Ocean Trade Routes University of Norway Chapter 4

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C. Barnhart and G. Laporte (Eds.), Handbook in OR & MS, Vol. 14Copyright 2007 Elsevier B.V. All rights reservedDOI: 10.1016/S0927-0507(06)14004-9

Chapter 4

Maritime Transportation

Marielle ChristiansenDepartment of Industrial Economics and Technology Management, Norwegian University

of Science and Technology, Trondheim, Norway

Department of Applied Economics and Operations Research, SINTEF Technology and

Society, Trondheim, Norway

E-mail: [email protected]

Kjetil FagerholtDepartment of Industrial Economics and Technology Management, Norwegian University


Department of Marine Technology, Norwegian University of Science and Technology,

Trondheim, Norway

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


Bjrn NygreenDepartment of Industrial Economics and Technology Management, Norwegian University



David RonenCollege of Business Administration, University of Missouri-St. Louis, St. Louis, MO, USA


1 Introduction

Maritime transportation is the major conduit of international trade, but theshare of its weight borne by sea is hard to come by. The authors have surveyed

the academic members of the International Association of Maritime Econo-mists and their estimates of that elusive statistic range from 65% to 85%.Population growth, increasing standard of living, rapid industrialization, ex-haustion of local resources, road congestion, and elimination of trade barriers,all of these contribute to the continuing growth in maritime transportation. Incountries with long shorelines or navigable rivers, or in countries consisting ofmultiple islands, water transportation may play a significant role also in do-mestic trades, e.g., Greece, Indonesia, Japan, Norway, Philippines, and USA.Table 1 demonstrates the growth in international seaborne trade during the

last couple of decades (compiled from UNCTAD, 2003, 2004).Since 1980 the total international seaborne trade has increased by 67% interms of weight. Tanker cargo has increased modestly, but dry bulk cargo has

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190 M. Christiansen et al.

Table 1.Development of international seaborne trade (millions of tons)

Year Tanker cargo Dry cargo Total

Main bulkcommodities1

Other

1980 1871 796 1037 37041990 1755 968 1285 40082000 2163 1288 2421 58722001 2174 1331 2386 58912002 2129 1352 2467 5948

20032 2203 1475 2490 6168

1Iron ore, grain, coal, bauxite/alumina, and phosphate.2

Estimates.

Table 2.World fleet by vessel type (million dwt)

Year Oiltankers

Bulkcarriers

Generalcargo

Containerships

Other Total

1980 339 186 116 11 31 6831990 246 234 103 26 49 6582000 286 281 103 69 69 8082001 286 294 100 77 69 8262002 304 300 97 83 60 8442003 317 307 95 91 47 857

increased by 85%. The Other dry cargo, which consists of general cargo (in-cluding containerized cargo) and minor dry bulk commodities, has more thandoubled.

The world maritime fleet has grown in parallel with the seaborne trade. Ta-

ble 2 provides data describing the growth of the world fleet during the sameperiod (compiled from UNCTAD, 2003, 2004).

The cargo carrying capacity of the world fleet has reached 857 million tonsat the end of 2003, an increase of 25% over 1980. It is worth pointing out thefast growth in the capacity of the container ships fleet with 727% increase dur-ing the same period. These replace general cargo ships in major liner trades. Toa lesser extent we see also a significant growth in the bulk carriers fleet. Thegap between the increase in total trade (67%) and in the world fleet (25%)is explained by two factors. First, the boom in construction of tankers during

the 1970s that resulted in excess capacity in 1980, and second, the increas-ing productivity of the world fleet, as demonstrated in Table 3 (compiled fromUNCTAD, 2003, 2004).


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Ch. 4. Maritime Transportation 191

Table 3.Productivity of the world fleet

Year World fleet

(million dwt)

Total cargo

(million tons)

Total ton-miles

performed (thousands ofmillions of ton-miles)

Tons carried

per dwt

Thousands of

ton-milesperformed per dwt

1980 6828 3704 16,777 54 2551990 6584 4008 17,121 61 2602000 8084 5871 23,016 73 2852001 8257 5840 23,241 71 2812002 8442 5888 23,251 70 2752003 8570 6168 24,589 72 287

Inconsistencies between these data and the Total in Table 1 are in the source. However, they do notaffect the productivity statistics presented in this table.

The utilization of the world fleet has increased from 5.4 tons carried perdeadweight ton in 1980 to 7.2 in 2003. At the same time the annual output perdeadweight ton has increased from 25.5 thousand ton-miles to 28.7.

These statistics demonstrate the dependence of the world economy onseaborne trade. A ship involves a major capital investment (usually millionsof US dollars, tens of millions for larger ships) and the daily operating cost ofa ship may easily amount to thousands of dollars and tens of thousands for thelarger ships. Proper planning of fleets and their operations has the potential

of improving their economic performance and reducing shipping costs. This isoften a key challenge faced by the industry actors in order to remain competi-tive.

The purpose of this chapter is to introduce the reader who is familiar withOperations Research (OR), and may be acquainted with other modes of trans-portation, to maritime transportation. The term maritime transportation refersto seaborne transportation, but we shall include in this chapter also other

water-borne transportation, namely inland waterways. The chapter discussesvarious aspects of maritime transportation operations and presents associateddecision making problems and models with an emphasis on ship routing andscheduling models. This chapter focuses on prescriptive OR models and as-sociated methodologies, rather than on descriptive models that are usually ofinterest to economists and public policy makers. Therefore we do not discussstatistical analysis of trade and modal-split data, nor ship safety and casualtyrecords and related topics. To explore these topics the interested reader mayrefer to journals dealing with maritime economics, such as Maritime Policyand Management and Maritime Economics and Logistics (formerly InternationalJournal of Maritime Economics).

The ocean shipping industry has a monopoly on transportation of large vol-

umes of cargo among continents. Pipeline is the only transportation mode thatis cheaper than ships (per cargo ton-mile) for moving large volumes of cargoover long distances. However, pipelines are far from versatile because they can


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Table 4.Comparison of operational characteristics of freight transportation modes

Operational characteristic Mode

Ship Aircraft Truck Train Pipeline

Barriers to entry small medium small large largeIndustry concentration low medium low high highFleet variety (physical &economic)

large small small small NA

Power unit is an integral partof the transportation unit

yes yes often no NA

Transportation unit size fixed fixed usually fixed variable NA Operating around the clock usually seldom seldom usually usuallyTrip (or voyage) length daysweeks hoursdays hoursdays hoursdays daysweeksOperational uncertainty larger larger smaller smaller smaller

Right of way shared shared shared dedicated dedicatedPays port fees yes yes no no noRoute tolls possible none possible possible possibleDestination change whileunderway

possible no no no possible

Port period spans multipleoperational time windows

yes no no yes NA

Vessel-port compatibilitydepends on load weight

yes seldom no no NA

Multiple products shippedtogether

yes no yes yes NA

Returns to origin no no yes no NA

NA not applicable.

move only fluids in bulk over fixed routes, and they are feasible and econom-ical only under very specific conditions. Other modes of transportation (rail,truck, air) have their advantages, but only aircraft can traverse large bodies of

water, and they have limited capacity and much higher costs than ships, thusthey attract high-value low-volume cargoes. Ships are probably the least reg-ulated mode of transportation because they usually operate in international

water, and very few international treaties cover their operations.Ship fleet planning problems are different than those of other modes of

transportation because ships operate under different conditions. Table 4 pro-vides a comparison of the operational characteristics of the different freighttransportation modes. We wish also to point out that ships operate mostly ininternational trades, which means that they are crossing multiple national juris-dictions. Actually, in many aspects aircraft are similar to ships. In both modeseach unit represents a large capital investment that translates into high dailycost, both pay port fees and both operate in international routes. However,

most aircraft carry mainly passengers whereas most ships haul freight. Evenaircraft that transport freight carry only packaged goods whereas ships carrymostly liquid and dry bulk cargo, and often nonmixable products in separate


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compartments. Since passengers do not like to fly overnight most aircraft arenot operated around the clock whereas ships are operated continually. In ad-dition, aircraft come in a small number of sizes and models whereas among

ships we find a large variety of designs that result in nonhomogeneous fleets.Both ships and aircraft have higher uncertainty in their operations due to theirhigher dependence on weather conditions and on technology, and because theyusually straddle multiple jurisdictions. However, since ships operate aroundthe clock their schedules usually do not have buffers of planned idleness thatcan absorb delays. As far as trains are concerned, they have their own dedi-cated right of way, they cannot pass each other except for at specific locations,and their size and composition are flexible (both number of cars and numberof power units). Thus the operational environment of ships is different fromother modes of freight transportation, and they have different fleet planning

problems.The maritime transportation industry is highly fragmented. The web site of

Lloyds Register boasts of listing of over 140,000 ship and 170,000 shipowner and manager entries. In order to take advantage of differences amongnational tax laws, financial incentives, and operating rules, the control struc-ture of a single vessel may involve multiple companies registered in differentcountries.

