Helicopter Routing in The

Helicopter routing in theNorwegian oil industryIncluding safety concerns for

passenger transport

Fubin Qian, Irina Gribkovskaia and yvind Halskau SrMolde University College, Specialized University in Logistics, Molde, Norway

Abstract

Purpose In the Norwegian offshore oil industry, helicopters have been used as a major mode oftransporting personnel to and from offshore installations for decades. Helicopter transportationrepresents one of the major risks for offshore employees. The purpose of this paper is to study the safetyof helicopter transportation in terms of the expected number of fatalities on an operational planning level.

Design/methodology/approach Based on an analysis of helicopter accidents, this paper proposesa mathematical model that can aid in the planning of routes for the fleet in order to minimize the expectednumber of fatalities.

Findings A theorem proven in this paper tells that hub-and-spoke configuration is the best way ofrouting helicopters in terms of minimizing expected number of fatalities. Computational results indicatethat the expected number of fatalities may be reduced at the expense of longer travel time byimplementing the proposed method into planning of routes for helicopter fleet.

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

Practical implications The suggested tool is able to provide decision makers with a set of solutionsfrom which they can choose the one with the best trade-off between travel time and transportation safety.

Originality/value The mathematical model and theoretical results for route planning with asafety-based objective are original.

Keywords Helicopters, Oil industry, Passenger transport, Air safety, Norway

Paper type Research paper

1. IntroductionIn the Norwegian offshore petroleum industry, offshore workers usually start their shiftby taking helicopters to the offshore installations and end it with another helicopterjourney 14 days later when their shift is completed. Staff members sometimes travel byhelicopters to execute some particular tasks on an irregular basis. Helicopters are alsoused for emergent equipment supply and other supplies in case of an unpredicted need.Although it is much more expensive to use helicopters than supply vessels for suchsupplies, the continuity of production at installations must be ensured since productioninterruption incurs huge costs for oil companies. Thus, helicopter transportation isprobably the only feasible means of delivering such supplies. Helicopters have steadilyreplaced standby vessels in oil fields to provide search and rescue (SAR) services at alltimes, for instance, there are two standby helicopters at the Ekofisk field in the North Sea.

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0960-0035.htm

The authors are grateful for the support from the project Helicopter Safety Study 3 (HSS-3),especially to Per Hokstad and Ivonne Herrera from SINTEF in Trondheim, Norway, and to thetwo referees for their valuable comments.

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International Journal of PhysicalDistribution & Logistics Management

Vol. 41 No. 4, 2011pp. 401-415

q Emerald Group Publishing Limited0960-0035

DOI 10.1108/09600031111131959

Casualties and patients may have to be transferred to onshore facilities quickly byhelicopters. In addition, helicopters make it possible to carry out a rapid evacuation ofoffshore personnel when there is bad weather predicted in the near future.

In the Norwegian Sea and the North Sea areas, helicopters have been used as amajor mode of transporting personnel to and from offshore installations in offshore oilindustry for decades. The helicopter provides speedy and flexible transport operations.Helicopter transport may be healthier and less hazardous in terms of reduced travelsickness and easier personnel transfer onto an installation compared with travel by sea(Morrison, 2001). Even though, travel by helicopter is still conceived by many offshoreworkers as an uncomfortable and risky part of their offshore work. Offshore personnelhave to experience heaviness and weightlessness during takeoff and landing, a lot ofnoise, strong vibration, even sometimes incident/accident. The hazards associated withhelicopter transportation of personnel have been considered as one of the major hazardrisk components for employees on offshore installations (Vinnem et al., 2006).

Helicopter accidents can occur close to an offshore installation when a helicopterstrikes an obstacle on it. For instance, a helicopters main rotor struck a flare tower whilelifting an underslung load at an installation on the Ekofisk field in the southern part ofthe North Sea in 1991. This was due to pilot error. That accident led to three fatalities.Accidents can also be induced by a mechanical failure. In 1976, a forced landing after tailrotor control failure ended up with the helicopter crashing onto the barge below thehelideck Forties field in the UK, which killed one passenger onboard. Also, there was aserious helicopter accident in 1986 which caused 45 fatalities due to component failurewhen the helicopter was flying over sea 1.5 miles away off Sumburgh in the UK. Piloterror is believed to be a main reason behind this accident. When a pilot attempted to takeoff while the helicopter was still tied down on the helideck, the resulting accident causedone injury and one fatality in the Gulf of Mexico region in 2002.

