KERTAS KERJA PERSIDANGAN
INVESTIGATING THE ROBUSTNESS OF THE TSP ROUTES
THROUGH THE RECOGNITION OF SPECIAL STRUCTURED MATRICES
DISEDIAKAN OLEH
AZMIN AZUZA BINTI AZIZ
JABATAN OPERASI DAN PENGURUSAN SISTEM MAKLUMAT
FAKULTI PERNIAGAAN
DAN PERAKAUNAN
UNIVERSITI MALAYA
"
1.0
Introduction
The optimal solution tours of routing problems are known to be exposed to uncertain and
unpredictable environments and thus vulnerable to changes. In a real life routing problem, for
instance, the number of clients or demands may vary after some time, which may cause the
current routing plan to be no longer optimal or lose its robustness. Due to this reason, achieving a
robust routing plan is vital to maintain the optimality of the solution tour. In this study, a robust
routing plan is defined as a routing strategy that can cope with reductions in the number of
clients, i.e. the solution tour is still optimal if one simply skips the clients that are removed from
the service.
This paper aims to investigate the robustness of the routing tours through the recognition of
special structured matrices. To be precise, the robustness ofTSP routes from various dimensions
of randomly generated instances is analysed by executing the linear programming and
combinatorial-based algorithms according to the special combinatorial structures of Kalmanson
and Burkard matrices. The percentage deviations between TSP shortest length and lower bounds
of Kalmanson and Burkard matrices are measured to approximate how far the TSP tour lengths
are from the lower bounds.
As such, this leads to the formulation of the specific objectives of this paper, which are:
(a) To identify which procedure provides better lower bounds to the TSP solutions
(b) To investigate how far the TSP tour lengths are from the lower bounds
(c) To examine the amount of reduction needed for a random TSP matrix to sufficiently
satisfy the Kalmanson and Burkard conditions
The paper is organized as follows: In the following section, we first define relevant theoretical
concepts in Section 2.0. Section 3.0 presents the proposed methodology where the objectives set
out above will be addressed in greater detail. The results of our computational experiments are
then reported in Section 4.0. Finally, we offer some concluding remarks in Section 5.0.
2.0
Relevant Theoretical Concepts
The TSP deals with the problem of finding an arbitrary permutation, 1t of cities from 1 through n
that minimize the total distance travelled. In Euclidean TSP, the cities are represented by points
which lie on two-dimensional plane. This forms a distance matrix with cities as the rows and
columns and the distances between them can be computed using Euclidean metric.
In this study, the robustness of TSP routes is analysed through the implementation of four
procedures. These procedures apply the definitions and theorems proposed by Deineko et al.
(1998) and Burkard et al. (1997), respectively. The fundamental idea underpinning these
procedures is known as the master tour, which was first formulated by Papadimitriou (1994).
2.1
The Master Tour
A master tour can be defined as follows: An optimal tour, 1t of a problem, X is called a master
tour if after removing any subset of points in X, 1t remains optimal. By way. of illustration of the
concept of master tour, consider the following example in Figure 1.
2
4
7
4
5
(a)
(b)
Figure 1: Illustration of the concept of a master tour
The illustration above shows a set of clients receiving a particular service scattered over seven
locations. The optimal TSP tour of Figure l(a) is IT = (1,2,3.4,5,6,7,1) and 1t is also a master
tour. Suppose that after few years, client 3 and client 6 leave the service. Hence, the new optimal
tour for Figure 1(b) can be obtained by simply removing these two clients from IT, that is IT' =
(1,2,4,5,7,1).
Based on the definition, recognizing a master tour for a particular TSP instance could guarantee
the robustness and optimality of a tour regardless of any changes, hence leading to costs,
distance and time savings. This concept may be useful in practice especially for services with
daily operations where the routing and scheduling design do not need to be performed every day
or frequently.
