An auction-based approach for prebooked urban logistics facilities

Accepted Manuscript

An Auction-based Approach for Prebooked Urban Logistics Facilities Kaidi Yang, Mireia Roca-Riu, Monica Menendez ´ ´ PII: DOI: Reference:

S0305-0483(17)31075-7 https://doi.org/10.1016/j.omega.2018.10.005 OME 1975

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Received date: Revised date: Accepted date:

7 November 2017 8 October 2018 9 October 2018

Please cite this article as: Kaidi Yang, Mireia Roca-Riu, Monica Menendez, An Auction´ ´ based Approach for Prebooked Urban Logistics Facilities, Omega (2018), doi: https://doi.org/10.1016/j.omega.2018.10.005

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Highlights • An auction-based unified approach proposed for prebooked urban logistics facilities • Heterogeneous carriers explicitly considered and social welfare optimized • Online readjustment formulation and robust formulations to handle uncertain arrivals • Theoretical and numerical analyses show advantages over the previous system

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• Treatment for uncertainty yields social welfare close to the theoretical upper bound

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An Auction-based Approach for Prebooked Urban Logistics Facilities Kaidi Yanga , Mireia Roca-Riua,∗, M´ onica Men´endezb,c,a a Institute

for Transport Planning and Systems (IVT), ETH Zurich, Zurich 8093, Switzerland of Engineering, New York University Abu Dhabi, United Arab Emirates c Tandon School of Engineering, New York University, USA

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Abstract

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The proper management of urban loading/unloading facilities is crucial for both the efficiency of the logistic services and urban traffic systems. Thanks to the advances in technologies, such management can be achieved through prebooked systems. In this paper, we propose a comprehensive and unified auction-based approach to optimize the use of prebooked loading and unloading facilities, based on a previous work of the authors. The new proposed approach assigns parking spots to companies and computes the price of the service, to maximize the social welfare of the system. It has the following advantages over the previous system: i) by employing an auction’s formulation based on the Vickrey-Clarke-Groves mechanism, the proposed system explicitly considers the heterogeneity in the penalty functions; ii) uncertainty in the carrier arrivals due to traffic condition is handled explicitly with an online readjustment and two robust formulations. The proposed auction system is tested on instances inspired by the logistic system in the city of Barcelona. Results show that compared to the previous work, the new approach exploiting the use of new information improves the social welfare by up to 22% with heterogeneous carriers. Results further show that the scenarios where demand is uniformly distributed over the time horizon present better utilities than the scenarios where demand is concentrated following a peak hour behavior. It is also shown that the proposed online readjustment formulation performs similar to the global offline optimum, and significantly improves the social welfare of the system when carriers fail to arrive at the allocated time due to traffic uncertainties.

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Keywords: Parking Slot Assignment Problem, Combinatorial Auctions, City Logistics, Mechanism Design, Urban Facilities

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1. Introduction

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Delivery operations in urban areas are crucial for cities (Ma, 2001). In Europe, more than 75% of the population lives in urban areas where flow of goods and services concentrates. The surging demand in both the traditional retail industry and the online shopping platforms calls for a more advanced delivery system. For example, Taobao, the largest online retailer in China owned by Alibaba, had 657 million delivery orders on a single day (Nov. 11, 2016). The proper management of urban loading/unloading facilities is crucial for both the efficiency of the logistic service and urban traffic systems. In this paper, we propose a comprehensive and unified auction-based approach to ∗ Corresponding

author Email addresses: [email protected] (Kaidi Yang), [email protected] (Mireia Roca-Riu), [email protected] (M´ onica Men´ endez)

Preprint submitted to Omega

October 10, 2018

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optimize the use of prebooked loading and unloading facilities. Urban logistic facilities are widely used in order to provide a space for efficient loading and unloading processes (Beziat, 2015). Many city councils offer some public space for logistic operations, where only commercial vehicles are allowed to park for a limited amount of time to perform their deliveries (e.g. Barcelona, Bilbao, Valencia, Paris, Lyon, Rome, Westminster, New York, amongst many others). However, even with these facilities, illegal parking is common at peak hours. Therefore, these facilities are not only important for delivery operations, but also for urban traffic systems. Logistic vehicles, both LGV and HGVs (Light/Heavy Goods Vehicle), act as bottlenecks and can create traffic congestion. These effects become severe if they have to wait, search for a parking space, or double park (Kong et al., 2016). Thanks to the evolution of information and communication technologies, new and intelligent systems have been recently developed and tested to efficiently manage delivery areas through the use of prebooked systems in the cities of Sevilla and Bilbao (Munuzuri et al., 2006; Freilot, 2011) and/or dynamic delivery parking spots (Roca-Riu et al., 2017). The facility offers a given number of parking spots for loading and unloading operations during a given time period, e.g. a day or a morning/afternoon time period. A set of carriers would like to use this facility at certain times and with different durations. In prebooked systems, designated time periods are assigned for each carrier to use the facility. On one hand, carriers are guaranteed an available spot within their preferred time interval. On the other hand, the efficient assignment avoids the negative effects when carriers can not find an available spot. Traditionally, these carriers are assigned parking spots and time slots following a first-come first-served criteria. However, this myopic criteria typically leads to a suboptimal performance of the system (Roca-Riu et al., 2015). The optimal assignment of carrier requests for prebooked systems has been formulated and studied in Roca-Riu et al. (2015) as the Parking Slot Assignment Problem. In this problem, carriers express in advance (e.g. one day) their preference in allocated time and duration for using the facility. Then the assignment from requests to parking spots and time slots is optimized. Ideally, the assignment should provide a solution that assigns carriers according to their preferences, and without exceeding the capacity of the facility. In summary, the problem lies on assigning a set of given requests with different durations on a facility with limited resources, taking into account the companies’ preferences. The problem, inspired by urban logistics facilities, can be also applied to similar problems of resource allocation, like the administration of public rechargeable points for electrical vehicles, the assignment of shared vehicles amongst its users, or the use of docks in a freight terminal. In Roca-Riu et al. (2015), it is assumed that each carrier reports a single time window of the desired starting times and a parking duration. Then the assignment from requests to parking spots and time slots is calculated using a mixed integer linear program (MILP). In this MILP, each request can be assigned to start at any time instant in any of the possible parking spots, as long as the capacity of the facility is not exceeded. If the parking spaces cannot satisfy all the carriers, a few different penalty functions are proposed, such as the number of unallocated requests, the total earliness or tardiness, etc. It should be noted that Roca-Riu et al. (2015) relies on two important assumptions. First, all the carriers are homogeneous in terms of penalty functions. Second, the time window that all the carriers request reflect their true preferences, i.e. they will not try to get a better assignment by requesting a larger or shorter time window and/or parking duration than their real preference. However, these two assumptions do not necessarily hold in all cases.

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As an extension and generalization of Roca-Riu et al. (2015), this paper proposes a unified approach from an auction’s perspective to relax the two aforementioned assumptions. The approach proposed in this paper has the following advantages over the previous one. First, we explicitly consider the heterogeneity in the penalty functions. Each carrier is assumed to have different valuations of starting times. Specifically, we employ an auction’s formulation based on the VickreyClarke-Groves mechanism (Nisan et al., 2007), which has some important properties: 1) carriers are motivated to participate in the auction, as they do not lose anything; 2) social welfare is maximized in the allocation problem, providing benefits to both the parking facility and the carriers; 3) truthfulness can also be satisfied by using the auction system. In other words, the carriers benefit from bidding their true valuations. Second, we explicitly handle the uncertainty in the carriers arrival due to traffic conditions, aiming to reduce the impact on both the carriers and the traffic around the parking facility. The remainder of the paper is structured as follows. Section 2 reviews the related work. Then, the problem is stated in a general form in Section 3. In Section 4, an auction-based formulation is proposed and the characteristics of the proposed auction’s formulation are studied. In Section 5, uncertainties in carrier arrival are explicitly handled by a readjustment formulation and two variants of the allocation problem. In Section 6, the algorithm used to solve the formulation is described, which uses an enhanced initial solution and lazy constraints. Section 7 compares the proposed auction’s formulation and the penalty formulation in Roca-Riu et al. (2015), showing the equivalence for homogeneous carriers under special conditions and quantifying the improvement for heterogeneous carriers. Section 8 evaluates the improvement of the solution algorithm, analyzes the properties of the proposed approach giving some managerial insights, and presents the results in cases of uncertainty. Finally, some conclusions are drawn in Section 9. 2. Related Work

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A short literature review is presented in this section on similar parking reservation problems (Section 2.1) and auction systems for resource allocation problems (Section 2.2).

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2.1. Parking reservation problems

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A similar problem to the Parking Slot Assignment Problem for logistics facilities is the parking reservation problem in urban traffic systems (Ayala et al., 2011, 2012; Geng and Cassandras, 2011; Mackowski et al., 2015; Lei and Ouyang, 2017; Shao et al., 2016; Guo et al., 2016; Du and Gong, 2016), e.g. parking reservation for parking spots. In such problems, each driver can reserve a parking spot from a parking lot, either in real-time or in advance. Ayala et al. (2011) studied a centralized and a distributed model for the parking spot competition within drivers. A pricing scheme was further applied in Ayala et al. (2012) to stimulate the selfish drivers to act in a favorable way for the entire system. However, those works did not take into account the desired parking durations and assumed a constant parking demand over time. Geng and Cassandras (2011) presented a dynamic model for the parking reservation problem based on a moving horizon scheme. Drivers send a request, obtain an assigned spot, and can then decide to accept or decline the assigned spot. However, the parking prices were assumed fixed. Mackowski et al. (2015) and Lei and Ouyang (2017) proposed a reservation system where the drivers can dynamically reserve a parking spot from a parking agency, and the agency can change the parking price over time. The problem was

