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)

Ireland Farmer Catchment Area

Location of Suppliers

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

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem

Extension of the Symmetric Travelling Salesman Problem Number of Nodes visited every other day Number of Nodes visited every day 1 – Period Problem Number of Tours 2 – Period Problem Number of Pairs of Tours

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)

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem Nearest Neighbour Solution for Every Day Farms

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem Nearest Neighbour +2-Interchange for Every Day Farms (Length = 687 Optimal)

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem Heuristic Solution Length 1750

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

INTEGER PROGRAMMING FORMULATION every other day visits both day visits every other day visits This is the 2-Matching Relaxation

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

Nodes 1-13 (+) Visited Both days Nodes 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

VARIABLE UPPER BOUND CONSTRAINTS (CUTS) Second pair of constraints may be violated by fractional solutions which satisfy first constraint. Would need to append such constraints. Will only append if violated.

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem LP Relaxation after violated VUB constraints appended Length = ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

+ + + * * * SUBTOURS COMBS * * * * * * *

SUBTOUR CONSTRAINTS all Where is set of farms visited every other day is set of farms visited every day Is set of edges with both ends in

SUBTOURS, COMBS AND AGGREGATED COMBS A violated subtour constraint * * A violated aggregated comb constraint * * * * ++ + (handle) (tooth 1) (tooth 2) (tooth 3)

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem Length = 1711

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem LP Relaxation after VUBs 2 nd set of subtours & Comb Constarints Length 1720

Nodes 1-13 (+) Visited Both days Nodes Visited Every Other Day + Every Day * Second Day 42 Node Problem Optimal Solution Length 1725

SOLUTION RESULTS LP Relaxation 1570 (B & B takes weeks and produces subtours) LP Relaxation after VUB constraints (B&B TAKES 40 MINUTES AND PRODUCES SUBTOURS) LP Relaxation after VUB, Subtour and Comb Constraints (B & B takes 7 minutes to produce subtours) LP Relaxation after VUB and further Subtour 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)

Solution Approach SOLVE LP RELAXATION AUTOMATICALLY APPEND VIOLATED VUB 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 No Yes No Yes

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 for each 3. Could generalise to more than 2 time periods.