Adaptive Constraint-Boundary Learning-Based Two-Stage Dual-Population Evolutionary Algorithm
Abstract
1. Introduction
- (1)
- This paper adopts a novel two-stage and dual-population evolutionary algorithm for CMOPs. The first stage aims at making the auxiliary population approach the Unconstrained Pareto Front (UPF) and assisting the main population to cross the infeasible regions. The second stage aims at making the main population and auxiliary population converge to the CPF from the feasible side and infeasible side, respectively.
- (2)
- A constraint boundary learning mechanism is designed that adaptively adjusts according to the proportion of feasible solutions, which can help the main population of the first stage fully explore the constraint space and further improve the exploration ability of the main population. Moreover, it can drive the main population to cross the infeasible regions as well.
- (3)
- A cascaded multi-criteria hierarchical ranking-based environment selection is proposed to update the auxiliary population of the second stage, including non-dominated sorting, reverse objective sorting, convergence sorting, and diversity sorting. Thus, this mechanism can discover different types of promising infeasible solutions and further assist the population to find the complete CPF.
2. Related Work and Motivation
2.1. Related Work
- (1)
- Multi-population-based CMOEAs: The main idea of this category is to utilize the cooperation of multiple populations to guide solutions to converge toward the CPFs. For instance, Li et al. [22] developed a two-archive algorithm C-TAEA, where these two archives evolve feasible solutions and infeasible solutions, respectively. Ming et al. [27] proposed a concept of collaborative multi-population assistance (CMOEMT), with the main algorithmic innovation being the enhancement of mating pools and environmental selection capabilities. Liu et al. [28] developed a bidirectional co-evolution algorithm with two populations for CMOPs (BiCO), where two populations can approach the CPFs from feasible sides and infeasible sides, respectively. In [24], a cooperative algorithm with two populations (CMOEA-PP) was designed. In CMOEA-PP, the propulsive population only searched corner and center solutions, while the normal population searched solutions in all feasible regions. Moreover, Yang et al. [29] developed a new dual-population algorithm (dp-ACS), where the main population is optimized by the designed adaptive constraint strength function.
- (2)
- Multi-stage-based CMOEAs: This category divides the evolution process into multiple stages, and each stage utilizes different strategies to deal with constraints. A typical example is PPS including push and pull search stages [10]. In the push stage, PPS ignores constraints and aims at making the population approach the UPFs. In the pull stage, an improved constraint method was designed to deal with constraints. In [18], a two-phase framework (ToP) was proposed. The first phase aims to search for promising feasible regions, while the second phase applies a CMOEA to obtain the final solutions. Similarly, Tian et al. [19] designed a two-stage algorithm (CMOEA-MS) that is capable of dynamically adjusting the priority relationships between objectives and constraints. In [30], a three indicator-based two-stage CMOEA (TSTI) was developed. Moreover, Ma et al. [11] proposed a multi-stage CMOEA (MSCMO) for CMOPs, in which constraints were added one after the other handled in different stages of evolution.
- (3)
- Multi-stage and multi-population-based CMOEAs: In order to maintain the balance between population diversity and convergence, some researchers have proposed the theoretical idea of two-stage two-population framework. In [31], a co-evolution framework for dual populations, called CCMO, is proposed. In simple terms, in two different stages, one population deals with CMOP, and the other population solves CMOP in the unconstrained case to further maintain the diversity of the algorithm. Ming et al. [7] proposed a two-population two-stage evolutionary algorithm (CTSEA). In phase 1, the algorithm ignores the constraints, finds feasible solutions, and then obtains a good PF. In stage 2, the CPF is obtained considering the constraints.
- (4)
- Multi-ranking-based CMOEAs: Recently, some researchers developed the multi-ranking technique to balance convergence, diversity, and feasibility. Ma et al. [32] utilized two rankings to design a fitness function and embedded them into NSGA-II [8], where one ranking is based on the CDP and the other ranking is based on the Pareto dominance. Zhou et al. [33] applied the multi-ranking technique to integrate the convergence, diversity, and feasibility indicators. Cheng et al. [34] proposed a method of using infeasible solutions to handle subpopulations, thereby identifying narrow feasible regions.
2.2. Motivations of This Work
3. The Proposed CL-TDEA
3.1. Framework of CL-TDEA
Algorithm 1 Framework of CL-TDEA. |
Input: N (population size) Output: P (final population)
|
3.2. The Adaptive Constraint Boundary Learning Mechanism
3.3. The Elite Mating Selection
Algorithm 2 The elite mating selection. |
Input: (main population), (auxiliary population), N (population size) Output: (mating pool)
|
3.4. The Cascading Multi-Criteria Hierarchical Ranking Environment Selection
Algorithm 3 Cascading multi-criteria hierarchical ranking environment selection. |
Input: U (combined population), N (population size) Output: (auxiliary population)
|
3.5. Exploration–Exploitation Balance
4. Experimental Studies
4.1. Experimental Settings
- (1)
- Population size: The population size N is fixed at 100 for all test instances across all compared algorithms.
