The Concept of Container Detention Applied to Inland Empty Container Repositioning Problem

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The Concept of Container Detention

Applied to Inland Empty Container

Repositioning Problem

Master’s Thesis, MSc Supply Chain Management, Faculty of Economics and Business,

University of Groningen, Netherlands

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Abstract

The concept of container detention has not been well researched in the field of empty container repositioning problem. Therefore, in this paper we aim at further studying the problem by developing a mathematical model with the objective of minimizing total costs. In order to solve the model, we employ the Mosel Xpress IVE to try to find out optimal routes. The test results indicate that container detention has an influence on the number of street-turns, and the transportation cost will be affected accordingly. We hope that this research could arouse an awareness of container detention in this field and help the shipping lines to make the transportation decision with optimal empty containers flows.

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Tables

Table 1: Classification of container routing problem research ... 8

Table 2: Set ... 13

Table 3: Parameters ... 13

Table 4: Decision Variables ... 13

Table 5: Test Instances Description ... 15

Table 6: Parameters ... 16

Table 7: Computational time for Free-time ONE day ... 16

Table 8: Optimal total cost and optimal transportation cost ... 17

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Figures

Figure 1: Overview of decision making in empty container repositioning ... 6

Figure 2: Street-turn approach ... 9

Figure 3: Cost reduction (instance 1, 2, 3) ... 18

Figure 6: The number of street-turns performed (instance 1, 2, 3)... 19

Figure 7: The number of street-turns performed (instance 4, 5, 6)... 19

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1. Introduction

The demand for container transportation is dramatically increasing, and the problem with empty containers accordingly emerges in maritime transportation management. Due to the trade imbalance, transport carriers need to reallocate the empty containers to fulfil the future demand for empty containers. According to the data provided, more than 40% of the land transportation is empty container movements (Crainic, Gendreau, Soriano, & Toulouse, 1993; Konings & Thijs, 2001). Unlike full container movements, empty container movements do not generate any revenues for the transportation. Even though the empty container movements cannot be completely avoided, minimizing these activities can provide future transportation with opportunities and reduce the operational expenses of maintaining containers fleet(Shi, Li, Yang, Li & Choi, 2012).

During last decades, many research have been focusing on the mathematical model for empty container repositioning(Bandeira, Becker & Borenstein, 2009; Deidda, Di Francesco, Olivo, & Zuddas, 2008; Jula, Chassiakos & Ioannou, 2006; Olivo, Zuddas, Di Francesco, & Manca 2005). In their studies, the concept of street-turn, as well as different objectives such as cost minimization and travel distance minimization, were addressed. Street-turn approach simply describes a special match back operation where the empty containers unloaded at importer is then directly moved to exporters for re-utilization. The benefits of street-turn are reflected in reducing congestions and the number of empty container movements.

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a Decision Support System (DSS) is developed to help shipping lines to make transport decisions on route planning at the reasonable computational time. The developed model is tested and validated through Mosel Xpress IVE for benchmark datasets that randomly generated from the system.

The remainder of this paper is organized as following: Section 2 discusses the literature related to empty container repositioning. In Section 3, problem description is explained in detail. In section 4, the structure of the mathematical model is extensively explained. Section 5 shows the study of the computational experiment. Finally, the conclusion is made in Section 6.

2. Literature review

2.1 Empty Container Repositioning

In addressing the empty container repositioning problem at the regional level, the decisions regarding planning levels have to be made, which can be categorized into three planning level, including strategic level, tactical level and operational level(Braekers, Janssens, & Caris, 2011). In the strategic level, general policies that establish the guidelines for decisions at the tactical level are determined. While, the tactical decisions set the framework for real-time decisions at the operational level. As shown in Figure 1, the hierarchical relationship exists among decisions at a different level. The problem complexity and data required for each level are also different.

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The problem that we intend to solve in this paper mainly belongs to the operational planning level, because the main issues of operational planning are fairly similar such as the scheduling of services and routing and assigning different resources of transportation such as containers, vehicles (Braekers et al., 2011).

