A Continuous Approach to the Design of Physical
Distribution Systems
ANTONIO G.N. NOVAES
1
, JOSÉ E. SOUZA DE CURSI
2
and ODACIR D. GRACIOLLI
1
1
Federal University of Santa Catarina, Brazil and
2
Institut National des Sciences Appliq
uées de Rouen, France
The problem of designing multi

delivery tours and defining the related districts, may become relatively
complex when associated with the servicing of a heterogeneous region, where the density of visiting points,
the amount of cargo,
and the delivery time vary over the served area. The optimization model presented in
this paper sets the district boundaries and seeks the best fleet of vehicles as to minimize total daily transport
costs. Both vehicle time and vehicle load are treated pr
obabilistically. Each district is related to a
characteristic function that takes into account distribution costs, time and capacity constraints, distribution
effort, and shape considerations (district slenderness). In a previous work (Novaes and Graciolli
, 1998) the
region under analysis was represented by a rectangular grid structure.
A
continuous approach
, however,
provides more accurate results and reduces the data preparing effort. The mathematical model developed to
solve this problem is a combinatio
n of a gradient method with random pertubations, and a hybrid genetic
algorithm
.
This method was
reported in the literature and
has been
a
pplied succesfully to
solve
topological
d
esign
problems in c
ivil and mechanical engineering.
The results of the model,
applied to a parcel delivery
problem in the city of São Paulo, Brazil, are compared with the previous findings.
Keywords
: Distribution, vehicle tours, districting
1. INTRODUCTION
The
vehicle routing
problem
(VRP) is the problem of designing a set of ro
utes from a
central depot to various demand points, each having service requirements, in order to
minimize the total distance covered. The total distance travelled is often substituted by a
cost function. When customer demands or some other element of the
problem are random
variables, we have the
stochastic vehicle routing problem
(SVRP) (Bastian and Rinnooy
Kan, 1992)
.
Common examples are stochastic demands and stochastic travel times. In
addition, sometimes the set of customers to be visited is not known
with certainty. The
primary objective of such models is to find optimal tours, i.e. the best sequence of visits in
order to minimize the total travelled distance or the total transport cost, respecting, at the
same time, service requirements
(Stewart and
Golden, 1983; Bertsimas, 1992
)
.
In
particular, when all
m
routes (vehicles) start and end at a common depot, one has the
m

travelling salesman problem with stochastic travel and/or service times
.
A recent review of the literature on the SVRP is provided b
y Gendreau, Laporte and
Séguin (1996). According to Gendreau
et al
(1996), a typical SVRP is usually modelled
either as a
chance constrained program
(CCP) or as a
stochastic program with recourse
2
(SPR). In CCPs, one seeks a solution for which the probabili
ty of failure is constrained to
be below a certain threshold. A CCP solution does not take into account the cost of
corrective actions in case of failure. In SPRs, the objective is to determine a first solution
that minimizes the expected cost of the secon
d stage solution. This cost is made up of the
cost of the first stage solution, plus the expected net cost of recourse. In fact, depending on
the problem under analysis, there are different possible recourse strategies that can be
investigated. For example
, whenever the load assigned to a specific route exceeds the lorry
capacity, one can assign an extra vehicle to carry the excess cargo.
The model described in this paper is intended to be used in the planning process stage,
in which a clear definition of t
he operational rules is still nonexistent. Thus, a CCP
approach seems more appropriate, permitting the inclusion of possible recourses to a later
stage. In fact, the present model can be used as a general framework for further extensions,
including SPR fea
tures and other related real

world applications. In our model we deal with
two kinds of failure. First, there is a
vehicle capacity constraint
, in which the probability
that the cargo load of a lorry exceeds its capacity is limited to a fixed level. Second
, there is
a
time constraint
, in which the probability that the daily cycle

time of a vehicle exceeds an
upper bound is limited to a pre

defined level.
We consider an urban region
of irregular shape, with the density of servicing points
varying over
, b
ut being nearly constant and Poisson distributed over distances
comparable with a district size (Newell and Daganzo, 1986b). The customer demands and
service times also vary over the region and are random variables. Each district is assigned
to a route whi
ch, in its turn, is assigned to a vehicle. The routes are both restricted by time
and capacity constraints in a stochastic way. We also assume that any one of the
n
points of
a random tour will require a visit with a probability
p
(Jaillet, 1988). The part
ition of the
region into districts is done considering an
equal

