DEA cardExtension card

Extension card · Classical

Slacks-Based Measure DEA (SBM) (Tone, 2001)

SBM is a form of DEA that, instead of a classical proportional contraction ratio, directly measures every input excess and output shortfall on its own scale. The result is again an efficiency score between 0 and 1, but this score rests not on a ratio but directly on the slack share.

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 slack is measured.

Scale equalisation. Classical DEA (CCR) seeks a single common contraction ratio (theta) for each unit; every input is contracted by the same percentage. SBM does not seek such a common ratio. It ratios each input's excess against its own value, ratios each output's shortfall against its own value, and then averages these. This is why SBM is called "slacks-based": it measures the contraction directly through slack, not proportionally.

Distance / score / combination. The SBM score is a fraction whose numerator carries the average proportional excess on the input side and whose denominator carries the average proportional shortfall on the output side: as the numerator shrinks and the denominator grows, the score falls. DecisionMind first linearises this fraction, since it cannot be solved directly in its raw form (the Charnes–Cooper transformation), and then solves it. This non-oriented form is the default; if desired, oriented forms that take only the input side or only the output side as the basis can also be used.

Result and defuzzification. The score (ρ) is again a single figure, as in classical DEA, and a value of 1 again means "no slack at all". The difference lies here: CCR's radial score looks at how much a unit could shrink all its inputs simultaneously, and can leave any remaining disproportionate slack entirely out of the score. SBM folds this remaining slack into the score itself; for this reason, the SBM score for the same data can come out lower (stricter) than CCR's radial score. There is no defuzzification; the calculation proceeds with crisp numbers throughout.

How to Read the Output

The reading is as on the DEA card: ρ is not a quality percentage; it is a relative result dependent on the set. The difference is that SBM's ρ value can differ from the radial score CCR gives for the same data, because SBM also accounts for disproportionate slack. The gap between the two scores for a given unit shows how much "hidden" slack, invisible to the radial measure, that unit carries.

So, instead of writing:

"This unit's DEA score is 0.75, meaning it uses 75 per cent of its resources efficiently"

the report should read:

"This unit's slacks-based score is 0.59; the classical radial score (0.75) shows only this unit's proportional contraction potential, whereas SBM comes out lower because it also folds a disproportionate shortfall in one output into the score"

When to Prefer This over the Base Method

SBM is preferred when where the slack accumulates, and by how much, on which measure matters more than a single common contraction ratio, and there is a concern that the classical radial score may hide slack on some measures. The input/output distinction and DEA's principle of not taking outside weights apply here exactly as they do there; SBM likewise does not require outside importance weights. The condition that the number of units be sufficient relative to the number of measures holds as on the DEA card.

Mistakes Specific to This Extension

Ignoring the small-sample rule. The number of units should be at least three times the sum of the number of inputs and outputs (the manifest carries this threshold as a warning); otherwise discriminatory power is lost in SBM just as it is in CCR.

Confusing the free and bounded weightings. SBM's input-side and output-side averages are divided by different measure counts (p inputs, q outputs); mixing these two into a single sum distorts the score. DecisionMind keeps this distinction fixed within the engine; the user chooses only the orientation (input-oriented, output-oriented or non-oriented).

Comparing the SBM score directly with the CCR score. For the same unit, the two scores measure different quantities; SBM coming out lower than CCR is not a calculation error, but the disclosure of hidden slack.

The governing principle is this:

Unlike the radial score, SBM's score folds disproportionate slack into itself; for the same unit the two scores can differ, and this difference is exactly what SBM is designed to measure.

Cases

Both cases are illustrative constructions. The first is a synthetic, hand-traceable table that DecisionMind's engine built by hand to include efficient, borderline and inefficient units together; it is not taken from a paper or book page.

1. Illustrative example: The input-output efficiency of three branches (DecisionMind's validation example)

Three branches (DMU1, DMU2, DMU3) convert two inputs (staff expense, materials expense) into two outputs (transaction volume, number of customers).

Branch	Staff expense	Materials expense	Transaction volume	Number of customers
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.)

The method solves a separate linear programme for each branch; this is a calculation that ratios the input excess and output shortfall directly against the measure's own value.

