Optimal Parking Slots Reservation and Allocation Problem for Periodic Parking Platforms with Preference Constraints

Various solutions, such as parking reservation systems, have been proposed to alleviate the difculty in fnding parking slots. In such systems, parking requests are submitted in advance by drivers, and the systems will reserve appropriate parking spots for drivers if their requests are accepted. However, the parking slots may be allocated unreasonably, which may lead to a waste of space and time resources. In addition, there is a game relationship between operator’s proft (OP) and users’ benefts (UB), which may afect the sustainable development of the system, if balanced improperly. Given the drivers’ arrival and departure time and their parking preference, the paper proposes a periodic reservation and allocation mode (PRAM) and establishes a dual-objective binary integer linear model to solve the reservation and allocation problem. Te model aims to maximize the comprehensive benefts of the operator and users and to take full advantage of parking resources. We proposed a TOPSIS-SA algorithm (Technique for Order Preference by Similarity to an Ideal Solution and Simulated Annealing algorithm) to solve our model. Numerical experiments show that our model performs better than the baseline models on all performance metrics such as total operating proft, users’ average walking distance, acceptance rate, and utilization of parking slots.


Introduction
With the acceleration of urbanization, car ownership in metropolises has increased rapidly, resulting in a growing shortage of parking slots in large cities.In addition, the opacity of parking information and the low utilization of parking slots also contribute to the parking problem.As a result, drivers often spend a lot of time searching for parking slots that meet their needs, which makes cruising trafc increase.Te desperate scramble for parking slots also adds to trafc congestion and environmental pollution.As found in a global parking survey by IBM, it took drivers more than 20 minutes on average to fnd a satisfactory parking slot [1].Shoup found that 30% of trafc congestion in road networks is caused when people are circulating around to fnd a parking spot, and vehicles in a small Los Angeles neighbourhood burn 47,000 gallons of gasoline looking for parking spots, producing 730 tons of carbon dioxide and a total of 945,000 miles extra travel within a year [2].
Due to the limitation of urban scale, parking problems are difcult to solve by continually building new parking facilities.Tus, from the perspective of parking demand management, various methods have been proposed including levying some road toll to the vehicle [3][4][5] on the basis of trafc fow studies [6][7][8], giving priority to the development of public transport [9,10] and developing high-occupancy vehicle lanes [11,12].However, with the continuous growth of car ownership, the above methods gradually show limitations.Terefore, the parking reservation systems, which can efectively improve the parking efciency and reduce parking cruising, have become an efective strategy to alleviate parking problems.Parking reservation was proposed and studied as a subsystem of the parking guidance system by Inabak [13].Parking reservation systems provide drivers with information inquiry and parking slots reservation services through SMS [14] or online network [15][16][17].Some reservation systems support recommending appropriate parking slots for drivers, making parking more convenient and efcient.Yang and Liu found out that trafc congestion at the bottleneck can be temporally relieved by combining reserved and unreserved parking spots [18,19].
Parking allocation plays a decisive role in optimizing the utilization of parking slots and improving the operating efciency of parking lots.In this way, parking difculties can be efectively alleviated through reasonable parking allocation.Many studies have focused on how to properly allocate parking slots.Te studies can be categorized into two categories.Te frst category of studies focuses on allocating slots based on user optimum.Shin proposed a smart parking guidance algorithm which supports drivers to fnd the most appropriate parking facility [20].To suggest the most suitable parking facilities, he considered several factors, such as driving distance and walking distance.Ten, he proposed a real-time parking reservation service and formulated a mixed-integer programming model to minimize the total travel cost of all users [21].In order to determine the optimal sites of street parking facilities in a working area and minimize the total queue delay, Du presented a method of mathematical programs with equilibrium constraints [22].Said used game theory to model the parking solution of the proposed reservation system and solved the main problems faced by car drivers in looking for available parking slots, such as parking fees, the amount of walking, parking duration, and so on [23].
Te second category of studies focuses on allocating slots based on system optimum.Operator's proft (OP) and the utilization of parking slots are considered most in the parking allocation models.Zhang and Liu considered operator's income and expenditure in the parking allocation models [24,25].Xue et al. further considered the impact of rejecting requests on the platform and added the penalty fee into their objective functions to avoid high rejection rate [26][27][28].To reduce cruising trafc, Gao et al. integrated shared parking into the ride-sourcing platform and proposed a platform proft maximization novel business model [29].Xiao and Xu considered a parking market with several kinds of parking players and established a model to pursue the balance between the profts of parking platforms [30].Zhao et al. and Tang et al. defned the utilization of parking slots as the ratio of the total occupied duration to the total available time of all the parking slots and maximized the utilization of parking slots in parking allocation [31,32].Similarly, Wang maximized the utilization of parking slots in allocation by considering the expectation of the total number of occupied parking hours in his model.Some scholars also studied how to achieve the idea of system optimum from other angles [33].In order to maximize the number of vehicles that can be accommodated in the area, a shared allocation model for night parking between residential area and nearby business district was established by Hu [34].Aiming at balancing the parking demand among multiple public facilities, Kim established a parking allocation model to alleviate the overloading caused by the imbalance of parking demands among multiple parking lots [35].Similarly, to solve the spatiotemporal imbalance in parking space utilization, Su considered the shared parking plan in the era of autonomous vehicles with parking autonomy [36].Considering the dynamic parking demand and drivers' probabilistic choice of each parking lot, Wang established a parking allocation model under dynamic parking fee [37].To minimize the total travel cost of the system, Zhang et al. examined optimal capacity allocation strategies under system optimum and user optimum, respectively [38].
Te real-time reservation and allocation mode (RRAM) is used in most user-optimum studies.In such a real-time mode, a user can submit his/her request at any time, and when the request is received, a parking slot will be allocated immediately.Te timeliness of allocation is guaranteed in such a mode.However, the platform will lose the global view and the parking resources will not be fully utilized.In particular, the problem of demand is getting worse in commuting situations.Te system needs enough time to develop a reasonable allocation scheme.Compared with RRAM, the periodic reservation and allocation mode (PRAM) which can guide the travel modes of citizens and adjust the distribution of parking demand is commonly required.More periodic allocation models for parking reservation services, such as same-day parking and dayahead allocation, have been developed, which guarantee the full utilization of parking resources and require users to submit requests in advance.
Moreover, on the one hand, the studies of user optimum ignored that each user has his/her own expectation and the allocation results may not meet users' expectations.As a result, users may cancel their parking reservations after receiving the allocation results.On the other hand, the allocation of both private shared parking slots and public parking slots considered only the user optimum or the system optimum in the aforementioned studies.Tey ignored that it is equally important to ensure the benefts of both the e-parking operators and the users in the current situation.Ensuring the OP is conducive to the long-term service provided by the platform, users' benefts (UB) should also be guaranteed so that users will use the parking allocation platform for a long time.
Terefore, to address the above limitations, in this paper, we investigate how to fulfll the periodic service, i.e., the prebooked allocation services, by proposing a PRAM.Ten, to avoid cancellations of user reservations due to dissatisfaction with the allocated parking slots, we establish the constraints on users' parking preference.Considering the game relationship between OP and UB, a dual-objective binary integer model is established to solve the parking resource optimization problem.However, we combine the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method and simulated annealing algorithm and propose a TOPSIS-SA algorithm which allows multiple objectives to approach their optimal levels to solve the model.Te algorithm allows multiple objectives to approach their optimal levels.At the same time, falling into local optima can also be efectively avoided.Te superiority of our model is validated by comparing it with two baseline models.In addition, sensitivity analyses are conducted and the matching of parking supply and demand is studied.We also 2 Journal of Advanced Transportation provide recommendations for the optimal penalty factor, the optimal supply-demand ratio, and the optimal minimum utilization threshold for platform operators that can be employed to alleviate parking problems and develop smart parking platforms.Te remainder of the paper is organized as follows.Section 2 presents the reservation mechanism and the allocation problem and establishes a dual-objective binary integer model to solve the problem.In addition, we provide two baseline models for comparison.Section 3 presents the solution algorithm for the optimal allocation model.Performance metrics and numerical experiments are presented in Section 4, and conclusions and suggestions for future research are given in Section 5. Te list of nomenclature in this paper can be found in Table 1.

