Extension card · Classical
Data envelopment analysis BCC (variable returns to scale) (Banker, Charnes & Cooper, 1984)
This is the form of DEA for situations where the units being compared differ in size and that size difference must not distort the efficiency comparison. The output remains an efficiency score, together with every unit's returns-to-scale classification (advantaged at small scale, or at large scale).
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?
Two things change; the input/output split, the separate optimisation for every unit, and the "no one can exceed a hundred per cent" constraint do not.
One equation is added to the constraint set. Base DEA (the CCR model), while searching for every unit's own weights, applies only the constraint that "no unit can exceed a hundred per cent"; this implicitly assumes constant returns to scale (doubling one unit's input doubles its output exactly). The BCC model adds a single equation to this: the sum of the reference weights (λ) a unit uses to express itself must equal exactly 1. This equation is called the convexity constraint. It prevents a small unit from appearing artificially "efficient" or "inefficient" by benefiting from a large unit's scale; every unit is compared only against the frontier formed by units close to it in size.
The efficiency score splits into two. Base DEA had a single theta. Here two thetas are computed: the BCC model's theta (pure technical efficiency, stripped of the scale difference) and the CCR model's theta (overall efficiency, which also includes the scale difference). The ratio of the two (CCR theta divided by BCC theta) gives scale efficiency (SE). How close SE sits to 1 shows how accurately the unit is operating at its own scale; SE below 1 shows that the unit's scale is far from the ideal size.
A returns-to-scale (RTS) classification is added. For every unit, the BCC model also gives one of the labels "increasing returns to scale" (a scaled-down version would be more efficient, IRS), "constant returns to scale" (the scale is already ideal, CRS) or "decreasing returns to scale" (a scaled-up version would be less efficient, DRS). This classification is read from the sum of the reference weights (λ) and does not exist at all in base DEA.
DecisionMind fixes, for this extension, the convexity constraint, the SE calculation from the ratio of the two thetas, and the RTS classification from the sum of λ. Input/output orientation (input- or output-oriented) is a user-selectable parameter; DecisionMind runs input-oriented by default.
How to Read the Output
Reading the BCC theta is the same as in base DEA (see the DEA card): a value of 1 does not mean "perfect," it means "in this set and at this scale, no weighting could show this unit to be more efficient."
The difference lies here. The number of units scoring theta=1 in BCC is always greater than or equal to the number scoring theta=1 in base DEA (CCR), because the BCC frontier is more "generous," evaluating units of different sizes separately. Being "efficient" is therefore easier in BCC than in CCR, and conflating the two results gives a mistaken impression.
Thus instead of writing:
"This branch came out efficient in BCC, so it really is the most efficient branch"
the report should read:
"This branch is efficient at its own scale (BCC, θ=1); if its scale efficiency (SE) is below 1, this branch can still lag behind in overall efficiency (CCR), which shows it needs to grow or shrink its scale"
When to Prefer This over the Base Method
When the units being compared differ greatly in size (a small clinic against a large hospital, a district branch against a regional centre, and the like). Base DEA's constant-returns-to-scale assumption can unfairly disadvantage or advantage a large unit; this extension removes that distortion.
If the units are already similar in size, the two models generally give close results and base DEA is sufficient; in addition, the wider "efficient" set BCC produces (frontier crowding) can further reduce discriminating power when working with a small number of units. Base DEA's exit conditions apply here too: where the aim is to measure efficiency rather than preference, and the number of units is sufficient relative to the number of criteria, this extension is suitable.
Mistakes Specific to This Extension
Confusing BCC efficiency (pure technical efficiency) with CCR efficiency (overall efficiency). BCC=1 does not mean the unit is operating at the optimal scale; it only means it could not be shown to do better at its own scale.
Using BCC with too few units. Because of the convexity constraint, the VRS frontier produces a wider "efficient" set than CCR; when the number of units is not at least three times the sum of inputs and outputs (as a rule of thumb, m ≥ 3·(p+q)), nearly every unit can come out efficient and discriminating power is lost.
Confusing input/output orientation between CCR and BCC. When computing scale efficiency, the two models must be solved with the same orientation (both input-oriented or both output-oriented); if one is solved input-oriented and the other output-oriented, the SE ratio becomes meaningless.
The governing principle is this:
BCC theta shows a unit's position at its own scale; reporting BCC theta alone, without scale efficiency (SE) and the returns-to-scale (RTS) label, conceals whether the unit is genuinely operating at the ideal size.