Branch	ρ (SBM)	Classical CCR score with the same data
DMU1	1.000	1.000
DMU2	1.000	1.000
DMU3	0.592	0.750

The result reads as follows: DMU1 and DMU2 have neither input excess nor output shortfall; both are fully efficient in SBM and CCR alike. DMU3 is not efficient; SBM identifies DMU2 as its reference and computes a 1.33-unit excess in materials expense, a 7-unit shortfall in transaction volume and a 0.33-unit shortfall in number of customers for DMU3. Because these slacks feed directly into the SBM score, DMU3's SBM score (0.592) is markedly lower than the same data's classical radial score (0.750).

The team's hesitation is this: what happens if DMU3's transaction volume rises from 9 to 10 (approaching DMU1's level)? This scenario was re-solved in Python; DMU3's score rises from 0.592 to 0.632 but it remains third. Lowering materials expense from 4 to 3 delivers a similarly sized improvement (0.671), and again does not lift it out of third place. An improvement in a single measure is not enough to bring DMU3 to the efficient frontier; several measures need to improve at once.

In the report: "DMU1 and DMU2 are fully efficient in both the slacks-based and classical radial measures. DMU3's slacks-based score (0.592) is lower than its classical radial score (0.750), because it carries excess on its second input alongside shortfalls on both its outputs at once. An improvement on a single measure is not enough to move DMU3 out of third place."

Source: This is a synthetic, hand-traceable validation example built by hand for DecisionMind's DEA-SBM (Tone, 2001) engine; it is not a reproduction of a paper or book table. The scores and slack values were independently recalculated in Python by this card's author, verified exactly against DecisionMind's manifest internal-audit record.

2. Librarianship: A provincial public library authority's branch comparison

A provincial public library authority will compare the efficiency of three branches. The authority has set two inputs (staff numbers and annual budget) and two outputs (number of books loaned and number of events held).

The method measures the input excess and output shortfall directly for each branch. Suppose the central branch comes out SBM-efficient because it is high on both outputs; a neighbourhood branch, holding few events relative to its budget, carries both a budget excess and an event shortfall.

The authority's hesitation is this: does the neighbourhood branch's low number of events stem from insufficient budget or from a lack of staff? SBM's slack table shows these two separately; the authority should decide which measure to target for improvement based on this distinction, not on a single overall score alone.

In the report: "The central branch is efficient on the slacks-based measure; the neighbourhood branch's low score stems from both a budget excess and an event shortfall. Improvement should be prioritised according to which of the two components in the slack table is larger."

3. What Not to Do

The first error is comparing DMU3's SBM score (0.592) with the same data's classical radial score (0.750) and concluding "the SBM engine disagrees with CCR, one of them must be wrong"; the two scores measure different quantities, and both are correct within their own definitions. The second error is saying only "DMU3 is inefficient" without looking at its slack table and without stating which measure needs improvement. The third error is collapsing the two inputs in the example (staff expense, materials expense) into a single "total expense" column without distinguishing them and feeding that into DEA; this erases the information showing which input is in excess.

Formulae

The computation step by step, with the formula behind each step.

1. Partition strictly positive observations into DEA inputs X and outputs Y; validate the requested SBM orientation.
$X \in R_{> 0}^{n \times p}, Y \in R_{> 0}^{n \times q}$
2. Solve the selected CRS SBM model per DMU: Charnes-Cooper linearized non-oriented SBM, input-oriented SBM, or output-oriented SBM.
$ρ^{*} = min \frac{1 - \frac{1}{p} \sum _{i} s _{i}^{-} / x _{i o}}{1 + \frac{1}{q} \sum _{r} s _{r}^{+} / y _{r o}}; ρ_{I}^{*} = min (1 - \frac{1}{p} i \sum s_{i}^{-} / x_{i o}); ρ_{O}^{*} = 1/ max (1 + \frac{1}{q} r \sum s_{r}^{+} / y_{r o})$
3. Identify SBM-efficient frontier DMUs using the solved orientation-specific efficiency scores.
$F_{S B M} = {o : ρ_{o}^{*} = 1}$
4. Rank DMUs by SBM efficiency descending with deterministic ID tie-breaking.
$rank ↓ ρ_{o}^{*}$

Sources

Tone, K. (2001). A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research, 130(3), 498–509. DOI: 10.1016/S0377-2217(99)00407-5

Tone, K., & Tsutsui, M. (2010). Dynamic DEA: A slacks-based measure approach. Omega, 38(3–4), 145–156. DOI: 10.1016/j.omega.2009.07.003

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