optimal selection of tests for fault diagnosis in multi
TRANSCRIPT
https://doi.org/10.1007/s10836-020-05854-9
Optimal Selection of Tests for Fault Diagnosis in Multi-Path Systemwith Time-delay
Zhexi Yao1,2 · Lingchao Zhu1,2 · Tao Zhang2 · JinboWang2
Received: 25 July 2019 / Accepted: 6 January 2020© Springer Science+Business Media, LLC, part of Springer Nature 2020
AbstractTest selection is an important process to simplify the system structure and build a diagnostic system. Test selection problemin a multi-path system with time delay is investigated in this paper. The fault detection rate (FDR) and fault isolation rate(FIR) of the multi-path system are introduced. A heuristic function is then established to evaluate the uncertainty of thetest, which can provide heuristic information to improve searching efficiency of the algorithm. An improved heuristic SSOalgorithm GSSO is proposed to solve the test selection problem. The grey system theory is introduced into GSSO to improvethe effect of optimization. The GSSO is compared with DPSO and SSO algorithms, and shows that the characteristics ofjumping out of the local optimal solution earlier and faster convergence speed.
Keywords Test selection · Swallow swarm optimization · Grey system theory
1 Introduction
Test selection problem for fault diagnosis is to find theminimal cost subset of tests that achieves a specifieddiagnostic resolution. It is an important part of thesystem testability design. The testability measured in thedevelopment process (design process) may greatly reducethe testing time, effort, rework and cost. The process oftest ability design is listed as follows: Firstly, the diagnosticdictionary matrix is constructed to denote the relationshipbetween faults and tests. In this step, the popular methodis the fault dictionary. Secondly, the tests are selected tomake the system satisfy the required measure. And the FDRand FIR are usually used, which are direct indicators indiagnosis. Thirdly, the diagnosis strategy is proposed andthe diagnosis system is built. Test selection of system levelin step 2 is the focus of this paper. It can be described aschoosing the best test sets to satisfy the required testabilitywith the lowest test cost.
Responsible Editor: X. Wen
� Zhexi [email protected]
1 University of Chinese Academy of Sciences, Beijing, China
2 Technology and Engineering Center for Space Utilization,Chinese Academy of Sciences, Beijing, China
Usually, test selection can be divided into perfect testingand imperfect testing depending on different premises. Thebasic premise of perfect testing is the relationship betweenthe faults and the tests is determined. It is obvious that theassumption of perfect testing lacks consideration for factorssuch as unreliability test, time delay and so on. However,the existing systems often work under complex environmentconditions. If only perfect test cases are considered, it willlead to false alarm and missing detection. Actually, thestudy of imperfect test is of great challenge, especiallyin the complex system with multi-path. Firstly, there aremore working modes and working paths in the system.The complex system is constructed by several subsystems,while each subsystem contributes to finish the final project.Depending on different situation, the system will makedifferent choices and get into different paths. Secondly,the time delay is a common phenomenon in most of thecomplex systems. The delay discussed in this paper isthe time delay between faults and tests. With time delaybeing considered, conventional FDR and FIR definitions areinapplicable. This paper focuses on the multi-path systemand discusses the influence of time delay.
An explicit example of multi-path system is demon-strated in Fig. 1. The system in Fig. 1 has two paths, Path1and Path2. With different switching conditions, the startmodesm1 chooses between Path1 and Path2. As Fig. 1 illus-trates, each path has two working modes, one is m1 andm2, the other is m1 and m3. Every working mode consists
Journal of Electronic Testing (2020) 36:75–86
/ Published online: 28 January 2020
Fig. 1 A simple example
of five subsystems: sub1, sub2, sub3, sub4 and sub5, whichmay work asynchronously. Our focus is the test selectionfor the whole system, and the test selection of each pathis considered. Since the modes vary in different paths, theduration of each subsystem as well as each fault will be dif-ferent. It is necessary to get the CETime(continuous existingtime) of different subsystem under consideration, which isrelated to the time delay between faults and tests. We give abrief and new constructive definition on CETime: it is a setincludes the continuous work duration of a subsystem or atest in a path. The CETime of different subsystems of Path1in Fig. 1 are shown in Fig. 2. With working mode changes,the CETime varies. And it is possible for some subsystemsto last across several working modes, such as sub5 in Path1.So the original test selection problem in the multi-path sys-tem can be divided based on different paths. And the newdefinition of test selection is described as: to find the bestarray of tests that give the satisfied FDR and FIR with thelowest test costs in different paths considering the CETime.
Considering the testability, it’s of great need to takeall situations into consideration and ensures entire pos-sible paths meet the testability requirement. So in thispaper, the method about test selection in the multi-pathsystem is discussed and an improved SSO algorithm isproposed. The main contributions of this article are asfollows:
– The calculation of FDR and FIR is proposed undermulti-path system with time delay.
– A GSSO algorithm is proposed, and the grey theory isintroduced to improve the performance.
– The results show good performance compared withother optimization algorithms.
The rest of this paper is organized as follows. Section2 summarizes related work. The discussion of test pointselection problem in the multi-path switching system withtime delay is presented in Section 3. The GSSO algorithmand its construction in test point selection are studied inSection 4. Sections 5 demonstrates the effectiveness of theproposed method with two experiments. Finally, Section 6concludes the whole paper.