- (2)
- Genetic operators: All methods except for ToP, IMTCMO, and POCEA adopt the simulated binary crossover (SBX) [44] and polynomial mutation (PM) [45] to generate the offspring. The crossover and mutation probability, respectively, are set as 1 and 1/D, and the corresponding distribution indexes both are set to 20. For ToP, IMTCMO, and POCEA, they adopt the differential evolution to reproduce the offspring, whose relevant settings follow their original papers.
- (3)
- Run times and terminal conditions: All methods on each test instance independently run 20 times. The terminal condition of each run is the maximum fitness evaluations () that is set to 100,000.
- (4)
- Parameters in compared algorithms: The parameters in these compared methods are the same as in their original papers. For our TDEA, the parameter , determining the computational resource cost of different stages, is set to 0.4.
- (5)
- Statistical test method: The Wilcoxon test at a 0.05 significant level is adopted to analyze the significant difference between CL-TDEA and its competitors [46].
4.2. Comparison on MW Test Suite
4.3. Comparison on LIRCMOP Test Suite
4.4. Comparison on ZXH_CF Test Suite
4.5. Comparison on C-DTLZ and DC_DTLZ Test Suite
4.6. Comparison on SDC Test Suite
4.7. Validation of Main Components in CL-TDEA
4.8. Applications of CL-TDEA on Three Real-World Problems
4.9. Discussion on Exploration–Exploitation Balance
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Symbol | Definition |
---|---|
A solution vector in the D-dimensional decision space | |
m | Number of objective functions |
The i-th objective function value of solution x | |
The i-th inequality constraint | |
The j-th equality constraint | |
Small tolerance parameter for equality constraints | |
Constraint violation degree of solution x on the j-th constraint | |
Total constraint violation of solution x | |
Feasible decision space | |
Main population in CL-TDEA | |
Auxiliary population in CL-TDEA | |
Constraint boundary at generation t | |
N | Population size |
Number of fitness evaluations | |
Maximum number of fitness evaluations |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MW1 | 100 | 2 | 15 | 3.3830e-2 (3.48e-2) − | NaN (NaN) | 5.2463e-2 (6.51e-2) − | 1.8263e-2 (3.83e-2) − | 1.3089e-2 (4.57e-3) − | 4.2037e-2 (6.54e-2) − | 9.0638e-2 (4.42e-2) − | 1.7055e-3 (2.97e-5) − | 1.6168e-3 (3.59e-5) |
MW2 | 100 | 2 | 15 | 1.8133e-2 (7.89e-3) ≈ | 1.7940e-1 (1.48e-1) − | 5.4455e-2 (3.51e-2) − | 2.2471e-2 (7.64e-3) ≈ | 1.3531e-1 (1.04e-1) − | 2.4023e-2 (8.46e-3) − | 4.0501e-2 (2.85e-2) − | 5.9994e-2 (3.53e-2) − | 2.2620e-2 (1.83e-2) |
MW3 | 100 | 2 | 15 | 1.2954e-2 (2.52e-3) − | 2.2097e-1 (3.23e-1) − | 5.1052e-2 (4.79e-2) − | 5.0488e-2 (1.37e-3) − | 2.7188e-2 (1.48e-2) − | 1.3648e-2 (5.31e-3) − | 1.6305e-2 (3.24e-3) − | 4.8148e-3 (1.13e-4) ≈ | 4.9172e-3 (2.58e-4) |
MW4 | 100 | 3 | 15 | 6.4495e-2 (2.06e-2) − | 7.2571e-1 (0.00e+0) ≈ | 1.2132e-1 (6.19e-2) − | 6.1061e-2 (2.08e-2) − | 5.4908e-2 (3.34e-3) − | 4.9339e-2 (7.63e-3) − | 1.1079e-1 (2.35e-2) − | 4.8393e-2 (1.35e-2) − | 4.0988e-2 (4.24e-4) |
MW5 | 100 | 2 | 15 | 1.4127e-2 (4.15e-3) − | 7.5416e-1 (0.00e+0) ≈ | 4.7386e-2 (9.07e-3) − | 3.2066e-2 (1.18e-1) − | 7.6855e-2 (2.34e-2) − | 6.9010e-2 (1.22e-1) − | 7.7276e-2 (2.06e-2) − | 1.8159e-3 (5.34e-4) − | 5.4565e-4 (5.47e-4) |
MW6 | 100 | 2 | 15 | 1.0959e-2 (6.47e-3) + | 7.3029e-1 (3.51e-1) − | 2.3935e-1 (3.09e-1) − | 2.3689e-2 (1.13e-2) ≈ | 5.7904e-1 (3.67e-1) − | 1.7844e-1 (2.16e-1) − | 3.6539e-2 (1.27e-2) − | 1.5146e-1 (1.88e-1) − | 2.0689e-2 (8.43e-3) |
MW7 | 100 | 2 | 15 | 6.7262e-3 (5.04e-4) − | 5.7367e-2 (9.94e-2) − | 3.1073e-2 (1.10e-2) − | 4.4436e-3 (3.77e-4) ≈ | 5.2727e-2 (6.31e-2) − | 3.1495e-2 (2.92e-2) − | 1.5659e-2 (3.63e-3) − | 4.3318e-3 (2.04e-4) ≈ | 4.3787e-3 (2.52e-4) |
MW8 | 100 | 3 | 15 | 5.4905e-2 (3.09e-3) − | 6.9878e-1 (4.11e-1) − | 5.5415e-1 (2.77e-1) − | 5.1137e-2 (6.93e-3) − | 3.0482e-1 (2.45e-1) − | 5.2374e-2 (1.45e-2) − | 7.0693e-2 (2.33e-2) − | 7.4128e-2 (3.75e-2) − | 4.5019e-2 (2.07e-3) |
MW9 | 100 | 2 | 15 | 8.7774e-3 (1.69e-3) − | 8.5441e-1 (8.52e-2) − | 9.8392e-2 (2.12e-1) − | 7.5498e-2 (2.17e-1) − | 1.0676e-1 (2.11e-1) − | 2.0160e-1 (2.60e-1) − | 9.5902e-2 (2.11e-1) − | 5.5130e-3 (9.16e-4) − | 4.3120e-3 (8.80e-5) |
MW10 | 100 | 2 | 15 | 2.6026e-2 (1.65e-2) ≈ | NaN (NaN) | 7.7929e-2 (5.48e-2) − | 1.1679e-1 (1.85e-1) − | 2.9326e-1 (1.81e-1) − | 3.6383e-2 (1.90e-2) − | 8.3408e-2 (5.89e-2) − | 1.7727e-1 (5.62e-2) − | 1.9151e-2 (1.82e-2) |
MW11 | 100 | 2 | 15 | 2.0545e-2 (7.72e-3) − | 5.2819e-1 (2.69e-1) − | 3.7307e-2 (8.00e-3) − | 6.0964e-3 (1.20e-4) − | 6.3878e-2 (7.87e-2) − | 1.4053e-2 (1.01e-2) − | 1.8706e-2 (4.97e-3) − | 5.9456e-3 (1.35e-4) ≈ | 5.9537e-3 (1.53e-4) |
MW12 | 100 | 2 | 15 | 6.1014e-2 (1.82e-1) − | 8.3690e-1 (0.00e+0) ≈ | 1.2301e-1 (2.26e-1) − | 9.0572e-2 (2.39e-1) ≈ | 4.7754e-2 (1.01e-1) − | 1.5524e-1 (2.82e-1) − | 2.6368e-2 (1.05e-2) − | 4.7654e-3 (1.32e-4) ≈ | 4.7235e-3 (1.34e-4) |
MW13 | 100 | 2 | 15 | 6.3400e-2 (3.47e-2) ≈ | 1.0845e+0 (7.97e-1) − | 4.5309e-1 (1.69e-1) − | 1.2221e-1 (6.58e-2) − | 4.7870e-1 (3.54e-1) − | 5.0814e-1 (4.96e-1) − | 1.0459e-1 (3.46e-2) − | 1.6385e-1 (9.68e-2) − | 5.3575e-2 (2.60e-2) |
MW14 | 100 | 3 | 15 | 4.9614e-1 (3.99e-1) − | 8.5539e-1 (8.65e-1) − | 1.8182e-1 (3.86e-2) − | 2.6988e-1 (2.13e-1) − | 5.8826e-1 (5.27e-1) − | 3.2031e-1 (1.88e-1) − | 7.1254e-1 (1.98e-1) − | 9.9032e-2 (1.80e-3) ≈ | 9.8363e-2 (1.52e-3) |
+/−/≈ | 1/10/3 | 0/9/3 | 0/14/0 | 0/10/4 | 0/14/0 | 0/14/0 | 0/14/0 | 0/9/5 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MW1 | 100 | 2 | 15 | 4.4028e-1 (5.14e-2) − | NaN (NaN) | 4.3091e-1 (5.71e-2) − | 4.6877e-1 (3.66e-2) − | 4.6778e-1 (8.81e-3) − | 4.4701e-1 (5.90e-2) − | 3.7795e-1 (3.83e-2) − | 4.8951e-1 (6.92e-5) − | 4.9004e-1 (1.44e-4) |
