The traditional data envelopment analysis (DEA) models for the observations containing ratio data as input or output may result incorrect efficiency scores. To overcome this shortcoming, a set of modified DEA models has been presented in this paper by taking into account the correct convexity of decision making units (DMUs) when a ratio variable is included in the assessment model. 1. Introduction Data envelopment analysis (DEA) has been proven as a powerful and attractive tool for researchers

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... n several fields, especially operations research, applied mathematics, information management and economics. The standard models of (DEA) assume that all inputs and outputs are controlled by managers or other users. But in real conditions there are also some non-discretionary (ND) inputs and outputs which are not under the control of managers and their efficiency evaluation should be paid attention to. Standard DEA models were originally presented by Charnes et al [1] and Banker et al [2] but they did not pay attention to (ND) inputs and outputs. Banker and Morey (BM) et al [3] suggested that these models be extended to a suitable model including (ND) factors. In addition, they presented constant (CRS) and variable (VRS) returns to scale models in the presence of (ND) and discretionary factor. They assumed that the factors are either fully discretionary or fully (ND). In DEA, the absolute data are discussed mostly, but there are some reports in which the researchers used the ratio data more than absolute data as input (input ratio) and output (output ratio). Hollingsworth and Smith et al [4] illustrate that when data are in the form of ratio, the CCR models of Charnes and Cooper and Rhodes et al [1] have no efficiency; instead its BCC models of Banker, Charnes and Cooper et al [4] should not be used. Emrouznejad and Amin et al [5] demonstrated that using the standard DEA models for the observations containing ratio data as input or output may result incorrect efficiency scores and then a set of modified DEA models taking into account the correct convexity of DMUs when a ratio variable is included in the assessment model, has been proposed by them. This paper develops (ND) and discretionary model of (BM) in input oriented when the one of