# An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets

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## Abstract

**:**

## 1. Introduction

## 2. Technical Background and Related Work

#### 2.1. Colored Petri Nets

#### 2.2. Literature Review

## 3. Proposed Mapping

- m = number of salesmen, and m > 1
- C = list of cities with C1, C2, C3, …, Cn
- X = starting point or depo
- W = cost matrix with Wij representing the time required to travel from city Ci to city Cj, for i,j = 1,2,3, …, n

- All salesmen start their tour from depot X
- The salesmen are not required to return to the depo after visiting the cities

#### 3.1. Mapping City

#### 3.2. Mapping Depo and Number of Salesman

#### 3.3. Mapping Time and Routes between Cities

## 4. Case Study

#### 4.1. Case One: mTSP with Symmetric Cost Matrix

#### 4.2. Case B: mTSP with Asymmetric Cost Matrix

#### 4.3. Case C: TSP with Symmetric Cost Matrix

#### 4.4. Simulation Results

#### 4.5. Reachability Analysis

## 5. Comparison with Other Approaches

## 6. Conclusion and Future Direction

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 11.**mTSP involving five connected cities, a depo and three salesmen with symmetric cost matrix.

**Figure 15.**mTSP involving five connected cities, a depo and three salesmen with asymmetric cost matrix.

Property | Description |
---|---|

Boundedness | It ensures that a place does not contain more tokens then its defined capacity. |

Terminating | It ensures that the model is terminating |

Dead Transition | It ensures that all the functionality of the system being modeled is used in principle |

Liveness | It ensures that a functionality of the modeled system can always be used again. It is mutually exclusive with the Terminating property |

Property | Result |
---|---|

Boundedness | |

Upper Bound | 1 |

Lower Bound | 0 |

Terminating | |

Dead Marking | None |

Dead Transition | None |

Liveness | |

Live Transitions | All |

Steps | Description |
---|---|

Step 1 | Create place d with an initial marking m. |

Step 2 a | For each city create three places: Pnv, Piv and Pv. |

Step 2 b | Set initial marking for each Pnv as 1. |

Step 2 c | For each city C, connect places Pnv and Piv through transition Ts and places Piv and Pv through transition Tc. |

Step 3 a | For each route defined from depo X to a city C, connect place d and place Piv of the city C through transition Ts. |

Step 3 b | Place an inscription on the arc connecting the transition Ts and the place Piv equals to the time taken to go from the depo to the city. |

Step 4 a | For each route defined from City(i) to City(j), connect the place Pv(i) and Piv(j) through transition Tij. |

Step 4 b | Place an inscription on the arc connecting Transition Tij and the place Piv(j) equal to the time required to go from City(i) to City(j). |

Step 4 c | Connect place Pnv(j) and transition Tij with an arc |

X | A | B | C | D | E | |
---|---|---|---|---|---|---|

X | 0 | 1 | 2 | 3 | 4 | 5 |

A | 1 | 0 | 6 | 7 | 8 | 9 |

B | 2 | 6 | 0 | 2 | 3 | 4 |

C | 3 | 7 | 2 | 0 | 7 | 8 |

D | 4 | 8 | 3 | 7 | 0 | 3 |

E | 5 | 9 | 4 | 8 | 3 | 0 |

X | A | B | C | D | E | |
---|---|---|---|---|---|---|

X | 0 | 1 | 2 | 3 | 4 | 5 |

A | 1 | 0 | 6 | 7 | 8 | 9 |

B | 2 | 1 | 0 | 2 | 3 | 4 |

C | 3 | 5 | 6 | 0 | 7 | 8 |

D | 4 | 9 | 1 | 2 | 0 | 3 |

E | 5 | 4 | 5 | 6 | 7 | 0 |

X | A | B | C | |
---|---|---|---|---|

X | 0 | 7 | 13 | 5 |

A | 7 | 0 | 4 | 6 |

B | 13 | 4 | 0 | 9 |

C | 5 | 6 | 9 | 0 |

Optimization Objective | Case A (Time Units) | Case B (Time Units) | Case C (Time Units) |
---|---|---|---|

miniSUM | 18 | 18 | NA |

miniMAX | 8 | 7 | 15 |

Liveness Properties | ||
---|---|---|

Dead Transition Instances | Dead Markings | |

Case One | None | Present |

Case Two | None | Present |

Case Three | None | Present |

^{1} B = Both Symmetric and Asymmetric Cost Matrix, S = Symmetric Cost Matrix Only | |||
---|---|---|---|

^{2} Y = Yes, N = No | |||

Approaches | Criterion A ^{1} | Criterion B ^{2} | |

Proposed Algorithm | B | Y | |

Exact Solutions | Laporte et al. [7] | B | Y |

Kara et al. [8] | B | Y | |

Iqbal Ali et al. [10] | B | Y | |

Heuristics | Carter et al. [12] | S | N |

Song et al. [14] | B | N | |

Wacholder et al. [15] | B | Y | |

Hosseinabadi [16] | S | N | |

Transformation | Hong et al. [17] | S | Y |

Rao [18] | S | Y | |

Jonker et al. [19] | S | Y |

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## Share and Cite

**MDPI and ACS Style**

Essani, F.H.; Haider, S.
An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets. *Algorithms* **2018**, *11*, 143.
https://doi.org/10.3390/a11100143

**AMA Style**

Essani FH, Haider S.
An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets. *Algorithms*. 2018; 11(10):143.
https://doi.org/10.3390/a11100143

**Chicago/Turabian Style**

Essani, Furqan Hussain, and Sajjad Haider.
2018. "An Algorithm for Mapping the Asymmetric Multiple Traveling Salesman Problem onto Colored Petri Nets" *Algorithms* 11, no. 10: 143.
https://doi.org/10.3390/a11100143