MW2 | 100 | 2 | 15 | 5.5749e-1 (1.36e-2) ≈ | 3.7181e-1 (1.54e-1) − | 5.1177e-1 (3.26e-2) − | 5.5054e-1 (1.29e-2) ≈ | 4.0848e-1 (1.04e-1) − | 5.4790e-1 (1.25e-2) − | 5.2371e-1 (3.92e-2) − | 4.9835e-1 (4.47e-2) − | 5.5091e-1 (2.61e-2) |
MW3 | 100 | 2 | 15 | 5.3036e-1 (5.69e-3) − | 3.8019e-1 (1.79e-1) − | 5.0679e-1 (3.05e-2) − | 5.2723e-1 (2.02e-3) − | 5.1197e-1 (2.20e-2) − | 5.3372e-1 (6.42e-3) − | 5.2490e-1 (6.74e-3) − | 5.4446e-1 (1.83e-4) + | 5.4369e-1 (5.38e-4) |
MW4 | 100 | 3 | 15 | 8.0781e-1 (3.59e-2) − | 1.1720e-1 (0.00e+0) ≈ | 7.5360e-1 (5.62e-2) − | 8.1638e-1 (2.74e-2) − | 8.2053e-1 (4.97e-3) − | 8.2816e-1 (1.25e-2) − | 7.4032e-1 (3.39e-2) − | 8.2920e-1 (1.99e-2) − | 8.4116e-1 (4.91e-4) |
MW5 | 100 | 2 | 15 | 3.1575e-1 (3.06e-3) − | 5.6763e-2 (0.00e+0) ≈ | 2.9680e-1 (5.95e-3) − | 3.1332e-1 (3.41e-2) − | 2.5604e-1 (2.97e-2) − | 2.9085e-1 (3.74e-2) − | 2.3863e-1 (2.70e-2) − | 3.2350e-1 (3.57e-4) − | 3.2444e-1 (1.79e-4) |
MW6 | 100 | 2 | 15 | 3.1273e-1 (9.65e-3) + | 7.1210e-2 (5.91e-2) − | 2.2186e-1 (7.25e-2) − | 2.9767e-1 (1.54e-2) ≈ | 9.8677e-2 (9.92e-2) − | 2.5440e-1 (6.20e-2) − | 2.7929e-1 (1.85e-2) − | 2.3016e-1 (6.84e-2) − | 3.0176e-1 (1.14e-2) |
MW7 | 100 | 2 | 15 | 4.0939e-1 (6.45e-4) − | 3.7013e-1 (3.75e-2) − | 3.9406e-1 (3.86e-3) − | 4.1204e-1 (7.67e-4) ≈ | 3.7044e-1 (3.95e-2) − | 3.9918e-1 (2.22e-2) − | 3.9896e-1 (4.10e-3) − | 4.1218e-1 (2.20e-4) ≈ | 4.1211e-1 (5.55e-4) |
MW8 | 100 | 3 | 15 | 5.1979e-1 (1.35e-2) − | 1.2738e-1 (1.54e-1) − | 2.0847e-1 (1.47e-1) − | 5.1916e-1 (1.92e-2) − | 2.8547e-1 (1.49e-1) − | 5.2514e-1 (1.37e-2) − | 4.7820e-1 (4.53e-2) − | 4.6974e-1 (6.71e-2) − | 5.3650e-1 (1.02e-2) |
MW9 | 100 | 2 | 15 | 3.9136e-1 (4.57e-3) − | 0.0000e+0 (0.00e+0) − | 3.2957e-1 (1.13e-1) − | 3.5645e-1 (1.22e-1) − | 3.1255e-1 (1.05e-1) − | 2.5257e-1 (1.48e-1) − | 3.2620e-1 (7.81e-2) − | 3.9469e-1 (2.81e-3) − | 3.9941e-1 (1.41e-3) |
MW10 | 100 | 2 | 15 | 4.2774e-1 (1.64e-2) ≈ | NaN (NaN) | 3.8532e-1 (3.36e-2) − | 3.7603e-1 (9.68e-2) − | 2.6992e-1 (8.49e-2) − | 4.1888e-1 (1.66e-2) − | 3.8526e-1 (3.58e-2) − | 3.3005e-1 (3.08e-2) − | 4.3590e-1 (1.79e-2) |
MW11 | 100 | 2 | 15 | 4.3958e-1 (3.89e-3) − | 3.0696e-1 (5.99e-2) − | 4.3233e-1 (3.57e-3) − | 4.4753e-1 (1.55e-4) − | 4.2041e-1 (2.57e-2) − | 4.4094e-1 (6.73e-3) − | 4.4071e-1 (1.88e-3) − | 4.4783e-1 (1.22e-4) ≈ | 4.4781e-1 (1.49e-4) |
MW12 | 100 | 2 | 15 | 5.5480e-1 (1.32e-1) − | 0.0000e+0 (0.00e+0) ≈ | 4.9722e-1 (1.90e-1) − | 5.3878e-1 (1.82e-1) ≈ | 5.6168e-1 (8.68e-2) − | 4.8489e-1 (2.14e-1) − | 5.7818e-1 (1.60e-2) − | 6.0465e-1 (3.98e-4) − | 6.0493e-1 (2.00e-4) |
MW13 | 100 | 2 | 15 | 4.5178e-1 (1.60e-2) ≈ | 1.6784e-1 (1.39e-1) − | 3.2082e-1 (5.20e-2) − | 4.3416e-1 (2.05e-2) − | 2.5860e-1 (1.24e-1) − | 3.3448e-1 (1.18e-1) − | 4.3310e-1 (1.60e-2) − | 3.8739e-1 (6.02e-2) − | 4.5439e-1 (8.90e-3) |
MW14 | 100 | 3 | 15 | 2.8965e-1 (1.83e-1) − | 2.4175e-1 (1.88e-1) − | 4.3891e-1 (1.65e-2) − | 4.0244e-1 (8.61e-2) ≈ | 2.9586e-1 (1.68e-1) − | 3.9108e-1 (8.24e-2) − | 2.2506e-1 (7.75e-2) − | 4.6953e-1 (1.42e-3) + | 4.6774e-1 (1.96e-3) |
1/10/3 | 0/9/3 | 0/14/0 | 0/9/5 | 0/14/0 | 0/14/0 | 0/14/0 | 2/10/2 |
Algorithm | IGD (Wilcoxon) | HV (Wilcoxon) | Average Rank | |||||
---|---|---|---|---|---|---|---|---|
-Value | -Value | IGD | HV | |||||
CL-TDEA | - | - | - | - | - | - | 1.25 | 1.33 |
C-TAEA | 96.0 | 9.0 | 94.0 | 11.0 | 5.50 * | 5.25 * | ||
ToP | 105.0 | 0.0 | 105.0 | 0.0 | 8.00 * | 8.00 * | ||
TiGE2 | 105.0 | 0.0 | 105.0 | 0.0 | 8.00 * | 8.00 * | ||
MSCMO | 103.0 | 2.0 | 89.5 | 1.5 | 6.75 * | 6.00 * | ||
POCEA | 105.0 | 0.0 | 105.0 | 0.0 | 8.00 * | 8.00 * | ||
CMOEA-MS | 105.0 | 0.0 | 105.0 | 0.0 | 8.00 * | 8.00 * | ||
CMOEMT | 105.0 | 0.0 | 105.0 | 0.0 | 8.00 * | 8.00 * | ||
IMTCMO | 85.0 | 6.0 | 87.5 | 17.5 | 4.50 * | 5.50 * |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
LIRCMOP1 | 100 | 2 | 30 | 3.0813e-1 (9.45e-2) − | 3.2853e-1 (2.16e-2) − | 2.0840e-1 (1.37e-2) − | 1.7366e-1 (6.64e-2) ≈ | 1.7640e-1 (1.36e-1) ≈ | 3.2481e-1 (4.71e-2) − | 3.3986e-1 (1.69e-2) − | 2.6521e-2 (8.07e-3) + | 1.4002e-1 (2.53e-2) |
LIRCMOP2 | 100 | 2 | 30 | 2.4246e-1 (9.34e-2) − | 2.8906e-1 (2.18e-2) − | 1.9456e-1 (1.79e-2) − | 1.7570e-1 (4.37e-2) − | 1.5439e-1 (1.02e-1) ≈ | 2.7883e-1 (4.45e-2) − | 2.7467e-1 (3.39e-2) − | 1.9316e-1 (4.29e-2) − | 1.4119e-1 (1.92e-2) |
LIRCMOP3 | 100 | 2 | 30 | 3.4241e-1 (1.64e-1) − | 3.4399e-1 (1.64e-2) − | 2.1517e-1 (1.30e-2) − | 1.7748e-1 (7.59e-2) ≈ | 2.0752e-1 (9.07e-2) ≈ | 3.2129e-1 (4.27e-2) − | 2.9253e-1 (4.21e-2) − | 1.5362e-2 (7.41e-3) + | 1.6735e-1 (4.45e-2) |
LIRCMOP4 | 100 | 2 | 30 | 2.9330e-1 (5.13e-2) − | 3.2278e-1 (1.21e-2) − | 2.1541e-1 (2.53e-2) − | 1.7877e-1 (6.06e-2) ≈ | 2.3693e-1 (9.22e-2) ≈ | 2.7799e-1 (4.96e-2) − | 2.8009e-1 (3.38e-2) − | 2.1581e-1 (4.24e-2) − | 1.8547e-1 (2.44e-2) |
LIRCMOP5 | 100 | 2 | 30 | 1.2229e+0 (2.19e-1) − | 1.1161e+0 (2.81e-1) − | 9.5812e-1 (4.11e-1) − | 7.1017e-1 (1.20e+0) − | 1.2393e+0 (9.41e-1) − | 3.7233e-1 (2.04e-1) − | 1.3044e+0 (1.84e-1) − | 1.6409e+0 (5.60e-1) − | 2.4234e-1 (6.37e-2) |
LIRCMOP6 | 100 | 2 | 30 | 1.4359e+0 (3.20e-1) − | 1.3467e+0 (4.70e-4) − | 9.6675e-1 (4.14e-1) − | 5.6711e-1 (4.75e-1) ≈ | 1.0393e+0 (6.30e-1) − | 5.0159e-1 (3.69e-1) ≈ | 1.3602e+0 (1.47e-2) − | 3.9458e-1 (4.69e-1) ≈ | 3.7238e-1 (2.40e-1) |
LIRCMOP7 | 100 | 2 | 30 | 8.2863e-1 (7.43e-1) − | 1.3804e+0 (6.23e-1) − | 3.2249e-1 (1.50e-1) − | 1.2684e-1 (5.15e-2) ≈ | 1.2660e+0 (1.05e+0) − | 1.3986e-1 (3.98e-2) ≈ | 1.0872e+0 (6.90e-1) − | 3.0093e-2 (4.74e-2) + | 1.2080e-1 (3.03e-2) |
