Edward Kent Jason Atkin Rong Qi 1

Contents Vehicle Routing Problem VRP in Forestry Commissioning Loading Bay Constraints Ant Colony Optimisation Handing Loading Bays Lower Bound Calculation Results Conclusions 2

Vehicle Routing Problem Graph of Points (Customers, with one point being the Depot) Vehicles Originate at the Depot Customers have a demand Vehicles have a capacity Objective: Fulfil all customer demand in the least cost

Vehicle Routing Problem Variants Time windows Heterogeneous Fleets Limited Route Distance Multiple Depots Methods of Solving Exact algorithms (Column generation, relaxations based on matchings and trees etc) Heuristics (SA, GA, Savings Algorithms, Tabu Search.. ) Aims: Save Fuel, Money, Time 4

VRP In Forestry commissioning Logs are cut using large chain-saw like machines into different sizes at different forests Different forests have different log cuts as well as different species of wood An ordering system is used where sawmills/power plants or pulp plants order specific cuts and species from particular forests. (consignments) Aim & Objective : Get the wood from the forests to the sawmills in the least amount of cost 5

VRP In Forestry commissioning Consignment: An a priori pairing of a forest to a sawmill Represents a sawmill’s order of a truck load of wood from a forest Trucks drive from the forest directly to the sawmill to deliver wood Time windows (Forest and sawmill) Re model problem into a Graph of Consignments (matrix given from distances between consignments) Asymmetric, non-euclidean, triangle rule does not apply 6

Loading Bay Constraints Many vehicles can arrive at a forest/sawmill at once Only a limited number of vehicles are allowed to be serviced simultaneously These constraints are non-linear Arrival time vehicle 1 > departure time of vehicle 2 OR Arrival Time vehicle 2 > departure time of vehicle 1 7

Ant Colony Optimisation A population based search that’s robust and versatile Example for solving TSP problems Set of cooperating agents “ants” One ant, one solution Ants build solution with decisions made based on Length of the arc between the cities Amount of “pheromone” on the arcs Visited before by the Ant Ants leave “pheromone” on the arcs – good solutions leave more pheromone than bad ones Pheromone evaporated at a rate of rho 8

Ant Colony Optimisation Over time, Pheromone on arcs strengthen on good solutions 9

Ant Colony Optimisation How to apply ACO to the Forestry Commissioning routing problem Consignments = Cities Ant is synonymous to a Vehicle Ants keep track of their own time Ant group represents a solution Consignments that will violate a constraint are invisible Return to Depot when no more consignments are available 10

Handling Loading Bays Handle Loading bays in 3 ways: Repair loading bay constraint violations Solution is created, ignoring loading bays Attempt re-arranging solution to abolish violations Avoid consignments that have no loading bays available Make consignments that have no loading bays invisible Ants will only go to “free” consignments Make vehicles wait for loading bays to be free Assign a penalty multiplier on the waiting time and driving time 11

Handling Loading Bays 12 Expected: (1) – Waiting/Queuing (2) – Avoid Queuing Calculating loading bay spread Cluster LB usage together Ratio of total usage and total length of cluster Determine how “busy” loading bays are.

Lower Bound Calculation a-TSP Lower Bound Relaxation Column generation on sub-tour elimination constraints CPLEX runs out of memory “Weak” lower bound Results in large optimality gap 13

Results 14

Results Consignments: 300 – 500 Vehicles constant at 40 per dataset (100 in the real world sets) Weak lower bounds means large gap 15

Results 2 out of 6 datasets could not be scheduled with the “avoid queuing” method. Time windows could not be met Consignments left too late Ignoring constraints gives a better objective Loading bays relaxed, less constrained No waiting recorded and no diversions Avoid queuing method causes bad objective values 16

Results Mann-Whitney U Test Results Objective: 5/6 datasets had better objectives when setting W2 to 2 vs setting W2 to 0. 4/6 datasets had better objectives when setting W2 to 1 vs setting W2 to 0. Similar results were found (objectives worse) when setting W1 to 2 ( making waiting time less penalised than driving time) Objectives are better when penalising waiting time with 1 or 2 Loading bay spread: Avoiding queuing produced more manageable solutions but worse objectives 17

Conclusion The forestry Commissioning routing problem was explained Re-formulated into a VRP model with time windows and loading bay constraints Ant colony optimisation heuristic was used Possible adaptations were explained, avoiding queuing or penalising waiting time in the ant’s visibility Results show best objectives from using the penalising waiting times method Set waiting time penalty to 1 or 2 18