effort criterion
in order to guarantee a
minimum of homogeneity among districts (Novaes and Graciolli, 1998).
The partitioning of the region into districts is based on a polar coordinate syst
em
centred at the depot. A number of circumferences with varying radii are set by the model,
forming concentric
rings
(Figure 1). Next, a set of radii are defined in each ring, whose
number and angles are also established by the model, forming the
distric
ts
. The latter are
associated with the individual vehicle tours. The term
partition
is used indifferently in the
3
text to indicate rings or districts. In addition, the model searchs for the best lorry capacity,
respecting the service requirements.
Many auth
ors treat SVRP with a mathematical programming formulation. Other
authors, instead of searching for specific optimal routes linking the servicing points, use
approximate formulas to estimate the travelled distances (Daganzo, 1984b; Han and
Daganzo, 1986;
Langevin and Soumis, 1989; Novaes and Graciolli, 1998). In our model we
are not interested in finding the optimal sequence of visits in each tour. This is usually done
in the operating phase. Our model is intended to be used mostly in the planning phase, i
n
which a more general framework of the system is sought. The effort is concentrated in
seeking a near optimal partition of the region supplied by the depot into districts (Han and
Daganzo, 1986; Novaes and Graciolli, 1998).
The paper is organized as fol
lo
ws: Section 2 discusses the methodology to estimate
time and distance from the depot to
the district and within it, defining the prob
lem
constraints as well. Section 3 introduces
continu
ous approximations and
presents analogies
among
physical distribution
problems and topological design problems enco
unte
re
d in
mechanical and civil engineering.
Section 4 discusses the districting process i
n general.
Section 5 define
s the
distribution effort functions used to balance
the
vehicle
tours
. Section
6
describes the
optimization model
,
which is based on a genetic method
associated with the
gradient method with random pertubations
(Souza de Cursi and Cortes, 1995). Section 7
presents
an
application
of the model to
an
urban distribution example
.
The f
ina
l
section of
th
e
paper
compares
the
obtained
results
with
a
previous
discrete grid

cell
formulation
(N
ovaes and Graciolli
, 1998)
.
Section 8
also compares the results and computing time
of
different model assumptions.
2. VEHICLE CYCLE CHARACTERISTICS
Vehicle travel with
in the districts is approximated assuming that the underlying road
network is equivalent to a Euclidean metric, with the real distances being estimated with
the aid of a mathematical function
(Daganzo, 1984b; Han and Daganzo, 1986; Novaes and
Graciolli, 19
98).
Following Beardwood
et al
(1959) and Stein (1978), the expected
4
distance
travelled by a vehicle within a district of area
A,
and
n
visiting points,
can be approximated as
,
(1)
where
is the point density. Expression (1) can be applied to most metrics (Novaes
and Graciolli, 1998) and presupposes that the points are uniformly and independently
scattered over the area, and the distric
t is fairly compact and fairly convex (Larson and
Odoni, 1981). The coefficient
can be expanded into three multiplicative factors (Novaes
and Graciolli, 1998). The first one depends solely on the adopted metric and routing
strategy
.
In our application the points are visited with a probability
p
, the value of
being
defined as in Jaillet and Odoni (1988).
The second factor is a corrective coefficient (
route
factor
) reflecting the road network impedance. The th
ird reflects the impact of the district
slenderness on the tour length. For more details the reader is referred to Novaes and
Graciolli (1998).
The vehicle starts from the depot, goes to the assigned district, does the delivery, and
comes back to the depot
when all the visits are completed, or when the maximum allowed
working time per day is reached, whichever occurs first. This complete sequence makes up
the vehicle cycle. In some practical circumstances more than one tour per day can be
assigned to the s
ame lorry. This implies extra line

haul costs, but depending on the cargo
characteristics, vehicle size restrictions, and other factors, multiple daily tours per vehicle
might sometimes be appropriate. For the sake of simplicity, we assume that the vehicle
s
perform just one cycle per day. The model can be easily modified to take into account
multiple daily cycles.
The total cycle length
D
is the sum of the line

