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Ind. Eng. Chem. Res. 2010, 49, 6104–6114
Operation Planning of Multiparcel Tankers under Fuel Price Uncertainty H. C. Oh† and I. A. Karimi*,‡ The Logistics InstitutesAsia Pacific, and Department of Chemical & Biomolecular Engineering, National UniVersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Maritime logistics is the workhorse of the growing global chemical trade. Fuel expenses claim up to 90% of the daily operating cost of a multiparcel tanker, and a tanker’s fuel consumption varies with the cube of its speed. This combined with high fuel prices makes it crucial that the tanker owners manage tanker speed and fuel purchases in a prudent manner. The volatility in fuel prices and their significant variations across refueling ports make this a difficult task, and systematic optimization can aid the decision-making process. Surprisingly, no literature model addresses this issue for multiparcel tankers. We propose a novel mixed-integer linear programming model that optimizes the operation of a multiparcel tanker in the presence of uncertain fuel prices. Given the route and cargo details of a tanker, it determines the optimal voyage speeds and a refueling plan that minimizes the expected total operating cost. We use a case study derived from industrial data to illustrate the application of our new model. 1. Introduction The global chemical trade achieved an impressive 14% average annualized growth between 2000 and 2006 to hit more than U.S. $1.24 trillion in 2006 as reported (Table 1) by the World Trade Organization.1 Maritime transport of bulk liquid cargos among chemical producers worldwide is critical in supporting this growth. As a result, the capacity of oil, chemical, and liquid gas tankers (300 gross tons and over) grew 3% annually between 2001 and 2005 to reach 368.4 million dead weight ton (dwt) at the beginning of 2005.2 However, it is not smooth sailing for the tanker owners. The shipping sector, which enjoyed a boom in the past few years, is now staring at a slower growth. In recent years, all ship owners faced the constant threat of weak voyage earnings due to the high fuel prices that almost doubled at one stage from 2006 to 2008. Given that fuel expenses can claim up to 90% of a tanker’s daily operating cost, and a tanker’s fuel consumption depends on the cube of its speed, a prudent refueling plan and sound management of vessel’s fuel consumption are crucial to the profitability of tanker owners. This is especially so in the current business environment, where the world is barely emerging from a global recession. Tanker owners naturally seek low-cost refueling options to reduce their total operating expenses. They may deviate slightly from their normal voyage plans, incur necessary port dues, or even delay the transit through a canal to refuel at a port with attractively priced fuel. Alternatively, they may resort to lowering voyage speeds, since the fuel consumption rate is proportional to the cube of cruising speed.3 Indeed, several major shipping companies4 like Torm, Orient Overseas Container Line Ltd. (OOCL), Maersk, and China Ocean Shipping Company (COSCO) recently lowered the cruising speeds of their vessels to cope with the rising fuel costs. Moreover, the fuel prices are also highly unpredictable and can exhibit significant variation across bunkering ports. Finally, the decisions of refueling and speed reduction must be made within the limits of various operational constraints such as the pickup/delivery laycans of cargos, tonnage limits of tankers, etc. All these make it difficult * To whom correspondence should be addressed. E-mail: cheiak@ nus.eds.sg. Tel.: +65 6516-6359. Fax: +65 6779-1936. † The Logistics Institute. ‡ National University of Singapore.
to determine optimal voyage speeds and an overall bunkering plan based on just the experience and judgment of individuals. The cubic correlation between fuel consumption rate and vessel speed, impact of speed on the schedule, and other key performance indicators (KPIs) of a vessel, and uncertainty in fuel prices make the tasks of refueling and managing a vessel’s speed to reduce overall operating costs complex. This clearly requires a systematic optimization approach. To the best of our knowledge, limited work exists in the open literature on these issues. In this paper, we first present a concise overview of the refueling practice in the shipping industry and describe relevant literature. Then, we develop a novel model for the operational planning of tankers in the face of uncertain fuel prices. Using a realistic problem of practical scale, we demonstrate the practical effectiveness of our new model and the importance of accounting for fuel price uncertainty and optimizing vessel speeds in the operational planning of tankers. While our work targets tanker operation, it can easily address the operational planning of other vessel types such as container ships, reefers, etc. 2. Ship Refueling The fuel for a ship is commonly known as marine fuel or bunker fuel. It is graded in terms of viscosity in centistokes (cst) at 50 °C. Most commercial marine vessels use fuels with 180 cst, 380 cst, and 500 cst viscosities with 380 cst being the most common. The fuel with lower viscosity generally commands a premium price due to the higher percentage of distillate Table 1. Exports of Various Manufacturing Clusters and Their Annual Growtha manufacturing cluster
annual change (%)
manufacturing exports (U.S. $ billions in 2006)
2000-2006
2005
2006
374 1248 1451
17 14 7
17 12 11
18 13 13
1016 219 311
10 5 8
7 5 7
10 7 12
iron and steel chemicals office and telecom equipment automotive products textiles clothing
a Data source: International trade statistics 2007 by World Trade Organizations.
