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A Comparative Analysis of Traveling Salesman Heuristic Implementations in GIS

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Given a set of cities, with m salesmen located at a single location - depot city ... consists of finding tours for all m salesmen, who all start and end at the depot, ... – PowerPoint PPT presentation

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Title: A Comparative Analysis of Traveling Salesman Heuristic Implementations in GIS


1
A Comparative Analysis of Traveling Salesman
Heuristic Implementations in GIS
  • GIS Masters Project - Summer 2006
  • Gabriela Voicu

2
Project Outline (I)
  • 1. Introduction
  • Define Traveling Salesman Problem (TSP)
  • Introduce Geographic Information Systems (GIS)
  • as a tool for solving the TSP
  • 2. Research Questions
  • 3. Literature Review
  • Who came up with the problem
  • Impact of technological progress
  • Mathematical Formulation of the TSP
  • TSP as a Combinatorial Optimization Problem
  • Various Solution Procedures for the TSP
  • Generalization of the problem and variations
  • TSP implementations within GIS
  • Lack of vendor specifications of methods used

3
Project Outline (II)
  • 4. Data Sources
  • Network dataset - street centerlines, City of
    Mesquite
  • School bus routing context
  • School attendance zones - Mesquite Independence
    School District (MISD)
  • Students information - that must be transported
    to the school
  • 5. Methodology
  • Implement two of the algorithms that have
    appeared in the literature review within ArcGIS
    using Visual Basic 6.0 Programming language
  • Perform tests across four different software
    packages ArcView 3.2 and ArcGIS 9.1with Network
    Analyst extension, ArcLogistics Route, GeoMedia
    with Transportation Manager extension and the two
    implemented algorithms
  • 6. Conclusions
  • All GIS implementations of the TSP studied here
    often generate sub-optimal solutions even for
    small number of instances (12)
  • Depending on the scope of the application, the
    current GIS implementations may prove not to be
    the appropriate approach

4
1. Introduction (I)
  • What is TSP?
  • The TSP is a classic tour problem in which a
    hypothetical salesman
  • must find the most efficient sequence of
    destinations in his territory,
  • stopping only once at each, and ending up at the
    initial starting location
  • What is the mathematical structure of TSP?
  • Graph where the cities are the nodes (vertices)
    of the graph
  • Find the cycle of minimum cost that visit each
    of the vertices
  • (S.S. Skiena 1997)

5
1. Introduction (II)
  • GIS as a tool for solving the problem
  • Integration of GIS with mathematical graph theory
  • Finding the most efficient route in which to
    visit locations in a network

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  • Methods used in GIS to solve the TSP are called
    heuristics
  • The term heuristic is used for algorithms which
    find solutions among all possible tours, but they
    do not guarantee that the best will be found

6
2. Research Questions
  • What are the heuristics for the TSP that have
    appeared in the literature?
  • What are the characteristics of these heuristics
    and the consequences of using them?
  • Do we know anything about the optimality of the
    solutions that they will provide?
  • Can we determine which (if any) of these
    heuristics have been implemented in GIS software?
  • How do these heuristics perform on a test
    database?

7
3. Literature Review
  • TSP origins
  • 1700s the Knights Tour problem
  • 1800s Sir William Rowan Hamilton (graph theory)
  • 1832 first published document that mentioned the
    TSP, a german manual
  • 1920s Karl Menger mentioned the TSP in Vienna
  • 1930s introduced in the United States at Harvard
    and Princeton Universities by Whitney Hassler (A.
    Schrijver 2004)
  • 1940s M Flood popularized the TSP at the RAND
    Corporation in California
  • In 1954, the most important progress on the TSP
    was made by Rand researchers Dantzig, Fulkerson
    and Johnson when they developed a new method for
    solving the TSP, the Cutting Plane method, which
    became a prototype in linear programming (Hoffman
    A. J. and Wolfe P. 1985)

