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Journal of ZheJiang University (Engineering Science)  2023, Vol. 57 Issue (11): 2133-2146    DOI: 10.3785/j.issn.1008-973X.2023.11.001
    
Dynamic multi-objective optimization algorithm based on individual prediction
Wan-liang WANG(),Zhong-kui CHEN,Fei WU,Zheng WANG,Meng-jiao YU
College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China
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Abstract  

A dynamic multi-objective optimization algorithm based on individual prediction (IPS) was proposed to quickly track the Pareto optimal front of the dynamic multi-objective optimization problem that changed with the environment. Firstly, the special points with good convergence and diversity were selected by the reference point relation algorithm, and the environment changes can be quickly responded to by predicting the special points set. Secondly, a feedback correction mechanism for population center point predication was proposed, and in the process of predicting the non-dominant solution set, the prediction step size was corrected to make the prediction more accurate. Finally, to avoid the algorithm falling into local optimal, a hybrid diversity maintenance mechanism was proposed, which introduced random individuals generated by Latin hypercube sampling and a precision controllable mutation strategy to improve the diversity of the population. The proposed algorithm was compared with the other four dynamic multi-objective optimization algorithms. Experimental results show that IPS can balance the diversity and convergence of the population, and the experimental results are better than that of the other four algorithms on the FDA, DMOP, and F5~F10 test suite.



Key wordsdynamic multi-objective optimization      reference point relation algorithm      special point      correction by feedback      diversity     
Received: 30 November 2022      Published: 11 December 2023
CLC:  TP 301  
Fund:  国家自然科学基金资助项目(51875524, 61873240);浙江大学CAD&CG国家重点实验室开放课题资助项目(A2210)
Cite this article:

Wan-liang WANG,Zhong-kui CHEN,Fei WU,Zheng WANG,Meng-jiao YU. Dynamic multi-objective optimization algorithm based on individual prediction. Journal of ZheJiang University (Engineering Science), 2023, 57(11): 2133-2146.

URL:

https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2023.11.001     OR     https://www.zjujournals.com/eng/Y2023/V57/I11/2133


基于个体预测的动态多目标优化算法

为了快速追踪随环境变化的动态多目标优化问题的Pareto前沿,提出基于个体预测的动态多目标优化算法(IPS). 利用参考点联系算法筛选出特殊点,该特殊点具有良好的收敛性和多样性,通过对特殊点集的预测快速响应环境变化. 提出针对种群中心点预测的反馈校正机制,在预测非支配解集的过程中,对预测步长进行反馈校正,从而使预测更加准确;为了避免算法陷入局部最优,提出混合多样性维持机制,引入由拉丁超立方抽样和精度可控的突变策略分别产生的随机个体,以提高种群的多样性. 将所提算法与其他4种动态多目标优化算法进行对比分析,实验结果表明,IPS能够平衡种群的多样性和收敛性,在FDA、DMOP、F5~F10系列问题上,实验结果优于其他4种算法.


关键词: 动态多目标优化,  参考点联系算法,  特殊点,  反馈校正,  多样性 
种类 PS PF
I 随时间变化 不变
II 随时间变化 随时间变化
III 不变 随时间变化
IV 不变 不变
Tab.1 Four different types of DMOPs
Fig.1 Flow chart of RM-MEDA
Fig.2 Schematic diagram of association between non-dominant individuals and reference points
Fig.3 Schematic diagram of special points prediction strategy
Fig.4 Step size exploration of decision space variables
Fig.5 Flow chart of IPS
问题 阶段 MIGD
PPS CKPS SPPS HPPCM IPS
1)注:?和?分别表示IPS算法的性能显著优于和等同于相应的算法.

FDA1
总阶段 0.0592(0.01680)?1) 0.0297(0.00636)? 0.0264(0.00630)? 0.0211(0.00518)? 0.0188(0.00423)
第1阶段 0.2743(0.08625)? 0.1168(0.03308)? 0.1032(0.03278)? 0.0813(0.02744)? 0.0692(0.02245)
第2阶段 0.0100(0.00117)? 0.0090(0.00013)? 0.0082(0.00021)? 0.0068(0.00076)? 0.0067(0.00029)
第3阶段 0.0063(0.00011) 0.0089(0.00020)? 0.0081(0.00022)? 0.0067(0.00069) 0.0070(0.00046)

FDA2
总阶段 0.0093(0.00120)? 0.0085(0.00056)? 0.0087(0.00044)? 0.0079(0.00087)? 0.0076(0.00075)
第1阶段 0.0219(0.00539)? 0.0183(0.00276)? 0.0170(0.00248)? 0.0172(0.00419)? 0.0160(0.00376)
第2阶段 0.0067(0.00058)? 0.0065(0.00021)? 0.0067(0.00025)? 0.0058(0.00028)? 0.0057(0.00009)
第3阶段 0.0059(0.00012)? 0.0060(0.00010)? 0.0068(0.00026)? 0.0056(0.00004) 0.0056(0.00004)