Mathematical Model
2.1.Reservation Mechanism and Variables.Figure 1 is used to describe the reservation mechanism.Te problem concerns a travel zone with multiple user destinations and heterogeneous parking lots.Suppose the platform provides a certain number of available parking slots in several parking lots.Parking slots have the same available time window, but the charge rate is diferent in diferent parking lots.Parking requests are submitted by users in advance (within the period provided by the platform, e.g., before 22 : 00 the previous night) through the parking reservation platform supported by map platforms.Users can search for their destinations and submit parking preferences.After receiving a request, the reservation and allocation system judges whether the user's demand for the parking duration and parking preference can be satisfed.If the demand cannot be satisfed, the system will inform the user on the application interface that there is no parking slot that meets the requirements.If the demand can be satisfed, the system will notify the user that the application has been submitted successfully and inform the user of the request delivery time.Te request will then be placed in the allocation pool.Te system will complete the parking allocation and update the time window supply of the parking slots.Finally, the allocation results will be presented to users within the specifed period so that those who are rejected have enough time to plan for other alternatives.
Our model is based on the following assumptions: (1) Users submit their parking duration based on the time interval given by the platform and they strictly abide by the parking duration they submitted (2) Users will not cancel the reservation due to personal factors such as itinerary change For simplicity, we divide the daily available time period T into a number of intervals (t min each interval) in our parking system.Let K be the total number of intervals.As a result, k � 1 is the frst time interval (e.g., 8 : 00-8 : 30) and k � K is the last time interval (e.g., 21 : 30− 22 : 00).Let J be the total number of the parking lots rented by the platform.Let N be the total number of the parking slots, and we introduce a binary indicator a nj which is defned to be 1 if parking slot n belongs to parking lot j and 0 otherwise.Tus, we have the parking slot distribution matrix A N×J � [a nj ], where n ∈ [1, N] and j ∈ [1, J].Similarly, s nk which is defned to be 1 if parking slot n is available in time interval k and 0 otherwise is introduced and the parking supply matrix Let M ′ denote the total number of parking requests and let t start m ′ and t end m ′ be the start and end time interval of request We further introduce a binary indicator d m ′ k which is defned to be 1 if parking request m ′ includes time interval k and 0 otherwise.Tus, we have the initial parking demand matrix Let (x j , y j ) be the central coordinate of parking lot j and let (x m ′ , y m ′ ) be the central coordinate of the destination of request m ′ .For simplicity, we take the linear distance between the parking lot j and the destination of request m ′ as the walking distance of request m ′ after parking.Let l m ′ j to be the walking distance of request m ′ which can be calculated as follows: Let f j (yuan/h) and p j (yuan/ da y) denote the charge rate of parking lot j and purchase price of parking lot j, respectively.We select the walking distance after parking and charge rate, which are the most concerned indicators of parking users, as the parking preferences of users.Users can submit the maximum acceptable walking distance l max m ′ and the maximum acceptable charge rate f max m ′ when they use the platform.Tus, the constraint of users' parking preferences can be expressed as follows: Since there are situations where the existing parking resources cannot satisfy the parking duration and parking preferences of some users, these requests will not be put into the allocation pool.Let M be the total number of requests in the allocation pool and update the index number of the requests in the allocation pool to m. Tus, we have the fnal parking demand matrix We then introduced a decision variable x mn which is defned to be 1 if request m is allocated to parking slot n and 0 otherwise.Tus, we have the parking slot allocation matrix Ten, based on the parking slot distribution matrix A N×J , we have the following parking lot allocation matrix, C M×J � [c mj ] � X M×N × A N×J , or the following equation: Clearly, c mj � 1 if request m is allocated to parking lot j; otherwise, c mj � 0.