Although ships are the least regulated mode of transportation, there aresignificant legal, political, regulatory, and economic aspects involved in mar-itime transportation. The control structure of a ship can be designed to hide

the identity of the real owner in order to minimize liability and taxes. Liabilityfor shipping accidents may be hard to pinpoint, and damages may be impos-sible to collect, because numerous legal entities from different countries areusually involved, such as: owner, operator, charterer, flag of registration, ship-

yards, classification society, surveyors, and contractors. That is in addition tothe crew that may have multiple nationalities and multiple native languages.

Only a small share of the world fleet competes directly with other modes oftransportation. However, in certain situations such competition may be im-portant and encouraged by government agencies. In short haul operations,relieving road congestion by shifting cargo and passengers to ships is often de-sirable and even encouraged through incentives and subsidies. A central policyobjective of the European Union for the upcoming years is to improve thequality and efficiency of the European transportation system by shifting traf-fic to maritime and inland waterways, revitalizing the railways and linking upthe different modes of transport. For further information regarding the Euro-pean transport policy see the European Commissions white paper EuropeanTransport Policy for 2010: Time to Decide (European Commission, 2004). Thissource provides information about many of the European Unions programs

where maritime transportation plays a prominent role.

Transportation planning has been widely discussed in the literature but mostof the attention has been devoted to aircraft and road transportation by trucksand buses. Other modes of transportation, i.e., pipeline, water, and rail, have


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attracted far less attention. One may wonder what the reason is for that lowerattention, especially when considering the large capital investments and op-erating costs associated with these modes. Pipeline and rail operate over a

dedicated right of way, have major barriers to entry, and relatively few op-erators in the market. These are some issues that may explain the lower levelof attention. It is worth mentioning that research on rail planning problemshas increased considerably during the last fifteen years. However, the issuesmentioned for pipeline and rail do not hold for water transportation. Severalexplanations follow for the low attention drawn in the literature by maritimetransportation planning problems:

Low visibility. In most regions people see trucks, aircraft, and trains, but notships. Worldwide, ships are not the major transportation mode. Most cargois moved by truck or rail. Moreover, research is often sponsored by largeorganizations. Numerous large organizations operate fleets of trucks, butfew such organizations operate ships.

Maritime transportation planning problems are less structured. In maritimetransportation planning there is a much larger variety in problem struc-tures and operating environments. That requires customization of decisionsupport systems, and makes them more expensive. In recent years wesee more attention attracted by more complex problems in transportationplanning, and this is manifested also in maritime transportation.

In maritime operations there is much more uncertainty. Ships may be delayed

due to weather conditions, mechanical problems and strikes (both onboard and on shore), and usually, due to their high costs, very little slack isbuilt into their schedules. This results in a frequent need for replanning.

The ocean shipping industry has a long tradition and is fragmented. Ships havebeen around for thousands of years and therefore the industry may be con-servative and not open to new ideas. In addition, due to the low barriersto entry there are many small, family owned, shipping companies. Mostquantitative models originated in vertically integrated organizations whereocean shipping is just one component of the business.

In spite of the conditions discussed above we observe significant growthin research in maritime transportation. The first review of OR work in shiprouting and scheduling appeared in 1983 (Ronen, 1983), and it traced papersback to the 1950s. A second review followed a decade later (Ronen, 1993),and recently a review of the developments over the last decade appeared(Christiansen et al., 2004). Although these reviews focused on ship routingand scheduling problems, they discussed also other related problems on allplanning levels. A feature issue on OR in water transportation was publishedby the European Journal of Operational Research (Ronen, 1991), and a spe-cial issue on maritime transportation was published by Transportation Science

(Psaraftis, 1999). A survey of decision problems that arise in container termi-nals is provided by Vis and de Koster (2003). The increasing research interestin OR-based maritime transportation is evidenced by the growing number of


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references in the review papers. The first review paper had almost forty refer-ences covering several decades. The second one had about the same number ofreferences most of which were from a single decade, and the most recent one

has almost double that number of references for the last decade. It is worthmentioning that a large share of the research in transportation planning doesnot seem to be based on real cases but rather on artificially generated data. Theopposite is true for maritime transportation, where the majority of problemsdiscussed are based on real applications.

We focus our attention on planning problems in maritime transportation,and some related problems. With the fast development of commercial aircraftduring the second half of the 20th century, passenger transportation by shipshas diminished to ferries and cruises. Important as they are, these are smalland specialized segments of maritime transportation. Therefore we shall focus

here on cargo shipping. Related topics that are discussed in other chapters ofthis volume are excluded from this chapter, namely maritime transportationof hazardous materials (Erkut and Verter, 2007) and operations of the land-side of port terminals (Crainic and Kim, 2007). We try to confine ourselves todiscussion of work that is relatively easily accessible to the reader. This chapteris intended to provide a comprehensive picture, but by no means an exhaustiveone.

This chapter is organized around the traditional planning levels, strategic,tactical, and operational planning. Within these planning levels we discuss the

three types of operations in maritime transportation (liner, tramp, industrial)and additional specialized topics. Although we try to differentiate among theplanning levels, one should remember the interplay among them. On the onehand, the higher-level or longer-term decisions set the stage for the lower-leveldecisions. On the other hand, one usually needs significant amount of detailsregarding the shorter-term decisions in order to make good longer-term deci-sions. We focus here on OR problems in maritime transportation, the relatedmodels, and their solution methods. Due to the fast development of computingpower and memory, information regarding the computing environment be-comes obsolete very quickly, and such information will only occasionally be

presented.The rest of the chapter is organized as follows: Section 2 defines terms used

in OR-applications in maritime transportation and describes characteristics ofthe industry. Sections 35 are dedicated to strategic, tactical, and operationalproblems in maritime transportation, respectively. In these sections we presentproblem descriptions, models and solution approaches for the three modesof operations in maritime transportation, namely liner, industrial, and tramp.We also address in these sections naval operations, maritime supply chains,ship design and management, ship loading, contract evaluation, booking or-

ders, speed selection, and environmental routing. The issue of robustness inmaritime transportation planning is addressed in Section 6. Important trendsand perspectives for the use of optimization-based decision support systems in


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maritime transportation and suggestions for future research are presented inSection 7, and some concluding remarks follow in Section 8.

2 Characteristics and terminology of maritime transportation

Maritime transportation planning problems can be classified in the tradi-tional manner according to the planning horizon into strategic, tactical andoperational problems.

Among the strategic problems we find:

market and trade selection, ship design,

network and transportation system design (including the determina-tion of transshipment points for intermodal services), fleet size and mix decisions (type, size, and number of vessels), and port/terminal location, size, and design.

The tactical problems include:

adjustments to fleet size and mix, fleet deployment (assignment of specific vessels to trade routes), ship routing and scheduling, inventory ship routing, berth scheduling, crane scheduling, container yard management, container stowage planning, ship management, and distribution of empty containers.

The operational problems involve:

cruising speed selection,

ship loading, and environmental routing.

Handling of hazardous materials poses additional challenges. However, thischapter concentrates on the water-side of maritime transportation. Land-sideoperations and hazardous materials are discussed in other chapters in this

volume. Before diving into discussion of OR models in maritime transporta-tion it is worthwhile to take a closer look at the operational characteristicsof maritime transportation and to clarify various terms that are used in thisarea. Figure 1 relates the demand for maritime transportation to its supply,

provides a comprehensive view of these characteristics and ties them together(adapted from Jansson and Shneerson, 1987). The following three sections de-scribe these characteristics, starting on the supply side.


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Fig. 1. Characteristic of maritime transportation demand and supply.

2.1 Ship and port characteristics

In this chapter we use the terms ship and vessel interchangeably. Althoughvessel may refer to other means of transportation, we shall use it in the tradi-tional sense, referring to a ship.