10.1021/ie901551j 2010 American Chemical Society Published on Web 05/26/2010
Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
fuel. Typically, ship owners purchase their marine fuel from spot markets (as individual transactions) or on a contract basis, where the purchases are made under forward contracts. They can purchase the fuel directly from major oil companies and independent suppliers at various ports, or indirectly through third parties like traders and brokers. While nearly every port involved in the ocean-going trade sells the marine fuel, a limited number of ports in strategic locations dominate the sales due to high ship-traffic or trade-volume. These ports are generally located along the major trade routes that allow the ships to make stopovers without a major deviation from their voyage schedule. Some examples of such ports are those located along canals such as Panama and Suez, along major straits such as Singapore, Gibraltar, Fujairah, and Istanbul, and in the middle of open sea routes such as Malta, Southern Africa, Canary Islands, and most Caribbean islands. The process of loading a marine fuel into a ship’s fuel tank is known in the maritime industry as bunkering, and ports that sell such a fuel are called bunkering ports. The marine fuel is delivered to ships in two ways. First, bunkering barges carrying the fuel can pull up alongside a ship and transfer marine fuel at 200 to 1500 t per hour. Marine and Energy Consulting Limited reported that ship-to-ship bunkering deliveries accounted for approximately 80% of the total marine fuel delivered in 2005. Second, it can also be delivered to ships through pipelines at berths or terminals. The average delivery rate of pipelines is 450 t per hour. In practice, ship operators make their refueling decisions after monitoring the market prices and trends through the use of trade publications/indices or brokers, and searching for the best possible prices on their trade routes. Prior to the arrival at a port, the ship owner or a broker working on behalf of the ship owner will typically contact the fuel suppliers at the port to get quotations. The refueling process will proceed, once the two parties reach an agreement on the bunkering price and refueling timelines. 3. Previous Work Since Merrill Flood’s pioneering work5 in the area of tanker routing and scheduling was published in 1954, many ship routing and scheduling models have appeared in the literature. Three papers4,6,7 have reviewed the status of ship routing and scheduling research so far. Over the years, the ship routing and scheduling models have become increasingly realistic and industrially relevant with several of them developed in response to the industry needs. For example, the high fuel prices in the 1970s escalated the operating costs of vessels, and shipping companies began to focus their attention on fuel-saving measures such as the reduction of cruising speeds. As a result, a string of papers for optimizing vessel speeds with routing and scheduling decisions appeared in the following decade. Benford8 reported an algorithm to maximize the profit of a ship that transports a given quantity of a single commodity from a loading port to a discharge port within a stipulated time interval. The decisions included the selection of ships and their speeds. Perakis9 addressed a similar problem. He used calculus to determine the optimal solution that reduced cost by 15% compared to that of Benford’s heuristic. Some work addressed the optimization of vessel speeds without considering the routing and scheduling decisions. Ronen10 presented three closed-form analytical models that determine the optimal speeds in three types of leg, namely income generating leg, positioning leg, and mixed leg. The results of these models are particularly useful to ship owners in their negotiation of delivery dates with clients or cargo
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owners. Perakis and Papdakis proposed nonlinear optimization algorithms to explicitly determine the full load and ballast speeds with the objective of minimizing the total fleet operating costs including the lay-up costs of unused vessels. They also reported a sensitivity analysis of the cost components. In the second part of that paper, Perakis and Papdakis12 addressed two extended versions of the above problem where one or more cost components are staircase functions of time, and the uncertain cost components have known distributions. Most subsequent work7,13–27 focused on the development of vessel routes and schedules without considering vessel speeds as variables. One class of this work involves ship routing and scheduling for given cargos of known quantities, loading/ unloading ports, and time windows. Al-Khayyal and Hwang25 defined it as a cargo routing problem, which Li et al.27 termed as a cargo service problem. Fagerholt and Christiansen21,22 addressed multiship pickup and delivery of dry bulk cargos, and Jetlund and Karimi23 focused on liquid bulk cargos. The primary aim of all this work is to meet the basic operational needs of the tanker companies that require short turnaround times to generate good routes and schedules for their fleets. The common approaches have been to use novel heuristics that can determine good solutions efficiently by (1) novel mathematical formulations19,23 that make the problem more tractable, or (2) using smart search algorithms.24 The other class of ship routing