Routing of supply vessels topetroleum installations

Bjrnar Aas, Irina Gribkovskaia, yvind Halskau Sr andAlexander Shlopak

The Norwegian School of Logistics, Molde University College, Molde, Norway

Abstract

Purpose In the Norwegian oil and gas industry the upstream logistics includes providing theoffshore installations with needed supplies and return flow of used materials and equipment.This paper considers a real-life routing problem for supply vessels serving offshore installations atHaltenbanken off the northwest coast of Norway from its onshore supply base. The purpose of thepaper is to explore how the offshore installations limited storage capacity affects the routing ofthe supply vessels aiming towards creating efficient routes.

Design/methodology/approach A simplified version of the real-life routing problem for onesupply vessel is formulated as a mixed integer linear programming model that contains constraintsreflecting the storage requirements problem. These constraints ensure that there is enough capacity atthe platform decks and that it is possible to perform both pickup and delivery services.

Findings The model has been tested on real-life-sized instances based on data provided by theNorwegian oil company Statoil ASA. The tests show that in order to obtain optimal solutions to thepickup and delivery problem with limited free storage capacities at installations, one has to include inthe formulation the new sets of constraints, the storage feasibility and the service feasibilityrequirements. In addition, two visits to some platforms are necessary to obtain optimality.

Research limitations/implications The main limitation is the present inability to solve largecases.

Originality/value The contribution of this paper is to provide a better insight into a real-liferouting problem which has a unique feature arising from the limited deck capacity at the offshoreinstallations that complicates the performance of service. This feature has neither been discussed normodeled in the vehicle routing literature before, hence the formulation of the problem is original andreveals some interesting results.

Keywords Delivery services, Problem solving, Optimization techniques, Storage management

Paper type Research paper

IntroductionIn terms of logistics, the production of oil and gas is divided into two parts: upstreamand downstream logistics. The activities aiming towards bringing oil and gas out tothe end customers are known as downstream logistics while the activities aimingtowards supplying the offshore installations with needed supplies is named upstreamlogistics.

The issue treated in this paper is related to upstream logistics in the oil and gasindustry in Norway. More specifically, we consider forward and backward flows ofcargo between the installations at Haltenbanken and the onshore supply base inKristiansund on the northwest coast of Norway. Haltenbanken is Norways secondbiggest area for oil and gas production and is located in The Norwegian Sea, about130 km from the Norwegian coastline. The production started in 1993 and todayseveral oil companies are operating various offshore installations in this area.

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International Journal of PhysicalDistribution & Logistics ManagementVol. 37 No. 2, 2007pp. 164-179q Emerald Group Publishing Limited0960-0035DOI 10.1108/09600030710734866

Even though upstream logistics in the oil and gas industry must be regarded as animportant issue for the industry, not much research has been done within this field. To thebest of our knowledge Fagerholt and Lindstad (2000) is the only exception. They exploresome of the challenges related to the supply of offshore installations and their mainobjective is to determine an efficient routing policy for the Norwegian Sea considering theeffect of not allowing installations to be closed for service during night periods.

The amount of money involved in the oil and gas industry is enormous.Transportation between the base and the platforms is expensive as well. Renting andoperating a supply ship costs approximately e18,000 per day and it is, therefore,important to try to minimize transportation costs as much as possible by planning theroutes for the supply vessels in an efficient way. On the other hand, if the platforms donot have the necessary supplies, they may have to close down the production ordrilling operations. The costs of such close downs will be very large. Hence, there willbe a trade off between an efficient planning of the supply vessels routes and theprobability of close downs due to lack of supplies or congestions of return goods.Operational planning of supply vessels is very complex due to uncertainties regardingdemands and weather conditions. An added complexity is that the offshoreinstallations in general have very limited free space for storing containers andequipment on the platform deck. This complicates the routing planning even more andhas to the best of our knowledge not been described in the literature before.

Taken together, this makes the problem of routing of supply vessels an interestingresearch subject. In particular, todays routing and dispatching of the supply vessels aredone almost only manually without any use of routing optimization tools. Hence, largesavings may be obtained by improving todays practice by trying to optimize the routingsof the vessels. The aim of this paper is to provide a mathematical formulation of thisrouting problem as a mixed integer programming model. The model takes into accountsuch aspects as delivery and pickup demands, vessel capacity, and free storage capacity atinstallations. In addition, it opens up for non-classical solutions where an installation canbe visited twice. However, it will not include stochastic aspects. As in practice the planningis done for only one vessel at a time, the model is formulated for a single vessel. Thesesimplifications and the size of the real-life applications make it possible to solve the modelto optimality. A comparative analysis of vessel routes achieved under differentassumptions based on realistic data is performed. It corroborates the necessity of inclusionin the problem formulation, the possibility for each installation to be visited twice; once fordelivery and once for pickup. The classical approach of insisting on one visit does notguarantee optimality of routes and may lead to an infeasible route.

