1.1. Research Background

China has consistently emphasized scientific and technological innovation as a fundamental pillar of its national development strategy. The country has set various strategic objectives and launched policy initiatives to foster scientific and technical advancement. Throughout the 14th Five-Year Plan, China is focused on enhancing its military capabilities, achieving breakthroughs in key technologies, and advancing the integration of equipment construction toward mechanized, informatized, and intelligent systems. This initiative aims to respond to the emerging global revolutions in science, technology, and military affairs. A particular emphasis is placed on the research and development of strategic and disruptive technologies and equipment [1, 2]. However, the structure of equipment research projects is inherently complex, characterized by challenging technological demands, substantial financial investments, lengthy development cycles, and numerous sources of uncertainty throughout the development process. Influenced by both internal and external factors, these elements can evolve into significant cost risks, which are deep-seated causes for the chronic overestimation of scientific research projects and procurement budgets, thereby imposing a heavy burden on military spending [3]. In this context, conducting a risk assessment for the costs associated with equipment research projects becomes crucial.

1.2. Research Motivation

1.3. Research Innovations

1.4. Research Structure

To fulfill the objectives of this research, the paper is structured as follows. Section 2 reviews related work in the field of risk assessment and introduces the concept of cost-risk assessment for equipment research projects. Section 3 develops a risk-assessment index system to identify risk factors across three dimensions: technical, economic, and scheduling. Section 4 elaborates on the research methodology, detailing the TDCM based on the FAHP-CRITIC combination of weights and the algorithm for sensitivity analysis. Section 5 applies the methodology to a MCM weapon system, evaluating its cost-risk profile and comparing the effects of different weighting methods on the assessment outcomes. Section 6 presents the main conclusions of the study, discusses the limitations of the model, and suggests areas for future improvement.

2.1. Literature Review

Risk represents the likelihood of incurring losses within a specific environmental and temporal context. It is generally perceived as a measure of uncertainty concerning potential future losses [8]. While risk cannot be completely eradicated, it is typically managed through the scientific identification and assessment of its impacts [9]. Risk-assessment methodologies are categorized into two main types: qualitative and quantitative. Qualitative approaches include expert grading, risk matrices, and the Delphi technique, whereas quantitative methods encompass Monte Carlo simulation, system dynamics modeling, and the development of indices.

Scholars have refined risk-assessment methodologies to tackle practical challenges, with these enhanced methods finding widespread application across diverse sectors such as project management, financial investment, environmental protection, and social security. Agarwal and Kansal [10–13] advocate for the identification of risk factors at the planning phase of large-scale infrastructure projects and the subsequent execution of preliminary risk assessments using multiindicator decision-making approaches. Hu et al. [14] and Zhu et al. [15] recommend the extraction of key features related to cost and financial risks from extensive datasets, enhancing the overall efficacy of models through the application of deep learning techniques. Liu et al. [16] and Lu et al. [17] have refined disaster risk-assessment models to rectify the lack of precision often found in traditional quantitative risk assessments. Yang et al. [18] and Guo et al. [19] devised a risk indicator system by amalgamating multiple characteristic parameters, organizing the evaluation framework into hierarchically structured levels to improve the assessment of complex system risks.

However, research projects focusing on equipment are often confidential, with limited data accessible through public channels. Researchers in this domain typically possess military backgrounds, which narrows the scope of representative and contemporary studies available. In addition, there is a noticeable deficiency in research concerning risk assessment within this field. The literature reviewed—primarily through keyword searches—was outdated and predominantly concentrated on employing simulations to evaluate cost risks. Bielecki [20] utilized a multiple regression model to forecast the escalation of project R&D costs. Xu, Wu, and Jia [21] applied statistical analyses to the outputs of Monte Carlo simulations to estimate combined cost and schedule risks. Wei et al. [22] developed a weighted regression measurement model based on entropy theory to improve the accuracy of equipment R&D cost predictions. Xu and Zou [23] enhanced the Monte Carlo model by integrating risk-driven theory for a more precise estimation of aircraft development cost risks. He and Huang [24] introduced a risk prediction model for the expenses of large-scale scientific research projects using gray correlation analysis. To develop a more accurate cost-risk assessment model, it is crucial to consider both inherent means and the actual conditions during equipment development, providing a valuable basis for formulating reasonable budgets and funding strategies [25].

