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1. Introduction
To model real-life problems and perform computations, we must deal with uncertainty and inexactness. These uncertainty and inexactness are due to measurement inaccuracy, simplification of physical models, variations of the parameters of the system, computational errors and so on. Grey system theory is an efficient and reliable tool that allows us to handle such problems effectively. Most of the real-world processes in decision problems are in the grey stage due to lack of information and uncertainty (Mujumdar and Karmakar, 2008). Deng developed the grey system theory and presented grey decision-making systems (Deng, 1989).
Linear programming problems with grey coefficients have been studied by several authors, such as Li et al. (2014), Nasseri and Darvishi (2018), Nasseri et al. (2016, 2017), Mahmodi et al. (2018), Liu et al. (2009), Razavi et al. (2012) and so on. Early applications of grey linear programming incorporated grey numbers into the objective function, constraint matrix, right-hand sides of constraints and all of the aforementioned (Rosenberg, 2009). Grey programming has the ability to deal with poor, incomplete or uncertain problems associated with the systems, and it has been widely used in many areas such as economics, agriculture, medicine, geography, industry and and so on. Huang and Moore (1993) developed and applied grey linear programming in water resource planning. Various methods have been developed for solving grey linear programing problems (Voskoglouo, 2018; Darvishi et al., 2018; Mahmodi et al., 2018, 2019; Li et al., 2014; Razavi et al., 2012). Nasseri et al. (2016) proposed a new approach for solving interval grey number linear programming problems without converting them to classical linear programming problems. The proposed method is established based on the primal simplex algorithm where the cost-coefficient row includes grey numbers.
In this paper, the duality theory for linear programming problems with grey parameters is introduced and established some duality results. The results will be also useful for post optimality analysis.
The rest of the paper is organized as follows: the fundamental notions of grey system theory, grey numbers and their ranking, which are needed in the next sections, are presented in Section 2. A definition of grey linear programming problems is given in Section 3, and then the duality of grey linear programming problems is...