LIRCMOP8 | 100 | 2 | 30 | 1.3794e+0 (7.07e-1) − | 1.5448e+0 (4.26e-1) − | 5.1108e-1 (3.28e-1) − | 2.0599e-1 (5.03e-2) − | 1.1036e+0 (1.04e+0) − | 1.9733e-1 (3.83e-2) ≈ | 1.3836e+0 (5.53e-1) − | 3.6623e-2 (5.69e-2) + | 1.6572e-1 (4.99e-2) |
LIRCMOP9 | 100 | 2 | 30 | 8.1715e-1 (2.97e-1) − | 5.9781e-1 (1.38e-1) ≈ | 7.2148e-1 (9.51e-2) − | 5.4604e-1 (2.37e-1) ≈ | 9.0584e-1 (3.05e-1) − | 6.9448e-1 (1.45e-1) − | 1.0183e+0 (1.42e-1) − | 3.7098e-1 (7.51e-2) + | 5.4696e-1 (1.22e-1) |
LIRCMOP10 | 100 | 2 | 30 | 7.2617e-1 (4.02e-1) − | 4.2407e-1 (5.83e-2) − | 8.5507e-1 (1.84e-1) − | 2.6753e-1 (1.36e-1) ≈ | 8.4783e-1 (3.40e-1) − | 4.7126e-1 (1.46e-1) − | 1.0074e+0 (9.14e-2) − | 2.2463e-2 (4.56e-2) + | 2.5525e-1 (1.07e-1) |
LIRCMOP11 | 100 | 2 | 30 | 6.2374e-1 (4.03e-1) − | 4.7327e-1 (1.14e-1) − | 7.9533e-1 (2.53e-1) − | 1.6791e-1 (1.34e-1) ≈ | 7.8143e-1 (3.27e-1) − | 3.5132e-1 (1.61e-1) − | 9.3467e-1 (1.50e-1) − | 1.1186e-1 (9.19e-2) ≈ | 1.1354e-1 (9.31e-2) |
LIRCMOP12 | 100 | 2 | 30 | 6.5436e-1 (4.89e-1) − | 3.1726e-1 (7.74e-2) − | 4.5723e-1 (9.00e-2) − | 4.1603e-1 (1.96e-1) − | 7.1277e-1 (3.84e-1) − | 3.9776e-1 (1.40e-1) − | 8.7013e-1 (2.83e-1) − | 1.7933e-1 (4.18e-2) ≈ | 1.6374e-1 (6.39e-2) |
LIRCMOP13 | 100 | 3 | 30 | 7.2146e-1 (6.26e-1) − | 1.3129e+0 (9.95e-2) − | 1.2939e+0 (3.17e-1) − | 9.6046e-2 (2.33e-3) ≈ | 7.9913e-1 (6.11e-1) − | 9.4959e-2 (2.80e-3) ≈ | 2.7971e-1 (2.22e-2) − | 1.3102e+0 (1.54e-3) − | 9.4607e-2 (1.15e-3) |
LIRCMOP14 | 100 | 3 | 30 | 1.1114e-1 (1.37e-3) − | 1.3028e+0 (6.03e-3) − | 1.3102e+0 (2.15e-1) − | 9.7539e-2 (2.42e-3) − | 9.6798e-1 (5.67e-1) − | 9.7374e-2 (2.81e-3) − | 2.4478e-1 (1.47e-2) − | 1.2842e+0 (4.32e-3) − | 9.5003e-2 (8.90e-4) |
0/14/0 | 0/13/1 | 0/14/0 | 0/5/9 | 0/10/4 | 0/10/4 | 0/14/0 | 6/5/3 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
LIRCMOP1 | 100 | 2 | 30 | 1.1153e-1 (2.82e-2) − | 1.0174e-1 (9.76e-3) − | 1.4146e-1 (6.14e-3) − | 1.4956e-1 (2.49e-2) − | 1.5953e-1 (5.44e-2) ≈ | 1.0830e-1 (1.23e-2) − | 1.0091e-1 (4.59e-3) − | 2.2698e-1 (6.49e-3) + | 1.6643e-1 (1.22e-2) |
LIRCMOP2 | 100 | 2 | 30 | 2.4258e-1 (5.15e-2) ≈ | 2.1439e-1 (1.74e-2) − | 2.6515e-1 (8.62e-3) − | 2.8703e-1 (1.63e-2) ≈ | 2.8226e-1 (5.06e-2) ≈ | 2.2503e-1 (2.03e-2) − | 2.1985e-1 (1.55e-2) − | 2.5735e-1 (2.07e-2) − | 2.8146e-1 (1.27e-2) |
LIRCMOP3 | 100 | 2 | 30 | 9.9568e-2 (2.44e-2) − | 9.3085e-2 (5.54e-3) − | 1.2988e-1 (6.94e-3) − | 1.3639e-1 (2.11e-2) ≈ | 1.3158e-1 (2.68e-2) ≈ | 9.8505e-2 (1.27e-2) − | 1.0345e-1 (1.17e-2) − | 2.0118e-1 (3.33e-3) + | 1.4366e-1 (1.63e-2) |
LIRCMOP4 | 100 | 2 | 30 | 1.9572e-1 (2.13e-2) − | 1.7739e-1 (1.21e-2) − | 2.2270e-1 (1.17e-2) − | 2.4669e-1 (2.45e-2) ≈ | 2.1458e-1 (4.01e-2) ≈ | 1.9938e-1 (2.30e-2) − | 1.9454e-1 (1.38e-2) − | 2.2162e-1 (1.89e-2) − | 2.3672e-1 (1.46e-2) |
LIRCMOP5 | 100 | 2 | 30 | 7.1968e-3 (3.22e-2) − | 2.0299e-2 (6.39e-2) − | 4.3036e-2 (6.80e-2) − | 1.2039e-1 (6.33e-2) − | 7.2046e-2 (1.16e-1) − | 1.3863e-1 (4.05e-2) − | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 1.7632e-1 (2.32e-2) |
LIRCMOP6 | 100 | 2 | 30 | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 3.8927e-2 (4.08e-2) − | 8.6392e-2 (5.33e-2) ≈ | 4.5566e-2 (5.91e-2) − | 8.7316e-2 (3.86e-2) ≈ | 0.0000e+0 (0.00e+0) − | 1.2314e-1 (7.09e-2) ≈ | 1.0191e-1 (2.84e-2) |
LIRCMOP7 | 100 | 2 | 30 | 1.3077e-1 (1.14e-1) − | 4.8767e-2 (1.01e-1) − | 1.9985e-1 (2.55e-2) − | 2.4612e-1 (1.65e-2) ≈ | 9.2152e-2 (1.16e-1) − | 2.4346e-1 (1.15e-2) ≈ | 8.6706e-2 (1.00e-1) − | 2.8471e-1 (1.80e-2) + | 2.4796e-1 (9.47e-3) |
LIRCMOP8 | 100 | 2 | 30 | 5.8184e-2 (9.46e-2) − | 1.9875e-2 (6.15e-2) − | 1.6923e-1 (5.11e-2) − | 2.2868e-1 (8.63e-3) − | 1.1464e-1 (1.07e-1) − | 2.3102e-1 (7.74e-3) − | 4.2404e-2 (7.75e-2) − | 2.8334e-1 (2.14e-2) + | 2.3944e-1 (1.13e-2) |
LIRCMOP9 | 100 | 2 | 30 | 1.9810e-1 (1.38e-1) − | 2.8476e-1 (8.05e-2) − | 2.3237e-1 (7.89e-2) − | 3.4950e-1 (8.95e-2) ≈ | 1.8196e-1 (1.10e-1) − | 2.6528e-1 (5.93e-2) − | 1.1293e-1 (6.34e-2) − | 4.1546e-1 (2.35e-2) + | 3.3724e-1 (6.00e-2) |
LIRCMOP10 | 100 | 2 | 30 | 2.8484e-1 (2.46e-1) − | 4.7631e-1 (6.28e-2) − | 1.4833e-1 (4.61e-2) − | 5.3428e-1 (1.14e-1) ≈ | 1.8828e-1 (1.65e-1) − | 4.2822e-1 (1.28e-1) − | 7.3399e-2 (2.80e-2) − | 6.9672e-1 (2.12e-2) + | 5.5210e-1 (7.44e-2) |
LIRCMOP11 | 100 | 2 | 30 | 3.7493e-1 (2.26e-1) − | 3.7436e-1 (8.42e-2) − | 2.1275e-1 (9.66e-2) − | 6.0356e-1 (9.57e-2) ≈ | 2.6700e-1 (1.47e-1) − | 4.6917e-1 (1.14e-1) − | 1.7346e-1 (7.21e-2) − | 6.2523e-1 (5.90e-2) ≈ | 6.3719e-1 (6.52e-2) |
LIRCMOP12 | 100 | 2 | 30 | 3.3161e-1 (1.95e-1) − | 4.6118e-1 (4.01e-2) − | 3.8103e-1 (5.36e-2) − | 4.6846e-1 (5.54e-2) − | 3.3562e-1 (1.20e-1) − | 4.3360e-1 (5.43e-2) − | 2.2078e-1 (1.02e-1) − | 5.2757e-1 (2.02e-2) ≈ | 5.4167e-1 (3.18e-2) |
LIRCMOP13 | 100 | 3 | 30 | 2.7248e-1 (2.80e-1) − | 5.3778e-3 (1.68e-2) − | 3.5692e-2 (1.17e-1) − | 5.5355e-1 (1.68e-3) + | 2.3106e-1 (2.54e-1) − | 5.5430e-1 (2.08e-3) + | 3.6032e-1 (1.87e-2) − | 1.6796e-4 (1.43e-4) − | 5.5131e-1 (1.76e-3) |
LIRCMOP14 | 100 | 3 | 30 | 5.4616e-1 (9.14e-4) − | 2.5125e-4 (2.75e-4) − | 1.5500e-2 (6.90e-2) − | 5.5389e-1 (1.73e-3) − | 1.5822e-1 (2.43e-1) − | 5.5415e-1 (1.90e-3) − | 4.0060e-1 (1.50e-2) − | 5.2185e-4 (2.74e-4) − | 5.5543e-1 (1.45e-3) |
0/13/1 | 0/14/0 | 0/14/0 | 1/5/8 | 0/10/4 | 1/11/2 | 0/14/0 | 6/5/3 |
Algorithm | IGD (Wilcoxon) | HV (Wilcoxon) | Average Rank | |||||
---|---|---|---|---|---|---|---|---|
-Value | -Value | IGD | HV | |||||
CL-TDEA | - | - | - | - | - | - | 1.64 | 1.64 |
C-TAEA | 105.0 | 0.0 | 105.0 | 0.0 | 6.50 * | 6.50 * | ||
ToP | 105.0 | 0.0 | 105.0 | 0.0 | 7.00 * | 7.00 * | ||
TiGE2 | 105.0 | 0.0 | 105.0 | 0.0 | 5.79 * | 5.79 * | ||
MSCMO | 99.5 | 5.5 | 83.0 | 22.0 | 3.07 | 3.07 | ||
POCEA | 105.0 | 0.0 | 83.0 | 22.0 | 6.00 * | 6.00 * | ||
CMOEA-MS | 105.0 | 0.0 | 103.0 | 2.0 | 4.43 | 4.43 | ||
