The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas

Alexandria Engineering Journal (2017) xxx, xxx–xxx

H O S T E D BY

Alexandria University

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ORIGINAL ARTICLE

The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas M.A. Dulebenets Department of Civil & Environmental Engineering, Florida A&M University-Florida State University, 2525 Pottsdamer Street, Building A, Suite A124, Tallahassee, FL 32310-6046, USA Received 13 May 2016; accepted 6 November 2016

KEYWORDS Marine transportation; Energy efficiency; Green vessel scheduling; Emission Control Areas; Emission limitations; Cargo transit time requirements

Abstract Increasing volumes of the international seaborne trade and new regulations on vessel emissions require liner shipping companies to enhance efficiency of their operations to remain competitive and in the meantime comply with the established restrictions on emissions that are produced by vessels. This paper presents a novel mixed integer nonlinear mathematical programming model for the green vessel scheduling problem with transit time requirements in a liner shipping route with ‘‘Emission Control Areas”, which not only enforces constraints on the quantity of emissions that are produced by vessels within ‘‘Emission Control Areas”, but also captures the cargo transit time requirements. The model’s objective aims to minimize the overall route service cost. A number of linearization techniques are proposed for linearizing the original nonlinear problem. The dynamic secant approximation procedure is used to solve the linearized problem. A set of numerical experiments are performed to evaluate the efficiency of the proposed solution approach and assess the effect of introducing both emission constraints and cargo transit time requirements on design of vessel schedules.
2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Carbon dioxide emissions from maritime transportation constitute approximately 2.2% of the overall world anthropogenic carbon dioxide emissions [1]. In order to decrease pollution levels the International Maritime Organization (IMO) enforced limitations on sulphur emissions (SOx) in the North E-mail addresses: [email protected], maxim.dulebenets@famu. edu Peer review under responsibility of Faculty of Engineering, Alexandria University.

Sea, the Baltic Sea, and the English Channel and designated those areas as ‘‘Sulphur Emission Control Areas” in 2008 [2]. Furthermore, IMO designated the United States (U.S.) – Canadian coastal zone as ‘‘Emission Control Area” (ECA) in 2010 with the main objective to reduce SOx, nitrogen oxides (NOx), and particulate matter emissions [2]. Restrictions on SOx and NOx emissions became effective in the U.S. Virgin Islands and Puerto Rico in 2014. Also there exist projections that IMO will introduce ECAs in Japan, Norway, and Mediterranean [3]. In 2011 IMO added a new chapter that contains amendments to MARPOL Annex VI, which was entitled as ‘‘Regulations on energy efficiency for ships”. The purpose of

http://dx.doi.org/10.1016/j.aej.2016.11.008 1110-0168
2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

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M.A. Dulebenets

the new amendments was to introduce mandatory measures against producing the greenhouse gas emissions [4], which are mainly represented by carbon dioxide (CO2), nitrous oxide (N2O), and methane (CH4). According to the regulation, vessels are obligated to obtain a specific ‘‘Energy Efficiency Design Index” (which is primarily dependent on the technical characteristics of vessels and the type of fuel used), while liner shipping companies must develop a ‘‘Ship Energy Efficiency Management Plan” [4]. Furthermore, the European Union aims to achieve a relatively challenging goal, which consists in reducing the greenhouse gas emissions by sixty per cent by 2050 as opposed to the greenhouse gas emissions that were produced in 2010 [5]. Scheduling of vessels in liner shipping received a substantial attention from the research community in the past years [6–14]; however, only a limited number of studies accounted for environmental considerations [15]. Qi and Song [16] focused on the vessel scheduling problem, capturing the impact of uncertainty in port times. The study did not explicitly model emissions, produced by vessels, but mentioned that emissions could be reduced by optimizing the schedule of vessels. Kontovas [17] presented a generic integer programming model for the green vessel routing and scheduling problem and elaborated on some aspects of the problem. The paper listed a number of different alternatives that could be used for modelling vessel emissions. Dulebenets et al. [18] proposed a novel mixed integer nonlinear programming model for the green vessel scheduling problem, considering constraints on the emissions produced by vessels at each voyage leg, belonging to the given liner shipping route, without modelling ECAs. Fagerholt and Psaraftis [19] and Fagerholt et al. [20] focused on the problem of routing vessels and sailing speed optimization within ECAs without explicitly modelling the vessel service at ports of the considered liner shipping route. Song et al. [21] developed a stochastic multiobjective model for the vessel scheduling problem with uncertain port times. The model aimed to minimize three objectives: (1) the total annual vessel operational costs; (2) the average vessel schedule unreliability; and (3) the total annual carbon dioxide emissions produced by all the vessels, which served a given liner shipping route. Considering a rapidly increasing attention of the society to the environmental concerns, liner shipping companies must focus on implementation of new strategies in development of their vessel schedules and improve the environmental sustainability. In order to reduce the emissions produced by oceangoing vessels liner shipping companies will need to decrease their sailing speed, as the vessel sailing speed is proportional to the amount of emissions produced [2,17]. In the meantime, liner shipping companies have to ensure that transit time requirements are met at each voyage leg, belonging to the given liner shipping route [22]. Depending on the type of cargo, transported at a given voyage leg, the required transit time will vary. For example, perishable and high value goods will require shorter transit time as compared to clothing. The contribution of this study to the state of the art is trifold: (1) A novel mixed integer nonlinear programming model for the green vessel scheduling problem in a liner shipping route with ECAs, which ensures that emission constraints are not violated within ECAs.

