Extension card · Classical
DEA Range-Adjusted Measure of Inefficiency (RAM) (Cooper, Park and Pastor, 1999)
RAM is a form of DEA that measures and sums, separately for each measure against its own value range, the input excess and output shortfall that a classical radial (percentage) score can overlook. The output is again an inefficiency score between 0 and 1, but this score captures not only the proportional contraction but also any remaining slack.
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 does not change: the input and output columns are crisp, non-negative numbers, as in classical DEA. The difference lies not in the data type but in how efficiency is calculated.
Scale equalisation. Classical DEA (CCR) seeks a single proportional contraction ratio (theta) for each unit; every input is contracted by the same ratio. RAM instead computes a separate "range weight" for each input and output: a measure's weight is the reciprocal of the difference (the range) between the largest and smallest value of that measure across units. A wide-ranging measure gets a small weight, and a narrow-ranging measure gets a large weight. This is an equalisation that makes the measures comparable with one another; classical DEA has no such step.
Distance / score / combination. RAM multiplies each unit's input excesses and output shortfalls (slack) directly by the range weights and sums them; it divides this sum by the number of measures and subtracts the result from 1. This differs from classical CCR's search for a single proportional (radial) contraction: RAM penalises the slack in each measure separately, on its own scale (which is why it counts as an "additive", or "non-radial", form of DEA). DecisionMind keeps this calculation fixed under the variable-returns-to-scale assumption (Σλ=1).
Result and defuzzification. The output is again a single figure, as in classical DEA (ρ), but "ρ = 1" is a stricter condition here: not only must the radial contraction be exhausted, but no input excess and no output shortfall may remain at all. Even if a unit scores fully efficient on CCR's percentage score, if it carries a non-radial slack, RAM will show it below 1. There is no defuzzification; both input and output remain crisp numbers throughout.
How to Read the Output
The reading is as on the DEA card: ρ is not a percentage or a quality score; it is a relative result dependent on the set. The difference is that ρ = 1 in RAM is a stronger claim than theta = 1 in classical CCR. A unit scoring theta = 1 in CCR may still be using one input in an imbalanced way relative to the others; the radial score cannot see this, because it looks only at how much all the inputs could be shrunk simultaneously. RAM sees such an imbalance directly in the same unit and pulls ρ below 1.
So, instead of writing:
"This unit scores fully efficient on CCR, so it must be entirely efficient"
the report should read:
"This unit is fully efficient on CCR's radial measure; if RAM's ρ value is also 1, there is no input excess or output shortfall either. If RAM comes out below 1, there is a residual slack that the radial score has overlooked"
When to Prefer This over the Base Method
RAM is preferred when the input and output slacks themselves, that is, how much saving potential exists on which measure, matter for the decision, and there is a concern that the classical radial score is hiding this slack. The input/output distinction and DEA's principle of not taking outside weights apply here exactly as they do there; RAM likewise does not ask for outside importance weights, deriving them itself from the range in the data. The condition that the number of units be sufficient relative to the number of measures also holds, as on the DEA card.
Mistakes Specific to This Extension
Using a constant column. If an input or output takes the same value across all units, that measure's range becomes zero and its range weight becomes undefined (division by zero). Such a measure should either be dropped or the data reviewed.
Assuming ρ = 1 is the same thing as CCR's theta = 1. As shown above, RAM's ρ = 1 condition is stricter; a unit may look efficient in CCR yet not come out efficient in RAM.
Making a constant-scale interpretation without noticing the variable-returns-to-scale (Σλ=1) assumption. DecisionMind's RAM member keeps this constraint fixed; this is what keeps units of very different sizes comparable, but the assumption must be stated when the interpretation is written up.
The governing principle is this:
RAM's ρ = 1, unlike the radial score, means that both the input excess and the output shortfall are separately zero; an interpretation that does not distinguish these two conditions loses RAM's one real contribution.
Cases
The first case is DecisionMind's validation example. The figures are constructed so that they can be followed by hand, rest on RAM's formal definition (Cooper, Park and Pastor, 1999), and have been verified to match the formulation in Mehdiloozad and Sahoo's (2016) handbook chapter (Ch. 12, §12.2.2, Eqs. 12.3–12.5). The second case is an illustrative construction.
