Multicriteria game approach to air-to-air combat tactical decisions for multiple UAVs

doi:10.23919/JSEE.2023.000115

Journal of Systems Engineering and Electronics ›› 2023, Vol. 34 ›› Issue (6): 1447-1464.doi: 10.23919/JSEE.2023.000115

• AUTONOMOUS DECISION AND COOPERATIVE CONTROL OF UAV SWARMS • Previous Articles Next Articles

Multicriteria game approach to air-to-air combat tactical decisions for multiple UAVs

Ruhao JIANG¹^,²(), He LUO¹^,²^,³(), Yingying MA¹^,²(), Guoqiang WANG¹^,²^,³^,*()

¹ School of Management, Hefei University of Technology, Hefei 230009, China
² Key Laboratory of Process Optimization & Intelligent Decision-making, Ministry of Education, Hefei 230009, China
³ Intelligent Interconnected Systems Laboratory of Anhui Province, Hefei 230009, China

Received:2022-08-30 Online:2023-12-18 Published:2023-12-29
Contact: Guoqiang WANG E-mail:jiangrh@mail.hfut.edu.cn;luohe@hfut.edu.cn;mayingying@mail.hfut.edu.cn;gqwang2017@hfut.edu.cn
About author:
JIANG Ruhao was born in 1993. He received his B.S. and M.S. degrees in mechanical engineering from Hefei University of Technology, in 2016 and 2019, respectively. He is currently pursuing his Ph.D. degree with the School of Management, HeFei University of Technology. His research interests include multi-aircrafts cooperative decision-making and game theory. E-mail: jiangrh@mail.hfut.edu.cn

LUO He was born in 1982. He received his B.S. and Ph.D. degrees from Hefei University of Technology, in 2004 and 2009, respectively. He is currently a professor in HeFei University of Technology. His research interests include intelligent decision-making, multi-agent system, and the applications of un-manned aerial vehicle. E-mail: luohe@hfut.edu.cn

MA Yingying was born in 1994. She received her B.S. and Ph.D. degrees from Hefei University of Technology, in 2016 and 2022, respectively. She is currently a post-doctoral in HeFei University of Technology. Her research interests include multi-UAVs cooperative target assignment in air combat and game theory. E-mail: mayingying@mail.hfut.edu.cn

WANG Guoqiang was born in 1982. He received his B.S. and M.S. degrees from University of Science and Technology of China, in 2004 and 2007, respectively, and Ph.D. degree from Hefei University of Technology, in 2016. He is currently an associate professor in HeFei University of Technology. His research interests include management and intelligent decision-making of unmanned aerial vehicle formation. E-mail: gqwang2017@hfut.edu.cn
Supported by:
This work was supported in part by the National Natural Science Foundation of China (71971075; 71871079; 71671059) and in part by the Anhui Provincial Natural Science Foundation (1808085MG213).

Abstract

Abstract:

Air-to-air combat tactical decisions for multiple unmanned aerial vehicles (ACTDMU) are a key decision-making step in beyond visual range combat. Complex influencing factors, strong antagonism and real-time requirements need to be considered in the ACTDMU problem. In this paper, we propose a multicriteria game approach to ACTDMU. This approach consists of a multicriteria game model and a Pareto Nash equilibrium algorithm. In this model, we form the strategy profiles for the integration of air-to-air combat tactics and weapon target assignment strategies by considering the correlation between them, and we design the vector payoff functions based on predominance factors. We propose a algorithm of Pareto Nash equilibrium based on preference relations using threshold constraints (PNE-PRTC), and we prove that the solutions obtained by this algorithm are refinements of Pareto Nash equilibrium solutions. The numerical experiments indicate that PNE-PRTC algorithm is considerably faster than the baseline algorithms and the performance is better. Especially on large-scale instances, the Pareto Nash equilibrium solutions can be calculated by PNE-PRTC algorithm at the second level. The simulation experiments show that the multicriteria game approach is more effective than one-side decision approaches such as multiple-attribute decision-making and randomly chosen decisions.

Key words: tactical decision, multicriteria game, preference relation, Pareto Nash equilibrium

Ruhao JIANG, He LUO, Yingying MA, Guoqiang WANG. Multicriteria game approach to air-to-air combat tactical decisions for multiple UAVs[J]. Journal of Systems Engineering and Electronics, 2023, 34(6): 1447-1464.

