^{1}

^{2}

^{3}

^{1}

^{3}

^{1}

^{3}

^{1}

^{2}

^{3}

In this paper, by taking the outsourcing transportation mode into account, a bilevel programming model is proposed to formulate the static bike repositioning (SBR) problem, which can be used to determine the number of bikes loaded and unloaded at each station and the optimal truck routes in bike sharing systems (BSS). The upper-level BSS providers determine the optimal loading and unloading quantities at stations to minimize the total penalties. The lower-level truck owner pursues the minimum transportation route cost. An iterated local search and tabu search are developed to solve the model. Computational tests on a set of instances from 20 to 200 bikes demonstrate the effectiveness of the model and algorithms proposed, together with some insightful findings.

Bike sharing systems (BSS) provide a carbon-free, green mode of transportation for urban users and reduce urban traffic congestion and air pollution, so they are very popular in urban environments at present [

To the best of our knowledge, only this paper addresses the problem of the balance between the BSS provider (e.g., Mobike) and truck owner (e.g., third-party logistics companies). In order to demonstrate a realistic BRP between the BSS provider and truck owner, in this paper, we develop a bilevel programming model to research the BRP, seeking to satisfy both of the decision makers, and to optimize customer satisfaction and the transportation cost between the upper- (BSS provider) and lower- (the truck owner) level decision makers.

The remainder of this paper is organized as follows. In the next section, we review related work on the BRP. In Section

The BRP has been constantly studied since the appearance of the seminal publication on it by Benchimol et al., in which the authors proved it to be NP-hard [

Generally, the models proposed for the BRP in the literature are classified into static and dynamic optimization problems according to the operational scenarios [

The publications on the static BRP are summarized in Table

Summary of the static BRP in the literature.

References | Number of trucks | Objectives | Pickup and dropoff quantities as decision variables |
---|---|---|---|

Benchimol et al. [ | 1 | Minimize total travel cost | No |

| |||

Cruz et al. [ | 1 | Minimize route cost | No |

| |||

Lin and Chou [ | ≥1 | Minimize total transportation time or distance | No |

| |||

Di Gaspero et al. [ | ≥1 | Minimize the weighted sum of the total absolute deviation from the target number of bikes, the total number of loading/unloading quantities, and the overall time required for all routes | Yes |

| |||

Di Gaspero et al. [ | ≥1 | Minimize the weighted sum of the total absolute deviation from the target number of bikes, and the travel times | Yes |

| |||

Angeloudis et al. [ | ≥1 | Minimize total travel time | Yes |

| |||

Dell’Amico et al. [ | ≥1 | Minimize total route cost | No |

| |||

Forma et al. [ | ≥1 | Minimize the weighted sum of the penalty cost and total travel time | Yes |

| |||

Alvarez-Valdes et al. [ | ≥1 | Minimize the weighted sum of the total service time and the coefficient of variations of the duration of all routes | Yes |

| |||

Ho and Szeto [ | ≥1 | Minimize the weighted sum of the penalty cost and total travel time | Yes |

| |||

Schuijbroek et al. [ | ≥1 | Minimize maximum tour length | Yes |

| |||

Pal and Zhang [ | ≥1 | Minimize the makespan of the rebalancing fleet (maximum tour length) | No |

| |||

Bulhões et al. [ | ≥1 | Minimize total route cost | No |

| |||

Szeto and Shui [ | ≥1 | Minimize first the positive deviation from the tolerance of total demand dissatisfaction and then service time in the form of the total service time or maximum service time of all vehicles | Yes |

| |||

Liu et al. [ | ≥1 | Minimize the weighted sum of the total number of unsatisfied customers, the inconvenience of getting a bike and total operational time | Yes |

| |||

Chemla et al. [ | 1 | Minimize route cost | No |

| |||

Nair et al. [ | 1 | Minimize total redistribution cost | Yes |

| |||

Erdoğan et al. [ | 1 | Minimize total route and handling cost | Yes |

| |||

Ho and Szeto [ | 1 | Minimize total penalty cost | Yes |

| |||

Erdoğan et al. [ | 1 | Minimize route cost | No |

| |||

Kadri et al. [ | 1 | Minimize the total waiting time of the imbalanced stations | No |

| |||

Li et al. [ | 1 | Minimize the sum of travel, imbalance, substitution, and occupancy costs | Yes |

| |||

Szeto et al. [ | 1 | Minimize the weighted sum of the total number of unsatisfied customers and the vehicle’s total operational time | Yes |

| |||

This paper | 1 | The upper-level model minimizes the total penalty cost; the lower-level model minimizes the travel cost | Yes |