2 RelatedWork
The test selection problem is an important part of testdesign, which has been widely concerned. Lots of researchwith different selection criteria about the test selectionproblem have been proposed in recent years. Slamani andKaminska propose fault sensitivity to select test and theexperiment shows the efficiency of the method [10, 15].Pinjala and Bruce applied the maximum isolated fault intest selection [14]. Starzyk introduces an entropy-basedapproach to select the near minimum test set [8]. Due tothe complexity of the system, the imperfect test [16, 21, 29]has higher requirements on testability design. An imperfecttest can be roughly described as there are uncertain or
Fig. 2 The continuous existing time of subsystems in Path1
J Electron Test (2020) 36:75–8676
time-dependent test results for a failure state. The existingresearches on imperfect test mainly focus on the reliabilityof the test and the correlation between faults. Imperfecttests introduce an additional element of uncertainty into thediagnostic and test selection process: the pass outcome of atest does not guarantee the normality of components undertest because the test may has missed a fault; in the same waythat a failed test outcome does not mean that one or moreof the implicated components are faulty as the false alarm.The consequences of an imperfect test lead to detectionmissed and false alarm. Thus, test selection must considerthis uncertainty in test outcomes. In the imperfect test, Wei,Wenchao develops a tabu search algorithm to deal withthe test sequencing problem under imperfect tests [24]. Ye,Xuerong, developed a joint distribution-based TME modelto evaluate the test selection, testability metric estimationfor unreliable tests. The efficacy of the developed TMEmodel was verified in [28].
Heuristic iterative search techniques have been widelyused in testability design. The main idea of the algorithmis to imitate the behavior of members in nature, andcontributes to the final subject. Heuristic algorithm includesthe genetic algorithm based on evolution, the particle swarmoptimization algorithm based on group migration, the neuralnetwork and so on. The genetic algorithm is a generalpurpose, population-based search algorithm in which theindividuals in the population represent samples from theset of all possibilities. What’s more, they are solutions ina problem space, strategies for a game, rules in classifiersystems, or arguments for problems in function optimization[23]. GA is based on the Darwinian principle, which hasthe reproduction and survival of the fittest and analogs ofnaturally occurring genetic operations such as crossoverand mutation. In recent years, GA is popular in the areaof testability design. Muhammad Bilal Bashir and AamerNadeem propose the improved GA to Mutation test. Theimproved GA is used to generate the test case automaticallyand minimizes the cost of mutation test [1]. The GA isused to the selection of test selection in analog circles by T.Golonek and J. Rutkowski, and the efficiency is improved[5]. Wan Jiuqing modeled the optimal test selection assearching for the test subset with the maximal diagnosticinformation in all tests of the circuit [7]. Shigang Zhanguses the GA and LRA algorithm to deal with the testabilitydesign in the imperfect test [30]. The initialization of thepopulation in GA is modified based on the imperfect tests.At the same time, the process of crossover and mutationis combined with the principle in Two-Value Champion.The quality of the population in the algorithm is improved.The variable mutation rate is given in this algorithm toreduce the possibility of trapped in the local optimum. C.Yang [26] proposed the grouped genetic algorithm in thesystem with multiple operation modes to solve the test
selection problem. Experiments are carried out to verifythe effectiveness of the algorithm, and it is proved thatthe algorithm can effectively reduce the running time.The PSO was proposed by Kennedy J and Eberhart R Cinspired by the foraging behaviors of birds [6]. The PSOhas been widely used in many fields due to its simplicityand efficiency. Shi X applied the DPSO algorithm to solvethe traveling salesman problem and verify the effectivenessof the algorithm [22]. Sen Deng used the DPSO into testselection of imperfect tests and took the uncertainty of testsinto account by adopting fitness function [4]. Chenglin Yang[27] proposed the combination of DPSO and the greedyalgorithm in sequential diagnosis. The test set with minimalcosts was chosen with the satisfaction of FDR and FIR.Ronghua Jiang [9] modified the DPSO with the multi-valuealgorithm. They constructed the fitness function with threegoals in testability design: the highest FDR and the minimalcosts. A tree dimension vector is constructed to representthe different values of those three goals.
The Swallow Swarm Optimization (SSO), inspired bysocial behavior simulation of Swallow, was originallydesigned and developed by Mehdi Neshat and GhodratSepidnam [17]. In detail, this algorithm is an efficientcombination of particle swarm optimization and artificialfish swarm algorithm (FSO). And with the comparison andcomputation, the algorithm will give out an array of sets thatsatisfy the limitation. Rutuparna Panda and Sanjay Agrawal[19] proposed an adaptive swallow swarm optimizationalgorithm in the optimal multilevel thresholding of brainMR images. The result was better than other methods in theexperiment. Safaa BOUZIDI and Mohammed Essaid RIFFI[3] solved the traveling salesman problem with the discreteswallow swarm optimization algorithm. They improved theSSO with discrete value and got a good result. Shuang Xuand Xingwei Wang [25] applied the SSO in Low EarthOrbit satellite networks, and proposed proportional fairnessbased resources allocation algorithm to get a good allocationin the resource, when considering the Inter-Satellite Links.A.Kaveh and T.Bakhshpoori [12] combined the PSO andSSO in the research of truss layout and size optimization,they proposed the HPSSO algorithm and got a good resultcompared with PSO.
Most of the methods mentioned above are used to handleoptimization problems. In reality, most of the optimizationproblems are with multidimensional constraints and withmany non-linear factors. And the heuristic iterative searchtechniques is easy to be trapped in the local optimum.What’s more, most complex system faces serious timedelay. In this paper, the calculations of FDR and FIR withthe influence of the time delay are discussed in differentpaths, and the grey system is introduced to reduce thepossibility of being trapped in the local optimum for theSSO algorithm.
J Electron Test (2020) 36:75–86 77
3 Test Selection in Multi-path SystemwithTime Delay
3.1 Problem Formulation
For the last few decades, particular attention has been paidto the problem of testability design in the complex system.The literature reviewed shows the lacks of focus on timedelay system about testability design. In this paper, theinfluence of time delay on test selection is studied.
3.2 Mathematical Description
The concept related to the formulation of test selection islisted as follows, including the array of working modes, thearray of tests, the array of fault states, and so on.