8
Chart showing solutions of increasingly complex
scenarios throughout time
Milestones in the solution of the TSP (Cook
2002), (Hoffman K 1996)
9
Mathematical Formulation of TSP (S. Vajda 1961)
The objective function is to minimize the sum of
all distances of all of the selected elements of
the tour
n the number of cities to be
visited i, j and k indices of cities that can
take integer values from 1 to n t
the time step in the route between the
cities Xijt 1 if the edge of the
network from i to j is used in step t of the
route, and 0 otherwise dij
the distance or cost from city i to city j
The variables are subject to the following
constraints
When a city is reached at time t, it must be
left at time t 1
For all cities, there is just one another city
which is being reached from it, at some time
  • For all values of t, exactly one arc
  • must be traversed

For all cities, there is only one another
city from which it is being reached, at some
time
10
TSP as a Combinatorial Optimization Problem
  • If n is the number of cities then (n-1)! is the
    total number of possible routes
  • n5 120 possible routes
  • n10 3,628,800 possible rotes
  • n20 2,432,902,008,176,640,000 possible
    routes
  • TSP is a NP-Complete combinatorial optimization
    problem non-deterministic polynomial time
  • Scientific challenge for solving TSP
  • 1 million prize for solving the TSP as one
    representative problem of a larger class of
    NP-complete combinatorial optimization problems
    has been offered by Clay Mathematics Institute of
    Cambridge (CMI)

11
Various Solution Procedures for the TSP
  • Exact Solution Procedures Integer Programming
  • Exact approaches require algorithms that generate
    both a lower bound and an upper bound on the true
    minimum value of the problem instance
  • The standard technique for obtaining lower bounds
    on the TSP problem is to use a relaxation that is
    easier to solve than the original problem
    (Hoffman K. 1996)
  • Cutting Plane
  • Procedure used to find integer solutions of a
    linear program
  • Branch and Bound
  • Algorithmic technique to find the optimal
    solution by keeping the best
  • solution found so far
  • Simplex
  • Method combined with Branch and Bound to solve
    the problem by
  • systematically moving from one solution to
    another until the optimal is
  • found, unable to solve large instances

12
Approximate Solution Procedures
  • Constructive Heuristics
  • Algorithms that try to build up feasible
    solutions step by step and then stop when a
    solution is found and never try to improve upon
    its solution (Hoffman K 1996)
  • Nearest Neighbor
  • Greedy heuristic
  • Minimum Spanning Tree
  • Insertion heuristics
  • Improvement Heuristics
  • Algorithms that start with an initial feasible
    solution and successively improve it through a
    sequence of exchanges (Evans J. 1992)
  • 2-opt algorithm
  • 3-opt algorithm
  • Lin-Kernighan algorithm
  • Iterated Lin-Kernighan

13
Metaheuristics
  • Set of concepts that can be used to define
    heuristic methods and to guide already existing
    heuristics to find their way out of local optima
    by continuing the search for better areas of the
    solution space (Voudouris and Tsang 1999)
  • Neural Networks, Genetic Algorithms, Simulated
    Annealing, Ant Colony, etc
  • Local Search (Neighborhood Search)
  • Iterative search procedure that, starting from an
    initial feasible solution, progressively improves
    it by applying a series of local modifications or
    moves
  • At each iteration, the search moves to an
    improving feasible solution
  • The search terminates when it encounters a local
    optimum with respect to the transformations that
    it considers (Gendreau 2003)

14
Tabu Search
  • Essential feature (use of memory) tabu list
  • To improve the efficiency of the exploration
    process, Tabu Search keeps track not only of the
    best solution visited so far but also of
    information related to the itinerary
  • As soon as non-improving moves are possible, the
    risk of cycling appears and the scope of tabu
    list is to forbid moves which might lead to
    recently visited solutions
  • Best improvement local search is performed only
    amongst those not included in the tabu list
    (Voudouris and Tsang 1999)
  • tabus may sometimes prohibit attractive moves
  • add criteria that will allow to cancel tabus
    aspiration criteria
  • aspiration criteria consists in allowing a move,
    if it results in a solution with an objective
    value better than that of the current best-known
    solution (Gendreau 2003)

15
Generalization of TSP the multiple TSP
  • The multiple traveling salesman problem (mTSP) is
    a generalization of the TSP, with more than one
    salesman
  • Given a set of cities, with m salesmen located at
    a single location - depot city and intermediate
    cities as remaining cities that are to be
    visited, the mTSP consists of finding tours for
    all m salesmen, who all start and end at the
    depot, such way that each intermediate city is
    visited exactly once and the total cost of
    visiting all cities is minimized (Bektas 2006)
  • The problem of scheduling buses is treated as a
    variation of the multiple TSP with additional
    side constraints (Angel R. D. 1972)

