Sustainability Assessment and Most Productive Scale Size: a Stochastic DEA Approach with Dual Frontiers

Abstract

Sustainability performance analysis is a critical and significant aspect for organizations due to growing challenges in competitive industries. Finding the most productive scale size (MPSS) is also necessary for reorganizing. Conventional data envelopment analysis (DEA) models analyze the performance from the optimistic viewpoint. However, evaluating the efficiency from the pessimistic outlook and integrating optimistic and pessimistic efficiencies lead to more rational and reliable results. Because of the presence of random data in many real-world applications and the importance of the efficiency examination from optimistic and pessimistic standpoints, a double frontiers approach based on stochastic DEA (SDEA) is proposed in the current paper to assess the sustainability performance of decision making units (DMUs) where random and triple bottom line factors including economic, social and environmental ones are presented. To illustrate, the sustainability performance and MPSS are measured using SDEA from two aspects, optimistic and pessimistic. Also, the best and worst stochastic efficiency gains are unified by the Hurwicz criterion. A case study on Iranian gas enterprises is applied to illuminate and demonstrate the proposed methodology. The findings reveal that the introduced approach can provide a fair identification of the MPSS sustainable company where random and undesirable measures are presented.

Appendix A

The deterministic equivalent form of stochastic model (4) is as follows:

$$\begin{array}{lll}{{OE}_{o}^{*}}^{\alpha }=Min& \frac{{\theta }_{o}^{\alpha }}{{\varphi }_{o}^{\alpha }}-\varepsilon \left(\sum\limits_{i=1}^{m}{s}_{i}^{-}+\sum\limits_{r=1}^{s}{s}_{r}^{+}+\sum\limits_{k=1}^{K}{\overline{s} }_{k}\right)& \\ subject\ to& \sum\limits_{j=1}^{n}{\lambda }_{j}{\overline{x} }_{ij}-{\theta }_{o}^{\alpha }{\overline{x} }_{io}+{s}_{i}^{-}-{\phi }^{-1}\left(\alpha \right){\sigma }_{i}^{I}\left({\theta }_{o}^{\alpha },\lambda \right)=0,& i=1,...,m,\\ & \sum\limits_{j=1}^{n}{\lambda }_{j}{\overline{y} }_{rj}-{\varphi }_{o}^{\alpha }{\overline{y} }_{ro}-{s}_{r}^{+}+{\phi }^{-1}\left(\alpha \right){\sigma }_{r}^{O}\left({\varphi }_{o}^{\alpha },\lambda \right)=0,& r=1,...,s,\\ & \sum\limits_{j=1}^{n}{\lambda }_{j}{\overline{z} }_{kj}-{\theta }_{o}^{\alpha }{\overline{z} }_{ko}+{\overline{s} }_{k}-{\phi }^{-1}\left(\alpha \right){\sigma }_{k}^{K}\left({\theta }_{o}^{\alpha },\lambda \right)=0,& k=1,...,K,\\ & \sum\limits_{j=1}^{n}{\lambda }_{j}=1,& \\ & {\lambda }_{j}\ge 0,\forall j, {\theta }_{o}^{\alpha },{\varphi }_{o}^{\alpha }\ge 0,& \end{array}$$

(10)

in which

$$\begin{aligned}{\left({\sigma }_{i}^{I}\left({\theta }_{o}^{\alpha },\lambda \right)\right)}^{2}&=\sum\limits_{j\ne o}\sum\limits_{b\ne o}{\lambda }_{j}{\lambda }_{b}Cov\left({\tilde{x }}_{ij},{\tilde{x }}_{ib}\right)\\&\quad+2\left({\lambda }_{o}-{\theta }_{o}^{\alpha }\right)\sum\limits_{j\ne o}{\lambda }_{j}Cov\left({\tilde{x }}_{ij},{\tilde{x }}_{io}\right)\\&\quad+{\left({\lambda }_{o}-{\theta }_{o}^{\alpha }\right)}^{2}Var\left({\tilde{x }}_{io}\right),\end{aligned}$$

$$ \begin{aligned}{\left({\sigma }_{r}^{O}\left({\varphi }_{o}^{\alpha },\lambda \right)\right)}^{2}&=\sum\limits_{j\ne o}\sum\limits_{b\ne o}{\lambda }_{j}{\lambda }_{b}Cov\left({\tilde{y }}_{rj},{\tilde{y }}_{rb}\right)\\&\quad+2\left({\lambda }_{o}-{\varphi }_{o}^{\alpha }\right)\sum\limits_{j\ne o}{\lambda }_{j}Cov\left({\tilde{y }}_{rj},{\tilde{y }}_{ro}\right)\\&\quad+{\left({\lambda }_{o}-{\varphi }_{o}^{\alpha }\right)}^{2}Var\left({\tilde{y }}_{ro}\right),\end{aligned}$$

$$\begin{aligned}{\left({\sigma }_{k}^{K}\left({\theta }_{o}^{\alpha },\lambda \right)\right)}^{2}&=\sum\limits_{j\ne o}\sum\limits_{b\ne o}{\lambda }_{j}{\lambda }_{b}Cov\left({\tilde{z }}_{kj},{\tilde{z }}_{kb}\right)\\&\quad+2\left({\lambda }_{o}-{\theta }_{o}^{\alpha }\right)\sum\limits_{j\ne o}{\lambda }_{j}Cov\left({\tilde{z }}_{kj},{\tilde{z }}_{ko}\right)\\&\quad+{\left({\lambda }_{o}-{\theta }_{o}^{\alpha }\right)}^{2}Var\left({\tilde{z }}_{ko}\right).\end{aligned}$$