Extension card · Classical
Network SBM DEA (Tone and Tsutsui, 2009)
This is the form of DEA that evaluates a multi-stage process with its own importance weight per stage. It measures input/output slack directly rather than proportionally, and can examine several periods together within a single window. Its output is again a network 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. Base CCR and the two-stage network DEA contract and expand input and output by a single shared proportional coefficient (theta). Here, each input and output can carry its own slack separately; the score rests on the ratio of these slacks, not on a single shared ratio. The process is still divided into stages, two stages on this card, and the stages are linked by a connecting product. Each stage can be given an importance weight.
Model constraints. In the base radial model, input and output slacks enter the score calculation only secondarily; the theta ratio is the primary determinant. In this slack-based measure, the score directly ratios the share of input slacks against the share of output slacks; this ratio is a fractional programme, which DecisionMind linearises through a transformation. The connecting product between stages is linked by an equality constraint that keeps it at the same level in both stages.
Score calculation. The score is the optimised value of a ratio whose numerator carries the weighted sum of input slacks and whose denominator carries the weighted sum of output slacks. The stage weights adjust which stage is treated as more decisive within this ratio; equal weighting is also possible. On this card, DecisionMind can also treat several periods together within a single window as one combined observation set; the window width is a user parameter.
Result and defuzzification. The score again lies between 0 and 1; a score of 1 means zero slack, that is, full efficiency. The difference lies here: the method reports not only a combined score, but also a separate stage efficiency for each stage. Which stage carries slack can be seen directly; this is a breakdown that the two-stage radial model cannot provide.
DecisionMind keeps the linearisation step and the free linking equality between stages fixed in this extension; on this card the number of stages is two, while the window width and the number of periods are user parameters.
How to Read the Output
As with base DEA, the score is relative to the set. The difference is that the score is no longer a proportional contraction, but a direct ratio of input and output slacks. Because of this, even small slacks can lower the score; a proportional reading, as with the radial model, cannot be made. The stage-level efficiencies must also be read separately; if the combined score is low, the stage responsible can be seen from the stage efficiencies.
Hence, instead of writing:
"This unit is the most efficient unit"
the report should read:
"This unit shows a score of 1 with zero slack in the two-stage network SBM model. The other unit, with a score of 0.921, carries slack on either the input or the output side; which stage this occurs in can be seen from the stage efficiencies"
When to Prefer This over the Base Method
If the process divides into several stages and the stages require different importance weights, this extension is appropriate. The same applies if input and output slacks need to be measured directly rather than proportionally. If the stages are of equal importance and a proportional score is sufficient, the simpler two-stage network DEA may be preferred.
If several periods need to be assessed together within a single window, window analysis comes into play. For single-period data, the window width equals the period, and window analysis has no effect. 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
Giving stage weights equal values without consideration. If the stages differ in strategic importance, equal weighting can distort the ranking. Weights should be set by expert judgement or by an explicit rationale.
Trying to solve the fractional programme directly. This model's raw form is fractional. DecisionMind converts it, through a transformation, into a linear programme; skipping this step produces either an incorrect result or a solver that fails to converge.
Locking the connecting product between stages to a fixed observed value. In network SBM, the linking activities are free; the intensity variables of the two stages equalise the connecting product internally, and it is not forced to an observed value. Treating this as fixed data breaks the model's degrees of freedom.
Comparing the radial theta directly, on the same scale, with this model's score. The theta of the two-stage network DEA and the score of this model are different mathematical definitions. Comparing the same unit's score across the two different models directly leads to the wrong conclusion.
The governing principle is this:
Network SBM exists to measure input and output slack directly and on a stage-by-stage basis. Any application that equalises the stages, or solves the fractional form without linearising it, removes this contribution.
Cases
The first case is DecisionMind's validation example; the fixture in the manifest is a small, hand-built table with two stages. The literature's real-world application (29 Taiwanese R&D organisations, 2005–2009, three stages) is not reproduced here; DecisionMind's engine currently runs two stages. The second case is an illustrative construction.
