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Abstract

The complexity of test and fault information within electronic devices makes their integrated diagnosis a challenging problem when designing equipment reliability. Current integrated diagnosis is analyzed for test optimization and test resource optimization. However, this neglects the connection between them. This paper proposes a design strategy for integrated diagnosis optimization based on the spatial mapping principle to quantitatively describe the constraint relationship between them. The integrated diagnosis optimization model is established by constructing the logical mapping relationship between test space, resource space, and fault space, and the optimal test configuration and test resource configuration are sought based on the grey wolf optimization algorithm. Seven high-dimensional benchmark functions and an integrated diagnosis model of electronic equipment are used to verify the efficiency of the algorithm proposed in this paper. The proposed algorithm is compared with the other four in terms of the algorithm’s optimization speed and accuracy. The results indicate that the electronic equipment after integrated diagnosis optimization has critical fault detection, fault detection, fault isolation, and false alarm rates of 100%, 99.99%, 98.99%, and 0.2993%, respectively. After the integrated diagnosis optimization, the number of tests of the equipment is reduced by 88.9%, and the test cost is saved by 89%. Compared with the other algorithms, grey wolf optimization achieves the best optimization results, reduces the number of tests by 42%–55%, and decreases the test cost by 77.63%–83.91%. This strategy not only considers the test optimization of equipment and test resources optimization but also dramatically reduces the test cost while improving the test efficiency.

Keywords

Integrated diagnosis test optimization resource optimization hierarchical analysis gray wolf optimization algorithm

Introduction

The vast majority of modern equipment is equipped with cutting-edge electronic technology, which improves its performance and combat effectiveness, significantly raises its lifetime cost, and creates significant challenges for maintenance and warranty.^1–3 The United States Institute of Defense Industry proposed the idea of an integrated diagnosis in the 1980s, which encouraged the US military to develop better electronic equipment to address existing testing, fault diagnosis, maintenance, and security issues. Over 20 years, the integrated diagnosis methodology system effectively improved US military equipment and its comprehensive security capabilities.^4,5 Integrated diagnosis is the organized process of design and management that maximizes the diagnostic capability of equipment considering every aspect that constitutes diagnostic capability, including personnel, training, technical information, maintenance aids, automated testing, built-in testing (BIT), manual testing, and maintenance.^6,7 Integrated diagnosis is not the combined use of multiple means for diagnosing a particular fault mode. Based on the needs of each aspect of the equipment system, integrated diagnosis ensures that faults are diagnosed quickly and efficiently by selecting the optimal test plan while minimizing the overall cost.⁸

Integrated diagnosis is a systematic endeavor. During the design phase, the apparatus has to select test resources such as BIT, automated, and manual testing, which highlights how quickly the equipment can make a test decision for satisfying the use and maintenance requirements. Furthermore, forming a high-efficiency maintenance management system while using it under the various mission requirements, maintenance levels, and security levels is necessary.

Briefly, integrated diagnosis involves considering test optimization and test resource allocation optimization. Electronic equipment with an unsatisfactory test design find it challenging to utilize external assistance such as diagnostic and infrared sensors in the fault diagnosis process, similar to that in mechanical faults. In addition, it is not possible to disconnect and disassemble the cables, circuits, and components at will. Therefore, electronic equipment in the test design must consider the subsequent maintenance and diagnosis. Given the complexity and integration of higher-level electronic equipment, using manual testing not only takes more time but also reduces the ability of the equipment to fight; using BIT will increase the costs associated with research and development for the overall design of the equipment in addition to affecting the structural performance, weight, and volume of the severe burden.^9,10 Therefore, it is necessary to consider both the needs for testability and diagnosability to perform integrated diagnostic design for electronic equipment. On the one hand, this means determining the diagnostic scheme under the various mission requirements, optimizing the allocation of test resources, optimizing fault detection and isolation points, and optimizing the test sequence. On the other hand, it involves evaluating the testability design, promptly making necessary adjustments and optimizations, and formulating the diagnostic strategy under the mission requirements.^11,12

Test and test resource optimizations are primary areas of contemporary integrated diagnosis research. Many tests must be conducted based on a thorough study of faults to assure fault diagnosis performance as equipment structure and function become more complicated.¹³ Test optimization focuses on finding the optimal set of tests that satisfy the testability and diagnosability requirements from the set of alternative system tests. This helps ensure that the cost of integrated diagnosis is minimized.¹⁴ The current research methods for test optimization are an information entropy-based ranking method and a search algorithm based on combinatorial optimization. Test optimization using the core concept of information entropy involves prioritizing tests with high weights until the test criteria are satisfied. The contribution of each test to fault diagnosis is measured using information entropy for calculating the test weights.¹⁵ Although this method can define the consequences of different tests, it is more subjective based on the information weights. Methods based on combinatorial optimization seek the optimal test set by defining the objective function and constraints. They are based on intelligent optimization algorithms, e.g. genetic algorithms (GAs),¹⁶ particle swarm algorithms,¹⁷ simulated annealing algorithms (SAAs),¹⁸ and bacterial foraging algorithms.¹⁹ However, it is difficult for this class of methods to achieve a good trade-off between search speed and global optimization. The best test set can be found by combining the two test optimization techniques, which define the objective function based on information entropy, and they enhance the intelligent optimization algorithm or combine multiple intelligent optimizations to get around the conflict between search speed and global optimality.^20,21

The optimal allocation of test resources is essential for the integrated diagnosis. The performance of relevant alternative test resources, test cost, and other factors should be analyzed for determining the optimal test set of the equipment. The experience of test experts and specific technical indicators should determine what test resources or combination of forms for tests to form a test resource allocation program.²² For the test, various feasible test resources or combinations are often available to complete. They require a reasonable choice among the many alternative test resources and multiple resources for comprehensive evaluation. The theory and technique of test resource selection are currently under-represented in the literature. Most literature consider costs associated with different test resource selection strategies, ignore several significant limitations, and skip over essential details in favor of a cursory overview of the procedure and methodology of test resource selection, which is sufficient information for developing a systematic overall program.²³ This leads to the irrational allocation of testing resources and maintenance level of equipment, which increases the cost of testing, fault diagnosis, and equipment maintenance, and seriously restricts equipment maintenance technology development. Therefore, for test resource optimization, it is necessary to comprehensively consider the target elements affecting the testability and diagnosability of the equipment for thoroughly assessing the effectiveness and costs and coordinate the trade-offs and compromises between the target factors.

The following problems exist in the research on integrated diagnosis at the present stage:

Allocating test resources for equipment at the current location is unreasonable, and there is no comprehensive analysis of the various cost factors affecting the allocation of test resources. Furthermore, there is a lack of in-depth research on the optimal allocation of test resources. The key to this problem is the lack of a theoretical basis for test resource allocation and inability to integrate all cost factors into the test resource optimization model.

The integrated diagnostic optimization modeling process considers only the test demand (whether the faults can be detected or isolated); however, the effect of the diagnosability demand (the difficulty of fault diagnosis) on the integrated diagnostic optimization is not considered.

The construction of traditional constraints on test optimization does not consider the effect of the fault of the test resources on the optimization results.

The integrated diagnostic optimization is a whole. The current stage of comprehensive diagnosis research studies tests optimization and resource optimization separately, weakening the constraint relationship between them.

The results of the above analysis indicate that the key to the integrated diagnosis problem lies in how to establish the relationship between test optimization and test resource optimization. For example, how to quantitatively describe the relationship between test, test resource, test cost, resource cost, and test resource fault rate. The spatial mapping principle has a unique advantage in dealing with different spatial mapping relationships. Based on the spatial mapping principle, this paper quantitatively describes the logical mapping relationships between each other's spaces and establishes an integrated diagnosis and optimization model by establishing the test space, resource space, fault space, and cost space. The integrated diagnosis problem is a combinatorial optimization problem, and bionic intelligent optimization algorithms play a great advantage in solving these optimization problems. Common bionic intelligent optimization algorithms include grey wolf optimization (GWO),²⁴ artificial bee colony,²⁵ ant colony,²⁶ cuckoo search,²⁷ bat,²⁸ firefly,²⁹ whale optimization,³⁰ squirrel search,³¹ chimpanzee,³² and osprey optimization algorithms.³³

In this paper, we use the GWO algorithm, and the main reason for this lies in its unique search mechanism. The search mechanism of the GWO algorithm is based on the leadership hierarchy of the grey wolf population, which can effectively avoid the algorithm from falling into local optimal solutions. In recent years, the GWO algorithm has been widely used in the fields of network clustering,³⁴ lifetime prediction,³⁵ feature selection,³⁶ path planning,³⁷ engineering problems,³⁸ community detection,³⁹ and so on. In summary, this paper presents an integrated diagnostic optimization approach based on integrated design principles, thereby aiming at issues present in the current integrated diagnosis research process. First, the mapping relationship between the fault space, test space, resource space, and their subspaces is constructed based on the spatial mapping principle. According to the logical mapping relationship between the test optimization and resource optimization vectors, the test and resource optimization models are integrated to establish an integrated diagnosis optimization model. Furthermore, based on the correlation and quantitative matrices, the constraint relationship of the integrated diagnosis optimization is constructed for ensuring that it meets the testability and diagnosability requirements of the equipment. Then, the mapping relationship between the cost types of test resources is established and the comprehensive evaluation cost matrix based on hierarchical analysis is obtained to ensure the reasonableness of the test resource allocation. Finally, the GWO algorithm will be designed to seek the optimal integrated diagnostic model of the equipment. This integrated diagnosis optimization strategy not only considers the test optimization of equipment and test resources optimization but also dramatically reduces the test cost while improving the test efficiency.

Integrated diagnosis optimization model

Characterization of the integrated diagnosis optimization

Integrated diagnosis optimization is an integrated system project based on the premise of meeting the testability and diagnosability needs of the equipment. This systems employs test optimization and resource allocation optimization as the primary means and reduces the cost of integrated equipment diagnosis as the optimization goal. Improving the reliability of the equipment and fault diagnosis ability is the ultimate goal.

Many tests are set up based on thoroughly investigating equipment faults to guarantee the degree of integrated diagnosis. However, tests do not have the same importance and are often redundant; therefore, as an essential content, test optimization begins the integrated diagnosis optimization that will be related to the efficiency of the entire integrated diagnosis.^40,41

Test resource configuration affects the structure and function of the equipment to a certain extent. In contrast, the complexity of the structure and function of the equipment seriously restricts the development of integrated diagnosis.^42,43 Test resource allocation involves optimizing different types of test resources such as online/offline through integrated diagnosis information and test information of different maintenance levels. Resource allocation requires clarifying the characteristics of test resources, understanding the fault diagnosis capabilities required for different maintenance levels, and analyzing the test resources as necessary for different maintenance levels. Diagnostic resources include BIT, automatic test equipment, manual test equipment, and general electronic equipment. Other types of test resources have different application scenarios, and their specific focuses are also different. The testability and diagnosability of the equipment are different for different maintenance levels. The allocation of test resources decides how to arrange various test resources at each maintenance level in a reasonable manner. The types of test resources vary depending on the type of equipment. For example, space satellites are mostly BIT-based, and the test resources are appropriately changed according to the corresponding application scenarios. Therefore, to make the study of general significance, this paper takes the test resource allocation of available equipment as the research object.

Test optimization and resource allocation optimization are mutual constraints and affect each other, and the test optimization constrains the selection of test resources. The choice of test resources constrains the test requirements, which further constrains the results of test optimization. The relationship between the two is shown in Figure 1.

Figure 1.

Constraint relationships between test and resource optimizations.

Spatial mapping principle

In a finite-dimensional linear space, given a basis, any vector can be uniquely represented by a basis vector. Thus, whenever the image of a basis vector can be determined under some linear transformation, the linear transformation can be completely determined.

Let $V_{1}$ and $V_{2}$ represent $n$ - and $m$ -dimensional linear spaces over the number field $F$ , respectively, and $T$ represents a linear map from $V_{1}$ to $V_{2}$ . Consider the basis $α_{1}, α_{2}, \dots, α_{n}$ and $β_{1}, β_{2}, \dots, β_{m}$ of $V_{1}$ and $V_{2}$ , respectively, such that

{\begin{matrix} T α_{1} = a_{11} β_{1} + a_{21} β_{2} + \dots + a_{m 1} β_{m} \\ T α_{2} = a_{12} β_{1} + a_{22} β_{2} + \dots + a_{m 2} β_{m} \\ ⋮ \\ T α_{n} = a_{1 n} β_{1} + a_{2 n} β_{2} + \dots + a_{m n} β_{m} \end{matrix}

(1)

Let

A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & \dots & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}],

(2)

where

A

represents the correlation matrix of

T

under bases

α_{1}, α_{2}, \dots, α_{n}

V_{1}

and bases

β_{1}, β_{2}, \dots, β_{m}

V_{2}

T

can be written in the matrix form as

T (α_{1}, α_{2}, \dots, α_{n}) = (T α_{1}, T α_{2}, \dots, T α_{n}) = (β_{1}, β_{2}, \dots, β_{m}) A

(3)

From the definition of

T

, it follows that

T

can be uniquely determined by

A

given the basis of the space. In turn, any

m \times n

-dimensional matrix

A

can be uniquely determined by a correlation mapping from

V_{1}

V_{2}

using the relation (1), when

V_{1}

and

V_{2}

are bases

α_{1}, α_{2}, \dots, α_{n}

and

β_{1}, β_{2}, \dots, β_{m}

, respectively. In this manner, a correlation mapping

T

from

V_{1}

V_{2}

and correlation matrix

A

are one-to-one correspondences when the basis of the space is provided.

The matrix A quantifies the logical mapping relationship between the linear spaces V and D . Matrix A quantifies the relationship between the tests, resources, faults, and costs in integrated diagnostics. Based on this principle, the relationship between tests, resources, faults, and costs in integrated diagnosis can be quantitatively described.

Integrated diagnosis space

Based on the demand of constructing an integrated diagnosis model based on spatial mapping, the test, fault, and test resources are built into the test, fault, and resource space. Defining the constraints between each other lays the foundation for constructing the integrated diagnosis optimization model.

Definition 1: (Fault space): Fault space $F = s p a n {f_{1}, f_{2}, \dots, f_{m}}$ is the space formed by the failure modes ${f_{1}, f_{2}, \dots, f_{m}}$ obtained from the testability analysis based on the failure mode effect and criticality analysis, combined with the relevant information of the fault.

Definition 2: (Test space): Test space $T = s p a n {t_{1}, t_{2}, \dots, t_{n}}$ is the space formed by the test set ${t_{1}, t_{2}, \dots, t_{n}}$ that satisfies the testability and diagnosability requirements obtained from the testability analysis.

Definition 3: (Resource space): The resource space $R = s p a n {r_{1}, r_{2}, \dots, r_{n}}$ is the space formed by test resources ${r_{1}, r_{2}, \dots, r_{n}}$ corresponding to tests ${t_{1}, t_{2}, \dots, t_{n}}$ in the test space.

(1) Correlation mapping $Φ : T \mapsto F$

Given an m-dimensional fault space

F

and n-dimensional test space

T

, define the correlation mapping

Φ : T \mapsto F

from

T

F

, take a set of bases

t_{1}, t_{2}, \dots, t_{n}

T

, and a set of bases

f_{1}, f_{2}, \dots, f_{m}

F

such that

{\begin{matrix} Φ t_{1} = d_{11} f_{1} + d_{21} f_{2} + \dots + d_{m 1} f_{m} \\ Φ t_{2} = d_{12} f_{1} + d_{22} f_{2} + \dots + d_{m 2} f_{m} \\ ⋮ \\ Φ t_{n} = d_{1 n} f_{1} + d_{2 n} f_{2} + \dots + d_{m n} f_{m} \end{matrix} .

