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Schriftenreihe Logistik
Band 6/2010
Why Size Does Matter - Estimating the
Length of Tours in ring-radial Transport
Networks
Nicholas Boone
Tim Quisbrock
Autor 3
Autor 4
Schriftenreihe Logistik
Band 6/2010
Why Size Does Matter - Estimating the
Length of Tours in ring-radial Transport
Networks
Nicholas Boone
Tim Quisbrock
Autor 3
Autor 4
Band 6/2010
Schriftenreihe Logistik
Fachbereich Produktion und Wirtschaft,
Hochschule Ostwestfalen-Lippe, Lemgo
Copyright © 2010 by Nicholas Boone & Tim Quisbrock
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Systemen.
ISBN 978-3-941645-06-6
1
Introduction and Research Objective
This paper analyses post-carriage delivery tours of a groupage depot. The depot serves as
the starting point for tours in a ring-radial transport network. Typically, tours are
constrained by restrictions with regard to driving and work time per driver and with regard
to the vehicle/trailer capacity. These tours can be modeled as a so-called Capacitated
Vehicle Routing Problem (CVRP).
Often, the costs for post-carriage transport dominate the other costs. It can be assumed that
drops that are farther away from the depot than others, on average, lead to tours with
longer first and last ‘legs’ (distance from depot to first and from last drop back to depot). In
order to analyse this effect, customers in the delivery area are divided into rings of
increasing radius around the depot. For each ring, the resulting tour length in km and the
total driving/working hours are evaluated and the cumulative effect is studied.
The objective of this paper is to develop an analytic estimate for the tour length of the
CVRP in ring-radial transport networks and to evaluate the theoretical results by means
of simulation experiments.
After a brief literature review, a concept for a ring-radial network is developed. First,
analytical estimates for the theoretical case of an uncapacitated ‘Mega-Tour’ and for the
CVRP are developed. Next, the estimates for the ‘Mega-Tour’ are compared to the
simulation results. Subsequently, a sensitivity analysis is performed with regard to key
simulation parameters, before the simulation results of the CVRP are compared to the
analytic estimates.
2
2
Existing Estimates for the Length of Tours in Literature
For n customers in a reasonably convex area A, the average distance between two
customer drops in a closed shortest path route is known to statistically converge to (cf. [1]):
δ  κ
A
n
(1)
The value of  depends on the specific shape and, especially, the convexity of the area
considered. It also varies for different travel metrics and depends on the quality of the tourbuilding process. (For an overview of various exact and approximate algorithms cf. [16]).
Typically, simulation studies yielded values of  around 0.7 to 1.15, depending on the
specific simulation setting (cf. [2], [3], [7], [9], [10], [13], [14], [15], [17]).
There are many possibilities to estimate the first and last leg of the tour. In the model of
BEARWOOD/HALTON/HAMMERSLEY, it is (implicitly) assumed that the average first and last
leg each be half the average distance between all drops (cf. [1]):
d 0,first  d last,0 
δ
 const.
2
(2)
Accordingly, the total length of a single tour Lkm with n customers converges to:
Lkm  n  δ  n  κ 
A
 κ  nA
n
(3)
Recently, FIGLIOZZI proposed an estimate for a known number m of capacitated tours (cf.
[9], also cf. [6]). The average first and last leg of each tour, respectively, is assumed to be
equal to the average distance of the drops from the depot, d0i :
d0,first  dlast,0  d0i
Then, the total length of all m tours Lkm with n customers is approximately:
(4)
3
Lkm  m2d 0i  (n  m)  κ 
3
A
n
(5)
Analytical Estimate for the Length of Tours in a ring-radial
Transport Network
3.1 Initial Situation and Model Assumptions
In the following, it is assumed that customer drop locations are uniformly and randomly
generated in a circular delivery area A:
A  π  r2
(6)
Each customer i has a determined pallet demand vi. Travel distances are measured under
a Euclidean travel metric (‘as the crow flies’). All vehicles are of the same type, travel at a
speed s and have a pallet space restriction of Rpall. Additionally, each driver is subject to
a restriction Rtime with regard to his or her total work hours (driving and drop times).
Total time for a drop at customer i consists of a fixed time tdrop,flat plus tLU minutes for
loading and unloading per pallet.
The following procedure is suggested, using FLEISCHMAN’s ring model as a loose analogy
(cf. [5], [11] and [19]):
 Step 1: Divide delivery area A into circles of increasing radius of rk around the depot for
k = 1 to kmax, with rk_max as the maximum distance of a customer drop from the depot.
 Step 2: Set k = 1.
 Step 3: Plan tours for all customers within a radius of r1 around the depot and calculate
all resulting tour lengths (in km, driving and working hours).
 Step 4: Set k = k + 1.
 Step 5: Plan tours for all customers within a radius of rk around the depot and calculate
all resulting tour lengths (in km, driving and working hours).
 Step 6: Repeat from Step 4 until k = kmax.
In this case, the expected number of customer drops n(rk) in a specific ring with the radius
rk can be determined as:
4
n(rk ) 
n(r
)
A(rk )
πr 2
r2
 n(rk_max )  2 k  n(rk_max )  2 k  n(rk_max )  2k_max  rk2
rk_max
rk_max
A(rk_max )
πrk_max
(7)
As one can easily see, expected customer drops grow proportionally to the increasing area
with radius r.
3.2 Estimating the Distances of Customer Drops in ring-radial
Transport Networks
In order to calculate tour lengths, we need to estimate both the distances between two
drops and the first and last legs of the tours.
Applying (1) to the ring-radial approach proposed here, the average distance between two
drops within tours up to rk can be estimated by:
δ(rk )  κ 
A(rk )
n(rk )
A(rk_max )
 κ
n(rk_max )
 const.
(8)
However, due to the assumed uniform drop distribution, δ(rk )  δ = constant for all radii.
There are various possibilities to estimate the first and last leg of the tour. The estimates
mentioned in chapter two, however, need to be adapted to the ring-radial approach
employed here.
Using (2), we receive:
d 0,first (rk )  d last,0 (rk ) 
δ(rk ) δ
  const.
2
2
(9)
Alternatively, one might estimate the first and last legs of tours in an analogy to (4) by
using the average distance of the drops to the depot. If the rings are close together (for
large values of k), it would be feasible to roughly estimate d 0i (rk ) as:
d 0i (rk ) 
rk
2
(10)
However, if the rings are too narrow and/or the total number of all drops is too small, the
probability of finding no drop at all in the first rings is non-negligible (cf. (11).
5

