parallel evolutionary multi-criterion optimization for block layout problems

Doshisha Univ., Japan

Parallel Evolutionary Multi-Criterion Optimization

for Block Layout Problems

○ Shinya Watanabe

Tomoyuki Hiroyasu

Mitsunori Miki

Intelligent Systems Design Laboratory，

Doshisha University， Japan


• Some of EMO can derive the good pareto optimum solutions.

• EMO need high calculation cost.

Evolutionary algorithms have potential parallelism.

PC Cluster Systems become very popular.

Background (1)

● ＥＭＯ・・・・Ｅｖｏｌｕｔｉｏｎａｒｙ　Ｍｕｌｔｉ－ｃｒｉｔｅｒｉｏｎ　 Optimizations

（ Ex. VEGA,MOGA,NPGA…etc）

Parallel ComputingParallel ComputingSW-HUB


• Some parallel models for EMO are proposed– There are few studies for the validity on parallel model.

• Divided Range Multi-Objective Genetic Algorithms (DRMOGA)– it is applied to some test functions and it is found that this

model is effective model for continuous multi-objective problems.

Background (2)

●Parallel ＥＭＯ Algorithms

DRMOGA hasn’t been applied to discrete problems.

PurposeThe purpose of this study is

to find the effectiveness of DRMOGA.


Multi-Criterion Optimization Problems(1)

●Multi-Criterion Optimization Problems (MOPs)

Design variables

Objective function

Constraints

Gi(x)<0 ( i = 1, 2, … , k)

F={f1(x), f2(x), … , fm(x)}

X={x1, x2, …. , xn}

In the optimization problems, when there are several objective functions,

the problems are called multi-objective or multi-criterion problems.

f2 (x) Feasible regionFeasible region

f1(x)

Weak pareto optimal solutions

Pareto optimal solutions


・・ Multi-objective GAMulti-objective GALike single objective GA , genetic operations such as evaluation, selection, crossover, and mutation, are repeated.

ff11(x)(x)

1st generation5th generation

10th generation

ff 22(x)

(x)

Pareto optimal solutions50th generation

Multi-objective GA (1)

30th generation


DGA model

Distributed GAs

A population is divided into subpopulations (islands)

SGA is performed on each subpopulation

Migration is performed for some generations

Exchange of individuals

1 island / 1PE

Migration


f(x)

f (x)1

2

Division 1

Division 2

MaxPareto Optimum solutionMin

f1(x)

f 2(x)

Division 1

f1(x)

f 2(x)

Division 2

Divided Range Multi-Objective GA(1)

1st The individuals are sorted by the values of focused objective function.

2nd The N/m individuals are chosen in sequence.

3rd SGA is performed on each sub population.

4th After some generations, the step is returned to first Step


・ Block Layout Problems with Floor Constraints(Sirai 1999)

Block Packing methodBlock Packing method

１

４

５

３

２

６

７

： Dead Space

Formulation of Block Layout Problems

i=1

n

f1=ΣΣcij dijj=1

n

f2=Total Area S

Objects

where

n:number of blocks

cij: flow from block i to block j

dij: distance from block i to block j


• Application models– SGA , DGA , DRMOGA

• Layout problems– 13, 27blocks

• Parameter

Numerical Example

Block No.verticalhorizontal1 18 242 36 183 18 424 42 185 36 426 24 367 24 548 30 369 48 18

10 36 2411 36 3612 54 2413 36 30

ex)13 blocks

GA parameter value

mutation rate 0.05migration interval (resorted interval)

20

migration rate

0.2

crossover rate 1.0Number of individuals100 (total 1600)

terminal condition 300generation

20


Cluster system for calculation

Spec. of Cluster (16 nodes)Processor Pentium (Deschutes)Ⅱ

Clock 400MHz# Processors 1 × 16Main memory 128Mbytes × 16Network Fast Ethernet (100Mbps)Communication TCP/IP, MPICH 1.1.2OS Linux 2.2.10Compiler gcc (egcs-2.91.61)


DGA DRMOGA

Results of 13 Blocks case

Real weak pareto solutions

13


Results of 13 Blocks case (SGA)

Local optimum solutions

Real weak pareto solutions

13



DRMOGA

A

B

27

DGA


A B

(f_1, f_2) = (38446, 49920) (f_1, f_2) = (42739, 43264)27


Results

• Most of the solutions were weak-pareto solutions.

• SGA, DGA and DRMOGA are applied to the layout problems– There are small difference between the results of three metho

ds.– When results of DRMOGA compared with those of DGA, ther

e isn’t big advantage.– SGA sometimes could not find the real weak pareto solutions.

These problems have little trade-off

relationships between the objective functions.


Resultsf (

x)

f (x)1

2

Division 1

Division 2

f (x)

f (x)1

2

The individuals can’t be divided into

two division by the value of the focused

objective function(f2(x)).

Can’t exchange individuals enough.


• The DRMOGA was applied to discrete problems ; The block layout problems.

– The test problems didn’t have definitely pareto solutions.

– The searching ability of DGA and DRMOGA were almost same in numerical examples.

– The mechanism of DRMOGA didn’t work effectively in these problems.

– SGA may be caught by local minimum.

Conclusion

The results of DRMOGA were compared with those of SGA and DGA


~ アルゴリズムの流れ~

①初期個体生成②個体を各島

に分配

④評価・選択・交叉

⑥全体シェ

アリン

グ⑤総個体数を調べ

る③終了判定⑦終了

⑦ へ


f2(x

)

f1(x)

f2(x

)

f1(x)

・ DGA( Island model)

・ DRMOGA

f2(x

)

f1(x)

f2(x

)

f1(x)

+ =

f2(x

)

f1(x)

f2(x

)

f1(x)

+ =

Divided Range Multi-Objective GA(2)


Results of 10 Blocks case (DRMOGA)

Real weak pareto set

Local optimum set

A

B


A B


DGA SGA



• Why are the results in this presentation different from the results in the paper?– In first, we selected GA parameters with no consideration. But we

investigated more suitable GA parameters, and in this presentation, we used new GA parameters. That’s why this results Is different from results in paper.

• What do you aim in this presentation?– Main purpose in this study is to investigate the effectiveness of DRMOGA

for Block layout problems. To my regret, this problem isn’t suitable for multi-criterion problems and we can’t get good results.

• How do you think about meaning of this presentation?– In other discrete problem, the effectiveness of DRMOGA hasn’t been

researched yet. And I think that in the problem that has obviously trade-off relationships, DRMOGA will get good results. Because in that problems , the characteristics of DRMOGA can work effectively.


•VEGA 　 Schaffer (1985)

•VEGA+Pareto optimum individuals Tamaki 　 (1995)

•Ranking 　 Goldberg (1989)

•MOGA Fonseca (1993)

•Non Pareto optimum Elimination Kobayashi (1996)

•Ranking + sharing Srinvas (1994)

•Others

Multi-objective GA (2)

Squire EMO


(f_1, f_2) = (838544, 14238) (f_1, f_2) = (879179, 13560)13


• Expression of solutions

Configuration of GA

• Genetic operationsSelection Pareto reservation strategyPareto reservation strategy

Crossover PMX methodPMX methodMutation 2 bit substitution method2 bit substitution method


Calculation Time

Case methodCalculation time(sec)

13blocksSGADGA 1.73E+01

DRMOGA 2.02E+01

27blocksSGA

DGA 5.28E+01DRMOGA 5.61E+01

1.08E+03

1.37E+03



SGA

27

parallel evolutionary multi-criterion optimization for block layout problems

Documents