M00300

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Presented on IInntteerrnnaattiioonnaallSSyymmppoossiiuummoonnCCoommppuuttaattiioonnaallSScciieennccee((IISSCCSS) )

May 15, 2012 - May 16, 2012, Universitas Gadjah Mada

Page 1

H.A Parhusip1_{, and Yohanes Martono}2

1_{Center of Applied Science and Mathematics, University of Satya Wacana Christian} Jl.Diponegoro 52-60, Salatiga-Central Java 50711

2_{Department of Chemistry, University of Satya Wacana Christian} Jl.Diponegoro 52-60, Salatiga-Central Java 50711

email: [email protected]

Abstract. Optimization of colour reduction for producing stevioside syrup is presented here. Some observations have been produced and used here to be analyzed by using logistic model to pose the optimization. The parameters in the logistic function are determined by least square which are found the same values for all sets of the observed data. There are 7 types of absorbances denoted by B, K, A, S, BK, BKA, and BKAS. The first four refer to Bentonite, Caoline, Active carbon and Silica gel respectively. The last three notations refer to the mixtures of B and K, B,K and A, and all types. Ant colony algorithm is the used optimization method to get the minimum colour reduction with a maximum steviode content. It is found that type A (active carbon) is the best absorbance to use for producing optimal stevioside content with minimum colour reduction.

Keywords:stevioside, logistic, least square, ant colony algorithm

1. INTRODUCTION

Most of this section is one section of applications in linear algebra which containts a summary of previous reseach in stevioside[1]. Leaves of Stevia rebaudiana Bertoni have been extracted in Chemisty Department of Science and Mathematics Faculty, SWCU in January – February 2012. Conventional extraction method for stevioside from S. rebaudiana leaves was conducted. Modification of extraction may be found in [2] where effects of pressure and temperature were evaluated. According to the used solvents, methanol produced higher extraction compared to ethanol and acethon [3] where behaviour of temperatures were also observed.

In the first research of our case[4], all experiments were dedicated to the relation of mass and time using a quadratic function. Procedures of minimization problem have been employed to find parameters in the objective function. We assume that the percentage of stevioside follows

 

   



₍ ₎2 ₍ ₎2

) ,

(t m t m

S := S model

where t and m denote time and mass respectively and the parameters



,





are determined due to the given data. Standard least square leads to minimize the residual function, i.e









mod , ,



 

 n

el i da ta

i S

R   =





2 1

2 2

, ( )



_n     

i i

da ta

i t m

S    .

The critical conditions require





0 

. Solving this nonlinear system, one yields





,



,





T= (0.4201. 0.8696.- 0.0688)T_{. The illustration of this function is depicted on} Figure 1. To get the maximum percentage stevioside from the given minimum mass has been the interest of the first research. This leads to a minimax problem in optimization,i.e

)

(

max

min

S x



(2)

As one kind of sweeteners, stevioside is promising. One good news of using stevioside is that it is not disturbing fertility and reproduction of a user which was investigated for mice [5].

Figure 1. S(t,m)(t0.4201)2(m0.8696)20.0688 [4]

Chemical composition of dried Stevia plants has been studied into detail [6] such as carbohydrates (61.93% d.w.), protein (11.41% d.w.), crude fiber (15.52% d.w.), minerals (K, 21.15; Ca, 17.7; Na, 14.93 and Mg, 3.26 mg/100 g d.w. and Cu, 0.73; Mn, 2.89; Fe, 5.89 and Zn, 1.26 mg/100 g d.w.) also essential amino acids. As the the strongest natural sweeteners,

“Fruit tea with Stevia” was produced and proved to be fast and friendly to the environment

by minimization of organic solvent consumption[7]. The near infrared reflectance spectroscopy (NIRS) was used to analyse stevioside concentration in Stevia leaves[8]. Although less accurate, The writers suggested that NIRS was a precise and simple method for routine stevioside determination in Stevia leaves. The other authors used hydrolysis and esterification for extraction of stevioside and then stevioside was evaporated to dryness and dissolved in methanol for quantitative analysis by high-performance liquid chromatography (HPLC). The recovery of steviosides from the leaves by methanol extraction was 90%[9].