Cases
The first case is DecisionMind's own verification table, illustrating the BCC model introduced by Banker, Charnes and Cooper (1984); the figures are not taken verbatim from a page in the book, but form an illustrative example built with three input-output pairs, independently verified with the engine's linear-programming solver. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's verification example): Comparing three units with BCC
Three units (A, B, C) are compared on a single input and a single output: A input=1/output=1, B input=3/output=4, C input=6/output=6. Input is lower is better, output is higher is better. DEA carries no "Weight" row; the method finds, for each unit on its own, the multipliers that show it in the most advantageous light.
| Unit | Input | Output |
|---|---|---|
| A | 1 | 1 |
| B | 3 | 4 |
| C | 6 | 6 |
The method solves the convexity-constrained (Σλ=1) linear program for every unit.
| Unit | BCC θ | CCR θ | Scale efficiency (SE) | Returns to scale (RTS) |
|---|---|---|---|---|
| A | 1.000 | 0.750 | 0.750 | Increasing (IRS) |
| B | 1.000 | 1.000 | 1.000 | Constant (CRS) |
| C | 1.000 | 0.750 | 0.750 | Decreasing (DRS) |
The result reads as follows: all three units are efficient at their own scale (BCC), because the VRS frontier makes all three points their own reference. But in overall efficiency (CCR) only B is efficient (θ=1.000); A's scale efficiency of 0.75 shows it needs to grow (increasing returns to scale), and C's scale efficiency, also 0.75, shows it needs to shrink (decreasing returns to scale).
The board's hesitation: what happens if a fourth unit is added to the set, one operating at a very small scale but with proportionally high output (D: input=0.5, output=0.7)? This scenario has been independently verified by re-solving the same linear-programming model in Python. Once D is added, A's BCC theta falls from 1.000 to 0.727; A is no longer efficient even at its own scale, because D now shows a better input-output ratio at a smaller scale than A. B and C remain efficient in BCC (θ=1.000) even after D is added.
In the report: "Among the current three units, all are efficient at their own scale (BCC θ=1.000); however, only B is also efficient in overall efficiency (CCR θ=1.000). A and C have a scale efficiency of 0.75 and require growth and shrinkage respectively. This result depends on the set of units compared; when a unit more efficient at a smaller scale than A is added, A's own-scale efficiency can also be lost."
Source: DecisionMind's DEA-BCC engine verification example; it follows the BCC/VRS model introduced by Banker, Charnes and Cooper (1984), and is not a table taken verbatim from the book's own page. This is an illustrative example. Theta, scale efficiency, RTS and the sensitivity scenario have been independently recomputed with the linear-programming solver and match the engine's output exactly.
2. Education: A district education office's comparison of schools
A district education office will compare resource use across five secondary schools in the district. Two of the schools are large, central schools with more than three hundred pupils; three are small neighbourhood schools with fewer than fifty. The office has set two inputs (number of teachers, annual budget) and two outputs (average exam-success score, graduation rate).
The office first tries base DEA (constant-returns-to-scale CCR); the result shows all the small neighbourhood schools as inefficient, because the constant-returns-to-scale assumption reflects the large schools' scale advantage against the small ones. The office cannot tell whether this result stems from the scale difference or a genuine management problem.
The office's hesitation is this: once it switches to the BCC model, two of the small schools come out efficient at their own scale; this shows that the small schools are not actually badly managed, they were simply being unfairly compared under the same scale assumption as the large schools. The office should base its resource-allocation decision on the BCC results, and its scale-expansion decision on the scale-efficiency (SE) results.
In the report: "Under the constant-returns-to-scale comparison, the small schools appear inefficient; once re-assessed with the variable-returns-to-scale model (BCC), two of these schools are efficient at their own scale. The result shows that the small schools are affected by scale difference rather than a management problem."
3. What Not to Do
The first mistake is looking at all three units coming out efficient in BCC (θ=1.000) in the illustrative example and concluding "all three are equally efficient"; B is also efficient in overall efficiency, A and C are not, and this difference shows up in the scale-efficiency column. The second mistake is solving the CCR and BCC models with different orientations (one input-, one output-oriented) and reporting the ratio of their thetas as scale efficiency; this ratio is meaningless. The third mistake is using eight input-output criteria with five units and concluding "everyone came out efficient with BCC, everyone is perfect"; this is a case where discriminating power has been lost to insufficient data, not a success of the method.
Sources
For the linear-programming model, intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-bcc
Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078–1092. DOI: 10.1287/mnsc.30.9.1078
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
Dyson, R. G., Allen, R., Camanho, A. S., Podinovski, V. V., Sarrico, C. S., & Shale, E. A. (2001). Pitfalls and protocols in DEA. European Journal of Operational Research, 132(2), 245–259. DOI: 10.1016/S0377-2217(00)00149-1