The rest of this paper is organized as follows. First, a detailed description of the realproblem is presented. Next, follows a review of existing literature on related problemsbefore a mathematical model for the simplified problem is presented in the followingsection. Thereafter, the model solutions for several data instances under differentassumptions are discussed. Finally, in the last section concluding remarks are givenand some issues for further research are discussed.

Case and problem descriptionThe routing problem for the supply vessels that serve the installations in theNorwegian Sea is a very complex problem which involves deterministic, stochastic anddynamic aspects. The goal is to design a sequence of installations to be visited by each

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supply vessel in such a way that the supply demands are delivered, the pickupdemands are collected, total transportation cost is minimized while satisfying theoperational constraints on the capacities of supply vessels and installations.

Problem characteristicsMany factors have to be taken into account in vessel routing planning, includinggeneral characteristics like nature of demand at installations, characteristics of vesselfleet and platforms, as well as operational characteristics reflecting restrictions onvessels capacities and on free storage space at platforms.

Demands. In order to operate efficiently, the installations need different suppliesbrought from the supply base. We divide all commodities to be delivered into two basiccategories. The first and major part is container commodities, i.e. commodities kept incontainers and baskets. These commodities can be of different kinds, sizes, weights andvolumes, but for routing planning one aggregated container commodity is normallyused, called average sized container. Commodities that have to be transported andstored as dry or liquid bulk constitute the second category bulk commodities.Each bulk commodity is kept at supply vessels and platforms in a predefined tank.

A substantial part of what is delivered to the installations must be sent back to thesupply base. The return goods mainly consist of waste in bulk or containers, rentedequipment and empty containers. It is important to remove these goods from theinstallations, especially containers, because too many containers on the platform deckmay jeopardize some platform operations. Hence, every installation can have bothdelivery and pickup demands. Over the last few years the annual demand fortransportation of supplies from the supply base in Kristiansund has been around100,000 tons. Approximately, 80 percent of this amount (measured in tons) constitutesthe return flow from installations to the base (Aas and Senkina, 2003).

It is absolutely necessary to serve an installation which has a demand for deliveriesand/or pickups. Delivery and pickup services not performed as ordered could stop or delaydrilling operations or production of oil and gas. The cost of delaying a drilling operationcan amount to approximately e5,000 per hour depending on the rig rates, and with todaysoil price, the value of an installations production can amount from 3 to 12 million euros perday depending on its production volume. Accumulation of pickups onboard installations(e.g. a congested drill deck) can make it very difficult, and dangerous, to perform drillingoperations. If this happens, the platform chief can decide that they have to wait until thepickups are collected before they can continue the operations as planned. In particular,some drilling operations require a lot of free deck space, for example, if they need to pull upthe drill pipe which could have a length of 5,000 meters.

It is not uncommon that the delivery and pickup demands can be changed on a veryshort notice due to unexpected events on platforms, more frequently on drillingplatforms than on production platforms. This stochastic nature of demands makes itdifficult to use pure deterministic approaches for routing planning.

The installations notify the supply base about required deliveries and amount ofreturn goods. The demands must be allocated to vessels so that they can be deliveredor collected as soon as possible. In order to satisfy installations requirements, atpresent time the base tries to use a fixed schedule where each installation is visitedregularly 2-3 times a week. However, this fixed schedule is not always possible tofollow because of uncertainties of demands. For example, the platform chief has the

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authority when needed to order a vessel to visit the installation immediately.The stochastic and dynamic nature of drilling and production makes it difficult to usefixed schedules. Observations show that the average length of a vessel trip is about54 hours, and the variation is large. A trip takes seldom less than 24 hours, but if theweather is bad it is not unusual that it takes five to six days to complete the trip.

Supply vessels. At the present time, there are three supply vessels serving theinstallations in the Norwegian Sea from the supply base in Kristiansund. Supplyvessels are a rather expensive means of transportation. The cost of renting andoperating one supply vessel for a year amounts to approximately e6.5 million, but it iscapable of operating 24 hours a day all year around.