A(rk ) 
Pn(rk )  0  1 

 A(rk_max ) 
n(rk_max )

r2 
 1  2 k 
 rk_max 
n(rk_max )
(11)
According to (7) the distribution function of the number of drops is a square function of r.
Therefore, the probability density function for the drop distribution between two radii is
triangular with a minimum at 0 and both the mode and maximum at rk. Hence the average
distance from the depot is calculated via:
d 0i (rk ) 
( Min  Mode  Max ) (0  rk  rk ) 2

  rk
3
3
3
(12)
To avoid extreme outliers, one might instead choose the median as an estimate:
(13)
d 0i (rk )  0.5  rk
If a saving heuristic is applied, typically, savings are low for drops very close to the depot
(cf. [4]). Therefore, another estimate for the first and last leg could be based on the
expected distance of the closest two customers to the depot. Using (7) we obtain:
n(r) 
r2
2
k_max
r
(14)
 n(rk_max )
and with that:
r2 
n(r)
2
 rk_max
n(rk_max )
(15)
r
n(r)
 rk_max
n(rk_max )
(16)
or:
Then the expected distance of the drop closest to the depot is:
d 0,first 
1
n(rk_max )
 rk_max 
rk_max
1
 rk_max 
n(rk_max )
n(rk_max )
(17)
The second closest drop is expected to be:
d last,0  d 0,second 
2
n(rk_max )
 rk_max  2 
The combined first and last legs are then expected to be:
rk_max
n(rk_max )
(18)
6
d 0,first  d last,0  (1  2 ) 
2
rk_max
n(rk_max )

(1  2 ) A(rk_max ) (1  2 )


δ
π
n(rk_max )
κ π
(19)
In which situation which of the estimates performs better has yet to be determined.
3.3 Estimating the Total Tour Length for the Case of an Uncapacitated
‘Mega-Tour’
The theoretical case of a ‘Mega’-Travelling Salesman tour without either pallet restrictions
per truck or a work time restriction for the drivers (Rpall = Rtime = infinite) can be
considered as a lower bound for the CVRP.
According to (8), the distance  between two tour drops is always constant. Deducting the
first leg and the last leg, leaves the following distance that has to be covered between
drops:
 n(rk ) - 1  δ   n(rk ) - 1  κ 
n(rk_max )  A(rk_max )