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formulated as a bi-level multi-period mathematical program with equilibrium constraints where the upper level optimizes the prices and the lower level considers the impact of the pricing on the parking demand. Mackowski et al. (2015) assumed that the drivers only send requests when they are close to the parking space. Lei and Ouyang (2017) relaxed this assumption and adopted an approximate dynamic programming algorithm to solve the problem. Another type of problem reflects the concept of the shared parking systems that enable the private parking spots to be used by public drivers. Shao et al. (2016) and Guo et al. (2016) considered a parking space sharing problem where an agency temporarily manages the residential private parking spots from the owners, and rent them to the public users to fully utilize the limited parking resources. A binary integer programming model to optimize the revenue of the agency was formulated in Shao et al. (2016). Then, Guo et al. (2016) considered the risk that the parking spots might still be occupied when the owners need them, and applied a simulation optimization method to solve the problem. Although looking similar, there are fundamental differences between the parking reservation problem for urban traffic systems and the Parking Slot Assignment Problem for logistics facilities. First, in urban traffic systems, the aim is to maximize their total revenue (Jakob et al., 2018). In contrast, the parking spots for logistic services are normally located in public space, thus efficiency and fairness are more important factors to be considered. Second, prebooked systems have higher potential for loading/unloading urban facilities than assignment upon arrival, a common approach used for parking in urban traffic systems. On one hand, the number of parking spots offered in logistic facilities is rather small compared to the parking supply for cars (Cao et al., 2017). Moreover, logistic carriers need to park at specific places, whereas cars are more flexible and can be indifferent across multiple parking locations (Cao and Menendez, 2015b, 2018). On the other hand, logistic vehicles searching for parking space on the urban road will cause more traffic congestion than cars (Kong et al., 2016). Therefore, it is desirable to guarantee an available parking space at their arrival time. Third, most of the literature on the parking reservation problem focuses primarily on the reservation of parking spots and often neglects the parking duration as part of the allocation problem (e.g Zou et al., 2015; Chen et al., 2015). However, the assignment of time slots cannot be neglected in the Parking Slot Assignment Problem, as the carriers typically park for a short period of time, even though their parking maneuvers tend to be more complicated and do take longer (Cao and Menendez, 2015a). For example, in the city of Barcelona, the average delivery time is around 18 minutes (Roca-Riu et al., 2017) and vehicles are allowed to park in dedicated areas for a maximum of 30 minutes. Similar restrictions can be found in other cities such as Valencia, Lyon, Rome, or Westminster (Nuzzolo et al., 2016; Beziat, 2015; Figliozzi, 2007). As a result, neglecting the parking duration renders a suboptimal assignment solution. Fourth, carriers in the Parking Slot Assignment Problem show interest in different starting times, a flexibility that allows a more efficient use of the parking facility. However this, at the same time, makes the problem more complex. The existing works in the parking literature do not normally deal with the uncertainty in the arrival times of vehicles. Vehicles with a parking reservation may not be able to arrive at the parking spot on time due to traffic congestion. To the best of the authors’ knowledge, there are only a few works that considered the arrival uncertainty in the parking literature. Liu et al. (2014) studied the expirable parking reservation system considering the uncertain arrival of vehicles. However, this work mainly evaluated the impact of such parking reservation system on the traffic system from a macroscopic perspective. Yan et al. (2011) proposed an intelligent parking system based on mobile phone information that reserves the parking spots en-route. This system handles

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the potential uncertainty in the vehicle’s arrival time at the parking spot using a simple rule-based strategy. Cao et al. (2016) tackles the uncertainty of the arrival time and charging time for electrical vehicles with reservation at some charging stations. The system periodically updates the selection of the charging station for the electrical vehicles en-route. However, both works assumed an online reservation system. Handling uncertain arrivals in these works does not have to consider the impact of the readjustment on the predetermined reservations. 2.2. Auction for resource allocation problems

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Auctions have been attracting increasing attention in the research community for resource allocation problems in fields of traffic systems, logistics, telecommunication, power grid, amongst others. These auctions systems are shown to be capable of allocating the system capacity efficiently. An extensively studied field is the spectrum auction in telecommunication where a government sells the rights to transmit signals over specific bands of the electromagnetic spectrum (Jia et al., 2009; Cramton, 2013). Different variants of auction systems have also been proposed to allocate cloud computing resources (Samimi et al., 2016; Lin et al., 2010). In energy systems, Tang and Jain (2011, 2015) hanlded the intermittent nature of the renewable energy and proposed stochastic resource auctions to integrate renewable energy into the power system. In traffic systems, auctions have been applied to solve problems such as airport time slot allocation (Rassenti et al., 1982), car sharing (Hara and Hato, 2017), intersection control (Carlino et al., 2013), congestion pricing (Teodorovi´c et al., 2008), tradable credit and tradable permit system (Nie, 2012; Wada and Akamatsu, 2013; Hara and Hato, 2012), and highway reservation (Su and Park, 2015). For example, Hara and Hato (2017) used combinatorial auctions to design a VCG-based tradable permit mechanism for a car sharing system. Su and Park (2015) evaluated a first-price auction system where vehicles bid for time slots for using the highway resources. In logistics, combinatorial auctions have been extensively applied in truckload procurement (Caplice and Sheffi, 2006; Remli and Rekik, 2013). A few papers also proposed auction-based parking management systems for urban traffic systems (Hashimoto et al., 2013; Ayala et al., 2012; Chen et al., 2015; Zou et al., 2015). Hashimoto et al. (2013) proposed an on-line auction system for parking reservation for electric vehicles, considering electricity trading. In the parking spot competition problem studied by Ayala et al. (2012), a vehicle-based auction was proposed to set incentive prices such that the Nash equilibrium defines the system optimum. However, both Hashimoto et al. (2013) and Ayala et al. (2012) cannot ensure truthful reporting from the drivers. A widely applied truthful mechanism is the VCG mechanism that, at the same time, achieves system optimum. Chen et al. (2015) proposed an online VCG auction system where the whole time horizon was divided into a few time intervals, and the VCG scheme was applied to each time interval to myopically optimize the system performance. It was also shown that the drivers have incentives to reserve as early as possible. However, Chen et al. (2015) did not consider the parking durations, and thus considered each time slot separately. Therefore, the optimal allocation within the time horizon of a whole day could not be guaranteed. The static and dynamic auction systems for parking reservation were studied in Zou et al. (2015). In the static auction system, the drivers were assumed to arrive and request the time slot at the same time. In the dynamic auction system, drivers were assumed to send parking requests when they were close to the parking facility. The requests included their reported arrival time, latest possible waiting time for a parking spot, departure time, and slot valuation. The auction system then assigned the time slot to the drivers and charged a certain price that enabled truth telling. However, the auction

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system proposed in Zou et al. (2015) can not be used for prebooked logistic facilities. First, the auction system does not allow prebooking, which, as stated in Section 2.1, is essential for logistic carriers to organize their routes. Spontaneously looking for parking spots leads to illegal parking in the case of logistic carriers, as it currently happens in many cities. Second, the static auction did not consider the parking durations, and each time interval was solved independently, which is not applicable in logistic facilities as the capacity is normally binding. Third, neither static nor dynamic systems considered the different valuation functions for starting at different times from the logistic companies. 2. Problem

3. Problem Statement

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We have a set N of n companies that are interested in using a loading and unloading facility We have a set N of n companies that are interested in using a single loading and unloading that offers c parking spots during the time horizon [0, T ] (See Figure 1a). Each company i ∈ N facility. This facility offers c parking spots during a time horizon (See Figure 1a). The time horizon needs to use the parking facility for a duration si starting within the time horizon and has its own cantime be preference. either one day, or part of a day. Each company i ∈ N needs to use the parking facility for a In duration , starting within time horizon. It different is assumed that along each company practise,sithese areas can servethis many companies with durations the day. Ashas an its own time preference to an start using the parking Such preference can be expressed in example, Figureon 1b when presents estimation of the numberfacility. of loading/unloading operations every 30 minutesforms, that take in an area center ofvector. Lyon. The durations of these that activities short different forplace example, withinathe valuation It is also assumed eachis company has at least in some cities (with lasting lesswe than 30 use minutes)[1]. Thus, assigning requests only one such request, and 90% for simplicity will exchangeably the termthe request ortocompany time slots is a combinatorial problem. to refer to an individual request from a company to use the facility. Cases where a company has The problem consists on finding a feasible pre-booking allocation of companies to time slots multiple independent requests for the same facility could be treated as if the requests were coming that takes into account the time preferences of the different companies. The objective is to find the from multiple companies. best possible return to the whole group of companies, i.e. maximizes social welfare, with the aim In practice, these loading/unloading areas can serve many companies with different parking of providing a fair system. durations along the day. As an example, Figure presents an estimation of the number of loadThe time horizon is considered as a discrete set Π1b r = {t0 , t1 , t2 , . . . , tr } of r+1 homogeneous time ing/unloading operations thatthe take place may every 30 minutes in an area isinmade the center Lyon, France. periods (t0 = 0, tr = T ) where requests start. This discretization for twoofreasons. The activities are generally with lasting less than 30 minutes Ondurations one side, of forthese operational purposes, the timeshort; can not be 90% treated as a continuous variable, at in some maximum one minute precision can be required. On the other hand, the formulation with time cities, like Barcelona, Paris, Valencia, Lyon or Amsterdam (Council; Dablanc and Beziat, 2015; discretization Figliozzi, 2007).has proved to work better in practice [7]. For notation convenience, Jr = {0, ..., r} will be the index set from the discretization set of the time horizon Πr . 400 400

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Time (b) Estimation of deliveries and pick-ups at the (b) Estimation of deliveries and pick-ups at the center of Lyon (France)[2]

(a) Vehicles interested to use a given

(a) Vehicles interested infacility using a given loading/unloading loading/unloading facility

center of Lyon (France)(David and Armetta, 2013)

Figure 1: Loading and unloading facilities problem

Figure 1: Loading and unloading facilities problem

The problem consists in finding a feasible prebooking allocation of the multiple requests to available time slots, taking into account the time preferences and the service durations of the 7

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different companies. The objective is to find the best possible assignment, and maximize the social welfare for the whole group of companies. Thus, we aim to optimize the system performance of all the companies, while providing a fair assignment. The time horizon [0, T ] is considered as a discrete set Πr = {t0 , t1 , t2 , . . . , tr } of r +1 homogeneously distributed time instants (t0 = 0, tr = T ) where the requests may start. Large discretization steps provide a less accurate assignment, and small discretization steps increase the computation complexity and would be more sensitive to the uncertainties in the system. The model also allows the discretization steps to be time varying. For notation convenience, we denote Jr = {0, ..., r} as the index set of the discrete time horizon Πr . In the end, the problem is to assign n requests to Πr time instants in a maximum of c different parking places simultaneously, taking into account the duration of the loading/unloading services. The problem is then a combinatorial optimization problem. 4. An Auction Formulation to Prebook Logistics Facilities