In terms of the problem objective functions addressed in the BRP, common measures are

The challenges that arise in the static BRP necessitate efficient methodologies. Exact methods such as the branch-and-cut algorithm have been used to solve the static BRP. However, such methods are intractable for dealing with medium and large realistic BRP. We refer the interested reader to the exact methods papers by Erdoğan et al. [

Summary of inexact solution methods for the static BRP in the literature.

References | Solution methods |
---|---|

Benchimol et al. [ | 9.5-approximation algorithm |

| |

Dell’Amico et al. [ | Destroy and repair algorithm |

| |

Cruz et al. [ | ILS |

| |

Lin and Chou [ | Classical heuristics |

| |

Di Gaspero et al. [ | Constraint programming with ACO |

| |

Di Gaspero et al. [ | Constraint programming with LNS |

| |

Raidl et al. [ | VNS |

| |

Rainer-Harbach et al. [ | VNS with VND |

| |

Rainer-Harbach et al. [ | GRASP with VNS |

| |

Angeloudis et al. [ | Multistage heuristic |

| |

Forma et al. [ | 3-step math heuristic |

| |

Alvarez-Valdes et al. [ | Heuristic based on minimum cost flow |

| |

Ho and Szeto [ | Hybrid LNS |

| |

Schuijbroek et al. [ | Cluster-first route-second heuristic |

| |

Pal and Zhang [ | Nested LNS with VND |

| |

Bulhões et al. [ | Branch-and-cut with ILS |

| |

Szeto and Shui [ | Enhanced ABC |

| |

Liu et al. [ | Enhanced CRO |

| |

Chemla et al. [ | Branch-and-cut method with TS |

| |

Ho and Szeto [ | Iterated TS |

| |

Li et al. [ | Combined hybrid GA |

| |

Szeto et al. [ | Enhanced CRO |

| |

This paper | ILS and TS |

To deal efficiently with the proposed bilevel model, a heuristic based on ILS that iteratively solves the upper-level and lower-level models is developed. The proposed model and heuristic are evaluated using a test instance, and the results show the effectiveness of the algorithm.

It appears that no previous work has addressed the BRP from this approach where the objectives of the different stakeholders are taken into consideration. In this paper, we intend to close a significant gap in the BRP literature and make a twofold contribution. First, the BRP is modeled as a mixed-integer, bilevel program, where the different objectives of the BSS provider and truck owner are considered. Second, we provide a mathematical formulation and heuristic algorithm to solve the introduced problem. While the bilevel model presented here is customized for the BRP under the situation that the BSS provider outsources the transportation work between bike stations to the third-party logistics companies, we hope that it will open a new line of research on the BRP.

In this section, first of all, we describe different aspects of the problem and then present its mathematical model. The basic idea of this bilevel model in the BRP is to design relocation transportation with the object of maximizing customer satisfaction with the BSS provider by considering the repositioning resources of third-party companies. We model this decision-making environment as a bilevel problem, so that, in this structure, the BSS provider is considered as the decision maker of the upper-level problem (ULP) and the truck owner as the decision maker of the lower-level problem (LLP) (see Figure

The conceptual structure of bilevel bike-sharing repositioning problem.

