# Multi-Objective Optimization of Friction Stir Welding Process Parameters of AA6061-T6 and AA7075-T6 Using a Biogeography Based Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Works

## 3. Developing the Mathematical Models

#### 3.1. Checking the Developed Model Accuracy

#### 3.2. Validation of the Regression Models

## 4. Multi Objective Optimization

#### 4.1. Biogeography Based Optimization

#### 4.1.1. Initialize the Habitats

#### 4.1.2. Migration

#### 4.1.3. Mutation

#### 4.1.4. Evaluating the HSI

#### 4.1.5. Update Habitat

#### 4.1.6. Stopping Criterion

#### 4.1.7. Implementation

Generate a set of habitats for a problem |

Evaluate the fitness value or HSI for each habitat |

while stopping criterion is not met |

Determine immigrating rate $\lambda $ and emigrating rate $\mu $ for each habitat |

Modify the habitats based on $\lambda $ and $\mu $ |

for$i=1$: N (population size) do |

Use $\lambda $ to probabilistically decide whether to modify to emigration |

if rand (0, 1) < ${\lambda}_{i}$ |

Select Habitat ${H}_{j}$ through roulette wheel method to emigration |

Perform migration on ${H}_{i}$ and ${H}_{j}$ |

Evaluate the fitness value or his |

Replace the new solution with ${H}_{i}$ |

end if |

if rand (0, 1) < ${P}_{Mutation}$ |

Apply mutation on ${H}_{i}$ |

Evaluate the fitness value or HSI for the newly generated solution |

end if |

end for |

Update habitats’ population |

end while |

#### 4.2. Objective Function, Decision Variables, and Constraints

- $\mathrm{Max}:ultimatetensilestrength\left(RS,TS,TA,TO\right)$
- $\mathrm{Max}:elongation\left(RS,TS,TA,TO\right)$
- $\mathrm{Max}:hardness\left(RS,TS,TA,TO\right)$
- Subject to:
- $800\le rotationalspeed\le 1600$
- $20\le traversespeed\le 180$
- $1\xb0\le tiltangle\le 3\xb0$
- $-2\le tooloffset\le 2$

#### 4.3. Decision Making Methods in Multi-Objective Optimization

#### 4.3.1. Shannon Entropy Method

#### 4.3.2. TOPSIS Decision Making Method

## 5. Results and Discussion

^{®}core™ CPU i7-4500U at 1.8 GHZ, 8 GB RAM computer with Windows 8.1. The MOBBO parameters are set in this study after a number of careful runs as follows; the habitat size (N) = 70, maximum migration and immigration rate of each habitat = 1, mutation probability = 0.05, and maximum iteration = 300.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Pareto frontier for: (

**a**) dual objective (UTS-E); (

**b**) dual objective (UTS-H); and (

**c**) dual objective (E-H) optimization.

No. | Parameters | Notation | Unit | Levels | ||||
---|---|---|---|---|---|---|---|---|

−2 | −1 | 0 | +1 | +2 | ||||

1 | Rotational speed | RS | rpm | 800 | 1000 | 1200 | 1400 | 1600 |

2 | Traverse speed | TS | mm/min | 20 | 60 | 100 | 140 | 180 |

3 | Tool offset | TO | mm | −2 | −1 | 0 | 1 | 2 |

4 | Tilt angle | TA | (°) | 1 | 1.5 | 2 | 2.5 | 3 |

Test Run | Process Parameters | Experimental Values | |||||
---|---|---|---|---|---|---|---|

$\mathit{R}\mathit{S}$ (rpm) | $\mathit{T}\mathit{S}$ (mm/min) | $\mathit{T}\mathit{A}$ (°) | $\mathit{T}\mathit{O}$ (mm) | $\mathit{U}\mathit{T}\mathit{S}$ (MPa) | $\mathit{E}$ (%) | $\mathit{H}$ (HV) | |

