# A Quantum Ant Colony Multi-Objective Routing Algorithm in WSN and Its Application in a Manufacturing Environment

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Energy Consumption Model

#### 3.2. Basic Ant-Based Routing (BABR) Algorithm

**Step 1**: At regular intervals, a forward ant k starts to move from the source node toward the destination. While moving, the identifiers of every visited node are recorded in a list, M

_{k}, and each forward ant avoids traversing a node that has been visited previously.

**Step 2**: At each node r, a forward ant selects the next hop node in accordance with a certain probability distribution:

_{s}is the actual energy level of node s), and μ, ν are weight parameters that signify the importance of pheromones versus heuristics.

**Step 3**: When a forward ant reaches the destination, a backward ant goes back along the links that the forward ant has visited. Before moving, the amount of pheromones that the ant will drop during the trip is computed:

_{k}is the distance traveled by the forward ant k.

**Step 4**: Whenever a node r receives a backward ant from a neighbor node, the routing table is updated:

**Step 5**: Once a backward ant returns to the source node, the next interval is continued.

## 4. The QACMOR Routing Method

#### 4.1. Mechanisms of QEAs

#### 4.1.1. Basic Elements of QEAs

^{n}kinds of states:

^{th}bit phase is ${\xi}_{i}=\mathrm{arctan}({\beta}_{i}/{\alpha}_{i})$. The position of ${\xi}_{i}$ in coordination is given in Figure 2.

#### 4.1.2. The Updating of Qubit in QEAs

_{i}degrees from the original vector, ${\left(\begin{array}{cc}{\alpha}_{i}& {\beta}_{i}\end{array}\right)}^{T}$ to ${\left(\begin{array}{cc}{\alpha}_{i}^{\prime}& {\beta}_{i}^{\prime}\end{array}\right)}^{T}$

_{i}is the rotation degree according to the following formula:

_{max}represents the predefined maximal times of calculation determined by the scale of the problem. The function $s({\alpha}_{i},{\beta}_{i})$ defines the direction:

^{th}qubit of the current and optimal solution, respectively. Finally, if $s({\alpha}_{i},{\beta}_{i})<0$, the θ

_{i}rotates clockwise—otherwise, it rotates counterclockwise.

#### 4.2. The QEAs in QACMOR

#### 4.2.1. Representing the Pheromone with Qubit

#### 4.2.2. Updating the Pheromone with Quantum Rotation Gate

^{th}bit of the j

^{th}individual ${q}_{j}$ of solution Q is described as follows:

^{th}bit of the j

^{th}individual of the current solution x and the best solution $b$, respectively. The schematic diagram in Figure 3 shows the rotation gate polar plot for a qubit individual.

#### 4.3. The QACMOR Algorithm

**Step 1:**The initialization step. Add every node and its neighbor nodes into the routing table. A forward ant is generated from source nodes which carry the information of source nodes, sink nodes, and passing nodes. The population is represented as $Q(t)=({q}_{1}^{t},{q}_{2}^{t},\cdots ,{q}_{j}^{t},\cdots ,{q}_{m}^{t})$ with the size of m individuals, where ${q}_{j}^{t}(j=1,2,\cdots ,m)$ is the j

^{th}individual in the t

^{th}iteration. The representation is shown as:

_{max}, and the initial value of the current iterations t is 0.

**Step 2:**Compute the binary solution P(t). $P(t)=({p}_{1}^{t},{p}_{2}^{t},\cdots ,{p}_{j}^{t},\cdots ,{p}_{m}^{t})$, ${p}_{j}^{t}(j=1,2,\cdots ,m)$ is a binary individual with n-length. The probable solution is obtained by measurement of Q(t). The value of element ${p}_{ji}$ in ${p}_{j}$ is determined by comparing ${\alpha}_{ji}$ of ${q}_{j}$ with w, 0 < w < 1.

**Step 3:**Generate the routing path. Assign m individuals into the source nodes at random. We used the state transition rule to generate the routing path of these individuals. In each step of the decision, an individual positioned on node r moves to the node s in line with Equations (4)–(6).

**Step 4:**Evaluate the solution and store the best solutions in B(t). The evaluation function of the routing tree is shown as follows:

_{1}, C

_{2}, C

_{3}, and C

_{4}are weight parameters, and ${E}_{r}(t)$ and ${\sigma}_{r}(t)$ are factors which describe the network load balance, and respectively represent the average value and standard deviation of the load for node r. Z

_{1}(t) is the energy consumption factor, K is an array which indicates the total number of leaf nodes extended from each node in the routing tree, λ is a parameter with a value from 2 to 4 which generally approaches 4, ${d}_{rs}$ is the distance of link (r,s), Tree(t) denotes the routing tree, Z

_{2}(t) is the time-delay factor, and Fd

_{k}is the distance traveled by the forward ant k.

**Step 5:**Update the pheromone according to the rules of the quantum rotation gate, after receiving back the ant.

**Step 6:**If the current iterations are less than the maximum iterations, return to Step 3.

## 5. Experimental Results

#### 5.1. Performance Evaluation

- (1)
- General property, such as communication distance, energy consumption, and hops.
- (2)
- Convergence rate, that is, the number of iterations needed to find an approximation to a fixed point.
- (3)
- Network lifetime, that is, the duration up to the time when data can no longer be forwarded due to the depletion of energy of the sensor nodes.

^{2}, and the total number of nodes was 50. Each link between a node and its accessible neighbors was denoted by a dotted line. Figure 5 shows the optimal path obtained by QACMOR, shown as a solid red line. Source nodes were numbered 16, 21, 22, 24, 30, 47, and 50, and the sink node was numbered 1. Notice that the value of t

_{max}should be greater than the number of iterations for the algorithm to converge.

#### 5.2. Case Study

_{1}, C

_{2}, C

_{3}, C

_{4}) as that listed in Table 2 in Experiment 3 in Section 5.1. As in that experiment, the comparison of the whole network lifetime was made by observing the number of dead nodes. Results in Figure 11 indicate that in terms of time elapsed before first node death or total network lifetime, QACMOR still has an advantage over BABR, even in harsh working conditions.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The look-up table of the quantum-inspired evolutionary algorithm (QEA) rotation angle [28].

${\mathit{x}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | $\mathit{f}\left(\mathit{x}\right)>\mathit{f}\left(\mathit{b}\right)$ | $\mathbf{\Delta}{\mathit{\theta}}_{\mathit{i}}$ | $\mathit{s}\left({\mathit{\alpha}}_{\mathit{i}},{\mathit{\beta}}_{\mathit{i}}\right)$ | |||
---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathit{i}}{\mathit{\beta}}_{\mathit{i}}>0$ | ${\mathit{\alpha}}_{\mathit{i}}{\mathit{\beta}}_{\mathit{i}}<0$ | ${\mathit{\alpha}}_{\mathit{i}}=0$ | ${\mathit{\beta}}_{\mathit{i}}=0$ | ||||

0 | 0 | False | 0 | 0 | 0 | 0 | 0 |

0 | 0 | True | 0 | 0 | 0 | 0 | 0 |

0 | 1 | False | 0 | 0 | 0 | 0 | 0 |

0 | 1 | True | 0.05π | +1 | −1 | 0 | $\pm 1$ |

1 | 0 | False | 0.01π | +1 | −1 | 0 | $\pm 1$ |

1 | 0 | True | 0.025π | −1 | +1 | $\pm 1$ | 0 |

1 | 1 | False | 0.005π | −1 | +1 | $\pm 1$ | 0 |

1 | 1 | True | 0.025π | −1 | +1 | $\pm 1$ | 0 |

Item | Experiment 1 | Experiment 2 | Experiment 3 |
---|---|---|---|

Number of nodes | 10, 20, 30, …, 100 | 50 | 50 |

Network range | 1000 m^{2} | 1000 m^{2} | 5000 m^{2} |

Initial energy | / | / | 0.5 J |

C_{1} | 0.5 | 0.5 | 0.5 |

C_{2} | 0.1 | 0.1 | 0.1 |

C_{3} | 0.1 | 0.1 | 0.1 |

C_{4} | 0.1 | 0.1 | 0.1 |

t_{max} | 400 | 400 | 400 |

Item | Value |
---|---|

Number of nodes | 60 |

Network range | 300 m × 280 m |

Initial energy | 0.5 J |

C_{1} | 0.5 |

C_{2} | 0.1 |

C_{3} | 0.1 |

C_{4} | 0.1 |

t_{max} | 500 |

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**MDPI and ACS Style**

Li, F.; Liu, M.; Xu, G.
A Quantum Ant Colony Multi-Objective Routing Algorithm in WSN and Its Application in a Manufacturing Environment. *Sensors* **2019**, *19*, 3334.
https://doi.org/10.3390/s19153334

**AMA Style**

Li F, Liu M, Xu G.
A Quantum Ant Colony Multi-Objective Routing Algorithm in WSN and Its Application in a Manufacturing Environment. *Sensors*. 2019; 19(15):3334.
https://doi.org/10.3390/s19153334

**Chicago/Turabian Style**

Li, Fei, Min Liu, and Gaowei Xu.
2019. "A Quantum Ant Colony Multi-Objective Routing Algorithm in WSN and Its Application in a Manufacturing Environment" *Sensors* 19, no. 15: 3334.
https://doi.org/10.3390/s19153334