Since positive impact of using stevioside, research on stevioside is becoming attractive and one needs to produce stevioside into easily and savely consumed such as syrup. The observation is done by reducing its colouring and having maximum stevioside content. Stevioside content here means that the amount of intensity of light absorbed by stevioside (dimensionless). This paper presents this optimization problem and a decision for choosing the best absorbance that can be used for further fabrication.

2.THE USED DATA

2.1 Materials

There are 3 samples taken from 3 populations where each set of samples is treated with the same conditions. We employ wavelength





291

nm and





215

nm provided by spectrophotometer to do coloumn chromatography of 7 types of absorbances. These types of absorbances denoted by B, K, A, S, BK, BKA, and BKAS. The first four refer to Bentonite, Caoline, Active carbon and Silica gel respectively. The last three notations refer to the mixtures of different absorbances taken from B, K, A or S. The experiment uses water for initial extraction and then uses ethanol 50% for futher extraction . Its temperature is mantained about

50

0C. The results of measurements are shown in the Table 1 and Table 2 respectively.

This paper adressess on determination of optimal stevioside that can be obtained by maitaining colour reduction in the syrup of stevioside.

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Table 1. Means and standard deviations of colour reduction measured in 291 nm wavelength. Observed by Y. Martono, at Chemisty Lab. Science and Mathematics Faculty, SWCU, January-February 2012.

Sample I Sample II Sample III

Type of absorbance

Mean colour reduction

Standard Deviation

Mean colour reduction

Standard Deviation

Mean colour reduction

Standard Deviation

Basic

Extraction 37.600 0.024 22.233 0.023 30.783 0.008

Neutral 4.343 0.010 9.210 0.019 5.283 0.012

B 1.105 0.040 0.572 0.046 1.259 0.010

K 1.301 0.020 0.661 0.089 1.248 0.021

A 1.470 0.086 0.778 0.093 1.481 0.095

S 1.326 0.024 0.675 0.065 1.206 0.014

BK 1.283 0.027 0.655 0.048 1.201 0.067

BKA 1.254 0.026 0.640 0.049 1.216 0.078

BKAS 1.1 0.039 0.569 0.047 1.142 0.031

Table 2. Means and standard deviations of stevioside content measured in 215 nm wavelength. Observed by Y. Martono, at Chemisty Lab. Science and Mathematics Faculty, SWCU, January-February 2012.

Sample I Sample II Sample III

Type of absorbance

Mean steoviside content

Standard Deviation

Mean steoviside content

Standard Deviation

Mean steoviside content

Standard Deviation

Basic extraction

6.267 0.080 0.000 0.000 1.700 0.031

Neutral 0.740 0.052 0.710 0 0.717 0.008

B 0.165 0.043 0.040 0.050 0.117 0.033

K 0.150 0.017 0.000 0.000 0.125 0.035

A 0.238 0.043 0.212 0.160 0.234 0.031

S 0.163 0.012 0.075 0 0.095 0.039

BK 0.187 0.070 0.000 0.000 0.111 0.056

BKA 0.170 0.011 0.000 0 0.132 0.062

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2.2 Procedure

Parameter Determination

Assume that the colour reduction (y) is a logictic continuous function of stevioside content (x) ,i.e

))

1 (

))

1 (

,

)

(

x y

x y K A Ae

K K x

y _kx









_ , x

where K and k must be determined. This model has been used for rate of growth diameter of Kailan [10] where t was the independent variable. In this paper, we claim that the stevioside content is the independent variable. We scale data into [0,1] by dividing each data by the maximum of each coloum. We employ least square method to find the best estimation of K and k. We get K=1.1549;k=1.6226 with error 6.7232%. Thus

))

1 (

))

1 (

,

1549

.

1 1549

.

1 )

(

₁_.₆₂₂₆

x y

x y K A Ae

y _x









_ . (1)

This function is illustrated in Figure 1 and each value of the approximation is shown in Table 3. One may observe that the approximation is good enough due to small error.