The static BRP considers a group of stations and a depot where the truck is originally located at the depot. If the optimal inventory (i.e., the inventory with the lowest penalty cost) at a station is smaller than the initial inventory at that station, then the station is assumed as a pick-up station. The initial inventory at pick-up station is too large and the extra bikes need to be taken away. On the other hand, if the optimal inventory at a station is greater than the initial inventory at that station, then the station is assumed as a drop-off station [

To facilitate the presentation of the essential ideas without loss of generality, the following basic assumptions are made in this paper:

The notations used throughout this paper are listed in Table

Notations for the bi-level model.

| Set of stations |

| |

_{ o } | Set of nodes, including the stations and the depot (indexed by 0) |

| |

| Indices of stations |

| |

| Initial inventory level at station |

| |

| Final inventory level at station |

| |

| Convex penalty function of |

| |

| The target optimal inventory at station |

| |

| Set of pick-up stations, i.e., |

| |

| Set of drop-off stations, i.e., |

| |

| Truck capacity |

| |

| Capacity of station |

| |

| Repositioning time |

| |

| Time required to load a bike from a station onto the truck |

| |

| Time required to unload a bike from the truck to a station |

| |

| A very large number |

| |

| Traveling time from station |

| |

| Binary variable that equals 1 if the truck travels directly from station |

| |

| Number of bikes on the truck when it travels directly from station |

| |

| The loading quantities at station |

| |

| The unloading quantitiesat station |

| |

| The vector of the upper-level decision variables, |

| |

| The vector of the lower-level decision variables, |

The penalty function used in this paper was first proposed by Raviv et al. [_{i}_{i}_{i}_{i}_{i }is _{i} is out of the domain _{i}_{i}

As described by Forma et al. [

In order to demonstrate the relationship between the BSS provider and the truck owner, a bilevel programming model modified on the basis of Ho and Szeto [

The upper-level model aims to determine the optimal

The upper-level objective function (

The lower-level model aims to determine the optimal

The lower-level objective function (

Constraint (

As shown in Table

ILS was introduced by Lourenço et al. in 2001 and iteratively alternates between local search (intensification, see Sections

_{0}≔

_{0})

A solution to the bilevel BRP with a single truck considered in this work can be represented as a sequence of visits to stations, starting and ending at the depot, along with the number of bikes loaded or unloaded at each visit.

A solution

Let_{h}, _{0},_{n+1},and_{0}=_{n+1}=0).

The elements of_{0}=_{n+1}=0.

The last elements of_{h}. In other words,_{h} gives the value of the load on the truck after visiting station_{h} of the route. The truck leaves the depot with no load; then, according to Hernández-Pérez and Salazar-González [

A feasible solution for the repositioning operation performed by the truck with the capacity of 10 bikes is shown in Figure

Solution representation.

In this paper, the initial feasible solution is quickly generated by a construction heuristic algorithm based on the penalty cost. The construction heuristic algorithm was first proposed by Ho and Szeto [

The approximate flow of the construction heuristic algorithm can be divided into two steps. Setp one, sort all stations in descending order by dividing them into pick-up and drop-off stations. The ranking criteria are

The neighborhood structure of local search is one of the most important components that determine the size of the search space and the quality of the final solution. The perturbation procedure is a diversification scheme that helps the search process to escape from local optima. In this paper, the local search in our proposed ILS is executed by a randomized variable neighborhood descent (RVND) procedure. RVND consists of systematically examining different types of neighborhoods in a random ordering of the neighborhoods. All possible moves of each neighborhood are examined and the search resumes from the best improving neighbor. If a neighborhood is not capable of finding an improved solution, the procedure then selects another one at random. The procedure ends when all neighborhoods fail to refine the current solution.

Focusing on the problem of determining the optimal truck route when the pickup and drop-off quantities are given by the upper-level model, the following neighborhood structures for the lower-level model were implemented.

Reinsertion (

Exchange (

2-opt: Two nonadjacent arcs are removed from the route and then two new ones are inserted into the route. In other words, a subsequence of the tour is reversed.

Insert: The move inserts a station which is not in the current solution into the current solution and then performs the repairing heuristic algorithm (see Section

Swap: The move replaces station

Delete: The move deletes a station from the current solution and then performs the repairing heuristic algorithm (see Section

The perturbation procedure randomly selects one of the following mechanisms:

Rand Reinsertion

Rand Exchange: Exchange

Rand Insert: The move inserts a station which is not in the current solution into the current solution and then performs the repairing heuristic algorithm (see Section

Rand Delete: The move deletes a station from the current solution and then performs the repairing heuristic algorithm (see Section

In this study, the bilevel BRP is solved as follows. With a given inventory level U, the truck route L can be found by solving the lower-level model using ILS (see Section

Local search and perturbation mechanism.