1) M = {M1, M2, M3 . . . M|M|}, is the set consisted ofall working modes.
2) P = {P1, P2, . . . , P|P |
}, where P is the set of all
working paths, —P— stands for the total paths.3) A = {α1, α2, . . . α|S|}; where A is the set of all
the failure probability, αi is the failure probability ofsubsystem i, and —S— is the total subsystems. It is
obvious that|S|∑
i=1αi = 1
4) F = {{f 11 , f 1
2 , ..., f 1|S1|}
, ...,{f
|S|1 , f
|S|2 , ..., f |S|
|S|S||}};
where F is the set of different failure mode in eachsubsystem, in which f i
j is the j failure mode of isubsystem.
5) B = {{
β11, β1
2, ..., β1|s1|}
, ...,{
β|S|1 , β|S|
2 , ..., β|S||S|S||
}};B
is the set of all the failure probability of different
modes in each subsystem, and|Si |∑
j=1βi
j = 1, i ∈[1, 2, 3, . . . , |S|]
6) T = [1, 2, ..., |T |]; where T is the set of all the tests tobe chosen, —T— is the total tests.
7) E = [e1, e2, ..., e|T |]; where E is a vector shows if atest is chosen. eq=1, if q is selected, 0 otherwise.
8) C = {c1, c2, ..., c|T |}; where C is the set of nominated
test costs,|T |∑
q=1|cq | = 1
9) K ∈ [0, 1]|T |×|M|; where K is the matrix of the testsand the working modes.kim = 1, if i test is included inmodem, 0 otherwise.
10) G ∈ [0, 1]|S|×|M|; where G is the matrix ofsubsystems and the working modes.gim=1, if isubsystem is included in modem, 0 otherwise.
11) D = [dijq ]′; where D is the diagnostic dictionary
matrix. 0 ≤ dijq ≤ 1, di
jqstands for the probability thatthe j fault in i subsystem can be detected by q test.
12) DL = [dlijq ]′, where DL is the matrix of time delay
between the faults and tests.0 ≤ dlijq ≤ 1, dlijqstandsfor the time delay of test q to detect j fault in isubsystem.
13) T R = {T r1, T r2, . . . T r|M|}; where T R is the set ofduration for each working mode.
14) SD = [SDijl]′;SDi
jl stands for the CETime of j failureof i subsystem in path l.
15) T D = [T Dql]′; T D is the set of all the CETime oftests.
Unlike conventional systems, more factors need to beconsidered in the testability design of complex systems. Pij
is an important part in FDR, which means the probabilitythat j fault of in i subsystem is detected. So withoutconsidering time delay, the P m
ij in FDR is listed as:
P mij = 1 −
|T |∏
q=1
(1 − dijq)
eq×gim×kqm
(1)
As is shown in Eq. 1, P mij denotes the possibility of j
fault in i subsystem can be detected in mode m withoutthe time delay. However, the time delay appears in most ofthe complicated systems. What’s more, with the switchingof the working path, the testability is supposed to beconsidered in different paths rather than in different modes.FDR needs to be redefined, the new equation of P l
ij can bedefined in Eq. 2:
P lij = 1 −
|T |∏
q=1
∏
T Y∈SDijl
⋂̃T Dql
(1 − fph(i, j, q) × dijq)
eq (2)
Where, the SDijl is the CETime of j failure of i subsystem
in path l and T Dql is the CETime of test q in the path l.
The⋂̃
denotes the set of modes, in which faults and testsboth exist. As a conclusion, the TY is the member of theset which consists of the intersection of each subset in SDi
jl
and T Dql . In a complex system, the tests with shorter timedelay are recommended. A punish function fph(i, j, q) isdefined for optimization choice, who gives the tests withshorter delay a higher possibility of being chosen. Thepunish function is described in Eq. 3:
fph(i, j, q) =
⎧⎪⎨
⎪⎩
1 − dlijq∑
r∈T Y
T RrT Y �= ∅anddlijq <
∑
r∈T Y
T Rr
0 else
(3)
Where dlijq is the time delay of test q to j failure in i
subsystem. The denominator computes the total durationthat the corresponding j failure and test q both exist.If there is no time delay (dlijq = 0), the punishfunctionfph(i, j, q) = 1, and the opportunity of q to beselected is not decrease. Then, the possibility of i subsystem
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been detected in path l is defined in Eq. 4:
rli =
|si |∑
j=1
(βijP
lij ) (4)
And the FDR in path l is defined:
FDRl =
|S|∑
i=1(αiγ
li )
|S|∑
i=1(αiGl(i))
(5)
Gl = G′(M1) ∪ G′(M2)... ∪ G′(M|M|) (6)
Where Gl is the union of the subsystems in different modes,which denotes to all subsystems in this path. The G′ is thetransposed matrix of G.
The regular calculation formula of FIR is listed in Eq. 7.But in multi-path system, the calculation of FIR also needsto be reconsidered. The reconstructed equation is listed inEqs. 8 and 9, where m is the m mode in the path l. FIR(m)
is the fault isolation rate of m mode in path l.