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In order to solve the optimal assignment of parking requests to time slots while taking companies’ time preferences into account, we propose an auction formulation where companies bid for time slots. We first model the time preference of company i regarding the possible starting times as a nonnegative valuation vector vi = [vi0 , · · · , vir ]. Each element vij ≥ 0 represents the value of company i for starting to use a parking spot at time instant tj , tj ∈ Πr . The companies do not need to reveal the valuation vector in the auction, the model just assumes that this vector exists and that each company can give values to different time instants. In other words, the parking facility does not need to know the valuation vectors in order to resolve the auction. In addition, the valuation vector is independent of the parking spot being used, since we assume that they are all equal. Instead, it is correlated with the benefits that the company obtains when the request is satisfied at the desired starting time. Such benefits depend on the convenience for a carrier to plan the delivery with its upstream suppliers and the downstream customers. For example, if a company is not interested in starting parking at a certain time instant at all, then the valuation for this time instant is 0. Note that this valuation vector accommodates any kind of valuation function shape for the companies. Figure 2 shows several valuation vectors with different shapes: Figure 2a is the valuation vector of one company that is only interested in starting the service between 9 and 11 in the morning, then its valuation function will be constant and equal to vmax within this time period, and zero otherwise; Figure 2b represents a company that has the maximum interest at 11h, and the valuation decreasing linearly before and after that time; and Figure 2c represents a company whose preferred time is also at 11h, but the valuation decreases quadratically over time. The proposed auction consists of two steps. In the first step, each company i ∈ N bids for a parking slot before the considered time horizon (e.g. one day before they need to park). Specifically, the company i reports a vector of bids bi = [bi0 , · · · , bir ] where each element bij represents the bid for starting parking at time tj , and a time duration si in minutes when it needs to use the facility. Note that the company could strategically have a bid vector different from its valuation vector. However, as we will later discuss it is desirable for each company to bid truthfully (i.e. bij = vij ). In the second step, two problems need to be solved: 1) an allocation problem which allocates each request to a time slot; 2) a pricing problem which defines how much each company has to pay for getting the assigned slot. Denote the assignment as a set of binary variables x = [xij , i ∈ N, j ∈ Jr ], 8

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(b) Linear Figure 2: Examples of valuation vectors

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where xij = 1 if the request of company i is assigned to the starting time tj , and xij = 0 otherwise. Denote the pricing for each company i as pi , which usually depends on the allocation x. The utility for a company is its valuation for getting the assigned starting time minus the price they have to pay for the given assignment. Specifically, the following function describes the utility function for company i that depends on the assignment x: X ui (x) = vij xij − pi (x) (1) j∈Jr

4.1. Design of the auction

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where pi is the price to be paid by company i. Each company is interested in maximizing its individual utility function, ui (x) .

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The aim is to design an auction (i.e. an allocation rule and a pricing rule) that fulfills the following desired properties.

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• The allocation rule maximizes social welfare. Here, the social welfare is the sum of the utilities obtained from all the companies with the given assignment, which represents the total utility of all the companies.

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• The auction is dominant strategy incentive compatible or, alternatively named, truthful. So the best strategy for each company is to bid its real valuation, i.e. to provide the true valuations (bij = vij ) and the true duration.

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• Under the allocation rule and the pricing rule, the utility of each company is non-negative. In other words, the companies do not suffer any loss by participating in this auction system, P i.e. pi ≤ j∈Jr vij xij

Fortunately, the Vickrey-Clarke-Groves (VCG) Mechanism for combinatorial auctions (Nisan et al., 2007) satisfies the above properties, and aids on the design of the desired auction. The allocation rule should maximize the social welfare, which is the sum of all the valuations obtained for each of the companies with a given assignment. However, as the true valuations are not known, the allocation problem instead optimizes the assignment according to the bids submitted.

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The optimal allocation x∗ is obtained solving the following integer program: XX max bij xij x

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xij ≤ 1, i ∈ N

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where Tij ⊆ Jr , with j ∈ Jr and i ∈ N , such that Tij = {k ∈ Jr |tk ≥ 0, tk < tj , tk > tj − si }. Notice that the allocation problem is always feasible, as there is always an all-zero solution. The allocation rule is defined by the solution of the integer programming problem (2)–(5). The starting times are assigned with the objective of maximizing total social welfare, while fulfilling the capacity restrictions of the parking spots. Constraints (3) impose that for every request at most one assignment is active for the whole time horizon, and Constraints (5) ensure that the decision variables x are binary. Constraints (4) guarantee that at a given time instant t, at most c requests (one per parking spot) are being served simultaneously. When we assign a request to a given starting time, that does block the parking spot for the duration of the service to the assigned request. For that reason, Constraints (4) check how many assignments are active at each time instant tj . In order to count the active assignments, it is not enough to add all assignment variables at time index j ∈ Jr for every request i ∈ N , but also the assignment variables from previous time indices k such that tk < tj and tk > tj − si , that could still be active as the assignments last for a duration of si . The previous features of the allocation rule show the combinatorial nature of the auction. The formulation simplifies the assignment to the starting time. However, this assignment is only valid if it stays active during the whole duration of the loading/unloading activity. Constraining the possibility of simultaneously using more than c parking spots converts the assignment in a combinatorial problem. It is later shown in Section 7.2 that this formulation is equivalent to Roca-Riu et al. (2015) when carriers are homogeneous and penalties are additive. The auction’s formulation proposed here, however, is a generalization, as it allows carriers to have different valuation vectors. Once the allocation rule has been defined, we define a payment rule such that it guarantees truthful bidding from carriers. As we will see, to guarantee truthful bidding, we ensure that each request pays its externality to the system. That is achieved by solving the following problem for each request i ∈ N : XX XX ∗ pi (x) = bhj yhj − bhj x∗hj (6) h6=i j∈Jr

h6=i j∈Jr

where x∗ is the optimal solution of the problem of the allocation rule, and y ∗ is the solution to the following problem, which is the same problem solved previously in the allocation rule, but without

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company i: max y

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where, as before, Thj ⊆ Jr and h ∈ N , with j ∈ Jr , such that Thj = {k ∈ Jr |tk ≥ 0, tk < tj , tk > tj − sh }. Problem (7)–(10) gives the solution to the maximum social welfare from parking slot allocation when i does not participate, and the other drivers’ welfare is maximized. Clearly, the solution to this problem does not depend on company’s i valuation vij , bid bij , and neither on its request duration si . Finally, the utility received by the company i is:

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where the first term is the maximum social welfare which is achieved under allocation x∗ , and the second term is independent of valuation vij , and bidding bij . Then, the payment scheme aligns each company’s utility maximization with the system welfare maximization objective. Note that this expression is obtained from Eq. (1), with the optimal solutions (x∗ , y ∗ ) from problems (2)–(5) and (7)–(10). The VCG mechanism (the allocation rule and pricing rule) ensures truthful bidding and nonnegative utilities for all companies. The general proof can be found in (Nisan et al., 2007). The concept is intuitive in a single-item auction, where the payment rule ensures that companies can not get better utility by bidding differently from their valuation. As already mentioned, the payment rule defined by the VCG mechanism is that each company pays the externality created to the system, i.e., the difference on social welfare triggered by the company participating on the auction. The allocation rule assigns the item to the company with the highest bid. The winning company has to pay a price equivalent to the difference in the social welfare of the rest of the companies comparing two situations: either the company is present in the auction or it is not. Note that the social welfare of the other companies when this company is present is 0. The social welfare when the company is not present is then the second highest bid, so this is the price that the winning company has to pay. Let us analyze the case from a given company perspective (i ∈ N ), with vi utility valuation, bi the bid and maxh6=i bh as the highest bid of all the companies except the company being analyzed. If the bid of the company (bi ) is lower than the highest bid of the rest (bi < maxh6=i bh ) the company loses the auction and receives 0 utility. If the bid is higher (bi > maxh6=i bh ), company i wins and receives as utility its valuation minus its price, which is equal to the highest bid of the rest (vi − maxh6=i bh ). Let us see that this value is the highest possible when the company bids its real 11

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valution, i.e. when bi = vi . On one hand, if the real valuation of the company is smaller than the highest bid of the rest of the companies (vi < maxh6=i bh ) the maximum utility that the company can get is zero. Therefore, there is no reason why the company could offer a different bid from its real valuation, as that would result in a lower utility. On the other hand, if the valuation is higher than the highest bid of the rest of the companies (vi > maxh6=i bh ), the utility that the company gets is (vi − maxh6=i bh ), which is achieved when company sends the real valuation as his bid (bi = vi ). In this case, the company bids truthfully and wins.

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4.2. Characteristics of the auction’s formulation

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The Parking Slot Assignment Problem is formulated into a combinatorial auction for two reasons: 1) the carriers can be interested in multiple starting times (i.e. can be multi-minded), and 2) the parking durations are explicitly considered. These two factors make this problem essentially different from previous work on the auctions of parking systems (e.g. Zou et al., 2015; Chen et al., 2015). Inspired by Hara and Hato (2017), we study the characteristics of the proposed auction’s formulation and the interpretation of VCG payments. Specifically, we study the characteristics for the single-minded and multi-minded carriers in Section 4.2.1 and Section 4.2.2, respectively. 4.2.1. Single-minded carriers

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In this scenario, we assume that all the carriers bid single-mindedly, i.e. they are only interested in one particular time slot and bid for it. Let us denote a ˆi ∈ Jr as the time slot carrier i is interested in, and bi as its bid for this time slot. The valuation function for carrier i is zero everywhere except for a single starting time at a ˆi . For single-minded auctions, the allocation problem can be formulated as optimization problem (12)-(14), where xi is 1 if the request of the company is assigned to its preferred starting time, and 0 if the carrier is not served. X max bi xi (12) i∈N

X

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i∈N

xi 1Ti (j) ≤ c, j ∈ Jr

xi ∈ {0, 1}

(13) (14)

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where 1Ti (j) represents the indicator function of set Ti = {j ∈ Jr |ai ≤ j < ai + si } . The LP relaxation of the allocation problem (12)-(14) can be written as max b T x

(15)

s.t. Ax ≤ c

(16)

x ≥0

(17)

where (·)T represents the transpose, c = [1, 1, · · · , 1, c, c, · · · , c]T , b = [b1 , b2 , · · · , bn ]T , and A is | {z } | {z } n

r

written as follows (here In represents the n-dimensional identity matrix).

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In

a2

a1

An×(n+r+1) = an a2 + s 2 − 1 a1 + s 1 − 1

a n + sn − 1

     0  . . .   0  . . .  1    .. .   1  . . .   1   .. .   1  0  . . .   0   .. . 0

0 . . . 1 . . . 1 . . . 1 . . . 1 . . . 0 0 . . . 0 . . . 0

...

...

...

...

...

... ...

...

...