(2) Capturing changes in the vessel schedule due to switching from Heavy Fuel Oil (HFO) to Marine Gas Oil (MGO) with low sulphur content within ECAs. (3) Consideration of the cargo transit time requirements in green vessel scheduling. The new mathematical model will allow liner shipping companies not only to improve the environmental sustainability and energy efficiency by reducing the amount of emissions produced within ECAs, but also to tackle important operational aspects (i.e., consideration of the cargo transit time requirements). The rest of the paper is organized in the following manner. The second section provides a description of the problem, which is studied in this paper. The third section presents the mathematical model, while the fourth section discusses the solution methodology adopted in this study. The fifth section demonstrates a set of computational experiments that were performed in this study, while the last section summarizes findings and discusses potential future research avenues. 2. Problem description This study focuses on modelling a typical liner shipping route that consists of I ¼ f1; . . . ; ng ports of call. The sequence of visited ports (which is generally referred to as a ‘‘port rotation” in the liner shipping literature) is assumed to be known. Construction of a port rotation is usually performed by the liner shipping company at the strategic level [23]. Each port of call is assumed to be visited once; however, the proposed methodology can be also implemented for the cases, when ports of call are visited multiple times in a given liner shipping route. In the latter case (i.e., when a given port of call is visited multiple times), an additional node will be introduced to the graph that represents the port rotation in order to capture every additional visit of a vessel to the same port. For example, the liner shipping route, which is presented in Fig. 1A, includes a total of 4 ports. We observe that ports 1 and 2 are visited twice; therefore, two additional nodes 10 and 20 must be introduced to the graph, and the total number of ports that have to be visited by vessels will become jIj ¼ 4 þ 2 ¼ 6 (see Fig. 1B). A vessel, serving a given liner shipping route, is assumed to sail between two consequent ports i and i þ 1 along voyage leg i. Certain legs of a liner shipping route are assumed to pass through ECAs. The subset of voyage legs, which pass through ECAs, will be referred to as I , while the rest of voyage legs (which pass outside ECAs) will belong to subset I0 . Note that I0 [ I ¼ I; I0 \ I ¼ Ø. A certain frequency of service should be provided by the liner shipping company at each port of call (generally weekly or biweekly service is negotiated between liner shipping companies and marine container terminal operators in practice). The marine container terminal operator establishes a specific arrival time window – TW [twei – start of TW at port i; twli – end of TW at port i ] at each port, during which a vessel is expected to arrive at the given port of the liner shipping route. Duration of a TW generally varies from one to three days depending on the port [24]. Once a given vessel arrives at the port, its service is assumed to start upon the arrival. If a vessel arrives at port i before the beginning of a TW, it will be required to wait for service at a dedicated area.

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

The green vessel scheduling problem with transit time requirements

3 v – average daily vessel sailing speed (knots); qðv Þ – daily bunker consumption by vessel when sailing at the designed speed (tons of fuel/day); v – design vessel sailing speed (knots); a; c – bunker consumption coefficients.

Figure 1

Schematic representation of a liner shipping route.

A monetary penalty (that is USD/h) will be incurred by the liner shipping company, if a vessel arrives at the given port of call after the end of TW [13,14]. Weekly container demand (that is TEUs) at each port of the liner shipping route is assumed to be known [23]. 2.1. Service of vessels at ports The liner shipping company is assumed to have a set of contractual agreements with marine container terminal operators, providing service of vessels at ports of call of the given liner shipping route. Based on those agreements, the marine container terminal operator at each port of call is able to offer a set of handling rates Si ¼ f1; . . . ; hi g 8i 2 I to the liner shipping company. Each handling rate has a specific handling productivity dis 8i 2 I; s 2 Si , which is measured in TEUs per hour. The vessel handling time pis 8i 2 I; s 2 Si (that is hours) is calculated based on the handling productivity, negotiated between the liner shipping company and the marine container terminal operator at the given port of call, and the container demand at the given port of call. Note that selection of a handling rate with a higher handling productivity at the given port of call will allow the liner shipping company reducing the vessel handling time, but in the meantime will incur additional port handling costs that have to be paid to the marine container terminal operator. 2.2. Bunker consumption The given liner shipping route is assumed to be served with a homogeneous vessel fleet. The term ‘‘homogeneous vessel fleet” is applied to those fleets that have vessels with the identical or similar technical characteristics. The latter practice (i.e., deployment of homogeneous vessel fleets for service of liner shipping routes) has been widely used in the published to date vessel scheduling literature [8,9,11,22,25]. The bunker consumption is generally assumed to be dependent on the vessel sailing speed and can be estimated based on the following power law relationship [25,26]:
a v qð vÞ ¼ qðv Þ  ¼ cð vÞa ð1Þ v where qðvÞ – daily bunker consumption by vessel (tons of fuel/day);

Technically, the values of bunker consumption coefficients a and c must be determined based on a comprehensive regression analysis, which has to be performed based on the data collected for every vessel in the fleet serving the given liner shipping route [25,26]. This study will rely on the most common values of the bunker consumption coefficients, which were identified from the published to date liner shipping literature [2,25]: a ¼ 3 and c ¼ 0:012. Once the sailing speed of a vessel between consequent ports of the liner shipping route is selected by the liner shipping company, it is assumed to remain constant throughout the voyage between those ports of call. The factors that may cause fluctuations of the vessel sailing speed during the voyage (e.g., weather, experience of the vessel crew, height of waves, and speed of wind) are not captured in this study. The bunker consumption by auxiliary vessel engines does not change substantially throughout the voyage, and the associated cost will be added to the weekly vessel operational cost. The bunker consumption fðvi Þ can be further calculated per nautical mile at voyage leg i using the following equation:  t 1 li 1 cðvi Þa1 i fðvi Þ ¼ qðvi Þ 8i 2 I ð2Þ ¼ cðvi Þa ¼ 24 li 24vi li 24 where li – length of voyage leg i, which connects ports i and i þ 1 (nmi); ti – sailing time between ports i and i þ 1 (h).

2.3. Inventory cost The container inventory cost is another important route service cost component, which has to be accounted for in design of vessel schedules by the liner shipping company. Generally, the overall container inventory cost is assumed to be dependent on the total transit time of containers, transported by vessels along the voyage legs of the given liner shipping route, and can be calculated based on the following relationship [11,18]: X IC ¼ l ti NCTi ð3Þ i2I

where IC – overall inventory cost (USD); l – unit inventory cost (USD per TEU per h); NCT i – quantity of containers transported at leg i (TEUs).

