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Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background
Accepted Manuscript Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background Chuanxu Wang, Junjun Chen PII: DOI: Reference:
S0360-8352(17)30480-1 https://doi.org/10.1016/j.cie.2017.10.012 CAIE 4949
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
10 June 2016 6 September 2017 11 October 2017
Please cite this article as: Wang, C., Chen, J., Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background, Computers & Industrial Engineering (2017), doi: https://doi.org/ 10.1016/j.cie.2017.10.012
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Strategies of refueling, sailing speed and ship deployment of 1
containerships in the low-carbon background Chuanxu Wanga*
Junjun Chena
a School of Economy and Management, Shanghai Maritime University, Shanghai, 201306, China *
Corresponding author ,Email: [email protected], Tel: (86)(21)38282490, Fax:(86)(21)38282490.
1
Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background2
Abstract: To solve the problems of refueling, ship speed determination and ship deployment of containerships, based on the nonlinear relationship between ship sailing speed, fuel consumption and carbon emissions, a corresponding nonlinear mixed integer programming model is developed by taking container transportation cost, fuel consumption cost and carbon emissions cost in multiple service routes as the total cost objective function, and ship refueling ports, refueling amounts, sailing speed and ship deployment quantities in the route as decision variables. The optimal solution can be obtained by transforming the nonlinear discrete mixed integer programming model into a nonlinear continuous programming model. The proposed model is applied to an Asia-Europe service route of China Shipping (Group) Company. The results show that the optimal refueling policy, speed selection and ship deployment can be determined using the proposed model and solution method. The impact of fuel price, container transportation quantity, ship tanker capacity and carbon tax rate on ship refueling strategy, speed and deployment is investigated. Some insights obtained from example analysis are provided. Keywords: carbon emissions, refueling strategy, speed determination, nonlinear programming 1 Introduction Shipping companies continuously balance reducing shipping costs against fuel price fluctuations, energy savings and emission reductions. Air pollution and carbon emissions are increasingly concerned by individual countries’ governments. Shipping companies need to pay for carbon emissions tax levied by governments. The costs of fuel and carbon emissions are essential components of shipping costs, and they are
2
mainly affected by ship speed. Maintaining effective strategies for ship refueling and speed pose key challenges in the operations management of shipping companies today. A number of researchers have investigated the ship speed decision problem. For example, Ronen (1982) obtained from empirical data that the daily fuel consumption was positively proportional to the third order of the ship speed, thereby presenting the ship speed optimization model. Corbett et al. (2009) used profit maximization function to estimate the economic speed of specific route; the effects of fuel tax and speed reduction on carbon emissions were analyzed. Alvarez (2009) studied the joint decision-making problem of containership deployment and route, and proposed the nonlinear
mixed integer programming model via introducing ship
speed
decision-making. Fagerholt et al. (2010) studied specific shipping routes with a series of ports with service time windows; they proposed the nonlinear continuous programming problem of ship speed optimization and developed the shortest route algorithm to solve the problem. According to management strategy optimization in liner shipping service, Meng and Wang (2011) adopted a mixed integer nonlinear programming model to determine the service frequency, containership deployment and ship speed for long-distance liner shipping routes, and solved the model using an efficient and exact branch-and-bound based optimal algorithm. Norstad et al. (2011) presented ship route and ship scheduling problem with ship speed optimization in tramp shipping. Ronen (2011) investigated the relationship between fuel price and sailing speed as well as ship deployment, and presented an optimization model for ship speed and deployment. Wang and Meng (2012) estimated the relationship between fuel consumption and ship speed based on the historic data of a liner shipping company, and considering transferring and container routing, they developed 3
a nonlinear mixed integer programming model of liner ship speed optimization. Doudnikoff and Lacoste (2014) targeted cost minimization by developing combinatorial optimization model of sailing speed for sulfur emission control area and non-sulfur emission control area, which was verified based on a North European shipping route. Considering the demurrage and dispatch in voyage charters, Wang and Xu (2015) established the ship speed decision-making problem under three different carbon emissions tax standards and analyzed the influence of carbon tax on shipowner's profit and carbon emissions. In recent decades, some scholars studied ship refueling strategies. For example, Besbes and Savin (2009) proposed random dynamic programming model of route and refueling optimization in liner and tramp shipping, targeting total profit maximization. Wang et al. (2014), using the Fuzzy-Delphi-TOPSIS method, developed the selection model of refueling ports of a liner company. Ghosh et al. (2015) considered the service contract between liner operator and fuel supplier, developed a dynamic programming model for optimizing ship refueling strategy to achieve minimization of total refueling cost. Sheng et al. (2015) used an inventory strategy (s, S) to establish a ship refueling optimization model with the uncertainties of fuel consumption and price, which was solved with an intelligent algorithm. Wang and Meng (2015) developed a mixed nonlinear programming model of a ship refueling strategy using the sum of ship cost, cargoes inventory cost and ship operation cost as the objective function. The approximation method was used to solve the model, and a case study was carried out for the Asia-Europe liner route of an international shipping company. Recently, several researchers investigated refueling and ship speed simultaneously. Yao et al. (2012) presented the decision problem for ship refueling port, refueling amounts and ship speed from the perspective of a single route. Taking an Asia-Europe 4
route for example, the influences of engine power, time window restriction and bunker tanker capacity on decision results were analyzed. In this paper, ship refueling strategy and ship speed optimization strategies are comprehensively addressed. When compared with the study by Yao et al. (2012), the main differences are that the previous study only considered fuel cost and profit loss cost during shipping, and not included ship deployment strategy. Meanwhile, the nonlinear function was processed with a subsection linear approximation method in their study, thereby transforming into a linear programming problem. Jointly considering the refueling cost, fuel inventory cost, containership inventory cost, costs of container loading and unloading, and carbon emission costs in containership transportation, we developed a nonlinear mixed integer programming model of optimizing the refueling port and amounts, ship speed and deployment strategies. Through model transformation, the existence of the optimal solution is theoretically proved, and the solution could be obtained by using Lingo 11.0 software. 2
Model Development
2.1 Problem description We formulate the problem of ship refueling, sailing speed determination and ship deployment of containerships by developing a nonlinear mixed integer programming model. Ronen(2011) indicates that ship fuel cost takes seventy five percent of ship variable costs in liner shipping. Ship fuel bunker management is important for shipping companies to reduce the costs and carbon emissions. The costs of fuel and carbon emissions are essential components of shipping costs, and they are mainly affected by ship speed. The model is developed by taking container transportation cost, fuel consumption cost and carbon emissions cost in multiple service routes as the 5
total cost objective function, and ship refueling ports, refueling amounts, sailing speed and ship deployment quantities in the route as decision variables. 2.2 Assumptions While formulating the mathematical model, the following basic assumptions are adopted in this study: (1) the containers are loaded containers of the same type; (2) only fuel consumption of the main engine is considered, except that during ship residence at ports; (3) ship route and order of ports are both determined; (4) the ships on the same route are of the same type, and that ship service frequency is once a week; (5) the plan period is one year. 2.3 Model Variables and Parameters 2.3.1 Variables : binary decision, if the ship is refueled while arriving at port , otherwise,
.
ship speed at segment
of route .
