# Performance Analysis of Multi-Objective Simulated Annealing Based on Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (1)
- decomposition strategy,
- (2)
- perturbation function, and
- (3)
- probability of acceptance.

#### 2.1. Decomposition Strategy

#### 2.2. Perturbation Function

#### 2.2.1. Classical Genetic Operators

#### 2.2.2. Differential Evolution Operators

#### 2.3. Probability of Acceptance

_{n}is the current solution, X

_{n+}

_{1}is the candidate solution, and T

_{k}is the current temperature of the algorithm.

#### 2.4. Multi-Objective Simulated Annealing Based on Decomposition Framework

_{i}) associated with the ith sub-problem. Additionally, the framework maintains a solution to explore the feasible surface of ith sub-problem. To update this solution, an evaluation is necessary with the new solution. Both evaluations of the new solution against the efficient and the exploring solution are conducted using the Tchebycheff approach. Tchebycheff approach avoids the use of Pareto dominance in the MOSA/D framework. Furthermore, the framework has a mechanism to escape from local optimums based on a logarithm compositive function. Finally, the perturbation functions are a novel part of the framework. Two algorithms are derived due to the perturbation functions: MOSA/D-CGO and MOSA/D-DE. We intend to observe the performance of both approaches in continuous multiple- and many-objective problems.

#### 2.4.1. MOSA/D-CGO Algorithm

_{current}solution serving as a guide in the search process, and the S

_{cand}solution, which is generated from disturbances on S

_{current}, the calculation of z (reference point). FE = N is updated because it is the size of the initial population (Lines 1 and 2). From Line 3, Algorithm 1 has three loops: the first, second, and third (Lines 3, 4, and 6, respectively).

Algorithm 1 MOSA/D-CGO | |

Input: MOP, Initial temperature Ti, Factor α, Markov chain L, Final temperatura Tf, Size of population N, Maximal function evaluations MFE | |

Output: Last generation of P | |

1 | To initialize: population P(N), weighted vectors v, solution S_{current}, solution S_{cand},reference point z, temperature T = Ti |

2 | FE = N |

3 | while (T ≥ Tf) y (FE ≤ MFE) |

4 | for i = 1 to N |

5 | S_{current} = P_{i} |

6 | for j = 1 to L |

7 | S_{cand} = PerturbationCGO(S_{current}, P) |

8 | p = BoltzmannProbability(S_{cand}, S_{current}, T) |

9 | if g(S_{cand}, v_{i}, z) < g(P_{i}, v_{i}, z) |

10 | P_{i} = S_{cand} |

11 | if g(S_{cand}, v_{i}, z) < g(S_{current}, v_{i}, z) or U(0,1) < p |

12 | S_{current} = S_{cand} |

13 | z = ObtainReferencePoint (S_{cand}) |

14 | j = j +1 |

15 | i = i +1 |

16 | FE = N × L |

17 | T = T × α |

18 | return P |

- Start a second loop that proceeds through all the sub-problems (Line 4),
- Update FE = N × L as the multiplication of the number of sub-problems by the annealing cycles (Line 16),
- The temperature T is updated. It is multiplied by the temperature drop factor α (Line 17).
- The second loop (from 1 to N sub-problems) has the following tasks (Line 4):
- The S
_{current}solution takes as its value the solution stored in P_{i}(Line 5), - The ith sub-problem is annealed for L cycles (Lines 6–14) in the third loop,
- The sub-problem counter i is updated.
- The third loop (annealing process) works as follows (Lines 6–14):
- S
_{cand}is obtained by perturbation of S_{current}(Line 7), - The Boltzmann probability p of S
_{cand}is computed based on Equation (6) (Line 8), - The Tchebycheff function g is calculated for S
_{cand}and P_{i}. If g (S_{cand}) is less than g (P_{i}), then S_{cand}is a better solution than P_{i}. Then P_{i}takes the value of S_{cand}(Lines 9–10). - The Tchebycheff function g is calculated for S
_{cand}and S_{current}. If g (S_{cand}) is less than g (S_{current}), then S_{current}takes the value of S_{cand}(Lines 11–12). - The reference point z is updated (Line 13).
- The Markov chain counter j is updated (Line 14)
- At the end of the main loop, the last generation of P is returned, and the algorithm ends (Line 18).

