# Indoor Traveling Salesman Problem (ITSP) Path Planning

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Indoor Traveling Salesman Problem (ITSP)

#### 3.1. Concepts and Modeling

#### 3.2. Procedures of ITSP Path Planning

- Step 1: Select intermediate destinations (nodes).Select departure location (${s}_{departure}$) and specified intermediate places (${s}_{intermediate}$). For instance, the ${s}_{departure}={s}_{1}$ and ${s}_{intermediate}=\{{s}_{4},{s}_{7},{s}_{9}\}$.
- Step 2: Compute navigation paths between every two selected nodes.This step takes the departure location and specified intermediate places as nodes to compute navigation paths between any two nodes based on Dijkstra. The lengths of the navigation paths are used as the distances between two nodes. In this paper, we set the $d({s}_{i},{s}_{j})=d({s}_{j},{s}_{i})$, i.e., the paths in this example are symmetric and we simply assume the route between the two intermediate destinations is bi-directional. In reality, $d({s}_{i},{s}_{j})$ may be not equal to $d({s}_{j},{s}_{i})$, e.g., one-way corridors such as escalators. In all cases, $d({s}_{i},{s}_{i})=d({s}_{j},{s}_{j})=0$. Continue the example in Step 1, this step can get six paths: ${s}_{1}\u21dd{s}_{4}$, ${s}_{1}\u21dd{s}_{7}$, ${s}_{1}\u21dd{s}_{9}$, ${s}_{4}\u21dd{s}_{7}$, ${s}_{4}\u21dd{s}_{9}$, ${s}_{7}\u21dd{s}_{9}$, and corresponding reverse paths. The symbol (⇝) means there are zero to several navigation nodes between two nodes. The travel distances are $d({s}_{1},{s}_{4})=d({s}_{4},{s}_{1})$, $d({s}_{1},{s}_{7})=d({s}_{7},{s}_{1})$, $d({s}_{1},{s}_{9})=d({s}_{9},{s}_{1})$, $d({s}_{4},{s}_{7})=d({s}_{7},{s}_{4})$, $d({s}_{4},{s}_{9})=d({s}_{9},{s}_{4})$, $d({s}_{7},{s}_{9})=d({s}_{9},{s}_{7})$.
- Step 3: Set up graph of desired intermediate destinations.Setting up a graph of all desired intermediate destinations is to take the travel distances of every two specified places as weights and all these places as nodes to make an undirected graph. For the example in Step 1, the undirected graph can be organized as a table (Table 1).
- Step 4: Select and sort the navigation paths based on B&B algorithm.Taking the undirected graph as the input, the orders of departure and desired intermediate destinations can be computed based on the B&B algorithm. Then, the orders are further used to select and sort navigation paths. For instance, the order could be <${s}_{1},{s}_{7},{s}_{9},{s}_{4},{s}_{1}$>, which means the following navigation paths will selected and sorted as ${s}_{1}\u21dd{s}_{7}$, ${s}_{7}\u21dd{s}_{9}$, ${s}_{9}\u21dd{s}_{4}$, and ${s}_{4}\u21dd{s}_{1}$.
- Step 5: Combine the navigation results of Dijkstra as the ITSP path.The last step is to combine the navigation results as the ITSP path. For instance, the ITSP path of the example is ${s}_{1}\u21dd{s}_{7}\u21dd{s}_{9}\u21dd{s}_{4}\u21dd{s}_{1}$. Then, the final navigation path becomes ${s}_{1}\to \dots \to {s}_{7}\to \dots \to {s}_{9}\to \dots \to {s}_{4}\to \dots \to {s}_{1}$. The symbol (→) means there is no other navigation node between the two nodes.

#### 3.3. Illustration

## 4. Implementation and Case Study

#### 4.1. Case Description and Data Preparation

#### 4.2. Navigation Network Derivation

#### 4.3. ITSP Path Planning

#### 4.4. Discussion

## 5. Conclusions and Future Work

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An indoor map and its navigation network. (

**a**) Indoor map used for illustration. (

**b**) Navigation network corresponding to the indoor map.

**Figure 2.**(

**a**) The navigation network of the indoor scene. (

**b**) Illustration of the undirected graph of all selected desired destinations.

**Figure 3.**The result of TSP and the optimal navigation path of ITSP. (

**a**) Results of sorted paths. (

**b**) The navigation path of ITSP.

**Figure 4.**Floor plan of the floor utilized for testing in the shopping mall. (

**a**) The floor plan edited in QGIS. (

**b**) The attributes of indoor elements.

**Figure 5.**The floor plan, QR codes, space subdivision, and navigation network derivation of the selected floor. Black squares are QR codes, green lines in the left figure are split lines, blue lines in the right figure are edges of navigation network.

