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Assessing the benefits of overbooking strategies on
inland freight transport systems: a simulation study
Master’s Thesis SCM MSc Supply Chain Management Faculty of Economics and Business
University of Groningen Netherlands
Co-Supervisor: Dr. Stefano Fazi Second Supervisor: Prof. dr. K.J Roodbergen
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ABSTRACT
In several transportation systems, the no-show problem has always been an issue, but also a possibility to increase revenues and capacity utilization. In container transportation systems, if containers cannot arrive at the terminal on time, it is hard for transport service providers to find last minute new customers to fill the vacant capacity. Similarly, to other industries, overbooking could potentially limit the drawbacks of the no-show problem. Although customer satisfaction and complexity in rescheduling are issues to tackle. However, inland transport systems provide a nice setting where the availability of different modalities may provide further flexibility in rescheduling. This research uses simulation to compare the environmental and financial performance of three cases (base model, overbooking model and synchromodality fare classes) under different overbooking scenarios. Results show that overbooking can significantly increase facility utilization and profit. On the other hand, in case of high penalty cost, environmental emissions will also be high due to the need of using trucks.
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Content
1. Introduction ... 4
2. Theoretical background ... 5
2.1 Inland transport system and no-show problem ... 5
2.2 Overbooking ... 7
3. Problem description ... 9
3.1 Case 1: Base model ... 10
3.2 Case 2: Overbooking ... 11
3.3 Case 3: Fare classes ... 12
4. Simulation design ... 13
4.1 Experiment setting and performance measurement ... 14
4.2 Validation and Verification ... 15
5. Simulation results ... 15
5.1 Environmental Performance ... 16
5.1.1 Utilization ... 16
1.1.1 Usage of truck ... 18
1.2 Financial Performance ... 19
2. Conclusion and further research ... 22
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1. Introduction
The container throughput of the main European seaports has seen a sharp increase in the last two decades (Notteboom and Rodrigue, 2009). This growth needs inland container transport system to provide higher capacity, flexibility, and performance than ever before. Moreover, an efficient hinterland transport system helps seaports to release containers faster and reduce terminal congestion (Franc and van der Horst, 2010). It is recognized that using high capacity transport modes, such as trains and barges, against trucks, could improve efficiency. However, their success is jeopardized by the lack of flexibility and low margins. For instance, trains have specific constraints such as sharing infrastructure with passenger trains (Behdani et al., 2016).
In inland transport systems, containers are assigned to specific routes and rides beforehand. However, some disturbances such as no-show or late cancellation might happen. Consequently, barges and trains will be affected since they are usually bound to tight schedules and also their economies of scale can be put at risk. There are many potential reasons for the occurrence of no-show. For example, transfer delay, including vessel delays, logistic constraints and lack of coordination (Culley et al., 1991), often leads to a no-show of booked services. The freights no-show caused by delays would waste unused capacity and increase re-scheduling costs. If freight misses the departure time, it has to wait for the next train while its booked spot remains unused. This problem occurs frequently in the air industry, where so-called “overbooking” is often performed to ensure a full flight. It aims to reduce the profit losses caused by accepted reservations which are canceled before flight departure or become no-show at the time of service (Feng et al., 2015).
Overbooking is a well-known topic in the area of revenue management. The hotel industry and the transport industry are the two classical examples of overbooking
practices. When the capacity is overbooked, two situations may occur. When all
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provider must find an alternative (Feng et al., 2015). In any case, this may also affect the image of the company. Another situation is that some customers do not show up, and capacity is better used in this case. A large number of researches worked on using overbooking in single transport systems such as air cargo industry (Hellermann et al., 2013; Moussawi and Cakanyildirim, 2005), rail freight transport industry (Feng et al., 2015) and ocean shipment (Xin, 2003; Li, Wu and Bu, 2006). All the previous studies only focus on applying overbooking to a single transport mode, such as railway and air flight. However, inland transport systems have typically various transport modes available and allow transport tasks to be switched among the different modes. Thus, inland transport systems can provide more flexibility to overbooking strategy compared to other logistics settings. However, no research is available on the subject. Hence, it is interesting to investigate whether overbooking can be better applied to inland transport which is easier to reschedule due to multiple alternative transport modes.
The aim of this paper is to investigate how overbooking can be applied to inland freight transport systems and to quantify the benefits. We propose a simulation study to compare the performances of no overbooking case, overbooking case and fare classes case that allow transport mode switch and explore how these can be handled operationally.