In this paper, the safety concerns are taken into account when helicopters are routedto perform transport service for personnel for offshore installations. In Section 2, afterdiscussing possible responses to risks of helicopter transportation in the offshore oilindustry, the accidents are categorized according to flight operation phases, which arefollowed by formulating an objective for minimizing the expected number of fatalities onan operational planning level. The mathematical formulation is provided in Section 3,which is followed in Section 4 by a description of and computational results for two setsof test instances arising in servicing offshore installations with helicopter fleets in theNorwegian Sea and in the North Sea. Finally, conclusions are given in Section 5.

2. Optimizing safety of passenger transportation with helicoptersIn this section, we first put forward the definition of risk and the responses to risk both ingeneral and in the helicopter transportation context. Then a categorization of helicopteraccidents is presented. Subsequently, we consider some perspectives on reducing theexpected number of fatalities.

2.1 Responses to riskRisk can be defined as the combination of the probability of an event and itsconsequences (ISO/IEC[1] 73). In the safety field, the consequences of an event aregenerally recognized as negative; therefore, management of the relevant risk has to befocused on prevention and mitigation of a safety threat.

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The responses to risk can vary in the range from the very easy to the extremelydifficult; Waters (2007) identifies several different types of them in supply chainmanagement:

. ignore or accept the risk;

. reduce the probability of the risk;

. reduce or limit the consequences;

. transfer, share or deflect the risk;

. make contingency plans;

. adapt to it;

. oppose a change; and

. move to another environment.

These listed responses range from the easy to difficult extreme. The easiest way ofresponding to a risk is to simply ignore the risk. While when the risky environment isthreatening the organizations existence, managers have to take more tough decisions,e.g. move to another environment in which there is no such severe risk.

It is impossible to simply ignore or accept the risk connected to transportation ofpassengers by helicopters. A number of efforts have been made to reduce the probabilityof accidents. They include introducing more technically reliable helicopters, optimizingdesign of the offshore helipad, restricting flights in adverse weather, etc. Meanwhile,several programs have been carried out to reduce or limit the consequences. Statoil usesa simulator to give offshore workers training courses on how to escape when a helicoptercrashes into sea. Standby helicopters have been used to perform SAR operations in orderto achieve quick response to all kinds of accidents.

Buying casualty insurance for the passengers can transfer the risk of hugemonetary loss for oil companies in case of a fatal accident. The helicopter operator andoil company do have all kinds of contingency plans prepared to deal with an event thatmight occur.

In the safety context, the remaining three responses are not suitable to be implementedin helicopter transportation. For instance, the passive response adapt to it is to adaptoperations to fit in the new circumstances when the situation is not that serious. Buthelicopter accident usually ends up with catastrophic consequences, and ordinaryoperations can hardly be adapted and must be stopped in this case. Activities associatedwith oil exploration and production in Norway must take place offshore. So it does notmake sense to talk about the response to move to another environment as well.

2.2 The expected number of fatalitiesHokstad et al. (2001) define the risk of offshore passenger transport as the product ofaccident frequency ( f ) and the average consequence (C) of an accident in helicoptertransportation serving offshore installations in the North Sea. Accordingly, risk R of thehelicopter transportation is quantified as R f C, where f accident frequency,i.e. mean number of accidents per million flight hours; and C accident consequence,i.e. mean number of fatalities in one accident. According to the terminology used by theInternational Civil Aviation Organization, a flight stage is the operation of an aircraftfrom takeoff to its next landing. By a flight, we mean a sequence of flight stages

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with the same flight number that starts from a helicopter base, serves some offshoreinstallations, and usually ends at the same base.

Using accident frequency f as an input in Hokstad et al. (2001), the expected number offatalities in a flight stage is calculated as follows. It is a product of accident frequency f,flight hours in this stage, number of passengers onboard, and probability of fataloutcome for an individual passenger if an accident happens.

