Selecting unique designs from complex design
spaces: a graph enumeration approach
Abhishek K. Shrivastava and Yu Ding
Department of Industrial and Systems Engineering
Texas A&M University
July 12, 2006
1
Outline
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Motivating problem
Redundancy or isomorphism in design
Related work
Graph/group theoretic approach
Summary and future work
2
Motivating Problem
Fixtures
Cell Phone Arrives
load
unload
Cell Phone Departs
Tests run on phone using
computer controller
• Equipments for running tests
• Buses for fixture-equipment
communication
Buses
EQ1
…
1
2
3
4
1
2
…
1
2
3
4
5
6
EQn
EQ2
m
Equipments
3
Phone Testing System: the Test Rack
 Test Rack
1
3
4
Fixtures
 number of fixtures,
 number of buses,
 number of units of each
equipment type, and
 mapping between these
5
6
7
8
Buses
(Which
fixture uses which equipment unit? Which
bus is used for this?)
1 2
3
4 5
6 7 8
9
…
EQ1
EQ2
EQ3
EQ4
1
1
1
1
…
…
…
…
EQ5
EQ6
…
EQn
1
2
1
…
…
 For a given phone model tests
are fixed and test rack is to be
designed
2
1
…
Equipments
4
Design Objective
 Objective: Find the configuration of test rack that
maximizes the testing throughput (phones tested per unit
time)
 given the maximum number of fixtures, buses and units of each
equipment
 Objective function/throughput is evaluated using the
ARENA® Simulation Software
 Discrete Event Simulation
 Over 1500 testing tasks modeled as “process” in ARENA
 11 types of summary statistics recorded, e.g. cycle times, fixture
utilizations, etc.
 Simulation may take 2- 60 minutes for one design configuration
5
Size of A ‘Real’ Design Space
 Test rack parameters
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
number of fixtures = 8
number of buses = 6
types of equipments = 6, number of units = {8, 8, 8, 4, 4, 1}
mapping between these
Number of variables = 1 + 1 + 6 + mapping variables
 Number of design points ~ about 10139
6
Redundancy in Design
 Given a test rack design, any reordering of fixtures,
buses or equipment units gives the same physical design
 E.g., the mapping (fixture1, bus1, EQ1 unit1) is equivalent to
(fixture2, bus1, EQ1 unit1)
Design
Space
 Unique design space
Unique
Designs
 No redundant designs
 Much smaller than the original design space
7
Effect of Redundancy on Optimization
 For optimization, one method people used is to
sequentially sample uniformly over the original space
and to select the most promising subspace for
exploration in subsequent steps.
 Uniform sampling in the original space is not uniform
over the unique design space. Number of unique designs
in the sample could be very small.
 For 4 fixtures, 3 equipments
(4,3,4 units) and 3 buses:
 # in Design space ~ 402 x106
 # of Unique designs ~ 106
 Ratio ~ 0.25% of original!
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Design
Space
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Unique
Designs
8
Related Work in Design of Experiments
 C.F.J. Wu, Y. Chen, “A Graph-Aided Method for Planning
Two-Level Experiments when Certain Interactions are
Important,” Technometrics, Vol. 34, pp. 162-175, 1992.
 Planning fractional factorial experiment for estimating main
effects and ‘given’ interactions
 Model each feasible 2n-k design as a graph
 Use a pattern vector to characterize the uniqueness of the graph
 Find all non-isomorphic graphs by comparing the pattern vectors
among the feasible graphs
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Related Work in Design of Experiments
 Jiahua Chen, D. X. Sun, C. F. J. Wu, “A Catalogue of
Two-Level and Three-Level Fractional Factorial Designs
with Small Runs,” International Statistical Review, Vol.
61, pp. 131-145, 1993.
 Generating minimum aberration 2 and 3 levels fractional factorial
designs
 Two 2n-k designs of same word-length and letter pattern are
equivalent if identical under some relabeling of the columns in
the 2(n-k) full factorial designs
 First, generate (n - k) independent columns, and sequentially add
additional k columns. For each added factor, exhaust all the
relabeling possibilities to eliminate the isomorphic designs.
10
Related Work – Evolutionary Computing
 S. Ronald, J. Asenstorfer, M. Vincent, “ Representational
redundancy in evolutionary algorithms,” In Proceedings
of the 1995 IEEE International Conference on
Evolutionary Computation, Vol. 2, pp. 631–636, 1995.
 Discussed the issues due to design space redundancy when
using evolutionary algorithms
 Identified two equivalence relations for a TSP (traveling
salesman problem) encoding in GA implementation
 Found canonical representation in the TSP problem and reduced
space by deleting redundant designs (still exhaustive
comparison).
11
Graph/Group Theoretic Approach
1. Model a design as a graph
 Collection of all graphs satisfying certain criteria makes up the
design space
2. Setup the permutation group for the design space
 by defining a symmetric group for every set of identical objects
3. Find the equivalence classes of the design space under
the permutation group defined above
4. Elements of the same equivalence class are equivalent
design points
 Collection of design points made by picking one element from
each equivalence class makes up the unique design space
Reference: Derek J.S. Robinson, “A Course in the Theory of Groups”, Number 80 in Graduate Texts in
Mathematics, Springer Verlag, 1991.
12
Necklace Problem
 We will use this example to explain the basic approach
and some key concepts
 Find all necklaces constructed from white and black
beads (total beads = 4, constant)
1
2
1
2
4
3
4
3
 Labeled graph – all nodes are assigned a specific label
 For labeled graphs the number of such necklaces is 24 = 16
 Above two graphs are distinct labeled graphs
 Unlabeled graph – nodes have no distinct identification
 Above two graphs are the same unlabeled graph (rotate by 90)
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Necklace Problem
 Permutation group for the unlabeled necklaces
 Should define equivalence relation after rotation or reflection.
 Dihedral Permutation Group D8
 Dihedral Group D8 on the set {1,2,3,4} is
 (), (2 4), (1 2)(3 4), (1 2 3 4), (1 3), (1 3)(2 4), (1 4 3 2),
(1 4)(2 3)
1
2
4
3
24)
3 4)
(1(1(2
2)(3
Anti-Clockwise
Reflection
90◦ Rotation
2
1
3
1
4
1
3
2
4
3
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Necklace Problem – Equivalence Classes
 Equivalence Classes (under the action of D8)
Representatives
0 []
1 [1], [2], [3], [4]
2 [1,3], [2,4]
2 [1,2], [2,3], [1,4], [3,4]
3 [1,2,3], [2,3,4], [1,3,4], [1,2,4]
4 [1,2,3,4]
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Counting the Number of Equivalence Classes
 The number of unique designs can also be found by
using the function counting series
 Function counting series – polynomial with the sum of
the coefficients equal to the number of equivalence
classes
 For a number of standard graph structures the function counting
series formulae exist in literature*
 Necklace function counting series: 1+x+2x2+x3+x4
 Sum of the coefficients = 1+1+2+1+1 = 6
 Exhaustive search goes through C162 = 120 comparisons.
 Some algorithms (GAP algorithm) from graph theory can
do better than exhaustive search.
*F. Harary and E. M. Palmer, “Graphical Enumeration”, Academic Press, 1973.
16
A Bipartite Graph Representation of Test Rack
Partition 1
bipartite
graph
Bus
•
•
•
Physical Constraints:
Each equipment unit maps to
only one bus

