Global synchromodal transport with dynamic and stochastic shipment matching

This paper investigates a dynamic and stochastic shipment matching problem, in which a platform aims to provide online decisions on accepting or rejecting newly received shipment requests and decisions on shipment-to-service matches in global synchromodal transportation. The problem is considered dynamic since the platform receives requests and travel times continuously in real time. The problem is considered stochastic since the information of requests and travel times is not known with certainty. To solve the problem, we develop a rolling horizon framework to handle dynamic events, a hybrid stochastic approach to address uncertainties, and a preprocessing-based heuristic algorithm to generate timely solutions at each decision epoch. The experimental results indicate that for instances with above 50% degrees of dynamism, the hybrid stochastic approach that considers shipment request and travel time uncertainties simultaneously outperforms the approaches that do not consider any uncertainty or just consider one type of uncertainties in terms of total profits, the number of infeasible transshipments, and delay in deliveries.


Introduction
Global container transportation is the movement of containers between inland terminals located in different continents by using ships, barges, trains, trucks or any combination of them (Yang et al., 2018). With the increasing volume of global trade, container transportation becomes more and more important in improving the efficiency of global supply chains. As the fastest-growing cargo segment, global containerized trade reached 152 million twenty-foot equivalent units (TEUs) in 2018 (UNCTAD, 2019). Traditionally, global container transportation is organized by multiple operators. For example, an inland operator in Asia transports containers from Chongqing Terminal to Shanghai Port; a shipping liner company manages the container transport from Shanghai Port to Rotterdam Port; an inland operator in Europe further transports containers from Rotterdam Port to Duisburg Terminal.
In the past decade, horizontal collaboration between shipping lines has been very popular by forming an alliance to improve the utilization of resources and increase service frequency and capacity (Lee and Song, 2017). Recently, port operators and shipping lines appear to be focusing more attention on vertical integration by expanding service networks to inland terminals, such as Maersk and W. Guo et al. Transportation Research Part E 152 (2021) 3 This paper contributes to the literature in the following ways: • We introduce the DSGSM problem for global synchromodal transport networks that integrates online acceptance and matching decisions, and considers stochastic spot requests and travel times simultaneously. • To solve the problem, we develop a rolling horizon framework to handle dynamic events and a hybrid stochastic approach to address uncertainties. The hybrid stochastic approach consists of a sample average approximation method addressing spot request uncertainty and a chance-constrained programming method addressing travel time uncertainty. Besides, we design a preprocessing-based heuristic algorithm to generate timely solutions at each decision epoch. The methodological framework for the DSGSM problem is shown in Fig. 3. • We evaluate the performance of the hybrid stochastic approach in comparison to the approaches that do not consider any uncertainty or only consider one type of uncertainties under a comprehensive set of experiments. The experimental results highlight the benefits of incorporating stochastic information in dynamic shipment matching processes in terms of total profits, the number of infeasible transshipments, and delay in deliveries. W. Guo et al. Transportation Research Part E 152 (2021) 102404 4 Table 1 The formulation characteristics and methodologies in related literature.  • Although we focus on the application in global synchromodal transportation, the developed methodologies can also be applied to the more specific fields of inland and maritime transportation.

Articles Formulation characteristics Methodologies
The remainder of this paper is structured as follows. We briefly review the relevant literature in Section 2. In Section 3, we provide a detailed problem description, followed by the methodology in Section 4. In Section 5, we present the experimental results. Finally, in Section 6, we provide concluding remarks and directions for future research.

Literature review
Global transportation consists of intercontinental transportation and inland transportation (Yang et al., 2018). In intercontinental transportation, containers are transported from export terminals to import terminals. In inland transportation, export containers are transported from inland origins to export terminals; import containers are transported from import terminals to inland destinations. While extensive studies investigated maritime transportation  and inland transportation (SteadieSeifi et al., 2014) in the literature, only a few studies investigated global transportation (Lee and Song, 2017). Furthermore, most of the existing global transport planning models (e.g., Meng et al., 2012;Liu et al., 2014;Tran et al., 2017;Yang et al., 2018;Wei and Dong, 2019) assumed that all the input information is static and deterministic. However, in reality, multiple dynamic events and uncertainties exist in global synchromodal transportation which highly affect the feasibility and profitability of transport plans.
In container transportation, the dynamic and stochastic models related to the DSGSM problem mainly include container booking and container routing problems. While the former considers the acceptance of booking requests to maximize revenue, the latter emphasizes the decisions on routing containers to minimize costs . The formulation characteristics and methodologies of dynamic and stochastic container booking and routing models are summarized in Table 1.