(20)
The following estimates for the tour as a whole merely differ on the first/last leg estimate.
Using (8), (9) and (14), the total length of the ‘Mega-Tour’ up to rk is calculated as:
δ
Lkm (rk )  2    n(rk ) - 1  δ  n(rk )  δ  n(rk )  κ  n(rk_max )  A(rk_max )
2
(21)
As the ‘Mega-Tour’ starts and ends at the depot, it also seems reasonable to assume, that,
for all radii, the first and last leg of the tour would be within the first ring.
d0,first (rk )  d last,0(rk )  d0i (r1 )
(22)
Therefore, the estimates for the first and last leg d0,first (rk )  dlast,0(rk ) of the tour mentioned
in section 3.2 based on (10), (12) or (13) have to be adapted (cf. table 1). (Both (9) and
(19) do not adapt to rk.)
7
Table 1. Alternative ‘Mega-Tour’ First and Last Leg Estimates (M_FLE)
First / Last Leg
Estimate
Estimate
according to
M_FLE_1
(9)
M_FLE_2
(10)
M_FLE_3
(12)
2
4
2   r1   r1
3
3
M_FLE_4
(13)
2  0.5  r1  2  r1
M_FLE_5
(19)
(1  2 )
δ
κ π
d0,first (rk )  d last,0 (rk )
2
δ
 δ  const.
2
2
r1
 r1
2
Using (12), for example, the total length of the ‘Mega-Tour’ up to rk is given by:
4
Lkm (rk )  r1   n(rk ) - 1  δ
3
(23)
For (19), the total length of the ‘Mega-Tour’ up to rk would be given by:
L km (rk ) 
 (1  2 )

(1  2 )
 δ   n(rk ) - 1  δ  
  n(rk ) - 1   δ
κ π
 κ π

(24)
3.4 Estimating the Total Tour Length for the Case of m Capacitated
Tours (CVRP)
In order to adapt the tour length estimate (5) for capacitated tours (cf. [14]), we need to
determine both the average first and last legs and the number of tours in the ring-radial
network:

A

Lkm ( rk )  m( rk )  d0,first ( rk )  d last,0 ( rk )  n ( rk )  m( rk )   κ 

n


(25)
For the first and last legs of the tours in the CVRP, we can use the following estimates (cf.
Table 2):
8
Table 2. Alternative First and Last Leg Estimates for the CVRP up to rk (C_FLE)
First / Last Leg
Estimate
Estimate
according to
C_FLE_1
(9)
C_FLE_2
(10)
C_FLE_3
(12)
C_FLE_4
(13)
2  0.5  rk  2  rk
C_FLE_5
(19)
(1  2 )
δ
κ π
d0,first (rk )  d last,0 (rk )
δ
 δ  const.
2
r
2  k  rk
2
2
4
2   rk   rk
3
3
2
However, it can be assumed that both the estimate according to (9) and to (19) are bad
estimates for the CVRP in a ring-radial network, as they do not adapt to the size of the
area. Regarding the number of tours m, typically, either the pallet spaces per vehicle (Rpall)
or the driving time (Rtime) would be more restrictive.
If only the pallet spaces were regarded, the number of tours required would be:
 v(r ) 
μ pall (rk )  roundup k 
R 
 pall 
(26)
If, instead, only the drivers’ working hours were restrictive, the amount of drops that an
average single driver can service is given by

d (r )  d0,last (rk )   t  t  v(rk )  
 R time  0,first k
 drop,flat LU n(r )  

s
k 


ν time (rk )  1 
δ
v(r ) 
  t drop,flat  t LU  k 
n(rk ) 
s
(27)
The total number of tours necessary can then be estimated as follows:
 n(rk ) 

