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

An under-approximation for the robust uncertain two-level cooperative set covering problem