A Decision Support Tool for Overbooking Strategies in Multimodal Inland Transport
Systems
Master’s Thesis
MSc Technology Operations Management University of Groningen
The Netherlands
Student: Robin Huijboom Student number: S3852695 Supervisor: Dr. Ir. S. Fazi Second Supervisor: Dr. C. Xiao
Date: July 10, 2020
Abstract
This paper considers an overbooking problem in the multimodal inland transport system.
Overbooking is a methodology in revenue management that allows an incoming booking to be accepted in exceedance of available capacity. This strategy is beneficial in systems where bookings are often late or cancelled, which is very common in the considered system. The transport of containers from a seaport to a final destination in a multimodal inland transport system is realized via truck, rail or inland shipping or a combination of those. Different from systems where the overbooking problem is widely researched, this system is more complex. In terms of possibilities the considered system has the ability to postpone and upgrade a booking, denials are not allowed, and instead of considering only cancelled and on-time bookings, containers can also arrive late but still need to be transported. We model this problem as a two- stage stochastic optimization problem to maximize profit. In the first stage, the optimal overbooking levels for the barge and train are set. In the second stage, the on-time and late arrivals are assigned to either a barge, train, truck, or warehouse in order to minimize the penalty costs. By means of numerical experiments, we show that the model is sensitive to several parameters and that the overbooking model can be applied in many settings. The practical implications of this study indicate that transport providers can use the proposed model to support the determination of the overbooking level in multimodal inland transport systems.
We conclude this thesis with directions for further research.
Keywords:
Overbooking, inland transport system, overbooking model, container transportation, multimodal
Word count: 10,328
Table of Content
1. Introduction ... 5
2. Theoretical Background ... 7
2.1 Multimodal Inland Transport System ... 7
2.2 Overbooking ... 9
3. Problem description ... 13
3.1 Problem setting ... 13
3.2 Assumptions ... 15
3.3 Base setting ... 15
4. Model formulation ... 18
5. Numerical experiments ... 22
5.1 Performance measurements ... 22
5.2 Effect of transport capacity, warehouse capacity, revenue, and costs ... 22
5.3 Effect of refund ... 28
5.4 Effect of on-time and late arrival rate ... 30
5.5 Overbooking in different instances ... 33
6. Discussion and managerial implications ... 36
7. Conclusion ... 38
8. Limitation and further research ... 39
References ... 40
Appendix 1, Instances of the sensitivity experiments ... 44
Acknowledgement
This thesis is written as a final part of the Master program Technology Operations Management (TOM) at the University of Groningen. The completion of this master’s thesis could not have been possible without the expertise of my first supervisor, Dr. Stefano Fazi. Due to the current pandemic, physical meetings were not possible, however, the quality of the numerous skype meetings were in no way inferior to normal meetings. I would like to thank Dr. Stefano Fazi for his time, effort, and guidance throughout this project. Next, I would like to thank my second supervisor C. Xiao for his feedback. Furthermore, I would like to thank my family and friends for supporting me during this project, especially in these extraordinary times.
1. Introduction
Shipment of goods carried via containers has increased significantly over the last decades, from 102 million metric tons in 1980 to 1.83 billion in 2017 (Wagner, 2019). To efficiently facilitate inland container transport, so-called booking platforms have become more and more available in recent years. These platforms receive a reservation of shippers (senders of containers) and decide to accept or reject the reservation based on the available capacity of barges, trains, and trucks at different operators. Because shippers do not have to communicate with several operators these platforms claim to be more efficient. Furthermore, these platforms include a third party, a transport provider, who allocate containers to different transport modes in the inland transport systems. The increase of shipment of goods has put inland transportation systems under high pressure due to high costs compared to maritime transport legs and societal impact, in terms of pollution and congestion.
The inland container transport typically entails the transportation from a seaport to a final destination via truck, rail or inland shipping or a combination of those. Due to the increasing capacity of the vessels (container ships) seaports require more unloading time, which results in congestion at the unloading docks (Buitendijk, 2019). Currently, inland transportation is dominated by truck transport, due to their fast and flexible nature. However, trucks cause high pollution and more congestion, due to the small capacity. To reduce these problems, transport modes with large capacities and less emission such as barges and trains are preferred. These modes of transport are costly and high utilization is required to obtain economies of scale.
However, it is not as straightforward to achieve high utilization due to the commonality of no shows and late arrivals of containers. This uncertainty is related to logistic problems or vessel delays due to bad weather (Bouchery et al., 2015). In those cases, the transport provider has little flexibility to reallocate the container since barges and trains are usually bound to tight schedules. Because of this, the transport provider can overbook barges and trains to guarantee a high level of utilization. However, it could occur, that the number of on-time containers is higher than the capacity of the barge or train, whereby a rebooking is required, which causes extra costs, together with temporarily storage. The basic trade-off to consider when deciding the overbooking level is either having a low utilization which results in low efficiency or having backlog containers which cause a penalty and extra cost.
