1. Ali Husseinzadeh Kashan Spring 2010
A New Solution Approach for Grouping Problems Based on Evolution Strategies
Ali Husseinzadeh Kashan
Spring 2010

2. Agenda Grouping problems and their applications
Grouping Genetic Algorithm (GGA)
Evolutionary Strategy (ES)
Proposed Grouping ES
Experimental Results
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

3. Grouping Problems
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)

4. Grouping Genetic Algorithm (GGA)
Two main representation schemes:
Number encoding: each item is encoded with a group ID, for example
Redundancy: example,
Individual 1: {2, 5}{1, 4}{3}
Individual 2: {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)

5. Grouping Genetic Algorithm (GGA)
Group encoding:
The Crossover: the general pattern
The Mutation: eliminate some existing groups; insert the missing items by a problem depended heuristic
Item Part
Group Part A B C ≡ : 3
4, 1
2 , 5 A B C C B A
Parent 1
Child 1: 6
4, 1
5, 3
7, 2 7
5, 6
3, 2 6
5, 6
4, 1
7, 2 7
5, 3
3, 2 6
5, 3
4, 1
7, 2 7
3, 6
2, 5
Parent 2
Child 1:
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

6. Evolutionary Strategy (ES)
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
Yti d = Xtikd + Zd ; d = 1,...,D, i = 1,...,
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

7. Evolutionary Strategy (ES)
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)

8. Evolutionary Strategy (ES)
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)

9. Grouping Evolutionary Strategies
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

10. GES: Initial Solution
Solution representation: solution X with DX groups as a structure whose length is equal to the number of groups
Xi:
The first solution is generated randomly
9, 10
6, 9, 4
1, 7
2, 3, 5
Xi1
Xi2
Xi3
Xi4
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

11. GES: Initial Solution 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)

12. GES: New Solution Generation
Similarity measure:
Distance/Dissimilarity measure:
Then, Gaussian mutation operator in GES is introduced as follows:
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

13. GES: New Solution Generation
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:
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

14. GES: New Solution Generation
Fixing the value of  t at a constant level  1, we only consider  t as the endogenous strategy parameter
Then,
Ultimately, the number of inherited items by each group of new solution is:
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

15. GES: New Solution Generation
Inheritance Phase:   
Xt: 7 1 5 10 2 11 9 4 8   3 6  12
ntid: 2 3 1   
Yt:   
Post assignment Phase:
Missed Items: 1 5 11 9 6 12      
Yt: 7 10 2 4 8 3
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

16. GES: New Solution Generation
Two type of constructive heuristic:
First-fit
Best-fit
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

17. GES: Experimental Results
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:
Implementation: MATLAB 7.3.0, Pentium 4, 3.2 GHz of CPU, 1 GB of RAM
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)

18. GES: Experimental Results
Problem
GES
GGA
min num of bins
Bins
Time
(Sec)
HARD0 56
103.7
517.4
HARD1 57
110.0
473.4
HARD2
105.8
446.1
HARD3
102.9
432.3
HARD4
110.5 58
452.9
HARD5
105.1
483.8
HARD6
104.0
440.4
HARD7 55
107.4
431.2
HARD8
106.2
465.7
HARD9
485.3
Average
56.4
56.7
462.8
Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES)