Model for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information

Abstract

Today, with the development of economic globalization, the quality competition between enterprises has extended to the supply chain. If the enterprises want to get sustainable development, the traditional closed “Vertical Integration” mode needs to be changed; “Horizontal Integration” has become the inevitable choice of enterprises. With the development of performance management and supply chain technology, the combination of the two and focusing on the green supply chain quality management under low-carbon agricultural economy environment have become the hot topics in both academic area and practical area. In this paper, we study on the multiple attribute decision making problems to estimate the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. Then, we propose the 2-tuple power Einstein weighted geometric (2TPEWG) operator for aggregating 2-tuple linguistic information, and then apply 2TPEWG operator for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. In the end, we propose an example to test the effectiveness of our proposed method.

Keywords

Multiple attribute decision making 2-tuple linguistic information power geometric operator 2-tuple power Einstein weighted geometric (2TPEWG) operator performance evaluation green supply chain

1 Introduction

In the 1990s, traditional purchase and logistics had already developed into a broadly-defined purchase and logistics with strategic significance progressively, and have formed the system of the supply chain management. Even to this day, the idea of supply chain management has been rooted in the hearts of the people already, and paid attention to by more and more people, and has got into use in a lot of industries, especially in manufacturing industry [1 –3]. It is obvious that the corporation have to change their traditional management model and pay attention to not only inner resource and competition ability but also outer resource and competition ability covering with the whole supply chain from the supplier to the customer. In other word, the competition focuses on the whole supply chain management [4 –6]. The concept of the supply chain was put forward to under the above background, the traditional evaluation method and idea concerning individual business enterprise is already not suitable to the request of supply chain, which seemed to be increasingly important, but the business enterprise evaluation is just the sub- problem among the evaluation of supply chain management [7 –10].

In recent years, there are many researchers have studied on the problem of supply chain performance evaluation [11]. Supply Chain Council, in 1996 presented 13 indexes about performance evaluation in supply chain operations reference (SCOR). Beamon [12] constructed an evaluation system via resources, output, and flexibility. Salvador et al. [13] studied why and how the SCM affected the organization’s time. Gunasekaran et al. [15] put forth the performance evaluation system utilizing different types of supply chain, planning, purchasing, and so on. Exploiting the SCOR model, Bolstorff and Rosenbaum [15] proposed the performance evaluation system for customers, internal processes, and shareholders. Bhagwat and Sharma [16] built up the performance evaluation system based on the balanced scorecard.

With more competitive globalization, shortening of product lifecycle and the improvement of customer’s expectation, the enterprises start to pay more attention to the green supply chain quality management under low-carbon agricultural economy environment. It changes the traditional competition pattern. The market competition translates into between the supply chains from individuals and single enterprise. The selection of every node in supply chain follows the principle of power-and power union, so gathers the economic resources which have most market competitive power. By supplementing each other and integrating resource and power of enterprises, exerting overall potency, core competitive power of each enterprise is fused into integral competitive power to enhance market competitive power greatly. Traditional performance management is focused on individuals and single enterprises, however the performance management tailored to green supply chain quality management under low-carbon agricultural economy environment is in great short. So developing whole evaluation research on performance evaluating has important academic and practical meaning. In this paper, we focus on the problem of investigate the multiple attribute decision making to estimate the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. Then, we propose the 2-tuple power Einstein weighted geometric (2TPEWG) operator for aggregating 2-tuple linguistic information, and then utilize 2TPEWG operator for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. At last, experiments are conducted to make performance evaluation.

2 Preliminaries

Assume that S ={ s_i |i = 1, 2, ⋯ , t } is a linguistic term set with odd cardinality. Any label, s_i represents a possible value for a linguistic variable, and the following conditions are satisfied [17 –22]:

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

Definition 1. [17 –22] Suppose that β refers to the result of an aggregation in a linguistic term set S. The condition β ∈ [1, t] is satisfied, where t is the cardinality of S. Suppose that i = round (β) and α = β - i mean two values, where i ∈ [1, t] and α ∈ [- 0.5, 0.5) are satisfied.

Definition 2. [17 –22] Suppose that S = {s₁, s₂, ⋯ , s_t} is a linguistic term set and β ∈ [1, t] refers to a number value which denotes the aggregation result of linguistic symbolic. Afterwards, Δ is utilized to obtain the 2-tuple linguistic information as follows. $Δ : [1, t] \to S \times [- 0.5, 0.5)$ (1) $Δ (β) = {\begin{matrix} s_{i}, i = round (β) \\ α = β - i, α \in [- 0.5, 0.5) \end{matrix}$ (2)

Where the symbol round(.) refers to the usual round operation, and α denotes symbolic translation’s value.