FDA3
总阶段 0.0843(0.01313)? 0.0383(0.00726)? 0.0363(0.00724)? 0.0216(0.00479) 0.0258(0.00647)
第1阶段 0.2590(0.06128)? 0.1222(0.03371)? 0.1323(0.03870)? 0.0862(0.02561) 0.1023(0.03297)
第2阶段 0.0436(0.01250)? 0.0189(0.00279)? 0.0137(0.00325)? 0.0064(0.00038) 0.0074(0.00060)
第3阶段 0.0419(0.00778)? 0.0179(0.00243)? 0.0133(0.00262)? 0.0061(0.00035) 0.0077(0.00067)

FDA4
总阶段 0.1314(0.00333)? 0.1184(0.00230)? 0.1095(0.00211)? 0.1020(0.00104)? 0.1010(0.00148)
第1阶段 0.1609(0.00920)? 0.1332(0.00845)? 0.1300(0.00688)? 0.1209(0.00692)? 0.1161(0.00498)
第2阶段 0.1258(0.00312)? 0.1148(0.00268)? 0.1050(0.00221)? 0.0969(0.00198) 0.0976(0.00167)
第3阶段 0.1230(0.00312)? 0.1149(0.00232)? 0.1043(0.00234)? 0.0980(0.00245)? 0.0973(0.00216)

DMOP1
总阶段 0.1296(0.24085)? 0.0076(0.00174)? 0.0099(0.00276)? 0.0068(0.00504)? 0.0064(0.00052)
第1阶段 0.5522(1.06835)? 0.0181(0.00901)? 0.0310(0.01421)? 0.0161(0.00382)? 0.0118(0.00272)
第2阶段 0.0528(0.09413)? 0.0050(0.00005) 0.0050(0.00070) 0.0047(0.00004) 0.0051(0.00002)
第3阶段 0.0057(0.00002)? 0.0051(0.00006) 0.0048(0.00054) 0.0046(0.00006) 0.0051(0.00005)

DMOP2
总阶段 0.0659(0.03602)? 0.0267(0.00616)? 0.0316(0.00584)? 0.0252(0.00504)? 0.0223(0.00254)
第1阶段 0.3008(0.16637)? 0.1006(0.03204)? 0.1263(0.03040)? 0.1021(0.02651)? 0.0848(0.01365)
第2阶段 0.0142(0.01391)? 0.0092(0.00012)? 0.0093(0.00026)? 0.0069(0.00047) 0.0075(0.00020)
第3阶段 0.0061(0.00003)? 0.0090(0.00020)? 0.0090(0.00030)? 0.0069(0.00042) 0.0073(0.00030)

DMOP3
总阶段 0.0482(0.04869)? 0.0277(0.00629) 0.0279(0.00828) 0.0242(0.00544) 0.0288(0.00431)
第1阶段 0.2173(0.00125)? 0.1065(0.03292) 0.1111(0.04314) 0.0958(0.02797) 0.1204(0.02258)
第2阶段 0.0099(0.00010)? 0.0091(0.00017)? 0.0082(0.00018)? 0.0078(0.00138)? 0.0071(0.00047)
第3阶段 0.0062(0.00954) 0.0089(0.00011)? 0.0081(0.00017)? 0.0066(0.00043) 0.0069(0.00017)
Tab.2 MIGD indicators for five strategies on FDA and DMOP
问题 阶段 MHVD
PPS CKPS SPPS HPPCM IPS

F5
总阶段 0.5124(0.07330)? 0.2769(0.01312)? 0.2782(0.01280)? 0.2745(0.00810)? 0.2725(0.01327)
第1阶段 1.4519(0.28264)? 0.4064(0.07001) 0.4155(0.06438)? 0.3752(0.03314) 0.4132(0.05877)
第2阶段 0.3309(0.05948)? 0.2466(0.00230)? 0.2455(0.00379)? 0.2508(0.00777)? 0.2379(0.00468)
第3阶段 0.2477(0.00173)? 0.2456(0.00171)? 0.2457(0.00174)? 0.2505(0.00681)? 0.2403(0.00508)

F6
总阶段 0.3222(0.03755)? 0.2659(0.00655)? 0.2693(0.01432)? 0.2652(0.00692) 0.2655(0.00412)
第1阶段 0.6239(0.18580)? 0.3462(0.03272)? 0.3647(0.07603)? 0.3470(0.03468)? 0.3455(0.02249)
第2阶段 0.2545(0.00503)? 0.2467(0.00117)? 0.2467(0.00093)? 0.2455(0.00147) 0.2459(0.00118)
第3阶段 0.2466(0.00144) 0.2470(0.00104) 0.2467(0.00095) 0.2459(0.00084) 0.2472(0.00165)