2.4. Emission estimation This study assumes that in order to improve the environmental sustainability of vessel schedules the liner shipping company has to comply with certain restrictions on the quantity of emissions that are produced at voyage legs, which pass through ECAs. Vessel technical characteristics substantially affect the

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

4

M.A. Dulebenets

quantity of emissions produced. This study assumes that the liner shipping company will be using vessels, which have two-stroke marine diesel engines, to serve ports of the given liner shipping route. The quantity of carbon dioxide emissions at voyage leg i (that is tons) can be calculated based on the total bunker consumption at that voyage leg and carbon dioxide emission factor FCO2 [2,17,27]: CO2i ¼ FCO2 fðvi Þli

8i 2 I

ð4Þ

The quantity of sulphur dioxide emissions, produced by oceangoing vessels, is primarily affected with the amount of sulphur in the fuel. Sulphur dioxide emissions at voyage leg i (that is tons) can be estimated based on the percentage of sulphur in the fuel (Pi 8i 2 I, %) and factor 0.02, which can be explained by the nature of a chemical reaction of oxygen with sulphur and underlines that only 2% of sulphur will be able to react with oxygen and produce sulphur dioxide emissions [17]: SO2i ¼ 0:02Pi fðvi Þli

8i 2 I

ð5Þ

The quantity of nitrogen oxide emissions at voyage leg i (that is tons) can be calculated using the total bunker consumption at that voyage leg and nitrogen oxide emission factor FNOx [1,17]: NOxi ¼ FNOx fðvi Þli

8i 2 I

ð6Þ

Based on the available literature this study will use the following emission factors [1,2,4,17,27]: FCO2 ¼ 3:082 tons of CO2 per ton of the fuel and FNOx ¼ 0:051 tons of NOx per ton of the fuel [17]. According to the existing IMO regulation [4], it is assumed that the vessels will use MGO with sulphur content Pi ¼ 0:1% 8i 2 I at voyage legs that pass through ECAs. At the rest of voyage legs the vessels will use HFO with sulphur content Pi ¼ 3:5% 8i 2 I0 . 2.5. Transit time requirements This study assumes that each vessel transports a set of different cargo types J ¼ f1; . . . ; wg between ports, belonging to a given liner shipping route. The required transit time at each leg will vary depending on the type of cargo transported. For example, perishable and high value goods will require shorter transit time as compared to clothing. Taking into account the emission limitations on voyage legs of a liner shipping route that pass through ECAs, the liner shipping company may not be able to meet the transit time requirements for certain cargo types. If the transit time requirement for cargo type j at voyage leg i is violated, a penalty ptij 8i 2 I; j 2 J will be imposed to the liner shipping company. 2.6. Decisions The problem studied herein belongs to the group of tactical level problems and will be further referred to as the green vessel scheduling problem with transit time requirements.

The following major decisions have to be made by the liner shipping company in this problem: (1) The quantity of vessels to be deployed at the given liner shipping route in order to provide the agreed service frequency at each port that has to be visited (assumed to be one week). (2) The sailing speed between consequent ports of call for vessels, which serve the given liner shipping route. (3) The handling rate at each port of the given liner shipping route. (4) The waiting time at each port of the given liner shipping route. (5) The vessel late arrival hours at each port of the given liner shipping route. There is a close relationship between the aforementioned decisions that must be made by the liner shipping company. The liner shipping company must take into consideration the limit on the quantity of vessels (q 6 qmax ), which are allocated for service of the given liner shipping route. Furthermore, lower and upper bounds on the vessel sailing speeds (vmin 6 vi 6 vmax 8i 2 I) must be accounted for in design of the vessel schedule. The minimum sailing speed vmin is typically set to decrease wear of the vessel’s engine [22]. The maximum sailing speed vmax is primarily affected with capacity of the vessel’s engine [2]. In order to decrease the total bunker consumption and associated vessel emissions, the liner shipping company may select lower vessel sailing speeds at voyage legs of the given liner shipping route (i.e., use the concept of ‘‘slow steaming”). The latter will further cause an increasing total container transit time at voyage legs of the given liner shipping route and will increase the quantity of vessels required to ensure that the weekly service is guaranteed at each port of call, belonging to the given port rotation. Moreover, reduction in vessel sailing speeds may lead to violation of the transit time requirements for certain cargo types. Availability of multiple handling rates at ports of the given liner shipping route provides more flexibility to the liner shipping company in terms of selection of port handling and vessel sailing times (for example, if the liner shipping company requests a handling rate that has a higher productivity at a given port of the liner shipping route, the vessel handling time will decrease, which will further allow selection of a lower speed when sailing to the next port of the liner shipping route). Furthermore, decision on the sailing speed at each voyage leg of a liner shipping route will be affected with the unit inventory costs imposed and the established restrictions on the emissions produced within ECAs. 3. Mathematical model This section presents notations, used throughout the paper, and the mathematical model for the green vessel scheduling problem with transit time requirements, which not only accounts for emissions produced at voyage legs of the given liner shipping route, but also captures the transit time requirements for each cargo type carried by vessels.

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

The green vessel scheduling problem with transit time requirements 3.1. Notations Sets I ¼ f1; . . . ; ng J ¼ f1; . . . ; wg Si ¼f1;...;hi g8i2I Decision variables vi 8i 2 I xis 8i 2 I; s 2 Si

5

3.2. Green vessel scheduling problem with transit time requirements set of ports in the port rotation set of cargo types set of handling rates available to the liner shipping company at port i vessel sailing speed at voyage leg i (knots) =1 if handling rate s is selected by the liner shipping company at port i (=0 otherwise)

Auxiliary variables q quantity of vessels assigned to the given liner shipping route (vessels) tai 8i 2 I time of vessel arrival at port i (h) time of vessel departure from port i (h) tdi 8i 2 I vessel waiting time at port i (h) wti 8i 2 I ti 8i 2 I sailing time of a vessel at voyage leg i (h) fðvi Þ 8i 2 I total bunker consumption by a vessel at voyage leg i (tons of fuel/nmi) hours of vessel late arrival at port i (h) lti 8i 2 I CO2i 8i 2 I quantity of carbon dioxide emissions produced by a vessel at voyage leg i (tons) SO2i 8i 2 I quantity of sulphur dioxide emissions produced by a vessel at voyage leg i (tons) NOxi 8i 2 I quantity of nitrogen oxide emissions produced by a vessel at voyage leg i (tons) rij 8i 2 I; j 2 J =1 if the transit time requirement is violated for cargo type j at voyage leg i (=0 otherwise) Parameters bi unit bunker cost at voyage leg i (USD/ton) weekly operational cost of a vessel (USD/week) cOC penalty for delayed arrival of a vessel at port i cLT 8i 2 I i (USD/h) l unit inventory cost (USD per TEU per h) length of voyage leg i (nmi) li 8i 2 I NCTi 8i 2 I quantity of containers transported at voyage leg i (TEUs) minimum sailing speed of a vessel (knots) vmin maximum sailing speed of a vessel (knots) vmax qmax maximum quantity of vessels that can be assigned for service of the given liner shipping route (vessels) twei 8i 2 I start of TW at port i (h) end of TW at port i (h) twli 8i 2 I tcis 8i 2 I; s 2 Si handling cost of a vessel at port i under handling rate s (USD) pis 8i 2 I; s 2 Si handling time of a vessel at port i under handling rate s (h) LCO2i 8i 2 I limitation on quantity of carbon dioxide emissions at voyage leg i (tons) LSO2i 8i 2 I limitation on quantity of sulphur dioxide emissions at voyage leg i (tons) LNOxi 8i 2 I limitation on quantity of nitrogen oxide emissions at voyage leg i (tons) FCO2 carbon dioxide emission factor (tons of carbon dioxide/ton of fuel) FNOx nitrogen oxide emission factor (tons of nitrogen oxide/ton of fuel) Pi 8i 2 I percentage of sulphur in the fuel used at voyage leg i (%) tmax 8i 2 I; j 2 J required transit time for cargo type j at voyage ij leg i (h) ptij "i 2 I, j 2 J penalty for violation of the transit time requirement for cargo type j at voyage leg i (USD) M1 large positive number

The mixed integer nonlinear green vessel scheduling problem GVSPT that accounts for the restrictions on emissions produced at voyage legs, which pass through ECAs, and the transit time requirements for different cargo types can be formulated as follows. GVSPT " X XX X min cOC q þ bi li fðvi Þ þ tcis xis þ cLT i lti i2I

i2I s2Si

X XX þl ti NCTi þ ptij rij i2I

#

i2I

ð7Þ

i2I j2J

Subject to : X xis ¼ 1 8i 2 I s2S

ð8Þ

i

li 8i 2 I vi

ti ¼

ð9Þ

tai P twei 8i 2 I X ðpis xis Þ þ wti þ ti P tweiþ1 8i 2 I; i < jIj tai þ

ð10Þ ð11Þ

s2Si

tai þ

X ðpis xis Þ þ wti þ ti  168q P twe1 8i 2 I;i ¼ jIj s2Si

tdi ¼ tai þ

X ðpis xis Þ þ wti 8i 2 I

ð12Þ ð13Þ

s2Si

lti P tai  twli 8i 2 I ¼ tdi þ ti

taiþ1

ð14Þ

8i 2 I; i < jIj

ð15Þ

8i 2 I; i ¼ jIj X XX X 168q P ti þ ðpis xis Þ þ wti

ta1

¼ tdi þ ti  168q i2I

i2I s2Si

v

ð17Þ

i2I

ð18Þ

q 6 qmax min

ð16Þ

6 vi 6 v

max

8i 2 I

ð19Þ

CO2i ¼ FCO2 fðvi Þli 8i 2 I

ð20Þ

SO2i ¼ 0:02Pi fðvi Þli 8i 2 I

ð21Þ

NOxi ¼ FNOx fðvi Þli 8i 2 I

ð22Þ

CO2i 6 LCO2i 8i 2 I SO2i 6 LSO2i 8i 2 I





ð23Þ ð24Þ

NOxi 6 LNOxi 8i 2 I

ð25Þ

ti  M1 rij 6 tmax ij

ð26Þ

8i 2 I;j 2 J

xis ; rij 2 f0; 1g 8i 2 I; j 2 J; s 2 Si

ð27Þ

;NCTi 2 N 8i 2 I

ð28Þ

max

q; q

min ; vi ;tai ; tdi ;wti ; ti ; fðvi Þ; lti ; CO2i SO2i ; NOxi ; bi ; cOC ;cLT i ; l; li ; v

vmax ; twei ;twli ; tcis ;pis ;LCO2i ;LSO2i ; þ LNOxi ;FCO2 ; FNOx ; Pi ; tmax 8i 2 I; j 2 J; s 2 Si ð29Þ ij ; ptij ; M1 2 R

In GVSPT the liner shipping company is aiming to minimize the overall route service cost (7), which consists of 6 components: (1) overall vessel weekly operational cost, (2) overall bunker consumption cost, (3) overall port handling cost, (4) overall late arrival penalty, (5) overall inventory cost, and (6)

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

6 overall penalty due to violation of the cargo transit time requirements. Note that the unit bunker cost bi differs by a voyage leg. A more expensive MGO with a low sulphur content has to be used by vessels at voyage legs that pass through ECAs (bi ¼ bMGO 8i 2 I , where bMGO – the unit MGO cost), while HFO with a regular sulphur content has to be used by vessels at the rest of the voyage legs (bi ¼ bHFO 8i 2 I0 , where bHFO – the unit HFO cost). Constraint set (8) indicates that the liner shipping company negotiates only one handling rate with the marine container terminal operator at each port of the given liner shipping route. Constraint set (9) computes the sailing time of a given vessel between consequent ports i and i þ 1. Constraint set (10) ensures that service of a vessel at port i cannot begin before the start of TW. Constraint sets (11) and (12) estimate the waiting time of a given vessel at port i, which is required to ensure a feasible arrival at the next port of the given liner shipping route. Constraint set (13) estimates the departure time of a given vessel from port i. Constraint set (14) computes the late arrival hours of a given vessel at port i. Constraint sets (15) and (16) calculate the arrival time of a given vessel at the next port of the given liner shipping route. Constraint set (17) ensures that service should be provided by the liner shipping company at each port of call of the port rotation with a weekly frequency (168 is a numerical value, which denotes the total number of hours in a week). The right-hand-side of the inequality provides an estimation of the total vessel turnaround time at the given liner shipping route, which is represented by the following three components: (a) the total sailing time; (b) the total port handling time; and (c) the total port waiting time. Constraint set (18) ensures that the total quantity of vessels to be deployed for service of the given liner shipping route will not exceed the quantity of available vessels. Constraint set (19) indicates that sailing speed of a given vessel at voyage leg i should be within the established bounds. Constraint set (20) computes the quantity of carbon dioxide emissions at voyage leg i. Constraint set (21) calculates the quantity of sulphur dioxide emissions at voyage leg i. Constraint set (22) estimates the quantity of nitrogen oxide emissions at voyage leg i. Constraint set (23) indicates that the quantity of carbon dioxide emissions at voyage leg i cannot exceed the established restrictions within ECAs. Constraint set (24) enforces that the quantity of sulphur dioxide emissions at voyage leg i cannot exceed the established restrictions within ECAs. Constraint set (25) indicates that the quantity of nitrogen oxide emissions at voyage leg i cannot exceed the established restrictions within ECAs. Constraint set (26) determines whether the transit time requirement for cargo type j is violated at voyage leg i. Constraint sets (27)–(29) show the nature of parameters and variables for GVSPT mathematical model. 4. Solution approach GVSPT is a nonlinear mathematical model due to (1) objective function; and (2) constraint sets (9), (20)–(22). This section discusses how GVSPT can be further reformulated as a liner problem. Replacing vessel sailing speed vi with its reciprocal yi ¼ v1i will further linearize constraint set (9). Denote GðyÞ as the bunker consumption function, which is calculated using vessel sailing speed reciprocal y. The nonlinear bunker consumption function GðyÞ may be further linearized using its piecewise linear secant approximating function Gm ðyÞ, where

