Models for real estate investment decision-making with hesitant fuzzy information

Abstract

In recent years, China’s real estate investment behavior with the market economy has been developing steadily and has become increasingly frequent. Real estate investment is a long construction period, huge investment, influence factors and complex activity, and the real estate enterprise is faced with increasingly fierce competition. How the scheme selection of more scientific, reasonable, comprehensive evaluation analysis of the investment plan, select the optimal scheme, is the key to win investment, is also the most critical problem facing investors. In this paper, we study on the multiple attribute decision making for real estate investment with hesitant fuzzy information. Inspired by the dependent aggregation, we propose the dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator, where the weights rely on the aggregated hesitant fuzzy arguments and can lower the influence of unfair hesitant fuzzy arguments on the aggregated results by allocating low weights to the “false” and “biased” ones and then utilize them to design this approach for multiple attribute decision making with hesitant fuzzy information. In the end, an example of real estate investment is proposed to testify our method.

Keywords

Multiple attribute decision making hesitant fuzzy information dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator real estate investment

1 Introduction

Since the reform in 1978, China began to implement the system of market economy which brings the upward trend in real estate investment. Real estate investment quota exploded. Especially, since the great changes of China’s housing system in the late 1990s. Large amounts of capital into the real estate market due to such Macro real estate investment environment change. The real estate investment’s prosperity has brought three aspects of reality: first is the contradiction between economic interests and social interests of the real estate development in the ongoing process of China’s urbanization process [1, 2]; Second is the contradiction between high housing prices and the housing need of low income families under the Non-equilibrium circumstance of real estate market; Third is the contradiction between the costs and benefits of real estate investment in this period of frequent policy-control and financial risks [3, 4]. All the three contradictions starts from the “real estate investment environment changes". The real estate investment environment is a combination of the macroeconomic environment, policies environment, monetary policy, land expropriation policies, municipal administration, residents, energy utilities, real estate location, cultural and natural environment etc, which is the basis of bring about its benefits and also the basis for analysis benefits’ distribution. All the three contradictions could be finally explained as “benefits implements and distributions which brings by real estate investment” [5]. “Interest” means the most basic economic and social interests, including all aspects of human social life among beneficiaries. Real estate investment environment have an important impact on investment decisions, benefits implement and distribution. Ultimately, the changes in the real estate investment environment brought changes to the “interests” of real estate investment [6, 7].

Real estate is one of the economic development foundation industries of all the countries around the world. In China the real estate economy has developed only for several decades. The market operating and trading are not normal enough. The real estate system needs to be improved gradually. The scientific and feasible research of the investment decision is not stringent enough, for the sustainable development of real estate industry and economic returns of real estate projects, and avoiding the investment gain more less than lose, these problems must be well analyzed and solved. How to overall analyze and comprehensively appraise the necessity and feasibility of the real estate project, and how to find out the most economic one in multiple-plans are the basis for government and investors to make decision [8, 9]. How to quantify the various influence factors and objectives of investment project and achieve the correct purpose of the comprehensive appraisal without subjective influence is one of the most important researches in real estate studies. Hesitant fuzzy set is a powerful tool to solve the fuzzy information in the domain of the real estate investment decision making [10 –24]. Thus, in this paper, we focus on the multiple attribute decision making for real estate investment with hesitant fuzzy information. Inspired by the dependent aggregation, we study on the dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator, where the associated weights only rely on the aggregated hesitant fuzzy arguments and can lower the affect of unfair hesitant fuzzy arguments on the aggregated results. In section 2, we discuss several concepts about the hesitant fuzzy sets. In Section 3 we propose the dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator. In Section 4 we concentrate on the MADM problem with hesitant fuzzy information based on the dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator. In Section 5, an example for real estate investment is given. In Section 6, the whole paper is concluded in detail.

2 Preliminaries

In the following, we introduce some basic concepts related to hesitant fuzzy sets.

