Data envelopment analysis

DEA is a powerful technique that is based on linear programming. It was developed for measuring the performance of a set of comparable entities, called DMUs, which convert multiple inputs into multiple outputs.^{26 32} In this method, each hospital is compared against the estimated efficient frontier comprising the best-performing hospitals.^{17 18}

DEA has been already the most commonly used technique for measuring the relative efficiency in healthcare.^{12 19} In systematic reviews, we can observe that DEA is the predominant method of public hospital efficiency assessment.^{12 14 19} DEA is widely applicable since does not require any a priori specification of the underlying functional form that relates the inputs with the outputs.⁹ In addition, use of DEA is justified by its ability to incorporate multiple inputs and outputs in different units of assessment.^{9 32}

Several DEA models have been developed to analyse the efficiency based on Farrell’s concept.¹⁰ The most well known and basis for the rest DEA models is the Charnes, Cooper and Rhodes (CCR) model developed by Charnes et al,³³ which assumes that production has Constant Returns to Scale (CRS) and the Banker, Charne, Cooper (BCC) model developed by Banker et al,³⁴ under the assumption of Variable Returns to Scale (VRS).^{9 12} The choice of CCR or BCC model depends on the context of the problem under examination, that is, the technology linking the inputs to outputs in the transformation process.⁹

Generally, the CCR model— whereby the efficiency frontier has a constant slope (CRS), which means that any change in the inputs results to a proportional change in the outputs.²⁶ CRS may be adopted when machines are involved in the process, which roughly means that the production can be doubled by doubling the levels of inputs. However, when employees (human factor) participate in the production process, then it is naive to expect that they could work at a constant rate. The CCR efficiency assessment by the may be affected if the DMUs are not operating on the optimal scale, since CRS does not distinguish between the scale and pure (managerial) technical efficiency.³⁵ If the efficiency analysis considers a managerial perspective, a BCC technology assumption will be appropriate to understand if a scale of operations or provider’s practice affects productivity.^{27 36} Scale efficiency (SE) is defined as a ratio of CRS to VRS efficiency scores and provides evidence whether the DMU is operating on the optimal scale size.^{12 20} Furthermore, the efficiencies of DMUs can be comprehensively analysed using both CRS and VRS assumption for more realistic changes in production process, and implications in the real world.^{9 26} Other systematic reviews^{20 25} have reported similar findings where studies used both CRS and VRS assumptions in efficiency measurements.

Rationally, the commonly used orientations in DEA analysis are input orientation (ie, minimisation of inputs with the given amount of outputs) and output orientation (ie, inputs are held constant and outputs are proportionally increased).²⁶ Previous empirical studies³⁵ have argued that hospitals have relatively little control over their outputs (eg, expanding surgical operations), but more control over the inputs (eg, medical devices), where they have the social responsibility to provide medical treatment through the public hospitals in general. Thus, most studies adopt input orientation for efficiency assessment of the hospitals.^{20 25 37} In a few studies, output orientation is adopted in response to the strategic health plans of the countries aiming to expand healthcare provision during a specific period.^{38 39} However, in our study, we aim to estimate the optimal levels of the resources without deteriorating the levels of the health services that the hospitals provide. In this way, we provide the central authorities with the potential savings that could be made in the health sector.

The efficiency of a hospital is defined as the ratio of the weighted sum of outputs (total virtual output) to the weighted sum of inputs (total virtual input), with the weights being obtained in favour of each evaluated unit by the optimisation process. Assume n DMUs, each using m inputs to produce s outputs. We denote the vector of inputs for DMU j is Xj=x1j,…,xmjT and the vector of outputs is Yj=y1j,…,yrjT. The model (1) is formulated and solved for each hospital in order to obtain its efficiency score. The variables η=(η₁,…,η_m) and ω=(ω₁,…,ω_s) are the weights associated with the inputs and the outputs respectively. These weights are calculated in a manner that they provide the highest possible efficiency score for each hospital _jo under evaluation.

The input-oriented BCC model that provides the efficiency for the hospital _jo under VRS assumption is given below:

Notice that by excluding the free of sign variable ω_ο from model (1), the CCR model is obtained. The fractional model (1) can be transformed to a linear programme by applying the Charnes and Cooper transformation (C-C transformation hereafter).⁴⁰ The transformation is carried out by considering a scalar t∈R+ such as tηXj0=1 and multiplying all terms of model (1) with t> 0 so that v=tη, u=t ω, u_ο=tω_ο. The linear equivalent of model (1) is formulated as:

Once an optimal solution v^*, u^*, u_ο^* of model (2) is derived, the input-oriented BCC-efficiency ej0* for the hospital_jo under evaluation is obtained directly from the objective function.

Banker et al determined the Returns to Scale (RTS) using the optimal value of the free variable u_o in the multiplier model (2).³⁴ Given the point (x0,y0) that lies on the efficient frontier, the RTS at this point are identified by the following three conditions:

Increasing Returns to Scale (IRS) prevail at (x0,y0) if and only if uo*§amp;lt;0 for all optimal solutions. Meaning the increase in all production factors (inputs) resulted in more production (outputs).

Decreasing Returns to Scale (DRS) prevail at (x0,y0) if and only if uo*§amp;gt;0 for all optimal solutions, meaning an equal increase in all production factors led to less production.

CRS prevail at (x0,y0) if and only if uo*=0 in any optimal solutions, where equal increase in all production factors led to the same amount of increase in production.

Improvement management software (PIM-DEA V.3.2) was used for DEA analysis.⁴¹