Operating Proft Optimum (PO).
From the perspective of OP, the platform operator aims to maximize the operating proft.However, the platform's blind pursuit of profts may lead to high rejection rates and low service levels.Terefore, we introduce a rejection penalty factor μ (yuan/request) to avoid a low service level.We further assume that the impact of the platform investment at the early stage and the expenses incurred in the operation management process on the OP can be negligible.Tus, from the perspective of OP, we can formulate the reservation and allocation problem as follows: x mn , d mk , s nk , a nj ∈ 0, 1 In equation ( 5), the frst term is the sum of parking fees paid by the users assigned to the parking slots, representing the total revenue of the platform; the second term is the total cost for purchasing the parking slots; and the third term is the product of the number of rejected users and the rejection penalty factor, representing the long-term loss due to request rejection, where (M −  M m�1  N n�1 x mn ) is the total number of rejected requests.In the constraint set, equations ( 6) and (7) are the constraints of users' parking preferences; equation (8) simply indicates that the parking duration of users must be within the system available interval; the time window constraint must be met in the process of parking allocation, which means the parking slots must be idle during the parking duration in the allocation scheme; therefore, equation (9) guarantees that any parking slot should accommodate at most one car in each time interval; our problem is a special assignment problem in which each request can only be accepted or rejected; hence, equation (10) guarantees that each request can be allocated to at most one parking lot; similarly, the parking slots constraint must be met in the process of parking allocation, which means the number of requests accepted in each time interval cannot exceed the total number of the parking slots in the allocation scheme, as is shown in equation (11); equation (12) simply implies x mn , d mk , s nk , and a nj are binary variables.

Walking Distance Optimum (WO).
From the perspective of UB, we choose to minimize the average walking distance, which can be formulated as follows: Te constraint set of equation ( 13) is the same as equation (5).

Te Optimal Allocation Model (OM).
We aim to pursue comprehensive benefts of operator and users under the game relationship between OP and UB.Tus, the optimal allocation model can be formulated as follows: ... Journal of Advanced Transportation

Operator
x mn ≤ 1

Baseline Models
For the purpose of comparison, we consider two baseline models of parking allocation which are frst-come-frst-serve (FCFS) and frst-book-frst-serve (FBFS).FCFS means that the platform processes requests according to the actual arrival order of the users.Te allocation algorithm for FCFS is shown in Algorithm 1.
To minimize the individual-level waiting time, the FBFS applies a myopic strategy.When a parking request is submitted by a user, the platform immediately allocates a parking spot.Te allocation algorithm for FBFS is shown in Algorithm 2.

Solution Algorithm for the Optimal Allocation Model.
Te optimal allocation model established in this paper belongs to the dual-objective multidimensional combinatorial optimization problems.We pursue the maximization of bilateral benefts under the game relationship between OP and UB.Terefore, the TOPSIS method, which pursues multiple objectives to approach their optimal level, is chosen to normalize our dual-objective optimal parking allocation model.Te idea of normalization can be described as follows.
We frst solve the two single-objective problems which are PO and WO separately.Ten, we assume that the optimal solutions of the two problems are max is called an ideal point in the solution set because it is always hard to reach.Tus, we assume ω which is the closest to ω * as an approximate optimal solution under the measure that we expect.One of the most direct methods is the shortest TOPSIS method, so we establish the following equation: Ten, we need to fnd the solution that is closest to the ideal point in the solution set.Tus, φ[ω(x mn )] needs to be minimized, which can be expressed by equation (16).Te optimal solution of equation ( 16) is taken as the optimal solution of OM.
As a result, the dual-objective multidimensional combinatorial optimization problem is transformed into three single-objective optimization problems.When solving the single-objective optimization problems, precise analytical algorithms such as enumeration method, dynamic programming method, and branch and bound method are efective methods in the case of small-dimensional problems.However, the time consumed will increase exponentially as the size of solving scale grows and the operational efciency will decrease rapidly, making it difcult to quickly obtain the optimal solution and easy to fall into the local optimal solution.Tus, we choose the simulated annealing algorithm, which is more efcient, as our solution algorithm.
Te simulated annealing algorithm is a kind of serial structure optimization algorithm that eventually tends to global optimal.By simulating the annealing process of solid in physics, the algorithm gives a time-varying probability jump that eventually tends to zero for the search process, thus efectively avoiding falling into local optimal.Te algorithm starts from a certain high initial temperature.With the continuous decrease of temperature parameters, it randomly fnds the global optimal solution of the objective function in the solution set combined with the probabilistic jump characteristics.Te basic process of TOPSIS-SA is shown in Algorithm 3.

