DEA Cross-Efficiency - peer appraisal using cross-evaluation matrix

Overview

DEA Cross-Efficiency - peer appraisal using cross-evaluation matrix

Output

Efficiency, higher is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

General MCDM

How it works

1. 1

   Identify inputs and outputs per DMU; same as CCR.

   $$X\in {\mathbb{R}}^{p\times m},Y\in {\mathbb{R}}^{q\times m}$$

   Sexton 1986, p.74 Sec.2

2. 2

   Solve CCR LP per DMU_o to obtain own efficiency θ*_o and weights.

   $$\max \frac{{\mathbf{u}}_o^T{Y}_o}{{\mathbf{v}}_o^T{X}_o}\text{s.t.}\frac{{\mathbf{u}}_o^T{Y}_j}{{\mathbf{v}}_o^T{X}_j}\le 1$$

   Sexton 1986, p.75 Eq.(1)

3. 3

   Cross-efficiency E_oj = (u_o^T Y_j)/(v_o^T X_j) using DMU_o's weights on DMU_j.

   $$E_{oj}=\frac{{\mathbf{u}}_o^T{Y}_j}{{\mathbf{v}}_o^T{X}_j}$$

   Sexton 1986, p.76 Eq.(2)

4. 4

   Average cross-efficiency ē_j = (1/m) Σ_o E_oj; descending ranking.

   $${\bar{e}}_j=\frac{1}{m}\sum _{o=1}^m{E}_{oj}$$

   Sexton 1986, p.76 Eq.(3)

Look elsewhere when

Assumptions to verify

• Clear input and output criteria distinction
• All DMUs in same industry/context

Edge cases and pitfalls

• Applying DEA-CROSSEFF without verifying this assumption.

  Requirement: Clear input and output criteria distinction

• Applying DEA-CROSSEFF without verifying this assumption.

  Requirement: All DMUs in same industry/context

• Using DEA-CROSSEFF when: Mixed input/output direction confusion.

  An alternative method is recommended in this situation.

• Using DEA-CROSSEFF when: Small sample (m < 3·(inputs+outputs)).

  An alternative method is recommended in this situation.

How to cite