Since we need the minimum of colour reduction, we pose an optimization problem, i.e minimize y(x). In this case, we use an ant colony algorithm proposed by Rao [11] which is one of modern algorithms in optimization. Its theory is developed by studying ant colonies behaviour in nature. As shown in nature, two ants will meet each other due to the present of pheromone left in paths which passed by ants. However, some pheromone will evaporate naturally. The percentage of pheromone in the paths and in the source of foods will be determined in the algorithm stochastically. The initial number of ants denotes the initial guesses of optimizers. Next section will show the used of an ant colony algorithm for our particular case, i.e minimize y(x) which is given by equation (1). One of advantages of using this algorithm is that it does not require gradient computational effort which is normally done in standard iteration methods such as Newton , Broyden methods and their modifications.

Table 3.

Data of colour reduction and its approximation

data

Approximation 1.105 1.2774

1.301 1.2596 1.470 1.3466 1.326 1.2751 1.283 1.3011 1.254 1.2830 1.100 1.0782

Figure 1. Comparison colour reduction given by data (darker histogram) and approximation.

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Ant colony algorithm(ACO) for logistic function

This algorithm has been used to solve some wellknown optimization problems succesfully, such as a travelling salesman problem [12] and quadratic optimization problem. The solutions are generated by the ants using local search algorithms. The experiment results show that the algorithm proposed in this study can substantially increase the convergence speed of the ACO. Our paper here review the standard ACO as follows.

Step 1.

Assume that there exist N ant colonies. The value N presents the initial guesses of our optimizers. Since an objective function can be multivariables, let us denote the number of variables to be n. Thus, each variable will have N colonies. One also assumes that the amout of pheromone



ij l =1 initially where index i denotes for the i-th variable, j denotes the j-th colony and superscript (l) denotes the l-th iteration. Thus l =1 in this step.

Step 2.

(a) Each colony may have different path to reach the source of foods. Therefore one needs to determine the probability of each colony to choose its path. Assume that this probability for

i-th variable and j-th colony is given by pijwhere



_  p

m l im l ij ij

1 ) ( ) (



; i=1,2,...,n; j=1,2,...,p (2)

(b) The chosen paths of k-th ant can be determined by random numbers (to guarantee the fairness in nature) and uniformly distributed over [0, 1]. Therefore we may have cumulative probability for each different path. Therefore we have N numbers of random numbers

r r

r₁, ₂,..., in the interval (0,1) for each value denotes probability of each ant. Step 3.

Compute the objective function for each colony, i.e

) ( (k)

k f X

f  , k = 1, 2, . . . ,N

We need to choose the best and the worst paths from all colonies, ie

 

k f

,..., 2 , 1

best



_

min

and

 

k f

,..., 2 , 1

worst



_

min

Step 4 .

Now we need to proceed a test of convergent. There exists no theorem to guarantee that this algorithm always converges. However there are many practical examples that this algorithm works well. It is only assumed that this algorithm converges if all ants pass the best path. If this convergency can not be achieved, all ants will go back to their home and start again to find foods. This procedure means that we restart the iteration by adding the number of iteration, started from l to be l +1. In this case, we need to renew the amount of pheromone, i.e









k k ij ij



() (old)

where



ij(old)denotes the content of pheromone on the previous iteration n and which is left after evaporation , i.e





( 1) )

old (

1 

 l





The symbol 



ijkdenotes the deposited pheromone by ants from the best path. Note that only 1 path ij (from p possibilities) for i-th variable. Evaporation rate of pheromone (



) is assumed in the interval (0.5,0.8) [6] and deposited pheromone 



ijkis formulated as

(6)

 

 

 

otherwise 0;

tour best global )

, ( ; worst

best

)

( f if i j

f k ij





By using new value of



ij(l), one may continue to step 2,3 and 4 until the solutions converge. Practically, one needs to replace this convergent condition by setting the maximum number of iteration. The used parameters are =2 and



=0.5 recommended by Rao [11]as the used literature here.

Some improvements in the ACO have been done by some authors. In [13], the all data come from UCI machine learning data-base were clustered and where probability in equation(2) was modified. The data of Car, Soybean, Voting, Zoo and Nursery were the five main collected data . We can not used this approach since the number of used data in this paper is considerable small. Rao does not include the ant algorithm for optimizing a problem subject to a given constraint[11]. One may refer to the modification of ant colony algorithm which considers an optimization problem with multiconstraints [14].