M.A. Dulebenets m – is the quantity of linear segments used in the piecewise secant approximating function [22]. Fig. 2 illustrates several piecewise linear secant approximations that have various quantities of linear segments (m ¼ 1; 3; 5; 10) for the nonlinear 2

bunker consumption function GðyÞ ¼ 0:012ðyÞ . The vessel sail24 ing speed lower and upper bounds were set to vmin ¼ 15 knots to vmax ¼ 25 knots respectively (i.e., 0:04 6 y 6 0:0667). The considered examples of piecewise linear secant approximations demonstrate that increasing quantity of linear segments m enhances the accuracy of approximating function for GðyÞ. Note that the proposed solution methodology (i.e., introduction of the piecewise approximation for linearization of the bunker consumption function) is viable due to convexity of the bunker consumption function. Let K ¼ f1; 2; . . . mg be the set of linear segments in the piecewise linear secant approximating function Gm ðyÞ. Let bik ¼ 1 if linear segment k is chosen for approximating the nonlinear bunker consumption function at voyage leg i (=0 otherwise). Denote stk ; edk ; k 2 K as values of the vessel sailing speed reciprocal at the start and the end of linear segment k (respectively); SLk ; INk ; k 2 K as the slope and the intercept of linear segment (respectively) k; and M2 ; M3 as sufficiently large positive numbers. Then mixed integer nonlinear GVSPT mathematical model can be reformulated as mixed integer linear programming model (that will be referred to as GVSPTL) as follows. GVSPTL ! " X X XX X OC min c q þ bi li Gk ðyi Þ þ tcis xis þ cLT i lti i2I

k2K

XX X ptij rij þl ti NCTi þ i2I

#

i2I s2Si

i2I

ð30Þ

i2I j2J

Subject to: Constraint sets (8), (10)–(18), (23)–(29) X bik ¼ 1 8i 2 I

ð31Þ

k2K

stk bik 6 yi 8i 2 I; k 2 K edk þ M2 ð1  bik Þ P yi 8i 2 I; k 2 K

ð32Þ ð33Þ

Gk ðyi Þ P SLk yi þ INk  M3 ð1  bik Þ 8i 2 I; k 2 K ti ¼ li yi 8i 2 I

ð34Þ ð35Þ

1=vmax 6 yi 6 1=vmin 8i 2 I X CO2i ¼ FCO2 li Gk ðyi Þ 8i 2 I

ð36Þ ð37Þ

k2K

SO2i ¼ 0:02Pi li

X Gk ðyi Þ 8i 2 I

ð38Þ

k2K

X NOxi ¼ FNOx li Gk ðyi Þ 8i 2 I

ð39Þ

k2K

In GVSPTL objective function (30) aims to minimize the overall route service cost. Constraint set (31) ensures that only one linear segment k should be selected by GVSPTL mathematical model for approximating the bunker consumption function at voyage leg i. Constraint sets (32) and (33) determine the range of vessel sailing speed reciprocal values, when linear segment k is selected by GVSPTL mathematical model for approximating the bunker consumption function at voyage leg i. Constraint set (34) computes the bunker consumption value, approximated using the piecewise linear secant function,

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

The green vessel scheduling problem with transit time requirements

Figure 2

7

Linear approximations for the bunker consumption function.

at voyage leg i. Constraint set (35) computes the sailing time of a given vessel between consequent ports i and i þ 1. Constraint set (36) indicates that a sailing speed of a given vessel at voyage leg i should be within the established bounds. Constraint sets (37)-(39) estimate carbon dioxide, sulphur dioxide, and nitrogen oxide emissions (respectively) at voyage leg i. Strict lower bounds for M2 and M3 can be defined as follows: 1 1 M2 ¼ vmin ; M3 ¼ SL1 vmax þ IN1 . Note that parameters M2 and M3 in constraint sets (33) and (34) can be substituted with  1  1 ; SL1 vmax þ IN1 . M ¼ max vmin There are two types of the secant approximation [22]: (1) the static secant approximation, and (2) the dynamic secant approximation. A static secant approximation (SSA) procedure assumes that the number of linear segments in the piecewise approximation remains fixed, and GVSPTL mathematical model must be solved only once. Along with SSA, there exists a dynamic secant approximation (DSA) procedure, which assumes that the number of linear segments in the piecewise

approximation is a variable, and GVSPTL mathematical model must be solved as a result of an iterative process, where the number of segments in the piecewise approximation will be continuously increasing until the desired accuracy of GVSPTL solution is obtained. The main advantage of using SSA against DSA is that GVSPTL mathematical model must be solved only once. However, there is a drawback associated with using the SSA method, specifically: if a large number of segments are used in the piecewise approximation, the required computational time can be substantial. This study will rely on the DSA procedure. The main DSA steps, required to solve GVSPTL mathematical model, are presented in Procedure 1. In step 1 the initial number of segments in the piecewise approximation is set to m ¼ 0, whereas the initial accuracy is set equal to D ¼ 100. Technically, the initial accuracy may be set equal to any value, which is greater than the desired accuracy (i.e., D > Dd ). Then, DSA starts an iterative process, where one linear segment is added to the piecewise secant

