7/30/2019 10.1.1.4.4671

1/4

Multiobjective Optimisation of the Fleet Size in the Road Freight

Transportation Company

Adam RedmerFaculty of Working Machines and Transportation

Poznan University of Technology, 3 Piotrowo Street, 60-956 Poznan, Polandfax: +48 61 665 27 36, e.mail: adam.redmer@put.poznan.pl

Piotr SawickiFaculty of Working Machines and Transportation

Poznan University of Technology

Jacek Zak

Faculty of Working Machines and TransportationPoznan University of Technology

Abstract

A fleet sizing problem in a road freight transportation company with heterogeneous fleet and its own

technical back-up facilities is considered in the paper. The mathematical model of the decision

problem is formulated in terms of multiobjective, non-linear, integer programming. The model is based

on queuing theory. Three optimisation criteria that focus on technical and economical aspects of the

problem are proposed. The solution procedure is composed of two general steps. In the first step a

sample of efficient solutions is generated. In the second step this set is reviewed and evaluated by the

Decision Maker. Evaluation of the solutions and selection of the most satisfactory fleet size is carried

out with an application of three MCDA methods: LBS, ELECTRE and UTA.

Introduction

The fleet sizing problem (FSP) consists in the definition of the most appropriate number of

vehicles to be maintained by an operator / carrier. In general, the problem is focused on the efficient

matching between supply of transportation capacity and demand for transportation services. The fleet

sizing problem has been a widely discussed topic in the literature. M. Turnquist and W. Jordan [14]

and P. Dejax and T. Crainic [4] present a comprehensive survey of different models of the problem.

In some publications [3][13] FSP is formulated as a static problem, in other reports it is presented

as a dynamic problem [1][6]. Some authors [8][7] consider FSP as an element of

a broader topic i.e. fleet composition problem (FCP). The vast majority of the FSP formulations has a

single criterion character. In some real life cases the FSP is combined with other fleet management

problems, such as vehicle assignment [1][2], vehicle routing and scheduling [5][11].

Problem formulation

In this paper the authors consider the FSP in a road freight transportation company managing a

heterogeneous fleet of vehicles. The fleet consists of different groups of vehicles and the optimal number for

each of them has to be determined. The transportation company has an open back-up facility to maintain its

own and external, commercial vehicles. Thus, the number of vehicles to operate in a transportation company

7/30/2019 10.1.1.4.4671

2/4

also influences on the utilisation of the back-up facility. In such circumstances contradictory objectives exist

and a multiobjective formulation of the problem sounds reasonable.

The decision problem is formulated as a multiobjective, integer, non-linear programming problem.

Its formulation, based on theM/M/n/0 queuing theory model, is presented below.

Decision variables

in total number of vehicles in each homogenous group i.

Criteria

iVU - utilisation index of vehicles of group i.

iigiii nkVUMax += 5.0 fori = 1, 2, 3, ... (1)

where:

i

mean arrival rate of incoming daily orders for vehicles of group i,

i mean service rate of transportation jobs carried out daily by vehicles of group i,

gik average availability ratio of vehicles in a homogeneous group i.

It is assumed, based on queuing theory, that mean arrival rate has Poisson distribution whereas

mean service rate is represented by Poisson or any other distribution (M/G/n/0 queuing theory model).

iH - total sales of subcontracted transportation orders assigned to group i.

...3,2,1,for

periodtime

unitsmonetary

iw

knk

HMi n i

nk

k

k

i

i

igi

nk

i

i

i

igi

igi

=

+

=

+

=

+

5.0

0

5.0

!!5.0

(2)

where:

w i total sales generated by i-th homogeneous group of vehicles in a certain time period

[monetary units / time period].

.avgBPU - average utilisation index of the basic posts in the back-up facilities.

( ) ( )

+

+= ==

fnaZfnaZBPUMaxi

q

k

k

iik

i

q

k

k

iikavg

00

. (3)

where:

ika

coefficient of a polynomial defined experimentally for a given transportation

company; the coefficient is correlated with the total annual mileage of vehicles and

their availability,

Z total external demand for maintenance jobs carried out on the basic posts of the

technical back-up facilities per time period [man-hours / time period],

f capacity of a basic post of the technical back-up facilities per time period [man-

hours / time period],

7/30/2019 10.1.1.4.4671

3/4

q degree of a polynomial.