μ time (rk )  roundup
 ν time (rk ) 
Then the number of tours required to service the area up to rk is given by:
(28)
9
m(rk) = max{pall(rk); time(rk)}
(29)
In reality, due to fact that customers are always served by exactly one vehicle, the actual
number of tours would be expected to be larger than the estimate based on an average.
4
Simulating the Length of Tours in ring-radial Networks
4.1 Simulation Parameter Setting and Results of the Theoretical
Estimates
Following assumptions were used for the simulation runs:
 Delivery area A: Customer drop locations are assumed to be uniformly distributed in
the circular area A around the depot. Based on typical maximum delivery areas for
groupage networks in Germany, the maximum distance of a customer drop i from the
depot is set at max{d0i} = 100 km.
 Size of the rings: Using equidistant radii, results of several pretests lead to dividing the
delivery area into kmax = 8 circles around the depot with Δrk+1 = (rk+1 – rk) = r1 = 12.5
km.
 Total number of customers: In order to avoid circles without drops, the total number of
randomly generated customer drops was set at n(rk_max) = 250 (cf. figure 1 for an
example). In this case, the expected amount of drops n(r1) = 3.91 and the probability of
finding no customer in the first ring is only 1.95 % (and much less for all other rings).
 Customer demand volume in pallets: It is assumed that every customer receives
vi = 1 pallet. If the goods are handled on pallets, it is realistic to assume that there is
hardly any difference in the process oriented costs incurred with regard to the actual
pallet weight (also cf. [12], [18]).
 Drop time: Total time per drop consists of tdrop,flat = 15 minutes plus tLU = 5 minutes for
loading/unloading per pallet. (For an empirical study concerning volumes and drop times
cf. [8].)
 Driving speed: due to the uniform distribution of drops with fairly large distances, an
average driving speed s = 50 km/h is assumed.
10
Fig. 1. Example of 250 uniformly distributed drop locations
Using (8) and κ  0.79, we receive the following estimate for the average distance
between two tour drops (cf. [9], but also cf. [14]):
A(rk_max )
δ  0.79 
n(rk_max )
(30)
 8.86 km.
In our simulation setting, total time for all drops (250 customers with one pallet each) is
83.3 hours (cf. the following table 3).
Table 3. Overview of delivery rings and expected customer drop time
k
rk
1
2
3
4
5
6
7
8
12.5
25.0
37.5
50.0
62.5
75.0
87.5
100.0
A(rk) in km² n(rk)
491
1,963
4,418
7,854
12,272
17,671
24,053
31,416
3.9
15.6
35.2
62.5
97.7
140.6
191.4
250.0
Total drop
time
1 h 18 min
5 h 13 min
11 h 43 min
20 h 50 min
32 h 33 min
46 h 53 min
63 h 48 min
83 h 20 min
11
4.2 Simulation Results for the ‘Mega-Tour’
All tours are planned based on CLARKE/WRIGHT’s Savings-algorithm and a subsequently
employed 2Opt-procedure (cf. [4]. In order to calculate a ‘lower bound’ for n capacitated
tours, the theoretical case of a ‘Mega’-Travelling Salesman tour was solved without either
pallet restrictions per truck or a work time restriction for the drivers (Rpall = Rtime =
infinite).
For the parameter set used here, all first and last leg estimates happen to range from  to
just under 2 (cf. table 4):
Table 4. Alternative Estimates for the first and last leg of a ‘Mega-Tour’
First / Last Leg
Estimate
d0,first (rk )  d last,0 (rk )
2
M_FLE_1
δ
 δ  const.
2
8.86 km
r1
 r1
2
12.5 km
2
M_FLE_2
M_FLE_3
2
4
2   r1   r1
3
3
16.67 km
M_FLE_4
2  0.5  r1  2  r1
17,68 km
M_FLE_5
(1  2 )
δ
κ π
15.27 km
The following table 5 shows the estimated values for the total length of the ‘Mega-Tour’:
Table 5. Alternative Estimates for the ‘Mega-Tour’
k
1
2
3
4
5
6
7
8
Total length
(km) between
drops
rk
12.5
26
25.0
130
37.5
302
50.0
545
62.5
856
75.0
1,237
87.5
1,686
100.0
2,205
First and last leg estimate (km)
of the ‘Mega-Tour’ based on:
M_FLE_1
M_FLE_2
M_FLE_3
M_FLE_4
M_FLE_5
8.86
12.5
16.67
17.68
15.27
12
As one can easily see, all estimates for the first and last leg of the ‘Mega-Tour’ are
dominated by the distances between drops. Theoretically, one would expect our ‘Mega-
tour driver’ to drive a total of approximately Lkm = 2,220 km in just over 44 hours, and the
stops would require just over 83 hours. The total required work hours would be Ltime = 127
hours. Figure 2 shows the results of one simulation run for the ‘Mega-Tour’.
Fig. 2. Example of a simulated ‘Mega-Tour’ for 250 uniformly distributed drops
In order to evaluate the estimate, 15 independent simulation experiments were run.
Overall, the analytical values for the distance between two drops in a tour, , are good
estimates for the actual simulation results, the quality of the estimate improving with the
number of drops n(rk) (cf. figure 3).
13
Fig. 3. Simulated distances between drops for the ‘Mega-Tour’
The simulation results for the distance of drops from the depot, d0i, are highly consistent
with the expected analytical values (cf. figure 4).
Fig. 4. Simulated average distance of drops from the depot, d0i, for the ‘Mega-Tour’
Overall, the estimates M_FLE_1 to 5 yield estimates for the first and last legs that, on
average, are too low compared to the simulation results (cf. figure 5).
14
Fig. 5. Average values for the combined first and last legs of the ‘Mega-Tour’ compared to
the analytical estimates M_FLE_1 to 5
This may well be due to the small sample size of only 15 experiments. Accordingly, figure
6 shows that the minimum and maximum values for the simulation experiments vary
strongly.
Fig. 6. Simulated values for the average first or last legs of the ‘Mega-Tour’
The following figure 7 compares the distances of the closest and second closest customer
drops with the estimate M_FLE_5. On average, the simulated actual distances are
slightly underpredicted.
15
Fig. 7. Simulated distance of two closest drops for the ‘Mega-Tour’ compared to the
estimated values.
The simulation results compared to the alternative estimates for the ‘Mega-Tour’ are
shown in the following table 6.
Table 6. Alternative Estimates for the ‘Mega-Tour’
rk
Total length Lkm of the ‘Mega-Tour’,
first and last leg estimate based on:
Simulated tour
length Lkm
M_FLE_1 M_FLE_2 M_FLE_3 M_FLE_4 M_FLE_5
min average max
12.5
35
38
42
43
34
26
42
63
25.0
138
142
146
147
138
121
161
202
37.5
311
315
319
320
311
243
324
386
50.0
553
557
561
562
553
497
585
643
62.5
865
868
873
874
865
786
889
974
75.0
1,245
1,249
1,253
1,254
1,245
1,137
1,260
1,360
87.5
1,695
1,699
1,703
1,704
1,695
1,626
1713
1,827
100.0
2,214
2,218
2,222
2,223
2,214
2,179
2,231
2,301
The CVRP is highly sensitive to restrictions concerning working hours of drivers and
capacity limits of the vehicles. Therefore, in the following, before determining the
simulation parameters for the CVRP, a sensitivity analysis is performed.
16
4.3 Sensitivity Analysis in Preparation of the CVRP
A maximum distance of max{d0i} = 100 km with one pallet each leads to the following
minimum required work time restriction of Rtime:

 v(rk_max ) 
 max{d0i }  
  t LU 
R time  2  
   t drop,flat  


s

 
 n(rk_max ) 

(31)
 100 km 
 2
  15 min  1  5 min  4 h, 20 min
 50 km/h 
For Rpall = infinite, rather obviously, the less restrictive Rtime becomes, the less net driving
time is needed (only two ‘legs’), to serve all customers in A (see figure 8).
Fig. 8. Net driving time for 250 drops related to the work time restriction per driver
without a pallet space restriction
Figure 9 shows the corresponding total tour length in km for all drops in A:
17
Fig. 9. Total tour length Lkm for 250 drops related to the work time restriction per driver
without a pallet space restriction
The total number of tours needed is highly correlated to tour length and driving time (cf.
figure 10):
Fig. 10. Total number of tours m to service 250 drops related to the work time restriction
per driver without a pallet space restriction
Similar results also apply to the case of Rtime = infinite and increasing values of Rpall: for
the case of Rtime = infinite, the less restrictive Rpall becomes, the shorter the total tour length
Lkm (cf. figure 11).
Fig. 11. Total tour length Lkm for 250 drops related to the pallet space restriction with
unlimited work time per driver
Correspondingly, the net driving time needed decreases strongly with growing values of
Rpall (cf. figure 12).
18
Fig. 12. Net driving time for 250 drops related to the pallet space restriction with unlimited
work time per driver
4.4 Simulation Results for the CVRP
Based on the sensitivity analyses and several further pretests, Rpall was set to 14 pallets
per truck, and Rtime was set to 7 hours per driver for the simulation of the CVRP.
Excluding test runs and sensitivity analyses, a total of 50 simulation experiments,
corresponding to 50 work days, were run with the afore mentioned parameters.
The simulation results for the distance of drops from the depot, d0i, are highly consistent
with the expected analytical values (cf. figure 13).
Fig. 13. Simulated average distance of drops from the depot, d0i for the CVRP
The estimates C_FLE_1 to 5 are compared to the average simulated distances (km) for the
first and last legs in figure 14. As expected, C_FLE_1 and C_FLE_5 are bad estimates, as
they do not adapt to rk.
19
Overall, C_FLE_2 seems to be a good estimate for the average distance of drops from the
depot in the case of ‘small’ radii. However, for ‘large’ values of rk, C_FLE_2
underestimates the actual distance. This seems to be due to the fact that, the farther away a
drop is from the depot, the less time is left to service further customers. This leads to
increasingly more tours and more km in the simulation.
Both C_FLE_3 and C_FLE_4 overestimate the average distance to the depot for all
regarded values of rk, For very large radii, however, C_FLE_3 is closest to the simulated
values.
Fig. 14. Average values for the combined first and last legs of the CVRP compared to the
analytical estimates C_FLE_1 to 5
As is shown in figure 15, the wing-span of the simulation values for the first/last leg
decreases for growing values of rk. The large wing-span for r1 is probably due to the fact
that r1 sometimes only contains one or two drops. This effect is ‘averaged-out’ for larger
radii.
20
Fig. 15. Wing-span of simulation values for the first/last legs up to rk for the CVRP
Table 7 shows the results of the total simulated tour length, Lkm, compared to the
alternative analytic estimates.
Table 7. Alternative Estimates for the CVRP (C_FLE)
Total length Lkm of the CVRP,
first and last leg estimate based on:
rk
Simulated tour
length Lkm
C_FLE_1 C_FLE_2 C_FLE_3 C_FLE_4 C_FLE_5
min
average
max
12.5
35
38
42
43
34
12
38
64
25.0
138
171
187
191
138
105
160
204
37.5
311
397
435
444
311
283
395
466
50.0
553
759
843
863
552
654
817
1,004
62.5
865
1.240
1.386
1.422
863
1,296
1,429
1,592
75.0
1.245
1.973
2.248
2.315
1.243
2,088
2,335
2,638
87.5
1.695
2.796
3.204
3.418
1.692
3,339
3,665
4,014
100.0
2.214
3.855
4.704
4.998
2.210
5,316
5,597
5,878
Overall, the analytical values underestimate the actual simulation results (cf. figure 16).
21
Fig. 16. Simulated length of tours Lkm of the CVRP for  = 0.79
This is mainly due to the fact that the analytically estimated distance between two
customers in a tour, , underestimates the actual simulation results (cf. figure 17).
Fig. 17. Simulated distances between drops  for the CVRP for  = 0.79
As mentioned, the factor for  is dependent on the shape and convexity of the area, the
