# Developing an economical model for reliability allocation of an electro-optical system by considering reliability improvement difficulty, criticality, and subsystems dependency

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## Abstract

The nature of electro-optical equipment in various industries and the pursuit of the goal of reducing costs demand high reliability on the part of electro-optical systems. In this respect, reliability improvement could be addressed through a reliability allocation problem. Subsystem reliability must be increased such that the requirements as well as defined requisite functions are ensured in accordance with the designers’ opinion. This study is an attempt to develop a multi-objective model by maximizing system reliability and minimizing costs in order to investigate design phase costs as well as production phase costs. To investigate reliability improvement feasibility in the design phase, effective feasibility factors in the system are used and the sigma level index is incorporated in the production phase as the reliability improvement difficulty factor. Thus, subsystem reliability improvement priorities are taken into consideration. Subsystem dependency degree is investigated through the design structure matrix and incorporated into the model’s limitation together with modified criticality. The primary model is converted into a single-objective model through goal programming. This model is implemented on electro-optical systems, and the results are analyzed. In this method, reliability allocation follows two steps. First, based on the allocation weights, a range is determined for the reliability of subsystems. Afterward, improvement is initiated based upon the costs and priorities of subsystem reliability improvement.

## Keywords

Reliability allocation Design and production cost Reliability improvement difficulty Sigma level Design structure matrix (DSM)## List of symbols

*R*_{sys}Whole reliability of system

- \(R_{\text{goal}}\)
Goal reliability

*R*_{i}Reliability of subsystem

*λ*_{goal}Goal failure rate

*λ*_{i}Failure rate of subsystem

*R*_{min}Minimum reliability

*R*_{max}Maximum reliability

- \(C_{ 1}\)
Design phase cost of system

- \(C_{ 2}\)
Production phase cost of system

- \(N\)
Number of subsystems

- \(A_{i}\)
Production cost of subsystem

- \(B 1_{i}\)
Reliability improvement cost of subsystem in the design phase

- \(B 2_{i}\)
Reliability improvement cost of subsystem in the production phase

- \(u_{ 1}\)
Predetermined budget for improvement in the design phase

- \(u_{ 1}\)
Predetermined budget for improvement in the production phase

- \(i\)
Interest rate

- \(T\)
Duration of the project for design and product development

- \(F_{{{\text{d}}_{i} }}\)
Feasibility factor of increasing the reliability for subsystem in the design phase

- \(F_{{{\text{p}}_{i} }}\)
Difficulty factor of increasing the reliability for subsystem in the production phase

- \(W_{i}\)
Allocation weight (criticality and dependency factors)

## Introduction

Product reliability evaluation follows product design process and is regarded as an inextricable part of this process (Liu et al. 2014). In different industries, the reliability optimization problem can be addressed by regarding a product as a system comprising a number of subsystems. This problem is stated using system structure and limitations as well as the characteristics and arrangement of subsystems and components. In fact, reliability improvement refers to the enhancement of reliability in such a way that the functions required by the system are ensured (Falcone et al. 2014). Reliability improvement problems include three procedures: (1) increasing subsystem reliability (reliability allocation), (2) use of redundant components in parallel (redundancy allocation problem), and (3) application of the two aforementioned procedures (reliability-redundancy allocation problem) (Mellal and Zio 2016). In these problems, the objective is to maximize reliability in the face of cost, weight, and volume limitations. Reliability allocation has an essential link with reliability design and serves as an important activity in product design and development. Hence, it is imperative to evaluate system behavior, function, and parameters by using failure effects and data, subsystem dependency, and the degree of reliability improvement. In fact, to determine subsystem reliability based on goal reliability, attention must be paid to improvement opportunities and priorities based on the real potential of reliability improvement (Yadav and Zhuang 2014). Besides, investigation of the relationship among subsystems and their importance links failure analysis process and reliability improvement feasibility to the product design and development process (Zhuang et al. 2014).

System reliability influences all system costs during the system’s life cycle. In order to reduce these costs, it is imperative to take account of various costs in reliability allocation and to reduce those costs by improving reliability in the design phase (Nguyen and Murthy 1988).

Reliability problems mostly focus on the design process. The most important system problems in the design process are the corollary of changes in production and assembly. Failure to control variability in the production process leads to increased costs and diminished system reliability. Hence, the production phase is crucial for system reliability and uncertainties existing in this stage and leads to the failure of the designed reliability required (Karaulova et al. 2012).