(4)

Let

D = [\begin{matrix} d_{11} & d_{12} & \dots & d_{1 n} \\ d_{21} & d_{22} & \dots & d_{2 n} \\ ⋮ & ⋮ & \dots & ⋮ \\ d_{m 1} & d_{m 2} & \dots & d_{m n} \end{matrix}],

(5)

where

D

represents the correlation matrix of

Φ

under the bases

t_{1}, t_{2}, \dots, t_{n}

T

and bases

f_{1}, f_{2}, \dots, f_{m}

F

. Furthermore,

Φ

can be written in the matrix form as

Φ {t_{1}, t_{2}, \dots, t_{n}} = {Φ t_{1}, Φ t_{2}, \dots, Φ t_{n}} = [f_{1}, f_{2}, \dots, f_{m}] D,

(6)

where

Φ

reflects the correlation between the test

t_{j} (1 \leq j \leq n)

and fault

f_{i} (1 \leq i \leq m)

. Therefore, the correlation matrix

D

is a digital description of the correlation between the fault and the test.

d_{i j} = {\begin{matrix} 1, & if t_{j} can test f_{i} \\ 0, & otherwise \end{matrix} .

(7)

From the definition of

Φ

, it follows that

Φ

can be uniquely determined by

D

given the basis of the space. In turn, any

m \times n

-dimensional matrix

D

can be uniquely determined by a correlation mapping from

T

F

by relation (4) when

T

and

F

are given the bases

t_{1}, t_{2}, \dots, t_{n}

and

f_{1}, f_{2}, \dots, f_{m}

, respectively. Thus, a correlation mapping

Φ

from

T

F

and a correlation matrix

D

are one-to-one correspondences when the basis of the space is given.

(2) Fault rate mapping $Γ : F \mapsto F$

Given an m-dimensional fault space

F

, construct a fault rate mapping

Γ : F \mapsto F

of the fault space itself. Take a set of bases

f_{1}, f_{2}, \dots, f_{m}

F

such that

{\begin{matrix} Γ f_{1} = λ_{f 1} f_{1} + 0 \cdot f_{2} + \dots + 0 \cdot f_{m} = λ_{f 1} f_{1} \\ Γ f_{2} = 0 \cdot f_{1} + λ_{f 1} f_{2} + \dots + 0 \cdot f_{m} = λ_{f 2} f_{2} \\ ⋮ \\ Γ f_{m} = 0 \cdot f_{1} + 0 \cdot f_{2} + \dots + λ_{f m} f_{m} = λ_{f m} f_{m} \end{matrix} .

(8)

Let

F_{λ} = [\begin{matrix} λ_{f 1} & 0 & \dots & 0 \\ 0 & λ_{f 2} & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & λ_{f m} \end{matrix}],

(9)

where

F_{λ}

represents the fault rate matrix of the fault rate mapping

Γ

under bases

f_{1}, f_{2}, \dots, f_{m}

F

, where

λ_{f i}

represents the fault rate of fault

f_{i}

. Therefore,

Γ

can be written in the matrix form as

\begin{aligned} Γ {f_{1}, f_{2}, \dots, f_{m}} & = {Γ f_{1}, Γ f_{2}, \dots, Γ f_{m}} \\ = [f_{1}, f_{2}, \dots, f_{m}] \cdot F_{λ} \\ = [f_{1}, f_{2}, \dots, f_{m}] \cdot d i a g [λ_{f 1}, λ_{f 2}, \dots, λ_{f m}] . \\ = [f_{1}, f_{2}, \dots, f_{m}] \cdot d i a g λ_{f}^{T} \end{aligned}

(10)

Analogously to

Φ

, it can be seen that, given bases

f_{1}, f_{2}, \dots, f_{m}

F

, the fault rate mapping

Γ

and fault rate matrix

F_{λ}

(or the fault rate vector

λ_{f}

) are in one-to-one correspondence.

(3) Cost mapping $Φ^{1} : T \mapsto R$

Given an n-dimensional test space

T

and n-dimensional resource space

R

, define a cost mapping

Φ^{1} : T \mapsto R

from

T

R

. Take a set of bases

t_{1}, t_{2}, \dots, t_{n}

T

and set of bases

r_{1}, r_{2}, \dots, r_{n}

R

, where

r_{i} = [r_{i 1}, r_{i 2}, \dots, r_{i l}]^{T}

c_{i} = [c_{i 1}, c_{i 2}, \dots, c_{i l}]^{T}

，

Let

{\begin{matrix} Φ^{1} t_{1} = c_{11} r_{11} + c_{12} r_{12} + \dots + c_{1 l} r_{1 l} = r_{1}^{T} \cdot c_{1} \\ Φ^{1} t_{2} = c_{21} r_{21} + c_{22} r_{22} + \dots + c_{2 l} r_{2 l} = r_{2}^{T} \cdot c_{2} \\ ⋮ \\ Φ^{1} t_{n} = c_{n 1} r_{n 1} + c_{n 2} r_{n 2} + \dots + c_{n l} r_{n l} = r_{n}^{T} \cdot c_{n} \end{matrix},

(11)

C^{T} = [c_{1}, c_{2}, \dots, c_{n}]^{T} = [\begin{matrix} c_{11} & c_{12} & \dots & c_{1 l} \\ c_{21} & c_{22} & \dots & c_{21} \\ ⋮ & ⋮ & \dots & ⋮ \\ c_{n 1} & c_{n 1} & \dots & c_{n l} \end{matrix}],

(12)

where

C

represents the cost matrix of the cost mapping

Φ^{1}

under bases

t_{1}, t_{2}, \dots, t_{n}

T

and bases

r_{1}, r_{2}, \dots, r_{n}

R

Φ^{1}

can be written in the matrix form as

\begin{aligned} Φ^{1} {t_{1}, t_{2}, \dots, t_{n}} & = {Φ^{1} t_{1}, Φ^{1} t_{2}, \dots, Φ^{1} t_{n}} \\ = [r_{1}^{T}, r_{2}^{T}, \dots, r_{n}^{T}] . * [c_{1}, c_{2}, \dots, c_{n}] . \\ = [r_{1}^{T}, r_{2}^{T}, \dots, r_{n}^{T}] . * C \end{aligned}

(13)

In the above equation,

. *

stands for the dot product, which represents the multiplication of the corresponding elements of the matrix of the same dimension.

Φ^{1}

shows that each test

t_{i} (1 \leq i \leq n)

has a set of test resources

r_{i} (1 \leq i \leq n)

of length l corresponding to it. In short, test

t_{i}

has l alternative test resources

{r_{i 1}, r_{i 2}, \dots, r_{i l}}

to choose from. The test cost of alternative test resources

r_{i j} (1 \leq i \leq n, 1 \leq j \leq l)

c_{i j}

. The test resources

r_{i}

correspond one-to-one with test cost

c_{i}

Analogously to $Φ$ , cost mapping $Φ^{1}$ is in one-to-one correspondence with cost matrix $C$ when the bases of $T$ and $R$ are given. Thus, the cost mapping relationships for different cost types, as well as cost matrix $C^{k}$ can be obtained depending on the test cost type.

When the test cost $c_{i j}$ of test resource $r_{i j}$ is expressed as fault rate $λ_{i j}$ ( $c_{i} = λ_{i} = [λ_{i 1}, λ_{i 2}, \dots, λ_{i l}]^{T}$ ), we take a set of bases $t_{1}, t_{2}, \dots, t_{n}$ of $T$ and set of bases $r_{1}, r_{2}, \dots, r_{n}$ of $R$ , and then, the resource fault rate matrix $C^{f}$ corresponds one-to-one with $Φ^{1}$ , which can be determined with the specific expression

C^{f} = [\begin{matrix} λ_{11} & λ_{21} & \dots & λ_{n 1} \\ λ_{12} & λ_{22} & \dots & λ_{n 2} \\ ⋮ & ⋮ & \dots & ⋮ \\ λ_{1 l} & λ_{2 l} & \dots & λ_{n l} \end{matrix}] .

(14)

The element

λ_{i j}

C^{f}

represents the fault rate of

r_{i j}

R

. At this point, the cost mapping

Φ^{1}

is called the resource fault rate mapping

Γ^{1}

(4) Quantitative evaluation mapping $Θ : F \mapsto F$

Given an m-dimensional fault space

F

, construct a quantitative evaluation mapping

Θ : F \mapsto F

of the fault space itself. Take a set of bases

f_{1}, f_{2}, \dots, f_{m}

F

such that

{\begin{matrix} Θ f_{1} = d_{(f_{1}, f_{1})} f_{1} + d_{(f_{1}, f_{2})} f_{2} + \dots + d_{(f_{1}, f_{m})} f_{m} \\ Θ f_{2} = d_{(f_{2}, f_{1})} f_{1} + d_{(f_{2}, f_{2})} f_{2} + \dots + d_{(f_{2}, f_{m})} f_{m} \\ ⋮ \\ Θ f_{m} = d_{(f_{m}, f_{1})} f_{1} + d_{(f_{m}, f_{2})} f_{2} + \dots + d_{(f_{m}, f_{m})} f_{m} \end{matrix} .

(15)

Let

D_{f} = [\begin{matrix} d_{(f_{1}, f_{1})} & d_{(f_{1}, f_{2})} & \dots & d_{(f_{1}, f_{m})} \\ d_{(f_{2}, f_{1})} & d_{(f_{2}, f_{2})} & \dots & d_{(f_{2}, f_{m})} \\ ⋮ & ⋮ & \dots & ⋮ \\ d_{(f_{m}, f_{1})} & d_{(f_{m}, f_{2})} & \dots & d_{(f_{m}, f_{m})} \end{matrix}],

(16)

where

D_{f}

represents the quantitative evaluation matrix of

Θ

under the bases

f_{1}, f_{2}, \dots, f_{m}

F

. Furthermore,

d_{(f_{i}, f_{i})}

represents the difficulty of detecting the fault

f_{i}

d_{(f_{i}, f_{j})} = d_{(f_{j}, f_{i})}

(i \neq j)

represents the difficulty of isolating

f_{i}

from

f_{j}

and

Θ

can be written in the matrix form

Θ {f_{1}, f_{2}, \dots, f_{m}} = {Θ f_{1}, Θ f_{2}, \dots, Θ f_{m}} = [f_{1}, f_{2}, \dots, f_{m}] D_{f}^{T} .

(17)

Analogously to

Φ

, the quantitative evaluation mapping

Θ

is in one-to-one correspondence with the quantitative evaluation matrix

D_{f}

given bases

f_{1}, f_{2}, \dots, f_{m}

F

Integrated diagnosis optimization model

Integrated diagnosis optimization involves meeting the requirements of testability and diagnosability as the basis to find the optimal test set and resource allocation set from the test space and resource space, and to ensure that the cost of the integrated diagnosis is minimal. The integrated diagnosis optimization can be reduced to two aspects: c $(c \leq n)$ base sheets are preferred from the test space to form a test subspace to meet the requirements of testability and diagnosability, and the test resources with the lowest integrated diagnosis cost are favored by the matching resource subspace of the test subspace.

Therefore, the integrated diagnosis optimization problem can be summarized as

Completeness of the test subspace, i.e. determining whether a set of bases in the test subspace satisfies the equipment diagnosability and testability requirements.

Based on expert experience and historical data, select appropriate bases among the test space to add or replace the bases of the test subspace to constitute a complete test subspace when the test subspace is incomplete.

Optimized test resources must be selected to ensure the testability needs of the equipment and minimize the overall cost.

Comprehensive analysis of the effect of each test cost type on the results of integrated diagnosis optimization.

Test optimization

Given the n-dimensional test space $T$ , define a test optimization mapping $Φ^{2} : T \mapsto T$ of the test space itself, and consider a set of bases $t_{1}, t_{2}, \dots, t_{n}$ of $T$ such that:

{\begin{matrix} Φ^{2} t_{1} = x_{11} t_{1} + 0 \cdot t_{2} + \dots + 0 \cdot t_{n} = x_{11} t_{1} \\ Φ^{2} t_{2} = 0 \cdot t_{1} + x_{22} \cdot t_{2} + \dots + 0 \cdot t_{n} = x_{22} t_{2} \\ ⋮ \\ Φ^{2} t_{n} = 0 \cdot t_{1} + 0 \cdot t_{2} + \dots + x_{n n} t_{n} = x_{n n} t_{n} \end{matrix} .

(18)

Let

X_{1} = [\begin{matrix} x_{11} & 0 & \dots & 0 \\ 0 & x_{22} & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & x_{n n} \end{matrix}] = d i a g (x_{11}, x_{22}, \dots, x_{n n}) = d i a g x_{1}^{T},

(19)

where

X_{1}

represents the test optimization matrix of the test optimization mapping

Φ^{2}

under the bases

t_{1}, t_{2}, \dots, t_{n}

T

Φ^{2}

can be written in the matrix form as

\begin{aligned} Φ^{2} {t_{1}, t_{2}, \dots, t_{n}} & = {Φ^{2} t_{1}, Φ^{2} t_{2}, \dots, Φ^{2} t_{n}} \\ = [t_{1}, t_{2}, \dots, t_{n}] \cdot X_{1} \\ = [t_{1}, t_{2}, \dots, t_{n}] \cdot d i a g x_{1}^{T} . \end{aligned}

(20)

Where

x_{i i} = {\begin{matrix} 1, i f t_{i} i s s e l e c t e d \\ 0, o t h e r w i s e \end{matrix} .

(21)

Analogous to

Φ

, the test optimization mapping

Φ^{2}

is in one-to-one correspondence with the test optimization vector

x_{1}

Select a new set of bases $a = {a_{1}, a_{2}, \dots, a_{c}}$ $(c = \sum_{i = 1}^{n} x_{i i})$ from bases $t_{1}, t_{2}, \dots, t_{n}$ in $T$ based on $x_{1}$ . Define a test subspace tensored by new bases $a_{1}, a_{2}, \dots, a_{c}$ as $T_{1} = s p a n {a_{1}, a_{2}, \dots, a_{c}}$ . The Pseudocode 1 for $a$ selection is

Pseudocode 1:

T_{1}

: = Find a(

T

x_{1}

)

Get the bases

t_{1}, t_{2}, \dots, t_{n}

T

i = 1

;

for each

x_{j j}

x_{1}

, do

x_{j j} = 1

a_{i} = t_{j}

;

i + +

;

end if

end for

T_{1} = s p a n {a_{i}}

Return

T_{1}

Pseudocode 2:

R_{1}

: = Find b(

R

x_{1}

)

Get the bases

r_{1}, r_{2}, \dots, r_{n}

R

i = 1

;

for each

x_{j j}

x_{1}

, do

x_{j j} = 1

b_{i} = r_{j}

;

i + +

;

end if

end for

R_{1} = s p a n {b_{i}}

Return R₁

Construct a $n \times c$ -dimensional test optimization matrix $X = [x_{1}, x_{2}, \dots, x_{c}]$ based on $x_{1}$ . The Pseudocode 3 for the construction of $X$ is

Pseudocode 3:

X

: = Find X(

x_{1}

)

Let

E = [e_{1}, e_{2}, \dots, e_{n}]

be an n-dimensional unit matrix;

i = 1

;

for each

x_{j j}

x_{1}

, do

x_{j j} = 1

x_{i} = e_{i}

;

i + +

;

end if

end for

X = [x_{i}]

Return

X

Given a test subspace $T_{1}$ , the test resource subspace $R_{1}$ , and fault space $F$ , analogously to $Φ$ construct the correlation mapping $Φ^{3}$ from $T_{1}$ to $F$ , and then, the correlation matrix corresponding one-to-one to $Φ^{3}$ is $D_{1} = D \cdot X$ . Similarly, construct the cost mapping $Φ^{4}$ from $T_{1}$ to $R_{1}$ , and then, the cost matrix corresponding one-to-one to $Φ^{4}$ is $C_{1} = C \cdot X$ .

