COMBINATORIAL IMPLEMENTATION OF A PARAMETER-LESS

EVOLUTIONARY ALGORITHM

Gregor Papa

Computer Systems Department, Joˇzef Stefan Institute, Ljubljana, Slovenia

Keywords:

Parameter-less, Evolution, Search, Combinatorial, Optimization.

Abstract:

The paper presents the combinatorial implementation of an adaptive parameter-less evolutionary-based search.

The algorithm is an extension of a basic numerical algorithm that does not need any predeﬁned control pa-

rameters values. These values are calculated by the algorithm itself, according to the progress of the search.

The efﬁciency of the proposed autonomous parameter-less algorithm is evaluated by two real-world industrial

optimization problems.

1 INTRODUCTION

Finding an appropriate parameter setup of an evolu-

tionary algorithm is a long standing research chal-

lenge (Eiben et al., 2007; Kang et al., 2006). Dif-

ferent optimization techniques, like genetic algorithm

(GA), differential evolution (DE), evolutionary strate-

gies (ES) or particle swarm optimization (PSO), re-

quire several important control parameters that need

to be set in advance to ensure effective optimization.

The issue of setting the values of various parameters

of an evolutionary algorithm is crucial for good per-

formance (Eiben et al., 2007). Furthermore, it has

been empirically and theoretically demonstrated, that

different values of parameters might be optimal at dif-

ferent stages of the evolutionary process (Stephens

et al., 1998; B¨ack, 1992). Therefore, particulary inter-

esting are the approaches that would solve any prob-

lem without any human intervention for setting the

suitable control parameters.

To skip the pre-setting of the control parameters a

new technique was developed, which is able to ﬁnd

near-optimal solutions relatively quickly. The pre-

sented algorithm is the extension of the original algo-

rithm (Papa, 2008). This version is adapted to com-

binatorial problems. The algorithm - Parameter-Less

Evolutionary Search (PLES) - is based on basic GA,

but this algorithm does not need any control parame-

ter, e.g., population size, number of generations, prob-

abilities of crossover and mutation, to be set in ad-

vance, but are calculated during the search progress.

2 PARAMETER-LESS SEARCH

The control parameters depend on the behavior and

convergence of the found solutions. The pseudo-code

of the algorithm is presented in Figure 1. In gen-

eral, solutions are selected, and recombined in the

PLES as they are in the GA, but the implementation

of these operators is different. The elitism, selection,

and crossover are implemented through forcing-of-

better-individuals function, while mutation is split be-

tween forcing of better individuals and moving of in-

dividuals. The control parameters are never set in ad-

vance and are not constant. They are determined each

time on the basis of statistical properties of each pop-

ulation. Besides, the algorithm varies the population

size. There are some constants used in calculation of

the control parameters, but they are treated as meta-

parameters. These constants ensure large enough ini-

tial population and other parameters in order the al-

gorithm to be operational. Furthermore, these meta-

parameters are never changed - not even for different

problems.

Setup. The chromosome that represents the solu-

tion is constructed upon the number of the variables of

the problem, i.e., for the n variables the chromosome

looks like the string of n values.

Initialization. The initial population size ensures

large enough population to be able to search within

the solution space.

307

Papa G..

COMBINATORIAL IMPLEMENTATION OF A PARAMETER-LESS EVOLUTIONARY ALGORITHM.

DOI: 10.5220/0003671403070310

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 307-310

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Parameter-Less Evolutionary Search

Set the initial population.

Evaluate the initial population.

While stopping criterion not met do.

Force better individuals to replace worse.

Move individuals.

Evaluate the current population.

Vary population size.

Figure 1: The pseudo-code of the PLES algorithm.