The trade-off for overbooking is well researched in other service industries where no-shows and late cancellations are quite common. For example, Ivanov (2015) and Toh & Dekay (2002), conducted a study on overbooking in the hotel industry, where they considered up- and downgrading between different room types. Furthermore, overbooking represents an important strategy in the airline industry (Lan et al., 2014). Y. Huang et al. (2013) developed an overbooking model to provide the optimal overbooking level which considers postponement to a later flight. Differently from those papers, overbooked containers in inland transport systems are more flexible and have two options, either to be upgraded or to be postponed. Little research has been conducted in overbooking in the freight container industry (Fen-ling et al., 2015; T. Wang et al., 2019) and both of these papers only consider one transport mode.
However, the inland transport system is more flexible and has the ability to switch between multiple transport modes. All papers concerning overbooking have found that overbooking results in higher revenue, therefore, it is interesting to investigate whether an overbooking strategy would be beneficial for a multimodal inland transport system. According to earlier research in a master’s thesis, the overbooking strategy is well applicable in inland transport systems (Wu, 2019). Nonetheless, the author simulates the behaviour and processes in the inland transport system but do not provide an optimal overbooking level. A valuable addition to his study would be an optimal overbooking level. Hence, current literature lacks knowledge on the benefits of overbooking level in multimodal inland transport systems.
This paper aims to find a proper overbooking strategy that maximizes the profit and minimizes the use of trucks by using model-based research. We formulate the problem as a stochastic two-stage programming model that incorporates four features: three modes of transport, upgrading, postponement, and an uncertain number of on-time, late, and cancelled containers.
By means of numerical experiments, we show the goodness of our proposed model and imply managerial insights.
The remainder of this thesis is structured as follows. Section 2 provides an overview of the existing literature on the multimodal inland transport system and overbooking strategies.
Section 3 contains the description of the problem. In Section 4 the mathematical model is presented. The numerical experiments and the results are given in Section 5. A discussion and managerial insights of the results are presented in Section 6. Our conclusion is given in Section 7, and this paper will end with research limitations and implications for further research.
2. Theoretical Background
In this section, prior literature is discussed. First, research on multimodal inland transport system is described. Second, the existing literature of overbooking is reviewed.
2.1 Multimodal Inland Transport System
As the container shipping industry is expanding, and the related supply chains have become more complex, it is crucial to make effective decisions. Therefore, many researchers have developed decision support tools for different aspects of the supply chain. We review the relevant literature concerning decisions in the multimodal inland transport systems from a strategic, tactical, and operational point of view.
The increasing demand for shipment of goods in containers generates drawbacks for the inland transport system in terms of CO2 emissions, traffic congestion, and shortage of capacity (Van Schijndel & Dinwoodie, 2000). These issues can be tackled by effective use of transport with high capacity, such as barges and trains. In most cases, the final destination is not accessible by barge or train due to the fixed stations and quays, therefore, multimodal transport is required.
Applying multimodal inland transport requires dry ports (Qiu & Lee, 2019; Roso, 2009) and hub and spoke systems (Konings et al., 2013). A dry port is an inland terminal that is connected to a seaport by railways. Lin et al. (2019) provide a bi-level programming model to optimally allocate inland terminals with railway characteristics, constrained by budget, classification capacity and a number of available tracks. With the upper-level aim to find an optimal set of investments plans for all candidate nodes and a lower-level aim to obtain a low-cost railcar reclassification plan. Hub and spoke systems allow container traffic in major routes to be combined and transported via barges to hub terminals. From here, the containers are transported to regional (spoke) terminals, by smaller transportation modes (Jiang et al., 2015).
Sun & Zheng (2016) provide a strategic planning model to identify potential hub locations. By aiming to minimize the total costs considering chartering and voyage cost of an arc, the optimal location of the hub can be identified.
The increasing number of internal terminals and dry ports allow to increase the use of barges and trains. With this increase in internal terminals and dry ports, traffic congestion and CO2
pollution can be reduced. Furthermore, barges and trains can favour economies of scales.
However, these modes of transport are not as flexible as trucks, in terms of the fixed departure
schedule. In addition, trucks are fast but relatively costly and generate high level of congestion and pollution. These characteristics must be taken into account when choosing a transportation mode. Several decision models are developed to support this operational decision in whether to transfer only by trucks or by a multimodal system. All papers found that a multimodal system can be beneficial, although sometimes adjustments must be made. Kurtulus & Cetin (2020) did a case study on a one short-distance inland container transportation. Their simulation showed that the success of the shift from trucks towards multimodal rail system is influenced by transit time, delays, frequency of the train, free time in the dry port, and mostly by transport costs. On the contrary, Meers et al. (2017) found that the reliability of the service has more influence on the success of a modal shift in short-distance routes than costs. These discrepancies may be caused by differences in the preferences of decision-makers located in different countries;
Kurtulus & Cetin (2020) did a case study in Turkey where Meers et al. (2017) did a case study in Belgium. Additionally, Van Fan et al. (2019) proposed a mathematical model to determine the transportation modes with lower energy consumption as well as emissions for a particular load and travelled distance. Their proposed graphical tool is feasible for a comparison of more than two transportation modes.