The risk-assessment methods discussed previously primarily focus on identifying and analyzing the risk factors of the target entity, constructing a robust and applicable risk-assessment index system and comprehensively calculating the quantitative values and weights of each risk factor to derive the final risk-assessment outcomes. The choice of weighting method is pivotal, directly influencing the generation of the final results. Commonly employed weighting methods include AHP [26], entropy weight method [27], grey relation analysis [28], and data envelopment analysis [29]. Nevertheless, these methods often exhibit discrepancies due to their subjective or objective weighting biases. To mitigate these discrepancies, scholars have proposed the integration of weights that consider both subjective and objective indicators, thereby rendering the weight determination process more scientific. The combined AHP weights are prevalently utilized, and Wang et al. [30–32] have enhanced these by incorporating fuzzy mathematical methods and other techniques to dynamically assess the indicator weights at each level. Furthermore, Wang et al. [33–35] have applied a combination of CRITIC methods to objectively and accurately delineate the interrelationships between indicators and their weight distribution.

Advancements in artificial intelligence have also facilitated the development of quantitative assessment tools such as neural network algorithms [36], Bayesian networks (BNs) [37], and cloud models [38, 39]. Specifically, cloud modeling—a probabilistic cognitive model of uncertainty—effectively translates qualitative concepts into quantitative data using the cloud generator algorithm. Chen et al. [40–43] have employed the cloud model for risk assessment, utilizing cloud parameter eigenvalues to depict the correlation between datasets. Wang et al. [44–46] have developed a two-dimensional comprehensive cloud diagram to represent the risk landscape from the perspective of underlying variable dimensions. The validity of this model was confirmed through the calculation of the degree of nearness.

2.2. Research Roadmap

3.1. Technical Level Risk (R₁)

Technical level risks are fundamentally predictive of the advancement required by the project’s technology, which is intrinsically linked to the equipment’s tactical and technical indices. As the tactical technical index increases, the associated challenges in achieving these indices also escalate. At this juncture, it becomes essential to evaluate the mix of old and new technologies, specifically the balance between technological innovation and inheritance. Should the existing technology fall short of meeting the target functionalities, it necessitates the development of new technologies, thereby incurring substantial research and development costs and person-hours dedicated to achieving crucial technological breakthroughs. Conversely, if the existing technology suffices for the target requirements, it is pertinent to assess the degree of alignment with the target technology and whether adjustments in materials or processes are required to integrate into the new project system. It is crucial to recognize the complexity of the technology involved, including whether a combination of multiple technological types is feasible within the system to fulfill the project’s needs. Ensuring technical coordination is vital to preclude cost escalations [49]. Moreover, the technological level of a project may be affected by factors such as the qualifications and credibility of the personnel within the research unit, the working environment, and the organizational structure. These elements can influence the total project cost, inclusive of considerations for equipment wear and tear. Table 1 delineates the criteria for a detailed assessment of technical level risk as elaborated above.

3.2. Economic Dimension Risk (R₂)

Economic dimension risks primarily arise from changes in procurement costs, which are influenced by external factors such as macroeconomic conditions, market sectors, and microeconomic elements. Inflation rates, price increases, and currency devaluations within the national economy directly affect project development and procurement costs. Moreover, shifts in international relations and the restructuring of global dynamics can lead to fluctuations in international exchange rates, which are crucial factors affecting the procurement costs of components that cannot yet be localized. State interventions in project developments, such as tax reductions or exemptions for research units, enhancements in the localization of support equipment, and incentives for outsourcing units to improve service levels, can indirectly influence transaction costs. In addition, the availability of capable contractors within the industry impacts the procurement method. For instance, if there is only one capable contractor, the military may be compelled to engage in single-source procurement. This scenario allows the contractor to exploit cost information asymmetries, potentially skewing costs in their favor and deviating from the actual costs of the equipment research project. The balance of supply and demand within the market determines pricing. When military demand exceeds what the research unit can normally supply, the military finds itself disadvantaged in the dynamic interplay between supply and demand, often resulting in higher costs to meet the research unit’s requirements. The detailed risk assessment at the economic level, based on the analysis above, is outlined in Table 2.

3.3. Progress Level Risk (R₃)

Progress level risk pertains to the uncertainty that the project may not advance as planned, according to either the overall project schedule or individual subproject timelines. Insufficient initial planning of one subsystem’s schedule can adversely affect the development of interconnected subsystems, leading to widespread project disruptions. Effective coordination between the schedules of these subsystems and the overarching project timeline is essential. Only through the realization of a well-configured schedule can an optimal solution be achieved. Delays or interruptions in the project can trigger a cascade of increased costs, including additional employee compensation, enhanced operation of production equipment, liquidated damages, and other consequential expenses. The detailed foundation for assessing progress level risk is illustrated in Table 3.

Cost risk is associated with the financial expenditures of equipment development projects. The intricacy of the equipment and the unpredictability of both internal and external conditions heighten the risk involved in developing equipment research projects, which in turn impacts the overall cost of delivery. To evaluate the likelihood of cost overruns in these projects, we have selected the probability of occurrence and the degree of impact as variables. A thorough assessment incorporating both dimensions provides a more comprehensive understanding of potential cost risks.

4.1. FAHP-CRITIC Combined Weight

4.1.1. FAHP Method

FAHP is an effective method that accounts for uncertainties in human judgment, often used in decision-making processes. This technique primarily utilizes two functions: the triangular membership function (TMF) and the trapezoidal membership function (TRMF). For this study, TMF was selected due to its widespread use and straightforward computational process. It translates qualitative data provided by experts into triangular fuzzy numbers (TFNs). To manage the inherent uncertainties of AHP, the FAHP approach extends the traditional AHP by employing fuzzified comparison ratios defined by TMF. The goal is to use TFNs to represent the weights of criteria on a nine-level judgment scale, thereby illustrating the relative importance of each criterion within the hierarchy [50]. Figure 3 illustrates a TFN, represented by real numbers, including parameters l, m, and u, where l < m < u. Here, l is the minimum value, m is the most likely value, and u is the maximum value related to the membership function $$. Figure 4 displays the linguistic variables used to determine the importance weight of criteria. The membership function of a TFN is represented by the following equation:

$$ (1)

where l and u are the respective lower and upper limits of the fuzzy number $$, with m denoting the midvalue of the fuzzy number.

The AHP approach was employed to determine the weights of criteria based on expert assessments. This method facilitates the evaluation of both qualitative and quantitative criteria through pairwise comparisons. To accommodate uncertainties, the standard AHP was enhanced by integrating the fuzzy set theory, resulting in FAHP. The FAHP method comprises the following essential steps:

•  

  Step 1: generation of fuzzy pairwise comparison matrices

•  

  Fuzzy pairwise comparison matrices were constructed at each hierarchical level using precise data collected from experts, who utilized a well-defined fuzzy linguistic scale, as depicted in Table 4. Subsequently, the extent analysis method was applied to compute priority weights using synthetic extent values. The assessment matrix for the fuzzy criteria was formulated by comparing various attributes linked to the overall objective. The correspondence between the TFNs and linguistic terms was established through the adaptation of Saaty’s scale to fuzzy contexts. When a TFN is denoted as $$, its reciprocal is represented as F⁻¹ = (1/u_i, 1/m_i, 1/l_i).

•  

  In the subsequent step, a fuzzy pairwise comparison matrix was constructed using linguistic terms. Experts applied the nine-point conversion scale to transcribe responses into fuzzy numbers, as illustrated in Table 4.

•  

  The resultant comparison matrix is formulated as follows:

  $$ (2)

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  The aggregated fuzzy pairwise comparison matrix, summarized in group decision-making contexts, is represented by the following equation.

  $$ (3)

•  

  Here, $$ and k denote the number of experts involved.

•  

  Step 2: computation of normalized weight values using the geometric mean (GM) method

•  

  The GM method was utilized to calculate normalized weights for distinct criteria and their relevant subcriteria. The GM method was chosen for its simplicity and effectiveness in determining the principal eigenvalue, thereby minimizing judgment inconsistencies. The procedure commenced with the calculation of the geometric mean for each row of the fuzzy comparison matrix, as outlined in the following equation.

  $$ (4)

•  

  Subsequently, the normalized weights for the j th row of the crisp comparison matrix were determined as detailed in the following equation.

  $$ (5)

•  

  Here, $$ represents the geometric mean of the fuzzy comparison values, and $$ symbolizes the criterion weights.

•  

  Step 3: consistency check

•  

  The consistency ratio (CR) with respect to each pairwise comparison matrix was utilized to control the outcomes of the AHP, requiring a value less than 0.1 for the matrix to be considered consistent. Consistency of a crisp comparison matrix ensures the consistency of the fuzzy pairwise comparison matrix. The consistency index (CI) and CR related to a comparison matrix are described as follows:

  $$ (6)

•  

  where λ_max is the largest eigenvalue, n is the matrix size, and the standard Random Index (RI) value is obtained using the matrix order.

•  

  Step 4: defuzzification of the fuzzy weights

•  

  The Center of Area (COA) method was utilized to defuzzify the weights, which were expressed as fuzzy numbers. This process is encapsulated in the following equation.

  $$ (7)

•  

  Step 5: normalization of the crisp weights

•  

  The crisp criteria weights were determined by normalizing the obtained crisp values. Finally, the subjective weight $$ of the i th evaluation indicator is calculated as follows.

  $$ (8)

Table 4. Fuzzy conversion scale.

Crisp scale	Linguistic terms	TFS scale	Reciprocal TFN scale
1	Equally preferred	(1, 1, 1)	(1/1, 1/1, 1/1)
2	Equally to moderately preferred	(1, 2, 3)	(1/3, 1/2, 1/1)
3	Moderately preferred	(2, 3, 4)	(1/4, 1/3, 1/2)
4	Moderately to strongly preferred	(3, 4, 5)	(1/5, 1/4, 1/3)
5	Strongly preferred	(4, 5, 6)	(1/6, 1/5, 1/4)
6	Strongly to very strongly preferred	(5, 6, 7)	(1/7, 1/6, 1/5)
7	Very strongly preferred	(6, 7, 8)	(1/8, 1/7, 1/6)
8	Very strongly to extremely preferred	(7, 8, 9)	(1/9, 1/8, 1/7)
9	Extremely preferred	(8, 9, 9)	(1/9, 1/9, 1/8)

4.2. TDCM Based on Combinatorial Weights

4.3. Sensitivity Analysis of BNs

5.1. Project Overview

Naval mines represent a prevalent method of covert maritime warfare, posing significant threats to both naval and civilian maritime navigation. The development of a new mine countermeasures (MCMs) weapon system is crucial for accurately identifying, tracking, and neutralizing naval mines, thereby enhancing the naval combat capabilities. This advancement is instrumental in protecting the safety of coastlines and port facilities, and by extension, ensuring the prosperity and stability of sea transportation channels. The development of an MCM weapon system is a multifaceted scientific endeavor that spans multiple technical domains and necessitates substantial resource investment. It is fraught with uncertainties and risks. For example, the creation of a specific MCM weapon system comprises six subsystems, each further divisible into three to six smaller subsystems, as illustrated in Figure 7. This complexity underscores the practical significance and exemplarity of conducting cost-risk assessments for such systems, providing valuable insights for other similar equipment development projects.