Containers are transported on the main deck of the supply vessel, and the supplyvessels have normally a deck capacity from 700 to 1,100 square meters. The containers,whether they are to be delivered or have been picked up, can be placed anywhere on thevessel deck. Bulk commodities are kept in tanks in the hull of the vessel. One part ofthese tanks is assigned for bulk commodities that are ordered and another part of thetanks is used for return goods. It is possible to switch some types of liquids betweendifferent tanks, but a tank usually needs to be thoroughly cleaned before a new type ofliquid can be poured into the tank which incurs additional cost.

A single supply vessel does not have sufficient capacity to serve all the installationsat the Haltenbanken area. On average, one supply vessel is usually capable of servingbetween three and six installations on a single route due to the limited deck capacity,and the average demands in context with the speed limitation of the vessels.

The vessels capabilities to perform the required services within a planned timehorizon are dependent on weather conditions. The time a vessel uses for travelingbetween the installations will not only depend on the actual distance and the normalvessel speed but of course, also on the weather. On arrival at a platform, a vessel mayhave to wait until the wave heights satisfy safety regulations. Sometimes the expectedwaiting time at a platform can be so long that it will be impossible for the vessel toperform the service in a reasonable time. All these stochastic factors will normally havean impact on the sequence of visits.

Platforms. In July 2004, there were ten offshore installations at Haltenbanken:Draugen (DRU), Njord A (NJA), Njord B (NJB),Asgard A (ASA),Asgard B(ASB),Asgard C (ASC), Heidrun (HEI), West Alpha (WAL), Scarabeo 5 (SCA),Transocean Searcher (TRS), served from the supply base in Kristiansund (FBK). Thelocations of the installations are shown on the map in Figure 1. To give an impressionof the distances, the supply base and HEI are around 135 nautical miles apart, whichgives a normal traveling time of approximately 11 hours.

The goods a platform needs in order to operate efficiently are stored in differentplaces: containers on the platform deck, bulk commodities in predefined tanks.

One part of the storage space on the platform deck may be a free space; another partwill be occupied by containers. Some of these containers are prepared for return to thebase; others are still in use as storage facilities or mobile workshops for specialoperations. When a platform notifies the base about the delivery and pickup demands,the size of the free space on the platform deck is also reported. It is clear that it will beimpossible to receive all the amount ordered if the space needed on the platform deck islarger than the available free space together with the space occupied by the returncontainers.

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On a platform, tanks assigned for bulk commodities in use are separate from tanksused for bulk commodities which will be returned to the base. In each of the tanks inuse there may be a free space. Obviously, when a platform notifies the base about thedemand for some bulk commodity, the volume ordered cannot exceed this free space inthe corresponding tank.

For some of the platforms, the storage capacity is not only related to the space, butalso to the weight. The main reason is that many of the platforms were built andconstructed to drill holes which were not as deep as the ones they drill today. Deeperholes require more heavy equipment, and for these platforms there are limitations onthe weight that can be stored on the deck.

Performance of service. When a vessel arrives at a platform which requires deliveryand pickup services, the process of performing these services during the same visit willbe different for the two categories of commodities.

An ordered bulk commodity will be pumped directly from the tank on the vesselinto the corresponding tank at the platform. In a similar fashion, a bulk commoditythat should be removed from the platform will be pumped from its tank on the platforminto the corresponding tank on the vessel. Hence, there will always be sufficient tankcapacity for performing these services both on the vessel and on the platform.

For containers, two possibilities can occur when delivery and pickup are performedduring the same visit. When the containers ordered need a space that is less or equal tothe free space at a platform deck, there will be no problem to perform the delivery.This is similar to the situation with the ordered bulk commodity. However, a problem

Figure 1.Map of the installationsand the supply base

ASA, B and C

NJA and B

Kristians und

DRU

SCA, WAL

TRS

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with the space on the platform can occur when there is not sufficient free space on theplatform deck to receive all containers ordered, unless some of the return containersare removed. In this case, the unloading of ordered containers has to be mixed with theloading of the return containers. This process is usually performed with so-calleddouble hives. This means that a crane first moves one container from the vessel ontothe platform and on return-hive takes another container from the platform onto thevessel, or vice versa. However, this is only possible if some free space is available eitheron the vessel or on the platform. If a fully loaded vessel arrives to a platform with nofree space, the platform cannot be served.

This is different from the situation when pickup and delivery services areperformed by a vehicle visiting a customer on land. At first, it will be not necessary tomix the loading and the unloading processes because it will be always possible to usesome extra space; for example, outside the storage area for temporary storing, thuscreating free space on the vehicle or inside the storage area. Hence, even if a fullyloaded vehicle arrives at a completely filled up warehouse, it will be possible to performboth services in the same manner.