Previous researchers have developed numerous models about empty containers repositioning at the operational level. Crainic, Dejax and Delorme (1989) analysed the problem of locating the empty vehicles in order to minimize the cost of depot and transportation, but they did not address the issues of space-time interdependency. The empty containers allocation problems in regard to dynamic and uncertainty demand were discussed in Crainic, Gendreau & Dejax (1993), in which a more general framework was addressed. Choong, Cole & Kutanoglu (2002) further adapted the model to manage the homogenous fleet of both empty containers and the loaded containers. They concluded that a longer period of planning horizon would stimulate the use of cheaper transport modes and lower the overall repositioning costs. Deidda, Di Francesco, Olivo, & Zuddas (2008) proposed a static optimization model with street-turn between importers, exporters and the design of vehicle routes of empty containers. Street-turn strategy describes that empty containers are moved directly from importers to exporters, without an intermediate stop at the port(Jula, Chassiakos, & Ioannou, 2003). Bandeira, Becker & Borenstein (2009) proposed a decision support system that modelled the routing of loaded containers and allocation of empty containers, the problem of which was formulated as a multi-depot vehicle scheduling problem. However, street-turn has not been addressed. As mentioned above, the optimization of regional empty container repositioning at the operational level cannot be solved by a single mathematical model due to the complexity of the problem. Therefore, the operational planning problem is divided into two separate problems, container allocation and routing problems(Braekers et al., 2011). Huth and Mattfled (2009) also discussed that the empty container repositioning should be treated as the integration of the routing problem with pickups and deliveries and dynamic resource allocation problems. Unlike operational decisions made for global empty container repositioning, the routing decisions have to be included when making decisions for regional/inland transportation. As it is highly related to the time factors and stochasticity inherent in the system, this paper mainly focuses on the container routing problem.

2.2 Container Routing Problem

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Model Classification Problems Single/Multi-commodity Objectives

Imai et al. (2007) VRPB Single commodity Min. cost

Chuang et al. (2007) TSP&VRP Multi-commodity Multi-objectives

Namboothiri et al.(2008) PDPTW Single commodity Multi-objectives

Vidović et al.(2011) VRPB Multi-commodity Min. cost

Table 1: Classification of container routing problem research

Imai, Nishimura, & Current (2007) formulated a container transportation problem as pick-up and delivery problem (PDP) from/to an intermodal terminal and developed a heuristic algorithm using Lagrangian relaxation to identify optimal solutions. In their study, they proposed single-stop routes where only one exporter must be served after one importer. Similar to the current study, their problem was strongly related to street-turn because delivery and pick-up can be served in a single route.

Chung, Ko, Shin, Hwang & Kim (2007) extended the container truck transportation problem with a number of models, which can be dealt with different cases involving multiple commodities and different types of trucks. The study also presented a heuristic algorithm to solve the model. The big difference compared to current paper is that truck and container are separated. In our case, the empty containers have to be considered as one of the transportation resources used to contain freight, with which the tractor and truck will stay with the container during loading and unloading operations.

Namboothiri and Erera (2008) continued the truck container pickups and deliveries studies in consideration of an appointment-based access control system. By adding a port access control constraint, the decision support system was capable of scheduling the port access for a specific time slot. Since the purpose of the model made in this paper is also to establish a decision support system, their research can give some hints on the model building.

Vidović, Radivojević, & Raković (2011) also studied drayage operation as container distribution and collecting processes. Their problem was closely related to VRPB for looking for optimal vehicle routes in container transportation. They extended the problem studied by Imai et al. (2007), by formulating the problem as a multiple matching problem that is to seek the optimal matching possibility of pickup and delivery nodes in single routes. Nevertheless, in their study, the issues related to time constraints were not respected. Since the major goal of the research was trying to build matching nodes during the drayage operation, the research can provide some insights into the problem we studied.

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container detention on container routing decision making will be provided to fill this gap and shed light on studies in the future.

3. Problem Description

At regional transport, there are different kinds of empty container movements presented among the port, consignees (importers) and local shippers (exporters). After unloading the loaded containers at the importers, the empty containers need to be transported back to the container terminals for future use. Meanwhile, the exporters also require empty containers to pack their cargo so that it can be delivered to the port. In this case, these two direct deliveries will generate two empty container movements as shown in figure 2-(a). However, there is one kind of carrier haulage services that can satisfy the demand of both importers and exporters in one-off delivery, which is called street-turn (See figure 2-(b)). When implementing street-turn, the empty containers can directly flow from the importer to a local exporter without stops at ports or depots. In this case, the assigned trucks are able to reach the importers and move back with loaded cargos from exporters to the departure port in a one-way trip, which allows the shipping company to save handling costs of carrying empty containers. Moreover, the trucks can serve both importers and exporters with fewer empty runs and waiting time for the assignment.