Test resource optimization

Given a c-dimensional test subspace $T_{1}$ , define a test resource optimization mapping $Φ^{5} : T_{1} \mapsto T_{1}$ from the test subspace itself. Consider a set of basis ${a_{1}, a_{2}, \dots, a_{c}}$ of $T_{1}$ such that

{\begin{matrix} Φ^{5} a_{1} = y_{11} \cdot a_{1} + 0 \cdot a_{2} + \dots + 0 \cdot a_{c} = y_{11} a_{1} \\ Φ^{5} a_{2} = 0 \cdot a_{1} + y_{22} a_{2} + \dots + 0 \cdot a_{c} = y_{22} a_{2} \\ ⋮ \\ Φ^{5} a_{c} = 0 \cdot a_{1} + 0 \cdot a_{2} + \dots + y_{c c} a_{c} = y_{c c} a_{c} \end{matrix} .

(22)

Let

Y_{1} = [\begin{matrix} y_{11} & 0 & \dots & 0 \\ 0 & y_{22} & \dots & 0 \\ ⋮ & ⋮ & \dots & ⋮ \\ 0 & 0 & \dots & y_{c c} \end{matrix}] = d i a g (y_{11}, y_{22}, \dots, y_{c c}) = d i a g y_{1}^{T},

(23)

where

Y_{1}

represents the test resource optimization matrix for

Φ^{5}

under bases

a_{1}, a_{2}, \dots, a_{c}

T_{1}

y_{i i} = {k | k \in N *, 1 \leq k \leq l},

(24)

where

y_{i i}

indicates that test

a_{i}

selects the k-th test resource

b_{i} (y_{i i})

with the corresponding test cost

c_{i k}

Therefore, $Φ^{5}$ can be written in the matrix form as

\begin{aligned} Φ^{5} {a_{1}, a_{2}, \dots, a_{c}} & = {Φ^{5} a_{1}, Φ^{5} a_{2}, \dots, Φ^{5} a_{c}} \\ = [a_{1}, a_{2}, \dots, a_{c}] \cdot Y_{1} \\ = [a_{1}, a_{2}, \dots, a_{c}] \cdot d i a g y_{1}^{T} . \end{aligned}

(25)

Analogous to

Φ

, the test resource optimization mapping

Φ^{5}

is in one-to-one correspondence with the test resource optimization vector

y_{1}

Analogous to

X

, a test resource optimization matrix

Y = [y_{1}, y_{2}, \dots, y_{c}]

is constructed based on

y_{1}

. Pseudocode 4 for the construction of

Y

Pseudocode 4:

Y

: = Find Y(

y_{1}

)

Let

E = [e_{1}, e_{2}, \dots, e_{l}]

be an l-dimensional unit matrix;

i = 1

;

for each

y_{j j}

y_{1}

, do

y_{j j} = k

y_{i} = e_{k}

;

i + +

;

end if

end for

Y = [y_{i}]

Return

Y

Determine the test cost

c_{1} = [c_{11}, c_{22}, \dots c_{c c}]

of the test resource

[\begin{matrix} b_{1} (y_{11}) & b_{1} (y_{22}) & \dots & b_{c} (y_{c c}) \end{matrix}]

selected from

R_{1}

y_{1}

based on

C

X

, and

Y

. Analogously to the constructed form of

c_{1}

based on

C^{f}

X

, and

Y

, the fault rate

λ_{1} = [λ_{11}, λ_{22}, \dots, λ_{c c}]

of the test resource

[\begin{matrix} b_{1} (y_{11}) & b_{1} (y_{22}) & \dots & b_{c} (y_{c c}) \end{matrix}]

selected by

y_{1}

is determined when the test resource cost represents the test resource fault rate. Pseudocode 5 for the construction of

c_{1}

Pseudocode 5:

c_{1}

: = Find c(

C

X

Y

)

M = (C \cdot X)_{g} * Y = [m_{1}, m_{2}, \dots, m_{c}]

;

i = 1

;

for

j = 1 : c

c_{i i}

represents a non-zero element in the vector

m_{j}

;

i + +

;

end for

c_{1} = [c_{i i}]

;

Return

c_{1}

Integrated diagnosis optimization model

The mapping architecture of the test space $T$ , fault space $F$ , resource space $R$ , and their subspaces are shown in Figure 2.

Figure 2.

Mapping relationships in the integrated diagnosis space.

Figure 2 shows that the integrated diagnosis optimization is based on the test optimization mapping $Φ^{2}$ , which uses the test optimization vector $x_{1}$ to construct the test subspace $T_{1}$ and resource subspace $R_{1}$ . Based on the test resource optimization mapping $Φ^{5}$ , the test resource optimization vector $y_{1}$ is used to determine the optimization test resources. The integrated diagnosis optimization problem is to construct mappings $Φ^{2}$ and $Φ^{5}$ (vectors $x_{1}$ and $y_{1}$ ).

On the one hand, $x_{1}$ selected the test set that met the testability and diagnosability requirements of the equipment. On the other hand, $y_{1}$ was constructed to select the test resource set that minimizes the integrated diagnosis cost. The specific steps of integrated diagnosis optimization are listed below.

Step 1: Initialize the test space $T$ , fault space $F$ , and resource space $R$ and construct the correlation mapping $Φ : T \mapsto F$ , cost mapping $Φ^{1} : T \mapsto R$ , and fault rate mapping $Γ : F \mapsto F$ . Obtain the correlation matrix $D$ , cost matrix $C^{j}$ , resource fault matrix $C^{f}$ , and fault rate vector $λ_{f}$ .

Step 2: Initialize the test-optimization mapping $Φ^{2} : T \mapsto T$ , construct the test optimization vector $x_{1}$ and construct the test optimization matrix $X$ according to Pseudocode 3.

Step 3: Based on the test optimization results, initialize the quantitative evaluation mapping $Θ$ to obtain the quantitative evaluation matrix $D_{f}$ .

Step 4: Based on $x_{1}$ , construct the test subspace $T_{1}$ according to Pseudocode 1 and determine the resource subspace $R_{1}$ corresponding to the test subspace $T_{1}$ according to Pseudocode 2.

Step 5: Initialize the test resource optimization mapping $Φ^{5} : T_{1} \mapsto T_{1}$ and construct test resource optimization vector $y_{1}$ .

Step 6: Based on $y_{1}$ , determine the resource optimization matrix $Y$ according to Pseudocode 4.

Step 7: Construct the correlation mapping $Φ^{3} : T_{1} \mapsto F$ and determine the correlation matrix $D_{1} = D \cdot X$ .

Step 8: Construct the cost mapping $Φ^{4} : T_{1} \mapsto R_{1}$ and determine the cost matrix $C_{1} = C \cdot X$ .

Step 9: Based on $X$ and $Y$ , determine the optimized test resource cost vector $c_{1}^{j} = [c_{11}^{j}, c_{22}^{j}, \dots c_{c c}^{j}]^{T}$ based on Pseudocode 5 when cost matrix $C^{k}$ represents the test resource cost. Determine the fault rate $λ_{1} = [λ_{11}, λ_{22}, \dots, λ_{c c}]^{T}$ of the optimized test resource based on Pseudocode 5 when matrix cost $C^{k}$ represents the test resource fault rate.

Step 10: Based on the type of the test resource cost, the test optimization vector $x_{1}$ , test resource optimization vector $y_{1}$ , and test resource cost vector $c_{1}^{j}$ , the integrated diagnosis optimization model for that type of test cost is determined as

min f_{j} (Φ^{2}, Φ^{3}, Φ^{4}, Φ^{5}) = f_{j} (x_{1}, y_{1}, c_{1}^{j}) = \sum_{i = 1}^{c} c_{i i}^{j} .

(26)

The integrated diagnosis must ensure the mapping relationship between the optimized test subspace and fault space to meet the testability and diagnosability requirements. The correlation matrix

D_{1}

must guarantee its completeness and satisfy testability constraints such as fault detection rate, fault isolation rate, and false alarm rate. The quantitative evaluation matrix

D_{f}

must satisfy the diagnosability constraints.

(1) Completeness constraints

The test subspace completeness is equivalent to the

m \times c

-dimensional correlation matrix

D_{1}

determined by

Φ^{3}

with the following properties:

The test ${a_{1}, a_{2}, \dots, a_{c}}$ of the test subspace $T_{1}$ can detect all fault ${f_{1}, f_{2}, \dots, f_{m}}$ in the fault space $F$ if and only if there are no non-zero rows in correlation matrix $D_{1}$ .

The fault in the fault space is single-fault distinguishable if and only if there is a difference between every row in correlation matrix $D_{1}$ .

As a result, the constraint problem of checking subspace completeness is as follows:

There are no non-zero rows in matrix $D_{1}$ , and each row is distinct from the others. Construct the $\frac{m (m + 1)}{2} \times c$ -dimensional matrix $\bar{D} = [\begin{matrix} D_{1} \\ {\bar{D}}_{1} \end{matrix}]$ . The Pseudocode 6 for the construction of matrix $\bar{D}$ is

Pseudocode 6:

\bar{D}

= Find D (

D_{1}

)

Initialize:

{\bar{D}}_{1}^{T} = [\begin{matrix} {\bar{d}}_{1} & {\bar{d}}_{2} & \dots & {\bar{d}}_{k} & \dots & {\bar{d}}_{(m (m - 1) / 2)} \end{matrix}]

;

D_{1}^{T} = [\begin{matrix} d_{1} & d_{2} & \dots & d_{m} \end{matrix}]

;

k = 1

;

for

i = 1 : (m - 1)

for

j = (i + 1) : m

{\bar{d}}_{k} = d_{i} \oplus d_{j}

；%

\oplus

is a dissimilarity symbol

k + +

;

end for

\bar{D} = [\begin{matrix} D_{1} \\ {\bar{D}}_{1} \end{matrix}]

Return

\bar{D}

Construct a vector $α = [1, 1, \dots, 1]^{T}$ in c -dimensional linear space and a vector $β = [1, 1, \dots, 1]^{T}$ in $\frac{m (m + 1)}{2}$ -dimensional linear space. Then, the completeness boundedness function can be defined as

\bar{D} \cdot α \geq β .

(27)

(2) Fault detection rate constraints

Fault detection rate (FDR) is defined as a ratio of the number of faults correctly detected by a defined method to the total number of faults, expressed as a percentage.

γ_{F D} = \frac{λ_{D}}{λ} = \frac{\sum λ_{D i}}{\sum λ_{i}} \times 100 % .

(28)

where

λ_{D}

λ

λ_{D i}

, and

λ_{i}

represent the total fault rate of the detected fault, total fault rate of all fault, fault rate of the i-th detected fault, and fault rate of the i-th fault, respectively.

Engineering factors that should be considered to determine the FDR include the failure rate of the unit under test for faults, detection method, and reliability constraints. For the correlation matrix $D_{1}$ , fault rate vector $λ_{f}$ , and test resource fault rate vector $λ_{1}$ , if a fault $f_{i}$ occurs based on $D_{1}$ , it obtains the tests associated with the fault $f_{i}$ . Furthermore, $f_{i}$ is undetectable only if all test resources assigned to these test points have faults; otherwise, $f_{i}$ is detectable. The alternative test resources $r_{j} = [r_{j 1}, r_{j 2}, \dots, r_{j l}]$ for test $t_{j}$ have faults independent of each other, whereas the faults of the test resources on each test point are also independent. Therefore, the probability that a fault $f_{i}$ can be correctly detected when it occurs is $1 - \prod_{j = 1}^{c} λ_{j j}^{d_{i j}}$ , and the constraint relation for FDR is

\frac{\sum_{i = 1}^{m} λ_{f i} (1 - \prod_{j = 1}^{c} λ_{j j}^{d_{i j}})}{\sum_{i = 1}^{m} λ_{f i}} \geq γ_{F D} .

(29)

In the above equation,

γ_{F D}

represents the fault detection rate requirement for the equipment.

(3) Fault isolation rate constraints

Fault isolation rate (FIR) is defined as the ratio of the number of detected faults correctly isolated to no greater than a given degree of ambiguity by a prescribed method to the total number of detected faults expressed as a percentage.

γ_{F I} = \frac{λ_{L}}{λ_{D}} = \frac{\sum λ_{L i}}{λ_{D}} \times 100 % .

(30)

where L,

λ_{L}

λ_{D}

, and

λ_{L i}

represent the fuzzy degree of fault isolation, sum of the fault rates of the fault modes that can be isolated to be less than or equal to L replaceable units, sum of the fault rates of all detected fault modes, and fault rate of the fault modes that can be isolated to less than or equal to the i-th fault mode in L replaceable units, respectively.

The factors that should be considered in determining the FIR are FDR, testing techniques, and acceptable isolation methods. Based on $D_{1}$ , obtain the tests related to fault $f_{i}$ . Furthermore, $f_{i}$ can be isolated only if the test resources assigned to these test points cannot be faulted at the same time; otherwise, $f_{i}$ cannot be isolated. Based on the two independence assumptions of FDR, the probability that $f_{i}$ can be correctly isolated when it occurs is $\prod_{j = 1}^{c} (1 - d_{i j} λ_{j j})$ , and the constraint relationship of FIR is

\frac{\sum_{i = 1}^{m} [λ_{f i} \prod_{j = 1}^{c} (1 - d_{i j} λ_{j j})]}{\sum_{i = 1}^{m} λ_{f i} (1 - \prod_{j = 1}^{c} λ_{j j}^{d_{i j}})} \geq γ_{F I},

(31)

where

γ_{F I}

represents the fault isolation rate requirement for the equipment.

(4) False alarm rate constraint

False alarm rate (FAR) is defined as the ratio of the number of false alarms occurring in a specified period to the total number of fault indications at the same time.

γ_{F A} = \frac{n_{A}}{N_{F}} = \frac{λ_{F A}}{λ_{F A} + λ_{D}} \times 100 %,

(32)

where

n_{A}

N_{F}

λ_{F A}

, and

λ_{D}

represent the number of false alarms, number of fault indications, probability of occurrence of false alarms, and sum of the failure rates of the detected failure modes, respectively. According to the US military standard MIL-STD-2165 definition, a false alarm refers to a situation where test resources indicate that the unit under test is faulty; however, the actual unit is not. Alternatively, a false alarm is caused by the detection module misreporting or falsely reporting a fault when the unit under test is not faulty. The factors that cause false alarms to occur are more complex and attributed to test module faults. The constraint relationship for FAR is

\frac{\sum_{i = 1}^{m} [(1 - λ_{f i}) \prod_{j = 1}^{c} λ_{i i}^{d_{i j}}]}{\sum_{i = 1}^{m} [(1 - λ_{f i}) \prod_{j = 1}^{c} λ_{i i}^{d_{i j}}] + \sum_{i = 1}^{m} [λ_{f i} (1 - \prod_{j = 1}^{c} λ_{i i}^{d_{i j}})]} \leq γ_{F A},

(33)

where

γ_{F A}

represents the fault false alarm rate requirement for the equipment.

(5) Quantitative constraints on diagnosability

The difficulty of fault detection and isolation cannot be quantified because the index constraints of testability can only qualitatively reflect whether a fault can be detected or isolated. This paper uses the quantitative evaluation method of diagnosability based on the maximum mean difference.44 Then, construct the quantitative evaluation mapping

Θ

to obtain the quantitative evaluation matrix

D_{f}

and establish the quantitative constraint relationship of diagnosability. The diagnosability indices can be divided into the fault detectability index (FDI) and fault isolability index (FII). The FDI characterizes the difficulty of fault detection, and the FII characterizes the difficulty of fault isolation.