PI =|s|∏
i=1
αi
|s|∏
k=1k �=i
[1 −|T |∏
q=1
(1 − dij − dkj + 2dij dkj )
eq
] (7)
FIRl = min(F IR(m)), m ∈ [1, 2, 3, . . . , |M|] (8)
FIR(m)=∑
i∈|S|j∈|Si|
αiβj
∏
i1∈|S|j1∈|Si|
|i1−i|+|j1−j|>0
[1−|T |∏
q=1
(1 − dijq − di1
j1q
+2dijq
di1j1q
)eq∗gim∗gi1m∗kqm]/|S|∑
i=1
(αiGl(i)) (9)
min|T |∑
q=1
cqeq (10)
subject to
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
FDR1 ≥ FDR∗1 , F IR1 ≥ FIR∗
1FDR2 ≥ FDR∗
2 , F IR2 ≥ FIR∗2
...FDR|P | ≥ FDR∗|P |, F IR|P | ≥ FIR∗|P |
The new definition of FIR is listed in Eqs. 8 and 9, Where|S| is the total subsystems. TheGl is defined in Eq. 6. Basedon the definition of FIR, the selected test needs to separatethe faults as much as possible, that is, the test set with theleast similarity of fault characterization needs to be selected.In the numerator of Eq. 9, the 1 − di
jq− d
j1q+ 2di
jqdi1
j1q
measures the difference between fault j and fault j1. If thevalue of di
jqand di
jqdiffer greatly, the outcome of 1− di
jq−
dj1q
+ 2dijq
di1j1q
is supposed to give a lower value. Specially,
the outcome of 1 − dijq
− dj1q
+ 2dijq
di1j1q
when dijq
= 0.1
and dijq
= 0.1 is larger than the outcome when dijq
= 0.1
and dijq
= 0.9. The numerator is the sum of possibility thateach fault to be distinguished with the tests selected . Thedenominator is the sum of possibility that each fault of thesubsystem in this mode appears. It is obviously that Eq. 9 isthe extension of Eq. 7 based on multi-path and multi-mode.The test selection equation of the total system is defined inEq. 10, which is decomposed by paths.
3.3 Illustrative Example
The system of Fig. 1 is used as an example to describethe formulas in section 3.2. The FDR* and FIR* for eachpath are 0.75 and 0.75. The detailed parameter is shown inTables 1, 2, 3, 4 and 5.
In Table 1, the G and K matrix is shown. It is clear thatmode m1 has subsystem 1 and 5, while m2 has subsystem2 and 5, m3 has subsystem 1, 3 and 4. In K matrix, m1 hastest t2 and t5, while m2 has t1, t3, and t5; m3 has t1, t2, andt4. The D matrix and DL matrix are listed in Tables 2 and 3.To simplify the system, it is supposed that each subsystemhas two kinds of failure state. Table 4 is the SD matrix ofsubsystems and tests, while Table 5 is the TD matrix ofsubsystems and tests. In this example, the duration for eachworking mode is set to 3.
Supposing the selected tests are {t1, t2, t5}, while the testcost is (c1 + c2 + c5) = 0.55. The calculation of FDR andFIR in Path1 is listed to verify our method.
P 111 = 1 −
∏
q=1|T |
∏
T Y∈SD111∩̃T Dq1
(1 − fph(1, 1, q) ∗ d11q)eq
P 111 means the possibility of the first fault in s1 is detected
in Path1. The T Y is listed in Table 6, which is calculatedbased on Tables 4 and 5.The T Y (1, 1) is generated as:SD1
11∩̃T D11 = {1} ∩ {2} = ∅. So after calculation ofall tests, all values of punish function are: fph(1, 1, 1) =
Table 1 The G matrix and K matrix of Fig. 1
m1 m2 m3
S1 1 0 1
S2 0 1 0
S3 0 0 1
S4 0 0 1
S5 1 1 0
t1 0 1 1
t2 1 0 1
t3 0 1 0
t4 0 0 1
t5 1 1 0
J Electron Test (2020) 36:75–86 79
Table 2 The D matrix of Fig. 1
t1 t2 t3 t4 t5
0.29 0.11 0.24 0.21 0.15
S1(0.85) F11(0.2) 0 0 0.3 0.9 0.8
F12(0.8) 0.9 0.9 0 0 0
S2(0.04) F21(0.5) 0 0.9 0 0 0
F22(0.5) 0.9 0 0 0 0.3
S3(0.05) F31(0.6) 0.9 0 0.9 0.7 0.9
F32(0.4) 0 0.5 0 0 0
S4(0.04) F41(0.3) 0 0 0.7 0.5 0
F42(0.7) 0 0.5 0 0 0
S5(0.02) F51(0.8) 0 0 0 0 0.9
F52(0.2) 0.9 0 0 0 7
0, fph(1, 1, 2) = 1, fph(1, 1, 5) = 1. The value of P 111 is:
P 111 = 1 − (1 − fph(1, 1, 1) ∗ d111) ∗ (1 − fph(1, 1, 2) ∗ d112)
∗ (1 − fph(1, 1, 5) ∗ d115) = 0.8.
In this way, the values P 1ij are calculated as:P 1
11 =0.8, P 1
12 = 0.9, P 121 = 0, P 1
22 = 0.3, P 151 = 0.3,
P 152 = 0.9233. Then according to (4), probabilities of the
subsystems are:r11 = 0.88, r12 = 0.15, r15 = 0.4247. TheFDR is calculated depending on (5), and is listed in Eq. 11.In Path1, the Gl is s1, s2, s5, which is the union of m1 andm2.
FDR1 =
5∑
i=1(αiγ
1i )
5∑
i=1(αiG1(i))
= 0.8379 (11)
The FDR in Path2 is computed in the same way, and getsthe result: FDR2 = 0.8960. FDR1 > FDR∗
1 , FDR2 >
FDR∗2 . We can get the conclusion that the selected tests
Table 3 The DL matrix and of Fig. 1
t1 t2 t3 t4 t5
0.29 0.11 0.24 0.21 0.15
S1(0.85) F111(0.2) 0 0 1 0 0
F12(0.8) 0 0 1 0 0
S2(0.04) F21(0.5) 4 0 0 0 0
F22(0.5) 4 0 0 0 0
S3(0.05) F31(0.6) 0 0 0 0 0
F32(0.4) 0 0 0 0 0
S4(0.04) F41(0.3) 0 0 0 0 0
F42(0.7) 0 0 0 0 0
S5(0.02) F51(0.8) 0 0 0 0 4
F52(0.2) 0 0 0 0 4
Table 4 The SD matrix of Fig. 1
Falut Path1 Path2
S1(0.85) F11(0.2) {1} {1,2}F12(0.8) {1} {1,2}
S2(0.04) F21(0.5) {2} /
F22(0.5) {2} /
S3(0.05) F31(0.6) / {2}F32(0.4) / {2}
S4(0.04) F41(0.3) / {2}F42(0.7) / {2}
S5(0.02) F51(0.8) {1,2} {1}F52(0.2) {1,2} {1}
t1, t2, t5 can make this system acceptable depending on therequired FDR*.