：ship refueling amounts at port i of route r. : number of ships deployed for route . : scheduled arrival time at port i of route r; : ship fuel inventory level while leaving port i of route r; : ship fuel inventory level while arriving at port i of route ;
2.3.2 Sets and parameters ：set of shipping routes; ：set of ports; ：set of ports where refueling is available; W: ship bunker tanker maximal capacity; 6
of route r,
: fixed cost of ship refueling; h: the total refueling times on all routes; : fuel price of port i in route r; : unit handling cost for loading and unloading container at port i of route r; : container transportation quantity during segment i of route r; : designed ship speed( nautical miles per hour); : ship travelling distance during segment i of route r ( nautical miles); : ship berthing time at port i of route r; ：
;
K: Maximum carbon emissions limit. 2.4 Model Description 2.4.1 Number of ships deployed It was assumed that the ships on the same route are of the same type, and that ship service frequency is once a week, then
( 1)
In Equation (1), the left-hand side is the sum of shipping time and berthing time, while the right-hand side represents the results of 7 days a week times 24 hours a day multiplied by the number of ships deployed for route r. 2.4.2 Ship fuel costs ⑴ Ship refueling costs (FC)
Ship fuel price could be discounted according to refueling amounts (Sheng et al., 2015). As previously defined, the fuel level upon arriving port i of route r is while the fuel inventory level when leaving port i of route r is amounts at port i of route r is
,
. The refueling
, and the single ship refueling cost at 7
port i of route r could be expressed as
( 2)
Where is the regular fuel price, , are the discounted fuel prices for different refueling amounts, . Q1 and Q2 are the breakpoints of refueling amounts for the discounted fuel prices, 10000 and 20000 represent the fixed refueling cost when fuel
price is discounted as p1 and p2. The refueling cost of all ships FC is ( 3)
⑵ Ship fuel inventory costs(HC) Because refueled bunker fuel occupies capital cost, there is fuel inventory cost. The fuel inventory cost of all ships is
(4)
2.4.3 Container loading and unloading costs(DC) The container loading and unloading costs at all routes DC are
(5)
2.4.4 Container inventory costs (IC) Ship speed increase means shorter transportation time and, therefore, low container inventory cost,
represents unit time container inventory cost. The total
inventory cost IC could be expressed as
(6)
2.4.5 Carbon emissions cost (EC) Because of increasingly severe environmental pollution, carbon emissions pollution was considered in this study. Carbon emission cost =carbon emissions 8
carbon tax rate (e). Ship carbon emissions are mostly from engine fuel consumption. Hughes (1996) stated that ship fuel consumption was positively proportional to the cube of ship speed; thus, the power function relationship was used in this study. Ship carbon emissions are equal to the daily fuel consumption of the main engine power (MF) at designed speed
multiplied by the carbon content ratio (86.41%, Corbett
(2009)) and carbon conversion rate (44/12, Corbett (2009)). Therefore, the carbon emission cost during each voyage EC could be expressed as:
(7)
2.4.6 Model formulation Based on the above conditions, the decision model can be obtained as
(8)
Subjected to the following constraints: (9) (10)
(11) (12) (13)
9
(14)
(15)
(16) (17) (18)
(19)
(20)
(21) (22) (23)
is integer
(24)
The objective function (8) is total cost, consisted of ship refueling cost, ship fuel storage cost, container loading and unloading cost, container inventory cost and carbon emission cost. Constraints (9) and (10) define the ship refueling amounts and the range respectively; Constraints
represent the minimal fuel
inventory at the beginning, and fuel inventory limits before and after every refueling; Constraint
reflects ship deployment; Constraint
constraints of all ships on all routes; Constraint
is carbon emission
represents the restriction about
the total times of refueling on all routes. Constraints (17) and (18) are the time constraints of ship from port
to port
. Constraints (19) and (20) is the fuel
consumption constraint of ship from port
to port
represents ship speed constraint. Constraints means that if ship is refueled at available ports, 10
; While constraint define binary variable, it
=1, and otherwise, 0.Constraint
(24) define integer variables.
3
Model Analysis and Solving
3.1 Model Transformation The above mentioned model is a typical 0-1 nonlinear mixed integer programming (0-1NMIP) problem; the objective function and partial constraint conditions are nonlinear functions of ship speed and discrete function because the model includes 0-1 decision variables, so it is hard to recognize concavity and convexity of objective function and identify the existence of optimal solutions of the model. In order to obtain the model solutions, the original model is converted to an equivalent model. First, the reciprocal ship speed variable ( see Objective function 8, Constraints 14, 15, 18, 19 and 20) is used as a new variable; second, linear transformation (Chen and Liao (2007)) is performed on constraints of 0-1 decision variables (see Constraints 22 and 23), thereby converting the original model to a continuous nonlinear programming problem. Based on such, concavity and convexity analysis is performed for the objective function and constraint conditions in order to recognize the presence of the optimal solution. In order to guarantee the convexity of the objective function and the left side function of constraint conditions in the model, it was defined that
, and
. The objective function (8) is converted to:
(25)
Constraint condition (14) is therefore turned into: (26) 11
Constraint condition (15) has became (27) Constraint condition (18) was changed to (28) Constraint condition (19) was converted to (29) Constraint condition (20) was converted to (30) Constraint condition (21) was converted to (31) The 0-1 decision variables in (22) and (23) are processed with a nonlinear method, so the equations (22) and (23) are changed to (32) (33) While all other constraint conditions remain the same. 3.2 Model Analysis To further analyze the objective function’s properties after conversion, the next theorems are introduced: Theorem 1: The objective function on Proof:
is a convex function of
.
At first, the first order derivative of
with respect to
is calculated. Although fuel cost function is subsection function, it does not affect derivation of 12
objective function in
,
are replaced by
The second order derivative of
, then
with respect to
is then calculated as
The third principal minor of the Hessian matrix of objective function is
Obviously, the above Hessian matrix is positive semi-definite, the objective 13
function
is convex function of
on
.
Theorem 2: The left side function of constraint conditions after conversion is a convex function. Proof: For constraint conditions ( 27), (28), (29), (30 )and (32), let
The first order derivatives of
and
,
with respect to
are calculated , so is the first order derivative of
with respect to
. The following results are derived:
The second order derivatives of
,
and
with respect to
are, respectively , ,
The second order derivative of
with respect to
14
is
Obviously, the left side functions of the above nonlinear constraint conditions are all convex functions. For the other constraint conditions, the left side functions are linear functions of , and, therefore, can be considered as convex functions as well. Hence, all left side functions of constraint conditions are convex functions. It is apparent that the converted model is a convex programming model. From the properties of convex programming, it is known that the optimal solution of the model exists, which can be obtained by solving the converted continuous model. 4. Example Analysis The AEX1 route of Asia-Europe Express (Figure 1) of China Shipping (Group) Company is used as an example. The calling ports of ship in this route are: Qingdao, Shanghai, Ningbo, Yantian, Singapore, Basheng, Felixstowe, Rotterdam, Hamburg, Zeebrugge, Rotterdam, Basheng, Yantian and Qingdao. Figure1
AEX1 service route
The containerships of all shipping segments are 8000TEU(Twenty feet Equivalent Unit). Table 1 lists the sailing distance among ports and ship berthing time at each port, while Table 2 lists the value of parameters in the model. Table 1. Inter-port distance and ship berthing time at each port No. 1 2 3 4 5 6 7 8 9 10
Port * Qingdao Shanghai Ningbo Yantian Singapore Basheng Felixstowe, Rotterdam Hamburg Zeebrugge
Distance 408 136 727 1445 204 8012 118 302 335 70
** (miles) Berthing time 1 1.5 2 2 1 1.5 1 1 1 1 15
***(hour)
11 Rotterdam 8088 2 12 Basheng 1649 2 13 Yantian 1105 2 14 Qingdao 2 * i represents port i, the distance between port i and port i+1, ** distance of segment i in route r, ***
The sailing
Berthing time at port i in route r.
Based on the data above, Lingo 11.0 software is employed to solve the model. The results are presented in Table 3. It can be seen that the total cost is $49,1994.6, and the number of ships deployed is 7. It can also be observed that different segments have varying ship speeds. From the results of refueling ports selection, considering the distance factor between different ports, refueling ports with a low fuel price should be chosen at the condition of assuring safety fuel level. More fuel is filled at ports with a lower fuel price. This strategy enables minimal total shipping costs, and is able to assure the ship’s arrival at its scheduled time. Table 2. Model parameters Parameters Symbol Unit fuel cost (dollars/ton) f Unit handling cost (dollars/TEU) Cri Unit container storage cost (dollars/day. TEU) Unit fuel storage cost per ton (dollars/ton) Ch Vmin Minimum ship speed ( nautical miles/h) Vmax Maximum ship speed ( nautical miles/h) v0 Designed ship speed ( nautical miles/h) Ship power (KW) MF Carbon tax rate (dollars/ton) e Container transportation quantity during segment i Xri (TEU) Ship bunker tanker maximal capacity (tons) W Initial ship fuel inventory level (tons) y11 Maximum carbon emissions limit (tons) C Maximum refueling times h 16
Value 1000 2 1 100 5 28 12 60 1 8000 5000 1000 40500 5
5 . Sensitivity Analysis
5.1 Impact of fuel price change While keeping the other factors constant, the fuel price is changed to analyze its influences on shipping costs, ship deployment and sailing speed. Figures 2 and Figure 3 show that with increase in fuel prices, the total shipping Table 3. Model calculation results
1
Qingdao
Fuel price Distance at refueling Section (miles) port (dollars/ton) 1 408 247
2
Shanghai
2
136
3
Ningbo
3
727
4
Yantian
4
1445
5
Singapore
5
204
6
Basheng
6
8012
7
Felixstowe, 7
118
8
Rotterdam
8
302
9
Hamburg
9
335
10
Zeebrugge
10
70
11
Rotterdam
11
8088
12
Basheng
12
1649
13
Yantian
13
1105
No. Port
Berthing Arrival Speed Refueling Refueling time time (knots) port amounts (days) (day) 1
28
0
1
1000
246
1.5
23
2
1
500
250
2
19
3
0
0
2
19
7
0
0
214
1
25
12
1
1243
227
1.5
19
13
0
0
1
18
32
0
0
192
1
28
34
1
4343
210
1
21
35
1
500
1
18
37
0
0
192
2
19
38
0
0
227
2
21
58
0
0
2
21
63
0
0
cost gradually increased correspondingly. Refueling amounts, average sailing speed, number of ships deployed and total sailing time kept constant at first. Figure 3 also shows that with the fuel price increase 27%, number of ships deployed and total sailing time ascended greatly, whereas refueling amounts and average sailing speed decreased significantly. The reason might be that when fuel price reached a certain level,
there
is
changing
in
refueling 17
ports,
resulting
in
corresponding
significant changes in refueling amounts, sailing time, average sailing speed and number of ships deployed. Figure 2
Impact of fuel price change on shipping cost and refueling amounts
Figure 3 Impact of fuel price change on sailing speed , sailing time and number of ship deployment
5.2 Impact of container transportation amounts Variations in container transportation amounts play a significant role in ship operations. While keeping the other factors constant, container transportation amounts is changed to analyze its relationship with shipping cost, number of ships deployed, refueling amounts, total sailing time and average sailing speed. The results are shown in Figures 4 and 5. It can be seen that with increasing container transportation amounts, total shipping cost increased as well, ship refueling amounts first increased abruptly, and then kept stable. In contrast, number of ships deployed and total sailing time first decreased and then kept constant. Meanwhile, the average sailing speed first increased with rising container transportation amounts and then kept at a stable level. Figure 4 Impact of container transportation amounts on shipping cost and refueling amounts Figure 5
Impact of container transportation amounts on sailing speed, sailing time and number of ship deployment
5.3 Impact of bunker tanker capacity Fuel tanker capacity depends on the ship scale, and it affects ship deployment, sailing speed, refueling amounts and the total shipping costs. In this section, while keeping the other factors constant, fuel tanker capacity is tested to analyze its influences on the above factors. The results are shown in Figures 6 and 7. It can be seen that with increasing fuel tanker capacity, i.e., rising ship scale, the total shipping cost decreased; refueling amounts first increased and then showed a 18
tendency of decrease and number of ships deployed first decreased and then kept unstable. In addition, the average sailing speed and total sailing time showed a stable level after a phase of increase and decrease, respectively. With small fuel tanker capacity, the corresponding ship scale is small, so more ships are needed, resulting in more costs. Figure 6 Impact of fuel tanker capacity change on shipping cost and refueling amounts Figure7 Impact of fuel tanker capacity change on sailing speed, sailing time and number of ship deployment
5.4 Impact of carbon tax rate Carbon tax is levied on the carbon emissions amounts derived from fuel consumption. Carbon tax rate is the carbon tax per carbon emissions amounts. Changes in the carbon tax rate definitely affect refueling strategy and sailing speed. While keeping all other factors constant, the influences of carbon tax rate on shipping cost, ship deployment, refueling capacity, total service time and average ship speed were analyzed. The results are shown in Figures 8 and 9. It can be seen that, as carbon tax rate increased, the total shipping cost gradually increased, and, after a period of stable condition, each of the refueling amounts, average sailing speed, total sailing time and number of ships deployed showed a trend of decrease, decrease, increase, and increase, respectively. This is due to increased carbon emission costs as a result of increased carbon tax rate, resulting in sailing speed reduction, and, therefore, more shipping time. Thus, more ships would be necessary.