CMOEMT | 105.0 | 0.0 | 105.0 | 0.0 | 7.50 * | 7.50 * | ||
IMTCMO | 53.0 | 52.0 | 50.0 | 55.0 | 3.07 | 3.07 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
ZXH_CF1 | 100 | 3 | 13 | 4.8756e-2 (1.50e-3) − | 9.7660e-2 (1.56e-2) − | 1.2369e-1 (1.36e-2) − | 4.9205e-2 (8.82e-4) − | 5.2941e-2 (1.79e-2) − | 4.9308e-2 (1.25e-3) − | 5.2649e-2 (1.43e-3) − | 5.4677e-2 (1.77e-3) − | 4.7541e-2 (3.39e-3) |
ZXH_CF2 | 100 | 3 | 13 | 9.6657e-2 (8.65e-2) + | 3.1536e-1 (1.70e-1) − | 2.6961e-1 (2.47e-1) − | 1.7290e-1 (2.02e-1) − | 1.5956e-1 (8.97e-2) − | 1.4038e-1 (2.63e-1) ≈ | 6.4230e-2 (1.64e-3) + | 6.0289e-2 (1.06e-3) + | 1.3447e-1 (3.16e-1) |
ZXH_CF3 | 100 | 3 | 13 | 7.5642e-2 (2.67e-3) − | 2.2845e-1 (5.99e-2) − | 1.2760e-1 (1.07e-2) − | 7.8090e-2 (1.78e-2) − | 1.8446e-1 (3.52e-2) − | 8.2462e-2 (3.86e-3) − | 6.7928e-2 (1.69e-3) − | 8.0494e-2 (2.65e-3) − | 6.2887e-2 (1.24e-3) |
ZXH_CF4 | 100 | 3 | 13 | 1.1208e-1 (6.64e-2) ≈ | 1.2052e+0 (2.55e-1) − | 3.0388e-1 (2.33e-1) − | 1.4814e-1 (1.13e-1) ≈ | 7.1163e-1 (4.22e-1) − | 2.5642e-1 (3.04e-1) − | 1.1381e-1 (7.27e-2) ≈ | 3.0602e-1 (4.02e-1) − | 1.2280e-1 (8.92e-2) |
ZXH_CF5 | 100 | 3 | 13 | 5.7842e-2 (3.25e-2) − | 1.1952e-1 (5.06e-2) − | 1.9909e-1 (3.17e-1) − | 1.8633e-1 (2.35e-1) − | 8.5074e-2 (1.55e-2) − | 1.5666e-1 (3.91e-1) − | 4.5666e-2 (1.87e-2) − | 3.6755e-2 (6.76e-4) + | 4.3152e-2 (4.60e-2) |
ZXH_CF6 | 100 | 3 | 13 | 3.6313e-2 (1.60e-3) − | 5.6768e-2 (7.17e-3) − | 5.9488e-2 (3.42e-3) − | 3.7145e-2 (1.53e-2) − | 7.0337e-2 (1.44e-2) − | 3.3786e-2 (9.61e-4) − | 3.5051e-2 (6.08e-4) − | 3.4249e-2 (8.42e-4) − | 3.1227e-2 (6.29e-4) |
ZXH_CF7 | 100 | 3 | 13 | 9.3481e-2 (6.01e-2) ≈ | 5.5079e-1 (2.46e-1) − | 2.2175e-1 (2.13e-1) − | 3.5776e-1 (2.13e-1) − | 6.2444e-1 (4.54e-1) − | 2.6985e-1 (2.08e-1) − | 7.5257e-2 (6.45e-2) ≈ | 1.9554e-1 (2.42e-1) − | 9.1350e-2 (8.85e-2) |
ZXH_CF8 | 100 | 3 | 13 | 5.6277e-2 (5.62e-3) − | 2.4528e-1 (3.62e-1) − | 7.3875e-2 (6.42e-3) − | 2.3968e-1 (1.34e-1) − | 1.3624e-1 (2.52e-2) − | 1.5169e-1 (2.76e-2) − | 4.0841e-2 (1.95e-3) − | 4.2737e-2 (1.05e-3) − | 3.4160e-2 (5.21e-4) |
ZXH_CF9 | 100 | 3 | 13 | 6.5038e-2 (1.40e-2) − | 5.1171e-1 (1.21e-1) − | 1.0999e-1 (1.97e-2) − | 7.4746e-2 (3.12e-2) − | 3.8132e-1 (1.95e-1) − | 1.4279e-1 (1.06e-1) − | 3.0373e-2 (9.33e-4) − | 2.9977e-2 (7.86e-4) − | 2.7075e-2 (7.43e-4) |
ZXH_CF10 | 100 | 3 | 13 | 1.2220e-1 (8.04e-2) − | 1.1566e+0 (4.25e-1) − | 1.6023e-1 (4.26e-2) − | 1.9627e-1 (1.66e-1) − | 1.1922e+0 (4.89e-1) − | 3.6530e-1 (3.20e-1) − | 9.2758e-2 (6.50e-2) − | 1.9639e-1 (1.98e-1) − | 6.1841e-2 (4.99e-2) |
ZXH_CF11 | 100 | 3 | 13 | 3.5532e-2 (9.70e-4) − | 5.1396e-1 (2.09e-1) − | 1.8884e-1 (3.04e-2) − | 9.0487e-2 (2.33e-2) − | 6.9402e-2 (6.93e-3) − | 2.9348e-2 (3.45e-4) − | 3.0692e-2 (5.31e-4) − | 2.8493e-2 (3.77e-4) − | 2.8135e-2 (3.63e-4 |
ZXH_CF12 | 100 | 3 | 13 | 7.5879e-2 (1.50e-1) − | 4.8357e-1 (1.01e-1) − | 2.4806e-1 (2.31e-1) − | 2.0889e-1 (3.48e-1) − | 1.1436e-1 (7.59e-2) − | 1.0819e-1 (2.19e-1) − | 2.9167e-2 (1.76e-3) + | 2.5987e-2 (8.12e-4) + | 5.9109e-2 (1.16e-1) |
ZXH_CF13 | 100 | 2 | 12 | 7.4464e-2 (1.04e-1) − | 9.5274e-1 (3.91e-1) − | 1.2942e-1 (8.57e-2) − | 2.0733e-1 (3.29e-1) ≈ | 7.8015e-1 (5.50e-1) − | 3.2273e-1 (2.31e-1) − | 9.4611e-2 (1.43e-1) − | 1.7974e-1 (2.36e-1) − | 3.4982e-2 (8.32e-2) |
ZXH_CF14 | 100 | 2 | 12 | 2.8541e-3 (1.22e-4) − | 1.4504e-1 (2.09e-1) − | 3.5773e-2 (5.30e-3) − | 2.3952e-3 (5.04e-5) ≈ | 1.6415e-2 (2.17e-3) − | 2.5264e-3 (5.04e-5) − | 3.3834e-3 (2.07e-4) − | 3.1063e-3 (7.60e-4) − | 2.4094e-3 (5.04e-5) |
ZXH_CF15 | 100 | 2 | 12 | 5.6692e-3 (7.02e-4) + | 1.8225e-1 (2.06e-1) − | 7.1716e-2 (8.93e-2) − | 1.5372e-1 (3.10e-1) ≈ | 3.5310e-2 (2.33e-2) − | 2.6988e-2 (5.14e-2) − | 2.0192e-2 (7.58e-2) − | 2.9271e-3 (7.37e-5) + | 1.9618e-2 (5.23e-2) |
ZXH_CF16 | 100 | 2 | 12 | 1.4255e-2 (5.13e-3) − | 2.7397e-2 (4.87e-2) − | 2.9742e-2 (7.27e-3) − | 2.6375e-3 (3.60e-5) ≈ | 1.5499e-2 (1.44e-3) − | 8.3846e-2 (5.56e-2) − | 2.6346e-3 (4.08e-5) ≈ | 2.7845e-3 (5.03e-5) − | 2.6207e-3 (4.63e-5) |
2/12/2 | 0/16/0 | 0/16/0 | 0/11/5 | 0/16/0 | 0/15/1 | 2/11/3 | 4/12/0 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
ZXH_CF1 | 100 | 3 | 13 | 8.3464e-1 (1.25e-3) + | 7.6055e-1 (2.29e-2) − | 7.5475e-1 (1.62e-2) − | 8.2885e-1 (1.13e-3) − | 8.0689e-1 (5.10e-2) − | 8.2610e-1 (2.35e-3) − | 8.2047e-1 (1.95e-3) − | 8.1946e-1 (2.36e-3) − | 8.3065e-1 (4.65e-3) |
ZXH_CF2 | 100 | 3 | 13 | 4.9353e-1 (1.25e-1) − | 2.3984e-1 (1.04e-1) − | 3.9404e-1 (1.45e-1) − | 4.1234e-1 (2.20e-1) − | 4.0154e-1 (8.37e-2) − | 4.8976e-1 (1.68e-1) ≈ | 5.2960e-1 (3.51e-3) + | 5.3594e-1 (2.71e-3) + | 5.0861e-1 (1.26e-1) |
ZXH_CF3 | 100 | 3 | 13 | 5.1163e-1 (2.57e-3) ≈ | 2.5602e-1 (5.77e-2) − | 4.4603e-1 (1.25e-2) − | 4.8755e-1 (2.93e-2) − | 4.2994e-1 (1.99e-2) − | 5.0770e-1 (2.85e-3) − | 4.9615e-1 (3.61e-3) − | 4.6812e-1 (5.09e-3) − | 5.1256e-1 (3.02e-3) |
ZXH_CF4 | 100 | 3 | 13 | 3.4425e-1 (9.58e-2) ≈ | 0.0000e+0 (0.00e+0) − | 1.7839e-1 (1.38e-1) − | 2.9902e-1 (1.22e-1) ≈ | 5.8954e-2 (1.29e-1) − | 2.5523e-1 (1.73e-1) − | 3.2821e-1 (9.97e-2) ≈ | 2.3009e-1 (1.38e-1) − | 3.1961e-1 (1.10e-1) |
ZXH_CF5 | 100 | 3 | 13 | 2.7618e-1 (4.36e-2) − | 1.9029e-1 (4.81e-2) − | 2.0341e-1 (9.31e-2) − | 1.9140e-1 (1.15e-1) − | 2.4109e-1 (1.24e-2) − | 2.7456e-1 (9.39e-2) ≈ | 2.8224e-1 (2.58e-2) − | 2.9592e-1 (9.99e-4) + | 2.9405e-1 (4.93e-2) |
ZXH_CF6 | 100 | 3 | 13 | 2.2440e-1 (1.01e-3) ≈ | 1.9209e-1 (8.92e-3) − | 2.0066e-1 (2.91e-3) − | 2.1862e-1 (1.60e-2) − | 1.8694e-1 (1.09e-2) − | 2.2296e-1 (1.36e-3) − | 2.1727e-1 (1.05e-3) − | 2.1801e-1 (1.06e-3) − | 2.2446e-1 (9.11e-4) |
ZXH_CF7 | 100 | 3 | 13 | 2.6190e-1 (9.00e-2) ≈ | 2.8320e-2 (5.66e-2) − | 1.7165e-1 (1.25e-1) − | 6.1354e-2 (5.78e-2) − | 7.0418e-2 (1.04e-1) − | 1.1377e-1 (1.12e-1) − | 2.9140e-1 (8.89e-2) ≈ | 2.1831e-1 (1.45e-1) − | 2.7669e-1 (1.13e-1) |
