Multi-objective optimization framework in the modeling of belief rule-based systems with interpretability-accuracy trade-off

doi:10.23919/JSEE.2024.000064

Journal of Systems Engineering and Electronics ›› 2025, Vol. 36 ›› Issue (2): 423-435.doi: 10.23919/JSEE.2024.000064

• SYSTEMS ENGINEERING • Previous Articles

Multi-objective optimization framework in the modeling of belief rule-based systems with interpretability-accuracy trade-off

Yaqian YOU(), Jianbin SUN(), Yuejin TAN(), Jiang JIANG()

College of Systems Engineering, National University of Defense Technology, Changsha 410073, China

Received:2022-08-16 Online:2025-04-18 Published:2025-05-20
Contact: Jianbin SUN E-mail:youyaqian13@nudt.edu.cn;sunjianbin@nudt.edu.cn;yjtan@nudt.edu.cn;jiangjiangnudt@nudt.edu.cn
About author:
YOU Yaqian was born in 1994. She received her B.E. degree in management engineering, M.E. and Ph. D degrees in management science and engineering from National University of Defense Technology, China, in 2017, 2019 and 2023, respectively. She is currently a lecturer at College of Information and Communication, National University of Defense Technology. Her research interests include the decision analysis under uncertainty and evidential reasoning. E-mail: youyaqian13@nudt.edu.cn

SUN Jianbin was born in 1989. He received his B.E. degree in management engineering, M.E. and Ph.D. degrees in management science and engineering from National University of Defense Technology, Changsha, Hunan, China, in 2012, 2014 and 2018, respectively. He is currently an associate professor of management science and engineering at National University of Defense Technology. He was a visiting scholar with the Lab for Information Retrieval and Knowledge Management, School of Information Technology, York University, Toronto, ON, Canada. His research interests include system of-systems engineering management, and decision analysis under uncertainty. E-mail: sunjianbin@nudt.edu.cn

TAN Yuejin was born in 1958. He received his B.S. degree from the Department of Mathematics, Hunan Normal University, M.S. degree from the Department of Systems Engineering and Mathematics, National University of Defense Technology, China, in 1981 and 1985, respectively. He is currently a professor of management science and engineering at National University of Defense Technology. He was a visiting scholar at Complex Systems Management Centre, Cranfield University from 1993 to 1994. His research interests include modeling and evaluation technologies of system of systems and complex systems engineering management. E-mail: yjtan@nudt.edu.cn

JIANG Jiang was born in 1981. He received his B.E. degree in systems engineering, M.E. and Ph.D. degrees in management science and engineering from National University of Defense Technology, Changsha, Hunan, China, in 2004, 2006, and 2011, respectively. He is currently an associate professor of management science and engineering at National University of Defense Technology. He was a visiting scholar at the Harvard University, Boston, MA, USA from 2018 to 2019. His research interests include evidential reasoning, uncertainty decision-making, and risk analysis. E-mail: jiangjiangnudt@nudt.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (71901212) and the Science and Technology Innovation Program of Hunan Province (2020RC4046).

Abstract

Abstract:

The belief rule-based (BRB) system has been popular in complexity system modeling due to its good interpretability. However, the current mainstream optimization methods of the BRB systems only focus on modeling accuracy but ignore the interpretability. The single-objective optimization strategy has been applied in the interpretability-accuracy trade-off by integrating accuracy and interpretability into an optimization objective. But the integration has a greater impact on optimization results with strong subjectivity. Thus, a multi-objective optimization framework in the modeling of BRB systems with interpretability-accuracy trade-off is proposed in this paper. Firstly, complexity and accuracy are taken as two independent optimization goals, and uniformity as a constraint to give the mathematical description. Secondly, a classical multi-objective optimization algorithm, nondominated sorting genetic algorithm II (NSGA-II), is utilized as an optimization tool to give a set of BRB systems with different accuracy and complexity. Finally, a pipeline leakage detection case is studied to verify the feasibility and effectiveness of the developed multi-objective optimization. The comparison illustrates that the proposed multi-objective optimization framework can effectively avoid the subjectivity of single-objective optimization, and has capability of joint optimizing the structure and parameters of BRB systems with interpretability-accuracy trade-off.

