# Enhancing System Safety and Reliability through Integrated FMEA and Game Theory: A Multi-Factor Approach

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## Abstract

**:**

## 1. Introduction

## 2. Comprehensive Review of Game Theory in the Literature

#### 2.1. The Concept of Game Theory

#### 2.2. Introducing the Different Classes of Strategies

#### 2.3. Nash Equilibrium

#### 2.4. Game Classification

#### 2.4.1. Static/Dynamic Games

#### 2.4.2. Zero-Sum Games

#### 2.4.3. Cooperative/Non-Cooperative Games

## 3. Proposed Methodology

- Pre-step: The process of collecting all necessary information about the system under investigation is a meticulous endeavor that forms the bedrock of our research. It involves a comprehensive review and assimilation of data, encompassing structural details, operational dynamics, historical performance metrics, and contextual factors that influence the system. This critical step requires a multi-faceted approach, engaging with various stakeholders for insights, examining relevant documentation, and utilizing analytical tools to capture the complexities of the system. Such thorough data collection ensures a robust foundation upon which meaningful analysis can be performed, hypotheses can be tested, and accurate conclusions can be drawn, ultimately contributing to the credibility and reliability of the research outcomes.
- Step 1: Assessing the Weight of Risk Factors: In this foundational stage, the focus is on quantifying the relevance of different risk factors. Utilizing the BWM methodology [102], we gauge the importance weights tied to the severity, occurrence, and detection of said risk factors.
- Step 2: Formulating the Payoff Evaluation Matrix: After ascertaining the importance weights, the next step is the formulation of the payoff evaluation matrix. This matrix is crafted based on insights and evaluations from an expert panel of decision-makers. To address the inherent uncertainties, this phase incorporates Pythagorean fuzzy uncertain linguistic variables.
- Step 3: Determining Risk Priorities of Failure Modes: The zenith of our framework is to present a detailed assessment of risk priorities associated with distinct failure modes. In this context, a zero-sum game methodology is leveraged to pinpoint the optimal strategies for both scenarios involving failure modes and those without.

- Severity (S): This refers to the potential impact or consequences of a failure mode on the system’s functionality, the environment, or the end-users. It measures the extent of harm or disruption that could result from the failure.
- Occurrence (O): This dimension assesses the likelihood or frequency of a particular failure mode occurring. It estimates the probability that the risk will materialize based on historical data, predictive models, or expert judgment.
- Detectability (D): Detectability evaluates how easily a failure mode can be discovered before it leads to an operational failure. This involves assessing the effectiveness of current detection processes or control measures.

