Extension card · Classical
Environmental DEA (Färe, Grosskopf, Lovell and Pasurka, 1989)
This is the form of DEA for situations where an undesirable output (such as a pollutant) cannot be disposed of freely, but can only be reduced proportionally alongside the inputs. Its output is again an environmental efficiency score between 0 and 1.
Base method
DEA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Classical →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Four things change; DEA's principle of finding its own weighting does not.
Input–output structure. In base DEA, every measure is either an input or an output, and output is generally treated as good. Here there is a third group: the undesirable output, for example carbon emissions. This measure is a by-product of production, and reducing it is desirable, but it cannot be cut freely as an input can; it is tightly bound to the production process. Where there is no desirable output at all, no undesirable output can be produced either.
Model constraints. In base CCR, every output is assumed to be freely disposable; a unit is still considered feasible if it chooses to produce less output. This assumption is dropped for the undesirable output. DecisionMind treats the undesirable output in this extension as a second input, tying it to the same contraction ratio as the other inputs. This is the linear-programming form of the weak disposability assumption.
Score calculation. In base CCR, theta contracts only the inputs, and outputs must remain at least at their observed level. Here, theta contracts both the inputs and the undesirable output by the same proportion; desirable outputs must still remain at least at their observed level. The method can optionally also take different importance weights across pollutants; in that case, each pollutant contracts at its own rate.
Result and defuzzification. The score is again a single figure between 0 and 1, and a score of 1 means the unit sits on the environmental frontier. The difference lies here: this score captures not only resource use but also pollutant generation. Even a unit that uses a great deal of resource but pollutes little may still see its score fall if its pollutant share is high.
DecisionMind keeps the constant-returns-to-scale assumption and the weak disposability of the undesirable output fixed in this extension. Pollutant weights come into play only if the user supplies them explicitly; if not supplied, the method computes the radial form.
How to Read the Output
As with base DEA, the score is not an absolute environmental-success percentage; it is relative to the set and the definition. The difference is that a low score can mean not only failing to produce a great deal with few resources, but also generating a disproportionately large amount of pollutant. The gap between two units' scores should not be interpreted without examining which share, resource use or pollutant, contributes more.
Hence, instead of writing:
"This plant is the most environmentally friendly plant"
the report should read:
"This plant sits on the environmental frontier for these three plants and this input–output–pollutant definition. Other plants would reach this score if they could deliver the same output with proportionally less input and less pollutant"
When to Prefer This over the Base Method
If the production process, alongside the desired output, generates an undesirable output (a pollutant, waste, or defect rate) that cannot be disposed of freely and cannot be separated from production, this extension is appropriate. If there is no undesirable output, or if it can be reduced entirely freely, base DEA is sufficient; forcing an undesirable output into this model adds needless complexity.
If there is a policy priority among pollutants (for example, a wish to reduce one pollutant more than another), the weighted form is preferred; otherwise the radial form suffices. The base method's exit condition still holds here: if the goal is to measure relative efficiency rather than preference, this family is appropriate.
Mistakes Specific to This Extension
Assuming the undesirable output is freely disposable. It is wrong to treat the pollutant either as an ordinary input or as something that could be produced not at all if desired. DecisionMind keeps the undesirable output bound to the same ratio as the inputs; relaxing this constraint unfairly advantages units that generate pollutants.
Adding the variable-returns-to-scale constraint directly to the constant-scale model. Adding this assumption directly stops the model being linear. DecisionMind uses a separate linearised form for this.
Giving a weighted interpretation without supplying pollutant weights. If the user has not specified weights, the method computes the radial form; interpreting the result as "one pollutant was given more importance" is wrong, because no weighting has been applied.
Removing the undesirable output from the table and running base CCR. Running CCR with only the input–output data, leaving out the pollutant column entirely, gives the same units a different, and generally higher, score. This score is no longer environmental but classical efficiency, and the report must be labelled accordingly.
The governing principle is this:
Environmental DEA exists to carry the undesirable output as a burden that cannot be separated from production and cannot be disposed of freely. Any application that ignores the pollutant, or freely zeroes it out like an ordinary input, removes this contribution.
Cases
The first case is DecisionMind's validation example; the fixture in the manifest is a small table built by hand. The literature's real-world application (OECD countries, 2000–2011) is not reproduced here. The second case is an illustrative construction.
