Author's Accepted Manuscript

Integrated maritime fleet deployment andspeed optimization: Case study from RoRoshipping

Henrik Andersson, Kjetil Fagerholt, Kirsti Hob-beslanda

PII: S0305-0548(14)00072-0DOI: http://dx.doi.org/10.1016/j.cor.2014.03.017Reference: CAOR3530

To appear in: Computers & Operations Research

Cite this article as: Henrik Andersson, Kjetil Fagerholt, Kirsti Hobbeslanda,Integrated maritime fleet deployment and speed optimization: Case studyfrom RoRo shipping, Computers & Operations Research, http://dx.doi.org/10.1016/j.cor.2014.03.017

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Integrated maritime fleet deployment and speed

optimization: Case study from RoRo shipping

Henrik Anderssona, Kjetil Fagerholta,b,∗, Kirsti Hobbeslanda

aNorwegian University of Science and Technology, Department of Industrial Economicsand Technology Management, Trondheim, Norway

bNorwegian Marine Technology Research Institute (MARINTEK), Trondheim, Norway

Abstract

When planning shipping routes, it is common to use a sequential approachwhere it is first assumed that each ship sails with a given service speed,and then later during the execution of the routes optimize the sailing speedsalong the routes. In this paper we propose a new modeling approach forintegrating speed optimization in the planning of shipping routes, as wellas a rolling horizon heuristic for solving the combined problem. As a casestudy we consider a real deployment and routing problem in RoRo-shipping.Computational results show that the rolling horizon heuristic yields goodsolutions to the integrated problem within reasonable time. It is also shownthat significantly better solutions are obtained when speed optimization isintegrated with the planning of shipping routes.

Keywords: Fleet deployment, speed optimization, rolling horizon heuristic

1. Introduction

Ships are usually operated in one of the three modes: industrial, tramp orliner. An industrial operator owns the cargo and controls the ships, tryingto minimize the cost of delivering the cargoes, similar to a private fleet. Ina tramp operation the ships follow the available cargoes (some of which maybe optional), trying to maximize profit, similar to a taxi service. In liner

∗Corresponding authorEmail addresses: henrik.andersson@iot.ntnu.no (Henrik Andersson),

kjetil.fagerholt@iot.ntnu.no (Kjetil Fagerholt)

Preprint submitted to Computers & Operations Research April 1, 2014

shipping, the ships follow a more or less fixed itinerary with given port callsaccording to a published schedule. Christiansen et al. (2013) provide a recentreview on ship routing and scheduling for the various modes.

When planning shipping routes, it is common to use a sequential approachwhere it is first assumed that each ship sails with a given service speed. Then,later during the execution of the routes, the sailing speed is optimized basedon the time windows that must be respected along the given ship routes.A ship has for all practical purposes a minimum and a maximum cruisingspeed which define the range of speeds at which it can actually sail. Fuelconsumption, and hence the cost of sailing, is strongly dependent on speed.As shown by Ronen (1982), a cubic function often provides a good estimationwithin the practical speed range of the relationship between fuel consumptionper time unit and speed for cargo ships (or a quadratic function per distanceunit).

In this paper, we propose a new modeling approach for integrating speedoptimization in the planning of shipping routes, as well as a rolling horizonheuristic for solving the combined problem. We also show that integratingthese two problems instead of solving them sequentially can give significantlyimproved solutions. As a case study, we consider a fleet deployment problemin Roll-on Roll-off (RoRo) shipping, a segment within liner shipping. Fleetdeployment is a tactical planning problem of assigning available ships in thefleet to voyages that must be performed repeatedly on given geographicaltrades. The results from fleet deployment, in addition to which ship will per-form which voyage, are sailing routes for the ships in the fleet, i.e. each shipis assigned a sequence of voyages to perform, possibly with ballast (empty)sailing between the last port call of one voyage and the first on the next. Thefleet deployment problem used as a case study originates from Wallenius Wil-helmsen Logistics (WWL), a major RoRo shipping company that transportscars, other types of rolling equipment, as well as cargo that can be placedon trolleys for loading and discharging. It should be emphasized that eventhough we use a fleet deployment problem from RoRo shipping in our casestudy, the proposed modeling approach and solution algorithm can be usedfor a wider range of integrated routing and speed optimization problems inmaritime transportation (even in the industrial or tramp shipping modes).

Most of the literature on fleet deployment originates from container ship-ping, which is the major segment of liner shipping, see for example Gelarehand Meng (2010), Meng and Wang (2010), Liu et al. (2011), Wang and Meng(2012) and Zacharioudakis et al. (2011). The models developed usually rely

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on two assumptions: 1) they assign each ship to a single route or rotationduring the whole planning horizon and do not deal with ship routing as such,and 2) they consider ship types and not individual ships. Even though theseassumptions are valid for most fleet deployment problems in container ship-ping, it can be too limiting for other important liner shipping segments, suchas RoRo shipping, where there is more flexibility in the planning of shiproutes. Fagerholt et al. (2009) propose a mathematical model for fleet de-ployment in liner shipping which do not rely on the assumptions given above.However, they do not integrate speed as a decision variable.

