Applying Duality Results to Solve the Linear Programming Problems with Grey Parameters

Linear programming problems have exact parameters. In most real-world, we are dealing with situations in which accurate data and complete information are not available. Uncertainty approaches such as fuzzy and random can be used to deal with uncertainties in real-life. Fuzzy and stochastic theories cannot be used if the number of experts and the level of experience is so low that it is impossible to extract membership functions or the number of samples is small. To solve these problems, the grey system theory is proposed. In this paper, a linear programming problem in a grey environment with resources in interval grey numbers is considered. Most of the proposed methods for solving grey linear programming problems become common linear programming problems. However, we seek to solve the problem directly without turning it into a standard linear programming problem for the purpose of maintaining uncertainty in the original problem data in the final solution. For this purpose, we present a method based on the duality theory for solving the grey linear programming problems. This method is more straightforward and less complicated than previous methods. We emphasize that the concept presented is beneficial for real and practical conditions in management and planning problems. Therefore, we shall illustrate our method with some examples in different situations.