**1. Introduction**

Public–private partnership (PPP), as a significant institutional innovation in infrastructure investment and public service delivery [1], is a long-term cooperation mechanism that advocates a relationship of "complementary advantages, benefit, and risk-sharing" between governmen<sup>t</sup> and private departments [2]. Since 2014, PPP has experienced a new boom under marked motivation from the central governmen<sup>t</sup> [3]. After seven years of rapid development in China, PPP has become an effective approach for stabilizing growth, facilitating innovation, regulating strategy, and increasing the welfare of individuals, promoting the integral expansion of the economy. According to the latest statistics from the China Public–Private Partnerships Center, as of January 2022, a totally of 10,254 projects, with a total investment volume of RMB 16.2 trillion, are collected in the managemen<sup>t</sup> database, of which 7714 projects with an investment of RMB 12.8 trillion have been contracted and landed, with an implementation rate of 78.9%. China has become one of the largest markets of PPP in the world. However, various stakeholders are involved in the PPP projects, as

**Citation:** Wang, H.; Lin, Q.; Zhang, Y. Risk Cost Measurement of Value for Money Evaluation Based on Case-Based Reasoning and Ontology: A Case Study of the Urban Rail Transit Public-Private Partnership Projects in China. *Sustainability* **2022**, *14*, 5547. https://doi.org/10.3390/ su14095547

Academic Editors: Pierfrancesco De Paola, Francesco Tajani, Marco LocurcioandFelicia DiLiddo

Received: 4 April 2022 Accepted: 2 May 2022 Published: 5 May 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

well as huge investments and long construction cycles. It is easy to encounter problems such as the irrational allocation of risks, the creation of explicit shares but real debts, an emphasis on construction while neglecting operation, and excessive financing leverage. Thus, a dialectical perspective should be taken in the practical application of PPP to clarify that it is not an almighty tool that is feasible for all projects.

Value for money (VFM) is defined as "the optimum combination of whole life costs and quality (or fitness for purpose) to meet the user requirement" [4]. In practice, the VFM of a PPP project can be expressed as the difference between the net present value (NPV) of the whole life-cycle cost (LCC) of a project procured by a traditional method (LCCPSC) and the NPV of the LCC of the same project procured through a PPP approach (LCCPPP) [5]. Ultimately, VFM is generated when the total net present value of PPP is less than the NPV of traditional procurement, indicating that the whole life cost of the proposed project can be reduced [6]. For both NPV of PPP and PSC, risk costs are engaged as the crucial issue and dilemma in the quantitative evaluation of VFM. At the early stages of popularization of VFM, there is a complete system, but not ye<sup>t</sup> an established sophisticated information platform that can accumulate specific data for types of projects, leading to the assessment and allocation of risk being widely dominated by experts whose subjective and unilateral nature properly induce some bias in the entire measurement of VFM. In addition, the topics of risk assessment and allocation have aroused a research fever that has facilitated the emergence of substantial-excellent studies during the period of rapid expansion of PPP. Almost of them are based on experts, but further mitigate the influence of individual subjectivity by employing different techniques that do not fundamentally improve the independence of specialists, and some methodologies are complex and not operable in practical projects. Moreover, many characteristics have emerged in China's PPP market, such as quantity reduction but quality increase, more sophisticated systems, stronger supervision, more transparent information, etc. These have provided a stable environment for PPP development, which considerably enhanced the general quality and maturity of PPP projects in the official database. Consequently, some items with high availability are accumulated in municipal engineering, transportation, and other PPP areas, creating friendly conditions to offer guidance for subsequent new projects. However, the utility has failed to be effectively utilized by existing accounts, especially in the risk cost estimation. Based on the current situation, this study seeks to explore a valid way to adopt useful historical projects to estimate risk costs in VFM evaluation, in parallel, decreasing the reliance on experts.

In this research, a risk costs estimation model for VFM evaluation was developed based on previous similar cases. The risk costs include the retained costs undertaken by governments and the transferable costs incurred by private sectors. An objective of this research is to improve the efficiency of risk cost assessment, as well as the utilization of old cases in the VFM evaluation phase, and to propose several suggestions for risk data accumulation in the process of project management. The accuracy of the VFM value determines whether the PPP can be successfully applied to an infrastructure project, while a more sophisticated data system will contribute greatly to industry development. The research was carried out as follows. A combination of case-based reasoning (CBR) and ontology is used to set up the estimation model, while the information from the China Public–Private Partnerships Center is used as an instruction in the process. The model has been organized into the following four submodules: (1) ontology model development, (2) attributes weighting, (3) similarity calculation, (4) VFM risk cost measurement. More specifically, an information ontology model for risk cost calculation was first developed from previous PPP projects that contained a series of attributes. Second, the ID3 algorithm of the decision tree was adopted in attributes weighting to identify similarities between the target and old cases. Third, similarities were calculated by the semantic similarity algorithm incorporated with principal component analysis (PCA) based on the ontology tree structure. Additionally, more than three most similar cases for the target case were retrieved, and the contributions were prioritized according to the degree of similarities. The

extracted cases were utilized to predict the risk cost of the target case. Deviations of data in similar cases which induced uncertain predicted values were considered in the process. A data revision step was taken to improve the accuracy of the estimation of risk costs. Finally, the retained cost and the transferable cost of risk were predicted from the revised data of similar historical projects. The outcomes were compared to the actual risk costs of the target cases documented in the official database for validation. This approach can increase the computational performance and availability for estimating costs of potential risk for governmen<sup>t</sup> and private sectors in the phase of VFM evaluation. Therefore, experts can be freed from repetitive work and devote their time to better implementing projects to optimize the application of the PPP model.

#### **2. VFM Risk Cost of PPP Project**

Risk is demonstrated as one of the most crucial drivers of VFM by numerous academics [7–10]. One of the most prominent tasks of PPP is to invite the private departments to share risks. It is necessary to confirm the risks and the risk costs borne by the governments and social capitals in the evaluation phase of VFM, thereby facilitating more detailed risk control in the subsequent steps. Compared to traditional procurement, where the governments take all the risks, PPP plays an effective role in sharing some of the risks with the private sector, called transferable risks, while the remaining risks taken by the governmen<sup>t</sup> itself are called retained risks. Furthermore, the associated costs of both are directly related to the achievement of value for money which requires accurate risk identification, assessment, and allocation. Optimization studies surrounding the evaluation are continuously active in the PPP field. There is extensive research on risk identification in all areas of PPP. Song et al. investigated ten key risks of PPP waste-to-energy (WTE) incineration projects in China [11]. Zhang et al. combined the 2-tuple linguistic representation model and DEMATEL to examine the risk factors and their interrelationships of EVskCI-PPP projects [12]. Similarly, the identification of other PPP fields, such as water supply, urban underground pipe gallery, sponge city, and construction projects is well undertaken by various surveys and academics [13–16]. Additionally, this determination is often done along with assessing the principal risks and classifying the related risks into different levels by using a series of methods. The Mann–Whitney U test was adopted to seek out the most important risk factors for PPP projects in China, including governmen<sup>t</sup> intervention, governmen<sup>t</sup> corruption, and poor public decision-making processes [17]. A combination of two-dimension linguistic variables and the cloud Choquet integral (CCI) is used to mitigate the subjectivity of experts [18]. Multi-organization fuzzy rough sets (MGFRSs) are incorporated with an improved DEMATEL method to deal with the influence of interrelationships on the ranking of risks [19]. Structural equation modeling (SEM) has been applied in ranking risks and identifying several risk paths by focusing on risk interaction and stakeholders' expectations [20]. Interpretative structural modeling (ISM), along with MICMAC analysis, were used to prioritize PPP risks [21]. Related research likewise provides valuable references for risk assessment, but they were all conducted on the premise of specialists' opinions. Specifically, the cost assessment is usually calculated by occurrence and impact, which heavily depend on expert judgments. Risk allocation, as an extremely significant part of VFM evaluation, affects the effective supervision and control of risks in the subsequent process of each PPP project. Optimal risk allocation, with its aim to achieve VFM [22], is perceived as the key to the success of the PPP model [23,24]. Thus, there are numerous studies on this theme. The Delphi questionnaire survey is conducted as the most prevalent tool [25,26] used to reduce the subjectivity of individuals; fuzzy synthetic evaluation, game theory, the artificial neural network, and other multi-attribute decision-making methods and intelligent technologies are used to obtain more precise results. Ke et al. found that the public sector preferred to retain most of the political, legal, and social risks, and share most of the microlevel risks and force majeure risks; the majority of microlevel risks were preferred to be retained by the private sector [27]. Ameyaw et al. adopted the fuzzy-set approach to examine the allocation of five key risk factors related