Figure 8 Impact of carbon tax rate change on shipping cost and refueling amounts Figure 9 Impact of carbon tax rate change on sailing speed, sailing time and number of ship deployment
6. Conclusions 19
With consideration taken toward carbon saving and emission reducing, a multi-port and multi-route containership refueling and sailing speed optimization model is constructed. The sum of fuel, carbon emission, shipping, loading and unloading costs is examined as an objective function. Selecting the Asia-Europe service route of the China Shipping (Group) Company as an example, an analysis of the model application is performed. The results show that the optimal refueling policy, speed selection and ship deployment can be determined using the proposed model and solution method. Specifically, the influence of fuel price, container volume, ship bunker fuel capacity and carbon tax rate on ship refueling strategy and ship speed is analyzed.
It can be shown that with increase in fuel prices, the total shipping cost
gradually increased correspondingly. Refueling amounts, average sailing speed, number of ships deployed and total sailing time kept constant at first. With the fuel price further increase, number of ships deployed and total sailing time ascended greatly, whereas refueling amounts and average sailing speed decreased significantly. With increasing container transportation amounts, total shipping cost increased as well, ship refueling amounts first increased abruptly, and then kept stable. In contrast, number of ships deployed and total sailing time first decreased and then kept constant. Meanwhile, the average sailing speed first increased with rising container transportation amounts and then kept at a stable level. With increasing fuel tanker capacity, the total shipping cost decreased; refueling amounts first increased and then showed a tendency of decrease and number of ships deployed first decreased and then kept unstable. In addition, the average sailing speed and total sailing time showed a stable level after a phase of increase and decrease, respectively. Changes in the carbon tax rate definitely affect refueling strategy and sailing speed. There is still room for further studies. First, since the fuel price is affected by 20
various factors, price uncertainties of fuel warrant attention in future investigations. Second, since container shipping demands differ at every port and are uncontrollable, future studies might focus on refueling and ship speed strategies with random container demands. References Alvarez, J.( 2009). Joint routing and deployment of a fleet of container vessels. Maritime Economics & Logistics .11:186–208. Besbes, O., Savin, S. (2009).Going bunkers: The joint route selection and refueling problem. Manufacturing & Service Operations Management.,11(4):694–711. Chen,G.Liao,X .( 2007).A penalty function algorithm for solving 0-1 nonlinear mixed integer programming. Communication on applied mathematics and computation. 21(1):110-115. Corbett, J.J., Wang, H., Winebrake, J.J. (2009). The effectiveness and costs of speed reductions on emissions from international shipping. Transportation Research Part D,14 (8):593–598. Doudnikoff ,M., Lacoste R., . (2014).Effect of a speed reduction of containerships in responseto higher energy costs in Sulphur Emission Control Areas. Transportation Research Part D.28:51-61. Fagerholt, K. Laporte,G., Norstad，I., (2010). Reducing fuel emissions by optimizing speed on shipping routes. Journal of the Operational Research Society .61:523 –529 Ghosh, S., Lee, L. H., Ng,S.H., (2015).Bunkering decisions for a shipping liner in an uncertain environment with service contract. European Journal of Operational Research.244:792–802. Hughes C．(1996).Ship performance：Technical, safety, environmental and commercial aspects .London: Lloyd’s of London Press. 21
Meng, Q., Wang, S., (2011). Optimal operating strategy for a long-haul liner service route. European Journal of Operational Research. 215 (1):105–114. Norstad, I., Fagerholt, K., Laporte, G.( 2011). Tramp ship routing and scheduling with speed optimization. Transportation Research Part C . 19: 853–865. Ronen,D.( 1982).The effect of oil price on the optimal speed of ships. Journal of the Operational Research Society. 33:1035–1040. Ronen, D.( 2011). The effect of oil price on containership speed and fleet size. Journal of the Operational Research Society .62 (1):211–216. Sheng,X., Chew , E.P., Lee,L.H., (2015). (s,S) policy model for liner shipping refueling and sailing speed optimization problem. Transportation Research Part E.76:76-92. Wang, S,
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22
Figure1
AEX1 service route
9000
5,20,000
6000
4,80,000
3000
4,40,000
0 initial value
+3%
+7%
+11%
+15%
+19%
+23%
+27%
Bunkering amounts
Total cost
5,60,000
Bunker fuel price(USD/ton) Bunkering amounts( Ton)
Figure 2
Total cost(USD)
Impact of fuel price change on shipping cost and refueling amounts
9
60
8
30
7
0
6 initial value
+3%
+7%
+11%
+15%
+19%
+23%
+27%
Bunker fuel price(USD/ton) Number of ship deployment Sailing speed (kn/h) 23
total sailing time(days)
Number of ship deployment
Sailing speed and total sailing time
90
Figure 3 Impact of fuel price change on sailing speed , sailing time and number of ship deployment
8000
600000
6000
400000
4000
200000
2000
0
Bunkering amounts
Total cost
800000
0 6000
7000
8000
9000
10000
11000
Container transportation amounts（TEU） Bunkering amounts( Ton)
Total cost(USD)
Figure 4 Impact of container transportation amounts on shipping cost and refueling amounts
9
60 8
45 30
7
15 0
6 6000
7000
8000
9000
10000
Container transportation amounts(TEU) Number of ship deployment
11000
Number of ship deployment
Sailing speed and total sailing time
75
total sailing time(days)
Sailing speed (kn/h)
Figure 5 Impact of container transportation amounts on sailing speed, sailing time and number of ship deployment
24
600000
8000 7000
500000 Total cost
5000
300000
4000 3000
200000
2000
100000
Bunkering amounts
6000
400000
1000
0
0
4000
5000
6000
7000
8000
9000
10000
Capacity of fuel tank(Ton) Bunkering amounts( Ton)
Total cost(USD)
Figure 6 Impact of fuel tanker capacity change on shipping cost and refueling amounts
9
70 60 8
50
40 30
7
20 10 0
Number of ship deployment
Sailing speed and total sailing time
80
6 4000
5000
6000
7000
8000
9000
10000
Capacity of fuel tank(Ton) Number of ship deployment total sailing time(days) Sailing speed (kn/h)
Figure7 Impact of fuel tanker capacity change on sailing speed, sailing time and number of ship deployment
25
800000
6000
600000
4000
400000
2000
200000
0
Total cost
Bunkering amounts
8000
0 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Carbon tax rate（USD/ton) Bunkering amounts( Ton)
Total cost(USD)
Figure 8 Impact of carbon tax rate change on shipping cost and refueling amounts 9
60 8
45 30
7
15 0
6 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Carbon tax rate(USD/ton) Number of ship deployment
2.4
Number of ship deployment
Sailing speed and totalsailing time
75
total sailing time(days)
Sailing speed (kn/h)
Figure 9 Impact of carbon tax rate change on sailing speed, sailing time and number of ship deployment
26
Highlights3 1 Problems of refueling, ship speed determination and ship deployment are considered. 2 Refueling ports and amounts, ship speed and deployment number are decision variables. 3 Proposed model is solved by transforming into a nonlinear continuous programming model. 4 Impact of fuel price, ship tanker capacity and carbon tax rate on decision is analyzed
27
S0360-8352(17)30480-1 https://doi.org/10.1016/j.cie.2017.10.012 CAIE 4949
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
10 June 2016 6 September 2017 11 October 2017
Please cite this article as: Wang, C., Chen, J., Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background, Computers & Industrial Engineering (2017), doi: https://doi.org/ 10.1016/j.cie.2017.10.012
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Strategies of refueling, sailing speed and ship deployment of 1
containerships in the low-carbon background Chuanxu Wanga*
Junjun Chena
a School of Economy and Management, Shanghai Maritime University, Shanghai, 201306, China *
Corresponding author ,Email: [email protected], Tel: (86)(21)38282490, Fax:(86)(21)38282490.