Constraints:

1iVU for i = 1, 2, 3, (4)

( )wioiii VVFH for i = 1, 2, 3, (5)

where:

Fi average fixed costs of utilisation of one vehicle (e.g. truck plus semi-trailer) in i-th

homogeneous group of vehicles [momentary units / vehicle / time period],

Vwi average share of the variable costs in total sales generated by i-th homogeneous

group of vehicles,

Voi average share of the costs of hiring external vehicles in total sales of subcontracted

transportation orders assigned to i-th homogeneous group of vehicles.

Subject toVoi > Vwi.

1. avgBPU (6)

Solution procedure

A two-step solution procedure has been proposed to solve the problem. In the first step a set of

efficient (Pareto optimal) solutions has been generated. As in many multiobjective problems this set

is quite large and a decision maker (DM) needs additional support to finally select the most satisfactory

solution. In the second step the set of efficient solutions is reviewed and evaluated. The DM expresses

his/her preferences and searches for the most desirable solution. Three different MCDA methods: LBS

[10], ELECTRE [12], UTA [9] have been applied in the second step of the solution procedure. These

methods are based on different methodological concepts and provide different ways of the expression

of the DMs preferences as well as reaching final compromise. LBS leads the DM to the final solution in

an interactive procedure, while ELECTRE and UTA generate final rankings of solutions.

Conclusions

The comparison of different MCDA methods is carried out. The analysis of their suitability to solve

the FSP is presented. Selected optimal solutions of the problem are compared with the present situation.

References

[1] G.J. Beaujon, M.A. Turnquist, A model for fleet sizing and vehicle allocation, Transportation

Science 25 (1), 19-45 (1991).

[2] T.G. Crainic, G. Laporte, Planning models for freight transportation, European Journal of

Operational Research, Vol. 97, 409-439 (1997).

[3] G.B. Dantzig, D.R. Fulkerson, Minimising the number of tankers to meet a fixed schedule,

Naval Research Logistics Quarterly 1, 217-222 (1954).

7/30/2019 10.1.1.4.4671

4/4

[4] of empty flow and fleet management models in freight

transportation, Transportation Science 21 (4), 227-248 (1987).

[5] M. Desrochers, T.W. Verhoog, A new heuristic for the fleet size and mix vehicle routing

problem, Computers Operations Research 18 (3), 263-274 (1991).

[6] Y. Du, R. Hall, Fleet sizing and empty equipment redistribution for center-terminaltransportation networks, Management Science 43 (2), 145-157 (1997).

[7] T. Etezadi, J.E. Beasley, Vehicle fleet composition, Journal of Operational Research Society

34, 87-91 (1983).

[8] J. Gould, The size and composition of a road transport fleet, Operational Research Quarterly

20, 81-92 (1969).

[9] E. Jacquet-Lagrze, J. Siskos, Assessing a set of additive utility functions for multicriteria decision-

making the UTA method, European Journal of Operational Research 10, 151-164 (1982).

[10] - an overview of methodology

and applications, European Journal of Operational Research 113 (2), 300-314 (1999).

[11] V. Katz, E. Levner, Minimizing the number of vehicles in periodic scheduling: The non-

Euclidean case, European Journal of Operational Research 107 (2) (1998) 371-377.

[12] Roy B., The Outranking Approach and the Foundations of ELECTRE Methods, in: Bana e

Costa C.A. (ed), 155-183, Readings in Multiple Criteria Decision Aid. Springer-Verlag, Berlin

pp. (1990).

[13] H.D. Sherali, C.H. Tuncbilek, Static and dynamic time-space strategic model and algorithms

for multilevel rail-car fleet management, Management Science 43 (2), 235-250 (1997).

[14] M.A. Turnquist, W.C. Jordan, Fleet sizing under production cycles and uncertain travel

times, Transportation Science 20 (4), 227-236 (1986).