The structure of this paper is shown as follows: Chapter 2 will review the literature related to inland freight transport and overbooking. The problem description and simulation design will be presented in chapter 3 and chapter 4. Chapter 5 will show experimental results and data analyses. Chapter 6 will conclude with the discussion and suggest directions for further research.
2. Theoretical background
2.1 Current status of inland transport systems
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cargos to reach their destinations on time. Hinterland transport service heavily depends on road transport, but the costs, traffic congestion, and growing environmental emissions are challenging the dominance of road for inland transport services (Frémont and Franc, 2010). In order to push the freight to hinterland efficiency, public authorities propose to use alternative transport modes with higher capacities. Therefore, trains and barges are selected by transport providers although they have less flexibility compared with trucks (Bouchery et al., 2015). In order to increase the flexibility, the intermodal transport plays an important role in European transport system which combines rail and inland waterway as the major part and uses the road transportation as little as possible (ECE, 2001).
Intermodal transport should provide more reliability and possibility of massifying goods flow. van Riessen et al. (2017) proposed the Cargo Fare Class Mix (CFCM) model of intermodal container transport system which provides customers standard and
express services with different lead times. Schönberger and Kopfer (2012) used revenue
management to suggest an inland freight transport capacity control strategy. It helps transport service providers to determine whether to reject the upcoming requirement or not.
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model to determine the optimal schedule for synchromodal transport system. Based on Behdani’s base model setting, F. Nab (2018) conducted a simulation study to analyze how revenue management provides flexibility to synchromodal transportation.
2.2 The no-show problem in transport systems
No-show problem is common in air and sea cargo transport industries. Typically, the price of ticket tends to go up at the last minute since it is easy to find customers who need these tickets. However, container ship has more difficulty in finding new customers to fill the vacant capacity because of the variety in the order size and the long loading time (iContainers, 2018). Capacities are sold before departure times for trains and barges, since the containers are allocated to certain routes and rides beforehand. Hence, no-show is very likely to occur.
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2.3overbooking
There are several papers on overbooking in the transport industry. The first passenger overbooking model was built to find the biggest expected gain for airlines by Shlifer
and Vardi (1975), and the first overbooking model of air cargo was researched by
Hendricks and Kasilingam (1992). Kasilingam (1997) compared the differences between air cargo overbooking and air passenger overbooking. Wannakrairot and Phumchusri (2016) built a two-dimensional air cargo overbooking models which used to find the optimal overbooking level in order to minimize the cost when considering the costs of spoilage and offloading. Hellermann et al. (2013) used a mathematical model to provide an optimal reservation policy to maximize the profit and analyzed the impact of overbooking in the air cargo industry. Xin (2003), Li et al. (2006) built mathematical models to deal with an overbooking problem of ocean shipment under stochastic capacity. Overbooking was also used in railway freight transport, a dynamic model created by Feng et al. (2015). It is shown that compared to the first come first service (FCFS), overbooking strategy can better improve revenue and reduce the vacancy capacity.
As discussed above, overbooking has been widely used in the air and sea freight transport industry. In the air cargo and railway transport, transport service providers have to reject orders and pay the penalty or postpone the order until the capacity is free if all booked customers show up. Inland freight transport is quite different from air cargo transport and ocean shipment industry because of its complexity (e.g. intermodal system, modal split, and modal switch). In an intermodal system, the overbooked order can be switched among different means of transportation, on one hand complicating the coordination, but on the other providing opportunity to handle extra demands.
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capacity wasteled by customer no-show in the inland transport system. This paper applies overbooking to the inland freight transport system and simulates the application of overbooking with simulation models.
The motivation to use simulation is due to the high complexity of the system, the large number of stochastic variables, and the interplay between several components. All in all, it would make a hard task to model the problem mathematically, hence simulation looks promising in these conditions.
The research questions and sub-questions are composed as follows: a) How can overbooking apply inland container transport system?
b) How can overbooking influences the performance of the inland transport system?
3. Problem description
The problem setting is based on the prior works of Behdani et al. (2016) and F. Nab (2018). The inland transport system consists of two main nodes: a sea terminal and an inland terminal. We consider three modalities (train, barge, and truck) for container transportation. To simplify, here we assume that both the sea terminal and inland terminal operate 24/7, and there is only one route between the two terminals. Each modality has a fixed capacity, revenue, cost and transfer time. The capacity of trains and barges is constant, and the number of trucks in the system is unlimited. Trains and barges depart according to fixed schedules, whereas trucks can leave at any time. There is a booking limit of customer booking capacity during the reservation period, which the number of orders enter the system will not exceed.
