# A Critical Analysis of a Tourist Trip Design Problem with Time-Dependent Recommendation Factors and Waiting Times

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## Abstract

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## 1. Introduction

- To propose a TTDP model with time-dependent recommendation factors, taking (and not taking) into account waiting times.
- To solve a set of test instances, in order to gain insights regarding the empirical behavior of the models and to provide reference values for future research.
- To evaluate whether or not considering waiting times provides any benefit in the tour’s overall interest.

## 2. Models for Time-Dependent TTDP

- $V=\{1,2,\dots ,n\}$: set of nodes (including all considered POIs) where 1 and n are the start and end nodes of the route, respectively. All nodes in between are POIs that may be included in the route.
- T: number of time periods.
- ${S}_{i}$: score of POI i.
- ${f}_{it}$: recommendation factor of POI i in period t.
- ${T}_{max}$: maximum duration time for the itinerary.
- ${t}_{ij}$: travel time (or distance) from POI i to POI j.
- ${v}_{i}$: visiting time at POI i.
- ${b}_{t}$: starting time of period t.
- ${e}_{t}$: ending time of period t.
- M: a very large constant.

- ${a}_{i}\in [0,{T}_{max}]$: arrival time at node i.
- ${X}_{ij}\in \{0,1\}$: 1 if there is a path from POI i to POI j in the route, 0 otherwise.
- ${Y}_{it}\in \{0,1\}$: 1 if the visit to the POI i starts in period t, 0 otherwise.

#### 2.1. TTDP-TDRF with Waiting Times

#### 2.2. TTDP-TDRF without Waiting Times

## 3. Description of the Test Instances (Samples)

## 4. Description of Computational Experiments

- To solve the 27 test instances under the two models considered (WT, NWT), thus obtaining the optimal solutions or reference values for future experimentation.
- To evaluate the implications of including or not including waiting times in the model considering:
- the quality of the solutions,
- the effort needed to obtain them,
- the similarity/differences between solutions.

WT | 18 | 22 | 12 | 11 | 28 | 20 | 9 | 15 | 30 | 21 | 4 | 3 | |||

NWT | 12 | 18 | 19 | 28 | 11 | 20 | 9 | 21 | 15 | 4 | 3 |

WT | 18 | 12 | 11 | 28 | 20 | 9 | 15 | 21 | 4 | 3 | ||||

NWT | 12 | 18 | 28 | 11 | 20 | 9 | 21 | 15 | 4 | 3 |

WT | 12 | 18 | 28 | 11 | 20 | 9 | 21 | 15 | 4 | 3 | |||||

NWT | 12 | 18 | 28 | 11 | 20 | 9 | 21 | 15 | 4 | 3 |

## 5. Analysis of Results

## 6. Discussion and Conclusions

- Although solutions obtained with the WT model are theoretically better than those obtained with the NWT model, in 78% of the test instances the best scores from both models are the same.
- In almost all cases, the solver obtained the solutions to the NWT model faster than to the WT model.
- The similarity analysis revealed that the solutions to both models are quite similar.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

No. | $\left|\mathit{V}\right|$ | Best Value | Execution Time | No. POIs | Travel Time | Visiting Time | Total Time |
---|---|---|---|---|---|---|---|

1 | 10 | 53 | 0.36 | 10 | 41.02 | 371 | 412.02 |

2 | 10 | 42 | 0.17 | 10 | 38.53 | 410 | 448.53 |

3 | 10 | 65 | 0.98 | 10 | 32.15 | 357 | 389.15 |

4 | 20 | 74.75 | 2324.51 | 12 | 35.93 | 443 | 478.93 |

5 | 20 | 83 | 2.80 | 13 | 32.33 | 444 | 476.33 |

6 | 20 | 85.5 | 99.71 | 11 | 30.13 | 449 | 479.13 |

7 | 30 | 85 | 965.87 | 13 | 51.92 | 428 | 479.92 |

8 | 30 | 106 | 25.97 | 14 | 42.57 | 437 | 479.57 |

9 | 30 | 96 | 2.14 | 12 | 33.40 | 440 | 473.40 |

10 | 40 | 111 | 465.69 | 13 | 39.67 | 436 | 475.67 |

11 | 40 | 96 | 156.00 | 12 | 46.95 | 432 | 478.95 |

12 | 40 | 112.75 | 171.67 | 13 | 31.48 | 445 | 476.48 |

13 | 50 | 114 | 154.23 | 13 | 39.12 | 440 | 479.12 |

14 | 50 | 112.5 | 3524.52 | 13 | 40.67 | 439 | 479.67 |

15 | 50 | 117.75 | 3600 | 14 | 35.50 | 444 | 479.50 |

16 | 60 | 113 | 156.37 | 14 | 30.65 | 448 | 478.65 |

17 | 60 | 112 | 88.82 | 14 | 27.87 | 451 | 478.87 |

18 | 60 | 116 | 159.27 | 15 | 42.95 | 437 | 479.95 |

19 | 70 | 124 | 1706.51 | 14 | 41.85 | 438 | 479.85 |

20 | 70 | 131 | 364.93 | 15 | 43.05 | 434 | 477.10 |

21 | 70 | 123.25 | 1119.97 | 14 | 42.78 | 436 | 478.78 |

22 | 80 | 131 | 3600 | 15 | 42.68 | 436 | 478.68 |

23 | 80 | 132 | 3600 | 15 | 39.85 | 440 | 479.85 |

24 | 80 | 130 | 3600 | 15 | 45.72 | 434 | 479.72 |

25 | 90 | 126 | 3600 | 15 | 47.77 | 432 | 479.77 |

26 | 90 | 133 | 3600 | 15 | 44.63 | 426 | 470.63 |

27 | 90 | 134 | 3600 | 15 | 54.07 | 423 | 477.07 |

No. | $\left|\mathit{V}\right|$ | Best Value | Execution Time | No. POIs | Travel Time | Visiting Time | Total Time |
---|---|---|---|---|---|---|---|