to PPPs in water supply infrastructure projects [26]. Li et al. proposed a bargaining game theory to prioritize risk allocation that considers the probability, severity, and impact of risk factors [28]. Artificial neural network (ANN) models were built up for risk allocation decision making based on the industry-wide questionnaire survey [29]. A neuro-fuzzy decision support system (NFDSS) was developed to assist the sharing process [30]. A genetic algorithm (GA) was applied to enhance efficiency [31]. Valipoura et al. presented a SWARA-COPRAS approach to utilize qualitative linguistic terms in the allocation of risks [32]. Parallelly, some relevant studies were carried out from particular perspectives. A framework with a deeper understanding of risk was afforded by the principal–agent theory (PAT) to ensure a more complete and optimal risk allocation across the whole life cycle of PPP projects [33]. Project finance contracts were also considered [34].

As we can conclude, it is no longer a dilemma to identify and reasonably allocate the risks owing to proven methodologies, in addition to estimating the risk costs borne by different parties. Deviations always arise due to the irregularity of data and the dependence of commonly used methods on experts when calculating the occurrence and degree of risk. In the past, the accumulation of historical data was not mature enough to apply to risk cost estimation in China. Currently, however, as the database keeps optimizing and expanding, it has often been overlooked as an important resource to be used in this field. While historical cases are used to predict total project costs and risk response strategies, the increasingly available data has the potential to be a valuable asset for risk costs calculation. This paper employed CBR and ontology to achieve the above purpose, while at the same time, enhancing the efficiency of using historical cases, supplementing the information on risk pre-management, and providing more useful support for later regulation.

#### **3. Case-Based Reasoning and Ontology**

CBR, or case-based reasoning, is an approach to problem-solving that originated from cognitive science [35,36] and which emphasizes solving new problems by reusing and if necessary, adapting the solutions to similar problems that were solved in the past [37]. As a computerized approach, CBR has a wide range of applications in various areas, such as fault detection, chemical prediction, disease inference, and rehabilitation practice [38–41]. In particular, it is commonly applied in cost prediction, accident pre-control, and strategic decision making in construction [24,42–44]. However, it has not been popularly applied because of the initial poor accumulation of available data. Now that the PPP mode in China has entered a stable development stage, as the information managemen<sup>t</sup> of the PPP official database has been strengthened, historical projects are expected to become powerful tools for new PPP projects, and applying CBR in this field is conducive to improving the efficiency of historical knowledge reuse.