#### 3.1. Computing the Importance Weight of Risk Factors Utilizing BWM

- (i)
- Identifying most of the minor significant risk factors. The most critical risk factor, ${RF}_{B}$, and the least important risk factor, ${RF}_{W}$, have to be determined by decision-makers’ opinions from the known $n$ risk factor. A fine approach is employed in the methodology proposed to discern the spectrum of risk factors within a system, extending from the most to the least significant. The criticality of these factors is determined through a qualitative analysis led by decision-makers who are well-versed in the intricacies of the system at hand. Recognizing the most critical risk factor denoted as ${RF}_{B}$, and the least significant one, ${RF}_{W}$, is pivotal in establishing a hierarchy of risks that guides the focus of risk management efforts.Considering less significant risk factors in the analysis is both strategic and practical. While these factors may have a lower impact on the system, their cumulative effect or impact under specific conditions can be non-trivial. By including these minor factors, decision-makers can ensure a comprehensive risk assessment, leaving no potential vulnerability unaddressed. This inclusion aligns with the principles of thoroughness and precaution in risk management, especially in complex systems where seemingly minor risks can propagate or interact with other factors to cause significant issues.To clarify the process, decision-makers typically leverage tools such as FMEA to evaluate and rank the criticality of risk factors. However, the initial identification of these factors often relies on their expertise and experiential judgment. The FMEA tool then provides a structured framework to analyze the identified factors, quantifying their severity, occurrence, and detectability to arrive at an RPN. This number assists in objectively determining the criticality of each risk factor.In practice, achieving consensus among decision-makers on the significance of risk factors can be challenging, mainly when relying on qualitative assessments. To mitigate this, our model incorporates mechanisms for reconciling differing opinions, such as employing a Delphi method or consensus-building workshops. These methods facilitate structured communication and negotiation, allowing for the emergence of a collective judgment on the risk factors’ criticality. In instances where consensus is elusive, the model adapts by assigning possible values to each risk factor, reflecting the spectrum of expert opinions. This range is then utilized in sensitivity analyses to determine how variations in risk criticality assessment could influence the system’s overall risk profile. Such an approach ensures that the model remains robust and applicable despite subjective variability, thus maintaining its utility and relevance in real-world risk management scenarios.
- (ii)
- Assessing the priority of the most critical risk factor relative to others. Next, the group of decision-makers collaboratively express their judgments concerning the significance of the primary risk factor compared to the remaining risk factors, utilizing the established nine-scale table in the existing literature. Additionally, we calculate the vector representing the best-to-others (BO) preference, which is defined as $RF,k=1,2,3,\dots ,l$, as follows:$${RF}_{BO}^{k}=({RF}_{B1}^{k},{RF}_{B2}^{k},\dots ,{RF}_{Bn}^{k})$$$${RF}_{Bj}=\frac{{RF}_{Bj}^{k}}{l},j=1,2,\dots ,n$$
- (iii)
- Computing the preference of the other risk factor over the most critical risk factor.Similarly, $l$ others-to-worst vector ${RF}_{OW},\mathrm{f}\mathrm{o}\mathrm{r}k=1,2,3,\dots ,l$ is computed by comparing to the other risk factor over the worst risk factor using nine-scale, as in the following equation:$${RF}_{OW}^{k}=({RF}_{1W}^{k},{RF}_{2W}^{k},\dots ,{RF}_{nW}^{k})$$$${RF}_{jW}=\frac{{RF}_{jW}^{k}}{l},j=1,2,\dots ,n$$
- (iv)
- Calculate the optimum risk factors’ importance weights.In BWM, the ratio of $\frac{{W}_{B}}{{W}_{j}}$ and $\frac{{W}_{j}}{{W}_{W}}$ is followed by $\frac{{W}_{B}}{{W}_{j}}={RF}_{Bj}$ and $\frac{{W}_{j}}{{W}_{W}}={RF}_{jW}$. For satisfying the above-mentioned conditions, a resolution must be determined by maximizing the value of $\left|\frac{{W}_{B}}{{W}_{j}}-{RF}_{Bj}\right|$ and minimizing the value of $\left|{RF}_{jW}-\frac{{W}_{j}}{{W}_{W}}\right|$.Therefore, the subsequent mathematical programming model determines the optimum risk factors’ weight:Model 1:$\mathrm{min}\mathrm{max}\left\{\left|\frac{{W}_{B}}{{W}_{j}}-{RF}_{Bj}\right|,\left|{RF}_{jW}-\frac{{W}_{j}}{{W}_{W}}\right|\right\}$,Subject to.${\sum}_{j=1}^{n}{w}_{j}=1$,${w}_{j}\ge 0$ $,j=1,2,\dots n.$Model 1 can be re-established into Model 2 as a linearization process:$\mathrm{min}\xi $Subject to.$\left|\frac{{W}_{B}}{{W}_{j}}-{RF}_{Bj}\right|\le \xi $,$\left|{RF}_{jW}-\frac{{W}_{j}}{{W}_{W}}\right|\le \xi $,${\sum}_{j=1}^{n}{w}_{j}=1$,${w}_{j}\ge 0,j=1,2,\dots n.$The optimum risk factors’ importance weights are computed by solving Model 2 and are signified as ${w}^{*}=\left({w}_{1}^{*},{w}_{2}^{*},\dots ,{w}_{n}^{*}\right)$.It is worth noting that in the final step, it is also possible to determine the aggregated optimal importance weights. This implies that the optimal importance weights for each risk factor are initially derived from individual decision-makers’ perspectives. Subsequently, factoring in the significance of each decision-maker’s input, we arrive at the aggregated importance weight for the risk factors.
- (v)
- Calculate the consistency ratio of resultsTo calculate the consistency value, first, it is essential to obtain the consistency ratio as follows:$$CR=\frac{{\xi}^{*}}{CI}$$

