KEY WORDS: Fuel consumption, budget, sailing speed, weather condition, robust optimization

Transcription

1 Du, Meng and Wang 0 0 ABSTRACT This paper proposes a practica fue budget probem which aims to determine a group of buner fue budget vaues for a iner container ship over a round voyage under uncertainties caused by severe weather conditions. The proposed probem hods a erne position in the ship fue efficiency management programs advocated by container shipping ines due to the downward pressure of soaring buner prices, according to our research coaboration with a goba container shipping ine in Singapore. We consider the synergetic infuence of saiing speed and weather conditions on ship fue consumption rate when estimating the buner fue budget of a ship over a round voyage. To address the adverse random perturbation of fue consumption rate under severe weather conditions, we empoy the state-of-the-art robust optimization techniques and deveop a robust optimization mode for the fue budget probem. The deveoped mode can be duaized into a mixed-integer inear programming mode which may be soved by commercia optimization sovers. However, agorithmic findings in the fied of robust optimization provide a poynomia time soution agorithm, and we retrofit it to accommodate the proposed ship fue budget probem. The case study on an Asia-Europe service demonstrates the computationa performance of the proposed soution agorithm, and the competence of the proposed robust optimization mode to produce fue budget vaues at different eves of conservatism possessed by the fue efficiency speciaists in container shipping ines. KEY WORDS: Fue consumption, budget, saiing speed, weather condition, robust optimization

2 Du, Meng and Wang 0 0 INTRODUCTION Buner fue prices have been soaring in the past years from about 00 USD/MT to around 00 USD/MT. For instance, the spot maret price of IFO 0 in Singapore increased from ower than 00 USD/MT in the first quarter of 00 to higher than 00 USD/MT at the same period of 0, and has remained above 00 USD/MT since then. High buner prices mae buner cost become a arge portion of the operating costs for a container shipping ine. Ronen () points out that buner cost wi account for three quarters of the tota operating costs of a arge container ship if the buner fue price exceeds 00 USD/MT. This poses considerabe downward pressure on the revenue of container shipping ines. To mae things worse, the current economic crisis has resuted in the sump of shipping demand which further crushes the profit margins of container shipping ines. To reieve the financia burden caused by the increasing buner cost, container shipping ines have been advocating ship fue efficiency management programs of various inds. In a ship fue efficiency management program, budgeting the fue consumption of each container ship in the feet over a panning horizon (say over a round voyage) is of significant importance. In fact, the buner fue budget probem for each container ship forms the basis of the entire ship fue efficiency management program. In the strategic or tactica eve, to aocate buner budget among various shipping routes, one needs to estimate the fue consumption of each container ship over each round voyage. In the operationa eve, fue efficiency speciaists in a container shipping ine have to ceary understand the fue consumption profie of each container ship over a round voyage at different operationa conditions to provide benchmars for impementing an as-and-inspection fue contro mechanism between captains on board and on-shore officers. 0 Actua 0 Estimated Fue consumption rate (MT/day) r F = 0.00 V Speed (nots) FIGURE Fue consumption rate of a 000-TEU container ship (S) at different speeds: R = 0.00 However, it is chaenging to precisey estimate the buner fue consumption of a container ship in a panning horizon, even over a round voyage, since the fue consumption of a ship in a time unit (say one day) is infuenced by many factors, such as its saiing speed, dispacement, trim, and weather/sea conditions experienced, in an extremey compicated way (). Among these factors, saiing speed is the main determinant. Figure iustrates a quantitative reationship between the fue consumption rate ( r F ) of a 000-TEU container

3 Du, Meng and Wang ship (abeed as Ship S hereinafter) and its saiing speed (V ), based on rea data coected from a goba container shipping ine. It can be seen that the saiing speed can expain up to 0% of the fue consumption. However, it shoud be noted that weather conditions wi aso significanty affect the fue consumption rate. Figure depicts the fue consumption rate of ship S in bow waves at different saiing speeds. We can observe that the fue consumption of ship S in one day increases dramaticay with wave heights when the ship experiences bow waves. In reaity, the infuence of saiing speed and that of weather conditions (wind, waves) are couped together in a sophisticated way (). Fue consumption rate (MT/day) Wave height: m Wave height: m Wave height: m Wave height: m Wave height: m Wave height: m Wave height: m Ship S Speed (nots) FIGURE Fue consumption rate of a 000-TEU container ship (S) in bow waves The infuence of saiing speed on fue consumption rate has recenty been we recognized by the maritime studies because it pays an important roe in iner shipping networ anaysis (-), incuding shipping networ design (), ship feet depoyment (), ship schedue design (), container assignment (0), and cargo booing and routing (). Notteboom and Vernimmen (), and Ronen () anayze the reationship between buner price, saiing speed, service frequency and the number of ships on a shipping route. Ávarez () taes the ships of different speeds as different ship types when examining the joint routing and depoyment of a feet of container ships, and quantifies the buner cost in the objective of his mode. Fagerhot et a. () discretize the arriva time (equivaenty the saiing speed) at each port ca and formuate the ship speed optimization over a singe voyage as a shortest path probem. Brouer et a. () impicity consider the saiing speed optimization in iner shipping networ design by experimentay evauating severa possibe vesse-speed-route combinations and seecting the most promising one. Wang and Meng (), and Qi and Song () minimize the buner fue consumption of ships by speed optimization under operationa uncertainties, such as random port durations and sea contingency. Goias et a. () and Du et a. () study the berth aocation probem considering fue consumption to evauate the performance of the virtua arriva poicy. Athough the infuence of weather conditions on ship fue consumption rate was reveaed severa decades ago from the viewpoint of nava architecture (; ), it is sedom considered by existing iner shipping studies. The weather routing probem (WRP) of ships exhibits the impact of weather on ship transit time and sea-eeping (-). Unfortunatey, it overoos the infuence of weather conditions on ship fue consumption. Lin et a. ()

4 Du, Meng and Wang capture the infuence of weather conditions on fue consumption during saiing in their three-dimensiona modified isochrones method. However, the propeer resoution speed of the ship aong the optima route is assumed to be constant. We note that the synergetic infuence of saiing speed and weather conditions on fue consumption of ships is usuay ignored. The uncertainties in ship fue consumption rates caused by variabe weather conditions are not captured by existing studies. Furthermore, more importanty, studies on budgeting ship fue consumption in a panning horizon, which intrinsicay requires to consider the synergetic infuence of saiing speed and weather conditions and the uncertainties in fue consumption resuting from variabe weather conditions, are not found. This poses a gap between industria needs and academic studies. Objectives and Contributions This study deas with the fue consumption budget probem of a singe container ship over a round voyage by incorporating the couped infuence of saiing speed and weather conditions and the uncertainties in fue consumption, utiizing the state-of-the-art robust optimization techniques (-). The robust optimization mode and the corresponding soution agorithm, which wi be presented in the subsequent sections, can produce different fue budget vaues refecting different conservatism eves of fue efficiency speciaists in container shipping ines. The contributions of this study are threefod: (a) this study proposes the fue consumption budget probem of a singe container ship over a round voyage, which is a new research topic in maritime studies; (b) it addresses the synergetic infuence of saiing speed and weather conditions on ship fue consumption which is sedom considered in iterature; and (c) this study taes an initiative to extend the appications of robust optimization approaches to iner shipping networ anaysis. The remainder of this paper is organized as foows. We first introduce the fue consumption budget probem for a singe container ship over a round voyage, and buid a nomina mathematica mode. Then, we proceed to deveop a robust optimization mode to address the fue consumption uncertainty over each saiing eg. Third, we give a poynomia time agorithm according to the theoretica findings of Bertsimas and Sim () on robust optimization. At ast, we report experimenta resuts and concude this study. FUEL CONSUMPTION BUDGET PROBLEM FOR A SINGLE CONTAINER SHIP AND THE NOMINAL MATHEMATICAL MODEL Probem Statement Consider a iner shipping service operated by a container shipping ine. A round voyage of a iner shipping service typicay consists of a sequence of port cas: N N N + th, namey the ast, port + +, in which the ( ) ca represents the same container port as the first ca. The voyage from the port ca is referred to as the saiing eg of the service, {,,..., N} th to ( + ) th. For each port ca EARLY LATE, each ship depoyed shoud compy with an arriva time window a, a and stay at this port with time duration p (hours). Meanwhie, denote the saiing distance of eg by d (n mie). Tae the LP service operated by American President Lines (APL) in Tabe for exampe, there are totay port cas: Ningbo (NTB) is the first port ca, and the subsequent Yangshan (YAN), Yantian (YAT) and Singapore (SIN) are the nd, rd and th port

5 Du, Meng and Wang 0 ca. Among saiing egs, Hamburg (HF) to Rotterdam (RTM) is the th one which is -nm ong. If we defined the departure time from Ningbo as time zero, the ship shoud arrive at Rotterdam after saiing over eg between time and (hours). After experiencing hours of maneuvering, anchoring, pioting and container handing, the ship wi eave Rotterdam and begins its ong-time saiing over eg to the Suez Canne (SUZ) which is virtuay considered as a port. TABLE Shipping schedue of service LP pubished by APL Saiing eg Port Time window (destination port) Origin Destination Distance Duration Eary arriva a Late arriva NTB YAN YAN YAT 00 YAT SIN 0 SIN SUZ 00 SUZ KLV 0 0 KLV SOU 0 SOU HF 0 0 HF RTM RTM SUZ 0 00 SUZ JED JED SIN 0 0 SIN YAT 0 0 YAT NTB 0 00 Note: a when an arriva time window is discretized on an houry basis, the eariest arriva time is Eary arriva pus. If we construct a shipping schedue with the arriva time at port ca {,,..., N, N+ } being a and define the departure time from the first port ca t = a + p = (so that the DEPART 0 th N saiing eg can be treated in the same way as other egs), then the transit time a+ a + p hours, and saiing speed v shoud be maintained at d ( a+ ( a + p) ) nots. Given the foowing power function reationship between fue consumption rate ( r F, MT/h) and its saiing speed V : c rf = c V () 0 t of the ship over saiing eg shoud be ( ) as iustrated in Figure, the tota buner fue consumption of this ship over the whoe round voyage can be cacuated as c N N c d F = c( v) t = c ( a ( a p) ) a ( a p) + + = = + + N () c c = c d a a + p = ( ) ( + ( ) ) N a = + It can be seen that the saiing schedue { } ship over a round voyage. determines the tota fue consumption of the

6 Du, Meng and Wang 0 0 The buner fue budge probem in this study attempts to find an optima saiing schedue to minimize the tota fue consumption of a ship over a whoe round voyage. Meanwhie, due to the adverse infuencee of weather conditions, the fue consumptionn rate of the ship over each eg might change (consider ony increase here for our purpose of budgeting fue consumption with upper imits) randomy, but within a pre-definabe interva c c ( v ) c, c ( ) v + +δ, wher re δ can be obtained by the historica weather records and the regression resuts simiar to those in Figure. Our objectivee is to construct robust saiing schedues to minimize the tota fue consumption of a ship over a round voyage under the uncertainties in ship fue consumption rates caused by weather conditions, whichh woud provide credibe fue consumption budget vaues of a ship over a round voyage in a more reaistic sense and thus some usefu benchmar vaues for ship fue efficiency speciaists in shipping ines. Nomina Mathematica Mode If we do not consider the perturbation (uncertainties) of the fue consumption rates during saiing, the buner fue budget probem can be easiy formuated and soved beow by foowing the eegant approach proposed by Fagerhot et a. (). We first discretize the EARLY LATE th arriva time window a, a at port ca into N vaues, denoted by i N A = { a}, and determining an arriva time at this port ca is nothing but to chose a vaue in i= A. With this discretization of arriva time windows, the nomina fue budget probem for a ship over a round voyage bois down to a shortest path probem shown in Figure. Let f i be the fue consumption rate of the ship over the in from the node representing to that for j a +, then the cost, namey the fue consumption, over this in is Finding a minima fue consumption schedue is to find a shortest path from the first node to one of the nodes in the ( N + th ayer. ) a f ( j ( i a + a p )) +. 0 FIGURE A shortest path mode for the nomina fue budget probem Mathematicay, if we define a binary variabe x indicating whether the in from the node for a i j to that for a + is chosen in the shortest path, the nomina fue budget probem can be formuated as a 0- integer programming mode:

7 Du, Meng and Wang ( ) N N NOMINAL j i = + = i= j= N + [NOMINAL] min + ( ) F f a a p x () subject to N N+ x =, =,..., N () i= j= N N+ ji x = x, =,,... N, i =,..., N j= j= () { 0, },,,,...,,...,,,..., x = N i= N j = N + where the objective function expressed by Eq. () cacuates the tota fue consumption aong a feasibe path in Figure. Constraints () impose that exacty one in is chosen for each saiing eg; constraints () ensure the fow conservation, and constraints () define the binary decision variabes. ROBUST OPTIMIZATION MODEL UNDER UNCERTAINTIES We now consider the uncertainties in ship fue consumption rates caused by random weather conditions, especiay the adverse infuence of bad weather in the reaistic buner fue budget probem. Due to the adverse infuence of bad weather, the rea fue consumption rate of the i j ship under consideration over the in from the node for a to that for a +, denoted by f, is assumed to randomy change in f, f + δ, where f is the nomina fue consumption rate, and δ > 0 refects the adverse infuence of weather conditions. However, the exact probabiity distribution of f is genaray hard to obtain (or to pass the statistica test for common types of probabiity distributions). Based on the experience of ship fue efficiency speciaists in the container shipping ine, the number of saiing egs on which the fue consumption rate of this ship perturbates above its nomina vaue basicay does not Γ,,..., N and its exceed Γ, among totay N saiing egs over a round voyage. { } specific vaue refects the estimation on the occurance of severe weather conditions, and thus represents the conservatism eve of the ship fue efficiency speciaists in the container shipping ine. Let A = {( i,, j) =,..., Ni ; =,..., N; j=,..., N + } denote the set of ins in Figure. To hedge against the worst case when the fue consumption rates over Γ among N saiing egs randomy increase, the objective fucntion shown in Eq. () shoud be retrofited as min ( + ( )) max δ + ( ) ( i,, j) A { SS A, S Γ} ( i,, j) S ( ) F = f a a + p x + a a + p x ROBUST j i j i To simpify the mathematica expression, we introduce: j i j i g = f a a + p, Δ = δ a a + p,, i, j A ( + ( )) + ( ) ( ) ( ) The robust optimization mode under uncertainties can be formuated as beow: ROBUST min F = [ROBUST] g x + max Δ x ( i,, j) A { SS A, S Γ} ( i,, j ) S subject to constraints ()-(). () () () ()

8 Du, Meng and Wang The second term of objective function () with the max operator is equivaent to a inear programming probem: max Δ x y (0) subject to Let,( i,, j) ( i,, j) A ( i j) 0 y,,, A () ( i,, j) A y Γ μ A and λ be the dua variabes with respect to of constraints () and (), respectivey. Soving the inear programming mode (0) - () is equivaent to soving its dua program: min Γ λ + μ () subject to ( i,, j) A x, (, i, j) ( i j) A () μ + λ Δ A () λμ, 0,,, Mode [ROBUST] can thus be rewritten as foows: ROBUST [ROBUST] min F = g x +Γ λ + μ ( i,, j) A ( i,, j) A subject to constraints ()-(), () and (). Compared to mode [ROBUST], mode [ROBUST] has more decision variabes. However, mode [ROBUST] becomes a mixed-integer inear programming (MILP) mode which coud be soved by a number of optimization sovers such as CPLEX and Gurobi. In fact, we can do better to sove the robust mode. As a component of the robust optimization theory, Bertsimas and Sim () prove that the robust counterpart of a poynomiay sovabe combinatoria optimization probem is aso poynomiay sovabe and propose the soution agorithm. We appy their theoretica findings and soution agorithm to mode [ROBUST], and describe them in next section. SOLUTION METHOD We rearrange the in index set A as O in the decreasing order of,( i,, j) namey, where = o, () () Δ A, Δ Δ... Δ O () O A. Based on this new index set,,,(,, ) g x i j A are repaced by o g and x o O respectivey. We define Δ = 0. The cosey-reated theoretica findings of O + Bertsimas and Sim () can be re-expressed by the foowing theorem, for our specific mode [ROBUST]. Theorem. Mode [ROBUST] can be optimay soved by soving totay O + nomina shortest path probems: F ROBUST = min G O () =,..., + where for a specific, the probem G is defined as

9 Du, Meng and Wang O G =Γ Δ + min go xo + ( Δo Δ) xo o= o= in which the first term is a constant, and the second term is a nomina shortest path probem. Proof. Foow the same process of Bertsimas and Sim (), which first eiminates the dua μ, i,, j A based on the structura property of optima soutions, and then λ variabes ( ) by empoying the fact that,(,, ) () x i j A are binary decision variabes. Remars for Theorem : (a) compared to the shortest path probem shown in Figure, the probem o,..., to G increases the cost (buner fue consumption) over in { } g o + ( Δo Δ ) whie it eaves the cost over other ins unchanged; (b) the shortest path probem in the second term of G is independent of the specific vaue of Γ, which supports O G + = the computationa merit that it ony requires soving a set of shortest path probems { } once when the robust fue consumption vaues at different eves of conservatism of industria fue efficiency speciaists are needed no matter how many possibe vaues of Γ are chosen; (c) if Δ =Δ +, the two optimization probems of G and G + wi be the same, which provides an additiona computationa advantage that the times for soving shortest path probems can be reduced to the tota number of different nonzero Δ pus ; and (d) a dummy termina node can be added into the shortest path probem invoved in G to faciitate using the Dstra s agorithm, athough the framewor proposed by Bertsimas and Sim (), and thus the derivation process to robust optimization modes [ROBUST] and [ROBUST], do not support using the dummy termina node and the dummy ins to it. Based on Theorem and the agorithm of Bertsimas and Sim () for a genera combinatoria optimization probem, the soution agorithm for our ship fue budget robust optimization mode can be designed as foows: Soution Agorithm i,, j in A in the decreasing order of its fue consumption Step. Sort the indexes/arcs ( ) ( ) deviation j ( i δ a+ a p) Δ = + and obtain a new index array O : Δ Δ... Δ O Step. For =,,..., O +, sove the shortest path probem G represented by (); * Step. Find = arg min G, and et the optima buner fue budget vaue of the ship over =,..., O + a round voyage be * * G and the robust ship schedue as the shortest path suggested by G. Let us anayze the computationa time compexity of the above soution agorithm. The time compexity of sorting in Step is ( og ( )) O A A ; Step soves shortest path probems with say the Dstra s agorithm O + = A + times, and thus needs

10 Du, Meng and Wang 0 0 computationa time of compexity OA N + = N ; Step finds the minimum among O + = A + vaues and thus consumes computationa time of compexity O( A ). Consequenty, the proposed soution agorithm is a poynomia time method. CASE STUDY We use the Asia-Europe service LP operated by APL in this case study, and the ship under consideration is assumed to be ship S shown in Figures and. The port rotation, port durations and arriva time windows are tabuated in Tabe. Each arriva time window is discretized on an houry basis, which is a fine time-resoution for a ong shipping voyage such as an Asia-Europe service generay asting for more than two months. For the infuence of different discretization granuarities on soution optimaity, the interested readers are referred to the wor of Fagerhot et a. (). The regression curve in Figure and the curve representing a wave height of m in Figure are utiized to define the ower and upper bound of f, f + δ in which the fue consumption rates of S perturbate. Computationa Performance Mode [ROBUST] is a mixed-integer inear programming probem which might be optimay soved by commercia optimization sovers such as CPLEX and Gurobi. To compare the computationa performance of the Branch and Cut (B&C) agorithm and that of the soution agorithm presented above, we sove mode [ROBUST] with both IBM ILOG CPLEX. and the proposed soution agorithm, in which process YALMIP () is used to formuate [ROBUST] in MATLAB. The time imit for the B&C agorithm in CPLEX is set to 00 seconds in view of the efficiency of the proposed soution agorithm. 0 0 Optimaity gap (%) Γ FIGURE Optimaity gaps when CPLEX terminates at the 00-s time imit

11 Du, Meng and Wang The whoe networ totay has + +=0 nodes and arcs over which there are totay 0 different deviation vaues of fue consumption ( Δ ). The B&C agorithm in CPLEX can sove the nomina mode [NOMINAL], i.e. Γ= 0, in ess than second. This can be easiy understood from the theoretica viewpoint because mode [NOMINAL] is a shortest path probem and it possesses the structura property of totay unimoduarity. However, when Γ, mode [ROBUST] seems much harder to sove and CPLEX cannot sove mode [ROBUST] to optimaity within 00 seconds except for Γ=. The optimaity gaps with different vaues of Γ are depicted in Figure. This is party because mode [ROBUST] oses the nice property of totay unimoduarity and much more dua variabes and reevant constraints enter the mode. The proposed soution agorithm needs to sove 0+= shortest path probems. It can sove mode [ROBUST] over this test case to optimaity in seconds according to our experiments, which fuy demonstrates its high computationa efficiency compared to commercia sovers and strongy underpins its industria appication in decision support systems. Robust Shipping Schedues and Price of Robustness The robust shipping schedues wored out by the proposed soution agorithm when Γ,,..., are shown in Tabe. We do not ist the resuts when Γ because the { } probabiity of a ship experiencing -meter bow waves over more than among egs is too ow in practice. Meanwhie, we pot the robust objective vaues, i.e. F ROBUST (fue consumption over a round voyage under uncertainties), and the nomina objective vaues of NOMINAL these robust shipping schedues, i.e. F of the robust schedues, in Figure. Γ TABLE Shipping schedues under different robustness protection eves () Shipping schedue YAN YAT SIN SUZ KLV SOU HF RTM SUZ JED SIN YAT NTB Note: unit: hour; departure time from NTB (first port ca) is considered as time zero. It can be seen that with the increase of the vaue of Γ, mode [ROBUST] pays more and more attention to the robust part (the second and third terms) of the objective function expressed by Eq. () to hedge against increased anticipated uncertainties, which causes the tota objective vaues to increase dramaticay. In other words, when the eves of conservatism of industria speciaists are ifted, the robustness of the shipping schedue is improved to hedge against the perturbation of ship fue consumption rates due to severe weather conditions, but we need to pay more to the ship fue budget and sacrifice the nomina optimaity. Ship fue efficiency speciaists in a shipping ine can choose a suitabe vaue of Γ based on their ris preference eve.