The size of the free space on a platform is a stochastic parameter because it maychange during lead time due to unpredictable situations at a platform like problemswith drilling operations or unplanned maintenance tasks. For example, drillingequipment (drill pipes and casings) may be drawn up from the well thus occupyingsome part of the free space.

Relevant optimization problems in the literatureThe problem of routing supply vessels belongs to the class of Vehicle RoutingProblems with Pickups and Deliveries (VRPPD) where deliveries to and collectionsfrom a set of customers are made with a fleet of vehicles based at the depot. In theproblem considered all deliveries originate at the depot, and all collections are destinedto the depot. This is different from the more general VRPPD involving deliveriesbetween customer locations; see review in Cordeau et al. (2006). The VRPPD is anNP-hard optimization problem with many applications. The VRPPD arises naturally inthe planning of reverse logistics operations (Dethloff, 2001; Tang and Galvao, 2002,2006). Other real-life applications encountered in the beverage industry are described inHalskau and Lkketangen (1998) and Prive et al. (2006).

Most known solution approaches for the VRPPD including models and algorithmshave been developed in contexts where the solution is constructed under the assumptionthat each customer is visited only once (Nagy and Sahli, 2005) for an elaboratedoverview. In such traditional solutions every route has the shape of a cycle. Another typeof VRPPD solution with a predetermined shape is a lasso solution, a term introduced byGribkovskaia et al. (2001), which generalizes the traditional solution. In a lasso solution,every vehicle first makes deliveries to a subsetS of customers, then visit other customerson its route to perform a combined delivery and pickup, and finally visit the customersfrom set S a second time to perform collections on the way back to the depot. Setting(S) 0 for every route yields a traditional cycle route. For some routes it may beadvantageous in terms of cost to perform two visits at some customers located close tothe depot having a large pickup demand compared to their delivery demand.

Recent research by Gribkovskaia et al. (2006) and Hoff et al. (2006) shows that it iseven more advantageous to design models and heuristics capable of constructing

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general solutions where some customers may be visited once while others may be visitedtwice, and no a priori shape is imposed on every route in the solution. Moreover, in suchsolutions customers visited twice can be visited by two different vehicles, where oneperforms a delivery while another performs a collection, and a precedence relationbetween these two services is not specified. General VRPPD solutions are less restrictedthan other solution types, and the set of feasible general solutions encompass traditionaland lasso solutions as subsets. A general solution approach allowing customers to bevisited once or twice gives more routing alternatives, but may require more time becauseone additional setup time period may be needed. In the problem under considerationthe setup periods are very small compared with travel time periods.

We define the real-life problem considered in this paper as a VRPPD with oneadditional characteristic, namely limited free storage capacity at customers.Gribkovskaia et al. (2001) use the terms delivery-feasibility, pickup-feasibility andload-feasibility when characterizing a feasible solution for the classical VRPPD problem.We find it convenient to use a new term storage-feasibility ensuring that there will beenough free storage space at any customer whenever the customer is visited on the route.As far as we are aware, VRPPD problems with limited storage capacities at customershave never been described in the literature on vehicle routing. The term storagecapacity at a customer can be found in the inventory routing literature(Christiansen et al., 2004; Campbell and Savelsbergh, 2004), but the meaning of thisterm is different from the one used in this paper. In inventory routing storage capacity atcustomer is the difference between the maximal storage capacity and the presentinventory level. In our definition, storage capacity at installation is the free storage spaceon the platform deck plus the space taken by containers that are going to be picked up.

Mathematical formulation of the problemAs described, the original problem is too complex to attack directly. In this paper, wedo not attempt to include in the optimization model all the aspects described in secondsection, but rather formulate a simplified model including constraints reflectingstorage feasibility requirements. First, some assumptions and simplifications are madein order to identify the core of the problem. Second, the simplified basic problem isformulated as a mixed integer linear programming problem. We introduce constraintsallowing up to two visits at each platform, imposing load feasibility, storage feasibility,and requirements to perform combined services. Then, the sub-tour eliminationconstraints and the objective function are presented.

Assumptions and simplificationsFor simplification purposes the decision problem is regarded as a cost minimizationdeterministic problem. This simplification is in accordance with the daily planningpractice at the base where all the problem characteristics like demands, free storagecapacities at platforms, traveling times and the set of platforms to visit by a vessel arepretended to be known by certainty by the start of the planning process.

Neither will the basic model consider the dynamic aspect of the problem, namelyplanning of routes for the three vessels in a multi-period time horizon. This will bepostponed for further research.

