Extension card · Classical
Dynamic Network DEA (Fukuyama and Weber, 2013)
This is the form of DEA that compares units across several consecutive periods rather than within a single year. It also accounts for an asset carried over between periods and an undesirable output that spills from one period into the next. Its output is a single combined efficiency score for each unit.
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 that no unit can exceed one hundred per cent does not.
Input–output structure. In base DEA, each unit converts its input into output within a single period. Here, the same unit is tracked across several consecutive periods; this card uses two periods. Each period has its own input, its own intermediate product (produced by the first stage and used by the second) and its own output. In addition, a "carry-over asset" is transferred from the end of one period to the start of the next; in the manifest's own example, this is the surplus reserve a bank holds.
Model constraints. In base CCR, a single linear programme is set up for each unit. Here, each period carries its own input–intermediate–output constraints, and the periods are linked by a carry-over equality. The level of the carry-over at the end of one period must equal the level at the start of the next. DecisionMind adds this link as a single equality constraint; the size of the carry-over is not fixed from outside, but is determined by the optimisation itself.
Score calculation. Base DEA computes a single theta. Here too a single, shared theta is computed, but this theta contracts the inputs of every period together within the same linear programme. The literature also defines a separate directional distance per period, along with a period-weighted sum of these; the form DecisionMind runs on this card is the shared theta that unifies the periods within one programme.
Result and defuzzification. The score again lies between 0 and 1, and theta = 1 means the unit sits on the joint efficient frontier across every period taken together. The difference lies here: even if a unit looks good in only one period and poor in another, the shared theta collapses to a single figure. If a period-by-period breakdown is wanted, the method's own per-period decomposition in the literature is required; this card reports only the combined theta.
DecisionMind keeps the number of periods, the carry-over equality and the linking of the two stages faithful to Fukuyama and Weber's linearised formulation in this extension.
How to Read the Output
As with base DEA, the score is not a quality percentage; it is relative only to this set and this definition. The difference is that the score is now the joint result of more than one period. A single poor period can drag down otherwise good periods; if the carry-over asset is managed well, a surplus in one period can offset the next. The report should state where the bottleneck period lies and, where possible, the share contributed by the carry-over asset, not only the total score.
Hence, instead of writing:
"This bank has performed efficiently and consistently in recent years"
the report should read:
"This bank scores a combined 0.852 in the joint two-period model; the score assesses the input–intermediate–output–carry-over chain of both periods together, and does not show either period's performance separately"
When to Prefer This over the Base Method
If only a single period's data exists for the units, or if there is no carry-over asset between periods at all, base DEA is sufficient. This extension is built for situations where the same units are tracked over several periods and a decision in one period (setting aside reserves, holding stock, accumulating investment) affects performance in the next. The input–output split must be clear, and the number of periods must be at least two.
The base method's exit condition still holds here: if the goal is to measure relative efficiency rather than a preference ranking, and the units are comparable, this family is appropriate.
Mistakes Specific to This Extension
Fixing the carry-over asset from outside. The carry-over asset is not a fixed piece of data; it is a decision variable that the linear programme itself finds. Fixing it externally solves the model incorrectly.
Failing to link the first and second stages. A period's intermediate product must equal the quantity entering that period's second stage. If this link is not made, the two stages become disconnected and the scores become inconsistent.
Trying to solve the non-linear form directly. The model's raw form is fractional and non-linear. DecisionMind converts it, through a transformation, into a linearised programme; skipping this step means the solver either fails to converge or finds the wrong result.
Running the two periods as two independent DEA models. This ignores the constraint introduced by the carry-over asset and the inter-period link. The result becomes indistinguishable from running base CCR twice, and the method's one real contribution, the link between periods, is lost.
The governing principle is this:
Dynamic network DEA exists to carry the carry-over asset between periods, and any delayed undesirable output, as a genuine constraint. Any application that decouples the periods or fixes the carry-over removes this contribution.
Cases
The first case is DecisionMind's validation example. The fixture in the manifest is a small table built to make the formula traceable by hand. The literature's real-world application (Japanese Shinkin banks, 2006–2012, seven periods) is not reproduced here. The second case is an illustrative construction.