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8

M.A. Dulebenets

Figure 3

The French Asia Line 2 route.

approximation in step 3. Then GVSPTL mathematical model is solved after updating the number of segments in step 4. Furthermore, a new vessel schedule VS and its objective function value Z are generated in step 4. In step 5 function EstObjðInputData; VSÞ further computes the true objective function value Z (i.e., the value of the nonlinear objective function at the solution VS, suggested by GVSPTL). In step 6 the accuracy of solution for GVSPTL mathematical model is updated. The DSA procedure exits the loop when the desired accuracy of solution for GVSPTL mathematical model is obtained. The DSA computational performance will be assessed based on a comparative analysis against SSA (details will be presented in the section of the paper devoted to numerical experiments).

Procedure 1. Dynamic Secant Approximation (DSA) DSAðInputData; Dd Þ in: InputData – liner shipping route characteristics; Dd – desired accuracy out: VS – vessel schedule 1: m 0; D 100 / Initialization 2: while D > Dd do 3: m m þ 1 / Increase the number of segments in the approximation 4: ½Z; VS GVSPTLðInputData; mÞ / Solve GVSPTL with m segments 5: Z EstObjðInputData; VSÞ / Estimate the nonlinear objective function value 6: D jZ  Zj=Z / Update the accuracy 7: end while 8: return VS

5. Numerical experiments This section discusses a set of computational experiments that were conducted to assess performance of the suggested solution approach and identify changes in the vessel schedule

that should be made by the liner shipping company to meet the established restrictions on the quantity of emissions produced by vessels within ECAs and the cargo transit time requirements. 5.1. Input data generation This study will focus on vessel scheduling at the French Asia Line 2 route (see Fig. 3), which is currently served by CMA CGM liner shipping company [28]. This liner shipping route provides a linkage between North Europe, Mediterranean Sea, Red Sea, and Asia. The French Asia Line 2 route is composed of 14 ports of call that have to be visited by vessels on a weekly basis (the distances between consequent ports of the liner shipping route in nautical miles are shown in parenthesis and were obtained using the world seaports catalogue1): 1. Ningbo, CN (146) ? 2. Shanghai, CN (628) ? 3. Xiamen, CN (317) ? 4. Yantian, CN (1,943) ? 5. Port Kelang, MY (9,124) ? 6. Le Havre, FR (279) ? 7. Rotterdam, NL (124) ? 8. Antwerp, BE (426) ? 9. Hamburg, DE (404) ? 10. Felixstowe, GB (166) ? 11. Rotterdam, NL (4,605) ? 12. Jeddah, SA (4,816) ? 13. Port Kelang, MY (1,893) ? 14. Chiwan, CN (866) ? 1. Ningbo, CN The available liner shipping and port operations literature [8,9,13,14,24,25,29–32] was used for generating the numerical data in order to conduct the computational experiments (see Table 1). The end of TW at each port of the considered liner shipping route was estimated based on the end of TW at preceding port, length of a voyage leg between consequent ports, and the vessel sailing speed bounds as follows: twliþ1 ¼ twli þ U½vminli;vmax  8i 2 I, where U – is a notation that was adopted in this study for uniformly distributed pseudorandom numbers. The quantity of containers to be transported at voyage leg i was set as follows: NCTi ¼ U½5; 000; 8; 000 8i 2 I TEUs. The weekly container demand NCi at large ports of the liner shipping route was assigned as U½500; 2; 000 TEUs. This study 1

https://www.searates.com.

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

The green vessel scheduling problem with transit time requirements Numerical data.

Table 1 Parameter

Value

Coefficients of the bunker consumption function: a; c Unit HFO cost: bHFO (USD/ton) Unit MGO cost: bMGO (USD/ton) Weekly operational cost of a vessel: cOC (USD/ week) Delayed arrival penalty: cLT (USD/h) i Unit inventory cost: l (USD per TEU per h) Carbon dioxide emission factor: FCO2 (tons of CO2 /ton of fuel) Per cent of sulphur in MGO: PMGO (%) Per cent of sulphur in HFO: PHFO (%) Nitrogen oxide emission factor: FNOx (tons of NOx/ton of fuel) Duration of TW (h) Minimum sailing speed of a vessel: vmin (knots) Maximum sailing speed of a vessel: vmax (knots) Maximum quantity of deployed vessels: qmax (vessels)

a ¼ 3; c ¼ 0:012 300 600 300,000 U½5; 000; 10; 000 1 3.082 0.1 3.5 0.051 U½24; 72 15 25 15

9

scis ¼ mhc  U½0; 50 8i 2 I; s 2 Si USD/TEU, where mhc – is the mean handling cost. Then the overall cost of handling containers at ports of call was calculated using the following equation: tcis ¼ scis NCi 8i 2 I; s 2 Si USD. The mean handling cost (mhc) was set equal to [700; 625; 550; 475] USD/TEU for four available handling rates respectively [29,30]. Furthermore, each marine container terminal operator was assumed to perceive the vessel handling cost in a different manner (i.e., vessel handling cost for exactly the same handling rate varies from one port to the other). The second (and random) term of the scis equation captures the latter operational aspect. All computational experiments were conducted using Dell T1500 Intel(T) Core i5 Processor with 2.00 GB of RAM. MATLAB 2014a was used to design a set of static secant approximations for linearization of the nonlinear bunker consumption function. GVSPTL mathematical model was coded in General Algebraic Modeling System (GAMS) and then solved using CPLEX optimization solver at each iteration of the DSA procedure. 5.2. Solution approach performance

