Bike-Sharing Static Rebalancing by Considering the Collection of

Hindawi Journal of Advanced Transportation Volume 2018, Article ID 8086378, 18 pages https://doi.org/10.1155/2018/8086378

Research Article Bike-Sharing Static Rebalancing by Considering the Collection of Bicycles in Need of Repair Sheng Zhang , Guanhua Xiang, and Zhongxiang Huang School of Traffic and Transportation Engineering, Changsha University of Science & Technology, Changsha 410114, China Correspondence should be addressed to Zhongxiang Huang; [email protected] Received 16 May 2018; Revised 23 July 2018; Accepted 28 August 2018; Published 13 September 2018 Academic Editor: Zhiyuan Liu Copyright © 2018 Sheng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Bike-sharing systems, which are used in many cities worldwide, need to maintain a balance between the availability of bicycles and the availability of unoccupied bicycle slots. This paper presents an investigation of the net flow of each bike-sharing station in Jersey City. The data was recorded at 1-minute intervals. The sum of the initial bicycle number and the minimum net flow value was determined to be the demand for static rebalancing, and this led to the proposal of a bike-sharing demand prediction method based on autoregressive integrated moving average models. Considering that the existence of bicycles in a state of disrepair may adversely affect demand prediction and routine planning, we present an integer linear programming formulation to model bike-sharing static rebalancing. The proposed formulation takes into account the problem introduced by the need to collect bicycles in need of repair. A hybrid Discrete Particle Swarm Optimization (DPSO) algorithm was proposed to solve the model, which incorporates a reduced variable neighborhood search (RVNS) functionality together with DPSO to improve the global optimization performance. The effectiveness of the algorithms was verified by a detailed numerical example.

1. Introduction Bike-sharing systems are designed to solve “the first and the last mile problem” and to provide a connection between multitransit modes. The most recent decade has witnessed the rapid worldwide development of bicycles as a transport mode driven by bike-sharing companies, such as Mobike and OFO. In this arrangement, the residents can rent or return bicycles at any public area, and this has greatly enhanced the convenience in the residents’ daily commutes. These systems have since spread all over the world. In contrast, the rapid development of bike-sharing has resulted in many problems, such as damaged bicycles, bike parking needs, and bicycle maintenance. In many cities, piles of bicycles are parked along public areas, and broken bicycles tend to be left carelessly. Among these violations, the parking problem is especially critical and has aroused people’s concern. These phenomena occur mainly because the operation management is not standardized and there is no effective rebalancing method. Scholars and policy makers are committed to determining the causes and solution of these problems. Reinforcing the standardization of the bike-sharing system management

is essential; however, more importantly, advanced methods should be used to rebalance the stations to their optimum levels of occupation to avoid the inefficiency due to full and/or empty stations. Today’s competitive environment also requires bike-sharing companies to increase the operational efficiency by adopting such measures. Bike-sharing systems are composed of several bike stations located at different sites across the city, and each station operates a number of bike slots from which bicycles can be collected or to which they can be returned. In a balanced situation, each bike station is required to have a certain number of empty slots, to allow arrivals, and a certain number of full slots, to allow departures. A certain amount of time after the beginning of the service, the users will have moved the bicycles among the stations of the system and the state will have deviated from the balanced one. Thus, these companies engage in the redistribution of bicycles, an operation known as rebalancing. Rebalancing, which is performed by vehicles with the necessary capacity, is normally required at the end of the day, when the system is closed or when the use of the system can be considered negligible. In this case, the rebalancing is considered to be static. Some bike-sharing

2

Journal of Advanced Transportation

operators may require the rebalancing to be performed when the system is open and in operation, incurring a situation known as dynamic rebalancing. In this paper, we present the static case, which is referred to as the Static Bike-sharing Rebalancing Problem (SBRP). As discussed elsewhere [1–4], SBRP is a vehicle route problem, modeled on a complete digraph 𝐺 = (𝑉, 𝐴), where 𝑉 = {0, 1, . . . , 𝑛} is the set of vertices, including the depot (vertex 0) and n stations (vertices 1,. . .,n), and A is the set of arcs between each pair of vertices. The traveling cost c𝑖𝑗 is associated with the arc (i,j)∈A. For each vertex i∈V, a request q𝑖 is given, with q0 =0. Requests can be either positive or negative. If q𝑖 ≥0, then i is a pickup station, from which q𝑖 bicycles should be removed; if q𝑖 Capacity or Free Space < P𝑖 O𝑗 = i − 1 end for i = 1, 2, . . . , N do repeat r←󳨀 r + 1; k ←󳨀 O𝑗 for m = k + 1 to j do Route(r) = X(𝑖) ⋃ 𝑅𝑜𝑢𝑡𝑒(𝑟) end end end n ←󳨀 n + 1 end Return Route end Algorithm 1: The initial particle swarm algorithm.

Input: Initial vehicle routes Output: Improved vehicle routes by VNS algorithm Begin Initialize sequence of three neighborhood structures (1-insert, 2-swap, 3-2-opt) and randomly produce a positive integer of no more than 3 Choose the 𝑘𝑡ℎ neighbor (S󸀠 ) of 𝑆󸀠 ∈ 𝑁(𝑆) from these three neighborhood structures and apply 𝑘𝑡ℎ neighborhood to P and obtain(P󸀠 ) if F(P󸀠 ) < 𝐹(𝑃) then P ←󳨀 𝑃󸀠 , F(P) ←󳨀 F(𝑃󸀠 ) and apply the remainder of the neighborhood structures in sequence one by one to (𝑃) and obtain(P󸀠 ) if F(P󸀠 ) < 𝐹(𝑃) then P ←󳨀 𝑃󸀠 , F(P) ←󳨀 F(𝑃󸀠 ) else delete the 𝑘𝑡ℎ neighborhood structure and apply the remainder of the neighborhood structures one by one to (𝑃) and obtain(P󸀠 ) return P and F(P) end Algorithm 2: Local search algorithm (VNS).