 0 . . . 0 . . . 0 . . . 1 . . . 1 . . . 1 1 . . . 1 . . . 0

                                                  

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We prove the following proposition for the total unimodality of the constraint matrix A .

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Proposition 1. The constraint matrix A for the LP relaxation of the allocation problem in the single-minded scenario is a totally unimodular matrix. Proof. The lower part of A is a matrix with consecutive ones. By Wolsey (1998), such matrix is a totally unimodular matrix. Also notice that the upper part is an identity matrix. As any totally unimodular matrix combined with the identity matrix is also totally unimodular. Therefore, the constraint matrix A is a totally unimodular matrix.

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Since the constraint matrix A is totally unimodular, the solution to the LP relaxation problem is integer, and thus yields an optimal solution to the original optimization problem. Therefore, the allocation problem for single-minded carriers can be solved by directly solving the LP relaxation problem. We can write the dual problem of the relaxation problem as follows. X X min µi + c γj (18)

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s.t. µi +

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ai +s i −1 X j=ai

γ j ≥ bi , i ∈ N

µi , γj ≥ 0, i ∈ N, j ∈ Jr

(19) (20)

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where µ is associated with the first n constraints in Constraints (16) (i.e., x ≤ 1 ) and γ is associated with the rest r constraints in Constraints (16), which are equivalent to Constraints (13). By strong duality, the optimum of the dual problem (18)-(20), equals the optimum of the LP relaxation (15)-(17), and thus equals the allocation problem (12)-(14). Dual variable µ can be interpreted as the utility of each carrier, and dual variable γ as the equilibrium price of each time slot, i.e. the price such that the demand equals the supply (Vohra, 2011). In fact, the VCG payments correspond to the “lowest” equilibrium prices. This can be summarized into Proposition 2. The proof is given in Appendix A. µ, γ ) such that γj represents the price for time slot Proposition 2. There exists a dual solution (µ j, and µi represents the utility of carrier i. 13

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4.2.2. Multi-minded carriers In multi-minded scenarios, the constraint matrix A ∈ Rnr×(n+r) is in most cases not totally unimodular. Interested readers can refer to Roca-Riu (2015) for the specific form of the constraint matrix A . Here, we have the following proposition. Proposition 3. Constraint matrix A is totally unimodular if and only if si = 1, i ∈ N .

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Proof. The necessity can be proved by a counter example provided in Roca-Riu (2015). Next, we show the sufficiency. If si = 1, i ∈ N , then the constraint matrix A becomes an assignment problem. Therefore the constraint matrix A is totally unimodular if si = 1, i ∈ N . This completes the proof.

i∈N

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Proposition 3 shows that the LP relaxation problem gives the optimal allocation if and only if every carrier has a unit parking time. The cases with unit parking time assume that the parking duration of each carrier is equal or smaller than the time discretization interval. The assumption of unit parking time results in an allocation problem similar to many works in the parking auction literature (e.g. Zou et al., 2015; Chen et al, 2015). In more general cases, if the parking duration is explicitly considered, the LP relaxation problem provides an upper bound for the social welfare. We can write the dual problem similar to the single-minded scenario, which yields the same optimum as the LP relaxation problem. X X min µi + c γj (21) j∈Jr

(22)

µi , γj ≥ 0, i ∈ N, j ∈ Jr

(23)

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s.t. µi + γj ≥ bij , i ∈ N, j ∈ Jr

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However, in general cases, the dual variables do not provide insights regarding the prices. Only in the special case of unit parking time are there dual variables µ and γ such that µ corresponds to the utility of carriers under VCG mechanism and γ is the equilibrium price of each time slot.

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Proposition 4. If all parking times are unit time (si = 1, i ∈ N ), there exist a dual solution µ, γ ) such that γj represents the price for a time slot, and µi represents the utility of carrier i (µ under VCG mechanism. The proof is similar to the proof of Proposition 2, and is omitted here for brevity.

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5. Auction’s formulation with uncertain arrival times

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In the proposed auction formulation of the prebooked system, it is assumed that carriers will arrive at the assigned time. However, due to the variability of travel times within the city (Noland and Polak, 2002), carriers may not be able to arrive at the exact assigned starting times. Delayed carriers may miss the parking spot, whereas early carriers might have to wait for the parking spot to become available, temporarily disturbing the traffic around the facility. In this section, we propose different approaches that deal with uncertain arrivals. Section 5.1 proposes a formulation to modify the allocation online, once early arrivals or delays are communicated to the system within a short time horizon. Section 5.2 proposes two new variants of the allocation formulation that already account for uncertainty in the arrival times. Section 5.3 defines two benchmark formulations for comparison purposes. Unfortunately, as it will be later discussed, the property of truthfulness is lost if carriers know that these modifications are implemented. 14

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5.1. Optimal online readjustment with uncertain arrival times

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In this subsection, we propose an online readjustment that modifies the allocation to handle the stochastic arrival times explicitly, where carriers might be reallocated to the facilty when they fail to arrive at their assigned time. Let us denote ξi as the earliness or delay of carrier i in number of time slots, i.e. the difference between the time slot in which carrier i will arrive at the parking facility and the assigned time slot. Here, ξi < 0 if carrier i arrives ahead of time, and ξi > 0 if carrier i is delayed. We assume carriers will be able to report their earliness or delay (or equivalently their real arrival times) at the latest at ∆T before their real arrival (e.g. 20min). This communication can be done through cell phones or on-board devices. At a given point in time, once the system receives new information reporting earliness or delays, a modification of the allocation can be proposed to reallocate early/delayed carriers. At each time slot k ∈ Jr , the system might receive new information about the real arrival times. For those who do not report their arrival time until time slot k, we assume that they will arrive at the assigned time (i.e ξi = 0) or the time k + ∆T , whichever later. Let us denote ξik as the system’s information of the earliness/delay for carrier i at time slot k. Let us further denote set Nek = {i ∈ N |ξik ≤ 0} as the set of carriers that are known to arrive early or expected to arrive on time, and set Nlk = {i ∈ N |ξik > 0} as the set of carriers that are known to arrive with delay. With the given information about arrival times, we formulate the optimal online readjustment problem as an optimization problem at each time slot. Every time there is an update in the real arrival times, the problem needs to be solved again, providing a readjustment of the allocation. The readjustment aims to find a new assigned time for early/delayed carriers, whereas carriers arriving at their assigned time can not be reallocated. Early carriers will be at the latest reallocated to their original assigned starting time. The readjustment problem uses previous variables xij from the allocation problem, which are now fixed representing the assigned times in advanced. The variables z = [zij , i ∈ N, j ∈ Jr ] k define the new readjustment allocation variables. Moreover, we use parameters z˜ij to indicate whether carrier i has started to use the parking facility at time slot j previous to time slot k. The optimization problem at time slot k can be written as follows:

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max z

X X

i∈Nek j∈Jr

s.t.

X

j∈Jr

i∈N j 0 ∈Tij

i∈Nlk j∈Jr

X

bij zij − λ

X

πi

i∈N

(25)

zij 0 ≤ c,

j ∈ Jr

(26)

j∈Jr

j 0 ≤j

j 0 ≤j

j 0 ≤j−ξik

zij 0 ≥

X

xij 0 ,

j 0 ≤j

j ∈ Jr , i ∈ Nek

πi ≥ 0, i ∈ N X X πi ≥ jzij − (j + ξik )xij , zij ∈ {0, 1},

j∈Jr

j ∈ Jr , i ∈ N

(28) (29) (30)

i∈N

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j∈Jr

(27)

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k z˜ij ≤ zij , j ∈ Jr , i ∈ N X X xij 0 , j ∈ Jr , i ∈ N zij 0 ≤

X

(24)

i∈N

xij ,

zij ≤

X X

X X

bij xij +

(31)

(32)

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where the objective function (24) is the social welfare in the scenario with uncertainty. The first two terms represent the valuations that the carriers obtain with the new reallocation. The third term represents a penalty associated with the waiting times of the carriers. We assume that a carrier arriving early or on time (i.e., i ∈ Nek ) receives its original assigned valuation (first term), and a carrier arriving late (i.e., i ∈ Nlk ) receives the valuation of the actual assignment (second term). Recall that the valuation of a carrier represents a measure of its convenience to plan the delivery with its upstream suppliers and the downstream customers. If a carrier arrives early or on time, it can always satisfy both the suppliers and the customers, and thus obtain the valuation of the original assignment. However, if a carrier arrives late because of the traffic conditions, it has to rearrange its delivery time with the downstream customers, and thus receives its valuation of the actual reassignment. The third term of the objective function represents a penalty associated P with the waiting times of the carriers. The penalty term i∈N πi is multiplied by the coefficient λ, where πi represents the waiting time of carrier i from its arrival time to the time it starts to use the facility (defined in Constraints (30) and (31)). This term includes both the cost of waiting time for the carriers and the externalities the waiting carriers impose on the traffic stream nearby. Notice that the carriers arriving early or late might have to wait to use the facility, whereas the carriers arriving on time will be allocated at its assigned time and have zero waiting time. Constraints (25) ensure that the carriers can only obtain a time slot in the readjustment if they have an assigned time. Constraints (26) impose the capacity limits for the readjustment. Constraints (27) controls that once a carrier starts to use the facility, its allocation is not modified in future time slots. Constraints (28) ensure that each carrier should be reallocated only after its arrival time. Constraints (29) represent that carrier arriving early or on time should be reallocated at the latest at its assigned time slot. Constraints (30) and (31) calculate the waiting time for each carrier. If P a carrier is not reallocated into the parking facility (i.e. j∈Jr zij = 0), the waiting time should P be 0. If a carrier is reallocated into the parking facility (i.e. j∈Jr zij = 1), the waiting time is calculated by the right hand side of (31). Constraints (32) ensure that the decision variables zij are 16

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binary. This formulation can be solved efficiently using CPLEX, yet most of the decision variables might already be fixed from previous solutions. The average solution time for the readjustment formulation at a time instant is less than 1 second, and this problem needs to be solved only when new information is sent to the system. 5.2. Robust variants of the allocation formulation

max (2)

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In this subsection, we handle the uncertain arrival times in the original allocation problem by proposing two formulation variants. Both variants aim to provide more robust solutions by adding buffer times. A buffer time is an additional time between two consecutive reservations, which helps protect the given schedule in case disruptions happen. The idea of buffer insertion comes from critical chain project scheduling introduced by Goldratt (1997). Similar ideas applied to other transportation problems are reflected in existing works, see for example Huang et al. (2018); Jovanovi´c et al. (2017); Mulder and Dekker (2019). The first variant, herein named the extension variant, extends the reserved time for the duration of the operation for a certain number of time slots δs. That is, instead of only reserving for a carrier the exact duration of the operation, some extra time slots are added. This way, if the carrier arrives between the assigned time instant j, and j + δs, the operation can still happen without affecting the rest of the assignments. The allocation formulation (2)-(5) is modified in the capacity constraints (4), by changing the duration of the operation with a new set Tij0 , and is formulated as follows.