Numerical Experiments
In this section, we conduct numerical experiments to conduct quantitative studies and illustrate the models and results.

Performance Metrics.
To evaluate the superiority of OM, four performance metrics including operating proft, average walking distance of users, utilization of parking slots, and acceptance rate are used to evaluate the model.
Operating proft and acceptance rate are the performance metrics that e-parking platform operators concern most.Tere are two kinds of operating proft which are total operating proft which has been given in equation ( 5) and actual operating proft ω 1 ′ which is the total operating proft without taking the loss of rejection into account.

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Journal of Advanced Transportation (1) Input: D M×K , S N×K , A N×J .

Journal of Advanced Transportation
Te average walking distance of users is used to measure whether the users' benefts are fully considered, which can be calculated as follows: Te utilization of parking slots is an important metric to measure whether the parking slots are fully utilized, which means the ratio of the total occupied duration to the total supply duration.It can be calculated as follows: Te acceptance rate refects the service level of the parking reservation platform.It is the ratio of the number of requests received to the number of requests in the allocation pool.It can be calculated as follows:

Computation Results on a Basic
Example.Suppose the time interval k is 0.5 h, and the daily available time period is T � 14 h (8 : 00-22 : 00).Tus, the total number of the time interval is K � 28.For simplicity, we suppose that a reservation and allocation cycle is 24 h, which means that users submit requests at least before 22 : 00 the previous night, and the system will allocate parking slots for all requests at once.Te allocation results will be presented to the users by 23 : 00 the previous night, giving rejected users enough time to plan for alternative options.In the base case, we suppose that our study area is a 500 m × 500 m region from (0, 0) to (500, 500) in the two-dimensional coordinate system.Suppose the platform buys 25 parking slots from parking lot 1 and 25 parking slots from parking lot 2 and receives 500 parking requests.Terefore, N � 50 and M ′ � 500.Te parking fee is assumed to be i 1 � 8 (yuan/h) per vehicle and i 2 � 4 (yuan/h) per vehicle.Te purchase cost is p 1 � 40 (yuan/day) per slot and p 2 � 20 (yuan/day) per slot, and the penalty factor is μ � 4 (yuan/request).Furthermore, we suppose that the arrival time of users follows a Poisson distribution, and the parking duration follows an exponential distribution, as usually considered in the literature [39].Te average parking duration is assumed to be 3 h.For simplicity, suppose that 50% of the users whose maximum acceptable walking distance is 300 m and the rest of users' maximum acceptable walking distance is 500 m.Suppose that 50% of the users whose maximum acceptable parking fee is 6 (yuan/h) and the rest of the users' maximum acceptable parking fee is 10 (yuan/h).Te central coordinates of parking lot 1 and parking lot 2 are assumed to be (100, 100) and (400, 400), respectively.Te travel destinations of users are randomly distributed in the study areas.
We use MATLAB R2016a to generate the experimental data including arrival time, parking duration, travel destination, and parking preference, as shown in Figures 2 and 3: PO and WO are also chosen to compare with OM. Figure 4 shows the assignment results of a basic example.In Figure 4, the x-axis and the y-axis represent the time interval and the parking slots, respectively.Occupied parking slots are indicated by coloured blocks and are labeled with "1" and free parking slots are indicated by white blocks and are labeled with "0." Table 2 presents the computation results of the fve strategies.
It can be clearly seen from the allocation results that the allocation scheme of OM signifcantly outperforms that of FCFS and FBFS on all four performance metrics.Compared with PO, OM can shorten the average walking distance by 21.90% while sacrifcing 10.44% of the proft.In terms of acceptance rate and utilization of parking slots, the performance of PO and OM is basically the same.Compared with WO, OM can increase the operating proft by 66.53% while increasing average walking distance by 24.40%.Meanwhile, OM's acceptance rate is 14% higher than WO's.Tus, OM can take both OP and users' walking distance into consideration and realize the comprehensive optimal of OP and UB.