At the same time, the FIR can be calculated accordingto Eqs. 8 and 9. All working modes in the two pathsneed to be considered separately when FIR is considered.When computing the FIR of m1 in Path1, the D matrix isreorganized and listed in Table 7. The faults and tests thatnot appearing in m1 are removed. The FIR is calculatedwithout considering time delay. The FIR of each path isequal to the minimum value of FIR of all modes underthis path. When the parameters in path l are substitutedinto the Eq. 9, the corresponding FIR values of all workingmodes in Path1 can be calculated as: FIR(m1) = 0.7567,FIR(m2) = 0.7231. The FIR of Path1 is calculated byEq. 8:FIR(Path1) = min(0.7567, 0.7231) = 0.7231.In the same way, the FIR of Path2:FIR(Path2) =min(0.7567, 0.6146) = 0.6146. FIR1 > FIR∗
1 , F IR2 <
FIR∗2 Therefore, the test set t1, t2, t5 cannot meet the
requirements of system testability based on the requirementof FIR*.
4 Test Selection Algorithm Based on GSSO
The problem of traditional test selection is a generalizationof the set-covering problem formulation studied in theliterature [2]. For the set-covering problem, Beasley andChu proposed a GA [11], which performs well in solving
Table 5 The TD matrix of Fig. 1
Test Path1 Path2
t1 {2} {2}t2 {1} {1,2}t3 {2} /
t4 / {2}t5 {1,2} {1}
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Table 6 The TY matrix of Fig. 1
t1 t2 t3 t5
T Y ∅ {1} ∅ {1}
most of the problems. As our problem can be seen as anextension of the set-covering problem, we propose a GSSObased on their research.
4.1 The SSO Algorithm
The SSO algorithm shares some common features with PSOand FSO, but with several significant differences. The majoridea of this new optimization algorithm is energized by theswallow swarm, and many studies had been conducted onvarious species of swallows. The main architecture of SSOis shown in Fig. 3. Just like the PSO, SSO has the basicsteps of the heuristic algorithm including the initializationof the initial population, calculation of fitness, updating ofparticle’s velocity and position, and selection of the bestparticle.
SSO has some common features with PSO [12, 13]. SSOimitates the behavior of swallow swarm and owns threekinds of particles: leader particle (Li), explorer particle(Xi), and aimless particle (Ai). At the beginning of thisalgorithm, all kinds of particles are initiated randomlywith the fitness values calculated. The Leader particles arecategorized into two types: Local Leaders (LL) that denoteto the local best in sub colonies and will contribute tothe evolution in these sub colonies; Head Leader (HL),which is the best particle in the entire group and willcontribute to the evolution of the entire colony. Theexplorer particles (ei), which own the largest part ofthe population, are the responsibilities of evolution. Andei will search for a better place for the entire colonydepending on the local leaders (LLi) and global leader(HLi). In each iteration k, the particles update their position(ei+1) and velocity (Vi+1) depending on Eqs. 12 and13. The aimless particles update their position based onEq. 16.
ei+1 = ei + Vi+1 (12)
Table 7 The calculation of FIR in m1
t2 t5
0.11 0.15
S1(0.85) F11(0.2) 0 0.8
F12(0.8) 0.9 0
S5(0.02) F51(0.8) 0 0.9
F52(0.2) 0 0.7
Fig. 3 The main architecture of SSO algorithm
Vi+1 = VHLi+1 + VLLi+1 (13)
VHLi+1 = VHLi+ αHLrand()(ebest − ei)
+ βHLrand()(HLi − ei) (14)
VLLi+1 = VLLi+ αLLrand()(ebest − ei)
+ βLLrand()(LLi − ei) (15)
Oi+1 = Oi + [rand({−1, 1}) ∗ rand(mins, maxs)
1 + rand()] (16)
Table 8 The Parameter of the DPSO algorithm
Parameter Value
Cognitive confidence coefficients c1= c2=1.5
Inertia parameter w = 1
Number of particles N= 50
Number of random particles NO=10
Velocity vmax=10, vmin=-10
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The calculation equation in detail is listed in the paper [17].If the explore particle has a better value of fitness than thelocal leader, it will exchange the place with local leadernearby after the computation. The global leader will bereplaced if another particle has a better position. The aimlessparticles are settled for a random search in the whole area.At the beginning of exploring, this kind of particles does nothave a good position in comparison with other particles, andthe amount of them is bad. Its behavior is not influenced byother particles. It gives a good chance to get out of the localoptimal solution.