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postponement system, late containers are temporarily stored in the terminal. If the next service has capacities, late containers are transported by next service. Otherwise, they are transported by truck.
The parameters of transporting cost, capacity and transit time are taken from Behdani et al. (2016), which are summarized in Table 1. Truck is the most expensive option, while both transport service providers and customers might choose trucks to make sure the containers can reach the destination on time. Costs for barges and trains are average costs and are based on the assumption that they never travel empty; hence variable costs based on economies of scale are not considered here for simplification. Finally, Barges’ and trains’ departure are based on fixed schedules, thus containers can be transported only if they arrive at the terminal before their departure time.
Table 1 Network parameters, based on Behdani et al. (2016)
MODE VARIABLE COST (€/CONTAINER) CAPACITY (CONTAINERS) NO. OF SERVICE (PER DAY) TRANSIT TIME (HOURS) BARGE 45 40 5 9 TRAIN 60 110 1 3 TRUCK 90 1 Infinite -
The transport price P is set as twice the amount of the transport cost. It means that for customers, the price of transport a container is 90€ by barge, 120€ by train and 180€ by truck. The number of late orders follows a uniform distribution and keeps a certain ratio to the number of coming orders. Three models are built to simulate how overbooking is applied in inland transport system. These models are described in the next subsections.
3.1 Case 1: Base model
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certain transport method. When the capacity is fully booked, base model starts to reject orders. There is a revenue loss because of the coming order rejection. Orders are held for a batch until the transport resource (barge and train) becomes available, and both barge and train services have a certain departure time. Some orders may not show up at the departure time, some of the orders will be canceled and others may not reach the terminal on time. If the transport service provider accepts the late order, late orders will be delayed to the next available service and tag the lower priority than the normal customers in order to ensure the service quality. Once the next service is fully loaded, these late orders will be transported by truck. Road transportation leads to higher cost, thus late customers should incur this higher transport cost. The logic flow chart of the base model is shown in Figure 1.
Figure 1: Logic flow chart of the base model
3.2 Case 2: Overbooking
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show. Thus, the overbooking policy is employed in this model. Most of the time, system allows overbooking which means that the number of accepted orders may exceed the system capacity. If the system fails to provide the promised capacity to customers, unfulfilled orders will be postponed to the next available service. Customer satisfaction and loyalty will be hampered if customer expectations are not met (Varini, 2011). Thus, this system will employ trucks to transport the overbooked order so that the containers could arrive at their destinations on time. Using trucks to transport containers induces higher transport costs while customer satisfaction will not be influenced. Customers pay for their booked transport services, and the higher road transport costs are borne by the service provider. Figure 2 shown the logic flow chart of the overbooking case.
Figure 2: Logic flow chart of overbooking modal
3.2 Case 3: Fare classes
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In the fare classes case, two types of customers are defined. The first fare class concerns the inflexible customers who have decided their desired transport means and do not accept any alternative. The transport service provider can only transport the containers in the chosen way. The second fare class is defined as flexible customers who only have the preference of transport methods, while service providers could switch the transport means if the preferred method does not have sufficient capacity. Customers whose orders transported by alternative means can get a discounted transport price. The system first sells capacity to inflexible customers until reaching the capacity limit, and then overbooks some capacities to flexible customers. If the system cannot transport flexible customers’ containers on time in their preferred ways, other transport means would be considered. Therefore, when both barges and trains are fully loaded, the service provider has to use trucks to deliver the orders on time. The logic flow chart of inflexible customers is similar to the base model shown in Figure 1, and the flow chart of flexible customers is presented in Figure 3.
Figure 3: Logic flow chart of fare classes modal
4. Simulation design
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inland transport system. Most prior studies about overbooking in cargo industry used mathematical models to maximize the profit or minimize the total cost (e.g. Wannakrairot and Phumchusri, 2016; Hellermann et al., 2013), while few studies simulated the overbooking issue (e.g. Feng et al., 2015). Simulation approach compares outputs under various scenarios, and shows what might happen in real life if different
rules and structures are applied(Dooley, 2002). Furthermore, simulation is usually used
in more complex systems because it could predict what will happen in the future, whereas other research approaches analyze past events. For these reasons, simulation is employed in this study.
Three models will be built to compare the performance of the system under different overbook rates. The first one is the basic model where the system rejects orders if all capacities are occupied. The second model shows the situation that the transport service provider overbooked the capacity of the system. The last one presents the overbooking policy used in synchromodal transport system. All the three models would be discussed under late order rebooking and postponement systems. Late order has to leave the system and rebook the capacity during the reservation period in the rebooking system. While in the postponement system, transporter postpones the late order to the next available service.
4.1 Simulation software and modeling details
Arena is a simulation and modeling software tool that helps users build the simulation model, run the experiment and get the statistic result report. Arena is widely applied in the supply chain industry, and it enables companies to model and evaluate everything related to global supply chain (Rockwell, 2019).
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and train orders). Since Arena software has a maximum entity limitation, each coming entity represents a batch in order to control the total number of the entities. Overbooking orders and late orders are created by the Separate modules, which randomly generate the number of those orders. The Decide modules are used to sort orders to different routes base on various customer demands and choices. Orders are held in Adjustable Batch modules until the batch sizes are fit for transport facilities’ capacity. Finally, orders leave the system from the Dispose after finishing the transport process. In terms of transport modes, transport resources have scheduled departures. The main processes of simulation models are shown in the appendix B (Using Fare classes model in postponement system as an example).
4.2Experimental setting and performance measurement
To analyze the influence of the overbooking and fare classes in the inland freight transport system, this paper studies three cases with different overbooking rates. The
overbooking rates Br is set between 0% and 40% of the system capacity. The
experiment simulates the operations of three cases in a month (30 days).
The performance measurement indicators of the freight transport system are summarized into several aspects: operational efficiency, service quality and environmental impact, etc. (Mckinnon, 2015). and airline overbooking models commonly use the reduction of the spoilage of the capacity and the revenues to measure the performance. Base on the literature of freight transport and overbooking, total cost and revenue, transport facility utilization and environmental impact are employed to measure the performance.
4.3Validation and Verification
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segments are finished, the final verification is performed. The key elements, for example, the number of containers in queues (NQ) and the current simulation times are displayed for verification. In addition, single containers are created and tracked through the whole models for testing the operation of the simulation. All testing of the different stages is performed under the same set of situations.
The validation of the conceptual model is done by building the models after the discussion with the supervisor and both the base model and the input data are based on the prior studies of Behdani et al. (2016) and Nab (2018). When the simulation models run, in order to get the validated results, the simulation system should run in a steady-state. A real system will not start empty and idle but a model will, then the results of this simulation experiment may be biased. Hence, a warm-up period is required. After testing the total entities WIP value of the system, output analyzer (the data analysis software of Arena) shows that after 25 hours the system reaches a steady-state (See Appendix A1). The warm-up period has to longer than 20 hours. The complexity of 6 models are different. In order to make sure all systems can reach steady-states, the warm-up periods should longer than 25 hours. Here set 100 hours as the warm-up time of the simulations so that models can start in steady-state. All experiments are replicated 10 times to ensure the reliability of the results.
5. Simulation results
This section will present the simulation results of the three presented models under various overbooking rates. Base on the output of the simulations, both environmental and financial performances are compared. The environmental performance is evaluated by the facility utilization and truck usage, and the financial performance is measured by profit, revenue and cost.
5.1 Environmental performance
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truck usage. Low facility utilization refers to the ineffective use and waste of resource, and a frequent use of truck transportation causes high emission and environmental damage.
5.1.1 Utilization
After running simulation experiments, the number of containers that transported by barges or trains is counted. Since the number of used transport resources during the simulation period is also known, the utilization of barge and train could be calculated.
Table 2. Utilization of late order rebooking
OVERBOOK RATE
0% 5% 10% 15% 20% 25% 30% 35% 40%
BARGE base model 71% 72% 74% 75% 75% 75% 75% 76% 75%
overbooking 71% 73% 75% 77% 78% 79% 81% 82% 85%
fare classes 71% 72% 72% 73% 76% 80% 82% 84% 87%
TRAIN base model 70% 70% 71% 72% 72% 73% 73% 73% 72%
overbooking 70% 71% 73% 74% 76% 77% 78% 79% 80%
fare classes 70% 74% 74% 74% 76% 81% 82% 84% 86%
Table 3 shows the barge and train utilization of the three cases in the late order rebooking system. When the overbooking rate is zero, the utilization of both barge and train are the same in all three models (71% for barge and 70% for train). Since the number of coming orders is lower than system's capacity when the overbooking rate equals zero, accepted orders can be fully transported in all cases. In the base model, utilizations of barge and train first experience a slight raise then remain stable when the overbooking rate grows. The number of coming orders of barge and train follow the uniform distribution, thus a higher booking limit will first allow more orders entering the system and increase the utilization of transport facilities. After the number of accepted orders reaches the system’s capacity, the system rejects further orders and the utilization of facilities stops increasing with the overbooking rate.
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increases as the overbooking rate increases. In these two models, the system accepts all coming orders. When some customers are late, overbooked orders can use the capacities booked by the no-show orders, which boosts the utilization of facilities. The fare classes model performs even better than the overbooking model when the overbook rate is high. For instance, the utilization of fare classes model is 87% for barge and 86% for the train when the overbooking rate equals 40%. The reason is that fare classes model allows orders to switch between barge and train, for example, the unoccupied capacity of barge can be filled by the overbooked train order and vice versa. Therefore, the utilization of fare classes model is further improved.
Table 3. Utilization of late order postponement
OVERBOOK RATE
0% 5% 10% 15% 20% 25% 30% 35% 40%
BARGE base model 83% 85% 87% 89% 89% 90% 88% 88% 89%
overbooking 82% 84% 86% 87% 88% 88% 89% 90% 91%
fare classes 82% 83% 85% 86% 92% 95% 96% 96% 98%
TRAIN base model 82% 84% 84% 84% 85% 85% 83% 85% 85%
overbooking 83% 84% 86% 87% 88% 89% 89% 90% 90%
fare classes 84% 86% 89% 89% 92% 93% 92% 95% 95%
If the transport service provider postpones late orders, the utilization of all the three models would be further increased. Late orders stay at the terminal and wait for capacities provided by the next service. Coming orders and overbooked orders have the priority to use capacities of barges and trains, after which the remained capacities are allocated to late orders from the last service. Given the highly efficient mechanism, utilization of fare classes model can reach 98% for the barge and 95% for the train when setting the overbooking rate to 40% (Table 4).
5.1.2 Usage of truck
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part analyzes the ratio of containers that transported by trucks.
Figure 4 Truck uasge of late order rebooking Figure 5 Truck usage of late order postponement
Figure 4 and Figure 5 show the truck usage of late order rebooking and postponement systems, respectively. In the late order rebooking system, base model rejects all the coming orders once the booking limit has been reached. Since barges and trains can transport all the accepted containers, no truck is used in the base model. In both overbooking model and fare classes model, the usage of the truck grows with the increase of the overbooking rate. However, the usage of truck in overbooking model is much higher than that in fare classes model. When 140% capacities are booked, 7.4% accepted containers would be transported by truck in overbooking model, while the number is only 1.1% in fare classes model. The reason of the huge difference is that the fare classes case is more flexible in choosing transport methods. In the overbooking case, barge and train transport systems are independent of each other. If capacities are overbooked and the number of show-up barge or train orders exceeds its corresponding capacity, all the overbooked containers will be transported by trucks. While fare classes model allows containers to switch between barges and trains, which means that trucks are used only if barge and train are fully loaded at the same time.
The usage of truck in late order postponement system is higher than that in rebooking system in all three cases, since late orders stay in the system and wait for capacities of
-2% 3% 8% 13% 18% 0% 5% 10% 15% 20% 25% 30% 35% 40% Tr u ck u as g e Overbooking rate
Truck usage for each
model
base model overbooking fare classes
0.0% 5.0% 10.0% 15.0% 20.0% 0% 5% 10% 15% 20% 25% 30% 35% 40% Tt ru ck u sa g e Overbooking rate
Truck usage for each
model
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next services in the postponement system. The system first allocates capacities to new coming orders, and then sells the rest to late customers from the last ride. With the rise of overbooking rate, less and less capacities are left to late orders. Therefore, late orders are more likely to be transported by trucks.
5.2 Financial performance
This part discusses the financial performances of different scenarios. Assuming that the transporter still needs to pay half of the transport cost if the coming order is rejected, and flexible customers get a 20% discount on the transport price. The total profit, cost and revenue of the three cases will be compared under various overbooking rates. Figure 6 and Figure 7 compare the total profit, while Figure 8 and Figure 9 analyse the cost (represented by columns) and revenue (represented by curves) in more detail.
Figure 6 Profit of late order rebooking Figure 7 Profit of late order postponement
200000 300000 0% 5% 10% 15% 20% 25% 30% 35% 40% Pr o fit Overbooking rate
Total Profit for each model
base model overbooking fare classes 200000 300000 400000 0% 5% 10% 15% 20% 25% 30% 35% 40% Pr o fit Overbooking rate
Total Profit for each model
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Figure 8 Cost and revenue of late order rebooking
Figure 9 Cost and revenue of late order postponement
In late order rebooking system, fare classes model always provides the highest profit under same overbooking rates. The profit of fare classes model climbs when the overbooking rate rises from 0% to 5%, then it remains stable until the overbooking rate reaches 20%, after which the profit grows smoothly. While in late order postponement system, the grey line keeps rising with the overbooking rate. Considering the discussion in 4.1 that the utilization of fare classes model is more than 90% when the overbooking rate exceeds 20%, it is clear that the growth in profit comes from accepting more coming orders and transporting them by trucks. Truck usage increases dramatically when the overbooking rate is over 20% as shown in Figure 5, which also proves that
0 200000 400000 600000 800000 0% 5% 10% 15% 20% 25% 30% 35% 40% Co st a nd re ve nu e Overbooking rate
Cost and revenue for each model
base model overbooking fare classes base model overbooking fare classes
0 200000 400000 600000 800000 0% 5% 10% 15% 20% 25% 30% 35% 40% co st an d r e ve n u e overbooking rate
Cost and revenue for each model
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higher ratios of truck usage contribute to the profit growth.
When the overbooking rate rises up, the profit of the overbooking case grows steadily no matter whether the system accepts late orders. Fare classes case brings the highest profit in late order rebooking system but its performance is worse than that of overbooking case in late order postponement system. We can find the reason why the overbooking model gains more profit than the fare classes model in late order postponement system by separating the profit into revenue and cost. Figure 8 and Figure 9 show that the cost of fare classes model is higher than that of overbooking model when there is no late order in the system. However, the cost of overbooking case is higher in postponement system. It proves that late orders in the overbooking model, which cannot be transported by next services, are transported by truck. A higher overbooking rate leads to more truck-transported late orders. Fare classes case provides more capacities to late orders because of its reasonable capacity allocation mechanism that transport means are switched flexibly. Transporting more orders by trucks results in a higher cost of overbooking model, however, trucks bring the highest profit for transporter (90€ /container) at the same time. Thus, the overbooking model generates more profit because it uses more trucks to transport late orders. The 20% price discount for flexible customers is another reason why the profit of the fare classes case is lower than that of the overbooking model.
With the increase of the overbooking rate, the curve of base model profit falls smoothly. When we look into cost and revenue, we can see that the base model’s revenue stays almost constant while the cost climbs with the overbooking rate. The cost climbs in the base model because orders are rejected if all capacities are booked. When there is a coming order boost, more profit will be lost since serviced customers stay the same while rejected orders increase.
23 Theoretical implications
This thesis researches how overbooking could be applied to inland transport system to reduce the wasted capacity due to no-show problem. Synchromodal fare classes policy is used to further enhance the performance. The results show that postponing late orders to the following service is much more efficient than letting late customers rebook the capacity. Although overbooking can significantly improve the utilization of the transport facilities, more trucks need to be used to transport the overbooked and late containers if overbooking rate is too high.
Practical implications
This paper shows that fare classes and synhromodality thought can bring more flexibility to inland transport system, and offers advices to transport service providers who want to use their transport facility more efficiently. Although this is a theoretical research based on assumed parameters, it presents how overbooking rate could influence the environmental and financial performances of the inland transport system. Transporters could balance these two performances according to their needs.
Limitation and further research
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Further research can use mathematical methods to find the optimal overbooking rate which would maximize the profit without using too many trucks at the same time. Other possible extensions are listed as follows: (1) The fare classes case only sells the overbooking capacity to flexible customers, it would also be interesting to study whether selling parts of the normal capacities to flexible customer could bring better performance, and what percentage of customers should be flexible customers. (2) Exploring the optimal price discount which can both attract more flexible customers and guarantee the high profit. (3) It might be interesting to take order split into consideration. For example, one batch of coming containers can be separated and transported by various transport modes in order to provide more flexibility to the system.
Appendix
Figure A1: Warm-up length
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Figure B2: Orders and late orders are hold and released waiting for signals.
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