Extended Experiments.
To further test the capabilities of OM and conduct sensitivity analysis, we expand the number of parking slots in each parking lot to 50 and vary the number of parking requests from 0 to 2000.Te rest of the experimental setup is consistent with the basic experiment.
From Figure 5, we can see that the performance of OM in the extended experiment is basically consistent with that in the basic experiment.OM's total operating proft is second only to PO's and signifcantly better than that of the other three methods; the acceptance rate of OM is basically consistent with that of PO, and obviously better than that of the other three methods; OM's average walking distance is second only to WO's, and obviously less than that of the other three methods; in terms of parking slots utilization, OM performs slightly better than the other four methods.
From Figure 6(a), we can see that the operator is at a defcit when there are a few parking requests.As the number of requests increases, the total operating proft increases linearly, and the total operating proft gradually turns from a loss to a proft when the number of requests is around 300.When the number of requests exceeds 500, the total operating proft presents diferent changing rules under μ � 0 and μ � 4, which means the efect of the penalty factor is gradually manifested.Te total operating proft starts to grow at a slower speed under μ � 4. Te total operating proft reaches its maximum when the number of requests approaches 800.Te total operating proft starts to decrease as the number of requests continues to increase because the parking supply is less than the parking demand, resulting in more and more users being rejected by the platform.Te penalty factor thus directly leads to a decrease in the total operating proft.However, in the absence of penalty term (μ � 0), the operating proft starts to grow at a slower speed and approaches the maximum when the number of requests is over 800, which means that it is hard to increase the OP by accepting more requests when the parking demand exceeds the parking supply.
From Figure 6(b), we can also see that the parking supply can meet the parking demand when the number of parking requests is below 800 and the inhibitory efect of the penalty factor is inconspicuous.However, as the number of parking requests continues to increase, the parking supply cannot meet the parking demand.As a result, the acceptance rate keeps decreasing under μ � 0 and the acceptance rate starts to decrease slower under μ � 4, which means the inhibitory efect of punishment factor is manifested.
From Figure 7, we can see that the average walking distance and the utilization of parking slots show the same trend under μ � 0 and μ � 4, which means that the two performance metrics are not afected by the penalty factor.Meanwhile, from Figure 7(a), we can see that the change of average walking distance tends to be stable after the number of requests reaches 1000, indicating that the average walking Journal of Advanced Transportation distance has reached the optimal level under the current supply and demand condition, which is difcult to be further optimized.
To study the efect of penalty factor on the platform, we vary the penalty factor from 0 to 5 in four situations of short supply (M ′ � 800, 1200, 1600, and 2000) and observe the change in total operating proft and acceptance rate which is shown in Figure 8.In Figure 8, it is obvious that the total operating proft decreases with the increase of penalty factor and the acceptance rate increases with the increase of penalty factor, which means that with the increase of the penalty factor, the restriction efect on the model's rejection of requests in pursuit of proft will also signifcantly increase.However, when the penalty is higher than 3.0, the acceptance rate starts to increase more slowly, while the total operating proft decreases linearly, indicating that it is difcult to continuously improve the restriction efect by increasing the penalty factor.Tus, we suggest that the value of penalty factor should be set within the range of 2.0 to 3.0 (yuan/ request).
For each combination of demand or supply, we can fnd the maximum proft by solving OM. Figure 9 plots the change of ω 1 in the two-dimensional space of parking demand and supply under diferent penalty factors, respectively.From Figure 9, the optimal ratio of N/M′ can be found.Tis is signifcant because it suggests the number of parking slots that the platform operator should purchase for a given or predicted number of parking requests.It also suggests the number of parking requests that the platform operator should accept for a given number of parking slots.Comparing Figure 9 (μ � 0) and (μ > 0), we can see that the optimal ratio changes slightly, indicating that the penalty    Journal of Advanced Transportation factor has little impact on the optimal number of parking slots for a given demand.However, when the rejection loss is neglected, the slope will be moderate for a given supply.Te minimum utilization threshold U min for parking slots is introduced to study the impact of improving the utilization of parking slots on the platform.From Figure 10, we can see that after U min is above 70%, with the increase of U min , the actual operating proft, the average walking distance, and the acceptance rate decrease.Te acceptance rate decreases more rapidly and the slope of average walking distance gets gentle when U min is above 80%.However, the acceptance rate decreases signifcantly when U min is above 90%.Tus, we can fnd out that more requests received by the platform can improve the actual operating proft, the utilization of parking slots, and the average walking distance, since an increase in the number of requests favors better  selection in parking allocation.However, at the same time, the acceptance rate will decrease, leading to a decrease in service level, which will decrease faster with the increase of demands.Terefore, we suggest that the value of U min should be set within the range of 70% to 80%.

