Measuring performance in the presence of noisy data with targeted desirable levels: evidence from healthcare units

Abstract

Noise in data is not uncommon in real-world cases, although it is commonly omitted from performance measurement studies. In this paper, we develop a stochastic DEA-based methodology to measure performance when the endogenous (e.g. efficiency) and exogenous variables (e.g. perspectives of patients’ satisfaction), which are incorporated in the assessment, are inversely related. This methodology identifies benchmark units that are not only efficient but are also assigned scores for their exogenous variables, which are at least equal to user-defined critical values. We apply the performance measurement methodology to the 14 largest Cypriot health centers. The advantages of our methodology are pointed out through comparative analysis with alternative stochastic and non-stochastic DEA approaches.

Appendix

1.1 Section A

Expression (6) can be rewritten as follows:

$$ \frac{{(b_{{k_{o} h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} }}{{(c^{*} - b^{*} )^{2} + (\eta^{*} - (\eta_{h} )^{ad} )^{2} }} = \frac{{(b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}{{(c^{*} - b^{*} )^{2} \cdot (\eta^{*} - (\eta_{h} )^{ad} )^{2} }} $$

(14)

Let $u_{1} = (b_{{k_{o} h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2}$ and $u_{2} = (b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2}$

Then, Eq. (14) is written as follows:

$$ \begin{aligned} u_{1} \cdot (c^{*} - b^{*} )^{2} \cdot (\eta^{*} - (\eta_{h} )^{ad} )^{2} & = u_{2} \cdot [(c^{*} - b^{*} )^{2} + (\eta^{*} - (\eta_{h} )^{ad} )^{2} ] \\ (\eta^{*} - (\eta_{h} )^{ad} )^{2} & = \frac{{u_{2} \cdot (c^{*} - b^{*} )^{2} }}{{u_{1} \cdot (c^{*} - b^{*} )^{2} - u_{2} }} \\ \left| {\eta^{*} - (\eta_{h} )^{ad} } \right| & = \left( {\frac{{u_{2} \cdot (c^{*} - b^{*} )^{2} }}{{u_{1} \cdot (c^{*} - b^{*} )^{2} - u_{2} }}} \right)^{1/2} \\ \end{aligned} $$

(15)

The acceptable real root of (15) is:

$$ (\eta_{h} )^{ad} = \eta^{*} + \left( {\frac{{u_{2} \cdot (c^{*} - b^{*} )^{2} }}{{u_{1} \cdot (c^{*} - b^{*} )^{2} - u_{2} }}} \right)^{1/2} $$

(16)

which satisfies the condition $(\eta_{h} )^{ad} > \eta^{*}$

By replacing $u_{1}$ and $u_{2}$ in (16) we obtain the following expression:

$$ (\eta_{h}^{{}} )^{ad} = \eta^{*} + \left( {\frac{{(b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \cdot (c^{*} - b^{*} )^{2} }}{{((b_{{k_{o} h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) \cdot (c^{*} - b^{*} )^{2} - (b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}} \right)^{1/2} $$

(17)

1.2 Section B

Let the expression (17)

$$ (\eta_{h}^{{}} )^{ad} = \eta^{*} + \left( {\frac{{(b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \cdot (c^{*} - b^{*} )^{2} }}{{((b_{{k_{o} h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) \cdot (c^{*} - b^{*} )^{2} - (b_{{k_{o} h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}} \right)^{1/2} $$

By replacing the undesirable exogenous variable $b_{{k_{o} h}}^{d}$ by the remaining undesirable exogenous variables (i.e. $b_{{(k \ne k_{o} )h}}^{d}$), and consequently replacing the critical value (i.e. $c^{*}$) by the adjusted value for the remaining undesirable exogenous variables (i.e. $(b_{{(k \ne k_{o} )h}}^{d} )^{ad}$), and by solving expression (17) for $(b_{{(k \ne k_{o} )h}}^{d} )^{ad}$, the obtained expression is as follows:

$$ \begin{aligned} & (\eta_{h}^{{}} )^{ad} = \eta^{*} + \left( {\frac{{(b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} }}{{((b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} - (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}} \right)^{1/2} \\ & ((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} = \frac{{(b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} }}{{((b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} - (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }} \\ & ((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} \cdot (((b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} - (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} ) \\ & \quad = (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \cdot ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} \\ & ((b_{{(k \ne k_{o} )h}}^{d} )^{ad} - b^{*} )^{2} \cdot (((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} \cdot ((b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) - (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} ) \\ & \quad = ((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} \cdot (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} \\ & (b_{{(k \ne k_{o} )h}}^{d} )^{ad} = b^{*} + \left( {\frac{{((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} \cdot (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}{{((\eta_{h}^{{}} )^{ad} - \eta^{*} )^{2} \cdot ((b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} + (\eta^{*} - \eta_{h} )^{2} ) - (b_{{(k \ne k_{o} )h}}^{d} - b^{*} )^{2} \cdot (\eta^{*} - \eta_{h} )^{2} }}} \right)^{1/2} \\ \end{aligned} $$

