Containership routing problem with consideration of
schedule reliability and environmental sustainability in
liner shipping
Master’s Thesis, MSc Supply Chain Management, Faculty of Economics and Business, University of Groningen, Netherlands
20th June 2016 Yue Hu Student number: S2844281 E-mail: y.hu.15@student.rug.nl Supervisor/ University: Dr. S. Fazi Co-assessor/ University:
Abstract
Schedule reliability in liner shipping industry is relative low and unreliable schedules have extensive harmful effects on customers and shipping lines. Ship operators often adopt speed adjustment to enhance the schedule reliability while influencing cost and carbon emission. If the strategy is not performed properly, it could affect the profit and environmental sustainability. Little literature has solved the containership routing problem under the restriction of cost, schedule reliability and environmental sustainability. In this thesis, in order to address the trade-off between above three issues, a multi-objective program is designed with consideration of the speed adjustment. This research considers a joint operational level problem and formulates the reliability and sustainability in the model in order to integrate with the cost objective by using fuzzy multi-objective programming. The main theoretical contribution is to provide a joint discussion of three aspects within a feeder network in liner shipping. This paper also contributes to managerial insight that aims to deliver more precise and efficient suggestions for shipping lines in order to cope with their business strategies. Numerical experiments are provided to illustrate the results, effectiveness and application of proposed model and solution approach.
Table of Contents
Abstract ... 1
1. Introduction ... 1
2. Literature Review ... 4
2.1 Liner shipping business ... 4
2.2 Containership routing problem ... 6
3. Problem description ... 9
4. Mathematical model ... 12
5. Solution approach ... 16
5.1 Fuzzy theory and membership function ... 16
5.2 Crisp formulation ... 17
5.3 Solution steps ... 17
6. Numerical experiments ... 19
6.1 Test instances ... 19
6.2 Results and analysis ... 20
6.2.1 Computational time and size limit ... 20
6.2.2 The effect of crisp formulation ... 21
6.2.3 Solutions with fuel price and penalty variance ... 23
7. Conclusion ... 26
1. Introduction
The cargo containerization, which refers to transport goods in standardized boxes from one place to another, is the core revolution in 20th century transportation technology that facilitated the global economy (Vis & De Koster, 2003; Levinson, 2010; Bernhofen et al., 2016). Although there were only 20% of container cargo transportations in the 1980s, the trend of container movements has grown rapidly and nowadays is expected to keep the growing path due to the economic globalization and increasing global trade volume (Hingorani et al., 2005; Agarwal & Ergun, 2008; Meng et al., 2013).
Ships perform container transportations across oceans (Vis & De Koster, 2003). Last few decades have witnessed a rapid growth of container shipping due to economies of scale (Dong & Song, 2009; Carlo et al., 2014). The liner shipping is the backbone of international container transportation because of the economic superiority (Christiansen et al., 2004; Mulder & Dekker, 2014; Karsten et al., 2015).
In liner shipping, ships follow the predetermined itinerary and schedule to fulfil demands of customers (Meng et al., 2013). However, the overall schedule reliability of liner shipping industry is relative low now (Yang et al., 2013). Uncertainties at ports (e.g. various container handling time) or sea (e.g. weather conditions) can result in delays in the ship round trips and influence the reliability of the schedule (Wang & Meng, 2012a). The unreliability would impact on the downstream parties in the end, which is harmful to financial aspect and the reputation of shipping lines (Song et al., 2015). A common way to reduce delay time is increase the speed of ships, but this would increase the fuel consumption (Corbett et al., 2009; Meng et al., 2013). The fuel consumption would not only increase the transportation cost, but also increase the total carbon emission.
Therefore, the trade-off among the cost, schedule reliability and sustainability for ship round trips is a vital consideration.
