Chinese Family Farm Business Risk Assessment Using a Hierarchical Hesitant Fuzzy Linguistic Model

Abstract

Chinese family farms are continuously expanding; they are also facing various business risks that lead to a shorter lifespan. This paper constructed a family farm business risk assessment model that combined a hesitant fuzzy linguistic term sets (HFLTS) model with a hesitant fuzzy weighted average (HFWA) operator. On the basis of the factor analysis, this study built a family farm indicator system that included the natural, technical, market, policy, society, and management risk. The HFLTS was used for the assessment of weights in pairwise comparison matrices, and the HFWA operator was used as an aggregation operator to calculate the business risk score of family farms. For our case study, a method comparison analysis was also performed to check the validity of the results obtained by our risk assessment model.

1. Introduction

A family farm—as a new type of agricultural business entity that engages in large-scale, intensive, and commercialized agricultural production [1]—represents an important form of modern agricultural micromanagement organization and plays a crucial role in promoting the development of China’s new urbanization and agricultural modernization. By the end of June 2022, China had more than 3.9 million family farms [2]. However, the National Credit Information Publicity System from 2022 shows that from 2013 to 2021, the average lifespan of family farms was only 3.3 years, with nearly half of them lasting less than 4 years. This leads to poor operating conditions for family farms, even facing closure due to factors such as large-scale diseases, significant fluctuations in agricultural product prices, a challenging agricultural risk environment, and even pandemics [3]. For instance, the number of purely breeding family farms in Chengdu, Sichuan Province, significantly decreased in 2023 compared with previous years due to the impact of environmental assessments, breeding bans, animal diseases, and other factors [4]. Methods to improve the resilience of family farms, enhance their ability to resist risks, and reduce losses when faced with business challenges are critical. The Chinese Communist Party Central Committee and the State Council unveiled the 2024 No.1 Document, which emphasized the need to strengthen the agricultural risk prevention capacity, and the Notice of the Ministry of Agriculture and Rural Affairs on the Implementation of the Promotion Action of New Agricultural Operators proposed the enhancing of market competitiveness and risk resistance of new-types of agricultural business entities. Family farms, as a new type of agricultural business entity, face more serious business risks than traditional small farmers [5]; strengthening risk protection for family farms is required for agricultural risk reduction. It is of significance to evaluate the various risks that may arise in the operation of China’s family farms.

Ullah et al. divided family farm risks into two categories [6]. The first is the business risk, including the production risk, market risk, system risk, and personal risk. The second is the financial risk from the different financing methods of agricultural enterprises. As for the business family farm risk, family farms face market risk, natural risk, policy risk, financing support, technical risk, etc., in the process of production and operation [7,8,9,10]. Gao et al. [11] and Zhang et al. [12] held that natural risk, market risk, and technical risk were the main business risks of agricultural business entities, while logistics risk and information risk were nonmajor business risks. Ullah et al. thought that family farm risks depended on the geographic location, government policies, the availability of risk management tools, and the type of agricultural product [6]. Bui and Le conducted a study of clam farms in northern Vietnam, categorizing family farm risks into natural physical risk, operational and technical risks, financial and market-related risks, social and political risks, and liability and consumer-related risks [13]. In addition, family farm business risks include product quality risk [14], scale risk [15], production risk [16,17], asset risk [18], price risk [19], and farm types [20]. The farmers’ own characteristics also affect the business risks of family farms. Hazell [21] found that factors such as the farmers’ risk attitude, entrepreneurial ability, and policy-environment cognitive behaviors have an important impact on coping with natural risks and market risks in production activities. Lu and Zhu identified land psychology and a low cognitive level as risk factors [22]; other research identified the risk identification ability of family farmers [23] and individual comprehensive ability [24] as risk factors.

Family farm business risk assessment studies in the literature can be summarized as follows.

Although Wu built a risk indicator system, the degree of risk was qualitatively measured [25]. Deng [26] and Liu et al. [27] established business risk evaluation indicator systems for family farms by using the survey data of 269 family farms in Heilongjiang Province, and conducted an empirical analysis of the business risks of family farms using a regression model. Chen [28] used a VaR method and weighted the moving average coefficient to measure the price risk and natural risk of family farms. Vaitkevicius used a hierarchical cluster analysis to build a risk evolution model to evaluate family farm risks, including economic, financial, production, and political risks [29]. The above method for assessing family farm business risks was mainly based on nonexpert scoring data; however, it is also necessary to construct a model to assess the family farm business risks using expert scoring data, while also considering the experts’ hesitation in providing risk evaluations.

