Maximizing Happiness using Ant Colonies

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Maximizing Happiness using Ant Colonies

• By Kiril Matev and John Galletly
• American University in Bulgaria

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Outline

• Problem Statement
• Problem Formalization
• Algorithm Overview
• Local Search Methods
• Test Specification and Results
• Summary

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Happiness and Conflict

• Max( Happiness ) ? Min( Conflict )
• Conflict is due to mismatches between
  characteristics of roommates and neighbors

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Representing Students

• A student is modeled as a point in N-dimensional
  space
• SleepHours (0-Early,1-Late)
• SmokingHabits (0-Smoker,1-Non-smoker)
• StudyHours (0-Early,1-Late)

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Quantifying Conflict

• The degree of conflict is depends on the level of
  mismatch of personal characteristics
• Vasil a (Early,Smoker,Early) lt0,0,0gt
• Ivan b (Late,Smoker,Late) lt1,0,1gt
• Conflict( a, b) a0-b0 a1-b1 a2-b2
  0 1 0 0 0 1 1 1 2

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Quantifying Conflict

• Some personal characteristics generate more
  conflict than others Smoking vs. StudyHours
• We need a way to accurately model these different
  levels of importance of personal characteristics

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Modeling Importance Levels

• Final grade in a course the weighted sum of
  assignments and exams
• Grade 0.4final 0.3midterm 0.3project
• Conflict the weighted sum of differences
  between characteristics
• Conflict( a, b) w0Smoking w1SleepHabit
• Conflict( a, b) 0.6 a0-b0 0.4 a1-b1

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Problem Statement

• Minimize the amount of personality conflict in a
  building where the level of conflict is measured
  by the weighted sum of differences among the
  characteristics of people living in the same or
  adjacent rooms

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Problem Specification

• Input
• A number of rooms with total capacity L
• lt101,102,..201,202,..301,302..gt
• N Students
• (Vasil lt0,2,1,2gt)
• A solution consists of an assignment of students
  to rooms
• Solution ltVasil, Ivan, Petar, Georgi..gt

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A Room Map Visualized

• The neighbors of rooms 1, 5 and 8 are shown.
• Double lines around 1,2 and 8,9 denote the fact
  the two rooms are in a single apartment

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Evaluating Solution Quality

• Solution1 ltVasil, Ivan, Petar, Georgi..gt
• Solution2 lt Petar, Ivan, Vasil, Georgi..gt
• TotalConflict 0
• Foreach room in building
• TotalConflict Room Conflict
• TotalConflict Neighbor Conflict
• End

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Problem Statement Diagram
9 Rooms with capacity 1, four grouped into
apartments 9 Students, with a one characteristic
on a scale 1,2,3
Neighborhood Structure
Optimal Assignment
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Recap

• Problem Statement
• Problem Formalization
• Algorithm Overview
• Local Search Methods
• Test Specification and Results
• Summary

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Example
Room1
Room2
Room3
Vasil
Vasil
Vasil
Ivan
Ivan
Ivan
Georgi
Georgi
Georgi

• Solution1 ltVasil, Ivan, Georgigt (red)
• Solution2 ltIvan, Georgi, Vasilgt (blue)
• A solution is a path from the first to the last
  column, without repetitions of names (one student
  one room)

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Problem Formalization

• We model the problem as a graph G (V,E)
• Vertices for every living space-student
  combination
• Edges connect adjacent living spaces (pheromone)

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Problem Formalization

• Problem can be formalized as looking for the
  least-cost path through a graph
• G (V,E) with a vertex for every living
  space-preference combination
• The cost of going from one node to another is
  equal to the amount of conflict between the two
  students (roommates or neighbors)

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How to Extend a Solution

• Choose the student causing least amount of
  conflict in the room and the neighborhood
• Conflict( Vasil, Ivan) 0.5
• Conflict( Vasil, Georgi) 0.3
• Selects Min(Conflict) Georgi