5.2. Project Data Collection

In accordance with the aforementioned system of assessment indicators, data were obtained through questionnaires. These questionnaires were anonymous and targeted professionals who were either deeply involved in or highly familiar with the development of MCM weapon systems. Distribution occurred via the Internet and mail, reaching experts and scholars specializing in risk assessment of equipment costs, scientific researchers involved in the development of MCM weapon systems, and operators of these systems. Participants were invited to evaluate the weighting factors, the probability of over-expenditure, and the impact of overexpenditure on each of the 12 secondary indicators. The questionnaire was designed in line with the predefined assessment indicators and included explanatory notes on the relevant indicators to guide the respondents. Scoring was required on a scale from 1 to 10, with 1 representing the least favorable outcome and 10 the most favorable. A total of 56 questionnaires were distributed, of which 47 were returned. Of these returns, 45 were deemed valid for analysis.

5.3. Calculating Index Weight

Using the FAHP-CRITIC combined weighting method, the collected weight coefficient data were input into equations (1)–(19). The resultant weights for all the indicators are displayed in Tables 6 and 7, with the distribution of weights for the secondary indicators depicted in Figure 8.

5.4. Computational Integrated Risk Cloud Digital Characterization

Data concerning the probability and impact of overexpenditure are incorporated into equation (21) to compute the cloud eigenvalues for the primary variables. Table 8 presents both the primary and secondary cloud eigenvalues, calculated by integrating previous results with their respective weights using equation (22). Subsequently, the numerical eigenvalues of the composite risk for the cloud were determined, revealing eigenvalues for the probability of occurrence of overruns for R as (6.855, 0.687, and 0.170) and for the degree of impact of overruns for R as (6.794, 0.676, and 0.162).

5.5. Comprehensive Risk Cloud Mapping

The MATLAB forward cloud generator, utilizing the standard cloud eigenvalues from Table 5 and the comprehensive cloud risk eigenvalues from Table 8, produces comparison diagrams of the common cloud and the total risk cloud, as depicted in Figure 9. These diagrams show that a specific type of MCM weapon system has an integrated risk cloud positioned between the III and IV standard risk clouds but closer to IV. This overlap is particularly evident from the top view, indicating that the cost risk for this MCM weapon system is at a Level IV, which implies a higher risk level.

The primary indicator cloud eigenvalues were entered into the MATLAB forward cloud generator to enhance the visualization of specific risks’ impact on the composite risk at the indicator level. This facilitated the creation of comparative maps for each primary indicator cloud, detailed in Figures 10, 11, and 12.

The figures indicate that the technical risk level is situated between Levels IV and V, with IV being more favorable. The economic risk level is positioned between Levels III and IV, while the schedule risk level also falls between Levels III and IV, leaning toward IV. Thus, the highest risk is posed at the technical level, followed by the schedule level, with the economic level presenting the lowest risk.

5.6. Calculating Nearness Degree

Equation (26) is employed to calculate the integrated risk cloud nearness degree and to determine the cost-risk level of a specific MCM weapon system. The nearness degrees of the comprehensive risk cloud to the five standard clouds are expressed as follows: D₁ = 0.1037; D₂ = 0.1896; D₃ = 0.3884; D₄ = 7.4536; and D₅ = 0.2224. These results rank the nearness degree as D₄ > D₅ > D₃ > D₂ > D₁, with the highest nearness degree to the IV standard cloud. It has been determined that the cost risk of developing this specific type of MCM weapon system is moderately high, aligning with the risk level indicated by the project’s development institute. Prioritizing cost-risk management during project development is essential, which may involve reducing the development proportion or pausing the project until further prestudy results are available.

5.7. Sensitivity Analysis

The BN topology was established based on the data from Table 2, with node parameters assigned according to the corresponding indicator’s risk level. Sensitivity analysis was conducted utilizing equations (27) and (28). The structure of the BN for a specific MCM weapon system was illustrated using GeNIe software, as shown in Figure 13.