In literatures, many studies have addressed the cost, reliability and sustainability issues in containership routing problems separately. Wang et al. (2014) analyzed the ship routing and scheduling with cargo allocation simultaneously and proposed an optimization model to minimize the transport cost of container shipments, the cost for waiting service in ports and the operation cost for maintain service for liner companies. Brouer et al. (2013a) proposed a general small-scale ship schedule recovery problem for multiple ships by providing a decision support tool for disruption, which aims to enhance the reliability of schedule. Song & Xu (2012) conducted a comparison of different service network in order to provide insight into network design problem from the perspective of carbon emissions. Song et al. (2015) presented the first and only one research that consider the three perspectives jointly. They formulated the problem of single service route with optimal speed of each ship and indicated the problem needs extensions with consideration of shipping networks.
In order to better optimize the containership routing problem, this paper would extend the work of Karlaftis et al. (2009) by adding reliability and sustainability objectives in a feeder network. Furthermore, based on the research of Song et al. (2015), this paper would adopt a new solution approach of fuzzy multi-objective programming (Zimmermann, 1978) that can consider the three issues jointly. The research question is formulated as follows:
How to balance the cost, schedule reliability and environmental sustainability of shipping networks jointly in containership routing problem？
For the purpose of solving the problem specifically, there are three sub-questions:
1. How to formulate schedule reliability and environmental sustainability in a mathematical model?
2. How to integrate the three issues in the context of shipping networks? 3. Is the model valid for practice?
integration phase, which would answer the sub-question two and contribute to a solution approach for multi-objectives problem. Consequently, the model derived from the integration phase would be examined with numerical experiments through existing data from previous literatures, which would answer the sub-question three.
The methods of this research are modelling and numerical experiments, which are included in three research phases. Modeling is first conducted to formulate the reliability and sustainability and combine them with cost, which would result in an integrated mathematical model. Computational experiment would provide the opportunity to analyze the trade-off among three objectives by adopting fuzzy multi-objective programming, which can solve multiple objectives problems.
By properly answering the research questions, the contribution of the paper is twofold. First, the theoretical contribution lies in the joint consideration of cost, reliability and sustainability of containership routing problems in complete shipping networks. This would fill the gap that no works have analyzed and achieved the three aspects in a shipping network. Then, the study aims to provide managerial insights by offering more precise suggestions for shipping lines to do operational level decisions in the ship round trips to cope with their own business strategy under multiple criteria as cost, environmental sustainability and schedule reliability.
2. Literature Review
2.1 Liner shipping business
There are three modes of shipping, which are tramp, industrial and liner shipping (Lawrence, 1972). The industrial shipping is widely used for mining and oil companies that control the ships in order to achieve the minimum transportation cost (Branchini et al., 2015). In tramp shipping, the ships do not follow the routes and schedule but received the requests from any ports for a considerable profit (Branchini et al., 2015). In liner shipping, ships follow the predetermined itinerary and schedule to fulfill the transportation demands of customers (Christiansen et al., 2004; Mulder & Dekker, 2014; Karsten et al., 2015). Among the three shipping ways, liner shipping of containers plays an important role in global trades, besides the container shipping is the main section in liner shipping now (Christiansen et al., 2013; Brouer et al., 2013a).
Figure 2.1. Four liner shipping network design categories (Meng et al., 2013)
Figure 2.2. Three decision-making levels (Meng et al., 2013)
2.2 Containership routing problem
In literatures, the containership routing problems can be solved as multiple vehicle routing problems (VRP) due to the similarities between ship routes and vehicle routes (Fagerholt, 2004). The single hub port of the designed feeder service network can be treated as the depot in VRPs (Wang & Meng, 2011). Brouer et al. (2013b) proved that the containership routing problem in liner shipping is a NP-hard problem according to the similarities. For review of ships routing and scheduling problem we refer to Meng et al. (2013).
nonlinear problem, which is minimizing the total transshipment cost and penalties of delay. Other than adopting the VRPs formulations, there are also researches that come up with other mathematical models. Lin & Tsai (2014) developed the ship routing problem with daily frequency, which indicate the ships would provide daily pickup and delivery services to customers. They attempted to minimize the total cost of voyaging cost, fixed cost of ships, idle cost of ships, loading and unloading cost at ports and delay costs. Overall, the studies that cover the cost aspects are of wide scope.
Even though uncertainties are harmful to schedules and it would affect the schedule reliability, these factors seldom addressed in ship schedule (Brouer et al., 2013a; Li et al., 2015). Wang & Meng (2012b) examined the ship routing in tactical-level with the uncertainties of port congestion and stochastic container handling time. Dong & Song (2012) considered the customer demands and stochastic inland transport time uncertainties to solve the container fleet sizing problem, which is to provide support for strategic decisions. Wang & Meng (2012a) formulated a tactical-level ship routes schedule with the uncertainties at sailing speed and port time. Kepaptsoglou et al. (2015) presented a containership routing problem with simultaneous pick-ups and deliveries and time deadlines while consider the weather condition as one influence factor to the transportation cost, which aims to increase the reliability of the schedule in the planning level.
Other than consider the reliability in schedule aspects, few literatures also come up with certain strategies to recovery the unreliable schedules. Brouer et al. (2013a) first proposed a general small-scale ship schedule recovery problem for multiple ships by providing a decision support tool for disruption, which is demonstrated in three recovery modes of speed change, omitting ports and swapping ports actions. They evaluated the recovery actions by measuring the trade-off between the increased bunker cost and the service level to customers and the influence on containers in remaining networks. Related to their work, Li et al. (2015) presented a detailed operation solution for single ship to recover the delay-caused schedules by considering the speeding up, port skipping and port swapping actions separately.
perspectives of cost and carbon emissions. They indicated the optimal speed adjustment is a dynamic process. The studies that focus on reduce the carbon emission and increase the sustainability are making the trade-off between the fuel consumption and sailing speed.
3. Problem description
Mansouri et al. (2015) argued that combining the carbon emission with other cost items has negative impact on later analysis since it ignoring the actual cost of carbon emission and the influence on environment. Therefore, multiple objectives are formulated in this study, which consists of minimizing the total transportation cost, total carbon emission and total number of delayed ports. This study would utilize the containership routing problem on the feeder network, which can enable the joint consideration of cost, reliability and sustainability in shipping network. We consider a feeder network, where a set of ships is based at the hub port (the depot). The shipping lines have determined the booking time at every. However, ships would deviate from the initial schedule due to uncertainties of ports (i.e. different ports have different service time for ships and each port has various demands). So operators can conduct speed adjustment for ships in travel arcs within an allowable range when implementing the schedule. For instance, operators can reduce the speed of ships when there is enough time to arrive the next port and can increase the speed when there is a delay or the possibility to delay. In order to clarify the problem, the main characteristics of the problem are listed below.
1. Start and end of each route
Since this study is formulated in the feeder network, each route would start from the hub port. When the ship cannot continue sailing (e.g. fulfill the request, no available travel time and etc.), the route would finish at the hub port as well.
2. Port requests fulfillment
For each port call, both delivery and pickup requests are considered. This paper would address the simultaneous deliveries and pickups, which require loading and unloading operations in every called port. Delivery containers will be transported from the hub port to feeder ports and pick up containers will be transported from feeder ports to the hub ports. Each ship would visit the port only once.
3. Number of ships and Fleet composition
This paper would study a set of homogeneous ships. The number of deployed ships cannot exceed the maximum available amount.
There are specific booking times for feeder ports, which represent the due date of the demands at the port. Since the ships can either arrive early or late at ports, this paper allows the early arrival and tardiness for demands. And only delay penalties are considered in this study.
5. Transportation capacity
The capacity of ships in this paper is measured in TEU (Twenty-foot Equivalent Unit). 6. Sailing speed
The actual sailing speed of ships during the service routes is the decision variable of the study. The range of sailing speed is from the slow steaming that has the lowest fuel consumption and the planed maximum sailing speed.
7. Container type
Only TEU containers are considered in this study. And the containers are available to loading or unloading once the ship arrives at the port.
8. Carbon emission objective
The carbon emission has positive correlation with the fuel consumptions and the empirical relationship for fuel consumption per nautical mile is given as (Corbett et al., 2009; Fagerholt et al., 2010):
The 0.0036, 0.1015 and 0.8848 are three empirical parameters to calculate the fuel consumptions (Fagerholt et al., 2010). The total carbon emission is calculated as (Corbett et al., 2009):
9. Cost objective
The total cost consists of bunker cost, penalty of delay cost and transportation cost. Therefore, the fuel consumption function would be adopted in the cost objective function as well.
