The consistency improvement of probabilistic linguistic hesitant fuzzy preference relations and their application in venture capital group decision making

Abstract

Driven by the entrepreneurial trend of mass entrepreneurship and innovation, venture capital(VC) has been widely concerned and valued by investors. There is no doubt that investment decision plays a critical role in venture capital, however, due to the complexity of the investment environment, it is often difficult for investors to make a definite judgement on an innovative solution. Consequently, to express more accurately the hesitation and ambiguity of investors in the decision-making process, this paper proposes the probabilistic linguistic hesitant fuzzy preference relation(PLHFPR) based on the probabilistic linguistic hesitant fuzzy set(PLHFS). Unlike hesitant fuzzy preference relation (HFPR), PLHFPR not only provides flexible linguistic expression for decision makers, but also gives the occurrence probability of each element in the PLHFPR. Considering that it is difficult for investors to give the exact probability of each element in the PLHFPR, a new probability calculation method is proposed based on the consistency analysis. What’s more, the convex consistency index(CCI) is defined to measure the consistency level of the PLHFPR by considering decision maker’s risk attitude. For the inconsistent PLHFPR, a weighted nonlinear programming model(WNPM) is constructed to derive an acceptable convex consistent PLHFPR and obtain the PLHFPR priority weight vector. Finally, an example about the venture capital is offered to verify the effectiveness of the proposed method.

Keywords

Venture capital Group decision making PLHFPR Consistency improvement Weighted nonlinear programming model

1. Introduction

With the transformation of China’s economic structure and the impact of internet and big data, innovation and entrepreneurship have become an irresistible trend. Accordingly, as an importantdriving force for innovation and entrepreneurship, VC has shown unprecedented importance, whether in internet plus projects, or in countless small and medium-sized entrepreneurial projects. For example, by March 2018, alibaba completed a $343 million investment in the OFO minibus. However, as its name implies, VC has high risk and uncertainty, so how to avoid risk as much as possible to obtain the maximum return has been a difficult problem for investors. For this, investment decision, as the most important link in VC, has become a hot research topic among scholars, such as the literature [1 –5].

In reality, venture capital problem is often regarded as group decision making (GDM) problem, while in the GDM problem, the assessment information given by experts has various forms, in which, linguistic term set(LTS) is one of the most commonly used forms, because it provides a flexible choice space for decision-makers(DMs). For example, when an investor compares two options, he only need to give one linguistic term, such as good or slight good, instead of give an exact value. Thus many literature [6 –10] have used it to study GDM problem. However, the traditional LTS only allows experts to express their preferences with a single linguistic term, while experts tend to give more than one linguistic term due to the complexity of the actual decision-making environment. To address this issue, Rodriguez et al. [11] introduced the concept of hesitant fuzzy linguistic term sets(HFLTSs) that permit a linguistic variable to have several linguistic terms. In order to further reflect the hesitancy of experts, Cheng and Meng [12] defined linguistic hesitant fuzzy sets(LHFSs), which not only gave the possible linguistic variable but also considered the possible membership degrees of each linguistic term. However, in the current research on LHFSs, all possible membership degrees of linguistic terms provided by experts have the same weight. For example, the quality evaluation information of the refrigerator given by experts in literature [12] can be expressed in the LHFS as {(s₆, 0.1, 0.2) , (s₇, 0.4, 0.5) , (s₈, 0.15, 0.2, 0.25)}, where 0.1 has the same importance as 0.2 for s₆, this is clearly not reasonable enough. In reality, due to the complexity of decision-making environment and the cognitive limitations of human thinking, DMs tend to have different degrees of preference for different membership degrees corresponding to the same linguistic term.

For example, when an investment team evaluates the quality of a company’s project based on the linguistic term set(LTS) S = {s₀:extremely poor, s₁:very poor, s₂:poor, s₃:fair, s₄:good, s₅:very good, s₆:extremely good}, one expert deems the company’s project quality is "very poor" with possible values of 0.6 and 0.7, but the intention to give a value of 0.6 is stronger than that of 0.7, while another expert deems the company’s project quality is "poor" with possible values of 0.7 and 0.8, but its intention to give a value of 0.7 is stronger than that of 0.8. The rest of the experts think that the company’s project quality is doubtlessly "fair" with the value 1. Then the assessment information given by the investment team can be expressed as {〈s₁, 0.6 (p₁) , 0.7 (p₂) 〉, 〈s₂, 0.7 (p₃) , 0.8 (p₄) 〉, 〈s₃, 1 (1) 〉}, where p_i (i = 1, 2, 3, 4) represents the satisfaction degree of the decision maker with the value given. So it’s easy to know p₁ > p₂, p₃ > p₄.

Therefore, in order to express more accurately the evaluation information of experts, this paper proposes the probabilistic linguistic hesitant fuzzy sets(PLHFSs). Considering that decision-makers are more willing to compare alternatives rather than give evaluation information directly in actual decision-making, we further propose PLHFPR based on PLHFS. The preference relation based on PLHFS is defined as PLHFPR, which not only expresses the possible membership degrees of linguistic terms, but also reflects the possible preference tendency(probability) of decision makers for different membership degrees. However, in practical decision-making, it is often difficult for decision-makers to give their preferences to different membership degrees with specific values (which can also be understood as the importance of different membership degrees). Therefore, inspired by the probabilistic calculation methods in literature [13, 27], this paper defines the score consistency of PLHFPR and presents a method for calculating the preference tendency(probability) of decision makers for different membership degrees.

As an important condition to ensure the rationality of decision-making results, the consistency of preference relations has been widely studied [28 –31], and many researchers have proposed corresponding improvement methods for inconsistent or unacceptable preference relations. For example, Zhang et al. [14] proposed the consensus reaching process for GDM with probabilistic linguistic preference relations(PLPRs), Zhou and Xu [13] presented the iterative optimization algorithm, Zhang et al. [24] gave a mixed 0-1 linear programming model based on average additive consistency of HFPR etc. However, these methods seldom take into account the risk attitude of investors, and most of them require complicated calculations. For this reason, to effectively solve the consistency problem of GDM based on PLHFPR, this paper puts forward the convex consistency index by considering the risk preference of investors comprehensively, and then presents the weighted nonlinear programming model.

Based on the above analysis, the main contributions of this paper are organized as follows:

We define a PLHFPR that can more fully reflect DMs hesitation and preference.

A convex consistency index is defined, it can fully reflect investor’s risk preferences.

A new weighted nonlinear programming model is established to solve the inconsistency problem of the PLHFPR.

We propose a new method for deriving the PLHFPR priority weight vector.

We develop a systematic GDM method to help investors make effective decisions.

