Abstract

Multi-state is a typical characteristic of engineered systems. Most existing studies of redundancy allocation problems (RAPs) for multi-state system (MSS) design assume that the state probabilities of redundant components are precisely known. However, due to lack of knowledge and/or ambiguous judgements from engineers/experts, the epistemic uncertainty associated with component states cannot be completely avoided and it is befitting to be represented as belief quantities. In this paper, a multi-objective RAP is developed for MSS design under the belief function theory. To address the epistemic uncertainty propagation from components to system reliability evaluation, an evidential network (EN) model is introduced to evaluate the reliability bounds of an MSS. The resulting multi-objective design optimization problem is resolved via a modified non-dominated sorting genetic algorithm II (NSGA-II), in which a set of new Pareto dominance criteria is put forth to compare any pair of feasible solutions under the belief function theory. A numerical case along with a SCADA system design is exemplified to demonstrate the efficiency of the EN model and the modified NSGA-II. As observed in our study, the EN model can properly handle the uncertainty propagation and achieve narrower reliability bounds than that of the existing methods. More importantly, the original nested design optimization formulation can be simplified into a one-stage optimization model by the proposed method.

References

1.
Wang
,
J.
, and
Li
,
M.
,
2016
, “
Redundancy Allocation for Multistate Systems With Component Dependencies and Load Sharing
,”
ASME J. Mech. Design
,
138
(
11
), p.
111403
. 10.1115/1.4034108
2.
Coit
,
D. W.
, and
Zio
,
E.
,
2018
, “
The Evolution of System Reliability Optimization
,”
Reliab. Eng. Syst. Saf.
,
192
, p.
106259
. 10.1016/j.ress.2018.09.008
3.
Pham
,
H.
,
Phan
,
H. K.
, and
Amari
,
S. V.
,
1995
, “
A General Model for Evaluating the Reliability of Telecommunications Systems
,”
Commun Reliability Maintainability, Supportability Int. J.
,
2
, pp.
4
13
.
4.
Shin
,
J.
, and
Lee
,
I.
,
2014
, “
Reliability-Based Vehicle Safety Assessment and Design Optimization of Roadway Radius and Speed Limit in Windy Environments
,”
ASME J. Mech. Design
,
136
(
8
), p.
081006
. 10.1115/1.4027512
5.
Harunuzzaman
,
M.
, and
Aldemir
,
T.
,
1996
, “
Optimization of Standby Safety System Maintenance Schedules in Nuclear Power Plants
,”
Nucl. Technol.
,
113
(
3
), pp.
354
367
. 10.13182/NT96-A35215
6.
Mo
,
Y.
,
Xing
,
L.
, and
Amari
,
S. V.
,
2014
, “
A Multiple-Valued Decision Diagram-Based Method for Efficient Reliability Analysis of Non-Repairable Phased-Mission Systems
,”
IEEE Trans. Reliab.
,
63
(
1
), pp.
320
330
. 10.1109/TR.2014.2299497
7.
Xiao
,
H.
, and
Peng
,
R.
,
2014
, “
Optimal Allocation and Maintenance of Multi-State Elements in Series-Parallel Systems With Common Bus Performance Sharing
,”
Comput. Ind. Eng.
,
72
, pp.
143
151
. 10.1016/j.cie.2014.03.014
8.
Mo
,
Y.
,
Xing
,
L.
,
Amari
,
S. V.
, and
Dugan
,
J. B.
,
2015
, “
Efficient Analysis of Multi-State k-out-of-n Systems
,”
Reliab. Eng. Syst. Saf.
,
133
, pp.
95
105
. 10.1016/j.ress.2014.09.006
9.
Xiahou
,
T.
, and
Liu
,
Y.
,
2019
, “
Reliability Bounds for Multi-State Systems by Fusing Multiple Sources of Imprecise Information
,”
IISE Trans.
10.1080/24725854.2019.1680910
10.
Levitin
,
G.
,
Lisnianski
,
A.
,
Ben-Haim
,
H.
, and
Elmakis
,
D.