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s.t. Constraints (3), (5) X X xij 0 ≤ c, j ∈ Jr

(33)

0 i∈N j 0 ∈Tij

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where Tij0 = {k ∈ Jr |tk ≥ 0, tk < tj , tk > tj − si + δs} The second variant, herein named the slackness variant, modifies the objective function of the allocation problem (2)-(5) to incorporate the value of slackness in the assignment. An assignment that preserves some slack between consecutive carriers is potentially more robust when facing early or delayed arrivals. In particular, we consider the slack as the gap between the departure time of the previous carrier and the assigned starting time of the following carrier. The objective function of the allocation problem is modified as follows:  XX X X ρXX  max bij xij + bij c − xij 0 (34) x n 0 i∈N j∈Jr

i∈N j∈Jr

s.t. Constraints (3)-(5)

i∈N j ∈Tij

(35)

where ρ is a weighting parameter associated with the level of uncertainty. The higher the uncertainty P P of the instance, the higher ρ should be. Note that the term c − i∈N j 0 ∈Tij xij 0 is the number of available parking spots at time j, what we name the slackness at time j. We introduce this slackness at the objective function weighted by the average value of the bids at that time slot. That is, the higher the interest from the companies to use a given time slot, the higher the weight of the slackness at that point.

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These formulations will provide optimal allocations with lower social welfare than the original formulation. However, the solutions provided will be more robust in case of earliness or delays in the real arrival times. Unfortunately, none of these modifications preserve truthfulness. For the readjustment of the allocation formulation, if carriers know they might be rescheduled when they arrive early or with delay, they could target another arrival time that satisfies better their preferences to obtain higher individual valuation. Also, with the variants of the allocation formulation, if carriers know there is some flexibility in the assignment, they can again target another arrival. 5.3. Benchmarks

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In order to evaluate the proposed online readjustment formulation and the allocation variants, two benchmark formulations are considered: 1) the global offline optimum of the readjustment; and 2) the optimum in retrospect. The global offline optimum of the readjustment assumes that delay/earliness of each carrier is known before the first carrier arrives at the facility. Using the global offline optimum, we can evaluate the performance of the online readjustment formulation. The online readjustment formulation has only partial information about the earliness/delays. The global offline optimum can be found solving the same formulation (24)-(32), without considering partial information at the different time slots. The earliness/delay of each carrier will be known, independent of the time slot, (i.e. ξi ) and Constraints (27) will not be needed. The optimum in retrospect is the optimum assuming that the earliness/delay of each carrier is known before the allocation. Therefore, the assignment can be optimized considering the earliness/delay of each carrier. For a given realization of the stochastic arrival, the optimum in retrospect provides an upper bound for all the variants of the allocation formulations and readjustment formulations, which can be calculated from the following optimization formulation assuming that the information of earliness/delay is accurate. X X X X X max bij xij + bij zij − λ πi (36) x,z

i∈Ne j∈Jr

i∈Nl j∈Jr

i∈N

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s.t. (25), (26), (28)-(32)

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Note that the difference between (36) and (24) is that (36) optimizes both the original allocation and the readjustment once the delays become known, whereas (24) optimizes only the readjustment of the allocation given that the original allocation can not be changed. Note that none of these two formulations can be used in practice, as this information would not be known before the earliness/delays actually happen. However, they are used as a reference, to compare how good the previous formulations could do in comparison to a theoretical optimum with full information. 6. Tailored Solution Algorithm

According to the discussion in Section 4.2, there may not be a polynomial time algorithm for the allocation problem in general cases (Lehmann et al., 2002). Moreover, we need to solve the pricing problem once for each company, i.e. n times for solving all the prices. Note that each of the pricing problems has the same structure as the allocation problem, with only one less variable. 18

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Hence, if the number of requests is large, the computation complexity of solving both problems is very high. Fortunately, we can exploit the similarity between the allocation and pricing problems to reduce the computation time. In this section, we discuss the implementation of improving the general solution algorithm for the mixed integer linear program based on the Branch and Bound (B&B) by exploiting two properties of the problem. In Section 6.1, we use the solution to the allocation problem as an initial solution to each of the pricing problems. In Section 6.2, we relax some constraints on the initial formulation and add them whenever they are needed within the solving procedure as lazy constraints, i.e. they are only active when the current solution violates some of these constraints. 6.1. Initial solution

6.2. Capacity constraints as lazy constraints

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The optimal solution to the allocation problem x∗hj , h ∈ N, j ∈ Jr provides obviously a feasible solution to the pricing problems. For the pricing problem of request i, it is obvious that x∗hj , h 6= i, j ∈ Jr is one feasible solution. That is to say, the solution obtained in the allocation problem when all companies participate in the auction, can serve as an initial solution to the n pricing problems that have to be solved. The effectivity of this strategy will be later evaluated in Section 8.1.

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When a formulation is solved with lazy constraints, this set of constraints is not generated on the inital formulation, but only later, in case the obtained solution violates some of them, they are added to the formulation. Lazy constraints have proven to be specially useful when solving problems with a large set of constraints, e.g. the subtour elimination constraints of exponential size for the travelling salesman problem (Laporte, 1992). Although the number of constraints of both the proposed allocation problem and the pricing problems are not large compared to the number of assignment variables, treating the capacity Constraints (4) and (9), from the allocation and the pricing problem respectively, as lazy constraints can still be useful to reduce the computation time. The reasons are two-fold. First, the companies are expected to have the highest time preferences in certain time periods, e.g. mornings or afternoons. In the other time periods, such as early morning or late night, the time preferences are usually quite low. As a result, the capacity constraints may only be active for a small number of time slots. Second, as the pricing problem has a similar structure to the allocation problem, it is very likely that the pricing problems have a similar set of active capacity constraints. Hence, the set of capacity constraints for the pricing problem can be initiated as the resulting set of active capacity constraints obtained when solving the allocation problem. Thanks to this, the pricing problems are expected to be solved faster. The application of the lazy constraints in this paper is summarized in Algorithm 1 (detailed below), which is used to solve both the allocation and the pricing problems. For the allocation problem, the initial set of capacity constraints can be either empty or user defined according to the structure of the valuation vectors. For example, if the valuation vectors express a high demand of service in a subset of time instants, the capacity constraints of these time instants can be included in the initial set. After solving the allocation problem, we use the output set of capacity constraints to initialize each pricing problem. In each of the algorithm iterations, we check if some constraints are violated, i.e. the obtained solution should not exceed the capacity of the loading/unloading area at any point in time. This is done by integrating the intermediate solution into the set of 19

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Algorithm 1 Algorithm with lazy constraints formulation

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capacity constraints. In order to improve the performance of the algorithm, at each algorithm ¯ be the iteration we may also add more constraints besides the single violated constraint. Let L ¯ whole set of capacity constraints of the given problem ((4) or (9)), and we use the notation of lj ∈ L, j ∈ Jr for one capacity constraint, as they are indexed in subset Jr . One possible way is to add all constraints between lj and lk as long as tj and tk are close enough, e.g. 0 < tj − tk < κ (κ is a positive parameter of this algorithm). We do this because if one constraint is active in the optimal solution, it is possible that the nearby constraints are also active. At each iteration a small subset of constraints can be added to the formulation, which might avoid the need to solve the problem once for every new violated constraint found.

˜ ⊂ L, ¯ where L ¯ is the whole set of capacity constraints Input: An initial set of capacity constraints, L ¯ j ∈ Jr is one capacity constraint, which is indexed in subset Jr . ((4), (9)). Note that lj ∈ L, Output: The optimal solution x∗ . ˜ 1: Initialize the current set of capacity constraints as L ← L; Solve the problem with L and obtain the optimal solution x ˆ of the relaxed problem. ¯\L ˜ | lj is violated in x while ∃ lj ∈ L ˆ do 4: Add lj to the constraint set, i.e. L ← L ∪ {lj } ¯\L ˜ | 0 < tj − tk < κ to the constraint set, i.e. L ← L ∪ {lk } 5: Add lk ∈ L 6: Solve the problem with L and obtain the optimal solution x ˆ of the relaxed problem.

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2: 3:

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x∗ := x ˆ The optimal solution of the relaxed problem x ˆ is the optimal solution.

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In this paper, we will evaluate the following four versions of the solution algorithm.

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• Branch and Bound (B&B): the standard B&B algorithm as implemented in CPLEX. The allocation problem and the pricing problems are solved independently without considering the relation between them.

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• B&B with Initial solution: the B&B algorithm implemented in CPLEX providing initial solutions to the pricing problems. The initial solutions are derived from the optimal solution to the allocation problem, as explained in Section 6.1

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• B&B with Lazy Constraints: the B&B algorithm implemented in CPLEX where capacity constraints are handled as lazy constraints. The initial set of lazy constraints of the pricing problem is derived from the allocation problem, as explained in Section 6.2.

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• B&B Combined: the B&B algorithm implemented in CPLEX with both the initial solution and the lazy constraints.

7. Comparison of Penalty and Auction’s Formulations In this section, we show that the proposed auction formulation provides a unified approach that generalizes the formulation of Roca-Riu et al. (2015). The proposed auction formulation is based on a valuation vector that each company has for the time of service. In contrast, the penalty formulation assumes that companies express their desired parking times only through the use of a time window. 20

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Section 7.1 describes the penalty formulation in a compact way. Section 7.2 shows that the penalty and auction’s formations are equivalent in some special cases, and Section 7.3 quantifies the improvement by using the auction’s formulation compared to the penalty formulation with different levels of information via a numerical experiment. 7.1. Penalty formulation

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In the penalty formulation (Roca-Riu et al., 2015), each company is interested in using the parking facilities during a time window [ai , di ]. However, when the assigned starting time is outside the requested time window, a penalty is incurred. This penalty expresses the potential inconvenience that the assignment represents for the company. As a generalization of the penalty concept, the objective of the penalty formulation can be represented as a function fi which takes the desired time window [ai , di ] and the assigned time tj as inputs. ( = 0, tj ∈ [ai , di ] φij = fi (ai , di , tj ) (37) > 0, tj 6∈ [ai , di ]

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where the penalty φij represents the inconvenience that company i experiences with starting time tj , and the values are given by function fi . The objective of the assignment is to minimize the overall penalty. We consider penalties to be additive, i.e. the overall penalty for all the companies is the sum of the penalties for each company. Recall that xij is a binary variable that is 1 when company i is assigned the starting time tj . The penalty formulation is then compactly expressed in Equations (38)– (40), where (38) is the objective function for minimizing penalties, and (39) guarantees that all the carriers are assigned. XX Φ(x, φ) = min φij xij (38) i∈N j∈Jr

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s.t.