Based on $D_{f}$ , the quantitative evaluation index of diagnosability is

F D I = \sum_{i = 1}^{m} d_{(f_{i}, f_{i})} .

(34)

F I I = \sum_{i = 1}^{m} \sum_{j = i + 1}^{m} d_{(f_{i}, f_{j})} .

(35)

In the actual fault diagnosis process, it is necessary to ensure that the diagnosability index of the equipment is guaranteed to be as maximum as possible. If the diagnosability requirements of the equipment are to be confirmed, the diagnosability indices need to be satisfied.

F D I \geq F D I_{r e q},

(36)

F I I \geq F I I_{r e q},

(37)

where

F D I_{r e q}

and

F I I_{r e q}

represent the diagnosability requirements of the equipment.

(6) Integrated diagnosis optimization models

According to the constraints and objective functions analyzed in the previous section, the integrated diagnosis optimization model can be obtained as

\begin{aligned} min \sum_{j = 1}^{M} μ_{j} f_{j} (Φ^{2}, Φ^{3}, Φ^{4}, Φ^{5}) = \sum_{j = 1}^{M} μ_{j} f_{j} (x_{1}, y_{1}, c_{1}^{j}) = \sum_{j = 1}^{M} \sum_{i = 1}^{c} μ_{j} c_{i i}^{j} \\ s . t . \\ \begin{matrix} \begin{matrix} {\bar{D}}_{1} \cdot α \geq β \\ γ_{F D C_{r t}} = 100 % \end{matrix} \\ \frac{\sum_{i = 1}^{m} λ_{f i} (1 - \prod_{j = 1}^{c} λ_{j j}^{d_{i j}})}{\sum_{i = 1}^{m} λ_{f i}} \geq γ_{F D} \\ \frac{\sum_{i = 1}^{m} [λ_{f i} \prod_{j = 1}^{c} (1 - d_{i j} λ_{j j})]}{\sum_{i = 1}^{m} λ_{f i} (1 - \prod_{j = 1}^{c} λ_{j j}^{d_{i j}})} \geq γ_{F I} \\ \begin{matrix} \frac{\sum_{i = 1}^{m} [(1 - λ_{f i}) \prod_{j = 1}^{c} λ_{i i}^{d_{i j}}]}{\sum_{i = 1}^{m} [(1 - λ_{f i}) \prod_{j = 1}^{c} λ_{i i}^{d_{i j}}] + \sum_{i = 1}^{m} [λ_{f i} (1 - \prod_{j = 1}^{c} λ_{i i}^{d_{i j}})]} \leq γ_{F A} \\ F D I \geq F D I_{r e q} \\ F I I \geq F I I_{r e q} \end{matrix} \end{matrix}, \end{aligned}

(38)

where M,

μ_{j}

, and

γ_{F D C_{r t}}

represent the number of equipment cost types, weight of each cost type, and equipment critical fault detection rate (usually 100%), respectively.

Comprehensive evaluation of testing resource cost types

The integrated diagnosis optimization model reveals that the choice of test resources affects the total cost of the integrated diagnosis and the testability and diagnosability constraints of the equipment, which impacts the test optimization outcomes. The three main strategies for solving the integrated diagnosis problem are as follows: (1) how to identify and quantify each cost type, (2) how to convey the cost weight of each cost type to the test in a reasonable manner, and (3) how to make the integrated diagnosis optimization model simpler and less challenging when there are a large number of cost types.

Therefore, it is essential to allocate test resources as efficiently as possible. There are several types of costs for test resources. Each cost type has a distinct evaluation weight from other cost types, and the degree of restriction on each cost type varies based on the test in the test space. All cost types of the equipment need to be thoroughly analyzed, and the restriction weights of tests on each cost type in the test space should be clarified. Furthermore, the evaluation weights for each cost type for resources in the resource space should be clarified. Moreover, the evaluation weight of the integrated evaluation should be obtained based on the evaluation weight and restriction weight of each cost type to each test resource and test to simplify the integrated diagnosis model. This will allow for a comprehensive evaluation of the effect of each cost type on integrated diagnosis optimization. The cost matrix for the comprehensive assessment is ultimately derived from the evaluation weights of each cost type for each test resource and the restriction weights of the cost types to the test to streamline the integrated diagnosis model.

Test resource cost types

Test resource costs are divided into several categories, which include expense cost $C^{1}$ , time cost $C^{2}$ , application cost $C^{3}$ , logistics cost $C^{4}$ , personnel cost $C^{5}$ , reliability and maintainability cost $C^{6}$ , physical constraints cost $C^{7}$ , and so on. Each cost type can be further refined, and the results are shown in Figure 3.

Figure 3.

Classification of the test cost types.

Test cost comprehensive evaluation

One must consider the integrated diagnosis requirements of the equipment when choosing test resources in a test space, thoroughly analyze the constraints of the cost types from various perspectives, and choose test resources based explicitly on the relationship between those costs. Therefore, it is crucial to thoroughly assess every cost type in the test area while choosing the test resources.

Definition 5 (Cost Space): The test cost space $C^{c} = s p a n {C_{1}, C_{2}, \dots, C_{M}}$ is the space formed by the M cost types ${C_{1}, C_{2}, \dots, C_{M}}$ that affects the test resources selection. Consider a set of bases ${t_{1}, t_{2}, \dots, t_{n}}$ of $T$ and a set of bases ${C_{1}, C_{2}, \dots, C_{M}}$ of $C^{c}$ and build the weight mapping $Φ^{6} : T \mapsto C^{c}$ from $T$ to $C^{c}$ as

{\begin{matrix} Φ^{6} t_{1} = μ_{11} C_{1} + μ_{12} C_{2} + \dots + μ_{1 M} C_{M} \\ Φ^{6} t_{2} = μ_{21} C_{1} + μ_{22} C_{2} + \dots + μ_{2 M} C_{M} \\ ⋮ \\ Φ^{6} t_{n} = μ_{n 1} C_{1} + μ_{n 2} C_{2} + \dots + μ_{n M} C_{M} \end{matrix} .

(39)

Let

μ = [\begin{matrix} μ_{11} & μ_{12} & \dots & μ_{1 M} \\ μ_{21} & μ_{22} & \dots & μ_{2 M} \\ ⋮ & ⋮ & \dots & ⋮ \\ μ_{n 1} & μ_{n 2} & \dots & μ_{n M} \end{matrix}],

(40)

where

μ

represents the cost matrix of

Φ^{6}

under the bases

{C_{1}, C_{2}, \dots, C_{M}}

C^{c}

. Furthermore,

Φ^{6}

can be written in the matrix form as

Φ^{6} {t_{1}, t_{2}, \dots, t_{n}} = {Φ^{6} t_{1}, Φ^{6} t_{2}, \dots, Φ^{6} t_{n}} = [C_{1}, C_{2}, \dots, C_{M}] μ^{T} .

(41)

Analogous to

Φ

Φ^{6}

and

μ

are one-to-one, given the bases of

T

and

C^{c}

A judgment matrix was constructed for characterizing the effect weights of each cost category on the test by allocating values according to Satty’s recommended 1–9 scale.⁴⁵ Scales 1–9 are shown in Table 1

Table 1.

Meanings of the scales 1–9.

Proportional scale	Meanings
1	The two factors are of equal importance when compared.
3	The former is slightly more important than the latter when compared.
5	The former is significantly more important than the latter when compared.
7	The former is more strongly important than the latter when compared.
9	The former is extremely more important than the latter when compared.
2, 4, 6, 8	Represents the intermediate value of the above neighboring judgments

Based on Table 1, define the ratio mapping relation $Φ^{7} : C^{c} \mapsto C^{c}$ of the cost space itself and consider any set of the basis ${C^{1}, C^{2}, \dots}$ of $C^{c}$ such that

{\begin{matrix} Φ^{7} C^{1} = a_{11} C^{1} + a_{12} C^{2} + \dots \\ Φ^{7} C^{2} = a_{21} C^{1} + a_{22} C^{2} + \dots \\ ⋮ \end{matrix} .

(42)

Let

A = [\begin{matrix} a_{11} & a_{12} & \dots \\ a_{21} & a_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}],

(43)

where

A

represents the ratio matrix of

Φ^{7}

under the base

{C^{1}, C^{2}, \dots}

C^{c}

. Furthermore,

Φ^{7}

can be written in the matrix form as

Φ^{7} {C^{1}, C^{2}, \dots} = {Φ^{7} C^{1}, Φ^{7} C^{2}, \dots} = [C^{1}, C^{2}, \dots] A^{T},

(44)

where

a_{i j}

represents the relative importance ratio scale of test cost

C^{i}

to test cost

C^{j}

and the ratio matrix

A

is satisfied as

a_{i j} > 0.

(45)

a_{j i} = 1 / a_{i j} .

(46)

a_{i i} = 1.

(47)

Figure 3 shows that the costs of testing resources are complex and diverse. Moreover, in test resource selection, the subjective judgment is unstable, and there are differences in preferences among decision makers. Therefore, it is not easy to accurately describe the degree of the difference of each cost type under the same criterion. Therefore, it is necessary to determine whether the ratio matrix

A

satisfies the consistency condition. Consequently, the consistency ratio (CR = CI/RI), which is a quantitative standard of hierarchical analysis, is used to quantify the degree of inconsistency. The judgment matrix has acceptable consistency when

C R \leq 0.1

; otherwise, it is not.

$C I = \frac{λ_{max} - n}{n - 1}$ represents the consistency indicator, where $λ_{max}$ represents the largest eigenvalue of the matrix. $R I$ represents the average random consistency indicator and is correlated with the complexity of the objective object, decision maker's comprehension of the problem’s variety, and potential of the contract for one-sidedness. The CI considers different values for matrix orders n, as listed in Table 2.

Table 2.

Average random consistency indicator.

n	1	2	3	4	5
RI	0	0	0.5149	0.8931	1.1185

n	6	7	8	9	10
RI	1.2494	1.3450	1.4200	1.4620	1.4874

n	11	12	13	14	15
RI	1.5056	1.5405	1.5583	1.5779	1.5894

Test cost types are categorized into seven major classes: $C^{1}, C^{2}, \dots, C^{7}$ . Each class $C^{i} (i = 1, 2, \dots, 7)$ can be further subdivided into $C^{j 1}, C^{j 2}, \dots$ . Mappings $Φ^{6}$ and $Φ^{7}$ are used to obtain the cost weights of each test in the test space based on hierarchical analysis to refine the impact weights of each cost type.

Now consider the test $t_{i}$ as an example to describe how to obtain the cost weights ${μ_{i 1}, μ_{i 2}, \dots, μ_{i M}}$ of $t_{i}$ .

Step 1: Divide the basis of the cost space into different basis groups and establish the cost subspace $C_{a} = s p a n {C^{1}, C^{2}, \dots, C^{7}}$ , $C_{b}^{j} = s p a n {C^{j 1}, C^{j 2}, \dots}$ , and $j = 1, 2, \dots, 7$ .

Step 2: For the orientation test $t_{i}$ , get matrix $A_{t_{i}}$ under bases $C^{1}, C^{2}, \dots, C^{7}$ and matrix $B_{t_{i}}^{j}$ under bases ${C^{j 1}, C^{j 2}, \dots}$ based on mapping $Φ^{7}$ ．

Step 3: Consider the eigenvector $W_{t_{i}} = {w_{t i}^{1}, w_{t i}^{2}, \dots, w_{t i}^{7}}$ corresponding to the largest eigenvalue $λ_{t_{i}}^{A}$ of $A_{t_{i}}$ as the weight of the cost type ${C^{1}, C^{2}, \dots, C^{7}}$ with respect to $t_{i}$ , which satisfies $A_{t_{i}} W_{t_{i}} = λ_{t_{i}}^{A} W_{t_{i}}$ .

Step 4: Consider the eigenvector $W_{t_{i}}^{j} = {w_{t_{i}}^{j 1}, w_{t_{i}}^{j 2}, \dots}$ corresponding to the largest eigenvalue $λ_{t_{i}}^{j}$ of $B_{t_{i}}^{j}$ as the weight of the test cost $C^{j}$ with respect to the test cost ${C^{j 1}, C^{j 2}, \dots}$ , which satisfies $B_{t_{i}}^{j} W_{t_{i}}^{j} = λ_{t_{i}}^{j} W_{t_{i}}^{j}$ .

Step 5: Based on the hierarchical analysis, the evaluation weight of test cost $C^{j k}$ relative to $t_{i}$ is $w_{t i}^{j} \cdot w_{t_{i}}^{j k}$ .

C_{l} = C^{j k}

is in the cost mapping, then

μ_{i l} = w_{t i}^{j} \cdot w_{t_{i}}^{j k}

Resource cost comprehensive evaluation

For each test resource in the test resource space, specifying its test cost for each cost type is important. Therefore, a comprehensive evaluation of each cost type of the test resource is required for test resource optimization.

There are l alternative test resources $r_{i} = {r_{i 1}, r_{i 2}, \dots, r_{i l}}$ available for $t_{i}$ . Establish a mapping $Φ^{8} : r_{i} \mapsto C^{c}$ from the resource subspace $r_{i}$ to the cost space $C^{c}$ . Consider ${r_{i 1}, r_{i 2}, \dots, r_{i l}}$ as a set of bases of $r_{i}$ and ${C_{1}, C_{2}, \dots, C_{M}}$ as a set of bases of $C^{c}$ such that

{\begin{matrix} Φ^{8} r_{i 1} = ξ_{11}^{i} C_{1} + ξ_{12}^{i} C_{2} + \dots + ξ_{1 M}^{i} C_{M} \\ Φ^{8} r_{i 2} = ξ_{21}^{i} C_{1} + ξ_{22}^{i} C_{2} + \dots + ξ_{2 M}^{i} C_{M} \\ ⋮ \\ Φ^{8} r_{i l} = ξ_{l 1}^{i} C_{1} + ξ_{l 2}^{i} C_{2} + \dots + ξ_{l M}^{i} C_{M} \end{matrix} .

(48)

Let

ξ^{i} = [\begin{matrix} ξ_{11}^{i} & ξ_{12}^{i} & \dots & ξ_{1 M}^{i} \\ ξ_{21}^{i} & ξ_{22}^{i} & \dots & ξ_{2 M}^{i} \\ ⋮ & ⋮ & \dots & ⋮ \\ ξ_{l 1}^{i} & ξ_{l 2}^{i} & \dots & ξ_{l M}^{i} \end{matrix}] .

(49)

The mapping

Φ^{8}

can be written in the matrix form as

Φ^{8} {r_{i 1}, r_{i 2}, \dots, r_{i l}} = {Φ^{8} r_{i 1}, Φ^{8} r_{i 2}, \dots, Φ^{8} r_{i l}} = [C_{1}, C_{2}, \dots, C_{M}] (ξ^{i})^{T} .

(50)

Analogously to

Φ

, the mapping

Φ^{8}

is one-to-one with the matrix

ξ^{i}

The matrix $ξ^{i}$ represents the test cost matrix of alternative test resources $r_{i} = {r_{i 1}, r_{i 2}, \dots, r_{i l}}$ for test $t_{i}$ . $ξ_{j k}^{i}$ represents the cost of the j-th test resource $r_{i j}$ of $t_{i}$ under the k-th cost type $C_{k}$ .

