An epsilon-based data envelopment analysis approach for solving performance measurement problems with interval and ordinal dual-role factors

Abstract

Data envelopment analysis (DEA) is a linear programming method for measuring the performance and efficiency of units called decision-making units (DMUs). In many real-world performance measurement problems, the input and output data are not precisely known. Furthermore, the data may include dual-role factors that can be considered an input and output factor simultaneously. We propose a novel DEA model in the presence of imprecise data and imprecise dual-role factors by developing a new pair of mixed binary linear epsilon-based DEA models. The proposed models estimate the lower and upper bound efficiency scores in the presence of interval input, output, and dual-role factors by considering a fixed and unified production frontier for all DMUs. We then extend our models by including the weak ordinal dual-role factors. In contrast to the existing methods that exclude the dual-role factors, we include the dual-role factors and find a strictly positive value for the lower bound of the weights of inputs, outputs, and dual-role factors. We present a case study to demonstrate the applicability and exhibit the superiority of our approach over the existing methods.

Appendix A. Proofs of the Theorems.

1.1 Proof of Theorem 1

It is easy to see that $({\varvec{v}},{\varvec{u}},\boldsymbol{\alpha },{\varvec{\beta}},{\varvec{\lambda}},{\varvec{\mu}},{\varvec{d}},{\varvec{b}},\varepsilon )=({0}_{m},{0}_{s},{0}_{K},{0}_{K},{0}_{T},{0}_{T},{0}_{K},{0}_{K},0)$ is a feasible solution for Model (10). Now, considering the first type constraints of Model (10) with the constraints of $\varepsilon \le {v}_{i},\forall i$ implies:

$$\sum_{i=1}^{m}\varepsilon {x}_{ij}^{U}\le \sum\limits_{i=1}^{m}{v}_{i}{x}_{ij}^{U}\le 1, j=1,2,\ldots,n\Rightarrow \varepsilon \le \underset{\mathit{ j}=1,\ldots,n}{\mathrm{min}}\left\{1/\sum\limits_{i=1}^{m}{x}_{ij}^{U}\right\}$$

Hence, we found an upper bound for the objective value of Model (10) which implies $0\le {\varepsilon }^{*}<\infty$. Now, we can show that ${\varepsilon }^{*}>0$.

Consider the LP Model (11). If we assume that $({{\varvec{x}}}_{j}^{L},{{\varvec{w}}}_{j}^{L},{{\varvec{\theta}}}_{t})\in {\mathbb{R}}^{m+K+T}$ and $({{\varvec{y}}}_{j}^{U},{{\varvec{w}}}_{j}^{U},1)\in {\mathbb{R}}^{s+K+T}$ are the input and output vectors for DMU_j, respectively, then, Model (11) will be identical to Mehrabian et al.’s (2000) model, and, ${\varepsilon }_{1}^{*}>0$ (Amin and Toloo, 2004).

$$\begin{array}{ll}{\varepsilon }_{1}^{*}=\mathrm{max}{\varepsilon }_{1}& \\ \mathrm{s}.\mathrm{t}.& \\ \begin{array}{c}\begin{array}{c}\begin{array}{c}\sum\limits_{k=1}^{K}{\beta }_{k}{w}_{kj}^{L}+\sum\limits_{t=1}^{T}{\mu }_{t}{\theta }_{t}+\sum\limits_{i=1}^{m}{v}_{i}{x}_{ij}^{L}\le 1\end{array}\end{array}\end{array}& j=1,\ldots,n\\ \sum_{r=1}^{s}{u}_{r}{y}_{rj}^{U}+\sum\limits_{k=1}^{K}{\alpha }_{k}{w}_{kj}^{U}-\sum\limits_{k=1}^{K}{\beta }_{k}{w}_{kj}^{L}+\sum\limits_{t=1}^{T}{\lambda }_{t}-\sum\limits_{t=1}^{T}{\mu }_{t}{\theta }_{t}-\sum_{i=1}^{m}{v}_{i}{x}_{ij}^{L}\le 0& j=1,\ldots,n\\ {\alpha }_{k},{\beta }_{k}\ge {\varepsilon }_{1}& k=1,\ldots,K\\ {\lambda }_{t},{\mu }_{t}\ge {\varepsilon }_{1}& t=1,\ldots,T\\ \begin{array}{c}{u}_{r}\ge {\varepsilon }_{1}\end{array}& r=1,\ldots,s\\ {v}_{i}\ge {\varepsilon }_{1}& i=1,\ldots,m\end{array}$$