,
1998
, “
Redundancy Optimization for Series-Parallel Multi-State Systems
,”
IEEE Trans. Reliab.
,
47
(
2
), pp.
165
172
. 10.1109/24.722283
11.
Levitin
,
G.
, and
Lisnianski
,
A.
,
1999
, “
Joint Redundancy and Maintenance Optimization for Multistate Series-Parallel Systems
,”
Reliab. Eng. Syst. Saf.
,
64
(
1
), pp.
33
42
. 10.1016/S0951-8320(98)00052-0
12.
Levitin
,
G.
,
2001
, “
Incorporating Common-Cause Failures Into Non-Repairable Multistate Series-Parallel System Analysis
,”
IEEE Trans. Reliab.
,
50
(
4
), pp.
380
388
. 10.1109/24.983398
13.
Wang
,
J.
, and
Li
,
M.
,
2015
, “
Redundancy Allocation Optimization for Multistate Systems With Failure Interactions Using Semi-Markov Process
,”
ASME J. Mech. Des.
,
137
(
10
), p.
101403
. 10.1115/1.4031297
14.
Wang
,
J.
, and
Li
,
M.
,
2015
, “
Redundancy Allocation for Reliability Design of Engineering Systems With Failure Interactions
,”
ASME J. Mech. Des.
,
137
(
3
), p.
031403
. 10.1115/1.4029320
15.
Du
,
M.
, and
Li
,
Y. F.
,
2020
, “
An Investigation of New Local Search Strategies in Memetic Algorithm for Redundancy Allocation in Multi-State Series-Parallel Systems
,”
Reliab. Eng. Syst. Saf.
,
195
, p.
106703
. 10.1016/j.ress.2019.10670
16.
Zaretalab
,
A.
,
Hajipour
,
V.
, and
Tavana
,
M.
,
2020
, “
Redundancy Allocation Problem With Multi-State Component Systems and Reliable Supplier Selection
,”
Reliab. Eng. Syst. Saf.
,
193
, p.
106629
. 10.1016/j.ress.2019.106629
17.
Tian
,
Z.
,
Zuo
,
M. J.
, and
Huang
,
H.
,
2008
, “
Reliability-Redundancy Allocation for Multi-State Series-Parallel Systems
,”
IEEE Trans. Reliab.
,
57
(
2
), pp.
303
310
. 10.1109/TR.2008.920871
18.
Zoulfaghari
,
H.
,
Hamadani
,
A. Z.
, and
Ardakan
,
M. A.
,
2015
, “
Multi-Objective Availability-Redundancy Allocation Problem for a System With Repairable and Non-Repairable Components
,”
Decis. Sci. Lett.
,
4
(
3
), pp.
289
302
. 10.5267/j.dsl.2015.4.007
19.
Wang
,
Y.
, and
Li
,
L.
,
2012
, “
Heterogeneous Redundancy Allocation for Series-Parallel Multi-State Systems Using Hybrid Particle Swarm Optimization and Local Search
,”
IEEE Trans. Syst. Man Cybern. A Syst. Hum.
,
42
(
2
), pp.
464
474
. 10.1109/TSMCA.2011.2159585
20.
Ouyang
,
Z.
,
Liu
,
Y.
,
Ruan
,
S. J.
, and
Jiang
,
T.
,
2019
, “
An Improved Particle Swarm Optimization Algorithm for Reliability-Redundancy Allocation Problem With Mixed Redundancy Strategy and Heterogeneous Components
,”
Reliab. Eng. Syst. Saf.
,
181
, pp.
62
74
. 10.1016/j.ress.2018.09.005
21.
Kulturel-Konak
,
S.
,
Smith
,
A. E.
, and
Coit
,
D. W.
,
2003
, “
Efficiently Solving the Redundancy Allocation Problem Using Tabu Search
,”
IIE Trans.
,
35
(
6
), pp.
515
526
. 10.1080/07408170304422
22.
Ouzineb
,
M.
,
Nourelfath
,
M.
, and
Gendreau
,
M.
,
2008
, “
Tabu Search for the Redundancy Allocation Problem of Homogenous Series-Parallel Multi-State Systems
,”
Reliab. Eng. Syst. Saf.