Experts’ scores can be evaluated using DEMATEL, VIKOR [30], and Fuzzy AHP [31,32,33] models. Uncertain knowledge is effectively dealt with using fuzzy logic and fuzzy set theory. However, when two or more sources of uncertainty exist at the same time, fuzzy sets are constrained. Buckley’s ordinary fuzzy AHP [34] has been transformed into a hesitant fuzzy linguistic term sets (HFLTS) AHP because Buckley’s fuzzy AHP is almost a unique method, without any criticism in the literature. Multiple experts can assign different membership degrees or compromise on a joint membership degree. HFLTS has several advantages, such as reducing the difficulty that decision-makers experience in decision linguistic terms, providing flexibility in evaluations, and allowing decision-makers to assign a weight to parameters. To the best of the authors’ knowledge, there are few studies that propose the application of a multicriteria HFLTS and the hesitant fuzzy weighted average (HFWA) method for family farm business risk assessments. In the existing research, this joint membership degree may be defined as a single or double linguistic term [35,36], but scant research uses multi-interval linguistic terms [37] such as “more than good” or “at most very good”.

In order to fill in the gaps of existing research for family farm business risk assessment, this study combined the HFLTS and HFWA operator to evaluate the family farm business risk, constructed an HFLTS model to obtain the indicator weights, and used an HFWA operator to calculate the comprehensive business risks score of family farms.

2. Family Farm Business Risk Indicator System

From an economic perspective, risk refers to uncertainty. Family farm business risks involve the uncertainties and unsustainability in the operation process or the losses caused by improper decision making by family farmers. These risks often manifest as uncertainties in income, costs, and losses. Uniform standard family farms risk measurements are currently lacking, and a comprehensive and systematic risk evaluation indicator system has not yet been developed. These scenarios are caused by two main reasons. First, the uncertainty inherent in agricultural production and management activities results in dynamic changes in risk. Second, the specific attributes and regional characteristics of agricultural production necessitate that the evaluation of family farm business risks must apply to different contexts. Therefore, the present study, which is based on the existing research [6,13,25,26,27,28,29] and field research of family farms, identifies 21 risk factor indicators to form a preliminary indicator system, as shown in

Table 1. Preliminary indicator system for family farm business risks.

No.	Risk Factor
1	Incidence of natural disaster
2	Incidence of diseases from pests
3	Technical adaptation degree
4	Technical staff satisfaction degree
5	Farm-scale
6	Potential market capacity
7	Product matching degree
8	Price fluctuations degree
9	Product quality
10	Policy support degree
11	Policy fluctuations degree
12	Product diversity
13	Social service system degree
14	Means of production value fluctuations degree
15	Frequency of land disputes
16	Contract performance degree
17	Risk cognition ability
18	Management decision-making ability
19	Organizational coordination ability
20	Financial management ability
21	Innovation ability

.

Many evaluation indicators are utilized in family farm business risk evaluation, and collinearity may exist among them. Therefore, we used factor analysis to screen the indicators. First, we conducted a questionnaire survey, followed by reliability analysis and exploratory factor analysis of the data using SPSS 26. The overall reliability test of the questionnaire showed that Cronbach’s α coefficient was above 0.7 (0.91), and the CITC was above 0.5 (0.74). The results of the exploratory factor analysis showed that the KMO test was higher than 0.6, and Bartlett’s sphericity test showed significance < 0.05. As shown in Figure 1, six categories have eigenvalues greater than 1, and the variance contribution rate is above 80%. Indicators with a correlation coefficient lower than 0.5 were excluded from the component matrix. The component matrix in

Table 2. Component matrix.

No.	Component
	1	2	3	4	5	6
6	0.817
4	0.753
3	0.702
9	0.682
5
16		0.584
11		0.560
21		0.501
7
12
13			0.786
18			0.632
15
1				0.769
2				0.651
17
10					0.575
14					0.528
8						0.663
1						0.524
21

Note: Null value represents absolute correlation coefficient < 0.5.

shows that the absolute values of the correlation coefficients of the six indicators, namely, farm scale (No. 5), product matching degree (No. 7), product diversity (No. 12), frequency of land disputes (No. 15), risk cognition ability (No. 17), and innovation ability (No. 21), are all lower than 0.5. According to the screening principle of the principal component analysis method, these indicators were eliminated. Finally, 15 evaluation indicators were obtained.