12 Du, Meng and Wang Objective vaue for fue consumption (MT) Nomina objective vaue of the robust soution Robust objective vaue 00 0 Robustness protection eve FIGURE Fue budget vaues of ship S over a round voyage at different robustness protection eves Γ Frequency distribution x 0- Γ = 0 Γ = Γ = Γ = Γ = Γ = Γ =.. Perturbation probabiity: 0. Perturbation probabiity: 0. Perturbation probabiity: 0. Perturbation probabiity: 0. Perturbation probabiity: Fue consumption over a round voyage (MT) FIGURE Distributions of fue consumption of ship S over a round voyage with different perturbation probabiities of buner consumption

13 Du, Meng and Wang Simuation Resuts To vaidate whether the proposed robust optimization mode can produce good fue budget vaues in rea shipping situation, we randomy generate 00 feasibe shipping schedues of service LP, and evauate the fue consumption impied by these schedues under uncertain weather conditions. To simuate the infuence of severe weather, we assume that the actua fue consumption rate of ship S over each networ in independenty perturbs, with α 0.,0.,0.,0.,0., from its nomina vaue f to f + δ. For each probabiity { } vaue of α, we generate 00 random scenarios and cacuate the fue consumption of ship S for each feasibe schedue over each random scenario (totay we have 00 00=0000 schedue-scenario combinations for each vaue of perturbation probabiity α ). The distributions of fue consumption of S, together with the fue budget vaues produced by our robust optimization modes, are poted as the curves/ines shown in Figure. It can be seen that when the perturbation probabiity α 0., the robust objective vaue with Γ= wi be a good budget vaue for buner fue consumption. Simiary, with Γ=,,, the proposed robust mode coud produce good budget vaues if the perturvation probabiity α caused by severe weather conditions is 0., 0. and 0., respectivey. Figure aso indicates the possibiity that actua fue consumption is higher than these budget vaues. This is impicated with the fact that these feasibe schedues (tested in experiments and adopted in practice) are not necessariy optima from the viewpoint of fue consumption management. We thus can see the importance of both robustness anaysis and optima schedue design, which is the spirit of robust optimization theory. CONCLUSIONS This paper has deat with the fue budget probem for a container ship over a singe round voyage, inspired by the iner shipping industria trend in impementing ship fue efficiency management programs. This study taes an initiative to examine this management issue with practica significance in iner shipping studies. To address the adverse infuence of the perturbation of ship fue consumption rates under severe weather conditions on buner fue budget estimation, we empoy the state-of-the-art robust optimization techniques deveoped by Bertsimas and Sim () and buid a robust optimization mode for the fue budget probem. Athough the robust optimization mode can be transformed to a MILP mode with the possibiity to be soved by commercia sovers, we utiize the agorithmic findings on a genera combinatoria probem by Bertsimas and Sim () and design a poynomia time agorithm based on soutions of mutipe shortest-path probems. A case study of the LP service operated by APL demonstrates the computationa competence of the proposed agorithm and shows that the proposed mode can wor out good fue budget vaues at different eves of conservatism under reaistic but uncertain situations. ACKNOWLEDGEMENTS We than three anonymous reviewers for their insightfu comments and suggestions. This study is supported by the research project Anaysis of Energy Consumption and Emissions by Shipping Lines funded by Singapore Maritime Institute. REFERENCES. Ronen, D. The effect of oi price on containership speed and feet size. Journa of the Operationa Research Society, Vo., No., 0, pp. -.. Carton, J. Marine Propeers and Propusion. Esevier Ltd, 0.

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