Ships come in a variety of sizes. The size of a ship is measured by its weightcarrying capacity and by its volume carrying capacity. Cargo with low weightper unit of volume fills the ships volume before it reaches its weight capacity.Deadweight (DWT) is the weight carrying capacity of a ship, in metric tons.That includes the weight of the cargo, as well as the weight of fuels, lube oils,supplies, and anything else on the ship. Gross Tons (GT) is the volume of theenclosed spaces of the ship in hundreds of cubic feet.

Ships come also in a variety of types. Tankers are designed to carry liquidsin bulk. The larger ones carry crude oil while the smaller ones usually carry

oil products, chemicals, and other liquids. Bulk carriers carry dry bulk com-modities such as iron ore, coal, grain, bauxite, alumina, phosphate, and otherminerals. Some of the bulk carriers are self-discharging. They carry their own


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unloading equipment, and are not dependent on port equipment for unloadingtheir cargo. Liquefied Gas Carriers carry refrigerated gas under high pressure.Container Ships carry standardized metal containers in which packaged goods

are stowed. General Cargo vessels carry in their holds and above deck all typesof goods, usually packaged ones. These vessels often have multiple decks orfloors. Since handling general cargo is labor intensive and time consuming,general cargo has been containerized during the last four decades, thus reduc-ing the time that ships carrying such cargo spend in ports from days to hours.Refrigerated vessels or reefers are designed to carry cargo that requires refrig-eration or temperature control, like fish, meat, and citrus, but can also carrygeneral cargo. Roll-onRoll-off(RoRo) vessels have ramps for trucks and carsto drive on and off the vessel. Other types of vessels are ferries, passenger ships,fishingvessels, service/supply vessels, barges (self propelled or pushed/pulled by

tugs), research ships, dredgers, naval vessels, and other, special purpose vessels.Some ships are designed as combination of the above types, e.g., ore-bulk-oil,general cargo with refrigerated compartments, passenger and RoRo.

Ships operate between ports. Ports are used for loading and unloading cargoas well as for loading fuel, fresh water, and supplies, and discharging waste.Ports impose physical limitations on the dimensions of the ships that maycall in them (ship draft, length and width), and usually charge fees for theirservices. Sometimes ports are used for transshipment of cargo among ships,especially when the cargo is containerized. Major container lines often oper-ate large vessels between hub ports, and use smaller vessels to feed containers

to/from spoke ports.

2.2 Types of shipping services

There are three basic modes of operation of commercial ships: liner, tramp,and industrial operations (Lawrence, 1972). Liners operate according to a pub-lished itinerary and schedule similar to a bus line, and the demand for theirservices depends among other things on their schedules. Liner operators usu-ally control container and general cargo vessels. Tramp ships follow the avail-able cargoes, similar to a taxicab. Often tramp ships engage in contracts ofaffreightment. These are contracts where specified quantities of cargo have tobe carried between specified ports within a specific time frame for an agreedupon payment per unit of cargo. Tramp operators usually control tankers anddry bulk carriers. Both liner and tramp operators try to maximize their profitsper time unit. Industrial operators usually own the cargoes shipped and controlthe vessels used to ship them. These vessels may be their own or on a timecharter. Industrial operators strive to minimize the cost of shipping their car-goes. Such operations abound in high volume liquid and dry bulk trades of

vertically integrated companies, such as: oil, chemicals, and ores. When any

type of operator faces insufficient fleet capacity the operator may be able tocharter in additional vessels. Whereas liners and tramp operators may give upthe excess demand and related income, industrial operators must ship all their


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cargoes. In cases of excess fleet capacity, vessels may be chartered out (to otheroperators), laid-up or even scrapped. However, when liners reduce their fleetsize they must reshuffle their itineraries and/or schedules, which may result in

reduced service frequency or withdrawal from certain markets. In both casesrevenues may drop. An interesting historical account of the development ofliner services in the US is provided by Fleming (2002, 2003).

Industrial operators, who are usually more risk-averse and tend not to char-ter-out their vessels, size their fleet below their long-term needs, and comple-ment it by short-term (time or voyage/spot) charters from the tramp segment.Seasonal variations in demand, and uncertainties regarding level of future de-mand, freight rates, and cost of vessels (both newbuildings and second-hand)affect the fleet size decision. However, when the trade is highly specialized(e.g., liquefied gas carriers) no tramp market exists and the industrial operator

must assure sufficient shipping capacity through long-term commitments. Theease of entry into the maritime industry is manifested in the tramp market thatis highly entrepreneurial. This results in long periods of oversupply of shippingcapacity and the associated depressed freight rates and vessel prices. However,certain market segments, such as container lines, pose large economies of scaleand are hard to enter.

Naval vessels are a different breed. Naval vessels alternate between deploy-ment at sea and relatively lengthy port periods. The major objective in navalapplications is to maximize a set of measures of effectiveness. Hughes (2002)provides an interesting personal perspective of naval OR.

2.3 Cargo characteristics

Ships carry a large variety of goods. The goods may be manufactured con-sumer goods, unprocessed fruits and vegetables, processed food, livestock,intermediate goods, industrial equipment, processed materials, and raw ma-terials. These goods may come in a variety of packaging, such as: boxes, bags,drums, bales, and rolls, or may be unpackaged, or even in bulk. Sometimes car-goes are unitized into larger standardized units, such as: pallets, containers, ortrailers. Generally, in order to facilitate more efficient cargo handling, goodsthat are shipped in larger quantities are shipped in larger handling units or inbulk. During the last several decades packaged goods that required multiplemanual handlings, and were traditionally shipped by liners, have been con-tainerized into standard containers. Containerization of such goods facilitatesefficient mechanized handling of the cargo, and thus saves time and money,and also reduces pilferage. Shipping containers come in two lengths, 20 feetand 40 feet. A 20 container carries up to approximately 28 tons of cargo with avolume of up to 1000 cubic feet. Most containers are metal boxes with an 88

cross-section, but other varieties exist, such as: refrigerated containers, open

top, open side, and half height. In addition there are containers of nonstan-dard sizes. Large containerships can carry thousands of Twenty feet EquivalentUnits (TEUs), where a 40 container is counted as two TEUs.


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In addition, goods that are shipped in larger quantities are usually shippedmore often and in larger shipment sizes. Cargoes may require shoring on theship in order to prevent them from shifting during the passage, and may require

refrigeration, controlled temperature, or special handling while on board theship. Different goods may have different weight density, thus a ship may be fulleither by weight or by volume, or by another measure of capacity.

2.4 Geographical characteristics

Shipping routes may be classified according to their geographical character-istics (and the corresponding type and size of vessel used): deep-sea, short-sea,coastal, and inland waterways. Due to economies of scale in shipping larger sizevessels are employed in deep-sea trades between continents whereas smaller

size vessels usually operate in short-sea and coastal routes, where voyage legsare relatively short. As mentioned above, smaller containerships are used onshort-sea routes that feed cargo to larger vessels that operate on long deep-searoutes. A similar picture can sometimes be observed with tankers where largecrude carriers used for long routes are lightered at an off shore terminal tosmaller vessels (often barges). Due to draft restrictions inland waterways areused mainly by barges. Barges are used to move cargoes between the hinter-land and coastal areas, often for transshipments to/from ocean-going vessels,or to move cargoes between inland ports.

2.5 Terms used in maritime transportation planning

Shippingrefers to moving cargoes by ships. The shipperis the owner of the transported cargo. A shipment is a specified amount of cargo that must be shipped to-

gether from a single origin to a single destination. Routing is the assignment of a sequence of ports to a vessel. Environ-

mental routingor weather routing is the determination of the best pathin a body of water that a vessel should follow.

Scheduling is assigning times (or time windows) to the various eventson a ships route.

Deployment refers to the assignment of the vessels in the fleet to traderoutes. The differentiation between deployment and scheduling is notalways clear cut. Deployment is usually used when vessels are desig-nated to perform multiple consecutive trips on the same route, andtherefore is associated with liners and a longer planning horizon. Lin-ers follow a published sailing schedule and face more stable demand.Schedulingdoes not imply allocation of vessels to specific trade routes,but rather to specific shipments, and is associated with tramp and

industrial operations. Due to higher uncertainty regarding future de-mand in these operations, their schedules usually have a shorter plan-ning horizon.


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A voyage consists of a sequence of port calls, starting with the portwhere the ship loads its first cargo and ending where the ship unloadsits last cargo and becomes empty again. A voyage may include multiple

loading ports and multiple unloading ports. Liners may not becomeempty between consecutive voyages, and in that case a voyage starts atthe port specified by the ship operator (usually a primary loading port).