travel metric involved and the quality of the tour planning algorithm.
CLARKE/WRIGHT’s Savings-procedure is based on a heuristic algorithm. Therefore, it
cannot guarantee achieving the global optimum like an exact algorithm. Typically, in its
basic form, it yields tour subareas that may overlap and, therefore, aren’t compact (cf.
figure 18) This does not apply to the case of the ‘Mega-Tour’ as there is only one tour (cf.
22
figure 2). Also, the capacity restrictions may lead to the termination of an individual tour,
so that an otherwise large saving cannot be achieved.
The simulated average distance between two tour drops is approximately 23% higher than
the analytical estimate for . Therefore, an adjusted value of  = 0.97 should yield better
estimates for the actual simulation results. Table 8 gives the corresponding values for  =
0.97.
Fig. 18. Example of the CLARKE/WRIGHT Savings-procedure applied to 250 random and
uniformly distributed customers
23
Table 8. Alternative Estimates for the CVRP (C_FLE) for  = 0.97
Total length Lkm of the CVRP,
first and last leg estimate based on:
rk
Simulated tour
length Lkm
C_FLE_1 C_FLE_2 C_FLE_3 C_FLE_4 C_FLE_5
min
average
max
12.5
43
44
48
49
47
12
38
64
25.0
170
199
215
219
179
105
160
204
37.5
383
463
501
510
396
283
395
466
50.0
681
877
960
980
703
654
817
1,004
62.5
1.065
1.426
1.572
1.607
1.095
1,296
1,429
1,592
75.0
1.533
2.238
2.513
2.580
1.581
2,088
2,335
2,638
87.5
2.087
3.159
3.568
3.780
2.148
3,339
3,665
4,014
100.0
2.726
4.330
5.174
5.467
2.804
5,316
5,597
5,878
As expected, the analytic estimates are closer to the actual simulation results (cf. figure
19).
Fig. 19. Simulated distances between drops for the CVRP for  = 0.97
Overall, C_FLE_3 and C_FLE_4 yield the best estimates for a factor  = 0.97.
24
5
Conclusions and Outlook
As was assumed in the introduction, drops that are farther away from the depot than
others, on average, lead to tours with longer first and last ‘legs’. Overall, the simulation
results show that the analytically derived estimates are good estimates for the total tour
lengths for in ring-radial transport networks.
However, the following limitations must be taken into account: The estimates are strongly
dependent on the value of  and probably can be improved. Therefore, further research
could explicitly take into account the quality of the tour planning algorithm and the
convexity of the area/subareas.
Possibly, the factor for  can be reduced, e.g. if an adapted Saving procedure with a higher
weight for the distance between two drops is employed, thus yielding more compact tour
subareas. Also, a more compact tour-building procedure, e.g. a ‘sweep-algorithm’ (cf. eg.
[16]) could be employed and might reduce the deviations of the simulation and the
estimate values.
The simulation results in this paper were based on a given set of parameters. A systematic
parameter study with regard to alternative customer densities, size of radii and different
metrics could yield further interesting findings. Finally, the effect of fluctuating customer
demand has yet to be analysed.
25
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26
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Gestaltung
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27
Appendix I: Notation
Term
Description
A(rk)
Circular delivery area around the subsidiary up to radius of rk
Ci
Total costs for customer drop i
C(rk)
Total costs up to a radius of rk
cdriv
Cost per hour for driving
Cdriv(rk)
Total driving time related costs up to a radius of rk
cdrop
Cost for drop per hour
Cdrop,i
Direct costs for drop i
Cdrop(rk)
Total direct drop related costs up to a radius of rk
ckm
Cost for driving per km
Ckm(rk)
Total driving distance related costs up to a radius of rk
cLU
Cost per hour for loading/unloading pallets
cover(rk) Marginal overhead cost per pallet from radius rk-1 to rk
Cover(rk)
Total overhead cost up to a radius of rk
Cpall,i
Total drop related pallet cost for customer i
Cpall(rk)
Total direct pallet related costs up to a radius of rk
ctour,flat
Fixed transfer price for indirect overhead costs per tour
Ctour(rk)
Total tour related overhead costs up to a radius of rk
C_FLE_1 Estimate (km) for combined first and last leg of CVRP No. 1
C_FLE_2 Estimate (km) for combined first and last leg of CVRP No. 2
C_FLE_3 Estimate (km) for combined first and last leg of CVRP No. 3
C_FLE_4 Estimate (km) for combined first and last leg of CVRP No. 4
C_FLE_5 Estimate (km) for combined first and last leg of CVRP No. 5