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i∈N

Constraint (4), (5)

(39) (40)

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Particularly, Roca-Riu et al. (2015) proposed the following types of penalties.

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• Number of unsatisfied instances, i.e. fi (ai , di , tj ) =

(

0,

tj ∈ [ai di ]

1,

otherwise

(41)

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• Earliness or tardiness, i.e. fi (ai , di , tj ) = min{ai − tj , 0, tj − di }

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• Earliness or tardiness with bound, i.e. fi (ai , di , tj ) = min{ai − tj , 0, tj − di },

tj ∈ [ai − q, di + q]

(43)

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7.2. Equivalence of formulations In this section, we provide the results of the theoretical comparison between the penalty formulation and the auction’s formulation. The penalties defined in (37) can be interpreted from a valuation’s perspective. In fact, given a penalty φij , the corresponding vij can be written as vij = vmax,i − αi φij

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where vmax,i represents the maximum valuation carrier i would achieve when assigned within its preferred time window; αi is a factor parameter that relates the inconvenience experienced by carrier i and monetary costs. Note that vmax,i and αi should satisfy that vij ≥ 0, i.e. the carriers do not suffer any loss by participating into the assignment system. Figure 3 illustrates how the three penalties proposed in Roca-Riu et al. (2015), which have been discussed in the previous section, can be converted into three different valuation functions: the binary (Figure 3a), trapezoid (Figure 3b), and the truncated trapezoid valuation (Figure 3c). Note that the auction’s formulation is a unified framework which does not require any assumption of the valuation vectors, giving freedom to carriers to express their time preferences. v(t)

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The penalty’s formulation receives less information from the carriers than the auction’s formulation. The penalty formulation lacks the carriers valuation of the parking time slots, i.e. the carriers do not share neither their maximum valuation nor the shape of the penalty function. Therefore, in general, true valuation cannot be obtained only with the penalty formulation. As a result, the optimal social welfare, cannot always be achieved through the penalty formulation. However, in some special cases, both formulations are equivalent and obtain the same allocation solution. These cases are summarized in Proposition 5.

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Proposition 5. The penalty formulation and the auction’s formulation are equivalent with conversion from Equation (44), if the system has the accurate information on the penalties φij and αi , and all the carriers have the same maximum valuation. The proof of Proposition 5 is straightforward given (44).

7.3. Numerical comparison for general cases Compared to the penalty’s formulation, the auction’s formulation requires more information, i.e. the shape of the penalty function and the maximum valuation of each carrier, and is able to 22

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exploit such information to obtain better solutions. However, notice that this additional information might be difficult to obtain and understand. Therefore, in this subsection, we aim to quantify the improvement of the auction’s formulation compared to the penalty’s formulation via a numerical experiment with heterogeneous carriers. The experiments of this section are performed with the same set of test instances as in Roca-Riu et al. (2015), denoted as set S1. This set has 60 test instances (numbered from 1 to 60), based on the patterns observed in an experimental study in the city of Barcelona, Spain. The number of parking places ranges from 2 to 8, and requests are expressed as time windows. The requests are distributed according to a triangular pattern around a peak hour that is located either in the center or at the beginning of the morning or afternoon interval. All details can be found in Roca-Riu et al. (2015), and the original instances can be downloaded from http://mrocariu.github.io/code/. For each instance, the valuation function for each carrier is chosen to be either binary, trapezoid, or truncated trapezoid, each type with an equal probability (1/3). The maximum valuation vmax,i is chosen as either 10 or 50, each with an equal probability of 1/2. Here, αi is chosen as vmax,i for the binary valuation, and 0.5 for the trapezoid and truncated trapezoid valuation. For each original instance, we generate 10 instances by randomly choosing a valuation function and a maximum valuation for each carrier according to these probabilities. We next conduct the numerical experiment when different levels of information are provided.

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1) No information (Penalty formulation). As originally formulated, the penalty formulation does not have access neither to the maximum valuation, nor to the exact shape of the penalty function. For this case, the penalty formulation is solved assuming that all the carriers have the same form of valuation (i.e. either all carriers have a trapezoid, a truncated trapezoid, or a binary valuation).

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2) Some information (Penalty formulation with known penalty shape). Carriers report the shape of the penalty function to the system, but the maximum valuation remains unknown. Note that the information on the shape of the penalty function can be easily incorporated into the penalty formulation to deal simultaneously with different shapes of penalty for each carrier. Also note that the three shapes of the penalty functions defined in (41)–(43) have different units and scales. To integrate these penalty functions into the same objective function, each shape is standardized to be between 0 and 1.

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3) Full information (Auction formulation). As we have proved in Section 7.2, the auction’s formulation is equivalent to the penalty formulation with information on the shape of the penalty and the maximum valuation.

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The comparison between these levels shows the combined value of the information and the auction’s formulation, respectively. Note that in the allocation solution of each tested instance, all carriers are assigned a starting time instant. The social welfare when solving the instances with the different levels of information are presented in Table 1, where the percentages of the loss compared to the full information (auction’s formulation) are in parenthesis. It can be seen from Table 1 that with no information, i.e. without knowing the shape of penalty function nor the maximum valuation, the social welfare suffers a loss up to 22% compared to the auction’s formulation, and a median from 5% to 9%. The median is used instead of the mean to 23

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Table 1: Comparison of the auction and the penalty formulations for heterogeneous carriers

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3120 (-2%) 1932 (-7%) 1755 (0%) 3074 (-7%) 2000 (-12%) 2601 (-3%) 1253 (0%) 1951 (-6%) 1919 (0%) 714 (0%) 2547 (0%) 3110 (0%) 4084 (-3%) 3930 (-1%) 1465 (-5%) 1689 (0%) 2260 (-4%) 1319 (0%) 2935 (0%) 605 (0%) 2159 (0%) 3124 (0%) 1437 (-1%) 3714 (0%) 1703 (-3%) 4063 (0%) 2587 (-5%) 3817 (-1%) 1513 (-3%) 5021 (0%) 2968 (0%) 6038 (-4%) 6655 (-2%) 1380 (0%) 670 (0%) 849 (0%) 668 (0%) 2163 (0%) 2161 (-2%)

3189 2097 1771 3326 2280 2686 1253 2094 1920 714 2552 3111 4211 3970 1542 1689 2354 1319 2935 605 2159 3156 1457 3750 1757 4063 2723 3891 1572 5025 2968 6304 6825 1380 670 849 668 2163 2222

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Trapezoid 2547 (-20%) 1739 (-17%) 1645 (-7%) 2840 (-14%) 1845 (-19%) 2435 (-9%) 1192 (-4%) 1822 (-12%) 1906 (0%) 687 (-3%) 2394 (-6%) 3008 (-3%) 3710 (-11%) 3557 (-10%) 1301 (-15%) 1670 (-1%) 2103 (-10%) 1225 (-7%) 2900 (-1%) 537 (-11%) 2138 (0%) 2855 (-9%) 1331 (-8%) 3361 (-10%) 1489 (-15%) 3943 (-2%) 2307 (-15%) 3483 (-10%) 1403 (-10%) 4708 (-6%) 2841 (-4%) 5597 (-11%) 6054 (-11%) 1361 (-1%) 668 (0%) 838 (-1%) 635 (-4%) 2101 (-2%) 2102 (-5%)

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Penalty Formulation No Truncated Trapezoid Binary 2843 (-10%) 2834 (-11%) 1774 (-15%) 1758 (-16%) 1671 (-5%) 1646 (-7%) 2831 (-14%) 2754 (-17%) 1838 (-19%) 1870 (-17%) 2443 (-9%) 2466 (-8%) 1198 (-4%) 1184 (-5%) 1842 (-12%) 1710 (-18%) 1912 (0%) 1890 (-1%) 687 (-3%) 668 (-6%) 2376 (-6%) 2338 (-8%) 2999 (-3%) 2984 (-4%) 3742 (-11%) 3656 (-13%) 3558 (-10%) 3612 (-9%) 1306 (-15%) 1294 (-16%) 1670 (-1%) 1664 (-1%) 2114 (-10%) 1996 (-15%) 1219 (-7%) 1236 (-6%) 2892 (-1%) 2876 (-2%) 575 (-5%) 580 (-4%) 2142 (0%) 2138 (0%) 2838 (-10%) 2712 (-14%) 1349 (-7%) 1280 (-12%) 3369 (-10%) 3376 (-9%) 1498 (-14%) 1372 (-21%) 3950 (-2%) 3906 (-3%) 2350 (-13%) 2188 (-19%) 3533 (-9%) 3364 (-13%) 1405 (-10%) 1330 (-15%) 4754 (-5%) 4648 (-7%) 2824 (-4%) 2800 (-5%) 5607 (-11%) 5400 (-14%) 6014 (-11%) 5978 (-12%) 1361 (-1%) 1346 (-2%) 668 (0%) 636 (-5%) 835 (-1%) 832 (-2%) 635 (-4%) 582 (-12%) 2101 (-2%) 2052 (-5%) 2108 (-5%) 2024 (-8%)

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reduce the impact of extreme values. In the middle level of information where the carriers report the shape of their penalty function but not the maximum valuation, the loss in the social welfare goes up to 12% with a median of 3%. Note that there are some instances, where the penalty formulation with known penalty gives the same optimum as the auction’s formulation. This is because these instances satisfy the second condition of Proposition 5. In general, the results show that the auction formulation provides the solutions with the highest social welfare making use of the whole information. Moreover, recall that the truthfulness of the penalty formulation cannot be ensured, as it does not involve bidding and pricing. 8. System Analysis

8.1. Efficiency of solution improvements

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In this section, we present the results obtained in a series of computational experiments to analyze the auction’s formulation studied in this paper. In Section 8.1, different solution approaches are compared in terms of efficiency and computational time. Section 8.2 analyzes some interesting features of the solutions provided with the auction approach. Finally, Section 8.3 addresses the uncertain arrivals and evaluates the online readjustment formulation and the variants of the allocation formulation. In all the experiments, the discretization steps are chosen as 5 minutes. That is reasonable for an application in real-life systems, and, as showed later, it is affordable in computational times. That being said, further analysis regarding the optimal discretization step would also be possible using more advanced techniques (Ge et al., 2015, 2014)

