Pareto-optimal peer evaluation in context-dependent DEA

Abstract

Context-dependent data envelopment analysis based on cross-efficiency evaluation has been proposed to present more meaningful measures of attractiveness and progress scores, cross-attractiveness, and cross-progress scores, by considering the distance between the decision-making unit (DMU) under evaluation and the entire evaluation context overall. Using an illustrative example, we show that the state of the art existing method does not guarantee to produce a non-dominated cross-attractiveness (cross-progress) scores vector. This raises concerns about the reliability and universal acceptance of the derived scores. Thus, we investigate the Pareto optimality of these peer evaluation scores and introduce a concept of optimality in line with the dominance notion. We subsequently propose a multi-objective model designed to produce all non-dominated cross-attractiveness (cross-progress) score vectors. We introduce two perspectives – a common base and an individualized approach. The common base method determines the same vector of weights for all units in the evaluation context to produce non-dominated scores. In contrast, the individualized based approach empowers each DMU within the evaluation context to assess the attractiveness (progress) score of DMUs under evaluation at a specific level based on their own distinct criteria. This methodology is aligned with the inherent desire of the DMUs within the evaluation context to have the most substantial impact on the evaluation of cross-attractiveness (cross-progress) for the DMUs under the assessment at a specific level. We illustrate our proposed methods with a real-world examples to yield non-dominated cross-attractiveness (cross-progress) scores.

Appendix A The counterpart models for Pareto-optimal cross-progress context

The counterpart of Model (11) in Pareto-optimal cross-progress context is given as follows:

$$\begin{aligned} \max & \frac{1}{n(E^{l_{o}-g})} \sum _{q\in E^{l_{o}}} w_{q} \sum _{j\in E^{l_{o}-g}} \frac{u_j^{l_{o}-g} y_q}{ v_{j}^{l_{o}-g} x_{q}}\nonumber \\ \mathrm {s.t.} & \frac{u_j^{l_{o}-g} y_p}{ v_{j}^{l_{o}-g} x_{p}}\le \frac{u_p^{l_{o}-g} y_p}{ v_{p}^{l_{o}-g} x_{p}},\,\,\,\forall j,p\in E^{l_{o}-g}, \nonumber \\ & \frac{u_j^{l_{o}-g} y_p}{ v_{j}^{l_{o}-g} x_{p}}\le 1,\forall p\in E^{l_{o}-g},\nonumber \\ & u_j^{l_{o}-g}, v_j^{l_{o}-g}\ge 0,\forall j\in E^{l_{o}-g}, \end{aligned}$$

(A1)

The counterpart of Theorem (2) in Pareto-optimal cross-progress context is provided below.

Theorem 5

There is a common weights optimal solution to Model (A1).

Proof

The proof of theorem is the same as the proof of Theorem 2 where $l_{o}+d$ should be replaced by $l_{o}-g$. $\square$

The counterpart of Model (13) in Pareto-optimal cross-progress context is provided as follows:

$$\begin{aligned} \max & \beta =\frac{1}{n(E^{l_{o}-g})} \sum _{q\in E^{l_{o}}} w_{q} \frac{u^{l_{o}-g} y_q}{ v^{l_{o}-g} x_{q}}\nonumber \\ \mathrm {s.t.} & \frac{u^{l_{o}-g} y_p}{ v^{l_{o}-g} x_{p}}\le 1,\forall p\in E^{l_{o}-g},\nonumber \\ & u^{l_{o}-g}, v^{l_{o}-g}\ge 0. \end{aligned}$$

(A2)

The counterpart of Model (14) in Pareto-optimal cross-progress context is given as follows:

$$\begin{aligned} \max & {u^{l_{o}-g}(\sum _{q\in E^{l_{o}}} y_q)}\nonumber \\ \mathrm {s.t.} & {v^{l_{o}-g}(\sum _{q\in E^{l_{o}}} x_q)}=1,\nonumber \\ & {u^{l_{o}-g} y_p}-{ v^{l_{o}-g} x_{p}}\le 0,\forall p\in E^{l_{o}-g},\nonumber \\ & u^{l_{o}-g}, v^{l_{o}-g}\ge 0. \end{aligned}$$

(A3)

The counterpart of Model (15) in Pareto-optimal cross-progress context is provided as follows:

$$\begin{aligned} \min & \theta \nonumber \\ \mathrm {s.t.} & \sum _{p\in E^{l_{o}-g}} \lambda _p x_{p} \le \theta \left( \frac{1}{n\left( E^{l_{o}}\right) }\sum _{q\in E^{l_{o}}} x_q\right) , \nonumber \\ & \sum _{p\in E^{l_{o}-g}} \lambda _{p} y_{p} \ge \frac{1}{n\left( E^{l_{o}}\right) } \sum _{q\in E^{l_{o}}} y_q,\nonumber \\ & \lambda _{p}\ge 0, \forall p \in S^{l_{o}-g}. \end{aligned}$$

(A4)

The counterpart of Model (16) in Pareto-optimal cross-progress context is as given follows:

$$\begin{aligned} \max & \frac{u_p^{l_{o}-g} y_p}{ v_p^{l_{o}-g} x_{p}}\nonumber \\ \mathrm {s.t.} & \sum _{q\in E^{l_{o}}} w_q \frac{u_p^{l_{o}-g} y_q}{ v_p^{l_{o}-g} x_{q}}=\alpha '^*,\nonumber \\ & \frac{u_p^{l_{o}-g} y_j}{ v_p^{l_{o}-g} x_{j}}\le 1,\forall j\in E^{l_{o}-g},\nonumber \\ & u_p^{l_{o}-g}, v_p^{l_{o}-g}\ge 0, \end{aligned}$$

(A5)

where $\alpha '^*$ is the optimal value of Model (A2).

The counterpart of Theorem 4 in Pareto-optimal cross-progress context is provided as follows:

Theorem 6

If $(u_p^{*l_{o}-g},v_p^{*l_{o}-g})$ is an optimal solution of Model  (A5) in the evaluation of $DMU_p, p \in E^{l_{o}-g}$, then the aggregate vector of $(u_p^{*l_{o}-g},v_p^{*l_{o}-g},\forall p \in E^{l_{o}-g})$ will be an optimal solution of Model (A1).

Proof

The proof of theorem is the same as the proof of Theorem 4 where $l_{o}+d$ should be replaced by $l_{o}-g$. $\square$