The reliability of the production process is a function of the capability of the critical variables of the process, complexity of activities, efficiency of time, and costs and capability of equipment and workforce to perform the defined missions without failure (Karaulova et al. 2012; Ávila 2015). Reliability estimation based on production phase data is an effective instrument to investigate the effect of production process on system reliability (Jinghuan et al. 2012). Variability of production processes and system performance can be quantitatively estimated, and the control and reliability improvement in systems can be investigated using process capability indexes (PCI) (Pearn * et al. 2005; Baril et al. 2011), which reflect the condition of reliability and quality in the production process (Jeang and Chung 2008). There are papers that they have investigated the performance of production process by using PCI like Cp, Cpk, sigma level and studied the product design based on the six sigma and reliability. Process performance evaluation using qualitative data is often based on the number of defective items. Used in this procedure is the sigma level index, to which reliability can be directly attributed. For example, for a sigma level of 3, the probability of the system’s performing a defined function within a day under existing conditions is 99.73% and the probability of failure is 0.27% and for achieving 100% reliability, it is unnecessary to optimize in 6 sigma level (Baril et al. 2011).

*W*

_{i}denotes allocation weight, subsystem reliability is determined through Eq. 1:

In a collection of papers including Yadav et al. (2006), Yadav (2007), and Itabashi-Campbell and Yadav (2009), allocation weights are linked to failure modes and effects analysis (FMEA) using risk priority numbers (RPN), where factors are multiplied by each other to compute RPN. In these papers, linear scaling is assumed and reliability is not allocated in accordance with criticality or potential for improvement. Kim et al. (2013) put forth an approach for computing weights, where exponential transformation function is used in place of the ordinal 10-item rate to compute failure severity, and where the exponential relationship between failure severity and failure effect is taken into consideration. The method focuses more intensely on subsystems with higher failure severity. Yadav and Zhuang (2014) considered weighted allocation for the improvement level of failure rate in series systems, in such a way that the degree of modified criticality is computed by considering a nonlinear relationship between failure severity and failure effect and between efforts for improvement and failure rate. In the paper by Zhuang et al. (2014), the degree of modified criticality is blended with functional dependency and based on the degree of significance of either, the allocation weight is computed so as to determine subsystem reliability. In this study, difficulty and complexity factors are set equal to 1.

In the study by Kumral (2005), the reliability variance of subsystems was added to the objective function of Mettas (2000) model. This model is solved by using the genetic algorithm designed for mining production systems, and the feasibility factor is initialized according to expert judgment. After incorporating failure rate and costs into the objective function of Mettas (2000) model, Zhang et al. (2007) defined an index known as subsystem importance, which is determined by using cost function derivative in relation to failure rate. Three ranges are defined for the values of importance, based on which three measures of reliability reduction, unimportant component, and reliability improvement are carried out. In the paper by Farsi and Jahromi (2012), effective feasibility factors such as complexity, criticality, state of art, operational profile, and availability were used for the reliability improvement feasibility factor, and the model was solved for a complex spatial system by using the genetic algorithm. Liu et al. (2014) added to the problem a new factor named manufacturing consistency, which is also known as PCI and is measured using the Cpk index. This index (as a variable) together with its costs is added to the model and is solved using the genetic algorithm after converting the primary model to the MDO model.

In the paper by Chen et al. (2013), optimization of the system is investigated by considering failure dependence and it is studied in K-out-of-n redundant systems in order to increase reliability by Mortazavi et al. (2016, 2017).

Investigation of previous research yields the conclusion that the investigated feasibility factors are considered as allocation weights. In this method, higher reliability is allocated to the subsystem with lower weight (higher reliability). Hence, these factors are not incorporated in the identification of the factors of subsystem reliability improvement priorities. Another point worth mentioning is the exclusion of system performance and costs from the production phase in allocation problems. Moreover, some studies have taken the intensity of subsystem interrelations into consideration, while they have ignored the type of relationships and their significance, despite the role of this issue in improving system design and modification.