Within this BN model, the “equipment research project cost risk” node was designated as the evidence node, with State 1 set at 100%. Inverse deduction was employed to identify the principal influencing nodes, as depicted in Figure 14. This figure demonstrates that the specific MCM weapon system exhibits a higher risk of increased costs associated with technical complexity (R₁₄), technical inheritance (R₁₃), technological advancement (R₁₁), international situation (R₂₂), and overall project progress (R₃₂), with the probability of these risks exceeding 69%.

Further analysis of the sensitivity results, shown in Figure 15, indicates that the nodes representing technical complexity (R₁₄), technological innovation (R₁₂), technical inheritance (R₁₃), technological advancement (R₁₁), and subsystem progress (R₃₁) exhibit heightened sensitivity to the target node when development cost risks are considered for the same MCM weapon system.

The combination of inverse deduction and sensitivity calculations highlights that technical complexity (R₁₄) exerts the most significant influence on the development costs of the specified MCM weapon system, followed by technical inheritance (R₁₃) and technological advancement (R₁₁). The project development team should prioritize these three areas and implement targeted control measures. It is crucial to leverage the experience gained from existing MCM weapon systems in technical development. Technological innovation should be pursued by fully integrating core technologies, attracting research talents, offering scientific and technological incentives, and fostering technological collaborations. Vigilant management of the project development plan is necessary to avoid undue additional costs due to delays related to technical research or other factors, thereby keeping potential cost risks within a reasonable range.

5.8. Discussion

The FAHP and CRITIC methods are individually utilized to compute the weights of various indicators. By aggregating the cloud eigenvalues of each indicator, integrated cloud eigenvalues R^′ and R^″ corresponding to these methods are derived. These values are subsequently compared with the integrated cloud eigenvalues R, which are calculated using the combined FAHP-CRITIC weighting method. The expected values of the integrated cloud eigenvalues are determined by inserting them into equation (26) to ascertain the degree of nearness, thereby affirming the efficacy and applicability of the proposed methodology. Due to space constraints, only the cloud feature expectation value is provided as an illustration; detailed results are presented in Tables 9 and 10.

The data in Tables 9 and 10 indicate that the rankings of nearness degree, as obtained by the three distinct methods—FAHP, CRITIC, and the integrated FAHP-CRITIC—are consistent. This consistency underscores the validity and reliability of the FAHP-CRITIC combined weighting approach. Moreover, the integrated FAHP-CRITIC method amalgamates subjective weights derived from the FAHP method with objective weights calculated via the CRITIC method. The resultant cloud eigenvalues and degrees of nearness are more stable compared with those derived from either the FAHP or CRITIC methods alone, further corroborating the practical utility of this approach in the context of complex equipment research projects.

Despite its advantages, the FAHP-CRITIC combined weighting approach exhibited several limitations during the case study: (1) the overall computational process is intricate, necessitating extensive mathematical operations and logical reasoning. (2) The requirements for data modeling and analysis are stringent; incomplete datasets can lead to erroneous assessment outcomes. (3) The absence of standardized cloud numerical feature divisions can result in significant discrepancies in the assessment outcomes for the same issue under different standards.

In response to these challenges, future research should focus on the following areas: (1) simplifying the calculation process of the TDCM by integrating machine learning and artificial intelligence technologies to automate feature extraction and risk assessment, thereby reducing subjectivity and human error and enhancing the efficiency of assessments. (2) Developing methodologies to manage incomplete data, incorporating real-time data for dynamic analysis to ensure that the most current risk information is promptly acquired for ongoing adjustments and updates. (3) Reducing reliance on historical data and promoting the practical implementation of the TDCM to accumulate research experience, establish international consensus through cooperation, and develop a unified standard for cloud numerical feature division to minimize discrepancies and ensure consistency in assessment results.