,
93
(
8
), pp.
1257
1272
. 10.1016/j.ress.2007.06.004
23.
Ghambari
,
S.
, and
Rahati
,
A.
,
2018
, “
An Improved Artificial Bee Colony Algorithm and Its Application to Reliability Optimization Problems
,”
Appl. Soft Comput.
,
62
, pp.
736
767
. 10.1016/j.asoc.2017.10.040
24.
Xiao
,
H.
, and
Peng
,
R.
,
2018
, “Trade-Off Between Redundancy, Protection, and Imperfect False Targets in Defending Parallel Systems,”
Recent Advances in Multi-State Systems Reliability
,
A.
Lisnianski
,
I.
Frenkel
, and
A.
Karagrigoriou
, eds.,
Springer
,
New York
.
25.
Tian
,
Z.
, and
Zuo
,
M. J.
,
2006
, “
Redundancy Allocation for Multi-State Systems Using Physical Programming and Genetic Algorithms
,”
Reliab. Eng. Syst. Saf.
,
91
(
9
), pp.
1049
1056
. 10.1016/j.ress.2005.11.039
26.
Taboada
,
H. A.
,
Espiritu
,
J. F.
, and
Coit
,
D. W.
,
2008
, “
MOMS-GA: A Multi-Objective Multi-State Genetic Algorithm for System Reliability Optimization Design Problems
,”
IEEE Trans. Reliab.
,
57
(
1
), pp.
182
191
. 10.1109/TR.2008.916874
27.
Mousavi
,
S. M.
,
Alikar
,
N.
,
Niaki
,
S. T. A.
, and
Bahreininejad
,
A.
,
2015
, “
Two Tuned Multi-Objective Meta-Heuristic Algorithms for Solving a Fuzzy Multi-State Redundancy Allocation Problem Under Discount Strategies
,”
Appl. Math. Model.
,
39
(
22
), pp.
6968
6989
. 10.1016/j.apm.2015.02.040
28.
Alikar
,
N.
,
Mousavi
,
S. M.
,
Tavana
,
M.
, and
Niaki
,
S. T. A.
,
2017
, “
A Multi-Objective Multi-State Series-Parallel Redundancy Allocation Model Using Tuned Meta-Heuristic Algorithms
,”
Int. J. Syst. Sci.: Operations Logistics
,
4
(
3
), pp.
275
296
. 10.1080/23302674.2016.1193254
29.
Sun
,
M. X.
,
Li
,
Y. F.
, and
Zio
,
E.
,
2017
, “
On the Optimal Redundancy Allocation for Multi-State Series-Parallel Systems Under Epistemic Uncertainty
,”
Reliab. Eng. Syst. Saf.
,
192
, p.
106019
. 10.1016/j.ress.2017.11.025
30.
Xiahou
,
T.
,
Liu
,
Y.
, and
Jiang
,
T.
,
2018
, “
Extended Composite Importance Measures of Multi-State Systems With Epistemic Uncertainty of State Assignment
,”
Mech. Syst. Signal Process.
,
109
, pp.
305
329
. 10.1016/j.ymssp.2018.02.021
31.
Liu
,
X.
,
Wang
,
X.
,
Xie
,
J.
, and
Li
,
B.
,
2019
, “
Construction of Probability Box Model Based on Maximum Entropy Principle and Corresponding Hybrid Reliability Analysis Approach
,”
Struct Multidisc Optim
,
61
(
2
), pp.
1
19
. 10.1007/s00158-019-02382-9
32.
Rubinstein
,
R. Y.
,
Levitin
,
G.
,
Lisnianski
,
A.
, and
Ben-Haim
,
H.
,
1997
, “
Redundancy Optimization of Static Series-Parallel Reliability Models Under Uncertainty
,”
IEEE Trans. Reliab.
,
46
(
4
), pp.
503
511
. 10.1109/24.693783
33.
Wattanapongskorn
,
N.
, and
Coit
,
D. W.
,
2007
, “
Fault-Tolerant Embedded System Design and Optimization Considering Reliability Estimation Uncertainty
,”
Reliab. Eng. Syst. Safety
,
92
(
4
), pp.