Throughout this chapter we use also the following definitions:

A cargo is a set of goods shipped together from a single origin to asingle destination. In the vehicle routing literature it is often referredto as an order. The terms shipment and cargo are used interchangeably.

Aload is the set of cargoes that is on the ship at any given point in time. A load is considered a full shipload when it consists of a single cargo

that for practical and/or contractual reasons cannot be carried withother cargoes.

A product is a set of goods that can be stowed together in the samecompartment. In the vehicle routing literature it is sometimes referredto as a commodity.

Aloading port is a pickup location (corresponds to a pickup node). An unloading port is a delivery location (corresponds to a delivery

node).

3 Strategic planning in maritime transportation

Strategic decisions are long-term decisions that set the stage for tactical andoperational ones. In maritime transportation strategic decisions cover a widespectrum, from the design of the transportation services to accepting long-termcontracts. Most of the strategic decisions are on the supply side, and theseare: market selection, fleet size and mix, transportation system/service networkdesign, maritime supply chain/maritime logistic system design, and ship design.Due to characteristics discussed earlier maritime transportation markets areusually competitive and highly volatile over time, and that complicates strategicdecisions.

In this section we address the various types of strategic decisions in maritimetransportation and present models for making such decisions. Section 3.1 thatdiscusses ship design is followed by Section 3.2 that deals with fleet size and mixdecisions. Section 3.3 treats network design in liner shipping, and Section 3.4handles transportation system design. Finally, Section 3.5 addresses evaluationof long-term contracts.

In order to be able to make strategic decisions one usually needs some tacti-cal or even operational information. Thus there is a significant overlap between

strategic and tactical/operational decisions. Models used for fleet size and mixdecisions and network design decisions often require evaluation of ship rout-ing strategies. Such routing models usually fall into one of two categories, arc


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flow models or path flow models. In arc flow models a binary variable is used torepresent whether a specific vessel v travels directly from port (or customer) ito port (or customer) j. The model constructs the routes that will be used by

the vessels, and the model has to keep track of both travel time and load oneach vessel. In path flow models the routes are predefined, one way or another,and a binary variable represents whether vessel v performs route r. A route isusually a full schedule for the vessel that specifies expected arrival times andload on the vessel along the route. Such a model can focus on the set of portsor customers to serve, and only feasible routes are considered.

3.1 Ship design

A ship is basically a floating plant with housing for the crew. Therefore, ship

design covers a large variety of topics that are addressed by naval architects andmarine engineers, and they include structural and stability issues, materials,on-board mechanical and electrical systems, cargo handling equipment, andmany others. Some of these issues have direct impact on the ships commercial

viability, and we shall focus here on two such issues, ship size and speed.The issue of the optimal size of a ship arises when one tries to determine

what is the best ship for a specific trade. In this section we deal with the opti-mal size of a single ship regardless of other ships that may be included in thesame fleet. The latter issue, the optimal size and composition of a fleet, is dis-cussed in Section 3.2. The optimal ship size is the one that minimizes the ship

operators cost per ton of cargo on a specific trade route with a specified cargomix. However, one should realize that in certain situations factors beyond costsmay dictate the ship size.

Ships are productive and generate income at sea. Port time is a necessaryevil for loading and unloading cargo. Significant economies of scale exist atsea where the cost per cargo ton-mile decreases with increasing the ship size.These economies stem from the capital costs of the ship (design, construc-tion, and financing costs), from fuel consumption, and from the operating costs(crew cost, supplies, insurance, and repairs). However, at port the picture is dif-ferent. Loading and unloading rates are usually determined by the land-sidecargo handling equipment and available storage space. Depending on the typeof cargo and whether the cargo handling is done by the land-side equipment orby the equipment on the ship (e.g., pumps, derricks), the cargo handling ratemay be constant (i.e., does not depend on the size of the ship), or, for dry cargo

where multiple cranes can work in parallel, the cargo handling rate may be ap-proximately proportional to the length of the ship. Since the size of the shipis determined by its length, width, and draft, and since the proportions amongthese three dimensions are practically almost constant, the size of the ship isapproximately proportional to the third power of its length. Therefore, in the

better case, cargo-handling rates will be proportional to the 1/3 power of theship size. However, when the cargo is liquid bulk (e.g., oil) the cargo-handlingrate may not be related to the size of the ship.


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A ship represents a large capital investment that translates into a large costper day. Port time is expensive and presents diseconomies of scale. Thus thetime of port operations caps the optimal size of ship. Generally, the longer a

trade route is, the larger the share of sea-days in a voyage, and the larger theoptimal ship size will be. Other factors that affect the optimal ship size are theutilization of ship capacity at sea (the trade balance), loading and unloadingrates at the ports, and the various costs associated with the ship. On certainroutes there may be additional considerations that affect the size of the ship,such as required frequency of service and availability of cargo.

A ship is a long-term investment. The useful life of a ship spans 2030 years.Thus, the optimal ship size is a long-term decision that must be based on ex-pectations regarding future market conditions. During the life of a ship a lotof market volatility may be encountered. Freight rates may fluctuate over a

wide range, and the same is true for the cost of a ship, whether it is a secondhand one or a newbuilding. When freight rates are depressed they may noteven cover the variable operating costs of the ship, and the owner has very fewalternatives. In the short run the owner may either reduce the daily variableoperating cost of the ship by slow steaming, that results in significant reductionin fuel consumption, or the owner may lay up the ship till the market improves.Laying up a ship involves a significant set-up cost to put the ship into lay up,and, eventually, to bring it back into service. However, laying up a ship signifi-cantly reduces its daily variable operating cost. When the market is depressed,owners scrap older ships. The value of a scrapped ship is determined by the

weight of its steel (the lightweight of the ship), but when there is high sup-ply of ships for scrap the price paid per ton of scrap drops. Occasionally, in a

very depressed market, a newly built vessel may find itself in the scrapping yardwithout ever carrying any cargo.

In the shorter run ship size may be limited by parameters of the specifictrade, such as availability of cargoes, required frequency of service, physicallimitations of port facilities such as ship draft, length, or width, and availablecargo handling equipment and cargo storage capacity in the ports. In the longerrun many of these limitations can be relaxed if there is an economic justifica-tion to do so. In addition there are limitations of ship design and constructiontechnology, as well as channel restrictions in canals in the selected trade routes.

The issue of long-run optimal ship size has been discussed mainly by econo-mists. Jansson and Shneerson (1982) presented a comprehensive model for thedetermination of long-run optimal ship size. They separated the ship capacityinto two components:

the hauling capacity (the ship size times its speed), and the handling capacity (cargo loaded or unloaded per time unit).

This separation facilitated the division of the total shipping costs into cost

per ton of cargo carried in the voyage that does not depend on the length of thevoyage, and cost per time unit. These two cost components are combined into acost model that conveys the cost of shipping a ton of cargo a given distance. The


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model requires estimation of output and cost elasticities. These elasticities,combined with the route characteristics and input prices, allow estimation ofthe optimal size of the ship. This model requires estimation of its parameters

through regression analysis. However, high shipping market volatility over timeresults in low reliability of such estimates. They demonstrated the use of themodel by calculating the optimal size of a coal bulk carrier for a specific trade.This work also inspects the sensitivity of the optimal ship size to four routecharacteristics: distance, port productivity, trade balance, and fuel costs. Mostof the elasticities that are necessary for this model were estimated from severaldatasets in their earlier work (Jansson and Shneerson, 1978). However, that

work calculated a single ship size elasticity of operating costs for each shiptype. In a later study, Talley et al. (1986) analyzed short-run variable costs oftankers and concluded that the ship size elasticity of operating costs may vary

according to the size of the ship of the specific type.Modern cargo handling equipment that is customized for the specific cargo

results in higher loading and unloading rates, and shorter port calls. Suchequipment is justified where there is a high volume of cargo. That is usually thecase in major bulk trades. Garrod and Miklius (1985) showed that under suchcircumstances the optimal ship size becomes very large, far beyond the capac-ity of existing port facilities. In addition, with such large ships the frequency ofshipments drops to a point where inventory carrying costs incurred by the ship-per start playing a significant role (the shipment size is the ship capacity). When

one includes the inventory costs in the determination of the optimal ship size,that size is reduced significantly. The resulting ship sizes are still much largerthan existing port facilities can accommodate, and thus the main limit on shipsizes is the draft limitation of ports. However, for a higher value cargo, or forless efficient port operations, smaller vessel sizes are optimal (see, for example,

Ariel, 1991). In short-sea operations competition with other modes may play asignificant role. In order to compete with other modes of transportation morefrequent service may be necessary. In such cases frequency and speed of ser-

vice combined with cargo availability may be a determining factor in selectingthe ship size.