To sum up, compared with the existing group decision-making methods, the main advantages of the proposed method are as follows:

At present, the study of using uncertain linguistic information to express group opinions has attracted wide attention, such as extended hesitant fuzzy linguistic term set(EHFLTS) proposed by Wang [25] and the proportional hesitant fuzzy linguistic term set(PHFLTS) proposed by Chen et al. [26]. However, to our knowledge, there are few studies on preference relations to express group opinions. Therefore, the PLHFPR proposed in this paper is of great significance. On the one hand, it can express group preference; on the other hand, it can more accurately reflect the hesitation and uncertainty of decision makers through probability description.

The general consistency improvement model needs to define the operation rules of the fuzzy preference relations and calculate various complex measures. In this paper, the consistency of PLHFPR is transformed into the consistency of its score matrix, which avoids the definition of complex measures and simplifies the calculation process.

Existing GDM methods involving probabilistic fuzzy preference relations(such as PLPR [14] and PHFPR [23]) all require DMs to provide subjective probability directly, and most of them fail to consider decision-makers’ risk attitude. The model proposed in this paper not only considers decision-makers’ risk attitude, but also gives a probability calculation method to measure the possible preferences of decision makers for different membership degrees. This is more in line with the actual decision-making situation and more easily adopted by DMs.

The remainder of this paper is organized as follows: Section 2 recalls some basic concepts, including HFLTS, hesitant fuzzy linguistic preference relation(HFLPR), probabilistic hesitant fuzzy set(PHFS). Section 3 introduces the concept of PLHFS and presents the PLHFPR. Section 4 defines the score consistency of a PLHFPR, then a goal programming model is proposed to derive the probability and obtain the priority weights of a PLHFPR. Section 5 defines the convex consistency index and establishes the weighted nonlinear programming model to obtain the acceptable convex consistent PLHFPR, then a new method is proposed to solve GDM problem with PLHFPR. In sections 6, an example about VC is given to demonstrate the proposed method. Finally, section 7 is concluding remarks.

2. Preliminaries

In this part, we main review some basic concepts of HFLTS and PHFS and point out the main disadvantages of these fuzzy sets.

2.1. HFLTS and PHFS

The full text makes X = {x₁, x₂, ⋯ , x_n} denotes a collection of the comparison objects. Rodriguez et al. [11] presented the concept of HFLTS to express the hesitation of DMs in terms of linguistic terms.

Definition 2.1. [11]. Let S = {s₀, s₁, ⋯ , s_2τ} be a linguistic term set, if H_s is an ordered finite subset of the consecutive linguistic terms of S, then we call H_s a HFLTS on S.

In order to make up for the equal weight defect of the HFS, Zhu [15] defined the PHFS.

Definition 2.2. [15]. Let X be a fixed set, a PHFS on X is in terms of a function that when applied to x returns to a subset of [0,1], which is expressed as $H = {< x, h_{x} (p_{x}) > | x \in X},$ (1) where h_x and p_x are two sets of some values in [0,1], h_x denotes the possible membership degrees of the element x ∈ X to the set H, p_x is a set of probabilities associated with h_x. h_x (p_x) is called the probabilistic hesitant fuzzy elemem(PHFE).

2.2. HFLPR

To solve the problem of GDM with the HFLTS, Herrera et al. [11] proposed the concept of HFLPR

Definition 2.3. [11] Let S = {s₀, s₁, ⋯ , s_2τ}, H_s is a HFLTS, the preference relation B = (b_ij) _n×n defined by the HFLTS is called the HFLPR, if it satisfies:

$b_{ij}^{σ (l)} + b_{ji}^{σ (l)} = s_{2 τ}, l = 1, 2, \dots, | b_{ij} |$ ;

|b_ij| = |b_ji|;

b_ii = s_τ;

$b_{ij}^{σ (l)} < b_{ij}^{σ (l + 1)}$ ;

$b_{ji}^{σ (l + 1)} < b_{ji}^{σ (l)}$ .

where b_ij ∈ H_s, b_ij is the hesitancy degree when x_i is prefered over x_j, |b_ij| is the cardinality of b_ij, and

b_{ij}^{σ (l)}

is the l-th linguistic term in b_ij.

Due to the HFLPR only considers possible linguistic terms without considering the possible membership degrees of each linguistic term, while the PHFS only considers the possible membership degrees and corresponding probabilities, which fails to provide flexible linguistic terms for DMs. Therefore, in order to reflect the preference information of investors more accurately and comprehensively, we propose the PLHFS by combining the HFLTS and PHFS.

3. PLHFS and PLHFPR

3.1. PLHFS

Definition 3.1. Let S = {s₀, s₁, ⋯ , s_2τ} be a linguistic term set, then a PLHFS on S is expressed by a mathematical symbol: $\begin{matrix} lh (p) & = {〈 s_{γ (l)}, h_{m}^{l} | p_{m}^{l} 〉 | m = 1, 2, \dots, M, \\ Σ_{m = 1}^{M} p_{m}^{l} \leq 1} . \end{matrix}$ (2) where $〈 s_{γ (l)}, h_{m}^{l} | p_{m}^{l} 〉$ is a PLHF element(PLHFE), which represents the lth linguistic term s_γ(l) in lh (p) and its corresponding possible membership degrees $h_{m}^{l}$ associated with its probability $p_{m}^{l}$ ; M is the number of possible membership degrees corresponding to the l-th linguistic term in lh (p); s_γ(l) ∈ S. For convenience, we abbreviate PLHFS to ${〈 s_{γ (l)}, h_{m}^{l} | p_{m}^{l} 〉}$ .

For the PLHFS lh (p), let $p_{m}^{N (l)} = p_{m}^{(l)} / Σ_{m = 1}^{M} p_{m}^{(l)} (m = 1, 2, \dots, M)$ , then we get the normalized PLHFS, for the convenience of discussion, we assume all the probabilities in the PLHFS are nomalized probabilities.

Example 3.1. As the case of project quality assessment mentioned in the introduction, the evaluation information of the decision-making team can be expressed as the PLHFS

lh (p) = {〈s₁, (0.6|p₁, 0.7|p₂) 〉, 〈s₂, (0.7|p₃, 0.8|p₄) 〉, 〈s₃, 1|1〉}.

Then the normalized PLHFS means that p₁ + p₂ = 1, p₃ + p₄ = 1.

3.2. The score function of PLHFS

Definition 3.2. Let $lh (p) = {〈 s_{γ (l)}, h_{m}^{l} | p_{m}^{l} 〉}$ be a PLHFS, then the score function of lh (p) is: $S (lh (p)) = s_{\bar{γ}}$ where $\bar{γ} = \frac{1}{L} \sum_{l = 1}^{L} γ (l) [\sum_{m = 1}^{M} h_{m}^{l} p_{m}^{l}]$ , and L is the number of linguistic terms in lh (p). According to the scoring function, we give the comparison rule between two PLHFSs lh (p) ₁ and lh (p) ₂:

If $s_{\bar{γ_{1}}} < s_{\bar{γ_{2}}}$ , then lh (p) ₁ < lh (p) ₂;

If $s_{\bar{γ_{1}}} = s_{\bar{γ_{2}}}$ , then lh (p) ₁ = lh (p) ₂.