Aamodt [45] stated that a case-based reasoning process can be represented by the three tasks of retrieval, reuse, and learning, which collapse several steps compared to the subsequent definition by the author. The full process is shown in Figure 1; the CBR consists of four primary processes: retrieval, reuse, revision, and retention. Case retrieval is responsible for looking for the most similar cases in the established case base that indicate corresponding data or solutions for the target case to reuse. If the proposed solutions are not well matched, it is necessary to make some revisions to obtain more credible results based on the initial solution generated from old similar cases. The revised solutions are then retained as useful old cases in the case base. Among the whole cycle, retrieval, as well as revision, are the critical steps to ensure the successful application of CBR [46,47]; an accurate retrieval method guarantees the availability of extracted historical cases with high similarity, while an effective revision process improves the accuracy of the final results.

**Figure 1.** The CBR cycle. Adapted from ref. [45].

Before implementing CBR, the most crucial task is the representation of case knowledge. The ontology technology, as one of the case knowledge representation approaches, is an explicit formal specification of a shared conceptual model with the goal to capture the knowledge of related domains, forming a consensus in a field, and improving the efficiency of information interchange [48]. While the ontology uses a hierarchical tree structure to represent the concept sets, as well as the semantic relationships, of the concepts [49]. The nodes of the tree are called classes, with edges between the nodes representing semantic relationships between concepts. From the top-down, concepts are classified from large to small, and the lower-level concepts are a subdivision of the upper-level concepts. It supports the definition of new concepts based on the existing vocabulary in a way that does not require the revision of the existing definitions [50].

Ontology has structured some mature knowledge models for biomedical science that contain substantial complex information and which are constantly being expanded. A comprehensive resource of computable knowledge about genes and their products, Gene Ontology (GO), has been established and developed by the Gene Ontology Consortium and is widely used in the biomedical community [51,52], while a human phenotype ontology (HPO) was introduced to bring together a standardized vocabulary of phenotypic abnormalities associated with more than 7000 diseases that were presented [53]. The Cell Ontology (CL) is an OBO Foundry candidate ontology covering the domain of canonical, natural biological cell types [54]. Given the better inherent capability of knowledge representation, ontology has been broadly used in other areas, such as multilingual interoperability, document management, and industrial resource forecasting [55–57], as well as construction engineering [49,58,59], where there are many stakeholders, complex situations, and considerable information. Previous studies reflect that ontology technology is competent to support more sophisticated information expressions in various domains. Furthermore, as a kind of knowledge integrator, it contributes to achieving a consensus of knowledge in different industries and effectively realizes easy interoperability of information, offering excellent backup for CBR.

Simultaneously, an excellent similarity estimation method, conceptual semantic similarity, is provided by the structure of the ontology model. Based on this structure, the ID3 algorithm is allowed to be well applied in attribute weighting that directly influences the overall similarity between target and old cases. With these conveniences, the objectivity of the entire CBR cycle can be substantially improved, and the reliance on experts can be effectively minimized, to some extent.

#### **4. VFM Risk Cost Measurement of a PPP Project Based on CBR and Ontology**

#### *4.1. Ontology Development*

In this paper, we used Protégé, an ontology development tool, to create the PPP information ontology model using a seven-step approach, whose detailed steps are illustrated in Figure 2.

**Figure 2.** Ontology Development Process.


above classes were applicable for all PPP industries and were allowed to be further expanded or subtracted according to the actual industries studied.


**Figure 3.** Major Classes of PPP Projects Information Ontology.

#### *4.2. Similar Case Retrieval*

#### 4.2.1. Attribute Weighting

It is necessary to assign weights to each major class, while every class in a PPP project information ontology has a different influence on VFM risk cost, which definitely shows the relative importance of each class. Given the tree structure of the ontology model, we applied the ID3 algorithm of decision tree [60] to determine class weights, whose core ideology is to measure feature weights by information gain. The larger the information gain is, the more information it contains, and the more important the attribute is.