Conclusions and Future Research
Considering the parking problems caused by the imbalance between parking supply and parking demand, we examined the regional parking allocation problem through an eparking-platform.A dual-objective binary integer linear programming model is proposed to allocate certain parking requests to specifc parking slots in order to pursue full use of parking resources and comprehensive optimal of OP and UB.We then propose a TOPSIS-SA algorithm to solve the model.Te OM method is compared with FCFS, FBFS, PO, and WO methods in numerical experiments.Te results show that the OM model signifcantly outperforms FCFS and FBFS on all four performance metrics.In addition, the OM method can decrease the average walking distance by 21.90% compared with the PO model and it can improve the operating proft by 66.53% compared with the WO model which means the OM model realize the comprehensive optimal of OP and UB.At the same time, the sensitivity analyses are conducted and the matching of parking supply and demand is studied.We fnd that the optimal penalty factor should be set within 2.0 to 3.0 and the optimal combination of supply and demand can be found in the contour plot (Figure 9).Moreover, the contour plot demonstrates that the penalty factor has little impact on the optimal number of parking slots for a given demand.In addition, the optimal minimum utilization threshold for parking slots is suggested to be within 70% to 80% so that the service level is guaranteed while the utilization of parking slots is improved.
In addition, this study considers that balancing the parking demand among multiple parking lots is conducive to alleviate the trafc congestion and improve the operating efciency of parking lots [35,36].It is of our interest to incorporate balancing the parking demand among multiple parking lots in our future research so that the trafc congestion caused by parking problems can be further alleviated and the parking efciency can be further improved.Moreover, the uncertainty in drivers' arrival/departure time is not taken into account in this study.However, the driver may arrive earlier or depart later, which may cause service failure [33].In this context, we consider pursuing the comprehensive optimal of OP and UB while addressing the parking unpunctuality.However, the optimal operation decisions under diferent operation objectives will be affected by such a trade-of.Tus, this idea should be examined in the further studies.In addition, case verifcation should be carried out in future research to realize the practical application of the parking reservation and allocation system.

Figure 2 :Figure 3 :
Figure 2: Te distribution of users' arrival time and parking time.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :
Figure 5: Change in performance metrics with the number of requests.

Figure 9 :
Figure 9: Change of ω 1 (10 3 ) in the two-dimensional space of parking demand and supply.

Table 1 :
Te meaning of notations in this paper.
m ) Te end time interval of request m ′ (m)

Table 2 :
Computation results of diferent strategies.