optimal noise control on plant using simulated …€¦ · optimal noise control on plant using...

OPTIMAL NOISE CONTROL ON PLANT USING SIMULATED ANNEALING

Tian-Syung Lan1 and Min-Chie Chiu2

lDepartment of Information Management, Yu Da College of BusinessMiaoli County, Taiwan 361, R.O.C.

Contact: [email protected] ofAutomatic Control Engineering, Chungchou Institute ofTechnology

Yuanlin, Changhua 51003, R.O.C.

Received January 2008, Accepted September 2008No. 08-CSME-03, E.I.C. Accession 3041

ABSTRACTNoise control is important and essential in a manufacturing factory, where the noise level is restricted

by the Occupational Safety and Health Act. Several researches on new techniques of single noise controlhave been well addressed and developed; however, the study of noise depression on the whole plant noiseby using optimum allocation planning is hardly found. An improper machine allocation will not onlyresult in the tremendous cost on noise control task, but also cause the harmful environment for theneighborhood; therefore, the approach of optimum and economic allocation of noise sources within aconstrained plant area becomes crucial and obligatory.

In this paper, a novel technique of simulated annealing (SA) is applied in the numerical optimization,and the multi-noise plant with various sound monitoring systems is also introduced. Before optimization,the single noise is tested and compared with the simulated data from SoundPlan, a commercial soundsimulation package, for the accuracy check of the mathematical model. The result reveals to be withingood agreements. The proposed SA optimization on the allocation of multi-noise plant surely provides aneconomic and effective methodology in reducing the sound accumulation around the plant boundary.

Keywords: sound transmission loss, space constraints, simulated annealing.

CONTROLE DU BRUIT OPTIMAL DANS LES USINES AU MOYENDU RECUIT SIMULE

RESUMELe contrOle du bruit est important et essentiel dans une usine de production, ou Ie niveau de bruit est

restreint par l'Occupational Safety and Health Act. Plusieurs recherches sur les nouvelles techniques decontrole de bruit unique ont ete bien abordees et elaborees; cependant, il y a peu d'etudes surI'affaiblissement du bruit dans I'ensemble des bruits de I'usine au moyen de la planification d'attributionoptimale. Vne attribution d'appareils inadequate entrainera non seulement des couts monumentaux sur latache de controle du bruit, mais elle causera aussi un environnement dangereux pour Ie voisinage. Parconsequent, une approche it I'attribution optimale et economique de sources de bruits it I'interieur d'unendroit contraint de I'usine devient cruciale et obligatoire.

Dans la presente, une nouvelle technique de recuit simule (RS) est appliquee it l'optirnisationnumerique et I'on presente une usine it bruits multiples avec divers systemes de surveillance du bruit.Avant I'optimisation, Ie bruit unique est teste et compare avec les donnees simulees de SoundPlan, unensemble de simulation du bruit commercial, pour en verifier I'exactitude du modele mathematique. Lesresultats sont it l'interieur de bonnes ententes. L'optimisation RS proposee dans l'attribution de bruitsmultiples de l'usine foumit certainement une methode economique et efficace de reduction des bruitsaccumules dans les limites de I'usine.

Mots-cles : perte de la transmission du bruit, contraintes d'espace, recuit simule

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1. NOMENCLATURE

This paper is constructed on the basis of the following notations.f : frequency (Hz)

F(X): objective function

ite':nax: maximum iteration

kk: cooling rate in SApH}): transition probability

rgk : distance between the g-th noise source and the k-th receiver (m)Rij: distance between the j-th noise source and the i-th receiver (m)SPL: sound pressure level (dB(A) re 2*10-5 Pa)STL: sound transmission loss (dB)SWL: sound power level (dB(A) re 10-12 Watt)1;: current temperature

1;+1 : next temperature

To: initial temperature

X: design variables'¥: sound absorption in air (dB)¢: humidity in air (%)

2. INTRODUCTION

As investigated by the Occupational Safety and Health Act (OSHA) of 1970, high noiselevels can be harmful to workers and can lead not only to psychological but also to physiologicalailments. As a result, the noise control work on equipment becomes important [1, 2]. However,the improper machine allocation will result in the extra extravagant cost and the deficientacoustic performance due to the sound accumulation effect even if some equipment has alreadybeen acoustically treated well individually. Therefore, the optimum allocation of plant noises isrequisite to be advanced.