From an operational point of view, there are many allocation and scheduling decision support systems. X. Wang (2016) proposed a stochastic model that assigned containers to different resources with the objective function to maximize the expected profit of a transport provider.
The decision support system of Fazi et al. (2015) facilitates the creation of schedules for barges by means of a heuristic approach. By defining different parameters, the model drives the decision and creates schedules with certain features, such as increasing the level of utilization of barges or reducing the number stops at the quays.
A crucial element in decision making is the availability of accurate and advanced information.
Dong et al. (2018) found that active interaction between stakeholders of the transport chain is an excellent innovation in inland transportation from a supply chain perspective.
Unfortunately, advanced and accurate information is rare in the container transportation industry (Bouchery et al., 2015). Currently, a transport provider receives reservations a week in advance. However in practice, some of these reserved containers are delayed or do not show up (Zhao et al., 2019). Cancelled and late arrivals have several explanations, such as vessel delays, broken-down trucks, traffic jams, port congestions, and bad weather. Another reason
for cancellation is that shippers book more capacity than they need to make sure their container is transported (Tirschwell, 2016).
2.2 Overbooking
Overbooking is a methodology in revenue management to deal with a large number of no shows and cancellation. Overbooking occurs when a company accepts more reservations than its physical capacity (Mookherjee & Friesz, 2008). To prevent revenue losses due to no shows and late cancellations, companies overbook their capacity (Moussawi-Haidar & Cakanyildirim, 2012). The number of overbooked services depends on the trade-off of either having low utilization which results in low efficiency or having denied services which cause for compensation costs.
Overbooking is a well-researched topic and widely used in service industries, like the airline and the hotel industry, to increase resource utilization (Mcgill & Van Ryzin, 1999). Beckmann (1958) proposed one of the earliest overbooking models for the airline industry, in which he considered a fixed probability of cancelled reservations. In this regard, Alstrup et al. (1989) developed a model which is more realistic since the model involves stochastic multistage optimal decision making and implemented two substitutional segments. Later, more models were proposed to find a proper overbooking level (Mou et al., 2019; Wannakrairot &
Phumchusri, 2016). Y. Huang et al. (2013) proposed a model which involve postponement of a passenger to a next flight instead of upgrading to a higher class. Ivanov (2015) developed a mathematical model with three types of rooms where upgrades and downgrades were taken into account to create high utilization. A multiple class approach could also be interesting for an overbooking model in a multimodal inland transport system. Transferring containers between different service levels must be considered in this study, for example, an overbooked container for barge transportation can be upgraded to train transportation, and the container will arrive before the due date. Differently from the hotel industry, containers cannot be denied and must be transported eventually. Karaesmen et al. (2004) developed an overbooking model to maximize the revenue by considering two periods; a reservation and a service period. In the reservation period, the decision of how many reservation requests to accept is made, and in the service period, the allocation of the on-time customers is realized. As previously described, there are two important stages in this study; accepting bookings and allocation of containers.
Therefore, the proposed model by Karaesmen et al. (2004) could be used as a starting point in this study.
Overbooking in the freight industry is less researched. Fen-ling et al. (2015) proposed a dynamic model for railway freight transport to determine the overbooking policy with different fare classes during the reservation period. They found that the overbooking strategy performs better in fields of utilization and revenue compared to an existing reservation strategy (First Come First Service). Similarly, T. Wang et al. (2019) found that an overbooking strategy can promote the profit in the liner container shipping industry. Differently from these papers, this study considers multiple modalities instead of one modal.
Our allocation problem is similar to the one tackled in Van Riessen et al. (2017), they proposed the Cargo Fare Class Mix (CFCM) model, with the purpose to set booking limits for each fare class at a tactical level, such that the expected revenue is maximised, considering the available capacity at the operational level. Another important factor similar to this study is that all accepted demand must be transported, and a penalty cost occurs when transportation by truck is required. Despite the similarities, the difference between their study and our study is that Van Riessen et al. (2017) assume that all reservations show up. However, as previous described many reserved containers are cancelled or delayed (Zhao et al., 2019).