Before calculating the information gain, it is first necessary to understand the concept of information entropy. In 1948, Shannon [61] defined information entropy as the probability of the occurrence of discrete random events; the more orderly a system is, the lower the information entropy, while the more chaotic the system is, the higher the information entropy. Its calculation formula is:

$$H(\upsilon) = -\sum\_{i=1}^{n} P(u\_i) \log\_2 P(u\_i) \tag{1}$$

$$P(u\_i) = \frac{|u\_i|}{|v|}\tag{2}$$

where *v* = a set of cases; the *<sup>P</sup>*(*ui*) represents probability of the occurrence of symbol *i*; |*ui*| = number of cases in symbol *i*; and |*v*| = a number of cases in the set *v*.

The information gain is specific from different attributes; for an attribute, the difference of information between the system with it and without it is the information gain, so the formula for calculating information gain is:

$$Gain(v, a) = H(v) - \sum\_{s \in valuc(a)} \frac{|v\_s|}{|v|} H(v\_s) \tag{3}$$

where *a* represents an attribute of a case; *value*(*a*) = a set of values taken by attribute *a*; *v* is a value of attribute *a*; *vs* = a set of cases with value *s* in *v*; and |*vs*| indicates the number of cases contained in *vs*.

#### 4.2.2. Conceptual Semantic Similarity

The PPP project information ontology contained both quantitative and qualitative information, which dictated that different methods should be applied for calculating similarities. For quantitative information, a concrete calculation mathematical formula was used. For qualitative information, it is difficult to compare two abstract concepts using a typical formula; therefore, this paper adopted the conceptual semantic similarity which is based on the tree structure of the ontology model to achieve the comparison of abstract concepts.

(1) For quantitative information, the similarity calculation formula is shown below:

$$\text{sim}(w\_{N\prime}w\_j) = 1 - \frac{|w\_N - w\_j|}{w\_{\text{max}} - w\_{\text{min}}} \tag{4}$$

where *wN* = value of an attribute for the target case; *wj* = value of an attribute for the *j*-th old case u; and *w*max, *w*min represent the maximum and minimum values for all the old cases included in the database.

(2) For qualitative information, we used an improved domain ontology similarity algorithm, which integrated a total of four dimensions of semantic similarity: semantic distance, node depth, node density, and semantic coincidence [62]. This algorithm ensured that the calculated value of each influencing factor was between [0, 1] and

the combined semantic similarity was always in the range of [0, 1], while the result was always 1 for the similarity calculation of the same node.

1 The formula for calculating similarity based on semantic distance is:

$$distance\\_sim(A,B) = \frac{2 \times (H-1) - L}{2 \times (H-1)}\tag{5}$$

where *H* = maximum depth of the ontology tree, while the depth of root node is defined as 1, and which increases by one unit for each additional level; and *L* = semantic distance between concepts *A* and *B*.

2 Similarity based on node depth incorporates the nearest common ancestor, which is calculated as:

$$depth\\_sim(A,B) = \frac{2 \times N\_{LCS}}{N\_A + N\_B + 2 \times N\_{LCS}}\tag{6}$$

where *NA*, *NB* = number of edges passed by concept *A* and *B* to the nearest common ancestor node, respectively. *NLCS* = a number of edges passed by the nearest common ancestor node to the root node.

3 Similarity based on node density takes into account the effect of node density in the ontology tree structure, and its similarity is calculated by the formula:

$$density\_-sim(A,B) = 1 - \frac{|2 \times wind(LCS) - wind(A) - wid(B)|}{\max(wid(Trec))}\tag{7}$$

where *wid*(*LCS*) = number of sibling nodes of the nearest common ancestor of concept *A* and *B*; *wid*(*A*), *wid*(*B*) = number of sibling nodes of concept *A* and *B* (including themselves); and max(*wid*(*Tree*)) = a maximum number of children nodes owned by each node in the concept tree.