Definition 1. [25] Suppose that X denotes a fixed set, and a hesitant fuzzy set (HFS) on is according to a function when using X to obtain a subset of [0, 1]. Furthermore, HFS can be formally described as follows. $E = (〈 x, h_{E} (x) 〉 | x \in X),$ (1) where h_E (x) refers to the values in [0, 1], which means the membership degree of the element x ∈ X to the set E. Moreover, h = h_E (x) is defined as a hesitant fuzzy element(HFE) and H refers to the set of all HFEs.

Definition 2. [26] Given HFE h, $s (h) = \frac{1}{# h} \sum_{γ \in h} γ$ is named as the score function of h, where # h refers to the number of the elements in h. If s (h₁) > s (h₂), then h₁ > h₂ as well; if s (h₁) = s (h₂), then h₁ = h₂.

In the following, Zhou et al. [27] proposed a series of Hamacher aggregation operators for HFEs based on the Hamacher operations [28].

Suppose that h_j (j = 1, 2, …, n) denotes a set of HFEs, and the following equations can be obtained.

The hesitant fuzzy Hamacher weighted average (HFHWA) operator

$\begin{matrix} {HFHWA}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \oplus_{j = 1}^{n} (ω_{j} h_{j}) \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - γ_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{j})}^{ω_{j}}}} \end{matrix}$ (2)

where ω = (ω₁, ω₂, …, ω_n) ^T means the vector of h_j (j = 1, 2, …, n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

The hesitant fuzzy Hamacher ordered weighted averaging (HFHOWA) operator

\begin{matrix} {HFHOWA}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} h_{σ (j)}) \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - γ_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{σ (j)})}^{w_{j}}}} \end{matrix}

(3)

where (σ (1) , σ (2) , …, σ (n)) refers to a permutation of (1, 2, …, n), and h_σ(j−1) ≥ h_σ(j) is satisfied for all j = 2, …, n, and w = (w₁, w₂, …, w_n) ^T denotes the aggregation vector and w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ are satisfied.

The hesitant fuzzy Hamacher hybrid average (HFHHA) operator

\begin{matrix} {HFHHA}_{w, ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \oplus_{j = 1}^{n} (w_{j} {\dot{h}}_{σ (j)}) \\ = \cup_{{\dot{γ}}_{σ (1)} \in {\dot{h}}_{σ (1)}, {\dot{γ}}_{σ (2)} \in {\dot{h}}_{σ (2)}, \dots, {\dot{γ}}_{σ (n)} \in {\dot{h}}_{σ (n)}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{γ}}_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{γ}}_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) {\dot{γ}}_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - {\dot{γ}}_{σ (j)})}^{w_{j}}}} \end{matrix}

(4)

where w = (w₁, w₂, …, w_n) and w_j is belonged, and $\sum_{j = 1}^{n} w_{j} = 1$ is satisfied. ${\dot{h}}_{σ (j)}$ refers to the jth largest one of the hesitant fuzzy arguments ${\dot{h}}_{σ (j)} ({\dot{h}}_{σ (j)} = (n ω_{j}) h_{j}, j = 1, 2, \dots, n)$ , ω = (ω₁, ω₂, …, ω_n) is vector of hesitant fuzzy arguments h_j (j = 1, 2, …, n), with ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , and n is the balancing coefficient.

3 Dependent hesitant fuzzy Hamacher average operator

This parts discusses how to design the dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator, and in this operator the weights rely on the aggregated hesitant fuzzy arguments and are able to lower the influence of unfair hesitant fuzzy arguments on the aggregated results.

Definition 3. [29] Let h_j (j = 1, 2, …, n) denotes a set of HFEs, the average value of the score function of h_j (j = 1, 2, …, n) is computed as $s (\bar{h}) = \frac{\sum_{j = 1}^{n} s (h_{j})}{n}$ (5)

Definition 4. Let h₁ and h₂ refer to two HFSs on X ={ x₁, x₂, …, x_n }, then the distance between h₁ and h₂ is calculated as follows. $d (h_{1}, h_{2}) = | s (h_{1}) - s (h_{2}) |$ (6)