10. Schedule reliability objective
Yang et al. (2013) defined the schedule reliability as:
4. Mathematical model
Let us consider a set of ports and a set of ships . We use the ports to represent the port calls. Each route starts at and ends at the hub port (port 1). The problem is to investigate the optimal routes to serve a set of port calls and the optimal sailing speed in each segment of a route.
All sets, parameters and decision variables are presented below.
N Set of ports, . Port 1 represent the hub port K Set of ships,
Table 4.1.Set
Cargo carrying capacity of ship. Maximum travel time for ship.
The sailing speed of slow steaming. The maximum sailing speed.
Sailing distances from port i to j. EF Carbon emission factor.
First empirical parameter of fuel consumption. Second empirical parameter of fuel consumption. Third empirical parameter of fuel consumption.
Service time at port. The unit fuel price.
The unit delay penalty. The unit transportation cost.
Pick up demands load in port i.
Delivery demands unload in port i.
M The arbitrarily large number. Booking date at port i. /{1}
The load of ship k after leaving the hub port. and integer. The load of ship k after leaving port j. and integer.
Fuel consumption for ship k from port i to j.
Table 4.3. Variables
Binary variable. Set 1 if ship k travels from port i to j; otherwise set 0.
Actual sailing speed of ship k when travel from port i to j.
Binary variable. Set 1 if arrive at port i late. Otherwise, set 0. Actual arrival time of ship k at port i.
Delay or early time when arrive at port i. Delay time when arrive at port i.
Table 4.4.Decision Variables
We formulate the model as follows: Objective function
The objective function (4-1) minimizes the total cost of fuel consumption, total transportation cost and delay penalty cost. Function (4-2) minimizes the amount of carbon emission (Corbett et al., 2009; Fagerholt et al., 2010) and (4-3) minimized the number of delay (Yang et al., 2013). Subject to:
5. Solution approach
Recent studies have shown more interests on the multiple objective optimizations in order to analyze the trade-offs in maritime shipping (Mansouri et al., 2015). By jointly considering the cost, schedule reliability and sustainability in the feeder networks, this paper provides a multiple objective model to address the operational level decisions in the context of the feeder network.
In order to prove that the model can give a result with a set of travel speed and actual visiting sequences with three objectives, a solution approach using fuzzy theory proved that it could give the solution to multiple objectives model (Kumar et al., 2004; Kumar et al., 2006).
5.1 Fuzzy theory and membership function
Bellman & Zadeh (1970) first proposed the fuzzy set and it is defined as:
The represents the membership function of A and of shows the degree of membership to which x belongs to A. And the scope of membership function is a set of non-negative numbers in the interval [0, 1]. A linear membership function is continuously increasing or decreasing over the range of parameter (Shaw et al., 2012).
The fuzzy objective is a subset of X that restricted by the membership function . The linear membership function for the fuzzy objectives is defined as:
and are the minimal and maximal value of objective j: . is the optimum
solution under the restrictions of all constraints. We can get the value of and by
solving the problem as a single objective problem and set the value of and as the
aspiration levels of objective j. Since no tolerance interval of constraint is considered in this paper and all objectives share a same set of constraints, the membership values of the objectives are in the interval [0, 1] and their equation can be calculated as follows:
5.2 Crisp formulation
The solution of fuzzy multi-objective problem is the intersection of all the fuzzy sets of objectives and constraints (Bellman & Zadeh, 1970; Shaw et al., 2012). We would adopt the solution approach of Zimmermann (1978) by transforming the fuzzy objectives into crisp formulation. The fuzzy solution for the J fuzzy objectives is the intersection of the objectives:
In the Eq. (5-3), and represent the membership functions of solutions and objectives, respectively.