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TSP Implementations within GIS
  • ArcGIS Network Analyst Tabu Search
  • Computes an origin-destination cost matrix
  • Applies an insertion algorithm to construct an
    initial solution
  • At each step, the insertion algorithm inserts the
    least-cost unvisited stop into the current
    partial solution
  • The initial solution is then improved upon by a
    tabu search process, where an existing solution
    is augmented by performing two-optimal and
    three-optimal moves (Rice 2006)
  • ArcLogistics Route Tabu Search
  • Constructs an initial route
  • Assigns stops to each route in the best sequence
  • Stops from one route are moved to another to try
    to find a best assignment of stops to routes

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4. Data Sources
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5. Methodology
21
Simplifying Assumptions U-Turn Allowed
  • ArcLogistics Route - no option or way of
    restricting U-Turns
  • ArcView 3.2 - turn table, via Avenue scripting
    language, but it does not seem to work on
    restricting a turn on the same edge
  • The Integraphs GeoMedia Transporter - allowing
    or restricting U-turns globally not consistent
  • ArcGIS with the Network Analyst 9.1 extension -
    option of allowing or restricting U-Turns
    globally but the algorithm implemented is written
    in such a way, that backtrack is allowed and no
    matter what option the user checks, U-Turns are
    still allowed in the network solution is
    CurbApproach

22
Enumerative Approach Implementation (I)
  • Construction of all (n-1)! possible solutions for
    the problem
  • Create a function that builds an array containing
    each possible route starting and ending at the
    initial location (school)
  • To construct all possible ordered combinations
    the method applied is to pass two arrays to a
    function that has 2 arguments
  • one array containing the initial location (1)
  • a second array containing all the stops (1,2,3,4)
  • the function calls itself recurrently, trying to
    increase the entry array size by one and adding
    to it a location that is not contained yet, in
    any particular tour
  • at the first step the function will return
    (1,2), (1,3), (1,4)
  • at the next step, the function will return the
    following (1,2,3), (1,2,4), (1,3,2), (1,3,4),
    (1,4,2), (1,4,3)
  • at the next step the returned array would contain
    all six possible combinations (1,2,3,4)
    (1,2,4,3) (1,3,2,4) (1,3,4,2) (1,4,3,2) and
    (1,4,2,3)
  • when the total number of locations in each tour
    equals the number of locations, the loop ends and
    the result contains all (n-1)!, possible
    combinations, each representing a valid tour

23
Enumerative Approach Implementation (II)
  • Iterate through all of the n2 of the distances to
    find the matching segment (line 12, 13, 14, etc)
    in each of the ordered stops array
  • When a match is found, the total distance for
    that particular tour is increased with the
    matching segment distance
  • After traversing the last tour, each of the
    arrays will contain a number of segments equal to
    the number of locations and their corresponding
    total distance
  • Loop through all of the results and choose the
    shortest tour as the optimal solution

24
Nearest Neighbor Algorithm Implementation
  • Uses the Network Analyst Closest Facility Solver
    to obtain all the distances between sets of stops
    (OD Matrix)
  • It uses each current location as origin and it
    finds the closest neighbor not traversed yet as
    destination
  • The location found as a destination not yet
    traversed becomes the next origin and it keeps
    checking all the stops till the last one, when it
    returns to the departing location
  • Any of the two approaches implemented return an
    array of lines defining a valid tour and the last
    step is to iterate through all the lines returned
    and draw only the ones existing in the solution

25
Enumerative (Brute Force) Implementation
Nearest Neighbor Implementation
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Comparative Analysis of Results
  • Optimal solutions were provided using Simplex
    software (with Optimization Programming Language)
    up to 26 stops
  • All GIS implementations of the problem start
    giving sub-optimal solutions when size 12 of the
    problem is being reached
  • ArcLogisticsRoute can be considered a special
    case since its using its own network dataset
  • Nearest Neighbor algorithm is scaled as the worse
    approach (18)
  • A meaningful comparison can be made between ESRI
    and Intergraph when Integraph proves its
    superiority in the solutions within 5