4 Similarity based on semantic coincidence considers the effect of the number of common ancestor nodes possessed by the two concepts, which is calculated as follows:

$$\text{coincidence\\_sim}(A, B) = \frac{\sum\_{\mathbb{C} \in \mathbb{T}\_A \cap \mathbb{T}\_B} F\_{A,B}(\mathbb{C})}{\sum\_{\mathbb{C} \in \mathbb{T}\_A \cap \mathbb{T}\_B} F\_{A,B}(\mathbb{C}) + \sum\_{\mathbb{C}\_A \in \mathbb{T}\_A - \mathbb{T}\_B} F\_A(\mathbb{C}\_A) + \sum\_{\mathbb{C}\_B \in \mathbb{T}\_B - \mathbb{T}\_A} F\_B(\mathbb{C}\_B)} \tag{8}$$

$$F\_{A,B}(\mathbb{C}) = \frac{2 \times Dep(\mathbb{C})}{Dep(A) + Dep(B)} \tag{9}$$

$$F\_A(\mathbb{C}\_A) = \frac{\operatorname{Dep}(\mathbb{C}\_A)}{\operatorname{Dep}(A)}\tag{10}$$

$$F\_B(\mathbb{C}\_B) = \frac{D\varepsilon p(\mathbb{C}\_B)}{D\varepsilon p(B)}\tag{11}$$

where *TA*, *TB* = set of nodes passed by concept *A* or *B* to the root node; *Dep*(*A*), *Dep*(*B*), *Dep*(*C*) = depth of concepts *A*, *B*, *C*, while *C* is the common ancestor of *A* and *B*; and *CA*, *CB* represent the ancestor nodes of *A*, the ancestor nodes of *B*, but excluding the common ancestor nodes of *A*, *B*, respectively.

5 Integrating the similarity of the above dimensions to find the combined similarity of each attribute is then calculated by the formula:

$$\begin{array}{ll}\text{semantic\\_sim}(A,B) = & \boldsymbol{a} \cdot \text{distance\\_sim}(A,B) + \boldsymbol{\beta} \cdot \text{depth\\_sim}(A,B) + \\ & \boldsymbol{\gamma} \cdot \text{density\\_sim}(A,B) + \boldsymbol{\phi} \cdot \text{coincidence\\_sim}(A,B) \end{array} \tag{12}$$

where *α*, *β*, *γ* and *φ* are the impact weights of semantic distance, node depth, node density, and semantic coincidence on the semantic similarity of the concepts, respectively, satisfying the condition *α* + *β* + *γ* + *φ* = 1. Generally, these four parameters are set manually; in order to overcome individual subjectivity and to allow flexible adjustment depending on the practical situation of different domains, principal component analysis (PCA) is introduced. It takes the contribution of principal components as the parameter value for weighting the semantic similarity in aggregate, which removes the artificial influence.

(3) Since the similarity between concept sets in qualitative information, it can be calculated based on the above four dimensions of similarity. Since a PPP project always contains multiple and variable numbers of "risk factors," the calculation of this attribute's similarity between two cases is actually a comparison between two sets of concepts of different sizes. In this paper, we use the "mean-maximum" algorithm to calculate the semantic similarity between concept sets, as proposed by Wang et al. [63] in Gene Ontology. It defines the semantic similarity between a concept *t* and a concept set *T* as the maximum semantic similarity between a concept *t* and any concept in the set *T*. That is

$$Sim(t, T) = \max \operatorname{semantic}\_{\text{-}sim}(t, t') \text{ } t' \in T \tag{13}$$

Therefore, given two concept sets *S* and *T* annotated by *S* = {*<sup>s</sup>*1,*s*2,...,*sm*} and *T* = {*<sup>t</sup>*1, *t*2,..., *tn*}, respectively, the similarity between the concept sets is defined as:

$$Sim(S, T) = \frac{1}{m + n} \left( \sum\_{1 \le i \le m} Sim(s\_{i\prime}, T) + \sum\_{1 \le j \le n} Sim(t\_j, S) \right) \tag{14}$$