The present study is an attempt to develop a model for subsystem reliability allocation aimed at maximizing system reliability and minimizing costs, such that reliability improvement priorities, criticality factors, and degree of subsystem interdependency are taken into consideration. In view of the importance of the reliability of the production phase, this study investigates design phase costs as well as production phase costs. For two reasons, these costs are investigated separately as two objective functions in the model: First, the amount of design and production cost is unequal and the smaller cost should not be affected by the larger cost. Second, the importance degree of costs is different in any system and project, and by separating design and production cost, the priorities of costs can be examined. In order to consider the priority of subsystem reliability improvement, the factor of reliability improvement difficulty is investigated. In this connection, the effective feasibility factors in the system are used for the factor of reliability improvement in the design phase, and the MEMV-OWA method is used to compute the weight of these factors. Besides, the sigma level index is computed so as to evaluate subsystem performance in the production phase and is incorporated into production cost function as a reliability improvement difficulty factor. In this study, another factor known as subsystem dependency was investigated using design structure matrix (DSM). This factor is incorporated into the model’s limitations together with modified criticality. Finally, the multi-objective model is converted to a single-objective model using goal programming (GP), and reliability allocation for the electro-optical system is investigated. The proposed model provides greater flexibility for design engineers who seek to regulate reliability objectives in accordance with reliability improvement difficulty, degree of criticality, and subsystem dependency.

The remainder of this paper is structured as follows: In “Statement of the problem” section, the main problem is defined, and the proposed model as well as the relationships pertaining to costs and each factor is presented. In “Goal programming” section, the GP model is presented. In “Practical example” section, the proposed model is solved for the electro-optical system and the results are analyzed. Finally, in “Conclusion” section, general conclusions and suggestions for further research are presented.

## Statement of the problem

### Proposed model

*s.t.*

### Design phase cost

### Reliability improvement feasibility factor in design phase

*f*_{i} denotes reliability improvement feasibility. Lower values of this factor represent greater reliability improvement difficulty and demonstrate the cases where subsystems enjoy higher reliability. As illustrated in Fig. 2, in this case, the cost function approaches infinity sooner (Mettas 2000). Most studies have assigned a value between 0 and 1 to this factor in accordance with expert judgment. In this paper, this factor is calculated by considering the features and behavior of the system and the opinion of the experts.

*φ*

_{i}is considered between 1 and 10 for each factor (Chang et al. 2009; Liaw et al. 2011). In the case of multiple experts, the average or mode of opinion is computed. The weight of selected factors is computed using MEMV-OWA model presented in Eq. 10, too (Chen et al. 2015):

*n*is the number of factors and \(\alpha\) is the degree of integration and position of OWA operator which adopts a value between 0 and 1. Values closer to 0 indicate pessimistic opinion and those closer to 1 indicate optimistic opinion of the expert. The models shows that non-certain data of decision-makers’ experience continue to be optimized as much as possible by maximizing entropy. Besides, minimizing the variance of the weighted vector is a potential way to prevent the overrating of decision-maker priorities (Chen et al. 2015). If \(w_{i}\) represents the feasibility factors weight obtained by solving the MEMV-OWA model and \(\varphi_{i}\) represents the values of each factor in each subsystem, the ultimate weight of feasibility of the design is determined by Eq. 11. In this case, the priorities of subsystem reliability improvement are investigated in a functional fashion.

### Production phase costs

Generally, an increase in reliability is accompanied by an increase in the related costs, as demonstrated in Figs. 2 and 3. Therefore, in reliability optimization problems, the objective is to maximize system reliability and minimize costs. In Eq. 5, production cost is converted to present value by using engineering economic equations until design and production cost can be compared.

### Reliability improvement difficulty factor in production phase

In Eq. 12, *v* represents the reliability improvement difficulty factor, an increase in which leads to an increase in reliability improvement difficulty. That is, the subsystem in question has a better performance (lower variability) in the production phase, making it more costly to improve its reliability. As demonstrated in Fig. 3, in this case, the costs approach infinity sooner. Most studies have assigned a value between 0 and 1 to this factor according to expert judgment.

### Modified criticality factor

*r*stands for the decrease rate of \(\lambda\) and \(E_{i}\) represents efforts for reliability improvement. \(\lambda_{i}\) is the summation of failure rate of various failure modes which is computed through Eq. 16.

### Dependency factor

Kinds of current relationships between subsystems or components of system

Interaction | Definition |
---|---|

Spatial | Functional requirement for alignment, orientation, serviceability, assembly |

Structural | Functional requirement for transferring design loads, forces |

Energy | Functional requirement for transferring heat, vibration, and electrical energy |

Material | Functional requirement for transferring air, oil, fuel, or water |

Information | Functional requirement for transferring signals or controls |

Intensity of existing relationships in the DSM

+ 2 | Relationship is necessary for functionality |

+ 1 | Relationship is beneficial for functionality |

0 | Relationships don’t affect functionality |

## Goal programming

GP is a multi-objective decision-making method which can conduct programming based upon the goals existing in each system and obtain satisfactory solutions for different goals. It is also possible to create deviations in the goals. GP incorporates goals, deviations, and goal and system limitations.

Equation 20 demonstrates the minimization of deviation from goals. This equation is rendered dimensionless by dividing it by the amount of deviation. Equation 21 relates to the maximization of reliability, where \(d_{ 1}^{ - }\) stands for negative deviation tendency. Equation 22 relates to the minimization of design cost, where \(d_{ 2}^{ + }\) represents positive deviation minimization. Equation 23 relates to the minimization of production costs, in which \(d_{ 3}^{ + }\) stands for the minimization of positive deviation. Equation 24 incorporates system limitations.

## Practical example

A questionnaire is used to select the effective factors among existing factors. Experts who know the system are asked to give a value between 0 and 100 for each factor. After calculating the frequency of each factor, Pareto chart is used for selecting effective factors. The effective feasibility factors in electro-optical systems include complexity, technology, operational time, environment and work condition, safety, repairability, and availability.

Values of each factor in each subsystem

Complexity | Technology | Operational time | Environment and work condition | Safety | Repairability | Availability | |
---|---|---|---|---|---|---|---|

Stabilator | 9 | 7 | 6 | 8.5 | 3 | 8.5 | 8 |

Site | 7 | 5 | 5 | 6.5 | 8 | 6 | 5.5 |

Panels | 7 | 5 | 6 | 2.5 | 4.5 | 7 | 7 |

Processor | 8 | 8 | 8 | 5.5 | 7 | 8 | 9 |

Sensors | 8 | 6.5 | 8.5 | 10 | 2.5 | 5 | 7 |

Power | 6 | 5 | 9 | 8 | 8 | 9 | 10 |

Result of calculating difficulty factor in each two phase

Subsystem | \(F_{\text{p}}\) | \(F_{\text{d}}\) |
---|---|---|

Stabilator | 0.5 | 0.44 |

Site | 0.41 | 0.36 |

Panels | 0.48 | 0.33 |

Processor | 0.25 | 0.45 |

Sensors | 0.35 | 0.43 |

Power | 0.39 | 0.41 |

As presented in Table 4, the variation of reliability improvement feasibility factors in the design phase is consistent with those of the difficulty factor in the production phase. That is, systems with lower \(F_{\text{d}}\) (where reliability improvement feasibility is lower, reliability is higher, and the improvement costs approach infinity sooner) exhibited a more efficient performance in the production phase, except for the stabilator subsystem.

Result of calculating criticality factor

Subsystem | \(s_{i}\) | \(e_{i}\) | \(C_{i}\) |
---|---|---|---|

Stabilator | 0.05 | 0.20 | 0.193 |

Site | 0.24 | 0.20 | 0.164 |

Panels | 0.53 | 0.13 | 0.080 |

Processor | 0.02 | 0.20 | 0.197 |

Sensors | 0.05 | 0.13 | 0.189 |

Power | 0.11 | 0.14 | 0.177 |

Final weights and costs of design and production phase

Subsystem | Di | \(W_{i}\) | \(A\) (million toman) | \(B 1\) (million toman) | \(B 2\) (million toman) |
---|---|---|---|---|---|

Stabilator | 0.149 | 0.171 | 10 | 30 | 20 |

Site | 0.075 | 0.119 | 15 | 20 | 5 |

Panels | 0.168 | 0.124 | 20 | 20 | 10 |

Processor | 0.223 | 0.210 | 50 | 100 | 150 |

Sensors | 0.260 | 0.224 | 10 | 20 | 20 |

Power | 0.126 | 0.151 | 20 | 50 | 80 |

Comparison of subsystems reliability

Subsystem | Reliability of subsystems | |
---|---|---|

Considering design cost | Considering design and production cost | |

Stabilator | 0.98603 | 0.98552 |

Site | 0.99197 | 0.99153 |

Panels | 0.99096 | 0.99018 |

Processor | 0.98338 | 0.98416 |

Sensors | 0.98796 | 0.98759 |

Power | 0.98759 | 0.98890 |

System | 0.93000 | 0.93011 |

### Comparison with previous methods

Comparison of results of proposed method with previous methods

Subsystem | Reliability of subsystems | |||||
---|---|---|---|---|---|---|

FOO method | Chang et al. (2009) | Yadav and Zhuang (2014) | Zhuang et al. (2014) | Farsi and Jahromi (2012) | Proposed method | |

