Benchmarking the performance of water companies for regulatory purposes to improve its sustainability

Efficiency estimation

To compute the efficiency scores of WCs based on the DEA-CSW approach, the methodology proposed by Wu et al.²⁵ was employed. It was assumed that there are n units (left( {j = 1,..,,d,..,,n} right)) ((WC = left{ {d|d,is,a,water,company} right})) and each WC uses m inputs (left( {i = 1,….,,m} right)) to produce s outputs (left( {r = 1,….,,s} right)).

To evaluate the efficiency of WC_d, the basic DEA-CCR model proposed by Charnes et al.¹⁷ was used (Model 1):

s.t.

where (u_{rd}) is the weight of the output r for the WC_d (observation evaluated) and (omega _{id}) is the weight of the input i for the water company evaluated (WC_d). Model (1) is an output-oriented DEA model because within a regulatory framework, the objective of WCs is to improve the quality of their services (outputs) keeping constant economic costs (inputs).

Model (1) selects the set of input and output weights that maximize the efficiency of WC_d. In other words, the efficiency score for the water company d in the DEA-CCR model ((E_d)) is the best that the WC_d can obtain. The WC_d is efficient if (E_d = 1) and is not efficient (i.e. has room for improvement) if (E_d ,<, 1). Based on Model (1), WCs cannot be evaluated and ranked on the same basis, because different weights to inputs and outputs are allocated in efficiency assessments.

Unlike to the traditional DEA-CCR approach shown in Model (1), in the common-weight DEA approach, a CSW is used to calculate the efficiency of units (WCs)^22,24. Under this approach, each unit can have its own upper and lower efficiency target²⁷. Model (1) shows that the CCR efficiency of WC_j is achieved when its most favourable weights are allocated; thus, under the CSW approach, the upper efficiency target for a unit is its CCR efficiency score (i.e., (E_j^{max} = E_j)). In contrast, the minimum efficiency score of each unit is 0. However, this value is only generated if the output weights of the CSW are all equal to 0, which would not be acceptable for any unit. Based on Wu et al.²⁵, the lower efficiency goal ((E_j^{min})) of each unit is calculated as:

where (left( {mu _{rd}^ ast ,varpi _{id}^ ast ,nabla i,r} right)) is (are) the most favourable set(s) of unit_d weights generated from Model (1). Equation (2) shows that the lower efficiency of a unit is obtained when it is forced to use a set of weights that is most favourable for another unit.

Considering the upper and lower efficiency goals of WCs, the DEA-CSW is defined as:

The model below (3) (DEA-CSW), which is non-linear, shows that when a CSW is chosen to evaluate the efficiency of WCs, it is guaranteed that all efficiency scores are between their upper and lower efficiency goals. Thus, in DEA-CSW model, different sets of common weights can be selected for the efficiency assessments of WCs. When efficiency scores are used to benchmark and rank WCs, a fundamental criterion to select the CSW is that they must be satisfied by all WCs. Thus, Wu et al.²⁵ proposed the concept of “satisfaction degree” of DMU_d for a weighting profile, which was measured as the distance from the proposed efficiency ratio to the efficiency ratio determined using CSW. Based on Wu et al.²⁵, each WC_d was assumed to have rational common-weights selection, allowing common weights to be selected that achieve the upper efficiency goal, (E_d^{max}). However, it is not possible to select a set of common weights that makes efficiency less or less equal to its lowest efficiency goal, (E_d^{min}). Based on this criterion, the satisfaction degree of WC_d based on the set of common weights selected from W^R (DEA-CSW) is defined as:

(psi _d in left[ {0,1} right]), (forall d). If (psi _d = 1) means that the selected CSW meets the upper efficiency target of the WC_d, (E_d^{max}). In other words, the WC_d obtains the most satisfactory situation when its most favourable weights are selected. By contrast, a value of 0 for (psi _d) means that the selected CSW gives WC_d its lowest efficiency, (E_d^{min}).

The methodological approach proposed by Wu et al.²⁵ defined common weights for units so as to maximize the satisfaction degree (Eq. 4) for all evaluated units. Moreover, to improve the willingness to accept the set of common weights defined, the CSW selected should not result in units that have satisfaction degrees with large differences among them. Thus, the multi-objective programming model (5) was used to define CSW:

s.t.

The model (5) maximises the satisfaction degrees of all WCs as follows:

s.t.

The model (6) allows us to generate the set of common weights for the evaluated WCs. Because this model is nonlinear, it cannot be directly solved. To overcome this limitation, Wu et al.²⁵ proposed two algorithms that allow CSW to be estimated. These algorithms are shown in the supplementary material.

Case study

The current study focused on the major WCs in Chile. In particular, efficiency was assessed for a sample of 23 WCs that provide both drinking water and sewerage services to around 95% of the urban population in Chile (13.5 million people). They are distributed across the 16 Chilean regions and therefore, our study covers the whole country from a territorial perspective. These 23 WCs included ten concessionary water companies, eleven private water companies, and one public water company. The water and sewerage industry in Chile was almost entirely privatised between 1998 and 2004²⁶. Nevertheless, all WCs are regulated using the same model, which is based on the efficient water company. Under this regulatory model, the “real” costs of each WC are compared with a virtual, efficient WC, defined by the urban water regulator (Superintendencia de Servicios Sanitarios -SISS-), and considered to be the benchmark²⁸. Moreover, the SISS also monitors the quality of the services provided by the WCs and established penalties when quality standards are not met. The customers can file complaints to both WCs and water regulator, which has to respond appropriately. This mechanism also contributes to monitor the quality of the service provided by the Chilean WCs.

Moreover, a basic prerequisite for applying DEA is that the selected input and output variables should have an isotonic relationship, which can be validated using correlation analysis. If the correlation between input and output variables is positive, this means the variables maintain an isotonic relationship and are appropriate to use in the DEA model. If the correlation is negative, other variables need to be selected^29,30.

The two inputs considered were: (i) operational expenditure (OPEX) expressed as CLP (On 25th February, the conversion rate was 1 US$ ≈ 704 CLP and 1 € ≈ 855 CLP.) per year, involving annual costs incurred by the WC to provide both drinking and sewerage services and; (ii) capital expenditure (CAPEX), which was measured as CLP per year, integrating the funds used by WCs to acquire, upgrade, and maintain physical assets. Following Molinos-Senante et al.³¹, two quality-adjusted outputs were considered to evaluate the efficiency of WCs in Chile. These outputs were estimated based on Eqs. (7) and (8).

where (y_1) is the quality-adjusted volume of supplied drinking water, estimated as the product of the volume of supplied drinking water ((VDW)) multiplied by the quality of drinking water ((QDW)). (y_2) is the quality-adjusted number of customers with access to wastewater treatment, which is the product of the number of customers with access to wastewater treatment ((CWW)) multiplied by the quality of the wastewater treated ((QWW)). Both quality indicators ((QDW) and (QWW)) are estimated by the Chilean water regulator for each WC, with values ranging from zero to one. A value of one indicates that the WC met all legal requirements regarding quality issues, and vice versa²⁶. In Eqs. (7) and (8), when a WC does not meet all quality requirements (i.e., its quality indicator is lower than one), then it is penalized in terms of output production. Hence, by multiplying the volume of drinking water ((VDW)) and the number of customers with access to wastewater treatment ((CWW)) by quality indicators ((QDW) and (QWW)) it is avoided favoring WCs that have lower costs but provide poor quality water and sanitation services.

To test that the selected outputs and inputs fulfil the isotonic condition, the Pearson correlation test was conducted. Our empirical results (Table 3) indicate that the variables used to estimate efficiency scores have strong positive correlation, indicating that the selected input and outputs can be used in the DEA model. Moreover, we have applied the methodology proposed by Lewis et al.³¹ (Eq. 9) to detect outliers because their presence distorts the efficiency results of the WCs. Nevertheless, none of the 23 WCs evaluated was identified as an outlier.

Table 4 provides an overview of the statistical data employed to compute the efficiency scores of the WCs evaluated in Chile.

Source: Resources - nature.com