A comparison table is made to show the differences and similarities between the proposed model in this study and the models proposed in prior literature (see Table 2.1). In a way, our problem could be similar to the one of Y. Huang et al. (2013). However, they did not consider overbooked passengers to transfer between classes. In this study, the model must consider transferring between classes because an overbooked container for the barge can be upgraded to the train. Another similar study to our problem is Karaesmen et al. (2004). Nonetheless, they did not consider postponement of the service. In our case, it is possible to postpone the container to the next barge, train, or truck. All these studies consider denied services and do not have a backlog of reservations. However, in our study, after the booking is accepted the container must be transported eventually thus denials are not possible. The allocation features in the study of Van Riessen et al. (2017) are similar to our problem except that they assume all reservations showed up. In our problem containers can either be late, cancelled or on-time, whereas all the existing studies only consider cancelled and on-time arrivals.
Table 2.1, Comparison table of existing overbooking models and proposed model
Multiple classes
Postpone
element Cancellation
Cancelled and late arrivals
Denials
Karaesmen et al. (2004) X X X
Ivanov (2015) X X X
Fen-ling et al. (2015) X X X
Y. Huang et al. (2013) X X X
Van Riessen et al. (2017) X X
Proposed model X X X X X
In this paper, we tackle an overbooking problem for multimodal inland transport systems.
Moreover, late arrivals and cancellations often happen in inland container transportation and no research has tackled this problem for this particular system. Many papers have found that a mathematical model can be applied as a decision tool for transport providers and overbooking is a promising strategy in inland container systems (K. Huang & Lu, 2015; Parizi & Ghate, 2015). Hence, it is interesting to research a support decision tool for determining the optimal overbooking level. By means of numerical experiments, we show the goodness of the proposed model. Our addition to current models is that items are allowed to transfer between classes, postponement is possible, and denials are not included. In addition, our study also considers containers to be on-time, late, or cancelled.
The aim of this study is to determine whether it is beneficial to set an overbooking level in a multimodal inland transport system with a dynamic approach. If this strategy is found to be useful, the developed model in this paper can be used as a decision support tool for a manager who wants to decide the overbooking level in a multimodal inland transport system. As discussed above, many similar studies did approach this problem with quantitative model- based research (Alavi Fard et al., 2019; Chen et al., 2015; Lan et al., 2014). A small number of overbooking studies applied the simulation-based method (Fen-ling et al., 2015). Model-based research consists of variables and addresses in a quantitative style the causal relationship between these variables. The objective of these models are based on assumptions to explain the (part of) real-life processes or can support (part of) the decision-making problems that are faced by managers in real-life processes (Karlsson, 2016). Because model-based research is
widely used in similar research and this research aims to support a decision, we have adopted quantitative model-based research.
3. Problem description
This section describes the problem setting of this research, the assumptions that have been made, and the values of the parameters in the base instance.
3.1 Problem setting
We consider a transportation system comprising containers arriving at a seaport that need to be moved to an inland terminal by means of train, truck or barge. Before departure, shippers (senders of containers) request a specific modality for their container at the transport provider.
The transport provider reviews the overbooking level of that modality and will assign the container to the requested modality or reject the booking request. A booking is accepted (i.e.
the container is assigned) when the booking is within the overbooking level of the considered modality. The overbooking level is determined per modality and is the first stage decision variable. Figure 3.1 visualizes the process of the first stage of the problem.
Figure 3.1 Process of the first stage of the problem
When a booking is accepted, the shipper arranges that the container arrives at the seaport.
However, the container can arrive on-time, late, or is cancelled. Cancelled containers are containers that do not need to be transported anymore and are excluded from further allocation.
The other containers need to be allocated; this is the second stage decision variable.
Containers that arrive on-time are containers which arrive before the departure time. When the number of on-time containers does not exceed the capacity of the booked modality the container is transported as assigned in the first stage. When the number of on-time containers
does exceed the capacity, the container is transported by another modality or stored in a warehouse for a later departure. To make sure the container arrives on schedule at the inland terminal, containers booked for a train cannot be transported by barge, since a train is faster than the barge. However, train containers can be upgraded (i.e. transported by another modality) to truck transportation, but only when the warehouse is fully utilized, in order to reduce truck transportation. Barge containers can be upgraded to either train or truck, although truck transportation is only considered when the capacity of the barge, train, and warehouse are satisfied.
Late arrivals are containers that arrive after the departure time but are not cancelled. We assume that the barge and train depart in the same stage, therefore late arrivals can either be stored in a warehouse and be transported later or they can be transported by truck. Because a truck does not have a fixed scheduled departure time, containers can always be transported. To minimize the use of truck transportation, the truck is only considered when the capacity of the warehouse is fully occupied.
Figures 3.2 and 3.3 visualize the second stage process of the problem, where 3.2 represents the allocation process of the barge containers and 3.3 the allocation of the train containers. The dotted lines represent the flow of the late containers and the numbers represents the sequence of allocation (1 is the first choice, and 4 is the last).
Figure 3.2 Allocation process of containers booked for a
barge. Figure 3.3 Allocation process of containers booked for a
train.