Partition 2
E1
•
•
•
FIX
•
•
•
•
•
•
E2
•
•
•
•
Each fixture maps to 1
unit of each equipment type

 An
equipment unit is mapped
to a bus only if it is mapped to
some fixture
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Symmetric Group For Individual Node Sets
 Let the following denote the node sets
 F: set of fixture nodes
 B: set of bus nodes
 E1,…, En: sets of equipment nodes
 Because the nodes within each of F, B, E1,…, En node
sets are identical, this type of equivalence relation is
defined by the symmetric permutation group, S|V| , on the
set V.
18
Permutation Group for Bipartite Graph
 Symmetric Group S3 on {1, 2, 3} is
 (), (2 3), (1 2), (1 2 3), (1 3 2), (1 3)
•1
•2
•3
(2 3)
•1
•1
•3
•2
•2
•3
(1 2 3)
•2
•3
•1
 Permutation group for the bipartite graph:
G = (S|F|·S|B|)×(S|E1|·…·S|En|)




Sk: Symmetric group on k objects;
‘×’ denotes the Cartesian product of the sets
‘·’ denotes the Direct product of the sets
G acts on the object set (F  B)×(E1 … En)
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Test Rack – Equivalence Classes
 Equivalence classes – set of all bipartite graphs that are
equivalent under the action of G
 Unique test racks – collection formed by picking one element
from each equivalence class
 A 2-fixture problem: 2 fixtures, 2 buses, 2 types of
equipment, 2 units per type.
 Number of designs in the original design space = 216 =
65,536
 Function Counting Series:
g2222(x) = 1 + 4x + 22x2 + 64x3 + 186x4 + 376x5 + 674x6 + 900x7 + 1034x8 +
900x9 + 674x10 + 376x11 + 186x12 + 64x13 + 22x14 + 4x15 + x16
 Number of unique designs = 5,488 (8% of original). All
unique designs can be identified by using permutation group
G and GAP algorithm.
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Summary and What’s next?
 Graph theoretic approach to find unique or nonisomorphic designs to reduce design space.
 Importance in large scale engineering system designs.
 Improve efficiency for getting unique designs and make
it scalable to large problems
 Incorporate physical constraints to further reduce the
number of unique designs
 Sequential uniform sampling over the unique design
space for design optimization.
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Backup Slides
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Algorithm for finding the Equivalence Classes
O:=All Graphs
P:= Permutation
Group
Initialize EqClasses to NULL
Yes
Is O empty?
No
END
p:= first element of O; Class:=[p];
a:= first permutation in P; delete p from O
a := next
permutation in
P;
p0 := a(p)
if p0 in O, delete p0 from O and add p0 to Class
Yes
Any more
permutations in P?
No
Add Class to
EqClasses
* The GAP Group, GAP –
Groups, Algorithms, and
Programming, Version 4.4;
2006. (http://www.gapsystem.org)
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