Figures/Tables 14

Fig 1

Table 1

Fig 2

Table 2

Table 3

Fig 3

Fig 4

Fig 5

Fig 6

Table 4

Fig 7

Table 5

Table 6

Table 7

References 48

1	SUN Z X, PIAO H Y, YANG Z, et al Multi-agent hierarchical policy gradient for air combat tactics emergence via self-play. Engineering Applications of Artificial Intelligence, 2022, 98, 104112.
2	XU X M, YANG R N, FU Y Situation assessment for air combat based on novel semi-supervised naive Bayes. Journal of Systems Engineering and Electronics, 2018, 29 (4): 768- 779.
3	JIANG Y Y, GAO Y, SONG W Q, et al Bibliometric analysis of UAV swarms. Journal of Systems Engineering and Electronics, 2022, 33 (2): 406- 425.
4	JEREMY J, FREDERICK L, DENNIS E. Deployment and flight operations of a large scale UAS combat swarm: results from DARPA service academies swarm challenge. Proc. of the International Conference on Unmanned Aircraft Systems, 2018: 12−15.
5	JOAO P, MARCOS R, ANDRE N, et al Machine learning to improve situational awareness in beyond visual range air combat. IEEE Latin America Transactions, 2022, 20 (8): 2039- 2045. doi: 10.1109/TLA.2022.9853232
6	YANG Z, SUN Z X, PIAO H Y, et al Online hierarchical recognition method for target tactical intention in beyond-visual-range air combat. Defence Technology, 2022, 18, 1349- 1361. doi: 10.1016/j.dt.2022.02.001
7	SHIN H, LEE J, KIM H, et al An autonomous aerial combat framework for two-on-two engagements based on basic fighter maneuvers. Aerospace Science and Technology, 2018, 72, 305- 315. doi: 10.1016/j.ast.2017.11.014
8	ISHWARYA M S, ASWANI K C Quantum-inspired ensemble approach to multi-attributed and multi-agent decision-making. Applied Soft Computing, 2021, 106, 107283. doi: 10.1016/j.asoc.2021.107283
9	ZHANG Q H, YANG C C, WANG G Y A sequential three-way decision model with intuitionistic fuzzy numbers. IEEE Trans. on Systems, Man, and Cybernetics: System, 2021, 51 (5): 2640- 2652.
10	SELVACHANDRAN G, QUEK S, HAN L, et al A new design of mamdani complex fuzzy inference system for multiattribute decision making problems. IEEE Trans. on Fuzzy Systems, 2021, 29 (4): 716- 730. doi: 10.1109/TFUZZ.2019.2961350
11	ALKAHER D, MOSHAIOV A Dynamic-escape-zone to avoid energy-bleeding coasting missile. Journal of Guidance, Control, and Dynamics, 2015, 38 (10): 1908- 1921. doi: 10.2514/1.G000776
12	LI S Y, CHEN M, WU Q X, et al Threat sequencing of multiple UCAVs with incomplete information based on game theory. Journal of Systems Engineering and Electronics, 2022, 33 (4): 986- 996.
13	ZHE Y A coalitional extension of generalized fuzzy games. Fuzzy Sets and Systems, 2020, 383, 68- 79. doi: 10.1016/j.fss.2019.06.010
14	BACKWELL D An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 1956, 6 (1): 1- 8. doi: 10.2140/pjm.1956.6.1
15	RETTIEVA A Dynamic multicriteria games with asymmetric players. Journal of Global Optimization, 2022, 83 (3): 521- 537. doi: 10.1007/s10898-020-00929-5
16	CLEMENTE M, FERNANDEZ F, PUERTO J Pareto-optimal security strategies in matrix games with fuzzy payoffs. Fuzzy Sets and Systems, 2011, 176 (1): 36- 45. doi: 10.1016/j.fss.2011.03.006
17	FERNANDEZ F, MONROY L, PUERTO J Multicriteria goal games. Journal of Optimization Theory and Applications, 1998, 99 (2): 403- 421. doi: 10.1023/A:1021726311384
18	RADJEF M S, FAHEM K A note on ideal Nash equilibrium in multicriteria games. Applied Mathematics Letters, 2008, 21 (11): 1105- 1111. doi: 10.1016/j.aml.2007.12.009
19	ZAPATA A, MARMOL A, MONROY L, et al A maxmin approach for the equilibria of vector-valued games. Group Decision and Negotiation, 2019, 28 (2): 415- 432. doi: 10.1007/s10726-018-9608-4
20	QUANT M, BORM P, FIESTRAS-JANEIRO G On properness and protectiveness in two-person multicriteria games. Journal of Optimization Theory and Applications, 2009, 140 (3): 499- 512. doi: 10.1007/s10957-008-9464-5
21	FAHEM K, RADJEF M S Properly efficient Nash equilibrium in multicriteria noncooperative games. Mathematical Methods of Operations Research, 2015, 82 (2): 175- 193. doi: 10.1007/s00186-015-0508-y
22	ZELENY M Games with multiple payoffs. International Journal of Game Theory, 1975, 4 (4): 179- 191. doi: 10.1007/BF01769266
23	MARCO G, MORGAN J A refinement concept for equilibria in multicriteria games via stable scalarizations. International Game Theory Review, 2007, 9 (2): 169- 181. doi: 10.1142/S0219198907001345
24	NOVIKOVA N M, POSPELOVA I I Scalarization method in multicriteria games. Computational Mathematics and Mathematical Physics, 2018, 58 (2): 180- 189. doi: 10.1134/S0965542518020112
25	COOK W D Zero-sum games with multiple goals. Naval Research Logistics, 1976, 23 (4): 615- 621. doi: 10.1002/nav.3800230406
26	NISHIZAKI I, NOTSU T Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game and corresponding mathematical programming problem. Journal of Optimization Theory and Applications, 2007, 135 (2): 217- 239. doi: 10.1007/s10957-007-9245-6
27	HONG J M, KIM M H, LEE G M On linear vector program and vector matrix game equivalence. Optimization Letters, 2012, 6 (2): 231- 240. doi: 10.1007/s11590-010-0237-3
28	CHANDRA S, AGGARWAL A On solving matrix games with pay-offs of triangular fuzzy numbers: certain observations and generalizations. European Journal of Operational Research, 2015, 246 (2): 575- 581. doi: 10.1016/j.ejor.2015.05.011
29	PENG H G, WANG X K, WANG T L, et al Multi-criteria game model based on the pairwise comparisons of strategies with Z-numbers. Applied Soft Computing, 2019, 74, 451- 465. doi: 10.1016/j.asoc.2018.10.026
30	ZYCHOWSKI A, GUPTA A, MANDZIUK J, et al Addressing expensive multi-objective games with postponed preference articulation via memetic co-evolution. Knowledge-Based Systems, 2018, 154, 17- 31. doi: 10.1016/j.knosys.2018.05.012
31	HAREL M, MOSHAIOV A, ALKAHER D Rationalizable strategies for the navigator–target–missile game. Journal of Guidance, Control, and Dynamics, 2020, 43 (6): 1129- 1142. doi: 10.2514/1.G004875
32	YOUSFI-HALIMI N, RADJEF M S, SLIMANI H Refinement of pure Pareto Nash equilibria in finite multicriteria games using preference relations. Annals of Operations Research, 2018, 267 (1): 607- 628.
33	HAMEL A H, LOHNE A A set optimization approach to zero-sum matrix games with multi-dimensional payoffs. Mathematical Methods of Operations Research, 2018, 88 (3): 369- 397. doi: 10.1007/s00186-018-0639-z
34	MA Y Y, WANG G Q, HU X X, et al Two-stage hybrid heuristic search algorithm for novel weapon target assignment problems. Computers & Industrial Engineering, 2021, 162, 107717.
35	YAO Z X, MING L, CHEN Z J, et al Mission decision-making method of multi-aircraft cooperatively attacking multi-target based on game theoretic framework. Chinese Journal of Aeronautics, 2016, 29 (6): 1685- 1694. doi: 10.1016/j.cja.2016.09.006
36	ZHAO K Y, LI L, CHEN Z Q, et al A survey: optimization and applications of evidence fusion algorithm based on Dempster-Shafer theory. Applied Soft Computing, 2022, 124, 109075. doi: 10.1016/j.asoc.2022.109075
37	SU M C, LAI S C, LIN S C, et al A new approach to multi-aircraft air combat assignments. Swarm and Evolutionary Computation, 2012, 6, 39- 46. doi: 10.1016/j.swevo.2012.03.003
38	ZHOU L P, WAN S P, DONG J Y A fermatean fuzzy ELECTRE method for multi-criteria group decision-making. Informatica, 2022, 33 (1): 181- 224.
39	JOSE R F, SALVATORE G, BERNARD R Electre-Score: a first outranking based method for scoring actions. European Journal of Operational Research, 2022, 297, 986- 1005. doi: 10.1016/j.ejor.2021.05.017
40	LOTFI A, DJAAFAR Z, KAMAL A, et al A novel epsilon-dominance Harris Hawks optimizer for multi-objective optimization in engineering design problems. Neural Computing and Applications, 2022, 34, 17007- 17036. doi: 10.1007/s00521-022-07352-9
41	SANDHOLM T, GILPIN A, CONITZER V Mixed-integer programming methods for finding Nash equilibria. Proc. of the 20th National Conference on Artificial Intelligence, 2005, 2, 495- 501.
42	SHANG K A survey on the hypervolume indicator in evolutionary multiobjective optimization. IEEE Trans. on Evolutionary Computation, 2021, 25 (1): 1- 20. doi: 10.1109/TEVC.2020.3013290
43	REDDY M J, KUMAR D N Multiobjective differential evolution with application to reservoir system optimization. Journal of Computing in Civil Engineering, 2007, 21 (2): 136- 146. doi: 10.1061/(ASCE)0887-3801(2007)21:2(136)
44	ZITZLER E, DEB K, THIELE L Comparison of multiobjective evolutionary algorithms: empirical results. Evolutionary Computation, 2000, 8 (2): 173- 195. doi: 10.1162/106365600568202
45	WU J H, ZHANG D Q, JIANG C, et al On reliability analysis method through rotational sparse grid nodes. Mechanical Systems and Signal Processing, 2021, 147, 107106. doi: 10.1016/j.ymssp.2020.107106
46	SHAW R L. Fighter combat: tactics and maneuvering. Maryland: The United States Naval Institute Annapolis, 1985.
47	HAN Y T, ZHOU Q B, DUAN F Q A game strategy model in the digital curling system based on NFSP. Complex & Intelligent Systems, 2021, 8, 1857- 1863.
48	HU X W, LUO P C, ZHANG X N, et al Improved ant colony optimization for weapon-target assignment. Mathematical Problems in Engineering, 2018, 2018, 6481635.

Variable	Description
$ R $	Our UAV formation
$ B $	Foe UAV formation
$ {r}_{m} $	The $ m $th tactic of $ R $, $ m\in M $
$ {b}_{n} $	The $ n $th tactic of $ B $, $ n\in N $
${R}_{{\rm{tac}}}$	A set of tactics for $ R $
${B}_{{\rm{tac}}}$	A set of tactics for $ B $
$ p $	$ R $’s UAV number $ p $, $ p\in P $
$ q $	$ B $’s UAV number $ q $, $ q\in Q $
$ {p}_{{r}_{m}}^{p} $	The position and state of $ p $ when $ {r}_{m} $ is selected
$ {p}_{{b}_{n}}^{q} $	The position and state of $ q $ when $ {b}_{n} $ is selected
${R}_{{\rm{pos}}}$	The position and state sets of $ R $’s UAVs for the sets of tactics
${B}_{{\rm{pos}}}$	The position and state sets of $ B $’s UAVs for the sets of tactics

Partial binary relation	Description	Constraint condition
${o}_{ {\tau }_{mc}{\tau }_{nd} }{P}_{ij}{o}_{ {\tau }_{ {m '}{c'} }{\tau _{ {n'} {d'} } }}$	$ {o}_{{\tau }_{mc}{\tau }_{nd}} $ is strongly preferred to ${o}_{ {\tau _{ {m'} {c'} }}{\tau _{ {n}' {d'}} } }$	${f}_{ij}\left({o}_{ {\tau }_{mc}{\tau }_{nd} }\right) > {f}_{ij}\left({o}_{ {\tau _{ {m} '{c'}} }{\tau _{ {n'} {d'}} } }\right)+{p}_{ij}$
${o}_{ {\tau }_{mc}{\tau }_{nd} }{Q}_{ij}{o}_{ {\tau _{ {m' }{c'}} }{\tau _{ {n'} {d'} }} }$	$ {o}_{{\tau }_{mc}{\tau }_{nd}} $ is weakly preferred to ${o}_{ {\tau _{ {m'} {c'} }}{\tau _{ {n'} {d'} }} }$	${q}_{ij} < {f}_{ij}\left({o}_{ {\tau }_{mc}{\tau }_{nd} }\right)-{f}_{ij}\left({o}_{ {\tau _{ {m}' {c'}} }{\tau _{ {n}' {d'}} } }\right)\le {p}_{ij}$
${o}_{ {\tau }_{mc}{\tau }_{nd} }{I}_{ij}{o}_{ {\tau _{ {m'} {c'}} }{\tau _{ {n}' {d'} }} }$	$ {o}_{{\tau }_{mc}{\tau }_{nd}} $ is indifferent to ${o}_{ {\tau _{ {m}' {c'}} }{\tau _{ {n}' {d'} }} }$	$-{q}_{ij} < {f}_{ij}\left({o}_{ {\tau }_{mc}{\tau }_{nd} }\right)-{f}_{ij}\left({o}_{ {\tau _{ {m'}{c'} }}{\tau _{ {n'} {d'} }} }\right)\le {q}_{ij}$

Number	Set 1	$ \eta $	Set 2	$ \eta $	Set 3	$ \eta $
1	4×4×8	0.6	16×16×8	0.6	8×12×8	0.7
2	8×8×8	0.7	16×16×10	0.7	8×16×8	0.7
3	12×12×8	0.7	16×16×12	0.8	12×18×8	0.7
4	16×16×8	0.7	16×16×14	0.8	12×24×8	0.7
5	20×20×8	0.7	16×16×16	0.8	16×24×8	0.7
6	24×24×8	0.7	16×16×18	0.8	16×32×8	0.7
7	28×28×8	0.7	16×16×20	0.8	20×30×8	0.7
8	32×32×8	0.7	16×16×22	0.9	20×40×8	0.7
9	36×36×8	0.7	16×16×24	0.9	24×36×8	0.7
10	40×40×8	0.7	16×16×26	0.9	24×48×8	0.7
11	50×50×8	0.7	16×16×30	1.0	30×60×8	0.7
12	60×60×8	0.7	16×16×40	1.0	40×70×8	0.7
13	70×70×8	0.7	16×16×60	1.0	50×80×8	0.7
14	80×80×8	0.7	16×16×80	1.0	60×100×8	0.6
15	100×100×8	0.7	16×16×100	1.0	70×120×8	0.6

Number	Foe’s tactical style	Opponent tactics set
1	Offensive	a, c, d
2	Neutral	b, d, e
3	Defensive	c, d, e

R’tactics	$\left({x}_{R}/\mathrm{k}\mathrm{m},{y}_{R}/\mathrm{k}\mathrm{m},{z}_{R}/\mathrm{k}\mathrm{m},{v}_{R}/\left(\mathrm{m}/\mathrm{s}\right),{\theta }_{R}/\mathrm{r}\mathrm{a}\mathrm{d},{\phi }_{R}/\mathrm{r}\mathrm{a}\mathrm{d}\right)$	B’tactics	$\left({x}_{B}/\mathrm{k}\mathrm{m},{y}_{B}/\mathrm{k}\mathrm{m},{z}_{B}/\mathrm{k}\mathrm{m},{v}_{B}/\left(\mathrm{m}/\mathrm{s}\right),{\theta }_{B}/\mathrm{r}\mathrm{a}\mathrm{d},{\phi }_{B}/\mathrm{r}\mathrm{a}\mathrm{d}\right)$
$ {r}_{{r}_{2}{w}_{2}} $	$p_{r_2}^1$=(−94.38, 12.27, 8.341, 314.5, 0.014, 0.181),	$ {b}_{{b}_{1}{v}_{1}} $	$p_{b_1}^1$=(103.82, 17.82, 8.24, 275.31, 0.087, 3.181),
$ {r}_{{r}_{2}{w}_{2}} $	$p_{r_2}^2$=(−73.44, −11.72, 8.633, 222.1, −0.023, 0.178)	$ {b}_{{b}_{1}{v}_{1}} $	$p_{b_1}^2$= (102.19, −15.42, 8.57, 255.82, −0.041, 3.078)
$ {r}_{{r}_{1}{w}_{2}} $	$p_{r_1}^1$=(−94.26, 22.53, 8.213, 262.2, 0.005, −0.117),	$ {b}_{{b}_{4}{v}_{1}} $	$p_{b_4}^1$= (103.82, 15.82, 8.24, 285.47, −0.037, 3.219),
$ {r}_{{r}_{1}{w}_{2}} $	$p_{r_1}^2$=(−101.92, −21.42, 8.335, 241.4, −0.031, 0.168)	$ {b}_{{b}_{4}{v}_{1}} $	$p_{b_4}^2$= (95.19, −15.42, 8.37, 225.82, −0.052, 3.183)

Multicriteria game approach to air-to-air combat tactical decisions for multiple UAVs

