鲁棒不确定两层合作集覆盖问题的欠逼近
An under-approximation for the robust uncertain two-level cooperative set covering problem
摘要：
研究了鲁棒不确定两层合作集覆盖问题。给定两种类型的设施，称为y设施和z设施。该问题是决定哪些设施的两种类型被选择，以满足需求节点合作以最小的成本。它通过引入“$Γ$ -鲁棒两级协作$α$ -覆盖”约束，结合了鲁棒、概率和协作覆盖的概念。此外，通过对约束的线性逼近，得到了RUTLCSCP的约束松弛版本，它也是RUTLCSCP的一个线性逼近鲁棒版本（RUTLCSCP-LA-RC），可以表示为一个紧凑的混合整数线性规划问题。我们证明了RUTLCSCP-LA-RC的解，$\varepsilon$ -下-近似解，在一定条件下也可以是RUTLCSCP的解.计算实验表明，在12类333个实例（共10125个实例）中，稍微违反RUTLCSCP约束的解都可以是有效的下近似解，而在其他实例中的可行解都被证明是最优的。
英文摘要：
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 "$Γ$-robust two-level-cooperative $α$-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.
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