Example 3.2. Let $\begin{matrix} lh (p)_{1} & = {〈 s_{1}, (0.6 | p_{11}, 0.7 | p_{12}) 〉, 〈 s_{2}, (0.7 | p_{13}, \\ 0.8 | p_{14}) 〉} \\ lh (p)_{2} & = {〈 s_{3}, (0.4 | p_{21}, 0.3 | p_{22}) 〉, 〈 s_{4}, (0.3 | p_{23}, \\ 0.2 | p_{24}) 〉} . \end{matrix}$ then $S (lh (p)_{1}) = s_{\bar{γ_{1}}}, S (lh (p)_{2}) = s_{\bar{γ_{2}}}$ ,where $\begin{matrix} \bar{γ_{1}} & = \frac{1}{2} [1 \times (0.6 p_{11} + 0.7 p_{12}) + 2 \times (0.7 p_{13} \\ + 0.8 p_{14})] \\ \bar{γ_{2}} & = \frac{1}{2} [3 \times (0.4 p_{21} + 0.3 p_{22}) + 4 \times (0.3 p_{23} \\ + 0.2 p_{24})] . \end{matrix}$ If $\bar{γ_{1}} < \bar{γ_{2}}$ , then $s_{\bar{γ_{1}}} < s_{\bar{γ_{2}}}, lh (p)_{1} < lh (p)_{2}$ ;

If $\bar{γ_{1}} = \bar{γ_{2}}$ , then $s_{\bar{γ_{1}}} = s_{\bar{γ_{2}}}, lh (p)_{1} = lh (p)_{2}$ .

Of course, we can use the variance function for further comparison, but this is not the focus of this paper, which is omitted here.

3.3. PLHFPR

Similar to the HFLPR, we present the definition of PLHFPR based on the PLHFS and its comparison rule.

Definition 3.3. Let P = (lh (p) _ij) _n×n be a matrix on the object set X = {x₁, x₂, ⋯ , x_n} for the LTS S = {s₀, s₁, ⋯ , s_2τ}, where $lh (p)_{ij} = {〈 s_{ij, γ (l)}, h_{ij, m}^{l} | p_{ij, m}^{l} 〉}$ is a PLHFS, s_ij,γ(l) ∈ S represents the preference degree of DMs for x_i over x_j, $h_{ij, m}^{l}$ denotes the possible membership degrees of the linguistic term s_ij,γ(l) ∈ S to the set lh (p) _ij. Probability $p_{ij, m}^{l}$ can reflect the distribution of $h_{ij, m}^{l}$ , and it can also be understood as the preference tendency of decision makers to the given membership degree $h_{ij, m}^{l}$ . P is called a PLHFPR, if it satisfies the following conditions:

$p_{ij, m}^{(l)} = p_{ji, m}^{(l)}$ ;

$h_{ij, m}^{(l)} + h_{ji, m}^{(l)} = 1$ ;

s_ij,γ(l) ⊕ s_ji,γ(l) = s_2τ;

lh (p) _ii = {s_τ, 1 (1)} = {s_τ};

|lh (p) _ij| = |lh (p) _ji|;

s_ij,γ(l) ≤ s_ij,γ(l+1), s_ji,γ(l) ≥ s_ji,γ(l+1).

for all i, j = 1, 2, ⋯ , n with i ≠ j, in which |lh (p) _ij| is the number of linguistic term in lh (p) _ij,

h_{ij, m}^{l}, p_{ij, m}^{l} \in [0, 1]

\sum_{m = 1}^{M} p_{ij, m}^{l} = 1

, and M is the number of possible membership degrees corresponding to the l-th linguistic term in lh (p) _ij.

Example 3.3. Below, we present a simple PLHFPR matrix based on the linguistic term set S = {s₀: extremely poor, s₁: very poor, s₂: poor, s₃: fair, s₄: good, s₅: very good, s₆: extremely good}. $P = (\begin{matrix} lh (p)_{11} & lh (p)_{12} & lh (p)_{13} \\ lh (p)_{21} & lh (p)_{22} & lh (p)_{23} \\ lh (p)_{31} & lh (p)_{32} & lh (p)_{33} \end{matrix})$ (3) where $\begin{matrix} lh (p)_{11} & = lh (p)_{22} = lh (p)_{33} = {s_{3}}; \\ lh (p)_{12} & = {〈 s_{2}, (0.3 | p_{12, 1}^{(1)}, 0.5 | p_{12, 2}^{(1)}, 0.7 | p_{12, 3}^{(1)}) 〉, \\ 〈 s_{3}, (0.4 | p_{12, 1}^{(2)}, 0.6 | p_{12, 2}^{(2)}) 〉}; \\ lh (p)_{21} & = {〈 s_{4}, (0.7 | p_{21, 1}^{(1)}, 0.5 | p_{21, 2}^{(1)}, 0.3 | p_{21, 3}^{(1)}) 〉, \\ 〈 s_{3}, (0.6 | p_{21, 1}^{(2)}, 0.4 | p_{21, 2}^{(2)}) 〉}; \\ lh (p)_{13} & = {〈 s_{3}, (0.3 | p_{13, 1}^{(1)}, 0.6 | p_{13, 2}^{(1)}) 〉, \\ 〈 s_{4}, (0.1 | p_{13, 1}^{(2)}, 0.2 | p_{13, 2}^{(2)}, 0.4 | p_{13, 3}^{(2)}) 〉}; \\ lh (p)_{31} & = {〈 s_{3}, (0.7 | p_{31, 1}^{(1)}, 0.4 | p_{31, 2}^{(1)}) 〉, \\ 〈 s_{2}, (0.9 | p_{31, 1}^{(2)}, 0.8 | p_{31, 2}^{(2)}, 0.6 | p_{31, 3}^{(2)}) 〉}; \\ lh (p)_{23} & = {s_{2}, (0.3 | p_{23, 1}, 0.8 | p_{23, 2})}; \\ lh (p)_{32} & = {s_{4}, (0.7 | p_{32, 1}, 0.2 | p_{32, 2})} . \end{matrix}$

By definition 3.3, it is easy to know that if we want to get the PLHFPR, we only need to know its upper triangular part or lower triangular part. Therefore, for the convenience of expression, we will only give the upper triangular part of PLHFPR for the next to be used.

For the use of preference relations to solve GDM problem, there have been considerable literature studies which show that it is extremely necessary to discuss consistency. Therefore, the consistency analysis of PLHFPR is given in the next section.

4. Consistency of PLHFPR

4.1. Score consistency of PLHFPR

According to the consistency property of fuzzy preference relation(FPR), Zhou and Xu [13] proposed the expected consistency of probabilistic hesitant fuzzy preference relation(PHFPR). Inspired by this, this paper proposes the score consistency of PLHFPR. Before we define it, let’s review the notion of consistency in a FPR.