6. Case Study We assume a fictitious small-scale bicycle system as an illustrative example to show the efficiency of the proposed model and solving method. The system contains 14 bike stations. The daily supply and demand, the number of bicycles in need of repair, and the adjacency matrix for distance

serving as cost matrix are given, separately, in Tables 3, 4, and 6. The hybrid algorithm based on DPSO and VNS for rebalancing bike sharing has the following process. 6.1. Generating the Initial Discrete Particle Swarm. Normally, the initial discrete particle swarm is produced randomly,

10


Input: Based data of bike system, DPSO parameters Output: The best optimization solution (routes of the bike system) Begin t ←󳨀 1 Initialize the population of the swarm, known as Pop and the relative parameters i.e., max iteration Apply the initial particle swarm algorithm to represent each particle in 𝑋𝑡 Calculate the personal best particle in 𝑋𝑡 and obtain the personal best of each particle, stored in 𝑃𝑡 Calculate the global best particle in 𝑋𝑡 and obtain the global best of each particle, stored in 𝐺𝑡 While (i is not more than the max iteration or stopping criterion is not satisfied) do Begin Update particles using Eq. (21) and Eq. (22) Apply the initial particle swarm algorithm to evaluate the updating particles(𝑋𝑡 ) Apply the Local search algorithm(VNS) to optimize the personal best particles(P𝑡𝑖 ) Update the Global best (𝐺𝑡 ) end Algorithm 3: Hybridization of DPSO with VNS. Table 3: Daily supply and demand. Station-ID Demand and Supply

1 12

2 -2

3 8

4 5

5 -7

6 4

7 -2

8 -8

9 2

10 1

11 1

12 -3

13 3

14 10

12 2

13 4

14 7

Table 4: Number of bicycles in disrepair. Station-ID Bicycles in disrepair

1 1

1

2 5

5

7

5

9 3

10 7

10

5

Swap option

11

6

4 2

1

5 8

7

11

6

2

7

8 11

4

4

6

2 1

2

1

9

9 10

5

3

7

8 6

5

3

2

4

8 2

6

10

11 2

9 1

10 0

11 3

6.2. Fitness Evaluation. The feasible solutions for this system are produced in Section 6.1, so the calculation of these feasible solutions is the main topic of this section. We utilize the fitness value of each particle as an optimized target; the fitness function is a standard for screening particles; hence we choose the reciprocal of the objective function as the fitness function. The specific calculation formula is

7

2-opt option

𝑗−1

8 11

7 3

𝑖 ∈ [1, 𝑝𝑜𝑝𝑠𝑖𝑧𝑒], 𝑗 ∈ [1, 𝑙𝑒𝑛𝑔𝑡ℎ], and obtain the initial particle 𝑋𝑖 = {𝑥𝑖𝑗 | 𝑥𝑖𝑗 ∈ [1, 20], 𝑖 ∈ [1, 𝑝𝑜𝑝𝑠𝑖𝑧𝑒], 𝑗 ∈ [1, 𝑙𝑒𝑛𝑔𝑡ℎ]}. According to the size of this fictitious bike-sharing system, we set 20 as the solution scale. Divide the particles according to the constraint condition, and obtain the pattern of particles that was revealed in Table 5. Numbers against asterisk mark denote the depot and the end of one route.

8

9

3

6 4

10

3

4

11 1

5 2

9

Insert option

8 6

4 2

1

9 10

3

3 4

4

Bike Station Depot

Figure 5: Three neighborhood structures.

𝐹𝑖𝑡𝑛𝑒𝑠𝑠 (𝑡) = 𝑐0,𝑋𝑡 (𝑖) + ∑ 𝑐𝑋𝑡 (𝑘),𝑋𝑡 (𝑘+1) + 𝑐𝑋𝑡 (𝑗),0

(23)

𝑘=𝑖+1

where 𝑋𝑡 (𝑖) represents the value of the 𝑖𝑡ℎ data of the 𝑡𝑡ℎ particle, 𝑐𝑖,𝑗 represents the vehicle cost (distance) from station i to j,

but choosing an appropriate initial swarm can expedite the convergence rate of the algorithm. In order to gain the favorable population, we initialize the discrete particle swarm randomly, belonging to the arrangement 𝑥𝑖𝑗 , 𝑥𝑖𝑗 ∈ [1, 20],

𝑓𝑖𝑡𝑛𝑒𝑠𝑠 is the fitness value of the particle. The following process of the first particle shows the steps to calculate the fitness value.