Table 2 shows that 15 evaluation indicators can reflect the family farm business risks. These 15 evaluation indicators were classified by referring to the existing literature. As shown in

Table 3. Indicator system for family farm business risks.

No.	First Layer Indicators	No.	Second Layer Indicators
A1	Natural risk	A11	Incidence of natural disaster
		A12	Incidence of diseases from pests
A2	Technical risk	A21	Technical adaptation degree
		A22	Technical staff satisfaction degree
A3	Market risk	A31	Potential market capacity
		A32	Product matching degree
		A33	Price fluctuations degree
A4	Policy risk	A41	Policy support degree
		A42	Policy fluctuations degree
A5	Social risk	A51	The social service system degree
		A52	Means of production value fluctuations degree
		A53	Contract performance degree
A6	Management risk	A61	Management decision-making ability
		A62	Organizational coordination ability
		A63	Financial management ability

, the 15 evaluation indicators are categorized into natural risk, technical risk, market risk, policy risk, social risk, and management risk. Each categorized subscale passed the reliability and validity tests. A₁–A₆ is the first layer indicator, and A₁₁–A₆₃ is the second layer indicator.

3. Family Farm Business Risk Assessment Model

3.1. Indicator Weight Calculation Model—HFLTS

On the basis of the constructed risk evaluation indicator system for family farms, we used the HFLTS model to measure the experts’ evaluated risk information and calculate the weights of the indicators. This method can effectively address the subjective issues in the multicriteria decision-making process. The first layer evaluation indicator set is $\left\{A_1,A_2,\cdots ,A_n\right\}$, and the second layer evaluation indicator set is $\left\{A_{11},A_{12},\cdots ,A_{ij}\right\}$. The experts set is $\left\{C_1,C_2,\cdots ,C_{k-1},C_k\right\}$. n represents the number of indicators, and k represents the number of experts. The weight of the indicator can be calculated through the following steps:

Step 1. The hesitant fuzzy sets (HFS) are defined using Equation (1) [38].

$$H=\left\{〈x,h\left(x\right)〉\left|x\in X\right.\right\}$$

(1)

where X is a fixed set, and HFS on X is the term of a function that returns a subset of [0, 1] when applied to X [39]. $h\left(x\right)$ is a set of some values in [0, 1], and denotes the possible membership degrees of the element $x\in X$ to the set H.

The semantics and syntax of the linguistic term set are defined using Equation (2) [40].

$$S=\left\{s_0,\cdots s_g\right\}$$

(2)

The set S = {no importance, very low importance, low importance, medium importance, high importance, very high importance, absolute importance} is defined in this paper, and the corresponding values are shown in

Table 4. The scale for the linguistic terms.

Level	Value
No importance (N)	0
Very low importance (VL)	1
Low importance (L)	2
Medium importance (M)	3
High importance (H)	4
Very high importance (VH)	5
Absolute importance (A)	6

[37]. On the basis of this language set S, experts can use expressions such as “less than”, “more than”, “at least”, “at most”, and “between and” to provide judgments, such as “less than VH”, “at least L”, “at most H”, and “between M and H”.

Step 2. A context-free grammar G_H is defined using Equation (3).

$$G_H=\left\{V_N,V_T,I,P\right\}$$

(3)

where V_N is the set of nonterminal symbols, V_T is the set of terminals’ symbols, I is the starting symbol, and P is the production rules defined in an extended Backus–Naur form [41].

Step 3. The hesitant fuzzy linguistic judgment matrix is constructed.

E_GH is a conditional function that converts the experts’ evaluated scores into the hesitant fuzzy linguistic sets $H_s$ through Equation (4), where the $\iota \iota$ is the language expression generated based on the context-free grammar G_H. The conversion rules are as follows: If $s_r=s_j$, then $E_{GH}$ is “s_j”. If $s_r<s_j$, then $E_{GH}$ is “less s_j”. If $s_r>s_j$, then $E_{GH}$ is “more than s_j”. If $s_r\ge s_j$, then $E_{GH}$ is “at least s_j”. If $s_r\le s_j$, then $E_{GH}$ is “at most s_j”. If $s_j\le s_r\le s_t$, then $E_{GH}$ is “between s_j and s_t”.

$$E_{GH}:\iota \iota \to H_s$$

(4)

Step 4. Conversion to the HFLTS envelope is conducted.

