A Refueling Scheme Optimization Model for the Voyage Charter with Fuel Price Fluctuation and Ship Deployment Consideration

Peng Jia; Weilun Zhang; E Wenhao; Xueshan Sun

doi:10.21078/JSSI-2017-267-12

Article Publicly Available

A Refueling Scheme Optimization Model for the Voyage Charter with Fuel Price Fluctuation and Ship Deployment Consideration

Peng Jia , Weilun Zhang , E Wenhao and Xueshan Sun

Published/Copyright: August 1, 2017

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From the journal Journal of Systems Science and Information Volume 5 Issue 3

Abstract

Due to the long operation cycle of maritime transportation and frequent fluctuations of the bunker fuel price, the refueling expenditure of a chartered ship at different time or ports of call make significant difference. From the perspective of shipping company, an optimal set of refueling schemes for a ship fleet operating on different voyage charter routes is an important decision. To address this issue, this paper presents an approach to optimize the refueling scheme and the ship deployment simultaneously with considering the trend of fuel price fluctuations. Firstly, an ARMA model is applied to forecast a time serials of the fuel prices. Then a mixed-integer nonlinear programming model is proposed to maximize total operating profit of the shipping company. Finally, a case study on a charter company with three bulk carriers and three voyage charter routes is conducted. The results show that the optimal solution saves the cost of 437,900 USD compared with the traditional refueling scheme, and verify the rationality and validity of the model.

Keywords: voyage charter; refueling scheme optimization; ship deployment; bunker fuel price prediction

1 Introduction

Refueling cost occupies a predominant share in the total operating costs of a voyage charter[1,2]. To decide ship refueling locations and volumes is of vital importance for the voyage charter company. Refueling scheme influences the sailing speed and carrying capacity of the ship, as well as involves with the voyage profit closely to determine the economy of voyage charter company. Recent studies on reducing fuel cost and improving profit in shipping industry are found as follows. Evans and Marlow[3] explained that fuel cost occupies a major proportion of ship variable operation costs, and the operation revenue can be increased by reducing fuel cost. Liu[4] suggested a management philosophy that ship owners should control fuel cost before, during, and after a voyage. Huang[5] designed a ship refueling scheme, and studied the relationship between fuel consumption and ship sailing speed. Psaraftis and Kontovas[6] made a survey of speed models in maritime transportation, studying how speed is to be incorporated in fleet and line management models to save cost.

Most of the literatures on ship refueling scheme are qualitative research, and there are a few quantitative studies on refueling schemes, only considering single voyage. Considering that there are many similarities between refueling schemes and inventory replenishment problems, inventory replenishment approaches (such as Syam[7]and Lee[8] ) could be used to solve refueling scheme problem. The former established an inventory replenishment model considering factors such as inventory size restriction, recharge times and recharge quantities, which provides a theoretical basis for inventory management[7]. The latter pointed out that by sharing information, logistics companies could monitor warehouse inventory status in real-time and work out the exact inventory plan, which guaranteed the reasonable inventory level and reduces storage costs when sales demand is meet[9]. Moreover, Cao[10] and Gao[11] also studied stock replenishment problem. The former has established a coordinating model between demand and inventory supply chain under shortage situation and put forward the thought of optimizing inventory scheme according to demand. The latter has set up an inventory control model based on variety selection, which combined commodities and goods shelves distribution with inventory control effectively when making stock replenishment plan. Wang[12] examined the fundamental properties of the containership sailing speed optimization problem which minimized the sum of ship cost, bunker cost, and inventory cost, and then developed a pseudo-polynomial-time solution algorithm for ship speed optimization on a single route with a continuous number of ships. Yao, et al.[13] developed an appropriate model to optimize the refueling ports, refueling amount, ship speeds jointly in order to obtain an optimal management strategy of bunker fuel for a single shipping liner service, and then studied the effects of bunker fuel prices, port arrival time window and ship bunker fuel capacity on the management strategy of bunker fuel.

Overall, quantitative research about operation economy of multiple voyages is not thorough, and studies on ship refueling beyond multiple voyages are rare. Jia[14], in consideration of refueling problem in continuous multiple voyages based on fuel price prediction, proposed an optimization model of the refueling scheme which is suitable for single-ship single-route scenario.

However, because a shipping company operates multiple ships at the same time, the refueling scheme decision that voyage charter companies pay much attention on is for the ship fleet rather than an individual ship. It is significant to explore an optimal set of refueling schemes for a ship fleet operating on different voyage charter routes for the charter company to maximize the operating revenue.

This paper presents an approach to optimize the refueling scheme and the ship deployment simultaneously with considering the trend of fuel price fluctuations. The rest of the paper is organized as follows. The problem description is presented in Section 2. Section 3 dedicates to formulate the model. A case study on the charter company with three bulk carriers and three voyage charter routes is introduced in Section 4 to examine the model. Finally, the conclusion is drawn in Section 5.

2 Problem Description

In the voyage charter industry, voyage charter companies charge freight from shippers who charter their ships to transport goods. During the operation, all the costs are born by the voyage charter company, except for loading/unloading costs at the port of call. Beenstock and Vergottis[15] analyzed operating costs in voyage charter transportation in 1989. They pointed out that fuel costs occupied a large share in the tramp shipping costs, and a model of operating cost control was established. Thus, in order to reduce costs, charter companies should control the fuel consumption and supply reasonably in continuous voyage charter operations. Refueling decision in the multiple consecutive voyages involves many factors that influence each other and determine the fuel supply scheme and the operation of the overall economy of the ship company. Figure 1 shows the relationship among various factors.

Figure 1

Relationship of factors influencing the refueling scheme

The refueling scheme is to determine a supply whether only to meet next single voyage, or to meet next multi-voyages. The refueling volume has a direct impact on the load capacity of a ship. Thus, over refueling may reduce the cargo volume of a ship, and it will result in the loss of operation profit. Under low fuel price refueling in large quantity once may reduce freight revenue, but can also decrease fuel costs. Thus, it is necessary for the voyage charter company to make refueling scheme decisions according to the fuel price fluctuations. In addition, the reasonable sailing speed of the ship can reduce fuel consumption, influence the operating costs, and then influence refueling quantity and time.

In general situation that the voyage charter company operates a ship fleet transporting cargo for several demands service at the same time, the refueling scheme for the individual ship and the ship deployment need to be determined simultaneously. It is obvious that the refueling scheme and ship deployment interact with each other. Here a nonlinear mixed integer optimization model is proposed to optimize fuel refueling scheme and ship deployment simultaneously, and to provide the optimal decision support for the voyage charter company.

3 Model Specification

3.1 Model Assumption

In addition to many factors related to ship refueling scheme, the factors involving in ship deployment are introduced. The decision becomes more complex. In order to abstract the core problem, the following basic assumptions have been put forward:

Before making the refueling scheme decision, the voyage of the ship and the freight in contract have been determined;
In each port of call the ship is to unload all the goods and load new goods;
During each voyage, the ship is loaded cargo and always runs at the current profit speed;
The shipping date of the contract is not considered;
The anchor days are known in the ports of call;
The types and quantity of the ships are known.

3.2 Fuel Price Forecast Model

In general, voyage charter operation is short term, therefore, short-term forecasting method is used to predict the fuel prices. Autoregressive moving average (ARMA) model is a linear model, which is the most common method of limited parameters used for describing stationary time series. Because of its high accuracy, thus ARMA model is used to predict fuel prices[16,17].

If the time series of fuel price s_t is to meet:

st=st−1+φ1st−1+⋯+φpst−p+εt−θ1εt−1−⋯−θqεt−q,

and for any value of t, E(ε_t) = 0, Var(ε_t) = σε2 > 0, ϕ₁ϕ₂ ⋯ ϕ_p are autoregressive coefficients, θ₁θ₂ ⋯ θ_q are moving average coefficients, ε_t is random variable that is independent and identically distributed), then the time series is called autoregressive moving average model that obeys (p, q) order, and the model is noted as ARMA (p, q).

