Multiobjective Clustering via Metaheuristic Optimization: An Application to Market Segmentation

1
Multiobjective Clustering via Metaheuristic
Optimization An Application to Market
Segmentation

• Rafael Caballero
• Manuel Laguna
• Rafael Martí
• Julián Molina

2
Clustering
i
aij
j
aij dissimilarity between i and j
3
Dissimilarity Matrix
Attributes for object i
Attributes for object j
ri1, ri2, , rin
rj1, rj2, , rjn
4
Multiobjective Problems

• Partitioning of objects using one partitioning
  criterion but multiple dissimilarity matrices
• Partitioning of objects using one dissimilarity
  matrix but more than one partitioning criteria

5
Objective Functions

• Partition diameter
• Unadjusted within-cluster dissimilarity
• Adjusted within-cluster dissimilarity
• Average within-cluster dissimilarity

6
Illustrative Example
7
Multiple Dissimilarity Matrices Market
Segmentation

• Data sources
• Performance drivers (descriptor variables)
• Performance measures (response variables)
• Find clusters that are homogenous with respect to
  the performance drivers and that at the same time
  help to explain the variation on the performance
  measures

8
Telecom Example (Brusco, Cradit and Stahl, 2002)

• Survey of 4400 business units
• Self-reported measures of firm technology and
  telecom activity
• LAN activity, desktop computing, remote access
  services, network transport activity, premise
  equipment inventory, telecommunication spending
  estimates
• Partitioning of market into homogenous clusters
  of business that were interested in network
  services and maintenance products

9
Clustering Procedure

• Variable selection (from 21 to 9)
• Remote data communication points, LAN segments,
  network size, trunks, toll-free-lines, high-speed
  digital lines, desktop computers, LAN nodes and
  network routers
• Application of bi-criterion clustering simulating
  annealing heuristic (SAH)
• Identification of homogenous high-spending
  segments

10
Mathematical Model
for l 1, , L
Maximize
where
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Scalar Objective Function
Minimize
The principal limitation of multiobjective
programming is the selection of an appropriate
weighting scheme. The severity of this problem
increases markedly when three or more criteria
are considered. (Brusco and Stahl, 2005)
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Multiobjective Metaheuristic Approach

• Scatter/Tabu search hybrid
• Approximation of Pareto front in a single run
• Use of compromise programming principles to guide
  the search
• Density approximation (distance to RefSet) used
  as stopping criteria

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Phase I Initial Tabu Searches
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Global Criterion Method (Compromise Programming)
Ideal Point
Approximated from
Anti-ideal Point
Use random weights until InitPhase searches fail
to produce a new efficient point
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Phase I Tabu Searches with a Global Criterion
2
x1
x2
5
x4
x6
1
7
6
3
4
x5
x3
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Phase II Scatter Search

• RefSet consists of
• Best single-objective solutions
• Diverse solutions (Max-min criterion using L?)
• An updated archive of efficient solutions is
  maintained throughout the search

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Phase II Improvement Method
Efficient frontier
xi
Ideal (xi , xj)
f2
Compromise point for (xi, xj)
xj
New trial solution
Search area
f1
wi 1 for compromise point
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Phase II RefSet Update

• Choose best p solutions according to the
  individual objective functions
• For each solution x ? \TabuRefSet calculate a
  normalized L? distance and find
• Create a list of eligible points
• Choose b-p eligible points (sequentially) to
  maximize

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Solution Representation
where xk (1 xk n) is the centroid of cluster
k
Then,
for k 1, , K
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Tabu Search Neighborhood

• Generate NSize (r, q) pairs such that q is not in
  x and r is between 1 and K
• Replace xr with q and evaluate the move with
  either one of the objective functions (initial
  p1 searches) or L? metric

21
Combination Method
Repair trial solution when the same object
appears twice
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Branch and Bound Results
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Centroid-based Solution Representation
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Experiments with Multiple Criteria
25
Experiments with Multiple Dissimilarity Matrices
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Application to Market Segmentation
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Conclusions

• Hybrid SS/TS approach seems to be an effective
  approach to multiobjective optimization problems
• Multiobjective problems in data analysis are a
  fertile application ground for metaheuristic
  optimization methods