estimation of a parameter The question is:
$x_i = \alpha + \omega_i, $ for $i = 1, \ldots, n.$
where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, Gaussian random variable with known variance $\sigma_i^2$. Note that $\sigma_i^2$ and $\sigma_j^2$, for $i \neq j$, may be distinct. We wish to estimate $\alpha$ from a weighted sum of $x_i$, i.e.
$$\hat{\alpha} = \sum^n_{i=1}b_ix_i$$
Determine $b_i$, $i= 1, \ldots, n$, such that $\hat{\alpha}$ is unbiased and the variance of $\hat{\alpha}$ is as small as possible.
I have tried to use the unbiased condition and get that: $\sum_{i=1}^nb_i = 1$ 
I don't know how to use the variance of $\hat{\alpha}$ is as small as possible condition. 
 A: The weights should be proportional to the reciprocals of the variances:
$$
b_k = \frac{1/\sigma_k^2}{\sum_{i=1}^n 1/\sigma_i^2}.\tag1
$$
This can be shown with Lagrange multipliers.
The variance of $\sum_{i=1}^n b_i x_i$ is $\sum_{i=1}^n b_i^2\sigma_i^2$.  The problem is to minimize that subject to the constraint $\sum_{i=1}^n b_i=1$.  The $i$th component of the gradient of the thing to be minimized is $2b_i\sigma_i^2$.  The vector whose components are those has to be a scalar multiple of the gradient of the function $\sum_{i=1}^n b_i$.  Notice that that happens with the weights in $(1)$.
A: For the unbiasedness, we have
$$
E\left(\hat{\alpha}\right)=E\left(\sum_{i=1}^nb_ix_i\right)=E\left(\sum_{i=1}^nb_i(\alpha+\omega_i)\right)=\alpha\sum_{i=1}^nb_i + E\left(\sum_{i=1}^nb_i\omega_i\right)=\alpha\sum_{i=1}^nb_i
$$
and we get that $\sum_{i=1}^nb_i=1$ as you say.
Now, what follows is to simply make this homoscedastic so that we can use the Gauss-Markov theorem. Divide through by $\sigma_i$:
$$
\frac{x_i}{\sigma_i}=\frac{\alpha}{\sigma_i}+\frac{\omega_i}{\sigma_i}\Rightarrow x_i^*=\alpha\frac{1}{\sigma_i}+\omega_i^*
$$
where $\omega_i^*\sim N(0, 1)$ (stars indicate variance adjusted). This satisfies the usual OLS conditions, so by the Gauss-Markov theorem OLS is efficient and unbiased. The estimator then is:
$$
\hat{\alpha}=\arg\min_{a}\sum_{i=1}^n(x_i^*-a\frac{1}{\sigma_i})^2\Rightarrow-2\sum_{i=1}^n\frac{(x_i^*-a\frac{1}{\sigma_i})}{\sigma_i}=0\\
\sum_{i=1}^n\frac{x_i}{\sigma_i^2}=\sum_{i=1}^n\frac{a}{\sigma^2_i}\\
\sum_{i=1}^n\frac{x_i}{\sigma^2_i}=a\sum_{i=1}^n\frac{1}{\sigma^2_i}\\
\frac{\sum_{i=1}^n\frac{x_i}{\sigma^2_i}}{\sum_{i=1}^n\frac{1}{\sigma^2_i}}=a
$$
so the weights are
$$
b_i=\frac{\frac{1}{\sigma^2_i}}{\sum_{i=1}^n\frac{1}{\sigma^2_i}}.
$$