1
Strategies of refueling, sailing speed and ship deployment of containerships in the low-carbon background2
Abstract: To solve the problems of refueling, ship speed determination and ship deployment of containerships, based on the nonlinear relationship between ship sailing speed, fuel consumption and carbon emissions, a corresponding nonlinear mixed integer programming model is developed by taking container transportation cost, fuel consumption cost and carbon emissions cost in multiple service routes as the total cost objective function, and ship refueling ports, refueling amounts, sailing speed and ship deployment quantities in the route as decision variables. The optimal solution can be obtained by transforming the nonlinear discrete mixed integer programming model into a nonlinear continuous programming model. The proposed model is applied to an Asia-Europe service route of China Shipping (Group) Company. The results show that the optimal refueling policy, speed selection and ship deployment can be determined using the proposed model and solution method. The impact of fuel price, container transportation quantity, ship tanker capacity and carbon tax rate on ship refueling strategy, speed and deployment is investigated. Some insights obtained from example analysis are provided. Keywords: carbon emissions, refueling strategy, speed determination, nonlinear programming 1 Introduction Shipping companies continuously balance reducing shipping costs against fuel price fluctuations, energy savings and emission reductions. Air pollution and carbon emissions are increasingly concerned by individual countries’ governments. Shipping companies need to pay for carbon emissions tax levied by governments. The costs of fuel and carbon emissions are essential components of shipping costs, and they are
2
mainly affected by ship speed. Maintaining effective strategies for ship refueling and speed pose key challenges in the operations management of shipping companies today. A number of researchers have investigated the ship speed decision problem. For example, Ronen (1982) obtained from empirical data that the daily fuel consumption was positively proportional to the third order of the ship speed, thereby presenting the ship speed optimization model. Corbett et al. (2009) used profit maximization function to estimate the economic speed of specific route; the effects of fuel tax and speed reduction on carbon emissions were analyzed. Alvarez (2009) studied the joint decision-making problem of containership deployment and route, and proposed the nonlinear
mixed integer programming model via introducing ship
speed
decision-making. Fagerholt et al. (2010) studied specific shipping routes with a series of ports with service time windows; they proposed the nonlinear continuous programming problem of ship speed optimization and developed the shortest route algorithm to solve the problem. According to management strategy optimization in liner shipping service, Meng and Wang (2011) adopted a mixed integer nonlinear programming model to determine the service frequency, containership deployment and ship speed for long-distance liner shipping routes, and solved the model using an efficient and exact branch-and-bound based optimal algorithm. Norstad et al. (2011) presented ship route and ship scheduling problem with ship speed optimization in tramp shipping. Ronen (2011) investigated the relationship between fuel price and sailing speed as well as ship deployment, and presented an optimization model for ship speed and deployment. Wang and Meng (2012) estimated the relationship between fuel consumption and ship speed based on the historic data of a liner shipping company, and considering transferring and container routing, they developed 3
a nonlinear mixed integer programming model of liner ship speed optimization. Doudnikoff and Lacoste (2014) targeted cost minimization by developing combinatorial optimization model of sailing speed for sulfur emission control area and non-sulfur emission control area, which was verified based on a North European shipping route. Considering the demurrage and dispatch in voyage charters, Wang and Xu (2015) established the ship speed decision-making problem under three different carbon emissions tax standards and analyzed the influence of carbon tax on shipowner's profit and carbon emissions. In recent decades, some scholars studied ship refueling strategies. For example, Besbes and Savin (2009) proposed random dynamic programming model of route and refueling optimization in liner and tramp shipping, targeting total profit maximization. Wang et al. (2014), using the Fuzzy-Delphi-TOPSIS method, developed the selection model of refueling ports of a liner company. Ghosh et al. (2015) considered the service contract between liner operator and fuel supplier, developed a dynamic programming model for optimizing ship refueling strategy to achieve minimization of total refueling cost. Sheng et al. (2015) used an inventory strategy (s, S) to establish a ship refueling optimization model with the uncertainties of fuel consumption and price, which was solved with an intelligent algorithm. Wang and Meng (2015) developed a mixed nonlinear programming model of a ship refueling strategy using the sum of ship cost, cargoes inventory cost and ship operation cost as the objective function. The approximation method was used to solve the model, and a case study was carried out for the Asia-Europe liner route of an international shipping company. Recently, several researchers investigated refueling and ship speed simultaneously. Yao et al. (2012) presented the decision problem for ship refueling port, refueling amounts and ship speed from the perspective of a single route. Taking an Asia-Europe 4
route for example, the influences of engine power, time window restriction and bunker tanker capacity on decision results were analyzed. In this paper, ship refueling strategy and ship speed optimization strategies are comprehensively addressed. When compared with the study by Yao et al. (2012), the main differences are that the previous study only considered fuel cost and profit loss cost during shipping, and not included ship deployment strategy. Meanwhile, the nonlinear function was processed with a subsection linear approximation method in their study, thereby transforming into a linear programming problem. Jointly considering the refueling cost, fuel inventory cost, containership inventory cost, costs of container loading and unloading, and carbon emission costs in containership transportation, we developed a nonlinear mixed integer programming model of optimizing the refueling port and amounts, ship speed and deployment strategies. Through model transformation, the existence of the optimal solution is theoretically proved, and the solution could be obtained by using Lingo 11.0 software. 2
Model Development
2.1 Problem description We formulate the problem of ship refueling, sailing speed determination and ship deployment of containerships by developing a nonlinear mixed integer programming model. Ronen(2011) indicates that ship fuel cost takes seventy five percent of ship variable costs in liner shipping. Ship fuel bunker management is important for shipping companies to reduce the costs and carbon emissions. The costs of fuel and carbon emissions are essential components of shipping costs, and they are mainly affected by ship speed. The model is developed by taking container transportation cost, fuel consumption cost and carbon emissions cost in multiple service routes as the 5
total cost objective function, and ship refueling ports, refueling amounts, sailing speed and ship deployment quantities in the route as decision variables. 2.2 Assumptions While formulating the mathematical model, the following basic assumptions are adopted in this study: (1) the containers are loaded containers of the same type; (2) only fuel consumption of the main engine is considered, except that during ship residence at ports; (3) ship route and order of ports are both determined; (4) the ships on the same route are of the same type, and that ship service frequency is once a week; (5) the plan period is one year. 2.3 Model Variables and Parameters 2.3.1 Variables : binary decision, if the ship is refueled while arriving at port , otherwise,
.
ship speed at segment
of route .