Room1
Room2
Vasil
Vasil
Ivan
Ivan
Georgi
Georgi
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Order of Graph Traversal

• Assume the optimal solution has Vasil and Ivan as
  roommates
• Random order
• Room1_1 ? Room2_1 ? Room1_2
• Vasil, Empty, Empty ? Vasil, Empty, Ivan
• Does not make use of all the previously made
  choices in the neighborhood

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Order of Graph Traversal

• Order going through adjacent rooms in sequence
• Room1_1 ? Room1_2 ? Room2_1
• Vasil, Empty, Empty ? Vasil, Ivan, Empty
• Makes optimal use of available knowledge about
  choices made in the neighborhood

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Why is Greedy not Sufficient?
Room101
Room102
Room103
Room203
Room202
Room201

• Assume a single dimension a ? 0,1
• The greedy solution would have extreme values at
  101 and 201
• lt 0, 0.1, 0.2, 0.4, 0.7, 1gt
• The optimal solution
• Floor 2 lt0, 0.2, 0.7gt
• Floor 1 lt0.1, 0.4, 1gt
• lt0.1, 0.4, 1, 0, 0.2, 0.7gt

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Local Search

• Look for a better solution by changing a small
  portion of the current solution
• In the context of the TSP

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Recap

• Problem Statement
• Problem Formalization
• Algorithm Overview
• Local Search Methods
• Test Specification and Results
• Summary

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Ant Colony Meta-heuristic

• Procedure AntMetaHeuristic
• Set parameters, initial pheromone levels
• While (termination condition not met) do
• Construct Ants Solutions
• Apply Local Search (optional)
• Update Pheromones
• End
• End

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Ant Colony System

• An extension of the original Ant System by Dorigo
  and Gambardella (1997)
• Was shown to be competitive with state-of-the-art
  algorithms in solving the TSP
• Ants consider both static (heuristic) and dynamic
  (pheromone level) factors

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Local Search Methods

• Information available to the heuristic is
  incomplete
• Local Search Methods
• Room local search exchanges all the occupants
  of neighboring rooms
• Solution1 ltVasil, Ivan, Petar, Georgi..gt
• Solution11 ltPetar, Georgi, Vasil, Ivan..gt
• Neighbor local search exchanges only one of the
  occupants of neighboring rooms
• Solution1 ltVasil, Ivan, Petar, Georgi..gt
• Solution11 ltVasil, Petar, Ivan, Georgi..gt

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Test Instances

• Synthetic problem sets used
• 100 Students, 64 double rooms 128 beds
• 200 Students, 123 double rooms 246 beds
• 300 Students, 169 double rooms 338 beds

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Test Specifications

• For each test, 20 runs, 1000 iterations each,
  with the total interference value at each
  iteration averaged over all the runs
• The algorithm was run for 10,000 iterations to
  find best-known solutions to track convergence

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Test Results
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Future Work

• Graph storage requirement
• Graph Edges S2 L
• Graph Edges Memory (500 Students, 500 living
  spaces) 5003 125,000,000 float values 4
  bytes/float 476 MB
• Decreasing the storage requirement certain
  groups of students which are close in space can
  be interchanged at negligible cost

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Summary

• Formalization of a real-world problem
• Ant Colony System coupled with a dynamic
  heuristic
• Local Search Methods
• Applications

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References

• Gambardella L. Dorigo M. Ant colony system A
  cooperative learning approach to the traveling
  salesman problem, 1997.
• Carbonaro A. Maniezzo V. An ANTS heuristic for
  the frequency assignment problem. Future
  Generation Computer Systems, 16(8)927935, 2000
• E. Bonabeau, M. Dorigo, G. Theraulaz. Swarm
  Intelligence From Natural to Artificial Systems,
  1999
• M. Dorigo and L. Gambardella. Ant colony system
  A cooperative learning approach to the traveling
  salesman problem, 1997.
• M. Dorigo and T. Stützle. Ant Colony
  Optimization, MIT Press, 2004.

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Maximizing Happiness

• Questions and Comments