#### 3.2. Constructing the Group of Payoff Evaluation Matrix Utilizing Pythagorean Fuzzy Uncertain Linguistic Variables

**Definition**

**1**

**.**Let us consider that there is a discourse universe, as follows:

**Definition**

**2**

**.**Let us take $\mathcal{X}$ as a discourse universe where $\left[{\beta}_{\theta \left(\mathcal{x}\right)},{\beta}_{\tau \left(\mathcal{x}\right)}\right]$ indicates an uncertain linguistic variable. A Pythagorean fuzzy uncertain linguistic variable $\stackrel{~}{\mathbb{P}}$ in $\mathcal{X}$ can be defined as follows:

**Definition**

**3**

**.**Let us take ${\stackrel{~}{\mathbb{P}}}_{1}=\u2329\left[{\beta}_{{\theta}_{1}},{\beta}_{{\tau}_{1}}\right],\left({\mathcal{u}}_{{\stackrel{~}{\mathbb{P}}}_{1}},{\mathcal{v}}_{{\stackrel{~}{\mathbb{P}}}_{1}}\right)\u232a$ and ${\stackrel{~}{\mathbb{P}}}_{2}=\u2329\left[{\beta}_{{\theta}_{2}},{\beta}_{{\tau}_{2}}\right],\left({\mathcal{u}}_{{\stackrel{~}{\mathbb{P}}}_{2}},{\mathcal{v}}_{{\stackrel{~}{\mathbb{P}}}_{2}}\right)\u232a$ as two different Pythagorean fuzzy uncertain linguistic variables. In such case, some important operational laws of Pythagorean fuzzy uncertain linguistic variables can be defined as follows:

**Definition**

**4**

**.**Let us take $\beta =\left\{{\beta}_{0},{\beta}_{1},\dots ,{\beta}_{g}\right\}$ as linguistic set terms and $\stackrel{~}{\mathbb{P}}=\u2329\left[{\beta}_{\theta},{\beta}_{\tau}\right],\left(\mathcal{u},\mathcal{v}\right)\u232a$ as Pythagorean fuzzy uncertain linguistic variables. Therefore, the score function of $\stackrel{~}{\mathbb{P}}$ can be determined as follows:

**Definition**

**5**

**.**Let us take ${\stackrel{~}{\mathbb{P}}}_{1}=\u2329\left[{\beta}_{{\theta}_{1}},{\beta}_{{\tau}_{1}}\right],\left({\mathcal{u}}_{{\stackrel{~}{\mathbb{P}}}_{1}},{\mathcal{v}}_{{\stackrel{~}{\mathbb{P}}}_{1}}\right)\u232a$ and ${\stackrel{~}{\mathbb{P}}}_{2}=\u2329\left[{\beta}_{{\theta}_{2}},{\beta}_{{\tau}_{2}}\right],\left({\mathcal{u}}_{{\stackrel{~}{\mathbb{P}}}_{2}},{\mathcal{v}}_{{\stackrel{~}{\mathbb{P}}}_{2}}\right)\u232a$ as two different Pythagorean fuzzy uncertain linguistic variables. In such a case, the “Hamming distance” between ${\stackrel{~}{\mathbb{P}}}_{1}$ and ${\stackrel{~}{\mathbb{P}}}_{2}$ can be determined as follows:

**Definition**

**6**

**.**Let us take $\stackrel{~}{\mathbb{P}}$ as the collection of Pythagorean fuzzy uncertain linguistic variables; ${\stackrel{~}{\mathbb{P}}}_{j}=\u2329\left[{\beta}_{{\theta}_{j}},{\beta}_{{\tau}_{j}}\right],\left({\mathcal{u}}_{{\stackrel{~}{\mathbb{P}}}_{j}},{\mathcal{v}}_{{\stackrel{~}{\mathbb{P}}}_{j}}\right)\u232a$, where $j=1,2,\dots ,n$, and the “Pythagorean fuzzy uncertain linguistic prioritized weighted averaging operator” is ${\stackrel{~}{\mathbb{P}}}^{\mathit{n}}\u27f6\stackrel{~}{\mathbb{P}}$. In such as case, the “Pythagorean fuzzy uncertain linguistic prioritized weighted averaging operator” can be defined as follows:

#### 3.3. Determining the Risk Priority of Failure Modes Utilizing the Zero-Sum Game

**Definition**

**7**

**Definition**

**8.**

## 4. Application of Study

- Elevating the cleaning routines, with a special focus on surfaces and tools that undergo frequent handling.
- Discouraging communal usage of equipment and supplies, thereby diminishing potential sources of contamination.
- Designing a dynamic communication blueprint that adjusts to different risk thresholds, ensuring every employee is adequately informed and aligned with the latest safety protocols.
- Curating a dedicated mental health support system, addressing the unique stresses and anxieties that may arise during such challenging times.
- Amplifying environmental sanitation measures, emphasizing the disinfection of objects and surfaces that are in regular use.
- Introducing protective installations, like clear Plexiglas barriers, at interaction points to reduce direct contact and safeguard both employees and visitors.
- Optimally selecting and distributing personal protective equipment (PPE) after a meticulous risk evaluation, ensuring it is utilized effectively and safely.
- Holding regular training workshops to impart knowledge about the correct methodologies for wearing and removing PPE without risking contamination.
- Enhancing on-site surveillance and audit mechanisms to ascertain strict adherence to all safety guidelines.
- Incorporating systematic temperature screenings and health evaluations at facility entrances, serving as preliminary checkpoints.
- Promoting the use of touchless technologies where possible, such as automatic doors and touch-free payment systems.
- Regularly updating and reviewing emergency response plans to address potential outbreak scenarios.
- Encouraging telecommuting and remote work options to reduce the density of people in a confined space.
- Facilitating virtual meetings and conferences as alternatives to in-person gatherings.
- Providing well-ventilated spaces and considering upgrading air filtration systems to capture potential viral particles.
- Educating and encouraging employees to stay home if they feel unwell or exhibit any symptoms.

## 5. Methodology Validations

- Assessment 1: To ensure the dependability of a decision-making tool, it is imperative that the agency consistently upholds the superiority of the best alternative. This means that the tool should never replace the top-ranked alternative with one that is ranked lower, unless this substitution is made while considering the relative importance of each criterion’s variation. In other words, the tool should prioritize the best option unless there is a compelling reason, based on the specific criteria and their importance, to choose an alternative that is not the highest-ranked overall.
- Assessment 2: Reliability in a decision-making tool necessitates adherence to the transitivity property. This property ensures that the tool maintains logical consistency in its decision-making process. If alternative A is preferred over alternative B, and alternative B is preferred over alternative C, then the tool should logically conclude that alternative A is preferred over alternative C. This consistency in decision outcomes is a fundamental characteristic of a reliable tool.
- Assessment 3: In a dependable decision-making tool, when a complex decision problem is dissected into smaller components using the same tool for alternative prioritization, the combined prioritization of alternatives at the component level must align with the original prioritization of the undivided decision problem. This means that breaking down the decision into smaller parts should avoid inconsistencies or contradictions in the overall decision. In our particular approach, which involves risk assessments for failure modes, it is important to note that these assessments are interdependent. Therefore, assessment three should be exclusively conducted using our introduced approach for evaluating risk factors to maintain the integrity and consistency of the decision-making process.