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In this section, the four different versions of the solution algorithm developed in Section 6 will be evaluated and quantified. Recall that we test the four following versions: B&B, B&B with initial solution (Initial), B&B with lazy constraints (Lazy) and the combined one (Com.). After exploratory testing, parameter κ is chosen as 10 for the Initial and Combined algorithms. Table 2: Computation time of the four solution algorithms (seconds)

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B&B 97 38 17

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Com. 37 37 20

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The four solution algorithms are tested with the S1 instance set with a discretization interval of 5min. Table 2 summarizes the median computation time of the different algorithms with the three valuation possibilities (trapezoid, truncated trapezoid, and binary). The computation time is the total time for solving both the allocation problem and all the pricing problems. All the instances can be solved to optimality within half an hour. The fastest algorithm for each form of valuation function is highlighted with shadings. The computation times presented are sufficient for a real application of a prebooking system, as we require the companies to submit the requests some time ahead (e.g. one day). The maximum number of requests handled is 235. It is expected that instances of larger size can also be solved if we allow longer computational times. For trapezoid valuation, Table 2 shows that the combined solution algorithm provides the optimal solution fastest in most of the scenarios, with 63% improvement compared to standard B&B and 50% improvement compared to B&B with initial solution. For 25

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trapezoid and truncated trapezoid valuation, the differences between all four algorithms are not significant. One possible explanation is that the binary and truncated trapezoid valuation functions enable a sparse structure of both the allocation and pricing problems, which is handled well with CPLEX software. For trapezoid, on the other hand, there is not much sparsity to exploit. 8.2. Analysis of the auction system

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This subsection discusses the properties of the auction system. To this end, we generate a set of new instances to control the relevant parameters. We use the allocation and pricing solutions of the new instances to evaluate the impact of the different instance features: the companies’ valuation vectors, the number of parking spots, and the parking duration. Particularly, we study how the values of the requests at the assigned time instant (assigned value), the prices charged from the requests, and the utilities of the requests change with the companies’ valuation, the number of parking spots, and the parking duration. First, the new instances are described in Section 8.2.1. Second, an analysis of the individual instances is conducted in Section 8.2.2. Third, an aggregated analysis is presented with the entire results in Section 8.2.3.

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In order to evaluate the new features that the unified approach offers, a new test instance set (S2) has been created. With the new approach, each company has its valuation vector over the different time intervals. If we combine the valuation vectors of all the companies together, we can observe the average valuation per time instant throughout the day. In this way, one can get a general idea of the demand for using a loading/unloading area at each time instant. All individual valuation vectors have the same shape with vectors of the following form. vij = vmax e−

(tj −τi )2 2σ 2

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where τi is the time instant with the highest valuation, σ is the shape parameter of the valuation vector, and vmax is the maximum valuation. In this paper, σ is equal for all requests and instances, and takes value of 60 min. In order to account the heterogeneity of the companies, vmax can take the value of either 10, 15 or 20. The combination of τi and vmax is determined such that the average valuation per instance adjusts to one of the two types of the average valuation patterns (uniform or peak). They are shown in Figure 4. The uniform pattern has a uniform valuation between 8:00-14:00, whereas the peak pattern has a peak between 10:00-12:00. For each average valuation pattern, we evaluate the system with different number of parking spots and different parking durations. As already mentioned, there are two average valuation patterns: uniform and peak. The number of parking spots is either 3 or 6; the parking duration can be 10 min, 20 min or 30 min. The instances with 3 parking spots have 60 requests and the instances with 6 parking spots have 120 requests. For each possible combination of the previous three features, 100 instances are randomly generated, so in total S2 has 1200 new instances. In particular, we will call Uniform and Peak the sets of instances with uniform and peak valuation patterns respectively, and number them from 1001...1600, and from 2001...2600. 8.2.2. Individual instance analysis. Figure 5 illustrates the allocation and pricing solutions of four instances, i.e. Instances 1001, 1002, 2001 and 2002, the first two of Uniform and the first two of Peak. Instances 1001 and 2001 26

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only have a vmax of 10 and 20, whereas 60% of the requests in Instances 1002 and 2002 have a vmax 15, and the rest 40% have a vmax 10 and 20. It can be seen by comparing Figure 5a and Figure 5b (also Figure 5c and Figure 5d) that despite the difference in valuation functions, the allocation and pricing solutions of different instances are similar overall if they have the same valuation pattern. This is particularly true for the prices. This is expected, as the valuation patterns determine how the valuation function of the requests distributes over the time horizon. For the same valuation pattern, the distributions of requests for different instances are very similar, which leads to the similar allocation and pricing solutions. A more general discussion will be given in Section 8.2.3. Nevertheless, apart from the difference in valuation functions in general, we can observe some randomness for the values of the requests, e.g. the second leftmost requests in the second slot for the peak demand pattern (Figure 5c and Figure 5d, marked in white). These two requests are assigned at a similar time slot, but with very different assigned values and utilities. This is due to the randomness of the requests. Although for peak demand pattern, most companies prefer the slots in the peak hour, still with a low probability, there are some companies with the highest value in the non-peak hour (the marked request in Figure 5c). This request can easily be allocated at the time slot with the highest value and charged with a low price, as the competition at this time slot is not intense. It is also shown that the allocation and pricing solutions for different valuation patterns are essentially different. For the peak valuation patterns, the requests concentrate in a short period of time (10:00-12:00). In this case, almost all companies would like to use the facility during 10:0012:00, creating an intense competition. Hence, the prices charged from the assigned requests tend to be higher during this period. On the other hand, the values of the requests are generally lower outside this period, as they are not assigned at their preferred time instants. For the uniform valuation patterns, the requests spread uniformly over a long period of time (7:00-15:00). As a result, the prices charged from the requests do not change much with the time, and the values of the requests are also distributed uniformly over the entire time horizon. Overall, the utilities of requests in the uniform demand pattern are higher than the utilities in the peak demand pattern. This shows that the concentrated time preference of the companies deteriorates the performance of the system.

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Figure 5: Plot of assigned individual instances (3 parking spots, parking durations are 20 minutes). Each colored rectangle represents a request. The heatmap colors are determined based on the price of each request. The color changes from black to white as the price increases. The numbers on each request represent the assigned value, the price, and the utility, respectively.

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Table 3: Average assigned values, prices and utilities for all set of instances

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U 0.0 0.9 6.7

P 1.0 6.0 5.9

Utility c=6

U 0.0 0.9 6.9

P 1.1 6.1 5.9

c=3 U 14.5 13.5 5.4

P 13.1 4.9 2.6

c=6 U P 14.6 13.0 13.5 4.8 5.3 2.5

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This is expected, since the valuation per parking spot determines the distribution of requests over the time horizon and the competition intensity of the requests. Hence, the allocation and pricing solutions are also expected to be similar. It can also be observed from Fig. 6a and Fig.6c that there are always peaks at 10, 15 and 20 for both utilities and values, especially at parking duration of 10 min. This is because the maximum of the valuation function vmax is either 10, 15 or 20. Comparing the different valuation patterns, we can see from Figure 6c that the uniform valuation pattern always leads to a higher utility than the peak valuation pattern, as the empirical PDFs of the uniform valuation pattern concentrates more to the right (higher utilities). The same property can be observed for the assigned values in Figure 6a, where companies receive higher assigned values in the scenarios with a uniform valuation pattern. This is expected, as the competition for the time instants is more intense for the peak demand pattern. Therefore, the requests assigned outside the preferred time period receive low values, and the requests assigned in the preferred time period suffer from high prices, as they have more externalities than in the scenarios with a uniform valuation pattern. This confirms that the concentrated requests tend to make the solution worse, as already seen in Section 8.2.2. However, for the prices in Figure 6b, there is no clear relation between the two valuation patterns. For the parking duration of 10 min and 20 min, the uniform valuation pattern renders a lower price than the peak demand pattern. However, for the parking duration of 30 min, the price for the uniform demand valuation is higher than for the peak demand pattern. This can be explained due to the impact of parking duration, which is detailed in the next paragraph. The above observation is consistent with the results in Table 3. The impact of parking duration can also be observed in Figure 6. Figure 6a shows that all empirical PDFs of values move to the left as the parking duration increases. This is because a higher parking duration makes the instances more saturated. Hence, with the increase of the parking duration, less requests can be allocated to the same time instant. It can also be observed from Table 3 that the average assigned values decrease with the increase of the parking duration. The same applies for the utility in Figure 6c. The increase of the parking duration reduces the utilities of the companies. In Figure 6b, the price distribution moves to the right (i.e. the price increases) with the increase of the parking duration for the uniform valuation pattern. However, for the peak valuation pattern, the prices first increase, and then decrease. For short parking durations, especially in scenarios with the uniform valuation, the instances are not very saturated. Therefore, most requests can be assigned at preferred time instants. Hence, the externality of each request is generally small. As the parking duration increases, the system becomes saturated. Then there are two cases. In the first case, the requests can still be assigned at preferred time instants, but they have more impact on the other requests. In this case, the assigned values of the requests are still quite high, and the externalities, i.e. the prices, increase. This corresponds to the scenarios with 30

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the uniform valuation pattern and the parking duration of 20 min and 30 min, and the case with the peak valuation pattern and the parking duration of 20 min. In the second case, most of the requests cannot be assigned at preferred time instants. Therefore, the assigned values of most of the requests are quite low. In this very saturated case, the oversaturation is caused by the joint influence of many requests. Hence, the externalities of any request cannot be high, as the removal of a single request cannot impose much improvement on the solution. This corresponds to the scenario with the peak valuation pattern and the parking duration of 30 min.