395
407
. 10.1016/j.ress.2005.12.011
34.
Muhuri
,
P. K.
,
Ashraf
,
Z.
, and
Lohani
,
Q. M. D.
,
2018
, “
Multi-Objective Reliability Redundancy Allocation Problem With Interval Type-2 Fuzzy Uncertainty
,”
IEEE Trans. Fuzzy Syst.
,
26
(
3
), pp.
1339
1355
. 10.1109/tfuzz.2017.2722422
35.
Zhang
,
E.
, and
Chen
,
Q.
,
2016
, “
Multi-Objective Reliability Redundancy Allocation in an Interval Environment Using Particle Swarm Optimization
,”
Reliab. Eng. Syst. Safety
,
145
, pp.
83
92
. 10.1016/j.ress.2015.09.008
36.
Gupta
,
R. K.
,
Bhunia
,
A. K.
, and
Roy
,
D.
,
2009
, “
A GA Based Penalty Function Technique for Solving Constrained Redundancy Allocation Problem of Series System With Interval Valued Reliability of Components
,”
J. Comput. Appl. Math.
,
232
(
2
), pp.
275
284
. 10.1016/j.cam.2009.06.008
37.
Fan
,
X.
, and
Zuo
,
M. J.
,
2006
, “
Fault Diagnosis of Machines Based on D-S Evidence Theory. Part 1: D-S Evidence Theory and Its Improvement
,”
Pattern Recognit. Lett.
,
27
(
5
), pp.
366
376
. 10.1016/j.patrec.2005.08.025
38.
Smets
,
P.
,
2005
, “
Decision Making in the TBM: The Necessity of the Pignistic Transformation
,”
Int. J. Approx. Reason.
,
38
(
2
), pp.
133
148
. 10.1016/j.ijar.2004.05.003
39.
Li
,
F.
,
Li
,
S.
, and
Denœux
,
T.
,
2018
, “
k-CEVCLUS: Constrained Evidential Clustering of Large Dissimilarity Data
,”
Knowl.-Based Syst.
,
142
, pp.
29
44
. 10.1016/j.knosys.2017.11.023
40.
Cao
,
L.
,
Liu
,
J.
,
Jiang
,
C.
,
Wu
,
Z.
, and
Zhang
,
Z.
,
2020
, “
Evidence-Based Structural Uncertainty Quantification by Dimension Reduction Decomposition and Marginal Interval Analysis
,”
ASME J. Mech. Des.
,
142
(
5
), p.
051701
. 10.1115/1.4044915
41.
Mourelatos
,
Z. P.
, and
Zhou
,
J.
,
2005
, “
A Design Optimization Method Using Evidence Theory
,”
ASME J. Mech. Des.
,
128
(
4
), pp.
901
908
. 10.1115/1.2204970
42.
Qiu
,
S.
,
Sallak
,
M.
,
Schön
,
W.
, and
Ming
,
H. X. G.
,
2018
, “
Extended LK Heuristics for the Optimization of Linear Consecutive-k-out-of-n: F Systems Considering Parametric Uncertainty and Model Uncertainty
,”
Reliab. Eng. Syst. Safety
,
175
, pp.
51
61
. 10.1016/j.ress.2018.01.016
43.
Dempster
,
A. P.
,
1967
, “
Upper and Lower Probabilities Induced by a Multivalued Mapping
,”
Annu. Math. Statistic
,
38
(
2
), pp.
325
339
. 10.1214/aoms/1177698950
44.
Shafer
,
G.
,
1976
,
A Mathematical Theory of Evidence
,
Princeton University Press
,
Princeton, NJ
.
45.
Smets
,
P.
, and
Kennes
,
R.
,
1994
, “
The Transferable Belief Model
,”
Artif. Intell.
,
66
(
2
), pp.
191
234
. 10.1016/0004-3702(94)90026-4
46.
Yin
,
L.
,
Deng
,
X.
, and
Deng
,
Y.
,
2019
, “
The Negation of a Basic Probability Assignment
,”
IEEE Trans. Fuzzy Syst.