Definition 4.1. [17] For the FPR R = (r_ij) _n×n, (i, j = 1, 2, ⋯ , n) ; r_ij ∈ [0, 1], if we have $r_{ij} = \frac{w_{i}}{w_{i} + w_{j}} .$ (4) for all i, j = 1, 2, ⋯ , n, and which satisfies: (1) r_ii = 0.5; (2) r_ij + r_ji = 1; (3) $\sum_{i = 1}^{n} w_{i} = 1$ , then we call the FPR R is consistent, where r_ij is the preference degree of the objectives x_i over x_j, and w_i ∈ [0, 1], w = (w₁, w₂, ⋯ , w_n) is the priority vector of R.

Similarly, we give the concept of score consistency for PLHFPR.

Definition 4.2. Let P = (lh (p) _ij) _n×n be a PLHFPR on the object set X = {x₁, x₂, ⋯ , x_n} for the LTS S = {s₀, s₁, ⋯ , s_2τ}. ${\begin{matrix} e_{ij} & = \bar{γ_{ij}} = \frac{1}{L} \sum_{l = 1}^{L} γ_{(l)} [\sum_{m = 1}^{M} h_{ij, m}^{l} p_{ij, m}^{l}], \\ e_{ji} & = 2 τ - e_{ij}, i \leq j, i, j = 1, 2, \dots, n . \end{matrix}$ (5) where L is the number of linguistic terms in lh (p) _ij, and the meaning of other symbols as defined in definition 3.3. If we have $\frac{e_{ij}}{2 τ} = \frac{w_{i}}{w_{i} + w_{j}}, (\forall i \leq j, i, j = 1, 2, \dots, n)$ (6) then we call the PLHFPR P is score consistent, where w = (w₁, w₂, ⋯ , w_n) is the priority vector of P, satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

Remark 4.1. It is easy to see from the above definition that when e_ij = τ, that is lh (p) _ij = {s_τ, 1 (1)}, we have w_i = w_j = 0.5, which means there’s no difference between x_i and x_j. In addition, $0 < \frac{e_{ij}}{2 τ} < 1$ is easily obtained from the expression of e_ij. And if $\frac{e_{ij}}{2 τ} = \frac{w_{i}}{w_{i} + w_{j}}$ is hold, then $\frac{e_{ji}}{2 τ} = \frac{w_{j}}{w_{i} + w_{j}}$ . This indicates that Equation (6) fully satisfies the consistency condition of the FPR, so we transform the consistency problem of PLHFPR into the consistency problem of its score matrix.

4.2. Solving probability and priority weightof PLHFPR

As analyzed in the introduction, in actual decision-making, decision makers often do not give their preferences for different membership degrees in the form of specific numerical values. Therefore, in this part, we give the probability calculation method and solve the priority weight of PLHFPR by establishing the goal programming model.

Based on Equation (6), we define the deviation function as: $ɛ_{ij} = | 2 τ w_{i} - (w_{i} + w_{j}) e_{ij} | .$ (7) Apparently, PLHFPR is consistent when ɛ_ij = 0, while the preference matrix given by experts is often inconsistent, so we want the deviation ɛ_ij to be as small as possible. Based on this principle, we set up the goal programming model to obtain the priority weights of the PLHFPR, and the model is established as follows:

min ∑_i<jɛ_ij = ∑_i<j|2τw_i - (w_i + w_j) e_ij| $s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i} = 1, w_{i} \geq 0, \\ i, j = 1, 2, \dots, n . \end{matrix}$ (8)

According to Remark 4.1, we can convert Equation (8) into follows.

min ∑_i<jɛ_ij = ∑_i<j| (w_i + w_j) e_ij - 2τw_i|

$s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i} = 1, w_{i} \geq 0, \\ i, j = 1, 2, \dots, n, j > i . \end{matrix}$ (9)

For the sake of simplicity, we can further convert Equation (9) into following optimization problem:

min $d = \sum_{i < j} (d_{ij}^{+} + d_{ij}^{-})$

$s . t . {\begin{matrix} (w_{i} + w_{j}) \cdot \frac{1}{L} \sum_{l = 1}^{L} γ_{(l)} [\sum_{m = 1}^{M} h_{ij, m}^{l} p_{ij, m}^{l}] \\ - 2 τ w_{i} - d_{ij}^{+} + d_{ij}^{-} = 0, \\ d_{ij}^{+}, d_{ij}^{-} \geq 0, \\ \sum_{i = 1}^{n} w_{i} = 1, w_{i} \geq 0, \\ i, j = 1, 2, \dots, n, j > i . \end{matrix}$ (10) where $d_{ij}^{+} = (w_{i} + w_{j}) e_{ij} - 2 τ w_{i}$ , $d_{ij}^{-} = 2 τ w_{i} -$ (w_i + w_j) e_ij. Obviously, if d = 0, we have $d_{ij}^{+} = d_{ij}^{-} = 0$ , that is $\frac{e_{ij}}{2 τ} = \frac{w_{i}}{w_{i} + w_{j}}$ . Thus, when d = 0, the PLHFPR is score consistent, otherwise, it is inconsistent.

By solving Equation (10), we can get the probability of each membership and the priority weights of PLHFPR.

Example 4.1. Continuing with the example 3.3, in order to obtain the priority weights of PLHFPR matrix P, we establish the following model by Equation (10):

min $d = d_{12}^{+} + d_{12}^{-} + d_{13}^{+} + d_{13}^{-} + d_{23}^{+} + d_{23}^{-}$ $s . t . {\begin{matrix} (w_{1} + w_{2}) * (0.3 p_{12, 1}^{(1)} + 0.5 p_{12, 2}^{(1)} + 0.7 p_{12, 3}^{(1)} + 0.6 p_{12, 1}^{(2)} \\ + 0.9 p_{12, 2}^{(2)}) - 6 w_{1} - d_{12}^{+} + d_{12}^{-} = 0, \\ (w_{1} + w_{3}) * (0.45 p_{13, 1}^{(1)} + 0.9 p_{13, 2}^{(1)} + 0.2 p_{13, 1}^{(2)} + 0.4 p_{13, 2}^{(2)} \\ + 0.8 p_{13, 3}^{(2)}) - 6 w_{1} - d_{13}^{+} + d_{13}^{-} = 0, \\ (w_{2} + w_{3}) * (0.6 p_{23, 1} + 1.6 p_{23, 2}) - 6 w_{2} - d_{23}^{+} + d_{23}^{-} = 0, \\ p_{12, 1}^{(1)} + p_{12, 2}^{(1)} + p_{12, 3}^{(1)} = 1, p_{12, 1}^{(2)} + p_{12, 2}^{(2)} = 1, p_{13, 1}^{(1)} + p_{13, 2}^{(1)} = 1, \\ p_{13, 1}^{(2)} + p_{13, 2}^{(2)} + p_{13, 3}^{(2)} = 1, p_{23, 1} + p_{23, 2} = 1, \\ p_{12, 1}^{(1)}, p_{12, 2}^{(1)}, p_{12, 3}^{(1)}, p_{12, 1}^{(2)}, p_{12, 2}^{(2)}, p_{13, 1}^{(1)} \geq 0, \\ p_{13, 2}^{(1)}, p_{13, 1}^{(2)}, p_{13, 2}^{(2)}, p_{13, 3}^{(2)}, p_{23, 1}, p_{23, 2} \geq 0, \\ d_{12}^{+}, d_{12}^{-}, d_{13}^{+}, d_{13}^{-}, d_{23}^{+}, d_{23}^{-} \geq 0, w_{1} + w_{2} + w_{3} = 1 . \end{matrix}$ The following results can be obtained by solving the model with LINGO:

$d_{12}^{+} = d_{12}^{-} = d_{13}^{+} = d_{13}^{-} = d_{23}^{+} = d_{23}^{-} = 0, w_{1} = 0.0871, w_{2} = 0.2434, w_{3} = 0.6695, p_{12, 1}^{(1)} = 0, p_{12, 2}^{(1)} = 0.0872, p_{12, 3}^{(1)} = 0.9128, p_{12, 1}^{(2)} = 0.0059, p_{12, 2}^{(2)} = 0.9941, p_{13, 1}^{(1)} = 1, p_{13, 2}^{(1)} = 0, p_{13, 1}^{(2)} = 0.7969, p_{13, 2}^{(2)} = 0.2031, p_{13, 3}^{(2)} = 0, p_{23, 1} = 0, p_{23, 2} = 1$ .

Remark 4.2. From above results we can see that w₁ < w₂ < w₃, that is, x₃ is better than x₂ and x₂ better than x₁. Moreover, it can also be seen x₁ < x₂ < x₃ from PLHFSs $lh (p)_{12} = {〈 s_{2}, (0.3 | p_{12, 1}^{(1)}, 0.5 | p_{12, 2}^{(1)}, 0.7 | p_{12, 3}^{(1)}) 〉, 〈 s_{3}, (0.4 | p_{12, 1}^{(2)}, 0.6 | p_{12, 2}^{(2)}) 〉}$ and lh (p) ₂₃ = {〈s₂, (0.3|p_23,1, 0.8|p_23,2) 〉}, which shows that the proposed method is not only reasonable but also more accurate. In addition, above results also present the probability of each membership, which accurately reflects the preference of DMs for each membership degree in the linguistic term.

5. Consistency improvement of PLHFPR

In this part, for the inconsistent PLHFPR, we discuss the consistency reaching process of PLHFPR and give a new algorithm for GDM.

5.1. Convex consistency index of PLHFPR

Consistency is always an important factor to ensure the rationality of GDM results. To solve the consistency problem, many literature [18 –22] have studied the consistency of preference relations based on various deviation degrees, but few literature can combine the risk attitude of investors to study the consistency of information given by experts. Thus, in order to obtain more reasonable acceptable consistency index, we define the convex consistency index(CCI) of PLHFPR.

Definition 5.1. $CCI = δ OCI + (1 - δ) PCI (0 \leq δ \leq 1)$ (11) where $OCI = \frac{2}{n \times (n - 1)} \sum_{i < j} ɛ_{ij}$ is the optimistic consistency index, which means that investors are optimistic. Conversely, PCI = max {ɛ_ij} is the pessimistic consistency index, which means that investors are conservative. δ and 1 - δ is the weight coefficient of OCI and PCI, respectively. If CCI ≤ CI(CI is the given threshold), then we call the PLHFPR is acceptable convex consistent PLHFPR.

5.2. Weighted nonlinear programming model

For an unacceptable consistent PLHFPR score matrix S, we want to get an acceptable consistent score matrix $\bar{S}$ that as close to S as possible. Thus, based on the definitions 4.2 and 5.1, the following weighted nonlinear programming model(WNPM) is established: $min \sum_{i < j} u_{ij} = \sum_{i < j} | γ_{ij} - \bar{γ_{ij}} |$ $s . t . {\begin{matrix} δ OCI + (1 - δ) PCI \leq CI, \\ \sum_{i = 1}^{n} w_{i}^{'} = 1, w_{i}^{'} \geq 0 . \end{matrix}$ (12)

where γ_ij and $\bar{γ_{ij}}$ are subscripts of s_{γ
_ij} and $s_{\bar{γ_{ij}}}$ , respectively. CI is the consistency threshold, which is generally determined by the investment decision team based on industry experience or by authoritative experts in the decision team.

To solve Equation (12), let’s introduce some parameters: $\begin{matrix} u_{ij}^{+} & = (γ_{ij} - \bar{γ_{ij}}) \lor 0, \\ u_{ij}^{-} & = (\bar{γ_{ij}} - γ_{ij}) \lor 0, \\ ɛ_{ij}^{+} & = {(w_{i}^{'} + w_{j}^{'}) \bar{γ_{ij}} - 2 τ w_{i}^{'}} \lor 0, \\ ɛ_{ij}^{-} & = {2 τ w_{i}^{'} - (w_{i}^{'} + w_{j}^{'}) \bar{γ_{ij}}} \lor 0 \end{matrix}$

Then Equation (12) can be converted into a goal programming model as follows:

$\min \sum_{i < j} (u_{ij}^{+} + u_{ij}^{-})$

$s . t . {\begin{matrix} δ \frac{2}{n \times (n - 1)} \sum_{i < j} (ɛ_{ij}^{-} + ɛ_{ij}^{+}) + (1 - δ) m \leq CI, \\ m \geq ɛ_{ij}^{-} + ɛ_{ij}^{+}, (i < j, i, j = 1, 2, \dots, n), \\ u_{ij}^{+} - u_{ij}^{-} = γ_{ij} - \bar{γ_{ij}}, \\ (w_{i}^{'} + w_{j}^{'}) \bar{γ_{ij}} - 2 τ w_{i}^{'} = ɛ_{ij}^{+} - ɛ_{ij}^{-}, \\ \sum_{i = 1}^{n} w_{i}^{'} = 1, w_{i}^{'} \geq 0 . \end{matrix}$ (13)

where $m = PCI = \max {ɛ_{ij}^{+} + ɛ_{ij}^{-}}$ . By solving Equation (13), we can get the acceptable consistent score matrix $\bar{S}$ and get it’s priority weight vector $w = (w_{1}^{'}, w_{2}^{'}, \dots, w_{n}^{'})$ .

Example 5.1. Let the PLHFPR matrix $P = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix})$ (14) where $\begin{matrix} a_{11} & = a_{22} = a_{33} = {s_{3}}; \\ a_{12} & = {〈 s_{1}, (0.6 | p_{12, 1}^{1}, 0.8 | p_{12, 2}^{1} 〉, \\ 〈 s_{2}, (0.4 | p_{12, 1}^{2}, 0.7 | p_{12, 2}^{2}) 〉}; \end{matrix}$ $\begin{matrix} a_{13} & = {〈 s_{3}, (0.6 | p_{13, 1}^{1}, 0.7 | p_{13, 2}^{1}, 0.8 | p_{13, 3}^{1}) 〉}; \\ a_{23} & = {〈 s_{4}, (0.2 | p_{23, 1}^{1}, 0.3 | p_{23, 2}^{1}) 〉, 〈 s_{5}, (0.6) 〉, \\ 〈 s_{6}, (0.7 | p_{23, 1}^{3}, 0.8 | p_{23, 2}^{3}) 〉} . \end{matrix}$

Applying Equation (10) and solving it, the following results can be obtained:

$w_{1} = 0.2222, w_{2} = 0.2593, w_{3} = 0.5185, d_{12}^{+} = d_{13}^{+} = d_{13}^{-} = d_{23}^{+} = d_{23}^{-} = 0, d_{12}^{-} = 0.8037 p_{12, 2}^{1} = p_{12, 2}^{2} = p_{13, 1}^{1} = p_{23, 1}^{1} = p_{23, 1}^{3} = 1, p_{12, 1}^{1} = p_{12, 1}^{2} = p_{13, 2}^{1} = p_{13, 3}^{1} = p_{23, 2}^{1} = p_{23, 2}^{3} = 0$ .

From above optimal solution, we can see that $d_{12}^{-} = 0.8037 \neq 0$ , that is d ≠ 0, which means the PLHFPR P is inconsistent. And it is easy to calculate the score matrix S of PLHFPR as follows. $S = (\begin{matrix} s_{3} & s_{1.1} & s_{1.8} \\ s_{4.9} & s_{3} & s_{3} \\ s_{4.2} & s_{3} & s_{3} \end{matrix})$

In order to get acceptable consistent score matrix $\bar{S}$ , we assume that the consistency threshold is 0.02, and the value of δ given by the decision-making team is 0.2. Based on this, the following WNPM can be established according to Equation (13).

min $u = u_{12}^{+} + u_{12}^{-} + u_{13}^{+} + u_{13}^{-} + u_{23}^{+} + u_{23}^{-}$

$s . t . {\begin{matrix} \frac{0.2}{3} (ɛ_{12}^{-} + ɛ_{12}^{+} + ɛ_{13}^{-} + ɛ_{13}^{+} + ɛ_{23}^{-} + ɛ_{23}^{+}) + 0.8 m \leq 0.02, \\ m \geq ɛ_{ij}^{-} + ɛ_{ij}^{+}, (i < j, i, j = 1, 2, 3), \\ u_{12}^{+} - u_{12}^{-} = 1.1 - \bar{γ_{12}}, \\ u_{13}^{+} - u_{13}^{-} = 1.8 - \bar{γ_{13}}, \\ u_{23}^{+} - u_{23}^{-} = 3 - \bar{γ_{23}}, \\ (w_{1}^{'} + w_{2}^{'}) \bar{γ_{12}} - 6 w_{1}^{'} = ɛ_{12}^{+} - ɛ_{12}^{-}, \\ (w_{1}^{'} + w_{3}^{'}) \bar{γ_{13}} - 6 w_{1}^{'} = ɛ_{13}^{+} - ɛ_{13}^{-}, \\ (w_{2}^{'} + w_{3}^{'}) \bar{γ_{23}} - 6 w_{2}^{'} = ɛ_{23}^{+} - ɛ_{23}^{-}, \\ w_{1}^{'} + w_{2}^{'} + w_{3}^{'} = 1, w_{i}^{'} \geq 0 . \end{matrix}$

The following results can be obtained by solving with LINGO:

$u_{12}^{-} = 0.1738, u_{12}^{+} = u_{13}^{+} = u_{13}^{-} = u_{23}^{+} = u_{23}^{-} = 0, ɛ_{12}^{+} = ɛ_{13}^{-} = ɛ_{23}^{+} = 0, ɛ_{12}^{-} = ɛ_{13}^{+} = ɛ_{23}^{-} = 0.02, m = 0.02, \bar{γ_{12}} = 1.2738, \bar{γ_{13}} = 1.8, \bar{γ_{23}} = 3, w_{1}^{'} = 0.1459, w_{2}^{'} = 0.4471, w_{3}^{'} = 0.4070 .$

Remark 5.1. From above results we can see that the ranking weight obtained through Equation (13) is $w_{1}^{'} < w_{3}^{'} < w_{2}^{'}$ when the consistency threshold given by the decision-making team is 0.02, this is clearly more consistent with the given preference information than the result of sorting without consistency improvement. Besides, through the consistency correction, we get the acceptable consistent score matrix as follows: $\bar{S} = (\begin{matrix} s_{3} & s_{1.2738} & s_{1.8} \\ s_{4.7262} & s_{3} & s_{3} \\ s_{4.2} & s_{3} & s_{3} \end{matrix})$

Summarizing above analyses, we present the concrete steps to solve the GDM problem with the PLHFPR.

5.3. A new algorithm for solving GDMwith PLHFPR

For the GDM problem of VC, there are a set of alternatives {x₁, x₂, ⋯ , x_n}, and a set of criteria {C₁, C₂, ⋯ , C_m}. Firstly, let a group of investment experts compare m criteria in pairs and give evaluation information. Secondly, according to the evaluation information given by experts, we construct the PLHFPR of criteria. Similarly, we can also get the PLHFPRs of n alternatives with respect to m criteria. The specific solution steps are given as follows:

Step 1. Identify the set of alternatives {x₁, x₂, ⋯ , x_n} and the set of criteria {C₁, C₂, ⋯ , C_m}.

Step 2. Let the investment team compare criteria set and alternative set in pairs, and get the PLHFPRs of criteria and alternative with respect to each criteria respectively. Let’s call it C and X_{c
_i} (i = 1, 2, ⋯ , m) respectively.

Step 3. Determine the consistency threshold CI and the value of parameter δ for the C and X_{c
_i}.

Step 4. By Equations (10) and (11), we calculate the convex consistency index(CCI) of PLHFPRs C and X_{c
_i}, if CCI ≤ CI, namely, PLHFPRs C and X_{c
_i} are acceptable convex consistent, then go to step 6. Otherwise, PLHFPR is unacceptable convex consistent and go the next step.

Step 5. Solve Equation (13), the acceptable consistent score matrices $\bar{C}$ and $\bar{X_{c_{i}}}$ are derived, and then, the priority weights $(w_{x_{i}}^{c_{j}})$ of the i-th alternatives with respect to criteria j and the criteria priority weight vector (w^{c
₁}, w^{c
₂}, ⋯ , w^{c
_m}) are obtained.

Step 6. Calculate the comprehensive ranking weight of each alternative by $w_{x_{i}} = \sum_{j = 1}^{m} w^{c_{j}} \cdot w_{x_{i}}^{c_{j}}$ (15)

Step 7. According to the comprehensive weight value to compare the alternatives, the best alternative has the highest value.

6. A case study

In this part, we will demonstrate the feasibility and effectiveness of the proposed method through actual investment cases.

In 2017, Sequoia Capital, a famous venture capital institution, made key investments in companies of the nature of medical health(x₁), cultural entertainment(x₂) and big data(x₃) respectively. Due to the high risk and uncertainty of venture capital, in order to minimize risks and maximize returns, investors often need to comprehensively investigate and evaluate the company from multiple aspects, such as team background, product technology, market prospect, profitability and organizational structure etc. To this end, the agency sent an investment team to comprehensively evaluate representative companies in x₁, x₂ and x₃. Moreover, the team experts were asked to make a pairwise comparison of the three companies in terms of profitability(c₁), innovation ability(c₂), development prospect(c₃) and team background(c₄). However, due to the complexity of factors to be considered and the cognitive limitations of investment experts, the investment team can only provide evaluation information in the form of PLHFPR. Based on historical investment experience, the investment team expects the consistency threshold to be 0.05, and through expert consultation, the determined risk preference value δ is 0.2. The PLHFPRs given by the investment team based on the linguistic term set S={s₀:extremely poor, s₁:very poor, s₂:poor, s₃:slightly poor, s₄:fair, s₅:slightly good, s₆:good, s₇:very good, s₈:extremely good} are as follows $C = (\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix})$ where $\begin{matrix} a_{11} & = a_{22} = a_{33} = a_{44} = {s_{4}}; \end{matrix}$ $\begin{matrix} a_{12} & = {〈 s_{4}, (0.6 | 1) 〉, 〈 s_{5}, (0.2 | p_{12, 1}^{2}, 0.3 | p_{12, 2}^{2}) 〉}; \\ a_{13} & = {〈 s_{2}, (0.4 | p_{13, 1}^{1}, 0.5 | p_{13, 2}^{1}, 0.6 | p_{13, 3}^{1}) 〉}; \\ a_{14} & = {〈 s_{7}, (0.7 | p_{14, 1}^{1}, 0.8 | p_{14, 2}^{1}) 〉}; \\ a_{23} & = {〈 s_{1}, (0.25 | p_{23, 1}^{1}, 0.3 | p_{23, 2}^{1}) 〉, \\ 〈 s_{2}, (0.35 | p_{23, 1}^{2}, 0.4 | p_{23, 2}^{2}) 〉, 〈 s_{3}, (0.6 | 1) 〉}; \\ a_{24} & = {〈 s_{6}, (0.8 | p_{24, 1}^{1}, 0.85 | p_{24, 2}^{1}) 〉}; \\ a_{34} & = {〈 s_{5}, (0.35 | p_{34, 1}^{1}, 0.5 | p_{34, 2}^{1}, 0.6 | p_{34, 3}^{1}) 〉, \\ 〈 s_{6}, (0.7 | 1) 〉}; \end{matrix}$

$X_{c_{1}} = (\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix})$ where $\begin{matrix} b_{11} & = b_{22} = b_{33} = {s_{4}}; \\ b_{12} & = {〈 s_{3}, (0.2 | p_{12, 1}^{1}, 0.3 | p_{12, 2}^{1}) 〉, 〈 s_{4}, (0.4 | p_{12, 1}^{2}, \\ 0.5 | p_{12, 2}^{2}) 〉}; \\ b_{13} & = {〈 s_{2}, (0.8 | p_{13, 1}^{1}, 0.85 | p_{13, 2}^{1} 〉, 〈 s_{3}, (0.2 | p_{13, 1}^{2}, \\ 0.3 | p_{13, 2}^{2}) 〉}; \\ b_{23} & = {〈 s_{3}, (0.45 | p_{23, 1}, 0.7 | p_{23, 2}, 0.8 | p_{23, 3}) 〉} . \end{matrix}$

$X_{c_{2}} = (\begin{matrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{matrix})$ where $\begin{matrix} c_{11} & = c_{22} = c_{33} = {s_{4}}; \\ c_{12} & = {〈 s_{3}, (0.8 | p_{12, 1}^{1}, 0.85 | p_{12, 2}^{1}) 〉}; \\ c_{13} & = {〈 s_{1}, (0.6 | p_{13, 1}^{1}, 0.8 | p_{13, 2}^{1} 〉, 〈 s_{2}, (0.2 | p_{13, 1}^{2}, \\ 0.3 | p_{13, 2}^{2}) 〉}; \\ c_{23} & = {〈 s_{2}, (0.4 | p_{23, 1}^{1}, 0.5 | p_{23, 2}^{1} 〉, 〈 s_{3}, (0.3 | p_{23, 1}^{2}, \\ 0.45 | p_{23, 2}^{2}) 〉}; \end{matrix}$ $X_{c_{3}} = (\begin{matrix} d_{11} & d_{12} & d_{13} \\ d_{21} & d_{22} & d_{23} \\ d_{31} & d_{32} & d_{33} \end{matrix})$ where $\begin{matrix} d_{11} & = d_{22} = d_{33} = {s_{4}}; \\ d_{12} & = {〈 s_{5}, (0.8 | p_{12, 1}^{1}, 0.9 | p_{12, 2}^{1}) 〉}; \\ d_{13} & = {〈 s_{3}, (0.6 | p_{13, 1}^{1}, 0.8 | p_{13, 2}^{1} 〉, \\ 〈 s_{4}, (0.2 | p_{13, 1}^{2}, 0.3 | p_{13, 2}^{2}) 〉}; \\ d_{23} & = {〈 s_{2}, (0.5 | p_{23, 1}^{1}, 0.6 | p_{23, 2}^{1} 〉, \\ 〈 s_{3}, (0.3 | p_{23, 1}^{2}, 0.45 | p_{23, 2}^{2}) 〉}; \end{matrix}$ $X_{c_{4}} = (\begin{matrix} f_{11} & f_{12} & f_{13} \\ f_{21} & f_{22} & f_{23} \\ f_{31} & f_{32} & f_{33} \end{matrix})$ where $\begin{matrix} f_{11} & = f_{22} = f_{33} = {s_{4}}; \\ f_{12} & = {〈 s_{6}, (0.6 | p_{12, 1}^{1}, 0.75 | p_{12, 2}^{1}) 〉}; \\ f_{13} & = {〈 s_{4}, (0.3 | p_{13, 1}^{1}, 0.5 | p_{13, 2}^{1} 〉, \\ 〈 s_{5}, (0.7 | p_{13, 1}^{2}, 0.9 | p_{13, 2}^{2}) 〉}; \\ f_{23} & = {〈 s_{2}, (0.6 | p_{23, 1}^{1}, 0.7 | p_{23, 2}^{1} 〉, \\ 〈 s_{3}, (0.1 | p_{23, 1}^{2}, 0.3 | p_{23, 2}^{2}) 〉}; \end{matrix}$

6.1. Decision making with proposed algorithm

According to algorithm 5.3, we can get the alternative priority weights and make the final decision through the following steps:

Step 1. Through above analysis,we have known the parameter δ = 0.2, and the consistency threshold CI = 0.05.

Step 2. Calculate the CCI of each matrix by Equations (10) and (11), and the specific results are shown in Table 1.

Table 1
Convex consistency index and priority weight

	C	X _{C ₁}	X _{C ₂}	X _{C ₃}	X _{C ₄}
CCI	1.3998	0	0	0	0
W	w^{c ₁}= 0.1104	$w_{x_{1}}^{c_{1}}$ = 0.0989	$w_{x_{1}}^{c_{2}}$ = 0.1001	$w_{x_{1}}^{c_{3}}$ = 0.2305	$w_{x_{1}}^{c_{4}}$ = 0.3642
	w^{c ₂}= 0.1378	$w_{x_{2}}^{c_{1}}$ = 0.3334	$w_{x_{2}}^{c_{2}}$ = 0.1420	$w_{x_{2}}^{c_{3}}$ = 0.1548	$w_{x_{2}}^{c_{4}}$ = 0.1214
	w^{c ₃}= 0.7174	$w_{x_{3}}^{c_{1}}$ = 0.5677	$w_{x_{3}}^{c_{2}}$ = 0.7579	$w_{x_{3}}^{c_{3}}$ = 0.6147	$w_{x_{3}}^{c_{4}}$ = 0.5144
	w^{c ₄}= 0.0344.

Step 3. It can be seen from Table 1 that only the CCI of the attribute matrix C is 1.3998 > 0.05, that is, the score matrix of C is an unacceptable consistent matrix. Thus, for matrix C, the priority weights after consistency improvement can be obtained by using model (13) as follows:

w′c₁ = 0.2072, w′c₂ = 0.2764, w′c₃ = 0.4646, w′c₄ = 0.0518.

Step 4. Combining the results of steps 2 and 3, the comprehensive ranking weights obtained through Equation (15) are as follows:

w_{x
₁} = 0.0804, w_{x
₂} = 0.1716, w_{x
₃} = 0.6258.

Therefore, the final ranking result is w_{x
₃} > w_{x
₂} > w_{x
₁}, namely, big data is the best investment direction for the team.

Remark 6.1. The above comprehensive priority vector is obtained from consistent PLHFPRs X_{C
₁}, X_{C
₂}, X_{C
₃}, X_{C
₄} and acceptable consistent score matrix $\bar{C}$ , which ensures the consistency level of ranking results is not less than 95%. In addition, for the problem of venture capital, it is necessary to consider the risk attitude of decision makers. In this paper, the convex consistency index of PLHFPR is used to combine the decision maker’s risk attitude with the objective programming model, which not only simplifies the solution process of priority weight but also ensures the rationality of the result.

Note that PLHFPR is a new type of preference relations, and there is no previous study about decision making with PLHFPRs. Therefore, no previous method can be used in this example. However, the proposed method for solving group decision problems with PLHFPR can be directly applied to LHFPR and PLPR environments, namely, PLHFPR has a more extensive application scenario as the extension of preference relation.

6.2. Comparison analyses

Considering that there is no previous research on PLHFS, this paper only gives a comparative analysis from the qualitative perspective, that is, the main advantages of the proposed method compared with the previous group decision method. Through the analysis above, the main advantages of the method presented in this paper can be obtained as follows:

Compared with HFLTS proposed by Rodriguez et al. [11] and PHFS proposed by Zhu [15], PLHFS proposed in this paper comprehensively reflects the subjective hesitant information of decision makers from two qualitative and quantitative perspectives, and gives the probability distribution of its hesitant information from an objective perspective. Moreover, it is difficult to fully express the decision maker’s hesitation only based on qualitative or quantitative information in actual decision-making problems, so the PLHFS mentioned in this paper has important practical significance. In addition, the PLHFS presented in this paper can not only express individual preference information, but also apply to the integration of group opinions, so it has a strong flexibility in actual decision-making problems.

Compared with the PHFS proposed by Zhu [15], the PLHFS presented in this paper not only adds the qualitative hesitant information of decision makers, but also gives the specific calculation method of probability based on the score consistency, which not only reflects the distribution of hesitant information of decision makers, but also is more consistent with the habits of decision makers in actual decision-making.

Compared with the probabilistic calculation method proposed by Zhou et al. [13], the CCI proposed in this paper fully takes into account the decision maker’s risk attitude, which is more in line with the actual decision situation. Therefore, the results obtained are more reasonable.

Compared with the consistency improvement method proposed in literature [1 , 18], the consistency improvement model proposed in this paper not only comprehensively considers the risk attitude of the decision-maker, but also makes use of the score consistency proposed to greatly simplify the calculation quantity without defining various measures or performing various complex aggregate calculations. Therefore, the proposed method is not only more reasonable but also has strong practical operation.

The PLHFPR proposed in this paper can be converted into other forms of preference relation according to the actual decision-making situation, so the method proposed in this paper is more general than the ordinary method.

7. Conclusions

In this paper, based on the concepts of PLHFS and PLHFPR, consistent PLHFPR is investigated and a new method for group decision making is proposed. Firstly, to accurately and reasonably reflect investors’ hesitation and uncertainty, we put forward the PLHFS and the PLHFPR. Secondly, in order to obtain the specific probability value of each membership degree and the priority weight of the alternatives, we set up the goal programming model based on the score consistency. Then, we present the CCI by combing the risk attitude of investors to solve the problem of inconsistency of PLHFPR. And then, a weighted nonlinear programming model is constructed. What’s more, we present a concrete algorithm to solve the GDM problem with the PLHFPR. Finally, through the analysis of a specific investment case, we demonstrate the practicability and accuracy of the proposed method.

However, there are still some areas to be studied in this article. For example, the determination of model parameters and the establishment of a model based on the PLHFPR under an incomplete information environment. Thus, in the next work, we will further study the multi-attribute GDM problem with the PLHFPR under the incomplete information environment. In addition, it is noted that the PLHFPR proposed in this paper mainly expresses collective preferences, so it will be an interesting research direction to use PLHFPR to express individual preferences in a complex decision-making environment and to discuss the consensus reaching process [32, 33] in GDM.

Footnotes

Acknowledgement(s)

The authors would like to thank the editors and anonymous reviewers for their insightful and constructive commendations that have lead to an improved version of this paper. This work was supported by National Natural Science Foundation of China (Grant No. 11661053, 11771198) and the provincial Natural Science Foundation of Jiangxi, China (Grant No. 20181BAB201003).

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The consistency improvement of probabilistic linguistic hesitant fuzzy preference relations and their application in venture capital group decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. HFLTS and PHFS

3. PLHFS and PLHFPR

3.1. PLHFS

3.3. PLHFPR

4.1. Score consistency of PLHFPR

5.1. Convex consistency index of PLHFPR

6.1. Decision making with proposed algorithm

Table 1 Convex consistency index and priority weight

7. Conclusions

Footnotes

Acknowledgement(s)

References

Table 1
Convex consistency index and priority weight