The upper bound $H_s^{+}$ of $H_s$ is given in Equation (5), and the lower bound $H_s^{-}$ of $H_s$ is given in Equation (6).

$$H_s^{+}=\max \left(s_i\right)=s_j,s_j\in H_s \mathrm{and} \forall i,s_i\le s_j$$

(5)

$$H_s^{-}=\min \left(s_i\right)=s_j,s_i\in H_s \mathrm{and} \forall i,s_j\le s_i$$

(6)

Then, each one $h_{ij}^k$ has a corresponding envelope $env\left(h_{ij}^k\right)=\left[h_{ij}^{k-},h_{ij}^{k+}\right]$, $h_{ij}^k\in H_s$.

Step 5. The preference relationship between pessimism and optimism is integrated.

The preference relationship between pessimism and optimism is determined by Equation (7).

$$\overline{x}=\Delta \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n{\Delta }^{-1}\left(s_i,{\alpha }_i\right)}\right)=\Delta \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n{\beta }_i}\right)$$

(7)

In Equation (6), β represents the resulting value obtained after aggregation, which can be converted from Δ: [0, μ] → S × [−0.5, 0.5) to the linguistic information of a binary feature, as shown in Equation (8). In Equation (8), round represents the rounding operator. $\left(s_i,\alpha \right)$ represents the information about the meaning of the language of the quadratic calculation, which shows the function ${\Delta }^{-1}$. This function obtains the corresponding value β ∈ [0, μ], as shown in Equation (9).

$$\Delta \left(\beta \right)=\left(s_i,\alpha \right),i=roud\left(\beta \right) \mathrm{and} \alpha =\beta -i$$

(8)

$${\Delta }^{-1}\left(s_i,\alpha \right)=i+\alpha =\beta$$

(9)

Step 6. The weight of the indicators is calculated.

Set vector $V^R$ is established as

$$V^R=\left(p_1^R,p_2^R,\cdots ,p_n^R\right)$$

(10)

In Equation (10), $p_i^R=\left[p_i^{-},p_i^{+}\right]$, n represents the risk indicator, $p_i^{+}$ is the optimistic collective preference of the risk indicator, and $p_i^{-}$ is the pessimistic collective preference of the risk indicator. The mean value of the interval vector is calculated and the result is normalized. Then, the weights of all indicators are obtained.

3.2. Risk Assessment Model—HFWA

The HFWA model was constructed to evaluate the family farm business risk. A nonempty set $X$ exists; $H=\left\{\left[x,\left.h_H\left(x\right)\right|x\in X\right]\right\}$ is the hesitant and fuzzy set on $X$, where $h_H\left(x\right)$ represents the set of possible membership degree values for set $x\in X$, and $h_H\left(x\right)\subseteq \left[0,1\right]$.

(1)

Statistical hesitant fuzzy evaluation sets

If the expert j evaluates indicator i by hesitating in the S evaluation levels of $h_H\left(x\right)$, then the hesitant fuzzy evaluation sets of indicator i is $f_{ij}=\left\{f_{ij}^1,f_{ij}^2,\cdots f_{ij}^s\right\}$.

(2)

Calculating each indicator score

The scores of each indicator are calculated by the hesitant fuzzy weighted average operator as follows [38]:

$$h_i=HFWA\left(h_1,h_2,\cdots ,h_n\right)=\underset{j=1}{\overset{n}{\oplus }}\left(h_j{w}_j\right)={\cup }_{r_1\in h_1,r_2\in h_2,\cdots r_n\in h_n}\left\{1-{\displaystyle \prod _{j=1}{\left(1-r_j\right)}^{w_j}}\right\}$$

(11)

In Equation (11), $h_j\left(j=1,2,\cdots ,n\right)$ is a group of hesitant fuzzy elements, $\oplus$ represents the summation of fuzzy elements, $w{\left(w_1,w_2,\cdots ,w_n\right)}^T$ is the weight of $h_j$, and ${\displaystyle \sum _{j=1}^n{w}_j=1,w_j=[0,1]}$. The evaluation score of hesitant fuzzy elements is $s\left(h\right)$, as shown in Equation (12), where $\#h$ is the number of the element h.

$$s\left(h\right)=\frac{1}{\#h}{\displaystyle \sum _{r\in h}r}$$

(12)

The indicator score $s\left(h_i\right)$ is calculated by the HFWA model. Thus, the score set $F=\left(f_1,f_2,\cdots ,f_n\right)$ is obtained.

