Returns to Scale in Decision-Making Units with Integer-Valued or Mixed Inputs and Integer-Valued or Real Outputs Using Radial Models in Data Envelopment Analysis.

Purpose: The current paper tries to determine the type of returns to scale in a decision-making unit under the condition that integer-valued inputs or outputs are present. Methodology: This paper introduces radial models for determining the value and type of Returns to Scale (RTS) in 4 scenarios, including single integer-valued input -- single real output (scenario one), mixed inputs -- exclusively real outputs (scenario two), exclusively integer-valued inputs -- exclusively real outputs (scenario three), and exclusively integer-valued inputs -- exclusively integer-valued outputs (scenario four); in each scenario, the values of the left RTS and right RTS are determined, and the RTS type is then determined on that basis. Finally, by presenting three examples based on two scenarios, namely single integervalued input -- single real output and single integer-valued input -- single integer-valued output, the new method is compared with previous methods using GAMS software, and the conclusions are provided. Findings: The type of returns to scale differs when integer-valued inputs or outputs are present as compared with the case where the inputs and outputs are assumed to have real values. Originality/Value: This study focuses on the value and type of returns to scale for integer-valued data. For this purpose, returns to scale was modeled in 4 scenarios using input-oriented radial models, and in the fourth scenario (exclusively integervalued inputs -- exclusively integer-valued outputs), the modeling was carried out for output orientation as well. The existence of a difference between the results produced by our proposed model and those of the classical model was demonstrated through two examples, one using hypothetical data and the other real-world data. [ABSTRACT FROM AUTHOR]