1 | 10 | 53 | 0.26 | 10 | 47.56 | 371 | 418.57 |

2 | 10 | 42 | 0.16 | 10 | 40.87 | 410 | 450.87 |

3 | 10 | 62 | 1.42 | 10 | 35.20 | 357 | 392.20 |

4 | 20 | 74.75 | 15.01 | 12 | 31.45 | 448 | 479.45 |

5 | 20 | 83 | 0.77 | 13 | 32.33 | 444 | 476.33 |

6 | 20 | 85.5 | 12.90 | 11 | 30.88 | 449 | 479.88 |

7 | 30 | 85 | 855.91 | 13 | 51.92 | 428 | 479.92 |

8 | 30 | 106 | 6.81 | 14 | 42.05 | 437 | 479.05 |

9 | 30 | 96 | 2.14 | 11 | 34.13 | 439 | 473.13 |

10 | 40 | 111 | 85.18 | 13 | 38.70 | 441 | 479.70 |

11 | 40 | 95.25 | 75.98 | 12 | 46.57 | 432 | 478.57 |

12 | 40 | 112.75 | 39.48 | 13 | 28.87 | 451 | 479.87 |

13 | 50 | 112 | 3308.19 | 13 | 39.52 | 440 | 479.52 |

14 | 50 | 112.5 | 305.14 | 13 | 40.67 | 439 | 479.67 |

15 | 50 | 117.75 | 3600 | 14 | 35.50 | 444 | 479.50 |

16 | 60 | 113 | 84.62 | 14 | 31.60 | 447 | 478.60 |

17 | 60 | 112 | 7.63 | 14 | 28.53 | 451 | 479.53 |

18 | 60 | 116 | 37.66 | 15 | 40.70 | 437 | 477.70 |

19 | 70 | 124 | 620.99 | 14 | 41.53 | 438 | 479.53 |

20 | 70 | 131 | 306.80 | 15 | 45.18 | 434 | 479.18 |

21 | 70 | 123.25 | 512.30 | 14 | 43.27 | 436 | 479.27 |

22 | 80 | 131 | 3600 | 15 | 42.08 | 437 | 479.08 |

23 | 80 | 133 | 434.04 | 15 | 34.70 | 445 | 479.70 |

24 | 80 | 130 | 1717.90 | 15 | 45.97 | 434 | 479.97 |

25 | 90 | 126 | 3600 | 15 | 46.70 | 432 | 478.70 |

26 | 90 | 131 | 3600 | 15 | 42.77 | 432 | 470.63 |

27 | 90 | 133 | 3600 | 16 | 40.98 | 439 | 479.98 |

No. | No. POIs of the Instance | No. POIs of the route (NWT) | No. POIs of the route (WT) | Similarity |
---|---|---|---|---|

1 | 10 | 10 | 10 | 0.40 |

2 | 10 | 10 | 10 | 0.70 |

3 | 10 | 10 | 10 | 0.30 |

4 | 20 | 12 | 12 | 0.42 |

5 | 20 | 13 | 13 | 1.00 |

6 | 20 | 11 | 11 | 0.73 |

7 | 30 | 13 | 13 | 0.92 |

8 | 30 | 14 | 14 | 0.79 |

9 | 30 | 11 | 12 | 0.50 |

10 | 40 | 13 | 13 | 0.38 |

11 | 40 | 12 | 12 | 0.67 |

12 | 40 | 13 | 13 | 0.46 |

13 | 50 | 13 | 13 | 0.92 |

14 | 50 | 13 | 13 | 1.00 |

15 | 50 | 14 | 14 | 1.00 |

16 | 60 | 14 | 14 | 0.64 |

17 | 60 | 14 | 14 | 0.79 |

18 | 60 | 15 | 15 | 0.60 |

19 | 70 | 14 | 14 | 0.71 |

20 | 70 | 15 | 15 | 0.53 |

21 | 70 | 14 | 14 | 0.71 |

22 | 80 | 15 | 15 | 0.67 |

23 | 80 | 15 | 15 | 0.53 |

24 | 80 | 15 | 15 | 0.73 |

25 | 90 | 15 | 15 | 0.60 |

26 | 90 | 15 | 15 | 0.20 |

27 | 90 | 16 | 15 | 0.31 |

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**Figure 1.**Illustrative example: itinerary (

**A**) has no waiting time and itinerary (

**B**) has waiting time.

**Figure 2.**Comparison of best values and execution time for every instance under WT and NWT models. (

**a**) Best values; (

**b**) Execution time.

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**MDPI and ACS Style**

Porras, C.; Pérez-Cañedo, B.; Pelta, D.A.; Verdegay, J.L.
A Critical Analysis of a Tourist Trip Design Problem with Time-Dependent Recommendation Factors and Waiting Times. *Electronics* **2022**, *11*, 357.
https://doi.org/10.3390/electronics11030357

**AMA Style**

Porras C, Pérez-Cañedo B, Pelta DA, Verdegay JL.
A Critical Analysis of a Tourist Trip Design Problem with Time-Dependent Recommendation Factors and Waiting Times. *Electronics*. 2022; 11(3):357.
https://doi.org/10.3390/electronics11030357

**Chicago/Turabian Style**

Porras, Cynthia, Boris Pérez-Cañedo, David A. Pelta, and José L. Verdegay.
2022. "A Critical Analysis of a Tourist Trip Design Problem with Time-Dependent Recommendation Factors and Waiting Times" *Electronics* 11, no. 3: 357.
https://doi.org/10.3390/electronics11030357