**Figure 6.**The shopping mall used for tests. (

**a**) All the floor plans of the shopping mall, in which the links between different levels are lifts/escalators/stairs. (

**b**) The derived navigation networks.

**Figure 8.**The ITSP path tests on the selected floor. The red star and circle represent the departure location, the blue location tags mark the shops that have stuff for kids/babies, the blue lines are the navigation paths, and the pink dots in the navigation path are the vertex of path segments. (

**a**) Departure is ‘LiftC_L5’; (

**b**) Departure is ‘EscalatorB_L5’.

**Figure 10.**The ITSP path tests on multi-floors. The red star represent the departure location, the blue location tags mark the shops that the customer is interested. (

**a**) Only Lifts; (

**b**) Stairs, escalators, and lifts.

${\mathit{s}}_{1}$ | ${\mathit{s}}_{4}$ | ${\mathit{s}}_{7}$ | ${\mathit{s}}_{9}$ | |
---|---|---|---|---|

${s}_{1}$ | 0 | $d({s}_{1},{s}_{4})$ | $d({s}_{1},{s}_{7})$ | $d({s}_{1},{s}_{9})$ |

${s}_{4}$ | $d({s}_{4},{s}_{1})$ | 0 | $d({s}_{4},{s}_{7})$ | $d({s}_{4},{s}_{9})$ |

${s}_{7}$ | $d({s}_{7},{s}_{1})$ | $d({s}_{7},{s}_{4})$ | 0 | $d({s}_{7},{s}_{9})$ |

${s}_{9}$ | $d({s}_{9},{s}_{1})$ | $d({s}_{9},{s}_{4})$ | $d({s}_{9},{s}_{7})$ | 0 |

Start | End | Path | Distance |
---|---|---|---|

R7 | R1 | R7 → N0 → N19 → N16 → N1 → R1 | 12.19 |

R7 | R2 | R7 → N0 → N7 → N13 → N2 → R2 | 13.24 |

R7 | R5 | R7 → N0 → N7 → N10 → N6 → N5 → R5 | 18.43 |

R7 | ATM | R7 → N0 → N19 → ATM | 8.19 |

R1 | R2 | R1 → N1 → N22 → N13 → N2 → R2 | 10.45 |

R1 | R5 | R1 → N1 → N22 → N13 → N10 → N6 → N5 → R5 | 19.81 |

R1 | ATM | R1 → N1 → N16 → ATM | 7.70 |

R2 | R5 | R2 → N2 → N10 → N6 → N5 → R5 | 14.50 |

R2 | ATM | R2 → N2 → N13 → N22 → N21 → N16 → ATM | 11.04 |

R5 | ATM | R5 → N5 → N6 → N10 → N7 → N20 → N19 → ATM | 18.73 |

R7 | R1 | R2 | R5 | ATM | |
---|---|---|---|---|---|

R7 | 0 | 12.19 | 13.24 | 18.43 | 8.19 |

R1 | 12.19 | 0 | 10.45 | 19.81 | 7.70 |

R2 | 13.24 | 10.45 | 0 | 14.50 | 11.04 |

R5 | 18.43 | 19.81 | 14.50 | 0 | 18.73 |

ATM | 8.19 | 7.70 | 11.04 | 18.73 | 0 |

Id | Name | Category |
---|---|---|

L5_6 | Sheriden | Baby and nursery, Clothing and accessories, Home |

L5_16 | Adairs | Baby and nursery, Home |

L5_21 | Bed Bath N’ Table | Clothing and accessories, Baby and nursery, Home |

L5_22 | Priceline | Baby and nursery, Discount and variety, Health and fitness |

L5_25 | Adairs Kids | Clothing and accessories, Home, Toys and hobbies stores |

L5_26 | Kidstuff | Books, stationary and gifts, Entertainment and activities, Sporting goods stores, Toys and hobbies stores |

L5_28 | Stokke | Baby and nursery |

L5_31 | Bonds Kids | Clothing and accessories |

L5_36 | Seed Kids | Books, stationary and gifts |

L5_37 | Purebaby | Baby and nursery, Clothing and accessories |

L5_38 | Cotton on Kids | Clothing and accessories |

L5_39 | Target | Baby and nursery, Toys and hobbies stores |

Departure | TSP Results |
---|---|

LiftC_L5 | LiftC_L5 ⇝ Seed Kids ⇝ Target ⇝ Kidstuff ⇝ Adairs Kids ⇝ Stokke ⇝ Bonds Kids ⇝ Priceline ⇝ Bed Bath N’ Table ⇝ Sheriden ⇝ Adairs ⇝ Purebaby ⇝ Cotton on Kids ⇝ LiftC_L5 |

EscalatorB_L5 | EscalatorB_L5 ⇝ Sheriden ⇝ Adairs ⇝ Seed Kids ⇝ Purebaby ⇝ Cotton on Kids ⇝ Target ⇝ Kidstuff ⇝ Adairs Kids ⇝ Stokke ⇝ Bonds Kids ⇝ Priceline ⇝ Bed Bath N’ Table ⇝ EscalatorB_L5 |

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## Share and Cite

**MDPI and ACS Style**

Yan, J.; Zlatanova, S.; Lee, J.; Liu, Q.
Indoor Traveling Salesman Problem (ITSP) Path Planning. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 616.
https://doi.org/10.3390/ijgi10090616

**AMA Style**

Yan J, Zlatanova S, Lee J, Liu Q.
Indoor Traveling Salesman Problem (ITSP) Path Planning. *ISPRS International Journal of Geo-Information*. 2021; 10(9):616.
https://doi.org/10.3390/ijgi10090616

**Chicago/Turabian Style**

Yan, Jinjin, Sisi Zlatanova, Jinwoo (Brian) Lee, and Qingxiang Liu.
2021. "Indoor Traveling Salesman Problem (ITSP) Path Planning" *ISPRS International Journal of Geo-Information* 10, no. 9: 616.
https://doi.org/10.3390/ijgi10090616