There are a few examples in the literature where ship routing have been in-tegrated with speed optimization, though in the industrial and tramp modes.Psaraftis and Kontovas (2013) provide an excellent taxonomy and survey ofspeed models in maritime transportation. Gatica and Miranda (2010) dealwith a heterogeneous fleet of vessels and minimize the cost of servicing aset of mandatory single trip cargoes while determining the speed for eachtrip. By discretizing the cargoes’ service time windows, they can use a net-work model for solving the problem. Norstad et al. (2011) consider a trampshipping problem with a mix of mandatory and optional spot cargoes withassociated service time windows for loading/discharging. They use a special-ized algorithm to optimize the speeds along a given route (Hvattum et al.(2013)) to evaluate moves in a local search heuristic. Norstad et al. (2011)also show that integrating speed optimization with ship routing gives otherand significantly improved solutions compared with the sequential approach.As far as we know, there exist no studies in the literature which use a similarmodeling approach and solution method as the one proposed in this paper.

The remainder of this paper is organized as follows: In the next sectionwe give a thorough description of the integrated fleet deployment and speedoptimization problem we use as a case study. Section 3 presents the math-ematical formulation for the problem with a special focus on the modelingof the ships’ speeds, while Section 4 describes the rolling horizon heuristicproposed for solving the problem. A computational study is performed inSection 5, while concluding remarks are given in the final section.

2. Problem description

This paper concerns fleet deployment, integrated with speed optimization,for the RoRo shipping company Wallenius Wilhelmsen Logistics (WWL).The shipping company controls a heterogeneous fleet of ships where the ships

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have different cargo capacities, sailing speed ranges and bunker consumptionprofiles. At the beginning of the planning period the ships have differentinitial positions, either in a port or somewhere at sea. Furthermore, theships can have different times for when they are available for new voyagessince they must first finish on-going voyages or dry-dockings (i.e. repairs).

Figure 1: Illustration of two trade routes sailed in sequence, Oceania to Europe via SouthAfrica, and South America to North America, with associated ballast sailing from Europeto South America.

A trade is defined as a transportation arrangement from one geographicalregion to another. A trade route consists of a number of loading ports in oneregion and a number of discharging ports in the other. In Figure 1 two traderoutes are illustrated by solid lines and the ports are shown as filled circles. Anumber of voyages have to be carried out on the different trade routes withina specific planning period, for example weekly or bi-weekly, depending ondemand and contractual obligations. Each such voyage must be sailed bya ship and is defined as a mandatory voyage. After a ship has sailed onevoyage on a trade route it often needs to reposition to start on the nextone due to trade imbalances. This repositioning means ballast sailing, i.e.sailing without cargo, which of course should be reduced as much as possible.The ballast sailing between the two trade routes in Figure 1 is illustratedby a stippled line. Differences in contractual requirements and a variety

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in the types of cargo transported on the various trade routes may restrictwhich vessels that can be assigned to a particular trade route, regarding bothcapacity and vessel type.

Each voyage and ballast sailing have given estimated durations dependingon the chosen speeds. This duration includes the sailing time between allports along the route and also the time spent in these ports. There is a timewindow, defined by an earliest and latest start time, associated with eachmandatory voyage which determines within what time interval the voyagemust start. For each voyage there is a penalty associated with a delayedstart of a voyage. This cost may vary between the trade routes and voyagesdepending on how strict the time windows should be. The earliest start timeis set as the day where the voyage preferably should start, and the lateststart time determines an upper limit for the delay. The longer the delay,the higher is the penalty cost. There are costs associated with serving themandatory voyages. Sailing a voyage incurs costs that include port, canaland fuel consumption costs, where the latter depends on the chosen speed.All costs depend on ship characteristics such as size and fuel consumption.

It is also possible to charter in available vessels from the market to servea voyage, though at a charter cost. These are denoted spot ships and areassumed to be available to serve any voyage during the planning period.There is no ballast sailing costs associated with chartering a spot vessel asthis is assumed to be included in the charter costs. It is also assumed thatwhen using a spot ship, the given voyage will start on the preferred day.

For each trade route, a monthly demand for transportation must be met.To fulfill these demand requirements on the various trade routes, there maysometimes be a need for additional voyages (depending on the size of theships assigned to the mandatory voyages on the given trade route). We callthese optional voyages, since there also exists an opportunity to use so-calledspace charter to fulfill these requirements. Space charter means that spacecan be bought from other shipping companies operating on the same traderoutes.

The objective of this integrated fleet deployment and speed optimizationproblem is to determine:

- the ship routes (i.e. which ship should perform which voyages and inwhat sequence),

- the ships’ sailing speeds along their routes,

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- the start time for each voyage,

- which optional voyages to sail,

- which voyages should be serviced by spot ships, and

- for which trades and for which months space charter should be used,

such that:

- all mandatory voyages are serviced within their given time windows,either by a ship from the shipping company’s fleet or by a spot ship,

- the monthly transportation demand is met on all trade routes.

The objective is to minimize total costs, which consist of the sum of the port,canal and sailing costs, the charter costs for spot ships, the space chartercosts, and the penalty costs for delayed start of the voyages.