11 Table 5: Initial Particles.

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th

6 7 2 10 8 2 9 14 3 11 2 9 10 1 8 3 4 6 11 9

3 1 10 9 9 3 2 7 7 13 10 10 6 10 14 11 1 10 1 11

11 13 4 0∗ x 0∗ 1 10 1 0∗ 13 0∗ 4 0∗ 6 10 14 14 0∗ 6

7 2 0∗ 6 0∗ 11 0∗ 8 0∗ 1 0∗ 5 0∗ 7 0∗ 0∗ 9 0∗ 10 0∗

∗

0 0∗ 5 13 12 1 6 0∗ 13 5 7 1 2 2 1 4 0∗ 11 8 7

14 14 3 2 2 6 11 2 9 14 9 11 7 3 7 7 2 12 14 10

8 6 8 1 10 14 0∗ 6 14 12 0∗ 6 14 9 9 12 10 4 0∗ 14

5 10 0∗ 0∗ 3 0∗ 13 12 8 0∗ 12 7 0∗ 13 4 0∗ 7 3 3 0∗

0∗ 0∗ 12 11 0∗ 5 10 0∗ 0∗ 8 14 0∗ 13 0∗ 0∗ 13 0∗ 0∗ 13 5

1 12 11 14 7 9 8 1 11 10 1 4 1 14 2 5 13 1 2 1

2 11 7 3 4 8 14 3 4 9 6 14 8 6 11 14 12 2 9 2

4 4 1 8 1 0∗ 3 4 10 4 5 12 5 0∗ 13 2 8 13 0∗ 0∗

0∗ 8 13 0∗ 11 13 7 0∗ 5 3 0∗ 13 3 8 10 6 0∗ 9 12 4

13 0∗ 0∗ 7 13 4 0∗ 5 0∗ 0∗ 11 0∗ 0∗ 5 0∗ 0∗ 6 0∗ 5 13

9 3 14 4 0∗ 12 4 11 6 7 8 8 12 4 3 8 3 7 4 3

10 9 6 5 5 7 12 9 12 6 3 2 9 12 5 9 5 8 7 8

12 5 9 12 6 10 5 13 2 2 4 3 11 11 12 1 11 5 6 12

Table 6: Cost Matrix. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0.00 0.26 0.18 0.21 0.35 0.43 0.40 0.29 0.23 0.29 0.27 0.20 0.08 0.20 0.27

2 0.26 0.00 0.11 0.16 0.40 0.44 0.45 0.46 0.13 0.17 0.45 0.34 0.30 0.22 0.29

3 0.18 0.11 0.00 0.07 0.30 0.36 0.36 0.36 0.18 0.12 0.34 0.23 0.20 0.12 0.19

4 0.21 0.16 0.07 0.00 0.24 0.29 0.29 0.32 0.25 0.08 0.30 0.20 0.20 0.06 0.13

5 0.35 0.40 0.30 0.24 0.00 0.08 0.06 0.20 0.48 0.24 0.20 0.17 0.28 0.19 0.11

6 0.43 0.44 0.36 0.29 0.08 0.00 0.06 0.27 0.54 0.27 0.28 0.25 0.36 0.25 0.17

7 0.40 0.45 0.36 0.29 0.06 0.06 0.00 0.22 0.54 0.29 0.23 0.21 0.33 0.25 0.16

8 0.29 0.46 0.36 0.32 0.20 0.27 0.22 0.00 0.50 0.36 0.02 0.12 0.21 0.26 0.22

9 0.23 0.13 0.18 0.25 0.48 0.54 0.54 0.50 0.00 0.29 0.48 0.38 0.30 0.29 0.37

10 0.29 0.17 0.12 0.08 0.24 0.27 0.29 0.36 0.29 0.00 0.35 0.25 0.27 0.11 0.14

11 0.27 0.45 0.34 0.30 0.20 0.28 0.23 0.02 0.48 0.35 0.00 0.10 0.19 0.24 0.21

12 0.20 0.34 0.23 0.20 0.17 0.25 0.21 0.12 0.38 0.25 0.10 0.00 0.12 0.14 0.13

13 0.08 0.30 0.20 0.20 0.28 0.36 0.33 0.21 0.30 0.27 0.19 0.12 0.00 0.17 0.21

10

12

14 0.20 0.22 0.12 0.06 0.19 0.25 0.25 0.26 0.29 0.11 0.24 0.14 0.17 0.00 0.08

15 0.27 0.29 0.19 0.13 0.11 0.17 0.16 0.22 0.37 0.14 0.21 0.13 0.21 0.08 0.00

Note: Number 1 represents the depot.

1st

6

3

11

7

0∗

14

8

5

0∗

Labeling method with list: Beginning from (0, 0) in the cost matrix, we obey these rules: Odd and even numbers of particles are indicated as rows and columns in the cost matrix, respectively. Further, when the adjacent number in the cost matrix is a cross, it is often the cost of the route. When all the numbers, exactly all the adjacent numbers, have been labeled,

1

2

4

13

0∗

9

this process is complete. Last, the consequence is equal to the sum of all the labeled points. We present a representation of the cost matrix of the first particle in Table 7. The fitness value is obtained by calculating the remaining particles in this way. The result is shown in Table 8. Numbers

12

Journal of Advanced Transportation Table 7: Cost calculation progress of routes 1, 2, 3, and 4.

Number 0 1 ∗ 0 0.00 0.26C $ 1 0.26 0.00 2 0.18 0.11C $ 3 0.21 0.16 4 0.35 0.40 5 0.43 0.44 6 0.40 0.45 7 0.29A # 0.46 8 0.23 0.13 9 0.29 0.17 10 0.27 0.45 11 0.20 0.34 12 0.08 0.30 13 0.20C $ 0.22 14 0.27 0.29

2 0.18 0.11 0.00 0.07 0.30 0.36 0.36 0.36 0.18 0.12 0.34 0.23 0.20 0.12 0.19

3 4 5 6 & 0.21 0.35 0.43B 0.40A # 0.16 0.40 0.44 0.45 0.07 0.30C $ 0.36 0.36 0.00 0.24 0.29 0.29A # 0.24 0.00 0.08 0.06 0.29 0.08 0.00 0.06 0.29 0.06 0.06 0.00 0.32 0.20 0.27 0.22 0.25 0.48 0.54B & 0.54 0.08 0.24 0.27 0.29 0.30 0.20 0.28 0.23 0.20 0.17 0.25 0.21 0.20 0.28 0.36 0.33 0.06 0.19C $ 0.25 0.25 0.13 0.11 0.17 0.16 A # Route 1 B& Route 2