ZXH_CF8 | 100 | 3 | 13 | 2.1152e-1 (5.81e-3) − | 1.0386e-1 (3.34e-2) − | 2.1263e-1 (5.15e-3) − | 8.6344e-2 (6.50e-2) − | 1.2063e-1 (1.91e-2) − | 1.2283e-1 (1.49e-2) − | 2.2296e-1 (2.93e-3) − | 2.2035e-1 (1.68e-3) − | 2.3158e-1 (6.55e-4) |
ZXH_CF9 | 100 | 3 | 13 | 2.3990e-1 (9.76e-3) − | 8.0970e-2 (3.67e-2) − | 2.4880e-1 (5.50e-3) − | 1.9252e-1 (5.01e-2) − | 1.1181e-1 (3.97e-2) − | 1.6162e-1 (6.18e-2) − | 2.6303e-1 (2.74e-3) − | 2.5758e-1 (2.02e-3) − | 2.6944e-1 (1.49e-3) |
ZXH_CF10 | 100 | 3 | 13 | 1.8661e-1 (1.01e-1) − | 1.2733e-2 (2.89e-2) − | 2.1093e-1 (1.07e-1) − | 1.3438e-1 (9.50e-2) − | 3.5459e-3 (5.36e-3) − | 1.2441e-1 (1.05e-1) − | 2.1794e-1 (1.08e-1) − | 1.6922e-1 (1.44e-1) − | 2.7501e-1 (9.87e-2) |
ZXH_CF11 | 100 | 3 | 13 | 3.8991e-1 (8.71e-3) − | 1.9864e-1 (6.11e-2) − | 2.8437e-1 (1.85e-2) − | 3.2902e-1 (3.57e-2) − | 3.5334e-1 (1.34e-2) − | 4.2558e-1 (2.18e-3) ≈ | 4.1834e-1 (3.44e-3) − | 4.2190e-1 (2.50e-3) − | 4.2663e-1 (2.29e-3) |
ZXH_CF12 | 100 | 3 | 13 | 6.7025e-1 (2.02e-1) − | 3.0386e-1 (1.08e-1) − | 4.5357e-1 (2.56e-1) − | 5.3642e-1 (2.56e-1) − | 5.8553e-1 (1.03e-1) − | 6.5299e-1 (2.46e-1) ≈ | 7.4573e-1 (6.03e-3) + | 7.4732e-1 (3.17e-3) + | 7.0009e-1 (1.74e-1) |
ZXH_CF13 | 100 | 2 | 12 | 2.4196e-1 (1.10e-1) − | 4.6817e-3 (1.40e-2) − | 1.8601e-1 (1.13e-1) − | 1.8976e-1 (1.27e-1) − | 2.7546e-2 (6.94e-2) − | 1.3589e-1 (1.25e-1) − | 2.3574e-1 (1.22e-1) − | 1.9168e-1 (1.37e-1) − | 2.8774e-1 (8.32e-2) |
ZXH_CF14 | 100 | 2 | 12 | 5.4203e-1 (5.07e-4) − | 3.8169e-1 (1.07e-1) − | 4.9882e-1 (9.92e-3) − | 5.4319e-1 (1.89e-4) ≈ | 5.1251e-1 (4.30e-3) − | 5.4317e-1 (1.18e-4) ≈ | 5.4114e-1 (3.36e-4) − | 5.4138e-1 (1.40e-3) − | 5.4315e-1 (1.58e-4) |
ZXH_CF15 | 100 | 2 | 12 | 6.6107e-1 (2.27e-4) + | 4.4797e-1 (1.43e-1) − | 5.9186e-1 (1.01e-1) − | 5.3860e-1 (2.38e-1) ≈ | 6.2683e-1 (2.55e-2) − | 6.4570e-1 (5.56e-2) + | 6.4283e-1 (8.01e-2) + | 6.6133e-1 (1.55e-4) + | 6.4002e-1 (6.81e-2) |
ZXH_CF16 | 100 | 2 | 12 | 7.6380e-1 (7.24e-3) − | 7.6813e-1 (1.59e-2) − | 7.5343e-1 (6.85e-3) − | 7.7822e-1 (9.42e-5) − | 7.6596e-1 (2.85e-3) − | 7.5334e-1 (1.94e-2) − | 7.7809e-1 (1.65e-4) − | 7.7762e-1 (1.32e-4) − | 7.7829e-1 (7.77e-5) |
2/10/4 | 0/16/0 | 0/16/0 | 0/13/3 | 0/16/0 | 1/10/5 | 3/11/2 | 4/12/0 |
Algorithm | IGD (Wilcoxon) | HV (Wilcoxon) | Average Rank | |||||
---|---|---|---|---|---|---|---|---|
-Value | -Value | IGD | HV | |||||
CL-TDEA | - | - | - | - | - | - | 1.50 | 1.44 |
C-TAEA | 108.0 | 28.0 | 98.0 | 22.0 | 5.00 * | 5.00 * | ||
ToP | 136.0 | 0.0 | 136.0 | 0.0 | 8.00 * | 8.00 * | ||
TiGE2 | 136.0 | 0.0 | 136.0 | 0.0 | 8.00 * | 8.00 * | ||
MSCMO | 135.0 | 1.0 | 134.5 | 1.5 | 7.50 * | 7.50 * | ||
POCEA | 136.0 | 0.0 | 136.0 | 0.0 | 8.00 * | 8.00 * | ||
CMOEA-MS | 136.0 | 0.0 | 115.0 | 5.0 | 6.50 * | 6.50 * | ||
CMOEMT | 84.0 | 52.0 | 78.0 | 42.0 | 3.50 | 3.50 | ||
IMTCMO | 98.0 | 38.0 | 90.5 | 29.5 | 4.00 | 4.00 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
C1_DTLZ1 | 100 | 3 | 7 | 2.3158e-2 (3.07e-4) − | NaN (NaN) | 3.1067e-1 (1.08e-1) − | 2.0558e-2 (5.80e-4) ≈ | 3.9621e-2 (2.26e-2) − | 2.1095e-2 (5.92e-4) − | 2.1772e-2 (3.15e-4) − | 2.1200e-2 (2.82e-4) − | 2.0173e-2 (1.72e-4) |
C1_DTLZ3 | 100 | 3 | 12 | 9.4902e-2 (1.13e-1) − | 2.6077e-1 (4.37e-1) − | 6.1809e+0 (3.18e+0) − | 4.5386e-1 (1.78e+0) ≈ | 8.0150e+0 (6.79e-3) − | 5.4781e-2 (1.54e-3) ≈ | 5.7978e-2 (1.32e-3) − | 6.0441e+0 (3.50e+0) − | 5.4636e-2 (7.46e-4) |
C2_DTLZ2 | 100 | 3 | 12 | 5.6604e-2 (1.15e-3) − | 1.1085e-1 (2.15e-1) − | 1.0147e-1 (1.18e-2) − | 4.3685e-2 (1.09e-3) ≈ | 5.1236e-2 (9.83e-4) − | 4.3628e-2 (1.44e-3) ≈ | 4.5200e-2 (6.12e-4) − | 4.5503e-2 (5.78e-4) − | 4.2961e-2 (6.18e-4) |
C3_DTLZ4 | 100 | 3 | 12 | 1.1172e-1 (2.04e-3) − | 1.4350e-1 (6.45e-3) − | 1.9444e-1 (1.17e-2) − | 1.7608e-1 (2.34e-1) ≈ | 1.1754e-1 (3.43e-2) − | 6.5367e-1 (5.54e-2) − | 1.0879e-1 (2.90e-3) − | 9.7809e-2 (1.77e-3) ≈ | 9.7680e-2 (1.37e-3) |
DC1_DTLZ1 | 100 | 3 | 7 | 3.0898e-2 (7.01e-2) − | 2.0815e-2 (6.46e-3) − | 4.9520e-1 (2.09e-1) − | 1.7235e-2 (9.70e-3) − | 3.6428e-2 (7.67e-3) − | 1.5678e-2 (5.65e-3) − | 1.2881e-2 (4.81e-4) − | 1.2287e-2 (2.82e-4) − | 1.1700e-2 (1.55e-4) |
DC1_DTLZ3 | 100 | 3 | 12 | 4.3182e-2 (1.39e-3) − | 8.1687e-1 (1.80e+0) − | 2.1081e+0 (7.60e-1) − | 2.0176e-1 (2.66e-1) − | 4.2008e-1 (6.07e-2) − | 5.0509e-2 (7.16e-2) ≈ | 3.5936e-2 (6.30e-4) − | 3.4730e-2 (5.00e-4) ≈ | 3.4671e-2 (6.40e-4) |
DC2_DTLZ1 | 100 | 3 | 7 | 2.3208e-2 (1.88e-4) − | NaN (NaN) | 3.1859e-1 (7.63e-2) − | 2.0659e-2 (5.52e-4) − | 1.4154e-1 (9.59e-2) − | 2.1261e-2 (5.66e-4) − | 2.2140e-2 (1.24e-3) − | 3.0650e-2 (3.26e-2) − | 2.0183e-2 (1.64e-4) |
DC2_DTLZ3 | 100 | 3 | 12 | 1.6441e-1 (1.99e-1) ≈ | NaN (NaN) | NaN (NaN) | 2.4699e-1 (2.31e-1) ≈ | NaN (NaN) | 4.6031e-1 (2.09e-1) ≈ | 5.6974e-2 (1.46e-3) ≈ | 5.6882e-1 (2.66e-3) − | 2.5790e-1 (2.57e-1) |
DC3_DTLZ1 | 100 | 3 | 7 | 9.3686e-3 (3.19e-4) − | 1.4582e+0 (2.19e+0) − | 1.1855e+0 (4.53e-1) − | 3.9023e-2 (3.54e-2) − | 2.0280e-1 (9.49e-2) − | 3.5516e-2 (9.15e-3) − | 7.3226e-3 (1.34e-4) − | 4.5143e-2 (6.85e-2) ≈ | 7.1387e-3 (1.17e-4) |
DC3_DTLZ3 | 100 | 3 | 12 | 2.6743e-2 (1.03e-3) ≈ | 6.8777e+0 (4.29e+0) − | 3.8042e+0 (9.81e-1) − | 8.6087e-1 (8.62e-1) − | 2.4949e+0 (5.47e-1) − | 2.6288e-1 (4.10e-1) ≈ | 2.1201e-2 (2.82e-4) + | 1.0104e+0 (3.75e-1) − | 2.6265e-1 (2.74e-1) |
0/8/2 | 0/7/0 | 0/9/0 | 0/5/5 | 0/9/0 | 0/5/5 | 1/8/1 | 0/7/3 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
C1_DTLZ1 | 100 | 3 | 7 | 8.3610e-1 (4.65e-3) − | NaN (NaN) | 2.8489e-1 (2.00e-1) − | 8.3959e-1 (1.90e-3) ≈ | 7.8952e-1 (4.70e-2) − | 8.3693e-1 (3.00e-3) − | 8.3466e-1 (1.93e-3) − | 8.3440e-1 (1.24e-3) − | 8.3946e-1 (1.80e-3) |