R01 | −1 | −1 | −1 | −1 | 236.56 | 8.3 | 60.2 |

R02 | 1 | −1 | −1 | −1 | 217 | 5.7 | 55.3 |

R03 | −1 | 1 | −1 | −1 | 246.32 | 9.5 | 68.2 |

R04 | 1 | 1 | −1 | −1 | 236.25 | 8 | 60.4 |

R05 | −1 | −1 | 1 | −1 | 232.58 | 7.5 | 58.5 |

R06 | 1 | −1 | 1 | −1 | 218.6 | 6.7 | 54.6 |

R07 | −1 | 1 | 1 | −1 | 247.26 | 9.8 | 70.8 |

R08 | 1 | 1 | 1 | −1 | 233.33 | 7.9 | 58.6 |

R09 | −1 | −1 | −1 | 1 | 229 | 7.5 | 57.2 |

R10 | 1 | −1 | −1 | 1 | 214.4 | 6.5 | 54.7 |

R11 | −1 | 1 | −1 | 1 | 250 | 7.3 | 65.3 |

R12 | 1 | 1 | −1 | 1 | 228 | 6.9 | 58.5 |

R13 | −1 | −1 | 1 | 1 | 228 | 7.2 | 58.7 |

R14 | 1 | −1 | 1 | 1 | 215 | 6.3 | 55.1 |

R15 | −1 | 1 | 1 | 1 | 233 | 7.2 | 59.8 |

R16 | 1 | 1 | 1 | 1 | 226.9 | 6.8 | 57.6 |

R17 | 2 | 0 | 0 | 0 | 219 | 6.3 | 55.7 |

R18 | −2 | 0 | 0 | 0 | 238 | 9.2 | 64.8 |

R19 | 0 | 2 | 0 | 0 | 241 | 8.9 | 66.5 |

R20 | 0 | −2 | 0 | 0 | 220 | 6.2 | 56.7 |

R21 | 0 | 0 | 2 | 0 | 220 | 6.1 | 55.6 |

R22 | 0 | 0 | −2 | 0 | 236.39 | 6.6 | 59.3 |

R23 | 0 | 0 | 0 | 2 | 215 | 6 | 54.3 |

R24 | 0 | 0 | 0 | −2 | 231.7 | 8.8 | 59.2 |

R25 | 0 | 0 | 0 | 0 | 248 | 7.5 | 71.5 |

R26 | 0 | 0 | 0 | 0 | 249 | 7.7 | 68.7 |

R27 | 0 | 0 | 0 | 0 | 254 | 7.2 | 69.2 |

R28 | 0 | 0 | 0 | 0 | 256 | 7.3 | 71.2 |

R29 | 0 | 0 | 0 | 0 | 252 | 6.9 | 69.6 |

R30 | 0 | 0 | 0 | 0 | 251 | 7.6 | 70.3 |

Responses | Source | Sum-of-Square | DF | Mean-Square | F-Ratio (Calculated) | F-Ratio (Tabulated) | p-Value |
---|---|---|---|---|---|---|---|

UTS | Regression | 4720.48 | 8 | 337.18 | 18.83 | 2.42 | 0.000 |

Residual | 268.58 | 21 | 17.91 | - | - | - | |

E | Regression | 30.52 | 6 | 2.18 | 16.27 | 2.53 | 0.000 |

Residual | 2.01 | 23 | 0.134 | - | - | - | |

H | Regression | 995.68 | 10 | 71.12 | 30.74 | 2.38 | 0.000 |

Residual | 34.71 | 19 | 2.31 | - | - | - |

No | Parameters | Experimental Value | Predicted Value | % of Error | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RS | TS | TA | TO | UTS | E | H | UTS | E | H | UTS | E | H | |

1 | 0.5 | 1.25 | −0.25 | −1 | 241.2 | 8.5 | 64.8 | 244.6 | 8.6 | 66.3 | 1.37 | 1.56 | 2.27 |

2 | −1 | 0.75 | −0.25 | 0.5 | 253.6 | 8.1 | 70.2 | 251.8 | 7.8 | 69.5 | 0.73 | 3.23 | 1.04 |

3 | 0.75 | −0.25 | 0.5 | −1 | 232.7 | 7.2 | 59.3 | 236.2 | 7.0 | 62.5 | 1.50 | 3.45 | 5.19 |

4 | −2 | 1.25 | −1.5 | −1.25 | 234.2 | 9.8 | 60.2 | 228.9 | 10.8 | 61.3 | 2.34 | 8.99 | 1.80 |

5 | 0.25 | −0.75 | 1.5 | 0 | 231.2 | 6.5 | 58.8 | 226.8 | 6.3 | 59.4 | 1.93 | 3.27 | 1.03 |

**Table 5.**The values of the design variables and objective functions of the optimum points specified using the decision making approach.