7 8 9 10 11 12 @ 0.29 0.23 0.29D 0.27 0.20 0.08D @ 0.46 0.13 0.17 0.45 0.34 0.30 0.36 0.18 0.12 0.34 0.23 0.20 0.32 0.25 0.08 0.30 0.20A # 0.20 0.20 0.48 0.24 0.20 0.17 0.28 0.27 0.54 0.27 0.28 0.25 0.36 0.22 0.54 0.29 0.23 0.21 0.33 0.00 0.50 0.36 0.02 0.12A # 0.21 0.50 0.00 0.29 0.48 0.38 0.30 0.36 0.29 0.00 0.35 0.25 0.27 @ 0.02 0.48 0.35D 0.00 0.10 0.19D @ 0.12 0.38 0.25 0.10 0.00 0.12 0.21 0.30 0.27 0.19 0.12 0.00 0.26 0.29 0.11 0.24 0.14 0.17 0.22 0.37 0.14 0.21 0.13 0.21 C $ Route 3 D@ Route 4

13 14 0.20 0.27B & 0.22 0.29 0.12 0.19 0.06 0.13 0.19 0.11 0.25 0.17 0.25 0.16 0.26 0.22 0.29 0.37B & 0.11 0.14 0.24 0.21 0.14 0.13 0.17 0.21 0.00 0.08 0.08 0.00

Table 8: Fitness value. 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th

6 7 2 10 8 2 9 14 3 11 2 9 10 1 8 3 4 6 11 9

3 1 10 9 9 3 2 7 7 13 10 10 6 10 14 11 1 10 1 11

11 13 4 0∗ x 0∗ 1 10 1 0∗ 13 0∗ 4 0∗ 6 10 14 14 0∗ 6

7 2 0∗ 6 0∗ 11 0∗ 8 0∗ 1 0∗ 5 0∗ 7 0∗ 0∗ 9 0∗ 10 0∗

0∗ 0∗ 5 13 12 1 6 0∗ 13 5 7 1 2 2 1 4 0∗ 11 8 7

14 14 3 2 2 6 11 2 9 14 9 11 7 3 7 7 2 12 14 10

8 6 8 1 10 14 0∗ 6 14 12 0∗ 6 14 9 9 12 10 4 0∗ 14

Initial Particles 5 0∗ 10 0∗ ∗ 0 12 0∗ 11 3 0∗ 0∗ 5 13 10 12 0∗ 8 0∗ ∗ 0 8 12 14 7 0∗ ∗ 0 13 13 0∗ 4 0∗ 0∗ 13 7 0∗ 3 0∗ 3 13 0∗ 5

against asterisk mark denote the depot and the end of one route. 6.3. Calculation of the Historical and Global Optimal Values of the Particle. The historical and global optimal values are the particle’s cognitive and social messages, which have a significant impact on the probability of finding the preferable particle. Hence, the definition for the historical optimal particle is as follows: particles with the best historical fitness

1 12 11 14 7 9 8 1 11 10 1 4 1 14 2 5 13 1 2 1

2 11 7 3 4 8 14 3 4 9 6 14 8 6 11 14 12 2 9 2

4 4 1 8 1 0∗ 3 4 10 4 5 12 5 0∗ 13 2 8 13 0∗ 0∗

13 0∗ 0∗ 7 13 4 0∗ 5 0∗ 0∗ 11 0∗ 0∗ 5 0∗ 0∗ 6 0∗ 5 13

0∗ 8 13 0∗ 11 13 7 0∗ 5 3 0∗ 13 3 8 10 6 0∗ 9 12 4

9 3 14 4 0∗ 12 4 11 6 7 8 8 12 4 3 8 3 7 4 3

10 9 6 5 5 7 12 9 12 6 3 2 9 12 5 9 5 8 7 8

12 5 9 12 6 10 5 13 2 2 4 3 11 11 12 1 11 5 6 12

fitness 4.89 4.70 4.58 3.25 4.21 5.24 6.35 4.25 4.68 4.78 5.21 5.36 4.25 6.25 5.48 5.36 7.21 4.87 4.98 4.56

and global optimal particle; the best particles with the best historical fitness are mentioned. According to the definition, we obtain all the particles’ historical optimal values and the particle swarm’s global value at each iteration, as shown in Table 9. In the first iteration, we initialize the original fitness value of the particles as the personal best and initialize the minimum of the personal best as the global best. The consequence is represented in Table 10.


13

Table 9: Historically optimal and globally optimal particles. 𝑥𝑡1,1 󸀠 𝑥𝑘2,1

𝑥𝑡1,2 𝑥𝑘2,2

𝑥𝑡1,3 𝑥𝑘2,3

⋅⋅⋅

𝑥𝑡1,𝑙𝑒𝑛𝑔𝑡ℎ−2

𝑥𝑡1,𝑙𝑒𝑛𝑔𝑡ℎ

⋅⋅⋅

𝑥𝑘2,𝑙𝑒𝑛𝑔𝑡ℎ−2


𝑥𝑘2,𝑙𝑒𝑛𝑔𝑡ℎ

𝑓2𝑘

⋅⋅⋅ 󸀠 x𝑙20,1

⋅⋅⋅ x𝑙20,2

⋅⋅⋅ x𝑙20,3

2nd Particle’s Personal Best

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅

⋅⋅⋅

⋅⋅⋅

x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ−2

x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ−1

x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ

⋅⋅⋅ 𝑙 f20

⋅⋅⋅ 20th Particle’s Personal Best


𝑓1𝑡

1st Particle’s Personal Best

min {𝑓𝑖𝑡 | 𝑖 ∈ [1, 𝑝𝑜𝑝𝑠𝑖𝑧𝑒] , 𝑡 ∈ [1, 𝑀𝑎𝑥𝑖𝑡𝑒𝑟𝑎𝑡𝑒]}

Global Best

Table 10: Initial fitness value, personal best, and global best of the particles.

1st 6 2nd 7 3rd 2 4th 10 5th 8 6th 2 7th 9 8th 14 9th 3 10th 11 11th 2 12th 9 13th 10 14th 1 15th 8 16th 3 17th 4 18th 6 19th 11 20th 9

3 1 10 9 9 3 2 7 7 13 10 10 6 10 14 11 1 10 1 11

11 13 4 0∗ x 0∗ 1 10 1 0∗ 13 0∗ 4 0∗ 6 10 14 14 0∗ 6

7 2 0∗ 6 0∗ 11 0∗ 8 0∗ 1 0∗ 5 0∗ 7 0∗ 0∗ 9 0∗ 10 0∗

0∗ 0∗ 5 13 12 1 6 0∗ 13 5 7 1 2 2 1 4 0∗ 11 8 7

14 14 3 2 2 6 11 2 9 14 9 11 7 3 7 7 2 12 14 10

8 6 8 1 10 14 0∗ 6 14 12 0∗ 6 14 9 9 12 10 4 0∗ 14

Initial Particles 5 0∗ 10 0∗ ∗ 0 12 0∗ 11 3 0∗ 0∗ 5 13 10 12 0∗ 8 0∗ ∗ 0 8 12 14 7 0∗ ∗ 0 13 13 0∗ 4 0∗ 0∗ 13 7 0∗ 3 0∗ 3 13 0∗ 5

1 12 11 14 7 9 8 1 11 10 1 4 1 14 2 5 13 1 2 1

2 11 7 3 4 8 14 3 4 9 6 14 8 6 11 14 12 2 9 2

4 4 1 8 1 0∗ 3 4 10 4 5 12 5 0∗ 13 2 8 13 0∗ 0∗

0∗ 8 13 0∗ 11 13 7 0∗ 5 3 0∗ 13 3 8 10 6 0∗ 9 12 4

13 0∗ 0∗ 7 13 4 0∗ 5 0∗ 0∗ 11 0∗ 0∗ 5 0∗ 0∗ 6 0∗ 5 13

9 3 14 4 0∗ 12 4 11 6 7 8 8 12 4 3 8 3 7 4 3

10 9 6 5 5 7 12 9 12 6 3 2 9 12 5 9 5 8 7 8

12 5 9 12 6 10 5 13 2 2 4 3 11 11 12 1 11 5 6 12

fitness 4.89 4.70 4.58 3.25 4.21 5.24 6.35 4.25 4.68 4.78 5.21 5.36 4.25 6.25 5.48 5.36 7.21 4.87 4.98 4.56

Personal best 4.89 4.70 4.58 3.25 (Global best) 4.21 5.24 6.35 4.25 4.68 4.78 5.21 5.36 4.25 6.25 5.48 5.36 7.21 4.87 4.98 4.56

Note: Numbers against asterisk mark denote the depot and the end of one route.

Table 11: Variable neighborhood operation. 󸀠

󸀠

󸀠

𝑥𝑡1,1

𝑥𝑡1,2

𝑥𝑡1,3

⋅⋅⋅

𝑥𝑘2,3

⋅⋅⋅ 󸀠 x𝑙20,1

⋅⋅⋅ 󸀠 x𝑙20,2

⋅⋅⋅ 󸀠 x𝑙20,3

󸀠

󸀠

𝑥𝑘2,1

𝑥𝑘2,2

󸀠

󸀠

󸀠

󸀠

󸀠


𝑥𝑡1,𝑙𝑒𝑛𝑔𝑡ℎ

⋅⋅⋅



󸀠

𝑥𝑘2,𝑙𝑒𝑛𝑔𝑡ℎ

󸀠 𝑓2𝑘

2nd Particle’s Personal Best

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅

⋅⋅⋅

⋅⋅⋅

⋅⋅⋅ 𝑙󸀠 f20

⋅⋅⋅ 20th Particle’s Personal Best

󸀠

󸀠 x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ−2

󸀠

𝑥𝑘2,𝑙𝑒𝑛𝑔𝑡ℎ−1 󸀠 x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ−1

After variable neighborhood operation Global Best 󸀠

󸀠 x𝑙20,𝑙𝑒𝑛𝑔𝑡ℎ

𝑓1𝑡

󸀠

1st Particle’s Personal Best

min {𝑓𝑖𝑡 | 𝑖 ∈ [1, 𝑝𝑜𝑝𝑠𝑖𝑧𝑒] , 𝑡󸀠 ∈ [1, 𝑀𝑎𝑥𝑖𝑡𝑒𝑟𝑎𝑡𝑒]}

Note: 𝑥𝑘𝑖,𝑗 is the result of the variable neighborhood operation.

6.4. Local Search through the Variable Neighborhood Operation. Our algorithm utilizes the variable neighborhood search algorithm given in Table 11 to enhance the ability to choose the solution randomly, ensure diversity of the individual, and overcome the disadvantage that the solution is a local optimum. First, we updated the individual particles’ historical best and global best. Then, in our implementation, three variable neighborhood structures are applied to the particle’s personal best mentioned at Section 5.4. Based on the particular variable neighborhood structure, we used the personal best as an operation by taking the

2-opt operation structure as an example. The steps and consequence are the following. Procedure 1. Produce a group of 2-opt structures randomly; i.e., in the first particle, there are three points, (3, 7), (14, 5), and (2, 10), respectively. Procedure 2. Subject the particle to the 2-opt operation. An example using the 1st particle is provided in Table 12. The remaining particles were subjected to operation according to the form procedure, and the final result is presented in Table 13.

14

Journal of Advanced Transportation Table 12: Particles subjected to neighborhood structure operation (2-Opt).

1st After 2-opt-1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th

6 6 7 2 10 8 2 9 14 3 11 2 9 10 1 8 3 4 6 11 9

3 3 1 10 9 9 3 2 7 7 13 10 10 6 10 14 11 1 10 1 11

11 7& 13 4 0∗ 14 0∗ 1 10 1 0∗ 13 0∗ 4 0∗ 6 10 14 14 0∗ 6

7 11 & 2 0∗ 6 0∗ 11 0∗ 8 0∗ 1 0∗ 5 0∗ 7 0∗ 0∗ 9 0∗ 10 0∗

0∗ 0∗ 0∗ 5 13 12 1 6 0∗ 13 5 7 1 2 2 1 4 0∗ 11 8 7

14 14 14 3 2 2 6 11 2 9 14 9 11 7 3 7 7 2 12 14 10

8 5& 6 8 1 10 14 0∗ 6 14 12 0∗ 6 14 9 9 12 10 4 0∗ 14

0∗ 0∗ 0∗ 12 11 0∗ 5 10 0∗ 0∗ 8 14 0∗ 13 0∗ 0∗ 13 0∗ 0∗ 13 5

5 8& 10 0∗ 0∗ 3 0∗ 13 12 8 0∗ 12 7 0∗ 13 4 0∗ 7 3 3 0∗

1 1 12 11 14 7 9 8 1 11 10 1 4 1 14 2 5 13 1 2 1

2 2 11 7 3 4 8 14 3 4 9 6 14 8 6 11 14 12 2 9 2

4 10 & 4 1 8 1 0∗ 3 4 10 4 5 12 5 0∗ 13 2 8 13 0∗ 0∗

13 9& 0∗ 0∗ 7 13 4 0∗ 5 0∗ 0∗ 11 0∗ 0∗ 5 0∗ 0∗ 6 0∗ 5 13

0∗ 0& 8 13 0∗ 11 13 7 0∗ 5 3 0∗ 13 3 8 10 6 0∗ 9 12 4

9 13 & 3 14 4 0∗ 12 4 11 6 7 8 8 12 4 3 8 3 7 4 3

10 4& 9 6 5 5 7 12 9 12 6 3 2 9 12 5 9 5 8 7 8

12 12 5 9 12 6 10 5 13 2 2 4 3 11 11 12 1 11 5 6 12

Note: Numbers against asterisk mark denote the depot and the end of one route. Numbers with & mark were subjected to operation by the 2-opt structure.

Table 13: Particles subjected to 2-Opt structure operation. After 2-opt-1st After 2-opt-2nd After 2-opt-3rd After 2-opt-4th After 2-opt-5th After 2-opt-6th After 2-opt-7th After 2-opt-8th After 2-opt-9th After 2-opt-10th After 2-opt-11th After 2-opt-12th After 2-opt-13th After 2-opt-14th After 2-opt-15th After 2-opt-16th After 2-opt-17th After 2-opt-18th After 2-opt-19th After 2-opt-20th

6 7 4& 10 8 2 9 14 3 11 2 9 10 1 8 3 4 6 11 9

3 13 & 10 & 9 14 & 3 6& 7 7 13 0& 10 2& 10 6& 11 9& 10 1 11

7& 1& 2& 0∗ 9& 0∗ 0& 10 13 & 0∗ 13 & 0∗ 0& 7& 14 & 10 14 & 11 & 0∗ 7&

11 & 2 0∗ 6 0∗ 11 1& 2& 0& 1 10 & 1& 4& 0& 0∗ 0∗ 1& 0& 10 0&

0∗ 0∗ 5 1& 12 1 2& 0& 1& 14 & 7 5& 6& 2 1 4 0∗ 14 & 8 6&

14 14 3 2& 2 6 11 8& 9 5& 9 11 7 3 7 7 2 12 9& 10

5& 4& 7& 13 & 10 14 0∗ 6 14 12 0∗ 6 14 9 9 12 10 4 2& 14

8& 11 & 11 & 0∗ 3 0∗ 13 12 8 0∗ 12 4& 0∗ 13 4 0∗ 7 3 13 & 0∗

0∗ 12 & 12 & 11 0∗ 5 10 0∗ 0∗ 8 14 0& 13 14 & 0∗ 13 0∗ 0∗ 3& 5

1 0& 0& 14 7 9 8 5& 11 10 1 7& 1 0& 2 5 13 1 0& 1

2 10 & 8& 3 4 4& 14 4& 4 9 6 14 5& 6 11 0& 12 2 14 & 8&

10 & 6& 1 8 11 & 0& 4& 3& 10 2& 5 12 8& 0∗ 0& 2& 8 13 0∗ 3&

9& 0∗ 0∗ 5& 13 & 8& 7& 1& 0∗ 6& 11 0∗ 0∗ 5 13 & 14 & 3& 7& 5 4&

0& 8 13 4& 1& 13 0& 0& 5 7& 0∗ 13 3 8 10 6 0& 9& 12 13 &

13 & 5& 14 0& 0∗ 12 3& 11 6 3& 8 8 12 4 3 8 6& 0& 4 0&

4& 9& 6 7& 5 7 12 9 12 0& 3 2 9 12 5 9 5 8 7 2&

12 3& 9 12 6 10 5 13 2 4& 4 3 11 11 12 1 11 5 6 12

Note: Numbers against asterisk mark denote the depot and the end of one route. Numbers with & mark were subjected to operation by the 2-opt structure.

Then, according to the constraint of the model, we check and recode particles. If the operated particles can satisfy the constraint, we accept these particles; otherwise, we recode the operated particles. The result of checking and recoding can be seen in Table 14.

6.5. Particle Updating Rules and Algorithm Termination Condition. The velocity and position of a particle are updated according to (21) and (22), after which iterative operation occurs. The particles are updated according to the relative operation relating to the position and velocity.


15 Table 14: Recoding particles after the 2-Opt operation.

After 2-opt-1st After 2-opt-2nd After 2-opt-3rd After 2-opt-4th After 2-opt-5th After 2-opt-6th After 2-opt-7th After 2-opt-8th After 2-opt-9th After 2-opt-10th After 2-opt-11th After 2-opt-12th After 2-opt-13th After 2-opt-14th After 2-opt-15th After 2-opt-16th After 2-opt-17th After 2-opt-18th After 2-opt-19th After 2-opt-20th

6 7 4 10 8 2 9 14 3 11 2 9 10 1 8 3 4 6 11 9

3 13 10 9 14 3 6 7 7 13 13 10 2 10 6 11 9 10 1 11

7 0∗ 2 0∗ 9 0∗ 1 10 13 0∗ 10 0∗ 4 7 14 0∗ 14 11 0∗ 7

0∗ 1 0∗ 6 0∗ 11 2 2 0∗ 1 0∗ 1 0∗ 0∗ 0∗ 10 1 0∗ 10 0∗

11 2 5 1 12 1 0∗ 0∗ 1 14 7 5 6 2 1 4 0 14 8 6

14 14 3 2 2 6 11 8 9 5 9 11 7 3 7 7 2 12 9 10

5 4 7 13 10 14 13 6 14 12 12 6 14 9 9 12 10 4 2 14

8 0∗ 11 0∗ 3 0∗ 10 12 8 0∗ 14 4 0∗ 13 4 0∗ 7 3 13 0∗

0∗ 11 12 11 0∗ 5 8 0∗ 0∗ 8 0∗ 0∗ 13 14 0∗ 13 0∗ 0∗ 3 5

1 12 0 14 7 9 0∗ 5 11 10 1 7 1 0∗ 2 5 13 1 0∗ 1

2 10 8 3 4 4 14 4 4 9 6 14 5 6 11 2 12 2 14 8

10 6 1 8 11 0∗ 4 3 10 2 5 12 8 5 13 0∗ 8 13 5 3

9 0∗ 0∗ 5 13 8 7 1 0∗ 6 11 0∗ 0∗ 8 0∗ 14 3 7 12 4

0∗ 8 13 4 1 13 0∗ 0∗ 5 7 0∗ 13 3 0∗ 10 6 0∗ 9 0∗ 13

13 5 14 0∗ 0∗ 12 3 11 6 3 8 8 12 4 3 8 6 0∗ 4 0∗

4 9 6 7 5 7 12 9 12 0∗ 3 2 9 12 5 9 5 8 7 2

12 3 9 12 6 10 5 13 2 4 4 3 11 11 12 1 11 5 6 12

1 1 14 1 9 0∗ 4 6 7 1 10 0∗ 3 0∗ 11 6 4 9 2 11

6 5 2 7 14 9 11 7 2 7 14 3 12 3 5 1 14 5 9 10

4 2 12 8 1 1 13 2 5 13 11 11 11 4 4 9 7 3 1 9


Table 15: Updated particles. 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th

11 7 1 5 8 8 1 11 6 6 4 4 14 1 6 14 2 8 6 14

2 8 11 14 12 7 7 8 13 4 2 5 2 2 14 10 13 13 5 1

9 0∗ 4 0∗ 2 5 12 14 0∗ 2 0∗ 12 7 8 13 8 3 1 11 2

0∗ 12 ∗

10 3 0∗ 0∗ 9 10 0∗ 9 2 0∗ 9 0∗ 0∗ 5 2 0∗ 0∗

3 10 7 9 0∗ 11 8 0∗ 1 9 8 0∗ 8 0∗ 2 2 0∗ 0∗ 3 3

5 13 9 13 4 10 10 3 14 12 13 1 4 6 7 5 1 7 13 12

8 6 10 4 5 12 5 13 12 8 1 8 1 11 12 12 8 14 4 4

12 4 6 6 7 6 6 10 0∗ 3 0∗ 14 10 14 8 11 11 6 14 0∗

13 0∗ 0∗ 0∗ 6 13 14 4 4 10 3 7 9 10 0∗ 3 6 0∗ 0∗ 13

0∗ 14 8 2 0∗ 4 0∗ 0∗ 9 0∗ 5 0∗ 0∗ 12 10 0∗ 0∗ 10 7 7

10 3 13 3 10 0∗ 3 5 11 14 12 6 5 0∗ 9 7 12 11 10 8

14 11 3 11 11 14 9 12 3 5 7 13 13 5 3 13 9 4 12 6

7 9 0∗ 0∗ 13 3 2 1 0∗ 11 0∗ 10 6 7 1 4 0∗ 12 0∗ 5

0∗ 0∗ 5 12 0∗ 2 0∗ 0∗ 8 0∗ 6 9 0∗ 13 0∗ 0∗ 10 0∗ 8 0∗


The results are provided in Table 15. Subsequently, the algorithm calculates the fitness value, personal best, and global best of the updated particles until |𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑛 +

1) − 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑛)|/|𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑛 + 1)| ≤ 0.05 or the iterative value exceeds the value of the original iterative step.

