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During the last four decades Data Envelopment Analysis (DEA) has attracted considerable attention in the OR community. Using DEA, the efficiency frontier is constructed based on assumptions concerning the production possibility set rather than a priori defining a functional relationship between inputs and outputs. In this contribution, we propose an algorithm to visualize the efficiency surface in a 3D diagram and to extract isoquants from the efficient hull based on different RTS assumptions which might be particularly helpful for presentation purposes. In doing so, we extend the existing literature which has concentrated on the visualization of production frontiers in 2D diagrams to the visualization of efficient rather than fully efficient hulls in 3D diagrams. Displaying a fully efficient hull, however, does not reflect all properties of the production possibility set as weakly efficient frontier segments are missing.

Starting with the pioneering work of Farrell and the subsequent publications on nonparametric efficiency measurement (see [

Nonparametric efficiency measurement approaches rely on assumptions concerning the production possibility set (PPS) rather than specifying a functional relationship between inputs and outputs to derive a production frontier, thereby offering a new way to efficiency studies and comparisons. Basically, the idea behind nonparametric efficiency measurement is to derive an unknown production frontier based on the analysis of observable input and output data of different decision making units (DMUs). In particular, using nonparametric approaches, the production frontier is constructed as an envelope to the PPS.

The basics of nonparametric efficiency measurement are often illustrated using diagrams to visualize the derivation of the PPS and the resulting production frontier. Many of these figures are based on the illustration of the production frontier for the single input single output case which, thus, reflects the maximum and thus efficient quantity of the single output that can be produced using a particular quantity of the single input. Extensions cover the visualization of the production possibility frontier for the two inputs single output and the two outputs single input case, respectively. These diagrams represent the efficient combination of two inputs to produce a particular quantity of a single output and the efficient combination of two outputs that can be produced given a particular quantity of a single input. Illustrations of production frontiers in 3D diagrams have been restricted to the portrayal of fully efficient hulls so far [

The purpose of this paper is, thus, to provide an algorithm which can be used to display the efficient hull of a set of DMUs in a 3D diagram, comprising fully as well as weakly efficient frontier segments. With the increasing use of DEA techniques, the visualization of the production surface in the two inputs one output case and the two outputs one input case, respectively, might be useful for presentation purposes. Particularly, the derivation of isoquants directly from the production surface using illustrative data sets is considered to be valuable regarding the discussion of results stemming from differing returns to scale (RTS) assumptions. The purpose of the paper, however, is not to minimize computer speed in producing production surfaces in 3D diagrams but to provide an algorithm which can be used to identify the respective production surface inclusive of the isoquants without using LP methods. In doing so, we consider both variable as well as constant RTS assumptions. For illustration purposes, approaches to derive isoquants for different input and output levels in combination with different RTS assumptions from the so-constructed production surface are presented.

Accordingly, the paper is structured as follows. First, we provide a brief introduction into nonparametric efficiency measurement and its underlying assumptions. Next, we present the algorithms suitable to visualize the efficient hull under constant returns to scale (CRS) and variable returns to scale (VRS) for the case of two inputs and one output. Finally, an algorithm to extract isoquants from the efficient hull for different output levels and different RTS assumptions is presented. All algorithms provided in this contribution have been implemented in Matlab. Some concluding remarks are given in the final section.

Nonparametric efficiency measurement is based on the analysis of observable input and output data of different DMUs. These input output correspondences constitute the PPS. To obtain an estimate of the unknown production frontier, particular assumptions concerning the PPS are made.

Consider a multiple input multiple output setting where DMU

All observed activities belong to

If activity

If activity

Using matrix notation

Given the respective PPS, the technical efficiency of a DMU

Given the optimal solution to (

Computationally, the difference between weak and full efficiency is obtained in a two-step procedure. First we solve (

According to the above definitions of inefficient, weakly and fully efficient DMUs, we can therefore identify three different areas of the PPS: inefficient DMUs lie inside the PPS, weakly and fully efficient DMUs are located on the frontier. To differentiate between weakly and fully efficient DMUs, it is necessary to further distinguish between weakly and fully efficient frontier segments. On fully efficient frontier segments, it is not possible to improve any input or output without worsening some other input or output, while, on weakly efficient frontier segments, it is possible to improve an input or output without worsening other input or output.

It is common to visualize the PPS and the resulting frontier, especially the differences between the two frontier segments, in 2D diagrams. We start accordingly with a VRS frontier based on the observation of 15 DMUs that use one input (

Data for the single input single output case.

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

9,50 | 3,75 | 3,75 | 5,75 | 8,25 | 8,00 | 11,50 | 14,50 | 12,00 | 6,67 | 6,25 | 9,25 | 8,00 | 12,50 | 4,25 | |

9,25 | 2,00 | 4,50 | 7,00 | 8,75 | 4,00 | 6,00 | 9,50 | 9,50 | 8,00 | 2,50 | 7,50 | 3,00 | 8,50 | 2,50 |

The resulting frontier in the input output space is the envelope to the smallest convex set containing the activities of all 15 DMUs (see Figure

Single input single output case, VRS.

Adding a second input

Data for the two inputs single output case.

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

9,50 | 3,75 | 3,75 | 5,75 | 8,25 | 8,00 | 11,50 | 14,50 | 12,00 | 6,67 | 6,25 | 9,25 | 8,00 | 12,50 | 4,25 | |

2,50 | 0,20 | 2,50 | 3,00 | 3,00 | 0,08 | 2,50 | 2,00 | 4,00 | 5,00 | 0,05 | 1,25 | 0,50 | 0,75 | 0,10 | |

9,25 | 2,00 | 4,50 | 7,00 | 8,75 | 4,00 | 6,00 | 9,50 | 9,50 | 8,00 | 2,50 | 7,50 | 3,00 | 8,50 | 2,50 |

Two inputs single output case, CRS.