haul distance (either way) and the local
travel distance given by (1). The total cycle time
T
, o
n the other hand, is the sum of the
line

haul time, the local travel and the total handling time. The latter is the sum of the times
spent in delivering the cargo at the customer´s locations. The expected value of
T
for a
generic district is
(2)
5
where
is the expected line

haul travel distance (one way) from the depot to the
district,
is the average line

haul speed,
is the average local speed,
p
is
the
probability that a customer be visited, and
is the expected stop time spent in one
delivery (Novaes and Graciolli, 1998).
Assuming statistical independence of the elements which form the c
ycle time, the
variance of
T
is given by
, (3)
where
is the line

haul travel time (one way), and
is the local travel time. Using the
central limit
theorem,
can be represented by the normal distribution
. We
assume that the cycle time cannot exceed a maximum of
working hours per day,
imposed by labour restrictions and company poli
cies. Let
be the unit normal
variate. Adopting a 98 percentile (monotail distribution),
= 2.06, and thus
(4)
is a restriction that must be respected. We also assume that the vehicle crew are paid
normal wages up to the limit of
working hours per day (8 hours). Above this level the
crew is entitled to receive an ove
rtime payment.
Let
be respectively the mean and the standard deviation of the
quantity
u
of product delivered per visiting point in the generic district. Then, assuming
statistical independence of the customer’s demands, the expecte
d value and the variance of
the total vehicle load
for one tour in the district is given by
and
. (5)
According to the central limit theorem
,
can be represented by the normal distribution
. If
W
is the lorry capacity, and adopting a 98 monotail percentile, another
restriction that must be respected is
.
(6)
Thus, given a servicing district and a vehicle of capacity
W
, the maximum number
of visiting points to be assigned to that district will be given by (Novaes and Graciolli,
1998)
6
. (7)
3. CONTINUOUS APPROXIMATIONS
A number of Operations Research problems, basically finite

dimensional in nature and
with a large number of variables, can be converted into problems in
volving continuous
functions, with good practical results. The paper by Newell (1973) is a seminal work in this
field. He developed a few interesting applications to transport and logistics based on
Professor’s William Prager work on continuum mechanics an
d plasticity (minimum weight
design). The exploitation of similarities between problems of different disciplines, although
not as frequent in the literature as one would expect, has led to an interesting logistics
application described in this paper. Many
problems in Mechanical and Civil Engineering
lead to situations where the unknowns are regions or shapes, and the main goal is the
partition of a given area or the construction of a lay

out. For instance, the minimum weight
design of mechanical structures
and automobile spare parts under plastic failure restrictions,
involves the determination of the optimal distribution of matter over a region. Thus, one has
to determine the part of the region corresponding to the structure and the part void of
matter. Su
ch a problem is known as the
topological design problem
(Haber
et al
, 1994).
The 2D formulation of such problems has close
resemblances with the
problem of
distributing cargo over a defined geographical region.
Moreover, the “equal

effort criterion” used t
o solve the Operational Research problem
of partitioning a region into districts (Novaes and Graciolli, 1998), is also employed to
solve matter allocation questions associated with topological optimization problems (Souza
de Cursi
,
1994
). Equally, the same
mathematical methods used to solve topological design
problems can also be employed successfully to solve physical distribution problems. In
particular, non

convex optimization methods involving discrete/continuous or micro/macro
approaches to solve topol
ogical design problems have been reported in
the lit
erature
(Allaire and Francfort, 1993
). In this framework, global optimization methods such as
evolutionary calcula
tions
(Schoenauer and Kane, in press
) have been introduced and new
hybrid algorithms have
been tested (Pogu and Souza de Cursi, 1994; Souza de Cursi and
Cortes, 1995). In this paper, it is shown how to use those optimization methods to solve
7
districting and fleet design questions associated with logistics physical distribution
problems.
Suppos
e initially a simplified one

dimensional distribution problem in which equal

capacity vehicles serve clients along a straight line of length
starting at the depot. The
straight line will be partitioned at points
, forming the districts (Fig. 2). We
wish to determine the vector
in such a way as to minimize total vehicle
cost. Clients, with stop times
are located at points
along the
x
axis.
The cumulative stop times, disregarding their integer nature and treating them as a
continuous variable on
x
, are given by
. (8)
Each vehicle departs from the d
epot, goes to the assigned district (a line segment), does
the delivery within it, and comes back to the depot when all the visits are completed. We
further assume that the problem is deterministic and time

