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The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland By Professor H P Williams,London School of Economics Dr Martin Butler, University College Dublin Appears in: M. Butler, H Paul Williams & l-A Yarrow Computational Optimization and Applications, 7(1997) 291-306

The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland By

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The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland By Professor H P Williams,London School of Economics Dr Martin Butler, University College Dublin. Appears in:M. Butler, H Paul Williams & l-A Yarrow - PowerPoint PPT Presentation

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Page 1: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

The 2 Period Travelling Salesman Problem

Applied to Milk Collection in Ireland

By

Professor H P Williams,London

School of EconomicsDr Martin Butler,

University College Dublin

Appears in: M. Butler, H Paul Williams & l-A Yarrow

Computational Optimization and Applications,

7(1997) 291-306

Page 2: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Ireland

Farmer Catchment Area

Page 3: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Location of Suppliers

Page 4: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

A MILK DISTRIBUTION PROBLEM

• Milk is to be collected from 41 farms using a vehicle based at a central depot.

• For 12 of the farms there is to be a daily collection.

• For the other 29 farms collection is to be every other day.

• 1. Decide which farms are to be visited on which days

• 2. Route the vehicle on these days.

• The objective is to minimise total distance travelled

Page 5: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

Page 6: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Extension of the Symmetric Travelling Salesman Problem

Number of Nodes visited every other day 1 29n

Number of Nodes visited every day 2 13n

1 2 42n n n

1 – Period Problem Number of Tours 49( 1)!1.67 10

2

nx

2 – Period Problem Number of Pairs of Tours

1

1 2 2 1

0 1

! ( 1)! (n ( ) 1)!1x x

2 ! ! 2 2

n

r

n n r n r

r n r

624.27 x10

22 2 1

2

(( 1)!) (2 1)!

8(2 1)!

n n n

n

Page 7: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

A HEURISTIC SOLUTION PROCEDURE(gives upper bound for total distance)

• Create a tour around “every day” farms (e.g. nearest neighbour heuristic)

• Apply an improvement heuristic (e.g. 2-interchange method)

• Duplicate tours

• Insert “every other day” farms into tours by an insertion heuristic (e.g. cheapest insertion method)

Page 8: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node ProblemNearest Neighbour

Solution for Every Day Farms

Page 9: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

Nearest Neighbour

+2-Interchange for Every Day Farms

(Length = 687

Optimal)

Page 10: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

Heuristic Solution

Length 1750

Page 11: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

OTHER HEURISTIC SOLUTION PROCEDURES

(give upper bounds for total distance)

1. (i) Partition Farms into 2 sets (e.g. Clustering, Distance from Depot etc)

(ii) Apply 1-Period TSP heuristic to each cluster +depot

2. (i) Grow 2 Spanning Trees from Depot. Each “every day” farm included in both trees

Each “every other day” farm included in one tree

(ii) Match odd degree nodes in each tree

(iii) Create Eulerian tours

(iv) “Short circuit” farms visited twice

Page 12: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

INTEGER PROGRAMMING FORMULATION

1 iff i,j on tour ijkx k, 1, 2,..., , , 1, 2i j n i j k

21 iff i visited on tour , 1,...,iky k i n n

. .ij ijk

i j ki j

Minimise c x

1 2 21, 1,..,i isubject to y y i n n every other day visits

22, 1,..., , 1, 2ijk jikj i j i

x x i n k

both day visits

2 22 0, 1,..., , 1, 2ijk jik ikj i j i

x x y i n n k

every other day visits

This is the 2-Matching Relaxation

2

29, 13, 42, 2 = 1131 2 1

( -1) 2 1780 (1

, 1, , = 0 1 22

n n n n n

n n n Variables

y yn n

Constraints and

To avoid symmetric alternate

solutions we set leading to a 112 x 1778 size model)

To create a ''correct'' model we wo -1 2 -1 2,199,023,255,551

n uld need to append

subtour elimination constraints, which is clearly impractical.

Page 13: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

SOLUTION APPROACH

1. Solve Linear Programming Relaxation of 2-Matching Relaxation. (2 secs, 993 iterations)

(Could then apply the Branch and Bound alogrithm to try to obtain optimal integer solution. This

(i) takes a prohibitive amount of time – weeks – to solve

(ii) only produces subtours)

2. Identify violated “VUB” Constraints (61)

3. Identify violated single day subtours (14)

4. Identify violated single day combs (1)

5. Identify violated aggregated combs (1)

6. Append these constraints (and those for other day)

Resolve LP Relaxation (2 secs, 536 iterations from starting basis)

Append violated VUB constraints (63)

Resolve LP Relaxation (13 secs, 2365 iterations)

7. Repeat 3,4,5

Page 14: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

LP Relaxation of 2 – Matching Relaxation

Length =1570

½

½

½

½

½

½

½

½

½

½

½ ½

½

½

½

½

½

½

½

½

½

½

½

½

½

½

Page 15: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

VARIABLE UPPER BOUND CONSTRAINTS(CUTS)

2 0ijk ijk ikj i j i

x x y

0ijk ikx y 2

2

1,....,

1,.... ,

i n n

j n n i j

0ijk jkx y 2

1,....,

1,....,

i n

j n n

Second pair of constraints may be violated by fractional solutions which satisfy first constraint.