Objective Functions | Solution Methods | Process Parameters | Mechanical Properties | Deviation Index | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{R}\mathit{S}$ (rpm) | $\mathit{T}\mathit{S}$ (mm/min) | $\mathit{T}\mathit{A}$ (°) | $\mathit{T}\mathit{O}$ (mm) | $\mathit{U}\mathit{T}\mathit{S}$ (MPa) | $\mathit{E}$ (%) | $\mathit{H}$ (HV) | |||

UTS-E-H | TOPSIS | 967.41 | 164.40 | 1.97 | −1.05 | 245.95 | 8.92 | 70.85 | 0.26 |

Shannon | 1002.14 | 149.73 | 1.92 | −0.74 | 252.23 | 8.19 | 72.11 | 0.12 | |

Ideal | - | - | - | - | 256.28 | 10.92 | 72.62 | 0 | |

Nadir | - | - | - | - | 216.93 | 7.13 | 63.02 | 1 | |

UTS-E | TOPSIS | 977.07 | 174.67 | 1.97 | −1.40 | 239.07 | 9.54 | 68.79 | 0.44 |

Shannon | 986.39 | 174.43 | 1.96 | −1.25 | 241.62 | 9.32 | 69.53 | 0.37 | |

Ideal | - | - | - | - | 256.29 | 10.92 | - | 0 | |

Nadir | - | - | - | - | 216.93 | 7.15 | - | 1 | |

UTS-H | TOPSIS | 1064.69 | 131.52 | 1.96 | −0.32 | 256.06 | 7.33 | 72.53 | 0.28 |

Shannon | 1066.10 | 130.53 | 1.96 | −0.31 | 256.11 | 7.30 | 72.51 | 0.24 | |

Ideal | - | - | - | - | 256.28 | - | 72.62 | 0 | |

Nadir | - | - | - | - | 255.49 | - | 72.31 | 1 | |

E-H | TOPSIS | 831.43 | 180.00 | 2.00 | −1.72 | 225.00 | 10.48 | 65.57 | 0.65 |

Shannon | 800.00 | 180.00 | 2.00 | −2.00 | 216.95 | 10.92 | 63.03 | 0.74 | |

Ideal | - | - | - | - | - | 10.92 | 72.62 | 0 | |

Nadir | - | - | - | - | - | 7.61 | 63.03 | 1 | |

UTS | - | 1080.44 | 126.54 | 1.89 | −0.24 | 256.29 | 7.15 | 72.26 | - |

E | - | 800.00 | 180.00 | 2.00 | −2.00 | 216.95 | 10.92 | 63.03 | - |

H | - | 1040.61 | 137.10 | 2.00 | −0.42 | 255.30 | 7.57 | 72.62 | - |

Decision Making Methods | Average Percentage Error | Max Percentage Error | ||||
---|---|---|---|---|---|---|

$\mathit{U}\mathit{T}\mathit{S}$ (MPa) | $\mathit{E}$ (%) | $\mathit{H}$ (HV) | $\mathit{U}\mathit{T}\mathit{S}$ (MPa) | $\mathit{E}$ (%) | $\mathit{H}$ (HV) | |

TOPSIS | 0.41 | 2.22 | 0.24 | 0.85 | 4.13 | 0.59 |

Shannon | 0.11 | 1.68 | 0.03 | 0.31 | 3.10 | 0.22 |

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## Share and Cite

**MDPI and ACS Style**

Tamjidy, M.; Baharudin, B.T.H.T.; Paslar, S.; Matori, K.A.; Sulaiman, S.; Fadaeifard, F.
Multi-Objective Optimization of Friction Stir Welding Process Parameters of AA6061-T6 and AA7075-T6 Using a Biogeography Based Optimization Algorithm. *Materials* **2017**, *10*, 533.
https://doi.org/10.3390/ma10050533

**AMA Style**

Tamjidy M, Baharudin BTHT, Paslar S, Matori KA, Sulaiman S, Fadaeifard F.
Multi-Objective Optimization of Friction Stir Welding Process Parameters of AA6061-T6 and AA7075-T6 Using a Biogeography Based Optimization Algorithm. *Materials*. 2017; 10(5):533.
https://doi.org/10.3390/ma10050533

**Chicago/Turabian Style**

Tamjidy, Mehran, B. T. Hang Tuah Baharudin, Shahla Paslar, Khamirul Amin Matori, Shamsuddin Sulaiman, and Firouz Fadaeifard.
2017. "Multi-Objective Optimization of Friction Stir Welding Process Parameters of AA6061-T6 and AA7075-T6 Using a Biogeography Based Optimization Algorithm" *Materials* 10, no. 5: 533.
https://doi.org/10.3390/ma10050533