The routing planning at the base at present is based on demand data available atthe time a vessel stays at the port ready for loading. Hence, the planning is done for one

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vessel at a time. To reflect this practical approach and for simplification we formulatethe problem considered as a Single Vessel Routing Problem with Pickups andDeliveries (SVRPPD) extended with capacity restrictions at customers.

Based on observations about the demands (see in case and problem descriptionsection), we find it reasonable to use a certain time horizon for scheduling of vessels.The relevant time horizon for a specific vessel route starts when the base hasaggregated the demands from the installations, and has a length of approximatelythree days to render the service.

Earlier in the paper two main commodity types are described. The bulkcommodities do not share common storage facilities. The sequence platforms arevisited in will not be influenced by the bulk commodities. Hence, it is not necessary toinclude bulk commodities in the model. The problem is then formulated for a singlecommodity, namely containers. Limitations on weight at platforms are disregarded inthe basic formulation. Demands, capacity of vessel and free storage space at platformsare measured in average sized containers.

In order to apply the general solution approach mentioned in the previous section itis assumed that each customer with registered demands is allowed to be visited once ortwice on the route. In the first case, both the pickup and the delivery are performedduring the same visit. In the second case, one visit is used for the delivery and the otherone for the pickup. However, neither the delivery demand nor the pickup demand canbe split between two visits.

It is also assumed that the total delivery demand and the total pickup demand donot exceed the vessel capacity, otherwise the problem is infeasible.

We will not consider the case when the total delivery demand and the total pickupdemand is equal to the vessel capacity, and all the platforms have no free storagespace, because then it will be impossible, by practical reasons, to perform serviceswhen arriving at any platform.

FormulationThis formulation is an extension of the model introduced in Gribkovskaia et al. (2006)for the SVRPPD allowing each customer to be visited once or twice.

The supply vessel routing problem is defined on the network where the nodescorrespond to the platforms (denoted by i or j) and the supply base (denoted by 0) inwhich a vessel of capacityQ is based. Each platform ihas a non-negative pickup demandpi and a non-negative delivery demand di. Initially, all delivery demands are located atthe base and ultimately all pickup demands must arrive at the base. Each platform has anon-negative free storage capacityCi expressed in the units of free space on the platformdeck. An arc (i, j) is defined for each pair of nodes i and j, associated with a non-negativelength cij representing the travel time spent from i to j, including setup time at j.

The problem consists of designing a minimum length vessel route starting andending at the supply base, making all pickups and deliveries, such that the vessel loadnever exceeds vessel capacity along the route, and at any visit there will be enoughstorage capacity at platform, and at arrival to a platform there will be available freespace either on the vessel or on the platform to perform combined services.This definition does not specify the number of visits made to a platform; this number isone if the pickup and delivery operations are combined during the same visit or two ifthese operations are performed separately in any order.

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To express the possibility of one or two visits at platforms, we associate with eachplatform i two nodes i and i n (a copy), where n is a number of platforms to beserved. We set pin pi. Two visiting options are allowed for each platform i.The pickup and delivery operations may be performed simultaneously, in which casenode i is visited and node i n is not visited. Otherwise, platform i is visited twice:delivery is made at node i and pickup at node i n. To indicate visiting options forplatform i binary variable yi is used, taking value 1 if pickup and delivery areperformed simultaneously at platform i in the optimal solution and taking value 0otherwise. To describe in what sequence platforms (or their copies) should be visited ona route, we use binary flow variables to indicate if one node is visited immediately afteranother node in the optimal solution. In other words, flow variable xij takes value 1if the vessel travels directly from node i to node j in the optimal solution and takesvalue 0 otherwise. Flow variables are not defined for arcs between nodes and theircopies reflecting the fact that the second visit at a platform is separated from the firstone on the route.

Visiting options. The following constraints impose that the first node associatedwith each platform is visited once, either for a combined pickup and delivery or for asingle delivery. They guarantee that all deliveries are performed at the first visit:

X2n

j0xij 1 ;i 0; 1; . . . ; n 1

X2n

i0xij 1 ;j 0; 1; . . . ; n 2

The next set of constraints ensures that the second node associated with each platformis visited only if a combined pickup and delivery does not occur at the first node. Inother words, a pickup is performed during the second visit only if it was not doneduring the first one:

X2n

j0xij 12 yi2n ;i n 1; n 2; . . . ; 2n 3

X2n

i0xij 12 yj2n ;j n 1; n 2; . . . ; 2n 4

Load feasibility. To guarantee that the current load of the vessel along the route cannotexceed the vessel capacity, we introduce an additional continuous variablewi representingan upper bound on the vessel load after visiting node, and impose conditions:

0 # wi # Q ;i 1; 2; . . . ; 2n 5The following constraint defines the vessel load upon leaving the base:

w0 Xn

i1di 6

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The next group of constraints defineswj in terms ofwiwhen j is visited immediately after i:

xij 1 ) wj $ wi 2 dj pjyj ;i 0; 1; . . . ; 2n; ;j 1; 2; . . . ; n 7

xij 1 ) wj $ wi pj12 yj2n ;i 0; 1; . . . ; 2n; ;j n 1;n 2; . . . ; 2n 8The binary conditions on x variables allow the constraints above to be linearized as:

wj $ wi 2 dj pjyj 2 12 xijQ ;i 0; 1; . . . ; 2n; ;j 1; 2; . . . ; n 7a

wj $ wi pj12 yj2n2 12 xijQ ;i 0;1; . . . ;2n; ;j n 1;n 2; . . . ;2n 8aConstraints (7) state that for the first node associated with the platform j, wj $wi 2 dj pj if pickup and delivery are performed simultaneously, and wj $ wi 2 djotherwise. Constraints (8) state that for the second node associated with platform j,wj $ wi if pickup and delivery are performed simultaneously, and wj $ wi pjotherwise. The first of the last two inequalities is redundant, because in the case of thecombined service at the first node it is optimal not to visit the second node.

Storage feasibility. The following constraints ensure that there is enough freestorage capacity at platform i to perform combined delivery and pickup or onlydelivery during the first visit:

di 2 piyi # Ci ;i 1; 2; . . . ; n 9Service feasibility. It is not always practically possible to perform the combined serviceat an installation. Such a situation may occur only when a fully loaded vessel arrives atan installation with no free storage space on the platform deck. To guarantee that thiswill not happen, it is required that there must be at least one unit of free space either onthe vessel or on the platform deck:

Q2 wi Ci $ 1 ;i 1; 2; . . . ; n 10In equation (10), the expression in brackets stands for the free space on the vessel uponleaving node i. This will be the same on arrival because the delivery and pickupdemands are equal.

Sub-tour elimination. Contrary to what happens in several routing problems(Desrochers and Laporte, 1991), constraints (7) and (8) are not sufficient to eliminatesub-tours. This is because in the pickup and delivery problem vessel load does notalways vary monotonically along the route. For example, sub-tours could be createdover subsets S of the set of nodes for which:

i[S

Xdi

i[S

Xpi:

To eliminate sub-tours the standard type of constraints (Dantzig et al., 1954) are used:

i; j[S

Xxij # jSj2 1 ;S , {0; 1; . . . ; 2n}; jSj $ 2 11

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Objective. The objective of the problem is to minimize the total travel time (related tothe length of the route):

minX2n

i0

X2n

j0cijxij 12

Parameter cij represents the length of arc (i, j) in the extended network.

Comparative analysis of optimal solutionsThe model described in the previous section was used to solve real-life sized instanceswith CPLEX. The data used is based on the reports from June 2004 provided by theNorwegian oil company Statoil. In this section, we illustrate on two examples the effectof introducing storage feasibility and service feasibility requirements on solutions.We also compare solutions under different assumptions on a number of visits.

Specification of parametersThe cost matrix in Table I provides the travel times in minutes between teninstallations and the base by the use of the supply vessels economical speed.For most of the supply vessels the economical speed is around 12 knots, where 1 knotequals to 1.852 kilometers per hour.

The demands for deliveries and pickups, the supply vessel capacity and free storagecapacities at installations are measured in number of averaged sized containers.One average sized container corresponds to a container that occupies eight squaremeters. The deck space of the supply vessel is 792 square meters which gives acapacity of 99 averaged sized containers.

The demands and the free storage capacities used in the following examples are notreal historical data, but still realistic demands. In our examples, not all the installationshave non-zero demands. This is in accordance with reality. Usually an installation doesnot need to be visited in every planning period.

Example 1In the first example, considered there are four installations that have registered theirdemands at the beginning of the planning horizon. The demands and the free storagecapacity on the platform deck for each installation are given in Table II.

From\To FBK NJA NJB DRU SCA WAL ASA ASB ASC TRS HEI

FBK 0 360 360 385 590 590 605 620 620 590 670NJA 0 0 80 235 240 245 255 260 235 310NJB 0 80 235 240 245 255 260 235 310DRU 0 255 250 260 265 270 230 310SCA 0 5 45 60 60 65 155WAL 0 50 65 75 70 155ASA 0 15 15 30 110ASB 0 10 30 95ASC 0 30 95TRS 0 100HEI 0

Table I.Cost matrix

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As can be seen, the vessel is fully loaded both on departure from and at arrival at thesupply base. Further, for each of the platforms the delivery and pickup demandsare identical. There is ample free space on all platforms except one to unload the wholequantity ordered.