2.2
Kalmanson matrices
Definition (Deineko et al., 1998). A symmetric n x n matrix C is a Kalmanson matrix (or simply
Kalmanson) if it fulfils the following conditions:
Ci,j+l
ci,l
+ Ci+l,j
+ ci+l,n
:5
:5
Ci,j
ci,n
+ ci+l,j+l
+ ci+l,l
for all 1:5 i :5 n - 3, i
for all 2 :5 i :5 n - 2
+2
:5 j :5 n - 1
(1)
(2)
These conditions are referred to as the Kalmanson conditions. It is known that the permutation of
(1,2, ... , n - 1, n) is the optimal tour for the TSP restricted to Kalmanson distance matrix
(Kalmanson, 1975) and these special structures can be recognized in O(n2) time (Deineko et aI.,
1998). The distance matrices of convex planar point set and tree are also Kalmanson matrices
(Woe ginger, n.d.). In 1998, Deineko et al. provided proof for the relatedness between Kalmanson
matrix and master tour through the following theorems (Deineko et al., 1998):
Theorem 1. A symmetric
Kalmanson matrix.
n x n matrix C possesses the master tour if and only if C is a
Theorem 2. For a symmetric n x n matrix C, it can be decided in O(n2log n) time whether C is a
permuted Kalmanson matrix.
Theorem 3. For a symmetric n x n matrix C, it can be decided in O(n2 log n) time whether C
possesses a master tour.
The theorems above show that the permuted Kalmanson matrix can be recognized in O(n2 log n)
time and is closely related to a master tour. The O(n2 log n) time complexity has then been
improved by Christopher et al. (1996) to O(n2) time. The proofs for these theorems can be found
in their respective paper.
2.3
Burkard matrices
The second approach applied the theorem proposed by Burkard et al. (1997), which is stated as
follows:
Theorem (Burkard et at, 1997). If the elements of a (not necessarily symmetric)
C satisfy
+ cq,p + Ci,j
+ Ci,q + Cq,j
*
for 1 ~ P < i j < k < q ~
then the identity permutation (1,2, ... , n - 1, n) is a master tour.
Ck,q
~ Ck,p
n
n x n matrix
(3)
The reader may refer to the paper by Burkard et al. (1997) for the proof of this theorem. For
brevity, the inequality (3) stated above and the matrix fulfilling this inequality will be referred to
as Burkard condition and Burkard matrix, respectively. It is therefore obvious from the theorem
that a matrix C possesses a master tour if C is a Burkard matrix.
The theorems presented in this section clearly demonstrate that the concept of a master tour is
closely related to the Kalmanson and Burkard matrices. To be specific, recognizing these special
structures in a TSP distance matrix will facilitate in finding a master tour which can guarantee
the robustness and optimality of the TSP solution.
3.0
Methodology
In this study, we utilised a pre-generated distance matrix developed by Chaudhuri S.P. (2010) to
perform the robustness testing procedures. This distance matrix was built up from 130 postcodes
which were randomly generated using Microsoft's Visual Basic for Applications (VBA) in
Microsoft® Excel 2007 (or simply Excel) with interaction of Microsoft® MapPoint 2009
(MapPoint). It is important to clarify here that these generated postcodes represent real and valid
points in the regions of Coventry. The distance between each pair of post codes in this matrix was
estimated using MapPoint software.
Of these 130 postcodes, we randomly generated various sets of test problems with the
dimensions ranging from five to fifty postcodes using a simple program in VBA. For each set, 30
instances with different list of postcodes were randomly generated to observe the trerids. Thus, a
total of 1380 random instances across 46 dimensions of distance matrices were attempted. Each
instance was optimized using MapPoint and the corresponding optimal distance matrix, called
the TSP matrix of this instance was obtained from the pre-generated distance matrix. The TSP
solution comprising total distance and the tour of the optimized instance are referred to as TSP
tour length and TSP tour hereafter. The TSP tour length was calculated from the optimal distance
matrix using formula (4) below:
n-l
Total distance
= 2>i,i+l
+ Cn,l
(4)
i=l
Subsequently, the TSP matrix was analyzed for its robustness through the recognition of special
structured matrices of Kalmanson and Burkard. Four procedures were proposed for this purpose,
namely, Combinatorial-based Kalmanson, Combinatorial-based Burkard and two approaches of
Linear Programming (LP)-based Kalmanson. It was not possible to conduct analysis on LPbased Burkard due to the complexity of Burkard inequality. In each procedure, the lengths or
total distances for the optimal tours of Kalmanson and Burkard matrices will be computed using
formula (4) above. As proven by Deineko et al. (1998) and Burkard et al. (1997) in the earlier
mentioned theorems, these optimal tours are indeed the master tours which are robust to changes.
3.1
Combinatorial-based Kalmanson procedure
Recognition of Kalmanson matrix requires O(n2) time (Deineko et aI., 1998). Due to the
symmetricity property, we are only required to check the Kalmanson conditions in each row as
well as on each element in the upper diagonal of the distance matrix. If the symmetric TSP
matrix does not hold the Kalmanson conditions, then some 'reductions' in the elements of the
matrix are necessary to transform this matrix into a Kalmanson matrix. This can be accomplished
by configuring conditions 1(a) and 2(a) stated earlier. Specifically, if any or both of these
conditions is/ are violated, then the second term of each condition is proposed to be recalculated
as follows:
Ci+l,j
=
Ci,j
+
ci+l,j+l
-
Ci,j+l
for all 1~ i ~
n -
3, i + 2 ~ j ~
for all 2 ~ i ~ n - 2
n -
1
(5)
(6)
3.2
Combinatorial-based Burkard procedure
The Burkard condition (inequality (3)) stated that i =1= j, which means that both i < j and i > j
are possible, indicating that the respective theorem of Burkard et al. (1997) only provides a
sufficient condition for permutation (1,2, ... , n - 1, n) to be a master tour. Since the theorem is
applicable to asymmetric cases, the condition was checked for each value ofp, i.], k and q of the
TSP matrix. Similar to the Kalmanson procedure, in the cases where the Burkard condition was
not satisfied, some 'reductions' in the elements of the TSP matrix are necessary by configuring
the inequality. The proposed recomputed value of Ck,q is as follows:
Ck,q
=
Ck,p
+
As stated earlier, the 'reduced'
Burkard matrix.
3.3
Ci,q
+
Cq,j -
cq,p -
Ci,j
(7)
TSP matrix satisfying the Burkard condition is referred to as the
Linear Programming-based Kalmanson procedure
The procedure based on linear programming concerns with a question of how much distance
should be 'reduced' from the elements of a TSP matrix such that the reduction is minimal and the
resulting matrix sufficiently satisfies the Kalmanson conditions. This can be represented
mathematically as:
where aij is the element of the TSP matrix, Oij is the distance to be reduced from aij and Cij is
the element of the Kalmanson matrix. This is in contrast with the combinatorial-based procedure
in which the latter deals with transforming the TSP matrix into Kalmanson matrix regardless of
the amount of reduction in each element.
In order to perform the robustness analysis using this procedure, the problem needs to be
mathematically formulated as a linear program. It is worth noting that the TSP instances of size
up to 20x20 can be easily solved using (standard) Excel built-in Solver. However, as our
instances extend far beyond this limit, an extension to the standard Excel's Solver is required
instead. Hence, to standardize the method, the LP-based procedures will be implemented in
Premium Solver Platform (PSP). PSP is an improved analysis tool developed by Frontline
Systems Inc. and capable of solving a wide variety of large scale LP problems. A review on the
software can be found in Albright (2001).
In this procedure, two approaches were proposed. The first approach aims to minimize the
maximum reduction value while fulfilling the Kalmanson conditions. The linear programming
formulation for this approach is given as follows.
Approach 1: LPI
Notation:
h = Maximum reduction value (in PSP: Set Target Cell & By Changing Cells)
aij = Distance of travel from postcode ito postcodej in TSP matrix (true value)
tJiJ
= Reduction value (in PSP: By Changing Cells)
CiJ
= Distance of travel from postcode ito postcodej in resulted Kalmanson matrix
Objective function:
Subject to:
Minimize h
(8)
s., s h
(9)
C,t,j .
= a·t,j . -
o· .
+ ci+l,j+l
ci,n + Ci+l,l Ci,j
(10)
t,j
-
Ci,j+l
-
Ci+l,j
~
0
S.t,j . >
0
-
(12)
(13)
h~ 0
(14)
ci,l -
ci+l,n
~
0
(11)
The objective function expressed in (8) requires the minimization of the maximum reduction
value, h. Constraint (9) ensures that each reduction value is at most h. It is imperative to note that
Oi,j
is adjusted according to constraint (10) such that the reduced TSP matrix satisfies the
Kalmanson conditions, denoted by constraints (11) and (12). Observe that these conditions are
rearranged from inequality (1 a) and (2a) which are the revised Kalmanson conditions. These
constraints guarantee that the resulted distance matrix is a Kalmanson matrix. Constraints (13)
and (14) specify the non-negativity value of Oi,j and h.
Nevertheless, focusing on minimizing the maximum reduction value as in (8) may result in a
shorter total distance of Kalmanson matrix. In fact, preliminary analysis on arbitrary data showed
that as the matrix dimension increases, the total distance of Kalmanson matrix would be inclined
towards negative value. Hence, in addition to the above, we also performed a second analysis of
LP-based procedure with the objective of maximizing the total distance of Kalmanson matrix.
