and 3.0, no matter how complex the environment is. In secondary level of our simulation strategy, we vary α in the range of 0.25 to 0.75(i.e in the neighborhood of 0.5) and β over the range 2.5 and 3.0 in steps of 0.1 and try to estimate their relation with two features of problem environment: i) the node density **and ii)** standard deviation **of lengths of smallest** 
arc associated with each node. We performed experiments on roadmaps with 250, 265, 280, 295, 310, 325 and 350 number of cities. For each of above roadmaps, we choose seven different distributions and recorded the values of α and β for best performance of our algorithm. Table Curve 3D V4.0**, a** 
curve fitting tool, is then employed to fit a curve through 49 data points for each of α and β and obtain an algebraic relation between α or β and the features of problem environment. The results are displayed in figures 16 and 17 along with the equations that relate the two sets of parameters. This exhaustive experimentation allows determination of optimum values of α and β when the features of problem environment are known in advance. 

![0_image_1.png](0_image_1.png)

Function: Chebyshev Series X,LnY Bivariate Polynomial Order 6 
[x': x scaled -1 to +1; y': ln(y) scaled -1 to +1; Tn(x') = cos(nacos(x')) 
x ≡ number of nodes in 200 sq unit area; y ≡**standard** 
deviation] 
α=a+bT1(x')+cT1(y')+dT2(x')+eT1(x')T1(y')+fT2(y')+gT3( x')+hT2(x')T1(y')+iT1(x')T2(y')+jT3(y')+kT4(x')+lT3(x')T
1(y')+mT2(x')T2(y')+nT1(x')T3(y')+oT4(y')+ 
pT5(x')+qT4(x')T1(y')+rT3(x')T2(y')+sT2(x')T3(y')+ 
tT1(x')T4(y')+uT5(y')+vT6(x')+aaT5(x')T1(y')+ 
abT4(x')T2(y')+acT3(x')T3(y')+adT2(x')T4(y')+ 
aeT1(x')+T5(y')+afT6(y') 
Co-efficient values: 
a=2.094, b=-5.892, c=-3.756,d=1.813, e= -8.864, f=1.257, g= 0.697, h= 2.269, i= 2.720, j=0.1556, k=0.9132, l=1.722, m= -1.423, n=0.232, o=0.743, p=1.270, q=1.345, r=0.412, s=0.738, t=1.575, u=0.774, v=0.604, aa=1.323, ab= -0.932, ac=-0.975, ad=-0.827,ae=-0.115, af=0.124 

![0_image_0.png](0_image_0.png)

Function: Sigmoid Series Bivariate Polynomial Order 6 
[ x': x scaled -1 to +1; y': y scaled -1 to +1; Si=2..n(x') = -1+2/(1+exp(-(x'+1-(i-1)*(2/n))/0.12)), S1(x') = 
x'] 
β=a+bS1(x')+cS1(y')+dS2(x')+eS1(x')S1(y')+fS2(y')+gS3(x'
)+hS2(x')S1(y')+iS1(x')S2(y')+jS3(y')+ 
kS4(x')+lS3(x')S1(y')+mS2(x')S2(y')+nS1(x')S3(y')+oS4(y')
+pS5(x')+qS4(x')S1(y')+rS3(x')S2(y')+ 
sS2(x')S3(y')+tS1(x')S4(y')+uS5(y')+vS6(x')+ 
aaS5(x')S1(y')+abS4(x')S2(y')+acS3(x')S3(y')+ 
adS2(x')S4(y')+aeS1(x')S5(y')+afS6(y') 
Co-efficient values: 
a=1.396, b=-0.106, c=1.427, d=1.203, e=-1.107, f=1.115, g=-0.214, h=1.415, i=-0.413, j=0.753, k=0.116, l=-0.360, m=1.666 , n=-0.867, o=0.293 p=-0.066, q=0.274, r=-0.229, s=0.572, t=-0.191 u=0.081, v=-0.047, aa=-0.158, ab=0.152, ac=-0.140, ad=0.502 , ae=-0.272 , af=-0.036 

## Vii. Conclusions And Future Work

The stability analysis and pheromone deposition approach presented in this paper are both entirely novel. The exponential deposition approach outperformed the classical one by a large margin and has lead to better solution quality and algorithm convergence. Our next venture includes studying the comparative behavior of the two kinds of deposition approach in other models of extended Ant System algorithm like the Rank-based Ant System, Ant Colony System and Elitist Ant System. 

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