3. Mathematical modeling

Before we start with the modeling, we need to explain how nodes andarcs should be interpreted. As illustrated in Figure 2, in this model a noderepresents a given voyage for a given trade route. A particular voyage may becharacterized by the trade route r which is sailed and what voyage number ithis voyage has in the series of voyages on that trade route. Thus, the pair isdenoted (r, i) and corresponds to a node in the model. The arcs between thenodes denote the ballast sailing between voyages. For example in Figure 2,the arc ((r, i), (q, j)) represents the ballast sailing from the last port call ofvoyage number i on trade route r to the first port call of voyage number jon trade route q. For a given ship k, the decision variable ykriqj is a binaryvariable that is equal to 1 if the ship performs voyages (r, i) and (q, j) insequence, and 0 otherwise. Figure 2 also illustrates the nodes o(k) and d(k),where node o(k) is the initial position, while d(k) is the artificial destinationof ship k (which corresponds to the last port of the last voyage performed bythe ship and is determined in the optimization).

Section 3.1 describes the modeling of the ships’ sailing speeds, while themathematical model for the problem in Section 2 is presented in Section 3.2.

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yo(k)rixo(k)ri

ykriqjxkriqj

yqjd(k)

o(k)

(r, i) (q, j)

d(k)

Figure 2: Illustration of nodes and arcs for a given ship k in the mathematical model

3.1. Modeling ships’ sailing speeds

As mentioned in the introduction, the bunker consumption for a givenship is a non-linear function of speed. According to Ronen (1982), bunkerconsumption per time unit can be approximated with an increasing convexcubic function of speed for speed values over a certain minimum. The func-tion of bunker consumption per distance is therefore a quadratic and convexfunction.

Figure 3 shows an illustrative graph of bunker consumption per nauticalmile as a function of speed (for a given payload). A ship has to maintain acertain minimum speed to sail safely and run at minimum cost. Furthermore,the installed machinery of the ship gives a maximum speed, which can beviewed as a theoretical upper limit that can be achieved in nice weatherconditions. Hence, the maximum speed used for planning of shipping routesand schedules should be somewhat lower.

To handle this non-linearity computationally, we have chosen to approxi-mate the non-linear fuel consumption function by discrete speed alternativesand linear combinations of these. The principle is illustrated for three speedalternatives which are shown by arrows in Figure 3. In the mathematicalmodel presented in full in Section 3.2 the weight for each speed discretiza-tion point for a given arc ((r, i), (q, j)) is represented by the speed variablexkriqjv, which represents the weight of speed alternative v for ship k alongthe arc. The speed variables are shown in the network in Figure 2. It may benoted that there are no speed variables associated with the arc to the ship’sartificial destination since it does not represent a real sailing.

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Figure 3: The non-linear function for fuel consumption per distance unit as a function ofspeed

8

Since the fuel consumption is a convex function of speed, it is obviousthat linear combinations of the alternatives that are not neighboring speeddiscretization points will give an higher or equal consumption and cost forthe same speed as can be achieved by linear combinations of two neighboringalternatives. This is also illustrated in Figure 3. This means that it is notnecessary to require a special ordered set of type 2 (SOS2) to achieve apiecewise linear approximation of the function, as it will automatically beenforced by the optimization.

It is easy to see that the piecewise linearization of the bunker consumptionfunction causes an overestimation because of the convexity. This is alsoclear from Figure 4, where the stapled line is strictly above the consumptionfunction for all values of speed between speed alternatives 1 and 2. However,there is an additional overestimation because of the non-linear relationshipbetween time t and speed v, where t = s/v, and s is the distance. Theweights of the speed alternatives for the chosen speed are used for calculatingthe corresponding sailing time. Because of the non-linearity this also yieldsan overestimation as illustrated in Figure 4 (though a minor one).

Here is an example to illustrate this issue. Suppose speed alternatives1 and 2 are 17 and 19 knots, respectively. Then, sailing a distance of 1000nautical miles takes 58.82 and 52.63 hours for the two speed alternatives,respectively. If the weights of the two speed alternatives are 0.3 and 0.7,respectively, the modeled speed is 18.40 knots. Interpolation of sailing timeyields 54.49 hours. However, traveling a distance of 1000 nautical miles in54.49 hours only requires a speed of 18.35 knots. This interpolation overes-timation is not substantial, but worth mentioning.

Proposition 1. There is an overestimation of consumption because the speedobtained by speed interpolation, v, is greater than the speed that is requiredfor feasibility v.

Proof. For α ∈ [0, 1], the time restrictions in the model interpolates dura-tion times in the following way

t = αt1 + (1− α) t2 (1)

where t1 and t2 are the durations obtained from using speed alternatives 1and 2, respectively. It follows from this that the speed for a given distance sis given by v

v =s

αt1 + (1− α) t2(2)

9

Figure 4: Illustration of how the model overestimates the bunker fuel consumption (exag-gerated)

10

Costs are however defined by interpolation of speed and v is given by

v = αv1 + (1− α) v2 (3)

Substituting for v1 and v2 gives

v = s

(α

t1+

1− α

t2

)(4)

To show that v ≥ v, consider v − v and show that

v − v = s

(α

t1+

1− α

t2

)− s

αt1 + (1− α) t2≥ 0 (5)

Multiply with the product of the denominators; t1t2 [αt1 + (1− α) t2] ≥ 0

α2t1t2 + α (1− α) t21 + α (1− α) t22 + (1− α)2t1t2 − t1t2 ≥ 0 (6)

Gathering terms yield

−2α (1− α) t1t2 + α (1− α) t21 + α (1− α) t22 ≥ 0 (7)

Since α (1− α) ≥ 0

−2t1t2 + t21 + t22 ≥ 0 (8)

And in fact

(t1 − t2)2 ≥ 0 (9)

Clearly, (t1 − t2)2 is non-negative for all t1 and t2.

As shown above, both errors corresponding to the piecewise linear ap-proximation of the fuel consumption function cause overestimations. Themagnitude of the errors will of course depend on the number of discretiza-tion point used. It should be mentioned that in the illustration of bunkerfuel consumption in Figure 4, the overestimations are exaggerated in orderto clearly show the distinctions.

Psaraftis and Kontovas (2013) discuss the importance of also consideringthe ship payload, as this also influences the fuel consumption and a ship’sloading conditions will generally vary whether it is a loaded or a ballast sailing

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leg. They also argue that the cubic approximation may not be appropriatefor some ship types and speeds. It should therefore be mentioned that thepiecewise linear approximation proposed here can handle any type of convexfuel consumption function, and not only cubic functions. Furthermore, inthe fleet deployment problem considered in this paper, the ships always sailthe pattern loaded - ballast - loaded - ballast, etc., where the loaded leg isthe service on a trade route, while the ballast leg represents the repositioningof the ship to the next loaded leg (possibly with zero distance). If we assumethat the ships are always fully loaded on the loaded legs, the piecewise linearapproximation can also easily incorporate that fuel consumption vary withship payload.

3.2. Mathematical model

Sets

K set of shipsKr subset of K, ships usable for service on trade route rR set of trade routesRk subset of R that can be serviced by ship kT set of months in the planning periodNM

r set of mandatory voyages of trade route rNO

r set of optional voyages of trade route rNr set of voyages on trade route r, NM

r ∪NOr

Nrt set of voyages on trade route r in month tAk set of arcs, two successive voyages, that can be sailed by

ship kV set of speed alternatives, ordered from low to high

Parameters

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CBo(k)riv cost of ballast sailing with speed v from initial position o(k)

to starting position of voyage (r, i) for ship kTBo(k)riv time spent for ship k sailing ballast from initial position o(k)

to starting position of voyage (r, i) with speed vCB

kriqjv cost of ship k sailing ballast with speed v from the last portof voyage (r, i) to reposition for service on voyage (q, j)

TBkriqjv time spent for ship k sailing ballast with speed v from the

last port of voyage (r, i) to reposition for service on voyage(q, j)

Ckriv cost of ship k performing voyage (r, i) with speed vTkriv time spent for ship k to sail voyage (r, i) with speed vEri earliest service start of voyage (r, i)Lri latest service start of voyage (r, i)Eo(k) earliest start from initial position o(k) for ship kCS

ri cost of chartering a spot ship for voyage (r, i)CSC

rt unit cost for space chartering on trade route r in month tCP

ri penalty cost per day the start of voyage (r, i) is delayedDrt transportation demand for trade route r in month tQk capacity of ship kQS capacity of a spot ship

Variables

xo(k)riv the weight of speed alternative v for the sailing from theinitial position o(k) of ship k to the first port of voyage (r, i)

yo(k)ri 1 if ship k sails directly from initial position o(k) to thestarting point of voyage (r, i), 0 otherwise

xkriqjv the weight of speed alternative v for voyage (r, i) and theballast sailing to voyage (q, j) by ship k

ykriqj 1 if ship k sails voyage (r, i) and the ballast sailing to voyage(q, j) directly afterwards, 0 otherwise

yrid(k) 1 if ship k sails voyage (r, i) as the last voyage, 0 otherwisesri 1 if voyage (r, i) is sailed by a spot ship, 0 otherwiseto(k) the starting time for ship k from initial position o(k)tri the start time of voyage (r, i)prt the amount of load on trade route r that is covered with

space charter in month t

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The model may be stated as follows:

min z =∑k∈K

∑r∈Rk

∑i∈Nr

∑v∈V

CBo(k)rivxo(k)riv

+∑k∈K

∑r∈Rk

∑i∈Nr

∑q∈Rk

∑j∈Nq

∑v∈V

(CB

kriqjv + Ckriv

)xkriqjv

+∑k∈K

∑r∈Rk

∑i∈Nr

Ckri1yrid(k)

+∑r∈R

∑i∈Nr

CPri(tri − Eri)

+∑r∈R

∑i∈Nr

CSr sri +

∑r∈R

∑t∈T

CSCr prt

(10)

yo(k)ri −∑v∈V

xo(k)riv = 0, k ∈ K, r ∈ Rk, i ∈ Nr (11)

ykriqj −∑v∈V

xkriqjv = 0, k ∈ K,((r, i), (q, j)

)∈ Ak (12)

sri +∑k∈K

∑q∈Rk

∑j∈Nq

ykriqj +∑k∈K

yrid(k) = 1, r ∈ R, i ∈ NMr (13)

sri +∑k∈K

∑q∈Rk

∑j∈Nq

ykriqj +∑k∈K

yrid(k) ≤ 1, r ∈ R, i ∈ NOr (14)

∑r∈Rk

∑i∈Nr

yo(k)ri ≤ 1, k ∈ K (15)

∑r∈Rk

∑i∈Nr

yrid(k) ≤ 1, k ∈ K (16)

yrid(k) − yo(k)ri +∑q∈Rk

∑j∈Nq

(ykriqj − ykqjri) = 0,

k ∈ K, r ∈ Rk, i ∈ Nr

(17)

∑i∈Nrt

⎛⎝QSsri +

∑k∈K

∑q∈R

∑j∈Nq

Qkykriqj

⎞⎠+

∑k∈K

∑i∈Nrt

Qkyrid(k) + prt ≥ Drt,

r ∈ R, t ∈ T

(18)

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yo(k)ri

(to(k) − tri +

∑v∈V

TBo(k)rivxkriv

)≤ 0,

k ∈ K, r ∈ Rk, i ∈ Nr

(19)

ykriqj

(tri − tqj +

∑v∈V

(TBkriqjv + Tkriv

)xkriqjv

)≤ 0,

k ∈ K, ((r, i) , (q, j)) ∈ Ak

(20)

to(k) ≥ Eo(k), k ∈ K (21)

Eri ≤ tri ≤ Lri, r ∈ R, i ∈ Nr (22)

yo(k)ri ∈ {0, 1} , k ∈ K, r ∈ Rk, i ∈ Nr (23)

ykriqj ∈ {0, 1} , k ∈ K, ((r, i) , (q, j)) ∈ Ak (24)

yrid(k) ∈ {0, 1} , k ∈ K, r ∈ Rk, i ∈ Nr (25)

sri ∈ {0, 1} , r ∈ R, i ∈ Nr (26)

xo(k)riv ≥ 0, k ∈ K, r ∈ Rk, i ∈ Nr, v ∈ V (27)

xkriqjv ≥ 0, k ∈ K, ((r, i) , (q, j)) ∈ Ak, v ∈ V (28)

prt ≥ 0, r ∈ R, t ∈ T (29)

The objective function (10) is to minimize the total cost; the sum of thecosts of initial ballast sailings, the costs associated with performing voyagesand the ballast sailings to successive voyages, the cost of performing the lastvoyages, the penalty cost of starting a voyage too late, the cost of hiringspot vessels on voyages, and the cost of space chartering. In the case wherethe shipping company also operates in a spot market, the revenues from thecargoes can no longer be considered fixed. In such a case one would rathermaximize the profit instead of minimizing costs. Constraints (11) and (12)yield the relationship between the flow variables and the speed-dependentvariables, such that the weights of the speed alternatives add up to either 1, ifthe voyage is performed by that ship, or 0 if not. Constraints (13) ensure thatall mandatory voyages are served, either by a spot vessel or by a ship from thefleet. Constraints (14) keep the optional voyages optional so that they maybe served either by a spot vessel, a ship in the fleet, or not at all. Constraints(15) - (17) are flow constraints for each ship. Together, constraints (15) and(16) ensure that each ship is either idle or must leave its initial position to avoyage (r, i) and arrive at its final position. Constraints (17) ensure that each

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ship that starts a voyage must also finish it. The exceptions where the nodesare a ship’s initial or final position are also considered in the constraints.Constraints (18) ensure that the transportation demand for each month isserviced either by the assigned ships capacity or by space chartering.

Constraints (19) ensure that the time spent from the ballast sailing frominitial position o(k) to a first voyage (r, i) does not exceed the start time forthe voyage (r, i). Similarly constraints (20) state that the time spent servingvoyage (r, i) together with the ballast sailing from (r, i) to (q, j) should notexceed the start time for voyage (q, j). In other words, they ensure thatthe ships arrive in time. Constraints (19) and (20) are linearized in theimplementation. Constraints (21) make sure that the start time for ship k isat a time where the ship is available, while constraints (22) define the timewindow in which a voyage must start. Constraints (23) - (29) define thedomain of the variables.

4. Rolling horizon heuristic

We can solve the mixed-integer programming (MIP) model presented inSection 3.2 with commercial MIP solvers for real instances only with shortplanning horizons, and we are not able to solve problems with the desiredplanning horizon of 6 - 10 months. Rolling horizon heuristics (RHH) haveproven to be a powerful method in several production planning problems,see for example Stauffer and Liebling (1997), Merce and Fontan (2003) andArujo et al. (2007). They have also sucessfully been used in maritime rout-ing problems, especially for problems with long planning horizons, see forexample Sherali et al. (1999) and Rakke et al. (2011). Therefore, we proposean RHH for the integrated fleet deployment and speed optimization problemdescribed in the two previous sections.

The principle of the RHH is to solve the problem iteratively through theplanning period. The rolling mechanism is illustrated in Figure 5. At itera-tion k, the subproblem for sub-horizon TWk is solved. The sub-horizon con-sists of the central section, SC(TWk) and the forecasting section SF (TWk).At each iteration, the solution corresponding to the first period(s) of SC(TWk)is frozen or fixed, and the central and forecasting sections are updated toSC(TW(k+1)) and SF (TW(k+1)). The sub-horizon moves forward a givennumber of periods at each iteration and a new subproblem is solved. Thepseudo-code of the rolling horizon heuristic is given in Algorithm 4.1. The al-gorithm moves forward at each iteration until the entire planning horizon has

16

Figure 5: Overview of the rolling horizon mechanism with sub-horizons and sections

17

been included, where K denotes the total number of periods in the planninghorizon.

Algorithm 4.1 Rolling Horizon Heuristic

k = 1while k < K doSolve the model for SC(TWk) and SF (TWk)Freeze flow and spot variables for the first period in SC(TWk)k = k + 1

end whileSolve the model for SC(TWK)Freeze all variables

In the central section a detailed plan is established. The length of thecentral section is one among several RHH design decisions that must be made.There is a trade-off between solution quality and time, as it is expected thata longer central section will give better solutions at the sacrifice of highersolution times. By including a forecasting section in the sub-horizon, thealgorithm will be less myopic and the solution of the central section will alsobe influenced by the constraints applying later in the planning horizon. Byextending the length of the forecasting section, the amount of informationabout future time periods considered in the analysis increases. It can beassumed that the RHH will give better results as the length of the forecastingsection increases, which was also the case in the study by Baker and Peterson(1979). However, as for the central section, extending the length of theforecasting section increases the solution time. We have tested a number ofalternatives regarding the length of the central and forecasting sections.

We have also tested simplification strategies for the forecasting section toachieve a better trade-off between solution quality and time. One obviousalternative is to relax the binary variables, which will reduce the computa-tional load. The drawback is that a continuous relaxation of binary variablesreduces the amount of information of future periods brought into the so-lution. Again, there is a balance of maintaining information and keepingthe problem size small. The relaxation of binary variables in the forecastingsection was successfully implemented by Bredstrom et al. (2004) and Rakkeet al. (2011). It is therefore chosen as the simplification strategy in the RHHpresented here.

There are also other design decisions that must be made regarding fixing

18

strategy for the fixed section. Merce and Fontan (2003) consider two alter-native fixing strategies for a multi-item capacitated lot-sizing problem. Thefirst strategy is to fix all decisions of the previous central section as opposedto the second strategy where only production decisions are frozen but quan-tities may be adjusted later on. The second strategy gave the best results. Inour algorithm we use a similar approach and fix only the binary variables, i.e.the sailing (except for the sailing to the artificial destination) and the spotship variables. The sailing speeds and time variables, which are representedby continuous variables, may however be adjusted later on.

5. Computational study

The mathematical model described in Section 3 has been implemented inMosel using Xpress IVE and solved using Xpress 23.01.06. The same soft-ware has been used for solving the subproblems in the rolling horizon heuristic(RHH). The RHH is also implemented in Mosel using Xpress IVE. All com-putational tests are performed using an HP dl165 G5 with 2 2.4GHz AMDOpteron 2431 - 6 core CPUs and 24Gb of RAM. Section 5.1 describes the testproblems used, while the performance of the RHH compared with only usingXpress is presented and discussed in Section 5.2. The effect of integratingfleet deployment with speed optimization is discussed in Section 5.3.

5.1. Test instances

Two test cases, L and S, have been developed and used in the testing.Test case L is based on real life data with nine trade routes and 53 ships,which reflects the real size of WWL’s planning problem. Test case S is areduced version of the real life problem where some trade routes and shipshave been removed. Most data, such as demand, costs, fuel consumptionsand initial positions, have been provided by WWL; otherwise it has beenestimated or calculated by the authors. Three speed discretization pointshave been used, i.e. the ships’ minimum and maximum planning speeds, anda service speed between those two. Tests showed that using three points gavea maximum error less than 3 %, which is well below inaccuracies in otherparameter values related to calculating the real fuel consumption. The shipsin the case study are various PCTCs (Pure Car and Truck Carriers) andRoRo-ships, with capacities ranging from approximately 5000 CEUs (CarEquivalent Units) to 8500 CEUs. The speed range used is 15 to 23 knots.Various planning horizons from four to 10 months have been tested, forming

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the test instances summarized in Table 1, where for example instance S6represent test case S with a planning horizon of six months. The number ofvoyages is given as an interval since there are optional voyages in addition tothe mandatory voyages denoted by the lower limit.

Table 1: Test instances. For the number of voyages, the lower limit is the number ofmandatory voyages that must be performed in the planning horizon, while the upper limitincludes the optional voyages that can additionally be used to meet the demand

Instance # Ships # Trade Routes # Voyages Planning HorizonS4 24 5 33 / 45 4 monthsS6 24 5 51 / 69 6 monthsS8 24 5 105 / 141 8 monthsS10 24 5 158 / 212 10 months

L4 53 9 52 / 72 4 monthsL6 53 9 79 / 109 6 monthsL8 53 9 159 / 219 8 monthsL10 53 9 239 / 329 10 months

5.2. Computational performance

The gaps from solving the various test instances with Xpress with one, twoand four hours of running time, respectively, are presented in Table 2. Thegap is here defined as the objective value of the best integer solution minusthe lower bound divided by the lower bound. We see that the gaps are high,especially for the instances with planning horizons longer than four months.This confirms that one needs a better algorithm to solve the integrated fleetdeployment and speed optimization problem.

In Table 3, the solutions from the RHH are compared with the bestinteger solutions and lower bounds from Xpress after four hours of runningtime. We found that using one month as both the length of the central andforecasting section, as well as the number of months to fix at each iteration,gave a good trade-off between solution quality and time, so these are thevalues used to produce the results in Table 3. Furthermore, we have used amaximum solution time for each subproblem of the RHH of 1800 seconds.

We can see from Table 3 that the RHH produces better solutions thanXpress for all instances except for the two smallest ones, S4 and S6, forwhich the RHH solutions have 2.1 and 0.2 % higher costs. Especially for the

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Table 2: Gaps when using Xpress after 1, 2 and 4 hours of running time, respectively

Instance 1 hour 2 hours 4 hoursS4 2.6 2.2 2.0S6 13.1 10.0 9.7S8 23.5 23.4 23.4S10 102.6 98.3 96.9

L4 5.3 4.9 4.2L6 186.6 19.8 19.8L8 218.1 212.9 209.6L10 244.8 238.0 230.8

Table 3: Comparison of solutions by the RHH with Xpress (Central section = 1 month,Forecasting section = 1 month, 1 month is fixed at each iteration, max time for subprob-lems = 1800 seconds)

Instance Cost compare Gaps from best bound Solution timewith Xpress, 4 hour (%) from Xpress, 4 hour (%) (s)

S4 102.1 4.2 31S6 100.2 9.9 91S8 90.8 12.0 168S10 57.8 13.8 260

L4 99.9 4.2 336L6 89.1 6.7 3 959L8 35.1 8.7 6 435L10 33.6 11.2 10 070

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largest instances L8 and L10, the RHH performs much better than Xpress,with approximately only a third of the costs of the Xpress solutions. Theseare also the ones that are closest to the real life problem size. We can alsocompare the solutions from the RHH with the lower bounds from Xpress,which indicates the maximum difference that can be possible between oursolutions and the optimal ones. From Table 3, we can see that the gapsincrease with longer planning horizons, though not dramatically. For theinstances L8 and L10, these gaps are 8.7 and 11.2 %, respectively. We cannotaddress whether these gaps come from a poor lower bound or because theRHH solutions have a potential for improvement.

The solutions from the RHH are also obtained within reasonable compu-tation time. Even for the largest instance, L10, the solution time is less thanthree hours. For budgeting purposes, WWL are also interested in solvingproblems with 12 - 18 months planning horizons, which would be impossi-ble using only Xpress. However, this should not be a problem for the RHHsince the computational time increases only mildly with increased planninghorizon.

We have also tested other RHH settings, where Table 4 summarizes theresults from two of these. In the first setting, we have used one and twomonths as the lengths of the central and forecasting sections, respectively.For the second we have used two and one months as the lengths of the centraland forecasting sections, respectively. For both settings we fix one month ateach iteration and use a maximum solution time for each subproblem of 1800seconds.

As we can see from Table 4, slightly improved solutions are on averageobtained with the new settings compared with the ones used in Table 3,however at much higher solution times.

We have also tested to increase the maximum time for solving each sub-problem to 3600 seconds, with the results that also the total computationaltime for the RHH increased. However, this did not on average give any im-provements on solution quality. This indicates that not much is gained byobtaining better solutions to the subproblems, and sometimes it might evenlead to poorer results overall.

5.3. Effect from integrating fleet deployment with speed optimization

As discussed in the introduction, the most common way to make fleetdeployment plans is to assume that each ship sails with a given service speed,and then later during the execution of the routes optimize the sailing speeds

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Table 4: RHH solutions with other settings (1 month is fixed at each iteration, max timefor subproblems = 1800 seconds); 1+1: Central section = 1 month, Forecasting section =1 month, 1+2: Central section = 1 month, Forecasting section = 2 month, 2+1: Centralsection = 2 month, Forecasting section = 1 month

Instance Time Quality1 + 1 1 + 2 2 + 1 1 + 1 1 + 2 2 + 1

S4 31 121 2005 100.0 98.6 98.2S6 91 3756 5622 100.0 99.2 97.7S8 168 7808 9347 100.0 100.2 98.2S10 260 11057 12827 100.0 99.5 98.2

L4 336 2233 5482 100.0 99.9 98.9L6 3959 7501 9107 100.0 100.1 98.9L8 6435 11134 11596 100.0 99.5 99.3L10 10070 14766 16386 100.0 99.9 99.2

based on the time windows that must be respected along the given shiproutes. We have tested the effect from integrating speed optimization intothe fleet deployment and compared this with the more common sequentialapproach for a subset of the test instances, i.e. instances S6, S8, L6 and L8.The results are summarized in Table 5.

Table 5: Effect from speed optimization

Instance A B CFixed service speed A posteriori Speed optimization

(mill. USD) (impr. in %) (impr. in % to B)S6 755.8 1.2 4.9S8 937.6 1.1 4.2

L6 767.1 5.6 4.5L8 927.2 5.6 4.1

The second column (marked with A) shows the planned costs of the so-lutions using a fixed speed (i.e. the ships’ service speeds). Then, column Bshows the improvement over A after optimizing the speeds along the ships’sailing routes, as is done in the sequential approach described above. How-ever, the additional cost improvement obtained from integrating speed opti-

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mization when planning the ship routes is between 4 and 5 %, as shown incolumn C. This is a significant cost saving that also shows that the routingsolutions are also changed when speed optimization is integrated in the fleetdeployment. This is also consistent with the results by Norstad et al. (2011)for a tramp ship routing and scheduling problem.

6. Concluding remarks

Fuel consumption, and hence the cost of sailing, is strongly dependenton the chosen speed of a ship, and a cubic function often provides a goodestimation of the relationship between fuel consumption per time unit andspeed for cargo ships. When planning shipping routes it is common to use asequential approach where it is first assumed that each ship sails with a givenservice speed, and then later during the execution of the routes optimize thesailing speeds based on the time windows that must be respected along theroutes.

In this paper we proposed a new modeling approach, using piecewiselinear approximation, for integrating speed optimization in the planning ofshipping routes. It was analyzed how this piecewise linear approximationoverestimates the fuel consumption. We also proposed a new rolling horizonheuristic (RHH) for solving the integrated problem. As a case study weconsidered a real fleet deployment problem in RoRo-shipping, a segmentwithin liner shipping.

The computational results showed that the RHH obtained good solutionswithin reasonable times and performed much better than solving the problemonly using a commercial mixed-integer programming solver for all instancesof realistic size. It should also be noted that even though the modeling ap-proach only gives an approximation (overestimation) of the fuel consumptionfunction, it represents a major improvement compared with the sequentialplanning approach, as demonstrated in the computational study. Further-more, for most practical cases, the errors from this approximation will notdominate over inaccuracies in parameter values, such as in fuel consumptionfunctions and sailing distances.

Finally, it should be emphasized that even though we used a fleet deploy-ment problem from RoRo-shipping as our case study, the proposed modelingapproach regarding how speed decisions have been integrated in the modelcan be applied for a wider range of integrated routing and speed optimiza-tion problems in maritime transportation. Furthermore, the rolling hori-

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zon heuristic can also be adapted to several planning problems of this type,though it will probably be best suited for cases where the planning horizonis long and because of that cannot be solved directly to optimality.

Acknowledgement

We want to thank WWL for explaining their fleet deployment problemand for providing data. This research was carried out with financial sup-port from the MARFLIX and DOMinant II projects, partly funded by theResearch Council of Norway. We also thank the reviewers for their valuablecomments and suggestions.

References

Arujo, S., Arenales, M., Clark, A., 2007. Joint rolling-horizon scheduling ofmaterials processing and lot-sizing with sequence-dependent setups. Jour-nal of Heuristics 13 (4), 337–358.

Baker, K., Peterson, D., 1979. An analytic framework for evaluating rollingschedules. Management Science 25 (4), 341–351.

Bredstrom, D., Lundgren, J., Ronnqvist, M., Carlsson, D., Mason, A., 2004.Supply chain optimization in the pulp mill industry – IP models, columngeneration and novel constraint branches. European Journal of OperationalResearch 156 (1), 2–22.

Christiansen, M., Fagerholt, K., Nygreen, B., Ronen, D., 2013. Ship routingand scheduling in the new millennium. European Journal of OperationalResearch 228 (3), 467–483.

Fagerholt, K., Johnsen, T., Lindstad, H., 2009. Fleet deployment in linershipping: a case study. Maritime Policy & Management 36 (5), 397–409.

Gatica, R., Miranda, P., 2010. A time based discretization approach for shiprouting and scheduling with variable speed. Networks & Spatial Economics11 (3), 465–485.

Gelareh, S., Meng, Q., 2010. A novel modeling approach for the fleet de-ployment problem within a short-term planning horizon. TransportationResearch Part E 46 (1), 76–89.

25

Hvattum, L., Norstad, I., Fagerholt, K., Laporte, G., 2013. Analysis of anexact algorithm for the vessel speed optimization problem. Networks 62 (2),132–135.

Liu, X., Ye, H.-Q., Yuan, X.-M., 2011. Tactical planning models for manag-ing container flow and ship deployment. Maritime Policy & Management38 (5), 487–508.

Meng, Q., Wang, T., 2010. A chance constrained programming model forshort-term liner ship planning problems. Maritime Policy & Management37 (4), 329–346.

Merce, C., Fontan, G., 2003. MIP-based heuristics for capacitated lotsizingproblems. International Journal of Production Economics 85 (1), 97–111.

Norstad, I., Fagerholt, K., Laporte, G., 2011. Tramp ship routing andscheduling with speed optimization. Transportation Research Part C19 (5), 853–865.

Psaraftis, H., Kontovas, C., 2013. Speed models for energy-efficient maritimetransportation: A taxonomy and survey. Transportation Research Part C26, 331–351.

Rakke, J., Stalhane, M., Moe, C., Christiansen, M., Andersson, H., Fager-holt, K., Norstad, I., 2011. A rolling horizon heuristic for creating a lique-fied natural gas annual delivery program. Transportation Research Part C19 (5), 896–911.

Ronen, D., 1982. The effect of oil price on the optimal speed of ships. Journalof the Operational Research Society 33 (11), 1035–1040.

Sherali, H., Al-Yakoob, S., Hassan, M., 1999. Fleet management models andalgorithms for an oil-tanker routing and scheduling problem. IIE Transac-tions 31 (5), 395–406.

Stauffer, L., Liebling, T., 1997. Rolling horizon scheduling in a rolling-mill.Annals of Operations Research 69, 323–349.

Wang, S., Meng, Q., 2012. Liner ship fleet deployment with container trans-shipment operations. Transportation Research Part E 48 (2), 470–484.

26

Zacharioudakis, P., Iordanis, P., Lyridis, D., Psaraftis, H., 2011. Liner ship-ping cycle cost modeling, fleet deployment optimization and what-if anal-ysis. Maritime Economics & Logistics 13 (3), 278–297.

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