GENETIC ALGORITHM FOR SOLVING A MULTI-OBJECTIVE

HELICOPTER ROUTING PROBLEM

Fubin Qian

Molde University College, Postboks 2110, N-6402 Molde, Norway

Keywords:

Helicopter routing problem, Pickup and delivery routing problem, Multi-objective optimization, Genetic

algorithm.

Abstract:

The petroleum industry uses helicopters to transport employees to and from the offshore installations. The

helicopter transportation represents a major risk for the employees. The helicopter routing problem is an

application of vehicle routing problem with combined pickup and delivery demands, which usually minimizes

the total cost of the routes and the ﬂeet size (the number of routes) in a classical form. It is also of interest

to minimize the transportation risk. In this paper, a multi-objective genetic algorithm is presented for the

helicopter routing problem. The algorithm uses a variation of the cluster-ﬁrst route-second method for routing

helicopters. We apply the proposed algorithm to instances derived from real data and evaluate its effectiveness

by comparing with ε-constraint approach with a state-of-the-art single-objective tabu search metaheuristic.

1 INTRODUCTION

In the offshore petroleum industry, helicopter has

been the main way of transporting personnel to and

from offshore installations for decades. It is very

costly to purchase and operate a helicopter. Take heli-

copter model Sikorsky S–92 as an example, which is

widely used in the petroleum industry, the unit pur-

chase cost is from US$ 13 million to US$ 14 mil-

lion and the direct operating cost is US$ 2,381 per

hour, comprising 1,194 ﬁxed and 1,175 variable costs

(2002)

. Helicopter transportation is perceived by

many offshore employees to be a risky part of their

work. Vinnem et al. (2006) claim that the hazards

associated with helicopter transportation of personnel

are among the main risks experienced by offshore em-

ployees.

Helicopter routing can be viewed as a vehicle

routing problem with combined pickups and deliv-

eries (VRPPD), in which each installation receives a

delivery originating at a common heliport and sends

a pickup quantity to the heliport. In literature, sev-

eral papers on helicopter routing for passenger trans-

portation have been published, and all of them focus

on minimizing transportation cost in terms of travel

distance or time (Galv˜ao and Guimaraes, 1990, Fi-

http://aviastar.org/helicopters eng/sik s-92.php. Last

accessed 30 November 2011.

ala Timlin and Pulleyblank, 1992, Sierksma and Ti-

jssen, 1998, Rosero and Torres, 2006, and Tang and

Galv˜ao, 2006). Moreno et al. (2006) and Menezes et

al. (2010) seek to minimize the ﬂight costs, the num-

ber of ﬂights, and the total number of offshore land-

ings in order to improve ﬂight safety. The objective

function in the optimization model uses weights to

balance these multiple goals.

The purpose of this paper is to conduct a study

of helicopter routing problem as a multi-objective

VRPPD, in which the risk, the cost and the num-

ber of ﬂights objectives are considered. We solve

the multi-objectivehelicopter routing problem using a

genetic algorithm (GA) by adapting the general pur-

pose multi-objective evolutionary algorithm NSGA-

II (Deb et al., 2002) to this particular application.

An instance from literature was studied and the re-

sults are evaluated by comparing with ε-constraint

approach with a state-of-the-art single-objective tabu

search metaheuristic.

The remainder of this paper is organized as fol-

lows. In Section 2, a multi-objective model of the he-

licopter routing problem is presented. The GA com-

ponents are provided in Section 3. A case study based

on a real-life setting are presented in Section 4, fol-

lowed by conclusions in Section 5.

458

Qian F..

GENETIC ALGORITHM FOR SOLVING A MULTI-OBJECTIVE HELICOPTER ROUTING PROBLEM.