restricted, i.e. the lorries leave the
depot with
a less

than

truckoad (LTL) route assignment.
Let
be the tour cycle time of
vehicle
j
. Its delivery district is limited by the points
and
. If
v
is the average
vehicle speed, the t
our length is
, (9)
and the cycle time
. ( 10)
Let H be the standard working time per day. Putting
int
o (10) and rearranging
the equation, produces
. (11)
Let
be the vehicle cost per tour ($/day). If
m
is the required number of
vehicles,
the total daily cost is
. Since the cost is a linear function of
m
,
given a certain type of
vehicle
the minimum cost solution is the one with the minimum number of lorries. A
simple graphical solution to the problem can b
e extracted from relation (11). The line AC in
8
Fig. 2 represents the right hand side of equation (11). One also draws, on the same graph,
the curve
S(x)
of cumulative stop times. The upper limit of the first district is found at the
point
where curve
S(x)
intercepts the line AC. Next, on
e draw
s
curve II, representing
, and seeking its interception with line AC. This point
is the upper limit of
the second district. Curve III is drawn ne
xt, representing
, and yielding
point
at the intersection with AC. The process continues until all the demand is covered.
The e
xtension of such a one

dimensional
vehicle routing problem to the two

dimensional
plane, however, is not trivial. First, it is necessary to define a rational
procedure for selecting the appropriate shape and orientation of the districts (Newell and
Daganzo, 1986a). Daganzo (1984b) pointed out that for regions in which the number of
vis
iting points is large compared with
, where
is the average number of points served
by one vehicle, it is usually possible to define the districts following some pattern of
concentric rings around the source (
the depot). The ring

radial partitioning pattern can be
further improved with the observation of some additional properties (Han and Daganzo,
1986). First, the districts should be elongated toward the depot. The number of visiting
points assigned to each
vehicle, on the other hand, will vary with cargo demands and
stopping times, and should decline with the distance from the depot.
4. DISTRICTING
The partitioning of the region
into districts will depend on a number of factors, such
as the distance fro
m the depot, the local density of servicing points, local and line

haul
traffic conditions, and the type and capacity of the vehicles. The exact definition of the
optimal district boundaries is a complex task, envolving topological aspects, as well as
urba
nistic, traffic and operational factors.
The partitioning process in our model follows a
ring

radial network centred at the depot. A generic district
comprises
visiting
points and is limited by radii
and
, and by angles
and
(Fig. 1). The district
area is
. (12)
9
Let us arbitrarily set
for the
x
axis (Fig. 3a). Let
be the cumulative function
of visiting points over the sector
(Fig.3a). Thus, the number of visiting points within a
generic district
is given by (Fig. 3b)
, (13)
with density
(14)
In a similar vein, let
be
the cumulative function of the time
spent
delivering cargo at the visiting points
i
= 1,2,... . The total time spent delivering cargo in
district
is
(15)
On t
he other hand, let
be the cumulative quantity of cargo delivered at the servicing
points. The total quantity of cargo
delivered in district
is
(16)
Let
NR
be the number of
rings in which the region will be partitioned (
NR
is one
the variables of the problem). The unknowns of the problem are represented by the vector
, comprising the
NR

1
radii. The radius
is made equal to the
distance of the farthest visiting point from the depot. For convenience, we add
Let
be th
e cost function (see Section 6.1
) to be minimized. We seek an optimal vector
such that
. (17)
At a certain stage of the optimization process, being defined a vector
,
one searchs separately for each ring the angles
that minimize a certain
characteristic functio
n, to be defined in Section 6
.
5. BALANCING THE DI
STRIBUTION EFFORT AMONG TOURS
10
Ideally all vehicles should produce the same output, i.e. they should visit an equal
number of client
s in each tour, delivering the same quantity of cargo, and presenting the
same cycle time. This is not possible due to the heterogeneous characteristics of the region,
and we need some criterion to get an adequate working balance among the vehicles. Two
di
fferent situations are considered: (a)
time restricted tours
and (b)
capacity restricted
tours
(Novaes and Graciolli, 1998).
5.1
Time restricted tours
This situation happens when constraint (4) is binding. With regard to the vehicle
distribution effort, th
e line

