Fairness based unique common equilibrium efficient frontier for evaluating decision-making units with fixed-sum outputs

Abstract

Data envelopment analysis, a non-parametric programming approach, has been extended to situations in which all total decision-making unit (DMU) outputs are fixed, and a secondary goal approach based on a minimum reduction strategy proposed to achieve an equilibrium efficient frontier to evaluate these fixed-sum output DMUs. However, the non-uniqueness of the equilibrium efficient frontier and the calculation burden of the iterative procedure have reduced the practicability of these approaches. Therefore, to address these problems, this paper developed a fairness based common equilibrium efficient frontier data envelopment analysis approach (CEEFDEA) that can guarantee the uniqueness of the common equilibrium efficient frontier and achieve such a frontier in only one step. Fairness is also included into the proposed CEEFDEA approach and the price of fairness is defined. One numerical example from previous studies and one case study focused on an efficiency evaluation of the Chinese appliance industry in 2019 are given to illustrate the effectiveness of the proposed approach. The results from the numerical example showed that the proposed CEEFDEA approach was able to achieve a fairer common equilibrium efficient frontier at the expense of a 5.85% increase in the adjustment proportion.

Appendices

Appendix A: Proof of Theorem 2

First of all, we prove that $\frac{{{C^{'}}}}{C}(v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)\;(\forall i,r,t,j)$ is a feasible solution to model (14). For convenience, we set $\lambda = \frac{{{C^{'}}}}{C}\;(\lambda > 0)$. Then, according to an optimal solution $(v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)$, we put $\lambda (v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)\;(\forall i,r,t,j)$ into all constraints of the model (14) with any positive constant ${{C^{'}}}$, and it follows that,

$$\begin{aligned} \begin{array}{l} \sum \limits _{r = 1}^s {\lambda u_r^*{y_{rj}}} + \sum \limits _{t = 1}^l {(\lambda w_t^*{f_{tj}} + \lambda a_{tj}^* - \lambda b_{tj}^*} ) - \sum \limits _{i = 1}^m {\lambda v_i^*{x_{ij}}} + \lambda \mu _0^*\\ = \lambda \left( \sum \limits _{r = 1}^s {u_r^*{y_{rj}}} + \sum \limits _{t = 1}^l {(w_t^*{f_{tj}} + a_{tj}^* - b_{tj}^*} ) - \sum \limits _{i = 1}^m {v_i^*{x_{ij}}} + \mu _0^* \right) = 0,\;\;\forall j\\ \sum \limits _{i = 1}^m {\lambda v_i^*{x_{ij}}} = \lambda \sum \limits _{i = 1}^m {v_i^*{x_{ij}}} \ge \lambda C\mathrm{{ = }}{C^{'}},\;\;\forall t\;\\ \sum \limits _{j = 1}^n {(\lambda a_{tj}^* - \lambda b_{tj}^*)} = \lambda \sum \limits _{j = 1}^n {(a_{tj}^* - b_{tj}^*)} \mathrm{{ = }}0\\ \lambda w_t^*{f_{tj}} + (\lambda a_{tj}^* - \lambda b_{tj}^*)\mathrm{{ = }}\lambda (w_t^*{f_{tj}} + (a_{tj}^* - b_{tj}^*)) \ge 0,\;\;\forall t,j\\ \lambda v_i^*,\lambda u_r^*,\lambda w_t^*,\lambda a_{tj}^*,\lambda b_{tj}^* \ge 0,\;\;\forall r,t,i,j\\ \lambda \mu _0^*\;is\;free. \end{array} \end{aligned}$$

(A.1)

Apparently, $\lambda (v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)\;(\forall i,r,t,j)$ satisfies all constraints in model (14) with a random positive constant $C^{'}$, therefore, the feasibility of the solution is proven.

Next, we prove that $\lambda (v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)\;(\forall i,r,t,j)$ is an optimal solution to model (14) with ${C^{'}}({C^{'}} > 0)$. Suppose that the above feasible solution is not an optimal solution, at least one nonnegative feasible solution can be found for model (14) with ${C^{'}}({C^{'}} > 0)$ ($(\lambda {v_i},\lambda {u_r},\lambda {w_t},\lambda {\mu _0},\lambda {a_{tj}},\lambda {b_{tj}})\;(\forall i,r,t,j)$) such that

$$\begin{aligned} \sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{\lambda {a_{tj}}}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{\lambda {b_{tj}}}}{{{f_{tj}}}}\right) }^2}\right) } } < \sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{\lambda a_{tj}^*}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{\lambda b_{tj}^*}}{{{f_{tj}}}}\right) }^2}\right) } }. \end{aligned}$$

(A.2)

Accordingly, it is easy to test that $({v_i},{u_r},{w_t},{\mu _0},{a_{tj}},{b_{tj}})\;(\forall i,r,t,j)$ is also a feasible solution to model (14) with ${C^{'}}({C^{'}} > 0)$.

