Data Envelopment Analysis

Concept

---

Theoretically, Data Envelopment Analysis (DEA) is a non-parametric mathematical optimisation method used to evaluate the relative technical efficiency of comparable decision-making units (DMUs) that utilise multiple inputs to produce multiple outputs. Developed by Charnes, Cooper and Rhodes, DEA is founded on linear programming and production frontier theory. Unlike parametric efficiency models, DEA constructs an empirical efficiency frontier directly from observed data without requiring assumptions about the underlying production function.

Mathematically, DEA estimates the efficiency of each decision-making unit by solving a constrained linear programming optimisation problem that maximises the ratio of weighted outputs to weighted inputs while ensuring that no unit can achieve an efficiency score greater than one under the same weighting scheme. The resulting efficiency score ranges from 0 to 1, with efficient units lying on the frontier and inefficient units located beneath it.

In practice, Data Envelopment Analysis is widely applied in health economics to evaluate the efficiency of hospitals, primary care organisations, health systems and public health programmes. It enables simultaneous assessment of multiple resource inputs and health outcomes, supports benchmarking, identifies best-performing organisations and estimates potential efficiency gains through improved resource allocation.

Purpose

---

Used to measure the relative technical efficiency of comparable organisations using multiple inputs and outputs, identify best practice, benchmark performance and support operational and health economic decision-making.

Mathematical Formulae

Primary Formula

Output-Oriented DEA Efficiency

Maximise

? = (�u?y??) � (�v?x??)

Subject to

(�u?y??) � (�v?x??) � 1

u? � 0

v? � 0

where:

• y? = outputs
• x? = inputs
• u? = output weights
• v? = input weights
• j = decision-making units

Supporting Formulae

Input-Oriented Efficiency

Minimise ?

Subject to

Y? � y?

X? � ?x?

? � 0

Technical Efficiency

TE = Observed Output � Frontier Output

Related Mathematical Methods

• Linear programming
• Production frontier analysis
• Technical efficiency
• Allocative efficiency
• Stochastic frontier analysis
• Productivity analysis

---

Example

Three hospitals use physicians, nurses and hospital beds as inputs while producing inpatient discharges, outpatient visits and surgical procedures as outputs.

DEA estimates the efficient production frontier and calculates an efficiency score for each hospital.

Hospital A receives:

Technical Efficiency = 0.92

This indicates that Hospital A could theoretically reduce its inputs by approximately 8% while maintaining the same level of outputs if it operated as efficiently as comparable frontier hospitals.

---

Excel Implementation

Function	Example Formula	Health Economics Application
SUMPRODUCT	=SUMPRODUCT(B2:D2,$H$2:$H$4)	Calculate weighted inputs.
SUMPRODUCT	=SUMPRODUCT(E2:G2,$I$2:$I$4)	Calculate weighted outputs.
Solver	Maximise =Output/Input	Solve the DEA linear programming optimisation problem.
IF	=IF(J2=1,"Efficient","Inefficient")	Classify decision-making units by efficiency.
RANK	=RANK(J2,$J$2:$J$101,0)	Rank hospitals by DEA efficiency score.

---

VBA (Optional)

Automate preparation of DEA input-output datasets, interface with Excel Solver for repeated optimisation, calculate efficiency scores for multiple decision-making units and generate benchmarking reports.

---

Sources

• Charnes A, Cooper WW, Rhodes E. Measuring the Efficiency of Decision Making Units. European Journal of Operational Research. 1978.
• Cooper WW, Seiford LM, Tone K. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software.
• Thanassoulis E. Introduction to the Theory and Application of Data Envelopment Analysis.
• Hollingsworth B. The Measurement of Efficiency and Productivity of Health Care Delivery.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.