Performance Measurement Using Data Envelopment Analysis (DEA)

Abstract

The 1980s brought many challenges to hospitals as they attempted to improve the efficiency of health care delivery through the fixed pricing mechanism of Diagnostic Related Groupings (DRGs). In the 1990s, the federal government extended the fixed pricing mechanism to physicians’ services through Resource-Based Relative Value Scale (RBRVS). Enacted more recently, the Hospital Value-Based Purchasing (HVPB) program demands higher performance and quality of care and reduces payments for those providers that cannot achieve certain performance levels. Although these pricing mechanisms attempted to influence the utilization and quality of services by controlling the amount paid to hospitals and professionals, effective cost control must also be accompanied by a greater understanding of variation in physician practice behavior and development of treatment protocols for various diseases.

Appendices

Appendix A

1.1 A.1 Mathematical Details

Fractional formulation of CCR [CRS] model is presented below:

Model 1

$$ Maximize\kern0.5em {\theta}_o=\frac{{\displaystyle \sum_{r=1}^s{u}_r{y}_{ro}}}{{\displaystyle \sum_{i=1}^m{v}_i{x}_{io}}} $$

$$ subject\kern0.5em to\kern0.5em \frac{{\displaystyle \sum_{r=1}^s{u}_r{y}_{rj}}}{{\displaystyle \sum_{i=1}^m{v}_i{x}_{ij}}}\le 1 $$

$$ {u}_r,{v}_i\ge 0\kern1em for\kern0.5em all\kern0.5em r\kern0.5em and\kern0.5em i. $$

This model can be algebraically rewritten as:

$$ Maximize\kern0.5em {\theta}_o={\displaystyle \sum_{r=1}^s{u}_r{y}_{ro}} $$

$$ subject\kern0.5em to\kern0.5em {\displaystyle \sum_{r=1}^s{u}_r{y}_{rj}}\le {\displaystyle \sum_{i=1}^m{v}_i{x}_{ij}} $$

With further manipulations we obtain the following linear programming formulation:

Model 2

$$ Maximize\kern0.5em {\theta}_o={\displaystyle \sum_{r=1}^s{u}_r{y}_{ro}} $$

subject to:

$$ {\displaystyle \sum_{r=1}^s{u}_r{y}_{rj}}-{\displaystyle \sum_{i=1}^m{v}_i{x}_{ij}}\le 0\kern1em j=1,....n $$

$$ {\displaystyle \sum_{i=1}^m{v}_i{x}_{io}}=1 $$

$$ {u}_r,{v}_i\kern0.5em \ge 0 $$

1.2 A.2 Assessment of the Weights

To observe the detailed information provided in Fig. 2.7, such as benchmarks and their weights (λ), as well as Σλ leading to returns to scale (RTS) assessments, a dual version of Model 2 is needed. The dual model can be formulated as:

Model 3

$$ Minimize\kern0.5em {\theta}_o $$

subject to:

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{x}_{ij}}\le \theta {x}_{io}\kern1em i=1,....m $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{y}_{rj}}\ge {y}_{ro}\kern1em r=1,....s $$

$$ {\lambda}_j\ge 0\kern1em j=1,....n. $$

In this dual formulation, Model 3, the linear program, seeks efficiency by minimizing (dual) efficiency of a focal DMU (“o”) subject to two sets of inequality. The first inequality emphasizes that the weighted sum of inputs of the DMUs should be less than or equal to the inputs of focal DMU being evaluated. The second inequality similarly asserts that the weighted sum of the outputs of the non-focal DMUs should be greater than or equal to the focal DMU. The weights are the λ values. When a DMU is efficient, the λ values would be equal to 1. For those DMUs that are inefficient, the λ values will be expressed in their efficiency reference set (ERS). For example, observing Fig. 2.7, H7 has two hospitals in its ERS, namely H1 and H3. Their respective λ weights are reported as λ₁ = 0.237 and λ₃ = 0.038.

Appendix B

2.1 B.1 Mathematical Details for Slacks

In order to obtain the slacks in DEA analysis, a second stage linear programming model is required to be solved after the dual linear programming model, presented in Appendix A, is solved. The second stage of the linear program is formulated for slack values as follows as:

Model 4

$$ Maximize\kern0.5em {\displaystyle \sum_{i=1}^m{s}_i^{-}}+{\displaystyle \sum_{r=1}^s{s}_r^{+}} $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{x}_{ij}}+{s}_i^{-}={\theta}^{*}{x}_{io}\kern1em i=1,....m $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{y}_{rj}}-{s}_r^{+}={y}_{ro}\kern1em r=1,....s $$

$$ {\lambda}_j\ge 0\kern1em j=1,....n $$

Here, θ^* is the DEA efficiency score resulting from the initial run, Model 2, of the DEA model. Here, s ⁻ _i and s ⁺ _r represent input and output slacks, respectively. Please note that the superscripted minus sign on input slacks indicates reduction, while the superscripted positive sign on output slacks requires augmentation of outputs.

In fact, Model 2 and Model 4 can be combined and rewritten as:

Model 5: Input-Oriented CCR [CRS] Model

$$ Minimize\kern0.5em \theta -\varepsilon \left({\displaystyle \sum_{i=1}^m{s}_i^{-}}+{\displaystyle \sum_{r=1}^s{s}_r^{+}}\right) $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{x}_{ij}}+{s}_i^{-}=\theta {x}_{io}\kern1em i=1,....m $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{y}_{rj}}-{s}_r^{+}={y}_{ro}\kern1em r=1,....s $$

$$ {\lambda}_j\ge 0\kern1em j=1,....n $$

The ε in the objective function is called the non-Archimedean, which is defined as infinitely small, or less than any real positive number. The presence of ε allows a minimization over efficiency score (θ) to preempt the optimization of slacks, s ⁻ _i and s ⁺ _r . Model 5 first obtains optimal efficiency scores (θ^*) from Model 2 and calculates them, and then obtains slack values and optimizes them to achieve the efficiency frontier.

2.2 B.2 Determination of Fully Efficient and Weakly Efficient DMUs

According to the DEA literature, the performance of DMUs can be assessed either as fully efficient or weakly efficient. The following conditions on efficiency scores and slack values determine the full and weak efficiency status of DMU:

Condition	θ	θ*	All s − i	All s + r
Fully efficient	1.0	1.0	0	0
Weakly efficient	1.0	1.0	At least one s − i  ≠ 0	At least one s + r  ≠ 0

When Models 2 and 4 run sequentially (Model 5), weakly efficient DMUs cannot be in the efficient reference set (ERS) of other inefficient DMUs. However, if only Model 2 is executed, then weakly efficient DMUs can appear in the ERS of inefficient DMUs. The removal of weakly inefficient DMUs from the analysis would not affect the frontier or the analytical results.

2.3 B.3 Efficient Target Calculations for Input-Oriented CCR [CRS] Model

In input-oriented CCR [CRS] models, levels of efficient targets for inputs and outputs can be calculated as follows:

Inputs: ${\overset{\frown }{x}}_{io}={\theta}^{*}{x}_{io}-{s}_i^{-*}\kern1em i=1,....m$

Outputs: ${\overset{\frown }{y}}_{ro}={y}_{ro}+{s}_i^{+*}\kern1em r=1,....s$

Appendix C

3.1 C.1 CCR [CRS] Output-Oriented Model Formulation

Since Model 5, as defined in Appendix B, combines the needed calculations for the input-oriented CRS model, we can adapt the output-oriented CRS model formulation using this fully developed version of the model.

Model 6: Output-Oriented CCR [CRS] Model

$$ Maximize\kern0.5em \phi -\varepsilon \left({\displaystyle \sum_{i=1}^m{s}_i^{-}}+{\displaystyle \sum_{r=1}^s{s}_r^{+}}\right) $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{x}_{ij}}+{s}_i^{-}={x}_{io}\kern1em i=1,....m $$

$$ {\displaystyle \sum_{j=1}^n{\lambda}_j{y}_{rj}}-{s}_r^{+}=\phi {y}_{ro}\kern1em i=1,....s $$

$$ {\lambda}_j\ge 0\kern1em j=1,....n $$

The output efficiency is defined by ϕ. Another change in the formula is that the efficiency emphasis is removed from input (first constraint) and placed into output (second constraint).

3.2 C.2 Efficient Target Calculations for Output-Oriented CCR [CRS] Model

In output-oriented CCR models, levels of efficient targets for inputs and outputs can be calculated as follows:

Inputs: ${\overset{\frown }{x}}_{io}={x}_{io}-{s}_i^{-*}\kern1em i=1,....m$

Outputs: ${\overset{\frown }{y}}_{ro}={\phi}^{*}{y}_{ro}+{s}_i^{+*}\kern1em r=1,....s$.