F7
总阶段 0.3596(0.03072)? 0.2690(0.01654)? 0.2664(0.00844)? 0.2672(0.01441)? 0.2614(0.00741)
第1阶段 0.8307(0.15396)? 0.3740(0.08647)? 0.3560(0.04440)? 0.3633(0.07469)? 0.3332(0.03972)
第2阶段 0.2492(0.00359)? 0.2443(0.00101) 0.2454(0.00071)? 0.2444(0.00074)? 0.2443(0.00105)
第3阶段 0.2463(0.00111)? 0.2439(0.00080) 0.2447(0.00093)? 0.2443(0.00075) 0.2444(0.00094)

F8
总阶段 0.3954(0.03307)? 0.4108(0.02681)? 0.3852(0.01921)? 0.3223(0.01427) 0.3277(0.00396)
第1阶段 0.6076(0.16045)? 0.5809(0.13203)? 0.5864(0.08519)? 0.4663(0.05926)? 0.4568(0.02104)
第2阶段 0.3504(0.01283)? 0.3684(0.05066)? 0.3374(0.00970)? 0.2889(0.00725) 0.2959(0.00706)
第3阶段 0.3397(0.01085)? 0.3725(0.02345)? 0.3373(0.01247)? 0.2872(0.01162) 0.2983(0.00646)

F9
总阶段 0.6408(0.07758)? 0.3705(0.03890)? 0.3411(0.01947)? 0.3353(0.04994)? 0.3113(0.02232)
第1阶段 1.6206(0.23861)? 0.7310(0.18668)? 0.6319(0.06566)? 0.6164(0.12440)? 0.5565(0.10975)
第2阶段 0.4945(0.09275)? 0.2950(0.01834)? 0.2697(0.01675)? 0.2695(0.05153)? 0.2593(0.00953)
第3阶段 0.3216(0.07873)? 0.2746(0.00704)? 0.2744(0.02416)? 0.2674(0.02417)? 0.2468(0.01399)