The linear programming formulation is stated below.
Approach 2: LP2
Notation:
au = Distance of travel from postcode ito postcodej in TSP matrix (true value)
tJiJ = Reduction value (in PSP: By Changing Cells)
CiJ
= Distance of travel from postcode ito postcodej in resulting Kalmanson matrix
Objective function:
n-I
Maximize L.J
"
C ',I'+1
+ C n, 1
(15)
i=1
Subject to:
C·t,j .
= a,t,j . - 0't,j ,
ci,j
+ ci+l,j+l
-
(10)
Ci,j+l
-
ci+l,j
~
0
(11)
(12)
(13)
The objective of this approach as expressed in (15) is to maximize the "total distance of the
Kalmanson matrix. This special structured matrix is obtained by reducing some elements of the
TSP matrix by biJ, as imposed by constraint (10). Constraints (11) and (12) ensure that the
resulting distance matrix satisfies Kalmanson conditions. Also, the non-negativity value of biJ is
imposed in constraint (13).
The robustness procedures described above involve the reduction of elements of a TSP matrix in
order to generate the special structured matrices. Thus, the total distances of Kalmanson and
Burkard matrices represent the lower bounds of the TSP tour length. To be specific, the four
procedures implemented in this study produce four lower bounds. Obviously, the quality of these
bounds is guaranteed as the special structured matrices possess a master tour. In order to
investigate the significant difference of the TSP tour length from the lower bound, we measured
the percentage deviation between the two using the formula:
Of.
70
d ev
=
(TTSP -T)
LB
X
100
(16)
TLB
where T TSP and T LB are the total distances of TSP matrix and special structured matrices
(Kalmanson or Burkard), respectively. However, in cases where the lower bounds are negative,
the denominator is changed to the minimum distance between two points in order to avoid
negative deviations. Specifically,
% dev =
(TTS~
-
TLB)
x 100
(16a)
mmdi,j
In addition, we also examined the reduction values in each instance across all dimensions and
procedures. The reduction value is the value to be deducted from the element of the TSP matrix
in order to transform it into the Kalmanson or Burkard matrix. Whilst this may draw an
understanding on the sufficient value required to obtain a master tour from an arbitrary matrix,
the analysis also assist in observing whether the reduction values influence the performance of
the lower bounds.
4.0
Results and Discussion
All the algorithms were coded in VBA programming language (Microsoft Visual Basic 6.5) with
a direct link to Microsoft® MapPoint 2009 and Microsoft® Excel Premium Solver Platform and
executed on a personal computer with Intel (R) Core 2 Duo 3.0 GHz processor with 3.21 GB
RAM.
In this section, the four lower bounds will be scrutinized and compared amongst each other as
well as with the TSP tour lengths. For this purpose, we will denote the total distances as follows:
Notation
TSP
SYMM
COMB-K
COMB-B
LPI
LP2
Description
Optimal total distance of asymmetric TSP matrix (from MapPoint)
Total distance of symmetric TSP matrix
Total
Total
Total
Total
distance
distance
distance
distance
of
of
of
of
Kalmanson matrix from combinatorial-based procedure
Burkard matrix from combinatorial-based procedure
Kalmanson matrix from Approach 1 of LP-based procedure
Kalmanson matrix from Approach 2 of LP-based procedure
Comparing the TSP and SYMM, we observed that the SYMM is relatively smaller than the TSP,
by a standard deviation of 0.77 miles. In further analysis, all the comparisons will be made based
on the TSP.
Figure 2 illustrates the performance of the computed lower bounds and the TSP across all
problem sizes while Figure 3 demonstrates the average deviations of the TSP from the lower
bounds. Owing to the large range of vertical axis and for easy observation, we 'truncate' the
vertical axis of Figure 2 into the range of [0, 100] miles. Also, Table 1 illustrates the
performance of the four lower bounds compared to TSP for small dimensions instances (median
values are considered).
100.-------------
-.
-TSP
-COMB-K
80
-COMB-B
-LPl
-LP2'
CII
-:
:..
60
• •
'W
.'
:00
•
~
• •
'.
'.
I1
-
t •
..;.:·II t
! d' ~.
40
.·1'
I;
I t
I
I
•
o
20
5
10
15
20
25
30
35
40
45
50
Problem size
Figure 2: Illustration ofTSP and the four lower bounds, namely COMB-K, COMB-B, LPI and
LP2 on instances with dimensions of 5 to 50 (30 instances each)
4500000%
4000000%
-LPl
3500000%
-LP2
3000000%
r:::::
0
'~
It!