PopSize

= 4

√

n+ 10log

(Range),

Range =

∑

j=1

(max

−min

+ 1)10

decplc

where n is the number of variables to be optimized,

max

and min

are the upper and the lower limit

of the j-th variable, respectively, and decplc

is the

number of decimal places of the j-th variable. In

the case of combinatorial optimization decplc

=0 and

(max

−min

+ 1) is equal to the number of possi-

ble combinations of the j-th variable, combinations

therefore

Range =

∑

j=1

(combinations

Stopping Criterion. The number of overall gener-

ations depends on the convergence speed of the best

solution found. Optimization proceeds while better

solution is found every few generations. But when

there is no improvement of the best solution for a

Limit (Limit = 10log

(PopSize

)) number of genera-

tions, the optimization process stops.

Variable Population Size. During the search pro-

cess the population size is changed, since the quality

of the solution might depend on the size of popula-

tion. The population size is changed every(

Limit

) gen-

erations, based on the average change of the standard

deviation (StDev) of ﬁtness values of solutions over a

last few generations.

PopSize

i+1

PopSize

StDev

i−1

+StDev

i−2

StDev

+StDev

i−1

The change of population is limited to

20% per change and is further limited to

[

PopSize

, 1.1 PopSize

]. The population shrink-

ing enables the search with smaller populations,

which is suitable to some types of the problem, or in

some stages of the search.

Forcing Better Solution. In everygenerationworse

individuals are replaced with better individuals. After

that, every s

(variable j of the solution i) is randomly

moved up to 20% of the difference between the vari-

able and the limit (upper or lower) of the variable.



(i−1)

+ rnd(max

−s

(i−1)

) ;rnd ≥ 0

(i−1)

+ rnd(s

(i−1)

−min

) ;rnd < 0

where rnd is a random number [-0.2,0.2].

In the case of combinatorial optimization the s

moved to one of the neighboring values. Here, 20%

of positions j are changed to the neighboring value

that is deﬁned as 20% of all possible combinations of

a variable value. If in s

(i−1)

the value of the variable

j is j

, then in s

the variable has the value j

(k+rnd·m)

where j = {j

, j

, . . . j

, . . . , j

}; variable j has m pos-

sible combinations.

Furthermore, in the case of ordered values the

move is performed by switching of randomly chosen

pairs of variables inside the solution. Again, up to

20% of variable pairs are switched.

Solution Moving. In the PLES, mutation is realized

through the moving of some positions in the chro-

mosome according to different statistical properties.

The Ratio of the parameters in the chromosome to be

moved is calculated on the basis of standard deviation

of the solutions in the previous generations as stated

in the following equation.

Ratio

StDev

i−1

StDev

i−3

where StDev

i−1

and StDev

i−3

are the standard devia-

tion of the solution ﬁtness of the previous generation,

and the standard deviation three generations ago, re-

spectively. Here Ratio

∈[0. .. n], and the Ratio

posi-

tions in the chromosome are selected to be moved.

The size of the move is calculated according to the

difference between the value of the parameter of the

global best solution and current solution, as presented

in the following equations

MoveRatio =



best

−s

(i−1)

average

(i−1)

−s

(i−1)



where s

is the value of the parameter j of the cur-

rent solution i, and s

best

is the value of parameter j of

the globally best solution, average

(i−1)

is the average

value of the parameter j in the previous generation.

Width = s

best

−s

(i−1)

= s

(i−1)

+ Direction ×MoveRatio×Width

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

308

where Direction is randomly selected (-1 or 1).

In the case of combinatorial optimization the s

rounded, since only integer values are accepted.

Solution Evaluation and Statistics. After recom-

bination operations the population is statistically eval-

uated. Here, the best, the worst, and average ﬁtness

value in the generation is found. Furthermore, the

standard deviation of ﬁtness values of all solutions in

the generation, and the average value of each variable

are calculated.