According to Eq. (A.2), we get

$$\begin{aligned} {\lambda ^2}\sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{{a_{tj}}}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{{b_{tj}}}}{{{f_{tj}}}}\right) }^2}\right) } } < {\lambda ^2}\sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{a_{tj}^*}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{b_{tj}^*}}{{{f_{tj}}}}\right) }^2}\right) } }. \end{aligned}$$

(A.3)

However, $(v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)\;(\forall i,r,t,j)$ is an optimal solution to the model (14) with ${C^{'}}({C^{'}} > 0)$.

Then

$$\begin{aligned} \sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{a_{tj}^*}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{b_{tj}^*}}{{{f_{tj}}}}\right) }^2}\right) < } } \sum \limits _{j = 1}^n {\sum \limits _{t = 1}^l {\left( {{\left( \frac{{{a_{tj}}}}{{{f_{tj}}}}\right) }^2} + {{\left( \frac{{{b_{tj}}}}{{{f_{tj}}}}\right) }^2}\right) } }, \end{aligned}$$

(A.4)

which contradicts with our assumption. So far, the theorem is proven.

Appendix B: Proof of Theorem 3

Theorem 3 is proven through contradiction.

Suppose $\theta = ({v_i},{u_r},{w_t},{\mu _0},{a_{tj}},{b_{tj}})$ is a solution to model (14). The corresponding objective value is ${\left\| \gamma \right\| ^2}$, where $\gamma = ({\frac{{{a_{tj}}}}{{{f_{tj}}}}}, {\frac{{{b_{tj}}}}{{{f_{tj}}}}} \left| {t \in T,j \in J} \right. )$. Then, it is assumed that $\theta _1^*$ and $\theta _2^*$ are the two optimal solutions to model (14), and ${\left\| \gamma _1^* \right\| ^2}$ = ${\left\| \gamma _2^* \right\| ^2}$ = $\varepsilon ^2$, where $\varepsilon ^2$ is the optimal value to model (14).

When $\theta = \frac{{\theta _1^* + \theta _2^*}}{2}$, it follows that $\theta$ is the solution to model (14)); therefore, we get:

$$\begin{aligned} \varepsilon \le \left\| \gamma \right\| \le \frac{1}{2}\left\| {\gamma _1^*} \right\| + \frac{1}{2}\left\| {\gamma _2^*} \right\| = \varepsilon \end{aligned}$$

That is,

$$\begin{aligned}\left\| \gamma \right\| = \frac{1}{2}\left\| {\gamma _1^*} \right\| + \frac{1}{2}\left\| {\gamma _2^*} \right\| \end{aligned}$$

So,

$$\begin{aligned}\gamma _1^* = \lambda \gamma _2^*,\left| \lambda \right| = 1\end{aligned}$$

If $\lambda = -1$, then $\gamma$ =0 is not the solution to the model (14). Therefore, $\lambda = 1$, $\gamma _1^* = \gamma _2^*$, and the solution $(v_i^*,u_r^*,w_t^*,\mu _0^*,a_{tj}^*,b_{tj}^*)$ is the unique optimal solution to model (14).

Appendix C: Proof of Theorem 4

First, it is necessary to prove it holds for the two-dimensional case, that is, that the unique optimal solution to model (13) is a Pareto solution to model (11).

Denote the unique optimal solutions to model (13) as $(x_1,y_1)$. If $(x_1,y_1)$ is not a Pareto solution to model (11), then there must be some solution ($x_2,y_2$) that is better than the solution $(x_1,y_1)$. As solution ($x_2,y_2$) is the feasible solution to model (13), so $x_2^2 + y_2^2 > x_1^2 + y_1^2$.

When $y_1 > x_1$, if $x_2 < x_1$, based on $x_2^2 + y_2^2 > x_1^2 + y_1^2$, then $y_2 > y_1$, so min-max $\{x_2,y_2\}>$ min-max $\{x_1,y_1\}$. Otherwise, if $x_2 > x_1$, as the feasible region for model (11) is a nonnegative convex set, we have the slope $\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ greater than $- 1/\frac{{{y_1}}}{{{x_1}}}$, i.e., $\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} > - 1/\frac{{{y_1}}}{{{x_1}}}$. Because $y_1 > x_1$, so $\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} > - 1$, then ${x_2} + {y_2} > {x_1} + {y_1}$. Therefore, regardless of $x_2 < x_1$ or $x_2 > x_1$, when $y_1 > x_1$, solution $(x_2,y_2)$ is not better than solution $(x_1,y_1)$, that is, the solution $(x_1,y_1)$ is a Pareto solution to model (11).

When $y_1 < x_1$, the same is true. A similar conclusion to the multidimensional case can be obtained from the two-dimensional case. Therefore, so far, the theorem is proven.