# Comparative Study of Type-2 Fuzzy Particle Swarm, Bee Colony and Bat Algorithms in Optimization of Fuzzy Controllers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

_{1}, C

_{2}and the inertia weight. The main difference between our proposed fuzzy PSO method is the inputs and outputs of the fuzzy system: while Niknam uses the best fitness and the number of generations of the best unchanged fitness as inputs for the fuzzy system, our inputs are the percentage of elapsed iterations and the diversity of the population.

_{1}, C

_{2}and inertia weight as in [52], but, in this case, these values are different for each particle, which is the main difference between our proposed fuzzy PSO method, as, in our approach, the values of C

_{1}, C

_{2}and constriction factor are the same for all particles and is used for all the swarm.

## 3. Bio-Inspired Optimization Methods

^{k}

_{i}is the set of feasible nodes (in a neighborhood) connected to node i with respect to bee k, and β is the probability to visit the following node. d

_{ij}indicates the distance of node i to node j, and for this algorithm indicates the total dance that a bee has in this moment. Finally, α is a binary variable that is used to find better solutions in the algorithm. Equation (2) represents the fact that a waggle dance will last for a certain duration, determined by a linear function, where K denotes the waggle dance scaling factor, Pf

_{i}denotes the profitability scores of bee i as defined in Equation (3) and Pf

_{colony}denotes the bee colony’s average profitability, as in Equation (4) and is updated after each bee completes its tour.

Algorithm 1. Bee Colony Optimization. | |

01. | Initialize parameters |

02. | Determine initial solutions |

03. | Evaluate the initial solutions |

04. | S← the best solution of the bees. |

05. | for each iteration do |

06. | for each bee do |

07. | Set an initial solution |

08. | Evaluate modified solutions generated by possible changes |

09. | By roulette wheel selection choose one of the modified solutions |

10. | Evaluate new solutions |

11. | Make a decision whether the bee is loyal |

12. | if the bee is not loyal then |

13. | Choice one of the loyal bees to be followed by the i-th bee. |

14. | if the best solution of the bees better the solution S |

15. | S← the best bee’s solution |

Algorithm 2. Fuzzy Bee Colony Optimization. | |

01. | Initialize parameters |

02. | Determine initial solutions |

03. | Evaluate the initial solutions |

04. | S← the best solution of the bees. |

05. | for each iteration do |

06. | Calculate iteration and diversity using Equations (10) and (11) |

07. | Use a fuzzy system to calculate the new Beta and Alpha parameters |

08. | for each bee do |

09. | Set an initial solution |

10. | Evaluate modified solutions generated by possible changes |

11. | By roulette wheel selection choose one of the modified solutions |

12. | Evaluate new solutions |

13. | Make a decision whether the bee is loyal |

14. | if the bee is not loyal then |

15. | Choice one of the loyal bees to be followed by the i-th bee. |

16. | if the best solution of the bees better the solution S |

17. | S← the best bee’s solution |

_{i}(t + 1) = x

_{i}(t) + v

_{i}(t + 1)

_{ij}(t + 1) = C[v

_{ij}(t) + c

_{1}r

_{1}(t)[y

_{ij}(t) − x

_{ij}(t)] + c

_{2}r

_{2j}(t)[ŷ

_{j}(t) − x

_{ij}(t)]]

_{i}(t) to the new position x

_{i}(t + 1) adding its new velocity v

_{i}(t + 1). Equation (6) represents the velocity of a particle i in its dimension j. This equation is an update of the velocity from actual velocity v

_{ij}(t) to its new velocity v

_{ij}(t + 1), adding the cognitive component c

_{1}r

_{1}(t)[y

_{ij}(t) − x

_{ij}(t)] and the social component c

_{2}r

_{2j}(t)[ŷ

_{j}(t) − x

_{ij}(t)]; this new velocity is also weighted with a constriction factor C.

Algorithm 3. Particle Swarm Optimization. | |

01. | Initialize size of the particle swarm n, and other parameters |

02. | Initialize positions and velocities for all particles randomly |

03. | While (end criterion is not met) do |

04. | Increment iteration counter |

05. | Calculate fitness value of each particle |

06. | Update new local and global best particle |

07. | Calculate new velocity of each particle |

08. | Update new position of each particle |

09. | End While |

_{1}, C

_{2}and Constriction factor from line 08.

Algorithm 4. Fuzzy Particle Swarm Optimization. | |

01. | Initialize size of the particle swarm n, and other parameters |

02. | Initialize positions and velocities for all particles randomly |

03. | While (end criterion is not met) do |

04. | Increment iteration counter |

05. | Calculate fitness value of each particle |

06. | Update new local and global best particles |

07. | Calculate iteration and diversity using Equations (10) and (11) |

08. | Calculate new C_{1}, C_{2}, and Constriction factor using a fuzzy system |

09. | Calculate new velocity of each particle |

10. | Update new position of each particle |

11. | End While |

_{i}at position x

_{i}, wavelength and frequency values are varied in each iteration, the BA is idealized in the pseudocode shown in Algorithm 5 [60,61].