3. RESULTS AND DISCUSSION 3.1 Analysis parameters of K and k

The obtained parameters K* and k* will be examined, i.e we need to guarantee that the sum square error is minimized which is formulated by following the standard least square,i.e





1 1,data 1,data data , 1 data

)

,

(



 

















kx i

i e y K Ky

Ky y

k K

R .

To present the analysis more conveniently, let us denote G



to be the vector coloum that each its component is the component in the summation of R,i.e G





g1,...,gn



Twith





_















i i _kx_i

e y K Ky

Ky y

k K g g

data , 1 data

, 1

data , 1 data

)

,

(

Thus we can write R













. We have known that

k R K

R R















with

 K

G G K

 

 

2 and

k R



 ₌ _G _G

 

 

2 . We need ₂

K R



, 2 2

k R



_and

k K



to compute since these are the entries of the Hessian R,i.e

















2 2 2

k R k

K R

k K

R K

H_R .

If this matrix is a positive definite matrix (i.e ₂

0 



and determinant of H_R



0

) then the optimal K,k are minimizers of R [15]. We have

 

G G

G G K

K K K

















2 .

(7)

 

G G G G k R k k k k

















₂

_.

2 2

; (3b)

 



G G G G



k K R K k K k



















2

. (3c)

All necessary formulas are shown below. The i-th component of _KG is

 

i KG





 



_





_





data , 1 data , 1 data , 1 data , 1 data , 1 data , data , data , 1 i i i kx kx i kx i i i K e y K Ky e y K Ky y y e y y K g G   









































data , 1 data , 1 data , 1 data , data , 1 data , 1 data , data , data , 1 i i i i kx kx i kx i kx i e y K Ky e y y y e y y e y y K    











Let us denote y1,datay,data e yi,data y1,data kx i i i    



. It is independent of K, but it is dependent of k, hence

 

_

_



_



_



data , 1 data , 1 data , 1 data , data , 1 data , 1 data , 1 i i i i kx kx i kx i kx i i i K e y K Ky e y y y e K e y K Ky K g G    





















. (4a)

Similarly,

 



_





_





data , 1 data , 1 data , 1 data , 1 data , data , 1 data , i i i i kx kx i i kx i kx i i i k e y K Ky e y K Ky y y x e y e Kx k g G    





























 









data , 1 data , 1 data , 1 data , data , 1 data , 1 data , data , data , 1 i i i i kx i kx i kx i kx i e y K Ky x e y K y y e y y e y y K    











Simplified, we get

 



_

_



_

i kx kx i i i i i i k e y K Ky e y y x y Kx k g G _           data , 1 data , 1 data , data , 1 data ,  +

















data , 1 data , 1 data , data , 1 2 data , 1 i i i kx i kx kx i i e y K Ky y y e e K y K x   









. (4b)

Therefore differentiating with respect to K and k once again for Eq.(a) and Eq.(b) respectively we get as follows.

 

2 K g K g K

G _i i i

K K









































data , 1 data , 1 data , 1 2 data , 1 data , 1 data , 1

2

i i i i kx kx kx kx i e y K Ky e y e y K Ky e y    













. (4c)

Futhermore

 





2 1 2

2 S S k g G i i k













(4d)

with





 









_            

   _ 2 

data , 1 data , 1 2 data , 1 data , data , 1 data , 1 i i i i kx kx i kx kx i i i i i e y K Ky Ke x Ke Ky e x y y x y Kx S

















data , 1 data , 1 data , data , 1 data , 1 2 2

2

i i i i kx i kx i kx e y K Ky y y e K y K e x S   











(8)

















data , 1 data , 1 data , data , 1 2 data , 1 2 2

2

i i i ii kx i kx i kx e y K Ky y y e K y K e x   











. Finally, 2 1 2 P P k K g_i







(4e)

with









data , 1 data , 1 data , 1 1 i i kx kx i i e y K Ky e y x P  











;

and



 











data , 1 data , 1 data , data , 1 2 data , 2 data , 1 2

2

i i i i kx i kx i kx i i i kx i e y K Ky y y e x Ke x y y e x P    





























data , 1 data , 1 data , 1 data , 1 data , data , 1 2 i i i i kx kx kx i kx i i e y K Ky e y K e y y y e K x            _.

All formulas in equations (4a)-(4e) are ready to use in equations (3a)-(3c). Finally, we are allowed to compute each component of the Hessian matrix R where the optimal values K

and k (K*=1.1549;k*=1.6226) are substituted. One yields

 

_ _ 0.0754 0.1947 -0.1947 0.5321 *) *, (K k R

H .

It is obviously the Hessian matrix R is positive definite which means that K*=1.1549 and

k*=1.6226 indeed minimize the error as we expect. 3.2Analysis of optimal solution

Since we have 3 sets of samples, we get 3 pairs values of (K,k) (see Table 4). We conclude that all sets of samples behave nearly the same since each pair of parameters have the same values with tolerable errors (less than 15%). Thus we have (K*,k*)=( 1.1549, 1.6226) for all samples. Note that all used data are already dimensionless. We continue to find the minimum of y(x) as the purpose of this research,i.e

Minimize

))

1 (

))

1 (

,

1549

.

1 1549

.

1 )

(

₁_.₆₂₂₆

x y x y K A Ae x

y _x









 subject to x≥ 0.04 (*)

where 0.04 is obtained from the given data. The ant colony algorithm is employed to the problem (*) and the optimal solution is shown in Table 5.

Table 4. Parameters in y(x) (Eq.1) for each set of samples

Data K K Error (%)

Sample I 1.1549 1.6226 6.7232 Sample II 1.1549 1.6226 14.5822 Sample III 1.1549 1.6226 7.7947

Table 5.The optimal solutions

Data x

optimal

minimum Sample

0.2380 1.0782 Sample

0.0400 0.5207 Sample

III

0.2340 1.2760

Though the set of samples II has the same value of (K,k) , we get significantly different optimal solutions compared to the sets of samples I and samples III. Therefore we will neglect the samples II for further analysis. We present the optimal solution as intervals, i.e x*=( 0.234, 0.238) and y*=(1.0782, 1.2760). According to the value of x*, the chosen absorbance to use is type A. The decision for choosing the best absorbance can not be relied on the optimal value of y* since the used algorithm firstly searchs the optimizer rather than

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the optimal function. The result is also practically reasonable compared to the Table 2 that the optimizer x* is produced by type A.

Though the gradient is it not required to be computed, it is good to calculate it since the objective function is only one variable. The first and second derivatives of the objective function are



_



2



_





kx kx

Ae K KkAe dx

and ₂ 2







2 



1











kx kx  kx kx 

Ae K Ae Ae

K Ae Kk dx

y d

. Therefore ₂

0 

dx y d

can be obtained if

2

kx





kx



1



1 

0

Ae K

Ae in the optimum point.

One may observe that in x*,

2

Aekx*





Aekx*



1



1

< 0 as we expected. The optimum result is still local optimizer, since the condition depends on the value of x*. This analysis validates the practical expectation where the obtained x* is also maximum value from the given data.

5. CONCLUSION

This paper has shown optimization on colour reduction of producing stevioside syrup by maximizing the stevioside content. The logistic function is used to present colour reduction as a function of stevioside content. The ant colony algorithm is revisited to introduce the optimization procedures. Three sets of data are analysed which produced by 7 types of absorbances. These data lead to nearly the same values of parameters in the logistic function. According to the results, one set is neglected since its optimizer differ significantly compared to the other two sets. These sets have reasonable optimizers. The Active carbon (type A) is considered to be the best absorbance. However, since its price is more expensive compared to Bentonite and Caoline, one needs to reconsider of using this result for future fabrication.

Acknowledgement

This paper is a short term research supported by Research Center in Satya Wacana Christian University.

REFERENCES

[1] Parhusip, H.A , 2012, Various Applications of Linear Algebra, to appear on proceeding

Seminar Aljabar 2012 UNDIP, Semarang.

[2] Choi, Y.H., Kirn, I., Yoon, K.D, Lee,S.J, Kim,C.Y, 2002, Supercritical Fluid Extraction and Liquid Chromatographic-Electrospray Mass Spectrometric Analysis of Stevioside from Stevia rebaudiana Leaves,Chromatographia,Vol 55,No.9.