1. Illustrative example: The joint efficiency of three branches tracked over two periods
A bank's three branches are tracked over two consecutive years. In each year, a branch's input is staff numbers and operating expense; these produce deposits and other funds (the intermediate product). The intermediate product produces loans and securities (the output). The surplus reserve remaining at the end of the first year (the carry-over asset) is transferred unchanged into the start of the second year.
| Branch | Input (year 1) | Intermediate (year 1) | Output (year 1) | Carry-over | Input (year 2) | Intermediate (year 2) | Output (year 2) |
|---|---|---|---|---|---|---|---|
| B1 | 2; 3 | 5; 4 | 10; 8 | 1.0 | 2; 3 | 5; 5 | 11; 9 |
| B2 | 3; 2 | 6; 3 | 12; 7 | 1.2 | 3; 2 | 6; 4 | 13; 8 |
| B3 | 4; 4 | 3; 3 | 8; 5 | 1.5 | 4; 4 | 3; 3 | 9; 6 |
The method solves a single linear programme for each branch; this programme sets up both years' input, intermediate product, output and the carry-over equality between them together.
| Branch | Combined score |
|---|---|
| B1 | 1.000 |
| B2 | 1.000 |
| B3 | 0.852 |
The result reads as follows. B1 and B2 sit on the joint efficient frontier across both years. B3 could have produced the same two-year input–intermediate–output–carry-over chain as its peers using roughly 85 per cent of the resources it actually used.
The branch's hesitation is this: would the score change if the surplus reserve B3 held had been larger? This scenario was tested by rebuilding the same linear programme in Python. When B3's carry-over asset was raised from 1.5 to 2.0, the combined score rose from 0.852 to 1.000, and B3 reached the fully efficient frontier. When the same asset was lowered from 1.5 to 1.0, the score fell to 0.568. This shows that B3's score depends heavily on the level of its carry-over asset.
In the report: "Three branches were compared using the joint two-year model. B1 and B2 sit on the efficient frontier in the combined score (1.000); B3 trails with 0.852. B3's score is quite sensitive to the level of the carry-over asset it holds."
Source: This is DecisionMind's validation example for the Dynamic Network DEA engine. It is based on Fukuyama and Weber's (2013, 2016) linearised two-period model, but it is not a table taken from the paper's own page. The figures were independently recalculated in Python.
2. Energy: The multi-year efficiency of a regional electricity distribution company
An electricity distribution company will compare the efficiency of three regional offices over the last three years. In each year, the input is maintenance staff and operating expense; these build network capacity (the intermediate product). Network capacity produces the amount of electricity distributed (the output). A grid investment made at the end of one year but not yet brought into service (the carry-over asset) is transferred to the following year. High outage duration in one year (the delayed undesirable output) is added as a burden to the following year's input side.
The method solves a single linear programme that sets up all three periods together. Suppose the result places the region with the highest investment but the lowest outages first as efficient in the combined score, and the region with low investment but high outages last with a low score.
The company's hesitation is this: how far does the rise in outages caused by last year's storm in the low-scoring region pull down this year's score? Because the method treats the delayed undesirable output as an input to the following period, an unusual event from a past year continues to depress the current year's score. The company should not permanently declare the region inefficient without decomposing this effect.
In the report: "Three regions were compared using the joint three-year model. The low-scoring region's score includes the delayed effect of last year's rise in outages. The region should not be considered permanently inefficient without decomposing this effect."
3. What Not to Do
The first error is running the two years of the illustrative example as two independent DEA models and never setting up the carry-over equality at all. This reduces the two years to a repetition of base CCR and erases the method's one real contribution, the link between periods. The second error is trying to supply the carry-over asset as a fixed number from outside; this value is a decision variable that the optimisation itself finds. The third error is reporting B3's score of 0.852 as "the branch is badly managed"; the score is valid only for these three branches, these two years and this definition of the carry-over.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/dea-dynamic-network
Fukuyama, H., & Weber, W. L. (2013). A dynamic network DEA model with an application to Japanese Shinkin banks. In Efficiency and Productivity Growth: Modelling in the Financial Services Industry (Ch. 9, pp. 193–213). Wiley. DOI: 10.1002/9781118541531.ch9
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. 241–274). Springer. DOI: 10.1007/978-1-4899-7705-2_10
Fukuyama, H., & Weber, W. L. (2015). Measuring Japanese bank performance: A dynamic network DEA approach. Journal of Productivity Analysis, 44(3), 249–264. DOI: 10.1007/s11123-014-0403-1
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