The gradient decent type algorithms have often been adopted in the optimal process.However, their accuracies are insufficient and tightly related to the starting point when anumerical work is performed [3]. During the past three decades, there has been a growinginterest in solving algorithm problems inspired by natural systems in physics and biology. TheSA, a stochastic relaxation technique oriented by Metropolis et al. [4] and developed byKirkpatrick et al. [5], imitates the physical process of annealing metal to reach the minimumenergy state. SA based optimization becomes excellent in seeking for a better solution withoutaccurate starting point and will get out of the local optimum. The SA has a better chance oflocating the global optimum in a near optimal manner. Besides, such location is achieved withoutthe complicated calculations in the first derivative of the objective function [6]. In this paper, SAis coupled with the equations of sound attenuation and minimal variation square in order tooptimize the allocation of equipment on constrained plant area. A quick and effectual method tominimize the sound impact along the plant boundary by using the strategy of reallocation ofequipments together and SA searching technique is thus provided in this study.

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3. THEORETICAL BACKGROUND

In this paper, an optimization of noise control for the m-noise plant (shown in Figure 1) isproposed. To evaluate the influence of equipment to the environment, an n-point monitoringstation uniformly around the boundary line has been introduced and shown in Figure 2,accordingly. For a sound wave propagating in a free sound field, the effects of distance decayand air absorption [7, 8, 9] have been considered. For a specified receiving point i, the totalsound pressure level- SPLi radiated and accumulated from m's equipments is expressed as

m SPLij /10

SPLj =10*log{~)0 } (la)j=l

SPLij = SWL j -'I'j(Rij,¢,f)-20*log{Rij)-1l (lb)

(R ..f

2 J 8'I'j(Rij,¢,f)=7.4 IJ¢ *10-

(1c)

where SPLij is the noise level at the i-th receiving point emitted from the j-th equipment, SWLj isthe sound power level of the j-th noise source, Rij is the distance from the j-th noise source to i-

th receiver, 'I'j (Rij' ¢, f) is the air's sound absorption at 20° C ,Jis the sound frequency, and

¢ is the humidity in air.

To minimize the noise influence along the plant boundary; an objective function FF, thetotal variation of sound level at receiving points with respect to the targeted noise level, is thenproposed and defined as below.

FF(x l ""'Xm;YI ,···,Ym; ZI , •.• , Zm; SPLtarget-l ,...,SPLtarget-m; SWL I , ... ,SWLm)16

= I (SPL i - SPLtarget_J2 (2)i=1

where FF is the function of coordinates, SWL (sound power level) and targeted SPL (soundpressure level) concurrently.

PLANT BOUNDARY

~.

•

~Q'2

EQ-5

~

f3EQ-j

&3Q.

m

Figure 1. Allocation plan of m-equipment plant


PH PT-2 PT-3 PT-(nl4)

• • • -- • • •PT-n. -- .PT-[(nl4)+1]

PT-(n-l). , •• - •r-I

Ri3 •I

~.I

I •• I

• .: Monitoring points

Rij •Ri4_ : the j-th Equipment

~Ri5• ,. •Ril

PT-(3n14)+1..PT-(nI2)--· . . -- ..

PT-(3n14) PT-i PT-(nI2)+l

Figure 2. Sound-monitoring system with n-receiving points

4. MODEL CHECK

To verify the reliability of SA optimization, a single-noise plant shown in Figure 3 isintroduced for pre-run purpose. The exemplified sound power level (SWL) ofEQ-l is set to 95dB(A) at 1000 Hz. By using SA optimization, the resultant allocation of equipment EQ-l and itsrelevant SPL at 1.5 meter of plant area is plotted in Figure 4. The resultant location ofEQ-l isoptimized at (24.97, 25.04, 1.5). As shown in Figure 4, the optimum location ofEQ-l isprecisely at the center of plant in which the averaged influence of sound from EQ-l to theboundary line is minimized. The SA optimization is then proved to be reliable.

Before performing SA optimal simulation on multi-noise plant, the accuracy check of thesimplified mathematical model of sound attenuation equation is performed with the simulateddata of SoundPlan, commercial sound simulating software [10]. By taking the SWL and optimalcoordinates of EQ-l into SoundPlan, the simulated noise contour map and noise levels ofreceiving points are plotted in Figure 5 and shown in Tablel.

~

Figure 3. Allocation plan of single-equipment plant

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Figure 4. The SPL distribution of optimized single-equipment plant

SPLI

•SPL20 •

SPLI9.