In liner trades, where there are numerous shippers, multiple ports, and awide variety of products shipped, the inclusion of the shippers inventory costsin the determination of the optimal ship size is more complex. Jansson andShneerson (1985) presented the initial model for this case. In addition to thecosts incurred by the ship owner/operator they included the costs of inventorythat are incurred by the shipper (including the cost of safety stocks). The size(and cost) of the safety stocks is a function of the frequency of sailings on theroute, and that frequency is affected by the ship size and the volume of trade.Numerous assumptions regarding the trade and the costs were necessary, and

the inclusion of the shippers costs reduced very much the optimal ship size.One could argue with the assumptions of the model, but the conclusions makesense.


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Whereas Jansson and Shneerson (1985) considered a continuous review in-ventory control system by the shippers, Pope and Talley (1988) looked at thecase of a periodic review system that is more appropriate when using a (sched-

uled) liner service. They found that optimal ship size is highly sensitiveto the inventory management model selected, the treatment of stockouts andsafety stocks, and the inventory management cost structure that prevails, andconcluded that rather than computing optimal ship size, it may be more ap-propriate to compute the optimal load size. As far as liner operations areconcerned we agree with this conclusion. The optimal ship size is a long-termdecision of the ship owner/operator who serves a large number of shippers.Each shipper may face different circumstances that may change over time,and therefore should be concerned with the optimal load (shipment) size. Theoptimal load size is a short-term decision that may change with the changing

circumstances.A historical perspective on the development of size, speed, and other char-

acteristics oflarge container ships is provided by Gillman (1999). Cullinane andKhanna (1999) present a more recent detailed study of the economies of scaleof large container ships. They take into account the considerable increase inport productivity, and take a closer look at the time in port. They find smallerdiseconomies of scale (in port) than earlier studies, and show that the opti-mal size of a container ship continues to increase with improvements in portproductivity. Taking advantage of these economies of scale to reduce shippingcosts per unit while maintaining frequency of service, requires larger volumeson the trade route. This is one of the major catalysts for industry consolidation.However, McLellan (1997) injects a dose of reality to the discussion and pointsout that there are practical limits to the size of large containerships imposedby port draft, container handling technology, space availability, and requiredinvestments in port and transportation infrastructure.

Whereas cargo ships come in a large variety of sizes, from under 1000 DWTup to more than 500,000 DWT, their designed speed varies in a much narrowerrange. When one excludes outliers the ratio between the designed speed of afast ship and a slow ship is about 2. The designed speed of a ship is a long-

term decision that affects its hauling capacity and is part of optimal ship sizeconsiderations. As a general rule the design speed of a ship increases by thesquare root of its length. This implies that the design speed is proportional tothe 1/6 power of the size of the ship. This relationship was confirmed statis-tically by Jansson and Shneerson (1978), and more recently by Cullinane andKhanna (1999).

3.2 Fleet size and mix

One of the main strategic issues for shipping companies is the design of anoptimal fleet. This deals with both the type of ships to include in the fleet, theirsizes, and the number of ships of each size.


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In order to support decisions concerning the optimal fleet of ships for anoperator, we have to consider the underlying structure of the operational plan-ning problem. This means that fleet size and mix models very often include

routing decisions. For the various fleet size and mix problem types discussed inthis section we can develop models that are based on the tactical models de-scribed thoroughly in Section 4.1. The objective of the strategic fleet size andmix problem is usually to minimize the fixed (setup) costs of the ships used andthe variable operating costs of these ships. In a tactical routing and schedulingproblem one usually minimizes only the operating costs of the ships. However,the routing decisions made in a strategic model can be later changed duringtactical planning.

In addition, the fleet size and mix decisions have to be based on an esti-mate of demand for the transportation services. The demand forecast is highly

uncertain, and stochastic techniques for considering the uncertain informationare relevant for solving such strategic planning problems. Issues of robust plan-ning are discussed in Section 6. In the literature, various demand patterns areconsidered where either the size of the cargoes or the frequency of sailing isspecified.

In tramp shipping, contract evaluation and fleet size issues are closely re-lated. A shipping company has to find the best split between fixed long-termcargo contracts and spot cargoes. This split should be based on estimation offuture prices and demand. When considering the fleet size and mix these issuesshould be included. This topic is further discussed in Section 3.5.

In Section 3.2.1 we describe the fleet planning problem for a homogeneousfleet where all the vessels are of the same type, size, and cost, while the fleetsize and mix for a heterogeneous fleet is the topic of Section 3.2.2.

3.2.1 Homogeneous fleet sizeIn this section, we want to focus on a simple industrial fleet size problem

for a fleet consisting of ships of the same type, size, and cost. In the end of thesection some comments regarding other studies are given.

In the fleet size planning problem considered here, a homogeneous fleet ofships is engaged in transportation of full shipload cargoes from loading ports tounloading ports. This means that just one cargo is onboard a ship at a time, andeach cargo is transported directly from its loading port to its correspondingunloading port.

All the required ship arrival times at the loading ports are fixed and known.Further, we also assume that the loading times and sailing times are known,such that the arrival times at the unloading ports can be easily calculated. Theunloading times and the sailing time from each unloading port to all loadingports are also known.

The demand is such that all cargoes, given by specified loading and unload-

ing ports, have to be serviced. The ships should be routed from the unloadingports to the loading ports in a way that minimizes the total cost of their ballastlegs. Since the fleet is homogeneous and all cargoes must be transported, the


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cost of the loaded legs is constant and we can leave it out. In addition, we wantto minimize the number of necessary ships, and we assume that the number ofships needed dominates the sailing costs.

In the mathematical description of the problem, letN

be the set of cargoesindexed by i and j. Cargo i is represented by a node in the network, and thisnode includes one loading port and one unloading port for cargo i. Since wehave full information about activity times, we can determine the feasible cargopairs (ij). If cargo i can be serviced just before cargo jby the same ship, suchan (i j)-pair is feasible and represents an arc in the network. However, if thetime between the loads is too long, the arc may be eliminated since using sucharcs would result in unacceptable high waiting times. Similarly, if the departingtime at node i plus the sailing time to j is greater than the given arrival timeat jthere will be no arc connecting the two cargoes. Let Ni andN

+i be the set

of all cargoes a ship can service immediately before and after servicing cargo i,respectively. Further, let V be the set of ships in the fleet indexed by v, andthis set includes an assumption on the upper bound on the number of shipsnecessary. For each possible ship, we define an artificial origin cargo o(v) andan artificial destination cargo d(v).

The operational cost of sailing from the unloading port for cargo i to theloading port of cargo j is denoted by Cij.

In the mathematical formulation, we use the following types of variables: thebinary flow variable xij, i N, j N

+i , equals 1, if a ship services cargo i just

before cargo j, and 0 otherwise. In addition, we define flow variables for theartificial origin and artificial destination cargoes: xo(v)j, v V, j N {d(v)},and xid(v), v V, i N {o(v)}. If a ship v is not operating, then xo(v)d(v) = 1.

The arc flow formulation of the industrial ship fleet size problem for onetype of ships and full ship loads is as follows:

(3.1)min

iN

jN+i

Cijxij vV

xo(v)d(v)

subject to

(3.2)

jN{d(v)}xo(v)j = 1 v V

(3.3)

iN{o(v)}

xid(v) = 1 v V

(3.4)

jN+i

xij+vV

xid(v) = 1 i N

(3.5)

iN

j

xij +vV

xo(v)j = 1 j N

(3.6)xij {0 1} v V i N

o(v)

j N+i

d(v)


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In the first term of the objective function (3.1), we minimize the costs of theballast legs of the ships. Since xo(v)d(v) = 1 if ship v is not operating, the secondterm in the objective function minimizes the number of ships in operation. The

first term is scaled in a manner that its absolute value is less than one. Thismeans that the objective (3.1) first minimizes the number of ships in use andthen as a second goal minimizes the operating costs of the ships. The secondterm in the objective function could easily be incorporated in the first term.However, the present form of the objective function is chosen to highlight thetwofold objective. Constraints (3.2) ensure that each ship leaves its artificialorigin cargo and either services one of the real cargoes or sails directly to itsartificial destination cargo. In constraints (3.3) each ship in the end of its routehas to arrive at its artificial destination cargo from somewhere. Constraints(3.4) ensure that the ship that services cargo i has to either service another

cargo afterward or sail to its artificial destination cargo, while constraints (3.5)say that the ship servicing cargo j has to come from somewhere. Finally, theformulation involves binary requirements (3.6) on the flow variables.

We can easily see that the formulation (3.1)(3.6) has the same structure asan assignment problem. Therefore the integrality constraints (3.6) are not acomplicating factor. The problem is easily solved by any version of the simplexmethod or by a special algorithm for the assignment problem.

When applying a simplex method, it would be possible to have just one com-mon artificial origin, o, and one common artificial destination, d, cargo. Then

xo(v)j, v V

, j N

{d(v)}, and xid(v), v V

, i N

{o(v)}, can be trans-formed into xoj, j N {d}, and xid, i N {o}. While the xoj and xidvariables remain binary the variable xod becomes integer.

For some problems, some of the cargoes may have a common loading portand/or a common unloading port. If the given starting times are such that sev-eral cargoes are loaded or unloaded in the same port at the same time, weassume that if this has any effect on the (un)loading times it is already ac-counted for in the specified data.

In a case with the same starting times in the same ports, we might change theformulation slightly. Constraints (3.4) can be considered as the constraints for

leaving the unloading port for cargo i, and (3.5) as the constraints for arrivingat the loading port for cargo j. We can then aggregate constraints for cargoes

with the same ports and starting times. This will give more variables at theleft-hand side of the constraints and a right-hand side equal to the numberof aggregated constraints. The corresponding flow variables from and to theartificial cargoes will become integers rather than binary.

If some of the cargoes have the same loading and unloading ports and thesame starting times then we can switch from indexing the variables by cargonumbers to indexing them by loading port, unloading port, and both loading

and unloading times. Then the variables can be integer rather than binary, andtheir number will be reduced. Dantzig and Fulkerson (1954) pioneered sucha model using a different notation for a problem with naval fuel oil tankers.


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They solved a problem with 20 cargoes by using the transportation model. Thenumber of ships was minimized and 6 ships were needed.

Later Bellmore (1968) modified the problem. An insufficient number of

tankers and a utility associated with each cargo were assumed. The problemwas to determine the schedules for the fleet that maximized the sum of theutilities of the carried cargoes, and it was shown to be equivalent to a trans-shipment problem.

Another homogeneous fleet size problem is considered in Jaikumar andSolomon (1987). Their objective is to minimize the number of tugs required totransport a given number of barges between different ports in a river system.They take advantage of the fact that the service times are negligible compared

with the transit times, and of the geographical structure of the port locationson the river, and develop a highly effective polynomial exact algorithm. This

problem has a line (or tree) structure, and this fact is exploited in the modeldefinition.

Recently Sambracos et al. (2004) addressed the fleet size issue for short-sea freight services. They investigate the introduction of small containers forcoastal freight shipping in the Greek Aegean Sea from two different aspects.First, a strategic planning model is developed for determining the homoge-neous fleet size under known supply and demand constraints where total fuelcosts and port dues are minimized. Subsequently, the operational dimension ofthe problem is analyzed by introducing a vehicle routing problem formulationcorresponding to the periodic needs for transportation using small containers.

Many simplifying assumptions are made in this study. They conclude that a 5 %cost saving may be realized by redesigning the inter-island links.

3.2.2 Heterogeneous fleet size and mixIn this section we extend the planning problem discussed in Section 3.2.1

and include decisions about the mix of different ship sizes.We study here one particular fleet size and mix problem, where a liner

shipping company wants to serve several customers that have a demand forfrequent service. The problem consists of determining the best mix of ships toserve known frequencies of demand between several origindestination portpairs. Many feasible routes are predefined, and just some of them will be usedin the optimal solution. The demand is given as a minimum required numberof times each port pair has to be serviced. The underlying real problem is apickup and delivery problem. However, with predefined routes in the model,the loading and unloading aspects are not visible but hidden in the routes.Since this is a pickup and delivery problem, the frequency demand applies toa pair of ports. The ships are heterogeneous so not all ships can sail all routes.The capacity of a ship determines, among other factors, which routes it cansail. A ship is allowed to split its time between several routes.

The planning problem consists of deciding: (1) which ships to operate and(2) which routes each ship should sail and the number of voyages along eachroute. The first part is a strategic fleet mix and size problem and the second


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part is a tactical fleet deployment problem. Fleet deployment problems arediscussed in Section 4.4. The second part is used here only to find the bestsolution to the first part. If the demand pattern changes later, the second part

can be resolved for the then available fleet.In the mathematical description of the problem, let V be the set of shipsindexed by v andRv the set of routes that can be sailed by ship v indexed by r.The set of origindestination port pairs is calledN indexed by i, and each suchpair needs to be serviced at least Di times during the planning horizon.

The cost consists of two parts. We define the cost of sailing one voyage withship v on route r as CVvr. The fixed cost for ship v during the planning horizonis called CFv. Each voyage with ship v on route rtakes TVvr time units, and Airis equal to 1 if origindestination port pair i is serviced on route r. The lengthof the planning horizon is T, and we assume that the ships are available for the

whole horizon. Let Uv be an upper bound on the number of voyages ship v cansail during the planning horizon.

Here we use the following types of decision variables: uvr, v V, r Rv,represents the number of voyages along route rwith ship v during the planninghorizon, and sv, v V, is equal to 1 if ship v is used.

The model for the strategic fleet size and mix problem with predefinedroutes can then be written as

(3.7)min

vV

rRv

CVvruvr +vV

CFvsv

subject to

(3.8)

rRv

uvr Uvsv 0 v V

(3.9)vV

rRv

Airuvr Di i N

(3.10)

rRv

TVvruvr T v V

(3.11)uvr 0 and integer v V r R

v

(3.12)sv {0 1} v V

Here (3.7) is the cost of sailing the used routes together with the fixed cost ofthe ships in operation. Constraints (3.8) ensure that the fixed costs for the shipsin operation are taken into account. Constraints (3.9) say that each port pairis serviced at least the required number of times, and constraints (3.10) ensurethat each ship finishes all its routes within the planning horizon. Finally, theformulation involves integer and binary requirements on the variables.

Fagerholt and Lindstad (2000) presented this model with different notation

and gave an example where the model was used to plan deliveries to Norwe-gian petroleum installations in the North Sea. Their problem had one loadingport and seven unloading installations. They managed to pre-calculate all the


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feasible routes and their integer program was solved by CPLEX. The modeldoes not ensure that services for a given port pair are properly spaced duringthe planning horizon. This aspect was treated manually after the model so-

lutions were generated. Fagerholt and Lindstad (2000) report that the modelsolution implemented gave annual savings of several million US dollars.Another study regarding fleet size and mix for liner routes was done by Cho

and Perakis (1996). The study was performed for a container shipping com-pany. The type of model and solution method is similar to the one used byFagerholt and Lindstad (2000). Xinlian et al. (2000) consider a similar prob-lem. They present a long-term fleet planning model that aims at determining

which ships should be added to the existing fleet, ship retirements, and the op-timal fleet deployment plan. Another study regarding the design of an optimalfleet and the corresponding weekly routes for each ship for a liner shipping

system along the Norwegian coast was presented by Fagerholt (1999). The so-lution method is similar to the one used by Fagerholt and Lindstad (2000). InFagerholt (1999) the solution method handled only instances where the dif-ferent ships that could be selected have the same speed. This is in contrast tothe work in Fagerholt and Lindstad (2000), where the ships can have differentspeeds. Yet another contribution within fleet size and mix for liner shipping isgiven by Lane et al. (1987). They consider the problem of deciding a cost effi-cient fleet that meets a known demand for shipping services on a defined linertrade route. The solution method has some similarities to the approach usedby Fagerholt and Lindstad (2000), but the method gives no proven optimal so-

lution since only a subset of the feasible voyage options are selected and theuser determines the combination of vessel and voyage. The method has beenapplied on the Australia/US West coast route. Finally, resource managementfor a container vessel fleet is studied by Pesenti (1995). This problem involvesdecisions on the purchase and use of ships in order to satisfy customers de-mand. A hierarchical model for the problem has been developed, and heuristictechniques, which solve problems at different decision levels, are described.