and scheduling problems, termed by Li et al.27 as an inventory service problem, involves maintaining material inventories at given ports rather than transporting given cargos to different ports. The aim is to ensure the continuity of operations at various sites at minimum cost. Doror and Ball14 defined this as a distribution problem where a central supplier ensures that no customer runs out of stock at any time. Miller et al.15 addressed a problem in which a supplier ships multiple chemicals from one port to multiple destinations to maintain certain inventory levels. Recently, Christiansen20 addressed a single-commodity problem without a central supplier. Heterogeneous ships transport the material among production and consumption ports to ensure adequate inventory level at each site. Al-Khayyal and Hwang,25 and Li et al.26,27 extended the work of Christiansen20 to address multiple materials, dedicated ship-compartments, unlimited jetties at each port, etc. Li et al.26,27 have developed novel mixed-integer linear programming formulations (MILP), included several additional reallife features, and rectified some issues with previous work. It is interesting to note that the above-mentioned papers did not account for the time needed for refueling and the uncertainty in fuel prices. In this paper, we aim to develop a methodology that (1) complements the existing decision-support tools for ship routing and scheduling, and (2) helps decision-makers manage vessel operation in the presence of uncertain fuel prices and relevant operating constraints. The key decisions in our methodology include refueling and vessel speeds. 4. Problem Statement A tanker operator is planning its operation at time zero. The tanker is either at a port or in high seas at this time. Its current voyage plan calls for K + 1 port visits (k ) 0, 1, ..., K) in a known sequence. If the tanker is already at a port at time zero, then visit k ) 0 refers to this port. If not, then it refers to the upcoming port at time zero. k ) 1 refers to the port after that and so on. A leg is the voyage between two successive ports. Thus, the voyage plan comprises K + 1 legs which we also denote using k. Leg k is the voyage between ports (k - 1) and k. The tanker ends leg k, when it reaches port k. If the tanker is
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Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
Figure 1. Schematic illustrating the bunkering options and price uncertainty.
in the high seas at time zero, then k ) 0 refers to its current incomplete leg, and if it is already at a port, then k ) 0 refers to a leg of length zero. Before entering (leaving) a port, it requires some time for port clearance. The tanker’s current voyage plan specifies the following information for each leg. 1. Cargos that the tanker will load defined by LCk ) {i | the tanker loads cargo i at the end of leg k}, 0 e k e K. 2. Cargos that the tanker will unload defined by UCk ) {i | the tanker unloads cargo i at the end of leg k}, 0 e k e K. 3. Cargo weights wi, i ∈ LCk ∪ UCk. 4. Pickup laycan of each cargo i ∈ LCk as given by EPTi (the earliest pick time for cargo i) and LPTi (the latest pick time for cargo i). 5. The ballast water usage during each leg. 6. Total weight (Wk) of cargo(s) that the tanker will carry during leg k. 7. The tanker maintains a constant speed during each leg. This speed may vary from leg to leg. Its current speed (V0) in leg k ) 0 is known and fixed. It has Sk discrete and known options (Vsk, 1 e k e K, s ) 1, 2, ..., Sk) for voyage speed during leg k. On the basis of the load that it will carry during leg k, its fuel consumption rate for speed option s can be computed on the basisof some correlation between fuel consumption rate, voyage speed, and vessel load. During its voyage, the tanker purchases fuel from the spot market only. At present, we do not consider the option of fuel purchase via forward contracts. At the end of leg k, the tanker has Ok options (o ) 1, ..., Ok) for refueling. These options differ in aspects such as fuel price, bunkering location, and voyage speed to the bunkering location, etc. The fuel prices for these options are uncertain except at the end of leg 0. Since the tanker may be at a port, or will reach the next port shortly after time zero, it is reasonable to assume that the fuel price of each refueling option after leg 0 is fixed and known. For example, consider Figure 1, where a tanker is at P4 at time zero and due to visit P11 and then P7. When at P4, the tanker has three refueling options (o1, o2, and o3). After it reaches P11, it has two refueling options (o4 and o5). The fuel prices for o1, o2, and o3 are fixed and known, but those for o4 and o5 are uncertain. For subsequent legs (k g 1), we postulate N discrete overall price scenarios (n ) 1, 2, ..., N) with known probabilities (πn, 1 e n e N). Several factors cause the variation in fuel prices within the refueling options. One is the overall price trend with respect to time. This will be highly correlated among various options. The other is the individual characteristic of each refueling option such as its geographical location, pricing policy, etc. We use a scenario-based approach to describe the fuel price uncertainty. We consider all the refueling options at all legs in