Statistics show that takeoff and landing operations contribute substantially to thehelicopter accidents. In the UK offshore oil industry, five out of eight fatal accidentsreported between 1976 and 2006[2] happened in the takeoff and landing phases. As aresult of these eight accidents, there were 95 fatalities among offshore workers and flightcrew traveling by offshore helicopters. European offshore helicopter data indicate thatfrom 1968 to 2000 there were 13 out of total 23 fatal and major injury accidents happenedonshore at a heliport and at an offshore helideck or within a 500 m zone of a helideck.As stated in Morrison (2001), four out of 13 above-mentioned accidents happenedonshore and nine offshore. These statistics imply that takeoff and landing operationscontribute a substantial fraction of fatalities and injuries when carrying out crewexchange activities with helicopter to and from offshore installations in oil industry.A recent summary internal report from SINTEF Trondheim, Norway shows that 22 outof 28 OGP[3] offshore accidents from 2000 to 2005 were takeoff and landing accidents.

For the purpose of this study, accidents were divided into three categories,i.e. takeoff and landing accidents, cruise accidents and others. A takeoff and landingaccident refers to an accident during the entire takeoff and landing operations.Specifically, takeoff operations consist of takeoff, initial climb and climb. Landingoperations consist of descent, approach, landing and parking operations. A cruiseaccident takes place during any flight stage over land or sea after completing takeoffoperations and before starting landing operations. Others contain any accidents notincluded in the first two categories. For example, a helicopter crashed into sea in poorvisibility when winching a casualty from a ship in the UK in 1981, that accidentincurred six fatalities. In this work, only the first two types of accidents are considered.

Based on the accident categorization stated above, we suggest looking into the safety ofhelicopter transportation in terms of the expected number of fatalities on an operationalplanning level. The operational planning mainly concerns execution of day-to-dayhelicopter operations. We consider takeoff and landing during a flight stage as a pair ofactivities that consists of the takeoff and the next landing. The takeoff and landingaccident frequency f1 is a mean number of takeoff and landing accidents per million pairsof takeoffs and landings. In a period, the expected number of fatalities (NF1) due to suchaccidents is a product of the accident frequency f1, the total number of person takeoffs andlandings (PTL) and the probability (Pr)1 of fatal outcome for an individual passengerinvolved in an accident. The total number of person takeoffs and landings (PTL) is asummation of persons exposed to pairs of takeoffs and landings over all flight stages, i.e.:

NF1 PTL f 1 Pr 1: 1The cruise accident frequency f2 is a mean number of cruise accidents per million flighthours. In a period, the expected number of fatalities (NF2) from such accidents is asummation over all flight stages of a product of the cruise accident frequency f2, the flighthours in a flight stage, the number of passengers on board in this flight stage and theindividual fatal outcome probability (Pr)2. If the person flight hours (PFH) is used to denote

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the summation over all flight stages of the number of persons on board multiplied by theflight hours of each flight stage, the expected number of fatalities (NF2) can be written asfollows:

NF2 PFH f 2 Pr 2: 2The total expected number of fatalities (TENF) is the sum of the expected number offatalities from these two types of accidents. According to equations (1) and (2), the TENFcan be expressed by the formula:

TENF NF1 NF2 PTL f 1 Pr 1 PFH f 2 Pr 2: 32.3 Perspectives on improving safety of helicopter transportationIn this paper, safer passenger transportation with helicopters means lower expectednumber of fatalities in a period. A lower expected number of fatalities in a planningperiod can be achieved by reducing the number of passengers in each flight stage.As shown in equation (3), the total expected number of fatalities in a period is calculatedas the summation of two terms in which PTL and PFH are multiplied to related riskfrequencies and fatal outcome probabilities. Accident frequencies f1 and f2 are assumedto be constant for a time period, given that the underlying conditions that have impactson accident frequency are not changed. Fatal outcome probabilities (Pr)1 and (Pr)2 arealso assumed to be constants. With given values of f1, f2, (Pr)1 and (Pr)2, it may bepossible to organize helicopter transportation in different ways to make sure that alltransportation requests are fulfilled and while PTL and PFH are reduced. Hence, thetotal expected number of fatalities can be reduced consequently.