If we disregard the storage capacity at the installations and solve the problem as apure VRPPD, i.e. use the model without constraints (9) and (10), we get the shortestroute FBK NJA ASC ASB WAL FBK with the travel time of 1,285 minutes.The shape of this route is a cycle. However, this route will be infeasible for the originalproblem, because on arrival at installation ASC the vessel will be fully loaded andthere will be no free space on the platform deck making it impossible to perform theservices.

In order to obtain feasible solutions to the original problem, constraints (9)and (10) were included. The optimal solution to the extended model will be the routeFBK NJA ASB ASC ASB WAL FBK with the travel time of 1,290. On thisroute the platform ASB is visited twice. At the first visit only the delivery is performedthus creating free space for 39 containers on the vessel. At the installation ASC visitednext on the route, both services can now be performed, leaving the same free space onthe vessel on departure. On the second visit to the installation ASB this space on thevessel is used to collect the pickup demand. Note that this route has a different shapefrom the previous one, and is only 5 minutes longer.

However, if one insists on exactly one visit at each installation, no feasible solutionto the original problem exists. This was proved by setting extra constraint

Pni1yi n

into the model. In the output report it was stated that the constraint (10) is not satisfied.

Example 2In this example, installations NJA, ASA, HEI and WAL have their demands identified.The data given in Table III shows that delivery and pickup quantities are not equal forall the installations as in Example 1. On the other hand, there is free storage space at allthe platforms. As in Example 1, the vessel has a full load on departure and when itreturns to the base.

If we solve the problem with our model without constraints (9) and (10), as we didfirst in Example 1, the shortest route will be FBK NJA HEI ASA WAL NJA FBK with the travel time of 1,430 minutes. This is a so-called lasso route,

Installations NJA ASB ASC WAL Total

Deliveries 10 39 40 10 99Pickups 10 39 40 10 99Free storage capacity 25 59 0 80

Table II.Data for Example 1

Installations NJA ASA HEI WAL Total

Deliveries 18 24 18 39 99Pickups 18 20 20 41 99Free storage capacity 15 50 80 80

Table III.Data for Example 2

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where installation NJA is visited twice. At the first visit the whole delivery should beperformed, while at the second visit the whole pickup should be collected. However,during the first visit it is impossible to deliver the whole quantity ordered because ofthe lack of free storage space on the platform deck.

Trying to achieve a feasible solution by requiring that both services should beperformed during the same visit at installation NJA, as well as for the other installations,we solve the model without constraints (9) and (10), but adding the constraint

Pni1yi n.

The route we get is FBK NJA ASA HEI WAL FBK with the travel time of 1,460minutes. It turns out that this route is feasible for the original problem. However, we do notknow whether this solution is optimal for the original problem or not. By applying ourmodel we prove that the previous solution is the optimal one.

These examples show that in order to obtain optimal solutions to the SVRPPD withlimited free storage capacities at customers, one has to include the possibility of twovisits at customers, the storage feasibility and the service feasibility requirements intothe formulation of the problem.

Conclusion and further researchIn this paper, we have described a real-life routing problem for supply ships servingoffshore installations from the supply base and provided a mathematical formulationof this routing problem as a mixed integer programming model. Because of thecomplexity of the problem, some assumptions and simplifications have been made toformulate the basic model including several core aspects of the problem such asdelivery and pickup demands as well as the limited free storage capacity at platforms.

The interest of the formulation is that it includes restrictions on storage capacity atcustomers. This has never been done in the vehicle routing literature before. Further, tobe able to identify all feasible solutions to the real problem, the model allows two visitsat some customers. With this assumption the model is capable of producing optimalroutes of different shapes which encompass known shapes as cycles and lassos.The classical approach of insisting on simultaneous service at every visit will neitherguarantee optimality nor feasibility of solution.

The small size of the real-life instances made it possible to feed the model directlyinto a standard optimization software for mixed integer programming. We usedCPLEX 9.0 to solve real-life sized instances to optimality. The examples show that thecost and the shape of the optimal route for a vessel can be very sensitive to the inputdata. Since, the use of supply vessels is very expensive, even a small increase in theutilization of these vessels can yield large savings. The main advantage of findingoptimal solutions for the mathematical problem described above is gaining insightabout possible shapes for the routes. Such insights can be important to have, when onewants to address and solve extended and more realistic cases. The gap between twodifferent solutions in the real problem can be substantial, measured in monetary units.