(3)

Calculating the final evaluation score

$$S=W\times F^T=\left(w_1,w_2,\cdots ,w_n\right){\left(f_1,f_2,\cdots ,f_n\right)}^T$$

(13)

The final risk score evaluation is shown in Equation (13). As shown in

Table 5. Risk assessment level.

Level	Very Low Risk	Low Risk	Medium Risk	High Risk	Very High Risk
Range	[0, 0.2)	[0.2, 0.4)	[0.4, 0.6)	[0.6, 0.8)	[0.8, 1]

, this study adopts five levels to evaluate the family farms’ business risk: “very low risk”, “low risk”, “medium risk”, “high risk”, and “very high risk”. A high value indicates a high business risk for family farms. Therefore, risk prevention measures can be implemented for the key risk factors in operations based on the evaluation value of S to prevent accidents.

4. Case Study

Fengshen Family Farm was established in December 2020 in Jianyang City, Sichuan Province, China. It operates as an integrated crop and livestock family farm and faces various risks. The owner of Fengshen Family Farm stated that the farm’s profit margins were low in 2022 and 2023. It receives minimal subsidies and relies mainly on its own accumulated assets for development. The farm has been affected by natural risk, leading to reduced yields and decreased quality. Local issues exist, such as the incomplete construction of a food safety and integrity system for agricultural products and the inadequate agricultural infrastructure. The farm also faces problems such as insufficient technical capabilities, weak adaptability, etc. Therefore, this study evaluates the business risks of this family farm through the proposed model and suggests corresponding management recommendations.

4.1. Calculate the Indicator Weight Based on HFLTS

(1)

Constructing the hesitant fuzzy linguistic judgment matrix

We invited four experts to evaluate the risk indicators for Family Farm A. The invited experts have long been engaged in family farm research and have similar professional backgrounds in the field. Thus, they were assigned with the same weight in the risk evaluation. On the basis of the rules defined in Table 4 and the conditional function E_GH, four experts used the language term set provided to evaluate the first layer indicator shown in Table 1 through the Delphi method. The hesitant fuzzy linguistic judgment matrix is shown in

Table 6. Expert evaluations for the first layer indicators.

Indicators	A1	A2	A3	A4	A5	A6
A1	-	at least H	M	M	M and H between	between M and VH
A2	less than M	-	at most L	at most L	M	more than M
A3	M	at least M	-	M and H between	more than M	between M and VH
A4	between L and M	more than H	between L and M	-	at least H	between H and VH
A5	at most M	M	less than M	between L and M	-	between VH and A
A6	between VL and M	at most L	between VL and M	less than L	at most L	-
A1	-	between H and VH	between M and H	at most M	M	VH
A2	between VL and L	-	L	at most L	between M and H	more than M
A3	at most M	at least M	-	between M and H	more than M	between M and H
A4	L	at least M	between L and M	-	H	between H and VH
A5	M	between H and VH	less than M	at most M	-	between H and VH
A6	less than M	less than M	between VL and L	between L and M	at most M	-
A1	-	M	H	M	between M and H	between VH and A
A2	M	-	M	at most L	between M and H	less than M
A3	between M and H	M	-	between M and H	more than M	between M and H
A4	M	at least M	M	-	at least M	M
A5	between L and M	between H and VH	less than M	at most M	-	at most M
A6	less than M	more than M	between VL and L	M	more than M	-
A1	-	M	between VL and L	M	M	L
A2	M	-	H	M	between M and H	between M and H
A3	at least M	at most M	-	at most H	between M and H	between M and H
A4	between L and M	M	M	-	between L and M	M
A5	M	between H and VH	between L and M	between M and H	-	M
A6	more than M	between L and M	between VL and L	M	M	-

.

(2)

Converting to HFLTS envelope

As shown in

Table 7. HFLTS envelope for the first layer indicators.

