Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai...
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Transcript of Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai...
![Page 1: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/1.jpg)
Traveling Salesman Problem
IEOR 4405 Production SchedulingProfessor Stein
Sally Kim James Tsai
April 30, 2009
![Page 2: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/2.jpg)
TSP Defined
Given a list of cities and their pairwise distances, find the shortest tour that visits each city exactly once
Well-known NP-hard combinatorial optimization problem
Used to model planning, logistics, and even genome sequencing
![Page 3: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/3.jpg)
Project Objectives
Perform a literature search of the TSP
Find interesting, real-life applications
Discover algorithms uncovering optimal solutions
![Page 4: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/4.jpg)
Fuzzy Multi-objective LP Approach
“Fuzzy Multi-objective Linear Programming Approach for Traveling Salesman Problem” (Rehmat, Amna; 2007)
Ideal solution would solve every TSP to optimality
Proven not only to be difficult, but also unrealistic
Impossible to have all constraints and resources in exact form – always vagueness
“Fuzzy Logic”: vague or imprecise data off which decisions are made
![Page 5: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/5.jpg)
Multi-objective LP
Takes a general linear multiple criteria decision making model and represents it as follows:
Find a vector xT = [x1, x2, … ,xn] which maximizes k objective functions, with n variables and m constraints
Opt Z = CX
s.t. AX <= b
Z = (z1, z2,…,zn) is the vector of objectives, C is a K x N matrix of constants and X is an Nx1 vector of decision variables, A is an M x N matrix of constants and b is a Mx1 vector of constants
![Page 6: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/6.jpg)
Fuzzy Multi-objective LP Approach
Modify the multi-objective LP formulation to:
Max Cx >=~Z0
s.t. AX<=~b
Where Z0=(z10,z2
0,…zn0) are aspiration levels and
>=~ are fuzzy inequalities
Consider a case of TSP with 3 objectives: minimize cost, time, and overall distance
![Page 7: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/7.jpg)
Ant Colony Optimization
“An interactive simulation and analysis software for solving TSP using Ant Colony Optimization algorithms” (Ugur, Aybars; 2008)
ACO is a population based probabilistic technique for solving NP-hard combinatorial problems
![Page 8: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/8.jpg)
Ant Colony Optimization
Simulation and analysis software are developed for solving TSP using ACO algorithm
Web-based tool employing virtual ants and interactive graphics to produce near-optimal solutions to the TSP
Artificial ants build solutions and exchange them with others via a communication scheme
![Page 9: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/9.jpg)
Ant Colony Optimization
ConstructSolutions: each ant starts at a particular state, then traverses the states one by one
ApplyLocalSearch: before updating the ant’s trail, a local search can be applied on each solution constructed
UpdateTrails: after the solutions are constructed and calculated, pheromone levels increase and decrease on paths according to favorability
![Page 10: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/10.jpg)
Ant Colony Optimization
Simulator TSPAntSim provides analysis of algorithms textually and graphically
Best tour-so-far represents the best found thus far
Tour best represents the best any tour length after
Standard deviation illustrates the evolution of the standard deviation of populations’ tour length
![Page 11: Traveling Salesman Problem IEOR 4405 Production Scheduling Professor Stein Sally Kim James Tsai April 30, 2009.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649f505503460f94c72ee2/html5/thumbnails/11.jpg)
Conclusions
While finding the exact solution is often desired in problems of optimality, this is sometimes not realistic
Relaxation and modification are some ways to approach a NP-hard problem that is otherwise difficult to solve