4.2 The Grey Absolute Correlation Degree Analysis
The SSO algorithm is not easy to fall into local optimum.Due to the global search characteristics of the algorithm, theglobal optimal solution can be obtained more effectively,but it also limits the speed of convergence of the algorithm.The grey absolute correlation degree analysis (GRA) isintroduced to optimize the algorithm. The definition of theGRA is listed in Eq. 17. Xi = (xi(1), xi(2), . . . , xi(n)) ,Xj = (xj (1), xj (2), . . . , xj (n)) are two sequences:
εij = 1 + |Si | + |Sj |1 + |Si | + |Sj | + |Sj − Si | (17)
|Si | = |n−1∑
k=2
x0i (k) + 1
2x0i (n)| (18)
|Sj | = |n−1∑
k=2
x0j (k) + 1
2x0j (n)| (19)
|Sj − Si | = |n−1∑
k=2
(x0j (k) − x0
i (k)) + 1
2(x0
j (n) − x0i (n))|
(20)
(x0i (1), x0
i (2), . . . , x0i (n))
= 7(xi(1) − xi(1), xi(2) − xi(1), . . . , xi(n) − xi(1))(21)
The εij in Eq. 17 is the GRA degree of Xi and Xj , whereSi , Sj and Sj − Si are defined in Eqs. 18, 19 and 20.And the Polyline sequence is calculated as: Xi − xi(1),which is defined in Eq. 21. It is clear that εij denotes thecomprehensive evaluation of the shape of two sequences.
The basic idea of GRA is to determine whetherthe connection between different sequences is tight ingeometry. In this method, the observations of the systemare converted into segmented continuous polylines throughlinear interpolation, then the correlation degree modelis constructed according to the geometric characteristicsof the polyline. The degree of correlation between thecorresponding sequences increases with the similarity ofpolygonal geometry. Grey correlation analysis is involved
in the process of each evolution of SSO algorithm. Priorityis given to sorting the correlation degree to accelerate theconvergence process of the algorithm.
4.3 The GSSO Algorithm for Test Selection
The main algorithm is listed in Algorithm 1. In Algorithm1, the inputs are the parameters of the system, FDR, andFIR. The overall process is similar to SSO algorithms. Thespecial part is the similarity evaluation to implement andfix to the local maximum. The process is listed in line 4in Algorithm 1. After the calculation of the algorithm, theoptimized test set is generated and output.
The fitness function of the test selection is listed inEq. 23.
T v = h ×|L|∑
l=1
(FDR∗l − FDRl)
+h ×|L|∑
l=1
(F IR∗l − FIRl) (22)
Fit = 1T∑
q=1cqeq + T v
(23)
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Table 9 The D matrix of verification experiment
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20
F1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
F2 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
F3 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
F4 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
F5 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
F6 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
F7 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0
F8 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
F9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
F10 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
F11 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0
F12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0
F13 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
F14 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
F15 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
where the element h is a positive constant, if FDRl >
FDRl∗ and FIRl > FIRl∗, 0 otherwise.Actually, the evaluation of grey similarity within the
system can be used to rank the importance of the selectedtests. Therefore, the similarity between subsystems ismeasured by GRA. In the calculation, the tests with smallerGRA will be selected. The detail steps are listed inAlgorithm 2. The test with the minimal value of GRA isadded to the test set.
5 Experiments
To verify the effectiveness of the proposed algorithm, twokinds of experiments are simulated in this paper: verifica-tion experiment and performance comparison experiment.In the verification experiment, the algorithm is applied tothe actual systems, and the required test sets are obtained.The performance comparison experiment is designed toshow the quality of the algorithm proposed. The GSSOalgorithm is compared with SSO and DPSO. The parame-ters of DPSO are listed in Table 8. The parameters of SSOare set as the same as the GSSO. The results and generationcurves are shown.
Table 10 The result of verification experiment
Algorithm FDR FIR Cost
DPSO [0.9892,0.9892] [0.9570,0.9570] 8
SSO [0.9892,0.9892] [0.9570,0.9570] 8
GSSO [0.9892,0.9892] [0.9570,0.9570] 8
5.1 Verification Experiment
The experiment is carried out on the system in [18].The delay matrix DL is set as zero to get a goodcomparison to the results in other methods [20]. Thereare 26 faults and 20 tests under the system. As thefocus of this paper is the test-ability to diagnose, thefailure rates are reorganized by merging the failures ownthe same performance in D matrix. The failure ratesare [10,10,5,5,100,100,100,100,5,5,5,5,5,5,5]*10-5. The Dmatrix is listed in Table 9. Each test cost is the same as astandard unit. The FDR* and FIR* are set as 0.95 and 0.95respectively.
As is listed in Table 10, the FDR and FIR are met byall algorithms. The tests selected are shown in Table 11.As the t14 and t17 own the same distribution and the testcost is equal, the results of all algorithms are the same. Asis shown from Fig. 4a, although the scale of the system issmall, the algorithm in this paper has certain advantagesover SSO and DPSO. It can be seen from the fitness functioncurves that GSSO can reach the optimal solution in a smallnumber of iterations. Figure 4b is the result after executingthe algorithm 20 times in the system. It can be seen thatthe GSSO algorithm can reach the optimal solution within
Table 11 The results of verification experiment
Algorithm Test
DPSO [1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0]
SSO [1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0]
GSSO [1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0]
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(a) (b) (c)
Fig. 4 The results of the experiments: a The fitness curves of the verification experiment b The fitness curves of 20 times c The fitness curves ofperformance comparison experiments
40 generations. Comparison with the listed outcome in [20,25], the result is persuasive in validating the effectiveness.
For the further validity of the proposed algorithm’seffectiveness. This paper designs an extra experiment tosimulate the multi-path system with time delay. To make acomparison with the experimental results in Table 10 andTable 11, the parameters of the system are introduced intothe multi-path system. There are four modes. The faultsof system become the subsystems of each mode. It can beseen that without delay, the tests satisfying the designedsystem are the same as the results in Table 10. The randomlygenerated delay matrix is shown in Table 12. And tomake comparison, the (F10, t16), (F11, t16), (F12, t16) and(F13, t16) is set to 8. The proposed algorithm is supposedto select the tests with shorter time delay, and satisfies theFDR* and FIR* with minimized costs. As the results are
shown in Table 13, the main difference between the selectedtests is the t15 and t16. It can be seen that t15 and t16 havethe same distribution in D matrix, while the time delay oft16 in DL matrix is larger than t15. Depending on the penaltyfunction in this paper, the algorithm chooses t15 as the finalresult. It can be seen that the method of this paper addstime delay into the problem of test selection. Moreover, theGSSO algorithm proposed in this paper has a good effect onsolving this complex problem.
