On some probabilistic problems of reliability theory with constraint

Let $F(x)$ and $G(x)$ be given distribution functions, and $\varphi(y)$ be a known Borel function satisfying (1). The problem under consideration is to minimize the functional
$$ \int\,d\pi(x)\int\varphi(y)\,dQ(y\mid x) $$
in
$$ Q(y\mid x)\in\mathfrak M(F,G)\cap\mathfrak L(F,G), $$
$\pi(x)$ being a given distribution function with the set of increase points imbedded into that of $F(x)$. Here $\mathfrak M(F,G)$ is the family of conditional distributions $Q(y\mid x)$ satisfying (2) and $\mathfrak L(F,G)$ consists of all $Q(y\mid x)$ with $\int\varphi(y)\,dQ(y\mid x)$ non-decreasing in $x$.