Indicators	A1	A2	A3	A4	A5	A6
A1	-	[H, A]	[M, M]	[M, M]	[M, H]	[M, VH]
A2	[N, L]	-	[N, L]	[N, L]	[M, M]	[H, A]
A3	[M, M]	[M, A]	-	[M, H]	[H, A]	[M, VH]
A4	[L, M]	[VH, A]	[L, M]	-	[H, A]	[H, VH]
A5	[N, M]	[M, M]	[N, L]	[L, M]	-	[VH, A]
A6	[VL, M]	[N, L]	[VL, M]	[N, VL]	[N, L]	-
A1	-	[H, VH]	[M, H]	[N, M]	[M, M]	[VH, VH]
A2	[VL, L]	-	[L, L]	[N, L]	[M, H]	[H, A]
A3	[N, M]	[M, A]	-	[M, H]	[H, A]	[M, VH]
A4	[L, L]	[M, A]	[L, M]	-	[H, H]	[H, VH]
A5	[M, M]	[H, VH]	[N, L]	[N, M]	-	[H, VH]
A6	[N, L]	[N, L]	[VL, L]	[L, M]	[N, M]	-
A1	-	[M, M]	[H, H]	[M, M]	[M, H]	[VH, A]
A2	[M, M]	-	[M, M]	[N, L]	[M, H]	[N, L]
A3	[M, H]	[M, M]	-	[M, H]	[H, A]	[M, VH]
A4	[M, M]	[M, A]	[M, M]	-	[M, A]	[M, M]
A5	[L, M]	[H, VH]	[N, L]	[N, M]	-	[N, M]
A6	[N, L]	[H, A]	[VL, L]	[M, M]	[H, A]	-
A1	-	[M, M]	[VL, L]	[M, M]	[M, M]	[L, L]
A2	[M, M]	-	[H, H]	[M, M]	[M, H]	[M, H]
A3	[M, A]	[N, M]	-	[N, H]	[M, H]	[M, VH]
A4	[L, M]	[M, M]	[M, M]	-	[L, M]	[M, M]
A5	[M, M]	[H, VH]	[L, M]	[M, H]	-	[M, M]
A6	[H, A]	[L, M]	[VL, L]	[M, M]	[M, M]	-

, we can obtain the corresponding envelope of each HFLTS by using Equations (5) and (6).

(3)

Integrating the preference between pessimism and optimism

Based on Equations (7)–(9), each envelope is integrated to obtain a pessimistic collective preference relationship $P_A^{-}$ and an optimistic collective preference relationship $P_A^{+}$, as shown in

Table 8. Pessimistic collective preference.

Indicators	A1	A2	A3	A4	A5	A6
A1	-	[H, −0.5]	[M, −0.25]	[M, +0.25]	[M, 0]	[H, −0.25]
A2	[L, −0.25]	-	[L, +0.25]	[VL, −0.25]	[M, 0]	[M, −0.25]
A3	[L, +0.25]	[L, +0.25]	-	[L, +0.25]	[H, −0.25]	[M, 0]
A4	[L, +0.25]	[L, +0.25]	[L, +0.25]	-	[H, −0.25]	[H, −0.5]
A5	[L, 0]	[H, −0.5]	[VL, −0.5]	[VL, +0.25]	-	[M, 0]
A6	[VL, +0.25]	[H, −0.25]	[VL, 0]	[L, 0]	[L, −0.25]	-

and

Table 9. Optimistic collective preference.

Indicators	A1	A2	A3	A4	A5	A6
A1	-	[H, +0.25]	[L, +0.25]	[M, 0]	[H, −0.5]	[H, 0]
A2	[L, +0.5]	-	[M, −0.25]	[L, +0.25]	[H, −0.25]	[H, +0.5]
A3	[H, 0]	[H, +0.5]	-	[H, 0]	[VH, +0.5]	[VH, 0]
A4	[M, −0.25]	[VH, +0.25]	[M, 0]	-	[H, −0.25]	[H, 0]
A5	[M, 0]	[H, +0.5]	[L, +0.25]	[M, +0.25]	-	[H, +0.25]
A6	[M, +0.25]	[M, +0.25]	[L, +0.25]	[L, +0.25]	[M, +0.5]	-

.

