Risk assessment in project management by a graph-theory-based group decision making method with comprehensive linguistic preference information

Abstract Risk assessment is a vital part in project management. It is possible that experts may provide comprehensive linguistic preference information in distinct forms with respect to different aspects of the risk assessment problem in investment management. It is a challenge to model and deal with comprehensive linguistic preference assessments in multiple forms given by experts. In this regard, this paper defines the generalised probabilistic linguistic preference relation (GPLPR) to represent different forms of linguistic preference information in a unified structure. Then, a probability cutting method is proposed to simplify the representation of a GPLPR. Afterwards, a graph-theory-based method is developed to improve the consistency degree of a GPLPR. A group decision making method with GPLPRs is then proposed to carry on the risk assessment in project management. Discussions regarding the comparative analysis and managerial insights are given.


Introduction
Risk assessment refers to the quantitative analysis of the impact on people's life and property caused by risk events.Risk assessment is an important task in determining project investment.Due to the lack of reliable historical data, risk assessments are usually based on the experience and knowledge of experts (Qiu et al., 2018).When giving assessments, it is easier for experts to give preference information based on pairwise comparisons of alternatives, compared with giving comprehensive assessments of each alternative directly.A preference relation is a matrix in which each element is a preference degree between two objects (Saaty, 1980).The preference relation is a good tool to reduce the difficulty in risk representation (Tang et al., 2018).
CONTACT Huchang Liao liaohuchang@163.comOriginally, the elements of a preference relation were represented in crisp numbers (Saaty, 1980).With the expertise of experts being enriched, experts could provide precise and complex preference information.In this regard, the preference information given by experts may be multiple values (Zhu et al., 2014) or intervals (Zhang & Pedrycz, 2019).For decision making problems associated with qualitative criteria, experts may prefer to give their preference information in single (Xu, 2004a), multiple (Wang & Xu, 2015;Zhu & Xu, 2014) or interval linguistic terms (Xu, 2004b).In addition, to address the different importance of linguistic terms, the probabilistic linguistic preference relation (PLPR) was presented (Zhang, et al., 2016).The PLPR assigns probabilities to single linguistic terms to represent the preference information; however, multiple or interval linguistic terms are ubiquitous in people's preference perceptions.When different forms of linguistic preferences such as single, multiple or interval linguistic representations occur simultaneously, as far as we know, there is no good method to represent such type of uncertain preference information.To solve this issue, this study introduces a generalised PLPR (GPLPR) to model single, multiple and interval linguistic preference assessments in a unified structure.
The consistency of a preference relation ensures that the pairwise assessments are logical.Failure to meet the requirement of consistency may lead to wrong decision results.The consistency of a preference relation was originally defined by transitivity, such as the additive transitivity (Herrera-Viedma et al., 2004).There were many studies about the consistency measurement and improving procedures for PLPRs (Gao et al., 2020;Zhang et al., 2016).For the GPLPR which is more complex than the PLPR, it is necessary to propose an effective method to deal with its consistency issue.Xu et al. (2013) introduced the graph theory (Boffey, 1982) to denote the consistency of preference relations intuitively.Wang and Xu (2015) pointed out that if the weights of a digraph are well defined, then the additive consistency can be explained intuitively by the graph theory.Based on the graph theory, scholars have defined the consistency indices for distinct preference relations, such as the fuzzy preference relation (Xu et al., 2013), extended hesitant fuzzy linguistic preference relation (Wang & Xu, 2015) and PLPR (Zhang et al., 2016).Inspired by the aforementioned work, this study develops a probability cutting method to simplify the consistency measurement process for GPLPRs and proposes a graph theory-based group decision making method with GPLPRs.
This study dedicates to conduct the following innovative work: 1.The GPLPR is proposed to facilitate experts to express assessments in risk assessment.2. The probability cutting method is proposed to simplify the representation of a GPLPR.3. The consistency index of a GPLPR based on the graph theory is introduced to check and improve the additive consistency of a GPLPR.Then, a graph theorybased group decision making method with GPLPRs is proposed.The method is further implemented in a case study regarding the risk assessment of investment projects to validate the efficiency of the proposed method.
This paper is organized as follows: Section 2 recalls the related work.Section 3 proposes a graph-theory-based group decision making method with generalised linguistic preference information.Section 4 applies the proposed method to deal with the risk assessment of investment projects, and the related discussions are given in Section 5. Section 6 ends the paper.

