Extension card · Classical
Two-Stage Network DEA (Fukuyama and Weber, 2010)
This is the form of DEA for situations where a unit converts input into output not in one step, but across two consecutive stages through an intermediate product. It can also produce an undesirable output in the second stage. 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. In base DEA, input converts directly into output, and the process in between is a closed box. Here, the process splits into two: the first stage uses the input to produce an intermediate product, for example deposits collected. The second stage uses this intermediate product to produce the desired output and, optionally, an undesirable output, for example loans and non-performing receivables. The intermediate product is both the output of the first stage and the input of the second.
Model constraints. Base CCR sets up a single set of constraints. Here, the two stages carry separate technology assumptions. In the first stage, the input, together with any delayed undesirable output, can be expanded proportionally together; in the second stage, the desired output together with the undesirable output can be contracted proportionally together. The two stages are locked together by a linking constraint: the quantity of the intermediate product produced in the first stage cannot be less than the quantity consumed in the second.
Score calculation. In base CCR there is a single set of intensity variables and a single theta. Here, each stage has its own set of intensity variables, but a single shared theta contracts both stages together. The method solves this within one linear programme; it does not run two separate DEA models and then multiply the results.
Result and defuzzification. The score is again a single figure between 0 and 1. The difference lies here: a low score does not on its own say whether the weakness comes from the first stage or the second; a separate stage-level reading is needed to make that distinction. DecisionMind reports the combined network score on this card.
DecisionMind keeps the linking constraint between the two stages fixed in this extension; which columns are the intermediate product is stated explicitly in the data entry.
How to Read the Output
As with base DEA, the score is relative to the set. The difference is that this score now measures, together, two consecutive transformations passing through an intermediate product, not a single one. A unit can be strong in the first stage and weak in the second; the combined score collapses these two into a single figure and does not show which stage is responsible.
Hence, instead of writing:
"This branch is the most efficient branch"
the report should read:
"This branch sits jointly on the efficient frontier for the two-stage process running from input to intermediate product and from intermediate product to output. Which stage contributes more to this result requires separate examination"
When to Prefer This over the Base Method
If there is an observable intermediate product between the input and the final output, for example deposits or a processed semi-finished good, this extension is appropriate. The quantity of this intermediate product must be recorded as a separate measure. If the process genuinely occurs in one step, or the intermediate product cannot be measured, artificially splitting the process in two adds needless complexity, and base DEA is sufficient.
If the second stage also has an undesirable output, for example non-performing receivables or defective products, this extension carries it as a separate constraint. If there is no undesirable output, only the two-stage structure is built. 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
Treating the intermediate product only as the first stage's output and not linking it to the second stage. This is the fundamental error the manifest warns against. If the linking constraint is not set up, the two stages become disconnected, any slack in the intermediate product goes nowhere, and the scores become inconsistent.
Running the two stages as two separate DEA models and multiplying the scores. This method jointly optimises the two stages within a single shared linear programme. Solving the stages separately and then multiplying produces a different, and generally incorrect, figure.
Adding the undesirable output to the second stage as an input. The undesirable output is not an input to the second stage; it is an output that contracts proportionally together with the desired output. Confusing this distinction sets up the second stage's constraint incorrectly.
Ignoring the intermediate product and running base CCR with only the input and the final output. This reverts the process to a closed box and entirely loses the information about which stage is the bottleneck; the result is a different, and generally more optimistic, score.
The governing principle is this:
Two-stage network DEA exists to carry the intermediate product as a genuine link. Any application that decouples the stages or ignores the intermediate product removes this contribution.
Cases
The first case is DecisionMind's validation example; the fixture in the manifest is a small table built by hand. The literature's real-world application (Japanese banks, 2002–2012) is not reproduced here. The second case is an illustrative construction.
1. Illustrative example: The two-stage efficiency of three branches via deposits
A bank's three branches are assessed through a two-stage process. In the first stage, staff and operating expense (input) produce deposits and other funds collected (the intermediate product). In the second stage, this intermediate product produces loans and securities (the output).
| Branch | Input | Intermediate | Output |
|---|---|---|---|
| B1 | 2; 3 | 5; 4 | 10; 8 |
| B2 | 3; 2 | 6; 3 | 12; 7 |
| B3 | 4; 4 | 3; 3 | 9; 6 |
The method jointly optimises the two stages within a single linear programme, contracting the input while observing both the intermediate-product link and the output target together.
| Branch | Network efficiency score |
|---|---|
| B1 | 1.000 |
| B2 | 1.000 |
| B3 | 0.536 |
The result reads as follows. B1 and B2 sit jointly on the efficient frontier in the two-stage process. B3 could have delivered the same intermediate-product-to-output chain as its peers using roughly 54 per cent of its input.
The branch's hesitation is this: would the score change if B3's intermediate product (its deposits) were a little higher? The same linear programme was rebuilt in Python; when B3's intermediate product was raised from 3; 3 to 4; 4, the score rose from 0.536 to 0.714, and the ranking did not change. This shows that an increase in the intermediate product improves the score, but does not on its own carry it to the frontier.
In the report: "Three branches were compared through the two-stage process running via deposits. B1 and B2 sit on the efficient frontier in network efficiency (score 1.000); B3 trails with 0.536. This gap arises from the joint model that assesses both stages together."
Source: This is DecisionMind's validation example for the Two-Stage Network DEA engine. It is based on Fukuyama and Weber's (2010) two-stage network framework, but it is not a table taken from the paper's own page. The figures were independently recalculated in Python.
2. Telecommunications: An operator's two-stage process from subscriber acquisition to revenue
A telecommunications operator will compare the efficiency of three regional offices. In the first stage, marketing expense and field staff (input) produce the number of new subscribers (the intermediate product). In the second stage, the subscriber count produces monthly revenue (output) and the number of complaints and cancellations (an undesirable output).
The method jointly optimises the two stages within a single linear programme, contracting the input and the complaint count while observing both the subscriber link and the revenue target together. Suppose the region that gains the most subscribers also receives the most complaints, and so trails in network efficiency. A region that gains fewer subscribers but has low complaints comes out on the efficient frontier.
The operator's hesitation is this: does the high-subscriber region's low score stem from marketing ineffectiveness or from a service-quality problem? The combined score does not separate the two. Distinguishing them requires a stage-level examination, assessing subscriber acquisition and the revenue-complaint balance separately.
In the report: "Three regions were compared through the two-stage process running from subscriber acquisition to revenue. The region that gained the most subscribers trails in network efficiency owing to its high complaint share. Which stage this result stems from should be clarified with a separate stage analysis."
3. What Not to Do
The first error is treating the intermediate product (deposits) in the illustrative example only as the first stage's output, never linking it to the second stage, and running two separate DEA models. This breaks the link between stages and makes the scores inconsistent. The second error is treating the complaint and cancellation count as an input to the second stage; it is an output that contracts proportionally together with revenue, not an input. The third error is reporting B3's score of 0.536 as "the branch is badly managed"; the score is valid only for these three branches and this two-stage definition.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-network
Fukuyama, H., & Weber, W. L. (2010). A slacks-based inefficiency measure for a two-stage system with bad outputs. Omega, 38(5), 398–409. DOI: 10.1016/j.omega.2009.10.006
Fukuyama, H., & Weber, W. L. (2016). Measuring bank performance: From static black box to dynamic network models. In Handbook of Operations Analytics Using Data Envelopment Analysis (Ch. 10, pp. 249–274). Springer. DOI: 10.1007/978-1-4899-7705-2_10
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