A rather special problem regarding the size of the US destroyer fleet is de-scribed in Crary et al. (2002), which illustrates the use of quantitative methodsin conjunction with expert opinion. These ideas are applied to the planningscenario for the 2015 conflict on the Korean Peninsula, one of two key sce-narios the Department of Defense uses for planning.

3.3 Liner network design

On all three planning levels the challenges in liner shipping are quite differ-ent from those of tramp or industrial. Liner ships are employed on more or lessfixed routes, calling regularly at many ports. In contrast to industrial or trampships a liner ship serves demand of many shippers simultaneously, and its pub-

lished route and frequency of service attract demand. The major challengesfor liners at the strategic level are the design of liner routes and the associatedfrequency of service, fleet size and mix decisions and contract evaluation for


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long-term contracts. The fleet size and mix decisions for the major market seg-ments, including liner operations, are discussed in Section 3.2, while contractevaluation will be treated in Section 3.5. Here we focus on the design of liner

routes. We split this section into three parts, where traditional liner operationsare discussed in Section 3.3.1, and the more complex hub and spoke networksare considered in Section 3.3.2. Finally, we comment upon shuttle services inSection 3.3.3.

3.3.1 Traditional liner operationsLiner routes and schedules are usually set up in a manner similar to bus

schedules. Before entering a particular market a liner shipping company hasto thoroughly estimate the demand, revenue and cost of servicing that market.Based on this information, the company has to design its routes and to publish

a sailing schedule.Most liner companies are transporting containers, so we use here the term

container(s) instead of cargo units or cargoes. We focus here on a problemwhere a liner container company is going to operate several different routesamong a set of ports ordered more or less along a straight line. Meaning thateven if a route skips a port in a contiguous sequence of ports the ship passesfairly close to the skipped port. This is usually the situation faced by longercontainer lines. The demands, as upper bounds on the number of transportedcontainers, are given between all pairs of ports. The fleet of ships is heteroge-

neous and the planning problem consists of designing a route for each ship ina manner that maximizes the total net revenue of the fleet. One route is con-structed for each ship and the ship sails as many voyages along that route as itcan during the planning horizon.

The mathematical model is based on an arc flow formulation. The portsare numbered from 1 to N, and there are some strict constraints on how theroutes can be constructed. Each route must have two end ports i and j, where1 i < j N. A route then starts in i and travels outbound to ports withhigher and higher number until the route reaches j, where it turns aroundand starts its inbound travel to ports with lower and lower number until the

route ends in i. A ship with i and j as end ports, does not necessarily call atall the ports between i and j, and it does not need to visit the same ports onthe outbound and inbound legs of the route. See Figure 2 for an illustration ofsuch routes.

When a ship arrives at one of its end ports it unloads all containers that areon board before it starts loading all the containers that it should load in thatport. This means that each container is loaded in its loading port and stays onboard the ship while the ship either sails a part of the outbound or inboundroute before it is unloaded in its unloading port.

In the mathematical description of the problem, letV

be the set of shipsindexed by v and N the set of linearly ordered ports indexed by i, j, k, i,or j. In addition we need the subsets N+i = {i + 1 N } of ports in the


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Fig. 2. Liner network design for traditional liner operations including some but not all routes.

line numbered after i and Ni = {1 i 1} of ports in the line numberedbefore i.

The revenue for transporting one container from port i to port j is RTij andthe cost of sailing directly from port i to port j with ship v is CTijv . Ship v hasa capacity that is measured in number of containers when it sails directly fromport i to port j, and it is represented by QTijv . Most often it will be sufficientnot to let capacity depend on the sailing leg (i j), but in rare cases capacitymay depend on weather conditions or other factors. The ship spends TTijv timeunits on that trip including the time for unloading and loading in port i. It ismeaningful to assume that this time does not vary with the number of contain-ers loaded and unloaded only if the number of such containers does not varyfrom call to call or that the unloading and loading time is very short comparedto the sailing time. The demand as an upper bound on the number of contain-ers transported from port i to port j during the planning horizon is denotedby DTij. The constant Sv is the maximum time ship v is available during theplanning period.

We use the following types of decision variables: eijv , v V, i N, j N,represents the number of containers transported from port i to port jby ship von each voyage during the planning horizon. Ship v does not necessarily saildirectly from port i to port j. If ship v sails directly from port i to port j on itsroute, then the binary variable xijv , v V, i N, j N, is equal to 1. Theinteger variable wv, v V, gives the number of whole voyages ship v managesto complete during the planning horizon. The binary variable yijv , v V,

i N\{N}, j N+

i , is equal to 1 if ship v is allocated to a route that starts inport i and turns around in port j. These two ports i and j are called end portsfor ship v.

A route design model for traditional liner operators can then be written as

(3.13)maxvV

iN

jN

wv(RTijeijv CTijv xijv )

subject to

xijv

iNi+1

jN+j1

eijv QTijv

0

(3.14)v V i N\{N} j N+i


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xijv

iN+

i1

jN

j+1

eijv QTijv

0

(3.15)v V i N\{1} j N

i

wveijv DTij

jN+i \N+

j

xijv

(3.16)v V i N\{N} j N+i

wveijv DTij

jNi \N

j

xijv

(3.17)v V i N\{1} j Ni

wveijv DTij

iNj \Ni

xijv

(3.18)v V i N\{N} j N+i

wveijv DTij

iN+j \N+i

xijv

(3.19)v V i N\{1} j Ni

(3.20)vV

wveijv DTij i N j N i = j

(3.21)wv

iN

jN

TTijv xijv

Sv v V

(3.22)

iN\{N}

jN+i

yijv 1 v V

yijv

jN+i \N+j

xijv 1

= 0

(3.23)v V i N\{N} j N+i

yijv

jN+i \N

+j

xjiv 1

= 0

(3.24)v V i N\{N} j N+i

yijv

iNk \Ni

xikv jN+k \N+j

xkjv

= 0

(3.25)v V i N\{N} j N+i k N+

i \N+

j1


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yijv

iN+

k\N

+j

xikv

jNk

\N

i

xkjv

= 0

(3.26)v V i N\{N} j N+

i k N+

i \N

+

j1(3.27)xijv {0 1} v V i N j N i = j

(3.28)eijv 0 v V i N j N i = j

(3.29)wv 0 and integer v V

(3.30)yijv {0 1} v V i N\{N} j N+

i

The objective function (3.13) maximizes the difference between the revenuefrom transporting containers and the cost of operating the ships. The capacityof the ship might vary from leg to leg of the voyage, and (3.14) and (3.15)

represent the capacity constraints for the possible outbound and inbound legs.To be able to transport containers from port i to port jon ship v, the ship needsto depart from i, either directly to jor to a port between them. In addition theship needs to arrive in j either directly from i or from a port between them.The four constraints, (3.16)(3.19), express these issues. The constraints for theoutbound and inbound parts of the voyage had to be given separately. Each ofthese constraints ensures that if none of the binary flow variables, xijv or xijv,is equal to 1, the number of containers transported by ship v from port i toport j during the planning horizon is zero. When the binary flow variables areequal to 1, the corresponding constraint is redundant. The demands as upper

bounds on the number of transported containers are expressed in (3.20), andthe upper bound on the number of voyages for each ship is expressed in (3.21).The connectivity of each route is expressed by (3.22)(3.26). Constraints (3.22)ensure that each ship can have only one pair of end ports (one starting port iand one turning port j). A ship that starts in port i and turns around in port j,needs to leave i for a port not farther away than j and it needs to arrive in ifrom a port not farther away than j. This is expressed in (3.23) and (3.24). Foreach port, k, numbered between i and j, the same ship must arrive in k thesame number of times, 0 or 1, as the number of times it departs from k, bothon the outbound part and on the inbound part of the route. This is taken careof by (3.25) and (3.26). The turning around in port j is taken care of by thefact that if port k is the last port ship v visits before it reaches port j, thenconstraints (3.25) say that the ship has to travel directly from port k to port j.

And if port k is the first port ship vvisits on the inbound part of its voyage afterleaving port j, then constraints (3.26) say that the ship has to travel directlyfrom port jto port k.