Average distance between two drops in a shortest path tour
(rk)
Average distance between two drops in a shortest path tour within rk
d0i
Distance of customer drop i from subsidiary
d0,first(rk)
Averge first ‘legs’ of all tours within radius rk
dlast,0(rk)
Average last ‘legs’ of all tours within radius rk
kmax
Maximum number of rings around subsidiary
Lkm(rk)
Total length of all tours in km
Lkm(rk)
Total length of all tours up to rk in km
Ltime(rk)
Total tour length (driving and drop-time) in hours up to a radius of rk
28
Term
Description
m
Total number of tours
m(rk)
Number of tours up to a radius of rk
M_FLE_1 Estimate (km) for combined first and last leg of ‘Mega-Tour’ No. 1
M_FLE_2 Estimate (km) for combined first and last leg of ‘Mega-Tour’ No. 2
M_FLE_3 Estimate (km) for combined first and last leg of ‘Mega-Tour’ No. 3
M_FLE_4 Estimate (km) for combined first and last leg of ‘Mega-Tour’ No. 4
M_FLE_5 Estimate (km) for combined first and last leg of ‘Mega-Tour’ No. 5
pall(rk)
Estimated number of tours (up to rk) due to Rpal
time(rk)
Estimated number of tours (up to rk) due to Rtime
n
Number of customer drops
n(rk)
Number of customer drops up to a radius of rk
time(rk)
Estimated number of drops per driver (up to rk) due to Rtime
r
Radius around the depot
rk
Radius of the kth circle around the subsidiary
Rpall
Pallet capacity restriction per truck
Rtime
Working time restriction per driver
s
Driving speed
tdriv(rk)
Total driving time up to a radius of rk
tdrop,flat
Time per drop (constant)
tLU
Time for loading/unloading per pallet
ttotal(rk)
Total working time of all drivers up to a radius of rk
v(rk)
Total drop pallets up to a radius of rk
vi
Customer demand volume in pallets for drop i
Die Autoren dieses Bandes
N ICHOLAS B OONE, geboren 26.06.1970 in Brighton (GB), Prof.
Dr. rer. pol., 1991 - 1995 Studium der Betriebswirtschaftslehre an
der Otto-Friedrich-Universität Bamberg mit den Schwerpunkten
Logistik, Statistik und Marketing (Thema der Diplomarbeit: „Loglineare Modelle in der Marketingforschung“). Zunächst als Assistant Brand Manager im Marketing bei Procter & Gamble, anschließend als wissenschaftlicher Mitarbeiter am Lehrstuhl für
Logistik der Universität Bamberg tätig. 2001 Promotion mit einer Arbeit über „Vernetzung dezentraler Lagersysteme im Großhandel“. Freiberufliche Tätigkeit in der Logistikberatung und in der Marketingforschung sowie als
Dozent an der Verwaltungs- und Wirtschaftsakademie (VWA) Nürnberg. Von 2002
bis 2006 Logistik-Consultant und Projektmanager Kontraktlogistik bei Dachser. Seit
März 2006 Professor für Logistik und Distribution an der Hochschule OstwestfalenLippe in Lemgo.
T IM Q UISBROCK, geboren 05.06.1982 in Bielefeld, Dipl.-Wirt.Ing. (FH), 2003 - 2006 Studium des Wirtschaftsingenieurwesens
insbesondere der Logistik an der Hochschule Ostwestfalen-Lippe
in Lemgo (Thema der Diplomarbeit: „Steuerungslogik für eine
Kommissionierung mit dynamischer Bereitstellung für die AfterSales-Distribution eines Automobilzulieferbetriebs“). Zunächst
bei der Hella Distribution GmbH und anschließend als Ingenieur für Materialflusssimulation und für Corporate Industrial Engineering bei der Hella Corporate Center
GmbH tätig (2006-2008). Seit März 2008 wissenschaftlicher Mitarbeiter am Lehrstuhl
für Logistik und Distribution der Hochschule Ostwestfalen-Lippe. Darüber hinaus
freiberufliche Tätigkeit in der Logistikberatung.
Errata
Formula 27 on page 8 gives the average amount of drops that a driver can manage due to
the time restriction:

d (r )  d0,last (rk )   t  t  v(rk )  
 R time  0,first k
 drop,flat LU n(r )  

s

k 

ν time (rk )  1 
δ
v(r ) 
  t drop,flat  t LU  k 
n(rk ) 
s
(27)
Due to a data link error, the analytic estimates in Table 7 on page 20 are too low (and,
obviously, also in table 8): While all other time-related values of formula 27 were
δ
calculated based on minutes,   was given in hours. Therefore, the values of ν time are too
s
large and the resulting estimates for μ time ( rk ) and, hence, for the number of tours, m( rk ) ,
are too low. The comparison of the correctly calculated estimates to the actual simulated
tour length shows a good fit for the estimate C_FLE_3 (cf. Table 7a).
Table 1a. Corrected Alternative Estimates for the CVRP (C_FLE)
Total length Lkm of the CVRP,
first and last leg estimate based on:
rk
Simulated tour
length Lkm
C_FLE_1 C_FLE_2 C_FLE_3 C_FLE_4 C_FLE_5
min
average
max
12.5
35
38
42
43
41
12
38
64
25.0
138
171
187
191
151
105
160
204
37.5
311
397
435
444
331
283
395
466
50.0
553
800
900
925
586
654
817
1,004
62.5
865
1.348
1.610
1.660
916
1,296
1,429
1,592
75.0
1.245
2.105
2.521
2.704
1.316
2,088
2,335
2,638
87.5
1.695
3.189
3.959
4.108
1.791
3,339
3,665
4,014
100.0
2.214
4.493
5.824
6.191
2.336
5,316
5,597
5,878
Therefore, the analytical values based on C_FLE_3 actually slightly overestimate the
actual simulation results (cf. figure 16).