1. Illustrative example: The two-stage handling efficiency of three ports
A port operator's three terminals are assessed through a two-stage process. In the first stage, the number of cranes and labour force (input) build handling capacity (the connecting product). In the second stage, this capacity produces container traffic and revenue (output). The stage weights are taken as equal (0.5; 0.5).
| Terminal | Input | Connection | Output |
|---|---|---|---|
| T1 | 2; 3 | 5; 4 | 10; 8 |
| T2 | 3; 2 | 6; 3 | 12; 7 |
| T3 | 4; 4 | 3; 3 | 9; 6 |
The method solves, for each terminal, the linear programme that optimises the weighted ratio of input and output slacks.
| Terminal | Network SBM score | Stage 1 efficiency | Stage 2 efficiency |
|---|---|---|---|
| T1 | 0.921 | 1.000 | 0.854 |
| T2 | 1.000 | 1.000 | 1.000 |
| T3 | 1.000 | 1.000 | 1.000 |
The result reads as follows. T2 and T3 are fully efficient with zero slack. T1's first stage has no slack, but its second stage does; this is why its combined score falls below 1.
The terminal's hesitation is this: would T1's output slack change if the stage weights changed? The same linear programme was re-solved in Python, giving more weight to the second stage (0.3; 0.7). T1's score fell from 0.921 to 0.893; when more weight is placed on the second stage, T1's slack penalises the score more heavily.
In the report: "Three terminals were compared using two-stage network SBM. T2 and T3 are fully efficient with zero slack (score 1.000); T1 trails at 0.921 because it carries slack in the second stage. This result is sensitive to the choice of stage weights."
Source: This is DecisionMind's validation example for the Network SBM DEA engine; it is based on Tone and Tsutsui's (2009) network SBM framework, but it is not a table taken from the paper's own page. The figures were independently recalculated in Python.
2. Logistics: A courier company's multi-year, two-stage efficiency
A courier company will compare the efficiency of three distribution centres over the last five years within a single window. In the first stage, vehicle fleet and staff numbers (input) build transport capacity in tonne-kilometres (the connecting product). In the second stage, this capacity produces the number of packages delivered and revenue (output). The company gives the first stage, fleet efficiency, slightly more importance than the second.
The method combines five years of observations within a single window and optimises, for each centre, the weighted ratio of input and output slacks. Suppose the centre with the largest fleet cannot fully translate its capacity into deliveries, carries slack in the second stage, and so receives a low combined score. A smaller centre that fully uses its capacity comes out fully efficient.
The company's hesitation is this: does the large-fleet centre's low score stem from fleet size, or from inefficient delivery routes? The combined score alone does not separate these. Reading the stage efficiencies separately shows the first stage (fleet use) is high and the second (delivery efficiency) is low, pointing to the problem lying on the route-planning side.
In the report: "Three distribution centres were compared within a shared five-year window. The centre with the largest fleet carries slack in the second stage; the first stage is efficient. This distinction shows that improvement should be sought in route planning rather than the fleet."
3. What Not to Do
The first error is trying to solve the fractional ratio in the illustrative example directly, without applying a transformation; this either fails to converge or produces an incorrect score. The second error is always taking the stage weights as equal without consideration; as seen in the T1 example, the choice of weights can change the score, and this choice must be justified. The third error is comparing T1's score of 0.921 directly with the theta of the two-stage radial network DEA; the two scores rest on different mathematical definitions and cannot substitute for one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-network-sbm
Tone, K., & Tsutsui, M. (2009). Network DEA: A slacks-based measure approach. European Journal of Operational Research, 197(1), 243–252. DOI: 10.1016/j.ejor.2008.05.027
Tone, K., & Tsutsui, M. (2014). Dynamic DEA with network structure: A slacks-based measure approach. Omega, 42(1), 124–131. DOI: 10.1016/j.omega.2013.04.002
Wu, Y.-C., Ting, I. W. K., Lu, W.-M., Nourani, M., & Kweh, Q. L. (2016). Capital stock and performance of R&D organizations: A dynamic DEA-ANP hybrid approach. In Handbook of Operations Analytics Using Data Envelopment Analysis (Ch. 7, pp. 169–196). Springer. DOI: 10.1007/978-1-4899-7705-2_7
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