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.

References

Albright, C. (2001). Premium Solver Platform for Excel. ORIMS Today, 28(3), 58-63. Burkard, R.E., Demidenko, V.M., & Rudolf, R. (1997). A General Approach for Identifying Special Cases of the Travelling Salesman Problem with a Fixed Optimal Tour. Optimierung und Kontrolle, 113, 1-12. Chaudhuri, S.P. (2010). Creation of special distance functions and their application in the realworld with TSP heuristics. Unpublished Master dissertation, Warwick Business School, University of Warwick, United Kingdom. Christopher, G., Farach, M., & Trick, M.A (1996). The structure of circular decomposable metrics. In J. Diaz, & M. Serna (Eds.), Proceedings of ESA IV, Lecture notes in Computer Science: Vol. 1136 (pp. 406-418). Springer-Verlag, New York. Deineko, V.G., Rudolf, R., & Woe ginger, G.J. (1998). Sometimes travelling is easy: The master tour problem. SIAM Journal on Discrete Mathematics, 11(1),81-93. Papadimitriou, C. H. (1994). Computational Complexity. Reading, MA: Addison Wesley. Woe ginger, G. J. (n.d.). Computational problems that can be solved without computation. Retrieved June 6, 2001, from http://citeseerx.ist.psu.eduJ