# Searching Strategies with Low Computational Costs for Multiple-Vehicle Bike Sharing System Routing Problem

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

- (1)
- The vehicles can load bicycles at the depot, because the depot has spare bicycles available.
- (2)
- All vehicles must leave the depot and return to the depot.
- (3)
- The number of bicycles does not change during the repositioning process.
- (4)
- Each port can be visited only once by one vehicle.

#### 2.1. Multiple-Vehicle Bike Sharing System Routing Problem (mBSSRP)

- (i)
- time limit constraint: vehicles must return to the depot within the time limit.
- (ii)
- capacity constraint: vehicles cannot load more bicycles than their capacity, and the capacity of a vehicle cannot be less than zero.

#### 2.2. Multiple-Vehicle Bike Sharing System Routing Problem with Soft Constraints (mBSSRP-S)

## 3. Heuristic Methods

#### 3.1. Construction Method for Generating an Initial Solution

- We randomly select p ports, where p denotes the number of available vehicles.
- Each vehicle constructs a subtour with the depot and one of the selected p ports.
- An unvisited port k, which is the farthest from the depot, is selected.
- The port k is inserted between ports i and j in the subtour where ${\mathrm{\Delta}}_{ikj}={t}_{ik}+{t}_{kj}-{t}_{ij}$ is minimized.
- Steps 3 and 4 are repeated until all ports are visited once by one vehicle.

#### 3.2. Local Search Methods for Improving the Initial Solution

#### 3.3. Method by Using the Tabu Search

#### 3.4. Dynamically Changing Weight of Penalties

## 4. Computational Cost Reduction Method

## 5. Numerical Experiments

#### 5.1. Results of Search Strategy before Obtaining a Feasible Solution

#### 5.2. Results of Search Strategy after Obtaining a Feasible Solution

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Flow of the proposed framework [20]. First, we construct an initial tour using the construction method. Then, we improve the initial tour using the inserting and the swapping method. Next, the Or-opt and the CROSS-exchange are controlled by the tabu search and aim to transition to a feasible solution of mBSSRP. The red rectangle corresponds to the computational cost reduction part. The details are described in Section 4. After feasible solutions of mBSSRP are obtained in the tabu search process, the 2-opt is executed. If we cannot obtain a feasible solution, the 2-opt method is not executed. Finally, to obtain better solutions, we executed the inserting and the swapping method again.

**Figure 2.**Example of the inserting method. Black squares represent depots and gray circles represent ports. (

**a**) A tour before the inserting method is executed. (

**b**) A partial tour $i-j$ is inserted into $k-l$ as the normal order (the orange rectangle). (

**c**) The partial tour $i-j$ is inserted into $k-l$ as the reverse order (the green rectangle).

**Figure 3.**Example of the swapping method. Black squares represent depots and gray circles represent ports. (

**a**) A tour before the swapping method is executed. (

**b**) Both partial tours $i-j$ and $k-l$ are exchanged with the normal order (the orange rectangles). (

**c**) Only a partial tour $i-j$ is exchanged in the reverse order (the green rectangle). (

**d**) Only a partial tour $k-l$ is exchanged in the reverse order (the green rectangle). (

**e**) Both partial tours $i-j$ and $k-l$ are exchanged with reverse order (the green rectangles).

**Figure 4.**Example of the Or-opt. Black squares represent depots and gray circles represent ports. The red and blue lines represent each tour. (

**a**) Tours before the Or-opt is executed. (

**b**) A partial tour $i-j$ is inserted into another tour $k-l$ in the normal order (the orange rectangle). (

**c**) The partial tour $i-j$ is inserted into another tour $k-l$ in the reverse order (the green rectangle).

**Figure 5.**Example of the CROSS-exchange. Black squares represent depots and gray circles represent ports. Orange rectangles express normal order and green rectangles express a reverse order. Red and blue lines represent each tour. (

**a**) Tours before the CROSS-exchange is executed. (

**b**) Partial tours $i-j$ and $k-l$ are exchanged with the normal order. (

**c**) Only the partial tour $k-l$ is exchanged in the reverse order. (

**d**) Only the partial tour $i-j$ is exchanged in reverse order. (

**e**) Partial tours $i-j$ and $k-l$ are exchanged in the reverse order.

**Figure 6.**Example of the 2-opt method. Black squares represent depots and gray circles represent ports. (

**a**) A tour before the 2-opt method is executed. (

**b**) A tour after the 2-opt method is executed.

**Figure 7.**An example of instance. A black square represents a depot, red circles represent pickup ports and blue circles represent delivery ports.

**Figure 8.**Relationship between time until a feasible solution is firstly obtained and the number of obtained feasible solutions. (

**a**) Search method 1A-S (search for all neighborhood solutions of the Or-opt and the CROSS-exchange). (

**b**) Search method 1C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange).

**Figure 9.**Transition of the best solutions found in case of instance No.1. The red lines show the transition of the best feasible solution from 50 initial feasible solutions, whereas the blue lines show their average. (

**a**) Search method 2C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with soft constraints). (

**b**) Search method 2C-H (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with hard constraints).

Paper | Static or Dynamic | Maximum Number of Port Size |
---|---|---|

Rainer-Harbach et al. (2013) | static | 90 |

Raidl et al. (2013) | static | 90 |

Papazek et al. (2013) | static | 700 |

Kloimüllner et al. (2014) | dynamic | 90 |

Dell’Amico et al. (2014) | static | 116 |

Dell’Amico et al. (2016) | static | 564 |

Chiariotti et al. (2018) | dynamic | 280 |

This study | static | 100 |

**Table 2.**Search strategy before finding a feasible solution of the mBSSRP. The first column shows local search names, the second column shows the types of the Or-opt or the CROSS-exchange operations, the third to fifth columns show the name of the search method and the sixth column shows the percentage of local search methods that are executed.

Local Search | Type | 1A-S | 1B-S | 1C-S | Executed Rate (One Decimal Point Omitted) |
---|---|---|---|---|---|

Or-opt | Normal | ✓ | ✓ | ✓ | 66% |

Reverse | ✓ | 4% | |||

CROSS-exchange | Normal | ✓ | ✓ | 20% | |

One-sided reverse | ✓ | 2% | |||

One-sided reverse | ✓ | 3% | |||

Reverse | ✓ | 4% |

**Table 3.**Search strategy after finding a feasible solution of the mBSSRP. The first column represents local search names, the second column shows the types of the Or-opt or the CROSS-exchange operations, the third to the eighth columns show the name of the search method and the bottom line indicates whether instances were solved with hard or soft constraints.

Local Search | Type | 2A-S | 2A-H | 2B-S | 2B-H | 2C-S | 2C-H |
---|---|---|---|---|---|---|---|

Or-opt | Normal | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

Reverse | ✓ | ✓ | |||||

CROSS-exchange | Normal | ✓ | ✓ | ✓ | ✓ | ||

One-sided reverse | ✓ | ✓ | |||||

One-sided reverse | ✓ | ✓ | |||||

Reverse | ✓ | ✓ | |||||

Constraint | Soft | Hard | Soft | Hard | Soft | Hard |

**Table 4.**Results of search methods before obtaining a feasible solution. (

**a**) Search method 1A-S (search for all neighborhood solutions of the Or-opt and the CROSS-exchange). (

**b**) Search method 1B-S (search only normal order insertion of the Or-opt). (

**c**) Search method 1C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange).

(a) | ||||

No. | Avg. [min] | Best [min] | Worst [min] | # of Feasible Solution |

1 | 243.76 | 230.85 | 259.07 | 50 |

2 | 238.16 | 230.87 | 252.95 | 50 |

3 | 228.47 | 218.98 | 237.25 | 50 |

4 | 242.56 | 232.38 | 252.36 | 50 |

5 | 249.06 | 240.95 | 260.69 | 50 |

6 | 270.81 | 260.50 | 284.04 | 31 |

7 | 238.52 | 225.12 | 252.72 | 34 |

8 | 213.15 | 207.43 | 226.01 | 50 |

9 | 241.64 | 231.13 | 255.18 | 49 |

10 | 243.28 | 234.40 | 252.54 | 45 |

Avg. | 240.94 | 231.26 | 253.28 | 45.9 |

(b) | ||||

No. | Avg. [min] | Best [min] | Worst [min] | # of Feasible Solution |

1 | 245.46 | 234.80 | 258.63 | 40 |

2 | 242.59 | 228.66 | 260.92 | 50 |

3 | 234.57 | 222.44 | 249.94 | 50 |

4 | 246.81 | 236.23 | 268.79 | 49 |

5 | 254.91 | 241.88 | 269.26 | 42 |

6 | 268.45 | 260.94 | 276.57 | 16 |

7 | 238.02 | 222.71 | 247.85 | 10 |

8 | 213.98 | 207.79 | 237.93 | 49 |

9 | 245.18 | 233.40 | 260.38 | 44 |

10 | 244.20 | 236.92 | 257.74 | 29 |

Avg. | 243.42 | 232.58 | 258.80 | 37.9 |

(c) | ||||

No. | Avg. [min] | Best [min] | Worst [min] | # of Feasible Solution |

1 | 241.63 | 233.74 | 251.90 | 50 |

2 | 236.61 | 227.33 | 246.43 | 50 |

3 | 228.31 | 217.29 | 242.36 | 50 |

4 | 242.15 | 233.15 | 254.30 | 50 |

5 | 249.59 | 241.04 | 259.27 | 50 |

6 | 269.55 | 256.99 | 284.94 | 32 |

7 | 236.17 | 225.00 | 252.39 | 30 |

8 | 213.14 | 205.85 | 224.93 | 50 |

9 | 239.44 | 228.72 | 252.01 | 50 |

10 | 239.80 | 229.91 | 253.65 | 44 |

Avg. | 239.64 | 229.90 | 252.22 | 45.6 |

**Table 5.**Results of the t-test to evaluate the difference of the average objective function values [min] between the search methods 1A-S and 1C-S.

1A-S | 1C-S | ||||
---|---|---|---|---|---|

No. | Avg. [min] | SD | Avg. [min] | SD | p |

1 | 243.76 | 5.72 | 241.63 | 4.51 | 0.041 |

2 | 238.16 | 5.08 | 236.61 | 4.34 | 0.104 |

3 | 228.47 | 4.36 | 228.31 | 6.42 | 0.886 |

4 | 242.56 | 4.92 | 242.15 | 4.54 | 0.668 |

5 | 249.06 | 4.64 | 249.59 | 4.38 | 0.558 |

6 | 270.81 | 6.42 | 269.55 | 7.13 | 0.464 |

7 | 238.52 | 6.27 | 236.17 | 6.96 | 0.163 |

8 | 213.15 | 3.39 | 213.14 | 4.00 | 0.985 |

9 | 241.64 | 6.19 | 239.44 | 5.25 | 0.061 |

10 | 243.28 | 4.62 | 239.80 | 5.14 | 0.001 |

Avg. | 240.94 | 5.16 | 239.64 | 5.27 | 0.393 |

**Table 6.**Results of the t-test to evaluate the difference time [s] until a feasible solution was obtained between the search methods 1A-S and 1C-S.

1A-S | 1C-S | ||||
---|---|---|---|---|---|

No. | Avg. [s] | SD | Avg. [s] | SD | p |

1 | 78.76 | 24.84 | 44.67 | 16.83 | 0.000 |

2 | 72.83 | 16.30 | 32.84 | 5.60 | 0.000 |

3 | 66.37 | 10.68 | 30.22 | 5.79 | 0.000 |

4 | 75.37 | 13.50 | 36.85 | 9.72 | 0.000 |

5 | 72.86 | 10.15 | 42.23 | 15.70 | 0.000 |

6 | 123.70 | 41.62 | 109.59 | 45.82 | 0.205 |

7 | 115.36 | 37.28 | 89.80 | 45.26 | 0.018 |

8 | 68.81 | 19.88 | 32.74 | 8.61 | 0.000 |

9 | 75.99 | 16.51 | 42.16 | 16.41 | 0.000 |

10 | 102.82 | 39.51 | 84.23 | 42.42 | 0.035 |

Avg. | 85.29 | 23.03 | 54.53 | 21.22 | 0026 |

**Table 7.**Results of search methods after obtaining a feasible solution. (

**a**) Search method 2A-S (search for all neighborhood solutions of the Or-opt and the CROSS-exchange with soft constraints). (

**b**) Search method 2A-H (search for all neighborhood solutions of the Or-opt and the CROSS-exchange with hard constraints). (

**c**) Search method 2B-S (search for only normal order insertion of the Or-opt with soft constraints). (

**d**) Search method 2B-H (search for only normal order insertion of the Or-opt with hard constraints). (

**e**) Search method 2C-S (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with soft constraints). (

**f**) Search method 2C-H (search for normal order insertion of the Or-opt and normal order exchange of the CROSS-exchange with hard constraints).

(a) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 241.81 | 232.13 | 254.94 |

2 | 238.45 | 227.14 | 245.73 |

3 | 228.09 | 218.60 | 244.40 |

4 | 241.43 | 231.95 | 251.71 |

5 | 247.58 | 240.39 | 256.60 |

6 | 266.76 | 257.19 | 281.07 |

7 | 234.84 | 222.53 | 251.80 |

8 | 212.02 | 205.77 | 220.14 |

9 | 240.48 | 232.09 | 251.94 |

10 | 241.65 | 230.99 | 256.75 |

Avg. | 239.31 | 229.88 | 251.51 |

(b) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 242.53 | 235.06 | 251.29 |

2 | 237.70 | 229.25 | 247.47 |

3 | 227.98 | 219.98 | 237.32 |

4 | 241.36 | 231.52 | 250.18 |

5 | 248.68 | 241.96 | 258.93 |

6 | 268.91 | 253.71 | 286.59 |

7 | 235.25 | 222.83 | 255.66 |

8 | 211.91 | 205.99 | 218.02 |

9 | 239.90 | 227.68 | 253.67 |

10 | 239.90 | 231.80 | 252.12 |

Avg. | 239.41 | 229.98 | 251.13 |

(c) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 244.20 | 232.75 | 259.46 |

2 | 239.29 | 228.66 | 249.19 |

3 | 227.79 | 217.16 | 238.15 |

4 | 242.41 | 234.88 | 254.57 |

5 | 251.11 | 242.17 | 264.62 |

6 | 270.60 | 253.39 | 284.24 |

7 | 236.22 | 226.03 | 251.93 |

8 | 211.75 | 206.02 | 223.83 |

9 | 243.22 | 231.37 | 258.92 |

10 | 241.93 | 230.33 | 258.83 |

Avg. | 240.85 | 230.28 | 254.37 |

(d) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 248.89 | 236.91 | 270.39 |

2 | 241.65 | 228.40 | 258.26 |

3 | 230.96 | 221.46 | 244.89 |

4 | 246.12 | 230.94 | 263.21 |

5 | 252.34 | 240.32 | 267.20 |

6 | 275.42 | 263.31 | 286.62 |

7 | 242.98 | 225.28 | 262.27 |

8 | 212.27 | 206.68 | 222.78 |

9 | 243.85 | 233.61 | 259.85 |

10 | 249.20 | 231.04 | 266.57 |

Avg. | 244.37 | 231.79 | 260.20 |

(e) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 240.88 | 232.42 | 254.18 |

2 | 235.86 | 228.96 | 243.84 |

3 | 226.88 | 216.81 | 235.18 |

4 | 240.18 | 231.96 | 254.29 |

5 | 246.75 | 241.68 | 254.52 |

6 | 264.58 | 255.89 | 276.92 |

7 | 234.09 | 222.51 | 254.30 |

8 | 211.11 | 205.87 | 216.95 |

9 | 237.62 | 227.48 | 249.52 |

10 | 238.92 | 230.84 | 249.83 |

Avg. | 237.69 | 229.44 | 248.95 |

(f) | |||

No. | Avg. [min] | Best [min] | Worst [min] |

1 | 241.36 | 232.87 | 255.56 |

2 | 236.58 | 227.65 | 249.19 |

3 | 225.96 | 216.81 | 233.52 |

4 | 239.99 | 231.78 | 247.36 |

5 | 247.18 | 240.11 | 255.19 |

6 | 267.60 | 254.18 | 281.03 |

7 | 235.12 | 222.29 | 255.66 |

8 | 210.42 | 206.12 | 215.98 |

9 | 238.72 | 230.49 | 247.28 |

10 | 237.86 | 230.42 | 253.19 |

Avg. | 238.08 | 229.27 | 249.40 |

**Table 8.**Results of the t-test to evaluate the difference of performance between methods 2C-S and 2C-H.

2C-S | 2C-H | ||||
---|---|---|---|---|---|

No. | Avg. [min] | SD | Avg. [min] | SD | p |

1 | 240.88 | 4.20 | 241.36 | 4.41 | 0.572 |

2 | 235.86 | 3.89 | 236.58 | 5.18 | 0.432 |

3 | 226.88 | 4.64 | 225.96 | 4.39 | 0.309 |

4 | 240.18 | 4.74 | 239.99 | 3.66 | 0.833 |

5 | 246.75 | 3.26 | 247.18 | 3.92 | 0.550 |

6 | 264.58 | 5.27 | 267.60 | 6.45 | 0.016 |

7 | 234.09 | 7.27 | 235.12 | 8.14 | 0.518 |

8 | 211.11 | 2.86 | 210.42 | 2.54 | 0.209 |

9 | 237.62 | 5.09 | 238.72 | 4.64 | 0.262 |

10 | 238.92 | 3.91 | 237.86 | 4.27 | 0.203 |

Avg. | 237.69 | 4.51 | 238.08 | 4.76 | 0.390 |

No. | Avg. [s] |
---|---|

1 | 38.82 |

2 | 31.54 |

3 | 31.71 |

4 | 36.01 |

5 | 34.17 |

6 | 77.78 |

7 | 68.04 |

8 | 29.10 |

9 | 35.56 |

10 | 59.94 |

Avg. | 44.27 |

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## Share and Cite

**MDPI and ACS Style**

Tsushima, H.; Matsuura, T.; Ikeguchi, T.
Searching Strategies with Low Computational Costs for Multiple-Vehicle Bike Sharing System Routing Problem. *Appl. Sci.* **2022**, *12*, 2675.
https://doi.org/10.3390/app12052675

**AMA Style**

Tsushima H, Matsuura T, Ikeguchi T.
Searching Strategies with Low Computational Costs for Multiple-Vehicle Bike Sharing System Routing Problem. *Applied Sciences*. 2022; 12(5):2675.
https://doi.org/10.3390/app12052675

**Chicago/Turabian Style**

Tsushima, Honami, Takafumi Matsuura, and Tohru Ikeguchi.
2022. "Searching Strategies with Low Computational Costs for Multiple-Vehicle Bike Sharing System Routing Problem" *Applied Sciences* 12, no. 5: 2675.
https://doi.org/10.3390/app12052675