Key words: belief rule-based (BRB) systems, interpretability, multi-objective optimization, nondominated sorting genetic algorithm II (NSGA-II), pipeline leakage detection

Yaqian YOU, Jianbin SUN, Yuejin TAN, Jiang JIANG. Multi-objective optimization framework in the modeling of belief rule-based systems with interpretability-accuracy trade-off[J]. Journal of Systems Engineering and Electronics, 2025, 36(2): 423-435.

Figures/Tables 10

Fig 1

Fig 2

Table 1

Fig 3

Fig 4

Fig 5

Fig 6

Table 2

Table 3

Table 4

References 35

1	YANG J B, LIU J, WANG J, et al. Belief rule-base inference methodology using the evidential reasoning approach−RIMER. IEEE Trans. on Systems Man & Cybernetics Part A Systems & Humans, 2006, 36(2): 266−285.
2	YANG J B, XU D L Evidential reasoning rule for evidence combination. Artificial Intelligence, 2013, 205, 1- 29. doi: 10.1016/j.artint.2013.09.003
3	CHANG L L, ZHOU Z J, CHEN Y W, et al. Belief rule base structure and parameter joint optimization under disjunctive assumption for nonlinear complex system modeling. IEEE Trans. on Systems, Man, and Cybernetics: Systems, 2017, 48(9): 1542−1554.
4	CHANG L L, DONG W, YANG J B, et al Hybrid belief rule base for regional railway safety assessment with data and knowledge under uncertainty. Information Sciences, 2020, 518, 376- 395. doi: 10.1016/j.ins.2019.12.035
5	FENG Z C, ZHOU Z J, HU C H, et al A new belief rule base model with attribute reliability. IEEE Trans. on Fuzzy Systems, 2018, 27 (5): 903- 916.
6	LIU J, MARTINEZ L, CALZADA A, et al A novel belief rule base representation, generation and its inference methodology. Knowledge-Based Systems, 2013, 53, 129- 141. doi: 10.1016/j.knosys.2013.08.019
7	YOU Y Q, SUN J B, CHEN Y W, et al Ensemble belief rule-based model for complex system classification and prediction. Expert Systems with Applications, 2021, 164, 113952. doi: 10.1016/j.eswa.2020.113952
8	FENG Z C, ZHOU Z J, HU C H, et al A safety assessment model based on belief rule base with new optimization method. Reliability Engineering & System Safety, 2020, 203, 107055.
9	FENG Z C, HE W, ZHOU Z J, et al A new safety assessment method based on belief rule base with attribute reliability. IEEE/CAA Journal of Automatica Sinica, 2020, 8 (11): 1774- 1785.
10	HE W, CHENG X Y, ZHAO X, et al An interval construction belief rule base with interpretability for complex systems. Expert Systems with Applications, 2023, 229, 120485. doi: 10.1016/j.eswa.2023.120485
11	CHANG L L, XU X J, LIU Z G, et al BRB prediction with customized attributes weights and tradeoff analysis for concurrent fault diagnosis. IEEE Systems Journal, 2020, 15 (1): 1179- 1190.
12	FU C, HOU B B, XUE M, et al. Extended belief rule-based system with accurate rule weights and efficient rule activation for diagnosis of thyroid nodules. IEEE Trans. on Systems, Man, and Cybernetics: Systems, 2023, 53(1): 251−263.
13	ZHENG J, YANG L H, WANG Y M, et al An explainable decision model based on extended belief-rule-based systems to predict admission to the intensive care unit during COVID-19 breakout. Applied Soft Computing, 2023, 149, 110961.
14	ZHU W, CHANG L L, SUN J B, et al Parallel multipopulation optimization for belief rule base learning. Information Sciences, 2021, 556, 436- 458. doi: 10.1016/j.ins.2020.09.035
15	MAMAGHANI A S, PEDRYCZ W. Structural optimization of fuzzy rule-based models: towards efficient complexity management. Expert Systems with Applications, 2020: 113362.
16	CAO Y, ZHOU Z J, HU C H, et al On the interpretability of belief rule-based expert systems. IEEE Trans. on Fuzzy Systems, 2020, 29 (11): 3489- 3503.