Definition 5. Let h_j (j = 1, 2, …, n) be a set of HFEs, and $s (\bar{h})$ the arithmetic mean of these hesitant fuzzy values. Afterwards, the standard deviation of the hesitant fuzzy values is defined as follows. $σ = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} {| s (h_{1}) - s (h_{2}) |}^{2}}$ (7)

Definition 6. Let h_j (j = 1, 2, …, n) mean a set of HFEs, then we call

$\begin{matrix} sim (s (h_{j}), s (\bar{h})) = 1 - \frac{| s (h_{j}) - s (\bar{h}) |}{\sum_{j = 1}^{n} | s (h_{j}) - s (\bar{h}) |}, \\ j = 1, 2, \dots, n . \end{matrix}$ (8) the similarity between the hesitant fuzzy values is defined as s (h_j) and the mean is represented as $s (\bar{h})$ .

In real-life situations, the hesitant fuzzy values h_j (j = 1, 2, …, n) often utilize the form of a set of n preference values generated by n different individuals. Several individuals can allocate unduly high or unduly low preference values to their preferred or repugnant objects. Therefore, we allocate very low weights to these “false” or “biased” opinions. Then, using Equation 2, we can define the HFHWA weights as follows.

$\begin{matrix} w_{j} & = & \frac{sim (s (h_{j}), s (\bar{h}))}{\sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))}, \\ j = 1, 2, \dots, n \end{matrix}$ (9)

That is to say w_j ≥ 0, j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ .

By (2), we have

$\begin{matrix} DHFHWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (\frac{sim (s (h_{j}), s (\bar{h})) h_{j}}{\sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))}) = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{sim (s (h_{j}), s (\bar{h})) / \sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))} - \prod_{j = 1}^{n} {(1 - γ_{j})}^{sim (s (h_{j}), s (\bar{h})) / \sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{sim (s (h_{j}), s (\bar{h})) / \sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{j})}^{sim (s (h_{j}), s (\bar{h})) / \sum_{j = 1}^{n} sim (s (h_{j}), s (\bar{h}))}}} \end{matrix}$ (10)

We define (10) a dependent hesitant fuzzy Hamacher weighted average (DHFHWA) operator. Consider that the aggregated value of the DHFHWA operator is independent of the ordering, hence, it is a neat operator.

The normal distribution is belonged to one of the most commonly observed ones, and it is often found in events which are the aggregation of many smaller and independent random events. Xu [30, 31] et al. proposed a normal distribution to obtain several dependent uncertain ordered weighted aggregation operators, where the associated weights rely on the aggregated arguments. Inspired by the idea, we provide an approach using the DHFHWA weights:

$w_{j} = \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s (h_{j}), s (\bar{h})))}^{2}}{2 σ^{2}}}, j = 2, \dots, n .$ (11)

where $s (\bar{h})$ and σ are the arithmetic mean and the standard deviation of these hesitant fuzzy arguments variables h_j (j = 1, 2, …, n).

Consider that w_j ≥ 0, j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ , then by (14), we have

$\begin{matrix} w_{j} & = & \frac{\frac{1}{\sqrt{2 π} σ} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} \frac{1}{\sqrt{2 π} σ} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}} \\ = & \frac{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}, \\ j = 1, 2, \dots, n . \end{matrix}$ (12) then by (2), we have

$\begin{matrix} DHFHWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{j = 1}^{n} (\frac{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}} h_{j}}}{\sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}) = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - γ_{j})}^{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{j})}^{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{j})}^{e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{((d (s (h_{j}), s (\bar{h}))))}^{2}}{2 σ^{2}}}}}} \end{matrix}$ (13)

From (10) and (13), we can see that all the weights of the DHFHWA operator rely on the aggregated hesitant fuzzy variables [32 –35].

4 Models for multiple attribute decision making with hesitant fuzzy information

Let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives and G ={ G₁, G₂, …, G_n } be a set of attributes. If the decision makers provide several values for the alternative A_i under the state of nature G_j with anonymity, these values can be considered as a hesitant fuzzy element h_ij. In the case where two decision makers provide the same value, then the value emerges only once in h_ij. Assume that the decision matrix H = (h_ij) _m×n is the hesitant fuzzy decision matrix, where h_ij (i = 1, 2, …, m, j = 1, 2, …, n) are in the form of HFEs.

Afterwards, we design a novel approach to tackle the MADM problems with hesitant fuzzy information. The method is made up of the following steps:

Step 1. Exploit the DHFHWA operator: $\begin{matrix} h_{i} = DHFHWA (h_{i 1}, h_{i 2}, \dots, h_{in}) \\ = \oplus_{j = 1}^{n} (ω_{j} h_{ij}) \\ = \cup_{γ_{i 1} \in h_{i 1}, γ_{i 2} \in h_{i 2}, \dots, γ_{in} \in h_{in}} \\ {\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{ij})}^{sim (s (h_{ij}), s ({\bar{h}}_{i})) / \sum_{j = 1}^{n} sim (s (h_{ij}), s ({\bar{h}}_{i}))} - \prod_{j = 1}^{n} {(1 - γ_{ij})}^{sim (s (h_{ij}), s ({\bar{h}}_{i})) / \sum_{j = 1}^{n} sim (s (h_{ij}), s ({\bar{h}}_{i}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{ij})}^{sim (s (h_{ij}), s ({\bar{h}}_{i})) / \sum_{j = 1}^{n} sim (s (h_{ij}), s ({\bar{h}}_{i}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{ij})}^{sim (s (h_{ij}), s ({\bar{h}}_{i})) / \sum_{j = 1}^{n} sim (s (h_{ij}), s ({\bar{h}}_{i}))}}} \end{matrix}$ (14)

to obtain the overall values h_i (i = 1, 2, …, m) of the alternative A_i, where $s ({\bar{h}}_{i}) = \frac{1}{n} \sum_{j = 1}^{n} s (h_{ij})$ .

Step 2. Compute the scores $S ({\tilde{h}}_{i})$ of the overall hesitant fuzzy preference value ${\tilde{h}}_{i} (i = 1, 2, \dots, m)$ to sort all the alternatives A_i (i = 1, 2, …, m) and then to choose the optimal ones.

Step 3. Sort all the alternatives A_i (i = 1, 2, …, m) and choose the optimal one(s) in terms of $S ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)$ .

5 Numerical example

As is well known that with the rapid development of real estate industry, real estate business men has transited from extensive management rely on their own sense to scientific project strategy planning. Moreover, differences still exist in understanding the meaning and research object of strategy planning of real estate investment. In this paper, we aim to construct integrated theory and framework of systematic strategy planning of real estate investment by analyzing real estate investment project and then proposing the planning methods and ways of key sub-systems in real estate investment to lower risks in real estate investment. Therefore, in this section we provide an example for real estate investment decision making with hesitant fuzzy information to explain the approach which is given in this paper. There is a panel with five possible real estate investment alternatives A_i (i = 1, 2, 3, 4, 5) to choose. The experts choose four attributes to estimate the five possible real estate investment alternatives: ➀G₁ is the environmental factor in real estate investment; ➁G₂ is the economic factors in real estate investment; ➂G₃ is the risk factors in real estate investment; ➃G₄ is the enterprise level in real estate investment. To avoid influence each other, the decision makers should estimate the five possible real estate investment alternatives A_i (i = 1, 2, 3, 4, 5) under the above four attributes in anonymity and the decision matrix H = (h_ij) _4×4 is proposed in Table 1, where h_ij (i = 1, 2, 3, 4, j = 1, 2, 3, 4) are represented as HFEs.

Next, we exploit the method which is designed for real estate investment with hesitant fuzzy information.

Utilize the DHFHWA operator to derive the overall preference values h_i (i = 1, 2, 3, 4, 5) of the real estate investment alternatives A_i. The results are illustrated in Table 2.

In terms of the aggregating results shown in Table 2, the ordering of the real estate investment alternatives are shown in Table 3. Note that >represents “preferred to”. The best real estate investment alternative is A₄.

6 Conclusion

Real estate project is a complicated system characterized by high investment input, high risk, long industrial chain and periodicity, many professions and connections involved. The decision making of real estate project depends mostly upon experience and qualitative analysis, lacking the professional and systematic guidance of risk management methods. The research on the internal rules and risk management methods is not enough from the government, academic institutions and enterprises. Therefore, it is very necessary to have a deep research on the risk management method of real estate project. Particularly, we study on the multiple attribute decision making for real estate investment with hesitant fuzzy information. To test the effectiveness of the proposed method, an example for real estate investment is proposed to testify the developed method. To extend the proposed research work, in the future, we will try to extend the proposed method in other fields by other models and approaches [36 –50].

References

Ben

, and Bernanke

M.G.

, Inside the black box: The credit channel of monetary policy transmission, Journal of Econometrics9 (1995)27–48.

Aoki

, Proudman

, and Vlieghe

, House prices, consumption, and monetary policy: A financial accelerator approach, Journal of Financial Intermediation13(4) (2004)414–435.

MatteoIa

, and Raoul

, The credit channel of monetary policy: Evidence from the housing market, Journal of Macroeconomics31 (2007)1–28.

Liu

, and He

, An efficiency test of monetary policy transmission channel in China’s real estate, South China Finance7 (2006)5–7.

Gao

, and Wang

, The efficiency of monetary policy transmission mechanism in China’s real estate market: 2000–2007, Finance & Trade Economics3 (2009)129–135.

Kyland

, and Prescott

, Time to build and aggregate fluctuations, Econometrical50(6) (1982)1345–1370.

Liang

, Bank credit payments constraint and China’s real estate prices, Studies of International Finance3 (2011)4–10.

Cogley

, and Nason

J.M.

, Effects of the Hodrick-Prefilter on trend and difference stationary time series Implications for business cycle research, Journal of Economic Dynamics and Control19(1-2) (1995)253–278scott.

, The dynamic stochastic general equilibrium model in the application of monetary policy analysis: Review and prospects, Financial Theory & Practice12 (2011)8–14.

10.

Z.S.

, and Xia

, On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligence Systems26(5) (2011)410–425.

11.

Z.S.

, and Xia

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences181(11) (2011)2128–2138.

12.

Wei

G.W.

, Zhao

X.F.

, Wang

H.J.

, and Lin

, Hesitant fuzzy choquet integral aggregation operators and their applications to multiple attribute decision making, Information: An International Interdisciplinary Journal15(2) (2012)441–448.

13.

Kahraman

, and Öztaysi

, and S.Ç. Onar, A multicriteria supplier selection model using hesitant fuzzy linguistic term sets, Multiple-Valued Logic and Soft Computing26(3-5) (2016)315–333.

14.

Wei

G.W.

, Hesitant fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems31 (2012)176–182.

15.

, Xiao

, Chen

, and Liu

, Interval type-2 hesitant fuzzy set and its application in multi-criteria decision making, Computers & Industrial Engineering87 (2015)91–103.

16.

Wei

G.W.

, and Zhao

X.F.

, Induced hesitant interval-valued fuzzy einstein aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems24 (2013)789–803.

17.

, and Xu

, Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information, IEEE Trans Cybernetics46(3) (2016)694–705.

18.

Zhang

, and Wei

G.W.

, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Applied Mathematical Modelling37 (2013)4938–4947.

19.

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 Systems46 (2013)43–53.

20.

García-Lapresta

J.L.

, and Pérez-Román

, Consensus-based clustering under hesitant qualitative assessments, Fuzzy Sets and Systems292 (2016)261–273.

21.

, and Wei

, GRA method for multiple criteria group decision making with incomplete weight information under hesitant fuzzy setting, Journal of Intelligent and Fuzzy Systems27 (2014)1095–1105.

22.

Wang

, and Xu

, Admissible orders of typical hesitant fuzzy elements and their application in ordered information fusion in multi-criteria decision making, Information Fusion29 (2016)98–104.

23.

Zhao

, Lin

, and Wei

, Hesitant triangular fuzzy information aggregation based on einstein operations and their application to multiple attribute decision making, Expert Systems with Applications41(4) (2014)1086–1094.

24.

Rodríguez

R.M.

, Bedregal

B.R.C.

, Bustince

, Dong

, Farhadinia

, Kahraman

, Martínez-López

, Torra

, Xu

Y.J.

, Xu

, and Herrera

, A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making, Towards high quality progress, Information Fusion29 (2016)89–97.

25.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems25 (2010)529–539.

26.

Xia

, and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52(3) (2011)395–407.

27.

Zhou

, Zhao

, and Wei

, Hesitant fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems26(6) (2014)2689–2699.

28.

Hamachar

(1978)276–288Uber logische verknunpfungenn unssharfer Aussagen undderen Zugenhorige Bewertungsfunktione, bernTral, Klir, Riccardi (Eds.), Progress in Cyatics and systems research, vol. 3, Hemisphere, Washington DC, pp.

29.

F.-Y.

, Yang

J.-Y.

, and Zhang

P.-D.

, An approach to multiple attribute decision making problems based on hesitant fuzzy set, Journal of Intelligent and Fuzzy Systems27(6) (2014)2749–2755.

30.

Z.S.

, Dependent OWA operators, Lecture Notes in Artificial Intelligence3885 (2006)172–178.

31.

Z.S.

, Dependent uncertain ordered weighted aggregation operators, Information Fusion9(2) (2008)310–316.

32.

Wei

, and Zhao

, Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making, Expert Systems with Applications39 (2012)5881–5886.

33.

Liu

, and Qi

, Some generalized dependent aggregation operators with 2-dimension linguistic information and their application to group decision making, Journal of Intelligent and Fuzzy Systems27(4) (2014)1761–1773.

34.

Liu

H.-C.

, Lin

Q.-L.

, and Wu

, Dependent interval 2-tuple linguistic aggregation operators and their application to multiple attribute group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems22(5) (2014)717–736.

35.

Liu

, Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making, J Comput Syst Sci79(1) (2013)131–143.

36.

Tian

S.-F.

, Zhou

S.-W.

, Jiang

W.-Y.

, and Zhang

H.-Q.

, Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation, Applied Mathematics and Computation218(13) (2012)7308–7321.

37.

Z.X.

, Chen

M.Y.

, Xia

G.P.

, and Wang

, An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications38 (2011)15286–15295.

38.

Wei

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems17(3) (2015)484–489.

39.

Park

J.H.

, Park

I.Y.

, Kwun

Y.C.

, and Tan

X.G.

, Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment, Applied Mathematical Modelling35 (2011)2544–2556.

40.

Ran

L.-G.

, and Wei

, Uncertain prioritized operators and their application to multiple attribute group decision making, Technological and Economic Development of Economy21(1) (2015)118–139.

41.

Y.J.

, Huang

, Da

Q.L.

, and Liu

X.W.

, Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator, Journal of Systems Engineering and Electronics21 (2010)990–994.

42.

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 Management15(2) (2014)277–298.

43.

Z.S.

, and Yager

R.R.

, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning48(1) (2008)246–262.

44.

Zhou

, and Wu

W.Z.

, On generalized intuitionistic fuzzy rough approximation operators, Information Sciences178 (2008)2448–2465.

45.

Merigo

J.M.

, Casanovas

, and Liu

P.D.

, Decision making with fuzzy induced heavy ordered weighted averaging operators, International Journal of Fuzzy Systems16(3) (2014)277–289.

46.

Meng

F.Y.

, Chen

X.H.

, and Zhang

, Generalized hesitant fuzzy generalized shapley-choquet integral operators and their application in decision making, International Journal of Fuzzy Systems16(3) (2014)400–410.

47.

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 Systems27 (2014)2153–2162.

48.

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 Systems55(3) (2013)679–684.

49.

Chen

Y.F.

, Peng

X.D.

, Guan

G.H.

, and Jiang

H.D.

, Approaches to multiple attribute decision making based on the correlation coefficient with dual hesitant fuzzy information, Journal of Intelligent and Fuzzy Systems26(5) (2014)2547–2556.

50.

, Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making, Applied Mathematical Modelling38(2) (2014)659–666.