Algorithm 5. Bat Algorithm. | |

01. | Initialize the bat population x_{i} (i = 1, 2, ..., n) and v_{i} |

02. | Initialize frequency f_{i}, pulse rates r_{i} and the loudness A_{i} |

03. | While (t < Max numbers of iterations) |

04. | Generate new solutions by adjusting frequency |

05. | and updating velocities and locations/solutions (Equations (7)–(9)) |

06. | if (rand > r_{i}) |

07. | Select a solution among the best solutions |

08. | Generate a local solution around the selected best solution |

09. | end if |

10. | Generate new solutions by flying randomly |

11. | if (rand < A_{i}&f(x_{i}) < f(x*)) |

12. | Accept the new solutions |

13. | Increase r_{i} and reduce A_{i} |

14. | end if |

15. | Rank the bats and find the current best x* |

16. | End While |

Algorithm 6. Fuzzy Bat Algorithm. | |

01. | Initialize the bat population x_{i} (i = 1, 2, ..., n) and v_{i} |

02. | Initialize frequency f_{i}, pulse rates r_{i} and the loudness A_{i} |

03. | While (t < Max numbers of iterations) |

04. | Normalize Iterations Equation (10) |

05. | Generate new solutions by adjusting frequency |

06. | and updating velocities and locations/solutions (Equations (7) to (9)) |

07. | if (rand > r_{i}) |

08. | Select a solution among the best solutions |

09. | Generate a local solution around the selected best solution |

10. | end if |

11. | Generate new solutions by flying randomly |

12. | if (rand < A_{i}&f(x_{i}) < f(x*)) |

13. | Accept the new solutions |

14. | Size Diversity Equation (11) |

15. | Assign Values r_{i} and A_{i} using fuzzy system |

16. | end if |

17. | Rank the bats and find the current best x* |

18. | End While |

## 4. Methodology for Dynamic Parameter Adaptation

## 5. Bio-Inspired Methods with Parameter Adaptation

_{1}, C

_{2}and the constriction factor C from Equation (6), which are the most important and have a big impact in the behavior of the algorithm, so controlling these parameters allows to control the entire algorithm.

_{1}, C

_{2}and constriction factor as outputs. In this case, the inputs were granulated into three triangular membership functions, and the outputs into five triangular membership functions. It contains nine rules that were designed to control the behavior of PSO.

_{i}), both are granulated into five fuzzy sets (Low, MediumLow, Medium, MediumHigh and High).

## 6. Problem Statement

_{1}and τ

_{2}denote the torques of the right and left wheel, respectively; $P\in {R}^{2}$ is the uniformly bounded disturbance vector; $M\left(q\right)\in {R}^{2x2}$ is the positive-definite inertia matrix; $C\left(q,\dot{q}\right)v$ is the vector of centripetal and Coriolis forces; and $D\in {R}^{2x2}$ is a diagonal positive-definite damping matrix. Equation (13) represents the kinematics of the system, where (x, y) is the position in the X–Y (world) reference frame; θ is the angle between the heading direction and the x-axis; and v and w are the linear and angular velocities, respectively.

## 7. Simulation Results

**Original PSO**is the original PSO method with linear decreasing inertia weight.

**PSO + T1FS**is the PSO method with parameter adaptation using the type-1 fuzzy system illustrated in Figure 3.

**PSO + IT2FS**is the PSO method with parameter adaptation using the interval type-2 fuzzy system illustrated in Figure 4.

^{−}^{3}.

^{−}^{3}.

^{−}^{4}, which can obtain a better fuzzy controller than compared to methods BA and BCO algorithms.

_{1}, C

_{2}and constriction factor, using iteration and diversity as inputs in an interval type-2 fuzzy system.

## 8. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Type-1 fuzzy system for dynamic parameter adaptation in PSO (Particle Swarm Optimization).