3 INDUSTRIAL PROBLEMS

3.1 Production Planning

Optimal planning of the production in the company

that produces different components for domestic ap-

pliances has to consider various constraints. The

most demanding production stage is the production

of cooking hot-plates. The production of the com-

ponents and parts for all the types of plates is more

or less similar, but the standard plate models of the

current range differ in size (height, diameter), con-

nector type and power characteristics. Many differ-

ent models exist due to the various demands of other

companies that use those plates for their own cooking

appliances. Orders for some particular models usu-

ally vary in quantities and deadlines. Orders from the

same company are usually also connected with the

same deadline. Therefore, their production must be

planned very carefully to fulﬁl all the demands (quan-

tities and deadlines), to maintain the speciﬁed amount

of different models in the stock, to optimally occupy

their workers, and to efﬁciently use all the produc-

tion lines. Also, not all the production lines are equal,

since each of them can produce only a few differ-

ent models. The production planning problem con-

sists of ﬁnding a production plan that satisﬁes produc-

tion time constraints and minimizes production costs

(Koroˇsec et al., 2010). The solving involves many

speciﬁc constraints that need to be considered. The

main problem is the exchange delay, caused by adapt-

ing production lines to different types of products and

supplying the appropriate parts. The manufacturing

processes of multiple types of products requires many

different steps, and different product parts for com-

pletion of each product type. We used the combinato-

rial version of the PLES algorithm to ﬁnd the optimal

production plan. For the optimal results also some lo-

cal search procedures were used with the algorithm

(Koroˇsec et al., 2010).

The cost function (Eq. (1)), as deﬁned in (Koroˇsec

et al., 2010) considers the number of delayed orders

orders

), exchange delay times (t

exchange

), overall pro-

duction time (t

overall

), and the sum of squared days of

delayed orders (n

days

f(P) = 10

·n

orders

+ 10

·t

exchange

overall

+ n

days

. (1)

3.2 Cooling Appliance Setting

Besides the built-in functionalities, the household ap-

pliances must have low power consumption energy-

efﬁcient optimal-performance. Optimal performance

means that the appliance is able to cool to the de-

sired temperature at the lowest possible power con-

sumption. Such optimization usually requires a lot

of long-term development measurements or a thor-

ough theoretical analysis of the cooling system and

the construction of a complex mathematical model for

its simulation (Angeli and Kountouriotis, 2011).

To determine the optimal performance of the re-

frigerator, we need to perform a set of development

measurements for each new type of the appliance

(Papa and Mrak, 2010). Thermal processes in cooling

systems are by nature very slow. One measurement

for determining the power consumption under stan-

dard conditions takes several days. To speed-up the

development process the temperature simulator was

used in connection with the optimizer. The simu-

lation tool simulates temperatures inside the cooling

appliance at different modes of regulation. The opti-

mizer uses an evolutionary heuristic search approach

to ﬁnd the optimal set of control parameters itera-

tively over evolving generations. The mixed com-

binatorial/numerical version of the PLES algorithm

was used to ﬁnd the optimal control parameters of an

appliance, that give an optimal performance with the

lowest possible power consumption.

The cost function (Eq. (2)), as deﬁned in (Papa

and Mrak, 2010), considers power consumption in the

calculation interval, the length of the calculation in-

terval, the mean and the reference temperature in C1

and C2 cabinets (meanT and refT), and the standard

deviation (σ

and σ

OFF

) of ON and OFF times for

the compressor.

f(P) =

2800consumption

interval

12000length

interval

+|meanT

−refT

+|meanT

−refT

1000

OFF

1000

. (2)

COMBINATORIAL IMPLEMENTATION OF A PARAMETER-LESS EVOLUTIONARY ALGORITHM

309

4 RESULTS

The PLES run 20-times for each problem. The exper-

iments were done on 2.2GHz computer, and each run

took approximately 5 minutes for production plan-

ning, and less than 1 minute for cooling appliance

optimization. However, time complexity was not the

subject of this evaluation.

4.1 Production Plan

The PLES algorithm was tested on two different real

order lists from production company. The Task 1 con-

sists of n = 711 orders for 251 products, while the

Task 2 consists of n = 737 orders for 262 products. In

both tasks m = 5 production lines are available. The

number of evaluations was limited to 500, 000.