SPL1B.

SPLI7.

SPL2

•SPL3

•SPL4

• SPL5

•• SPL6

• SPL7

• SPUl

.SPL9

• SPLIO

PLANT BOUNDARY

Figure 5. Noise contour map of optimized single-equipment plant by SoundPlan

Table 1. Accuracy check for theoretical model and SoundPlan at receivin points [dB(A

S~~~,~~&~ SPL~ ~:7' ...'~;'y SPL6 S~~?~~f~~P~?I§~f'f,~I~)t~:l~~I~~L1~ ~~t:,!~~~:161~~~~Z:~~;.~~~~t~3 ~&wgSoundPlan 51.6 54.3 55.6 54.3 51.7 51.9 54.1 55.7 54.2 51.7 52.1 54.3 55.6 53.9 51.6 52.1 54.1 55.7 54.4 52.0

Theoretical 52.8 54.8 55.7 54.7 52.8 52.8 54.7 55.6 54.7 52.8 52.8 54.7 55.6 54.7 52.8 52.8 54.8 55.7 54.8 52.8Modelvariation +1.2 +0.5 +0.1 +0.4 +0.1 +0.9 +0.6 -0.1 +0.5 +1.1 +0.7 +0.4 +0.0 +0.8 +1.2 +0.7 +0.7 +0.0 +0.4 +0.8

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Figure 5 indicates that the simulated SPLs around plant fence are approximately in 52~65

dB(A) which is similar to that in Figure 4. As depicted in Table 1, the accuracy comparisonshows good agreement between the theoretical and simulated data for the models. The variationswith respect to twenty receiving points are among 0 ~ 1.2 dB(A); therefore, the proposedmathematical model is found satisfactory. Consequently, the model linked with numericalmethods is used for the allocation optimization ofmulti-equipment plant in the following section.

5. CASE STUDY

The noise controls of four-equipment plant and eight-equipment plant are introduced as thenumerical cases. The restricted plant area is 50 meter in length and 50 meter in width. Two typesofmonitoring systems, twenty points and thirty-six points ofmonitoring stations, are submittedinto the optimization process. To access the numerical optimization precisely and quickly, the SAmethod is adopted, accordingly. A series of cases and the related information are thereafterillustrated as below.5.1 Case I: Four-equipment plant

The allocation plan of four-equipment plant is depicted in Figure 6. The related soundpower levels of four equipments with 1.5 meter in height are given asEQ-l: SWL]=95 dB(A); EQ-2: SWL]=100 dB(A); EQ-3: SWL]=98 dB(A); EQ-4: SWL]=85dB(A).

The targeted noise level of 65 dB(A) is presumed. Additionally, the constrained conditionsin plant are

0~x]~50; 0~y]~50; 0~x2~50; 0~Y2~50; 0~x3~50; 0~Y3~50; 0~x4~50; 0~Y4~50;rgk ~ 2

PLANT BOUNDARY

~Q'2 ~EQ-4

Figure 6. Allocation plan of four-equipment plant

5.2 Case II: Eeight-equipment plantThe allocation plan of four-equipment plant is depicted in Figure 7. The related sound

power levels of four equipments with 1.5 meter in height are given asEQ-1: SWL]=95 dB(A);EQ-2: SWL1=100 dB(A);EQ-3: SWL]=98 dB(A);EQ-4: SWL]=85 dB(A);EQ-5: SWL]=95 dB(A);EQ-6: SWL]=100 dB(A);EQ-7: SWL]=98 dB(A);


EQ-8: SWL1=85 dB(A);The targeted noise level of70 dB(A) is presumed. In adciition, the constrained conditions in

plant are

0~xl~50;0~Yl~50;0~X2~50;0~Y2~50;0~X3~50;0~Y3~50;0~X4~50;0~Y4~50;0~X5~50;0~

Y5~50; 0~x6~50; 0~Y6~50; 0~x7~50; 0~Y7~50; 0~x8~50; 0~Y8~50;rgk ~ 2

PLANT BOUNDARY

~Q'2

EQ-S

~

SOm

PLANT BOUNDARY~

Figure 7. Allocation plan of eight-equipment plant

6. OPTIMIZATION

The simulated annealing (SA) algorithm, a local search process, simulates the softeningprocess (annealing) of metal. The basic concept behind SA was first introduced by Metropolis etat. [4] and developed by Kirkpatrick et at. [5]. In the physical system, annealing is the process ofheating and keeping a metal at a stabilization temperature and cooling it slowly. Slow coolingallows the particles to keep their state close to the minimal energy state. In this state, theparticles have a more homogeneous crystalline structure. Conversely, a fast cooling rate resultsin the higher distortion energy stored inside the imperfect lattice. The purpose of SA is to avoidstacking local optimal solutions during optimization.