#### 5.1. Validity Examination of the Proposed Approach Using Assessment 1

#### 5.2. Validity Examination of the Proposed Approach Using Assessments 2 and 3

- {$F$7, $F$23, $F$14, $F$11, $F$16, and $F$17}
- {$F$1, $F$12, $F$15, $F$4, and $F$3}
- {$F$9, $F$22, $F$21, $F$10, and $F$5}
- {$F$5, $F$13, $F$19, $F$20, $F$22, and $F$8}

## 6. Sensitivity Analysis

- Nodes: There are 23 nodes, labeled $F$1 to $F$23. These likely represent specific factors or features. The central positioning of some nodes (like $F$1, $F$2, and $F$3) might suggest their importance or centrality in the network.
- Connection Types:
- ○
- Severity-based (orange): These connections are the most prominent in the figure. Notably, $F$2 appears to have the most Severity-based connections.
- ○
- Occurrence-based (gray): These are less prevalent than the Severity-based connections but are still significant. $F$4 and $F$5, for instance, seem to have multiple Occurrence-based connections.
- ○
- Linguistic-based (red): These connections are the least common. They primarily involve nodes like $F$6, $F$7, $F$8, and $F$.

- Clusters and Sub-networks: The nodes and their connections can be divided into distinct clusters or sub-networks. For example, $F$6 to $F$9 forms a cluster primarily connected by Linguistic-based relations. Similarly, nodes $F$10 to $F$15 are interconnected, primarily with Severity-based connections.Analysis:
- Central Nodes: $F$1, $F$2, and $F$3 appear to be central nodes given their location and the number of connections. This might indicate their importance in this network or their role as primary or overarching factors.
- Diversity of Relations: The multiplicity of connection types suggests that the network is examining the relationships between nodes from different perspectives or criteria. The preponderance of Severity-based connections might indicate that the severity of relations or factors plays a dominant role in this context.
- Peripheral Nodes: Nodes like $F$16 to $F$23 are on the periphery, with fewer connections. This could mean they are secondary or less influential factors in this network.
- Potential Hierarchies: The central nodes’ connections to the peripheral nodes might suggest a flow of influence or a hierarchical structure. For example, $F$2’s connections might indicate its influence over multiple other factors.
- Linguistic Relations: The presence of Linguistic-based connections, especially around $F$6 to $F$9, might imply a subset of factors that are related based on language, semantics, or terminologies.

## 7. Conclusions

- Assumption of Rationality: The model assumes that all decision-makers behave rationally and that their judgments are consistent. This may only sometimes hold in real-world scenarios due to cognitive biases and emotional factors.
- Complexity and Comprehensibility: The integration of Game Theory and Pythagorean fuzzy logic increases the complexity of the FMEA process, which may require additional training for stakeholders to utilize the model effectively.
- Data Dependence: The model’s effectiveness is highly dependent on the accuracy and completeness of the input data. Any gaps or inaccuracies in the initial data can significantly affect the reliability of the risk assessment outcomes.
- Static Nature of Analysis: While the model excels in capturing a snapshot of risk factors and their interactions, it may need to fully account for the dynamic nature of healthcare systems where risks can evolve rapidly.
- Scope of Application: The current implementation of the model is tailored to healthcare systems and may require modifications to be effective in other industries or contexts.
- Consensus Building: The model presumes a consensus among decision-makers when determining the weights of risk factors, which can be challenging to achieve in practice.
- Resource Limitations: The application of this advanced FMEA framework demands certain computational resources and expertise, which might only be readily available in some healthcare settings.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The Pythagorean fuzzy uncertain linguistic variables payoff matrix by four decision-makers.