While the PLES does not need any parameter,

the comparing GA used the following parameters:

the population size N = 100; the number of gener-

ations was 5, 000; the replacement rate r = 0.2; the

crossover probability p

= 0.7; the mutation proba-

bility p

= 0.005.

In Table 1 best, mean, worst, and standard devia-

tions of solutions are presented for each task.

Table 1: Results of optimization for Task 1 and Task 2

PLES GA

Task 1 Best 1.309×10

1.308×10

Mean 1.327 ×10

1.340×10

Worst 1.416×10

1.610×10

StD 3.230 ×10

6.526×10

Task 2 Best 1.576×10

1.611×10

Mean 1.664 ×10

1.748×10

Worst 1.813×10

1.914×10

StD 7.390 ×10

7.235×10

When comparing with the previous approach of

production planning (Koroˇsec et al., 2010), the ex-

pert’s manual plan for those two tasks had about four-

times more delayed orders. This is signiﬁcantly worse

than results obtained by both of the algorithms. Any-

way, the PLES algorithm shows faster convergence

and proves its search ability to ﬁnd the solution with-

out the predeﬁned control parameter settings.

4.2 Cooling Appliance

The PLES algorithm was used to speed-up the devel-

opment process of a new type of a cooling appliance.

The PLES was able to ﬁnd the optimal set of cooling

appliance control parameters setting in all runs.

Optimization based on stochastic search method,

might give several possible solutions. Criteria for

choosing the appropriate one is not only the smallest

energy consumption and the desired temperature, but

it is also necessary to verify the behavior of the com-

ponents. Namely, frequent on/off switching of them

shortens their life cycle.

5 CONCLUSIONS

The paper presented an adaptive parameter-less evo-

lutionary search for combinatorial problems. The ef-

ﬁciency of the proposed parameter-less algorithm was

evaluated by two real-world industrial optimization

problems. Following the previously presented numer-

ical results, it was shown, that the combinatorial im-

plementation of the algorithm had faster convergence

than comparing algorithm and also it proved its search

ability to ﬁnd the solution without the predeﬁned con-

trol parameters. Furthermore, it was shown that the

combinatorial implementation of the algorithm is as

effective as the numerical one.

REFERENCES

Angeli, D. and Kountouriotis, P.-A. (2011). A stochastic

approach to ”dynamic-demand” refrigerator control.

Control Systems Technology, IEEE Transactions on,

PP(99):1 –12.

B¨ack, T. (1992). The interaction of mutation rate, selec-

tion, and self-adaptation within a genetic algorithm.

In M¨anner, R. and Manderick, B., editors, Proceed-

ings of the 2nd Conference on Parallel Problem Solv-

ing from Nature. North-Holland, Amsterdam.

Eiben, A., Michalewicz, Z., Schoenauer, M., and Smith, J.

(2007). Parameter control in evolutionary algorithms.

In Lobo, F., Lima, C., and Michalewicz, Z., edi-

tors, Parameter Setting in Evolutionary Algorithms,

volume 54 of Studies in Computational Intelligence,

pages 19–46. Springer Berlin / Heidelberg.

Kang, Q., Wang, L., and di Wu, Q. (2006). Research on

fuzzy adaptive optimization strategy of particle swarm

algorithm. International Journal of Information Tech-

nology, 12(3):65–77.

Koroˇsec, P., Papa, G., and Vukaˇsinovi´c, V. (2010). Appli-

cation of memetic algorithm in production planning.

In Proc. Bioinspired Optimization Methods and their

Applications, BIOMA 2010, pages 163–175.

Papa, G. (2008). Parameter-less evolutionary search. In

GECCO, pages 1133–1134.

Papa, G. and Mrak, P. (2010). Optimization of cooling ap-

pliance control parameters. In Proceedings of the 2nd

International Conference on Engineering Optimiza-

tion, EngOpt2010.

Stephens, C. R., Olmedo, I. G., Vargas, J. M., and Wael-

broeck, H. (1998). Self-adaptation in evolving sys-

tems. Artif. Life, 4:183–201.

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

310