The algorithm starts by generating a random initial solution. The scheme of the SA is avariation of the hill-climbing algorithm. All downhill movements for improvement are acceptedfor the decrement of the system energy. Simultaneously, SA also allows movement resulting inworse-quality solutions (uphill moves) than the current solution in order to escape from the localoptimum. At higher temperatures, the uphill movement changes well. On the other hand,changes occurring when going uphill are declined when the temperature drops.

To imitate the evolution of the SA algorithm, a new random solution is chosen from theneighborhood of the current solution. If the change in objective function (or energy) is negative,the new solution will be acknowledged as the new current solution. If not, the transition property(Ph(T)) of accepting the increment will be calculated by the Boltzmann's factor (Ph(T)=exp(~ / CT)); wherein the ~ , C and Tare the difference of the objective function,

Boltzmann constant and current temperature, correspondingly. To reach an initial transitionprobability of 0.5, the original temperature (To) is selected to be 0.2 [11]. If the chance is greaterthan a random number in the interval of [0, 1], the new solution will then be accepted.Otherwise, it is rejected. The algorithm repeats the perturbation of the current solution and themeasurement of the change in the objective function. Each successful substitution of the newcurrent solution will lead to the decay of the current temperature as


Tnew = kk * Toldwhere kk is the cooling rate. The process is repeated until the predetennined number (itermax) ofthe outer loop is reached.

The flow diagram of SA optimization is described and shown in Figure 8. As indicated,the objective function in SA is represented by (FF). The control parameters in SA are the kk andItermax.. The design variables {X }={Xl, ... ,Xm} consist of {[Xl, Yd, ... ,[xm,ym]} for mequipment plant.

J(=Xo;T=To .

Iter"",., kk

Select Random

r-------.I 'pointXn '·'from

neighborhoodofXil .

No

Program

terminate

Figure 8. Flow diagram of SA optimization

7. RESULTS AND DISCUSSION

7.1 ResultsThe accuracy of the SA optimization depends on the cooling rate (kk) and the number of

iteration (itermax) [12]. To investigate the influences of the cooling rate and the number ofiteration, an assessment of SA parameters of the cooling rate and the iteration is then preceded infollowing cases.7.1.1 Case I: Four-equipment Plant7.1.1.1 20-point monitoring systemA. cooling rate

Under the twenty-point monitoring system (n=20), six kinds of cooling rates (0.9, 0.92,0.94,0.96,0.98 and 0.99) are adopted at the maximal iteration number (itermax) of5000 with theinitial temperature (To) of 0.2. The results are summarized in Table 2. As indicated in Table 2, thebest result occurred at the cooling rate of 0.94. In the six cases, the calculations of SA

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optimization (run on an IBM PC - Pentium IV) are 0.59-0.68 minutes around. The relatedannealing response curve with a cooling rate of 0.94 is demonstrated in Figure 9.

Table 2. Results of FF with various cooling rates (kk) in four-equipment plant[20' .. ]-POlOt momtonn2 system

Common Control Results Elapsedparameters parameters timeTo itermax

kk xl(m) Yl(m), x2(m) y;(m) x3(m) Y3(m) xiril) Y4 FF t(Min.)

0.2 5000 0.9 13.07 23.34 22.48 26.64 32.54 30.81 9.11 20.46 1145.8 0.680.2 5000 0.92 27.69 25.6 32.62 21.91 25.07 17.82 13.43 31.71 1116.3 0.630.2 5000 0.94 21.49 22.49 25.04 30.9 27.32 24.52 35.75 18.58 1086.4 0.590.2 5000 0.96 37.52 14.41 24.36 29.18 26.56 23.3 33.39 9.6 1147.6 0.640.2 5000 0.98 21.54 29.16 26.67 24.47 29.83 31.8 36.72 4.23 1126.5 0.590.2 5000 0.99 34.31 31.27 23.92 24.73 27.71 22.0 31.66 37.72 1096.2 0.64

c 7(0)0""":;:'''''

f!:::SOOOIQ)'

~-5coo

g400J

Figure 9. Annealing response curve with a cooling rate of 0.94[To=0.2,itermax=5000,four-equipment plant, 20-point monitoring system]

B. iterationUnder the twenty-point monitoring system, four kinds ofmaximal iteration (500,5000 and

50,000) are tested at the cooling rate of 0.94. The results are summarized in Table 3. As indicatedin Table 3, the best result occurred at the iteration number of 5000. It is observed that theminimal state (optimum) will be achieved sufficiently at the iteration number of 5000. Using thesecond optimal design set, the SPL distribution within the plant area is then plotted in Figure 10.