3.2 Assumptions
For simplification, some assumptions are made. We assume there is one inland terminal which can receive containers that are transported by barge, train, or truck from a seaport. The reason for this assumption is that there must be a possibility to upgrade containers, terminals which only can receive one of the transport modes would give a problem with the destination of a container. With the second assumption, the inland terminal is considered as the final destination of the container, further transportation is done by the customer itself. Furthermore, to minimize the use of truck containers, bookings can only be made for barge or train. When the barge container is fully occupied and train capacity not, containers booked for the barge can be transported by train, this is called an upgrade. Another form of upgrading is assigning containers that are booked for a barge or a train to a truck. Typical in inland transportation systems, a due date is set for a container to be at the final destination, commonly these due dates are ambitious and not very rigid. However, we assume a due date by given the constraint that the backlog must be transported by the preferred modality. Furthermore, we assume that the demand is infinite, i.e. there is no upper bound for the optimal overbooking level. We are interested in the behaviour of the optimal overbooking level, an upper bound might disturb this. Because the demand is infinite there must be a capacity constraint otherwise the result will be infinite and no optimal can be found, therefore we assume a fixed capacity for the barge, train, and warehouses. The number of trucks is unlimited because a container is a perishable product that must be transported eventually. To diminish truck transportation, we assume a large fine when assigning a container to a truck. This fine must be higher than the revenue, hence, the truck is considered as last resort.
3.3 Base setting
Since no real data were available, the values of the parameters had to be based on publicly accessible data and assumptions derived from papers. However, as all data are based on realistic assumptions, the results of the model can be assumed to be valid for the overbooking level decision in the real market. In this section, the base setting is defined that is used in this research
The study of Behdani et al. (2016) provide the capacity and costs for barge, train, and truck transportation. The fixed cost of a barge is €45 per container and one barge has a capacity of 40. However, in their study, they suggest multiple barges, for the base setting we assume three
barges with a total capacity of 120. The cost of train transportation of one container is €60 and has a capacity of 110 containers. The fixed costs are made despite the actual number of containers on the modality, this suggests the spoilage cost; the cost of unused capacity. To realize a profit, the revenue must be higher than the costs. We assume that 50% of revenue is used to cover the fixed cost, therefore the revenue is €90 for a barge container and €120 for a train container. The revenue of the train must be greater than the revenue for a barge container, otherwise, it is not possible to upgrade from barge to train.
The values of the penalty costs define the sequence of the allocation; the lowest penalty cost is first in the sequence and the highest cost is last. This is because the objective is to maximize profit and therewith minimize costs. We assume the sequence of allocation represented in section 3.1. The penalty cost for allocating a barge container to a train is equal to the difference in revenue. The model must compensate for the revenue that they could have retrieved from a train container booking, also known as opportunity cost. The penalty cost for storing a container is randomly picked and between the upgrading cost and the penalty cost for truck transportation. Using a truck cost €90 according to Behdani et al. (2016). As the number of trucks must be minimized, we assume a high penalty cost of €90 plus the corresponding revenue. According to Fen-ling et al. (2015), 8% of the freight charges are refundable, therefore we assume a refund rate of 0.08 of the corresponding revenue. Additionally, we assume that the capacity of the warehouse is 30% of the capacity of the modalities. A random number of containers between one and the warehouse capacity is set as the backlog. Table 3.1 summarizes the base setting.
Table 3.1 Data for the base setting
Data
Capacity (uj) Barge (b) 120
Train (t) 110
Warehouse Barge (h) 36 Warehouse Train (k) 33
Backlog (Gi) Barge (B) 11
Train (T) 20
Fixed costs (Fi) Barge (B) €7200
Train (T) €6600
Revenue (Ri) Barge (B) €90
Train (T) €120
Penalty costs (Cij) CBt 30
CBl 180
CBh €86
CTl 210
CTk €107
Refund (D) 0.08
The decisions in this problem are made under different circumstances by means of on-time, late, and cancelled bookings. To address this uncertainty, we set up different scenarios. We formulated three scenarios for the on-time arrivals. According to a maritime analysis from IHS of the market from 2016 till 2018, the percentage of on-time vessels globally varies from 65%
and 85%. Therefore, the lowest scenario is 65%, the medium 75% and the highest is 85%, the probability of the scenarios are 0.26, 0.58, 0.16, respectively (Mongelluzzo, 2018). The remaining containers can either be late or cancelled. From the remaining containers, we assume that 70% are late arrivals. We take this as the medium scenario and set the values of the low and high scenario with 10% more and less. The probability for the low, medium and high scenarios is 0.25;0.5;0.25, respectively. Table 3.2 summarizes the nine scenarios of the on- time, late, and cancelled bookings.
Table 3.2 Scenarios of on-time, late and cancelled bookings
Scenario (!) On-time (#!) Late ($!) Cancelled (%!) Probability (&!)
1 0.65 0.6 0.4 0.065
2 0.75 0.6 0.4 0.145
3 0.85 0.6 0.4 0.04
4 0.65 0.7 0.3 0.13
5 0.75 0.7 0.3 0.29
6 0.85 0.7 0.3 0.08
7 0.65 0.8 0.2 0.065
8 0.75 0.8 0.2 0.145
9 0.85 0.8 0.2 0.04
4. Model formulation
Our problem can be classified as a two-stage stochastic problem. As described in the previous section, this study considers two decision moments with two decision variables; the overbook limit in the first stage and the allocation in the second stage. The first decision variable depends on additional information that becomes available at a later moment; the actual number of on- time, late, and cancelled containers. To capture the interplay between decisions and uncertainty, a stochastic programming model is recommended (Higle, 2005). Therefore, this study proposes a two-stage stochastic programming model with the objective function to maximize the profit.
Consider I containers denoted by i = Barge (B), Train (T), that can be booked. Let J be the set of locations, denoted by j = barge (b), train (t), truck (l), warehouse barge (h), and warehouse train (k). The set of scenarios is called W, denoted by 1 … ( .
Each container receives revenue of Ri, has a fixed cost, fi, and a backlog of Gi, where Ri, Fi, and Gi are positive integer. The capacity of the location j is uj. We define Cij as the costs for assigning container i to location j. About the cancelled containers refund Dis paid. The number of on-time containers i in scenario ( is defined by )"#. The number of on-time containers depends on the percentage of accepted bookings that are on-time in scenario ( is set as *". From the remaining containers, the containers that are not on-time, late containers i in scenario ( is defined as +"#, and depends on the percentage of the remaining containers, set as ,". Let
-" be the percentage of the remaining containers that are cancelled in scenario (, and let ."#
be the number of cancelled containers i in scenario (. The probability of the scenarios is set as /".
The first decision variable is the overbooking level of container i and denoted as 0#. The overbooking level must include the backlog because the capacity of the barge and train capacity decreases with backlog Gi (see constraint 3). The revenue Ri is retrieved from the number of accepted bookings for container i, defined as 1#. To characterize the allocation of containers i to location j in scenario ( we set 2"#$% for on-time containers, and 2"#$& for late containers. All the sets, parameters, variables, and data are summarized in Table 4.1.
Table 4.1 Elements of the model
Set
I Set of containers Barge (B), Train(T)
J Set of locations barge (b), train (t), truck (l), warehouse barge (h), warehouse train (k)
3 1…(
Data
4# Revenue of container i, ∀ 6 ∈ 8 9# Fixed costs of container i, ∀ 6 ∈ 8 :# Backlog of container i, ∀ 6 ∈ 8
;$ Capacity of location j, ∀ < ∈ =
>#$ Cost for placing container i to location j, ∀ < ∈ = , ∀ 6 ∈ 8
@ Refund rate for cancelled containers
*" Rate of accepted bookings arrive on-time in scenario (, ∀ 3 ∈ (
," Rate of remaining containers arrive late in scenario (, ∀ 3 ∈ (
-" Rate of remaining containers are cancelled in scenario (, ∀ 3 ∈ (
Parameters
)"# Number of on-time containers of container i in scenario (, ∀ 6 ∈ 8, ∀ 3 ∈ (
+"# Number of late containers of container i in scenario (, ∀ 6 ∈ 8, ∀ 3 ∈ (
."# Number of no-show containers of container i in scenario (, ∀ 6 ∈ 8, ∀ 3 ∈ (
/" Probability of scenario (, ∀ 3 ∈ (
Variables
1# Number of accepted booking request of container i, ∀ 6 ∈ 8 0# Overbooking limit of container i, ∀ 6 ∈ 8
2"#$% Number of on-time containers, i, assigned to location j in scenario (,
∀ 6 ∈ 8, ∀ < ∈ =, ∀ 3 ∈ (
2"#$& Number of late containers i, assigned to location j in scenario (,
∀ 6 ∈ 8, ∀ < ∈ =, ∀ 3 ∈ (
We propose the following overbooking model formulation:
A10 B(4#1# − 9#;$)
#∈(
− E" F(2, G) (1)
Where,
F(2, G) = A6I >)*2")* + >+*2"+*+ >),2"),+ (2")-& + 2")-% )>)- + (2"+.& + 2"+.% )>+. + @ B ."# 4#/"
" ∈ 0
# ∈ 1
(2)
Subject to
1# = 0# − :# , ∀ 6 ∈ 8
(3) 0# = B ()"# + +"# + ."# + :# ) /"
" ∈ 0
, ∀ 6 ∈ 8 (4)
)"# = *"1# , ∀ 6 ∈ 8, ∀ ( ∈ 3 (5)
+"# = ,"(1# − )"#) , ∀ 6 ∈ 8, ∀ ( ∈ 3 (6)
."# = -"(1#− )"#) , ∀ 6 ∈ 8, ∀ ( ∈ 3 (7)
," + -" = 1 , ∀ ( ∈ 3 (8)
)") = 2")2% + 2"),% + 2")-% + 2")*% , ∀ ( ∈ 3 (9)
+") = 2")-& + 2")*& , ∀ ( ∈ 3 (10)
)"+ = 2",,% + 2",.% + 2",*% , ∀ ( ∈ 3 (11)
+"+ = 2"+.& + 2"+*& , ∀ ( ∈ 3 (12)
2")2% ≤ ;2− :), ∀ ( ∈ 3 (13)
2"+,% + 2"),% ≤ ;, − :+ , ∀ ( ∈ 3 (14)
2")-& + 2")-% ≤ ;- , ∀ ( ∈ 3 (15)
2"+.& + 2"+.% ≤ ;. , ∀ ( ∈ 3 (16)
2"#$% , 2"#$& , 1# , 0# ≥ 0 , 6IO,P,Q , ∀ 6 ∈ 8, ∀ < ∈ =, ∀ ( ∈ 3 (17)
The objective function (1) maximizes the expected profit on revenue, fixed cost and variable costs. E3 denotes the mathematical expectation with respect to G, where the random vector G"
depends on the scenario (. In this notation is E3 the expected variable costs, while F(2, G) is the variable cost with the on-time, late, and cancelled containers in scenario ( (2). The second
outsourcing cost for train containers transported by truck (second term), the opportunity cost for assigning a barge container to train transportation (third term), the outsourcing cost for barge container assigned to the barge warehouse (fourth term), the outsourcing cost for train containers assigned to the train warehouse (fifth term), and the refund cost (last term).
The number of accepted bookings differs from the overbooking level, revenue is achieved from accepted bookings and not from the backlog. Constraint (3) ensures that the number of accepted bookings does not exceed the overbooking level including the backlog, this is because the available capacity of the barge and train decreases with the backlog. Constraint (4) ensures that the sum of the on-time, late, and cancelled containers and backlog are equal to the booking level of the container. In constraint (5) the number of on-time containers is defined. The number of late (6) and cancelled (7) containers are expressed by a rate multiplied by the remaining containers, containers that are not on-time. The sum of the late and cancelled container rates must be equal to 1 (8), to ensure that all remaining containers are either late or cancelled.
All the on-time containers must be assigned to one of the locations, the barge container can be assigned to all the locations, the train containers can be assigned to all the locations except to the barge, we assume that downgrading is not possible (9) (11). The late containers can only be assigned either to the warehouse or to truck transportation (10) (12). The capacity of the locations cannot be exceeded (13) (14) (15) (16). The backlog must be transferred by the preferred means of transport, therefore the available capacity of the barge and train decreases with the backlog (13) (14). All the decision variables are nonnegative and integer (17).
5. Numerical experiments
In this section, we first explain the performances measurements and then the experiments with the results are defined. The experimentation is in twofold. First, the sensitivity of the model is analysed and secondly, the overbooking performance in various settings are examined. The instances of the experiments can be found in Appendix 1. CPLEX 12.1 is used to assess the sensitivity and the performance of the overbooking in different scenarios.
5.1 Performance measurements
By comparing the financial, environmental performances and decision variables we assess how sensitive the model is to fluctuations in the parameters and data. We focus on the behaviour of the overbooking limit, which is represented by the number of containers that are booked over the available transport capacity (barge and train). The financial performance is evaluated by the profit given in euros. The environmental performance is evaluated by the use of trucks, given in percentage of the accepted containers, and utilization of the modalities and warehouses.
5.2 Effect of transport capacity, warehouse capacity, revenue, and costs
In order to assess the sensitivity of the model to the parameters; transport capacity, warehouse capacity, revenue, and costs, we conducted nine instances for each parameter, where the parameter is adjusted in steps of 20% of the base setting (see Section 3.3). The results of this sensitivity analysis are reported in Tables 5.1 – 5.4.
