  In real applications, there are situations where we need to model some
problems based on uncertain data. This leads us to define an uncertain model
for some classical geometric optimization problems and propose algorithms to
solve them. In this paper, we study the $k$-center problem, for uncertain
input. In our setting, each uncertain point $P_i$ is located independently from
other points in one of several possible locations $\{P_{i,1},\dots,
P_{i,z_i}\}$ in a metric space with metric $d$, with specified probabilities
and the goal is to compute $k$-centers $\{c_1,\dots, c_k\}$ that minimize the
following expected cost $$Ecost(c_1,\dots, c_k)=\sum_{R\in \Omega}
prob(R)\max_{i=1,\dots, n}\min_{j=1,\dots k} d(\hat{P}_i,c_j)$$ here $\Omega$
is the probability space of all realizations $$R=\{\hat{P}_1,\dots,
\hat{P}_n\}$$ of given uncertain points and $$prob(R)=\prod_{i=1}^n
prob(\hat{P}_i).$$
  In restricted assigned version of this problem, an assignment $A:\{P_1,\dots,
P_n\}\rightarrow \{c_1,\dots, c_k\}$ is given for any choice of centers and the
goal is to minimize $$Ecost_A(c_1,\dots, c_k)=\sum_{R\in \Omega}
prob(R)\max_{i=1,\dots, n} d(\hat{P}_i,A(P_i)).$$ In unrestricted version, the
assignment is not specified and the goal is to compute $k$ centers
$\{c_1,\dots, c_k\}$ and an assignment $A$ that minimize the above expected
cost.
  We give several improved constant approximation factor algorithms for the
assigned versions of this problem in a Euclidean space and in a general metric
space. Our results significantly improve the results of \cite{guh} and
generalize the results of \cite{wang} to any dimension. Our approach is to
replace a certain center point for each uncertain point and study the
properties of these certain points. The proposed algorithms are efficient and
simple to implement.
