Research on fuzzy evaluation of performance in green supply chain based on environmental economics

Abstract

Along with the development of human being, the environmental problems have attracted more and more attention, “The Green” will become the main trend in the 21th century, and “The Green Action” will be an important department and a decisive factor to any country and enterprise which take part in the competition of the world market. Based on the above analysis, it is an important mode for the modern enterprise. We propose “The Green Idea” to the management of supply system, and then construct “The Green Supply Management” to reduce the resource cost and the environmental affect. In the study of “The Green Supply”, how to estimate the performance with high accuracy is of great importance. Furthermore, high quality performance evaluation may let the enterprise to answer the questions, to discover the solutions, and then to enhance the quality of the operation and improvement of “The Green Supply”. Therefore, it is very crucial to enhance “The Green Supply” by making the performance evaluation with high accuracy. In this paper, we study on the multiple attribute decision making problems to estimate the performance in green supply chain based on environmental economics with fuzzy information. Then, we have developed the fuzzy Hamacher correlated averaging (FHCA) operator. We have exploited the FHCA operator to multiple attribute decision making to estimate the wireless network security with fuzzy information. In the end, an example to estimate the performance in green supply chain based on environmental economics has been proposed to prove the effectiveness of the proposed approach.

Keywords

Multiple attribute decision making triangular fuzzy number performance in green supply chain environmental economics

1 Introduction

In the middle and late of the 20th century, the over-consumption of resources and the environmental pollution problems are becoming more and more serious under the rapid development of the world economy. Resources and environment will be the biggest challenge facing human beings in the 21st century. The rising contradiction among economic growth, resources, environment and global frequent extreme weather disasters force human beings to deeply reflect the traditional development mode of the industrial age, which is at the cost of over-consumption of resources, and destruction of ecological balance. It is under this circumstance that the strategy of sustainable development which satisfies the requirements of the present with no compromising the ability of future generations to satisfy their own needs is becoming common consensus among countries in the world. And then the global “Greening Wave” and “Greening Revolution” swept over the world. “Greening” constantly penetrated into all aspects of economic and social life. Under the double dilemma of resources and environment in the 21st century, the extensive growth mode mainly characterized as “three highs and one low” (high inputs, high consumption, high pollution, and low efficiency) has come to the end. Accelerating the change in the pattern of economic growth and promoting the “greening” of supply chain become extremely urgent, especially for the manufacturing industry which is closely connected to resources and environment. Greening supply chain, is a continuously optimizing process, which targeted at sustainable development and based on circular economy principle, inside the system which is composed of suppliers, manufacturers, distributors, retailers, recyclers, consumers, environment, regulation and culture, a comprehensive consideration of the environmental influences and resources efficiency is taken in the whole process from the development of resources to the consumption of products, which including acquisition, processing, packing, storage, transportation, sale, use, scrap processing and recycling, through the right control of logistics, cash flow, information flow, and knowledge flow to achieve less negative influences to environment, higher utilization efficiency of resources and better overall performance in the life cycle of products. Green supply chain management is a kind of modern management mode, which strengthens resources and environmental awareness in the supply chain, strengthens cooperation among upstream and downstream enterprises and communication among departments inside the enterprise, a comprehensive consideration of resources efficiency and environmental benefits is taken in the whole process from product design, material selection, manufacturing, sales and recovery, through planning, organizing, leading, coordinating and controlling of logistics, information flow, fund flow and knowledge flow among participation bodies in the whole supply chain to improve both environmental and economic performance in the enterprise, consequently to achieve sustainable development in the enterprise and its supply chain.

The problems of evaluating the performance in green supply chain based on environmental economics with fuzzy information is the classical multiple attribute decision making problems [1 –21]. Particularly, we study on the multiple attribute decision making problems to estimate the performance in green supply chain based on environmental economics with fuzzy information. Afterwards, we have designed the fuzzy Hamacher correlated averaging (FHCA) operator. We have utilized the FHCA operator to multiple attribute decision making to test the wireless network security with fuzzy information. Next, an example to testify the performance in green supply chain based on environmental economics has been given to show the developed method.

2 Preliminaries

We simply illustrate several basic concepts and operational rules which are corresponding to triangular fuzzy numbers.

Definition 1. [22] A triangular fuzzy numbers $\tilde{a}$ is defined using a triplet (a^L, a^M, a^U). The membership function $μ_{\tilde{a}} (x)$ is defined as follows. $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, & a^{L} \leq x \leq a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, & a^{M} \leq x \leq a^{U}, \\ 0, & x \geq a^{U} . \end{matrix}$ (1) where 0 < a^L ≤ a^M ≤ a^U, a^L and a^U stand for the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M for the modal value.

Definition 2. [22] Basic operational laws related to triangular fuzzy numbers: $\begin{matrix} \tilde{a} \oplus \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \oplus [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}] \\ \tilde{a} \otimes \tilde{b} & = & [a^{L}, a^{M}, a^{U}] \otimes [b^{L}, b^{M}, b^{U}] \\ = & [a^{L} b^{L}, a^{M} b^{M}, a^{U} b^{U}] \\ λ \otimes \tilde{a} & = & λ \otimes [a^{L}, a^{M}, a^{U}] \\ = & [λ a^{L}, λ a^{M}, λ a^{U}], λ > 0 . \\ \frac{1}{\tilde{a}} & = & [1 / a^{U}, 1 / a^{M}, 1 / a^{L}] \end{matrix}$

Definition 3. [23 –33] If A is a triangular fuzzy variable [a^L, a^M, a^U], then the expected value of A is $E (A) = \frac{1}{3} (a^{L} + a^{M} + a^{U})$ (2)

T-norm and t-conorm refer to key notions in the fuzzy set domain [34]. Roychowdhury et al. [35] provided some definitions and conditions of t-norm and t-conorm. Utilizing the t-norm (T) and t-conorm (T*), a generalized union and a generalized intersection of intuitionistic fuzzy sets are illustrated in the reference [36]. Afterwards, Hamacher [37] presented a generalized t-norm and t-conorm.

Hamacher product ⊗ denotes a t-norm and Hamacher sum ⊕ is a t-conorm, where $T (a, b) = a \otimes b = \frac{ab}{γ + (1 - γ) (a + b - ab)}$ (3) $T * (a, b) = a \oplus b = \frac{a + b - ab - (1 - γ) ab}{1 - (1 - γ) ab}$ (4)

Particularly, and γ is equal to 1, then Hamacher t-norm and t-conorm can be represented as follows. $T (a, b) = a \otimes b = ab$ (5) $T * (a, b) = a \oplus b = a + b - ab$ (6)

Equations 5 and 6 denote the algebraic t-norm and t-conorm respectively. If the parameter γ is equal to 2, the Hamacher t-norm and t-conorm can be represented as follows. $T (a, b) = a \otimes b = \frac{ab}{1 + (1 - a) (1 - b)}$ (7) $T * (a, b) = a \oplus b = \frac{a + b}{1 + ab}$ (8) which are named as the Einstein t-norm and t-conorm [34].

3 Fuzzy Hamacher correlated aggregating operators

But, the proposed aggregation operators with picture fuzzy information is designed to use the assumption that the attributes of decision makers are not dependent to each other. We also can see that the operators are designed to utilize the assumption that attributes of decision makers are not related to others. Therefore, the independence axiom cannot be satisfied, and then it is very crucial to solve the proposed problem.

Assume that μ (x_i) (i = 1, 2, ⋯ , n) means the weight of the elements x_i ∈ X (i = 1, 2, ⋯ , n), where μ denotes a fuzzy measure, which is defined in Definition 4.

Definition 4. [38] A fuzzy measure μ on the set X is a set function μ : μ (X) → [0, 1] satisfying the following axioms and μ (X) is the set of all subsets of X:

μ (φ) = 0, μ (X) = 1;

A ⊆ B implies μ (A) ≤ μ (B), for all A, B ⊆ X;

μ (A ∪ B) = μ (A) + μ (B) + ρμ (A) μ (B), for all A, B ⊆ X and A ∩ B = φ, where ρ ∈ (- 1, ∞).

Especially, if ρ = 0, then the condition (3) reduces to the axiom of additive measure:

μ (A ∪ B) = μ (A) + μ (B), for all A, B ⊆ X and A ∩ B = φ.

If all the elements in X are independent, and we have $μ (A) = \sum_{x_{i} \in A} μ ({x_{i}}), for all A \subseteq X .$

Definition 5. [39] Assume that f is a positive real-valued function on X, and μ is a fuzzy measure on X. Then, the discrete Choquet integral of f corresponding to to μ is given as follows. $C_{μ} (f) = \sum_{i = 1}^{n} f_{σ (i)} [μ (A_{σ (i)}) - μ (A_{σ (i - 1)})]$ (9) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that f_σ(i-1) ≥ f_σ(i) for all j = 2, ⋯ , n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

There, we find that the discrete Choquet integral refers to a linear expression up to a reordering of the elements.

Using Definition 5, we can design the fuzzy Hamacher correlated aggregation (FHCA) operator using the Choquet integral [40].

Definition 6. Assume that α_j (j = 1, 2, ⋯ , n) is a collection of PFNs on X, and μ refers to a fuzzy measure on X, then we name

$\begin{matrix} {FHCA}_{μ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{α}}_{σ (j)}) \end{matrix}$ (10) the fuzzy Hamacher correlated aggregation (FHCA) operator, where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Utilizing the operation of fuzzy values, the FHCA operator is transformed using Equation 11.

$\begin{matrix} {FHCA}_{μ} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{α}}_{σ (j)}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) α_{σ (j)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - α_{σ (j)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}) \end{matrix}$ (11) whose aggregated value is also a fuzzy number.

Especially, if μ ({ x_σ(j) }) = μ (A_σ(j)) - μ (A_σ(j-1)), i = 1, 2, ⋯ , n, then FHCA operator reduce to FHWA operator. If μ (A) = ∑_{x_j∈A}μ ({ x_j }), for all A ⊆ X, where |A| refers to the number of the elements in the set A, then w_j = μ (A_σ(j)) - μ (A_σ(j-1)), i = 1, 2, ⋯ , n, where w = (w₁, w₂, ⋯ , w_n) ^T, w_j ≥ 0, i = 1, 2, ⋯ , n, and $\sum_{j = 1}^{n} w_{j} = 1$ , then, FHCA operator reduce to FHOWA operator.

4 Research on fuzzy evaluation of performance in green supply chain based on environmental economics

For a multiple attribute decision making problems to estimate the performance in green supply chain based on environmental economics with triangular fuzzy information. We suppose that A ={ A₁, A₂, ⋯ , A_m } refers to a set of alternatives, G ={ G₁, G₂, ⋯ , G_n } refers to a 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 $A = {({\tilde{a}}_{ij})}_{m \times n} = {(a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U})}_{m \times n}$ is the multiple attribute decision making matrix, where ${\tilde{a}}_{ij} = (a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U})$ is preference values, which can be used to construct triangular fuzzy numbers for the alternative A_i ∈ A which is related to the attribute G_j ∈ G.

Afterwards, we use the FHCA operator to design a method to multiple attribute decision making problems for evaluating the performance in green supply chain based on environmental economics with triangular fuzzy information, which can be described as following:

Step 1. Utilize the decision information $A = {({\tilde{a}}_{ij})}_{m \times n} = {(a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U})}_{m \times n}$ , and the FHCA operator $\begin{matrix} {\tilde{r}}_{i} & = & (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) = {FHCA}_{μ} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = & \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{r}}_{σ (ij)}) \\ = & (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{L})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{M})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) r_{σ (ij)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - r_{σ (ij)}^{U})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}) \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . ∥ \end{matrix}$ (12)

Step 2. To compute the expected value of the overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ $E (r_{i}) = \frac{1}{3} (r_{i}^{L} + r_{i}^{M} + r_{i}^{U}) .$ (13)

Step 3. Sort the alternatives A_i (i = 1, 2, ⋯ , m) and choose the optimal one in terms of the expected values E (r_i) (i = 1, 2, ⋯ , m).