3.3 The Relationship Model Between Speed and Fuel Consumption

Profit speed refers to the speed at which a ship sails when it targets for the maximum average profit every working day[18]. There is no empty voyage when a ship operates at profit speed, thus the computation formula of profit speed derived theoretically is as follows:

16(tL)×v3+v2=8RkL.(1)

v is the profit speed, t is for berthing time during the voyage, R represents voyage revenue, L denotes voyage distance, k is the function coefficient of the ship, k=24×10−6×B×g×Δ23/c,B signifies fuel price, g is as the fuel consumption rate of main engine (g/(kwh)), Δ represents ship tonnage, and c is the navy constant[19]. The host oil consumed a day is C_e = k₂ × v³, where k₂ is the quotient of ship function coefficient k and the fuel price B. The relationship between fuel consumption with the speed is that C = k₂ × L × v².

3.4 Core Model

Assume that a charter company operates j routes, one ship is to operate N consecutive voyages on a route, and there are M ports to berth in N voyages (annular route). y_kj is 0-1 variable, signifying that ship k operates on route j. x_kji (which can be 0) represents the fuel recharge volume when ship k is docked in voyage i on route j. Whereby, y_kj and x_kji are decision variables. So with the goal of maximizing the total operating profit of the charter company, refueling scheme and ship deployment simultaneously optimization model is formulated as follows:

maxZ=∑k=1m∑j=1n∑i∈Rj(ykj×([min(Dji,Sk−xkji−qkji)×rkji]−(xkji×Bkji)−fk×(Livkji×24+t)))(2)

s.t.ykj=1,route j is operated by ship k,0,others,(3)

∑k=1m∑j=1nykj≤Q(4)

∑i∈Rj(qkji+xkji)≥∑i∈RjCkji,k=1,2,⋯,m,j=1,2,⋯n,(5)

∑i∈Rjxkji≥∑i∈RjCkji,k=1,2,⋯,m,j=1,2,⋯,n,(6)

Ckji=k2×Lji×vkji2,k=1,2,⋯,m,j=1,2,⋯,n,(7)

qkji+xkji≤Mk,k=1,2,⋯,m,j=1,2,⋯,n,(8)

qkji=qkj(i−1)+xkj(i−1)−Ckj(i−1),k=1,2,⋯,m,j=1,2,⋯,n,(9)

16tkjiLjivkji3+vkji2=8×{min(Dji,Sk−xkji−qkji)×rkji}k2×Bkji×Lji,k=1,2,⋯,m,j=1,2,⋯,n,(10)

qkjO=qkjZj=0,k=1,2,⋯,m,j=1,2,⋯,n.(11)

D_ji — cargo to be delivered at port i on route j;

r_kji — freight rate in contract at port i when ship k is operated on route j;

q_kji — remaining oil volume at port i when ship k is operated on route j;

B_kji — fuel price at port i when ship k is operated on route j;

f_k — fixed cost of ship k everyday;

S_k — total deadweight of ship k;

C_kji — fuel consumption from port i to the next destination port when ship k is operating on route j;

M_k — the maximum capacity of tanker of ship k;

L_ji — voyage distance from port i to the next port of destination on route j;

t_kji — anchor time of ship k at port i on route j;

Q — number of routes;

R_j — a set of voyages on route j;

Z_j — number of ports of call on route.

Objective function (2) is the charter companys total profit, which is the difference between total revenue and fuel plus fixed costs. Constraint (3) indicates that if ship k operates on route j, y_kj is 1, and the vice, y_kj is 0. Constraint (4) means that the number of operating routes is less than the given number of routes. Constraint (5) ensures that refueling volume is sufficient to satisfy the next voyage. Constraint (6) illustrates that the total refueling volume of consecutive voyages is equal to the total fuel volume. Constraint (7) explains the relationship between fuel consumption and profit speed. Constraint (8) means that the total amount of remaining fuel at port i and loaded fuel should not exceed the capacity of the oil tank. According to the refueling volume of the last voyage, constraint (9) calculates the remaining fuel volume of the current voyage. The calculation method of profit speed is given in Equation (10). Constraint (11) ensures that there is no fuel in the oil tank at both the beginning and the end of the voyage.