16

Journal of Advanced Transportation Table 16: Parameters of the algorithm.

Parameters’ name of Hybridization algorithm of DPSO with VNS Population of particles (N) The length of per particle (L) Weight Value (C) Iteration Learning Value (𝐶𝑃 ) Social Value (𝐶𝑔 )

Value 20 14 0.6 50 1.5 2

Table 17: Dispatching plan.

2400 2200 2000 1800 1600 1000 1200 1000 800 600 2000

Routes 9-2-1-8 3-13-14 10-11-12 4-6-5-7

Cost of per route 5.4 km 4.7 km 6.2 km 9.33 km 25.63 km

Depot

11

9

R4

2

R2

R1

1

12

13 14

8 3

10

Coordinate Y

Coordinate Y

Route-ID R1 R2 R3 R4 Total Cost

7 4

R3

5 6

2500

3000 3500 Coordinate X

4000

4500

Figure 6: Dispatching route of the hybridization algorithm of DPSO with VNS.

When the conditions of convergence are met, the operation is complete. Otherwise, return to process in Section 6.1 and implement the iterative operation again. 6.6. Computational Result and Analysis. The parameters of the hybridization algorithm of DPSO with VNS are included in Table 16. The dispatching plan is presented in Table 17 and the dispatching routes are shown in Figure 6. Computation using the hybridization algorithm of DPSO with VNS produces four routes for dispatching, route 1= [9-2-1-8], route 2= [3-13-14], route 3= [10-11-12], and route 4 = [4-6-5-7]. Furthermore, the cost per route scaled by the distance corresponds to 5.4 km, 4.7 km, 6.2 km, and 9.33 km. The total cost is 25.63 km. In order to test the performance of the hybridization algorithm of DPSO with VNS, Goksal et al. made an overall comparison with Reactive Tabu Search (RTS), Large Neighborhood Search (LNS), Particle Swarm Optimization (PSO), Ant Colony System (ACS), Parallel Iterative Local Search (PILS), and Adaptive Memory Methodology (AMM). Since they are the most effective heuristics proposed in the literature. According to the indexes of average deviation to the best known solution, average computation time, scaled

2400 2200 2000 1800 1600 1000 1200 1000 800 600 2000

Total Distance = 27.07, Iteration = 201 Depot R4

9 2

R2 14

R1

1

13

8 3

10

11

12

R3 4

7 5 6

2500


4000

4500

Figure 7: Result of DPSO algorithm.

average computation time over 40 instances, and number of best known solutions found, the comprehensive performance of hybridization algorithm of DPSO with VNS is the best. For the computational results of the comparison, see Goksal et al. [21]. In this study, we implement Discrete Particle Swarm Optimization (DPSO) and Genetic algorithm (GA) separately over the case study data set to get a more comprehensive view of the performance of the hybridization algorithm of DPSO with VNS. The results are provided in Figures 7 and 8. DPSO is considered in the comparison with the purpose of presenting the performance of VNS. We can see from Figure 7 that the cost of the solution without VNS is 27.07 km. The result of the hybridization algorithm of DPSO with VNS (25.63 km) dropped by 5.3%. GA [15] is a population-based heuristic which can be used to solve the problem like SBRP. Y. Li et al. reported the performance in their study. Thus, in order to make a direct comparison, we run a GA program over the same data set of case study. From Figure 8, we can see that the total cost is 26.33 km, while the hybridization algorithm of DPSO with VNS brings a drop by 2.7%. This analysis indicates that the hybridization algorithm of DPSO with VNS outperforms the other existing heuristics in the SBRP literature.

Coordinate Y

Journal of Advanced Transportation 2400 2200 2000 1800 1600 1000 1200 1000 800 600 2000

17

Acknowledgments

Total Distance = 26.33, Iteration = 169 Depot

11 10

9

R4

2

R1

1

13

R2

8 3

12 14

7 4

R3

5 6

2500


4000

4500

Figure 8: Result of GA.

7. Concluding Remarks We quantify the station state using the bicycle net flow by calculating the difference between the inflow and the outflow. The maximum daily demand can be determined by the minimum value on the state curve. It should be noted that this minimum daily demand reveals the imbalance among the bicycle arrival and departure, rather than the real demand of the static rebalancing. An extended VRPSPD program is developed to model the bike-sharing static rebalancing problem considering the number of bicycles that need repair. We employed a hybrid DPSO-VNS algorithm to solve the proposed model. The computational result shows that the effectiveness of the algorithm is attributed to a combination of the following reasons. First, the idea of vehicle orientation ensures that each route only covers a restricted area. Second, the quality of the solution is improved by the cheapest insertion heuristic and 2-opt method, both of which are applied during the route construction. Third, the DPSO mechanism is capable of generating diverse solutions and maintaining the best solution found during the iterative process. Moreover, this approach is flexible, because it is observed that it can be applied to the related problems by modifying the initial solution and a feasibility check of neighborhood methods. The approach can be easily extended and adjusted by adding or replacing these neighborhood methods. Some aspects may improve the performance of the proposed algorithm, such as the parameter optimization and programming implementation. Although the PSO parameter set used in this study was obtained from a preliminary experiment, it may not be the best one. In addition, the implementation of the algorithm may be further improved. These aspects would be further investigated in future studies.

Data Availability The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research has been substantially supported by research grants from the National Natural Science Foundation Council of China (51408058, 51338002), grants from the China Scholarship Council (201608430161), and Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety (Changsha University of Science & Technology), Ministry of Education, under grant number kfj130301.

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