Related work
Before introducing our theoretical model, we provide a short review on the risk assessment in project management.Then, we introduce the PLPR and GPLTS to facilitate further presentation.

A short review for the risk assessment in project management
Investment risk refers to the uncertainty in future investment, which may lead to the loss of profit or even loss of principal.The idea of risk assessment before decision making was first put forward by Athenians (Bernstein, 1996).It was pointed out that there is a significant relationship between risk management methods and the success of projects (Acharyya, 2008;Voetsch et al., 2004).Risk assessment refers to the comprehensive analysis of the possibility of risk and the degree of loss, combined with other factors to comprehensively analyze investment projects.When deciding whether to invest or not, it is necessary to first carry out detailed and systematic risk assessments, and then make decisions according to such assessments.Risk assessment has attracted the attention of many scholars in the past decades (see Table 1 for details).
It can be seen from Table 1 that experts tend to give linguistic evaluations when evaluating project risks, which may be single term, a set of multiple terms or interval linguistic term.Probability has been introduced to express the preference information given by experts (Liang et al., 2021;Zhang et al., 2016), and sometimes the probability may be incomplete.In addition, to reduce the difficulty of project selection, preference relations have been introduced to establish evaluation procedures (Tang et al., 2018;Zeng et al., 2007;Zhang et al., 2016).However, there is still no model which can not only express linguistic preferences with multiple forms, but also allow incomplete information.How to establish a risk assessment process with incomplete linguistic preference assessments in multiple forms is an unsolved problem.

Probabilistic linguistic preference relation
where t is a positive integer (Herrera et al., 1996).A symmetric set of discrete LTS S ¼ s a a ¼ Às, :::, 0, :::, s j g f was proposed to intuitively express the meanings of linguistic terms, satisfying: (1) s i >s j , if i>j; (2) negðs a Þ ¼ s Àa , where s is a positive integer (Xu, 2004a).A symmetric continuous LTS can be defined as S ¼ s a a 2 ½Àq, q j g ð q>sÞ: È To further deal with uncertainty, Xu (2004b) proposed an uncertain LTS as S ¼ s s ¼ ½s i , s j , À s i j s É : n For any three uncertain linguistic terms s ¼ ½s i , s j , s1 ¼ ½s i1 , s j1 , s2 ¼ ½s i2 , s j2 2 S, and l, l 1 , l 2 2 ½0, 1, the following operations were defined (Xu, 2004b) For n alternatives X ¼ x 1 , x 2 , :::, x n f g , let S ¼ s a a ¼ Às, :::, 0, :::, s j g f be an LTS.Then, an uncertain linguistic preference relation can be defined as (Xu, 2004b) ii ¼ s 0 for all i, j ¼ 1, 2, :::, n: Since different linguistic assessments may be with different preference intensities, Pang et al. (2016) introduced the concept of PLTS which uses probabilities to express the preference degrees of linguistic terms.To apply PLTSs to preference relations, Zhang et al. (2016) proposed the concept of PLPR as D ¼ ðh ij Þ nÂn & X Â X, where h ij ðpÞ ¼ fs t ðp t Þjs t 2 S, p t !0, t ¼ 1, 2, :::, T, P T t¼1 p t 1g with p t >0 and P T t¼1 p t 1, and s t ðp t Þ is the tth linguistic term s t associated with the probability p t , and T is the number of different linguistic terms in h ij ðpÞ arranged in ascending order.The PLPR is a good tool to model single linguistic terms and their probabilities.However, when experts give different forms of linguistic preference relations at the same time, such as single term, a set of multiple linguistic terms or interval linguistic term, the PLPR fails to model such preference relations.In this regard, this paper proposes a GPLPR to model preference relations with several forms, which will be described in Section 3 for details.