_{current}, P) is shown in Algorithm 2. The algorithm’s input is the current solution S

_{current}and the population P. At Line 1, all solutions are initialized. Next, parent1 takes the value of S

_{current}(Line 2). Next, a random solution is selected from P as parent2 (Line 3). Then the SBX is applied to parent1 and parent2, producing the new solution child (Line 4). Next, the mutation operator is applied to the solution child (Line 5), producing the candidate solution. Finally, the candidate solution is returned to the main algorithm.

Algorithm 2 PerturbationCGO | |

Input: Current solution S_{current}, population P | |

Output: Candidate Solution S_{cand} | |

1 | Initialize solutions: parent1, parent2, child |

2 | parent1 = S_{current} |

3 | parent2 = RandomSelection(P) |

4 | child = SBX(parent1, parent2) |

5 | S_{cand} = PolynomialMutation(child) |

6 | return S_{cand} |

#### 2.4.2. MOSAD-DE Algorithm

Algorithm 3 MOSA/D-DE | |

Input: MOP, Initial temperature Ti, Factor α, Markov chain L, Final temperatura Tf, Size of population N, Maximal function evaluations MFE | |

Output: Last generation of P | |

1 | To initialize: population P(N), weighted vectors v, solution S_{current}, solution S_{cand},reference point z, temperature T = Ti |

2 | FE = N |

3 | while (T ≥ Tf) y (FE ≤ MFE) |

4 | for i = 1 to N |

5 | S_{current} = P_{i} |

6 | for j = 1 to L |

7 | S_{cand} = PerturbationDE(S_{current}, P) |

8 | p = BoltzmannProbability(S_{cand}, S_{current}, T) |

9 | if g(S_{cand}, v_{i}, z) < g(P_{i}, v_{i}, z) |

10 | P_{i} = S_{cand} |

11 | if g(S_{cand}, v_{i}, z) < g(S_{current}, v_{i}, z) or U(0,1) < p |

12 | S_{current} = S_{cand} |

13 | z = ObtainReferencePoint (S_{cand}) |

14 | j = j +1 |

15 | i = i +1 |

16 | FE = N × L |

17 | T = T × α |

18 | return P |

_{current}, P). Its pseudocode is shown in Algorithm 4. The algorithm begins with the initialization of all solutions used in the algorithm (Line 1). Next, the solution target takes the values of S

_{current}(Line 2). The next step is to select three random solutions from the population P (Lines 3, 4, and 5). In Line 6, RandOneMutation() (based on Equation (12)) produces the mutant solution. However, the mutant solution may exceed the limit values of its decision variables. For this reason, a repair operation is necessary over the mutant solution (Line 7). Then, the binomial crossover is executed to generate the candidate solution (S

_{cand}) (Line 8). Finally, the candidate solution (S

_{cand}) is returned to Algorithm 3.

Algorithm 4 PerturbationDE | |

Input: Current solution S_{current}, population P | |

Output: Candidate Solution S_{cand} | |

1 | Initialize solutions: target, mutant, sol1, sol2, sol3 |

2 | target = S_{current} |

3 | sol1 = RandomSelection(P) |

4 | sol2 = RandomSelection(P) |

5 | sol3 = RandomSelection(P) |

6 | mutant = RandOneMutation(ind1, ind2, ind3) |

7 | mutant = Repair(mutant) |

8 | S_{cand} = BinomialCrossover (target, mutant) |

9 | return S_{cand} |

## 3. Experimental Setup

**Benchmark problems.**The experimental design used the DTLZ [25] and CEC2009 [18]. From DTLZ benchmark, DTLZ1 to DTLZ7 were selected with 3, 5, and 10 objectives for both algorithms MOSA/D-CGO and MOSA/D-DE. The number of variables of decision $n$ was calculated by $n=m+k$, where $m$ is the number of objectives and $k$ is set to 5, 10, and 20 for DTLZ1, DTLZ2-6, and DTLZ7, respectively. The total number of experimental instances is 42 (7 problems $\times $ 3 objective configurations $\times $ 2 algorithms). From CEC2009 benchmark, UF1 to UF7 were selected with 2 objectives and UF8 to UF10 with 3 objectives. For all problems, the number of decision variables was 30. The total number of experimental instances was 20 (7 problems $\times $ 1 objective configurations $\times $ 2 algorithms + 3 problems $\times $ 1 objective configurations $\times $ 2 algorithms).

