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The set covering problem (SCP) is an NP-complete problem that has many important industrial applications. Since industrial applications are typically large in scale, exact solution algorithms are not feasible for operations research (OR) practitioners to use when called on to solve real-world problems involving SCPs. However, the best performing heuristics for the SCP reported in the literature are not usually straightforward to implement. Additionally, these heuristics usually require the fine-tuning of several parameters. In contrast, simple greedy or even randomized greedy heuristics typically do not give as good results as the more sophisticated heuristics. In this paper, the authors present a compromise; a straightforward to implement, population-based solution approach for the SCP. It uses a randomized greedy approach to generate an initial population and then uses a genetic-based two phase approach to improve the population solutions. This two-phase approach uses transformation equations based on a Teaching-Learning based optimization approach developed by Rao, Savsani and Vakharia (2011, 2012) for continuous nonlinear optimization problems. Empirical results using set covering problems from Beasley's OR-library demonstrate the competitiveness of this approach both in terms of solution quality and execution time. The advantage to this approach is its relative simplicity for the practitioner to implement.

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As generalizations of the classic set covering problem (SCP), both the set K-covering problem (SKCP) and the set variable (K varies by constraint) K-covering problem (SVKCP) are easily shown to be NP-hard. Solution approaches in the literature for the SKCP typically provide no guarantees on solution quality. In this article, a simple methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, that iteratively uses any general-purpose integer programming software (Gurobi and CPLEX in this case) is discussed. This approach is shown to quickly generate solutions that are guaranteed to be within a tight tolerance of the optimum for 135 SKCPs (average of 67 seconds on a standard PC and at most 0.13% from the optimums) from the literature and 65 newly created SVKCPs. This methodology generates solutions that are guaranteed to be within a specified percentage of the optimum in a short time (actual deviation from the optimums for the 135 SKCPs was 0.03%). Statistical analyses among the five best SKCP algorithms and SSIT demonstrates that SSIT performs as well as the best published algorithms designed specifically to solve SKCPs and SSIT requires no time-consuming effort of coding problem-specific algorithms-a real plus for OR practitioners.Contribution/Originality: This study documents a methodology that iteratively uses integer programming software to efficiently generate solutions that are guaranteed to be very close to the optimums for the set Kcovering problem. A significant benefit of this methodology is that no problem specific algorithm needs to be coded by the user.