A simplified example illustrates how PTL and PFH can be affected by differentoperational planning of helicopter routes. Suppose that on the graph in Figure 1 twoinstallations (nodes 1 and 2) need to be served from the heliport denoted by node 0.Numbers marked on the edges are the flight hours between the nodes. For the sake ofsimplicity, the problem involves only pickup service from the installations to theheliport. The numbers of passengers to be collected is shown by the numbers above thenodes. Assume that we have a helicopter with a capacity that is larger than the totalnumber of passengers to be collected, i.e. all pickups can be performed in one flight. In thesolution in Figure 1(a), the installations are visited clockwise, and the numbers insquares on each flight stage show the number of passengers onboard. In this solution,PTL is 0 3 5 8 and PFH is 0*0:2 3*0:2 5*0:2 1:6. In the solution inFigure 1(b), where installations are visited counter-clockwise, PTL is 7 and PFH is 1.4.

Figure 1.Helicopter routes with

opposite flying direction(b) Counter-clockwise solution

0.2

233

(a) Clockwise solution

0.2

0.2

0.2

1

0.2

0.21 2

0

3

5 0

2

5

0 0

2

2

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The latter solution is better in terms of the total expected number of fatalities due to itslower PTL and PFH.

As seen in the example, it is of great importance to consider PTL andPFH minimization for helicopter route planning in order to improve the safetyperformance of passenger transportation. It necessitates optimization model that is ableto accommodate such factors as PTL andPFH in the minimization objective, rather thanminimize merely travel costs as in Fiala Timlin and Pulleyblank (1992), Sierksma andTijssen (1998) and Rosero and Torres (2006). We have not seen any publications in thefield of vehicle routing including passenger transportation safety. This motivates us tocarry out this research.

3. Mathematical formulationIn the helicopter routing problem considered in this paper, a set of helicopters with givencapacities have to serve a set of offshore installations from an onshore heliport. For eachinstallation, some workers should be delivered from the heliport (delivery request) andsome should be brought back to the heliport (pickup request). The objective is to find aset of helicopter routes so that the total expected number fatalities is minimized. Thisproblem belongs to the class of vehicle routing problems with pickups and deliveries(VRPPD) from the depot; see Desaulniers et al. (2001) and Berbeglia et al. (2007).In helicopter routing problem, one route corresponds to a flight consisting of a series ofconnecting flight stages that starts from an onshore heliport, visits some offshoreinstallations to pickup and/or deliver some passengers, and ends at the same heliport.

The mathematical-programming model of the helicopter routing problem withpassenger safety concern formulated in this section is an adaptation of the VRPPDmodel from Tang and Galvao (2006) and the VRP model of Kara et al. (2007). In thefirst model, total distance traveled is minimized, while in the second model the objectiveis to minimize the summation of the products of travel distance and total vehicle weightincluding load over all arcs. The cumulative function in the objective of delivery manproblem (DMP), see for instance Lucena (1990), Fischetti et al. (1993) or Mendez-Diaz et al.(2008), is somewhat similar to the PFH in our objective function.

Mathematically, the helicopter routing problem can be defined on a directed graphG (V, A), where V {0; . . . ; n} is a set of nodes, and A {i; j : i; j [ V ; i j} is aset of arcs. Node 0 corresponds to the depot (onshore heliport) and the rest of nodescorrespond to the customers (offshore installations). Each arc (i, j) is associated with atravel time tij. Each installation i has a number pi of workers to be picked up andbrought back to the onshore heliport, and a number di of workers to be delivered to theinstallation from the onshore heliport. The offshore installations are served by ahomogeneous fleet composed of m helicopters; and each of them has a capacity Q.We define the following decision variables:

xij 1 if the arci; jis on a tour; i; j [ A0 otherwise

(

lij number of passengers on helicopter if it goes from i to j; i; j [ A0 otherwise

(

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uij pickup load on helicopter if it goes from i to j; i; j [ A0 otherwise

(

vij delivery load on helicopter if it goes from i to j; i; j [ A0 otherwise

(

The mathematical model is presented below:

min f 1Pr 1i;j[A

Xlij f 2Pr 2

i;j[A

Xtijlij 4

subject to:

0;i [A

Xx0i m 5

i;0[A

Xxi0 m 6

i; j[A

Xxij 1; ;j [ V \{0} 7

i; j[A

Xxij 1; ;i [ V \{0} 8

i; j[A

Xuij 2

j;i [A

Xuji pi; ;i [ V \{0} 9

u0i 0; ;i [ V 10uij $ pixij; ;i; j [ A 11

j;i [A

Xvji 2

i; j[A

Xvij di; ;i [ V \{0} 12

vi0 0; ;i [ V 13vij $ djxij; ;i; j [ A 14

lij uij vij; ;i; j [ A 15lij # Qxij; ;i; j [ A 16

xij [ {0; 1}; uij; vij $ 0; ;i; j [ A: 17Objective function (4) minimizes the total expected number of fatalities. Constraints (5)and (6) ensure that m helicopters are used, while equations (7) and (8) tell that eachoffshore installation is visited by one and only one helicopter.

Constraints (9) balance the outflow and inflow of the pickup load for each node. Theseconstraints also eliminate subtours (Desrochers and Laporte, 1991). In constraints (10),it is indicated that each helicopter has no initial pickup load onboard when leaving

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the depot. The lower bounds for the pickup flow on any arc are generated byconstraints (11). Corresponding to constraints (9)-(11), constraints (12)-(14) are used tocontrol the delivery load.

Constraints (15) calculate the onboard load of a helicopter as a sum of the pickupload and the delivery load; the load fluctuation will depend on the sequences of thevisits. Constraints (16) ensure that the helicopter capacity will not be violated along theroute. Binary and non-negativity requirements on the variables are imposed byconstraints (17).

By replacing constraints (5) and (6) with the new constraint (18), it is possible todetermine the minimal required number of helicopters needed in the solution. Thisconstraint ensures that the number of helicopters departing from the heliport is equalto the number of helicopters returning back to the heliport:

0;i [A

Xx0i

i;0[A

Xxi0 18

Lemma 1 and Theorem 1 presents some properties of the solution structure in theEuclidean space.Lemma 1. Suppose that nodes 1 and 2 are served from the depot 0, tij is the travel

time between i and j, where i, j 0, 1, 2. Parameters pi and di denote pickup and deliverydemands for each node i 1, 2, respectively. The solution with a round trip involvingthe two nodes always has a larger PTL and a larger or equivalent PFH compared to thesolution with direct service to each node.Proof. If each node is directly served from the depot 0, person takeoff and landing

PTL1 will be p1 d1 p2 d2, and person flight hours PFH1 will be p1 d1t01 p2 d2t02.

Suppose that the round trip follows the sequence 0-1-2-0, then the correspondingPTL2 is: d1 d2 d2 p1 p1 p2 d1 2d2 2p1 p2;which is evidently greater than PTL1. The PFH2 is:

d1 d2t01 d2 p1t12 p1 p2t20:Let d be PFH1 minus PFH2, i.e.:

d p1t01 2 t02 2 t12 d2t02 2 t01 2 t12:The value of dwill always be less than or equal to 0 since triangular inequalities hold inthe Euclidean space. Thus, we have PFH 1 # PFH 2. ATheorem 1. In the case of unlimited number of helicopters, serving each installation

directly is always better than any other solution in terms of the expected number offatalities in the Euclidean space.Proof. Theorem 1 can be regarded as a general form of Lemma 1, and is applicable to

the solutions which have two or more installations served in one route.In general, there are two ways to serve installation i with pickup demand pi and

delivery demand di: either by a direct service from the heliport, or by an indirect servicevia some intermediate node(s). Direct service will contribute PFHdsi dit0i piti0 toPFH and PTLdsi di pi to PTL, while indirect service will contribute:

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PFHisi di t0u1 tu1u2 tul i pi tiv1 tv1v2 tvk0

and:

PTLisi l 1di k 1pi:Here, uj ( j 1,2, . . . ,l ) and vj ( j 1,2, . . . ,k) are the intermediate node(s) demands di andpi travel via, respectively, where l k $ 1. The last inequality indicates that at least oneof the demands of installation i is fulfilled via some intermediate node. By applyingtriangular inequality k times to t0u1 tu1u2 tul i , we have:

t0u1 tu1u2 tul i $ t0u2 tu2u3 tul i $ $ t0i:Similarly, we get tiv1 tv1v2 tvk0 $ ti0. Thus, we have PFHdsi # PFHisi . It isevident that PTLdsi is strictly less than PTL

isi due to the fact that l k $ 1. Summing

PFH and PTL over all served installations, the solution with each installation directlyserved will not have larger PFH and PTL than any solution with indirect service. A

Theorem 1 shows that hub-and-spoke configuration is the best way of routinghelicopters in terms of the minimum total expected number of fatalities. For thepassenger, direct delivery is also preferred since it means less discomfort and stress withheavy noise, heaviness and weightlessness, vibration during takeoff and landing. Themore takeoffs and landings he experiences, the higher potential risk it is for him. It isalso confirmed by historical data that takeoff and landing operations contribute themain part of helicopter accidents. Performing routes with more than one stop, the totalexpected number of fatalities (TENF) will increase. However, this will also lead to thedecreased number of flying hours. In a hub-and-spoke solution with heliport as the hub,the cost in terms of the total number of flying hours will be at its maximum. Hence, thereis a trade-off between reducingTENF and keeping cost low. The hub-and-spoke solutionmay not be always feasible if the planning time horizon is too short and/or there are notenough helicopters available.

4. Computational experimentsThe mathematical model was coded in AMPL and the helicopter routing problem wassolved optimally with CPLEX 9.0 solver. All the experiments were performed on apersonal computer with an Inter(R) Pentium(R) IV 3.0 GHz CPU and 1.0 GB of RAM, withthe operating system Microsoft Windows XP Professional Version 2002.

4.1 Test instancesWe have generated two sets of problem instances based on the geographical location ofthe installations at two offshore operation regions in the Norwegian Sea and the NorthSea. The first set of instances (A-instances) is based on the real network, which consistsof ten production installations; see Aas et al. (2007) for details. We aggregated twoinstallations NJA and NJB in one since they are closely located. Helicopter fleet is routedfrom an onshore heliport at Kristiansund (FBK). Geographical data for the second set ofinstances (B-instances) are provided by the largest Norwegian oil company Statoil. Thenetwork is composed of 15 nodes, including one onshore heliport and 14 offshoreinstallations. Distance data are transformed into flying time in hours based on theaverage speed of 250 km/h.

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The helicopter fleet is assumed to be homogeneous and each helicopter canaccommodate 19 passengers and is operated by two pilots. Delivery and pickup demandsare generated employing the same mechanism for generating random demands as inDethloff (2001). For each installation i, delivery demand di is generated as an integeruniformly distributed in interval [0, 18]. Correspondingly, pickup demand pi for eachinstallation i is calculated by formula pi 0:5 adi , where parameter a is uniformlydistributed in interval [0, 1]. Each demand is then rounded to the nearest integer. If thepickup amount pi happens to be greater than or equal to the helicopter capacity 19, it willbe set to pi 2 19. Since the demands cannot be split in our model, they are reset in such away that all the demands are below the helicopter capacity. In practice, this resetting canbe realized by performing a direct flight to these installations with extraordinarily highdemands. Regarding the number of helicopters, we start our analysis with the minimumnumber of helicopters required for each instance. The problems are then solved withone added helicopter, and we continue this increment of the number of helicopters untilall the installations can be served directly from the onshore base.

In order to compare the solutions, we conducted three runs with different objectivefunctions for every problem instance. First, the objective is to minimize the total traveltime as in:

mini; j[A

Xtijxij: 19

Second, TENF from equation (4) is used as the objective function to minimize the totalexpected number of fatalities. Third, TENF includes contribution from the crew of twopilots, so the term:

f 1Pr 1i[V 1

Xcm c

0@

1A f 2Pr 2c

i; j[A

Xtijxij 20

is added to the objective function (4), where V1 denotes a node set consisting ofthe customer nodes with positive demand, either pickup or delivery, and parameter cdenotes the onboard crew size. In equation (20), the first term is a constant for anyspecific instance, since the number of served installations is fixed and each willcontribute c units of PTL to the objective function, while the second term is proportionalto the total travel time.