The related SPL and its variation with respect to targeted sound pressure level of 65 dB(A)is summarized and listed in Table 4. As indicated in Table 4 that the SPL at twenty receivingpoints are 59.4-65.7 dB(A). The variation with respect to the destiny of65 dB(A) is -0.7 -+5.6.Observably, under the allocation optimization, the SPL along plant's boundary has beenminimized uniformly.


Table 3. Results of FF with various maximal iterations (itermax) in four-equipment plant[20 ]-POlOt momtonn2 system

Common Control Results Elapsedparameters parameters timeTo "kp', itermcd: ~l(in) , YICm)" ,xz(m),; Yl(m);" \:x~(m)F k!:'~~(ri)), ~~(O:t)~} :Ew:r~ (!'FE:!: t(M~)

0.2 0.94 500 13.0 23.37 23.43 28.96 27.7 31.74 33.95 13.3 1127.2 0.130.2 0.94 5000 21.49 22.49 25.04 30.9 27.32 24.52 35.75 18.58 1086.4 0.590.2 0.94 50,000 22.94 16.73 26.87 18.78 32.15 21.52 35.81 33.24 1112.4 19.66

..... ; .

80 ....

70 .

.... ~ ...

...... ..~.... ....

~ """ .

Figure 10. SPL distribution of four-equipment plant with respect to the best design set[xl=2I.49,YI=22.49,x2=25.04,Y2=30.9,x3=27.32,Y3=24.52,x4=35.75,y4=18.58]

Table 4. Variations of twenty receiving points with respect to SPLtarget = 65 dB(A)[xl=2I.49,YI=22.49,X2=25.04,Y2=30.9,X3=27.32,' 3=24.52,x4=35.75,Y4=I8.581

SPLI SPL2 SPL3 SPL4 SPL5 SPL6 SPL7 SPL8 SPL9 SPLIO SPLII SPL12I~~~I~

SPLI4 SPLlS SPLI6 SPL17 SPLI8 SPLI9 SPL20cp·, ;,' ,;;; i::" , .'!!' 'c,! .! ,@},;

Theoretical 59.4 61.2 62.6 62.6 61.2 61.2 63.9 65.7 64.7 62.3 62.3 64.4 64.8 62.9 60.4 60.4 61.7 61.9 61.0 59.4SPLTargeted 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0SPLvariation +5.6 +3.8 +2.4 +2.4 +3.8 +3.8 +1.1 -0.7 +0.3 +2.7 +2.7 +0.6 +0.2 +2.1 +4.6 +4.6 +3.3 +3.1 +4.0 +5.6

7.1.1.2 36-point monitoring systemTo receive more accurate control along the plant bOUfidary, thirty-six monitoring point

have been Ufliformly set along the fence. Six kinds of cooling rates (0.9, 0.92, 0.94, 0.96, 0.98and 0.99) are also adopted at the maximal iteration number (itermax) of5000 with the initialtemperature (To) of 0.2. The results are summarized in Table 5.

As indicated in Table 5, the best result occurred at the cooling rate of 0.92. In the six cases,the calculations of SA optimization (fUfi on an IBM PC - Pentium IV) are 0.92~1.64 minutes.The related armealing response curve with a cooling rate of 0.92 is demonstrated in Figure 11.

Using the second optimal design set, the SPL and its variation with respect to targetedsOUfid pressure level of 65 dB(A) is summarized and listed in Table 6. The SPL distribution


within plant area is then plotted in Figure 12. As indicated in Table 6 that the SPL at twentyreceiving points are 59.6~65.4 dB(A). The variation with respect to the destiny of 65 dB(A) is0.4 -+5.4. Evidently, under the allocation optimization, the SPL along plant's boundary has beenfurthermore minimized uniformly.