The above model is a mixed-integer nonlinear programming model, and it is solved by Lingo software package. In order to prove the practicability and validity of this model, the actual data are used to validate.

4 Case Study

4.1 Data Collection

It is assumed that a charter company possesses three bulk ships with net deadweight of 70000t, 80000t and 90000t, oil tank capacity of 3500t, 5000t and 7500t, and main engine consumption rate are g₁ = 100 g/(kw⋅h), g₂ = 126g/(kw⋅h), g₃ = 150g/(kw⋅h) respectively. The main motor power at the speed of 24 knot is 20000kw, 24000kw and 27000 kw, and the fixed costs are 6500 USD/Day, 8000 USD/Day and 9000 USD/Day respectively.

The charter company signed a continuous voyage charter party with three routes to operate. Specific voyage is as follows: Route 1, Ningbo-Xiamen-Keelung-Nagasaki-Ningbo. Route2, Ningbo-Singapore-Brisbane-Auckland-Osaka-Pusan-Ningbo. Route 3, Ningbo-Hong Kong-Ho Chi Minh-Singapore-Manila-Kaohsiung-Ningbo. Figure 2 shows the location of each port and route of each voyage. Table 1 is about the voyage distance, freight and cargo demand. Table 2 indicates the anchor time on each route.

Figure 2

Location of Ports and Routes of each voyage

Table 1

Sailing distance and freight rate of all kinds of ship types sailing in every voyage involved in each route

Voyage			Ningbo-	Xiamen-	Keelung-	Nagasaki-
Voyage			Xiamen	Keelung	Nagasaki	Ningbo
	Voyage		610	228	632	483
	distance(nm)		610	228	632	483

Route1	Freight	Ship1	7	4	6.8	5.7
	Freight
	(USD/t)	Ship2	8	6	7.5	7.5
	(USD/t)
		Ship3	9.5	8	8	8.5

	Cargo		75630	54370	68990	74230
	demand(t)		75630	54370	68990	74230

Voyage			Ningbo-	Singapore-	Brisbane-	Auckland-	Osaka-	Pusan-
Voyage			Singapore	Brisbane	Auckland	Osaka	Pusan	Ningbo

	Voyage		2212	3728	6466	4556	370	514
	distance(nm)		2212	3728	6466	4556	370	514

Route2	Freight	Ship1	9	15.5	21.5	17.5	4.5	5
	Freight
	(USD/t)	Ship2	11	17	23	19	6	7
	(USD/t)
		Ship3	12	18.5	24	21	7	8

	Cargo		62780	71300	42900	68320	52500	67420
	demand(t)		62780	71300	42900	68320	52500	67420

Voyage			Ningbo-	Hong Kong-	Ho Chi Minh-	Singapore-	Manila-	Kaohsiung-
Voyage			Hong Kong	Ho Chi Minh	Singapore	Manila	Kaohsiung	Ningbo

	Voyage		875	911	632	1304	543	5654
	distance(nm)		875	911	632	1304	543	5654

Route3	Freight	Ship1	7.8	8.5	7	9.5	5	6
	Freight
	(USD/t)	Ship2	8.5	9.6	7.8	10.5	6	7.2
	(USD/t)
		Ship3	9.2	10	8.3	11	6.8	7.9

	Cargo		73450	54360	65380	47890	43670	36720
	demand(t)		73450	54360	65380	47890	43670	36720

Table 2

Anchor time of all kinds of ship types at every port involved in each route (days)

Port		Ningbo	Xiamen	Keelung	Nagasaki
	Ship1	1	2	2	1

Route1	Ship2	1	2	2	2

	Ship3	2	1	1	2

Port		Ningbo	Singapore	Brisbane	Auckland	Osaka	Pusan

	Ship1	4	4	2	3	4	3

Route2	Ship2	3	4	3	2	2	3

	Ship3	3	3	4	2	2	4

Port		Ningbo	Hong Kong	Ho Chi Minh	Singapore	Manila	Kaohsiung

	Ship1	2	2	1	3	2	3

Route3	Ship2	1	3	2	2	2	2

	Ship3	3	2	2	3	3	2

4.2 Fuel Price Forecast

It is assumed that fuel price in each port is abide with that in Rotterdam market. 180 CST oil price data from January 4th, 2008 to April 9th, 2010, with 118 terms measured in weeks, are collected. During the prediction, unit root test to fuel price is conducted firstly. Then the stationarity time series of fuel price is judged and a stationary series is obtained by differential treatment. Then, to analyze stationary sequence based on autocorrelation and to decide the model structure and order that are suited for time series. Finally, the ARMA model is estimated.

4.2.1 Unit Root Test and Autocorrelation Analysis

The DickyCFuller method is used to test the unit root of original data. The result of the t-test is bigger than the threshold of confident value (5%), indicating that the original data are not stationary. Therefore, the second test of unit root after having conducted the first-order difference to original data is essential. Table 3 shows the unit root test results of original fuel price data after the first-order difference. Now the result of the t-test is smaller than the threshold of confident value (5%), which denotes that the difference sequence of the sample data is stationary. Besides, the DurbinCWaston test statistic is 2.01, thus, the data are not auto-correlated. Table 4 shows that for both autocorrelation coefficient and partial autocorrelation coefficient, there is no apparent truncation. Thus, the difference sequence is fit for ARMA model[20,21].

Table 3

The results of unit root test against the differentiated data

	t-Statistic	Prob.^*
Augmented Dickey-Fuller test statistic	−7.223348	0.0000
Test critical values 5% level	−1.943612
Durbin-Watson stat	2.01

Table 4

Sample autocorrelation of fuel price difference sequence

4.2.2 Parameters Estimation

According to the shape of the autocorrelation figure, a comparatively good prediction structure ARMA (4, 4) is obtained. The values of parameters and statistics are provided in Table 4. As shown in Table 5, the autoregressive correlations are 0.561705, 0.557372, 0.534768 and −0.871121, respectively, and the moving average coefficients are −0.584769, −0.531571, −0.555494 and 0.976241, respectively. t-test shows that each coefficient is significantly not zero, and their standard errors are relatively small. The Durbin-Watson statistical result is 1.877004, which indicates that the sample data are not auto-correlated. Fig. 3 compares the actual oil price and the oil prices predicted by using the ARMA model. Obviously, the prediction is of high accuracy.

Figure 3

Comparison of the real and forecasted data

Table 5

Difference sequence of a good forecast model parameter estimation results

Variable	Coefficient	Std. Error	t-Statistic	Prob.
AR(1)	0.561705	0.039350	14.27448	0.0000
AR(2)	0.557372	0.040682	13.70056	0.0000
AR(3)	0.534768	0.038080	14.04345	0.0000
AR(4)	−0.871121	0.035940	−24.23843	0.0000
MA(1)	−0.584769	0.021530	−27.16008	0.0000
MA(2)	−0.531571	0.028040	−18.95757	0.0000
MA(3)	−0.555494	0.018388	−30.20998	0.0000
MA(4)	0.976241	0.002631	371.0026	0.0000
Durbin-Watson Stat	1.877004
Inverted AR Roots	0.93−0.25i	0.93+0.25i	−0.65−0.72i	−0.65+0.72i
Inverted MA Roots	0.95−0.30i	0.95+0.30i	−0.66−0.74i	−0.66+0.74i

4.3 Fuel Supply Model and Result Analysis

Based on the predicted fuel price and data related to routes and voyages, the model is worked out. The results can be seen from Table 6.

Table 6

Solution results of the optimization model

Route 1: Ship of 80000t, total revenue: 3,717,258USD

Voyage	Ningbo-	Xiamen-	Keelung-	Nagasaki-
Voyage	Xiamen	Keelung	Nagasaki	Ningbo
q_kji(t)	0	1211.5130	798.2933	0
x_kji(t)	4370	0	0	989.1498
v_kji(nm/h)	11.3	12.4	11.7	12.3
C_kji(t)	3158.487	413.2197	798.2933	989.1498
B_kji(USD/t)	443.3525	509.8409	523.5202	473.8013

Route 2: Ship of 90000t, total revenue: 4,369,442USD

Voyage	Ningbo-	Singapore-	Brisbane-	Auckland-	Osaka-	Pusan-
Voyage	Singapore	Brisbane	Auckland	Osaka	Pusan	Ningbo

q_kji(t)	0	1237.743	424.6501	0	0	0
x_kji(t)	1391.166	226.5442	227.1684	1120.575	431.8117	1535.682
v_kji (nm/h)	11.8	12.3	13.4	12.6	11.7	13.1
C_kji(t)	153.4228	1039.637	651.8188	1120.575	431.8117	1535.682
B_kji(USD/t)	538.81	521.9898	521.9061	473.8013	473.8013	438.9343

Route 3: Ship of 70000t, total revenue: 4,162,463USD

Voyage	Ningbo-	Hong Kong-	Ho Chi Minh-	Singapore-	Manila-	Kaohsiung-
Voyage	Hong Kong	Ho Chi Minh	Singapore	Manila	Kaohsiung	Ningbo

1_kji(t)	0	1563.202	856.0417	0	0	0
x_kji(t)	3500	0	0	704.2724	549.99	460.3103
v_kji (nm/h)	13.1	12.7	11.6	12.3	12.3	13.2
C_kji(t)	1936.798	707.1603	856.042	704.2724	549.99	460.3103
B_kji(USD/t)	410.2305	523.6559	523.6559	473.7727	473.7727	473.8013

As it can be seen, ships with net deadweight of 80000t, 90000t and 70000t are assigned to route 1, route 2 and route 3 respectively. In route 1, the ship is refueled at the first and the last port where the fuel price is relatively low. Thus the scheme is reasonable. In route 2, due to the long operating distance of each voyage, in order to guarantee the operation, the ship is refueled properly at each port and refueling volume varies with the fluctuation of fuel price. In route 3, the ship is refueled large amounts of oil when fuel prices are low, and when fuel prices increase, it is no longer supplied but to consume the storage in the fuel tank. The obtained operating and refueling scheme optimization saves the cost of 437,900 USD compared with the traditional refueling scheme (that is to fill up the tank each refueling time).

5 Conclusions

The refueling location and volume in the voyage charter is an important decision that the charter company faces, which is directly related to the operating profit of company. In this paper, a mixed-integer nonlinear programming model is established to simultaneously optimize the refueling scheme and ship deployment considering fuel price fluctuation. Based on the predicted fuel price, the above optimization model of fuel supply scheme is solved and the optimal solution is obtained. Finally, the case study is introduced to prove the applicability of the model. The results show that, considering the fuel price trend of fluctuations, to optimize the ship deployment and the refueling scheme on different routes simultaneously can effectively save operating costs and improve the company’s operating income. In addition, some assumptions in the model, for example, not considering the shipping date restrictions in the contract, are to be relaxed in the further research.

Supported by the National Natural Science Foundation of China (71303026), China Postdoctoral Science Foundation (2015M580128), Liaoning Natural Science Foundation (2015020074), Higher Education Development Fund (for Collaborative Innovation Center) of Liaoning (20110116102) and Teaching Reform Fund of Dalian Maritime University (2014Y18)

References

[1] Zhao G. International shipping management (2nd Edition). Dalian: Dalian Maritime University Press, 2006.Search in Google Scholar

[2] Min D Q. Water transport business management (2nd Edition). Dalian: Dalian Maritime University Press, 2008.Search in Google Scholar

[3] Evans J J, Marlow P B. Quantitative methods in maritime economics (2nd Edition). Fairplay Publications, 2001.Search in Google Scholar