Accident frequencies f1 and f2 are determined according to the historical data in thefollowing manner. The fatal accident rates are 2.3 and 2.0 per million flight hours in theNorth Sea in periods 1995-2002 and 1996-2005[2], respectively. In these two reports, fatalaccidents are not categorized as described in Section 2.2. We assume fatal accident rateto be 2.3 per million flight hours. Owing to incomplete accident information in the entireNorth Sea area, accident statistics from the UK is used since the UK is a major oilproducer in the North Sea. Among the eight relevant accidents from 1976 to 2006, therewere five takeoff and landing accidents and three cruise flight accidents. Averaged timeof a flight stage is some 0.45 h. Thus, one flight hour corresponds to 1/0.45 2.22 flightstages, which are equivalent to the number of takeoff and landing pairs. Frequency f1 oftakeoff and landing accident is calculated as 2.3*(5/8)*(1/2.2) 0.65 per million flightstages. Frequency f2 of cruise accident is then 2.3*(3/8) 0.86 per million flight hours.While these values of accident frequencies should not be considered too literally

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since the historical data are far from being sufficient to statistically support such anestimate of high accuracy. The number of accidents is too small, which provides apoor basis for making accurate predictions. Probabilities (Pr )1 and (Pr )2 are assumedto be equal to 1.

4.2 Computational resultsComputational results are shown in Figures 2 and 3 for the 50 A-instances and the50 B-instances, respectively. For comparison purposes, the average statistics over50 instances, i.e. average total expected number of fatalities (TENF), average flying hours,average CPU time, is presented when the problem is solved with different objectives.The values in parentheses give the averageTENF of onboard crew. All A-instances can besolved within 1 s, and B-instances within 5 s. The computational time decreases when thenumber of used helicopters increases in both sets of instances.

In row MIN, the results when using minimal required number of helicopters arepresented corresponding to each objective function, i.e. travel time (line TT), expectednumber of fatalities (line TENF), and expected number of fatalities including crew(line TENFc). The same set of statistics is given for solutions where one extra helicopteris used for every instance (row PLUS 1); two extra helicopters used (row PLUS 2), etc.We keep this increment of the number helicopters used until all installations get directservices (row PLUS 5 in Table I). In row PLUS 7 of Table II, one out of 50 B-instanceshas solution where two customers are served in one route and others are served directly.As averaged statistics will not change significantly when we allow one extra helicopterto be used only for one instance out of 50, we choose not to present the statistics.

Compared to TT solutions, the TENF and TENFc solutions tend to consist of flightswith smaller expected number of fatalities at a price of some extra flying hours in eachscenario from MIN to PLUS 5 for A-instances and to PLUS 7 for B-instances. Thetotal travel time of the TENFc solution is in between those from the TT and TENFsolutions, and it is fairly close to the latter since they have similar objective functions.In A-instances, a reduction in the expected number of fatalities can reach up to13.95 percent, while the travel time correspondingly increases by 53.28 percent if wecompare two extreme solutions, i.e. the TT solution in row MIN and the TENF solution

Figure 2.Number of extra

helicopters vs totalexpected number

of fatalities of TENFsolution of A-instances

200

210

220

230

240

250

0 1 2 3 4 5

TEN

F*10

e6

8

9

10

11

12

13

14

15

Number of extra helicopters

Flyi

ng h

ours

TENFFlying hours

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in row PLUS 5 in Table I. The reduction of TENF is 17.41 percent and the travel timeincreases by 53.01 percent for B-instances.

As the number of used helicopters increases, the total expected number of fatalities(TENF) decreases accordingly while the total travel time increases in all three solutions.TENF is determined by two variables, i.e. PTL and PFH according to equation (4). PTLcan be calculated as the average number of passengers onboard times the total number ofpairs of takeoffs and landings. The total number of pairs is fixed for each specific instance,and is equal to the cardinality of set V1. As a result of having more helicopters used, thetotal number of flight stages in solution will increase and the average number ofpassengers onboard will decrease. This explains the considerable decrement inPTL fromscenario MIN to the last scenario for both sets of instances. Similar interpretation applies

No. of helicopters Objective TENF 1026 Flying hours CPU secondsMIN TT 239.34 9.29 0.34