haul time is wasted. Let
be the fraction of the working
available time (
) which is spent inside the district
. For that, we take out the line

haul displacement time twice from
(either
way
), and divide the result by
, leading to
. (18)
The visits in district
will con
sume a total time equal to
, where
is
the average local travelling time in district
, given by
.
(19)
Since the tours are time restricted, the number
of lorries necessary to serve district
is proportional to the ratio of the time
by the productive net time
, (19)
We define a “distribution effort function” that measures the relative difficulty to
perform the delivery work in district
, proportional to the required number of vehicles
(Novaes and Graciolli, 1998):
11
. (20)
5.2 Capacity restricted tours
This situation happens when constraint (6)
is binding. If
W
is the lorry capacity and
is the expected quantity of cargo to be delivered in district
, then the required
number of vehicles is
.
(21)
Thus, the distribution effort for the capacity constrained problem is
, (22)
5.2
Balancing criterion
Take all th
e districts
,
,
contained in a generic ring
i
. Let
be the
average effort associated with district
i.
In order to guarantee
a minimum of homogeneity
among districts we add a penalty to
the cost function to be minimized (see S
ection 6.1
),
represented by
, (23)
where
is a constant.
Two other penalties are additionally incorporate
d into the model. The first, to guarantee
that restriction (4) is respected, the other to satisfy constraint (6), namely
, (25)
12
where
and
are constants.
6. THE OPTIMIZATION MODEL
6.1 Objective function
One of the
objectives of our model is to define the best vehicle size for the
distribution problem under analysis. Due to the discrete availability
of vehicle types in the
automobile market, the vehicle capacity
W
is taken as an exo
genous variable, i.e. the model
runs separately for different values of
W
. The analysis of the results leads to the l
east

cost
vehicle capacity. Next
, adopting the pre

sele
cted value of
W
, a more detailed analysis of the
distr
icting process is performed
. Vehicle operating expense
s are grouped into mileage cost
(fuel, tires, maintenance), represented by
($/km), and hourly cost
(crew wages, lorry
deprec
iation, insurance), represented by
($/hr). In addition to these operating costs, a
marginal overtime cost
is incorporated to the objective function whenever the daily
vehicle cycle time exceeds
(
Novaes and Graciolli, 1998). Total shipment

handling and
war
ehousing costs
are assumed to be independent of vehicle size, and were not
incorporated in the objective function. Inventory costs of items in transit to
customers
were
not considered a
s well, since they are constant. We seek the (uniform) fleet of lorries,
represented by the number
m
of vehicles and capacity
W
, that minimizes the total daily
distribution cost.
The objective function to be minimized, which is implicit in
X
, is
where
NR
is the number of rings,
is the number of districts in ring
i
,
is the tour
total distance for district
,
and
is the expected d
aily overtime for the tour in
(Novaes and Graciolli, 1998).
13
6.2 The optimization method
In order to avoid convergence to local minima when minimizing (26), we use a
population based method
(genetic method)
, associated with the g
radient method with
random pertubations
.
Information on the gradient is introduced as a kind of genetic
mutation, and convergence to a global minimum is ensured by choosing suitable random
pertubations with a fixed parameter (namely, a simulated annealing
pertubation). This
mixed algorithm prevents convergence to local minima and, on the other hand, increases
the computation speed when compared with the simple gradient method (Pogu and Souza
de Cursi, 1994; Souza de Cursi and Cortes, 1995).
Let
, with
, represent the solution vector obtained at
the
stage of the optimization process. The recursive equation of original gradient
method with a fixed parameter
is
,
(27)
where
is the gradient of
F
at the point
X
.
Instead of applying relation
(27)
in a
deterministic way, we define a population
, with
NS
elements, formed by partial
solutions of the problem, namely
. The rules to select such a population
will be discussed later.
A random pertubation
, which decreases slowly enough in order to avo
id local
minima, is added to (27). The solution vector in stage
k
is then calculated among the
NR
elements of the population
(
Souza de Cursi and Cortes, 1995
)
:
.
(28)
The random p
ertubation
in (28) is taken as
,
(29)
where
,
(30)
14
with
c
and
d
are constants. We take
, independent of
k
, where
Z
is a
random sample
of
the unit normal variate, w
hich can be easily obtained by using the log