Stabilator | 0.98379 | 0.98696 | 0.98611 | 0.98769 | 0.99000 | 0.98552 |

Site | 0.99423 | 0.98927 | 0.98816 | 0.99138 | 0.99000 | 0.99153 |

Panels | 0.99733 | 0.99011 | 0.99419 | 0.99101 | 0.99000 | 0.99018 |

Processor | 0.98578 | 0.98659 | 0.98582 | 0.98490 | 0.99000 | 0.98416 |

Sensors | 0.97777 | 0.98719 | 0.98636 | 0.98384 | 0.99000 | 0.98759 |

Power | 0.98908 | 0.98776 | 0.98726 | 0.98907 | 0.99000 | 0.98890 |

System | 0.93000 | 0.93000 | 0.93000 | 0.93000 | 0.94148 | 0.93011 |

The two methods FOO and MEMV-OWA allocate higher reliability to the subsystems with lower weights. Lower weights are determined for the subsystems with higher reliability. Therefore, these two methods fail to take into consideration difficulty and subsystem reliability improvement priorities, which makes them unable to assist managers and designer in rational decision-making.

In the modified criticality method and in the case of taking dependency into consideration, the factors of costs and reliability improvement feasibility are not investigated according to the properties and conditions of subsystems. The difference between the reliability values of these two methods and those of the proposed method lies in the incorporation of costs, feasibility factors, and system performance in the production phase in order to improve reliability.

The model developed by Farsi and Jahromi (2012) offers reliability close to maximum reliability, which leads to an exponential increase in costs.

### Sensitivity analysis

Value of subsystems reliability with different goal reliability

Number | Subsystem | \(R_{\text{goal}} = 0. 8 7\) | \(R_{\text{goal}} = 0.91\) | \(R_{\text{goal}} = 0.97\) |
---|---|---|---|---|

1 | Stabilator | 0.98044 | 0.98368 | 0.99499 |

2 | Site | 0.98396 | 0.98976 | 0.99622 |

3 | Panels | 0.98745 | 0.98890 | 0.99596 |

4 | Processor | 0.96731 | 0.97773 | 0.99357 |

5 | Sensors | 0.97482 | 0.98096 | 0.99364 |

6 | Power | 0.96845 | 0.98544 | 0.99525 |

Design cost (million toman) | 398 | 1087 | 1843355 | |

Production cost (million toman) | 505 | 510 | 515 |

Effect of reliability improvement difficulty factor on the subsystems reliability

Subsystem | Real difficulty factor | Same difficulty factor | Level of reliability improvement |
---|---|---|---|

Stabilator | 0.98552 | 0.99828 | − 0.01276 |

Site | 0.99153 | 0.99116 | 0.00037 |

Panels | 0.99018 | 0.98992 | 0.00027 |

Processor | 0.98416 | 0.98074 | 0.00342 |

Sensors | 0.98759 | 0.98121 | 0.00638 |

Power | 0.98890 | 0.98666 | 0.00224 |

System | 0.93011 | 0.93000 |

## Conclusion

The reliability of the final product is determined in the design phase and achieved in the production phase. The production phase is crucial for system reliability and the uncertainties existing in this stage. In a realistic reliability allocation method, the potential for reliability allocation must be investigated. Taking subsystem dependency into consideration in reliability allocation modifies allocation weights and links failure analysis process and reliability improvement difficulty to the product design and development process. In this paper, in order to allocate the reliability of subsystems, a multi-objective model was developed by maximizing whole system reliability and minimizing the costs of design and production phases. Four factors including reliability improvement in the design phase and production phase, criticality, and subsystem dependency were investigated. After converting the multi-objective model to GP and solving the model for the electro-optical system, the results were evaluated and analyzed. These results are consistent with the priorities of reliability improvement for each subsystem, and the difficulty factor used in the objective function investigates reliability improvement difficulty. These factors and costs tend to improve reliability considering the characteristics and function of subsystems, and improvement is subjected to criticality and dependency restrictions. In fact, in this method, reliability allocation is implemented in two stages: First, considering the allocation weight, a range is determined for the reliability of subsystems, and, afterward, improvement is implemented by considering the priorities of subsystem reliability improvement and the related costs. The proposed model provides greater flexibility for the engineers who seek to identify improvement opportunities and regulate reliability objectives with respect to reliability improvement feasibility, degree of criticality, and subsystem dependency. It is recommended that future studies investigate the effect of learning on production costs, study failure dependency among subsystems, investigate the proposed model for parallel and series–parallel systems, and examine other indices as factors of reliability improvement in the production phase and compare the result with this paper.

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