C1_DTLZ3 | 100 | 3 | 12 | 5.1571e-1 (1.20e-1) − | 3.8957e-1 (1.78e-1) − | 0.0000e+0 (0.00e+0) − | 5.2777e-1 (1.24e-1) − | 0.0000e+0 (0.00e+0) − | 5.5818e-1 (2.49e-3) − | 5.5149e-1 (6.36e-3) − | 1.0815e-1 (2.16e-1) − | 5.6066e-1 (1.07e-3) |
C2_DTLZ2 | 100 | 3 | 12 | 5.0709e-1 (1.82e-3) − | 4.3501e-1 (1.04e-1) − | 4.5745e-1 (8.49e-3) − | 5.1683e-1 (1.64e-3) ≈ | 5.0229e-1 (3.67e-3) − | 5.1607e-1 (1.22e-3) ≈ | 5.0965e-1 (2.19e-3) − | 5.0482e-1 (2.57e-3) − | 5.1623e-1 (1.37e-3) |
C3_DTLZ4 | 100 | 3 | 12 | 7.8570e-1 (1.11e-3) − | 7.5308e-1 (5.29e-3) − | 7.6600e-1 (3.16e-3) − | 7.6208e-1 (7.91e-2) ≈ | 7.7804e-1 (1.27e-2) − | 5.1934e-1 (5.39e-2) − | 7.8220e-1 (2.72e-3) − | 7.8859e-1 (1.43e-3) + | 7.8728e-1 (9.87e-4) |
DC1_DTLZ1 | 100 | 3 | 7 | 5.9650e-1 (1.34e-1) − | 5.8569e-1 (2.55e-2) − | 8.5395e-2 (7.45e-2) − | 6.0932e-1 (4.00e-2) − | 5.3982e-1 (2.26e-2) − | 6.1898e-1 (2.01e-2) − | 6.2632e-1 (2.73e-3) − | 6.2804e-1 (1.73e-3) − | 6.3263e-1 (7.08e-4) |
DC1_DTLZ3 | 100 | 3 | 12 | 4.6132e-1 (2.49e-3) − | 1.8706e-1 (1.70e-1) − | 0.0000e+0 (0.00e+0) − | 3.0680e-1 (2.05e-1) − | 2.8557e-1 (6.85e-2) − | 4.6759e-1 (2.65e-2) ≈ | 4.7131e-1 (2.41e-3) − | 4.6921e-1 (1.72e-3) − | 4.7388e-1 (8.28e-4) |
DC2_DTLZ1 | 100 | 3 | 7 | 8.3817e-1 (3.78e-4) − | NaN (NaN) | 2.7052e-1 (1.37e-1) − | 8.4128e-1 (1.04e-3) − | 5.2596e-1 (2.59e-1) − | 8.3898e-1 (1.23e-3) − | 8.3631e-1 (4.94e-3) − | 8.1233e-1 (8.27e-2) − | 8.4230e-1 (3.59e-4) |
DC2_DTLZ3 | 100 | 3 | 12 | 4.3140e-1 (2.13e-1) ≈ | NaN (NaN) | NaN (NaN) | 3.4596e-1 (2.45e-1) ≈ | NaN (NaN) | 1.2253e-1 (2.24e-1) − | 5.5394e-1 (5.50e-3) ≈ | 1.0616e-2 (1.34e-3) − | 3.4126e-1 (2.75e-1) |
DC3_DTLZ1 | 100 | 3 | 7 | 5.2341e-1 (2.79e-3) − | 3.4257e-3 (1.53e-2) − | 0.0000e+0 (0.00e+0) − | 4.0398e-1 (1.03e-1) − | 1.1866e-1 (1.39e-1) − | 4.1284e-1 (3.87e-2) − | 5.3501e-1 (1.58e-3) + | 4.3237e-1 (1.82e-1) − | 5.3408e-1 (1.72e-3) |
DC3_DTLZ3 | 100 | 3 | 12 | 3.6003e-1 (4.54e-3) ≈ | 4.9278e-3 (2.20e-2) − | 0.0000e+0 (0.00e+0) − | 1.1254e-1 (1.40e-1) ≈ | 0.0000e+0 (0.00e+0) − | 2.3868e-1 (1.80e-1) ≈ | 3.6757e-1 (1.64e-3) + | 0.0000e+0 (0.00e+0) − | 2.0169e-1 (1.87e-1) |
0/8/2 | 0/7/0 | 0/9/0 | 0/5/5 | 0/9/0 | 0/7/3 | 2/7/1 | 1/9/0 |
Algorithm | IGD (Wilcoxon) | HV (Wilcoxon) | Average Rank | |||||
---|---|---|---|---|---|---|---|---|
-Value | -Value | IGD | HV | |||||
CL-TDEA | - | - | - | - | - | - | 1.00 | 1.00 |
C-TAEA | 36.0 | 19.0 | 36.0 | 19.0 | 5.50 | 5.50 | ||
ToP | 45.0 | −1.0 | 45.0 | −1.0 | 8.00 * | 8.00 * | ||
TiGE2 | 55.0 | 0.0 | 55.0 | 0.0 | 9.00 * | 9.00 * | ||
MSCMO | 50.0 | 5.0 | 47.0 | 8.0 | 6.50 * | 6.50 * | ||
POCEA | 55.0 | 0.0 | 55.0 | 0.0 | 9.00 * | 9.00 * | ||
CMOEA-MS | 45.0 | 0.0 | 55.0 | 0.0 | 7.50 * | 7.50 * | ||
CMOEMT | 36.0 | 19.0 | 35.0 | 20.0 | 5.00 | 5.00 | ||
IMTCMO | 45.0 | 0.0 | 54.0 | 1.0 | 6.00 | 6.00 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
SDC1 | 100 | 2 | 50 | 2.0356e+1 (1.38e+0) − | 1.6076e+1 (2.63e+0) − | 5.7731e+0 (1.41e+0) − | 5.1095e+0 (1.57e+0) − | 2.8451e+1 (1.33e+0) − | 6.0785e+0 (1.76e+0) − | 8.2242e+0 (1.67e+0) − | 1.7155e+0 (2.11e+0) + | 2.3192e+0 (2.20e+0) |
SDC2 | 100 | 2 | 50 | 5.1595e-1 (2.54e-1) ≈ | 6.0530e-1 (5.91e-2) − | 1.0341e+0 (1.09e-1) − | 3.6558e-1 (2.04e-1) ≈ | 6.5775e-1 (5.33e-2) − | 4.8002e-1 (1.87e-1) − | 4.8952e-1 (1.39e-1) − | 7.1633e-2 (3.74e-2) + | 3.2001e-1 (5.82e-2) |
SDC3 | 100 | 2 | 50 | 1.5318e+0 (1.66e-1) ≈ | 2.5726e+0 (4.54e-1) − | 1.0613e+0 (7.94e-2) + | 2.0493e+0 (2.17e-1) − | 2.5110e+0 (0.00e+0) ≈ | 1.8924e+0 (8.40e-2) − | 2.0244e+0 (1.33e-1) − | 2.3601e+0 (3.59e-1) − | 1.4360e+0 (1.83e-1) |
SDC4 | 100 | 2 | 50 | 6.5800e+2 (6.23e+2) − | 2.8840e+2 (3.15e+2) − | 1.8043e+2 (1.62e+2) − | 3.9434e+2 (4.71e+2) − | 2.2452e+3 (2.09e+3) − | 5.5305e+2 (2.55e+2) − | 9.6260e+2 (9.55e+2) − | 8.2912e+2 (1.53e+3) ≈ | 4.8915e+0 (5.11e+0) |
SDC5 | 100 | 2 | 50 | 1.4161e+3 (4.11e+2) − | 8.1365e+2 (2.18e+2) − | 5.0335e+2 (1.04e+2) − | 1.6931e+2 (6.62e+1) + | 6.5360e+2 (2.94e+2) − | 3.4468e+2 (1.10e+2) ≈ | 5.7207e+2 (1.34e+2) − | 3.7417e+2 (2.52e+2) ≈ | 2.8685e+2 (9.38e+1) |
SDC6 | 100 | 2 | 50 | 6.2495e+3 (2.56e+3) − | 2.3169e+3 (9.36e+2) − | 1.9951e+3 (1.28e+3) − | 3.3566e+3 (2.83e+3) − | 1.3278e+4 (5.59e+3) − | 2.3105e+3 (2.41e+3) − | 2.1836e+3 (1.66e+3) − | 1.7278e+3 (1.75e+3) ≈ | 9.7488e+2 (9.48e+2) |
SDC7 | 100 | 2 | 50 | 6.5597e+0 (3.60e+0) − | 1.3826e+1 (7.50e-1) − | 6.9116e+0 (1.11e+0) − | 2.3190e+0 (9.99e-1) ≈ | 1.1307e+1 (8.83e-1) − | 8.5498e+0 (2.90e+0) − | 1.0722e+1 (2.42e+0) − | 8.6289e+0 (3.94e+0) − | 2.0232e+0 (1.41e+0) |
SDC8 | 100 | 2 | 50 | 7.7485e-1 (6.93e-1) + | 4.2101e+2 (2.15e+2) − | 3.8620e+1 (1.24e+1) − | 2.9899e+0 (3.85e+0) + | 1.1217e+2 (4.87e+1) − | 3.6790e+0 (2.22e+0) ≈ | 8.4963e+1 (8.79e+1) − | 8.3762e+0 (5.63e+0) − | 4.4728e+0 (5.11e+0) |
1/5/2 | 0/8/0 | 1/7/0 | 2/4/2 | 0/7/1 | 0/6/2 | 0/8/0 | 2/3/3 |
Problem | N | m | D | CL-TDEA1 | CL-TDEA2 | CL-TDEA3 | CL-TDEA4 | CL-TDEA5 | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|
LIRCMOP1 | 100 | 2 | 30 | 1.7402e-1 (1.81e-2) − | 1.9915e-1 (4.36e-2) − | 3.1374e-1 (6.74e-2) − | 2.5050e-1 (6.68e-2) − | NaN (NaN) | 1.4002e-1 (2.53e-2) |
LIRCMOP2 | 100 | 2 | 30 | 1.3403e-1 (2.04e-2) ≈ | 1.9647e-1 (3.00e-2) − | NaN (NaN) | 2.4836e-1 (6.87e-2) − | NaN (NaN) | 1.4119e-1 (1.92e-2) |
LIRCMOP3 | 100 | 2 | 30 | 1.8411e-1 (2.66e-2) ≈ | 2.2180e-1 (4.83e-2) − | NaN (NaN) | 3.0086e-1 (7.62e-2) − | NaN (NaN) | 1.6735e-1 (4.45e-2) |
LIRCMOP4 | 100 | 2 | 30 | 1.7869e-1 (2.15e-2) ≈ | 2.0611e-1 (2.90e-2) − | NaN (NaN) | 2.5554e-1 (5.82e-2) − | NaN (NaN) | 1.8547e-1 (2.44e-2) |
LIRCMOP5 | 100 | 2 | 30 | 1.2190e+0 (7.77e-3) − | 3.3365e-1 (2.14e-1) − | 3.0953e-1 (6.89e-2) − | 6.9053e-1 (4.96e-1) − | 3.0566e-1 (5.23e-2) − | 2.4234e-1 (6.37e-2) |