5.2 Performance Comparison Experiments
The influence of the number of tests on the algorithmperformance is studied. In this section, the number of testsis set to 200. The system is constructed by 20 subsystemswith 4 failure types. The D matrix and the delay matrix DL
Table 12 The DL matrix of verification experiment
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20
F1 0 0 0 0.65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.9 0
F2 0 0 0 0 0 0 0 0 0 1.7 0 0 0 0 0 0 0 2.8 0,0
F3 0 0 2.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
F4 0 0 0 0 0 0 4.22 0 0 0 0 0 0 0 0 0 0 0 0 0
F5 0 0 0 0 0 0 4.22 0 0 0 0 0 0 0 0 0 0 0 0 0
F6 0 2.47 0 0 3.58 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
F7 0 0 0 0 0 0 0 0 0 0 0 0 0 2.73 0 0 0 0 0 0
F8 0 0 0 0 0 2.36 0 0 0 0 0 0 0 0 0 0 0 0 0 0
F9 0 0 0 0 0 0 0 0 1.89 0 0 0 0 0 0 0 0 0 0 4.3
F10 0 0 0 0 0 0 0 0 0 0 0 0 1.48 0 0 8 0 0 0 0
F11 0 0 4.8 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0.84 0
F12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 2.38 0.56 0
F13 0 0 0 0 0 0 0 4.62 0 0 0 0 0 3.72 0 8 0 0 0 0
F14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.79 0 0 0
F15 0 0 0 0 0 0 0 2.16 0 0 1.15 0 0 0 0 0 0 0 0 0
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Table 13 The comparison of tests selected
Condition Test
Without delay [1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0]
With delay [1,1,1,1,0,0,0,0,0,1,1,0,0,0,1,0,0,0,1,0]
are generated randomly. And there are 4 working modes,m1, m2, m3, m4. There are two paths m1, m2, m3 and m1,m2, m4. The FDR* and FIR* are 0.9 and 0.9. After 2000iterations, the outcome is listed in Table 14 and the curvesof fitness got are shown in Fig. 4c.
In Table 15, FDR* and FIR* are met by all threealgorithms. In terms of test cost, GSSO in this paperachieves a lower test cost of 0.355, compared with theother two algorithms. In Fig. 4c, it can be seen that inthe case of a large test set, the proposed algorithm has agreat advantage over other algorithms. In this experiment,the maximum number of cycles is 2000, and the algorithmconverges within 400 times. Owing to the introduction ofGRA in local values, the particles get better fitness earlier.The GSSO get better fitness than the other two algorithmsafter 200 iterations. What’s more, it is clear that GSSOhas a good quality of jumping out of local optimum. Allthe above experiment results demonstrate that the newproposed method has better accuracy and quality in findingthe optimum test set. It is an effective and feasible methodfor minimizing the cost of the test set.
6 Conclusion
The time delay in the complex system is always animportant factor affecting system reliability. In the systemswith multi-path, this paper studies the test selection ofimperfect tests with time delay. The FDR and FIR of thesystem are discussed and their specific connotations areclarified.Then, the test selection problem was formulatedas an optimization problem with multi-constricts. It can bedescribed as minimizing the test cost subject to specifiedlower bounds on FDR* and FIR* in different paths. Thisformulation was extended to realistic problems with timedelay.
The test selection problem is NP-hard. The heuristicalgorithm of SSO is used to slove this problem. The
Table 14 The result of the performance comparison experiment
Algorithm FDR FIR Cost
DPSO [0.9892,0.9892] [0.9570,0.9570] 8
SSO [0.9892,0.9892] [0.9570,0.9570] 8
GSSO [0.9892,0.9892] [0.9570,0.9570] 8
Table 15 The result of the performance comparison experiment
Algorithm FDR FIR Cost
DPSO [0.9600,0.9636] [0.9003,0.9003] 0.365
SSO [0.9695,0.9711] [0.9026,0.9009] 0.385
GSSO [0.9593,0.9534] [0.9040,0.9000] 0.355
grey system theory is combined to SSO algorithm andthe performance is improved. The effectiveness andperformance of the proposed algorithm are verified inthe experiment. The experimental results show that thealgorithm inherits the advantages of SSO and is moresuitable for solving the test selection problem. As theexistence of random particles in this algorithm, it is alsorobust enough to avoid premature convergence of heuristicalgorithms.
References
1. Bashir MB, Nadeem A (2017) Improved genetic algorithm toreduce mutation testing cost. IEEE Access 5:3657–3674
2. Bhushan Mani R (2000) Raghunathan: Design of sensor networkbased on the signed directed graph of the process for efficient faultdiagnosis. Ind Eng Chem Res 39(4):999–1019
3. Bouzidi S, Riffi ME (2017) Discrete swallow swarm optimizationalgorithm for travelling salesman problem. In: Proceedings of the2017 International Conference on Smart Digital Environment, pp80–84
4. Deng S, Jing B, Zhou H (2017) Heuristic particle swarmoptimization approach for test point selection with imperfect test.J Intell Manuf 28(1):37–50
5. Golonek T, Rutkowski J (2007) Genetic-algorithm-based methodfor optimal analog test points selection. IEEE Trans Circuits SystII Express Briefs 54(2):117–121
6. Kennedy J, Eberhart R (1995) Particle swarm optimization.In: Proceeding of IEEE International Conference on NeuralNetworks, vol 4, pp 1942–1948
7. Jiuqing W, Fan Z (2011) Diagnostic information based test pointsselection for analog circuit diagnosis. In: Proc IEEE InternationalConference on Cybernetics and Intelligent Systems, pp 42–46
8. Starzyk JA, Liu D, Liu ZH, Nelson DE (2004) Entropy-based optimum test points selection for analog fault dictionarytechniques. IEEE Trans Instrum Meas 53(3):754–761
9. Jiang R, Wang H, Tian S, Long B (2010) Multidimensional fitnessfunction dpso algorithm for analog test point selection. IEEETrans Instrum Meas 59(6):1634–1641
10. Saab K, Hamida NB, Kaminska B (2001) Closing the gap betweenanalog and digital testing. IEEE Trans Comput Aided Des IntegrCircuits Syst 20(2):307–314
11. Al-Sultan KS, Hussain MF, Nizami JS (1996) A genetic algorithmfor the set covering problem. J Oper Res Soc 47(5):702–709
12. Kaveh A, Bakhshpoori T, Afshari E (2014) Hybrid pso andsso algorithm for truss layout and size optimization consideringdynamic constraints. Comput Struct 143:40–59
13. Kaveh A, Bakhshpoori T, Afshari E (2015) An efficient hybridparticle swarm and swallow swarm optimization algorithm. StructEng Mech 54(3):453–474
14. KK P, CK B (2003) An approach for selection of test pointsfor analog fault diagnosis. In: Proceeding of the 18th IEEE
J Electron Test (2020) 36:75–86 85
International Symposium on Defect and Fault Tolerance in VLSISystems, pp 278–294
15. Slamani M, Kaminska B, Quesnel G (1994) An integratedapproach for analog circuit testing with a minimum numberof detected parameters. In: Proceeding of International TestConference, pp 631–640
16. Shakeri M, Pattipati KR, Raghavan V, Patterson-Hine A (1998)Optimal and near-optimal algorithms for multiple fault diagnosiswith unreliable tests. IEEE Transactions on Systems, Man, andCybernetics-Part C: Applications and Reviews 28(3):431–431
17. Neshat M, Sepidnam G, Sargolzaei M (2013) Swallow swarmoptimization algorithm: a new method to optimization. NeuralComput Applic 23(2):429–454
18. Pan JL, Ye XH, Xue Q (2009) A new method for sequentialfault diagnosis based on ant algorithm. In: Proc 2009 SecondInternational Symposium on Computational Intelligence andDesign, pp 44–48
19. Panda R, Agrawala S, Samantarayb L, Abrahamc A (2017) Anevolutionary gray gradientalgorithm for multilevel thresholdingof brain mr images using soft computingtechniques. Appl SoftComput 50:94–108
20. Pattipati K, Alexandridis M (1990) Application of heuristic searchand information theory to sequential fault diagnosis. IEEE TransSyst Man Cybern 20(4):872–887
21. Ruan S, Zhou Y, Yu F, Pattipati KR (2009) Dynamic multiple-fault diagnosis with imperfect tests. IEEE Transactions onSystems, Man, and Cybernetics - Part A: Systems and Humans39(6):1224–1236
22. Shi X, Zhou Y, Wang L, Wang Q (2006) A discrete particleswarm optimization algorithm for travelling salesman problem.Computational Methods pp 1063–1068
23. Tanese R, Holland JH, Stout QF (1989) Distributed geneticalgorithms for function optimization. University of Michigan AnnArbor p 164
24. WeiW, Li H, Leus R (2017) Test sequencing for sequential systemdiagnosis with precedence constraints and imperfect tests. DecisSupport Syst 103:104–116
25. Xu S, Wang X, Huang M (2015) Proportional fairness basedresources allocation algorithm for leo satellite networks. In: 201514th International Symposium on Distributed Computing andApplications for Business Engineering and Science (DCABES),pp 18–24
26. Yang C, Chen F, Tian S (2017) Grouped genetic algorithm basedoptimal tests selection for system with multiple operation modes.J Electron Test 33(4):415–429
27. Yang C, Yan J, Long B, Liu Z (2014) A novel test optimiz-ing algorithm for sequential fault diagnosis. Microelectron J42(6):719–727
28. Ye X, Chen C, Kang M, Zhai G, Pecht M (2018) Ajoint distribution-based testability metric estimation model forunreliable tests. IEEE Access 6:42566–42577
29. Ying J, Kirubarajan T, Pattipati K, Patterson-Hine A (2000) Acomparative study of iterative and non-iterative feature selectiontechniques for software defect prediction. A Hidden MarkovModel-Based Algorithm for Fault Diagnosis with Partial andImperfect Tests 30(4):463–473
30. Zhang S, Pattipati KR (2013) Optimal selection of imperfect testsfor fault detection and isolation. IEEE Trans Syst. Man Cybern:Syst 43(6):1370–1384
Publisher’s Note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.
Zhexi Yao is a Ph.D. candidate at School of Computer Science andTechnology, University of Chinese Academy of Sciences. Her researchinterests include software analysis and testing, machine learning.
Lingchao Zhu is a Ph.D. candidate at School of Computer Science andTechnology, University of Chinese Academy of Sciences. His researchinterests include machine learning, complex system and aerospacecontrol.
Tao Zhang received Ph.D.at Tsinghua University, Beijing, China, in2001. He is a Professor at Technology and Engineering Center forSpace Utilization, Chinese Academy of Sciences. His main researchinterests include high reliability software testing and validation, highreliability electronic information system analysis and design methods,and complex system simulation.
Jinbo Wang Received Ph.D.at University of Chinese Academy ofSciences, Beijing, China, in 2008. He is a Professor at Technologyand Engineering Center for Space Utilization, Chinese Academy ofSciences. His main research interests include high reliability softwaretesting and validation.
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