Quantitatively characterizing the test cost of test resources is the foundation of optimum resource allocation. Both a qualitative analysis and fuzzy language may be used for calculating the expenses for the type of costs that cannot be quantified. A score set of {10, 9, 8, 7, 5, 4, 3, 2, 1} corresponds to the ten evaluation indicators arranged under the degree of effect of the cost type to aid in assessing qualitative indicators. Scores are assigned by pertinent experts in compliance with the guidelines above to create the fuzzy assessment matrix $ξ^{i}$ , as per the test resource evaluation criteria.

Assessment metrics for the test cost must be normalized by transforming the evaluation matrix $ξ^{i}$ into an affiliation matrix $S^{i} = (s_{j k}^{i})$ because the qualitative and quantitative metrics differ in certain features and quantiles. Furthermore, $s_{j k}^{i}$ denotes the relative affiliation of the k-th cost type $C_{k}$ of the j-th test resource $r_{i j}$ of $t_{i}$ . The affiliation function is

For positive indicators (bigger is better):

s_{j k}^{i} = {\begin{matrix} 1, & ξ_{j k}^{i} \leq min (ξ_{\cdot k}^{i}) \\ \frac{max (ξ_{\cdot k}^{i}) - ξ_{j k}^{i}}{max (ξ_{\cdot k}^{i}) - min (ξ_{\cdot k}^{i})}, & min (ξ_{\cdot k}^{i}) < ξ_{j k}^{i} < max (ξ_{\cdot k}^{i}) \\ 0, & ξ_{j k}^{i} \geq max (ξ_{\cdot k}^{i}) \end{matrix} .

(51)

For negative indicators (smaller is better):

s_{j k}^{i} = {\begin{matrix} 1, & ξ_{j k}^{i} \geq max (ξ_{\cdot k}^{i}) \\ \frac{ξ_{j k}^{i} - min (ξ_{\cdot k}^{i})}{max (ξ_{\cdot k}^{i}) - min (ξ_{\cdot k}^{i})}, & min (ξ_{\cdot k}^{i}) < ξ_{j k}^{i} < max (ξ_{\cdot k}^{i}) \\ 0, & ξ_{j k}^{i} \leq min (ξ_{\cdot k}^{i}) \end{matrix} .

(52)

In the above equation,

min (ξ_{\cdot k}^{i})

and

max (ξ_{\cdot k}^{i})

denote the minimum and maximum values of the test cost under the kth cost type

C_{k}

for alternative test resources of test

t_{i}

, respectively.

Comprehensive evaluation of the cost matrix $C$

Many types of costs need to be considered in the optimal allocation of test resources; the restriction weights of tests on each cost type in the test space are different, and the evaluation weights of each cost type of test resources in the resource space are also different. Cost matrix $C$ is evaluated comprehensively for analyzing the effect of each cost type on the integrated diagnosis optimization results based on mapping $Φ^{6}$ , $Φ^{7}$ , and $Φ^{8}$ . The specific steps are listed below.

Step 1: Establish the weight mapping relationship $Φ^{6} : T \mapsto C^{c}$ between the test resource cost space $C^{c}$ and test space $T$ , analyze the restriction weight of each cost type relative to the test, and obtain the weight matrix $μ$ .

Step 2: Based on the ratio mapping relationship $Φ^{7} : C^{c} \mapsto C^{c}$ and utilizing Steps 1–5 in Test cost comprehensive evaluation section, conduct a comprehensive evaluation of the weight matrix $μ$ .

Step 3: Sequentially establish the cost mapping relationship $Φ^{8} : r_{i} \mapsto C^{c}$ between the test resource subspace $r_{i}$ and test resource cost space $C^{c}$ and perform an analysis to obtain the evaluation weights $ξ^{i}$ of the test resources in $r_{i}$ for each cost type $i = 1, 2, \dots, n$ .

Step 4: Based on the affiliation function, normalize matrix $ξ^{i}$ to obtain affiliation matrix $S^{i}$ , $i = 1, 2, \dots, n$ .

Step 5: Based on the weight matrix $μ^{T} = [\begin{matrix} μ_{1 \cdot} & μ_{2 \cdot} & \dots & μ_{n \cdot} \end{matrix}]$ and affiliation matrix $S = [\begin{matrix} S^{1} & S^{2} & \dots & S^{n} \end{matrix}]$ , the cost matrix $C$ of the cost mapping $Φ^{1} : T \mapsto R$ can be optimized as

C = [C_{1 \cdot}, C_{2 \cdot}, \dots, C_{n \cdot}] = [S^{1} \cdot μ_{1 \cdot}, S^{2} \cdot μ_{2 \cdot}, \dots, S^{n} \cdot μ_{n \cdot}] .

(53)

Comprehensively measure the effect of each cost type on the optimization of test resources by integrating the influencing factors of each cost type into the cost matrix. Furthermore, it optimizes the integrated diagnosis model and reduces the difficulty of integrated diagnosis optimization. Then, the optimization function of the integrated diagnosis optimization model can be simplified as

min f (Φ^{3}, Φ^{4}, Φ^{5}) = f (x_{1}, y_{1}) = \sum_{i = 1}^{c} c_{i i} .

(54)

Integrated diagnosis optimization based on the GWO algorithm

The analysis in Comprehensive evaluation of testing resource cost types section indicated that the integrated diagnosis optimization problem is a combinatorial optimization problem that can be described by the ensemble coverage model. The ensemble coverage problem is a complex NP problem for large and complex equipment. The complexity of the equipment system increases with an increase in ensemble size, and therefore, it is necessary to find an effective algorithm to obtain the global optimal solution of the integrated diagnosis problem quickly and accurately.

GWO algorithm

Gray wolves prey through stalking, chasing, encircling, harassing, and other behaviors. The GWO algorithm is a new bionic intelligent optimization algorithm⁴⁶ designed to achieve algorithmic optimality seeking by mimicking the social hierarchies and prey behaviors of gray wolves.

In nature, the gray wolf is an apex predator that lives in packs of 5–12 wolves, with a strict hierarchy among them. The wolves in the gray wolf family can be divided into four categories based on their social status: $α$ wolf, $β$ wolf, $δ$ wolf, and $ω$ wolf. The $α$ wolf is the top wolf in the society, also called “head wolf.” The ɑ wolf is the high ruler and manager of the wolf pack, responsible for deciding the time, place, and tactics of the wolf pack hunting. The second tier is the $β$ wolf, the deputy leader of the head wolf, and it is responsible for continuing to lead the pack if the head wolf dies. On the third level is the $δ$ wolf, who is responsible for scouting, sentry, and guarding. At the bottom tier is the $ω$ wolf, which obeys the other dominant wolves in the pack.

In the GWO algorithm, the solution with the optimal fitness value, second best solution, and third generation of solutions in the gray wolf population are regarded as the $α$ , $β$ , and $δ$ wolves, respectively, to construct the social hierarchy of gray wolves, and the remaining solutions are considered as $ω$ wolves. The $α$ , $β$ , and $δ$ wolves are responsible for guiding. The $ω$ wolf is accountable for following the $α$ , $β$ , and $δ$ wolves and completing the hunting optimization by searching, encircling, and attacking the prey.

In the D-dimensional search space, assume that the population $X = (X_{1}, X_{2}, \dots, X_{N})$ consists of N individual gray wolves. Define the position of the i-th gray wolf in the D-dimensional space as $X_{i} = (X_{i}^{1}, X_{i}^{2}, \dots, X_{i}^{D})$ . Define the historical optimal solution of the gray wolf population as the $α$ wolf, second optimal solution as the $β$ wolf, third optimal solution as the $δ$ wolf, and other solutions as the $ω$ wolf.

The process of the gray wolf approaching and surrounding the prey is reflected in the algorithm as the process of the gray wolf arriving at the optimal solution. The position update formula of the i-th gray wolf is

X_{i}^{d} (t + 1) = X_{p}^{d} (t) - A_{i}^{d} | C_{i}^{d} X_{p}^{d} (t) - X_{i}^{d} (t) | .

(55)

In the above equation, t represents the current iteration number, $X_{p} = (X_{p}^{1}, X_{p}^{2}, \dots, X_{p}^{D})$ represents the position of the prey, and $A_{i}^{d} | C_{i}^{d} X_{p}^{d} (t) - X_{i}^{d} (t) |$ represents the encircling steps $A_{i}^{d}$ and $C_{i}^{d}$ , respectively.

A_{i}^{d} = 2 a \cdot {rand}_{1} - a,

(56)

C_{i}^{d} = 2 \cdot {rand}_{2},

(57)

In the above equation,

{rand}_{1}

and

{rand}_{2}

represent random numbers on the interval

[0, 1]

. Furthermore, a represents the convergence factor, which varies with the number of iterations, i.e.

a = 2 (1 - \frac{t}{t_{max}}),

(58)

where t and

t_{max}

represents the current and maximum iteration numbers, respectively.

The formula for updating the location of each type of gray wolf is

{\begin{matrix} X_{i, α}^{d} (t + 1) = X_{α}^{d} (t) - A_{i, 1}^{d} | C_{i, 1}^{d} X_{α}^{d} (t) - X_{i}^{d} (t) | \\ X_{i, β}^{d} (t + 1) = X_{β}^{d} (t) - A_{i, 2}^{d} | C_{i, 2}^{d} X_{β}^{d} (t) - X_{i}^{d} (t) | \\ X_{i, δ}^{d} (t + 1) = X_{δ}^{d} (t) - A_{i, 3}^{d} | C_{i, 3}^{d} X_{δ}^{d} (t) - X_{i}^{d} (t) | \end{matrix} .

(59)

X_{i}^{d} (t + 1) = \sum_{j = α, β, δ} w_{j} X_{i, j}^{d} (t + 1),

(60)

where

w_{j} (j = α, β, δ)

denotes the weight coefficients of

α

β

, and

δ

w_{j} = \frac{f (X_{j} (t))}{\sum_{j = α, β, δ} f (X_{j} (t))},

(61)

where

f (X_{j} (t))

represents the fitness value of gray wolf j.

Integrated diagnosis optimization steps

Integrated diagnosis optimization steps based on the GWO algorithm are as follows:

Step 1: Population initialization, which includes the number of populations N = 500, maximum number of iterations $t_{max} = 20$ , and parameters a, A, and C.

Step 2: Randomly initialize the position $X_{i}$ of individual gray wolves based on the range of the test optimization vector $x_{1}$ and test resource optimization vector $y_{1}$ .

Step 3: Suppose the individual gray wolf $X_{i}$ satisfies the constraints of equation (38). In that case, the fitness value of $X_{i}$ is calculated based on equation (54). Otherwise, the fitness value of $X_{i}$ is infinite. Then, find the $α$ , $β$ , and $δ$ wolves according to the fitness value.

Step 4: Update the position of the gray wolf individual $X_{i}$ according to the position update formula.

Step 5: Update the parameters a, A, and C.

Step 6: Evaluate whether the maximum number of iterations is reached. If satisfied, the algorithm stops and returns the value of $X^{α}$ as the optimal solution; otherwise, it returns to Step 4.

In the GWO algorithm, two randomly adjusted parameters, A and C, allow the algorithm to search and mine. When A is greater than 1 or less than −1, the algorithm starts a global search; furthermore, when C is greater than 1, it passes the search capability for the algorithm. In contrast, when

| A | > 1

and

| C | < 1

, the algorithm performs a local search. Based on Equations (56)–(58), it can be seen that the parameters of the GWO algorithm are adaptive, which can effectively prevent the algorithm from falling into local extremes due to the parameter settings. This is the advantage of GWO over other algorithms.

Simulation verification

Optimizing the design of an integrated diagnosis architecture for a specific type of equipment with 22 a priori faults and 36 preparatory tests is necessary. Each test preferably has nine test resources to choose from, where each test type has three alternative test resources. The mapping relationship of the integrated diagnosis space of the equipment is shown in Annex 1.

Comprehensive evaluation of the cost matrix $C$

Each cost category has been thoroughly examined, and test $t_{1}$ is being used as an example of a thorough assessment of its test cost.

Comprehensive evaluation of test costs

The oriented test

t_{1}

and matrix

A_{t_{1}}

of mapping

Φ^{7}

under the basis

{C^{1}, C^{2}, \dots, C^{7}}

of the cost space

C^{c}

$t_{1}$	$C^{1}$	$C^{2}$	$C^{3}$	$C^{4}$	$C^{5}$	$C^{6}$	$C^{7}$	$W_{t_{i}}$
$C^{1}$	1	1/2	1/3	1/4	1/5	1/6	1/8	0.0291
$C^{2}$	2	1	1/2	1/3	1/4	1/5	1/7	0.0414
$C^{3}$	3	2	1	1/2	1/3	1/4	1/6	0.0621
$C^{4}$	4	3	2	1	1/2	1/3	1/5	0.0947
$C^{5}$	5	4	3	2	1	1/2	1/4	0.1438
$C^{6}$	6	5	4	3	2	1	1/3	0.2146
$C^{7}$	8	7	6	5	4	3	1	0.4144

λ_{t_{1}}^{A} = 7.2441

，

C I = 0.0407

C R = 0.0302

The oriented test

t_{1}

and matrix

B_{t_{1}}^{1}

of mapping

Φ^{7}

under the basis

{C^{11}, C^{12}, \dots, C^{15}}

of the cost space

C^{c}

$C^{1}$	$C^{11}$	$C^{12}$	$C^{13}$	$C^{14}$	$C^{15}$	$W_{t_{1}}^{1}$
$C^{11}$	1	1/3	1/3	4	5	0.1663
$C^{12}$	3	1	1	6	6	0.3569
$C^{13}$	3	1	1	7	7	0.3765
$C^{14}$	1/4	1/6	1/7	1	2	0.0579
$C^{15}$	1/5	1/6	1/7	1/2	1	0.0424

λ_{t_{1}}^{1} = 5.1348

，

C I = 0.0346

，

C R = 0.0309

The oriented test

t_{1}

and matrix

B_{t_{1}}^{2}

of the mapping

Φ^{7}

under the basis

{C^{21}, C^{22}, C^{23}, C^{24}}

of the cost space

C^{c}

$C^{2}$	$C^{21}$	$C^{22}$	$C^{23}$	$C^{24}$	$W_{t_{1}}^{2}$
$C^{21}$	1	2	2	1/3	0.2044
$C^{22}$	1/2	1	1	1/5	0.1071
$C^{23}$	1/2	1	1	1/6	0.1022
$C^{24}$	3	5	6	1	0.5864

λ_{t_{1}}^{2} = 4.0042

，

C I = 0.0014

，

C R = 0.0016

The oriented test

t_{1}

of matrix

B_{t_{1}}^{3}

of mapping

Φ^{7}

under the basis

{C^{31}, C^{32}, C^{33}}

of the cost space

C^{c}

$C^{3}$	$C^{31}$	$C^{32}$	$C^{33}$	$W_{t_{1}}^{3}$
$C^{31}$	1	2	3	0.5396
$C^{32}$	1/2	1	2	0.2970
$C^{33}$	1/3	1/2	1	0.1634

λ_{t_{1}}^{3} = 3.0092

，

C I = 0.0046

，

C R = 0.0089

The oriented test

t_{1}

and the matrix

B_{t_{1}}^{4}

of mapping

Φ^{7}

under the basis

{C^{41}, C^{42}, C^{43}}

of the cost space

C^{c}

$C^{4}$	$C^{41}$	$C^{42}$	$C^{43}$	$W_{t_{1}}^{4}$
$C^{41}$	1	1/3	1/6	0.0914
$C^{42}$	3	1	1/4	0.2176
$C^{43}$	6	4	1	0.6910