Table 5. Results of FF with various cooling rates (kk) in four-equipment plant[36-point monitorin2 systeml

Common Control Results Elapsedparameters parameters time

To itermax kk ..... xl(m) YI(m) . x2(m) YI(m) x3(m) Y3(m) x4(m) Y4 e. fE.." t(Min.)

0.2 5000 0.9 13.07 23.34 22.48 26.64 32.54 30.81 9.11 20.46 2172.3 1.640.2 5000 0.92 27.69 25.6 32.62 21.91 25.07 17.82 13.43 31.71 2113.7 1.200.2 5000 0.94 15.83 32.75 19.81 21.26 23.73 29.65 7.44 14.95 2139.1 1.310.2 5000 0.96 26.58 37.82 21.37 33.72 23.77 24.92 28.96 18.17 2156.4 1.340.2 5000 0.98 30.87 24.47 25.87 20.39 14.52 27.18 8.89 15.45 2163.6 0.920.2 5000 0.99 21.54 29.16 26.67 24.47 29.83 31.8 36.72 4.23 2125.9 0.98

Obective Function values yersus Iterations •10000,---------------------.,

Figure 11. Annealing response curve with a cooling rate of 0.92[To=O.2, itermax=5000, four-equipment plant, 36-point monitoring system]

Table 6. Variations of thirty-six receiving points with respect to SPLtarget = 65 dB(A)[XI=27.69,YI=25 .6,x2=32.62,Y2=21.91 ,x3=25.07,Y3=17.82,x4=13.43,Y4=31.71]

Theoretical 59.6 60.6 61.4 62.1 62.5 62.5 62.1 61.5 60.7 60.7 62.0 63.1 64.3 64.9 64.9 64.4 63.4 62.2SPLTargeted 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0SPLvariation +5.4 +4.4 +3.6 +2.9 +2.5 +2.5 +2.9 +3.5 +4.3 +4.3 +3.0 +1.9 +0.7 +0.1 +0.1 +0.6 +1.6 +2.8

SPL19 SPL20 SPL21 SPL22 SPL23 SPL24 SPL25 SPL26 SPL27 SPL28 SPL29 SPL30 SPL31 SPL32 SPL33 SPL34 SPL35 SPL36,; . . ..,.,. .<;. .' '.'. . . ,. 'i\;, <,"hi;" iT """ X.;;;,> 1'::"11'+

Theoretical 62.2 63.5 64.6 65.4 65.3 64.6 63.4 62.1 60.9 60.8 61.6 62.2 62.5 62.4 62.0 61.4 60.5 59.6SPLTargeted 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0SPLvariation +2.8 +1.5 +0.4 -0.4 -0.3 +0.4 +1.6 +2.9 +4.1 +4.2 +3.4 +2.8 +2.5 +2.6 +3.0 +3.6 +4.5 +5.4


Figure 12. SPL distribution of four-equipment plant with respect to the best design set[xl=27.69,Yl=25.6,X2=32.62,Y2=2I.9I,X3=25.07,y3=I7.82,x4=I3.43,Y4=3I.7I]

7.1.2 Case II: Eight-equipment PlantAccording to the simulated results shown in section 7.1.1, it is noticeable that more

monitoring points around plant boundary will result in more accurate noise control. Besides, theiteration number of 5000 will be sufficient to reach the minimal objective value. Therefore, the36-point monitoring system together with iteration of 5000 and cooling rate of 0.92 are appliedin the optimization of eight-equipment plant. The optimal result is summarized and shown inTable 7.

Table 7. Results of FF with various maximal iterations (itermax) in eight-equipment plantTo=O.2, kk=0.92, itermax=5000, 36- oint monitorin s stem

Results

Using the optimal design set, the SPL and its variation with respect to targeted soundpressure level of70 dB(A) is summarized and listed in Table 8. The SPL distribution within plantarea is then plotted in Figure 13. As indicated in Table 8 that the SPL at thirty-six receivingpoints are 62.9~70.2 dB(A). The variation with respect to the destiny of 65 dB(A) is -0.2 ---+7.1.It is found that the variation for each receiving point has been minimized more homogeneouslyunder the allocation optimization.