[4] Liu T Y. Shipping fuel cost management of shipping companies. Journal of the World Shipping, 2006, 6: 36–37.Search in Google Scholar

[5] Huang T. Shipping cost analysis and control. Dalian: Dalian Maritime University, 2002.Search in Google Scholar

[6] Psaraftis H N, Kontovas C A. Speed models for energy-efficient maritime transportation: A taxonomy and survey. Transportation Research Part C: Emerging Technologies, 2013, 26: 331–351.10.1016/j.trc.2012.09.012Search in Google Scholar

[7] Syam S, Shetty B. Coordinated replenishments from multiple suppliers with price discounts. Naval Research Logistics, 1998, 45: 579–598.10.1002/(SICI)1520-6750(199809)45:6<579::AID-NAV3>3.0.CO;2-#Search in Google Scholar

[8] Lee H L, So K C, Tang C. The value of information sharing in a two-level supply chain. Management Science, 2000, 46(5): 626–643.10.1287/mnsc.46.5.626.12047Search in Google Scholar

[9] Lee H L, Whang S. Information sharing in a supply chain. International Journal of Manufacturing Technology and Management, 2000, 1: 79–93.10.1504/IJMTM.2000.001329Search in Google Scholar

[10] Cao Z H, Zhou Y W. Supply chain coordination model with demand influenced by shortage for deteriorating items. Journal of Systems Engineering, 2011, 1: 50–59.Search in Google Scholar

[11] Gao J J, Meng Z Q. Joint decision model of variants selection shelf-space allocation and inventory control. Journal of Systems Engineering, 2009, 5: 615–620.10.1109/SOLI.2007.4383939Search in Google Scholar

[12] Wang S. Fundamental properties and pseudo-polynomial-time algorithm for network containership sailing speed optimization. European Journal of Operational Research, 2016, 250(1): 46–55.10.1016/j.ejor.2015.10.052Search in Google Scholar

[13] Yao Z, Ng S H, Lee L H. A study on bunker fuel management for the shipping liner services. Computers & Operations Research, 2012, 39(5): 1160–1172.10.1016/j.cor.2011.07.012Search in Google Scholar

[14] Jia P, Sun X S, Yang Z Z. Optimization research of refueled scheme based on fuel price prediction of the voyage charter. Journal of Transportation Systems Engineering and Information Technology, 2012, 12(5): 110–116.10.1016/S1570-6672(11)60229-8Search in Google Scholar

[15] Beenstock M, Vergottis A. An econometric model of the world shipping market for dry cargo. Freight and Shipping, Applied Economics, 1989, 21: 339–356.10.1080/758522551Search in Google Scholar

[16] Kızılkaya A, Kayran A H. Matrix-based computation of the exact Cramer-Rao bound for the ARMA parameter estimation realized with relatively short data records: Frequenz. Frequenz, 2004, 58(11–12): 278–281.10.1515/FREQ.2004.58.11-12.278Search in Google Scholar

[17] Wang Z L. Time series analysis. Beijing: China Statistical Press, 2000, 157–201.Search in Google Scholar

[18] Yin X Y, Zhang J, Xie X L. Research on the optimal speed of tramp vessels. Navigation of China, 2012.Search in Google Scholar

[19] Xie X L. Shipping operation and management (2nd Edition). Dalian: Dalian Maritime University Press, 2006.Search in Google Scholar

[20] Stopford M. Maritime economics (2nd Edition). Routledge, 2007.10.4324/9780203891742Search in Google Scholar

[21] Friedlander B, Porat B. A spectral matching technique for ARMA parameter estimation. IEEE Trans. Acoust. Speech Signal Process, 1984, 32: 338–343.10.1109/ICASSP.1983.1171965Search in Google Scholar

Received: 2016-6-7

Accepted: 2017-2-7

Published Online: 2017-8-1

Articles in the same Issue

https://doi.org/10.21078/JSSI-2017-267-12

Keywords for this article

voyage charter; refueling scheme optimization; ship deployment; bunker fuel price prediction