[3] Buchori, L., 2007, Pembuatan Gula Non Karsinogenik Non Kalori Dari Daun Stevia,

Reaktor, Vol. 11 No.2, Desember 2007, hal.57-60.

[4] Parhusip, H. A dan Y. Martono, 2011, Kadar Steviosida Maksimum pada Waktu dan Massa yang Minimum, prosiding Sem. Nas FSM, ISSN:2087-0922,Vol. 2, No. 1,645-650, FSM.

[5] Yodyingyuad, V and Bunyawong, S,1991, Effect of stevioside on growth and reproduction , Hum. Reprod. Vol.6,158-165.

[6] Arab.A.E.A,Arab, A.A.A,Salem, M.F.A, 2010, Physico-chemical assessment of natural sweeteners steviosides produced from Stevia rebaudiana bertoni plant, African Journal of Food Science Vol. 4,No.5,pp. 269- 281, Academic Journals.

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[7] Vanek, T., Nepovı, A., Vali, P., 2001, Determination of Stevioside in Plant Material and Fruit Teas, Journal of Food Composition and Analysis,Vol. 14, No. 4, pp. 383–388. [8] Nishiyama ,P., Alvarez,M., Vieira,L.G.E., 1992, Quantitative analysis of stevioside in

the leaves of Stevia rebaudiana by near infrared reflectance spectroscopy, Journal of the Science of Food and Agriculture,Vol. 59, No. 3, pp. 277–281.

[9]

Rajasekaran , T., Giridhar , P., Ravishankar, G.A.,2007, Production of

steviosides in ex vitro and in vitro grown Stevia rebaudiana Bertoni,

Journal of

the Science of Food and Agriculture

,

Vol. 87, No. 3, pp. 420–424.

[10] Parhusip, H.A.,2010. Crown Diameter of Phythoremidiation Agent by Logistic Model, proceeding on International Conference on Biotechnology and Climate Change by Logistic Model, UNS.

[11] Rao, S.S, 2009, Engineering Optimization, Theory and Practice, John Wiley & Sons, Inc., Hoboken, New Jersey,pp.714-722.

[12] Hlaing, Z.C.S, Khine, M.A, 2011, An Ant Colony Optimization Algorithm for Solving Traveling Salesman Problem, International Conference on Information Communication and Management, IPCSIT, 16, IACSIT Press, Singapore.

[13] Dai,W., Liu,S, and Liang,S, 2009, An Improved Ant Colony Optimization Cluster Algorithm Based on Swarm Intelligence., Journal of Software, Vol.4, No.4, Academic Publisher.

[14] Wang, H, Shi,Z., Ma, J.,A, 2006, Modified Ant Colony Algorithm for Multi-constraint Multicast Routing, IEEE,Vol.1,4244-0463.

[15] Peressini, A.L, Sullivan, F.E., Uhl,J. 1988. The Mathematics of Nonlinear Programming, Springer Verlag, New York, Inc.

(11)

Presented on IInntteerrnnaattiioonnaallSSyymmppoossiiuummoonnCCoommppuuttaattiioonnaallSScciieennccee((IISSCCSS)) May 15, 2012 - May 16, 2012, Universitas Gadjah Mada

(1)

 

 

 

otherwise 0;

tour best global )

, ( ;

worst best

)

( f if i j

f k ij





By using new value of



ij(l), one may continue to step 2,3 and 4 until the solutions converge.

Practically, one needs to replace this convergent condition by setting the maximum number of iteration. The used parameters are =2 and



=0.5 recommended by Rao [11]as the used literature here.

3. RESULTS AND DISCUSSION 3.1 Analysis parameters of K and k

The obtained parameters K* and k* will be examined, i.e we need to guarantee that the sum square error is minimized which is formulated by following the standard least square,i.e





1 1,data 1,data

data , 1 data

)

,

(



 

















kx i

e y K Ky

Ky y

k K

R .