λ_{t_{1}}^{4} = 3.0536

，

C I = 0.0268

，

C R = 0.0521

The oriented test

t_{1}

and matrix

B_{t_{1}}^{5}

of mapping

Φ^{7}

under the basis

{C^{51}, C^{52}, C^{53}, C^{54}}

of the cost space

C^{c}

$C^{5}$	$C^{51}$	$C^{52}$	$C^{53}$	$C^{54}$	$W_{t_{1}}^{5}$
$C^{51}$	1	1/4	4	1/3	0.1447
$C^{52}$	4	1	6	2	0.4903
$C^{53}$	1/4	1/6	1	1/5	0.0572
$C^{54}$	3	1/2	5	1	0.3078

λ_{t_{1}}^{5} = 4.1367, C I = 0.0456, C R = 0.0510

The oriented test

t_{1}

and matrix

B_{t_{1}}^{6}

of mapping

Φ^{7}

under the basis

{C^{61}, C^{62}, C^{63}, C^{64}}

of the cost space

C^{c}

$C^{6}$	$C^{61}$	$C^{62}$	$C^{63}$	$C^{64}$	$W_{t_{1}}^{6}$
$C^{61}$	1	1/5	1/6	1/4	0.0603
$C^{62}$	5	1	1	2	0.3657
$C^{63}$	6	1	1	1/2	0.2728
$C^{64}$	4	1/2	2	1	0.3013

λ_{t_{1}}^{6} = 4.2179, C I = 0.0726, C R = 0.0813

The oriented test

t_{1}

and matrix

B_{t_{1}}^{7}

of mapping

Φ^{7}

under the basis

{C^{71}, C^{72}, \dots, C^{76}}

of the cost space

C^{c}

$C^{7}$	$C^{71}$	$C^{72}$	$C^{73}$	$C^{74}$	$C^{75}$	$C^{76}$	$W_{t_{1}}^{7}$
$C^{71}$	1	1/3	1/5	1/7	1/4	1/2	0.0405
$C^{72}$	3	1	1/3	1/5	1/2	2	0.0958
$C^{73}$	5	3	1	1/3	2	3	0.2200
$C^{74}$	7	5	3	1	3	5	0.4287
$C^{75}$	4	2	1/2	1/3	1	2	0.145
$C^{76}$	2	1/2	1/3	1/5	1/2	1	0.0700

λ_{t_{1}}^{7} = 6.1483, C I = 0.0297, C R = 0.0237

Consequently,

μ_{1 \cdot} = [\begin{matrix} μ_{11} & μ_{12} & μ_{13} \end{matrix}]

μ_{11} = [\begin{matrix} 0.0048 & 0.0104 & 0.0109 & 0.0017 & 0.0012 & 0.0085 & 0.0044 & 0.0042 & 0.0243 & 0.0335 \end{matrix}]

μ_{12} = [\begin{matrix} 0.0185 & 0.0102 & 0.0087 & 0.0206 & 0.0654 & 0.0208 & 0.0705 & 0.0082 & 0.0442 & 0.0129 \end{matrix}]

μ_{13} = [\begin{matrix} 0.0785 & 0.0585 & 0.0646 & 0.0168 & 0.0397 & 0.0912 & 0.1777 & 0.0601 & 0.0290 \end{matrix}]

Integrated evaluation of resource costs

The values for the nine test resources corresponding to test $t_{1}$ can be given by quantitative calculation for costs that are easy to describe quantitatively. Qualitative evaluation indexes can describe the costs that are easy to describe qualitatively.

The fuzzy evaluation matrix $ξ^{1}$ of the test cost of alternative test resources for the test $t_{1}$ obtained by converting the quantitative evaluation indexes into the corresponding ratios is

ξ^{1} = [\begin{matrix} 3 & 4 & 7 & 7 & 7 & 4 & 7 & 5 & 7 & 4 & 4 & 9 & 2 & 4 & 4 & 7 & 5 & 7 & 9 & 2 & 7 & 4 & 6 & 7 & 7 & 9 & 8 & 2 & 9 \\ 9 & 2 & 4 & 4 & 5 & 2 & 5 & 5 & 5 & 2 & 5 & 5 & 9 & 2 & 5 & 5 & 2 & 8 & 5 & 4 & 4 & 9 & 2 & 8 & 4 & 5 & 9 & 5 & 5 \\ 2 & 3 & 5 & 5 & 2 & 3 & 6 & 2 & 6 & 6 & 2 & 6 & 4 & 6 & 6 & 2 & 6 & 5 & 6 & 7 & 5 & 8 & 6 & 9 & 5 & 2 & 4 & 4 & 8 \\ 4 & 9 & 6 & 6 & 3 & 2 & 3 & 6 & 2 & 3 & 3 & 4 & 7 & 9 & 9 & 6 & 4 & 6 & 3 & 9 & 6 & 5 & 4 & 5 & 6 & 1 & 6 & 8 & 4 \\ 6 & 8 & 2 & 1 & 2 & 1 & 2 & 3 & 1 & 5 & 1 & 7 & 8 & 8 & 2 & 8 & 9 & 2 & 2 & 6 & 2 & 2 & 9 & 4 & 2 & 4 & 9 & 9 & 8 \\ 9 & 4 & 1 & 3 & 1 & 3 & 0 & 2 & 2 & 1 & 4 & 9 & 5 & 4 & 8 & 9 & 5 & 4 & 8 & 3 & 1 & 1 & 6 & 8 & 9 & 6 & 2 & 5 & 5 \\ 5 & 1 & 4 & 2 & 2 & 6 & 1 & 4 & 3 & 2 & 5 & 5 & 6 & 5 & 7 & 2 & 2 & 7 & 7 & 2 & 4 & 8 & 2 & 5 & 2 & 2 & 1 & 2 & 5 \\ 3 & 3 & 8 & 2 & 8 & 6 & 6 & 9 & 4 & 8 & 6 & 2 & 5 & 6 & 3 & 4 & 1 & 9 & 4 & 4 & 8 & 4 & 5 & 9 & 4 & 3 & 4 & 5 & 5 \\ 5 & 6 & 9 & 3 & 6 & 9 & 2 & 2 & 9 & 4 & 9 & 1 & 2 & 2 & 4 & 7 & 2 & 5 & 3 & 5 & 6 & 5 & 6 & 8 & 1 & 6 & 5 & 4 & 8 \end{matrix}]

The normalized optimized matrix

ξ^{1}

is given by

S^{1} = [\begin{matrix} S_{1}^{1} & S_{2}^{1} & S_{3}^{1} \end{matrix}] .

S_{1}^{1} = [\begin{matrix} 0.0357 & 0.0968 & 0.1622 & 0.2500 & 0.2222 & 0.111 & 0.2188 & 0.1500 & 0.0476 & 0.1081 \\ 0.2500 & 0.0323 & 0.0811 & 0.1250 & 0.1481 & 0.0370 & 0.1563 & 0.1500 & 0.0925 & 0.1622 \\ 0 & 0.0645 & 0.1081 & 0.1667 & 0.0370 & 0.0741 & 0.1857 & 0 & 0.0714 & 0.0541 \\ 0.714 & 0.2581 & 0.1351 & 0.2083 & 0.0741 & 0.0370 & 0.0938 & 0.2000 & 0.1667 & 0.1351 \\ 0.1429 & 0.2258 & 0.0270 & 0 & 0.0370 & 0 & 0.0625 & 0.0500 & 0.1905 & 0.0811 \\ 0.2500 & 0.0968 & 0 & 0.0833 & 0 & 0.0741 & 0 & 0 & 0.1667 & 0.1892 \\ 0.1071 & 0 & 0.0811 & 0.0417 & 0.0370 & 0.1852 & 0.0313 & 0.1000 & 0.1429 & 0.1622 \\ 0.0357 & 0.0645 & 0.1892 & 0.0417 & 0.2593 & 0.1852 & 0.1857 & 0.3500 & 0.1190 & 0 \\ 0.1071 & 0.1613 & 0.2162 & 0.0833 & 0.1852 & 0.2963 & 0.0625 & 0 & 0 & 0.1081 \end{matrix}] .

S_{2}^{1} = [\begin{matrix} 0.1190 & 0.2051 & 0 & 0.0714 & 0.0667 & 0.1563 & 0.1481 & 0.1429 & 0.2414 & 0 \\ 0.0925 & 0.1026 & 0.2333 & 0 & 0.1000 & 0.0938 & 0.0370 & 0.1714 & 0.1034 & 0.0833 \\ 0.1667 & 0.1282 & 0.0667 & 0.1429 & 0.1333 & 0 & 0.1852 & 0.0857 & 0.1379 & 0.2083 \\ 0.1429 & 0.0769 & 0.1667 & 0.2500 & 0.2333 & 0.1250 & 0.1111 & 0.1143 & 0.0345 & 0.2917 \\ 0.1905 & 0.1538 & 0.2000 & 0.2143 & 0 & 0.1875 & 0.2963 & 0 & 0 & 0.1667 \\ 0.1190 & 0.2051 & 0.1000 & 0.0714 & 0.2000 & 0.2188 & 0.1481 & 0.0571 & 0.2069 & 0.0417 \\ 0.0952 & 0.1026 & 0.1333 & 0.1071 & 0.1667 & 0 & 0.0370 & 0.1429 & 0.1724 & 0 \\ 0.0714 & 0.0256 & 0.1000 & 0.1429 & 0.0333 & 0.0625 & 0 & 0.2000 & 0.0690 & 0.0833 \\ 0 & 0 & 0 & 0 & 0.0667 & 0.1563 & 0.0370 & 0.0857 & 0.0345 & 0.1250 \end{matrix}] .

S_{3}^{1} = [\begin{matrix} 0.1765 & 0.0811 & 0.1429 & 0.1111 & 0.1935 & 0.2759 & 0.1795 & 0 & 0.2381 \\ 0.0882 & 0.2162 & 0 & 0.1481 & 0.0968 & 0.1379 & 0.2051 & 0.1154 & 0.0476 \\ 0.1176 & 0.1892 & 0.1429 & 0.1852 & 0.1290 & 0.0345 & 0.1769 & 0.0769 & 0.1905 \\ 0.1471 & 0.1081 & 0.0714 & 0.0370 & 0.1613 & 0 & 0.1282 & 0.1308 & 0 \\ 0.0297 & 0.0274 & 0.2500 & 0 & 0.0323 & 0.1034 & 0.2051 & 0.2692 & 0.1905 \\ 0 & 0 & 0.1429 & 0.1481 & 0.2581 & 0.1724 & 0.0256 & 0.1154 & 0.0476 \\ 0.0882 & 0.1892 & 0 & 0.0370 & 0.0323 & 0.0345 & 0 & 0 & 0.0476 \\ 0.2059 & 0.0811 & 0.1071 & 0.1852 & 0.0968 & 0.0690 & 0.0769 & 0.1154 & 0.0476 \\ 0.1471 & 0.1081 & 0.1429 & 0.1081 & 0 & 0.1724 & 0.1026 & 0.0769 & 0.1905 \end{matrix}] .

Comprehensive evaluation of the cost matrix

From $C = [C_{1 \cdot}, C_{2 \cdot}, \dots, C_{n \cdot}] = [S^{1} \cdot {(μ_{1 \cdot})}^{T}, S^{2} \cdot {(μ_{2 \cdot})}^{T}, \dots, S^{n} \cdot {(μ_{n \cdot})}^{T}], C_{1}$ corresponds to the test $t_{1}$ as

C_{1} = {[\begin{matrix} 0.1479 & 0.1184 & 0.1111 & 0.1232 & 0.1383 & 0.1088 & 0.0655 & 0.0878 & 0.0991 \end{matrix}]}^{T} .

The comprehensive evaluation cost matrix for this equipment is

C = [\begin{matrix} C_{11} & C_{12} & C_{13} \end{matrix}] .

C_{11} = [\begin{matrix} 0.1429 & 0.2173 & 0.0113 & 0.1477 & 0.0076 & 0.2186 & 0.0780 & 0.0761 & 0.1265 & 0.1677 & 0.1578 & 0.0597 \\ 0.1184 & 0.0369 & 0.1578 & 0.1404 & 0.1304 & 0.1647 & 0.1183 & 0.1293 & 0.0931 & 0.1530 & 0.2127 & 0.0344 \\ 0.1111 & 0.0235 & 0.0576 & 0.1136 & 0.2049 & 0.0961 & 0.0511 & 0.0239 & 0.0481 & 0.0081 & 0.1167 & 0.0630 \\ 0.1232 & 0.0822 & 0.0904 & 0.1407 & 0.1555 & 0.0488 & 0.0589 & 0.2006 & 0.0532 & 0.0165 & 0.0714 & 0.0986 \\ 0.1383 & 0.0437 & 0.1172 & 0.1404 & 0.0443 & 0.1094 & 0.1252 & 0.1947 & 0.1258 & 0.0766 & 0.0232 & 0.1181 \\ 0.1088 & 0.1081 & 0.2016 & 0.0375 & 0.0857 & 0.1232 & 0.0538 & 0.1810 & 0.0280 & 0.1272 & 0.1342 & 0.1025 \\ 0.0655 & 0.0750 & 0.0894 & 0.0270 & 0.1071 & 0.0308 & 0.1672 & 0.0577 & 0.1781 & 0.1568 & 0.1710 & 0.1961 \\ 0.0878 & 0.2101 & 0.2103 & 0.2104 & 0.2281 & 0.1506 & 0.1993 & 0.1316 & 0.1401 & 0.0977 & 0.0930 & 0.1161 \\ 0.0991 & 0.2032 & 0.0645 & 0.0360 & 0.0363 & 0.0578 & 0.1481 & 0.0050 & 0.2071 & 0.1965 & 0.0199 & 0.2114 \end{matrix}]

C_{12} = [\begin{matrix} 0.1420 & 0.0494 & 0.1451 & 0.0104 & 0.0604 & 0.1004 & 0.0547 & 0.0797 & 0.0652 & 0.2010 & 0.0670 & 0.0195 \\ 0.2132 & 0.1474 & 0.0637 & 0.0512 & 0.2281 & 0.0782 & 0.0277 & 0.1613 & 0.0244 & 0.1456 & 0.1398 & 0.1090 \\ 0.0536 & 0.1863 & 0.0595 & 0.2099 & 0.0669 & 0.1184 & 0.1407 & 0.1596 & 0.1927 & 0.2770 & 0.1150 & 0.1100 \\ 0.1505 & 0.0760 & 0.1736 & 0.1377 & 0.1903 & 0.1190 & 0.1403 & 0.0491 & 0.1324 & 0.0609 & 0.1413 & 0.1786 \\ 0.0644 & 0.1722 & 0.1862 & 0.0444 & 0.0469 & 0.1248 & 0.1063 & 0.1176 & 0.0983 & 0.0295 & 0.1708 & 0.1006 \\ 0.1496 & 0.1490 & 0.1305 & 0.0992 & 0.0715 & 0.1168 & 0.1534 & 0.1116 & 0.1311 & 0.0306 & 0.1781 & 0.0861 \\ 0.1548 & 0.0015 & 0.0521 & 0.1767 & 0.0227 & 0.1737 & 0.1785 & 0.1036 & 0.1117 & 0.0177 & 0.3520 & 0.1393 \\ 0.0151 & 0.1329 & 0.0181 & 0.0588 & 0.1433 & 0.0384 & 0.0811 & 0.1667 & 0.1327 & 0.1128 & 0.0254 & 0.1573 \\ 0.0567 & 0.0853 & 0.1712 & 0.2148 & 0.1699 & 0.1303 & 0.1534 & 0.0507 & 0.1115 & 0.1250 & 0.1275 & 0.1079 \end{matrix}]

C_{13} = [\begin{matrix} 0.0988 & 0.0877 & 0.0544 & 0.1409 & 0.1499 & 0.1027 & 0.1542 & 0.1354 & 0.0943 & 0.1071 & 0.1428 & 0.0669 \\ 0.0426 & 0.1798 & 0.0396 & 0.0704 & 0.0326 & 0.1890 & 0.1125 & 0.2196 & 0.0857 & 0.0451 & 0.1121 & 0.1736 \\ 0.1666 & 0.0964 & 0.1653 & 0.0403 & 0.1681 & 0.2084 & 0.0431 & 0.0141 & 0.1433 & 0.1218 & 0.1789 & 0.2011 \\ 0.0745 & 0.1669 & 0.0865 & 0.1457 & 0.1930 & 0.1994 & 0.1124 & 0.1354 & 0.0407 & 0.0153 & 0.1118 & 0.0102 \\ 0.0126 & 0.1791 & 0.1785 & 0.1771 & 0.1003 & 0.0804 & 0.0335 & 0.0660 & 0.0890 & 0.1144 & 0.0033 & 0.1962 \\ 0.2146 & 0.1080 & 0.0787 & 0.0611 & 0.1724 & 0.1348 & 0.0125 & 0.1917 & 0.1610 & 0.1208 & 0.0233 & 0.1157 \\ 0.0690 & 0.0048 & 0.1547 & 0.1253 & 0.1146 & 0.0227 & 0.1933 & 0.0442 & 0.1027 & 0.1334 & 0.1663 & 0.0158 \\ 0.1257 & 0.0808 & 0.0789 & 0.0818 & 0.0302 & 0.0282 & 0.1273 & 0.1025 & 0.1511 & 0.1628 & 0.0933 & 0.1563 \\ 0.1955 & 0.1036 & 0.1626 & 0.1555 & 0.0390 & 0.0344 & 0.2112 & 0.0911 & 0.1320 & 0.1795 & 0.1628 & 0.0615 \end{matrix}]

Optimization analysis of the integrated diagnosis based on the GWO algorithm

GWO algorithm performance analysis

Seven test functions are selected to test its performance to verify the performance of the GWO algorithm. The test functions are listed in Table 3.

