2.1. CDW Management and Circular Economy

CDW circular-economy research consistently emphasizes recovery, reuse, recycling, secondary-material production, and market uptake ^1,3,4,5,6. López Ruiz et al. ⁵ organize circular strategies across the construction life cycle and identify recirculation of recovered materials as a key pathway. Purchase et al. ⁴ highlight technical, regulatory, economic, social, and market barriers, whereas Papamichael et al. ³ emphasize that CDW management must be evaluated across the broader material cycle. These studies support product-oriented indicators but also caution against equating recovered quantity with circularity when quality and market functionality are not represented.

2.2. DEA in Environmental and Waste Management

While the CCR model estimates overall technical efficiency under constant returns to scale, the BCC model permits variable returns to scale and isolates pure technical efficiency ^7,8, and their ratio reveals scale efficiency. DEA has been applied to municipal waste services and environmental systems to compare performance across jurisdictions and identify cost, operational, and scale inefficiencies ^10,11,12,13. Because DEA constructs a frontier from observed data, the results are sensitive to variable selection, dimensionality, data quality, and atypical observations; therefore, transparent specifications and diagnostic reporting are important ¹⁵.

2.3. Dynamic Productivity and Research Gap

The Malmquist productivity index extends frontier analysis across time by decomposing productivity change into efficiency and technological changes ¹⁴. Recent studies of waste-management applications have revealed that recycling and sustainable-waste productivity can evolve through different combinations of catch-up and frontier movements ^16,17. Nevertheless, most waste DEA studies have focused on municipal solid waste or aggregate environmental performance rather than on subnational CDW systems. The present study therefore combines static DEA, slack diagnostics, and Malmquist decomposition, while explicitly incorporating recycled product output and residual waste.

3.1. Study Units, Analytical Sample, and Pooled Frontier

The original database covered 19 Taiwanese counties and cities for the period 2023–2025. Preprocessing excluded county-year observations for which treated/reused CDW, recycled product output, and residual waste were simultaneously zero. Chiayi City had all-zero outputs in the available years and was therefore not included in the final DEA dataset, and for 2023, Hsinchu County was excluded under the same criterion. The final static DEA dataset comprised 53 county-year observations from 18 geographic units (17 for 2023 and 18 for both 2024 and 2025). Each county-year observation period was evaluated against the common pooled frontier.

With \({m=3}\) inputs and \({s=3}\) outputs, the commonly used DEA dimensionality heuristic \(n \geq \max\{{m*s}, {3(m+s)}\}\) proposed by Cooper et al. ¹⁸ gives a minimum of 18 observations ¹⁵. The pooled DEA contains 53 county-year observations and therefore exceeds this heuristic, although the observations represent only 18 geographic systems over a short time period.

3.2. Variables, Construction, and Official Data Sources

Data for 2023–2025 were compiled from the official information systems maintained by the Resource Circulation Administration, Ministry of Environment, Taiwan. CDW received, treated/reused CDW, recycled product output, and residual waste were obtained from statistical data in the Industrial Waste Report and Management Information System ¹⁹. The licensed treatment/reuse capacity was compiled from the Waste Clearance and Disposal Organization Service Management Information System and the Resource Recycling Management Information System. The treatment-cost input was estimated as the relevant treatment/output quantity multiplied by the corresponding unit treatment fee obtained from the Waste Clearance and Disposal Organization Service Management Information System. The resulting annual cost was calculated in New Taiwan dollars per year (NT$/year). Before DEA estimation, this variable was divided by \(10^{6}\); the values supplied to DEA are therefore numerically expressed in million NT$/year (this unit rescaling does not change the economic definition of the variable). Accordingly, this variable represents an estimated treatment-cost input rather than an audited financial expenditure by individual local governments or facilities. “CDW received” denotes CDW received during the reporting year, whereas “treated/reused CDW” denotes the total material entering treatment or reuse processes, including non-CDW materials and inventories carried over from previous periods. “Recycled product output” and “residual waste” are the corresponding quantities generated by these processes. Treated/reused CDW is approximately equal to recycled product output plus residual waste. These related indicators describe processing throughput, product recovery, and residual generation rather than independent material quantities (Table 1).

3.3. DEA Model Specification

An input-oriented DEA model was adopted because the analytical question was whether the observed outputs could be achieved with lower resource inputs. The CCR model assumes constant returns to scale and estimates overall technical efficiency, whereas the BCC model adds a convexity constraint and estimates pure technical efficiency under variable returns to scale ^7,8.

For the evaluated decision-making unit \(o\), let \(\theta\) denote the input-oriented efficiency score. The input-oriented CCR envelopment model is given as follows:

The BCC model adds \({\sum_{j} \lambda_{j} =1}\). Scale efficiency is calculated as \({\mathit{SE} = {\theta_{\mathit{CCR}}}/{\theta_{\mathit{BCC}}}}\). In the implemented input-oriented CCR/BCC models, \({0<\theta \leq 1}\) and \({\theta = 1}\) denotes frontier efficiency. Super-efficiency was calculated only as an auxiliary frontier diagnostic and was not substituted for the reported CCR/BCC scores.

3.4. Undesirable Output Transformation, Zero Treatment, and Extreme Observations

Residual waste was defined as an undesirable output. Because the implemented standard DEA output matrix treats larger output values as preferable, residual waste was converted to a desirable-form indicator using the reciprocal transformation \({b^{*} = {1/(b+c)}}\), where \(b\) denotes residual waste and \({c=1}\). Therefore, a lower residual quantity produces a higher transformation value. Consequently, the DEA model operates on the transformed indicator rather than directly on the raw residual waste. Non-positive CDW received and estimated-cost values were replaced by 0.0001, capacity values below 1 were set to 1, and the cost variable was divided by \(10^{6}\). No observation was removed solely because it was a statistical outlier. Super-efficiency diagnostics were used to identify influential frontier observations; infinite super-efficiency solutions were flagged and set to missing only for summary reporting, without altering the underlying CCR/BCC results.