Failure Modes Tag | Decision-Makers | Severity | Occurrence | Detection |
---|---|---|---|---|

$F$1 | $DM1$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}\right],(0.6,0.4)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{C}}\right],\left(\mathrm{0.7,0.3}\right)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{F}}\right],(0.85,0.15)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{C}}],(0.25,0.75)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{C}}],(0.25,0.75)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | |

$F$2 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$DM4$ | $\u2329{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | $\u2329{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$F$3 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | |

$F$4 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{E}}],(0.45,0.55)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | |

$F$5 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{G}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{A}}],(0.15,0.85)\u232a$ | |

$F$6 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | |

$F$7 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | |

$F$8 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{D}}],(0.35,0.65)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{D}}],(0.35,0.65)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | |

$F$9 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{F}}],(0.7,0.3)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{E}}],(0.55,0.45)\u232a$ | |

$F$10 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{E}}],(0.55,0.45)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$F$11 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{E}}],(0.55,0.45)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$12 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$13 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8))\u232a$ | $\u2329[{\mathsf{\phi}}_{G},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$14 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{E}}],(0.55,0.45)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$F$15 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$16 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{B}}],(0.65,0.35)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$F$17 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{E}}],(0.65,0.35)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}],(0.85,0.15)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$18 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}\right],(0.8,0.2)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{F}},{\mathsf{\phi}}_{\mathrm{G}}],(0.8,0.2)\u232a$ | |

$F$19 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ |

$DM2$ | $\u2329{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$20 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM3$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{E},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}],(0.6,0.4)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{F}}],(0.65,0.35)\u232a$ | |

$F$21 | $DM1$ | $[{\mathsf{\phi}}_{\mathrm{A}}$$,{\mathsf{\phi}}_{\mathrm{C}}$], (0.25, 0.75) | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $[{\mathsf{\phi}}_{\mathrm{A}}$$,{\mathsf{\phi}}_{\mathrm{C}}$], (0.25, 0.75) | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | |

$DM3$ | $\u2329\left[{\mathsf{\phi}}_{3},{\mathsf{\phi}}_{4}\right],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{D}}\right],\left(\mathrm{0.7,0.3}\right)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}\right],(0.85,0.15)\u232a$ | |

$F$22 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{D}}\right],\left(\mathrm{0.7,0.3}\right)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}\right],(0.85,0.15)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{B}},{\mathsf{\phi}}_{\mathrm{C}}],(0.3,0.7)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM3$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}\right],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $[{\mathsf{\phi}}_{\mathrm{A}}$$,{\mathsf{\phi}}_{\mathrm{C}}$], (0.25, 0.75) | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ | |

$F$23 | $DM1$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{C}}],(0.25,0.75)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{G}},{\mathsf{\phi}}_{\mathrm{G}}\right],(0.85,0.15)\u232a$ |

$DM2$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | |

$DM3$ | $\u2329\left[{\mathsf{\phi}}_{\mathrm{D}},{\mathsf{\phi}}_{\mathrm{E}}\right],(0.6,0.4)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $[{\mathsf{\phi}}_{\mathrm{A}}$$,{\mathsf{\phi}}_{\mathrm{C}}$], (0.25, 0.75) | |

$DM4$ | $\u2329[{\mathsf{\phi}}_{\mathrm{C}},{\mathsf{\phi}}_{\mathrm{D}}],(0.4,0.6)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{A}},{\mathsf{\phi}}_{\mathrm{B}}],(0.2,0.8)\u232a$ | $\u2329[{\mathsf{\phi}}_{\mathrm{E}},{\mathsf{\phi}}_{\mathrm{G}}],(0.75,0.25)\u232a$ |

**Table A2.**Aggregated normalized single-weighted Pythagorean fuzzy uncertain linguistic variables payoff matrix.

Failure Modes Tag | Severity | Occurrence | Detection |
---|---|---|---|

$F$1 | $\u2329\left[{\mathsf{\phi}}_{0.1103},{\mathsf{\phi}}_{0.1362}\right],(0.2213,0.2224)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0343},{\mathsf{\phi}}_{0.0289}\right],(0.1556,0.6121)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0190},{\mathsf{\phi}}_{0.0180}\right],(0.1217,0.6397)\u232a$ |