The fully efficient DMUs are now D, A, L, N, O, and F. Among the inefficient DMUs, we find DMU G which obviously has to reduce its inputs proportionately to be located on the frontier. DMU K, however, is now revealed as being weakly efficient, that is, DMU K has no possibility of proportionately reducing its inputs without reducing the output. DMU K, however, uses the same level of input

Obviously, in the case of two inputs and one output, most illustrations fall back on isoquants to explain and visualize how nonparametric efficiency measurement works. Dividing input

Nevertheless, there has been little attempt to visualize the idea behind nonparametric efficiency measurement using production surfaces in 3D diagrams as illustrations of production surfaces in 3D diagrams are hardly found in the literature. If at all, these production surfaces have been restricted to the presentation of fully efficient hulls, similar to those provided in [

Fully efficient hull.

Efficient hull.

Visualizing the idea behind nonparametric efficiency measurement in 3D diagrams with a fully efficient hull as in Figure

The main idea of our first algorithm is to display the efficient hull in a 3D diagram for a set of DMUs that produce a single output with two inputs with the PPS satisfying all properties in (

Accordingly, we proceed as follows. First, we show that the fully efficient hull for a set of DMUs is part of the convex hull of the same set. To derive the convex hull, we fall back on the well known Quickhull algorithm (see [

The convex hull

Hyperplane

Based on the above definitions, we can state the following.

A fully efficient DMU according to (

Without loss of generality, we assume that DMU

DMU

DMU

Ad (a): If DMU

Ad (b): If DMU

The fully efficient hull is part of the convex hull. The hyperplanes that constitute the fully efficient hull are called fully efficient hyperplanes

The hyperplanes that constitute the weakly efficient hull are called weakly efficient hyperplanes

The efficient hull is the combination of the fully and weakly efficient hull.

Based on the definitions in Section

Compute the convex hull with Quickhull. Identify all triangles (Quickhull uses triangles to produce the convex hull in 3D) with normal vector components

Each triangle consists of three vectors. Project the remaining triangles to the

Assign the remaining vectors to one of the following groups according to the subsequent procedure.

Start with the minimum output DMU, and go clockwise up to the first vector with normal vector components

Vectors with normal vector components

Vectors with normal vector components

Vectors with normal vector components

Go to the maximum output DMU, and go clockwise up to the first vector which has already been assigned to one of the groups 3(a), 3(b), or 3(c) in Step

clockwise direction: project the vectors between the maximum output DMU and the first vector which has already been assigned to one of the above groups 3(a), 3(b), or 3(c) to the

anticlockwise direction: project the vectors between the maximum output DMU and the first vector which has already been assigned to one of the above groups 3(a), 3(b), or 3(c) to the

Produce the efficient hull in a 3D diagram.

To visualize algorithm 1_VRS, we display some intermediate results graphically. Using Quickhull, the convex hull assuming VRS for the set of 15 DMUs is displayed in Figure

Convex hull, VRS.

Fully efficient hull, VRS.

2D hull.

Efficient hull, VRS.

Efficient hull, VRS.

To take CRS into consideration, we add the origin to the set of DMUs before we proceed with the following Algorithm 1_CRS.

Find the maximum

Compare the unity normal vector of each DMU vector with the unity normal vector of the maximum

If the unity normal vector component

If the unity normal vector component

Compute the convex hull with Quickhull for these vectors. Identify all triangles with normal vector components

Project the remaining triangles to the

inside the 2D hull and

with normal vector components

Divide the remaining two vectors according to the type of the normal vector into the

Add two new triangles by joining

the origin (0,0,0), the point

the origin (0,0,0), the point

Produce the efficient hull in a 3D diagram.

Figure

Scaled-up triangles, CRS.

Fully and all weakly efficient triangles.

Efficient hull, CRS.

After having constructed the efficient hull for a set of DMUs using Algorithms 1_VRS and 1_CRS, respectively, we now proceed to the illustration of the extraction of the resulting isoquants for different output levels under different RTS assumptions. In doing so, we proceed as follows according to Algorithm 2.

Define

Cut off the efficient hull as obtained from Algorithm 1_VRS and/or 1_CRS at level

at least one vertex satisfies

at least one vertex satisfies

For each intersecting triangle, compute the two intersection points at level

Display the isoquant(s) in a 2D diagram.

Following Algorithm 2, we now display four different isoquants, one for the output level

Input isoquants at

Input isoquants at

Input isoquant CRS at

The algorithms introduced above can also be used to visualize the derivation of output isoquants given the two outputs one input case as illustrated in Figure

Output isoquants at

Deriving a production frontier based on assumptions concerning the production possibility set rather than a priori specifying a functional relationship between inputs and outputs to perform efficiency comparisons among different DMUs has attracted considerable attention during the last decades. Most of the ideas inherent in nonparametric efficiency measurement, as well as the results of nonparametric efficiency measurement, have been explained using different types of curves, mostly restricted to illustrations in 2D diagrams. In this contribution, we present algorithms which can be used to determine and visualize a production frontier in the form of an efficient hull in a 3D diagram in the case where multiple DMUs use two inputs to produce a single output. The algorithms introduced can easily be adjusted to the two outputs single input case. In deriving efficient production surfaces for the case of CRS as well as VRS, we fall back on the Quickhull algorithm rather than solving a DEA model. Additionally, based on the so-constructed hulls, we introduce an algorithm to directly derive isoquants for particular output (input) levels and CRS and VRS as well. The next steps could be the implementation of the above algorithms in available DEA software programs and the optimization of computer speed in producing efficient production surfaces and isoquants.