The real problem introduced in this paper is very complex and several extensionsmust be included in the basic model before it completely represents the real-lifeproblem. Some of these aspects are discussed below and they will be relevant subjectsfor further research.

In the present model only the container commodity is treated. Bulk commodities willnot represent a difficulty as long as routing planning is done for only one vessel ata time. However, if the routing is done for several vessels in a dynamic way, and the

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bulk demands from the platforms exceed the tank capacities for a single vessel, bulkcommodities must be included in the model.

Six of the total ten installations in the Norwegian Sea can be served 24 hours a day,while the remaining four are closed between 07:00 p.m. and 07:00 a.m. (closed at night).The reason for this practice is that it is usually not necessary for installations that onlyproduce, to receive supplies during night hours. An installation closed during the nightneeds fewer crane operators which represents a cost saving. So for these fourinstallations, time window constraints have to be added.

In routing planning only customers with some (positive) demand are considered.In the problem discussed in this paper, the vessel capacities and the limited free storagespace at the platform decks cause major problems. It is important to have some freespace in order to make efficient routes. It is possible to create free space on a vesselwhen visiting a platform that does not have present demands, but does have a freestorage space, by temporarily storing some units of demand destined for anotherplatform. Such possibility for intermediate storing is an issue for further research.

In this paper, only splits of services are considered, i.e. a supply vessel is allowed tovisit an installation twice, doing first a delivery and thereafter a pickup. Split deliveriesinvolving the use of several vehicles have been treated in the vehicle routing literature(Brenninger-Gothe, 1989). If the routing planning in the present problem is done formore than one vessel, splitting of both delivery and pickup demands between differentvessels can be considered. Such an approach may reduce the overall costs and createmore free space on the vessels.

As mentioned in the beginning of the paper, a single supply vessel is not largeenough to service all the installations on one trip. Hence, in practice more than onesupply vessel is needed. Vessels arrive at and departure from the supply base atdifferent times. Routing decisions concerning one vessel will most likely influence therouting of the other vessels as well. The vessel routing problem in this dynamic settingcan be considered as a multitrip vehicle routing problem (Brandao and Mercer, 1998;Taillard et al., 1996). Additionally, it is usually enough to visit an installation two orthree times a week. In this context, the problem can be looked at as a periodical vehiclerouting problem (Cordeau et al., 1997).

To be able to solve extended models that include some or all aspects mentionedabove, in the future we are going to develop efficient heuristic algorithms. For themodel presented in this paper we will develop a tabu-search heuristic able to generatesolutions for larger sized instances.

The model presented in this paper is deterministic. However, there are manystochastic elements surrounding the routing of supply vessels. These are brieflydiscussed in the sections Demands and Supply vessels. Incorporating stochasticparameters will necessitate a very different approach than used in this paper. However,a stochastic approach could make a more robust planning of the routes and over timedecrease the costs.

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About the authorsBjrnar Aas holds a Master degree in Logistics (2003) from Molde University College, Molde,Norway and is currently a PhD candidate in Logistics at the same institution. His PhD researchstudies the issues of supplying offshore oil and gas installations with needed supplies. This workis done in collaboration with the Norwegian oil company: Statoil ASA. His current researchinterests include routing of supply vessels and logistics planning under uncertainty. Bjrnar Aasis the corresponding author and can be contacted at: bjornar.aas@himolde.no

Irina Gribkovskaia is an Associate Professor of Quantitative Logistics at Molde UniversityCollege, Molde, Norway. She received her Candidate of Science (PhD equivalent) degree inControl Theory in 1997 at the Byelorussian State University, Minsk, Belarus. Her currentresearch interests include distribution and routing, and scheduling with batching. Her recentworks have appeared in Operations Research Letters, European Journal of Operational Research,International Journal of Physical Distribution & Logistics Management. E-mail: irina.gribkovskaia@himolde.no

yvind Halskau sr. is an Associate Professor at Molde University College, Molde, Norway.He received his Droecon at the Norwegian School of Business and Administration. His mainresearch interests are vehicle routing planning and inventory theory. He has published inInternational Journal of Physical Distribution & Logistics Management and European Journal ofOperational Research. E-mail: oyvind.halskau@himolde.no

Alexander Shlopak is a former Master student in Logistics at Molde University College.This paper represents the core of his MSc thesis. E-mail: alexander.shlopak@himolde.no

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