DOI: 10.5220/0003830904580461

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 458-461

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 THE MULTI-OBJECTIVE

HELICOPTER ROUTING

PROBLEM

The risk for passenger transportation is deﬁned as the

expected number of fatalities for passengers. Risk

is decomposed into take-off and landing risk (TL

risk

)

and cruise risk (C

risk

Risk = TL

risk

= PTL· f

· p

+ PFH · f

· p

, (1)

where PTL is the total number of person take-offs and

landings, f

is the probability of an accident during

a combined take-off and landing operation, deﬁned

as the mean number of take-off and landing accidents

per million pairs of take-offs and landings, p

is the

probability of a fatal outcome for an individual per-

son involved in a take-off and landing accident; PFH

is the total person ﬂight hours, f

is the probability

of an accident during one cruise hour, deﬁned from

statistics as the mean number of cruise accidents per

million ﬂight hours, p

is the probability of a fatal

outcome for an individual involved in a cruise acci-

dent.

In this paper, we deal with the helicopter routing

problem as a multi-objective vehicle routing problem

under a general routing policy, in which the risk, the

cost and the number of ﬂights objectives are consid-

ered. In a general solution, each installation is al-

lowed to be visited twice if necessary, once for de-

livery and once for pickup, and these visits to an in-

stallation may take place in two different ﬂights. The

cost objective is measured in terms of travel time of

the routes. The risk objective contains the passenger

risk.

3 A GENETIC ALGORITHM

3.1 GA Components for Helicopter

Routing Problem

Non-dominated sorting genetic algorithm II (NSGA-

II) was proposed by Deb et al. (2002) for multi-

objective optimization. In the past years this al-

gorithm has become very popular in solving multi-

objective vehicle routing problems. It is adapted to

solve vehicle routing problem with route balancing

(Jozefowiez, Semet and Talbi, 2007a), covering tour

problem (Jozefowiez, Semet and Talbi, 2007b) and

traveling salesman problem with proﬁts (Jozefowiez,

Glover and Laguna, 2008). Maruta and Itai (2005)

employ this algorithm to a vehicle routing problem

where both the number of vehicles and the maximum

routing time among them are minimized. It is also ap-

plied to a capacitated arc routing problem which seeks

to minimize the total cost of the routes and minimize

the cost of the longest trip (Lacomme, Prins and Se-

vaux, 2006).

3.1.1 Chromosome and Route Construction

Heuristics

One way of encoding a VRP solution is using giant

tour chromosome, in which a chromosome is a se-

quence of n client nodes, without trip delimiters (Prins

2004). An optimal splitting procedure Split is also

proposed to retrieve the best VRP solution respecting

the sequence. It is regarded as a route-ﬁrst, cluster-

second heuristic for the VRP.

In our implementation, a solution chromosome S

is a permutation of n installations, which is interpreted

as the order in which the installations are routed by

some constructive heuristics. The installations are in-

serted into a route sequentially along the S until no

more installation can be inserted without violating ca-

pacity constraint. The rest installations are inserted

into a new route in the same manner, and so on. This

process ﬁnishes when the last installation in the chro-

mosome is inserted.

To construct each route, we adapted a Generalized

Insertion Procedure (GENI) initially developed for

Traveling Salesman Problem in Gendreau, Hertz and

Laporte (1992). We implement two versions of GENI

heuristics by modifying the original GENI heuristic.

In the ﬁrst version GENIc, we implement the least

cost insertion which does not incur any capacity vi-

olations to obtain cost efﬁcient solutions. In the sec-

ond version GENIr, we evaluate the increment in risk

instead of cost for each possible insertion, and imple-

ment the least risk insertion to yield solutions with

small risk values.

The installations appeared in the routes may not

follow the installation sequence in the chromosome

due to the insertion procedure. In this regard, our

scheme is different from the chromosome represen-

tation and Split procedure in Prins (2004), in which

the installations appeared in the route respect the

sequence in the chromosome. Our scheme can be

viewed as a cluster-ﬁrst, route-second heuristic, since

only the consecutive installations in chromosome be

inserted in the same route.

3.1.2 Initial Chromosomes

We apply a modern version of Fisher-Yates shufﬂe al-

gorithm (Durstenfeld, 1964) for generating N random

GENETIC ALGORITHM FOR SOLVING A MULTI-OBJECTIVE HELICOPTER ROUTING PROBLEM

459

chromosomes. The Fisher-Yates shufﬂe is an algo-

rithm for generating a random permutation of a ﬁnite

set. This algorithm is unbiased, so every permutation

is equally likely.