a combined manner to define the N price scenarios. The N scenarios themselves and their probabilities can be identified on the basis of historical price data for various options. Readers may refer to Smith and Maccardle,28 and Borison and Hamm29 for more details on how such scenarios and their probabilities can be derived. The tanker operator wishes to determine where to refuel, how much to refuel, and the voyage speed during each leg to minimize the expected total cost of operation, which involves refueling expenses, port dues, and daily operating cost. The operational plan must satisfy all the relevant operational constraints such as cargo pickup laycans, fuel tank capacity, minimum fuel level, maximum vessel speed, and vessel tonnage limit. In this work, we assume the following. 1. In line with practice in the industry, cargos have no delivery laycans, or the delivery laycans are not critical. 2. No refueling after the last leg (k ) K). That decision belongs to the next operational planning exercise. 3. The tanker uses at most only one refueling option after each leg. 4. The fuel prices for all options at the end of leg 0 are fully known. 5. The tanker uses one constant speed for each leg, and the selection of this speed is independent of uncertain fuel prices. Allowing the speed to depend on future fuel prices will result in a more complex model, as the number of scenarios will increase drastically and model solution may become impractical. Moreover, implementing such a strategy in practice may be a significant challenge for shipping operations. 6. The time required for port clearance is the same for entering or leaving a port. 7. Cargo loading and unloading are sequential (one cargo at a time) at all ports. 8. The loading/unloading sequence is generally fixed a priori, as it affects ship stability. 9. The tanker’s current schedule satisfies all the laycan constraints pertinent to the cargos to be loaded at the end of leg 0. 5. Problem Formulation Using a scenario-based approach, we combine together the various scenarios for all options at all legs to identify N overall fuel price scenarios (n ) 1, 2, ..., N). Knowing the probabilities of various price scenarios for each option, we can compute the probability of each overall scenario. Since there will be some correlation among the fuel prices for various options, the total number of overall scenarios may not necessarily be too high. The selection of refueling option will be scenario-dependent except for leg 0. Therefore, we define
Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
xo )
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{
1 if the tanker uses refueling option o after leg 0 0 otherwise
xokn )
{
1 if the tanker uses refueling option o after leg k in the event of scenario n 0 otherwise
1 e n e N,
1 e k < K,
1 e o e Ok
Figure 2. Schematic of a leg k and tanker activities.
where Ok is the number of refueling options available to the vessel after leg k. To ensure at most one refueling option after each leg, we use O0
∑x
o
(1a)
e1
o)1
the end of leg 0, and Qokn (1 e k e K - 1, 1 e o e Ok, 1 e n e N) denote the weight that it purchases from option o at the end of leg k during scenario n, and RRok be the refueling rate for option o at the end of leg k, then the above activities will determine the time at which the tanker reaches the port after leg k as follows.
Ok
∑
O0
1 e k < K,
xokn e 1
1eneN
(1b)
T1 g T0 + χ1 +
o)1
∑ o)1
[ ]
S
1 Qo + D1 [Vs1-1ys1] + RRo0 s)1
∑
O0
∑t
o0xo
o)1
(4a)
Since the voyage speed is independent of price scenarios, we define
{
1 e k e K,
T2n g T1 + χ2 +
)1
∑
∑t
T(k+1)n g Tkn + χk+1 +
∑ o)1
1ekeK
(2)
Sk+1
D(k+1)
Let Dk (0 e k e K) be the distance that the tanker travels during leg k. Let T0 ) D0/V0 denote the time at which the tanker ends its journey for leg 0 and arrives at port 0, but does not enter it yet. Similarly, let T1 denote the time at which it ends the journey for leg 1, and Tkn denote the time at which it ends leg k (1 e k e K) in scenario n. Note that the geographical locations of various refueling options affect the tanker schedule. Furthermore, since the refueling options after leg 0 are deterministic, T1 is scenario-independent. Clearly, the tanker must reach (not enter) each port k in time to load all pickup cargos, that is, before the cargo laycans expire. T1 e ( min LPTi) - τ20
(3a)
i∈LC1
Tkn e ( min LPTi) - τ(k+1)0
2ekeK
(3b)
where, τk0 is the administrative time required to enter port (k 1) during leg k. Note that the bounds in eq 3a are simple variable bounds rather than nontrivial constraints. Figure 2 shows some of the activities of the tanker during leg k. After reaching a port, the tanker may request port clearance for entry, unload cargos, load cargos, refuel at that port, clean tanks, await port clearance for exit, clean tanks, sail to another port for refueling, and then sail to the next port in its voyage plan. The refueling activity at a port may require additional time. If the tanker does not refuel at the current port, then it may need to travel additional distance to refuel. If refueling is possible at the current port, then cargo loading/ unloading and refueling may often proceed simultaneously, and may not require additional time. Let Qo denote the weight of fuel that the tanker purchases from option o (1 e o e O0) at
∑
o1xo1n
o)1
(4b) Ok
s)1
i∈LCk
∑
O1
1 e s e Sk
Sk
sk
S
2 Qo1n + D2 [Vs2-1ys2] + RRo1 s)1
o)1
Since a tanker uses only one speed during each leg, we also have,
∑y
[ ]
O1
1 if the tanker speed is Vsk during leg k ysk ) 0 otherwise
[ ]
Qokn + RRok
[Vs(k+1)-1ys(k+1)] +
s)1
Ok
∑t
okxokn
o)1
2ek
