# A QUBO Model for the Traveling Salesman Problem with Time Windows

^{*}

## Abstract

**:**

## 1. Introduction

**Contribution**. In this paper, we give the first, to the best of our knowledge, QUBO formulation for the TSP with Time Windows. The existence of an Ising or QUBO formulation for a problem is the essential precondition for its solution on the current generation of D-Wave computers. For the vanilla TSP, there exists such a formulation, as presented in an elegant and comprehensive manner in [33], which has enabled the actual solution of TSP instances on the D-Wave platform (see [36,37,38]). In contrast, prior to this work, the TSPTW had not been cast in the QUBO framework. This can be attributed to the extra difficulty of expressing the time window constraints of TSPTW. We hope and expect that the formulation presented here will lead to the experimental execution of small-scale TSPTW instances.

## 2. Related Work

#### Previous Work on TSPTW

## 3. The Classical Formulation of the TSPTW

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- minimize the sum of the arc traversal costs along the tour; and
- minimize the time to return to the depot.

**arrival time**at the ith customer and the time at which

**service**starts at the ith customer, which are denoted by ${A}_{{p}_{i}}$ and ${D}_{{p}_{i}}$, respectively. At this point, we make the important observation that ${D}_{{p}_{i}}$ is the

**departure time**from the ith customer in the case of

**zero service time**. The assumption of zero service time is widely used in the literature in order to simplify the problem, and, thus, we too follow this assumption in our presentation.

#### 3.1. Small Scale Examples for the TSPTW

#### 3.1.1. A Four-Node Example

#### 3.1.2. A Five-Node Example

## 4. A QUBO Formulation for the TSPTW

- For each i, $1\le i\le n+1$, exactly one of the binary variables ${x}_{u,v}^{i}$ is 1, where u and v range freely from 0 to n, with the proviso that $u\ne v$. As a matter of fact, when $i=1$ and $i=n+1$, we can be more precise. In the former case, exactly one of the binary variables ${x}_{0,v}^{1}$ is 1, where v ranges from 1 to n, and, in the latter case, exactly one of the binary variables ${x}_{v,0}^{n+1}$ is 1, where v ranges from 1 to n.
- In addition to the above constraints, for each u, $1\le u\le n$, exactly one of the binary variables ${x}_{u,v}^{i}$ is 1, where i ranges from 2 to $n+1$ and v ranges from 1 to n. Obviously, when $i=n+1$, the only relevant decision variable is ${x}_{u,0}^{n+1}$.
- Symmetrically, we also have the constraint that for every v, $1\le v\le n$, exactly one of the binary variables ${x}_{u,v}^{i}$ is 1, where i ranges from 1 to n and u ranges from 1 to n. Again, we point out that, in case $i=1$, the only relevant decision variable is ${x}_{0,v}^{1}$.

**Example**

**1.**

**Definition**

**5.**

**time margin**of the customer at position i is ${l}_{v}-{A}_{{p}_{i}}$.

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QPU | Quantum Processing Unit |

QUBO | Quadratic Unconstrained Binary Optimization |

TSP | Traveling Salesman Problem |

TSPTW | Traveling Salesman Problem with Time Windows |

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**Figure 1.**The above graph depicts the network ${G}_{1}$ consisting of three nodes plus the depot (node 0).

**Figure 2.**The above graph depicts an example of a tour consisting of four nodes plus the depot (node 0).

**Table 1.**Quantitative characteristics (qubits and connectivity) of the Chimera topology of previous D-Wave generations (from [22]).

D-Wave One | D-Wave Two | D-Wave 2X | D-Wave 2000Q | |
---|---|---|---|---|

Size | $4\times 4$ | $8\times 8$ | $12\times 12$ | $16\times 16$ |

Qubits | 128 | 512 | 1152 | 2048 |

Node # | X | Y | Ready Time | Due Date |
---|---|---|---|---|

0 | 2 | 2 | 1 | 30 |

1 | 2 | 3 | 14 | 15 |

2 | 3 | 3 | 12 | 25 |

3 | 3 | 4 | 4 | 5 |

Node 0 | Node 1 | Node 2 | Node 3 | |
---|---|---|---|---|

Node 0 | 0 | 1 | 1.41 | 2.23 |

Node 1 | 1 | 0 | 1 | 1.41 |

Node 2 | 1.41 | 1 | 0 | 1 |

Node 3 | 2.23 | 1.41 | 1 | 0 |

Ordering | Node 0 | Node 3 | Node 2 | Node 1 |
---|---|---|---|---|

cost | 0 < 1 | 3.23 < 4 | 5 < 12 | 13 < 14 |

feasibility | yes | yes | yes | yes |

Node # | X | Y | Ready Time | Due Date |
---|---|---|---|---|

0 | 2 | 2 | 1 | 30 |

1 | 2 | 3 | 14 | 15 |

2 | 3 | 3 | 12 | 25 |

3 | 3 | 4 | 4 | 5 |

4 | 2 | 1 | 8 | 10 |

Node 0 | Node 1 | Node 2 | Node 3 | Node 4 | |
---|---|---|---|---|---|

Node 0 | 0 | 1 | 1.41 | 2.23 | 1 |

Node 1 | 1 | 0 | 1 | 1.41 | 2 |

Node 2 | 1.41 | 1 | 0 | 1 | 2.23 |

Node 3 | 2.23 | 1.41 | 1 | 0 | 3.16 |

Node 4 | 1 | 2 | 2.23 | 3.16 | 0 |

Ordering | Node 0 | Node 3 | Node 4 | Node 2 | Node 1 |
---|---|---|---|---|---|

cost | 0 < 1 | 2.23 < 4 | 7.16 < 8 | 10 < 12 | 12 < 14 |

feasibility | yes | yes | yes | yes | yes |

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## Share and Cite

**MDPI and ACS Style**

Papalitsas, C.; Andronikos, T.; Giannakis, K.; Theocharopoulou, G.; Fanarioti, S.
A QUBO Model for the Traveling Salesman Problem with Time Windows. *Algorithms* **2019**, *12*, 224.
https://doi.org/10.3390/a12110224

**AMA Style**

Papalitsas C, Andronikos T, Giannakis K, Theocharopoulou G, Fanarioti S.
A QUBO Model for the Traveling Salesman Problem with Time Windows. *Algorithms*. 2019; 12(11):224.
https://doi.org/10.3390/a12110224

**Chicago/Turabian Style**

Papalitsas, Christos, Theodore Andronikos, Konstantinos Giannakis, Georgia Theocharopoulou, and Sofia Fanarioti.
2019. "A QUBO Model for the Traveling Salesman Problem with Time Windows" *Algorithms* 12, no. 11: 224.
https://doi.org/10.3390/a12110224