Ant Colony Optimization Chapter 5 Ant Colony Optimization for NP- Hard Problems Ben Sauskojus

NP-Hard Problem Types Routing Problems Routing Problems Assignment Problems Assignment Problems Scheduling Problems Scheduling Problems

Routing Problems Agents visiting locations Agents visiting locations Objective depends on order locations are visited Objective depends on order locations are visited

Routing Problems Sequential Ordering Problem (SOP) Sequential Ordering Problem (SOP) –Generalized asymmetric TSP –Has precedence constraints –Application: ACS based. Top performers use local search (3-opt)

Routing Problems Vehicle Routing Problem (VRP) Vehicle Routing Problem (VRP) – evolutions.com/dnaappletsample.html evolutions.com/dnaappletsample.htmlhttp:// evolutions.com/dnaappletsample.html –Capacitated (CVRP) »Each customer needs a specific amount of goods

Vehicle Routing Problem Objectives Objectives –Each customer is served by one vehicle –Vehicles start and end at Depot –Vehicles cannot deliver more than overall capacity Subproblems Subproblems –TSP –Bin packing problem

Vehicle Routing Problem Application: AS-rank based. Application: AS-rank based.

Vehicle Routing Problem Time Window (VRPTW) Time Window (VRPTW) –Each customer has a time window in which they must be served Objectives Objectives –Minimize the number of vehicles (routes) –Minimize travel time Application Application –Multiple ACS (Two layered colonies)

Assignment Problems Assign a set of items to resources Assign a set of items to resources Two assignment Types Two assignment Types –Assignment order –Assignment to specific resources

Quadratic Assignment (QAP) Assigning facilities to locations Assigning facilities to locations Objectives Objectives –Minimize the sum of the products between flows and distances

Quadratic Assignment (QAP) Example Example –Facilities are ‘Bathrooms’ –‘Main work Area’ –‘Parking Lot’

General Assignment (GAP) Tasks are assigned to Agents Tasks are assigned to Agents Each Agent has limited capacity Each Agent has limited capacity Each Task consumes some of an Agent’s capacity Each Task consumes some of an Agent’s capacity Assigning tasks incurs a cost Assigning tasks incurs a cost Objectives Objectives –Find a feasible task assignment of minimum cost

General Assignment (GAP) Application: MMAS-based Application: MMAS-based –Only one ant –Only feasible solutions get pheromone –Pheromone has nothing to do with solution quality

General Assignment (GAP)

Scheduling Problems Allocating scarce resources to tasks over time Allocating scarce resources to tasks over time Definition: An operation is a job that has to be processed on more than one machine. Example: building a car Definition: An operation is a job that has to be processed on more than one machine. Example: building a car Note: Processing time are fixed and job cannot be interrupted Note: Processing time are fixed and job cannot be interrupted

Single-Machine Total Weighted Tardiness (SMTWTP) Jobs have to be processed sequentially on a single machine Jobs have to be processed sequentially on a single machine Each job has: Each job has: –Processing time –Weight –Due date

Single-Machine Total Weighted Tardiness (SMTWTP)

Pheromone trails refer to the desirability of scheduling a job to the i-th position Pheromone trails refer to the desirability of scheduling a job to the i-th position Application: ACS based and is one of the best algorithms for the problem Application: ACS based and is one of the best algorithms for the problem

Job Shop, Open Shop, Group Shop Given: Given: –A set of Operations –A set of Machines that can only do specific operations –A set of Jobs which consist of operations Each operation has a processing time Each operation has a processing time

Job Shop, Open Shop, Group Shop Job Shop (JSP) Job Shop (JSP) –Precedence constraints which induce a total ordering –Example: Robbing a bank Open Shop (OSP) Open Shop (OSP) –No precedence constraints –Example: Employee scheduling,Cleaning house

Job Shop, Open Shop, Group Shop Group Shop (GSP) Group Shop (GSP) –Operations are arranged in group. –Groups must be completed in some order –Operations inside groups can be done in any order –Example: Commercial Cleaning

Job Shop, Open Shop, Group Shop Objectives Objectives –Minimize the completion time of the last task (Makespan) Applications Applications –AS based used for JSP (performs poorly) –AntQ based for OSP (performs poorly) –MMAS based for GSP (performs well)

Job Shop, Open Shop, Group Shop Pheromones Pheromones –JSP and OSP pheromones refer to the desirability of scheduling operation j directly after i –GSP pheromones refer to the desirability of scheduling operation j sometime after i

Resource-Constrained Project Scheduling (RCPSP) Given Given –Activities with precedence constraints, processing times, and resource requirements –Non-reusable resources Objectives Objectives –Assign to each activity a start time minimizing makespan

Resource-Constrained Project Scheduling (RCPSP)