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} = \\ 2 TLDG W {BM}_{w}^{t_{j}} {(r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'})} \end{matrix}$ (26)

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

If t_j = 0 (j = 4, 5, ⋯ , k), then the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM) operator reduces to the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator. $\begin{matrix} {2 TLDGWBM}_{w}^{t_{1}, t_{2}, t_{3}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = {2 TLGWBM}_{w}^{t_{1}, t_{2}, t_{3}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ (\sum_{i, j, k = 1}^{n} w_{i} w_{j} w_{k} {(Δ^{- 1} (r_{i}, a_{i}))}^{t_{1}} \\ {{(Δ^{- 1} (r_{j}, a_{j}))}^{t_{2}} {(Δ^{- 1} (r_{k}, a_{k}))}^{t_{3}})}^{\frac{1}{t_{1} + t_{2} + t_{3}}} \end{matrix}$ (27)

If t_j = 0 (j = 3, 4, ⋯ , k), the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM) operator reduces to the following: $\begin{matrix} 2 TLG W {BM}_{w}^{t_{1}, t_{2}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\sum_{i, j = 1}^{n} {(w_{i} w_{j} Δ^{- 1} (r_{i}, a_{i}))}^{t_{1}} {(Δ^{- 1} (r_{j}, a_{j}))}^{t_{2}})}^{\frac{1}{t_{1} + t_{2}}} \end{matrix}$ (28)

If t_j = 0 (j = 2, 3, ⋯ , k), the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM) operator reduces to the following: $\begin{matrix} 2 TLG W {BM}_{w}^{t_{1}, 0, \dots} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\sum_{i = 1}^{n} {(w_{i} Δ^{- 1} (r_{i}, a_{i}))}^{t_{1}})}^{\frac{1}{t_{1}}} \end{matrix}$ (29)

5 An approach to multiple attribute decision making with linguistic information

In this section, we shall utilize the developed operators to multiple attribute decision making.

For a multiple attribute decision making problems with linguistic information, let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, G ={ G₁, G₂, ⋯ , G_n } be the set of attributes, whose weight vector is ω = (ω₁, ω₂, ⋯ , ω_n),with ω_j ≥ 0, j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $R = {(r_{ij})}_{m \times n}$ is the multiple attribute decision making matrix, where r_ij ∈ S is an attribute values, which take the form of linguistic variable, given by the decision maker for the alternative A_i ∈ A with respect to the attribute G_j ∈ G.

In what follows, we shall apply the 2TLGWBM operator to solve the MADM problems with linguistic variables.

Step 1. Transforming linguistic decision matrix $R = {(r_{ij})}_{m \times n}$ into 2-tuple linguistic decision matrix $\tilde{R} = {(r_{ij}, 0)}_{m \times n}$ .

Step 2. We utilize the decision information given in matrix $\tilde{R}$ , and the 2TLGWBM (2TLDGWBM)operator $\begin{matrix} {\tilde{r}}_{i} = 2 TLG W {BM}_{w}^{p, q, r} {(r_{i 1}, 0), (r_{i 2}, 0), \dots, (r_{in}, 0)} \\ = Δ (\sum_{m, s, t = 1}^{n} w_{m} w_{s} w_{t} {(Δ^{- 1} (r_{im}, 0))}^{p} \\ {{(Δ^{- 1} (r_{is}, 0))}^{q} {(Δ^{- 1} (r_{it}, 0))}^{r})}^{\frac{1}{p + q + r}} \end{matrix}$ (30)

OR $\begin{matrix} {\tilde{r}}_{i} = 2 {TLDGWBM}_{w}^{t_{j}} {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} \\ = Δ {(\sum_{m_{1}, m_{2}, \dots, m_{k} = 1}^{n} (\prod_{j = 1}^{k} ω_{m_{j}} {(Δ^{- 1} (r_{im}, a_{im}))}^{t_{j}}))}^{1 / - \sum_{j = 1}^{k} t_{j}} \end{matrix}$ (31) to derive the overall preference values ${\tilde{r}}_{i}$ i = 1, 2, ⋯, m) of the alternative A_i.

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with ${\tilde{r}}_{i} (i$ = 1, 2, ⋯, m). If any alternative has the highest ${\tilde{r}}_{i}$ value, then, it is the most important alternative.

Step 4. End.

6 Numerical example

In this section, we present an empirical case study of evaluating the management performance for the transnational corporations. The management performance of five possible transnational corporations A_i (i = 1, 2, 3, 4, 5) is evaluated according to the following four attributes: G₁ is the costs of management; G₂ is the contribution to organization performance; G₃ is the effort to transform from current system; G₄ is the outsourcing management performance, their weight vector is w = (0.3, 0.2, 0.2, 0.3). The management performance of five possible transnational corporations A_i (i = 1, 2, ⋯ , 5) are to be evaluated using the linguistic variables by the decision maker under the above four attributes, as listed in the following Table 1.

Table 1
Decision matrix R

G ₁ G ₂ G ₃ G ₄

A ₁ G G M EG

A ₂ VP EP EG G

A ₃ P VG VG P

A ₄ VP EP VP M

A ₅ M P G M

	G ₁	G ₂	G ₃	G ₄
A ₁	G	G	M	EG
A ₂	VP	EP	EG	G
A ₃	P	VG	VG	P
A ₄	VP	EP	VP	M
A ₅	M	P	G	M

Then, we utilize the proposed approach to get the most desirable transnational corporations.

Step 1. Transforming linguistic decision matrix $R = {(r_{ij})}_{5 \times 4}$ into 2-tuple linguistic decision matrix $\tilde{R} = {(r_{ij}, 0)}_{m \times n}$ . The results are shown in Table 2.

Table 2

Decision matrix $\tilde{R}$

	G ₁	G ₂	G ₃	G ₄
A ₁	(G,0)	(G,0)	(M,0)	(EG,0)
A ₂	(VP,0)	(EP,0)	(EG,0)	(G,0)
A ₃	(P,0)	(VG,0)	(VG,0)	(P,0)
A ₄	(VP,0)	(EP,0)	(VP,0)	(M,0)
A ₅	(M,0)	(P,0)	(G,0)	(M,0)

Step 2. Utilizing the information given in the matrix ${\tilde{R}}_{k} = {({\tilde{r}}_{ij} (t_{k}))}_{5 \times 4} (k = 1, 2, 3)$ , and 2TLGWBM (2TLGDWBM)operator, we get the values ${\tilde{z}}_{i} = (r_{i}, a_{i})$ of the transnational corporations A_i (i = 1, 2, 3, 4, 5).

Step 3. According to the aggregating results shown in Table 3, the ordering of the transnational corporations are shown in Table 4. Note that > means “preferred to”. Thus, the best transnational corporation is A₁.

Table 3

The overall preference values of the transnational corporations

	A ₁	A ₂	A ₃	A ₄	A ₅
2TLGWBM	(VG,–0.4864)	(M,0.3243)	(M,0.4497)	(P, –0.3542)	(M,0.0497)
2TLDGWBM	(VG, –0.4864)	(M,0.3243)	(M,0.4497)	(P, –0.3542)	(M,0.0497)

Table 4

Ordering of the transnational corporations

	Ordering
2TLGWBM	_{A₁ >A₃ >A₂ >A₅ >A₄}
2TLDGWBM	_{A₁ >A₃ >A₂ >A₅ >A₄}

5 Conclusion

In this paper, we investigate the multiple attribute decision making problems for evaluating the management performance for the transnational corporation with 2-tuple linguistic information. Motivated by the ideal of generalized weighted Bonferroni mean and dual generalized weighted Bonferroni mean, we develop the 2-tuple linguistic generalized Bonferroni mean (2TLGBM) operator and the 2-tuple linguistic dual generalized Bonferroni mean (2TLDGBM) operator for aggregating the 2-tuple linguistic information. For the situations where the input arguments have different importance, we then define the 2-tuple linguistic generalized weighted Bonferroni mean (2TLGWBM) operator and the 2-tuple linguistic dual generalized weighted Bonferroni mean (2TLDGWBM) operator, based on which we develop the procedure for multiple attribute decision making under the 2-tuple linguistic environments. At last, a numerical example for evaluating the management performance for the transnational corporation is provided to illustrate the proposed method. The result shows the approach is simple, effective and easy to calculate. In the future, we shall continue working in the extension and application of the developed operators to other domains.

References

Herrera

and Herrera-Viedma

, Choice functions and mechanisms for linguistic preference relations, European Journal of Operational Research 120 (2000), 144–161.

Herrera

and Herrera-Viedma

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

Herrera

, Herrera-Viedma

and Martinez

, A fusion approach for managing multi-granularity linguistic term sets in decision making, Fuzzy Sets and Systems 114 (2000), 43–58.

Herrera

and Herrera-Viedma

, Aggregation operators for linguistic weighted information, Ieee Transactions on Systems Man and Cybernetics Part a-Systems and Humans 27 (1997), 646–656.

Cabrerizo

F.J.

, Pérez

I.J.

and Herrera-Viedma

, Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information, Knowledge-Based Systems 23(2) (2010), 169–181.

Herrera

, Martínez

and Sánchez

P.J.

, Managing non-homogeneous information in group decision making, European Journal of Operational Research 166(1) (2005), 115–132.

