Extension card · Classical
Super-Efficiency DEA (Andersen and Petersen, 1993)
Super-efficiency DEA is a form that allows units classical DEA rates as equally "efficient" (theta=1) to be ranked amongst themselves as well. Each unit is re-evaluated with its own data excluded from the reference set; this brings out a degree of superiority even among efficient units.
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?
Cells. The data type is unchanged: the input and output columns are crisp, non-negative numbers. The difference is not in the data type but in how the reference set is built.
Scale equalisation. In classical DEA (CCR), each unit is evaluated against a frontier formed by all units, including itself; this is why every unit on the frontier scores theta=1 and no distinction is made among them. Super-efficiency excludes each unit in turn from the reference set and builds a new frontier from the remaining units. An efficient unit is thereby no longer compared against itself, only against the others.
Distance / score / combination. This exclusion can push an efficient unit's score above 1: the score now answers the question "by how much could this unit have expanded its inputs relative to the frontier formed by the others, and still have remained on the frontier?" For an inefficient unit, this exclusion has no effect at all, because its own data was never defining the frontier in the first place; its score remains identical to the classical CCR score.
Result and defuzzification. The output is again a single figure (theta), but its upper bound is no longer 1. Values above theta=1 show how far and "safely" the unit sits from the frontier; values below 1 are read exactly as in classical DEA. There is no defuzzification; the calculation proceeds with crisp numbers.
How to Read the Output
For an inefficient unit, the reading is as on the DEA card. The difference lies among units scoring theta=1: classical DEA cannot distinguish these units, whereas super-efficiency ranks them by looking at how far above 1 theta rises. A unit with a higher theta stands further from the frontier formed by the other units in the reference set; this is a numerical answer to the question "how much more resource could it use and still remain on the frontier?"
So, instead of writing:
"Both A and B come out efficient in DEA, so both are equally good"
the report should read:
"A and B are both efficient with theta=1 in classical DEA; in super-efficiency, B's score (1.80) is higher than A's (1.71). This shows B stands a little further than A from the frontier formed by the other units, though the two values are close to one another"
When to Prefer This over the Base Method
Super-efficiency is preferred when more than one unit scores theta=1 as efficient in classical DEA and a priority order is also needed among these units. The input/output distinction and DEA's principle of not taking outside weights apply here exactly as they do there. For inefficient units, super-efficiency adds nothing to classical DEA; it should be used only when distinguishing among the frontier units is required.
Mistakes Specific to This Extension
Forgetting that the linear programme can become infeasible for some efficient units in the input-oriented model. If a unit's input-output composition differs radically from all the other units, excluding it may leave no combination able to define its frontier at all. DecisionMind flags this case in the data check; for such a unit, the super-efficiency score cannot be interpreted.
Mistaking a score above 1 for a percentage. Theta=1.80 does not mean the unit is "80 per cent more efficient"; it means the unit could expand its inputs by 80 per cent relative to the frontier formed by the others and still remain on the frontier.
Trying to "improve" an inefficient unit's score using super-efficiency. Super-efficiency exists only to distinguish among efficient units; an inefficient unit's score is identical to its classical DEA score and is not changed by this method.
The governing principle is this:
Super-efficiency ranks the efficient units that classical DEA cannot distinguish, by excluding each in turn from its own reference set; a value above 1 is not a measure of quality, but a measure of distance from the frontier.
Cases
Both cases are illustrative constructions. The first is a synthetic, hand-traceable table that DecisionMind's engine built by hand to include both efficient and inefficient units together; it is not taken from a paper or book table.
1. Illustrative example: Ranking the efficiency of three hospitals (DecisionMind's validation example)
Three hospitals (DMU1, DMU2, DMU3) convert two inputs (number of beds, staff expense) into two outputs (number of treatments, satisfaction score).
| Hospital | Number of beds | Staff expense | Number of treatments | Satisfaction score |
|---|---|---|---|---|
| DMU1 | 2 | 3 | 10 | 8 |
| DMU2 | 3 | 2 | 12 | 7 |
| DMU3 | 4 | 4 | 9 | 9 |
| Input/Output | Input | Input | Output | Output |
(There is no "Weight" row in the DEA family.)
