How to calculate weight of positive and negative values. We have used formula formula to calculate weight as,

$$ w_1 = \frac{s_1}{s_1 + s_2 + s_3};$$
$$ w_2 = \frac{s_2}{s_1 + s_2 + s_3};$$
$$ w_3 = \frac{s_3}{s_1 + s_2 + s_3};$$

However, their is possibility of negative and positive numbers. Even all can be negative or positive. How to calculate weight in this type of situation.

For us -/+ are according to strict algebraic rules. i.e bigger the negative number smaller will be its value.

Thanks.
Let the smallest of the numbers $s_1$, $s_2$, $s_3$ be $m$. Let $t_i=s_i-m+1$ for each $i$. Then the $t_i$ are all positive, and satisfy the same order relatins as the $s_i$, for example, if $s_1\lt s_2$, then $t_1\lt t_2$. So you can use your formula to assign weights to the $t_i$, and then assign those same weights to the $s_i$. 

Whether you will like the weights you get, I can't say. Experiment a bit and you may find some other constraint you want your weights to satisfy, then we can refine the technique.
I'm a little confused by your question. When you say the weight can be positive or negative, do you mean just the value can be negative or positive because of the measurement technique, or is it actually a negative weight? I would assume the first (for example, if you slow down really fast in an elevator, and measure your weight in that frame, you'll actually have 'negative' weight). 

With that said,  I think what you're after is the RMS value (root mean squared). It's a common technique used to measure velocity since velocity is a vector and can have negative components, but often we care only about it's magnitude. If that is the case for your weight, then do the following. 

$w1 = (s1^2/(s1^2 + s2^2 + s3^2))^{1/2}$

$w2 = (s2^2/(s1^2 + s2^2 + s3^2))^{1/2}$

$w3 = (s2^2/(s1^2 + s2^2 + s3^2))^{1/2}$

If you indeed just want the average and weight can be negative, do exactly as what the formulas you provided for us tell you to do.