X5= ,y5= ,X6= ,y6= ,X7= ,Y7= ,X8= ,Y8=SPLI SPLZ SPL3 SPL4 SPL5 SPL6 SPL7 SPL8 SPL9 SPLJO SPLlI SPLlZ'SPLI3 SPLI4 SPLI5 SPLI6 SPLI7 SPLI8

""

;

Theoretica~ 6Z.9 63.6 64.4 65.0 65.4 65.7 65.7 65.Z 64.3 64.4 66.0 67,6 68.4 67.9 66.9 66.0 65.Z 64.3SPLTargeted 70,0 70.0 70,0 70.0 70.0 70.0 70,0 70.0 70.0 70.0 70.0 70.0 70.0 70.0 70.0 70.0 70.0 70.0SPLvariation +7.1

j~+5,6 +5.0 +4,6 +43 +43 +4.8 +5.7 +5.6 +4,0 +Z.4 +1.6 +Z.I +3.1 +4,0 +4.8 +5,7

!.}/,LI ~ SPLZI SPL22 IW ~~~ urJ....· .,PLZ5 SPL2'6 SPL27 8PL2& SPL29 SPBO SPLll' SP£32 SPL33 SPL34 SPL35 SPL36': , "

) ,. ~ i

Theoretical 64.4 65,6 67.0 68,5 69,8 70,Z 69.5 68.0 66.4 66.3 67.5 68.Z 68.0 67.3 66.Z 65,0 63.9 6Z.9SPLTargeted 70,0 70.0 70,0 70,0 70.0 70,0 70.0 70.0 70.0 70.0 70.0 70,0 70.0 70.0 70.0 70.0 70.0 70.0SPL ,,0,

variation +5,6 +4.4 +3.0 +1.5 +O.Z -0.2 +0.5 +Z.O +3.6 +3.7 +Z.5 +1.8 +Z.O +2.7 +3.8 +5.0 +6.1 +7.1

Table 8. Variations of thirty-six receiving points with respect to SPLtarget = 70 dB(A)in eight-equipment plant[xI=29.01,y1=18.11,xz=34.04,Yz=10.22,x3=42.92,Y3=39.34,x4=36.47,y4=26.36,

2727 1167 3489 3225 4344 3775 2955 191~

Figure 13. SPL distribution of eight-equipment plant with respect to the best design set[xI=29.01,YI=18.11,xz=34.04,yz=10.22,x3=42.92,Y3=39.34,x4=36.47,y4=26.36,x5=27.27,y5=11.67,x6=34.89,Y6=32.25,X7=43.44,Y7=37.75,x8=29.55,Y8=19.14]

7.2 DiscussionAs described in Section 6.1, both the cooling rate (kk) and the maximal iteration (itermax)

play essential roles in the SA optimization. The excellent value of the cooling rate (kk) is foundat 0.92~0.94. In addition, the sufficient accuracy of the SA will be achieved at the iteration(itermax) of 5000 being selected in the optimization process.

As Tables 4 and 6 reveal that the variation ofreceiving point in 36-point monitoringsystem is consistently less than that in 20-point monitoring system for the four-equipment plant.It denotes that more monitoring points will result in higher accuracy ofnoise control valuesalong the plant's border line.

By using the parameters of itermax and cooling rate together with the best 36-point

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monitoring system, the optimized allocation for eight-equipment plant has been performed andshown in Table 7. The related variation with respect to 36 receivers is shown in Table 8. It isindicated in Table 8 that the variation range is among -0.2 ~+7.1 dB(A). It is recognizable thatthe noise control of receiving points with the targeted noise level of70 dB(A) or below is exactlyachieved.

As the optimized SPL distribution line shown in Figures 10, 12 and 13, the noise sourcesare reasonably position toward to the center ofplants. Additionally, the noise levels along plantboundary are depressed equivalently.

8. CONCLUSION

It has been shown that both SA can be utilized in the optimization of plant's noise controlby adjusting the locations of equipments within the constrained plant area. Both the cooling rateand the iteration number play essential roles in SA optimization. By increasing the monitoringpoints around the plant, the variation of controlled receivers' noise with respect to the specifiednoise limit can be equally depressed.

Noise control is important and essential in a manufacturing factory. In this paper, a noveltechnique of simulated annealing (SA) is applied in the numerical optimization, and the multinoise plant with various sound monitoring systems is also introduced. The proposed optimizationapproach to the allocation of multi-noise plant definitely provides an innovative scheme usingsimulated annealing in determining the best allocation plan for multi-noise plant under boundaryconstraints with profound insight.

ACKNOWLEDGEMENTS

The authors would also like to thank the anonymous referees who kindly provided thesuggestions and comments to improve this work.

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