  This paper investigates the robust uncertain two-level cooperative set
covering problem (RUTLCSCP). Given two types of facilities, which are called
y-facility and z-facility. The problem is to decide which facilities of both
types to be selected, in order to cover the demand nodes cooperatively with
minimal cost. It combines the concepts of robust, probabilistic, and
cooperative covering by introducting "$\Gamma$-robust two-level-cooperative
$\alpha$-cover" constraints. Additionally, the constraint relaxed verison of
the RUTLCSCP, which is also a linear approximation robust counterpart version
of RUTLCSCP (RUTLCSCP-LA-RC), is developed by linear approximation of the
constraints, and can be stated as a compact mixed-integer linear programming
problem. We show that the solution for RUTLCSCP-LA-RC,
$\varepsilon$-under-approximate solution, can also be the solution for RUTLCSCP
on some conditions. Computational experiments show that the solutions in 333
instances (10125 instances in total) with 12 types which tinily violate the
constraints of RUTLCSCP, can be an efficient under-approximate solutions, while
the feasible solutions in other instances are proven to be optimal.

Optimization and Control