1. Illustrative example: Comparing three plants' input, output and carbon emissions
Three production plants use two inputs (energy, capital) to produce two desirable outputs (production volume, revenue) and one undesirable output (carbon emissions).
| Plant | Energy (input) | Capital (input) | Production (output) | Revenue (output) | Carbon (undesirable) |
|---|---|---|---|---|---|
| P1 | 2 | 3 | 10 | 8 | 1.0 |
| P2 | 3 | 2 | 12 | 7 | 1.2 |
| P3 | 4 | 4 | 8 | 6 | 3.0 |
The method solves, for each plant, a linear programme that contracts the inputs and carbon by the same ratio while keeping desirable outputs at least at their observed level.
| Plant | Environmental efficiency score |
|---|---|
| P1 | 1.000 |
| P2 | 1.000 |
| P3 | 0.500 |
The result reads as follows. P1 and P2 sit on the environmental frontier. P3 could have delivered the same production and revenue as its peers using roughly half of its inputs and its carbon emissions.
The plant's hesitation is this: what would the score be if P3's carbon emissions were halved (from 3.0 to 1.5)? The same linear programme was rebuilt and tested in Python; the score rose from 0.500 to only 0.536. By contrast, when P3's inputs (energy and capital) were halved, the score rose to almost 1.000. This shows that P3's low score stems mainly from its input use, and the effect of its carbon share is smaller.
In the report: "Three plants were compared by assessing input, output and carbon emissions together. P1 and P2 sit on the environmental frontier (score 1.000); P3 trails with 0.500. P3's low score stems mainly from its input use; the effect of its carbon share is smaller."
Source: This is DecisionMind's validation example for the Environmental DEA engine. It is based on the weak disposability framework of Färe, Grosskopf, Lovell and Pasurka (1989), but it is not a table taken from the paper's own page. The figures were independently recalculated in Python.
2. Textiles: Comparing the environmental performance of three production plants
A textile group will compare the environmental efficiency of three production plants. The inputs are energy consumption and labour hours; the desirable output is the quantity of fabric produced; the undesirable output is the volume of untreated wastewater discharged.
The method solves, for each plant, a linear programme that contracts the inputs and the wastewater by the same ratio while keeping production volume at least at its observed level. Suppose the plant with the oldest technology receives the lowest score; the cause is both high energy consumption and a high wastewater share.
The group's hesitation is this: should the low-scoring plant invest only in treatment, or should it also change its production process? The score alone does not say which share is more influential; separating the input and undesirable-output shares requires their own examination.
In the report: "Among the three plants, the one with the oldest technology falls below the environmental frontier. This result stems from both energy use and wastewater share. Which share is more influential should be determined by a separate sensitivity analysis."
3. What Not to Do
The first error is running CCR on the illustrative example using only energy, capital, production and revenue, leaving the carbon column out of the table entirely. This produces a different figure, and it is no longer an environmental efficiency score but a classical one. The second error is placing carbon in the input column as though it could be reduced freely. The undesirable output cannot be separated from production; it is contracted together with the inputs under an equality constraint. The third error is reporting P3's score of 0.500 as "P3 is a dirty plant"; the score is valid only for these three plants and this input–output–pollutant definition.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-env
Färe, R., Grosskopf, S., Lovell, C. A. K., & Pasurka, C. (1989). Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach. Review of Economics and Statistics, 71(1), 90–98. DOI: 10.2307/1928055
Zhou, P., Poh, K. L., & Ang, B. W. (2016). Data envelopment analysis for measuring environmental performance. In Handbook of Operations Analytics Using Data Envelopment Analysis (Ch. 2, pp. 31–62). Springer. DOI: 10.1007/978-1-4899-7705-2_2
Tyteca, D. (1997). Linear programming models for the measurement of environmental performance of firms: Concepts and empirical results. Journal of Productivity Analysis, 8(2), 183–197. DOI: 10.1023/A:1013296909029
Zhou, P., Ang, B. W., & Poh, K. L. (2007). A mathematical programming approach to constructing composite indicators. Ecological Economics, 62(2), 291–297. DOI: 10.1016/j.ecolecon.2006.12.020
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. DOI: 10.1016/0377-2217(78)90138-8