Herrera-Viedma

, Martinez

, Mata

and Chiclana

, A consensus support system model for group decision-making problems with multigranular linguistic preference relations, Ieee Transactions on Fuzzy Systems 13 (2005), 644–658.

Herrera

, Herrera-Viedma

and Martínez

, A fuzzy linguistic methodology to deal with unbalanced linguistic term sets, IEEE Transactions on Fuzzy Systems 16(2) (2008), 354–370.

Herrera-Viedma

and Lopez-Herrera

A.G.

, A Review on Information Accessing Systems Based on Fuzzy Linguistic Modelling, International Journal of Computational Intelligence Systems 3 (2010), 420–437.

10.

Herrera-Viedma

, Peis

, Morales-del-Castillo

J.M.

, Alonso

and Anaya

, A fuzzy linguistic model to evaluate the quality of Web sites that store XML documents, International Journal of Approximate Reasoning 46 (2007), 226–253.

11.

Martinez

, Ruan

, Herrera

and Herrera-Viedma

, Linguistic decision making: Tools and applications Preface, Information Sciences 179 (2009), 2297–2298.

12.

Moreno

J.M.

, del Castillo

J.M.M.

, Porcel

and Herrera-Viedma

, A quality evaluation methodology for health-related websites based on a 2-tuple fuzzy linguistic approach, Soft Computing 14 (2010), 887–897.

13.

Merigó

J.M.

, Casanovas

and Martínez

, Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of Evidence, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18(3) (2010), 287–304.

14.

Merigo

J.M.

and Casanovas

, Decision Making with Distance Measures and Linguistic Aggregation Operators, International Journal of Fuzzy Systems 12 (2010), 190–198.

15.

Merigo

J.M.

, Gil-Lafuente

A.M.

, Zhou

L.G.

and Chen

H.Y.

, Generalization of the linguistic aggregation operator and its application in decision making, Journal of Systems Engineering and Electronics 22 (2011), 593–603.

16.

Liao

X.W.

, Li

and Lu

, A model for selecting an ERP system based on linguistic information processing, Information Systems 32 (2007), 1005–1017.

17.

D.F.

and Sun

, Fuzzy LINMAP method for multiattribute group decision making with linguistic variables and incomplete information, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 15 (2007), 153–173.

18.

D.F.

, multiattribute group decision making method using extended linguistic variables, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 17 (2009), 793–806.

19.

Liu

P.D.

, A novel method for hybrid multiple attribute decision making, Knowledge-based Systems 22(5) (2009), 388–391.

20.

Zhou

L.G.

and Chen

H.Y.

, A generalization of the power aggregation operators for linguistic environment and its application in group decision making, Knowledge-based Systems 26 (2012), 216–224.

21.

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.

22.

Wei

G.W.

, Extension of TOPSIS method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Knowledge and Information Systems 25 (2010), 623–634.

23.

Wei

G.W.

, GRA-Based Linear-Programming Methodology for Multiple Attribute Group Decision Making with 2-Tuple Linguistic Assessment Information, Information-an International Interdisciplinary Journal 14 (2011), 1105–1110.

24.

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 (2011), 4824–4828.

25.

Wei

G.W.

, Lin

, Zhao

X.F.

and Wang

H.J.

, Models for multiple attribute group decision making with 2-tuple linguistic assessment information, International Journal of Computational Intelligence Systems 3 (2010), 315–324.

26.

Wei

G.W.

, A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information, Expert Systems with Applications 37(12) (2010), 7895–7900.

27.

Wei

G.W.

, Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making, Computers & Industrial Engineering 61 (2011), 32–38.

28.

Y.J.

and Wang

H.M.

, Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment, Applied Soft Computing 11(5) (2011), 3988–3997.

29.

Herrera

and Martinez

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

30.

Herrera

and Martinez

, An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 8 (2000), 539–562.

31.

Bonferroni

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

32.

Yager

R.R.

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

33.

Beliakov

, James

, Mordelova

, Rúckschlossovä

and Yager

R.R.

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

34.

Xia

, Xu

and Zhu

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

35.

Zhu

, Xu

Z.S.

and Xia

M.M.

, Hesitant fuzzy geometric Bonferroni means, Information Sciences 205(1) (2012), 72–85.

Research on the management performance appraisal for the transnational corporation with 2-tuple linguistic information

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 1 Decision matrix R G 1 G 2 G 3 G 4 A 1 G G M EG A 2 VP EP EG G A 3 P VG VG P A 4 VP EP VP M A 5 M P G M

References

Table 1
Decision matrix R

G ₁ G ₂ G ₃ G ₄

A ₁ G G M EG

A ₂ VP EP EG G

A ₃ P VG VG P

A ₄ VP EP VP M

A ₅ M P G M