Ali Husseinzadeh Kashan
Spring 2010
Grouping problems and their applications
Grouping Genetic Algorithm (GGA)
Evolutionary Strategy (ES)
Proposed Grouping ES
Experimental Results
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
2
Partitioning a set (V) of n items into a collection of
mutually disjoint subsets (groups, Vi) such that:
Partition the members of set V into D (1≤ D ≤ n)
different groups where each item is exactly in one group
Ordering of groups is not relevant
well-known problems as grouping problems:
graph (vertex/edge) coloring, bin packing, batch-processing
machine scheduling, line-balancing, various timetabling
problems, cell formation problem, etc.
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
3
Two main representation schemes:
Number encoding: each item is encoded with a group ID, for
example 2 1 3 2 1
Redundancy: example,


Individual 1: 2 1 3 2 1
Individual 2: 1 2 3 1 2
{2, 5}{1, 4}{3}
{1, 4}{2, 5}{3}
Group encoding: items belonging to the same group are placed
into the same partition, for example {2, 5}{1, 4}{3}
Search operators can work on groups rather than items
Groups are the meaningful building blocks of solutions
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
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Group encoding:
2 , 5 4, 1
3
Item Part
≡
B A C B A
Group Part
: A B C
The Crossover: the general pattern
Parent 1 7, 2 5, 3 4, 1
6
Child 1: 7, 2 4, 1 5, 36
6
Parent 2 3, 2 4, 1 5, 6
7
3, 52 3,
5, 63 4, 1
Child 1: 2,
7
The Mutation: eliminate some existing groups; insert the
missing items by a problem depended heuristic
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
5
Darwin’s theory: the most important features of the
evolution process are inheritance, mutation and selection
Main steps of (μ+)-ES:
Initial solutions:  t = Xt1 , Xt2 , ..., Xtμ 
Repeat until (Termin.Cond satisfied) Do
Mutation: create a set Qt =  Yt1 , Yt2 , ..., Yt  of solutions via
mutation
 New population ( t +1): the μ best of the μ+ candidate solutions
in  t  Q t are selected.


Replace the current best solution if it is better than the best solution
found so far
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
6
Xti = xti1, xti2, ..., xtid a solution of current population
Yti = yti1, yti2, ..., ytid an offspring obtained via mutation
Zd = t Nd (0, 1)
 t : distance of an offspring candidate solution from the parent
 t is varied on the fly by the “1/5 success rule”
This rule resets  t after every k iterations by



 =  / a if ps > 1/5
 =  . a if ps < 1/5
=
if ps = 1/5
where ps is the % of successful mutations,
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
7
Difficulty with developing the grouping version of ES:
ES owns a Gaussian mutation to produce new real-valued solution
vectors during the search process. To introduce GES, we should
develop a new comparable mutation which works based on the role of
groups, while keeping the major characteristics of the classic ES
mutation. The paper is going to cover this issue.
Originally, ES has been introduced for optimizing non-linear functions
in continuous space. But grouping problems are all discrete. We will
show how we can keep the new mutation in continuous space while
using the consequences in discrete space.
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
8
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
9
Solution representation: solution X with DX groups as a
structure whose length is equal to the number of groups
Xi: 2, 3, 5
1, 7
6, 9, 4 9, 10
The first solution is generated randomly
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
10
Yti d = Xtd + Zd ; d = 1,...,D, i = 1,...,
(1)
The key idea is to use appropriate operators in the place
of arithmetic operators
Indeed, we have to determine how many items of current
groups (X td) must be inherited by the new groups (Y tid)
By reshaping (1) in the form of Yti d - Xtd = Zd,
Substitution of “-” operator with an appropriate one in
grouping problem
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
11
Similarity measure:
t
t
|Y

X
t
t
id
d|
Jaccard ' s Similarity (Yid ,X d )  t
|Yid  X dt |
Distance/Dissimilarity measure:
t
t
|Y

X
t
t
id
d|
Jaccard ' s Distance (Yid ,X d )  1  t
|Yid  X dt |
Then, Gaussian mutation operator in GES is introduced
as follows:
Distance ( yidt , xdt )  z d
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
12
Zd values are unrestricted in sign but the range of
distance measure is only real values in [0, 1]
Appropriate source of variation:


With 0 and 1 as the lower and upper bound of candidate PDF
With flexible PDF that provides different chances for getting a
specific value in [0, 1] by means of some controllable
parameter(s)
The new mutation operator of GES:
Distance ( yidt , xdt )  Beta d ( t ,  t ),
d  1,..., D, i  1,..., 
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
13
Fixing the value of  t at a constant level  1, we only
consider  t as the endogenous strategy parameter
Then, Distance ( yidt , xdt )  Beta d ( t ,  ),
d  1,..., D, i  1,..., 
Ultimately, the number of inherited items by each group
of new solution is:
t
n
Distance ( yidt , xdt )  1  idt  Beta d ( t ,  )
|x d|
 nidt  (1  Beta d ( t ,  ))|x dt |
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
14
Inheritance Phase:
Post assignment Phase:
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
15
Two type of constructive heuristic:


First-fit
Best-fit
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
16
one-dimensional bin packing problem:



set of n items,
size of jth item is sj,
objective is to pack all items into the minimum number of bins
(groups) of capacity B
Comparisons: The GGA proposed by Falkenauer (a
steady-state order-based GA and its overall procedure)
Benchmark: ten problem instances via the URL:
http://www.wiwi.uni-jena.de/Entscheidung
Implementation: MATLAB 7.3.0, Pentium 4, 3.2 GHz of
CPU, 1 GB of RAM
Ali Husseinzadeh Kashan
Grouping Evolutionary Strategy (GES)
17
GES
Problem
GGA
min num of bins
56
Time
(Sec)
517.4
HARD0
56
Time
(Sec)
103.7
HARD1
57
110.0
57
473.4
57
HARD2
57
105.8
57
446.1
57
HARD3
56
102.9
56
432.3
56
HARD4
57
110.5
58
452.9
58
HARD5
56
105.1
57
483.8
57
HARD6
57
104.0
57
440.4
57
HARD7
55
107.4
55
431.2
55
HARD8
57
106.2
57
465.7
57
HARD9
56
102.9
57
485.3
57
Average
56.4
105.8
56.7
462.8
56.7
Ali Husseinzadeh Kashan
Bins
Bins
Grouping Evolutionary Strategy (GES)
56
18