(4)

Calculating the weight of indicators

The pessimistic and optimistic collective preference relationships for each indicator are summarized in Table 8 and Table 9, respectively. The corresponding language interval vectors are obtained through Equation (10). Based on Table 4, the linguistic term intervals are assigned, the mean value of each interval is calculated, and the results are normalized. The final weights of the first layer indicators are shown in

Table 10. The linguistic intervals and weight for first layer indicators.

Indicators	Linguistic Intervals	Numerical Interval	Mean Value	Weight
A1	[(M, −0.46);(M, −0.17)]	[2.54, 2.83]	2.69	0.18
A2	[(L, −0.25);(M, −0.37)]	[1.75, 2.63]	2.19	0.15
A3	[(L, +0.25);(H, −0.37)]	[2.25, 3.83]	3.04	0.20
A4	[(L, +0.46);(H, +0.13)]	[2.46, 3.13]	2.79	0.19
A5	[(L, −0.25);(M, −0.12)]	[1.75, 2.88]	2.31	0.16
A6	[(VL, +0.25);(M, +0.46)]	[1.25, 2.46]	1.85	0.12

.

4.2. Calculate the Risk Score Based on HFWA

(1)

Statistical hesitant fuzzy evaluation sets

On the basis of Table 5, the expert group provides a score of the family farm business risks, i.e., $h_H\left(x\right)$ = {0, 0.1, 0.2, …, 0.9, 1.0}. The higher the score is, the higher the risk of the indicator is. The final score of the hesitant fuzzy sets is obtained, as shown in

Table 11. Expert evaluation score for the second layer indicators.

Indicators	Weight	Experts Evaluation Score
		1	2	3	4
A11	0.079	0.5	0.6	(0.4, 0.5)	(0.5, 0.6)
A12	0.070	(0.5, 0.7)	0.7	0.6	0.5
A21	0.059	0.5	(0.4, 0.5)	0.3	(0.4, 0.6)
A22	0.064	(0.4, 0.5, 0.6)	0.5	(0.5, 0.6)	0.6
A31	0.068	0.3	0.5	0.4	0.5
A32	0.072	0.5	(0.5, 0.6)	(0.4, 0.6)	(0.5, 0.6)
A33	0.071	0.3	(0.2, 0.3, 0.4)	(0.3, 0.4)	0.4
A41	0.073	0.6	0.5	0.5	(0.4, 0.5, 0.6)
A42	0.071	(0.4, 0.5)	0.4	(0.5, 0.6)	0.5
A51	0.065	0.4	0.6	0.4	(0.5, 0.6)
A52	0.066	0.3	(0.1, 0.3)	0.3	(0.2, 0.3)
A53	0.067	0.2	(0.3, 0.4)	0.4	0.3
A61	0.056	(0.5, 0.7)	(0.5, 0.7)	(0.4, 0.5)	0.6
A62	0.057	(0.3, 0.4)	0.3	0.3	(0.4, 0.5)
A63	0.062	(0.5, 0.6)	0.5	(0.5, 0.6)	0.5

.

(2)

Calculating each indicator score

Given that the expert weights are equal, the default expert weights can be set as w = {0.25, 0.25, 0.25, 0.25}^T. The calculation process of Equation (11) is as follows (with the calculation of A₁₁ taken as an example):

$$\begin{array}{l}{h}_1=HFWA\left(h_{11},h_{12},h_{13},h_{14}\right)=HFWA\left\{0.5,0.6,\left(0.4,0.5\right),\left(0.5,0.6\right)\right\}=\underset{j=1}{\overset{4}{{\displaystyle \oplus }}}\left(h_i{w}_j\right)\\ =U_{r_{11}\in h_{11},r_{12}\in h_{12},r_{13}\in h_{13},r_{14}\in h_{14}}\left\{1-{\left(1-r_{11}\right)}^{0.25}{\left(1-r_{12}\right)}^{0.25}{\left(1-r_{14}\right)}^{0.25}\right\}=\end{array}$$

$$\left\{0.505,0.527,0.532,0.553\right\}$$

Calculated using Equation (12), the score of A₁₁ is 0.529. All the scores of the second layer indicators are shown in

Table 12. The integrating score for the second layer indicators.