Would need to append 1 1 1 2( 1) 1189n n n n

such constraints.

Will only append if violated.

Page 16: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

LP Relaxation after violated VUB constraints appended

Length = 1696.5

½

½ ½

½

½

½ ½

½

½ ½

½

½ ½

½

½

½

½ ½

½

Page 17: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

+

+

+

*

**

SUBTOURS

COMBS

+ +

++

+

+

**

*

*

*

**

1 2* , V V

Page 18: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

SUBTOUR CONSTRAINTS

2V

1

2( )

1kek v

e E S v S V

x S V y

all

2

depot, 2

, 1,2

S V S

S V k

1( )

1 all S , 2

S,k 1,2

ek vk ke E S v S V

x y y

V S

Where is set of farms visited every other day1V

is set of farms visited every day

1 2V V V

( )E S Is set of edges with both ends in S

Page 19: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

SUBTOURS, COMBS AND AGGREGATED COMBS

1

2

A violated subtour constraint

*

*

+1

134,2

32

2

4,1

y

y

1

110,1

12

2

0,1

y

y

34

10

39

10,34,2 10,39,2 34,39,2 34,2 0x x x y

A violated aggregated comb constraint

* *

* *

+ +

+

5,6,1 5,8,1 6,8,1 5,6,2 5,8,2 6,8,2

5,25,1 5,25,2

8,28,1 8,28,2

6,26,1 6,27,1 6,26,2 6,27,2 26,27,2

8

x x x x x x

x x

x x

x x x x x

(handle)

(tooth 1)

(tooth 2)

(tooth 3)

1

2

1

2

1

21

2

1

2 1

21

2

1

2

1

2

1

2

1

21

2

1

2

2.82.5 5 8

6

26 27

Page 20: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

21

2

1

2

1

2

1

21

21

2

1

2

1

2

1

2

1

21

2

1

2

1

21

2

1

2

1

2

1

2

1

21

21

2

1

2

1

2

1

2

1

2

1

2

1

2

Length = 1711 1

2

Page 21: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

1

3

1

3

1

31

3

1

3 1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

31

3

1

3

1

3

1

3

1

3

1

3

1

3 1

3

1

3

1

3

1

3

1

3

1

31

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

2

1

2

1

2

1

2

1

2

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

32

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

2

3

1

6

1

6

1

6

3

6

5

6

LP Relaxation after VUBs

2nd set of subtours & Comb Constarints

Length 1720 5

6

Page 22: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Nodes 1-13 (+) Visited Both days Nodes 14-42 Visited Every Other Day

+ Every Day

* Second Day

42 Node Problem

Optimal Solution

Length 1725

Page 23: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

SOLUTION RESULTS

LP Relaxation 1570

(B & B takes weeks and produces subtours)

LP Relaxation after VUB constraints 1696.5

(B&B TAKES 40 MINUTES AND PRODUCES SUBTOURS)

LP Relaxation after VUB, Subtour and Comb 1711.5

Constraints

(B & B takes 7 minutes to produce subtours)

LP Relaxation after VUB and further Subtour 1720.83

Constraints

B & B takes 13 seconds to produce optimal solution 1725

Heuristic Solution 1750

A total of 63 VUB, 18 Subtour, 1Comb Constraint and 2 Aggregated Comb Constraints were used.

Resultant model has 149 Constraints (excluding (VUB’s), 1778 Variables and solves in a total of 26 seconds (On a 486 PC)

Objective (1/10 miles)

Page 24: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

Solution Approach

SOLVE LP RELAXATION

AUTOMATICALLY APPEND VIOLATED VUB CONSTRAINTS

APPEND VIOLATED CONSTRAINTS

APPEND VIOLATED CONSTRAINTS

SOLVE LP RELAXATION

DRAW GRAPH OF SOLUTION

IS SOLUTION INTEGER?

IDENTITY INEQUALITIES VIOLATED BY SOLUTION

VIOLATED INEQUALITIES IDENTIFIED

ARE THERE VIOLATED SUBTOURS?

STOP AN OPTIMAL 2-TOUR FOUND

APPLY BRANCH AND BOUND ALGORITHM

Yes

NoNo

Yes

No

Yes

Page 25: The 2 Period Travelling Salesman Problem Applied to Milk Collection in Ireland     By

FURTHER CONSIDERATIONS

1. Could identify further cuts (facets?) and avoid use of Branch and Bound.

2. . Can include capacity constraints limiting each day’s collection

(Constraints on i ik

y for each )k

3. Could generalise to more than 2 time periods.