Generalised probabilistic linguistic term set and its normalization process
In PLTSs, each linguistic term is associated with a probability, but in many cases, experts may also give a set of multiple linguistic terms or interval linguistic terms.To model the probabilistic linguistic assessments with multiple forms of linguistic expressions, Fang et al. (2021) introduced the GPLTS as: (1) where S ¼ s a a ¼ Às, :::, 0, :::, s j g f is a discrete LTS, K is the number of linguistic terms in fs a k q 1 g, and #GLðpÞ is the number of different linguistic expressions in GLðpÞ, satisfying #GLðpÞ Note 1.The motivation of introducing the GPLTS is not to propose a complicated representation model for decision making, but to model the precise linguistic assessments given by experts directly and comprehensively.Due to the continuous changes of objects and the limited cognition of human beings, only key states of change can be perceived by experts.Thus, the differentiation of S and S can be achieved, where S is used to express experts' assessments and S is applied to describe the state of objects.
Note 2. In a GPLTS, there may be incomplete probability which is not assigned to any linguistic term in the GPLTS.In this case, we need to normalize the original GPLTS.In the original study of GPLTSs (Fang, et al., 2021), there was no clear explanation about why it is necessary and possible to normalize GPLTSs.Thus, we further explain the normalization process in detail in Appendix A.

A graph-theory-based group decision making method with generalised linguistic preference information
Risk assessment in project management is a complex decision-making process.Since preference relations do not require to identify criteria, researchers (Gou et al., 2021;Tang et al., 2018;Zeng et al., 2007;Zhang et al., 2016) have introduced different forms of preference relations for risk assessment.Preference relations for risk assessment involve several linguistic representations such as single linguistic term (Liang et al., 2021;Zeng et al., 2007), a set of multiple linguistic terms (Tang et al., 2018), or interval linguistic term (Lan et al., 2021), but existing methods cannot deal with these representations at the same time.In this regard, this paper proposes a GPLPR to model different linguistic representations at the same time, a probability cutting method to simplify the representation of a GPLPR, and a graph theory-based method to check and improve the additive consistency of GPLPR.Then, a graph theory-based group decision making method with GPLPRs is developed, which can be demonstrated intuitively as Figure 1.As can be seen, the contributions of this paper, marked with red words, mainly include modelling several linguistic representations of preference relation, simplifying the representations, checking and improving the additive consistency, and calculating the utility values of alternatives.In the following, the way to model several linguistic representations of preference relation is described in Section 3.1, and the way to simplify the representations is discussed in Section 3.2.Section 3.3 addresses the process to check and improve the additive consistency.Section 3.4 shows the process to calculate the utility values and generate the ranking of alternatives.

Establish the generalised probabilistic linguistic preference relation
For a practical risk assessment problem with n alternatives X ¼ fx 1 , x 2 , :::, x n g, experts are invited to give the preference degree between each pair of alternatives based on the LTSs S ¼ s a a ¼ Às, :::, 0, :::, s j g f and S ¼ s a a 2 ½Às, s j g : È The experts may use different forms of linguistic expressions, such as single linguistic terms, multiple linguistic terms or interval linguistic terms.To represent multiple linguistic forms comprehensively, we define the GPLPR as follows: Definition 1.A GPLPR on X ¼ fx 1 , x 2 , :::, x n g is denoted as where K is the number of linguistic terms in fs a k , ij q 1 g, and #GL ij ðpÞ is the number of different linguistic expressions in Especially, if all s ij (for i, j ¼ 1, 2, :::, n) are single linguistic terms, then the GPLPR degenerates into a PLPR; if all s ij (for i, j ¼ 1, 2, :::, n) are a set of multiple linguistic terms, then the GPLPR degenerates into a hesitant fuzzy linguistic preference relation (Zhu & Xu, 2014) or extended hesitant fuzzy linguistic preference relation (Wang & Xu, 2015); if all s ij (for i, j ¼ 1, 2, :::, n) are interval linguistic terms, then the GPLPR degenerates into an uncertain linguistic preference relation.The introduction of the GPLPR makes it possible to model different linguistic expressions in one model.