defines ‘‘large ports” based on their total throughput. Specifically, if based on the annual throughput a given port of call was included in the list of top 20 world container ports, provided by the World Shipping Council [31], it was considered as a ‘‘large port”. The other ports in the port rotation were classified as ‘‘smaller ports”. The weekly container demand at smaller ports of call was initialized as U½200; 1; 000 TEUs. This study assumes that the following handling productivities (dis ) were offered by the marine container terminal operators to the liner shipping company at large ports of call: [125; 100; 75; 50] TEUs/h. At smaller ports of call the liner shipping company was allowed to select either [100; 75; 60; 50] TEUs/h or [75; 70; 60; 50] TEUs/h. The assumption regarding the handling productivities may be further supported by the fact that marine container terminal operators, serving large ports of call, are generally able to allocate more handling equipment for service of vessels, which further allows them to provide more flexibility in terms of selection of the handling rate alternatives to the liner shipping company. In the meantime, large ports typically handle more TEUs as compared to smaller ports, which can also increase productivity. The North Sea and the English Channel were designated as ECAs for the considered liner shipping route based on the regulations that were established by IMO [4]. The limitations on emissions produced at legs, which pass through ECAs, were assigned as follows: LCO2i ¼ FCO2 LSO2i ¼ 0:02Pi

a1 cðU½vmin ;vmax Þ

a1 cðU½vmin ;vmax Þ

li 8i 2 I 24 a1 cðU½vmin ;vmax Þ FNOx li 8i 2 I 24

24 

As mentioned in section 4 of the paper, a computational performance of the dynamic secant approximation (DSA) procedure was evaluated based on a comparative analysis against the static secant approximation (SSA) procedure. The number of segments in the SSA approximation was set to m ¼ 100 segments. A total of 10 instances were generated based on the data, which were described in Section 5.1 and shown in Table 1, by altering vessel arrival TWs at ports and duration of TWs. The desired accuracy for DSA was set to Dd ¼ 0:1%. GVSPTL was solved using DSA and SSA for each one of the developed problem instances. Results of the conducted analysis are demonstrated in Table 2, where the following information is provided for DSA and SSA procedures: (1) instance number; (2) number of linear segments (m) used in the piecewise approximation to solve GVSPTL mathematical model; (3)  objective gap D ¼ ðZZZÞ , where Z is the GVSPTL objective  function value, and Z is the value of the nonlinear objective function at the solution, which was suggested by GVSPTL; and (4) CPU time (average over 5 replications).

Table 2

Performance of the solution approach.

Instance

Number of segments

Objective Gap, %

CPU time, sec

SSA

DSA

SSA

DSA

SSA

DSA

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10

100 100 100 100 100 100 100 100 100 100

7 9 7 7 7 7 8 7 7 8

0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0005 0.0002 0.0003

0.0891 0.0796 0.0848 0.0845 0.0787 0.0782 0.0725 0.0776 0.0747 0.0991

14.2 14.7 13.8 14.0 13.7 14.2 14.7 13.9 13.8 14.2

1.6 2.5 1.7 1.8 1.8 1.7 2.2 1.7 1.8 2.2

Average:

100

7

0.0003

0.0819

14.1

1.9

li 8i 2 I tons; tons;

LNOxi ¼ tons. A total of 5 cargo types were assumed to be carried by vessels at the considered liner shipping route. The required transit time for cargo type j at voyage leg i was assigned as follows: tmax ¼ U½vminli;vmax  8i 2 I; j 2 J h. A penalty for violating the ij transit time requirement for cargo type j at voyage leg i was generated as follows: ptij ¼ U½100; 000; 800; 000 8i 2 I; j 2 J USD. The vessel handling cost per TEU scis at port i under handling rate s was estimated as follows:

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M.A. Dulebenets

Computational experiments show that on average over the generated problem instances DSA required 7 linear segments in the piecewise secant approximation in order to solve GVSPTL mathematical model with the desired accuracy, which is substantially smaller as opposed to the number of segments in SSA. Furthermore, the DSA objective gaps were found to be larger than the SSA objective gaps. The latter finding can be justified by the fact that the SSA procedure used significantly more linear segments in the piecewise secant approximation, adopted for the bunker consumption function

Figure 4

linearization. The average over 5 replications and 10 problem instances DSA CPU time comprised 1.9 s to solve GVSPTL mathematical model. The average over 5 replications and 10 problem instances SSA CPU time comprised 14.1 s to solve GVSPTL mathematical model. Hence, it can be concluded that the DSA procedure was able to solve GVSPTL mathematical model significantly faster with the desired solution accuracy as opposed to the SSA procedure for all the considered problem instances. The latter finding confirms efficiency of the adopted solution methodology.

Objective function and its components: VSPL vs. GVSPL vs. GVSPTL.

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

The green vessel scheduling problem with transit time requirements 5.3. Managerial insights The objective function value and its components were calculated for GVSPTL vessel schedules for all the generated problem instances and were compared to the following vessel schedules: (1) VSPL vessel schedules, which did not impose the constraints on emissions produced within ECAs and the cargo transit time requirements (were obtained from solving GVSPTL with relaxation of constraint sets (23)-(26), (37)(39)); and (2) GVSPL vessel schedules, which captured the emission limitations within ECAs, but did not impose the cargo transit time requirements (were obtained from solving GVSPTL with relaxation of constraint set (26)). Results of a comparative analysis are demonstrated in Fig. 4, which showcases the following route service cost components: the overall route service cost – Z, the overall weekly vessel operational cost – OC, the overall bunker consumption cost – BC, the overall port handling cost – PC, the overall late arrival penalty – LP, the overall inventory cost – IC, and the overall penalty due to violated cargo transit time requirements – VP. It can be observed that all the route service cost components, excluding the bunker consumption cost, are greater (or equal for some problem instances) for GVSPL and GVSPTL vessel schedules as opposed to VSPL vessel schedules. Additional emission constraints within ECAs cause reduction in the bunker consumption cost (i.e., the liner shipping company is required to decrease the vessel sailing speed at voyage legs, which pass through ECAs, in order to meet the emission limitations) for vessel schedules, proposed by GVSPL and GVSPTL. However, the bunker consumption cost for GVSPTL vessel schedules is higher as opposed to GVSPL vessel schedules, which indicates that the liner shipping company has to increase sailing speed at certain voyage legs to meet the cargo time requirements. Furthermore, results from the numerical experiments showcase that the liner shipping company is not able to satisfy the transit time requirements for certain cargo types, which yields an additional penalty due to violation of the transit time requirements for GVSPTL vessel schedules. We also observe that port handling costs are generally higher for GVSPL and GVSPTL vessel schedules as opposed to VSPL vessel schedules. The latter finding may be supported by the fact that introduction of emission constraints would require the liner shipping company to negotiate handling rates that have a higher handling productivity with marine container terminal operators, which will further allow the liner shipping company saving time at ports of call of the given liner shipping route. The port time savings may be further utilized by the liner shipping company in order to increase the sailing time of vessels within ECAs (i.e., select a lower speed when sailing to the consequent port of the given liner shipping route and decrease the total production of emissions within ECAs). Furthermore, GVSPL and GVSPTL produce vessel schedules, which have a significantly higher total late arrival penalty as opposed to the late arrival penalty associated with VSPL vessel schedules. Increasing late arrival penalty values for GVSPL and GVSPTL vessel schedules can be justified by the fact that the liner shipping company may be required to violate the arrival TWs at ports of call, which were originally established by the marine container terminal operators, in order to satisfy the emission restrictions posed within ECAs. However, the late

11

arrival penalty of GVSPTL vessel schedules is lower as opposed to GVSPL vessel schedules, since GVSPTL vessel schedules also account for the cargo transit time requirements. Due to increase in the vessel sailing time the overall inventory cost appears to be higher for vessel schedules that are proposed by GVSPL and GVSPTL as opposed to VSPL vessel schedules. Moreover, GVSPL and GVSPTL vessel schedules showcase larger values of the vessel weekly operational cost for a number of problem instances (i.e., I1, I7, and I8). The latter finding may be supported by the fact that the increasing sailing time for GVSPL and GVSPTL schedules requires the liner shipping company to deploy more vessels to provide service at ports of call of the given liner shipping route with a weekly frequency, which was negotiated with marine container terminal operators. The conducted numerical experiments demonstrate that on average over all generated problem instances the overall route service cost may increase by 15.9% from imposing emission constraints and by 36.5% from introducing both emission constraints and transit time requirements. The mathematical model, suggested in this paper, can not only further assist liner shipping companies with design of efficient vessel schedules, but in the meantime account for the regulations on emissions produced and the transit time requirements and make necessary alterations in vessel schedules accordingly. 6. Conclusions and future research Attention of the society to environmental concerns has been consistently increasing over the last years, which further requires the liner shipping companies to develop innovative strategies in design of their vessel schedules that will improve energy efficiency and environmental sustainability overall. In the meantime, liner shipping companies have to satisfy the transit time requirements for the goods, carried by vessels. This paper presents a novel mixed integer nonlinear programming model for the green vessel scheduling problem, which not only introduces additional limitations on emissions that are produced by vessels at voyage legs, which pass through ‘‘Emission Control Areas”, but also captures the cargo transit time requirements. The model’s objective minimizes the overall route service cost. A number of linearization techniques were applied to linearize the original nonlinear problem. The dynamic secant approximation procedure was used to solve the linearized problem. Computational experiments were undertaken for the French Asia Line 2 route, which was served by CMA CGM liner shipping company. Results from computational experiments demonstrated that the suggested solution approach was superior to the static secant approximation procedure and was able to obtain substantially faster solutions with the desired accuracy. As for the managerial insights, it was found that introduction of the emission limitations resulted in decreasing vessel sailing speeds at certain voyage legs of the liner shipping route, which reduced the overall bunker consumption cost, while the transit time constraints yielded an additional penalty due to violation of the transit time requirements for certain cargo types. Hence, the proposed mathematical model may be further used as an efficient practical tool by liner shipping companies in construction of green vessel schedules, improving energy efficiency, and consideration of important operational aspects (e.g., cargo

Please cite this article in press as: M.A. Dulebenets, The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.11.008

12 transit time requirements). The future research avenues will focus on the following aspects: (a) apply the proposed mixed integer nonlinear mathematical model for various liner shipping routes; (b) model deployment of a heterogeneous vessel fleet (i.e., the liner shipping company deploys vessels that have different technical characteristics for service of the given liner shipping route); (c) develop a multi-objective mathematical model for capturing the conflicting objectives; (d) take into consideration the uncertainty in vessel bunker consumption; and (e) consider alternative penalty functions for violation of the cargo transit time requirements. Acknowledgement This work was partially supported by the Department of Civil Engineering at the University of Memphis (Memphis, TN) and the Department of Civil and Environmental Engineering at the Florida A&M University - Florida State University (Tallahassee, FL). Any opinions, findings, conclusions, or recommendations are those of the author and do not necessarily reflect the views of the aforementioned organizations. References [1] IMO. Prevention of GHG emissions from ships. Third IMO GHG Study 2014 – Final Report, 2014. [2] H. Psaraftis, C. Kontovas, Speed models for energy-efficient maritime transportation: a taxonomy and survey, Transp. Res. Part C 26 (2013) 331–351. [3] Marine Urea, MARPOL regulation, , (accessed 10 February 2016). [4] IMO, Air Pollution, Energy Efficiency and Greenhouse Gas Emissions, , (accessed 10 February 2016). [5] The European Union, EU action on climate, , (accessed 15 February 2016). [6] K. Fagerholt, Ship scheduling with soft time windows: an optimization based approach, Eur. J. Oper. Res. 131 (2001) 559– 571. [7] T. Chuang, C. Lin, J. Kung, M. Lin, Planning the route of container ships: a fuzzy genetic approach, Expert Syst. Appl. 37 (2010) 2948–2956. [8] S. Wang, Q. Meng, Liner ship route schedule design with sea contingency time and port time uncertainty, Transp. Res. Part B 46 (2012) 615–633. [9] S. Wang, Q. Meng, Robust schedule design for liner shipping services, Transp. Res. Part E 48 (2012) 1093–1106. [10] B. Brouer, J. Dirksen, D. Pisinger, C. Plum, B. Vaaben, The vessel schedule recovery problem (VSRP) – a MIP model for handling disruptions in liner shipping, Eur. J. Oper. Res. 224 (2013) 362–374. [11] S. Wang, A. Alharbi, P. Davy, Liner ship route schedule design with port time windows, Transp. Res. Part C 41 (2014) 1–17. [12] A. Alhrabi, S. Wang, P. Davy, Schedule design for sustainable container supply chain networks with port time windows, Adv. Eng. Inform. 29 (2015) 322–331.

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