3.5. Input-Correlation Diagnostic

To address possible redundancy among the inputs, Pearson’s correlations were calculated using the 53 final county-year observations. Correlation analysis was used as a redundancy diagnostic, rather than as a regression-style multicollinearity test.

3.6. Malmquist Productivity Index

The Malmquist productivity index was estimated using the same input/output structure. \({\mathit{MI}>1}\) indicates a productivity improvement and \({\mathit{MI}<1}\) a productivity decline; \(\mathit{EC}\) reflects movement toward or away from the frontier; and \(\mathit{TC}\) represents a shift in the frontier ¹⁴. \(\mathit{MI}\) was interpreted based on \(\mathit{EC}\) and \(\mathit{TC}\). A diagnostic review identified four zero or non-finite transitions (for Kaohsiung City in 2024 and 2025 and Taitung County in 2024 and 2025). The estimation procedure removed baseline rows with missing \(\mathit{MI}\) values and converted infinite \(\mathit{MI}\)/\(\mathit{EC}\)/\(\mathit{TC}\) values into missing values. However, it did not automatically remove finite zero-valued \(\mathit{MI}\) or \(\mathit{TC}\) estimates. These transitions are reported separately and are not interpreted as literal zero productivity changes. Consequently, Hsinchu County had one interpretable transition in the diagnostic summary, whereas most other regions had two transitions.

3.7. Software and Analytical Scope

All the DEA, slack, super-efficiency, and Malmquist calculations were implemented in R using the Benchmarking package. Slack analysis was performed with respect to the BCC solution using slack(x, y, e=dea_bcc); thus, the reported slacks were BCC-based input excesses and output shortfalls after radial adjustment. The analysis used dea(..., RTS="crs", ORIENTATION="in") for CCR, dea(..., RTS="vrs", ORIENTATION="in") for BCC, sdea(...) for super-efficiency, slack(...) for slack diagnostics, and malmquist(...) for productivity analysis. No second-stage regression was performed, because the objective was benchmarking rather than causal estimation and because conventional inference using estimated DEA scores requires strong assumptions and specialized procedures ^20,21,22.

4.1. Descriptive Statistics and Input Correlation

The input-correlation diagnostic yielded \({r=0.572}\) for CDW received versus licensed capacity, \({r=0.578}\) for CDW received versus estimated treatment costs, and \({r=0.275}\) for licensed capacity versus estimated treatment costs. This indicates that the relationship between the received volume and licensed capacity was moderate rather than near-perfect. Because these three inputs represent conceptually distinct dimensions—realized processing burden, authorized infrastructure, and estimated treatment cost—all three were retained (Table 2).

4.2. Static DEA Performance Under CCR and BCC

For 2023, 2024, and 2025, the mean CCR efficiencies were 0.509, 0.455, and 0.461 and the mean BCC efficiencies were 0.639, 0.598, and 0.573, respectively. Mean scale efficiency ranged from 0.794 to 0.833 (Table 3). The persistent gap between the CCR and BCC scores indicates that scale-related effects contributed materially to overall inefficiency.

Taitung County has a mean BCC score of 1.000, followed by Keelung City (0.999), Yunlin County (0.918), Kaohsiung City (0.855), Changhua County (0.851), and Pingtung County (0.820). The lowest mean BCC scores were observed in Taipei City (0.158), New Taipei City (0.161), and Taichung City (0.270). These scores are relative to the pooled frontier and should not be interpreted as absolute measures of environmental quality (Table 4 and Fig. 1).

4.3. Slack Analysis

Positive treated-/reused-output slack occurred in 79.2% of the observations. On the input side, CDW-received slack occurred in 62.3%, estimated treatment-cost slack in 45.3%, and licensed-capacity slack in 35.8% of the observations. Recycled-product-output slack was less frequent (9.4%) but was substantial for the selected observations (Table 5). Because the slack magnitudes are expressed in different native or transformed units, the raw magnitudes should not be compared directly across variables.

4.4. Dynamic Productivity Change

Across valid transitions, the mean \(\mathit{MI}\) was 1.060 for 2024 and 1.090 for 2025 (Table 6). For 2024, mean \(\mathit{TC}\) (1.350) exceeded mean \(\mathit{EC}\) (0.805), whereas for 2025, both mean \(\mathit{EC}\) (1.064) and \(\mathit{TC}\) (1.032) exceeded 1 (Fig. 2). Hsinchu County had the largest valid county mean \(\mathit{MI}\) (1.983), with \({\mathit{EC}=1.831}\) and \({\mathit{TC}=1.083}\). However, this estimate is based on only one valid transition and should therefore be interpreted as a single-period change rather than as a stable three-year trend. Changhua County had \({\mathit{MI}=0.790}\), \({\mathit{EC}=0.891}\), and \({\mathit{TC}=0.886}\) across the two valid transitions, indicating a decline in both components relative to the frontier (Table 7).

Four transitions were not interpreted because their \(\mathit{MI}\) and \(\mathit{TC}\) scores were zero or non-finite: Kaohsiung City (2024, 2025) and Taitung County (2024, 2025). They are reported as numerical/undefined cases rather than as genuine zero productivity changes.