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Running Time
  • As the number of stops increases, the
    computational time increases in an extremely
    rapid manner (asymptotic increase)

32
6. Conclusions
  • This research concludes that GIS implementations
    of the TSP tested often generate suboptimal
    solutions
  • As soon as the problem reaches size 12, the
    studied GIS implementations start giving
    approximate solutions within 14 above the
    optimal solution
  • Depending on the scope of the application, the
    current GIS implementations may prove not to be
    the appropriate approach, especially when real
    world applications involve significant costs
  • In contrast, optimal solutions can be generated
    with desktop linear programming software (up to
    26)
  • This research suggests that advancements in the
    solution of the TSP could benefit from the
    implementation of GIS and optimal solution
    software

33
7. References (I)
  • Angel R. D., C. W. D., Noonan R. and Whinston A.
    (1972). "Computer assisted school bus
    scheduling." Management Science 18 279 - 288
  • Applegate D., B. R., Chvtal V., and Cook W.
    (1998). "On the Solution of Traveling Salesman
    Problems." Documenta Mathematica Extra Volume
    ICM(III) 645-656
  • Bektas, T. (2006). "The multiple traveling
    salesman problem an overview of formulations and
    solution procedures." Omega, Volume 34, Issue 3,
    June 2006, Pages 209-219 34(3) 209-219
  • Cook, W. J. (2001, apr 2005). "The Traveling
    Salesman Problem ", from http//www.tsp.gatech.edu
    /methods/talks/cook01/slide3.html
  • Cook, W. J. (2002, jan 2005). "Milestones in the
    Solution of TSP Instances." from
    http//www.tsp.gatech.edu/history/milestone.html
  • Dantzig G, F. D. R., Johnson S. M. (1954).
    "Solution of a Large Scale Traveling Salesman
    Problem." Journal of the Operations Research
    Society of America 2 393 - 410
  • Danztig G (1949). Application of the simplex
    method to a transportation problem. Acticity
    Analysis of Production and Allocation -
    Proceedings of a Conference, Chicago, Illinois
    Wiley - New York
  • Evans J., M. E. (1992). Optimization Algorithms
    for Networks and Graphs. New York, Basel, Hong
    Kong, Marcel Dekker, Inc.
  • Flood, M. M. (1956). "The traveling-salesman
    problem." Operations Research 4 61-75
  • Gendreau, M. (2003). An Introduction to Tabu
    Search. Handbook of Metaheuristics. G. A. K. Fred
    W Glover. Boston, Dordretcht, London, Springer

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7. References (II)
  • Glover, F. (1990). "Tabu Search A Tutorial."
    Interfaces 20 74-94
  • Hoffman A. J. and Wolfe P. (1985). History. The
    Traveling Salesman Problem - A Guided Tour of
    combinatorial Optimization. Chichester, John
    Wiley Sons 1-17
  • Hoffman K, P. M. (1996). "Traveling Salesman
    Problem." Encyclopedia of Operations Research
    76-83.
  • J. R. Evans, E. M. (1992). Optimization
    Algorithms for Networks and Graphs. New York,
    Basel, Hong Kong, Marcel Dekker, Inc.
  • Lawler, E. S. (2000). Combinatorial Optimization
    - Networks and Matroids, Dover Publications
  • Marinakis, Y., A. Migdalas, et al. (2005). "A
    Hybrid Genetic - GRASP Algorithm Using Lagrangean
    Relaxation for the Traveling Salesman Problem."
    Journal of Combinatorial Optimization 10(4)
    311-326
  • Schrijver, A. (2004). "On the history of
    combinatorial optimization." from
    http//homepages.cwi.nl/lex/files/histco.pdf
  • Skiena, S. S. (1997). "The Algorithm Design
    Manual." from http//www2.toki.or.id/book/AlgDesig
    nManual/BOOK/BOOK/BOOK.HTM
  • Vajda S. (1961). Mathematical Programming Addison
    - Wesley Company, Inc.
  • Voudouris, C. and E. Tsang (1999). "Guided Local
    Search and Its Application to the Traveling
    Salesman Problem." European Journal pf
    Operational Research 113(2) 469-499

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Lets assume that the optimal tour for n4 is
1gt3gt4gt2gt1
j1 j2 j3 j4
i1 i2 i3 i4
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