TENF 235.18 9.42 0.46TENFc 235.19 (34.36) 9.41 0.45

PLUS 1 TT 227.30 10.53 0.16TENF 220.37 10.81 0.18TENFc 220.43 (38.02) 10.80 0.14





Table I.Computational resultsfor A-instances

Figure 3.Number of extrahelicopters vs totalexpected number offatalities of TENF solutionof B-instances

250

260

270

280

290

300

0 1 2 3 4 5 6 7Number of extra helicopters

TEN

F*10

e6

9

10

11

12

13

14

15

Flyi

ng h

ours

TENFFlying hours

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to the changes in PFH, given that PFH can also be calculated as the average number ofpassengers onboard times the total travel hours. PFH is reduced due to reduced averagenumber of passengers onboard, although the total travel hours are moderately increased.

For A-instances, Figure 2 shows that the average expected number of fatalities (TENF) isreduced sharply when the first extra helicopter is allowed to be used for each instance in theTENF solution. The rate of this reduction of TENF decreases as the number of extrahelicopters increases. This can be explained as following. For a particular instance, themaximal number of helicopters used will be equal to cardinality of setV1, which means thatall installations with positive pickup or delivery demand will be served directly. Althoughone extra helicopter is allowed to be used from one scenario to the next one, it will notnecessarily be used in each instance since some instances may already have maximalrequired number of helicopters. As the number of extra helicopters increases, the number ofinstances in which this extra helicopter is actually used will decrease. This comparabletrend for B-instances is shown in Figure 3.

5. ConclusionsThe problem of routing a helicopter fleet serving offshore installations to pickup anddeliver offshore workers is studied. The mathematical integer-programming model isproposed for planning of helicopter routes minimizing the expected number of fatalities.Computational results show that the developed tool can be used to generatea set of solutions from which decision makers can choose the one with the best trade-offbetween travel time and passenger transportation safety.

No. of helicopters Objective TENF 1026 Flying hours CPU secondsMIN TT 306.96 9.47 4.32

TENF 293.51 9.90 6.89TENFc 293.55 (45.10) 9.86 6.15








Table II.Computational results

for B-instances

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Notes

1. The International Organization for Standardization/the International ElectrotechnicalCommission Guide.

2. UK Offshore Public Transport Helicopter Safety Record 1976-2002 and UK Offshore PublicTransport Helicopter Safety Record 1977-2006.

3. The International Association of Oil & Gas producers.

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Desaulniers, G., Desrosiers, J., Erdmann, A., Solomon, M.M. and Soumis, F. (2001), VRP with pickupand delivery, in Toth, P. and Vigo, D. (Eds), The Vehicle Routing Problem, Society forIndustrial and Applied Mathematics, Philadelphia, PA, pp. 225-42.

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About the authorsFubin Qian is a PhD research fellow at Molde University College (Specialized University inLogistics in Norway). He obtained his Masters degree in Applied Mathematics from SoochowUniversity, China and Masters degree in Logistics from Molde University College, Norway.His research interests include transportation safety, vehicle routing problems, metaheuristicsand transportation simulation. His articles appeared in Numerical Mathematics: A Journal ofChinese Universities (English Series) and Applied Mathematics: A Journal of Chinese Universities(English Series). He taught courses at Molde University College. Fubin Qian is the correspondingauthor and can be contacted at: [email protected]

Irina Gribkovskaia is a Full Professor in Quantitative Logistics at Molde University College,Specialized University in Logistics, Molde, Norway. She received her PhD degree at theByelorussian State University, Minsk, Belarus. Her current research interests include vehiclerouting, mathematical modelling of practical logistics problems and scheduling with batching. Herrecent works have appeared in European Journal of Operational Research, Operations ResearchLetters, Journal of the Operational Research Society, Computers & Operations Research, Networks,Omega, IIE Transactions and International Journal of Physical Distribution & LogisticsManagement.

yvind Halskau Sr was born in 1944. He holds a Masters in Mathematics and a PhD inEconomics. He has been teaching logistics/operations research for the last 22 years at MoldeUniversity College both at Bachelor, Master and PhD level. In the context of research, his maininterests lie within vehicle routing problems and inventory theory.

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