1.3 Section C

The first non-stochastic performance measurement application draws on program (1) (see Sect. 2), which replaces program (3) in the algorithm, and the following program, which is used instead of program (11):

$$ \begin{aligned} & \hbox{min} \theta \\ & \begin{array}{*{20}l} {s.t.} \hfill & { \mathop \sum \limits_{j = 1}^{n} \lambda_{j} X_{ij} \le \theta X_{io} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} \ge y_{ro} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} B_{tj} \ge \theta B_{to} } \hfill \\ {} \hfill & {\lambda_{j} \ge 0} \hfill \\ \end{array} \\ \end{aligned} $$

(18)

Concerning the incorporation of DDF in the algorithm presented in Sect. 2 of this study, program (19) is used instead of program (3)

$$ \begin{aligned} & \hbox{max} \beta \\ & \begin{array}{*{20}l} {s.t. } \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{ij} + \beta g_{x} \le x_{io} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} - \beta g_{y} \ge y_{ro} } \hfill \\ {} \hfill & {\lambda_{j} \ge 0} \hfill \\ {} \hfill & {g_{x} = 1,\;g_{y} = 0} \hfill \\ \end{array} \\ \end{aligned} $$

(19)

where $\beta$ expresses inefficiency, and $g_{x}$ and $g_{y}$ are the actual input and output direction vectors, respectively.

Moreover, program (20) replaces program (11)

$$ \begin{aligned} & \hbox{max} \beta \\ & \begin{array}{*{20}l} {s.t.} \hfill & { \mathop \sum \limits_{j = 1}^{n} \lambda_{j} X_{ij} + \beta g_{X} \le X_{io} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} - \beta g_{y} \ge y_{ro} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} B_{tj} - \beta g_{B} \le B_{to} } \hfill \\ {} \hfill & {\lambda_{j} \ge 0} \hfill \\ {} \hfill & {g_{x} = 1,\;g_{y} = 0,\;g_{B} = - 1} \hfill \\ \end{array} \\ \end{aligned} $$

(20)

where $g_{X}$ and $g_{B}$ stand for the direction vectors of all inputs (i.e. adjusted and non-adjusted, or actual) and all exogenous variables (i.e. adjusted and non-adjusted), respectively. The exogenous variables are freely disposable, like the inputs, as they cannot be regulated.

The third non-stochastic approach, the GDDF, is written as follows:

$$ \begin{aligned} & min\frac{{1 - \frac{1}{m}\mathop \sum \nolimits_{i = 1}^{m} \beta g_{i} /x_{io} }}{{1 + \frac{1}{s}\mathop \sum \nolimits_{r = 1}^{s} \beta g_{r} /y_{ro} }} \\ & \begin{array}{*{20}l} {s.t. } \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} x_{ij} + \beta g_{x} \le x_{io} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} - \beta g_{y} \ge y_{ro} } \hfill \\ {} \hfill & { \lambda_{j} \ge 0} \hfill \\ {} \hfill & {g_{x} = 1,\;g_{y} = 0} \hfill \\ \end{array} \\ \end{aligned} $$

(21)

where $\beta g_{i} /x_{io}$ denotes the proportional decrease in inputs, and $\beta g_{r} /y_{ro}$ is the proportional increase in outputs. Program (21) is used in the performance measurement algorithm instead of program (3).

Moreover, program (11) is replaced by the following program:

$$ \begin{aligned} & min\frac{{1 - \frac{1}{m}\mathop \sum \nolimits_{i = 1}^{m} \beta g_{i} /X_{io} }}{{1 + \frac{1}{s + w}\left( {\mathop \sum \nolimits_{r = 1}^{s} \beta g_{r} /y_{ro} + \mathop \sum \nolimits_{t = 1}^{w} \beta g_{t} /B_{to} } \right)}} \\ & \begin{array}{*{20}l} {s.t. } \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} X_{ij} + \beta g_{x} \le X_{io} } \hfill \\ {} \hfill & {\mathop \sum \limits_{j = 1}^{n} \lambda_{j} y_{rj} - \beta g_{y} \ge y_{ro} } \hfill \\ {} \hfill & { \mathop \sum \limits_{j = 1}^{n} \lambda_{j} B_{tj} - \beta g_{B} \le B_{to} } \hfill \\ {} \hfill & {\lambda_{j} \ge 0} \hfill \\ {} \hfill & {g_{X} = 1,\;g_{y} = 0,\;g_{B} = - 1} \hfill \\ \end{array} \\ \end{aligned} $$

(22)

where $\beta g_{i} /X_{io}$ expresses the proportional reduction in inputs, and $\beta g_{r} /y_{ro}$ and $\beta g_{t} /B_{to}$ are the proportional increase in outputs and proportional decrease in exogenous variables, respectively.