1. Illustrative example: The input-output inefficiency of four production units (DecisionMind's validation example)
Four production units (A, B, C, D) convert one input (resource use, less is better) into one output (production volume, more is better).
| Unit | Input (C1) | Output (C2) |
|---|---|---|
| A | 2 | 1 |
| B | 4 | 5 |
| C | 5 | 3 |
| D | 6 | 4 |
| Input/Output | Input | Output |
(There is no "Weight" row in the DEA family: RAM derives the range weights automatically from the input and output columns' own maximum-minimum difference.)
The method first finds the range weights: the input column's range is 6−2=4, the output column's range is 5−1=4; both weights are 1/4=0.25. It then solves, for each unit, a linear programme that weighs and sums the input excess and output shortfall using these weights.
| Unit | ρ (RAM) | Input excess | Output shortfall |
|---|---|---|---|
| A | 1.000 | 0 | 0 |
| B | 1.000 | 0 | 0 |
| C | 0.625 | 1 | 2 |
| D | 0.625 | 2 | 1 |
The result reads as follows: A and B have neither input excess nor output shortfall; both are RAM-efficient. C and D share the same ρ value (0.625), but their slacks sit in different places: C's input excess is small and its output shortfall is large; for D it is the exact opposite. Both project onto the same peer, B.
The team's hesitation is this: what happens if D's output rises from 4 to 4.4 (roughly a 10 per cent improvement)? This scenario was re-solved with the same linear-programming model in Python. The result shows D's ρ rising from 0.625 to 0.675, overtaking C (0.625); D's chronic input-side excess, still the highest of the four units, is masked by this improvement. This shows that ρ in RAM is more sensitive to a change in a single measure than C is.
In the report: "A and B are RAM-efficient among the current four units (ρ=1.000); C and D share the same ρ value (0.625), but their slack concentrates in different measures: C trails on the output side, D on the input side. This ranking can shift with a small improvement in D's output."
Source: DecisionMind's DEA-RAM validation example; RAM's formal definition is based on Cooper, Park and Pastor (1999). The range weights, inefficiency scores and slacks were independently recalculated in Python by this card's author, verified exactly against DecisionMind's manifest internal-audit record and against the formulation in Mehdiloozad and Sahoo's (2016) handbook chapter (Ch.12 §12.2.2).
2. Waste Management: A municipality's comparison of refuse collection units
A municipality will compare the efficiency of four district refuse collection units. The municipality has set one input (vehicle-and-staff expense) and one output (the amount of waste collected).
The method finds each unit's range weights and weighs the input excess against the output shortfall together. Suppose the unit with the lowest expense produces the highest amount of waste and so comes out RAM-efficient; two mid-sized units carry both an expense excess and a collection shortfall and so are not efficient.
The municipality's hesitation is this: even if the slack of the inefficient units sums to the same ρ value, where it concentrates can differ. If one unit's problem is mainly excess staff expense while another's is low collection capacity, applying the same improvement prescription to both, for example only increasing the number of vehicles, would be wrong.
In the report: "One of the four units examined is RAM-efficient; the other three fall within the same ρ range, but the source of their slack differs. Improvement priority should be set by examining each unit's own input excess and output shortfall separately."
3. What Not to Do
The first error is interpreting C and D's shared ρ value (0.625) in the illustrative example as "both have the same problem"; C's problem is an output shortfall, D's is an input excess, and this distinction cannot be seen without looking at the slack table. The second error is treating A and B's RAM-efficient result (ρ=1.000) as the same information as CCR's radial score and reporting it, as on the classical DEA card, only as "theta=1"; RAM's ρ=1 additionally guarantees that the slack is zero, which is a separate claim. The third error is treating the input column (resource use) as a measure that should be maximised, like an output, and reversing its direction label; the input/output distinction rule on the DEA card applies exactly the same way in RAM.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-ram
Cooper, W. W., Park, K. S., & Pastor, J. T. (1999). RAM: A range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. Journal of Productivity Analysis, 11(1), 5–42. DOI: 10.1023/A:1007701304281
Mehdiloozad, M., & Sahoo, B. K. (2016). Identifying the global reference set in DEA: An application to the determination of returns to scale. In Handbook of Operations Analytics Using Data Envelopment Analysis (Ch. 12, pp. 306–336). Springer. DOI: 10.1007/978-1-4899-7705-2_12
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
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