(11)

Now, let ${({\varvec{v}}}^{*},{{\varvec{u}}}^{*},{\boldsymbol{\alpha }}^{*},{{\varvec{\beta}}}^{*},{{\varvec{\lambda}}}^{*},{{\varvec{\mu}}}^{*},{\varepsilon }_{1}^{*})$ be the optimal solution of Model (11). If $\sum_{i=1}^{m}{v}_{i}^{*}{x}_{ij}^{U}\le 1, j=1,\ldots,n$, then it is evident that $({{\varvec{v}}}^{*},{{\varvec{u}}}^{*},{0}_{K},{{\varvec{\beta}}}^{*},{0}_{T},{{\varvec{\mu}}}^{*},{\varepsilon =\varepsilon }_{1}^{*})$ together with ${{\varvec{d}}}_{K}={0}_{K}, {{\varvec{b}}}_{T}={1}_{T}$ is a feasible solution for Model (10). Otherwise, let ${\widehat{v}}_{i}=\frac{min\left\{{x}_{ij}^{L}\left|{x}_{ij}^{L}\ne 0\right., j=1,\ldots ,n\right\}}{max\left\{{x}_{ij}^{U}, j=1,\ldots ,n\right\}}{v}_{i}^{*}, i=1,\ldots ,m$, in this case ${\widehat{v}}_{i}>0, i=1,\ldots ,m$, and we can conclude:

$$\sum_{i=1}^{m}{\widehat{v}}_{i}{x}_{ij}^{U}\le \sum_{i=1}^{m}{v}_{i}^{*}{x}_{ij}^{L}\le \sum_{k=1}^{K}{\beta }_{k}^{*}{w}_{kj}^{L}+\sum_{t=1}^{T}{\mu }_{t}^{*}{\theta }_{t}+\sum_{i=1}^{m}{v}_{i}^{*}{x}_{ij}^{L}\le 1, j=1,\ldots ,n$$

Now, select $\delta \ge 0$ such that the following relations are established:

$$\sum_{r=1}^{s}{u}_{r}^{*}{y}_{rj}^{U}+\sum_{k=1}^{K}{\alpha }_{k}^{*}{w}_{kj}^{U}-\sum_{k=1}^{K}{\beta }_{k}^{*}{w}_{kj}^{L}+\sum_{t=1}^{T}{\lambda }_{t}^{*}-\left({\mu }_{1}^{*}+\delta \right){\theta }_{1}-\sum_{t=2}^{T}{\mu }_{t}^{*}{\theta }_{t}-\sum_{i=1}^{m}{\widehat{v}}_{i}{x}_{ij}^{L}\le 0, j=1,\ldots ,n$$

Let $\varepsilon =min\left\{{\varepsilon }_{1}^{*},{\widehat{v}}_{i}, i=1,\ldots ,m\right\}>0$, in this case $({\widehat{{\varvec{v}}}}^{*},{{\varvec{u}}}^{*},{0}_{K},{{\varvec{\beta}}}^{*},{0}_{{\varvec{T}}},{\mu }_{1}^{*}+\delta ,{\mu }_{2}^{*},\ldots ,{\mu }_{T}^{*},\varepsilon )$ together with ${{\varvec{d}}}_{K}={0}_{K}, {{\varvec{b}}}_{T}={1}_{T}$ is a feasible solution for Model (10), and the proof is completed.

