Multiobjective Optimisation of the Fleet Size in the Road Freight Transportation Company Adam Redmer Faculty of Working Machines and Transportation Poznan University of Technology, 3 Piotrowo Street, 60-956 Poznan, Poland fax: +48 61 665 27 36,
e.mail:
[email protected]
Piotr Sawicki Faculty of Working Machines and Transportation Poznan University of Technology Jacek Zak Faculty of Working Machines and Transportation Poznan 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
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 the M/M/n/0 queuing theory model, is presented below. Decision variables
ni – total number of vehicles in each homogenous group i. Criteria §
VUi - utilisation index of vehicles of group i. Max VU i = λ i
( k
gi
)
⋅ n i + 0.5 ⋅ µ i for i = 1, 2, 3, ...
(1)
where:
λi
–
mean arrival rate of incoming daily orders for vehicles of group i,
µi
–
mean service rate of transportation jobs carried out daily by vehicles of group i,
kgi
–
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). §
H i - total sales of subcontracted transportation orders assigned to group i. k ⋅n +0 .5 λ gi i i µ i Min Hi = k g i ⋅n i +0 .5 λ i k gi ⋅ n i + 0.5 !⋅ ∑ µ i k =0
⋅ wi k !
k
monetary units time period
for
i =
1, 2, 3, ...
(2)
where:
wi
–
total sales generated by i-th homogeneous group of vehicles in a certain time period [monetary units / time period].
§
BPU avg . - average utilisation index of the basic posts in the back-up facilities.
Max BPU avg .
q = Z + ∑∑ a ik ⋅ n ik i k =0
(
)
f
q Z + ∑∑ a ik ⋅ n ik i k =0
(
)
f
(3)
where: –
a
ik
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 [manhours / time period],
q
–
degree of a polynomial.
Constraints: §
VU i ≤ 1 for i = 1, 2, 3, …
(4)
§
Hi ≥ Fi (Voi − Vwi ) 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 to Voi > Vwi . §
BPUavg. ≤ 1
(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 DM’s 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.
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