Definition 3. [17 –22] Suppose that S = {s₁, s₂, ⋯ , s_t} refers to a linguistic term set and (s_i, α_i) is a 2-tuple. Function Δ^-1 is defined based on a 2-tuple (s_i, α_i), which can 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)

Xu et al. [23] proposed power geometric (PG) operator based on the PA operator [24 –31] and geometric mean [32 –42]. $PG (a_{1}, a_{2}, \dots, a_{n}) = \prod_{i = 1}^{n} {(a_{i})}^{(1 + T (a_{i})) / \sum_{i = 1}^{n} (1 + T (a_{i}))}$ (5)

where $T (a_{i}) = \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} Sup (a_{i}, a_{j})$ , and Sup (a, b) is the support for a from b, which satisfies the following three properties:

Sup (a, b) ∈ [0, 1];

Sup (a, b) = Sup (b, a);

Sup (a, b) ≥ Sup (x, y), if |a - b| < |x - y|.

3 2TPEWG operator

As the the power geometric (PG) operator [23] are the exact values, in this study, we should modify the power geometric (PG) operator and Einstein operations [43 –45] to 2-tuple linguistic assessment information. In the following, we shall develop 2-tuple power Einstein weighted geometric (2TPEWG) operator as follows:

Definition 4. Supposet that x = {(r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)} is a set of 2-tuple and ω = (ω₁, ω₂, ⋯ , ω_n) ^T denotes the weighting vector of 2-tuple (r_i, a_i) (i = 1, 2, ⋯ , n) and ω_i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ , then we define the 2-tuple power Einstein weighted geometric (2TPEWG) operator as follows:

$\begin{matrix} (\tilde{r}, \tilde{a}) & = & 2 {TPEWG}_{ω} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = & Δ (\otimes_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}) \\ = & Δ (\frac{γ \prod_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}}{\begin{matrix} {\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - Δ^{- 1} (r_{i}, a_{i})))}^{ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i})))} \\ + (γ - 1) \prod_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}} \end{matrix}}) \\ \tilde{r} \in S, \tilde{a} \in [- 0.5, 0.5) \end{matrix}$ (6) where $T (r_{i}, a_{i}) = Δ (\sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} ω_{j} Sup ((r_{i}, a_{i}), (r_{j}, a_{j})))$ (7) and Sup ((r_i, a_i) , (r_j, a_j)) is the support for (r_i, a_i) from (r_j, a_j), with the conditions:

Sup ((r_i, a_i) , (r_j, a_j)) ∈ [0, 1];

Sup ((r_i, a_i) , (r_j, a_j)) = Sup ((r_j, a_j) , (r_i, a_i));

Sup ((r_i, a_i) , (r_j, a_j)) ≥ Sup ((r_s, a_s) , (r_t, a_t)), if d ((r_i, a_i) , (r_j, a_j)) ≤ d ((r_s, a_s) , (r_t, a_t)), where d is a distance measure.

Especially, if $ω = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the 2TPEWG operator reduces to a 2-tuple power Einstein average (2TPEG) operator:

$\begin{matrix} (\tilde{r}, \tilde{a}) & = & 2 {TPEWG}_{ω} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = & Δ (\otimes_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} ω_{i} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}) \\ = & Δ (\otimes_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{(1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}) \\ = & Δ (\frac{γ \prod_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{(1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}}{\begin{matrix} {\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - Δ^{- 1} (r_{i}, a_{i})))}^{(1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} (1 + Δ^{- 1} (T (r_{i}, a_{i})))} \\ + (γ - 1) \prod_{i = 1}^{n} {(Δ^{- 1} (r_{i}, a_{i}))}^{(1 + Δ^{- 1} (T (r_{i}, a_{i}))) / \sum_{i = 1}^{n} (1 + Δ^{- 1} (T (r_{i}, a_{i})))}} \end{matrix}}) \\ \tilde{r} \in S, \tilde{a} \in [- 0.5, 0.5) \end{matrix}$ (8) where $T (r_{i}, a_{i}) = Δ (\frac{1}{n} \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} Sup ((r_{i}, a_{i}), (r_{j}, a_{j})))$ (9)

We can obtained the the following properties for the 2TPEWG operator.

Theorem 1. (Idempotency) If ((r₁, a₁) , (r₂, a₂) , …, (r_n, a_n)) = (r, a), then

$\begin{matrix} 2 {TPEWG}_{ω} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = (r, a) \end{matrix}$ (10)

Theorem 2. (Commutativity).

$\begin{matrix} 2 {TPEWG}_{ω} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ = 2 {TPEWG}_{ω} ((r_{1}^{'}, a_{1}^{'}), (r_{2}^{'}, a_{2}^{'}), \dots, (r_{n}^{'}, a_{n}^{'})) \end{matrix}$ (11) 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)).

Theorem 3. (Boundedness).