trigonometric
gen
erat
or of the Gaussian distribution.
6.3 The
genetic method
We
follow
Souza de Cursi
and Cortes
(1995)
.
We start
with a random population
formed by
np
elements
. At stage
, the
population
is
obtained from
by
:
a) generating two populations
and
(parents)
that
will produce
the
offsprings (see S
ection
6.4
);
b
)
from
and
,
g
enerate
nc
offsprings
from
. A si
mple way to do this
consists
in usin
g
nc
random combinations in the form
, where
. Elements laying out of the feasible set are
rejected;
c
) generati
n
g mutations of the elements taken from
by applying (28)
,
which
leads
to a set
;
d
) selecting the best
np
elements
from
.
6.4 The
model
As mentioned,
t
he vehicle capacity
W
is taken a
s an exo
genous variable
i
n the
model
. Thus, given a certain
value of
W
, the model runs as follows:
a)
The model
makes
an
exhaustive search on
NR
(the number of rings)
, over a given
interval
.
b) For a given
NR
,
two
parent
set
s
of
NR
radii
,
as defined in section 4
,
respectively
and
,
are
initially
generated.
The
set
is
obtained by randomly splitti
ng the
interval (0,
)
into
NR
rings.
Th
e
second parent
set
is obtained
by evenly
dividing
the interval
(0,
)
into
rings of equal
thickness
.
15
c) For a generic
ring
i
, the number
of districts
and
one of
the
p
a
rent
population
s
ar
e
determined as follo
ws:
(i)
let
be the angle as
defined in Section 4,
such that
,
where
and
are the extremes of the
interval
,
for ring
i
;
(ii)
starting at
, the model increases
until one of the
res
trictions
(4)
and (6) is binding. This
angle
is referred as
.
(iii)
s
tarting
at
and
s
earching
over
, the model check
s
restrictions
(4) and (6)
again,
until one of them is
binding,
and
yiel
ding
.
(iv)
t
he
process continues
until
all points
located
in the ring
are covered.
The number
of districts in ring
i
is given by the number of
partitions resultin
g from (ii) and (iii) above. The
set
form the
first parent population
of the
genetic method
used in conjunction with
the angles in
ring
i
.
d) For a generic ring
i,
the second
parent popu
lation set
is obtained by randomly splitti
ng the
interval
into
districts
.
e)
The genetic t
ransformations involving the angles
in each ring
i
(
i
= 1,...,
NR
)
are
imbbeded within
the
process of optimizing the
radii
7
.
AN
APPLICATION TO A PARCEL DISTRIBUTION PROBLEM
The described methodology was applied to the example described in Novaes and
Graciolli (1998), extracted from a parcel delive
ry problem in the city of São Paulo, Brazil.
The region
, with an area of 666 sq. km, comprises a total of 6385 servicing points. For
each demand point the following data are recorded: (a) the
x,y
ground coordinates (b) the
Euclidean distance from the de
pot, (c) the mean and the standard variation of the stopping
time spent for one visit (e) the mean quantity of cargo delivered per visit. In the example
p
= 1, meaning that the clients are visited daily. The depot location is shown in Fig. 7. Line
haul
average speed is 35 km/hr. Local average speed is around 24 km/hr. The normal daily
crew working time is
, with a maximum of
. A set of four lorries
16
available in the Brazilian market, with capacity 500, 750, 1,
000, and 1,500 kg, were
analysed. Overtime work has an additional cost of US$ 8.00 per hour. The vehicle cargo
load is measured in kilograms.
First, we apply the model to define the least

cost type of vehicle. To do this we
empirically adopt a number of s
ectors to partition region
. Due to the expressive
differences in cost among vehicles, the effect of changing the number of sectors at this
stage is negligible. Later, the solution will be refined, with the model searching for the best
number of sectors.
Table 1 shows the results of such an analysis. One can see that the 500

kg lorry fleet has the smallest total daily cost. Furthermore, the analysis has shown that the
fleet is time restricted for this parcel delivery problem, meaning that constraint (10) i
s
binding.
Table 1