Algorithm | ||||||
---|---|---|---|---|---|---|

Neighborhood | ILS_LM | ILS_BL | TS_BL | |||

Local search | LS_LM | 2-opt | LS_BL | Insert | LS_BL | Insert |

Relocation | Swap | Swap | ||||

Exchange | Remove | Remove | ||||

| ||||||

Perturbation | Perb_LM | Rand Exchange | Perb_BL | Rand Insert | ||

Rand Relocation | Rand Delete |

Algorithm

ConsIter _LM≔0;

ConsIter _LM≔ ConsIter _LM +1;

Algorithm _{0} by applying the initialization procedure (see Section

_{0}≔

_{0}; ConsIter_BL≔0;

_{0}, LS_BL);

ConsIter _BL≔0;

ConsIter _BL≔ ConsIter _BL +1;

_{0}≔

The TS algorithm was introduced by Glover in 1986 and is an extension to the local search algorithm. It has been successfully applied to tackle a wide variety of routing problems, such as the traveling salesman problem, capacity vehicle routing problems, pickup and delivery problem, BRP [

TABU1: for a removal move, the tabu list records the station number which is removed from the current solution. When generating a neighborhood, if the station selected to be removed from the current solution exists in the tabu list, the move is marked as tabu. The length of the tabu list is denoted as_{1}.

TABU2: for an insertion move, the tabu list records the station number which is inserted into the current solution. When generating a neighborhood, if the station selected to be inserted into the current solution exists in the tabu list, the move is marked as tabu. The length of the tabu list is denoted as_{2}.

For a swap move, if station

In fact, after generating an initial solution, the following two steps are repeated until the stopping criterion is met. The stopping criterion of TS is achieving a predetermined number of iterations with no improvement (MaxConsIter_TS).

Generating a new solution: a temporary solution

Evaluating the new solution: in this step, the objective value of the new solution is compared with that of the best solution found. If the new solution improves the best solution, it is accepted and the best solution will be updated; otherwise, the new solution is accepted only if its move is not in the tabu list.

In this TS_BL procedure, let

_{0}≔

_{0}; _{0}; ConsIter_TS≔0;

obtained by applying the three local search moves).

Select a solution

non-tabu solution or a solution better than the best one so far (i.e.,

and update the tabu.

ConsIter_TS≔0;

ConsIter_TS≔ ConsIter_TS+1;

Sometimes the neighbor solution may no longer remain feasible with reference to the loading and capacity constraints after implementing the Insert and Delete neighborhood moves. Hence, the solution can be repaired by adjusting the pick-up/drop-off quantities at one of the stations on the new route visited by the truck.

Let

Let

Let

There are two rules to choose the station by adjusting the quantities to generate the feasible new solution:

Set

If we choose station

The heuristic algorithm presented in Section

The proposed algorithm ILS_BL relies mainly on two parameters, namely, the maximum consecutive iterations without improvement in the ILS_LM (MaxConsIter_LM) and the maximum consecutive iterations without improvement in the ILS_BL (MaxConsIter_BL). The TS_BL relies on four parameters, namely, the maximum consecutive iterations without improvement in the ILS_LM (MaxConsIter_LM), the maximum consecutive iterations without improvement in the TS_BL (MaxConsIter_TS), and the lengths of the two tabu lists (_{1 }and_{2}). The value of the parameters MaxConsIter_LM and MaxConsIter_BL in ILS, and the parameter MaxConsIter_TS in TS were defined as a function of

As pointed out by Forma et al. [_{i}. In the ‘‘heavy’’ case, the standard deviation is equal to 0.6_{i}. The ‘‘real’’ case corresponds to real initial inventories that were observed in the system at midnight on a random day.