Classical DEA first evaluates these three hospitals: DMU1 and DMU2 are efficient with theta=1.000, DMU3 is inefficient with theta=0.750. Classical DEA draws no distinction between DMU1 and DMU2. Super-efficiency re-evaluates each hospital against a reference set with its own data excluded.
| Hospital | theta (super-efficiency) | Classical DEA (theta) |
|---|---|---|
| DMU2 | 1.800 | 1.000 |
| DMU1 | 1.714 | 1.000 |
| DMU3 | 0.750 | 0.750 |
The result reads as follows: although DMU1 and DMU2 appear equally efficient in classical DEA, in super-efficiency DMU2 (1.800) comes out higher than DMU1 (1.714); DMU2 could expand its inputs by more, relative to the frontier formed by the other two hospitals, and still remain on it. DMU3's score is unchanged (0.750), because DMU3 was never defining the frontier in the first place; excluding it has no effect on the result.
The team's hesitation is this: what happens if DMU1's number of beds falls from 2 to 1.8 (a 10 per cent reduction)? This scenario was re-solved in Python; DMU1's score rises from 1.714 to 1.905, overtaking DMU2 (1.800). This shows that the ranking between the two efficient hospitals is sensitive to a small change in a single input, while DMU3's ranking (remaining third) is unaffected by this change.
In the report: "DMU1 and DMU2 are equally efficient in classical DEA; super-efficiency distinguishes between them, placing DMU2 (1.800) ahead of DMU1 (1.714). This ranking can shift with a small improvement in DMU1's number of beds; DMU3's third-place position is unaffected by this change."
Source: This is a synthetic, hand-traceable validation example built by hand for DecisionMind's DEA-SUPEREFF (Andersen and Petersen, 1993) engine; it is not a reproduction of a paper or book table. The scores were independently recalculated in Python by this card's author, verified exactly against DecisionMind's manifest internal-audit record.
2. Sports Facilities: A municipality's annual incentive ranking among neighbourhood sports centres
A municipality will distribute an annual incentive among three neighbourhood sports centres and wants to base the distribution on an efficiency ranking. The municipality has set two inputs (staff numbers, operating expense) and two outputs (number of members, number of events held).
Two centres score theta=1.000 as efficient in classical DEA, one is inefficient. Because the municipality wants a priority order between the two efficient centres as well, super-efficiency is applied. Suppose the centre with the smaller budget overtakes the larger-budget centre in the super-efficiency score; this means the smaller centre sits further from the frontier, that is, it uses its resources more intensively relative to the others.
The municipality's hesitation is this: is the super-efficiency ranking on its own sufficient to determine the incentive amount? If the two efficient centres' scores are close to one another (for example, 1.80 against 1.71), the gap between them can reverse with a small change in a single measure. The municipality should also consider splitting the incentive equally if the scores are close.
In the report: "The two sports centres are equally efficient in classical DEA; super-efficiency places one ahead of the other. However, because the two scores are close, the incentive distribution should take account not only of this ranking but also of how small the gap between them is."
3. What Not to Do
The first error is treating DMU2's super-efficiency score (1.800) being higher than DMU1's (1.714) in the illustrative example as an absolute statement that "DMU2 is definitely better managed"; the gap is small and, as shown above, can reverse with a small change in a single input. The second error is comparing DMU3's super-efficiency score (0.750) with its classical DEA score (0.750) and concluding "the method changed nothing, so it must be working incorrectly"; DMU3 was never defining the frontier, so the score staying unchanged is the expected result. The third error is presenting a score above 1 (for example, 1.80) as a "success rate" of one hundred and eighty per cent; this score is not a percentage but a measure of distance from the frontier.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-supereff
Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39(10), 1261–1264. DOI: 10.1287/mnsc.39.10.1261
Tone, K. (2002). A strange case of the cost and allocative efficiencies in DEA. Journal of the Operational Research Society, 53(11), 1225–1231. DOI: 10.1057/palgrave.jors.2601438
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
Cook, W. D., & Seiford, L. M. (2009). Data envelopment analysis (DEA) – Thirty years on. European Journal of Operational Research, 192(1), 1–17. DOI: 10.1016/j.ejor.2008.01.032