Dynamic and stochastic container booking models
Container booking control, also called slot allocation and capacity control, is one of the primary research topics in revenue management and is widely adopted by the airline industry (Meng et al., 2019). Container booking control aims to maximize revenue in a stochastic environment by effectively deciding on the acceptance of booking requests. According to the network structure, studies on container booking control can be divided into two groups: single-leg level and network level. While the single-leg level models (e.g., Lee et al., 2009;Wang, 2017) consider services operating on a single corridor, the network level models study services that operate on a network with the possibility of transshipments (Meng et al., 2019).
Most network container booking control models study static environments with the main strategies of booking limits and bid-price. A booking limit represents the maximum number of containers that should be allocated to a service. For example, Zurheide and Fischer (2012) proposed a slot allocation model for a liner shipping network to determine the booking limits for different booking classes (e.g. OD pair, container type, and service segment). van Riessen et al. (2017) investigated a cargo fare class mix problem in an intermodal network to maximize revenue by determining the booking limits on each fare class. Under a bid-price strategy, the decision of whether to accept or reject a booking is made based on the lowest acceptable profit value or the marginal costs for the next unit of capacity. Zurheide and Fischer (2014) developed a slot allocation model for a liner shipping company to decide the opportunity cost of a container slot as the bid-price and proved that the bid-price strategy outperforms the booking limit strategy due to the better utilization of capacity for profitable requests.
With the development of information technologies and digitalization in the container industry, researchers and industries have increasingly shifted their attention to dynamic models (Meng et al., 2019). These models can better reflect the online container booking processes and therefore better manage resource capacity. Bilegan et al. (2014) designed a load acceptance management system for rail container transport planning to dynamically accept requests or reject them in favor of future requests with potentially higher profit. Wang et al. (2016) developed a probabilistic mixed integer programming optimization model to make acceptance decisions with the objective to maximize the expected revenue of a barge carrier over a given planning horizon. Wang et al. (2017) investigated a dynamic resource allocation problem, in which an intermodal operator attempts to determine the policy that characterizes the optimal quantities of each service product allowed to be sold during each time interval within a finite selling horizon.
In comparison to the DSGSM problem proposed in this paper, the above-mentioned container booking models focus on the acceptance decisions of requests before the transport process by setting booking limits or bid-price to maximize revenue. However, we take into account the synchronization of shipments with specific time windows and services with specific time schedules during the transport process. Besides, our work considers the dynamic and stochastic information of shipment requests and travel times in a global synchromodal network, and re-optimizes the transport plan when disturbances (i.e., infeasible transshipments) happen.

Dynamic and stochastic container routing models
In the literature, container routing models have been well investigated at the strategic and tactical levels under a static context . Most of the studies integrate the container routing decision with other decisions such as empty container repositioning (e.g., Song and Dong, 2012) and service network design (e.g., Dong et al., 2015;Demir et al., 2016). Recently, with the increasing interest towards synchromodality, several dynamic container routing models in synchromodal transportation have been proposed. Li et al. (2015) proposed a rolling horizon approach to control and reassign container flows in an inland synchromodal freight transport network with dynamic transport demand and traffic conditions. Qu et al. (2019) proposed a mixed-integer  programming model to reschedule services and reroute shipment flows under the framework of synchromodality when unexpected dynamic events cause deviations from original plans. Guo et al. (2020b) investigated a dynamic shipment matching problem in which a platform provides online matches between shipment requests and transport services in an inland synchromodal network. The recent developments in information technologies and data analytics have facilitated the utilization of stochastic information in online decision-making processes (Ritzinger et al., 2015). With regards to dynamic and stochastic container routing problems, van Riessen et al. (2016) proposed a decision tree to instantaneously allocate incoming containers to inland services by analyzing the solution structure of an optimization model on historical data of transport demand. Rivera and Mes (2017) proposed an adaptive approximate dynamic programming algorithm to assign the newly arrived shipment requests to a barge or trucks incorporating the probability distributions of future requests, to achieve cost minimization over a multi-period horizon. Guo et al. (2020a) proposed an anticipatory optimization approach to create online matches between shipment requests and transport services in an inland synchromodal network by incorporating the probability distributions of future requests. Guo et al. (2020c) developed a chanceconstrained programming model to address travel time uncertainty in a global synchromodal shipment matching problem.
Compared with the DSGSM problem proposed in this paper, the majority of the dynamic and stochastic container routing models focus on the routing/matching decisions for booking requests in inland networks to minimize total costs without the consideration of acceptance decisions. Furthermore, none of them consider the dynamic and stochastic shipment requests and travel times simultaneously in global synchromodal transportation.

Summary
According to the above literature review, the majority of the dynamic and stochastic container booking and routing studies focus on inland networks. In this paper, we investigate global networks that integrate inland and intercontinental transportation. Besides, most of the above-mentioned studies only consider container volume uncertainty. This paper considers that all the attributes of a shipment request are uncertain, including shipments' origin, destination, container volume, time window, and fare class. Although Rivera and Mes (2017) considered shipment request uncertainty, their model neither considers the time schedules of transport services nor the transshipment operations between different modalities. To the best of our knowledge, most related to our work are the papers of Guo et al. (2020a) that addressed shipment request uncertainty and Guo et al. (2020c) that addressed travel time uncertainty in synchromodal transportation with the consideration of re-optimization procedures when disturbances happen. However, none of them consider stochastic shipment requests and travel times simultaneously. This paper develops a hybrid stochastic approach to address shipment request and travel time uncertainties integrally in online decision-making processes under the field of global synchromodal transportation.

Problem description
We consider a platform owned by a global operator that receives contractual and spot shipment requests from shippers, and receives ship, barge, train, and truck services from carriers, as shown in Fig. 2. We define the global operator as the coordinator that Future requests received at decision epoch t. The probability distributions {π k , πo, π d , πu, πa, πe, π l , πp, π c D } are assumed known.
collaborates with shippers, carriers and terminal operators to provide integrated transport planning in a global synchromodal network. The global operator does not typically own any of the transport services used to move a shipment from its origin to its destination or any of the terminals used for transshipments. Instead, the global operator enters into contracts for transport services with carriers and for loading/unloading and storage operations with terminal operators. The contract with carriers specifies the services that are available to the global operator with specific modalities, OD pairs, time schedules, available capacities, and costs. The global operators combine these services into itineraries to provide integrated transport for shipments. The global operator publishes the fare classes for each OD pair with specified freight rates, lead times, and delay costs. Shippers choose the fare classes for their shipments based on the value and urgency of commodities. After that, they initiate requests to the platform with specific OD pairs, container volumes, time windows, and fare classes, and leaves the choices of services to the platform. The notation used in this paper is shown in Table 2.

Terminals
Let N be the set of terminals. Each terminal i ∈ N is characterized by its loading/unloading cost c m i , loading/unloading time f m i with mode m ∈ M = {ship, barge, train, truck}, and storage cost per container per hour c S i . We assume terminal operators provide unlimited loading/unloading and storage capacity to the global operator.

Shipment requests
Let R be the set of requests. Each request r ∈ R is characterized by its container type k r (i.e., dry or reefer), origin terminal o r , destination terminal d r , container volume u r , announce time a r (i.e., the time when the platform receives the request), release time e r (i. e., the time when the shipment is available for transport process), and fare class including freight rate p r , lead time L r , and delay cost c D r . The due time of request r is represented as, l r = e r + L r . Requests R consist of two groups: contractual requests R 0 and spot requests R t . For a contractual request r ∈ R 0 , the global operator has long-term contracts with shippers. Therefore, the announce time of contractual request r is a r = 0. All the information {k r , o r , d r , u r , e r , l r , p r , c D r } is known in advance. On the contrary, for a spot request r ∈ R t , the platform receives the request from spot markets during time interval (t − 1, t]. The information of the spot request {k r , o r , d r , u r , e r , l r , p r , c D r } is unknown before its announce time a r . However, the probability distributions of spot requests {π k , π o , π d , π u , π a , π e , π l , π p , π c D } are assumed to be available to the platform. In addition, shippers require their shipments to be transported as a whole, and ask to receive the transport plan as soon as possible. Besides, we do not consider cancellation of requests from shippers. The requests accepted by the platform will not be rejected in the future.

Transport services
Let S be the set of services. Each ship, barge or train service s ∈ S ship ∪ S barge ∪ S train is characterized by its mode m s ∈ M, origin terminal o s , destination terminal d s , free capacity U tk s in terms of container type k ∈ K = {dry, reefer} at decision epoch t, total free capacity U t s , scheduled departure time D s , scheduled arrival time A s , estimated travel time t s , travel cost c s , and generation of carbon emissions ε k s for container type k. Let t s , D s and A s be the actual travel, departure and arrival time of service s which are unknown before their realization. Moreover, different services with the same mode might be operated by the same vehicle. For two successive services operated by the same vehicle, transshipment is unnecessary at the intermediate terminal. Let ξ sq be equal to 0 if services s and q are operated by the same vehicle, and service s is the preceding service of service q, 1 otherwise. Each truck service s ∈ S truck is characterized by its origin terminal o s , destination terminal d s , free capacity U tk s in terms of container type k ∈ K at decision epoch t, total free capacity U t s , estimated travel time t s , travel cost c s , and generation of carbon emissions ε k s for container type k. Let t s be the actual travel time of service s which is unknown before its realization. Each truck service consists of a fleet of trucks that have flexible departure times. We define D rs as a variable that indicates the departure time of service s ∈ S truck with shipment r ∈ R. We assume the platform receives real-time information once a service s ∈ S departs from or arrives to a terminal.
In practice, travel time uncertainties are quite common resulting from weather conditions and traffic congestions (Demir et al., 2016). The probability distributions of travel times are assumed to be available to the platform through the analysis of historical data and external factors. In the literature, different distributions have been used to characterize travel times, such as normal distributions (e.g., Li et al., 2010;Ehmke et al., 2015;Shi et al., 2018) and Gamma distributions (e.g., Taş et al., 2014b;Long et al., 2018). Researches on fitting continuous distributions to travel time variations tend to present inconclusive and inconsistent results due to different traffic conditions and different service areas (Rahman et al., 2018). The main objective of this paper is not to find the best fit distributions of travel times but rather to investigate the benefits of incorporating travel time uncertainty in online decision-making processes under the field of global synchromodal transportation. Thanks to the properties of normal distributions, the sum and difference between two independent normal random variables are still normal distributed. These properties are critical for checking the feasible transshipments between different services. Therefore, we assume the travel times Due to travel time uncertainties, the actual departure and arrival time of service s ∈ S are also uncertain. The distribution of the departure time of service s is based on the distribution of the arrival time of its preceding service; the distribution of the arrival time of service s is based on the distributions of the departure and travel time of service s. For vehicle v ∈ V⧹V truck , we define the itinerary of vehicle v as the sequence of services that the vehicle operated, and define I n v as the n th service of vehicle v. Therefore, the departure time of service s = I n v follows normal distribution given by: Similarly, the arrival time of service s = I n v follows normal distribution given by:

Objectives and infeasible transshipments
The objective of the platform is to maximize total profits over the planning horizon T by dynamically optimizing acceptance and matching decisions over a global synchromodal network. In practice, the first-come-first-served (FCFS) strategy has been widely adopted in the container industry (Meng et al., 2019). Under such a strategy, decisions are made based on deterministic information only. An illustrative example of online matching processes under the FCFS strategy is shown in Fig. 4. At decision epoch t = 8, the platform accepts request r1, and matches r1 with ship service s1 and barge service s2 which are the cheapest services. At decision epoch t = 9, the platform accepts request r2, and matches r2 with rail service s3 and truck service s4. At decision epoch t = 10, the platform receives reefer request r3 which is very profitable. However, the platform has to reject request r3 since no capacity is available. To make better decisions over the planning horizon, the platform needs to consider the stochastic information of future requests.
On the other hand, travel time uncertainty of services in a global synchromodal network may lead to infeasible transshipments in addition to the commonly studied outcome of late or early delivery at destinations (e.g., Li et al., 2010;Taş et al., 2014a;Rodrigues et al., 2019). An illustrative example is shown in Fig. 5. A shipment is planned to be transported by a train service from its origin terminal to port A, by a ship service from port A to port B, and by two barge services from port B to its destination terminal according to fixed time schedules. The outcomes of travel time uncertainty in global synchromodal transportation include late delivery at destination terminal under realization 1 which causes delayed costs, early delivery at destination terminal under realization 2 which causes storage costs, and infeasible transshipment at port B under realization 3 which requires reoptimization.

Methodology
Methodologies for dynamic and stochastic problems can be divided into two categories: methods based on preprocessed decisions which determine the values and policies of decision making before the execution of the transport plan, such as approximate dynamic programming; and methods based on online decisions which focus on the computation when a dynamic event occurs with respect to the current system state and the available stochastic information, such as rolling horizon approaches (Ritzinger et al., 2015). The methodology proposed for the DSGSM problem belongs to the class of online decision methods. The methodology consists of a rolling horizon framework (RHF) that handles dynamic events, a hybrid stochastic approach (HSA) that addresses two types of uncertainties integrally, and a preprocessing-based heuristic algorithm (P-HA) that generates timely solutions at each optimization run, as shown in Fig. 3. Specially, we use the RHF to adapt the matching platform to new states upon receiving new requests or real-time travel times; we use a chance-constrained programming method (CCP) to generate chance constraints regarding infeasible transshipments caused by travel time uncertainty; we use a sample average approximation method (SAA) to sample requests appearing in the near future. Due to the computational complexity, we adopt the P-HA to solve the optimization problem at each decision epoch.

Rolling horizon framework
RHF is known as an efficient periodic reoptimization approach that has been applied in many fields, such as routing problems (Arslan et al., 2019) and scheduling problems (Silvente et al., 2015). Compared with the FCFS strategy, the RHF can handle multiple dynamic events that appear in a system simultaneously, which is quite common in global synchromodal transport, as illustrated in , 1, …, T} 2: receive requests R t , actual departure time Ds of service s ∈ S +t ⧹S truck , actual arrival time As of service s ∈ S − t ⧹S truck , actual departure time Drs of service s ∈ S +t ∩ S truck , and actual arrival time Ars of service s ∈ S − t ∩ S truck 3: for request r ∈ R 0 ∪ … ∪ R t− 1 do

4:
if I R r = ∅ then 5: go to r = r +1 6: else 7: for terminal i ∈ N do 8: if request r just arrived terminal i, service s ∈ {S +1 ∪ … ∪ S +t |os = i} has already departed; or request r has already arrived terminal i, service s ∈ {S +t |os = i} just departed, but the time for transshipment operations is not enough then We define y t r as the binary variable which is 1 if request r ∈ R t is accepted, 0 otherwise. Let x t rs be the binary variable which is 1 if request r ∈ R t ∪ R t is matched with service s ∈ S, 0 otherwise. Here, R t is the set of accepted requests that require reoptimization at decision epoch t due to infeasible transshipments. Based on the actual arrival and departure times of matched services {s ∈ S|x rs = 1} of accepted request r ∈ {R 0 ∪ … ∪ R t− 1 |y r = 1}, the RHF checks which requests need reoptimization due to infeasible transshipments. An infeasible transshipment happens in two situations: first, the accepted shipment r has just arrived terminal i, the matched service s ∈ {S +1 ∪ … ∪ S +t |x rs = 1, o s = i} has already departed; second, the accepted shipment r has already arrived terminal i, and the matched service s ∈ {S +t |x rs = 1, o s = i} has just departed from terminal i, but the remaining time is not enough for transshipments. The platform thus cancels the capacity bookings on the matched services which depart after decision epoch t for reoptimization request r ∈ R t . The RHF then updates the free capacity regarding dry and reefer slots of these services.
The optimization model used at each decision epoch is developed based on the HSA (presented in Section 4.2). The RHF uses the P-HA (presented in Section 4.3) to generate acceptance and matching decisions based on the input parameters and the optimization model. The platform thus books capacities on the matched services based on the matching decisions. After that, the RHF updates free capacities of services and itineraries of requests, and calculates the actual profits AP t generated at each decision epoch. At the end of the planning horizon, the RHF calculates the total actual profit generated over planning horizon T.

Hybrid stochastic approach
In this section, we develop the HSA to address uncertainties, which consists of the CCP that handles travel time uncertainty and the SAA that handles shipment request uncertainty.

Chance-constrained programming model
In the literature, different stochastic programming models have been developed to handle travel time uncertainty (Gendreau et al., 2016). In general, stochastic programming can be either formulated as a CCP model or a stochastic programming model with recourse (SPR) (Li et al., 2010). While CCP models ensure the feasibility of stochastic constraints, SPR models define recourse actions to induce an expected penalty cost on objective functions. Typically, SPR models define a delay cost for late delivery, a storage cost for early delivery, and a large penalty cost for infeasible transshipments without the consideration of reoptimization after disturbances. Since the extra costs caused by reoptimization procedures at later stages are hard to estimate, we develop a CCP model to address travel time uncertainty. The CCP model does not take into account the correction costs caused by the reoptimization of requests.
Under CCP, each stochastic constraint will hold at least with probability α, where α is referred to as the confidence level provided by the platform. A high α means the matches have a low probability causing infeasible transshipments. The confidence level α also controls the problems' tightness and computational complexity. The objective is to maximize the total profits which consist of the planned profits at decision epoch t including freight rates, travel costs, transfer costs, storage costs, delay costs and carbon tax. The optimization model under the CCP at decision epoch t is: ∑ r∈R tk ∪R tk x t rs u r ⩽U tk s , ∀s ∈ S, k = reefer, e r + f ms P{e r + f ms or ⩽D s + B(1 − x t rs )}⩾α, ∀r ∈ R t ∪ R t , s ∈ S + or ⧹S +truck or , where c T ri is the planned transfer cost of request r at terminal i; E(w ri ) is the estimated storage time of request r at terminal i; ) is the estimated delay in delivery of request r at destination terminal d r ; P is the probability measure; z t rsq is a binary variable which equals to 1 if request r has to transfer between service s and q, 0 otherwise; E(D s ) = μ + s , E(Ã s ) = μ − s , E(t s ) = μ s .

Constraints (2) and
(3) ensure that new request r ∈ R t will not be accepted by the platform if there is no matching possibility. Constraints (4,5) ensure that at most one service transports new request r ∈ R t departing from its origin or arriving to its destination.
Constraints (6,7) ensure that reoptimization request r ∈ R t must be transported by one service departing from its origin and by one service arriving to its destination. Constraints (8)-(11) are designed to eliminate subtours of shipments' itineraries. Subtours could be formulated when services that arrive at the shipment's origin and depart from the shipment's destination are selected or when multiple services that depart from or arrive at a transshipment terminal are selected. Constraints (12) ensure flow conservation at transshipment terminals. Constraints (13) ensure that the total container volumes of requests matched with service s do not exceed its free capacity at decision epoch t. Constraints (14) ensure that the total volumes of reefer containers matched with service s cannot exceed its free capacity on reefer slots. In practice, dry containers can use reefer slots, but reefer containers cannot use dry slots (Meng et al., 2019).
Constraints (15) ensure that the departure time of service s minus loading time must be earlier than the release time of request r, if request r will be transported by service s depart its origin terminal. Here, B is a large enough number which ensures the time compatibility between shipment r and service s when binary variable x t rs equals 1, but leaves the constraints "open" if x t rs is 0. Constraints (16)-(20) ensure that the possibility of feasible transshipment at terminals will be higher than the confidence level α. Constraints (21)-(23) calculate the loading costs at origin terminals, the unloading costs at destination terminals, and the loading and unloading costs at transshipment terminals. Constraints (24)-(26) ensure that binary variable z t rsq equals 1 if x t rs = 1 and x t rq = 1, 0 otherwise. Constraints (27)-(34) calculate the storage time at origin, transshipment, and destination terminals. Constraints (35,36) calculate delay in deliveries at destination terminals.
To solve the CCP model, the traditional methods is to convert the chance constraints into their corresponding deterministic equations. Based on the properties of normal distributions, constraints (16)-(20) can be linearized as: where ϕ − 1 (α) is the inverse function of standardized normal distributions.

Sample average approximation method
Based on model P1, in this section, we present the SAA that samples requests which appear in the near future. At decision epoch t, a sample {ω 1 , ω 2 , …, ω γ , …, ω Γ } of Γ scenarios is generated according to the probability distributions {π k , π o , π d , π u , π a , π e , π l , π p , π c D } of shipment requests. Each scenario includes a prediction of spot requests arrived between stage t +1 and stage t + H, ω γ = {ω γ(t+1) , ω γ(t+2) ,…,ω γ(t+H) }. Here, H is the prediction horizon that is just long enough to obtain good decisions at decision epoch t. The expected cost in the future is approximated by the sample average function Γ − 1 ∑ Γ γ=1 . Let ŷ γh r be the binary variable which equals 1 if sample request r ∈ ω γh is accepted, and x γh rs be the binary variable which equals 1 if sample request r ∈ ω γh is matched with service s ∈ S under scenario γ ∈ {1, .., Γ} at stage h ∈ H = {t + 1, …, max{T, t + H}}. We define ĉ T ri as the transfer cost and Ê (w ri ) as the waiting time of sample request r ∈ ω γh at terminal i,Ê

Preprocessing-based heuristic algorithm
Due to the computational complexity, we design the P-HA to solve model P2 at each decision epoch. The P-HA is adapted from the heuristic algorithm designed by Guo et al. (2020b) in which travel times are considered deterministic. The P-HA consists of three steps: preprocessing of feasible paths, preprocessing of feasible matches, and binary integer linear programming.

Preprocessing of feasible paths
We define a path p as a combination of services in sequence. A path p is feasible only if the services inside a combination satisfy time-spatial compatibility. Specifically, for two consecutive services s i , s i+1 within path p, the destination of service s i must be the same as the origin of service s i+1 ; the arrival time of service s i plus unloading time must be earlier than the departure time of service s i+1 minus loading time at transshipment terminal d si with confidence level α. We define N path as the largest number of services in a path. Let P denote the set of feasible paths, and P l ij represent the set of feasible paths with l services that depart from terminal i and arrive to terminal j.
The pseudocode of preprocessing of feasible paths is shown in Algorithm 2. The algorithm starts with determining the feasible paths for each OD pair with just one service, and subsequently combines these paths with a single service to create feasible paths with two services, three services, and so on. To examine whether a new path [s 1 , …, s l− 1 , s] ∈ P l ij consisting of feasible path p = [s 1 , …, s l− 1 ] ∈ P l− 1 ios and service s ∈ S − j is feasible, we check the transshipment feasibility between service s l− 1 and service s by using constraints (38)-(41) with x rs l− 1 = 1, x rs = 1. After that, we check whether feasible path p ∈ P has subtours, and remove paths with subtours. if origin os = i and destination ds = j then 4: for service s ∈ S do 9: if origin os ∕ = i and destination ds = j then 10: for feasible path p = [s1, …, s l− 1 ] ∈ P l− 1 ios do 11: if P{feasible transshipment between service s l− 1 and s} ⩾α then 12: for service s ∈ p do 17: if ds = i or os = j then 18: P l ij ← P l ij ⧹{p} 19: for service s ∈ p, q ∈ p do 20: if os = oq then 21: P l ij ← P l ij ⧹{p}

Preprocessing of feasible matches
A match between request r ∈ R and path p = [s 1 , …, s l ] ∈ P is feasible if it satisfies time-spatial compatibility. Specifically, the origin terminal of shipment request r should be the same as the origin of service s 1 ; the destination of request r should be the same as the destination of service s l . The release time of request r should be earlier than the departure time of service s 1 minus loading time at origin terminal o r with confidence level α. We denote N match as the maximum number of feasible matches. Let Φ r be the set of feasible paths for request r, and c rp be the total costs of matching request r with path p including travel costs, transfer costs, storage costs, delay   ,2,15,9] costs, and carbon tax. The pseudocode of preprocessing of feasible matches is shown in Algorithm 3. For request r and path p = [s 1 , …, s l ] ∈ P l ordr , the transshipment feasibility between r and p is checked by using constraints (15) and (37) with x rs1 = 1. For each request r, if the number of feasible matches in Φ r exceeds N match , Φ r will be replaced by the set of N match cheapest matches.

Binary integer programming model
Based on the above preprocessing procedures, the objective function is updated to maximize the total profits for the matching of requests with feasible paths. Let z t rp be a binary variable equal to 1 if request r ∈ R t ∪ R t is matched with path p ∈ P, and 0 otherwise. Let ẑ γh rp be the binary variable equal to 1 if request r ∈ ω γh , γ ∈ {1, …, Γ}, h ∈ H is matched with path p ∈ P, and 0 otherwise. Model P2 will be translated into a binary integer programming (BIP) model: subject to where Φ rs = {p ∈ Φ r |s ∈ p}. Constraints (46,47) ensure that at most one feasible path will be assigned to each new request r ∈ R t if r is accepted. Constraints (48) ensure that one feasible path will be assigned to each reoptimization request r ∈ R t . Constraints (49,50) ensure that at most one feasible path will be assigned to each sample request r ∈ ω γh if r is accepted. Constraints (51,52) ensure that the total container volumes of requests assigned to service s ∈ S does not exceed its free capacity regarding total slots and reefer slots.

Numerical experiments
In this section, we evaluate the performance of the HSA on the DSGSM problem in comparison to a deterministic approach (DA) which does not consider spot request and travel time uncertainties, a stochastic approach (SA1) which only considers travel time uncertainty, and a stochastic approach (SA2) which only considers spot request uncertainty. While the DA is similar to the approach presented in Guo et al. (2020b), the SA1 is similar to the approach presented in Guo et al. (2020c) and the SA2 is similar to the approach of Guo et al. (2020a). All of the above approaches are implemented under the RHF and supported by the P-HA that generates timely solutions at each decision epoch. The methodological differences among these approaches are summarized in Table 3 {25,22.5,20,17.5,15,12.5}     delay cost (unit: €/TEU-h) c D r = 0.005 * p r for r ∈ R; mean of travel times μ s = t s for s ∈ S; standard deviation of travel times σ s = 0.1 * t s for s ∈ S⧹S truck , σ s = 0.5 * t s for s ∈ S truck . Regarding the P-HA, the default settings are as follows: the largest number of services in a path N path = 7, the maximum number of feasible matches for each request N match = 300.

A small network
To test the impact of different policies in global synchromodal transport, we first consider a small network G1 presented by Guo et al. (2020c), as shown in Fig. 7. It consists of two terminals in Europe and three terminals in Asia that are connected by Suez Canal Route (SCR), Northern Sea Route (NSR), and Eurasia Land Bridge (ELB). Compared with the SCR, the NSR has a shorter travel time but a higher travel cost caused by ice-breaking fees (Lin and Chang, 2018). With the implementation of IMO 2020 regulations, shipping liner companies are required to use low-sulfur fuels on the sea, which in turn increases about 60% of travel costs in the SCR and the NSR (Lian et al., 2020). As an alternative, the ELB becomes more and more competitive thanks to its shortest travel time. However, without subsidies from governments, the ELB is still the most expensive route.
We use the same service data designed by Guo et al. (2020b) which consists of 18 services: 8 in Asia, 6 in Europe and 4 connecting Asia and Europe. We consider 6 contractual requests received by the system before the planning horizon. The detailed request data is shown in Table 4. Compared with reefer shipments (requests 1, 3, 5), dry shipments (requests 2, 4, 6) have longer lead times, lower freight rates, and lower delay costs.
The effects of policies are tested under instance G1-6-0 without spot request and travel time uncertainties, i.e., μ s = t s , σ s = 0, ∀ s ∈ S,R t = ∅, ∀t ∈ {1, …,T}. Therefore, we set α = 0.5,H = 0,Γ = 0. Under the benchmark case, the travel costs are designed under the consideration of ice-breaking fees in the NSR, IMO 2020 regulations in the SCR and the NSR, and without subsidies from governments. Table 5 shows that the itineraries of dry shipments (i.e., requests 2, 4, 6) are more sensitive to different policies than reefer shipments (i.e., requests 1, 3, 5). Under the same policy, it is obvious that reefer shipments are more likely to be assigned on the ELB Fig. 10. Performance of the SA2 in comparison to the DA.

Table 10
Performance of the hybrid stochastic approach. compared to dry shipments due to the time sensitivity of reefer cargoes. When government subsidies cover above 50% of ELB transport costs, all dry shipments will be switched from maritime to railway transportation. With the government subsidies and the IMO 2020 regulations, the ELB becomes more competitive in the global transport market than the SCR and the NSR. Liner shipping companies need to take actions to meet these challenges, such as adjusting sailing routes and adopting low sulfur fuel. Besides, due to the existence of the ice-breaking fee, the SCR is still more competitive than the NSR. To attract more demands on the NSR, liner shipping companies can adopt smart navigation methodologies to avoid the ice-breaking fee.

A realistic network
In this section, we test the behavior of the methodologies under a realistic network G2 with 8 terminals and 106 services. The hinterland-related data is adapted from Guo et al. (2020b); the intercontinental-related data is adapted from Guo et al. (2020c). The topology of network G2 is shown in Fig. 8. We generate several instances to represent different characteristics of requests under network G2. The probability distributions of spot requests are shown in Table 6. To ensure demand and supply balance, the total number of shipment requests is set to 300. We use G2 − n 1 − n 2 to represent an instance under network G2 with n 1 contractual requests and n 2 spot requests.

Performance of the P-HA
The performance of the P-HA is tested under 8 instances without dynamic events and uncertainties. The P-HA is based on preprocessing procedures which may lead to suboptimal solutions. However, when N path is large enough and N match is unlimited, the P-HA generates optimal solutions. We use N path = 7 and N match =unlimited as the benchmark and vary N path from 3 to 7 and N match from 100 to unlimited. We also report the results generated by an exact approach in which optimization model P2 is solved by CPLEX directly. We denote 'N.var' as the number of variables, 'N.con' as the number of constraints, 'Obj' as the total profits, and 'CPU' as the computation time in seconds. We use 'gaps' to represent the gaps in total profits between the P-HA with different settings, i.e., gaps= Total profits(N path ,N match )− Total profits(7,unlimited) Total profits (7,unlimited) . Table 7 shows that the exact approach can only solve the first three small instances within 24 h. Increasing the largest number of services in a path to 5 and the maximum number of feasible matches to 200, the P-HA can get optimal solutions for these instances within 13.03 s. Under the same setting, the P-HA can generate 'very good' solutions within 4 min for large instances in which the largest gap is just − 0.01%. By using the P-HA, the system has the flexibility to choose proper N path and N match values to achieve the trade-off between computational complexity and solution quality.

Performance of the SA1 for travel time uncertainties
In this section, we aim to investigate the performance of the SA1 in addressing travel time uncertainties under the impact of different confidence levels, degrees of dynamism, and travel time deviations.
To investigate the impact of different confidence levels, we use instance G2-150-150 under 20 realizations of travel times. The realizations of travel times are generated based on Monte Carlo Simulation by sampling their probability distributions with a fixed lower bound of 0.9 * t s for s ∈ S. Fig. 9 shows that with the same confidence level, different solutions are generated under different realizations of travel times. From Fig. 9(a), we can see that on average, the SA1 has the best performance in total profits when α = 0.7. Fig. 9(b) shows that in general, the higher the confidence level, the lower the delays in deliveries. Fig. 9(c) indicate that with a higher confidence level, the platform will choose 'suboptimal' decisions that have lower probability of infeasible transshipments. Fig. 9(d) shows that the higher the confidence level, the higher the number of rejections. Although we use profit maximization as the objective function, infeasible transshipments, delays, and rejections are also important performance indicators which reflect the platform's service level. By playing with the confidence level, decision makers can decide on how risk-averse they are and manage the trade-off between operational efficiency and service level.
To investigate the influence of confidence level on instances with different degrees of dynamism (DODs), we design the following four instances: G2-225-75, G2-150-150, G2-75-225, G2-0-300. We define DOD as the ratio between the number of spot requests and the number of total requests. We use confidence level 0.5 as the benchmark and denote 'gaps' as the gaps in total actual profits, i.e., gaps= Total profits(α)− Total profits(0.5) Total profits(0.5) . Besides, we present the average results generated under 20 realizations of travel times. Table 8 shows that for all the instances, the SA1 has the best performance in total profits with confidence level 0.7. With confidence level 0.7, the SA1 has the largest improvements in total profits under instance G2-150-150 with 50% DOD. It is also interesting to see that for all the instances, the higher the confidence level, the lower the number of infeasible transshipments and the higher the number of rejections. Furthermore, the computational complexity decreases with the increasing confidence level and increases with the increasing DOD.
To test the impact of different standard deviations of travel times, we set α = 0.7 for the SA1 and use average results generated based on 20 realizations of actual travel times. Let 'gaps' be the gaps in total profits between the SA1 and the DA. Table 9 shows that travel time deviations have a large impact on the performance of the SA1 in increasing total profits. The larger the standard deviation, the better the performance of the SA1 in comparison to the DA. This is reasonable since with higher standard deviations, the variability in travel times is very high and the estimations are not as accurate. In comparison, instance G2-75-225 has the best performance in improving the total profits that achieves 67.85% with standard deviations σ s = 0.2 * t s for s ∈ S⧹S truck and σ s = t s for s ∈ S truck .
Companies that have the main source of uncertainty as travel time with larger variations may benefit more from the stochastic approach SA1.

Performance of the SA2 for spot request uncertainties
In this section, we aim to investigate the performance of the SA2 in addressing spot request uncertainties under the impact of different numbers of scenarios (Γ) and lengths of the prediction horizon (H). We use Γ = 0, H = 0 as the benchmark. Let 'gaps in total profits'= Total profits(Γ,H)− Total profits(0,0) Total profits (0,0) . Here, the total profits are the average total profits generated under 20 realizations of travel times.
In case of sample requests instability, we replicate the optimization process 10 times for all instances. Fig. 10(a) shows that under instance G2-150-150, the performance of the SA2 in total profits increases as the number of scenarios and the length of the prediction horizon grow up. We set Γ = 10, H = 12 for the SA2 and use the DA as the benchmark. Fig. 10(b) shows that the SA2 outperforms the DA in all the instances, and the gap between the SA2 and the DA grows with the increasing DOD. Companies running on a larger percentage of spot requests are expected to benefit more from the stochastic approach SA2.

Performance of the HSA for travel time and spot request uncertainties
To investigate the benefits of incorporating the stochastic information of both travel times and spot requests, we compare the performance of the HSA with the DA, the SA1, and the SA2 under default settings. We use the DA as the benchmark and report the average results under 20 realizations of travel times and 10 samples of spot requests. We use 'gaps' to represent the gaps in total profits between different approaches, which is given by (objective value-benchmark value)/benchmark value. Table 10 shows that the HSA has better performance in total profits than the DA in all the instances and than the SA1 and SA2 in highly dynamic instances, namely above 50% DODs. On average, while the SA1 has 3.05% improvements and the SA2 has 5.01% improvements in comparison to the DA, the HSA has the largest improvements, namely 7.03%. In addition, for all the instances, the HSA generates lower infeasible transshipments and delays than other approaches. Furthermore, we notice that the number of rejections increases with the increasing DOD and achieves the highest under instance G2-0-300 which has no contractual requests. This is reasonable since the system cannot reject contractual requests. Moreover, we observe that the HSA has the best performance in instance G2-0-300 as stochastic information pays off when it is highly dynamic. Besides, it is obvious that increasing the confidence level, the computational complexity will decrease thanks to the chance constraints. However, increasing the number of scenarios and the length of the prediction horizon, the computational complexity will increase dramatically caused by the increasing size of sample requests.

Managerial insights
In this section, we summarize the key managerial insights derived from the above experimental results.
• Thanks to the horizontal and vertical collaboration among carriers, global operators have more flexibility to choose different routes for shipments which is critical when disturbances happen (e.g., Suez Canal blockage in March 2021). However, the increased integration in global transportation gives rise to large instances which generally cannot be solved by commercial software. Heuristic algorithms that can generate timely and high-quality solutions, like the one designed in this paper, are essential to global operators. • Travel time uncertainty, as the main source of uncertainties in global transportation, not only affects operational efficiency but also the feasibility of transport plans. Stochastic approaches that address travel time uncertainties are critical for global operators to improve the robustness of transport plans and ensure the on-time delivery of shipments under traffic disruptions. • Due to future request uncertainty and service capacity limitations, the capacity assigned to current requests will be unavailable for future requests which might be more profitable. Stochastic approaches that evaluate the impact of current decisions on the future state and behavior of the system play a key role in increasing the total profits of global companies over a given planning horizon. • In practice, travel time uncertainty and shipment request uncertainty always happen simultaneously. Stochastic approaches that consider these two types of uncertainties integrally can help global companies achieve better performance in profits, feasible transshipments, and on-time deliveries in transport planning.

Conclusions and future research
In this paper, we investigated a dynamic and stochastic shipment matching problem in global synchromodal transportation. We considered a platform that aims to provide online acceptance and matching decisions for contractual and spot shipment requests in a global synchromodal transport network. The platform receives requests and travel times dynamically, but the probability distributions of dynamic events are available. To solve the problem, we developed a rolling horizon framework to handle dynamic events, a hybrid stochastic approach (HSA) to address shipment request and travel time uncertainties, and a preprocessing-based heuristic algorithm to generate solutions at each decision epoch.
We conducted extensive experiments to validate the performance of the HSA in comparison to the approaches that do not consider any uncertainty or only consider one type of uncertainties. The experimental results indicate that the HSA is highly effective in increasing total profits, reducing infeasible transshipments and delays on instances with a large degree of dynamism. Thanks to the flexibility of the HSA, global companies are able to adapt the transport system to their business characteristics by tuning the parameters accordingly. This way they can manage the trade-off between computational complexity and solution quality, and the tradeoff between operations efficiency and service level.
This research can be extended in several directions. First, in this paper, we considered a centralized platform that provides integrated decisions for global shipments. To ensure the fairness among players, profit-sharing mechanism design is a promising research direction. Second, in practice, a large number of entities are involved in global container transport and they may not all be willing to give authority to a centralized platform. The coordination mechanism among them and incentives to stimulate cooperation are part of future research. Third, we assumed that the platform publishes fixed fare classes for container bookings. Future research can consider dynamic pricing strategies for online platforms to realize the balance between supply and demand. Fourth, in this paper, we assumed travel times follow normal distributions, future research will be conducted to relax this restriction.