To present the analysis more conveniently, let us denote G



to be the vector coloum that each its component is the component in the summation of R,i.e G





g1,...,gn



Twith





_















i i _kx_i

e y K Ky

Ky y

k K g g

data , 1 data

, 1

data , 1 data

)

,

(

Thus we can write R













. We have known that

k R K

R R















with

 K

G K

 

 

2 and

k R



 ₌ _G _G

 

 

2 . We need ₂

K R



, 2

k R



_and

k K



to compute since these are the entries of the Hessian R,i.e

















2 2 2

k R k

K R

k K

R K

H_R .

If this matrix is a positive definite matrix (i.e ₂

0 



and determinant of H_R



0

) then the optimal K,k are minimizers of R [15]. We have

 

G G

G G K

K K K

















2 .

(2)

 

G G G G k R k k k k

















₂

_.

2 2

; (3b)

 



G G G G



k K R K k K k



















2

. (3c)

All necessary formulas are shown below. The i-th component of _KG is

 

i KG

 

 



_





_





data , 1 data , 1 data , 1 data , 1 data , 1 data , data , data , 1 i i i kx kx i kx i i i K e y K Ky e y K Ky y y e y y K g G   









































data , 1 data , 1 data , 1 data , data , 1 data , 1 data , data , data , 1 i i i i kx kx i kx i kx i e y K Ky e y y y e y y e y y K    











Let us denote y1,datay,data e yi,data y1,data kx i i i    



. It is independent of K, but it is dependent

of k, hence

 

_

_



_



_



data , 1 data , 1 data , 1 data , data , 1 data , 1 data , 1 i i i i kx kx i kx i kx i i i K e y K Ky e y y y e K e y K Ky K g G    





















. (4a)

Similarly,

 



_





_





data , 1 data , 1 data , 1 data , 1 data , data , 1 data , i i i i kx kx i i kx i kx i i i k e y K Ky e y K Ky y y x e y e Kx k g G    





























 









data , 1 data , 1 data , 1 data , data , 1 data , 1 data , data , data , 1 i i i i kx i kx i kx i kx i e y K Ky x e y K y y e y y e y y K    











Simplified, we get

 



_

_



_

i kx kx i i i i i i k e y K Ky e y y x y Kx k g G _           data , 1 data , 1 data , data , 1 data ,  +

















data , 1 data , 1 data , data , 1 2 data , 1 i i i kx i kx kx i i e y K Ky y y e e K y K x   









. (4b)

Therefore differentiating with respect to K and k once again for Eq.(a) and Eq.(b) respectively we get as follows.

 

2 K g K g K

G _i i i

K K









































data , 1 data , 1 data , 1 2 data , 1 data , 1 data , 1

2

i i i i kx kx kx kx i e y K Ky e y e y K Ky e y    













. (4c)

Futhermore

 





2 1 2

2 S S k g G i i k













(4d)

with





 









_            

   _ 2 

data , 1 data , 1 2 data , 1 data , data , 1 data , 1 i i i i kx kx i kx kx i i i i i e y K Ky Ke x Ke Ky e x y y x y Kx S

















data , 1 data , 1 data , data , 1 data , 1 2 2

2

i i i i kx i kx i kx e y K Ky y y e K y K e x S   











(3)

















data , 1 data

, 1

data , data , 1 2

data , 1 2

2

i i i

kx i kx

i kx

e y K Ky

y y e K y

K e x

  











Finally,

2 1 2

P P k K

g_i







(4e)

with









data , 1 data

, 1

data , 1 1

i i

kx kx i

e y K Ky

e y x P

 











;

and



 











data , 1 data

, 1

data , data , 1 2 data

, 2 data , 1 2

2

i i

i kx

i kx i i i

kx i

e y K Ky

y y e x Ke

x y

y e x P

  





























data , 1 data

, 1

data , 1 data

, 1 data , data , 1

i i

kx kx

i kx

i i

e y K Ky

e y K e y y y e K x



 



 



  _.

All formulas in equations (4a)-(4e) are ready to use in equations (3a)-(3c). Finally, we are allowed to compute each component of the Hessian matrix R where the optimal values K

and k (K*=1.1549;k*=1.6226) are substituted. One yields

 

_ _

0.0754 0.1947

-0.1947

0.5321

*) *, (K k R

H .

It is obviously the Hessian matrix R is positive definite which means that K*=1.1549 and

k*=1.6226 indeed minimize the error as we expect.

3.2Analysis of optimal solution

Minimize

))

1 (

))

1 (

,

1549

.

1 1549

.

1 )

(

₁_.₆₂₂₆

x y

x y K A Ae

y _x









 subject to x≥ 0.04 (*)

where 0.04 is obtained from the given data. The ant colony algorithm is employed to the problem (*) and the optimal solution is shown in Table 5.

Table 4. Parameters in y(x) (Eq.1) for each set of samples

Data K K Error (%)

Sample I 1.1549 1.6226 6.7232 Sample II 1.1549 1.6226 14.5822 Sample III 1.1549 1.6226 7.7947

Table 5.The optimal solutions

Data x

optimal

minimum Sample

0.2380 1.0782 Sample

0.0400 0.5207 Sample

III

0.2340 1.2760

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the optimal function. The result is also practically reasonable compared to the Table 2 that the optimizer x* is produced by type A.

Though the gradient is it not required to be computed, it is good to calculate it since the objective function is only one variable. The first and second derivatives of the objective function are



_



2



_





kx kx

Ae K KkAe dx

and ₂ 2







2 



1











kx kx  kx kx 

Ae K Ae Ae

K Ae Kk dx

y d

. Therefore ₂

0 

dx y d

can be obtained if

2

kx





kx



1



1 

0

Ae K

Ae in the optimum point.

One may observe that in x*,

2

Aekx*





Aekx*



1



1

5. CONCLUSION

Acknowledgement

This paper is a short term research supported by Research Center in Satya Wacana Christian University.

REFERENCES

[1] Parhusip, H.A , 2012, Various Applications of Linear Algebra, to appear on proceeding

Seminar Aljabar 2012 UNDIP, Semarang.

[3] Buchori, L., 2007, Pembuatan Gula Non Karsinogenik Non Kalori Dari Daun Stevia,

Reaktor, Vol. 11 No.2, Desember 2007, hal.57-60.

[4] Parhusip, H. A dan Y. Martono, 2011, Kadar Steviosida Maksimum pada Waktu dan Massa yang Minimum, prosiding Sem. Nas FSM, ISSN:2087-0922,Vol. 2, No. 1,645-650, FSM.

[5] Yodyingyuad, V and Bunyawong, S,1991, Effect of stevioside on growth and reproduction , Hum. Reprod. Vol.6,158-165.

(5)

the leaves of Stevia rebaudiana by near infrared reflectance spectroscopy, Journal of the Science of Food and Agriculture,Vol. 59, No. 3, pp. 277–281.

[9]

Rajasekaran , T., Giridhar , P., Ravishankar, G.A.,2007, Production of

steviosides in ex vitro and in vitro grown Stevia rebaudiana Bertoni,

Journal of

the Science of Food and Agriculture

,

Vol. 87, No. 3, pp. 420–424.

[10] Parhusip, H.A.,2010. Crown Diameter of Phythoremidiation Agent by Logistic Model, proceeding on International Conference on Biotechnology and Climate Change by Logistic Model, UNS.

[11] Rao, S.S, 2009, Engineering Optimization, Theory and Practice, John Wiley & Sons, Inc., Hoboken, New Jersey,pp.714-722.

[13] Dai,W., Liu,S, and Liang,S, 2009, An Improved Ant Colony Optimization Cluster Algorithm Based on Swarm Intelligence., Journal of Software, Vol.4, No.4, Academic Publisher.

[14] Wang, H, Shi,Z., Ma, J.,A, 2006, Modified Ant Colony Algorithm for Multi-constraint Multicast Routing, IEEE,Vol.1,4244-0463.

[15] Peressini, A.L, Sullivan, F.E., Uhl,J. 1988. The Mathematics of Nonlinear Programming, Springer Verlag, New York, Inc.

(6)

Presented on IInntteerrnnaattiioonnaallSSyymmppoossiiuummoonnCCoommppuuttaattiioonnaallSScciieennccee((IISSCCSS)) May 15, 2012 - May 16, 2012, Universitas Gadjah Mada