：ship refueling amounts at port i of route r. : number of ships deployed for route . : scheduled arrival time at port i of route r; : ship fuel inventory level while leaving port i of route r; : ship fuel inventory level while arriving at port i of route ;
2.3.2 Sets and parameters ：set of shipping routes; ：set of ports; ：set of ports where refueling is available; W: ship bunker tanker maximal capacity; 6
of route r,
: fixed cost of ship refueling; h: the total refueling times on all routes; : fuel price of port i in route r; : unit handling cost for loading and unloading container at port i of route r; : container transportation quantity during segment i of route r; : designed ship speed( nautical miles per hour); : ship travelling distance during segment i of route r ( nautical miles); : ship berthing time at port i of route r; ：
;
K: Maximum carbon emissions limit. 2.4 Model Description 2.4.1 Number of ships deployed It was assumed that the ships on the same route are of the same type, and that ship service frequency is once a week, then
( 1)
In Equation (1), the left-hand side is the sum of shipping time and berthing time, while the right-hand side represents the results of 7 days a week times 24 hours a day multiplied by the number of ships deployed for route r. 2.4.2 Ship fuel costs ⑴ Ship refueling costs (FC)
Ship fuel price could be discounted according to refueling amounts (Sheng et al., 2015). As previously defined, the fuel level upon arriving port i of route r is while the fuel inventory level when leaving port i of route r is amounts at port i of route r is
,
. The refueling
, and the single ship refueling cost at 7
port i of route r could be expressed as
( 2)
Where is the regular fuel price, , are the discounted fuel prices for different refueling amounts, . Q1 and Q2 are the breakpoints of refueling amounts for the discounted fuel prices, 10000 and 20000 represent the fixed refueling cost when fuel
price is discounted as p1 and p2. The refueling cost of all ships FC is ( 3)
⑵ Ship fuel inventory costs(HC) Because refueled bunker fuel occupies capital cost, there is fuel inventory cost. The fuel inventory cost of all ships is
(4)
2.4.3 Container loading and unloading costs(DC) The container loading and unloading costs at all routes DC are
(5)
2.4.4 Container inventory costs (IC) Ship speed increase means shorter transportation time and, therefore, low container inventory cost,
represents unit time container inventory cost. The total
inventory cost IC could be expressed as
(6)
2.4.5 Carbon emissions cost (EC) Because of increasingly severe environmental pollution, carbon emissions pollution was considered in this study. Carbon emission cost =carbon emissions 8
carbon tax rate (e). Ship carbon emissions are mostly from engine fuel consumption. Hughes (1996) stated that ship fuel consumption was positively proportional to the cube of ship speed; thus, the power function relationship was used in this study. Ship carbon emissions are equal to the daily fuel consumption of the main engine power (MF) at designed speed
multiplied by the carbon content ratio (86.41%, Corbett
(2009)) and carbon conversion rate (44/12, Corbett (2009)). Therefore, the carbon emission cost during each voyage EC could be expressed as:
(7)
2.4.6 Model formulation Based on the above conditions, the decision model can be obtained as
(8)
Subjected to the following constraints: (9) (10)
(11) (12) (13)
9
(14)
(15)
(16) (17) (18)
(19)
(20)
(21) (22) (23)
is integer
(24)
The objective function (8) is total cost, consisted of ship refueling cost, ship fuel storage cost, container loading and unloading cost, container inventory cost and carbon emission cost. Constraints (9) and (10) define the ship refueling amounts and the range respectively; Constraints
represent the minimal fuel
inventory at the beginning, and fuel inventory limits before and after every refueling; Constraint
reflects ship deployment; Constraint
constraints of all ships on all routes; Constraint
is carbon emission
represents the restriction about
the total times of refueling on all routes. Constraints (17) and (18) are the time constraints of ship from port
to port
. Constraints (19) and (20) is the fuel
consumption constraint of ship from port
to port
represents ship speed constraint. Constraints means that if ship is refueled at available ports, 10
; While constraint define binary variable, it
=1, and otherwise, 0.Constraint
(24) define integer variables.
3
Model Analysis and Solving
3.1 Model Transformation The above mentioned model is a typical 0-1 nonlinear mixed integer programming (0-1NMIP) problem; the objective function and partial constraint conditions are nonlinear functions of ship speed and discrete function because the model includes 0-1 decision variables, so it is hard to recognize concavity and convexity of objective function and identify the existence of optimal solutions of the model. In order to obtain the model solutions, the original model is converted to an equivalent model. First, the reciprocal ship speed variable ( see Objective function 8, Constraints 14, 15, 18, 19 and 20) is used as a new variable; second, linear transformation (Chen and Liao (2007)) is performed on constraints of 0-1 decision variables (see Constraints 22 and 23), thereby converting the original model to a continuous nonlinear programming problem. Based on such, concavity and convexity analysis is performed for the objective function and constraint conditions in order to recognize the presence of the optimal solution. In order to guarantee the convexity of the objective function and the left side function of constraint conditions in the model, it was defined that
, and
. The objective function (8) is converted to:
(25)
Constraint condition (14) is therefore turned into: (26) 11
Constraint condition (15) has became (27) Constraint condition (18) was changed to (28) Constraint condition (19) was converted to (29) Constraint condition (20) was converted to (30) Constraint condition (21) was converted to (31) The 0-1 decision variables in (22) and (23) are processed with a nonlinear method, so the equations (22) and (23) are changed to (32) (33) While all other constraint conditions remain the same. 3.2 Model Analysis To further analyze the objective function’s properties after conversion, the next theorems are introduced: Theorem 1: The objective function on Proof:
is a convex function of
.
At first, the first order derivative of
with respect to
is calculated. Although fuel cost function is subsection function, it does not affect derivation of 12
objective function in
,
are replaced by
The second order derivative of
, then
with respect to
is then calculated as
The third principal minor of the Hessian matrix of objective function is
Obviously, the above Hessian matrix is positive semi-definite, the objective 13
function
is convex function of
on
.