$F$2 | $\u2329\left[{\mathsf{\phi}}_{0.1348},{\mathsf{\phi}}_{0.1114}\right],(0.3088,0.0568)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0317},{\mathsf{\phi}}_{0.0276}\right],(0.1493,0.4959)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0091},{\mathsf{\phi}}_{0.0212}\right],(0.0371,0.8538)\u232a$ |

$F$3 | $\u2329\left[{\mathsf{\phi}}_{0.1399},{\mathsf{\phi}}_{0.1076}\right],(0.2984,0.0860)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0279},{\mathsf{\phi}}_{0.0327}\right],(0.1190,0.6178)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0156},{\mathsf{\phi}}_{0.0213}\right],(0.0887,0.7127)\u232a$ |

$F$4 | $\u2329\left[{\mathsf{\phi}}_{0.1399},{\mathsf{\phi}}_{0.1076}\right],(0.2963,0.0710)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0279},{\mathsf{\phi}}_{0.0327}\right],(0.0940,0.6951)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0156},{\mathsf{\phi}}_{0.0213}\right],(0.0973,0.7703)\u232a$ |

$F$5 | $\u2329\left[{\mathsf{\phi}}_{0.1374},{\mathsf{\phi}}_{0.1059}\right],(0.3141,0.0435)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0253},{\mathsf{\phi}}_{0.0326}\right],(0.1549,0.5120)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0180},{\mathsf{\phi}}_{0.0176}\right],(0.0909,0.7923)\u232a$ |

$F$6 | $\u2329\left[{\mathsf{\phi}}_{0.1395},{\mathsf{\phi}}_{0.1037}\right],(0.3501,0.0060)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0325},{\mathsf{\phi}}_{0.0282}\right],(0.1429,0.5274)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0167},{\mathsf{\phi}}_{0.0193}\right],(0.1161,0.7056)\u232a$ |

$F$7 | $\u2329\left[{\mathsf{\phi}}_{0.1450},{\mathsf{\phi}}_{0.0723}\right],(0.3171,0.0348)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0304},{\mathsf{\phi}}_{0.0306}\right],(0.0690,0.9679)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0188},{\mathsf{\phi}}_{0.0192}\right],(0.1104,0.6853)\u232a$ |

$F$8 | $\u2329\left[{\mathsf{\phi}}_{0.1403},{\mathsf{\phi}}_{0.0988}\right],(0.3171,0.0348)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0199},{\mathsf{\phi}}_{0.1234}\right],(0.0690,0.7421)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0175},{\mathsf{\phi}}_{0.0204}\right],(0.1083,0.5454)\u232a$ |

$F$9 | $\u2329\left[{\mathsf{\phi}}_{0.1393},{\mathsf{\phi}}_{0.0984}\right],(0.3210,0.0314)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0280},{\mathsf{\phi}}_{0.0344}\right],(0.1345,0.5646)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0205},{\mathsf{\phi}}_{0.0137}\right],(0.1248,0.6666)\u232a$ |

$F$10 | $\u2329\left[{\mathsf{\phi}}_{0.1219},{\mathsf{\phi}}_{0.1378}\right],(0.2851,0.0759)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0279},{\mathsf{\phi}}_{0.0312}\right],(0.1080,0.6692)\u232a$ | $\u2329\left[{\mathsf{\phi}}_{0.0202},{\mathsf{\phi}}_{0.0160}\right],(0.1253,0.6574)\u232a$ |

$F$11 | $\u2329\left[{\mathsf{\phi}}_{0.0970},{\mathsf{\phi}}_{0.1387}\right],(0.1703,0.2438)\u232a$ |