5 Numerical example

Nowadays, the green revolution is on the rise in the worldwide scale, which takes resource conservation and environmental protection as its goal. So the concept of green supply chain is known to the public gradually. The traditional economic developing mode of steel makers is one kind of extensive development that at the expense of the environment, which has no consideration for synchronous economic and social development and harmonious human and nature relationship. This kind of condition leads to resource scarcity and environmental degradation with the constant quickening of urbanization and industrialized process, which has become one of the main bottlenecks that restrict steel maker’s sustainable development. However, the steel industry is typical resource and energy-intensive industry whose whole operation process will make certain impact on environment, from iron ore and coal’s exploration and transportation to steel products’ manufacture, application, final discard and recycling. It is necessary to conduct targeted research on steel makers which as large power consumption and large polluters because its peculiar position in manufacturing. Hence, Process equipment, manufacturing process and product’s greening and green supply chain’s functional extension like products manufacture, energy conversion, and waste disposal can be achieved only by combining green supply chain management theory and steel maker’s status and development and taking management performance evaluation of green supply chain as management method. Therefore, to establish a set of comprehensive, efficient and feasible green supply chain performance evaluation system and mode is the key to promote steel makers to carry out green supply chain management. In this section, we propose a case study to testify the performance in green supply chain based on environmental economics. The performance quality of 5 possible green supply chain systems A_i (i = 1, 2, 3, 4, 5) is tested in terms of 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 A_i (i = 1, 2, ⋯ , 5) are to be tested utilizing the triangular fuzzy numbers by the decision maker based on the proposed four attributes, which is illustrated as follows.

Next, we exploit the proposed method to test the security of wireless network.

Step 1. Assume that the fuzzy measure of attribute of G_j (j = 1, 2, ⋯ , n) and attribute sets of G as follows: $\begin{matrix} μ (G_{1}) = 0.35, μ (G_{2}) = 0.30, μ (G_{3}) = 0.25, \\ μ (G_{4}) = 0.20, μ (G_{1}, G_{2}) = 0.60, \\ μ (G_{1}, G_{3}) = 0.55, μ (G_{1}, G_{4}) = 0.50, \\ μ (G_{2}, G_{3}) = 0.65, μ (G_{2}, G_{4}) = 0.50 \\ μ (G_{3}, G_{4}) = 0.45, μ (G_{1}, G_{2}, G_{3}) = 0.80 \\ μ (G_{1}, G_{2}, G_{4}) = 0.85, μ (G_{1}, G_{3}, G_{4}) = 0.70 \\ μ (G_{2}, G_{3}, G_{4}) = 0.75, μ (G_{1}, G_{2}, G_{3}, G_{4}) = 1.00 \end{matrix}$

Step 2. Aggregate all fuzzy value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ exploiting the FHCA operator to obtain the fuzzy values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the green supply chain systems A_i. $\begin{matrix} {\tilde{r}}_{1} = (0.174, 0.199, 0.226) \\ {\tilde{r}}_{2} = (0.172, 0.193, 0.205) \\ {\tilde{r}}_{3} = (0.212, 0.233, 0.243) \\ {\tilde{r}}_{4} = (0.197, 0.215, 0.232) \\ {\tilde{r}}_{5} = (0.165, 0.176, 0.201) \end{matrix}$

Step 3. Sort all the green supply chain systems A_i (i = 1, 2, ⋯ , 5) according to the expected value E (r_i) (i = 1, 2, ⋯ , 5): A₃ ≻ A₄ ≻ A₁ ≻ A₂ ≻ A₅, and then the most suitable green supply chain systems is A₃.

6 Conclusion

In this paper, we study on the problem of multiple attribute decision making with fuzzy information. Firstly, we proposed the fuzzy Hamacher correlated averaging (FHCA) operator. Secondly, we used the FHCA operator to multiple attribute decision making to test the wireless network security. In the end, an example is illustrated to test the wireless network security. Experimental results demonstrate that the proposed approach can evaluate the performance of green supply chain with high accuracy.

References

Z.S.

and Yager

R.R.

, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning 48(1) (2008), 246–262.

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.

D.J.

, Wu

Y.Y.

and Lu

, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowledge-Based Systems 30 (2012), 57–66.

Liu

P.D.

, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Transations on Fuzzy Systems, In press.

Wang

W.Z.

and Liu

X.W.

, Intuitionistic fuzzy geometric aggregation operators based on einstein operations, International Journal of Intelligent Systems 26 (2011), 1049–1075.

Wei

G.W.

, Some geometric aggregation functions and their application to dynamic multiple attribute decision making in intuitionistic fuzzy setting, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems 17(2) (2009), 179–196.

Wei

G.W.

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

Wei

G.W.

, Wang

H.J.

and Lin

, Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision making with incomplete weight information, Knowledge and Information Systems 26(2) (2011), 337–349.

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

10.

, Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets, Applied Mathematical Modelling 34 (2010), 3864–3870.

11.

Wei

G.W.

, Wang

J.M.

and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems 55(3) (2013), 679–684.

12.

D.F.

, TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems 18 (2010), 299–311.

13.

D.F.

, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Applied Soft Computing 11 (2011), 3402–3418.

14.

D.F.

, The GOWA operator based approach to multiattribute decision making using intuitionistic fuzzy sets, Mathematical and Computer Modelling 53 (2011), 1182–1196.

15.

Ran

L.-G.

and Wei

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

16.

Wei

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

17.

Wei

G.W.

and Zhao

, Methods for probabilistic decision making with linguistic information, Technological and Economic Development of Economy 20(2) (2014), 193–209.

18.

Wei

, Lin

, Zhao

and Wang

, An Approach to multiple attribute decision making based on the induced Choquet integral with fuzzy number intuitionistic fuzzy information, Journal of Business Economics and Management 15(2) (2014), 277–298.

19.

Singh

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

20.

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.

21.

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.

22.

Van Laarhoven

P.J.M.

and Pedrycz

, A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems 11 (1983), 229–241.

23.

Zétényi

, Fuzzy Sets in Psychology North-Holland, Amsterdam, 1988.

24.

Zhao

, Lin

and Wei

G.W.

, Fuzzy prioritized operators and their application to multiple attribute group decision making, Applied Mathematical Modelling 37(7) (2013), 4759–4770.

25.

Wei

G.W.

, Zhao

, Lin

and Wang

, Generalized triangular fuzzy correlated averaging operator and their application to multiple attribute decision making, Applied Mathematical Modelling 36(7) (2012), 2975–2982.

26.

Wei

G.W.

, FIOWHM operator and its application to multiple attribute group decision making, Expert Systems with Applications 38(4) (2011), 2984–2989.

27.

Z.S.

, Fuzzy harmonic mean operators, International Journal of Intelligent Systems 24 (2009), 152–172.

28.

Merigó

J.M.

, The Fuzzy Probabilistic Weighted Averaging Operator and its Application in Decision Making, 2009 Ninth International Conference on Intelligent Systems Design and Applications, pp. 485–490.

29.

Z.S.

, A priority method for triangular fuzzy number complementary judgement matrix, System Engineer-Theory & Practice 23(10) (2003), 86–89.

30.

Wang

X.R.

and Fan

Z.P.

, Fuzzy ordered weighted averaging (FOWA) operator and its application, Fuzzy Systems and Mathematics 17(4) (2003), 67–72.

31.

Z.S.

, A fuzzy ordered weighted geometric operator and its application to in fuzzy AHP, Journal of Systems Engineering and Electronics 31(4) (2002), 855–858.

32.

Z.S.

and Wu

, Method based on fuzzy linguistic judgement matrix and fuzzy fuzzy induced ordered weighted averaging (FIOWA) operator for decision making problems with a limited set of alternatives, Fuzzy Systems and Mathematics 18(1) (2004), 76–80.

33.

Z.S.

and Da

Q.L.

, Method based on fuzzy linguistic scale and FIOWGA operator for decision making problems, Journal of Southeast University(English Edition) 19(1) (2003), 88–91.

34.

Deschrijver

, Cornelis

and Kerre

E.E.

, On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transactions on Fuzzy Systems 12 (2004), 45–61.

35.

Roychowdhury

and Wang

B.H.

, On generalized Hamacher families of triangular operators, International Journal of Approximate Reasoning 19 (1998), 419–439.

36.

Deschrijver

and Kerre

E.E.

, A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms, Notes on Intuitionistic Fuzzy Sets 8 (2002), 19–27.

37.

Hamachar

, Uber logische verknunpfungenn unssharfer Aussagen undderen Zugenhorige Bewertungsfunktione, in: Tral

Klir

Riccardi (Eds.), Progress in Cybernatics and Systems Research, vol. 3, Hemisphere, Washington DC, 1978, pp. 276–288.

38.

Wang

, Klir

, Fuzzy Measure Theory, Plenum Press, New York, 1992.

39.

Grabisch

, Murofushi

and Sugeno

, Fuzzy Measure and Integrals, Physica-Verlag, New York, 2000.

40.

Choquet

, Theory of capacities, Ann Inst Fourier 5 (1953), 131–295.