Operating costs for different types of vehicles
Vehicle cargo
Capacity (kg)
Number
of
ring
s
Number of
vehicles
Total cost
(US$/day)
500
10
84
4,038.00
750
8
64
4,987.00
1,000
6
64
5,767.
00
1,500
6
63
6,807.00
Next, assuming a lorry of 500

kg capacity, we varied the number of partitioning sectors,
from 5 to 13. The division of the region into 10 sectors was the partitioning scheme with
the smallest cost and, thus, the one adopted.
Figure 3 shows the final partitioning scheme
produced by the model. The final results are shown in Table 2.
Table 2

Optimal Results
Num扥r潦灡rtiti潮o湧 ring
s: ††††
Num扥r潦摩
stricts (ehicles)
㠹
潴al潰orating c潳t
U摡y)
㌬㤵㜮ㄴ
17
8.
PERFORMANCE EVALUATION AND CONCLUSIONS
The model program was written in Pascal and run in a 4 194 MHZ WorkStation. A
sensitivity analysis was performed in order to check the effects o
f changing the genetic
algorithm parameters in the processing time and in the resulting vehicle operating costs.
The results are sumarized in Table 3. One can see that the processing time is very sensitive
to the variations on the genetic algorithm paramet
ers. The basic model formulation required
a running time of 10min 22sec, whereas the last formulat
ion shown in Table 3 spent
almost
47
hours to be performed. The reduction in total cost,
however, was minimal, only 0.57
%
for the latter case. These results
show that the model can be applied to similar cases with
reasonable computer running times.
Table 3

Cost reductions for diverse model assumptions
Sectors
Angles
Results
# of
iterations
Population
Size
Replica

tions
#
of
iterations
Population
Size
Replica

tions
# of
vehicles
Running
Time
(h:m:s)
Cost
Reduction
4
4
4
20
4
4
90
00:10:22
Basic run
10
4
4
20
4
4
90
00:55:48
0.21 %
10
10
10
20
4
4
90
04:07:06
0.40 %
30
10
10
20
4
4
88
12:03:13
0.51 %
30
10
10
30
8
8
8
9
46:48.36
0.57 %
Next, we compare the results shown in Table 2 with the grid

cell formulation
presented in Novaes and Graciolli (1998).
The grid

cell
approach
yielded a fleet of eighty

nine
500

kg capacity lorries, with a tot
al daily cost of US$ 4,145.
37
, meaning a 3.4% cost
reduction
and a fleet reduction of one vehicle
with the new methodology.
In addition to
these savings,
the c
ontinuous formulation
has a number of advantages over the initial
discrete formulation. First, the data preparation is less
cumbersome. It is only required
beforehand the preparation of a file containing, for each visiting point (client), the
appropriate information (ground coordinates, expected delivery time and standard variation,
18
cargo quantities, visiting frequency, etc.).
In the previous formulation, the region
has to
be first divided into small cells (a raster structure), followed by a detailed recording of each
cell contents. Additionally, the redution of the input data speeds

up the computations as
well.
The discrete
grid

cell formulation, on the other hand, may present connectivity
problems when partitioning the region
into districts. In order to cope with this problem,
one is forced to try other partition criteria and iterate the process experimentally. This tends
to break the homogeneity of the districting process. The continuous formulation, on the
other hand, authomatically eliminates unconnected partitions.
One point that requires further improvement in the continuous formulation is the
treatment of the dist
ricts located at the edge of region
(see Fig. 3). A part of such districts
are void of delivery points, requiring a somewhat different method to compute the area and
other district attributes.
REFERENCES
Allaire, G. and Francfort, G. (1993) A numeric
al algorithm for topology and shape
optimiza
tion, in
Bendsoe, M. et al (eds.)
Topology Design of Structures
, NATO
ASI
Series E
, pp. 239

248, Kluwer, Dordrecht.
Bastian, C. and Rinnooy Kan, A.K. (1992) The stochastic vehicle routing proble
m
revisited,
European Journal of Operational Research
, vol 56, pp. 407

412.
Beardwood, J., Halton, J.H. and Hammersley, J.M. (1959) The shortest path through many
points,
Proceedings Cambridge Philosophical Society
, vol 55, pp 299