Table 5.1 Result sensitivity analysis adjusting barge capacity (ub) and train capacity (ut)
Overbooked % Overbooked Utilization
Instance ub ut Barge Train Barge Train Profit % by Truck barge train Warehouse barge
Warehouse train
1 24 22 38 35 158% 159% 2600 0% 100% 100% 96% 97%
2 48 44 40 37 83% 84% 7800 0% 100% 100% 94% 98%
3 72 66 40 37 56% 56% 12958 0% 100% 100% 90% 92%
4 96 88 43 39 45% 44% 18072 0% 99% 100% 93% 93%
5 120 110 48 43 40% 39% 22860 1% 98% 99% 98% 98%
6 144 132 46 45 32% 34% 27447 2% 97% 98% 94% 98%
7 168 154 53 46 32% 30% 31783 3% 97% 97% 98% 98%
8 192 176 61 52 32% 30% 35998 5% 97% 97% 100% 100%
9 216 198 69 60 32% 30% 40053 7% 97% 97% 100% 100%
Table 5.2 Result sensitivity analysis adjusting barge warehouse capacity (uh) and train warehouse capacity (uk)
Overbooked Utilization
Instance uh uk Barge Train Profit % by truck Barge Train Warehouse Barge
Warehouse Train
1 7 7 37 30 18518 15% 97% 97% 100% 100%
2 14 13 37 30 19794 10% 97% 97% 100% 100%
3 22 20 37 30 21237 5% 97% 97% 100% 99%
4 29 26 38 35 22095 2% 97% 98% 96% 98%
5 36 33 48 43 22860 1% 98% 99% 98% 98%
6 43 40 51 47 23413 0% 99% 100% 92% 91%
7 50 46 60 51 23726 0% 100% 100% 93% 88%
8 58 53 68 59 23983 0% 100% 100% 93% 90%
9 65 60 74 68 24256 0% 100% 100% 92% 92%
Table 5.3 Result sensitivity analysis adjusting barge revenue (RB) and train revenue (RT)
Overbooked Utilization
Instance RB RT Barge Train Profit % by truck Barge Train Warehouse Barge
Warehouse Train
1 18 24 11 10 689 0% 83% 85% 61% 56%
2 36 48 35 30 5852 0% 96% 97% 79% 72%
3 54 72 36 30 11340 0% 96% 97% 81% 72%
4 72 96 39 38 16953 0% 97% 98% 85% 91%
5 90 120 48 43 22860 1% 98% 99% 98% 98%
6 108 144 48 47 28897 2% 98% 100% 98% 99%
7 126 168 48 47 35011 2% 98% 100% 98% 99%
8 144 192 48 47 41299 10% 100% 100% 100% 100%
9 162 210 - - - - - - - -
Table 5.4 Result sensitivity analysis adjusting warehouse cost for barge (CBh) and train (CTk) and truck costs for barge (CBl) and train (CTl)
Overbooked Utilization
Instance CBh CTk CBl CTl Barge Train Profit % by truck Barge Train Warehouse Barge
Warehouse Train
1 17 21 108 138 60 51 28603 6% 100% 100% 100% 100%
2 34 43 126 156 52 47 27056 3% 99% 100% 100% 99%
3 52 64 144 174 48 47 25623 2% 98% 100% 98% 99%
4 17 21 162 192 48 43 24227 1% 98% 99% 98% 98%
5 86 107 180 210 48 43 22860 1% 98% 99% 98% 98%
6 103 128 198 228 42 38 21515 0% 98% 98% 91% 91%
7 120 150 216 246 39 38 20323 0% 97% 98% 85% 91%
8 138 171 234 264 37 30 19277 0% 97% 97% 81% 72%
9 155 193 324 378 37 30 18251 0% 97% 97% 81% 72%
From the results reported in Table 5.1, we note in instance 1, a low capacity, the overbooked containers are exceeding the transport capacity. This can be explained by the greater capacity of the warehouse compared to the capacity of the barge and train capacity; therefore, the overbooked containers are assigned to the warehouse. Furthermore, the results indicate that as the capacity increases, the optimal overbooking level increases. However, the percentage of overbooked containers remains the same in instances 7 till 9. This suggests that a percentage of overbooking is more profitable even if the costs rise because these overbooked containers are being transported by trucks, while the barge and train are not fully occupied. Accepting more containers could increase the utilization of the barge and train but results in extra costs because accepting more containers also increases the number of late containers which always result in extra costs. Hence, to minimize outsourcing costs, the optimal overbooking level is not higher.
Correspondingly, when increasing the warehouse capacity, the optimal overbooking level increases as well (see Table 5.2). The outsourcing costs are lower than the revenue, therefore, accepting a booking that must be outsourced is more profitable than not accepting a booking.
However, the optimal overbooking level is not always sensitive to the warehouse capacity. At a low warehouse capacity (instances 1 till 3), the number of overbooked containers remains the same. These overbooked containers must be transported by truck. For example, looking at the first instance, 15% of the accepted bookings are assigned to the truck. This suggests that in order to maximize profit, it is most profitable to accept bookings executed with truck transportation which result in penalties relative to deny bookings resulting in spoilage costs.
From the results reported in Table 5.3, we see that the revenue has a positive correlation with the optimal overbooking level. This is because as the revenue increases, the impact of the costs on the profit decreases. As a result, the optimal overbooking level increases in order to maximize profit. The increasing optimal overbooking level is noticed in the utilization of the barge, train, and warehouses, which is lower in circumstances with low revenue. However, the barge and train are not fully occupied. In this case, accepting more containers means more late containers which always results in costs. Due to the smaller impact of the costs, more containers can be accepted and transported by truck. It is interesting that there is a maximum revenue of
€167 and €194, for the barge and train, respectively. This is because the difference between revenue and outsourcing cost of allocating a container to a truck is small. With an infinite
