Master Tour Routing

• Vladimir Deineko, Warwick Business School

Outline

Vehicle routingMaster tour problemTravelling Salesman Problem with Kalmanson matricesQuadratic Assignment Problem/ Special CaseSummary

3Warwick Business School

Vehicle routing problem

PostCode 15A15

PostCode 14A14

PostCode 13A13

PostCode 12A12

PostCode 11A11

PostCode 10A10

PostCode 9A9

PostCode 8A8

PostCode 7A7

PostCode 6A6

PostCode 5A5

PostCode 4A4

PostCode 3A3

PostCode 2A2

PostCode 1A1Given a set of customers

Find a tour with the minimal total length

PostCode 15A15

PostCode 14A14

PostCode 13A13

PostCode 12A12

PostCode 11A11

PostCode 10A10

PostCode 9A9

PostCode 8A8

PostCode 7A7

PostCode 6A6

PostCode 5A5

PostCode 4A4

PostCode 3A3

PostCode 2A2

PostCode 1A1Given a set of today’s customers

???

4Warwick Business School

Vehicle routing problem

PostCode 15A15

PostCode 14A14

PostCode 13A13

PostCode 12A12

PostCode 11A11

PostCode 10A10

PostCode 9A9

PostCode 8A8

PostCode 7A7

PostCode 6A6

PostCode 5A5

PostCode 4A4

PostCode 3A3

PostCode 2A2

PostCode 1A1Given a set of customers

Find a tour with the minimal total length

PostCode 15A15

PostCode 14A14

PostCode 13A13

PostCode 12A12

PostCode 11A11

PostCode 10A10

PostCode 9A9

PostCode 8A8

PostCode 7A7

PostCode 6A6

PostCode 5A5

PostCode 4A4

PostCode 3A3

PostCode 2A2

PostCode 1A1Given a set of today’s customers

???

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The travelling salesman problem (TSP)

city3 city2 city5

city1

city6city4

Find a cyclic permutation (tour) that minimizes

n

i

iic1

))(,(

An optimal TSP tour < 1, 2, 3,…, n, 1> is called the master tour, if it is an optimal tour and it remains to be an optimal after deleting any subset of cities.

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cij cik

clk

(cmn )= clj

TSP with the master tour

j

i

k

l

+

clkcij + clj + cik

cjlcik + cil + cjk

A matrix is called Kalmanson matrix if for all i<j<k<l the inequalities below are satisfied.

7Warwick Business School

cij cik

clk

(cmn )= clj

Specially structured matrices

j

i

k

l

+If (cmn ) is a Kalmanson matrix, then

<1,2,…,n> is an optimal TSP tour

<1,2,…,n> is the master tour for the TSP with (cmn )

then (cmn ) is a Kalmanson matrix

Recognition of specially structured matrices

dij dik

dlk(dmn )= dlj

Is there a permutation to permute rows and columns in the matrix so that the new permuted matrix (cmn) with cmn= d(m)(n) is a Kalmanson matrix?

cij cik

clk

(cmn )= d(m)(n) = clj

X

1 2 3 4 5 6

1 1213 10 15 17

2 12 1 2 3 5

3 13 1 3 4 6

4 10 2 3 5 7

5 15 3 4 5 8

6 17 5 6 7 8

10

2

51

3+ −

4835155

4613133

8657176

3152122

5372104

15131712101

536241

K

Recognition of specially structured matrices

dij dik

dlk(dmn )= dlj

cij cik

clk(cmn )= clj

Permuted Kalmanson matrices can be recognized in O(n2) time

+

Site 1

Site 2

Site 4

Site 3

Related problems: Quadratic Assignment Problem (QAP)

Site 1

Site 2

Site 4

Site 3

Site 1 Site 2 Site 3 Site 4Site 1 0 d(1,2) d(1,3) d(1,4)Site 2 d(1,2) 0 d(2,3) d(2,4)Site 3 d(1,3) d(2,3) 0 d(3,4)Site 4 d(1,4) d(2,4) d(3,4) 0

Distance matrix d(i,j)=

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Office1

Office2

Office3

Office4

Site1

Site2

Site3

Site4

1 23

Site 1 Site 2 Site 3 Site 4Site 1 0 d(1,2) d(1,3) d(1,4)Site 2 d(1,2) 0 d(2,3) d(2,4)Site 3 d(1,3) d(2,3) 0 d(3,4)Site 4 d(1,4) d(2,4) d(3,4) 0

Distance matrix d(i,j)=

Office1 Office2 Office3 Office4Office1 0 c(1,2) c(1,3) c(1,4)Office2 c(1,2) 0 c(2,3) c(2,4)Office3 c(1,3) c(2,3) 0 c(3,4)Office4 c(1,4) c(2,4) c(3,4) 0

Frequency of contacts c(i,j)=

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),())(),((,

jidjicji

Quadratic Assignment Problem

Find a permutation that minimizes

total distance traveled in the allocation problem above

),())(),((,

jidjicji

Quadratic Assignment Problem (QAP)

Find a permutation that minimizes

NP-hard little hopes to find a polynomial algorithm

The hardest solved instances n=22 (30?)

Heuristics (approximate algorithms)

Identify solvable cases

15

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Site 1 Site 2 Site 3 Site 4Site 1 0 d(1,2) d(1,3) d(1,4)Site 2 d(1,2) 0 d(2,3) d(2,4)Site 3 d(1,3) d(2,3) 0 d(3,4)Site 4 d(1,4) d(2,4) d(3,4) 0

Distance matrix d(i,j)=

Office1 Office2 Office3 Office4Office1 0 1 5 1Office2 1 0 1 5Office3 5 1 0 1Office4 1 5 1 0

Frequency of contacts c(i,j)=

Quadratic assignment problem: Solvable case

If the distance matrix d(i,j) is a Kalmanson matrix, and the frequencies c(i,j) are proportional to the distances along a circle, then the identity is an optimal permutation for the QAP

Related specially structured matrices

cij cik

clk(cmn )= clj

+

Kalmanson matrices

K*= <1,2,…,n> is an optimal TSP tour

cij cik

clk(cmn )= clj

Demidenko matrices * is a pyramidal tour O(n2) time

(cmn )=+

Relaxed Kalmanson matrices

S* is in a special set of N-permutations

O(n4) time

1 1

n

2 34 5

6 78

11

Specially Structured Matrices & Heuristics

n-1

k+1

k-1

j+1

1 1

n

j-1

k

j

1

2

3

4

5

6

1 2 3 4 5 6

x

x

xx

x

? ? ????

x

x

xx

Know how to solve the TSP with the matrices like

1' 2' 3' 4' 5' 6'

x

x

xx

x

+ + ++++

1'

2'

3'

4'

5'

6'

x

x

xx

1

2

1 2

x

x

xx

x

+ + +???

3' 4' 5' 6'

3'

4'

5'

6'

can be transformed to

x

x

xx

n-1

k+1

k-1

j+1

1 1

n

j-1

k

j

n-1

k+1

k-1

j+1

1 1

n

j-1

k

j

n-1

k+1

k-1

j+1

1 1

n

j-1

k

j

n-1

k+1

k-1

j+1

1 1

n

j-1

k

j

localsearch?

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Warwick Business School

Summary

• Master tour exists only for the TSP with Kalmanson matrices

• If distances are calculated along the unique paths in a tree, then the corresponding matrix is the Kalmanson matrix

•Kalmanson matrices ( the master tour case) can be recognised in O(n2) time