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

Share this article

Figures/Tables 14

References 48

Related Articles 6

Recommended Articles

Metrics

Comments

Tactic	$ {b}_{{b}_{1}{v}_{1}} $	$ {b}_{{b}_{4}{v}_{1}} $
$ {r}_{{r}_{2}{w}_{2}} $	${{{\boldsymbol{F}}} }_{\mathit{R} }\left({r}_{ {r}_{2}{w}_{2} },{b}_{ {b}_{1}{v}_{1} }\right)$=(0.49825,0.40916,0.410481,0.285469, 0.268969,0.281608,0.290261,0.455265) ${{{\boldsymbol{F}}} }_{\mathit{B} }\left({r}_{ {r}_{2}{w}_{2} },{b}_{ {b}_{1}{v}_{1} }\right)$= (0.002112,0.0617146,0.059379,0.251632, 0.268914,0.255821,0.246304,0.00378254)	${{{\boldsymbol{F}}} }_{\mathit{R} }\left({r}_{ {r}_{2}{w}_{2} },{b}_{ {b}_{4}{v}_{1} }\right)$= (0.5,0.491106,0.485832,0.288779, 0.499388,0.499952,0.49992,0.499755) ${{{\boldsymbol{F}}} }_{\mathit{B} }\left({r}_{ {r}_{2}{w}_{2} },{b}_{ {b}_{4}{v}_{1} }\right)$= (0,6.22E−13,2.18E−08,0.117967, 0,0,0,0)
$ {r}_{{r}_{1}{w}_{2}} $	${{{\boldsymbol{F}}} }_{\mathit{R} }\left({r}_{ {r}_{1}{w}_{2} },{b}_{ {b}_{1}{v}_{1} }\right)$= (0.49825,0.420225,0.120395,0.200223, 0.268969,0.282472,0.2271,0.473773) ${{{\boldsymbol{F}}} }_{\mathit{B} }\left({r}_{ {r}_{1}{w}_{2} },{b}_{ {b}_{1}{v}_{1} }\right)$= (0,0.0428182,0.376873,0.234746, 0.268914,0.254891,0.30651,0.0000731)	${{{\boldsymbol{F}}} }_{\mathit{R} }\left({r}_{ {r}_{1}{w}_{2} },{b}_{ {b}_{4}{v}_{1} }\right)$= (0.5,0.492209,0.422766,0.303366, 0.499388,0.499953,0.499891,0.499857) ${{{\boldsymbol{F}}} }_{\mathit{B} }\left({r}_{ {r}_{1}{w}_{2} },{b}_{ {b}_{4}{v}_{1} }\right)$= (0,1.16E−14,0.0387535,0.230959, 0,0,0,0)

Tactics	Approaches	Offensive	Neutral	Defensive
${W}_{{\rm{HV}}}$	MG-ACTDMU MADM Random	0.01721 0.0154 0.00669	0.01836 0.01742 0.00777	0.01912 0.01787 0.00725
$ {P}_{w} $	MG-ACTDMU MADM Random	0.36 0.22 0.11	0.42 0.19 0.13	0.38 0.27 0.07
${P}_{{\rm{CER}}}$	MG-ACTDMU MADM Random	0.8461 1.2978 1.6296	0.9469 1.3804 1.7625	0.8983 1.1800 1.8243

[1]	Wei Zhou and Zeshui Xu. Modeling and applying credible interval intuitionistic fuzzy reciprocal preference relations in group decision making [J]. Systems Engineering and Electronics, 2017, 28(2): 301-314.
[2]	Qi Yue1, Zhiping Fan, and Lihua Shi. New approach to determine the priorities from interval fuzzy preference relations [J]. Journal of Systems Engineering and Electronics, 2011, 22(2): 267-273.
[3]	Shengbao Yao and Wan’an Cui. Method for multi-attribute group decision-making based on the compromise weights [J]. Journal of Systems Engineering and Electronics, 2010, 21(4): 591-597.
[4]	Zeshui Xu1,2,*. Interactive group decision making procedure based on uncertain multiplicative linguistic preference relations [J]. Journal of Systems Engineering and Electronics, 2010, 21(3): 408-415.
[5]	Yejun Xu1,?,Qingli Da2,and Xinwang Liu2. Some properties of linguistic preference relation and its ranking in group decision making [J]. Journal of Systems Engineering and Electronics, 2010, 21(2): 244-249.
[6]	Xiaohui Yu and Qiang Zhang. Fuzzy Nash equilibrium of fuzzy n-person non-cooperative game [J]. Journal of Systems Engineering and Electronics, 2010, 21(1): 47-56.