Since this research is a minimization problem, the highest membership degree is the optimum solution of the problem according to Eq. (5-2). We define the as the membership value of objective j. Then the fuzzy programing model can be transferred into the crisp formulation as follows (Kumar et al., 2006; Shaw et al., 2012):
Subject to: 5.3 Solution steps
A solution could be the Pareto-optimal to the fuzzy multi-objective problem if there is no objective that achieves the highest membership value (Jiménez & Bilbao, 2009). Therefore, if no objective achieves the same solution as its optimal solution, the solution is the Pareto-optimal solution in this paper. The procedure of fuzzy multi-objective programming applying in this paper, which aims to find the optimal sailing speed and ship deployment, is described as below:
Step 2: Set the aspiration level of the fuzzy goals.
2.1: Select the first objective along with the constraint set of containership routing problem. Solve this problem as a single objective problem and get the minimal and maximal value of the objective function. Treat the 2 values as the aspiration levels of first goal, which are and .
2.2: Repeat Step 2.1 for all the other objectives of problem.
Step 3: Define the membership function of each fuzzy goal in the containership routing problem formulation with the maximum and minimum limits according to Eq. (5-2).
Step 4: Construct the equivalent crisp formulation of the problem according to Eq. 4) to (5-7).
6. Numerical experiments
6.1 Test instances
The instances are generated from the work of Karlaftis et al. (2009). The proposed mathematical model and solution approach are applied for scheduling a vessel fleet of 2 ships from the hub port, which is port A in this research, to a set of 4 feeder ports: B, C, D and E. The maximum and minimum sailing speed in our experiment are 26 knots and 14.1knots (Song et al., 2015). The empirical parameters to calculate the fuel consumption are derived from the work of Fagerholt et al. (2010). We will adopt the value of 3170(kg/ton-fuel) as the carbon emission factor in this study (Corbett et al., 2009; Song et al., 2015). Table 6.1 summarizes the value of parameters. Table 6.2 and 6.3 summarize the distances between hub and feeder ports and delivery and pickup demands for each port respectively. We set the classes for fuel price and penalty cost as Song et al. (2015) and Qi & Song (2012) to reveal the impact of different cost on multi-objectives in this research. Table 6.4 summarizes the total six instances applied in this study. The optimization model is solved in LINGO 11.0. All numerical experiments are performed on an i5-4210U CPU with 1.70 GHz clock and 4 GB RAM.
Parameters Values
Cargo carrying capacity of ship. Q=100 TEU
Maximum travel time for ship. T=30h
The sailing speed of slow steaming. =14.1knots
The maximum sailing speed. =26knots
First empirical parameter of fuel consumption. =0.0036 Second empirical parameter of fuel consumption. =0.1015 Third empirical parameter of fuel consumption. =0.8848
Carbon emission factor. EF=3170(kg/kg-fuel)
The unit fuel price. =€300, €600/ton
The unit delay penalty. =€100, €1000, €10000/port
The unit transportation cost. =€5/h
A B C D E A 158 174 124 97 B 158 38 108 100 C 174 38 142 109 D 124 108 142 88 E 97 100 109 88
Table 6.2. Distances between ports in miles (port A is the hub port)
Destination Delivery demand (TEU) Pickup demand (TEU) Service time (h) Booking time (h) B 7 12 1.8 6.1 C 57 70 6.1 9.5 D 1 5 1.3 5 E 43 10 3.1 9.5
Table 6.3. Delivery and pickup demands, service time and booking time per port
Instances 1 2 3 4 5 6
Ports (#) 5 5 5 5 5 5
Ships (#) 2 2 2 2 2 2
Unit fuel price (€) 300 600 300 600 300 600
Unit delay penalty (€) 100 100 1000 1000 10000 10000
Table 6.4. Test instances descriptions 6.2 Results and analysis
6.2.1 Computational time and size limit
Table 6.5 shows the computational time for the aspiration levels, which are and , of
Objectives Sustainability Reliability Cost
Computational time (sec) 343 54 803 ~ 1133
Table 6.5. Aspiration levels-computational time
Instances 1 2 3 4 5 6
Computational time (sec) 491 533 371 511 414 457
Table 6.6. Crisp Formulation-computational time
6.2.2 The effect of crisp formulation
Consider the fuel cost to be €300 per ton and delay penalty to be €1000 per hour, the results are given in Table 6.7 and membership values of three objectives under different solutions are compared in Figure 6.1. The “Best cost” column presents the three objectives under the best cost solution, the “Best sustainability” column gives the three objectives under the best carbon emission solution, the “Best reliability” column proposes the three objectives under the best schedule reliability solution and the “Crisp formulation” column gives the three objectives with the restriction of crisp formulation.
Total cost (€) Carbon emission(kg) Delayed port calls (#) Total delay time (h) Membership value Route for ship 1 Route for ship 2
Best cost 41103 418771 4 12.63 1.98 A-B-C-A A-D-E-A
Best sustainability 55489 394086 4 17.97 1.98 A-B-C-A A-D-E-A
Best reliability 1033101 1090006 0 0 2.26 A-B-C-A A-D-E-A
Crisp formulation 73148 658316 1 10.64 2.46 A-B-C-A A-E-D-A
Table 6.7. Comparison of crisp formulation and single objective
Figure 6.1. Membership value comparison
It can be observed from Table 6.7 that the sailing itinerary generated from three single objective models is the same. Only the crisp formulation gives a different route. For the “Best cost” and “Best sustainability” solution, they have a bad schedule reliability of 0% since all the port calls are delayed. Although the “Best reliability” solution achieves the 100% schedule reliability, its cost and carbon emissions are the highest compared to other solutions. Besides, according to Figure 6.1, the “Best cost”, “Best sustainability” and “Best reliability” solutions have the highest membership value of 1.00 in total cost, carbon emission and delayed port calls respectively. But they all have one or two other objectives that have low membership value. Therefore it is obvious that only consider one objective is not sufficient to meet the decision maker’s goal since other critical criteria will suffer.
Figure 6.2. Speed adjustment of different model
It can be concluded that the performances of three objectives are balanced with the help of crisp formulation, which enables more integrated suggestions for decision makers when there are more criteria considered.
6.2.3 Solutions with fuel price and penalty variance
Both fuel price and delay penalty are critical to the operational cost of shipping. However, the fuel unit cost varies greatly in recent years and we do not have accurate data about the delay penalties. Therefore, multiple scenarios settings are generated, which enable us to have sensitive analysis of results in respect of fuel price and delay penalties. The categories of fuel price and penalty cost are inspired by Song et al. (2015) and Qi & Song (2012). The solutions of six instances are listed in Table 6.8. Figure 6.3 and Figure 6.4 presents the membership values in different scenario settings.
Instances Fuel price (€/ton) Penalty cost (€/h) Total cost (€) Carbon emission (kg) Delayed port calls (#) Route for ship 1 Route for ship 2 1 300 100 48680 491036 2 A-C-B-A A-E-D-A 2 600 100 125735 657670 1 A-B-C-A A-E-D-A 3 300 1000 73148 658316 1 A-B-C-A A-E-D-A 4 600 1000 135246 660640 1 A-B-C-A A-E-D-A 5 300 10000 93928 990653 0 A-B-C-A A-D-E-A 6 600 10000 187670 990593 0 A-B-C-A A-D-E-A
Table 6.8. Impacts of decision variables on three objectives
13.00 18.00 23.00 1 2 3 4 Sp e e d o f sh ip s Visiting Sequences
Speed adjustment Ship 2
Best Cost Best Sustainability Best Reliability Crisp Formulation 13.00 18.00 23.00 1 2 3 4 Sp e e d o f sh ip s Visiting Sequences
Speed adjustment Ship 1
Figure 6.3. Instances membership value comparison with fuel price €300
Figure 6.4. Instances membership value comparison with fuel price €600
A couple of interesting patterns can be concluded. First, multi-objectives are rather insensitive to the fuel prices since Figure 6.3 and Figure 6.4 have similar curves and variation tendencies. Second, all the solutions are fairly sensitive to the delay penalty. With the increase in penalty cost, the total delay times and delayed port calls are decrease (see Figure 6.5).
Figure 6.5. Delay time comparison
Cost Sustainability Reliability
delay €100 0.94 0.94 0.50 delay €1000 0.89 0.83 0.75 delay €10000 0.99 0.61 1.00 0.50 0.60 0.70 0.80 0.90 1.00 Val u e
Membership value with fuel price €300
Cost Sustainabilit y Reliability delay €100 0.83 0.83 0.75 delay €1000 0.86 0.83 0.75 delay €10000 0.97 0.66 1.00 0.50 0.60 0.70 0.80 0.90 1.00 Val u e
Membership value with fuel price € 600
0 5 10 15 20 25 100(€/h) 1000(€/h) 10000(€/h) hour Penalty cost
Total delay time
Third, under the high delay penalty, the cost and reliability objectives are dominated by the delays simultaneously. This can be explained as follows. When delay penalty is high, the delay cost will play an important role in total cost. Although increasing the speed of ships will result in higher fuel costs, there will be fewer delays. Therefore, the reduction in delay penalties can compensate for the increase in fuel consumption costs. For example, the instance 5 has a high delay penalty of €10000/h. It has a high fuel consumption cost of €93753, which is almost 99% of the total cost. If the instance 5 takes the schedule of instance 1, which has a lower delay penalty of €100/h, it will contribute to grow in delay costs for €200000. In this case, reducing delays can benefit both total cost and reliability. Fourth, under low delay penalty, the cost and sustainability objectives are dominated by the fuel consumptions simultaneously. For example, the instance 1 has a low penalty of €100/h and 20 hours delay time. The delay penalty only stands for 4.1% of total cost and the fuel consumption is the major component of total cost. If the instance 1 takes the schedule of instance 5, which have higher delay penalties and actual sailing speed, the delay cost will be reduced to 0 but the fuel consumption cost will increase to €93753. Since fuel consumption is critical to both cost and carbon emissions, keep slow steaming is beneficial to both cost and sustainability under this circumstance.
Figure 6.6 presents the speed adjustment of ships in instance 1 and 5, which is in line with our analysis. For instance 5, ships keep higher speed before the last port call (sequence 3 in Figure 6.6) to avoid delay penalties, which result in low delay cost and high reliability. For instance 1, ships are sailing with relative low speed to prevent high fuel consumptions, which contribute to low fuel cost and carbon emission.
Figure 6.6. Speed adjustment of ships
13.00 15.00 17.00 19.00 21.00 23.00 25.00 27.00 1 2 3 4 Visiting Sequences
Speed adjustment Ship 2
Instance 1 Instance 5 13.00 15.00 17.00 19.00 21.00 23.00 25.00 27.00 1 2 3 4 Visiting Sequences
Speed adjustment Ship 1
7. Conclusion
In this study, a containership routing problem with consideration of schedule reliability and sustainability is formulated and solved, which is adapted from the feeder network design problem. A fleet of ships starts from the hub port in order to fulfil the delivery and pick up demands in feeder ports. The specific booking times are considered to compute the schedule reliability, which is defined as the proportion of port calls that the ships arrive on time. Besides, the fuel consumption equation is applied to calculate both the fuel consumption cost and the carbon emissions, which address the cost and environmental sustainability simultaneously. The contribution of this paper is twofold. From the theoretical perspective, the proposed model formulates the schedule reliability, environmental sustainability and total cost in a feeder network jointly. The practical contribution is the adoption of fuzzy multi-objectives programming, which enables the measurement and balance of the three objectives at the same time. This can serve as a decision support system to decision makers when there is more than one criterion that requires to be considered, which can give more precise suggestions.
The model formulated in this work integrates three critical issues of shipping lines, which are schedule reliability, environmental sustainability and total cost. A solution approach for fuzzy multi-objective programming that called crisp formulation is used to solve the multi-objectives model. According to the result, it can be concluded that the crisp formulation can balance multiple objectives and generate reasonable solutions. Besides, the result also shows that the objectives are sensitive to the delay penalty and rather insensitive to the fuel prices.
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