3.1.3 Crossover

A classical Order Crossover (OX) is used to construct

child chromosomes (Prins, 2004).

3.1.4 Mutation

Three mutation operators are implemented: Swap, In-

version and Relocation.

3.1.5 Stopping Criterion

The genetic algorithm is terminated after running a

predeﬁned number of generations MaxGen.

4 COMPUTATIONAL STUDY

The genetic algorithm is tuned experimentally. The

computational results are generated based on the fol-

lowing setting of parameters: Size of the popula-

tion: N = 256; Probability of Swap mutation 0.5;

Probability of Inversion mutation 0.1; Probability

of Relocation mutation 0.1; Probability of the least

risk insertion 0.85; Maximum number of generations

MaxGen = 10, 000.

4.1 ε-constraint Approach

As far as we know, there is no other approach in the

literature that addresses the helicopter routing prob-

lem as a multi-objective problem. To assess the

efﬁciency of the proposed genetic algorithm, an ε-

constraint approach is used to transform the multi-

objective problem to a single-objective one. In our

case, the risk objective is optimized and the other

objectives are considered as constraints. The single-

objective problem is then solved with an adapted ver-

sion of tabu search heuristic, with which high quality

solutions are reported in literature.

4.2 Case Study

The algorithms were coded in C using Pelles for

Windows, version 5.00.8. All experiments were

performed on a personal computer with an inter(R)

Core(TM)2 Duo CPU E8400 @ 3.00GHz, 2.99GHz,

and 3.25 GB of RAM, with the operating system Mi-

crosoft Windows XP Professional Version 2002, Ser-

vice Pack 3.

An test instance from literature is presented here

to demonstrate the solutions from the GA approach.

The GA approach identiﬁed 143 solutions for this in-

stance. The risk value and the travel time varies in

[718.24, 837.73] and [34.04, 53.09], respectively. The

number of routes varies in [14, 24]. These solutions

are illustrated in Figure 1, in which the y axis rep-

resents the risk value and the x axis represents the

travel time. The GA solutions are well dispersed

(symbol ×). There are 26 solutions found by the ε-

constraint approach (symbol +). For this instance, the

ε-constraint outperforms the GA approach approach

in terms of solution quality. For each solution found

by the ε-constraint approach, we can ﬁnd one or sev-

eral dominated GA solutions. On the other hand, the

GA approach uses less time to ﬁnd more solutions as

compared to the ε-constraint approach. The GA ap-

proach found these solutions in 2688 CPU seconds,

while the ε-constraint approach use 5067 CPU sec-

onds to identify the relatively small amount of solu-

tions.

Figure 1: Risk and travel time objectives of the case in-

stance.

Figure 2: Inter-relationship among the three objectives of

the case instance.

The inter-relationship among the three objectives

including the number of ﬂights of the GA solutions

are illustrated in Figure 2. We use symbol + for the

travel time and symbol × for the risk at each given

number of ﬂights (x axis) in the ﬁgure. At a given

number of ﬂights, both the risk and cost objectives

may vary to some extent. The general trend is that as

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

460

the number of ﬂights increases, the risk decreases but

the travel time increases.

In the GA solutions with low risk, each ﬂight ﬁrst

performs deliveries, and then it starts with pickups af-

ter ﬁnishing all the deliveries. All installations are

visited twice, expect those at which the helicopters

perform their last delivery and their ﬁrst pickup.

In the GA solutions with low travel time, Hamil-

tonian routes are often identiﬁed. The installations in

Hamiltonian routes are visited only once to simulta-

neously perform pickup and delivery.

5 CONCLUSIONS

We have addressed a helicopter routing problem aris-

ing in the transportation of offshore employees as

a multi-objective problem, in which the risk, the

cost and the number of ﬂights objectives are consid-

ered. A genetic algorithm is applied to the problem

and it is evaluated by comparing with ε-constraint

approach with a single-objective tabu search meta-

heuristic. Preliminary case study shows that the ge-

netic algorithm can generate high quality solution, in

terms of the spread and convergence of solutions.

ACKNOWLEDGEMENTS

Thanks are due to the referees for their valuable com-

ments.

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