**Parameter settings**. In general, for both algorithms, the setting of the parameter is defined in Table 1. The parameter setting for the CGO perturbation function is defined in Table 2. Finally, the parameter setting for the DE perturbation function is defined in Table 3.

**Performance indicator.**The indicators used to measure the performance of both algorithms were hypervolume (HV) and Inverted Generational Distance (IGD). HV allows measurement of convergence and diversity with a single value. Therefore, to calculate HV as a reference point is necessary. The reference points were obtained from determining both algorithms’ maximum objective value for DTLZ1 to DTLZ7 with 3, 5, and 10 objectives (Table 4). The reference points from UF1 to UF7 (2 objectives) and UF8 to UF10 (3 objectives) are shown in Table 5. IGD allows measurement of convergence and diversity with a single value. To calculated IGD is necessary set of points of the real Pareto front.

## 4. Results

#### 4.1. DTLZ Benchmark Analysis

#### 4.2. CEC2009 Benchmark Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Decomposition strategy. (

**a**) A MOP is divided into N sub-problems ${\lambda}^{1},\dots ,{\lambda}^{N}$; (

**b**) N random solutions are generated; (

**c**) over a finite number of generations, solutions are simultaneously improved.

**Figure 3.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ1 approximate PF using MOSA/D-CGO; (

**b**) DTLZ1 approximate PF using MOSA/D-DE; (

**c**) DTLZ3 approximate PF using MOSA/D-CGO; (

**d**) DTLZ3 approximate PF using MOSA/D-DE.

**Figure 4.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ2 approximate PF using MOSA/D-CGO; (

**b**) DTLZ2 approximate PF using MOSA/D-DE.

**Figure 5.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ4 approximate PF using MOSA/D-CGO; (

**b**) DTLZ4 approximate PF using MOSA/D-DE.

**Figure 6.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ5 approximate PF using MOSA/D-CGO; (

**b**) DTLZ5 approximate PF using MOSA/D-DE.

**Figure 7.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ6 approximate PF using MOSA/D-CGO; (

**b**) DTLZ6 approximate PF using MOSA/D-DE.

**Figure 8.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) DTLZ7 approximate PF using MOSA/D-CGO; (

**b**) DTLZ7 approximate PF using MOSA/D-DE.

**Figure 9.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF1 approximate PF using MOSA/D-CGO; (

**b**) UF1 approximate PF using MOSA/D-DE.

**Figure 10.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF2 approximate PF using MOSA/D-CGO; (

**b**) UF2 approximate PF using MOSA/D-DE.

**Figure 11.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF3 approximate PF using MOSA/D-CGO; (

**b**) UF3 approximate PF using MOSA/D-DE.

**Figure 12.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF4 approximate PF using MOSA/D-CGO; (

**b**) UF4 approximate PF using MOSA/D-DE.

**Figure 13.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF5 approximate PF using MOSA/D-CGO; (

**b**) UF5 approximate PF using MOSA/D-DE.

**Figure 14.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF6 approximate PF using MOSA/D-CGO; (

**b**) UF6 approximate PF using MOSA/D-DE.

**Figure 15.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF7 approximate PF using MOSA/D-CGO; (

**b**) UF7 approximate PF using MOSA/D-DE.

**Figure 16.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF8 approximate PF using MOSA/D-CGO; (

**b**) UF8 approximate PF using MOSA/D-DE.

**Figure 17.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF9 approximate PF using MOSA/D-CGO; (

**b**) UF9 approximate PF using MOSA/D-DE.

**Figure 18.**The plots of the approximate PFs with the highest HV value against real PF. (

**a**) UF10 approximate PF using MOSA/D-CGO; (

**b**) UF10 approximate PF using MOSA/D-DE.

Parameter | Setting |
---|---|

Maximal function evaluations (MFE) | 100,000 |

Size of the population (N) | 100 |

Chain of Markov (L) | 20 (DTLZ), 2 (CEC2009) |

Initial temperature (Ti) | 1 |

Final temperature (Tf) | 0.0000001 |

Temperature factor (α) | 0.98 |

Parameter | Setting |
---|---|

Crossover probability | 1 |

SBX distribution index | 15 |

Mutation probability | 1/number of objectives |

Mutation distribution index | 20 |

Parameter | Setting |
---|---|

Scale Factor (F) | 0.5 |

Crossover rate (Cr) | 0.8 |

Test Problem | 3 Objectives | 5 Objectives | 10 Objectives |
---|---|---|---|

DTLZ1 | 216.790132039933, 245.572126619941, 265.114937485812 | 80.8714391297344, 227.104643547038, 235.570177240396, 201.976698417331, 344.81892944792 | 4.06142015762104, 2.25156894368627, 28.0355439457454, 18.0131641679773, 48.6777918069853, 55.3772497318585, 133.939834807773, 139.713524060566, 222.454058357031, 321.944734395031 |

DTLZ2 | 2.38622940286774, 2.47201723363377, 2.6806935648516 | 2.05896735099775, 2.07725455911758, 2.30808795675219, 2.57449333535658, 2.9380095369211 | 0.6594305076443, 0.648350703222635, 0.840342066735919, 1.24805308667332, 1.54612890881046, 1.70777551383235, 1.8764060443055, 1.97914511371776, 2.29490194623458, 2.65985264422613 |

DTLZ3 | 822.509995384644, 799.103543751443, 1242.39097615921 | 689.016436482976, 547.613912205914, 500.989149866526, 827.81074606278, 1089.39809110161 | 49.243814823296, 78.5147608980575, 133.64045805898, 150.770000538526, 217.750852407648, 340.827375270017, 565.932643419409, 495.059846407958, 829.140836249926, 1271.6885408782, |

DTLZ4 | 2.58950883612183, 1.65645180809823, 2.02042410538188 | 2.66379101299125, 2.13852567972279, 2.04131611330391, 1.99090557186032, 1.98125761314518, | 2.64822143939513, 2.20030288801471, 2.2491760776445, 2.083066369613, 2.1065144928458, 2.09386422171755, 1.99635654481651, 2.24556199410806, 2.01705020385728, 2.14178980883573 |

DTLZ5 | 2.06970298570428, 2.11909090597952, 2.74503917772623 | 1.31770044511346, 1.68921445015483, 3.31646087578869, 3.41224588178889, 3.49994217680484 | 0.291869064968389, 0.49338116154924, 0.525290602300518, 0.611762301869089, 0.751594578115702, 1.10935039999348, 2.68788397225164, 3.29311486089528, 3.41214316344024, 3.49975337563819 |

DTLZ6 | 9.71802048756533, 9.50035002572186, 10.5676062054648 | 9.56330574133902, 9.59158225806973, 10.477256602242, 10.9414071660156, 10.923652672406, | 3.6503932212918, 3.29696637156103, 5.62792594695437, 7.16542587886431, 7.97964101213103, 9.77412868397095, 9.89622372940959, 10.6364664206205, 10.9073924572044, 10.9367989501483 |

DTLZ7 | 1.0, 0.999999999998507, 24.2793251715135 | 1.0, 0.999999994782373, 0.999999999986268, 0.999999999997808, 42.0189690066051, | 1.0, 0.999999999990235, 0.999999991586879, 1.0, 0.999999998676653, 1.0, 0.99999999981814, 1.0, 0.999999997160675, 21.0 |

Test Problem | 2 Objectives | 3 Objectives |
---|---|---|

UF1 | 4.056791172 3.587159682 | - |

UF2 | 2.700068047 2.02284611 | - |

UF3 | 3.643565528 4.753315586 | - |

UF4 | 1.205666356 1.221462823 | - |

UF5 | 9.178972446 10.07250673 | - |

UF6 | 12.40456172 13.75189469 | - |

UF7 | 4.169101474 3.206875182 | - |

UF8 | - | 9.214592744 10.62998938 11.46084736 |

UF9 | - | 9.853718763 12.22418277 8.259884555 |

UF10 | - | 9.853718763 12.22418277 8.259884555 |

PROBLEM | M | MOSA/D-CGO | MOSA/D-DE |
---|---|---|---|

DTLZ1 | 3 | 1.411396 × 10^{7} (7.856170 × 10^{1}) | ↑1.411407 × 10^{7} (2.077398 × 10^{1}) |

5 | 3.013240 × 10^{11} (7.546308 × 10^{4}) | ↑3.013241 × 10^{11} (1.109078 × 10^{4}) | |

10 | 1.667917 × 10^{16} (2.419205 × 10^{13}) | ↑1.668377 × 10^{16} (3.215349 × 10^{8}) | |

DTLZ2 | 3 | 1.508330 × 10^{1} (2.781792 × 10^{−}^{2}) | ↑1.517692 × 10^{1} (7.637901 × 10^{−}^{3}) |

5 | 7.361979 × 10^{1} (7.759884 × 10^{−}^{1}) | ↑7.387519 × 10^{1} (3.847436 × 10^{−}^{2}) | |

10 | ↓2.600539 × 10^{1} (4.958777 × 10^{−}^{1}) | 2.569999 × 10^{1} (1.980545 × 10^{−}^{1}) | |

DTLZ3 | 3 | 8.163884 × 10^{8} (6.516296 × 10^{4}) | ↑8.165865 × 10^{8} (4.513141 × 10^{2}) |

5 | 1.689445 × 10^{14} (8.081423 × 10^{12}) | ↑1.704707 × 10^{14} (9.718498 × 10^{7}) | |

10 | 1.699675 × 10^{24} (4.226203 × 10^{22}) | ↑1.707961 × 10^{24} (8.685232 × 10^{18}) | |

DTLZ4 | 3 | 7.958441 × 10^{0} (1.283030 × 10^{−}^{2}) | ↑8.002655 × 10^{0} (1.583281 × 10^{−}^{2}) |

5 | ↓4.522733 × 10^{1} (3.629265 × 10^{−}^{2}) | 4.509505 × 10^{1} (8.641444 × 10^{−}^{2}) | |

10 | ↓2.327196 × 10^{3} (3.100255 × 10^{0}) | 2.298456 × 10^{3} (1.139568 × 10^{1}) | |

DTLZ5 | 3 | 9.944000 × 10^{0} (1.078512 × 10^{−}^{2}) | ↑9.990017 × 10^{0} (2.501031 × 10^{−}^{2}) |

5 | ---7.652259 × 10^{1} (3.121247 × 10^{−}^{1}) | ---7.657619 × 10^{1} (1.983684 × 10^{−}^{1}) | |

10 | 3.412050 × 10^{0} (2.043577 × 10^{−}^{2}) | ↑3.428953 × 10^{0} (8.094812 × 10^{−}^{3}) | |

DTLZ6 | 3 | ↓9.009791 × 10^{2} (1.712226 × 10^{1}) | 7.219506 × 10^{2} (8.185913 × 10^{0}) |

5 | ↓1.053329 × 10^{5} (4.977173 × 10^{3}) | 7.647568 × 10^{4} (1.856713 × 10^{3}) | |

10 | ↓4.179011 × 10^{8} (6.692261 × 10^{6}) | 2.876860 × 10^{8} (9.403331 × 10^{6}) | |

DTLZ7 | 3 | 1.536594 × 10^{1} (6.362204 × 10^{−}^{1}) | ↑1.699378 × 10^{1} (2.933636 × 10^{−}^{1}) |

5 | ---2.219751 × 10^{1} (1.606481 × 10^{0}) | ---2.263389 × 10^{1} (9.949435 × 10^{−}^{1}) | |

10 | 0.000000 × 10^{0} (0.000000 × 10^{0}) | ↑5.649215 × 10^{−}^{6} (1.130646 × 10^{−}^{5}) |

Problem | M | MOSA/D_CGO | MOSA/D-DE |
---|---|---|---|

UF1 | 2 | 1.207938 × 10^{1} (3.053312 × 10^{−}^{1}) | ↑1.359271 × 10^{1} (8.924848 × 10^{−}^{2}) |

UF2 | 2 | 4.420144 × 10^{0} (1.124631 × 10^{−}^{1}) | ↑4.898329 × 10^{0} (4.114705 × 10^{−}^{2}) |

UF3 | 2 | 1.272886 × 10^{1} (2.770899 × 10^{−}^{1}) | ↑1.446032 × 10^{1} (1.560741 × 10^{−}^{1}) |

UF4 | 2 | 6.199354 × 10^{−}^{1} (7.645704 × 10^{−}^{3}) | ↑6.226053 × 10^{−}^{1} (4.375474 × 10^{−}^{3}) |

UF5 | 2 | 5.794553 × 10^{1} (2.163053 × 10^{0}) | ↑7.537953 × 10^{1} (9.461970 × 10^{−}^{1}) |

UF6 | 2 | 1.449117 × 10^{2} (2.789804 × 10^{0}) | ↑1.558762 × 10^{2} (9.956338 × 10^{−}^{1}) |

UF7 | 2 | 1.023408 × 10^{1} (3.651312 × 10^{−}^{1}) | ↑1.159760 × 10^{1} (2.062088 × 10^{−}^{1}) |

UF8 | 3 | 1.021735 × 10^{3} (2.236911 × 10^{1}) | ↑1.112894 × 10^{3} (4.871298 × 10^{−}^{1}) |

UF9 | 3 | 8.871173 × 10^{2} (1.462057 × 10^{1}) | ↑9.214801 × 10^{2} (8.179234 × 10^{0}) |

UF10 | 3 | 2.676289 × 10^{2} (5.962540 × 10^{1}) | ↑8.332120 × 10^{2} (1.796126 × 10^{1}) |

Problem | M | MOSA/D_CGO | MOSA/D-DE |
---|---|---|---|

DTLZ1 | 3 | 1.698462 × 10^{0} (5.231658 × 10^{−}^{1}) | ↑2.567411 × 10^{−}^{2} (3.988398 × 10^{−}^{4}) |

5 | 1.848446 × 10^{0} (7.371114 × 10^{−}^{1}) | ↑1.362515 × 10^{−}^{1} (8.662087 × 10^{−}^{3}) | |

10 | 2.605653 × 10^{0} (8.858904 × 10^{−}^{1}) | ↑2.746888 × 10^{−}^{1} (2.055289 × 10^{−}^{2}) | |

DTLZ2 | 3 | 1.338869 × 10^{−}^{1} (1.050833 × 10^{−}^{2}) | ↑6.539183 × 10^{−}^{2} (8.887102 × 10^{−}^{4}) |

5 | ↓3.879495 × 10^{−}^{1} (2.383225 × 10^{−}^{2}) | 4.315983 × 10^{−}^{1} (1.731880 × 10^{−}^{2}) | |

10 | ↓8.086104 × 10^{−}^{1} (3.491915 × 10^{−}^{2}) | 8.740702 × 10^{−}^{1} (3.886488 × 10^{−}^{2}) | |

DTLZ3 | 3 | 3.943222 × 10^{1} (1.066523 × 10^{1}) | ↑8.927530 × 10^{−}^{2} (8.582940 × 10^{−}^{3}) |

5 | 3.527925 × 10^{1} (1.256126 × 10^{1}) | ↑2.402407 × 10^{0} (1.052387 × 10^{0}) | |

10 | 3.857988 × 10^{1} (8.728455 × 10^{0}) | ↑1.452693 × 10^{1} (3.864857 × 10^{0}) | |

DTLZ4 | 3 | 2.193973 × 10^{−}^{1} (2.708875 × 10^{−}^{2}) | ↑7.370770 × 10^{−}^{2} (2.095534E × 10^{−}^{3}) |

5 | 4.264948 × 10^{−}^{1} (2.618424 × 10^{−}^{2}) | ↑3.453491 × 10^{−}^{1} (1.259533 × 10^{−}^{2}) | |

10 | ---7.485737 × 10^{−}^{1} (2.873640 × 10^{−}^{2}) | ---7.565568 × 10^{−}^{1} (2.740685 × 10^{−}^{2}) | |

DTLZ5 | 3 | 3.692489 × 10^{−}^{2} (4.482185 × 10^{−}^{3}) | ↑8.796775 × 10^{−}^{3} (4.662547 × 10^{−}^{4}) |

5 | ↓2.173594 × 10^{−}^{1} (2.151767 × 10^{−}^{2}) | 3.198228 × 10^{−}^{1} (3.323426 × 10^{−}^{3}) | |

10 | ↓8.274774 × 10^{−}^{1} (3.498018 × 10^{−}^{2}) | 9.278413 × 10^{−}^{1} (8.544034 × 10^{−}^{3}) | |

DTLZ6 | 3 | ↓2.360560 × 10^{0} (4.875849 × 10^{−}^{1}) | 5.755759 × 10^{0} (1.295398 × 10^{−}^{1}) |

5 | ↓3.233184 × 10^{0} (5.565340 × 10^{−}^{1}) | 7.249152 × 10^{0} (1.815938 × 10^{−}^{1}) | |

10 | ↓4.311238 × 10^{0} (3.579400 × 10^{−}^{1}) | 7.249090 × 10^{0} (1.507697 × 10^{−}^{1}) | |

DTLZ7 | 3 | 2.640245 × 10^{0} (5.499560 × 10^{−}^{1}) | ↑6.763103 × 10^{−}^{1} (4.596304 × 10^{−}^{2}) |

5 | 5.536790 × 10^{0} (6.953019 × 10^{−}^{1}) | ↑1.562072 × 10^{0} (2.362084 × 10^{−}^{1}) | |

10 | 5.536790 × 10^{0} (6.953019 × 10^{−}^{1}) | ↑1.562072 × 10^{0} (2.362084 × 10^{−}^{1}) |

Problem | M | MOSA/D_CGO | MOSA/D-DE |
---|---|---|---|

UF1 | 2 | 3.118639 × 10^{−}^{1} (3.337199 × 10^{−}^{2}) | ↑1.245048 × 10^{−}^{1} (6.952854 × 10^{−}^{3}) |

UF2 | 2 | 1.632088 × 10^{−}^{1} (1.376776 × 10^{−}^{2}) | ↑4.207095 × 10^{−}^{2} (4.246406 × 10^{−}^{3}) |

UF3 | 2 | 5.070647 × 10^{−}^{1} (3.574368 × 10^{−}^{2}) | ↑3.410425 × 10^{−}^{1} (1.406431 × 10^{−}^{2}) |

UF4 | 2 | ↓8.882224 × 10^{−}^{2} (3.714504 × 10^{−}^{3}) | 9.322984 × 10^{−}^{2} (2.313839 × 10^{−}^{3}) |

UF5 | 2 | 2.573650 × 10^{0} (3.112357 × 10^{−}^{1}) | ↑1.614455 × 10^{0} (1.012664 × 10^{−}^{1}) |

UF6 | 2 | 1.323509 × 10^{0} (1.554659 × 10^{−}^{1}) | ↑7.905724 × 10^{−}^{1} (4.148593 × 10^{−}^{2}) |

UF7 | 2 | 4.111049 × 10^{−}^{1} (4.643987 × 10^{−}^{2}) | ↑1.801766 × 10^{−}^{1} (2.685634 × 10^{−}^{2}) |

UF8 | 3 | 8.631345 × 10^{−}^{1} (1.315110 × 10^{−}^{1}) | ↑2.057906 × 10^{−}^{1} (1.069731 × 10^{−}^{2}) |

UF9 | 3 | 8.061542 × 10^{−}^{1} (1.206518 × 10^{−}^{1}) | ↑2.987048 × 10^{−}^{1} (1.581723 × 10^{−}^{2}) |

UF10 | 3 | 6.233193 × 10^{0} (9.715169 × 10^{−}^{1}) | ↑1.485037 × 10^{0} (1.454003 × 10^{−}^{1}) |

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## Share and Cite

**MDPI and ACS Style**

Vargas-Martínez, M.; Rangel-Valdez, N.; Fernández, E.; Gómez-Santillán, C.; Morales-Rodríguez, M.L.
Performance Analysis of Multi-Objective Simulated Annealing Based on Decomposition. *Math. Comput. Appl.* **2023**, *28*, 38.
https://doi.org/10.3390/mca28020038

**AMA Style**

Vargas-Martínez M, Rangel-Valdez N, Fernández E, Gómez-Santillán C, Morales-Rodríguez ML.
Performance Analysis of Multi-Objective Simulated Annealing Based on Decomposition. *Mathematical and Computational Applications*. 2023; 28(2):38.
https://doi.org/10.3390/mca28020038

**Chicago/Turabian Style**

Vargas-Martínez, Manuel, Nelson Rangel-Valdez, Eduardo Fernández, Claudia Gómez-Santillán, and María Lucila Morales-Rodríguez.
2023. "Performance Analysis of Multi-Objective Simulated Annealing Based on Decomposition" *Mathematical and Computational Applications* 28, no. 2: 38.
https://doi.org/10.3390/mca28020038