Simplify the GPLPR: a probability cutting method
To ensure that preference information is logical and non-random, it is essential to check the consistency of the preference relation.Due to the probability information assigned to different forms of linguistic information, the consistency processing for PLPRs and GPLPRs is much difficult.To simplify the consistency checking process of preference relations, the a-cut method (Liao et al., 2019) was introduced.However, since the threshold a may be different values, the information gained by the a-cut method may be misleading.In this subsection, motivated by the idea to measure the consistency by the distance between PLTSs (Fang et al., 2021;Wu & Liao, 2019;Xu et al., 2013), we propose a probability cutting method to address the consistency of GPLPRs.
The main idea of the probability cutting method is to adjust the probability distribution of the GPLPR to the same structure, and then divide the GPLPR into several preference relations with linguistic information.The consistency of a GPLPR can be achieved by dealing with the consistency of all the obtained preference relations with linguistic information.For a normalized GPLPR, we define its consistency as follows: Definition 2. Given a normalized GPLPR, D ¼ ð s ij ðpÞÞ nÂn is said to satisfy the additive consistency if s ij ðpÞ ¼ s ie ðpÞ s ej ðpÞ, for all i, e, j ¼ 1, 2, :::, n, i 6 ¼ j: If D ¼ ð s ij ðpÞÞ nÂn satisfies the additive consistency, then D ¼ ðs ij ðpÞÞ nÂn satisfies the additive consistency.

Deal with the consistency of GPLPRs based on the graph theory
Graph theory is an important method to solve decision-making problems.In the research of preference relationship persistence, Wang and Xu (2015) and Zhang et al. (2016) applied the preference relation graph (P-graph) and symmetric preference relation graph (S-P-graph) to deal with the consistency of the extended hesitant fuzzy LPR and PLPR respectively.In this regard, it may be a good attempt to use the graph theory to intuitively deal with the consistency of GPLPRs.
For a GPLPR D ¼ ðGL ij ðpÞÞ nÂn & X Â X, we normalize it by the method presented in Section 3.1 to get 2, :::, Rg, and its dimension reduction as pÃ ¼ fp r jr ¼ 1, 2, :::, Rg can be obtained, respectively.After that, R uncertain linguistic preference relations as ) can be generated.To calculate conveniently and embody the symmetry of a preference relation, the uncertain linguistic preference relation is introduced as and in the left lower triangle, the preference relationship s Às s þ ij s À ij s s : D is called an additively consistent uncertain linguistic preference relation if sie s ej for any i, e, j ¼ 1, 2, :::, n, i 6 ¼ j: To deal with consistency, the weighted S-P-graph of D is defined as G SÀUL ðV, AÞ, where V ¼ fv 1 , v 2 , :::, v n g is the set of vertices and A ¼ fðv i , v j Þji 6 ¼ j, i, j ¼ 1, 2, :::, nÞg is the set of arcs.ðv i , v j Þ represents a directed line segment from v i to v j with the weight wðv i , v j Þ ¼ ½r À ij , r þ ij , where r À ij and r þ ij represent the subscripts of s À ij and s þ ij , respectively.Example 2 is given to show the S-P-graph of D: Example 2. Given an uncertain linguistic preference relation on S as D 2 , then, its S-P-graph is shown as Figure 2.
As can be seen from Figure 2, there are multiple paths from v i to v j : For a consistent uncertain linguistic preference relation, the average length of the path lenðv i , ðv i , v j Þ, v j Þ should be equal to the average length of the path lenðv i , ðv i , v e Þ, v e , ðv e , v j Þ, v j Þ, where i, e, j ¼ 1, 2, :::, n, i 6 ¼ j: Then, we define the additively consistent uncertain linguistic preference relation as: The proof of Theorems 2 is shown in Appendix C. Based on the additively consistent uncertain linguistic preference relation, according to the similarity between sij in D ¼ ðs ij Þ nÂn and s ij in D ¼ ð s ij Þ nÂn , the consistency index of sij can be calculated by Eq. ( 4): where r À ij and r þ ij are the subscripts of sÀ ij and sþ ij , and r À ij and r þ ij are the subscripts of s À ij and s þ ij , respectively.The higher the consistency index is, the more logical the preference relation given by the expert is.Especially , for i, j ¼ 1, 2, . . ., n, i 6 ¼ j According to the obtained CI ij of sij , we can get the consistency level of The strict additive consistency is a strong condition for evaluating preference relations.Due to the complexity of decision-making problems, such conditions are usually difficult to meet.Therefore, we introduce the weak consistency of uncertain linguistic preference relations.
Definition 5. Let D ¼ ðs ij Þ nÂn be an uncertain linguistic preference relation.If when s À ik !s a and s À kj !s a , for i, j, k 2 f1, 2, :::, ng⏧i 6 ¼ j 6 ¼ k, there is s þ ij !s a , then D is said to satisfy the weak consistency.
To intuitively represent the weak consistency of uncertain linguistic preference relations, an uncertain linguistic preference graph (UL-graph) is proposed, which is a weighted digraph G UL ðV, AÞ: V ¼ fv 1 , v 2 , :::, v n g is the set of vertices and A ¼ fðv i , v j Þji<j, i, j ¼ 1, 2, :::, nÞg is the set of arcs, where ðv If there is no circular triad in the UL-graph, then D is said to satisfy the weak consistency.
For a weakly consistent D ¼ ðs ij Þ nÂn , it conforms to logical relations and can be used to solve decision-making problems.If D ¼ ðs ij Þ nÂn does not satisfy the weak consistency, there is a need to find circular triads whose arcs are denoted by A TRI ¼ fðv m i , v m j Þjm is the number of arcs in A TRI }.In A TRI , we need to find the arc with the worst consistency, i.e., ðv m i , v m j Þ with min m fCI m ij g, and replace its weight with the average weight of the other two arcs in its circular triad.Then, we check circular triad again and process it until there is no circular triad in the UL-graph.Finally, the D ¼ ðs ij Þ nÂn satisfying weak consistency is obtained.
The weak consistency of uncertain linguistic preference relations ensures that the assessments satisfy the logic and avoids the influence on the accuracy of the decision result due to the misjudgements.For ease of application, we summarize the procedure in Algorithm I and give an example as Example 3.
Algorithm I: Weak consistency checking and improving for uncertain linguistic preference relations Step 1.Let p ¼ 0 and D ðpÞ ¼ D ¼ ðs ij Þ nÂn be an uncertain linguistic preference relation.
Step 2. Build the S-P-graph and UL-graph of D ðpÞ : Step 3. If D ðpÞ satisfies the weak consistency, then go to Step 7; else go to Step 4.
Step 4. Find out all the circular triads to form A TRI : Next, we calculate the consistency index CI ij of ðv m i , v m j Þ by Eq. ( 4) and select the arc with the lowest additive consistency level and go to Step 5.
Step 5. Replace the weight of the arc ðv m i , v m j Þ found in Step 4 with n k¼1, k6 ¼i, j ðs ik s kj Þ=ðn À 2Þ, and change the corresponding uncertain LTSs in D ðpÞ : Step 6.Let D ðpþ1Þ ¼ D ðpÞ , p ¼ p þ 1: Then, go to Step 3.
Example 3. Given an uncertain linguistic preference relation on S 2 in Example 1 as D 3 :  3 and 4