No. | Inputs | Outputs | ||
---|---|---|---|---|

Iteration | Diversity | Beta | Alpha | |

1 | Low | Low | High | Low |

2 | Low | Medium | MediumHigh | Medium |

3 | Low | High | MediumHigh | MediumLow |

4 | Medium | Low | MediumHigh | MediumLow |

5 | Medium | Medium | Medium | Medium |

6 | Medium | High | MediumLow | MediumHigh |

7 | High | Low | Medium | High |

8 | High | Medium | MediumLow | MediumHigh |

9 | High | High | Low | High |

No. | Inputs | Outputs | |||
---|---|---|---|---|---|

Iteration | Diversity | C_{1} | C_{2} | Constriction | |

1 | Low | Low | High | Low | High |

2 | Low | Medium | MediumHigh | Medium | MediumHigh |

3 | Low | High | MediumHigh | MediumLow | Medium |

4 | Medium | Low | MediumHigh | MediumLow | MediumHigh |

5 | Medium | Medium | Medium | Medium | Medium |

6 | Medium | High | MediumLow | MediumHigh | MediumLow |

7 | High | Low | Medium | High | Medium |

8 | High | Medium | MediumLow | MediumHigh | MediumLow |

9 | High | High | Low | High | Low |

No. | Inputs | Outputs | ||
---|---|---|---|---|

Iteration | Diversity | Beta | Pulse Rate | |

1 | Low | Low | Low | High |

2 | Low | Medium | Low | High |

3 | Low | High | Low | MediumHigh |

4 | Medium | Low | MediumLow | MediumHigh |

5 | Medium | Medium | MediumLow | Medium |

6 | Medium | High | Medium | MediumLow |

7 | High | Low | MediumHigh | MediumLow |

8 | High | Medium | MediumHigh | MediumLow |

9 | High | High | High | Low |

No. | Inputs | Outputs | ||
---|---|---|---|---|

Linear Error | Angular Error | Torque 1 | Torque 2 | |

1 | Negative | Negative | Negative | Negative |

2 | Negative | Zero | Negative | Zero |

3 | Negative | Positive | Negative | Positive |

4 | Zero | Negative | Zero | Negative |

5 | Zero | Zero | Zero | Zero |

6 | Zero | Positive | Zero | Positive |

7 | Positive | Negative | Positive | Negative |

8 | Positive | Zero | Positive | Zero |

9 | Positive | Positive | Positive | Positive |

Parameter | Original PSO | Proposed PSO |
---|---|---|

Population | 30 | 30 |

Iterations | 100 | 100 |

C_{1} | 1 | Dynamic |

C_{2} | 3 | Dynamic |

Constriction | 1 | Dynamic |

Inertia weight | Linear decreasing | 1 |

Parameter | Original BA | Proposed BA |
---|---|---|

Population | 30 | 30 |

Iterations | 100 | 100 |

Frequency min | 0 | 0 |

Frequency max | 2 | 2 |

Loudness | 0.5 | 0.5 |

Beta | Random [0, 1] | Dynamic |

Pulse rate | 0.5 | Dynamic |

Parameter | Original BCO | Proposed BCO |
---|---|---|

Population | 30 | 30 |

Iterations | 100 | 100 |

Alpha | Random [0, 1] | Dynamic |

Beta | Random [2, 4] | Dynamic |

Food Number | 20 | 20 |

Limit | 100 | 100 |

MSE | Original BCO | BCO + T1FS | BCO + IT2FS |
---|---|---|---|

Average | 14.61 | 9.8394 | 9.1062 |

Best | 8.84 × 10^{−3} | 1.50 × 10^{−3} | 2.80 × 10^{−3} |

Worst | 106.81 | 61.9165 | 65.4639 |

Standard Deviation | 23.34 | 14.69 | 16.6722 |

MSE | Original BCO | BCO + T1FS | BCO + IT2FS |
---|---|---|---|

Average | 16.54 | 14.61717 | 14.7402 |

Best | 19.46 | 2.3864 | 2.2093 |

Worst | 61.90 | 73.9592 | 73.9592 |

Standard Deviation | 2.24 | 17.3484 | 17.3346 |

MSE | Original PSO | PSO + T1FS | PSO + IT2FS |
---|---|---|---|

Average | 2.4166 | 3.0426 × 10^{−1} | 3.2236 × 10^{−2} |

Best | 1.3936 × 10^{−1} | 4.4127 × 10^{−3} | 1.9081 × 10^{−4} |

Worst | 6.6763 | 1.5119 | 3.3309 × 10^{−1} |

Standard Deviation | 2.9784 | 3.1949 × 10^{−1} | 7.1146 × 10^{−2} |

MSE | Original PSO | PSO + T1FS | PSO + IT2FS |
---|---|---|---|

Average | 6.4302 | 3.1108 | 2.3630 |