'>CIJ
'tJ
'*
-COMB-K
2500000%
-COMB-B
2000000%
1500000%
1000000%
500000%
0%
5
10
15
20
25
30
35
40
45
50
Problem size
Figure 3: Average deviation of the TSP from the lower bounds for instances with dimensions of
5 to 50
Table 1: Performance of lower bounds for small problem sizes
Dimension
TSP
LP1
LP2
COMB K
COMB B
5
24.261
22.101
23.646
23.044
24.261 *
6
27.949
26.209
26.933
25.817
27.850*
7
31.681
26.967
29.142
27.517
29.630*
8
28.847
24.442
27.455*
26.122
26.868
9
38.739
30.186
37.049*
33.843
35.080
10
38.482
29.483
35.198*
32.693
29.565
11
39.376
29.792
37.428*
30.556
30.394
12
39.407
29.248
37.633*
31.914
28.390
13
41.688
27.018
39.771 *
27.831
16.848
The symbol * indicates the best lower bound for the TSP
It is apparent from Figure 2 that the TSP increases as dimension gets larger. The lower bounds of
COMB-K, COMB-B, LPI and LP2 are relatively close for instances with small dimensions. As
supported by the values in Table 1, Burkard procedure performs extremely well with smaller
dimensions instances where it manages to offer similar outcomes to TSP for 29 out of 30 (97%)
instances of size 5x5. The results also show that at least 73%,37% and 20% of 6x6, 7x7 and 8x8
Burkard matrices, respectively, are identical to TSP matrices, thus producing the same total
distances. Despite its advantage in exploiting asymmetric data, COMB-B gives the worst lower
bounds amongst other procedures for instances with higher dimensions, specifically when the
problem size is greater than llxll. The average deviation of COMB-B as demonstrated in
Figure 3 also increases drastically as the problem size increases. In the cases 'when the dimension
increases, it is observed that LP2 clearly outperforms the other three procedures, where the total
distances appear to be the closest to the TSP. This finding is supported by the nearly constant
rate of the average deviations in Figure 3 that is about 6.4% from the LP2 as well as the optimal
lower bounds in Table 1. It is also worth noting that 30% of 5x5 instances have robust solutions
according to the linear programming approach (LPI and LP2).
On the contrary, it appears that the performance of COMB-K, COMB-B and LPI deteriorate
much from the TSP as the matrix size increases. Indeed, the lower bounds of these procedures
are getting worse as the values tend to drop to negative. This is shown by the large average
deviations of the TSP from the three lower bounds.
In order to address the third objective, which is to investigate the amount of reduction required
for a random TSP matrix to adequately fulfil the Kalmanson and Burkard conditions, we first
make use of the following notations:
Notation
Description
MinX
MaxX
MaxX-K
MaxX-B
the maximum
the maximum
the maximum
the maximum
reduction value, Xij
reduction value, Xij
reduction value, Xij
reduction value, Xij
associated with LPI
associated with LP2
associated with COMB-K
associated with COMB-B
The plot for reduction values of LPl, LP2, COMB-K and COMB-B for instances across all
problem sizes is depicted in Figure 4.
20.----- __ ~~------------MaxX-K
t
-MaxX-B
..
t".o
,
~~------_,
,
".
.
"
.
15
'II
:;:,
1
c
0
:e
10
:;:,
"'C
&!
5
5
10
15
20
25
30
35
40
45
so
Problem size
Figure 4: Reduction values ofLPl, LP2, COMB-K and COMB-B for instances with dimensions
of5 to 50.
The proposed recognition algorithms which provide four lower bounds in this study could be
applied to a wide range of test problems. For the combinatorial-based procedure, although no
limitation on the problem size is imposed, there is a tendency to obtain worse lower bounds if the
dataset used is large. On the contrary, the LP-based procedure could provide good lower bounds,
however the application maybe restricted up to a certain problem size and this depends on the
capability of the software used. The results obtained from the robustness procedures could
provide a recommendation to the decision makers on which routes to be implemented. The
exercise will allow the decision makers to comprehend the underlying characteristics of the
routes and possibly make necessary modifications to ensure the optimality and robustness of the
solutions regardless of any variations in the inputs. As such, these modifications could be done
by taking into account the geographical regions of these services as specified in the current
practice.
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