Table

Characterization of the numerical instances.

| Initial cost | Ideal cost | |||||
---|---|---|---|---|---|---|---|

Light level | Δ | Real level | Δ | Heavy level | Δ | ||

20 | 159.9751 | 24.6652 | 159.4013 | 24.0914 | 251.0328 | 115.7229 | 135.3099 |

| |||||||

40 | 304.3676 | 40.6803 | 317.6306 | 53.9433 | 467.3612 | 203.6739 | 263.6873 |

| |||||||

60 | 458.1260 | 60.9716 | 462.8977 | 65.7433 | 707.0593 | 309.9049 | 397.1544 |

| |||||||

80 | 594.52439 | 68.0321 | 601.1442 | 74.6524 | 870.232 | 343.7424 | 526.4918 |

| |||||||

100 | 734.5029 | 86.7962 | 737.1299 | 89.4232 | 1048.1935 | 400.4868 | 647.7067 |

| |||||||

120 | 898.2554 | 118.6006 | 884.1184 | 104.4636 | 1286.6367 | 506.9819 | 779.6548 |

| |||||||

140 | 1057.5950 | 140.8475 | 1042.6900 | 125.9425 | 1510.8252 | 594.0777 | 916.7475 |

| |||||||

160 | 1203.5684 | 157.8831 | 1205.3941 | 159.7088 | 1699.9332 | 654.2479 | 1045.6853 |

| |||||||

180 | 1378.5382 | 195.3434 | 1388.4729 | 205.2781 | 1986.7625 | 803.5677 | 1183.1948 |

| |||||||

200 | 1566.6917 | 227.475 | 1548.7675 | 219.5508 | 2300.6803 | 971.4636 | 1329.2167 |

Table

Results comparison of different workloads (

| Light level | Real level | Heavy level | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Cpu | | | Best | Avg | Cpu | | | Best | Avg | Cpu | | | |

20 | 135.3294 | 135.3517 | 6.0980 | 99.9209 | 99.8305 | 138.1661 | 138.1661 | 5.9453 | 88.1443 | 88.1443 | 154.5146 | 154.5998 | 7.3875 | 83.4030 | 83.3310 |

| |||||||||||||||

40 | 270.2953 | 271.4314 | 5.7781 | 83.7563 | 80.9635 | 271.2084 | 271.824 | 4.0347 | 86.0574 | 84.9162 | 356.7850 | 357.7206 | 6.8385 | 54.2908 | 53.8314 |

| |||||||||||||||

60 | 421.6972 | 422.4598 | 4.4906 | 59.7472 | 58.4964 | 415.5485 | 418.1919 | 3.3344 | 72.0213 | 68.0005 | 593.7192 | 594.7404 | 5.5156 | 36.5725 | 36.2430 |

| |||||||||||||||

80 | 557.5409 | 558.8367 | 5.4187 | 54.3611 | 52.4564 | 553.2614 | 554.6676 | 4.8453 | 64.1410 | 62.2573 | 759.5554 | 761.2930 | 3.7969 | 32.1982 | 31.6927 |

| |||||||||||||||

100 | 699.3996 | 700.4135 | 3.2047 | 40.4434 | 39.2752 | 694.8413 | 696.0872 | 2.6360 | 47.2904 | 45.8971 | 936.6365 | 939.3733 | 5.8656 | 27.8554 | 27.1720 |

| |||||||||||||||

120 | 854.5804 | 856.4121 | 3.7313 | 36.8253 | 35.2809 | 839.5803 | 842.9962 | 3.0297 | 42.6350 | 39.3651 | 1181.2395 | 1184.1772 | 4.3281 | 20.7891 | 20.2097 |

| |||||||||||||||

140 | 1011.7422 | 1012.9844 | 3.3391 | 32.5549 | 31.6730 | 998.4752 | 999.0857 | 2.2234 | 35.1071 | 34.6224 | 1402.5349 | 1405.9053 | 3.0641 | 18.2283 | 17.6610 |

| |||||||||||||||

160 | 1157.7237 | 1159.6947 | 2.2297 | 29.0371 | 27.7887 | 1153.7849 | 1154.4173 | 1.4031 | 32.3146 | 31.9186 | 1585.4935 | 1592.0343 | 3.6047 | 17.4918 | 16.4921 |

| |||||||||||||||

180 | 1328.6067 | 1331.4690 | 2.0203 | 25.5609 | 24.0956 | 1335.9792 | 1337.0632 | 1.8750 | 25.5720 | 25.0439 | 1878.4544 | 1881.6833 | 2.5594 | 13.4784 | 13.0766 |

| |||||||||||||||

200 | 1504.3961 | 1505.5343 | 2.0094 | 22.9896 | 22.4892 | 1495.6188 | 1497.4245 | 1.8703 | 24.2079 | 23.3855 | 2190.1434 | 2192.0859 | 3.1266 | 11.3784 | 11.1784 |

| |||||||||||||||

Avg | | | | | | | | | |

To demonstrate the effectiveness of ILS_BL, comparisons of the performance of the ILS_BL and TS_BL were conducted. Table

Results comparison of ILS_BL and TS_BL (

| TS_BL | ILS_BL | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Cpu | | | Best | Avg | Cpu | | | | | |

20 | 138.4328 | 138.4328 | 7.4031 | 87.0373 | 87.0373 | 138.1661 | 138.1661 | 5.9453 | 88.1443 | 88.1443 | 1.1070 | 1.1070 |

| ||||||||||||

40 | 271.7736 | 272.4146 | 5.4016 | 85.0096 | 83.8213 | 271.2084 | 271.8240 | 4.0347 | 86.0574 | 84.9162 | 1.0478 | 1.0949 |

| ||||||||||||

60 | 419.9921 | 422.2647 | 4.8875 | 65.2623 | 61.8055 | 415.5485 | 418.1919 | 3.3344 | 72.0213 | 68.0005 | 6.7590 | 6.1950 |

| ||||||||||||

80 | 559.8575 | 560.9618 | 5.1563 | 55.3053 | 53.8260 | 553.2614 | 554.6676 | 4.8453 | 64.1410 | 62.2573 | 8.8357 | 8.4313 |

| ||||||||||||

100 | 698.6127 | 699.7694 | 3.0641 | 43.0729 | 41.7794 | 694.8413 | 696.0872 | 2.6360 | 47.2904 | 45.8971 | 4.2175 | 4.1177 |

| ||||||||||||

120 | 843.4800 | 846.9870 | 3.4813 | 38.9020 | 35.5448 | 839.5803 | 842.9962 | 3.0297 | 42.6350 | 39.3651 | 3.7330 | 3.8203 |

| ||||||||||||

140 | 1002.6325 | 1003.3431 | 2.3906 | 31.8062 | 31.2420 | 998.4752 | 999.0857 | 2.2234 | 35.1071 | 34.6224 | 3.3009 | 3.3804 |

| ||||||||||||

160 | 1155.8360 | 1156.7478 | 1.6354 | 31.0303 | 30.4594 | 1153.7849 | 1154.4173 | 1.4031 | 32.3146 | 31.9186 | 1.2843 | 1.4592 |

| ||||||||||||

180 | 1337.7577 | 1339.0367 | 2.0235 | 24.7056 | 24.0825 | 1335.9792 | 1337.0632 | 1.8750 | 25.5720 | 25.0439 | 0.8664 | 0.9614 |

| ||||||||||||

200 | 1498.2623 | 1499.3265 | 2.2625 | 23.0039 | 22.5192 | 1495.6188 | 1497.4245 | 1.8703 | 24.2079 | 23.3855 | 1.2040 | 0.8663 |

| ||||||||||||

Avg | | | | | | | | |

In this section, we are interested in evaluating the impact of the truck capacity, the loading and unloading time per bike, and the repositioning time.

Table

Results comparison of different vehicle capacities

| | | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Cpu | | | Best | Avg | Cpu | | | | | |