17	ZHOU Z J, CAO Y, HU G Y, et al New health-state assessment model based on belief rule base with interpretability. Science China Information Sciences, 2021, 64 (7): 1- 15.
18	YOU Y Q, SUN J B, GUO Y, et al Interpretability and accuracy trade-off in the modeling of belief rule-based systems. Knowledge-Based Systems, 2021, 236, 107491.
19	MING Z C, ZHOU Z J, CAO Y, et al A new interpretable fault diagnosis method based on belief rule base and probability table. Chinese Journal of Aeronautics, 2023, 36 (3): 184- 201. doi: 10.1016/j.cja.2022.08.003
20	YANG J B, LIU J, XU D L, et al. Optimization models for training belief-rule-based systems. IEEE Trans. on Systems, Man, and Cybernetics Part A: Systems and Humans, 2007, 37(4): 569−585.
21	CHANG R, WANG H W, YANG J B An algorithm for training parameters in belief rule-bases based on the gradient and dichotomy methods. Systems Engineering, 2007, 25 (S): 287- 291.
22	HU G Y, ZHOU Z J, ZHANG B C, et al A method for predicting the network security situation based on hidden BRB model and revised CMA-ES algorithm. Applied Soft Computing, 2016, 48, 404- 418. doi: 10.1016/j.asoc.2016.05.046
23	WANG Y M, YANG L H, FU Y G, et al Dynamic rule adjustment approach for optimizing belief rule-base expert system. Knowledge-Based Systems, 2016, 96, 40- 60. doi: 10.1016/j.knosys.2016.01.003
24	CHANG L L, ZHOU Z J, CHEN Y W, et al Akaike information criterion-based conjunctive belief rule base learning for complex system modeling. Knowledge-Based Systems, 2018, 161, 47- 64. doi: 10.1016/j.knosys.2018.07.029
25	ZADEH L A Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. on Systems, Man, and Cybernetics, 1973, (1): 28- 44.
26	ANTONELLI M, DUCANGE P, LAZZERINI B, et al Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity and partition integrity. Soft Computing, 2011, 15 (12): 2335- 2354. doi: 10.1007/s00500-010-0665-0
27	FERRANTI A, MARCELLONI F, SEGATORI A, et al A distributed approach to multi-objective evolutionary generation of fuzzy rule-based classifiers from big data. Information Sciences, 2017, 415, 319- 340.
28	DEB K, PRATAP A, AGARWAL S, et al A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. on Evolutionary Computation, 2002, 6 (2): 182- 197. doi: 10.1109/4235.996017
29	ALONSO J M, MAGDALENA L, GONZÁLEZ-RODRÍGUEZ G Looking for a good fuzzy system interpretability index: an experimental approach. International Journal of Approximate Reasoning, 2009, 51 (1): 115- 134. doi: 10.1016/j.ijar.2009.09.004
30	YOON K P, HWANG C L. Multiple attribute decision making: an introduction. Berlin: Sage Publications, 1995.
31	XU D L, LIU J, YANG J B, et al Inference and learning methodology of belief-rule-based expert system for pipeline leak detection. Expert Systems with Applications, 2007, 32 (1): 103- 113. doi: 10.1016/j.eswa.2005.11.015
32	PEARL J. The mediation formula: a guide to the assessment of causal pathways in nonlinear models. BERZUINI C, DAWID P, BERNARDINELLI L, ed. Causality: Statistical Perspectives and Applications. New York: John Wiley & Sons, 2012.
33	CHEN Y W, YANG J B, XU D L, et al Inference analysis and adaptive training for belief rule based systems. Expert Systems with Applications, 2011, 38 (10): 12845- 12860. doi: 10.1016/j.eswa.2011.04.077
34	YOU Y Q, SUN J B, JIANG J, et al A new modeling and inference approach for the belief rule base with attribute reliability. Applied Intelligence, 2019, 50 (9): 976- 992.
35	TANG X L, XIAO M Q, LIANG Y J, et al Online updating belief-rule-base using Bayesian estimation. Knowledge-Based Systems, 2019, 171, 93- 105. doi: 10.1016/j.knosys.2019.02.007

Number	FlowDiff	PressureDiff	N	Complexity	MSE	Uniformity
1	2	2	2	8	0.6177	1.0000
	3	2	2	12	0.5187	0.9456
	6	3	2	36	0.3077	0.7256
	5	3	2	30	0.3240	0.7430
2	2	2	2	8	0.4837	1.0000
	6	3	2	36	0.2686	0.7212
	5	3	2	30	0.2840	0.8421
3	2	2	2	8	0.4933	1.0000
	5	3	2	30	0.3074	0.7036
	6	3	2	36	0.2899	0.6974
	3	4	2	24	0.4102	0.6770
4	2	2	2	8	0.5924	1.0000
	6	3	2	36	0.2691	0.6752
	3	3	3	27	0.3548	0.7318
	3	3	2	18	0.4068	0.8341
	5	3	2	30	0.2829	0.7910
5	6	3	2	36	0.2893	0.6623
	2	3	2	12	0.5137	0.9995
	7	2	2	28	0.2919	0.7223
	6	2	2	24	0.3147	0.7554
6	2	2	3	12	0.5707	0.7865
	6	3	2	36	0.2908	0.6635
	4	2	2	16	0.3504	0.9289
	5	3	2	30	0.2953	0.7751
7	2	2	2	8	0.5141	1.0000
	3	2	2	12	0.3809	0.8664
	6	3	2	36	0.2720	0.6710
	5	3	2	30	0.2889	0.7664
8	2	2	2	8	0.5377	1.0000
	6	3	2	36	0.3294	0.5752
	3	2	2	12	0.4203	0.8376
9	6	3	2	36	0.3152	0.5541
	5	3	2	30	0.3541	0.8623
	2	2	3	12	0.4095	0.7617
10	4	3	2	24	0.2839	0.6770
	5	3	2	30	0.2444	0.6515
	6	3	2	36	0.2349	0.5658
	2	2	2	8	0.4554	1.0000

Occurrence	FlowDiff	PressureDiff	N	Complexity	MSE	Uniformity
7	2	2	2	8	0.5302	1.0000
3	3	2	2	12	0.4251	0.8832
2	2	2	3	12	0.4115	0.7741
1	2	3	2	12	0.4937	0.9995
1	4	2	2	16	0.3554	0.9289
1	3	3	2	18	0.4534	0.8341
1	3	4	2	24	0.3959	0.6770
1	6	2	2	24	0.3096	0.7554
1	4	3	2	24	0.3318	0.6770
1	3	3	3	27	0.3791	0.7318
1	7	2	2	28	0.2861	0.7223
8	5	3	2	30	0.3031	0.7669
10	6	3	2	36	0.2875	0.6511

Number	Year	FlowDiff	PressureDiff	NOR	N	Complexity	MSE	Uniformity
1	2007 [31]	8	7	56	5	280	0.4049	0.7566
2	2011 [33]	8	7	56	5	280	0.3990	0.6273
3	2016 [23]	3	2	6	5	30	0.5040	0.8936
3	2016 [23]	7	2	14	5	70	0.4450	0.8127
4	2019 [34]	8	7	56	5	280	0.3466	0.5145
5	2019 [35]	3	2	6	5	30	0.4356	0.9401
5	2019 [35]	3	2	6	5	30	0.3900	0.8223
6	2021 [18]	6	2	12	2	24	0.3360	0.8044
7	This paper	2	2	4	2	8	0.5302	1.000
		5	3	15	2	30	0.3031	0.7669
		6	3	18	2	36	0.2875	0.6511

Method		MSE
BP		0.3924
DT		0.6373
LR		0.5933
RF		0.2149
BRB systems	(2, 2, 2)	0.5302
	(5, 3, 2)	0.3031
	(6, 3, 2)	0.2875

[1]	Sixing LIU, Changbao PEI, Xiaodong YE, Hao WANG, Fan WU, Shifei TAO. Efficient sampling strategy driven surrogate-based multi-objective optimization for broadband microwave metamaterial absorbers [J]. Journal of Systems Engineering and Electronics, 2024, 35(6): 1388-1396.
[2]	Licong ZHANG, Chunlin GONG, Hua SU, Da Ronch ANDREA. Design methodology of a mini-missile considering flight performance and guidance precision [J]. Journal of Systems Engineering and Electronics, 2024, 35(1): 195-210.
[3]	Jiaxin HU, Leping YANG, Huan HUANG, Yanwei ZHU. Optimal reconfiguration of constellation using adaptive innovation driven multiobjective evolutionary algorithm [J]. Journal of Systems Engineering and Electronics, 2021, 32(6): 1527-1538.
[4]	Zining WANG, Min LIN, Xiaogang TANG, Kefeng GUO, Shuo HUANG, Ming CHENG. Multi-objective robust secure beamforming for cognitive satellite and UAV networks [J]. Journal of Systems Engineering and Electronics, 2021, 32(4): 789-798.
[5]	Shiyun LI, Sheng ZHONG, Zhi PEI, Wenchao YI, Yong CHEN, Cheng WANG, Wenzhu ZHANG. Multi-objective reconfigurable production line scheduling for smart home appliances [J]. Journal of Systems Engineering and Electronics, 2021, 32(2): 297-317.
[6]	Jianjiang WANG, Xuejun HU, Chuan HE. Reactive scheduling of multiple EOSs under cloud uncertainties: model and algorithms [J]. Journal of Systems Engineering and Electronics, 2021, 32(1): 163-177.
[7]	Zhen XU, Enze ZHANG, Qingwei CHEN. Rotary unmanned aerial vehicles path planning in rough terrain based on multi-objective particle swarm optimization [J]. Journal of Systems Engineering and Electronics, 2020, 31(1): 130-141.
[8]	Yan'gang LIANG, Zheng QIN. A decision support system for satellite layout integrating multi-objective optimization and multi-attribute decision making [J]. Journal of Systems Engineering and Electronics, 2019, 30(3): 535-544.
[9]	Jiale GAO, Qinghua XING, Chengli FAN, Zhibing LIANG. Double adaptive selection strategy for MOEA/D [J]. Journal of Systems Engineering and Electronics, 2019, 30(1): 132-143.
[10]	Jiting Li, Sheng Zhang, Xiaolu Liu, and Renjie He. Multi-objective evolutionary optimization for geostationary orbit satellite mission planning [J]. Systems Engineering and Electronics, 2017, 28(5): 934-945.
[11]	Ying Zhang, Rennong Yang, Jialiang Zuo, and Xiaoning Jing. Enhancing MOEA/D with uniform population initialization, weight vector design and adjustment using uniform design [J]. Journal of Systems Engineering and Electronics, 2015, 26(5): 1010-1022.
[12]	Haipeng Ren* and Yang Zhao. Immune particle swarm optimization of linear frequency modulation in acoustic communication [J]. Systems Engineering and Electronics, 2015, 26(3): 450-456.
[13]	Lixia Han, Shujuan Jiang, and Shaojiang Lan. Novel electromagnetism-like mechanism method for multiobjective optimization problems [J]. Journal of Systems Engineering and Electronics, 2015, 26(1): 182-.
[14]	Wei Jingxuan & Wang Yuping. Multi-objective fuzzy particle swarm optimization based on elite archiving and its convergence [J]. Journal of Systems Engineering and Electronics, 2008, 19(5): 1035-1040.

Multi-objective optimization framework in the modeling of belief rule-based systems with interpretability-accuracy trade-off

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

Share this article

Figures/Tables 10

References 35

Related Articles 14

Recommended Articles

Metrics

Comments