,
27
(
1
), pp.
135
143
. 10.1109/TFUZZ.2018.2871756
47.
Zhang
,
J.
,
Qiu
,
Y.
,
Li
,
M.
, and
Xu
,
M.
,
2017
, “
Sequential Multi-Objective Optimization for Lubrication System of Gasoline Engines With Bilevel Optimization Structure
,”
ASME J. Mech. Des.
,
139
(
2
), p.
021405
. 10.1115/1.4035493
48.
Muhuri
,
P. K.
, and
Nath
,
R.
,
2019
, “
A Novel Evolutionary Algorithmic Solution Approach for Bilevel Reliability-Redundancy Allocation Problem
,”
Reliab. Eng. Syst. Saf.
,
191
, p.
106531
. 10.1016/j.ress. 2019.106531
49.
Dempe
,
S.
, and
Zemkoho
,
A. B.
,
2012
, “
On the Karush–Kuhn–Tucker Reformulation of the Bilevel Optimization Problem. Nonlinear Analysis: Theory
,”
Methods Appl.
,
75
(
3
), pp.
1202
1218
. 10.1016/j.na.2011.05.097
50.
Lisnianski
,
A.
, and
Levitin
,
G.
,
2003
,
Multi-State System Reliability: Assessment, Optimization and Applications
,
World Scientific
,
Singapore
.
51.
Levitin
,
G.
, and
Lisnianski
,
A.
,
2001
, “
A New Approach to Solving Problems of Multi-State System Reliability Optimization
,”
Quality Reliab. Eng. Int.
,
17
(
2
), pp.
93
104
. 10.1002/qre.388
52.
Denœux
,
T.
,
Li
,
S.
, and
Sriboonchitta
,
S.
,
2018
, “
Evaluating and Comparing Soft Partitions: An Approach Based on Dempster-Shafer Theory
,”
IEEE Trans. Fuzzy Syst.
,
26
(
3
), pp.
1231
1244
. 10.1109/TFUZZ.2017.2718484
53.
Simon
,
C.
, and
Weber
,
P.
,
2009
, “
Evidential Networks for Reliability Analysis and Performance Evaluation of Systems With Imprecise Knowledge
,”
IEEE Trans. Reliab.
,
58
(
1
), pp.
69
87
. 10.1109/TR.2008.2011868
54.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evol. Comput.
,
6
(
2
), pp.
182
197
. 10.1109/4235.996017
55.
Destercke
,
S.
, and
Sallak
,
M.
,
2013
, “
An Extension of Universal Generating Function in Multi-State Systems Considering Epistemic Uncertainties
,”
IEEE Trans. Reliab.
,
62
(
2
), pp.
504
514
. 10.1109/TR.2013.2259206
56.
Li
,
C.
,
Chen
,
X.
,
Yi
,
X.
, and
Tao
,
J.-Y.
,
2011
, “
Interval-Valued Reliability Analysis of Multi-State Systems
,”
IEEE Trans. Reliab.
,
60
(
1
), pp.
323
330
. 10.1109/TR.2010.2103670
57.
Zand
,
A. D.
, and
Damghani
,
K. K.
,
2015
, “
Design of SCADA Water Resource Management Control Center by a Bi-Objective Redundancy Allocation Problem and Particle Swarm Optimization
,”
Reliab. Eng. Syst. Safety
,
133
, pp.
11
21
. 10.1016/j.ress.2014.07.020
58.
Farhang-Mehr
,
A.
, and
Azarm
,
S.
,
2003
, “
An Information-Theoretic Entropy Metric for Assessing Multi-Objective Optimization Solution Set Quality
,”
ASME J. Mech. Des.
,
125
(
4
), pp.
655
663
. 10.1115/1.1623186
59.
Huang
,
H. Z.
,
Tian
,
Z.
, and
Zuo
,
M. J.
,
2005
, “
Intelligent Interactive Multi-Objective Optimization Method and Its Application to Reliability Optimization
,”
IIE Trans.
,
37
(
11
), pp.
983
993
. 10.1080/07408170500232040
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