20 | 138.1661 | 138.1661 | 5.9453 | 88.1443 | 88.1443 | 138.1661 | 138.1661 | 6.9360 | 88.1443 | 88.1443 | 0 | 0 |

| ||||||||||||

40 | 271.2084 | 271.824 | 4.0347 | 86.0574 | 84.9162 | 273.2792 | 274.2790 | 5.6156 | 82.2186 | 80.3651 | -3.8388 | -4.5511 |

| ||||||||||||

60 | 415.5485 | 418.1919 | 3.3344 | 72.0213 | 68.0005 | 419.4120 | 419.7309 | 3.9047 | 66.1447 | 65.6596 | -5.8766 | -2.3409 |

| ||||||||||||

80 | 553.2614 | 554.6676 | 4.8453 | 64.1410 | 62.2573 | 556.7589 | 558.0334 | 4.9063 | 59.4560 | 57.7487 | -4.6850 | -4.5086 |

| ||||||||||||

100 | 694.8413 | 696.0872 | 2.6360 | 47.2904 | 45.8971 | 694.9719 | 697.3861 | 3.4297 | 47.1444 | 44.4446 | -0.1460 | -1.4525 |

| ||||||||||||

120 | 839.5803 | 842.9962 | 3.0297 | 42.6350 | 39.3651 | 841.0893 | 843.126 | 4.3875 | 41.1905 | 39.2408 | -1.4445 | -0.1243 |

| ||||||||||||

140 | 998.4752 | 999.0857 | 2.2234 | 35.1071 | 34.6224 | 1001.7933 | 1004.6089 | 3.2047 | 32.4725 | 30.2369 | -2.6346 | -4.3855 |

| ||||||||||||

160 | 1153.7849 | 1154.4173 | 1.4031 | 32.3146 | 31.9186 | 1160.7397 | 1162.2074 | 1.7469 | 27.9599 | 27.0409 | -4.3547 | -4.8777 |

| ||||||||||||

180 | 1335.9792 | 1337.0632 | 1.8750 | 25.5720 | 25.0439 | 1336.9554 | 1340.2226 | 2.7328 | 25.0964 | 23.5048 | -0.4756 | -1.5391 |

| ||||||||||||

200 | 1495.6188 | 1497.4245 | 1.8703 | 24.2079 | 23.3855 | 1497.3002 | 1498.4470 | 3.2828 | 23.4421 | 22.9198 | -0.7658 | -0.4657 |

| ||||||||||||

| | | | | | | | |

Table

Results comparison of different unloading and loading service times (

| | | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Cpu | | | Best | Avg | Cpu | | | | | |

20 | 138.1661 | 138.1661 | 5.9453 | 88.1443 | 88.1443 | 138.1661 | 138.1661 | 6.8984 | 88.1443 | 88.1443 | 0 | 0 |

| ||||||||||||

40 | 271.2084 | 271.824 | 4.0347 | 86.0574 | 84.9162 | 264.9843 | 265.3725 | 5.8828 | 97.5956 | 96.8760 | 11.5382 | 11.9598 |

| ||||||||||||

60 | 415.5485 | 418.1919 | 3.3344 | 72.0213 | 68.0005 | 409.9184 | 411.0709 | 4.7121 | 80.5851 | 78.8321 | 8.5638 | 10.8316 |

| ||||||||||||

80 | 553.2614 | 554.6676 | 4.8453 | 64.1410 | 62.2573 | 545.789 | 547.3909 | 5.9654 | 74.1506 | 72.0048 | 10.0096 | 9.7475 |

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100 | 694.8413 | 696.0872 | 2.6360 | 47.2904 | 45.8971 | 683.9921 | 684.5961 | 3.7500 | 59.4228 | 58.7474 | 12.1324 | 12.8503 |

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120 | 839.5803 | 842.9962 | 3.0297 | 42.6350 | 39.3651 | 832.4907 | 833.3217 | 3.7250 | 49.4217 | 48.6262 | 6.7867 | 9.2611 |

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140 | 998.4752 | 999.0857 | 2.2234 | 35.1071 | 34.6224 | 987.7048 | 988.9890 | 4.0281 | 43.6590 | 42.6393 | 8.5519 | 8.0169 |

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160 | 1153.7849 | 1154.4173 | 1.4031 | 32.3146 | 31.9186 | 1141.8162 | 1142.4030 | 3.6500 | 39.8086 | 39.4411 | 7.4940 | 7.5225 |

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180 | 1335.9792 | 1337.0632 | 1.8750 | 25.5720 | 25.0439 | 1322.5730 | 1324.8880 | 2.0813 | 32.1027 | 30.9749 | 6.5307 | 5.9310 |

