Abstract
This article enhances properties and applications associated with the characteristic equation (CE) developed to find an optimal and other ranked-optimal solutions of linear integer programming model. These enhanced properties have applications in the analysis of the multi-objective linear integer programs. The paper also identifies why the CE approach is not possible for some special linear programming (LP) models and creates a challenge for further investigation.
Highlights
In the context of a pure integer programming model, Kumar et al (2007) developed a conceptual idea for determination of an integer optimal solution to an integer program
Further assume that we have identified some of the ranked-optimal integer solutions with respect to the objective z1, i.e. kth ranked − optimal solution denoted by z11 ≥ z12 ≥ ⋯ ≥ z1k
3.3 Why a characteristic equation (CE) does not exist for Special linear programming (LP) like an Assignment and Transportation Models? The concept of CE is based on the fact, that by varying the value of the slack variables in a linear inequality, one can move constraints to various parallel positions and the CE helps to find those integer values of the slack variables to reach to the desired integer point directly
Summary
In the context of a pure integer programming model, Kumar et al (2007) developed a conceptual idea for determination of an integer optimal solution to an integer program. Kumar et al (2007) developed a linear hyperplane form the simplex output and reached directly to the optimal integer solution for the given linear integer program. The linear descending hyperplane was renamed by Kumar and Munapo (2012) as a characteristic equation (CE) for integer optimal solutions. These properties have applications in multi-objective linear integer programs, see AlHasanai et al (2020). The characteristic equation was developed from the LP output (2), i.e. from the objective row in the final simplex tableau These coefficients in the final simplex table are non-negative quantities i.e. am+1,1, am+1,2,..., am+1,n ≥ 0 but they can be integer, zero or fractional values, one can re-write equation (2) in the form of equation (3):.
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