CCR Calculations Example

This page illustrates the calculations of the CCR model through a little example (2 inputs, one output, 7 variables). The model described below is the same as the model CCR Input Oriented of the OSDEA library.

Basic knowledge of DEA is assumed in order to read this page. Most DEA basic concepts are covered in the dea page.

It is possible to download an Excel file which illustrates the CCR calculation of this example. The Excel file can be downloaded here. The Excel Solver needs to be enabled in Excel in order to run the optimisations.

The CCR Model – Fractional Form

The CCR model was the first developed by Charnes Cooper and Rhodes in 1978 ^[1]. The model in its fractional form is as follows:

The variables of this fractional problem are ‘u’ and ‘v’ which are respectively the output and input weight vectors (u=(u₁,u₂,…,u_s ) while v=(v₁,v₂,…,v_m)). The model was expressed in a sum (Σ) notation in the CCR paper although for clarity purposes the formulation has here been expanded. The notation in the CCR paper (‘s’ as number of outputs, ‘m’ as number of inputs, ‘n’ DMUs and DMU_O as the DMU under examination) is generally accepted and used in DEA’s literature. Because the above model formulation only measures DMU_O’s efficiency, the problem will have to be solved n times to measure all the DMUs’ efficiency. Linear programming optimisation techniques are used to find the optimal solutions (e.g. the simplex algorithm).

The first constraint ensures that the maximum possible value for the ratio is 1. The last two constraints restrict all the inputs and outputs to be non-negative. Some inputs are allowed to be equal to 0 although at least one input (or output) will need to have a positive value per input vector (or output vector). The input and output vectors are said to be semi-positive.

The formulation above measures the efficiency of DMU_O. As illustrated by the first constraint, the set of weights used in the objective function (the ‘max’ line) will be used with each of the other DMU’s values as per illustrated by the first constraint. This constraint specifies that the set of weights assigned to DMU_O and used with any other DMU’s values, must be lower or equal to 1. This particular constraint is responsible for enveloping the data and identifying the efficient frontier.

The CCR Model – Linear Form

The fractional problem illustrated above is not easy to solve. Rather than solving this problem directly, it is easier to transform it into an equivalent linear problem. This method was first described by Charnes and Cooper ^[2] in 1962. This gives the following linear equivalent model:

The CCR Model in its linear extended multiplier Form

For computational reasons, this model is generally solved in its dual form (see Wikipedia on duality) illustrated below (in its vectorial form; i.e. sums where replaced by vectorial products):

The CCR Model in its linear dual vector Form

DLPo has a feasible solution θ=1,λ_o=1,λ_j=0 (j≠o). This implies that θ^o is lower or equal to 1. Similarly, because Y ≥ 0,Y ≠ 0, the second constraint implies that λ is positive. Because X is also positive, the first constraint forces θ to be strictly positive. This consequently implies 0 < θ^* ≤ 1. The optimisation process tries to reduce the inputs in a radial manner while staying in the production possibility set. Thus, it is possible to say that some activities in (Xλ, Yλ) outperform (θx₀,y_O) when θ^* < 1 ^[3].

This linear dual model can be expressed with slacks as follows:

The CCR Model in its linear dual vector Form with slacks

Where e is a vector with all elements equal to 1.

It is possible to solve the LP problem in two phases, the first phase aiming at minimising x_o by minimising θ, and a second phase trying to maximise s^– and s⁺. An optimal solution θ^*,s^-*,s^+* obtained after solving Phase II is called the max slack solution. The solution obtained is not systematically unique (the score obtained is an optimum so is unique; however, the lambdas and weights might not be unique).

This is this formulation which OSDEA solves in the CCR Input oriented model (the solution to the output oriented model is deduced from the input oriented model). This is also this formulation which the Excel file illustrates.

The dataset

For this example, the dataset used is as follows:

The objective is to minimise the inputs while assuming the output is constant (slacks allowed). Obviously there would be degree of freedom using this small dataset but this dataset is only used for demonstration purposes.

It is possible to plot the data as follows:

Where the blue line represents the efficiency frontier. Unlike in the DEA page, where the objective was to maximise two outputs with one input, the frontier is defined by the DMUs which show the minimum amount of input 1 and input 2 in respect of their output. From the figure above, we can determine that the DMU D, E & F are efficient (on the efficiency frontier and with no slack).

The Calculations

From the dataset and the CCR_DLPo model above, the computation for DMU_a can be written as follows:

For the computations, we re-arrange the above linear problem to the equivalent linear problem below. This clearly reflects the matrices which will be computed.

Phase I calculations example for DMUa (extended form)

The result of this optimisation gives the optimised θ^* which is unique for DMU_a. This θ^* is set as an additional constraint for the second phase which aims at maximising the slacks. This constraint corresponds to the last row in the following figure.

Solving the last problem gives the slacks for DMU_a.

In order to get the score and slacks of all DMUs, it is necessary to solve this problem n times (n being the number of DMUs in the dataset; in this instances 7).

The Excel File

The calculations of the last two figures are illustrated in the Excel file. The file contains instructions as how to solve the problem.

References

↑ Cooper, William W.; Seiford, Lawrence M.; Tone Karou Data Envelopment Analysis – A Comprehensive Text with Models, Applications References and DEA-Solver Software. Second Edition, Springer, 2007, pp. 46-47
↑ Charnes, A. & Cooper, W. W. Programming with Linear Fractional Functionals. 9, Naval Research Logistics Quarterly, 1962, pp. 181-185
↑ Cooper, William W.; Seiford, Lawrence M.; Tone Karou Data Envelopment Analysis – A Comprehensive Text with Models, Applications References and DEA-Solver Software. Second Edition, Springer, 2007, pp. 44