Rana and Vickson (1988) presented a model for routing of one ship. Later(Rana and Vickson, 1991) they enhanced the model to a fleet of ships, and thislatter model is the same as the one presented here with a different notation,

and with constraints (3.14) and (3.15) written linearly. The solution methodused by Rana and Vickson can be summarized as follows. They started withreducing the nonlinearities in the model. If we look carefully at constraints


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(3.14)(3.26) we see that constraints (3.20) are the only type of constraints thatis summed over v. All the other constraints are written separately for eachship. The authors exploited this fact to apply Lagrangian relaxation to con-

straints (3.20). Then the problem decomposes into one problem for each ship.However, they needed to iterate or optimize over the Lagrangian multipliers.In solving the problem for each ship they solved it for different fixed values forthe number of voyages. In this way, they got mixed linear integer subproblems,

which they solved to near optimality by using Benders decomposition. Theygive results for problems with 3 ships and between 5 and 20 ports. On averagetheir solutions are about 2% from the upper bounds.

All the nonlinearities in (3.13)(3.26) consist of products of two variables orone variable and a linear expression in other variables. Apart from the terms

with wveijv , all the nonlinear terms consist of products where at least one vari-

able is binary. So by first expressing wv by binary variables, we can remove theproduct terms by defining one new variable and three new constraints for eachproduct term as described by Williams (1999) in Chapter 9.2. We might then,over a decade after the publication of that paper (Rana and Vickson, 1991),be able to solve small instances of the underlying problem by using standardcommercial software for mixed integer programming.

A rather special liner shipping problem is described by Hersh and Ladany(1989). However, the structure of the problem has some similarities to theproblem described here. A company leasing a luxury ocean liner for Christ-mas cruises from Southern Florida is confronted with the problem of deciding

upon the type of cruises to offer. The decision variables in the problem includethe routing of the ship, the duration of the cruises, the departure dates, andthe fare schedules of the cruises.

3.3.2 Hub and spoke networksContainers are usually both faster and cheaper to load and unload than

the general cargo that is stuffed in them. This means that containers can ef-ficiently be loaded and unloaded several times between their origin and theirfinal destination. One type of maritime transportation systems for containersis the so-called hub and spoke network or a trunk line and feeder system. In suchsystems we have a trunk line operating between the major ports (hubs) anda system of feeder ships working in the geographical region around each hubport visited by the trunk line. The ports feeding containers to a hub are thespokes. Thus, a container is typically loaded and unloaded three times. Firsta feeder ship transports the container from its initial loading port to a trunkline hub port. Then a trunk line ship transports the container to another trunkline hub port, and finally another feeder ship takes the container to its finalunloading port. Such networks are further described in the chapter by Crainicand Kim (2007) on intermodal transportation in this handbook.

Here we study a short-sea application of a feeder system around one trunkline hub port with a homogeneous fleet of feeder ships. We model the trans-portation of containers between one hub port and a set of feeder ports (spokes)


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in one geographical region. Each container is either loaded or unloaded in thehub.

The demands both to and from a spoke port are assumed to increase with

the number of visits in the port during the planning horizon. These demandsare upper bounds on the number of containers available for transportation, butthe shipping company is not obliged to satisfy the total demand.

The planning problem consists of choosing which of a possible huge set ofpredefined routes to use and how many voyages to sail along the chosen routes,

while maximizing the net revenue. Figure 3 illustrates the problem with onehub and several spokes. The designed routes might be overlapping.

In the mathematical description of the problem, let R be the set of pre-defined routes indexed by r and N be the set of ports, excluding the hub,indexed by i. Further, let Nr be the set of ports, excluding the hub, visited

on route r. The routes that visit port i are given by the set Ri. The ports calledafter port i on route r belong to the set N+ir and the ports called before and

including port i on route r belong to the set Nir . Let M be the set of possiblecalls at the same port during the planning horizon indexed by m.

We assume that there are fixed revenues, RLi and RUi, for carrying onecontainer to and from port i. The cost consists of three parts. We call the fixedcost of operating a ship during the planning horizon CF. The cost of sailing one

voyage along route r is CVr and the cost of unloading (loading) one containerin port i on route r is CUir (CLir). Since the fleet is homogeneous and the unitcosts are specified before we know the loading pattern along the routes, we willnormally have CUir and CLir independent ofr. The time each ship is availableduring the planning horizon is called the shipping season S. The sailing timefor one voyage along route r is TVr and the capacity measured in number ofcontainers of a ship is Q. The demand is specified in the following way: DUim

Fig. 3. Liner network design for a hub and spoke system. Example of three overlapping routes.


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(DLim) is the incremental demand for unloading (loading) in port i when thenumber of calls at that port increases from m 1 to m.

In the mathematical formulation, we use the following types of variables: the

integer variable s represents the number of ships in operation and ur, r R

,represents the number of voyages along route r during the planning horizon.The number of containers unloaded and loaded in port i on route r during theplanning horizon is given by qUir and qLir, r R, i Nr, respectively. Theinteger number of calls at port i is hi, i N, and finally, the binary variablegim, i N, m M, is equal to 1 if port i is called at least m times during theplanning horizon.

A liner network design model for a network with one hub and several spokesis as follows:

max

rR

iNr

(RUi CUir)qUir

(3.31)+

rR

iNr

(RLi CLir)qLir

CFs

rR

CVrur

subject to

(3.32)rR

TVrur Ss 0

(3.33)iNr

qUir Qur 0 r R

(3.34)

jNir

qLjr +

jN+ir

qUjr Qur 0 r R i Nr

(3.35)

rRi

ur hi = 0 i N

(3.36)

mM

gim hi = 0 i N

(3.37)gi(m1) gim 0 i N m M(3.38)

rRi

qUir

mM

DUimgim 0 i N

(3.39)

rRi

qLir

mM

DLimgim 0 i N

(3.40)qUir qLir 0 r R i Nr

(3.41)hi s ur 0 and integer r R i N

(3.42)gim {0 1} i N m M

The objective function (3.31) maximizes the net revenue over the planninghorizon. We calculate the number of needed ships in (3.32) in a way that might


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be too simple. The constraints ensure that the total available sailing time forthe total fleet of ships is larger than the sum of the voyages times. We havenot verified that the available time of the ships can be split in such a manner

that each ship can perform an integer number of voyages during the planninghorizon. Constraints (3.33) and (3.34) take care of the capacity when the shipsleave the hub and the spokes on the route. Constraints (3.35) and (3.36) usethe number of voyages along the routes to calculate the number of calls at eachport. The precedence constraints (3.37) for the gim variables are not neededif the incremental increase in the demand diminishes with increasing num-ber of calls. The numbers of containers unloaded and loaded in the ports arebounded by the demand constraints (3.38) and (3.39). Finally, the formula-tion involves binary, integer and nonnegativity requirements on the variablesin (3.40)(3.42).

Bendall and Stent (2001) presented this model using a different notationand equal costs for loading and unloading containers. Their paper does notprovide any information regarding how the model is solved. From the size oftheir practical example and the lack of information about the solution method,

we conclude that they used some standard software for integer programming.After solving the stated model, they use heuristic methods to find a schedulefor each ship. They report results for an application with Singapore as the huband 6 spokes in East-Asia. The routes are different from the impression thatthe mathematical model gives, because they had 6 single spoke routes, one foreach spoke and 2 routes with 2 spokes each. The demand data was for one

week and it was assumed that the transportation pattern would be replicatedfor many weeks.

If we cannot guarantee that the incremental demand diminishes with in-creasing number of visits, then (3.35)(3.39) can be reformulated in the fol-lowing way. Some of the symbols will be redefined to avoid defining too manynew ones. Now, let DUim (DLim) be the unloading (loading) demand in port i

when the number of calls in port i is m, and gim is equal to 1 if port i is calledexactly m times during the planning horizon.

These changes result in the following new or revised constraints:

(3.43)

mM

mgim

rRi

ur = 0 i N

(3.44)

mM

gim = 1 i N

(3.45)

rRi

qUir

mM

DUimgim 0 i N

(3.46)

rRi

qLir

mM

DLimgim 0 i N

Here (3.43) has replaced (3.35) and (3.36) and (3.44) is used instead of(3.37). After changing the meaning of the symbols, the last two constraints


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above, (3.45) and (3.46), are unchanged from the original formulation. Thisreformulation might be useful when branching on gim for one value ofi and all

values ofm as one entity. Some solvers include this possibility, and this set of

variables is then defined as a special ordered set of type one (SOS1 or S1). Fora definition of such sets, see Chapter 9.3 in Williams (1999). For such sets somesolvers will do binary branching by setting some of the variables equal to zeroin one branch and setting the other variables equal to zero in the other branch.Such branching often results in a more evenly balanced branching tree. This inturn usually results in fewer branches to investigate.

3.3.3 Shuttle servicesFerries are often used to provide a shuttle service between a pair

Ocean Trade Routes University of Norway Chapter 4

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