Figure 2: Street-turn approach (Jula et al., 2003)

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In fact, the problem addressed in this paper can be transformed into vehicle routing problem with backhauls and time windows (VRPBTW), because all deliveries must be accomplished before any pick-ups made in the street-turn formulation (Deidda et al., 2008; Parragh, Doerner, & Hartl, 2008). Additionally, in order to satisfy the requirements of multi-customers, the shipping lines has to acquire relevant information such as locations of customers, pick-up and delivery with time window restriction, transportation costs, and so on (Furio, Andres, Adenso-Diaz & Lozano, 2013). In order to clarify the problem, the significant features are discussed in the following assumptions:

1. Starting and ending points

The delivery system of street-turn will be initiated from the ports. As the focus is on operational level of logistics planning, the set of ports in the distribution network will be fixed. In this case, there is only one port in the system, and every route starts and finishes at the port.

2. Customer visits

In this paper, every importer and every exporter are visited exactly once, by exactly one vehicle, which means he demand for each customer is delivered by only one vehicle. In this case, the demand of each customer is quantified as only one container, no matter how much weight of goods are and how big the cargo is.

3. Customer requests

Two kinds of customers are considered in this case. Firstly, the importers require transportation services, and the loaded containers must be dispatched from port to their location. The empty containers will be available as soon as the full containers are unloaded, then they can be delivered to exporters to be reutilized. Secondly, the exporters also need the empty containers to load their cargos. The empty containers will be moved to the exporters to be loaded and shipped to the port terminals. Both importers’ and exporters’ requests are associated with supply and demand of empty containers.

4. The number of trucks

The activities that involved in the street-turn approach are performed by trucks. The routes of the trucks are determined by the number of the customers served. Therefore, the number of trucks is unknown, which will depend on the data from the benchmark.

5. Container type and size

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6. Routing time

Each customer is served with specific time interval (time window). Therefore, a vehicle is not allowed to begin service before or after the time windows. Also, each customer has a manoeuvring time that spent by the truck to unload and load the goods. Hence, the total routing time of a truck consists of travelling time that is proportional to the distance between two nodes and manoeuvring time.

7. Transportation capacity

An assigned truck can serve not more than one importer during the delivery and not more than one exporter during the pickup. In the original individual trips, the truck can go empty in one leg and be full in the other leg. For street-turn approach, it can go in full in both legs. As each vehicle is carrying only one container, the capacity for one truck can be regarded as one, which means there is only two status of the containers during the transportation: loaded and empty. Therefore, the demand at any nodes that exceeds the vehicle capacity will not be an issue in this study.

8. Sequence of deliveries and pick-ups

As mentioned above, the vehicle has to complete the delivery before any pickups. In a single route, the importer is served before the exporter.

9. Container Detention

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empty. Therefore, in this paper container detention can be operationalized into unified charges for the delay.

4. Mathematical Model

In this section, we describe in detail the container detention with the perspective of application to the container routing under the study. We first give a brief description of the model. Then we show model formulation.

We address the problem of seeking best routes to serve a set of importers and exporters by homogeneous container fleet, composed by truck. We assume both loaded and empty containers are intermediately available. The routes start at the port, which is also the terminal for container storage. The routes are made considering the street-turn and time windows of customers and container detention. The detention period is counted as soon as the truck leaves the port. A truck has to visit different customers through either direct deliveries or street-turn. An additional cost for street-turn routes will be charged in the model. It entails the service costs generated by occupying the truck for a longer time. This cost is in direct proportion to routing time that the truck takes to perform the street-turn. The objective is to minimize the total cost of the travel.

4.1 Model formulation

Let us consider a set P = {0, 1, … , 𝑛 + 𝑚} of nodes, including a set of N = {1, 2, … , 𝑛}of importers and a set of M = {𝑛 + 1, 𝑛 + 2, … , 𝑛 + 𝑚}, and a set K = {1,2, … , 𝑘} of trucks. The cost per hour of travelling is set as 𝐶𝑖𝑗𝑘 > 0 (€/h) with i, j ∈ P, k ∈ K. The penalty cost

for delay is 𝑝𝑘 > 0 (€/day) with k ∈ K and the service cost for street-turn is 𝑓𝑘(€/h) with

k ∈ K. The objective is to find the best tours that satisfy the detention time restriction of the containers and minimize the total costs.

We define a distance matrix 𝑡𝑖𝑗, where i, j ∈ P, calculated with travelling time by truck. The

port has index 0, when the truck is assigned to the same nodes, the distance between them is equal to a large value M. The manoeuvring time for loading and unloading the goods is defined as S. We also define the detention free-time as D and variable 𝛽𝑘 that calculate the

tardiness of truck k at the port. The variable 𝑇𝑖𝑘 records the time when truck k arrives at the

location of customer i. The route is defined by the binary decision variable 𝑥𝑖𝑗𝑘 ∈ {0,1}, with

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P set of node, {𝟎, 𝟏, … , 𝒏 + 𝒎}, index 0 indicates the port

N set of importers, indexed 1, 2, …, n M set of exporters, indexed n+1, n+2,…,n+m

K set of trucks, indexed 1, 2,…k Table 2: Set

𝒆𝒊 earliest arrival time of node i

𝒍𝒊 latest arrival time of node i

𝒕𝒊𝒋 travelling time for arc (i,j)

𝑫 The detention free-time days allowed for the container

S Manoeuvring time for load or unload the goods 𝒇𝒌 Service cost for street-turn per hour for truck k

𝒑𝒌 Penalty cost for delay per day for truck k

𝑪𝒊𝒋𝒌 Transportation cost by truck k from customer i to customer j , i, j= 0, 1 …, n, n+1,

n+2, … n+m, k= 1, 2, …, K 𝑴 a large value

Table 3: Parameters

𝒙𝒊𝒋𝒌 Binary variable, equals to 1 if truck k travels from customer i to j, i,j= 0, 1, …, n,

n+1, …, n+m, otherwise 0 𝑻𝒊𝒌 Arrival time of vehicle k at node i

𝜷𝒌 Delay of container for truck k Table 4: Decision Variables

We formulate the model as follows:

Objective function

Minimize total cost

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∑ 𝑥0𝑗𝑘 𝑛+𝑚 𝑗=1 ≤ 1, ∀𝑘 = 1, 2, … , 𝐾 (4) 𝑇𝑖𝑘− 𝑇𝑗𝑘+ 𝑆 + 𝑡𝑖𝑗 − (1 − 𝑥𝑖𝑗𝑘)𝑀 ≤ 0, ∀ 𝑖, 𝑗 = 1, 2, … , 𝑛 + 𝑚, 𝑘 = 1, 2, … , 𝐾 (5) 𝑒𝑖 ≤ 𝑇𝑖𝑘 ≤ 𝑙𝑖, ∀𝑖 = 1, 2, … , 𝑁 + 𝑀, 𝑘 = 1, 2, … , 𝐾 (6) 𝑇𝑖𝑘+ 𝑆 + 𝑡𝑖0 ≤ 𝐷 + ∑ 𝑥0𝑗𝑘 𝑛+𝑚 𝑗=1 ∗ (𝑒𝑗− 𝑡0𝑗) + 𝛽𝑘+ (1 − 𝑥𝑖0𝑘)𝑀, ∀ 𝑖 = 1, 2, … , 𝑛 + 𝑚, 𝑘 = 1, 2, … , 𝐾 (7) ∑ ∑ 𝑥𝑖𝑗𝑘 𝑛+𝑚 𝑗=0 𝑛+𝑚 𝑖=0 ≤ 3, ∀𝑘 = 1, 2, … , 𝐾 (8) ∑ ∑ 𝑥𝑖𝑗𝑘 𝑛+𝑚 𝑗=𝑛+1 𝑛 𝑖=1 ≤ 1, ∀𝑘 = 1, 2, … , 𝐾 (9) ∑ ∑ 𝑥𝑖𝑗𝑘 𝑛 𝑗=1 = 0 𝑛 𝑖=1 , ∀𝑘 = 1,2, … , K (10) ∑ ∑ 𝑥𝑖𝑗𝑘 𝑛+𝑚 𝑗=𝑛+1 = 0 𝑛+𝑚 𝑖=𝑛+1 , ∀𝑘 = 1,2, … , K (11) ∑ ∑ ∑ 𝑥𝑖𝑗𝑘 𝐾 𝑘=1 𝑛 𝑗=1 𝑛+𝑚 𝑖=𝑛+1 = 0 (12) 𝑥𝑖𝑗𝑘∈ {0,1}, ∀𝑖, 𝑗 = 1, 2, … , 𝑛 + 𝑚, 𝑘 = 1, 2, … , 𝐾 (13) 𝑇𝑖𝑘 ≥ 0, ∀ i = 1, 2, … , n + m, k = 1, 2, … , K (14) 𝛽𝑘 ≥ 0, ∀ k = 1, 2, … , K (15)

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Constraint (8) (9) (10) (11) describe that every route can cover a mix of delivery, delivery and pickup and pure pickup nodes, if required. Constant (12) ensures that if the route requires delivery and pick-up, delivery precedes pick-up.

5. Numerical experiments

5.1 Overview of Test Instances

In order to test the proposed model formulation, several experiments are performed to measure the effect of detention time using datasets that are randomly generated from Mosel Xpress IVE. Besides, the dataset is modified by choosing 40%, 50%, and 60% of the customers as exporters, and leaves other attributes the same. Table 5 summarizes the total nine instances applied in this paper. The optimization model is solved in Mosel Xpress IVE. All computational experiments are performed on a personal computer with Intel Core i7 2.00 GHz CPU and 4GB RAM

Instances 1 2 3 4 5 6 7 8 9

Nodes 15 15 15 20 20 20 22 22 22

# of exporter nodes 6 7 9 9 10 11 10 11 12

Table 5: Test Instances Description

5.2 Parameter Generation and Description

In this paper, the datasets of customer locations are designed to be randomly generated from Mosel. X-axis and y-axis of the nodes are uniformly distributed between (0, 500) kilometres. The travelling time 𝑡𝑖𝑗 is calculated as [√(𝑥𝑖− 𝑥𝑗)

2

+ (𝑦𝑖− 𝑦𝑗) 2

]/50, where 50 stands for the average traveling speed (km/h) of the truck (Jie, 2010). Furthermore, we assume that the time windows of customers will be sparsely distributed. Given that the nested time windows will have more possibility to generate street-turns which might lead to less sensitive results, we assume the earliest arrival time 𝑒𝑖 as uniform distribution between (0, 480) hours and

latest arrival time as 𝑙𝑖 = 𝑒𝑖+ 𝑢𝑛𝑖𝑓𝑜𝑟𝑚(0, 15).

In fact, the detention cost is charged on the basis of the number of days, the number of the free days are set as (1, 5, 10, 18, 19) *24 hours in the time unit. We also assume that the manoeuvring time is 1 hour (Rieck & Zimmermann, 2010). We could not base our analysis on real value for the service cost 𝑓𝑘, so it can be considered as a dummy parameter. In reality,

there is no provision stating that a precise service cost for taking fleet in a longer time. However, such cost has to be considered in order to achieve street-turn in such a long distance. Hence, we set the 𝑓𝑘 equal to 5€ per hour which is a relatively small value. Therefore, it will

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information in the Netherlands (Hapag-Lloyd, 2014), we made 𝑝𝑘 as 50€ per day. Finally,

the unitary cost of truck is set as 75€ per hour (Fazi, 2014). The actual parameters of all the instances are resumed in Table 6.

𝒕𝒊𝒋 𝒆𝒊 , 𝒍𝒊 𝑪𝒊𝒋𝒌 𝒑𝒌 𝒇𝒌 S D Value Uniform distributed Uniform distributed 75€/h 50€/day 5€/h 1 hour (1, 5, 10, 18 ,19) *24 hours Table 6: Parameters

5.3 Result and Analysis

5.3.1 Computational time and size limit

Based on the results of these nine instances, we find out that with the number of nodes increase, the length of computational time increases. The results show that the model can be solved within 22 nodes to optimality in reasonable computational time between 1.4s and 35.1s. However, when the number of nodes exceeds 22, the instances cannot be solved, due to the fact that solved statement is not depicted by Mosel. Table 7 summarizes the calculated time of all the nine instances.

Instances 1 2 3 4 5 6 7 8 9

Nodes 15 15 15 20 20 20 22 22 22

Computational time (seconds)

2.2 1.6 1.4 2.4 2.2 2.1 35.1 26.3 18.5

Table 7: Computational time for Free-time ONE day 5.3.2 Total costs

In order to test the effect of container detention on empty container flow, totally nine instances are tested with numbers of free days in 1, 5, 10, 18 and 19 days respectively. The total cost and the transportation cost for each instance are shown in Table 8. The proposed lower bound does not take into account container detention constraint. To set the lower bound is necessary since it can benchmark the performance of the container detention in the model. In fact, with the setting of the lower bound, the routes will completely depend on the locations and time windows of the customers. In such case, the delay of container detention will be independent.

The number of days for free-time

Instance Node(#

exporters)

1 day 5 days 10 days 18 days 19 days Lower bound

1 15(6) 10305 9825 9934 9300 9463 8775 9231 8475 9231 8475 9231 8475

2 15(7) 10305 9825 10105 9825 9715 9075 9552 9000 9552 9000 9552 9000

3 15(9) 10086 9450 9886 9450 9520 8700 9357 8625 9357 8625 9357 8625

4 20(9) 14952 13650 14790 13350 14766 13350 14766 13350 14766 13350 14766 13350

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6 20(11) 16135 15375 16023 15075 16023 15075 16023 15075 16023 15075 16023 15075

7 22(10) 15054 13500 14654 13500 14358 12750 14358 12750 14358 12750 14358 12750

8 22(11) 15178 13650 14778 13650 14655 13275 14655 13275 14655 13275 14655 13275

9 22(12) 15157 12975 14557 12975 14184 12600 14184 12600 14184 12600 14184 12600

Table 8: Optimal total cost and optimal transportation cost 5.3.3 Routes

In order to better compare results from the instances, Table 9 indicates the number of the street-turn routes in different scenarios. Similarly, the results are based on different days of detention free-time and lower bound for the benchmark.

The number of days for free time

Instance # of

exporters

1 day 5 days 10 days 18 days 19 days Lower bound 1 6 1 2 3 4 4 4 2 7 1 1 3 3 3 3 3 9 2 2 4 4 4 4 4 9 5 6 6 6 6 6 5 10 4 5 5 5 5 5 6 11 3 4 4 4 4 4 7 10 6 6 8 8 8 8 8 11 6 6 7 7 7 7 9 12 7 7 8 8 8 8

Table 9: The number of street-turns 5.3.4 Discussion

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Figure 3: Cost reduction (instance 1, 2, 3)

Based on the results of costs and routes from the benchmark, it is obvious that the street-turn route is the most significant factor influencing the transportation plan. According to the results showing in figure 6, 7 and 8, the number of street-turns will change as the variation of

7500 8000 8500 9000 9500 10000 1 5 10 18 19 Tra n sp o rta tio n co sts

# days of detention free-time

instance1 instance2 instance3 12000 12500 13000 13500 14000 14500 15000 15500 16000 1 5 10 18 19 Tra n sp o ra tio n co sts

instance4 instance5 instance6 12000 12200 12400 12600 12800 13000 13200 13400 13600 13800 1 5 10 18 19 Tra n sp o ra tio n co sts

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free-time. For each instance, the number of the street-turns is relatively small in the condition of short detention free time, and it starts to increase as the detention free time becomes longer until it reaches the lower bound.

All in all, we can draw from the results shown above that the container detention has an influence on the number of street-turns so that the transportation cost will be affected.

Figure 6: The number of street-turns performed (instance 1, 2, 3)

0 1 2 3 4 5 1 5 10 18 19 # o f stree t-u rn

instance1 instance2 instance3 2 3 4 5 6 7 1 5 10 18 19 # o f stree t-tu rn

instance4 instance5 instance6 5.5 6 6.5 7 7.5 8 8.5 1 5 10 18 19 # o f stree t-tru n

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6. Conclusion

In this paper, a mathematical model with the focus on container detention is formulated and solved, generating routes for container drayage operations in the inland transportation network. The proposed model provides optimal routes with the objective of total cost minimization, which is subject to several constraints regarding transport flows, time issues. The contribution of this paper is twofold: Firstly, from the theoretical perspective, the proposed model extends in literature with the concept of container detention in the empty container repositioning problem. Secondly, from the practical point of view, the model helps solve the routing problems, which can be served as a decision support system for the shipping lines to make their decisions in planning the routes and allocating the empty container flow. The model formulated in this paper takes into account the concept of container detention, which can influence transport planning during drayage operation to some extent. Furthermore, the notion of container detention is quite novel in the empty container repositioning problem. In general, existing literature in container routing problems mainly focuses on the delivery and pickup issues when it comes to the real-life constraints of the model. This study aims to attract researchers to pay more attention to the container detention. Finally, based on the results of testing several instances, it can be concluded that the transportation cost of routing will reduce due to the increase of the number of street-turns when the detention free-time extends. The variation will remain unchanged until the detention free-time becomes long enough.

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