F10
总阶段 0.7720(0.06904)? 0.3612(0.01334) 0.3361(0.02204) 0.4034(0.05090)? 0.3927(0.03353)
第1阶段 1.5730(0.10988)? 0.7354(0.05476)? 0.6343(0.10452)? 0.6220(0.11651)? 0.5665(0.09724)
第2阶段 0.6341(0.13136)? 0.2782(0.01809) 0.2652(0.00560) 0.3423(0.04547) 0.3631(0.01942)
第3阶段 0.5294(0.08679)? 0.2666(0.01608) 0.2653(0.00905) 0.3606(0.06961)? 0.3398(0.03363)
Tab.3 MHVD indicators for five strategies on F5 to F10
问题 $ ({\tau _t},{n_t}) $ MIGD
PPS CKPS SPPS HPPCM IPS
FDA2 (20,10) 0.1194(0.00213)? 0.0087(0.00039)? 0.0097(0.00106)? 0.0087(0.00908)? 0.0082(0.00034)
(25,10) 0.0093(0.00120)? 0.0085(0.00056)? 0.0087(0.00044)? 0.0079(0.00087)? 0.0076(0.00075)
(30,10) 0.0081(0.00046)? 0.0075(0.00024)? 0.0084(0.00046)? 0.0072(0.00074) 0.0073(0.00296)
FDA4 (20,10) 0.1485(0.00453)? 0.1327(0.00086)? 0.1400(0.04446)? 0.1094(0.00244)? 0.1082(0.00119)
(25,10) 0.1314(0.00333)? 0.1184(0.00230)? 0.1095(0.00211)? 0.1020(0.00104)? 0.1010(0.00148)
(30,10) 0.1221(0.00174)? 0.1132(0.00066)? 0.1067(0.00127)? 0.0962(0.00129) 0.0964(0.00165)
DMOP1 (20,10) 0.1272(0.21435)? 0.0139(0.00559)? 0.0148(0.00274)? 0.0082(0.00175) 0.0084(0.00118)
(25,10) 0.1296(0.24085)? 0.0076(0.00174)? 0.0099(0.00276)? 0.0068(0.00504)? 0.0064(0.00052)
(30,10) 0.1093(0.19550)? 0.0060(0.00037) 0.0071(0.00103)? 0.0062(0.00120)? 0.0061(0.00044)
DMOP2 (20,10) 0.0805(0.01495)? 0.0408(0.00221)? 0.0353(0.00597) 0.0374(0.00724) 0.0384(0.00904)
(25,10) 0.0659(0.03602)? 0.0267(0.00616)? 0.0316(0.00584)? 0.0252(0.00504)? 0.0223(0.00254)
(30,10) 0.0391(0.01263)? 0.0224(0.00188)? 0.0196(0.00325)? 0.0196(0.00645)? 0.0168(0.00458)
F6 (20,10) 0.1212(0.10641)? 0.0284(0.00389)? 0.0291(0.00198)? 0.0252(0.00519)? 0.0248(0.00261)
(25,10) 0.0525(0.02698)? 0.0210(0.00376)? 0.0219(0.00705)? 0.0189(0.00492)? 0.0188(0.00281)
(30,10) 0.0432(0.02731)? 0.0152(0.00167)? 0.0155(0.00201)? 0.0127(0.00207) 0.0142(0.00106)
F9 (20,10) 0.8790(0.27757)? 0.1495(0.03041)? 0.2328(0.10508)? 0.1030(0.01668)? 0.0864(0.01512)
(25,10) 0.5884(0.20324)? 0.1476(0.06633)? 0.1019(0.02075)? 0.0832(0.03398)? 0.0677(0.01341)
(30,10) 0.5169(0.16636)? 0.0872(0.01821)? 0.1089(0.05834)? 0.0658(0.01456) 0.0731(0.02148)
Tab.4 MIGD indicators of five strategies on some test questions
问题 $ ({\tau _t},{n_t}) $ MHVD
PPS CKPS SPPS HPPCM IPS
FDA2 (20,10) 0.0329(0.00083)? 0.0324(0.00033)? 0.0333(0.00155)? 0.0326(0.00105)? 0.0315(0.00044)
(25,10) 0.0320(0.00113)? 0.0319(0.00074)? 0.0322(0.00061)? 0.0313(0.00068)? 0.0311(0.00045)
(30,10) 0.0310(0.00031)? 0.0310(0.00030)? 0.0319(0.00046)? 0.0307(0.00040) 0.0309(0.00036)
FDA4 (20,10) 0.4382(0.01399)? 0.3787(0.00302)? 0.4177(0.17410)? 0.2934(0.01041) 0.2986(0.00405)
(25,10) 0.3799(0.01168)? 0.3294(0.00692)? 0.3024(0.00749)? 0.2876(0.00799)? 0.2728(0.00496)
(30,10) 0.3459(0.00825)? 0.3116(0.00252)? 0.2909(0.00533)? 0.2473(0.00487) 0.2544(0.00545)
DMOP1 (20,10) 0.2308(0.13815)? 0.1510(0.00239)? 0.1455(0.00316) 0.1510(0.00152)? 0.1503(0.00066)
(25,10) 0.2199(0.12614)? 0.1489(0.00106) 0.1460(0.00146) 0.1506(0.00059)? 0.1502(0.00031)
(30,10) 0.2193(0.12482)? 0.1492(0.00061) 0.1473(0.00145) 0.1508(0.00054)? 0.1504(0.00044)
DMOP2 (20,10) 0.2462(0.01526)? 0.1699(0.00150)? 0.1684(0.00273)? 0.1754(0.00443)? 0.1644(0.00216)
(25,10) 0.2253(0.04903)? 0.1644(0.00351)? 0.1677(0.00252)? 0.1676(0.00185)? 0.1639(0.00175)
(30,10) 0.1896(0.02220)? 0.1642(0.00065)? 0.1619(0.00201)? 0.1656(0.00429)? 0.1593(0.00404)
F6 (20,10) 0.3912(0.09846)? 0.2744(0.00899)? 0.2776(0.00544)? 0.2811(0.01186)? 0.2729(0.00377)
(25,10) 0.3222(0.03755)? 0.2659(0.00655)? 0.2693(0.01432)? 0.2652(0.01061) 0.2655(0.00412)
(30,10) 0.3066(0.03760)? 0.2598(0.00432) 0.2602(0.00464) 0.2611(0.00575)? 0.2603(0.00321)
F9 (20,10) 0.8048(0.14654)? 0.3967(0.02728)? 0.4101(0.03460)? 0.3551(0.01192)? 0.3419(0.02853)
(25,10) 0.6408(0.07758)? 0.3705(0.03890)? 0.3411(0.01947)? 0.3353(0.04994)? 0.3113(0.02232)
(30,10) 0.6098(0.06508)? 0.3442(0.01784)? 0.3293(0.02125)? 0.3079(0.01038)? 0.3014(0.01270)
Tab.5 MHVD indicators of five strategies on some test questions
Fig.6 Solution set obtained by five strategies in process of solving DMOP2
Fig.7 Solution set obtained by five strategies in process of solving F9
Fig.8 IGD trend comparison of IPS(0) and IPS(9) over number of changes for 20 runs on F6 and F9
Fig.9 IGD trend comparison of IPS and IPS(N) over number of changes for 20 runs on F6 and F9
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