$\begin{matrix} min_{i} (r_{i}, a_{i}) \\ \leq 2 {TPEWG}_{ω} ((r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})) \\ \leq max_{i} (r_{i}, a_{i}) \end{matrix}$ (12)

4 Model for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information

In the following, we utilize the 2-tuple power Einstein weighted geometric (2TPEWG) operator for evaluating the green supply chain performance under low-carbon agricultural economy environment. Suppose that A ={ A₁, A₂, ⋯ , A_m } is a discrete set of alternatives, G ={ G₁, G₂, ⋯ , G_n } is 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×n is the evaluating matrix, where r_ij ∈ S is an attribute values, which construct the 2-tuple linguistic variables. Afterwards, we use the 2TPEWG operator to construct the model for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information as following:

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

Step 2. Aggregating all 2-tuple linguistic preference value (r_ij, 0) (j = 1, 2, ⋯ , n) by using the 2-tuple power Einstein weighted geometric (2TPEWG) operator:

$\begin{matrix} {\tilde{r}}_{i} & = & 2 {TPEWG}_{ω} ((r_{i 1}, 0), (r_{i 2}, 0), \dots, (r_{in}, 0)) \\ = & Δ (\otimes_{i = 1}^{n} {(Δ^{- 1} (r_{ij}, 0))}^{ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0))) / \sum_{i = 1}^{n} ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0)))}) \\ = & Δ (\frac{γ \prod_{i = 1}^{n} {(Δ^{- 1} (r_{ij}, 0))}^{ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0))) / \sum_{i = 1}^{n} ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0)))}}{\begin{matrix} {\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - Δ^{- 1} (r_{ij}, 0)))}^{ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0))) / \sum_{i = 1}^{n} ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0)))} \\ + (γ - 1) \prod_{i = 1}^{n} {(Δ^{- 1} (r_{ij}, 0))}^{ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0))) / \sum_{i = 1}^{n} ω_{j} (1 + Δ^{- 1} (T (r_{ij}, 0)))}} \end{matrix}}) \\ i = 1, 2, \dots, m . ∥ \end{matrix}$ (13) to derive the 2-tuple linguistic values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i, where ω = (ω₁, ω₂, ⋯ , ω_n) is the weighting vector of the attributes.

Step 3. Sort all the alternatives A_i (i = 1, 2, ⋯ , m) in terms of the overall 2-tuple linguistic values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ .

5 Numerical example

To make performance evaluation, we propose a numerical example for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. The performance evaluation of five possible green supply chain systems under low-carbon agricultural economy environment A_i (i = 1, 2, 3, 4, 5) is evaluated in terms of the following 4 attributes: G1 is the debt paying ability; G2 is the operation capability; G3 is the earning capacity; G4 is the development capability. The five possible green supply chain systems under low-carbon agricultural economy environment A_i (i = 1, 2, ⋯ , 5) are to be tested via the linguistic term set S. The linguistic evaluating matrix are listed in Table 1:

Afterwards, we use the 2TPEWG operator to construct the model for evaluating the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information as following:

Step 1. Converting linguistic evaluating information to 2-tuple linguistic evaluating matrix in Table 2.

Step 2. By utilizing the evaluating information given in Table 2, and the 2-tuple power Einstein weighted geometric (2TPEWG) operator, ω = (0.2, 0.4, 0.3, 0.1) is the weighting vector of the attributes, we get the overall values of the five possible green supply chain systems under low-carbon agricultural economy environment: $\begin{matrix} {\tilde{r}}_{1} & = & (VP, 0 . 27), {\tilde{r}}_{2} = (P, 0 . 18), {\tilde{r}}_{3} = (VG, - 0 . 16) \\ {\tilde{r}}_{4} & = & (M, 0 . 18), {\tilde{r}}_{5} = (P, - 0 . 19) \end{matrix}$

Step 3. Sorting all the five possible green supply chain systems under low-carbon agricultural economy environment A_i (i = 1, 2, ⋯ , 5) in terms of the 2-tuple linguistic preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ of the five possible green supply chain systems under low-carbon agricultural economy environment: A₃ ≻ A₄ ≻ A₂ ≻ A₅ ≻ A₁, and thus the most desirable five possible green supply chain system under low-carbon agricultural economy environment is A₃.

6 Conclusion

At present, a crucial problem is the lack of the more prominent and effective performance evaluation system and improves the incentive and restraint mechanisms in green supply chain management, that it is difficult to form the interests of the community for long-term stability. For green supply chain management under low-carbon agricultural economy environment, there seems to be numbers of evaluation indicators, but how to select the key performance indicators is essential. How to build an efficiency for green supply chain management under low-carbon agricultural economy environment and use the scientific method of performance evaluation, that is related to the success of establishing and developing green supply chain management partnership under low-carbon agricultural economy environment and the achievement of corporate strategic alliances. Therefore, the study of green supply chain performance evaluation under low-carbon agricultural economy environment and its method is of great importance. We propose a novel method to estimate the green supply chain performance under low-carbon agricultural economy environment with 2-tuple linguistic information. Then, we propose the 2-tuple power Einstein weighted geometric (2TPEWG) operator for aggregating 2-tuple linguistic information, and then exploit 2TPEWG operator to make performance evaluation. At last, we design an example to testify the performance of our method.

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