LIRCMOP6 | 100 | 2 | 30 | 1.3179e+0 (1.24e-1) − | 3.8971e-1 (2.40e-1) ≈ | 3.1085e-1 (7.82e-2) ≈ | 8.6103e-1 (5.06e-1) − | 3.4025e-1 (8.24e-2) ≈ | 3.7238e-1 (2.40e-1) |
LIRCMOP7 | 100 | 2 | 30 | 3.2288e-1 (4.68e-1) − | 1.1829e-1 (4.32e-2) ≈ | NaN (NaN) | 1.7345e-1 (3.59e-2) − | 9.2216e-1 (8.12e-2) − | 1.2080e-1 (3.03e-2) |
LIRCMOP8 | 100 | 2 | 30 | 7.2249e-1 (6.69e-1) − | 1.7971e-1 (7.06e-2) ≈ | NaN (NaN) | 1.4767e-1 (3.58e-2) ≈ | 7.1204e-1 (2.47e-1) − | 1.6572e-1 (4.99e-2) |
LIRCMOP9 | 100 | 2 | 30 | 4.8492e-1 (7.23e-2) ≈ | 5.8239e-1 (1.36e-1) ≈ | 7.5287e-1 (2.21e-1) − | 4.0686e-1 (2.78e-2) + | 8.9659e-1 (3.31e-1) − | 5.4696e-1 (1.22e-1) |
LIRCMOP10 | 100 | 2 | 30 | 5.8448e-1 (2.66e-2) − | 4.7916e-1 (1.96e-1) − | 1.5707e-1 (6.66e-2) + | 3.2829e-1 (1.34e-1) ≈ | 2.9394e-1 (1.54e-1) ≈ | 2.5525e-1 (1.07e-1) |
LIRCMOP11 | 100 | 2 | 30 | 3.7220e-1 (7.70e-2) − | 1.0067e-1 (5.17e-2) ≈ | 8.6697e-1 (3.51e-1) − | 3.7326e-1 (1.06e-1) − | 7.2460e-1 (3.58e-1) − | 1.1354e-1 (9.31e-2) |
LIRCMOP12 | 100 | 2 | 30 | 1.9633e-1 (6.33e-2) − | 2.0542e-1 (1.14e-1) ≈ | 9.8682e-1 (2.15e-3) − | 2.4258e-1 (3.93e-2) − | 2.1365e+0 (2.82e+0) − | 1.6374e-1 (6.39e-2) |
LIRCMOP13 | 100 | 3 | 30 | 1.3186e+0 (2.28e-3) − | 9.9335e-2 (2.00e-3) − | 9.4006e-2 (1.42e-3) ≈ | 1.3235e+0 (2.68e-1) − | 9.4718e-2 (1.46e-3) ≈ | 9.4607e-2 (1.15e-3) |
LIRCMOP14 | 100 | 3 | 30 | 1.2739e+0 (1.25e-3) − | 1.1089e-1 (3.44e-3) − | 6.2796e-1 (1.37e-1) − | 1.3301e+0 (6.71e-2) − | 2.3823e-1 (3.08e-2) − | 9.5003e-2 (8.90e-4) |
0/10/4 | 0/8/6 | 1/6/2 | 1/11/2 | 0/7/3 |
Problem | N | m | D | CL-TDEA1 | CL-TDEA2 | CL-TDEA3 | CL-TDEA4 | CL-TDEA5 | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|
LIRCMOP1 | 100 | 2 | 30 | 1.5264e-1 (8.07e-3) − | 1.4526e-1 (1.66e-2) − | 1.1550e-1 (1.82e-2) − | 1.2696e-1 (1.71e-2) − | NaN (NaN) | 1.6643e-1 (1.22e-2) |
LIRCMOP2 | 100 | 2 | 30 | 2.8248e-1 (1.26e-2) ≈ | 2.6048e-1 (1.78e-2) − | NaN (NaN) | 2.3666e-1 (2.92e-2) − | NaN (NaN) | 2.8146e-1 (1.27e-2) |
LIRCMOP3 | 100 | 2 | 30 | 1.3672e-1 (1.15e-2) ≈ | 1.2379e-1 (1.34e-2) − | NaN (NaN) | 1.0316e-1 (1.90e-2) − | NaN (NaN) | 1.4366e-1 (1.63e-2) |
LIRCMOP4 | 100 | 2 | 30 | 2.3642e-1 (1.18e-2) ≈ | 2.2969e-1 (1.14e-2) ≈ | NaN (NaN) | 2.0553e-1 (2.53e-2) − | NaN (NaN) | 2.3672e-1 (1.46e-2) |
LIRCMOP5 | 100 | 2 | 30 | 0.0000e+0 (0.00e+0) − | 1.4906e-1 (4.15e-2) − | 1.5000e-1 (2.63e-2) − | 9.6903e-2 (9.12e-2) − | 1.5306e-1 (2.13e-2) − | 1.7632e-1 (2.32e-2) |
LIRCMOP6 | 100 | 2 | 30 | 2.1829e-3 (9.76e-3) − | 9.8675e-2 (2.60e-2) ≈ | 1.1345e-1 (1.12e-2) ≈ | 4.3432e-2 (4.34e-2) − | 1.0833e-1 (1.24e-2) ≈ | 1.0191e-1 (2.84e-2) |
LIRCMOP7 | 100 | 2 | 30 | 2.1095e-1 (7.34e-2) − | 2.4991e-1 (1.39e-2) ≈ | NaN (NaN) | 2.2459e-1 (1.27e-2) − | 8.6916e-2 (4.32e-2) − | 2.4796e-1 (9.47e-3) |
LIRCMOP8 | 100 | 2 | 30 | 1.4709e-1 (1.02e-1) − | 2.3807e-1 (1.38e-2) ≈ | NaN (NaN) | 2.3825e-1 (8.64e-3) ≈ | 1.2051e-1 (4.24e-2) − | 2.3944e-1 (1.13e-2) |
LIRCMOP9 | 100 | 2 | 30 | 3.6047e-1 (4.09e-2) ≈ | 3.0463e-1 (7.08e-2) ≈ | 2.9408e-1 (3.97e-2) ≈ | 4.0702e-1 (1.88e-2) + | 2.7325e-1 (7.49e-2) − | 3.3724e-1 (6.00e-2) |
LIRCMOP10 | 100 | 2 | 30 | 4.2451e-1 (2.20e-3) − | 3.7136e-1 (1.59e-1) − | 6.2007e-1 (2.45e-2) + | 5.2770e-1 (5.39e-2) ≈ | 5.4520e-1 (6.90e-2) ≈ | 5.5210e-1 (7.44e-2) |
LIRCMOP11 | 100 | 2 | 30 | 5.0634e-1 (3.87e-2) − | 6.3910e-1 (3.39e-2) ≈ | 4.1054e-1 (9.90e-2) − | 5.3192e-1 (7.44e-2) − | 4.4893e-1 (1.01e-1) − | 6.3719e-1 (6.52e-2) |
LIRCMOP12 | 100 | 2 | 30 | 5.1664e-1 (3.63e-2) ≈ | 5.1973e-1 (5.34e-2) ≈ | 3.4785e-1 (9.87e-3) − | 5.0057e-1 (1.19e-2) − | 2.8206e-1 (1.33e-1) − | 5.4167e-1 (3.18e-2) |
LIRCMOP13 | 100 | 3 | 30 | 1.2765e-4 (1.30e-4) − | 5.4564e-1 (1.77e-3) − | 5.5272e-1 (2.10e-3) + | 2.3042e-2 (1.03e-1) − | 5.5299e-1 (2.25e-3) + | 5.5131e-1 (1.76e-3) |
LIRCMOP14 | 100 | 3 | 30 | 5.1431e-4 (2.20e-4) − | 5.3629e-1 (3.27e-3) − | 2.5006e-1 (5.92e-2) − | 2.3209e-3 (9.61e-3) − | 4.4678e-1 (2.23e-2) − | 5.5543e-1 (1.45e-3) |
0/9/5 | 0/7/7 | 2/5/2 | 1/11/2 | 1/7/2 |
Problem | N | m | D | C-TAEA | ToP | TiGE2 | MSCMO | POCEA | CMOEA_MS | CMOEMT | IMTCMO | CL-TDEA |
---|---|---|---|---|---|---|---|---|---|---|---|---|
DBD | 100 | 2 | 4 | 8.1515e-1 (3.28e-2) − | NaN (NaN) − | 8.6262e-1 (1.93e-4) − | 5.7254e-1 (9.38e-2) − | 8.2594e-1 (1.57e-2) − | 8.5752e-1 (4.72e-3) − | 5.5756e-2 (1.54e-2) − | 7.9041e-1 (2.78e-2) − | 8.6326e-1 (1.99e-4) |
WRM | 100 | 2 | 4 | 4.1104e-1 (1.51e-2) − | 2.7366e-1 (2.23e-3) − | 3.4154e-1 (1.31e-4) − | 2.5424e-1 (2.73e-2) − | 4.2242e-1 (3.32e-3) − | 3.2516e-2 (1.34e-2) − | 4.2004e-1 (2.38e-3) − | 3.7432e-1 (2.18e-4) − | 4.3459e-1 (2.60e-4) |
CSI | 100 | 2 | 4 | 4.0938e-1 (2.35e-4) ≈ | 2.0983e-1 (7.58e-5) − | 3.1372e-1 (1.47e-4) − | 5.4684e-2 (1.07e-3) − | 3.9519e-1 (3.36e-3) − | 3.1208e-1 (1.55e-4) − | 3.7601e-2 (9.81e-5) − | 4.0964e-1 (2.79e-4) ≈ | 4.0886e-1 (3.26e-4) |
0/3/0 | 0/2/1 | 0/3/0 | 0/3/0 | 0/3/0 | 0/3/0 | 0/3/0 | 0/2/1 |
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Xiu, X.; Yu, F.; Wang, H.; Song, Y. Adaptive Constraint-Boundary Learning-Based Two-Stage Dual-Population Evolutionary Algorithm. Mathematics 2025, 13, 3206. https://doi.org/10.3390/math13193206
Xiu X, Yu F, Wang H, Song Y. Adaptive Constraint-Boundary Learning-Based Two-Stage Dual-Population Evolutionary Algorithm. Mathematics. 2025; 13(19):3206. https://doi.org/10.3390/math13193206
Chicago/Turabian StyleXiu, Xinran, Fu Yu, Hongzhou Wang, and Yiming Song. 2025. "Adaptive Constraint-Boundary Learning-Based Two-Stage Dual-Population Evolutionary Algorithm" Mathematics 13, no. 19: 3206. https://doi.org/10.3390/math13193206
APA StyleXiu, X., Yu, F., Wang, H., & Song, Y. (2025). Adaptive Constraint-Boundary Learning-Based Two-Stage Dual-Population Evolutionary Algorithm. Mathematics, 13(19), 3206. https://doi.org/10.3390/math13193206