Best | 4.4098 | 2.2788 | 1.9009 |

Worst | 9.1806 | 4.2922 | 3.0982 |

Standard Deviation | 2.7437 | 4.7588 × 10^{−1} | 3.3354 × 10^{−1} |

MSE | Original BA | BA + T1FS | BA + IT2FS |
---|---|---|---|

Average | 4.0223 | 36.8036 | 20.4708 |

Best | 4 × 10^{−3} | 1.47 × 10^{−2} | 1.6437 |

Worst | 9.3143 | 97.9869 | 34.3176 |

Standard Deviation | 2.9013 | 34.5750 | 12.2588 |

MSE | Original BA | BA + T1FS | BA + IT2FS |
---|---|---|---|

Average | 12.9050 | 6.7172 | 13.2668 |

Best | 5.4808 | 5.0436 | 9.808 |

Worst | 19.1657 | 8.4160 | 15.6879 |

Standard Deviation | 4.5601 | 1.3102 | 2.9829 |

MSE | Original BA | BCO + IT2FS | PSO + IT2FS |
---|---|---|---|

Average | 4.0223 | 9.1062 | 3.2236 × 10^{−2} |

Best | 4 × 10^{−3} | 2.80 × 10^{−3} | 1.9081 × 10^{−4} |

Worst | 9.3143 | 65.4639 | 3.3309 × 10^{−1} |

Standard Deviation | 2.9013 | 16.6722 | 7.1146 × 10^{−2} |

MSE | BA + T1FS | BCO + T1FS | PSO + IT2FS |
---|---|---|---|

Average | 6.7172 | 14.61717 | 2.3630 |

Best | 5.0436 | 2.3864 | 1.9009 |

Worst | 8.4160 | 73.9592 | 3.0982 |

Standard Deviation | 1.3102 | 17.3484 | 3.3354 × 10^{−1} |

**Table 16.**Comparison against Shi et al. [49].

Rosenbrock Function | ||||
---|---|---|---|---|

Population Size | Dimensions | Iterations | Shi et al. [49] | Our Approach |

20 | 10 | 1000 | 66.01409 | 2.5431 |

20 | 1500 | 108.2865 | 15.8462 | |

30 | 2000 | 183.8037 | 38.8491 | |

40 | 10 | 1000 | 48.76523 | 1.4956 |

20 | 1500 | 63.88408 | 9.5468 | |

30 | 2000 | 175.0093 | 27.2133 | |

80 | 10 | 1000 | 15.81645 | 2.5656 |

20 | 1500 | 45.99998 | 8.5468 | |

30 | 2000 | 124.4184 | 22.8496 |

**Table 17.**Comparison against Shi et al. [49].

Rastrigin Function | ||||
---|---|---|---|---|

Population Size | Dimensions | Iterations | Shi et al. [49] | Our Approach |

20 | 10 | 1000 | 4.955165 | 1.8491 |

20 | 1500 | 23.27334 | 5.0213 | |

30 | 2000 | 48.47555 | 16.8492 | |

40 | 10 | 1000 | 3.283368 | 1.1334 |

20 | 1500 | 15.04448 | 5.8704 | |

30 | 2000 | 35.20146 | 12.2354 | |

80 | 10 | 1000 | 2.328207 | 0.5849 |

20 | 1500 | 10.86099 | 3.2136 | |

30 | 2000 | 22.52393 | 9.6935 |

**Table 18.**Comparison against Shi et al. [49].

Griewank Function | ||||
---|---|---|---|---|

Population Size | Dimensions | Iterations | Shi et al. [49] | Our Approach |

20 | 10 | 1000 | 0.091623 | 0.0654 |

20 | 1500 | 0.027275 | 0.0192 | |

30 | 2000 | 0.02156 | 0.0114 | |

40 | 10 | 1000 | 0.075674 | 0.0568 |

20 | 1500 | 0.031232 | 0.0149 | |

30 | 2000 | 0.012198 | 0.0097 | |

80 | 10 | 1000 | 0.068323 | 0.0628 |

20 | 1500 | 0.025956 | 0.0181 | |

30 | 2000 | 0.014945 | 0.0110 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Olivas, F.; Amador-Angulo, L.; Perez, J.; Caraveo, C.; Valdez, F.; Castillo, O.
Comparative Study of Type-2 Fuzzy Particle Swarm, Bee Colony and Bat Algorithms in Optimization of Fuzzy Controllers. *Algorithms* **2017**, *10*, 101.
https://doi.org/10.3390/a10030101

**AMA Style**

Olivas F, Amador-Angulo L, Perez J, Caraveo C, Valdez F, Castillo O.
Comparative Study of Type-2 Fuzzy Particle Swarm, Bee Colony and Bat Algorithms in Optimization of Fuzzy Controllers. *Algorithms*. 2017; 10(3):101.
https://doi.org/10.3390/a10030101

**Chicago/Turabian Style**

Olivas, Frumen, Leticia Amador-Angulo, Jonathan Perez, Camilo Caraveo, Fevrier Valdez, and Oscar Castillo.
2017. "Comparative Study of Type-2 Fuzzy Particle Swarm, Bee Colony and Bat Algorithms in Optimization of Fuzzy Controllers" *Algorithms* 10, no. 3: 101.
https://doi.org/10.3390/a10030101