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8.3. Performance under uncertain arrivals

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In this subsection, we evaluate the readjustment formulation and the variants of allocation formulation proposed in Section 5. Here, we conduct numerical experiments based on the dataset S2 generated in Section 8.2.1. The uncertainty in the arrivals is generated according to Table 4. Two uncertainty levels are analized: moderate and high. For each instance in S2, we generate 5 instances with random earliness/delay for each carrier. The total cost of unit waiting time λ is chosen as 0.2/min. It is assumed that carriers report the arrival information ∆T = 20min before their arrival. This time is chosen such that 1) it allows sufficient flexibility for readjustment and 2) the reports can be typically accurate enough (with an error of less than 5 min). Table 4: Distribution of the earliness/delay

-3 0.05 0.14

-2 0.1 0.14

-1 0.15 0.14

0 0.4 0.16

1 0.15 0.14

2 0.1 0.14

3 0.05 0.14

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Earliness/delay (time slots) Moderate Uncertainty Level High Uncertainty Level

8.3.1. Performance of the readjustment formulation

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The performance of the readjustment formulation is evaluated by comparing the resulting social welfare of: 1) the formulation without readjustment where carries are only allowed to use the facility from their assigned starting time instant, 2)the formulation with readjustment, and 3) the global offline readjustment optimum. The average values of the social welfare for each set of scenarios (with a given valuation, number of parking spots and parking duration) are summarized in Table 5. The percentages in the parentheses of column Online Readjustment show the improvement of the online readjustment formulation compared to the formulation without readjustment. While the parentheses of column Offline Optimum show the percentages of improvement of the global offline optimum compared to the online readjustment formulation. Compared to the formulation without readjustment, the online readjustment formulation significantly improves the social welfare of the system. In the scenarios with high uncertainty level, the improvement ranges from 14% to 37%, whereas in the scenarios with moderate uncertainty level, the improvement ranges from 28% to 58%. The benefits of the readjustment formulation are therefore clear. It can also be seen from Table 5 that the online readjustment formulation can, in most cases, provide a social welfare similar to the global offline optimum. That is probably due to the fact that the impact of the early or delayed carriers is mainly local. 8.3.2. Performance of the variants of the allocation formulation We evaluate the two robust variants proposed in Section 5.2, the Extension and the Slackness variants. For the extension variant, we assume that the time window is extended by 1 time slot. 31

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Table 5: Performance of the online readjustment formulation

Valuation Uniform

10 Peak Uniform Moderate

20 Peak Uniform 30 Peak Uniform 10 Peak Uniform

High

20 Peak Uniform 30

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Parking Spot 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6

Without Readjustment 640 1267 620 1230 632 1251 489 963 536 1056 381 750 545 1124 529 1093 538 1112 424 879 460 958 338 694

Online Readjustment 860 (34%) 1732 (37%) 793 (28%) 1622 (32%) 778 (23%) 1625 (30%) 580 (19%) 1220 (27%) 631 (18%) 1327 (26%) 432 (14%) 926 (23%) 859 (58%) 1727 (54%) 786 (49%) 1609 (47%) 775 (44%) 1614 (45%) 578 (36%) 1213 (38%) 621 (35%) 1325 (38%) 434 (28%) 926 (33%)

Offline Optimum 860 (0% ) 1732 (0% ) 793 (0% ) 1624 (0% ) 781 (0% ) 1631 (0% ) 583 (0% ) 1226 (0% ) 637 (1% ) 1339 (1% ) 438 (1% ) 934 (1% ) 859 (0% ) 1727 (0% ) 789 (0% ) 1612 (0% ) 780 (1% ) 1623 (1% ) 584 (1% ) 1223 (1% ) 632 (2% ) 1340 (1% ) 443 (2% ) 935 (1% )

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Uncertainty Level

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For the slackness variant, we choose the parameter ρ as 0.1. The results are summarized in Table 6, where the column Static Optimum represents the optimal social welfare if all the carriers arrive exactly on time (Deterministic Arrivals). Under Stochastic Arrivals, the columns Original, Extension, and Slackness are obtained by solving the online readjustment formulation on the allocation solutions obtained by the original allocation formulation, the Extension Variant formulation and the Slackness Variant formulation. Shaded cells indicate the highest utility. The last column shows the optimal social welfare obtained with the benchmark formulation of the optimal in retrospect. It is shown that the Extension Variant performs the best in the cases where si =10min and carriers have a uniform valuation, but significantly worse than the other two variants in the rest of the cases. This is because when the carriers have very high competition over the same time slots, extending the assigned time window would significantly deteriorate the allocation solution, and such deterioration outweighs the improvement in robustness. However, tests have shown that the performance of the Extension Variant improves as λ increases. Recall that λ represents the cost of the waiting time for carriers and the externalities the waiting carriers impose on the traffic system nearby. If waiting carriers are costly for the system, it would be more beneficial to use a robust variant. It is also demonstrated that on average, the slackness variant performs slightly better than the rest. For the scenarios where the carries do not present very high competition (e.g. with si = 10min and uniform valuation), the three allocation formulations give an optimum close to the optimum in retrospect (e.g. within 1%). For scenarios, where the carriers have high competition over some

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Table 6: Comparison of the three variants of the allocation problem

Uniform 10 Peak Uniform Moderate

20 Peak Uniform 30 Peak Uniform 10 Peak Uniform

High

20 Peak Uniform 30 Peak

Parking Spot 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6 3 6

Optimum in Retrospect 874 (1%) 1747 (1%) 842 (6%) 1690 (4%) 858 (10%) 1722 (6%) 644 (11%) 1303 (7%) 704 (11%) 1431 (8%) 489 (12%) 997 (8%) 874 (2%) 1747 (1%) 842 (7%) 1685 (5%) 856 (10%) 1717 (6%) 641 (11%) 1296 (7%) 702 (13%) 1430 (8%) 489 (12%) 989 (7%)

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Valuation

Stochastic Arrival Online readjustment Original Extension Slackness 860 866 861 1732 1736 1733 793 747 792 1622 1508 1623 778 779 781 1625 1593 1629 580 554 581 1220 1136 1224 631 612 633 1327 1296 1331 432 419 436 926 890 926 859 860 859 1727 1728 1727 786 742 788 1609 1503 1611 775 773 775 1614 1583 1618 578 554 578 1213 1130 1215 621 612 621 1325 1287 1320 434 418 434 926 886 924

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Deterministic Arrival Static Optimum 874 1747 845 1693 863 1726 658 1320 728 1459 507 1017 874 1747 845 1693 863 1726 658 1320 728 1459 507 1017

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time slots (e.g. with si = 30min and peak valuation), the three variants can give an optimum up to 13% less than the optimum in retrospect. Recall that the optimum in retrospect is not achievable in reality, as it requires the accurate information of earliness/delay before solving the allocation problem. This is expected as the earliness and delay information is essential for the allocation problem, and the online readjustment formulation has insufficient information compared to the optimum in retrospect. Also note that the numerical experiment assumes that the earliness/delay of each carrier is independent. However, in reality, the arrival times depend on the traffic conditions, hence their earliness/delay could be correlated. In such cases, we expect the difference between the three allocation formulations and the optimum in retrospect to be even smaller. It can also be seen that such difference is in general smaller in the instances with more parking spots. This is because as the number of parking spots increases, there is more room for the online readjustment. 9. Conclusions In this paper we proposed a new approach for the optimal management of urban loading/unloading facilities for logistic carriers. The approach, which is based on combinatorial auctions, optimizes the use of prebooked facilities with the goal to maximize the social welfare of the system. The new formulation of the problem enhances participation, provides a fair system, and guarantees that the carriers will express their true valuations. We proved that the flexibility of the new approach allows 33

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carriers to use any valuation function to express their time preferences over the offered spots, and it is valid to generalize a previous approach. We studied different improvements to the standard solution algorithm. Results showed that the combined approached with an initial solution and the use of lazy constraints for the capacity constraints provides the optimal solution the fastest in most scenarios. The features of the solutions were also analyzed, individually and at the system level. The solutions for the problems where demand is uniformly distributed over the time horizon have better properties (higher valuation and utilities, and lower prices) than the ones where demand is concentrated in time. In any case, the optimal solution provides the maximum social benefit. This shows the benefits of using combinatorial auctions to optimally manage the prebooked urban loading/unloading facilities, which may become very popular with the introduction of technological and communication systems to improve the efficiency in urban areas. Additionally, we addressed the uncertainties in arrival times by developing an online readjustment formulation and two robust variants of the allocation problem. It is shown that the online readjustment formulation significantly outperforms the formulation without readjustment and yields a similar social welfare to the global offline optimum. The results also show that the Extension Variant performs better when the competition is not high, and the Slackness Variant performs slightly better in most cases. All the three variants can yield a social welfare close to the optimum in retrospect. One future research direction to explore is the study of a dynamic prebooked system, where the system does not have the full information of carriers when making the decision. Also, more sophisticated yet practically feasible approaches to handle uncertainties can be explored. Acknowledgements

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This work was supported by the Hasler Foundation with the CALog project. Appendix A. Proof for Proposition 2

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In order to prove Proposition 2, we prove the following two conclusions ai +s i −1 X µ, γ ) = {i|j ∈ Ti , µi + µ∗ , γ ∗ ) 1) Denote setIj (µ γj = bi }. There exists a dual solution (µ j=ai

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µ∗ , γ ∗ )| ≥ c + 1 or γj = 0, where | · | represents the number of such that for each j ∈ Jr , either |Ij (µ elements. µ∗ , γ ∗ ) is the dual solution satisfying 1), then the utility for carrier i satisfies ui = µi . 2) (µ

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µ, γ ), by Complementary Slackness ui (1 − xi ) = 0, Proof. 1) For any optimal dual solution (µ aX i +si X    γj xi 1Ti (j) − c = 0, and xi µi + γj − bi = 0. Therefore, for each j ∈ Jr , it holds i∈N

j=ai

µ, γ )| ≥ c or γj = 0. In the case where |Ij (µ µ, γ )| = c, We decrease γj by a small ∆j and that |Ij (µ increase the corresponding µi by ∆j . Then all the constraints still hold and the dual objective value µ∗ , γ ∗ )| = c + 1 or γj∗ = 0. Then remain unchanged. We can continue this procedure until either |Ij (µ µ∗ , γ ∗ ) is still an optimal dual solution. (µ 2) Denote Vsg [N ] as the optimal social welfare allocating carriers in set N . Let x∗ denote an µ∗ , γ ∗ ). For optimal solution to the primal problem, satisfying the Complementary Slackness with (µ 34

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µ∗ , γ ∗ )| = c + 1. In the former case, let x each j ∈ Ti , we have either γj∗ = 0 or |Ij (µ ˜ = x∗ . In µ∗ , γ ∗ ), and x the latter case, let x ˜˜i = 1, ∀˜i 6= i ∈ Ij (µ ˜˜i = x˜∗i otherwise. Then it is not difficult to ∗ ∗ µ−i , γ −i ) satisfy Complementary Slackness. Hence (µ µ∗−i , γ ∗−i ) is optimal for validate that x ˜−i and (µ Vsg [N \{i}]. Hence, if xi = 0, we have ui = 0 = µ∗i . If xi = 1, we have pi = bi − Vsg [N ] + Vsg [N \{i}] =

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