Abstract
In most standard data envelopment analysis (DEA) models, data are deterministic. However, real-world applications are subject to stochastic data as production costs, which depend on many factors such as environmental and social elements. So, this shortcoming requires the generalization of DEA models to stochastic data. Also, the standard DEA models are often used for positive inputs and outputs, while in many real-world situations, inputs or outputs may take negative values. This article is intended to extend the semi-oriented radial measure model for dealing with negative and stochastic data in the DEA framework. Some new chance-constrained optimization models and their deterministic equivalent models are introduced to evaluate the production units. Besides, some numerical examples, including an empirical application on 61 bank branches, were used to evaluate the proposed approach.
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Appendix A
Appendix A
To convert the chance-constrained optimization Model 2 to the deterministic Model 3, we proceed as follows. The first constraint, in Model 2, for positive input variables is stated as Pr{\(~\mathop \sum \limits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p}\) ≤ 0} ≥ 1-\(~\alpha\). Since \(\tilde{x}_{j}^{p}\) elements are distributed normally, vector \(\mathop \sum \limits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p}\) has a random variable with normal distribution in all elements. So we may write:
Pr \(\left\{ {\frac{{\mathop \sum \nolimits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p} - E\left( {~\mathop \sum \nolimits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p} } \right)}}{{\sqrt {~var\left\{ {\mathop \sum \nolimits_{{j = 1}}^{n} \tilde{x}_{j}^{p} \lambda _{j} - ~\tilde{x}_{k}^{p} } \right\}} }} \le \frac{{ - E\left( {\mathop \sum \nolimits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p} ~} \right)~}}{{\sqrt {var\left\{ {\mathop \sum \nolimits_{{j = 1}}^{n} \tilde{x}_{j}^{p} \lambda _{j} - ~\tilde{x}_{k}^{p} } \right\}} }}} \right\}\) \(\ge 1 - \alpha\). For simplicity of presentation, we denote \(\sqrt {~var\left\{ {\mathop \sum \limits_{{j = 1}}^{n} \tilde{x}_{j}^{p} \lambda _{j} - ~\tilde{x}_{k}^{p} } \right\}}\) by \(~\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)\). So we have Pr \(\left\{ {\frac{{\mathop \sum \nolimits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p} - \mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} }}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}} \le \frac{{ - \mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} ~}}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}}.} \right\}\) ≥ \(1 - \alpha\). Equivalently, we may write Pr \(\left\{ {\tilde{z} \le \frac{{ - ~\mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} ~~}}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}}} \right\}\) \(\ge 1 - \alpha\), in which, \(\widetilde{{{\text{z~}}}} = \frac{{\mathop \sum \nolimits_{{j \epsilon J}} \tilde{x}_{j}^{p} \lambda _{j} - \tilde{x}_{k}^{p} - \mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} }}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}}\) has a standard normal distribution with zero mean and unit variance. This argument leads to \(\varPhi \left[ {\frac{{ - \mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} ~~}}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}}} \right] \ge 1 - \alpha\), in which \(\varPhi\) is a normal cumulative distribution function. Notice that this expression has no stochastic elements. We know that \(\varPhi\) is a reversible function. Thus, we can write it as \(\frac{{ - \mathop \sum \nolimits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} + x_{k}^{p} ~~}}{{\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right)}} \ge \varPhi ^{{ - 1}} \left( {1 - \alpha } \right)\). So we have \(\mathop \sum \limits_{{j \epsilon J}} x_{j}^{p} \lambda _{j} - \varPhi ^{{ - 1}} \left( \alpha \right)\sigma _{{\text{i}}}^{{\text{I}}} \left( \lambda \right) \le ~x_{k}^{p}\). For the other constraints, the required calculations are similar and straightforward.
To derive equations related to variance values, note that:
For other constraints, the variance equations are obtained similarly.
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Babaie Asil, H., Kazemi Matin, R., Khounsiavash, M. et al. A modified semi-oriented radial measure to deal with negative and stochastic data: an application in banking industry. Math Sci 16, 237–249 (2022). https://doi.org/10.1007/s40096-021-00416-2
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DOI: https://doi.org/10.1007/s40096-021-00416-2