Indicators	Score Integration	Score
A11	{0.505, 0.527, 0.532, 0.553}	0.529
A12	{0.584, 0.634}	0.609
A21	{0.404, 0.431, 0.462, 0.486}	0.446
A22	{0.505, 0.532, 0.527, 0.553, 0.553, 0.557}	0.541
A31	{0.431}	0.431
A32	{0.477, 0.577, 0.477, 0.577, 0.477, 0.577, 0.477, 0.577}	0.527
A33	{0.304, 0.352, 0.352, 0.330, 0.326, 0.376}	0.340
A41	{0.505, 0.527, 0.553}	0.528
A42	{0.452, 0.505, 0.452, 0.505}	0.479
A51	{0.482, 0.510}	0.496
A52	{0.229, 0.300, 0.229, 0.300}	0.265
A53	{0.304, 0.330}	0.317
A61	{0.505, 0.634, 0.505, 0.634, 0.505, 0.634, 0.505, 0.634}	0.569
A62	{0.326, 0.381, 0.326, 0.381}	0.354
A63	{0.500, 0.553, 0.500, 0.553}	0.526

.

(3)

Calculating the final evaluation score

The score values of each indicator of the first layer were further calculated using Equation (13), as shown in

Table 13. First layer indicators and target layer score.

Indicators	First Layer Indicators’ Score	First Layer Indicators’ Weight	Target Layer Score
A1	0.567	0.181	0.474
A2	0.495	0.147
A3	0.433	0.204
A4	0.504	0.188
A5	0.358	0.155
A6	0.484	0.125

. The score results of the first layer indicators for the family farm business risk are 0.567, 0.495, 0.433, 0.504, 0.358, and 0.484. The final score of the target layer, i.e., Fengshen Family Farm business risk, is 0.474. Several suggestions for risk control measures can be drawn from these results. According to the evaluation level in Table 5, the family farm business risk is moderate. Thus, the government and farmers should pay special attention to this family farm. In particular, Table 11 and Table 12 show that in the operation process of this family farm, the probability of occurrence is relatively high for natural risk and policy risk, but relatively low for social risk. Therefore, the government should provide additional policy support, such as tax incentives and loan discount subsidies. Given that the incidence of natural risk is high, farmers should focus on availing insurance and diversifying farm products. The top risks among the second layer indicators are the incidence of diseases from pests, management’s decision-making ability, and technical staff satisfaction degree. Thus, farmers should purchase insurance against diseases from pests and natural disaster. They should also hire professional and technical personnel to manage and operate the family farms and improve technical efficiency. Additionally, farmers should enhance their learning to improve their business decision-making abilities.

We also compared the HFTS method in this study with the Buckley’s ordinary fuzzy AHP method [34].

Table 14. Comparison with ordinary fuzzy AHP.

Method	First Layer Indicators’ Weight						Target Layer Score
	A1	A2	A3	A4	A5	A6
Buckley’s AHP	0.180	0.148	0.201	0.185	0.152	0.133	0.475
HFLTS	0.181	0.147	0.204	0.188	0.155	0.125	0.474

Note: When calculating the weight by Buckley’s AHP, replace “-” with “M” in the judgment matrix.

shows that the HFLTS and Buckley’s AHP method yield similar results; the method comparison analysis demonstrates the validity of the obtained result by HFLTS.

5. Conclusions

A risk assessment should be made to enhance the resistance of family farm business risk and improve the family farm lifecycle. This study, which presents a new approach to family farm business risk assessment, combined the HFLTS and HFWA operator models. The family farm business risk indicators, including natural, technical, market, policy, society, and management risk, were obtained through the factor analysis. The HFLTS model was constructed with the consideration of experts’ hesitation in the evaluation to calculate the weight of each risk indicator. Then, the HFWA operator model was constructed to calculate the score of family farm business risks. The proposed method was applied to a Chinese family farm for a case study. Moreover, the method comparison analysis was also presented to check the validity of the obtained result.

This study contributes to Chinese family farm business risk assessment in two ways:

(1)

A family farm business risk indicator system was built and the key factors were identified based on factor analysis.

(2)

The HFLTS and HFWA model were constructed for family farm business risk assessment, and a solution was provided for when experts hesitate between several linguistic expressions in family farm business risk assessment.

In future work, we will improve this approach using the expanded fuzzy sets or triangular fuzzy numbers. Given that this study evaluated only one family farm, sensitivity analysis is lacking. Thus, comparisons across multiple objects and sensitivity analysis will be made in future research.