Generate the ranking of alternatives
In group decision making process, if a GPLPR satisfies the acceptable additive consistency, the preference information needs to be aggregated to obtain an overall assessment matrix and then generate the ranking the alternatives.In this subsection, the classical arithmetic average operator is introduced to illustrate the information fusion of GPLPR.Definition 6.For n normalized GPLTSs GL i ðpÞ ¼ f½U q i , V q i ðp q i Þjq ¼ 1, 2, :::, # GL i ðpÞgði ¼ 1, 2, :::, nÞ, the arithmetic average operator of them can be defined as: The preference information of alternatives can be aggregated by this operator, and obtain the corresponding comprehensive assessment.Then, according to the score function and deviation function of the comprehensive assessment, the alternatives can be ranked.

Definition 7. Let
GLðpÞ ¼ f½U q , V q ðp q Þjq ¼ 1, 2, :::, # GLðpÞg be a normalized GPLTS.Then, 1) the score function of GLðpÞ is Eð GLðpÞÞ ¼ s a , where 2 Á p q with r q U and r q V being the subscribes of U q and V q , respectively.2) the deviation function of GLðpÞ is rð GLðpÞÞ ¼ s r , where r ¼ For any two normalized GPLTSs GL 1 ðpÞ and GL 2 ðpÞ :1) Based on the above analysis of GPLPR, Algorithm II for group decision making with GPLPR is developed.

Algorithm II: A graph theory-based method for group decision making with GPLPRs
Step 1. Determine the GPLPR D ¼ ðGL ij ðpÞÞ nÂn & X Â X: Step 2. Get the normalized GPLPR Step 4. Reduce the dimension of DÃ to obtain R matrices in the uncertain linguistic preference relation D rÃ ¼ ðs r ij Þ nÂn ðr ¼ 1, 2, :::, RÞ: In each probability p r ðr ¼ 1, 2, :::, RÞ, we can apply Algorithm I to deal with the additive consistency of D rÃ ¼ ðs r ij Þ nÂn : Then, we can get DrÃ ¼ ðs r ij Þ nÂn which satisfies the weakly additive consistency, and then calculate the corresponding consistency index CI rÃ : Step 5.According to D rÃ ¼ ðs r ij Þ nÂn in p r ðr ¼ 1, 2, :::, RÞ, we can transform the preference relation from a uncertain linguistic preference relation to a GPLPR to get D ¼ ðs ij Þ nÂn which satisfies the weakly additive consistency, and the consistency level is CIð DÞ ¼ P R r¼1 CI rÃ : Step 6. Aggregate the preference information in D by Eq. ( 6), and then rank the alternatives.End the algorithm.

Case study: risk assessment of investment projects
This section solves a case study about risk assessment of investment projects, and then conducts comparative analyses to illustrate the advantages of the proposed method.

Case description
Suppose that there is a company, whose main business projects are to establish industry, material supply and marketing industry, import and export business, and real estate development.For the sustainable operation and development, the company needs to choose new projects for investment.Whether for social responsibility or high profits, the purpose of investment is for the development of the company.However, as long as new projects are invested, the risks are inevitable.Now, the company plans to invest in one of four projects x i ði ¼ 1, 2, 3, 4Þ: For these projects, there is no significant difference in profit and other benefits.Thus, decision makers mainly consider the risk situation to choose the optimal investment project.The basic information of these four projects is described in Table 2.
Next, the graph theory-based group decision making method with GPLPR is applied to deal with this risk assessment of investment projects.
For four investment projects x i ði ¼ 1, 2, 3, 4Þ, ten experts are invited to compare their risk levels in pairs based on the LTS S ¼ fs À3 ¼ very low, s À2 ¼ low, s À1 ¼ a little low, s 0 ¼ fair, s 1 ¼ a little high, s 2 ¼ high, s 3 ¼ very high}.The evaluations given by experts are shown as Table A.1.Then, such evaluations can be expressed by GPLPR as Table 3.

Solve the case study by the proposed method
It can be seen from Table 3 that there are incomplete evaluations.To model the incomplete evaluations, the envelope of GPLTS is introduced.In this regard, the normalized GPLPR is obtained as Table A.2. Based on this, the rearranged probability set pÃ ¼ ð0:3, 0:3, 0:4Þ T can be obtained by cutting the probability of D: Then, the adjusted GPLPR can be generated as Table 4.
Source: The Authors.
Table 5.Three ULPRs for risk assessment.
Source: The Authors.
large space or have limited funds, single person tends to choose single apartment with small space.In addition, the investment risk of constructing green community and sanatorium community is similar, and the risk of the former is slightly lower than that of the later.The risk of constructing smart community is the highest.The possible reason is that the construction of smart community needs a lot of money, and the domestic complete smart home is not perfect and popular.Based on this, the target population may have concerns about the function and quality of this community.

Discussions
To further understand the probability cutting method of GPLPRs, we have some discussions and analyses on it.

Necessity of the proposed additive consistency of GPLPR
If the consistency of D is not processed, the original GPLPR is aggregated directly with Eq. ( 4), and the following results can be obtained: and Eðx 4 Þ ¼ s À1:225 : Then, we have x 1 >x 3 >x 2 >x 4 : Thus, x 4 with the lowest risk is still our best choice.
It can be seen that the processing of consistency has an impact on the score values of the original preference relations, and may affect the final decision-making results.In this case, since the consistency of the original GPLPR is high enough, the results satisfying the consistency may not change much compared with the original results.However, while in the decision-making problem of ranking a large number of alternatives, the influence of the consistency may be obvious.When experts compare a large number of alternatives in pairs, it is likely to cause the problem of logical inconsistency, so it is necessary to deal with the consistency, so as to get a logical preference relationship.

Comparative analysis
To illustrate the effectiveness of the proposed consistency checking method, we compare it with three methods, including the subscript calculation method (Zhang et al., 2016), the improved subscript calculation method (Liang et al., 2020) and the linear programming method (Xie et al., 2019).The characteristics of the four consistency checking methods are listed in Table 7.
By applying the three methods to deal with the same case study in Section 5.1, we can get the comparative analysis of the results as shown in Table 8.The detailed steps for the three methods are shown in Appendix E.
Based on the comparative analysis, we can find that the proposed method in this paper has the following four advantages: 1. Feasible in complex probabilistic linguistic information.These four methods can measure and improve the additive consistency of a PLPR, but only the proposed method can further deal with the complex additive consistency of UPLPR and GPLPR.2. Easy to understand.In the calculation process, the methods proposed by Zhang et al. (2016) and Liang et al. (2020) both used the subscript calculation of linguistic terms to transform the original preference information into virtual linguistic terms, which is convenient to the calculation but difficult to explain the meaning.
In addition, the method proposed by Xie et al. ( 2019) transformed the original linguistic information into numerical information through semantic functions, which is also difficult to intuitively display the meaning of the transformed information.In the process of calculation, the proposed method always keeps preference information in the set linguistic category, so the meaning can be easily understood.3. Keeping the original information.For the preference relation with unacceptable additive consistency, the other three methods need to change all the elements of the original information, while the proposed method only needs to change some key elements which have the highest deviation from other linguistic terms in the original preference information.4. Simple calculation.Compared with the methods proposed by Zhang et al. (2016) and Liang et al. (2020) which need to get acceptable consistent preference relations based on virtual linguistic terms, and the method proposed by Xie et al. (2019) which needs to set the deviation thresholds between alternatives, the proposed method is simple and does not need to set any parameter.

Managerial implications
For practicing managers, this study has some implications.First, it proposes a GPLPR for the risk assessment of investment projects.This model allows experts to give pairwise comparison information of projects, which reduces the difficulty of giving assessments.At the same time, due to the diversity of linguistic expressions allowed by a GPLPR, it can directly model the original linguistic assessments given by experts, which makes the original information not lost in the process of modelling.For managers, the effective use of information can provide reliable evidence for decision making, which is conducive to the development of enterprises.Second, the proposed probability cutting method can improve the efficiency of decision making.It is important for managers to ensure that the obtained information is logical and non-random.Based on this, it is a vital part to measure and improve the consistency of preference relations.But for the complex GPLPR, handling the consistency is not an easy task.In this regard, the probability cutting method can reduce the difficulty of consistency processing and improve the efficiency of the whole decision-making process.
Overall, an effective method for risk assessments has been developed.Managers can use the proposed method to assess their projects on risk and select the best investment project.Based on the implementation of accurate assessments, managers can have a clear understanding of the actual situation of each project, which plays a great role in the selection of projects.Choosing an appropriate investment project may affect the development of the enterprise in the future.Therefore, it is important for managers to select an appropriate investment project by using the correct decision-making method.

Conclusion
This paper proposed a graph theory-based group decision making method with GPLPRs to deal with the risk assessment problem of project investment.It could be noted that the existing probabilistic preference models mainly include linguistic preference relations, PLPRs and uncertain linguistic preference relations, which do not contain various forms of linguistic preference.In addition, for incomplete probabilities, these models usually assigned the remaining probability proportionally to the existing linguistic expressions.This way might ignore the uncertainty caused by incomplete probability, and produce unreasonable results.Therefore, in this paper, we first proposed the GPLPR, which can cover several linguistic probabilistic preference expressions, and then introduced a method to handle the incomplete probability.To simplify the measuring the consistency of GPLPRs, a probability cutting method was proposed.It was proved that the consistency of the GPLPR can be achieved by processing the consistency of the cut linguistic preference relations separately.Besides, based on the graph theory, we proposed a method to measure and improve the consistency of GPLPRs.Finally, a graph theory-based group decision making method with GPLPRs was developed and verified by carry on the risk assessment of project investment.
In the future, the probability cutting method should be applied to deal with probabilistic preference relations with hesitant fuzzy linguistic information, extended hesitant fuzzy linguistic information, intuitive fuzzy linguistic information or other linguistic expressions in the existing methods.In addition, while aggregating the consistency of preference relations, it may be interesting to combine different aggregation operators to support the proposed model.Besides, it will also be interesting to apply the proposed method to deal with more practical decision-making problems.

Disclosure statement
No potential conflict of interest was reported by the authors.

D. Tables
1) The calculation process of the consistency checking method based on subscript calculation proposed by Zhang et al. (2016).
To deal with the additive consistency of PLPRs, Zhang et al. (2016) proposed a subscript calculation method based on the graph theory to directly measure and improve the additive consistency of PLPR.Next, we will use this method to solve the same case study in Section 5.1.The detailed steps are shown as follows.
Step 1. Establish the judgement matrix with PLPR.In the method of Zhang et al. (2016), the assessments given by experts should be PLPRs and the subscript of linguistic terms should be interval.Since the subscript calculation method is a consistency checking tool for PLPR, it is necessary to transform the original GPLPR to PLPR for the convenience of comparison.Based on this, to facilitate the comparative analysis of the two methods, there is a need to transform the GPLPR D ¼ ðGL ij ðpÞÞ 4Â4 to a PLPR D 0 ¼ ðL ij ðpÞÞ 4Â4 by Eq.( 7).

#
Step 4. Measure the consistency index of D 0 by Eq. ( 8) as CIðD 0 Þ ¼ 0:0113: Zhang et al. (2016) used the deviation degree between D 0 and D C 0 to measure the consistency index of D 0 : Next, it is necessary to compare the consistency index CIðD 0 Þ with the maximum acceptable deviation degree , then the consistency level of D 0 is acceptable; otherwise, the consistency level of D 0 is unacceptable.According to reference (Dong et al., 2008), for the judgment matrix with n ¼ 4, the maximum acceptable deviation degree is Step 5. Rank alternatives.Experts have no preference on the four projects, so the weight vector is w ¼ ð1=4, 1=4, 1=4, 1=4Þ T : By the probabilistic linguistic weighted averaging operator (Zhang et al., 2016), the comprehensive preference values of alternatives can be obtained as: PV 1 ¼ fs 0 , s 0:375 , s 0:9 g, PV 2 ¼ fs À0:225 , s À0:15 , s 0:425 g, PV 3 ¼ fs À0:35 , s 0:075 , s 0:4 g, PV 4 ¼ fs À1:075 , s À0:3 , s À0:075 g: Then, the scores of the comprehensive preference values of x i ði ¼ 1, 2, 3, 4Þ are generated as EðPV 1 Þ ¼ s 0:425 , EðPV 2 Þ ¼ s 0:0167 , EðPV 3 Þ ¼ s 0:0417 , EðPV 4 Þ ¼ s À0:4833 : Therefore, the order of alternative projects is obtained as x 1 >x 3 >x 2 >x 4 : 2) The proposed calculation process of the consistency checking based on subscript calculation proposed by Liang et al. (2020).Liang et al. (2020) pointed out that there is a deficiency in the original formula for calculating consistency index based on subscript calculation, that is, when the subscript values of linguistic terms are the same, the consistency index is zero whatever their respective probability values are, which ignores the role of probability values in consistency measurement.Based on this, Liang et al. (2020) proposed a consistency measurement method, which takes the subscript value and probability value of linguistic terms into account.
In the original subscript calculation method, the calculation formula of consistency index CIðD 0 Þ is replaced by Due to CIðD 0 Þ ¼ 0:0681< C IðD 0 Þ ¼ 0:173, D 0 satisfies acceptable consistency (Saaty, 1980).In addition, other steps are consistent with the subscript calculation method.In this regard, the order of alternative projects is obtained as x 1 >x 3 >x 2 >x 4 : 3) The calculation process of the consistency checking based on linear programming proposed by Xie et al. (2019).

Figure 1 .
Figure 1.Framework of the graph theory-based group decision making method with GPLPRs for risk assessment.Source: The Authors.

Figure 2 .
Figure 2. The S-P-graph of D 2 : Source: The Authors.
Let p ¼ 0 and D ðpÞ 3 ¼ D 3 : Step 2. Build the S-UL-graph and UL-graph of D ð0Þ 2 as Figures

Figure 5 .
Figure5.Three UL-graphs by the rearranged probability set pÃ ¼ ð0:3, 0:3, 0:4Þ T : Note: The bold line is the circular triad.The interval terms in the bracket are the evaluation after consistency correction.Source: The Authors.

Table 1 .
Researches on risk assessment of investment projects.

Table 2 .
The basic information of projects.

Table 3 .
Decision matrix for risk assessment in GPLPR.

Table 4 .
Decision matrix for risk assessment in adjusted GPLPR.D Ãx 1

Table 6 .
The GPLPR with the weakly additive consistency.

Table 8 .
Comparative analysis of the results obtained by four consistency checking methods.