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200 | 1495.6188 | 1497.4245 | 1.8703 | 24.2079 | 23.3855 | 1484.8130 | 1486.3200 | 2.0156 | 29.1297 | 28.4432 | 4.9218 | 5.0577 |

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Avg | | | | | | | | |

Table

Results comparison of different repositioning times (

| | | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg | Cpu | | | Best | Avg | Cpu | | | | | |

20 | 138.1661 | 138.1661 | 5.9453 | 88.1443 | 88.1443 | 138.1661 | 138.1661 | 7.1078 | 88.1443 | 88.1443 | 0 | 0 |

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40 | 271.2084 | 271.824 | 4.0347 | 86.0574 | 84.9162 | 263.8787 | 264.0471 | 6.1563 | 99.6452 | 99.3330 | 13.5878 | 14.4168 |

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60 | 415.5485 | 418.1919 | 3.3344 | 72.0213 | 68.0005 | 404.8966 | 405.6153 | 4.8651 | 88.2236 | 87.1304 | 16.2023 | 19.1299 |

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80 | 553.2614 | 554.6676 | 4.8453 | 64.1410 | 62.2573 | 541.4491 | 542.3618 | 6.3177 | 79.9641 | 78.7415 | 15.8231 | 16.4842 |

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100 | 694.8413 | 696.0872 | 2.6360 | 47.2904 | 45.8971 | 674.1251 | 675.9928 | 4.1979 | 70.4569 | 68.3683 | 23.1665 | 22.4712 |

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120 | 839.5803 | 842.9962 | 3.0297 | 42.6350 | 39.3651 | 822.3390 | 823.2306 | 4.5764 | 59.1396 | 58.2861 | 16.5046 | 18.9210 |

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140 | 998.4752 | 999.0857 | 2.2234 | 35.1071 | 34.6224 | 976.7542 | 977.8183 | 3.6290 | 52.3539 | 51.5090 | 17.2468 | 16.8866 |

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160 | 1153.7849 | 1154.4173 | 1.4031 | 32.3146 | 31.9186 | 1131.2477 | 1132.9226 | 2.2207 | 46.4260 | 45.3773 | 14.1114 | 13.4587 |

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180 | 1335.9792 | 1337.0632 | 1.8750 | 25.5720 | 25.0439 | 1309.3586 | 1310.8218 | 2.8672 | 38.5401 | 37.8273 | 12.9681 | 12.7834 |

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200 | 1495.6188 | 1497.4245 | 1.8703 | 24.2079 | 23.3855 | 1474.0601 | 1475.6522 | 2.6250 | 34.0274 | 33.3022 | 9.8195 | 9.9167 |

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Avg | | | | | | | | |

The bike repositioning problem plays an important role in bike sharing systems. In this paper, the bilevel model we have presented offers a new research direction and field for studying BRP under the situation that BSS providers outsource the transportation between bike stations to the third-party logistics companies (the truck owners). In this bilevel model, the upper-level decision maker is the BSS provider whose goal is to minimize the total penalties cost caused at all stations, while the truck owner as the lower-level decision maker aims to minimize the travel cost. We proposed the ILS and TS to solve the bilevel model.

Computational experiments are set up to illustrate the performance of the proposed algorithm and the properties of the problem. In particular, the results show that ILS outperforms TS in its solution quality and convergence speed. Moreover, the results also demonstrate that the size of the repositioning truck, the value of loading/unloading time for a single bike, and the length of repositioning time can significantly affect the total penalties cost of BSS provider.

This study can be extended in several directions in the future: first, multiple repositioning trucks used by the third-party logistics companies can be considered to make this study more practical. Second, this problem will be more practical and flexible if some assumptions are relaxed. For example, it will also be more practical to relax the assumption that a station will be visited by a vehicle more than once. Third, it would be interesting to investigate the impact of temporary loading/unloading bikes at balanced stations.

The data used in this study can be accessed from the web site

The authors declare that they have no conflicts of interest.

The authors acknowledge the National Natural Science Foundation of China (Grant no. 71271220) and the Hengyang Social Science Foundation Project (Grant no. 2017D031).