1.2 Proof of Theorem 2

We show Model (8) is feasible. The proof for Model (9) is similar. Let $({{\varvec{v}}}^{\boldsymbol{*}},{{\varvec{u}}}^{\boldsymbol{*}},{\boldsymbol{\alpha }}^{\boldsymbol{*}},{{\varvec{\beta}}}^{\boldsymbol{*}},{{\varvec{\lambda}}}^{\boldsymbol{*}},{{\varvec{\mu}}}^{\boldsymbol{*}},{{\varvec{d}}}^{\boldsymbol{*}},{{\varvec{b}}}^{\boldsymbol{*}},{\varepsilon }^{*})$ be the optimal solution of Model (10). If $\sum_{i=1}^{m}{v}_{i}^{*}{x}_{ip}^{L}=1$, then it is evident that $({{\varvec{v}}}^{\boldsymbol{*}},{{\varvec{u}}}^{\boldsymbol{*}},{\boldsymbol{\alpha }}^{\boldsymbol{*}},{{\varvec{\beta}}}^{\boldsymbol{*}},{{\varvec{\lambda}}}^{\boldsymbol{*}},{{\varvec{\mu}}}^{\boldsymbol{*}},{{\varvec{d}}}^{\boldsymbol{*}},{{\varvec{b}}}^{\boldsymbol{*}},{\varepsilon }^{*})$ is a feasible solution to Model (8). Otherwise, let ${x}_{hp}^{L}>0,h\in \{1,2,\ldots,m\}$ and select $\eta >0$ such that $\sum_{\begin{array}{c}i=1\\ i\ne h\end{array}}^{m}{v}_{i}^{*}{x}_{ip}^{L}+({v}_{h}^{*}+\eta ){x}_{hp}^{L}=1$, in this case, we can conclude that:

$$\sum_{r=1}^{s}{u}_{r}^{*}{y}_{rj}^{U}+\sum_{k=1}^{K}{\alpha }_{k}^{*}{w}_{kj}^{U}-\sum_{k=1}^{K}{\beta }_{k}^{*}{w}_{kj}^{L}+\sum_{t=1}^{T}{\lambda }_{t}^{*}-\sum_{t=1}^{T}{\mu }_{t}^{*}{\theta }_{t}-(\sum_{\begin{array}{c}i=1\\ i\ne h\end{array}}^{m}{v}_{i}^{*}{x}_{ip}^{L}+({v}_{h}^{*}+\eta ){x}_{hp}^{L})\le \sum_{r=1}^{s}{u}_{r}^{*}{y}_{rj}^{U}+\sum_{k=1}^{K}{\alpha }_{k}^{*}{w}_{kj}^{U}-\sum_{k=1}^{K}{\beta }_{k}^{*}{w}_{kj}^{L}+\sum_{t=1}^{T}{\lambda }_{t}^{*}-\sum_{t=1}^{T}{\mu }_{t}^{*}{\theta }_{t}-\sum_{i=1}^{m}{v}_{i}^{*}{x}_{ij}^{L}\le 0, j=1,\ldots ,n$$

As a result, the new vector $({v}_{1}^{*},\ldots,{v}_{h}^{*}+\eta ,\ldots,{v}_{m}^{*},{{\varvec{u}}}^{\boldsymbol{*}},{\boldsymbol{\alpha }}^{\boldsymbol{*}},{{\varvec{\beta}}}^{\boldsymbol{*}},{{\varvec{\lambda}}}^{\boldsymbol{*}},{{\varvec{\mu}}}^{\boldsymbol{*}},{{\varvec{d}}}^{\boldsymbol{*}},{{\varvec{b}}}^{\boldsymbol{*}},{\varepsilon }^{*})$ satisfies all constraints of Model (8), which completes the proof.