#### *4.3. Risk Cost Measurement*

4.3.1. Preliminary VFM Risk Cost Calculation

After obtaining the weights of each attribute and the similarities between the target with each historical case in all attributes, the general similarity can be figured out based on the following equation:

$$Sim\left(V\_{\mathcal{N}\prime}, V\_{\mathcal{S}\dot{\jmath}}\right) = \sum\_{i=1}^{n} \omega\_i \left(sim\_i \left(V\_{\mathcal{N}i\prime}, V\_{\mathcal{S}\dot{\jmath}i}\right)\right) \tag{15}$$

where *ωi* = weight of the *i*-th attribute; *SimVN*, *VSj* = similarity between the target case and the *j*-th historical case; and *simiVNi*, *VSji* = similarity between the target case and the *j*-th historical case in the *i*-th attribute. According to the general similarity between the target case and the historical cases, the nearest historical cases can finally be selected as the candidates using these principles:

1 The general similarity between selected historical cases and the target case should not be less than 70%;

2 The number of selected historical cases should not be less than three;

3 The higher the similarity between historical cases and the target cases, the higher their contribution to the target case.

When selecting the moderate historical cases, the retained cost of risk and the total cost of risk for the target case can be estimated from Equations (16) and (17); the difference between *R* and *R*0 is the transferable cost of risk.

$$R\_0 = \sum\_{j\geq 3} R\_{j0} \text{Sim}(V\_{N\prime} V\_{Sj}) \quad \text{Sim}(V\_{N\prime} V\_{Sj}) \geq 70\% \tag{16}$$

$$R = \sum\_{j \ge 3} R\_j \text{Sim}(V\_{N\prime}, V\_{Sj}) \quad \text{Sim}(V\_{N\prime}, V\_{Sj}) \ge 70\% \tag{17}$$

#### 4.3.2. Case Revision

Considering that China's PPP project managemen<sup>t</sup> database is still in the process of perfection, and there is no normative constraint on the measurement of risk cost, the data in some cases may deviate from reality. Therefore, after retrieving several cases with high similarity, some formula revisions were required for those historical cases that have highly similar characteristics of each attribute, but large variations with other extracted cases in risk costs, and the corrected risk costs were used to calculate the final risk cost of the target project.

The revision was based on the PPP value after deducting the retained cost of risk, the PSC value after deducting the total cost of risk, as well as the contribution of each case to improve the accuracy of the result; the formula is shown as follows:

$$R\_{i0r} = \sum\_{j} \frac{PPP\_i - R\_{i0}}{PPP\_j - R\_{j0}} R\_{j0} \* \frac{\omega\_j}{\sum\_{j} \omega\_j} \tag{18}$$

$$R\_{ir} = \sum\_{j} \frac{PSC\_i - R\_i}{PSC\_j - R\_j} R\_j \* \frac{\omega\_j}{\sum\_j \omega\_j} \tag{19}$$

where *Ri*0, *Ri*, *PPPi*, *PSCi* are retained cost, total cost, PPP value and PSC value of the *i*-th historical case that need to be revised; *Ri*<sup>0</sup>*<sup>r</sup>*, *Rir* = revised value of retained cost and total cost; *Rj*0, *Rj*, *PPPj*, *PSCj* are retained cost, total cost, PPP value, and PSC value of the *j*-th historical case that stand still, while *<sup>ω</sup>j* = weight of the *j*-th historical case. After the revisions for the selected historical cases are completed, the risk costs with higher degrees of acceptance for the target case can be measured by the following formulas

$$R\_0 = \sum R\_{i0r} Sim(V\_{N\prime}, V\_i) + \sum R\_{j0} Sim(V\_{N\prime}, V\_j) \quad i+j \ge 3 \tag{20}$$

$$R = \sum R\_{ir} \text{Sim}(V\_{N\prime}, V\_i) + \sum R\_j \text{Sim}(V\_{N\prime}, V\_j) \quad i + j \ge 3 \tag{21}$$