Table 3.

Test functions.

Function	Dimension	Range
$F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	30	[−100, 100]
$F_{2} (x) = max \| x \|$	30	[−100, 100]
$F_{3} (x) = \sum_{i = 1}^{n - 1} 100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}$	30	[−30,30]
$F_{4} (x) = \sum_{i = 1}^{n} i \cdot x_{i}^{4} + rand$	30	[−1.28, 1.28]
$F_{5} (x) = \sum_{i = 1}^{n} {x_{i}^{2} - 10 \cdot \cos (2 π x_{i})} + 10 n$	30	[−5.12, 5.12]
$F_{6} (x) = - 20 e^{- 0.2 \sqrt{\frac{\sum_{i = 1}^{n} x_{i}^{2}}{n}}} - e^{\frac{\sum_{i = 1}^{n} \cos (2 π x_{i})}{n}} + 20 + e$	30	[−32, 32]
$F_{7} (x) = \frac{\sum_{i = 1}^{n} x_{i}^{2}}{4000} - \prod_{i = 1}^{n} \cos \frac{x_{i}}{\sqrt{i}} + 1$	30	[−600, 600]

Functions $F_{1} (x)$ , $F_{2} (x)$ , and $F_{3} (x)$ are single-peak functions used to verify the optimization speed of the algorithm. Furthermore, $F_{4} (x)$ , $F_{5} (x)$ , $F_{6} (x)$ , and $F_{7} (x)$ are multi-peak functions that demonstrate the global convergence ability and convergence accuracy of the algorithm. A total of 30 repetitive tests are conducted for each test function to verify the stability of the algorithm. Four different algorithms, GWO, GA,⁴⁷ SAAs),⁴⁸ genetic simulated annealing algorithm,⁴⁹ and sine cosine algorithm (SCA)⁵⁰ are used sequentially for simulation verification. The test results are shown in Figures 4 –10.

Figure 4.

$F_{1} (x)$ test results: (a) $F_{1} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 5.

$F_{2} (x)$ test results: (a) $F_{2} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 6.

$F_{3} (x)$ test results: (a) $F_{3} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 7.

$F_{4} (x)$ test results: (a) $F_{4} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 8.

$F_{5} (x)$ test results: (a) $F_{5} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 9.

$F_{6} (x)$ test results: (a) $F_{6} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 10.

$F_{7} (x)$ test results: (a) $F_{7} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figures 4(a)–10(a) show images of each function; Figures 4(b)–10(b) show the fitness curves of each function; Figures 4(c)–10(c) show the simulation of each function for 30 repetitions of the test, in which Avg, Std, Best, and Worst represent the mean, variance, optimal solution, and worst solution, respectively. Figures 7(b)–10(b) show that the convergence speed of the GWO algorithm is significantly greater than the other algorithms. Furthermore, Figures 5(b), 8(b), and 9(b) shows that the convergence accuracy of the GWO algorithm is significantly more potent than the other algorithms. The stability of the GWO algorithm is considerably more powerful than other algorithms, as seen from (c) of Figures 4 –10. The mean and variance are kept low.

The structural and functional complexity of the equipment leads to the integrated diagnosis optimization problem as a large-scale NP problem. The dimensions of the independent variables for the test functions are increased to 60 and 90, respectively, to verify the superiority of the GWO algorithm in solving large-scale NP puzzles. Thirty test repetitions are carried out, and the simulation results are shown in Figures 11 and 12.

Figure 11.

$F_{4} (x)$ test results at 60 dimensions: (a) fitness curves and (b) stabilization analysis.

Figure 12.

$F_{4} (x)$ test results at 90 dimensions: (a) fitness curves and (b) stabilization analysis.

Comparing Figures 7, 11, and 12 reveals that the GWO algorithm maintains good convergence speed and accuracy with an increase in the dimension of the independent variables, and the stability of the algorithm remains unchanged. The convergence speed and accuracy of the GWO algorithm are significantly more potent than that of the other algorithms. In addition, it has certain advantages in solving large-scale NP problems. Therefore, since the parameters of the GWO algorithm are adaptive, it has a more significant advantage over other non-adaptive algorithms in terms of the algorithm’s speed and accuracy of the algorithm's optimization search, the stability of the algorithm, and the solution of complex problems.

Optimization analysis of integrated diagnosis based on the GWO algorithm

The integrated diagnosis optimization model of the equipment is shown in equation (51), and the constraints are shown in equation (35), such that $γ_{F D C_{r t}} = 1$ , $γ_{F D} \geq 0.99$ , $γ_{F I} \geq 0.98$ , and $γ_{F A} \leq 0.003$ . The serial numbers of critical faults are $f_{6}$ , $f_{9}$ , $f_{10}$ , $f_{11}$ , and $f_{14}$ . If the equipment satisfies the detectability and isolability rate requirements, its quantitative diagnosability requirement is satisfied for this electronic equipment. The simulation process does not consider the characteristics of the equipment.

The GWO, SA, GA, GASA, and SCA algorithms simulate and analyze the integrated diagnosis optimization problem, and 30 repetitive simulation tests are conducted. The specific simulation results are shown in Figure 13. Figure 13(a) indicates that the GWO algorithm is significantly more potent than the other algorithms in terms of the convergence speed and accuracy of the integrated diagnosis optimization. For the overall convergence cost, the GWO algorithm has the most minor convergence cost, and the remaining three algorithms are comparable. Figure 13(b) shows that the stability of the GWO algorithm is stronger than that of the other three algorithms. The results of the GWO-based integrated diagnosis optimization are shown in Figure 13(c), where the preferred tests are $t_{20}, t_{22}, t_{30}, t_{36}$ ; preferred test resources are $r_{20, 9}, r_{22, 7}, r_{30, 7}, r_{36, 7}$ ; the critical fault detection rate is 100%; the fault detection rate is 99.99%; the fault isolation rate is 98.99%; and the false alarm rate is 0.2993%.

Figure 13.

Integrated diagnosis optimization results: (a) fitness curves, (b) stabilization analysis, and (c) GWO optimization results.

The optimal results of the integrated diagnosis of each algorithm in 30 tests are listed in Table 4. The results of the SA-based integrated diagnosis optimization are that the preferred tests are $t_{1}, t_{2}, t_{6}, t_{8}, t_{20}, t_{28}, t_{29}, t_{34}, t_{36}$ and the preferred test resources are $r_{1, 7}, r_{2, 3}, r_{6, 7}, r_{8, 3}, r_{20, 3}, r_{28, 6}, r_{29, 3}, r_{34, 4}, r_{36, 9}$ . The results of the GA-based integrated diagnosis optimization indicate that the preferred tests are $t_{8}, t_{9}, t_{20}, t_{22}, t_{27}, t_{30}, t_{36}$ , and preferred test resources are $r_{8, 9}, r_{9, 3}, r_{20, 6}, r_{22, 8}, r_{27, 8}, r_{30, 5}, r_{36, 7}$ . The results of the GASA-based integrated diagnosis optimization are that the preferred tests are $t_{2}, t_{7}, t_{20}, t_{21}, t_{22}, t_{29}, t_{31}, t_{36}$ , and the preferred test resources are $r_{2, 7}, r_{7, 1}, r_{20, 6}, r_{21, 2}, r_{22, 6}, r_{29, 9}, r_{31, 3}, r_{36, 7}$ . The results of the SCA-based integrated diagnosis optimization are that the preferred tests are $t_{4}, t_{7}, t_{10}, t_{12}, t_{16}, t_{20}, t_{22}, t_{36}$ , and the preferred test resources are $r_{4, 3}, r_{7, 1}, r_{10, 4}, r_{12, 6}, r_{16, 8}, r_{20, 9}, r_{22, 7}, r_{36, 7}$ .

Table 4.

Optimization results of the integrated diagnosis.

Algorithm	$γ_{F D C_{r t}}$	$γ_{F D}$	$γ_{F I}$	$γ_{F A}$	Success rate	p
GWO	100%	99%	98.99%	0.30%	100%	3.35e-07
SA	100%	99%	98.27%	0.30%	100%	1.73e-06
GA	100%	99%	98.78%	0.30%	90%	1.73e-06
GASA	100%	99%	98.57%	0.30%	77%	1.67e-06
SCA	100%	99%	98.90%	0.30%	92%	2.21e-07

For the number of preferred tests, the GWO algorithm optimizes four tests, which is significantly less than the other three. Compared with before the integrated diagnosis optimization, the number of tests of the device is reduced by 88.9%, and the test cost is saved by 89%. The GWO reduces the number of tests by 42%–55% and the test cost by 77.63%–83.91% compared with the other algorithms, which is the best in the optimization results.

In terms of constraints, all four algorithms satisfy the conditions of equipment diagnosability and are not substantially different from each other. The success rates of the algorithms are shown in Table 4. The success rates of GWO and SA are 100%, which is better than that of the other two algorithms. A non-parametric statistical test, Wilcoxon’s rank-sum test, was accomplished at 5% significance level to see whether the obtained results of GWO differ from other benchmark trainers in a statistically significant way. Therefore, the obtained P-values are shown in the table of results as well. Logically, P-values less than 0.05 are considered as powerful evidence against the null hypothesis.

In summary, the GWO has the best convergence rates and accuracy among other benchmark methods. Figure 13(b) shows that GWO is caused by other algorithms in terms of average performance and stability performance. The results of the P-values in Table 4 produced from the Wilcoxon’s rank-sum test indicate that the superiority is statistically significant.

Equipment integrated diagnosis optimization analysis

This section starts with the comprehensive evaluation of the test cost, integrating various influencing factors. According to the hierarchical analysis method, the test cost of each alternative test resource in lays the foundation for integrated diagnosis optimization. This section verifies the superiority of the proposed GWO algorithm through simulation. Finally, the GWO algorithm is used to obtain the results of the integrated diagnosis optimization of the equipment, further verifying the effectiveness and reasonableness of the GWO algorithm.

Conclusion

Based on the principle of spatial mapping, the integrated diagnosis optimization space of equipment is constructed for the integrated diagnosis optimization problem of the equipment. The mapping relationships of test space, resource space, fault space, and its subspace are established. Based on the logical mapping relationship of the integrated diagnosis space, the test optimization model and resource optimization model are found. Based on the logical connection between the subspace mapping variables, the test optimization and resource optimization models were integrated, and the integrated diagnosis optimization model was designed. Based on the hierarchical analysis method, the logical mapping relationship of the cost space was established to make the test resource allocation more reasonable. The cost matrix was evaluated to simplify the integrated diagnosis optimization model, which made the integrated diagnosis optimization process more sensible and practical. A simulation was used to verify the effectiveness of the proposed algorithm and model based on the GWO algorithm for the integrated diagnosis optimization model of the electronic equipment.

The integrated diagnosis optimization method of the equipment proposed in this paper effectively solves the current problem of the lack of correlation between test and resource optimization by establishing an integrated diagnosis optimization model based on the principle of spatial mapping. Based on the hierarchical analysis method and mapping relationship of the cost space, the integrated evaluation of the effect of each type of cost on the results of integrated diagnosis optimization makes the equipment test resource allocation more reasonable. The electronic equipment after integrated diagnosis and optimization has a critical fault detection rate of 100%, fault detection rate of 99.99%, fault isolation rate of 98.99%, and false alarm rate of 0.2993%. Compared with the before integrated diagnosis and optimization, the number of tests for the equipment has been reduced by 88.9%, and test costs have been saved by 89%.

In this paper, the proposed integrated diagnosis optimization has to be performed on the premise that the fault information of the device is known to address the inability to deal with unknown faults of the device. This integrated diagnosis optimization does not consider the effect of generic and multipurpose test resources on the results of the integrated diagnosis optimization. Therefore, in future, the research focuses on the following aspects: integrated diagnosis optimization for unknown faults, considering the effect of generic tests and multi-purpose test resources on the integrated diagnosis optimization, and a dynamic integrated diagnosis optimization strategy capable of reconfiguring tests by scheduling on-line test resources according to test resource faults.

Footnotes

Authors’ contribution

Concept and design: Xian-Jun Shi and Xu-ping Gu; data collection and analysis: Xu-ping Gu; drafting of the article: Xu-ping Gu; critical revision of the article for important intellectual content: Xu-ping Gu and Xian-Jun Shi; study supervision: Xian-Jun Shi. All the authors approved the final version of this article.

Data availability

All data that support the findings of this study are included in this manuscript and its supplementary information files.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Xu-ping Gu

Annex 1. Mapping relationships in the integrated diagnosis space

It is known that, the test space $T = s p a n {t_{1}, t_{2}, \dots, t_{36}}$ , fault space $F = s p a n {f_{1}, f_{2}, \dots, f_{22}}$ , and resource space $R = s p a n {r_{1}, r_{2}, \dots, r_{36}}$ , where $r_{i} = {r_{i 1}, r_{i 2}, \dots, r_{i 9}}$ . Then, the correlation matrix $D$ corresponding one-to-one to the correlation mapping $Φ : T \mapsto F$ is The fault rate vector $λ_{f}$ corresponding to the fault rate mapping $Γ : F \mapsto F$ is

\begin{aligned} λ_{f} = [\begin{matrix} 0.00185 & 0.00923 & 0.185 & 0.00185 & 0.00185 & 0.00923 & 0.00185 & 0.00923 & 0.185 & 0.185 & 0.185 \end{matrix} \\ {\begin{matrix} 0.00185 & 0.00923 & 0.185 & 0.00923 & 0.00185 & 0.00923 & 0.00185 & 0.00185 & 0.00185 & 0.00185 & 0.00185 \end{matrix}]}^{T} \end{aligned}

The resource fault rate matrix

C^{f}

corresponding to the resource fault rate mapping

Γ^{1}

C^{f} = [\begin{matrix} C_{1}^{f} & C_{2}^{f} & C_{3}^{f} \end{matrix}]

C_{1}^{f} = [\begin{matrix} 0.003 & 0.004 & 0.001 & 0.003 & 0.003 & 0.004 & 0.002 & 0.003 & 0.003 & 0.002 & 0.004 & 0.004 \\ 0.002 & 0.005 & 0.003 & 0.003 & 0.002 & 0.002 & 0.003 & 0.002 & 0.002 & 0.002 & 0.005 & 0.007 \\ 0.004 & 0.005 & 0.001 & 0.002 & 0.004 & 0.001 & 0.004 & 0.003 & 0.003 & 0.005 & 0.009 & 0.006 \\ 0.002 & 0.003 & 0.003 & 0.004 & 0.004 & 0.002 & 0.002 & 0.001 & 0.001 & 0.006 & 0.005 & 0.005 \\ 0.004 & 0.004 & 0.002 & 0.006 & 0.002 & 0.003 & 0.003 & 0.005 & 0.002 & 0.002 & 0.006 & 0.004 \\ 0.001 & 0.004 & 0.003 & 0.002 & 0.001 & 0.001 & 0.001 & 0.002 & 0.003 & 0.002 & 0.004 & 0.006 \\ 0.001 & 0.007 & 0.004 & 0.003 & 0.004 & 0.002 & 0.002 & 0.006 & 0.004 & 0.004 & 0.009 & 0.003 \\ 0.002 & 0.001 & 0.003 & 0.005 & 0.005 & 0.003 & 0.003 & 0.004 & 0.005 & 0.006 & 0.007 & 0.002 \\ 0.003 & 0.003 & 0.002 & 0.001 & 0.002 & 0.004 & 0.002 & 0.002 & 0.006 & 0.002 & 0.005 & 0.004 \end{matrix}]

C_{2}^{f} = [\begin{matrix} 0.002 & 0.003 & 0.003 & 0.002 & 0.004 & 0.003 & 0.002 & 0.003 & 0.002 & 0.003 & 0.003 & 0.004 \\ 0.004 & 0.002 & 0.002 & 0.003 & 0.002 & 0.002 & 0.001 & 0.002 & 0.001 & 0.002 & 0.003 & 0.006 \\ 0.001 & 0.004 & 0.003 & 0.001 & 0.005 & 0.005 & 0.002 & 0.001 & 0.003 & 0.001 & 0.002 & 0.002 \\ 0.003 & 0.005 & 0.002 & 0.003 & 0.006 & 0.003 & 0.006 & 0.006 & 0.002 & 0.002 & 0.002 & 0.007 \\ 0.005 & 0.006 & 0.003 & 0.002 & 0.004 & 0.006 & 0.002 & 0.005 & 0.001 & 0.003 & 0.002 & 0.005 \\ 0.006 & 0.001 & 0.001 & 0.003 & 0.003 & 0.002 & 0.006 & 0.002 & 0.003 & 0.006 & 0.002 & 0.004 \\ 0.002 & 0.003 & 0.003 & 0.004 & 0.002 & 0.002 & 0.004 & 0.003 & 0.004 & 0.003 & 0.003 & 0.009 \\ 0.003 & 0.002 & 0.001 & 0.003 & 0.003 & 0.001 & 0.002 & 0.002 & 0.003 & 0.004 & 0.002 & 0.005 \\ 0.003 & 0.002 & 0.004 & 0.001 & 0.002 & 0.001 & 0.006 & 0.003 & 0.001 & 0.003 & 0.005 & 0.002 \end{matrix}]

C_{3}^{f} = [\begin{matrix} 0.003 & 0.004 & 0.001 & 0.001 & 0.002 & 0.005 & 0.002 & 0.003 & 0.002 & 0.002 & 0.002 & 0.004 \\ 0.002 & 0.004 & 0.003 & 0.003 & 0.002 & 0.002 & 0.003 & 0.002 & 0.001 & 0.002 & 0.001 & 0.004 \\ 0.003 & 0.003 & 0.002 & 0.002 & 0.003 & 0.003 & 0.004 & 0.002 & 0.003 & 0.003 & 0.005 & 0.004 \\ 0.003 & 0.002 & 0.001 & 0.003 & 0.004 & 0.002 & 0.002 & 0.001 & 0.002 & 0.003 & 0.004 & 0.005 \\ 0.002 & 0.004 & 0.005 & 0.002 & 0.001 & 0.002 & 0.003 & 0.005 & 0.005 & 0.004 & 0.006 & 0.004 \\ 0.003 & 0.006 & 0.005 & 0.003 & 0.001 & 0.001 & 0.006 & 0.003 & 0.005 & 0.004 & 0.003 & 0.003 \\ 0.002 & 0.003 & 0.003 & 0.002 & 0.002 & 0.003 & 0.005 & 0.003 & 0.003 & 0.002 & 0.003 & 0.003 \\ 0.003 & 0.002 & 0.001 & 0.004 & 0.001 & 0.005 & 0.006 & 0.001 & 0.002 & 0.006 & 0.001 & 0.002 \\ 0.003 & 0.002 & 0.002 & 0.005 & 0.003 & 0.006 & 0.002 & 0.001 & 0.001 & 0.001 & 0.003 & 0.002 \end{matrix}]

The maximum mean difference yields a quantitative evaluation matrix

D_{f}

corresponding to the quantitative evaluation mapping

Θ

. Assuming that the test optimization vector is

x_{1} = {[\begin{matrix} 1 & 1 & \dots & 1 \end{matrix}]}_{1 \times 36}

, the matrix

D_{f}

D_{f} = [\begin{matrix} D_{f 1} & D_{f 2} \end{matrix}]

, where

D_{f 1} = [\begin{matrix} 0.9874 & 0.7452 & 0.7685 & 0.8542 & 0.7621 & 0.7852 & 0.6843 & 0.8542 & 0.7456 & 0.7456 & 0.6524 \\ 0.7452 & 0.9853 & 0.6587 & 0.4856 & 0.7856 & 0.6524 & 0.6231 & 0.7451 & 0.6235 & 0.4251 & 0.5326 \\ 0.7685 & 0.6587 & 0.9474 & 0.5862 & 0.5472 & 0.6254 & 0.7485 & 0.6524 & 0.9586 & 0.7548 & 0.6285 \\ 0.8542 & 0.4856 & 0.5862 & 0.9863 & 0.8547 & 0.2658 & 0.7459 & 0.7451 & 0.5368 & 0.8542 & 0.6258 \\ 0.7621 & 0.7856 & 0.5472 & 0.8547 & 0.9374 & 0.4265 & 0.5286 & 0.7584 & 0.9652 & 0.4251 & 0.6584 \\ 0.7852 & 0.6524 & 0.6254 & 0.2658 & 0.4265 & 0.9862 & 0.2563 & 0.7456 & 0.5685 & 0.6285 & 0.6254 \\ 0.6843 & 0.6231 & 0.7485 & 0.7459 & 0.5286 & 0.2563 & 0.9663 & 0.6524 & 0.7451 & 0.6251 & 0.3251 \\ 0.8542 & 0.7451 & 0.6524 & 0.7451 & 0.7584 & 0.7456 & 0.6524 & 0.9862 & 0.7451 & 0.5264 & 0.2654 \\ 0.7456 & 0.6235 & 0.9586 & 0.5368 & 0.9652 & 0.5685 & 0.7451 & 0.7451 & 0.9874 & 0.4521 & 0.2658 \\ 0.7465 & 0.4251 & 0.7548 & 0.8542 & 0.4251 & 0.6258 & 0.6251 & 0.6251 & 0.4512 & 0.9761 & 0.2548 \\ 0.6254 & 0.5326 & 0.6258 & 0.6258 & 0.6584 & 0.6254 & 0.3251 & 0.2654 & 0.285 & 0.2584 & 0.9855 \\ 0.5658 & 0.3562 & 0.5478 & 0.2548 & 0.5284 & 0.2351 & 0.6589 & 0.2548 & 0.2653 & 0.5489 & 0.2659 \\ 0.2653 & 0.2546 & 0.4578 & 0.6254 & 0.6584 & 0.2896 & 0.6524 & 0.7456 & 0.5489 & 0.6521 & 0.3659 \\ 0.8452 & 0.5148 & 0.5685 & 0.2358 & 0.6589 & 0.8214 & 0.2653 & 0.5784 & 0.2547 & 0.2478 & 0.4256 \\ 0.6251 & 0.5624 & 0.5878 & 0.4584 & 0.2352 & 0.1943 & 0.2514 & 0.7562 & 0.6598 & 0.4526 & 0.7658 \\ 0.2358 & 0.1487 & 0.7956 & 0.5695 & 0.1254 & 0.597 & 0.2135 & 0.5478 & 0.2358 & 0.2458 & 0.8526 \\ 0.6254 & 0.8451 & 0.5847 & 0.5246 & 0.5628 & 0.2413 & 0.2189 & 0.5362 & 0.5984 & 0.2654 & 0.6359 \\ 0.1254 & 0.2568 & 0.5685 & 0.6524 & 0.2654 & 0.2584 & 0.8754 & 0.7418 & 0.5426 & 0.2659 & 0.8542 \\ 0.8562 & 0.5476 & 0.6258 & 0.2658 & 0.2548 & 0.1635 & 0.9521 & 0.7469 & 0.2654 & 0.5487 & 0.6258 \\ 0.2654 & 0.2358 & 0.6548 & 0.2485 & 0.2659 & 0.2143 & 0.6985 & 0.7452 & 0.2578 & 0.5689 & 0.9852 \\ 0.7548 & 0.5487 & 0.5847 & 0.6254 & 0.5487 & 0.1254 & 0.2597 & 0.5628 & 0.5462 & 0.5489 & 0.6358 \\ 0.6254 & 0.6859 & 0.6584 & 0.5416 & 0.5624 & 0.2658 & 0.5498 & 0.2654 & 0.2654 & 0.9754 & 0.7549 \end{matrix}]

D_{f 2} = [\begin{matrix} 0.5658 & 0.2653 & 0.8452 & 0.6251 & 0.2385 & 0.6254 & 0.1254 & 0.8562 & 0.2654 & 0.7548 & 0.6254 \\ 0.3562 & 0.2546 & 0.5148 & 0.5624 & 0.1478 & 0.8451 & 0.2568 & 0.5476 & 0.2358 & 0.5487 & 0.6859 \\ 0.5478 & 0.4578 & 0.5685 & 0.5878 & 0.7956 & 0.5874 & 0.5685 & 0.6258 & 0.6548 & 0.5847 & 0.6584 \\ 0.2548 & 0.6254 & 0.2358 & 0.4584 & 0.5659 & 0.5246 & 0.6254 & 0.2658 & 0.2485 & 0.6254 & 0.5416 \\ 0.5284 & 0.6584 & 0.6589 & 0.2352 & 0.1254 & 0.5628 & 0.2654 & 0.2548 & 0.2659 & 0.5487 & 0.5624 \\ 0.2351 & 0.2896 & 0.8214 & 0.1943 & 0.2597 & 0.2413 & 0.2584 & 0.1635 & 0.2143 & 0.1254 & 0.2658 \\ 0.6589 & 0.6524 & 0.2653 & 0.2514 & 0.2135 & 0.2189 & 0.8576 & 0.9521 & 0.6985 & 0.2597 & 0.5498 \\ 0.2548 & 0.7456 & 0.5784 & 0.7562 & 0.5478 & 0.5362 & 0.7418 & 0.7469 & 0.7452 & 0.5628 & 0.2498 \\ 0.2563 & 0.5489 & 0.2547 & 0.5698 & 0.2358 & 0.5984 & 0.5426 & 0.2654 & 0.2578 & 0.5462 & 0.2654 \\ 0.5489 & 0.6521 & 0.2478 & 0.4526 & 0.2458 & 0.2654 & 0.2659 & 0.5487 & 0.5689 & 0.5489 & 0.8754 \\ 0.2659 & 0.3659 & 0.4256 & 0.7658 & 0.8562 & 0.6359 & 0.8542 & 0.6258 & 0.9852 & 0.6358 & 0.7549 \\ 0.9852 & 0.2784 & 0.9584 & 0.5736 & 0.5846 & 0.2196 & 0.5463 & 0.2485 & 0.2354 & 0.8456 & 0.7462 \\ 0.2784 & 0.9685 & 0.2451 & 0.7456 & 0.9563 & 0.2547 & 0.6254 & 0.6258 & 0.6248 & 0.5639 & 0.5482 \\ 0.9584 & 0.2451 & 0.9552 & 0.9876 & 0.6254 & 0.8563 & 0.5476 & 0.3529 & 0.6254 & 0.6358 & 0.3658 \\ 0.5736 & 0.7456 & 0.5476 & 0.2458 & 0.2485 & 0.6358 & 0.4967 & 0.6254 & 0.7652 & 0.4683 & 0.2486 \\ 0.5846 & 0.9563 & 0.6254 & 0.6358 & 0.9552 & 0.6257 & 0.4562 & 0.8546 & 0.8746 & 0.6358 & 0.8459 \\ 0.2196 & 0.2547 & 0.8563 & 0.6358 & 0.6257 & 0.9563 & 0.5476 & 0.6258 & 0.5476 & 0.7658 & 0.5474 \\ 0.5463 & 0.6254 & 0.5476 & 0.4976 & 0.4562 & 0.5476 & 0.9745 & 0.2584 & 0.3268 & 0.3548 & 0.3149 \\ 0.2485 & 0.6358 & 0.3529 & 0.6354 & 0.8546 & 0.6258 & 0.2584 & 0.9843 & 0.5246 & 0.3258 & 0.2548 \\ 0.2354 & 0.6248 & 0.6254 & 0.7652 & 0.8746 & 0.5476 & 0.3268 & 0.5246 & 0.9871 & 0.3254 & 0.3163 \\ 0.8456 & 0.5639 & 0.6358 & 0.4683 & 0.6358 & 0.7658 & 0.3584 & 0.3254 & 0.2354 & 0.9356 & 0.2147 \\ 0.7462 & 0.5482 & 0.3658 & 0.2486 & 0.8459 & 0.5474 & 0.3149 & 0.2163 & 0.2163 & 0.2147 & 0.9853 \end{matrix}]

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Integrated diagnosis optimization design of the electronic equipment based on spatial mapping

Abstract

Keywords

Introduction

Integrated diagnosis optimization model

Characterization of the integrated diagnosis optimization

Spatial mapping principle

Integrated diagnosis space

Integrated diagnosis optimization model

Test optimization

Test resource optimization

Integrated diagnosis optimization model

Comprehensive evaluation of testing resource cost types

Test resource cost types

Test cost comprehensive evaluation

Resource cost comprehensive evaluation

Comprehensive evaluation of the cost matrix C

Integrated diagnosis optimization based on the GWO algorithm

GWO algorithm

Integrated diagnosis optimization steps

Simulation verification

Comprehensive evaluation of the cost matrix C

Comprehensive evaluation of test costs

Integrated evaluation of resource costs

Comprehensive evaluation of the cost matrix

Optimization analysis of the integrated diagnosis based on the GWO algorithm

GWO algorithm performance analysis

Optimization analysis of integrated diagnosis based on the GWO algorithm

Equipment integrated diagnosis optimization analysis

Conclusion

Footnotes

Authors’ contribution

Data availability

Declaration of conflicting interests

Funding

ORCID iD

Annex 1. Mapping relationships in the integrated diagnosis space

References

Comprehensive evaluation of the cost matrix $C$

Comprehensive evaluation of the cost matrix $C$