327.
Be
rtsimas, D. J. (1992) A vehicle routing problem with stochastic demand,
Operations
Research
, 40, pp. 574

585.
Bertsimas, D. J., and Howell, L. H. (1993) Further results on the probabilistic traveling
salesman problem,
European Journal of Operation
al Research
, 65, pp 68

95.
Daganzo, C.F. (1984a) The length of tours in zones of different shapes,
Transportation
Research

B
, vol 18B, pp 135

146.
Daganzo, C.F. (1984b) The distance traveled to visit N points with a maximum of C stops
p
er vehicle: an analytic model and an application,
Transportation Science
, vol. 18,
4, pp 331

350.
19
Daganzo, C.F. (1985) Supplying a single location from heterogeneous sources,
Transportation Research

B
, vol 19B, pp 409

419.
De Cursi, J.E.
and Cortes, M.B. (1995) General genetic algorithms and simulated
annealing pertubation of the gradient method with a fixed parameter, in: Topping,
B.H.V. (ed.)
Developments in Neural Networks and Evolutionary Computing for
Civil and
Structural Engineering,
Civil Comp Press, pp 189

198.
Gendreau, M., Laporte, G. and Séguin, R. (1996) Stochastic Vehicle Routing,
European
Journal of Operational Research
, vol 88, pp. 3

12.
Haber, C. S., Jog, C.S., and Bendsoe, M.P. (1994) Topology des
ign with optimized, self

adaptive materials,
International Journal of Numerical Methods in Engineering
, vol.
37, pp. 1323

1350.
Han, A.F.W and Daganzo, C.F. (1986) Distributing nonstorable items without
transshipments”,
Transportation
Research Record
1061, TRB, 32

41, Washington,
DC.
Jaillet, P. and Odoni, A. (1988) The probabilistic vehicle routing problem, in: B.L. Golden
and A.A. Assad (eds.)
Vehicle Routing: Methods and Studies
, North

Holland,
Amsterdam.
La
mbert, V., Laporte, G. and Louveax, F.V. (1993) Designing collection routes through
bank branches,
Computers & Operations Research
20, 783

791.
Laporte, G. (1992) The traveling salesman problem: an overview of exact and approximate
algor
ithms,
European Journal of Operational Research
, vol 59, pp. 231

247.
Laporte, G., Louveax, F.V. and Mercure, H. (1989) Models and exact solutions for a class
of stochastic location

routing problems,
European Journal of Operations Research
39,
71

78.
Larson R.C. and Odoni, A.R. (1981)
Urban Operations Research
, Prentice

Hall,
Englewood Cliffs.
Newell, G.F. (1973) Scheduling, location, transportation, and continuum mechanics; some
simple approximations to optimization problems
, SIAM J. Appl. Math., Vol. 25, Nº 3,
pp. 346

360.
Newell, G.F. and Daganzo, C.F. (1986a) Design of multiple

vehicle tours

I A ring

radial
network,
Transportation Research

B
, Vol. 20, 5, pp.345

363.
20
Novaes, A. G. and Graciolli, O. D. (1998
) Designing multi

vehicle delivery tours in a grid

cell format,
European Journal of Operational Research
(in press).
Pogu, M. and De Cursi, J.E.S. (1994) Global optimization by random pertubation of the
gradient method with a fixed parameter,
Journal of Global Optimization
, Vol. 5, pp.
159

180.
Schoenauer, M. and Kane, C.
,
Topological optim
um design using genetic algorithms,
Control and Cybernetics
, special issue on Optimal Desi
gn (in press).
Souza de Cursi, J.E. and Cortes, M.B
.S. (1995) General genetic algorithms and simulated
annealing pertubation of the gradient method with a fixed parameter,
Developments in
Computational Techniques for Structural
Engineering
, pp. 189

198, BHV Topping,
UK.
Souza de Cursi,
J.E. (1994) Allegement d’une piece elastique homogene soumise a des
constraintes planes, Research Report nº 1, Institut de Mécanique, Institut National des
Sciences Appliquées de Rouen, France.
Stein, D. M. (1978) An asymptotic probabilistic a
nalysis of a routing problem,
Mathematics of Operations Research
, vol 3, pp 89

101.
Stewart, W.R., Jr., and Golden, B.L. (1983) Stochastic vehicle routing: A comprehensive
approach,
European Journal of Operations Research
14, 371

385.
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο