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.

The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley’s OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums—far better than any published results for these 270 MKP test instances.

The generalized assignment problem (GAP) is a NP-hard problem that has a large and varied number of important applications in business and industry. Approximate solution approaches for the GAP do not require excessive computation time, but typically there are no guarantees on solution quality. In this article, a methodology called the simple sequential increasing tolerance (SSIT) matheuristic that iteratively uses any general-purpose integer programming software is discussed. This methodology uses a sequence of increasing tolerances in conjunction with optimization software to generate a solution that is guaranteed to be within a specified percentage of the optimum in a short time. SSIT requires no problem-specific coding and can be used with any commercial optimization software to generate bounded solutions in a timely manner. Empirically, SSIT is tested on 51 GAP instances (24 medium and 27 large) in the literature. The performance of CPLEX versus Gurobi on these 51 GAP test instances is also statistically analyzed.

Vasko et al. [1] defined the Cable-Trench Problem (CTP) as a combination of the Shortest Path and Minimum Spanning Tree Problems. Specifically, let be a connected weighted graph with specified vertex (referred to as the root), length for each , and positive parameters and . The Cable-Trench Problem is the problem of finding, for given values of and , a spanning tree of such that is minimized, where is the total length of the spanning tree and is the total path length in from to all other vertices of . Consider the ratio R = t/g. For R large enough the solution will be a minimum spanning tree and for R small enough the solution will be a shortest path. This is the first article to present a methodology that iteratively uses integer programming software (CPLEX in this article) to efficiently generate all optimal spanning trees (GEAOST) for a CTP (for all values of R). An example will illustrate how sensitive the spanning trees solution can be to small changes in edge lengths. Also, GEAOST will be used to generate all optimal spanning trees for graphs based on a real-world radio astronomy application. How the sequence of all optimal spanning trees can be used for sensitivity analysis will be demonstrated.

The generalized assignment problem (GAP) is a NP-hard problem that has a large and varied number of important applications in business and industry. Approximate solution approaches for the GAP do not require excessive computation time, but typically there are no guarantees on solution quality. In this article, a methodology called the simple sequential increasing tolerance (SSIT) matheuristic that iteratively uses any general-purpose integer programming software is discussed. This methodology uses a sequence of increasing tolerances in conjunction with optimization software to generate a solution that is guaranteed to be within a specified percentage of the optimum in a short time. SSIT requires no problem-specific coding and can be used with any commercial optimization software to generate bounded solutions in a timely manner. Empirically, SSIT is tested on 51 GAP instances (24 medium and 27 large) in the literature. The performance of CPLEX versus Gurobi on these 51 GAP test instances is also statistically analyzed.