Theorem 2: The left side function of constraint conditions after conversion is a convex function. Proof: For constraint conditions ( 27), (28), (29), (30 )and (32), let
The first order derivatives of
and
,
with respect to
are calculated , so is the first order derivative of
with respect to
. The following results are derived:
The second order derivatives of
,
and
with respect to
are, respectively , ,
The second order derivative of
with respect to
14
is
Obviously, the left side functions of the above nonlinear constraint conditions are all convex functions. For the other constraint conditions, the left side functions are linear functions of , and, therefore, can be considered as convex functions as well. Hence, all left side functions of constraint conditions are convex functions. It is apparent that the converted model is a convex programming model. From the properties of convex programming, it is known that the optimal solution of the model exists, which can be obtained by solving the converted continuous model. 4. Example Analysis The AEX1 route of Asia-Europe Express (Figure 1) of China Shipping (Group) Company is used as an example. The calling ports of ship in this route are: Qingdao, Shanghai, Ningbo, Yantian, Singapore, Basheng, Felixstowe, Rotterdam, Hamburg, Zeebrugge, Rotterdam, Basheng, Yantian and Qingdao. Figure1
AEX1 service route
The containerships of all shipping segments are 8000TEU(Twenty feet Equivalent Unit). Table 1 lists the sailing distance among ports and ship berthing time at each port, while Table 2 lists the value of parameters in the model. Table 1. Inter-port distance and ship berthing time at each port No. 1 2 3 4 5 6 7 8 9 10
Port * Qingdao Shanghai Ningbo Yantian Singapore Basheng Felixstowe, Rotterdam Hamburg Zeebrugge
Distance 408 136 727 1445 204 8012 118 302 335 70
** (miles) Berthing time 1 1.5 2 2 1 1.5 1 1 1 1 15
***(hour)
11 Rotterdam 8088 2 12 Basheng 1649 2 13 Yantian 1105 2 14 Qingdao 2 * i represents port i, the distance between port i and port i+1, ** distance of segment i in route r, ***
The sailing
Berthing time at port i in route r.
Based on the data above, Lingo 11.0 software is employed to solve the model. The results are presented in Table 3. It can be seen that the total cost is $49,1994.6, and the number of ships deployed is 7. It can also be observed that different segments have varying ship speeds. From the results of refueling ports selection, considering the distance factor between different ports, refueling ports with a low fuel price should be chosen at the condition of assuring safety fuel level. More fuel is filled at ports with a lower fuel price. This strategy enables minimal total shipping costs, and is able to assure the ship’s arrival at its scheduled time. Table 2. Model parameters Parameters Symbol Unit fuel cost (dollars/ton) f Unit handling cost (dollars/TEU) Cri Unit container storage cost (dollars/day. TEU) Unit fuel storage cost per ton (dollars/ton) Ch Vmin Minimum ship speed ( nautical miles/h) Vmax Maximum ship speed ( nautical miles/h) v0 Designed ship speed ( nautical miles/h) Ship power (KW) MF Carbon tax rate (dollars/ton) e Container transportation quantity during segment i Xri (TEU) Ship bunker tanker maximal capacity (tons) W Initial ship fuel inventory level (tons) y11 Maximum carbon emissions limit (tons) C Maximum refueling times h 16
Value 1000 2 1 100 5 28 12 60 1 8000 5000 1000 40500 5
5 . Sensitivity Analysis
5.1 Impact of fuel price change While keeping the other factors constant, the fuel price is changed to analyze its influences on shipping costs, ship deployment and sailing speed. Figures 2 and Figure 3 show that with increase in fuel prices, the total shipping Table 3. Model calculation results
1
Qingdao
Fuel price Distance at refueling Section (miles) port (dollars/ton) 1 408 247
2
Shanghai
2
136
3
Ningbo
3
727
4
Yantian
4
1445
5
Singapore
5
204
6
Basheng
6
8012
7
Felixstowe, 7
118
8
Rotterdam
8
302
9
Hamburg
9
335
10
Zeebrugge
10
70
11
Rotterdam
11
8088
12
Basheng
12
1649
13
Yantian
13
1105
No. Port
Berthing Arrival Speed Refueling Refueling time time (knots) port amounts (days) (day) 1
28
0
1
1000
246
1.5
23
2
1
500
250
2
19
3
0
0
2
19
7
0
0
214
1
25
12
1
1243
227
1.5
19
13
0
0
1
18
32
0
0
192
1
28
34
1
4343
210
1
21
35
1
500
1
18
37
0
0
192
2
19
38
0
0
227
2
21
58
0
0
2
21
63
0
0
cost gradually increased correspondingly. Refueling amounts, average sailing speed, number of ships deployed and total sailing time kept constant at first. Figure 3 also shows that with the fuel price increase 27%, number of ships deployed and total sailing time ascended greatly, whereas refueling amounts and average sailing speed decreased significantly. The reason might be that when fuel price reached a certain level,
there
is
changing
in
refueling 17
ports,
resulting
in
corresponding
significant changes in refueling amounts, sailing time, average sailing speed and number of ships deployed. Figure 2
Impact of fuel price change on shipping cost and refueling amounts
Figure 3 Impact of fuel price change on sailing speed , sailing time and number of ship deployment
5.2 Impact of container transportation amounts Variations in container transportation amounts play a significant role in ship operations. While keeping the other factors constant, container transportation amounts is changed to analyze its relationship with shipping cost, number of ships deployed, refueling amounts, total sailing time and average sailing speed. The results are shown in Figures 4 and 5. It can be seen that with increasing container transportation amounts, total shipping cost increased as well, ship refueling amounts first increased abruptly, and then kept stable. In contrast, number of ships deployed and total sailing time first decreased and then kept constant. Meanwhile, the average sailing speed first increased with rising container transportation amounts and then kept at a stable level. Figure 4 Impact of container transportation amounts on shipping cost and refueling amounts Figure 5
Impact of container transportation amounts on sailing speed, sailing time and number of ship deployment
5.3 Impact of bunker tanker capacity Fuel tanker capacity depends on the ship scale, and it affects ship deployment, sailing speed, refueling amounts and the total shipping costs. In this section, while keeping the other factors constant, fuel tanker capacity is tested to analyze its influences on the above factors. The results are shown in Figures 6 and 7. It can be seen that with increasing fuel tanker capacity, i.e., rising ship scale, the total shipping cost decreased; refueling amounts first increased and then showed a 18
tendency of decrease and number of ships deployed first decreased and then kept unstable. In addition, the average sailing speed and total sailing time showed a stable level after a phase of increase and decrease, respectively. With small fuel tanker capacity, the corresponding ship scale is small, so more ships are needed, resulting in more costs. Figure 6 Impact of fuel tanker capacity change on shipping cost and refueling amounts Figure7 Impact of fuel tanker capacity change on sailing speed, sailing time and number of ship deployment
5.4 Impact of carbon tax rate Carbon tax is levied on the carbon emissions amounts derived from fuel consumption. Carbon tax rate is the carbon tax per carbon emissions amounts. Changes in the carbon tax rate definitely affect refueling strategy and sailing speed. While keeping all other factors constant, the influences of carbon tax rate on shipping cost, ship deployment, refueling capacity, total service time and average ship speed were analyzed. The results are shown in Figures 8 and 9. It can be seen that, as carbon tax rate increased, the total shipping cost gradually increased, and, after a period of stable condition, each of the refueling amounts, average sailing speed, total sailing time and number of ships deployed showed a trend of decrease, decrease, increase, and increase, respectively. This is due to increased carbon emission costs as a result of increased carbon tax rate, resulting in sailing speed reduction, and, therefore, more shipping time. Thus, more ships would be necessary.
Figure 8 Impact of carbon tax rate change on shipping cost and refueling amounts Figure 9 Impact of carbon tax rate change on sailing speed, sailing time and number of ship deployment
6. Conclusions 19
With consideration taken toward carbon saving and emission reducing, a multi-port and multi-route containership refueling and sailing speed optimization model is constructed. The sum of fuel, carbon emission, shipping, loading and unloading costs is examined as an objective function. Selecting the Asia-Europe service route of the China Shipping (Group) Company as an example, an analysis of the model application is performed. The results show that the optimal refueling policy, speed selection and ship deployment can be determined using the proposed model and solution method. Specifically, the influence of fuel price, container volume, ship bunker fuel capacity and carbon tax rate on ship refueling strategy and ship speed is analyzed.
It can be shown that with increase in fuel prices, the total shipping cost
gradually increased correspondingly. Refueling amounts, average sailing speed, number of ships deployed and total sailing time kept constant at first. With the fuel price further increase, number of ships deployed and total sailing time ascended greatly, whereas refueling amounts and average sailing speed decreased significantly. With increasing container transportation amounts, total shipping cost increased as well, ship refueling amounts first increased abruptly, and then kept stable. In contrast, number of ships deployed and total sailing time first decreased and then kept constant. Meanwhile, the average sailing speed first increased with rising container transportation amounts and then kept at a stable level. With increasing fuel tanker capacity, the total shipping cost decreased; refueling amounts first increased and then showed a tendency of decrease and number of ships deployed first decreased and then kept unstable. In addition, the average sailing speed and total sailing time showed a stable level after a phase of increase and decrease, respectively. Changes in the carbon tax rate definitely affect refueling strategy and sailing speed. There is still room for further studies. First, since the fuel price is affected by 20
various factors, price uncertainties of fuel warrant attention in future investigations. Second, since container shipping demands differ at every port and are uncontrollable, future studies might focus on refueling and ship speed strategies with random container demands. References Alvarez, J.( 2009). Joint routing and deployment of a fleet of container vessels. Maritime Economics & Logistics .11:186–208. Besbes, O., Savin, S. (2009).Going bunkers: The joint route selection and refueling problem. Manufacturing & Service Operations Management.,11(4):694–711. Chen,G.Liao,X .( 2007).A penalty function algorithm for solving 0-1 nonlinear mixed integer programming. Communication on applied mathematics and computation. 21(1):110-115. Corbett, J.J., Wang, H., Winebrake, J.J. (2009). The effectiveness and costs of speed reductions on emissions from international shipping. Transportation Research Part D,14 (8):593–598. Doudnikoff ,M., Lacoste R., . (2014).Effect of a speed reduction of containerships in responseto higher energy costs in Sulphur Emission Control Areas. Transportation Research Part D.28:51-61. Fagerholt, K. Laporte,G., Norstad，I., (2010). Reducing fuel emissions by optimizing speed on shipping routes. Journal of the Operational Research Society .61:523 –529 Ghosh, S., Lee, L. H., Ng,S.H., (2015).Bunkering decisions for a shipping liner in an uncertain environment with service contract. European Journal of Operational Research.244:792–802. Hughes C．(1996).Ship performance：Technical, safety, environmental and commercial aspects .London: Lloyd’s of London Press. 21
Meng, Q., Wang, S., (2011). Optimal operating strategy for a long-haul liner service route. European Journal of Operational Research. 215 (1):105–114. Norstad, I., Fagerholt, K., Laporte, G.( 2011). Tramp ship routing and scheduling with speed optimization. Transportation Research Part C . 19: 853–865. Ronen,D.( 1982).The effect of oil price on the optimal speed of ships. Journal of the Operational Research Society. 33:1035–1040. Ronen, D.( 2011). The effect of oil price on containership speed and fleet size. Journal of the Operational Research Society .62 (1):211–216. Sheng,X., Chew , E.P., Lee,L.H., (2015). (s,S) policy model for liner shipping refueling and sailing speed optimization problem. Transportation Research Part E.76:76-92. Wang, S,
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22
Figure1
AEX1 service route
9000
5,20,000
6000
4,80,000
3000
4,40,000
0 initial value
+3%
+7%
+11%
+15%
+19%
+23%
+27%
Bunkering amounts
Total cost
5,60,000
Bunker fuel price(USD/ton) Bunkering amounts( Ton)
Figure 2
Total cost(USD)
Impact of fuel price change on shipping cost and refueling amounts
9
60
8
30
7
0
6 initial value
+3%
+7%
+11%
+15%
+19%
+23%
+27%
Bunker fuel price(USD/ton) Number of ship deployment Sailing speed (kn/h) 23
total sailing time(days)
Number of ship deployment
Sailing speed and total sailing time
90
Figure 3 Impact of fuel price change on sailing speed , sailing time and number of ship deployment
8000
600000
6000
400000
4000
200000
2000
0
Bunkering amounts
Total cost
800000
0 6000
7000
8000
9000
10000
11000
Container transportation amounts（TEU） Bunkering amounts( Ton)
Total cost(USD)
Figure 4 Impact of container transportation amounts on shipping cost and refueling amounts
9
60 8
45 30
7
15 0
6 6000
7000
8000
9000
10000
Container transportation amounts(TEU) Number of ship deployment
11000
Number of ship deployment
Sailing speed and total sailing time
75
total sailing time(days)
Sailing speed (kn/h)
Figure 5 Impact of container transportation amounts on sailing speed, sailing time and number of ship deployment
24
600000
8000 7000
500000 Total cost
5000
300000
4000 3000
200000
2000
100000
Bunkering amounts
6000
400000
1000
0
0
4000
5000
6000
7000
8000
9000
10000
Capacity of fuel tank(Ton) Bunkering amounts( Ton)
Total cost(USD)
Figure 6 Impact of fuel tanker capacity change on shipping cost and refueling amounts
9
70 60 8
50
40 30
7
20 10 0
Number of ship deployment
Sailing speed and total sailing time
80
6 4000
5000
6000
7000
8000
9000
10000
Capacity of fuel tank(Ton) Number of ship deployment total sailing time(days) Sailing speed (kn/h)
Figure7 Impact of fuel tanker capacity change on sailing speed, sailing time and number of ship deployment
25
800000
6000
600000
4000
400000
2000
200000
0
Total cost
Bunkering amounts
8000
0 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Carbon tax rate（USD/ton) Bunkering amounts( Ton)
Total cost(USD)
Figure 8 Impact of carbon tax rate change on shipping cost and refueling amounts 9
60 8
45 30
7
15 0
6 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Carbon tax rate(USD/ton) Number of ship deployment
2.4
Number of ship deployment
Sailing speed and totalsailing time
75
total sailing time(days)
Sailing speed (kn/h)
Figure 9 Impact of carbon tax rate change on sailing speed, sailing time and number of ship deployment
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Highlights3 1 Problems of refueling, ship speed determination and ship deployment are considered. 2 Refueling ports and amounts, ship speed and deployment number are decision variables. 3 Proposed model is solved by transforming into a nonlinear continuous programming model. 4 Impact of fuel price, ship tanker capacity and carbon tax rate on decision is analyzed
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