Output-specific inputs in DEA: An application to courts of justice in Portugal

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Output-specific inputs in DEA: An application to courts of justice in PortugalR Maria Conceição A. Silva CEGE, Católica Porto Business School, Rua Diogo Botelho, 1327, Porto 4169-005, Portugal

a r t i c l e

i n f o

Article history: Received 6 November 2016 Accepted 26 July 2017 Available online xxx

a b s t r a c t This paper addresses the efficiency assessment of production units in cases where some characteristics of the production process are known. In particular we focus on the existence of direct linkages between inputs and outputs, where certain outputs are produced from specific inputs and not jointly produced from all inputs. Our aim is to use and empirically compare alternative forms of reflecting the linkages between inputs and outputs. The alternatives to be compared to reflect the linkages between inputs and outputs are: the use of separate assessments; the use of ratios between linked outputs and inputs; and the use of differences between linked outputs and inputs. These alternatives are presented and contextualised within existing procedures for dealing with output-specific inputs, and results are discussed and illustrated empirically in the context of evaluating Portuguese courts’ efficiency. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Efficiency measurement through frontier methods (such as Data Envelopment Analysis, DEA) assume that all inputs are jointly used in the production of all outputs. In real situations, however, this may not be the case since certain inputs may be associated to the production of a single output or to a subset of the outputs. One of the earliest examples in the literature considering the link between specific inputs and specific outputs is that of Thanassoulis et al. [27], where the authors considered the output ‘survival rate of babies at risk’, using the denominator (babies at risk) on the input side and the numerator (number of babies at risk surviving) on the output side. Given that these two variables were intrinsically linked, the authors introduced weight restrictions in the model linking the weights of these input and output variables. In education contexts, when value added of students is computed, it is usual to consider attainment on entry on the input side and attainment on exit of a certain educational stage on the output side (see e.g. [21,22]). In this circumstance, if on exit we consider two subjects (e.g. reading and mathematics) and on entry the same subjects are considered, it is clear that previous attainment in reading is mostly responsible for the output attainment on exit in reading and the previous attainment in mathematics is mostly responsible for the attainment on exit in mathematics. Another situation, depicted in the empirical application of this paper, is that of courts’

R

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efficiency where cases handled by the court depend on the cases that were assigned to that court. As a result, outputs can be defined as the cases of various types that were resolved within a period of analysis, and inputs can be, among others, the number of cases that entered the court in the same period of analysis. If we specify the type of cases (e.g. family cases) it is clear that the output number of family cases resolved is mainly linked to the input number of family cases entered, and not with the remaining workload of the court. An additional, and recent, example of this specification of inputs and outputs can be found in Peyrache and Zago [19] where on the input side pending cases divided into criminal and civil cases were considered and on the output side the same type of resolved cases were chosen. Other examples can be found in Salerian and Chan [24], who analysed a railway application where the input ‘number of passenger cars’ was related to the output ‘passenger kilometres’, but not to the output ‘net tonnes of freight’, and the input ‘number of freight wagons’ was related to the output ‘net tonnes of freight’ but not to the output ‘passenger kilometres’. Recently Cherchye et al. [4] proposed a method for handling this type of links between inputs and outputs and applied it to electric utilities taking into account the fact that the input ‘fuel consumption’ does not influence the output ‘non-fossil energy generated’, but has an impact in the remaining outputs (see also [5]). We will use in this paper the same terminology as that used in Cherchye et al. [4], where ‘Joint inputs’ are the inputs used in the production of all outputs; ‘Output-specific inputs’ are those inputs that only contribute to the production of a specific output, and ‘Sub-joint inputs’ are those inputs that contribute to the

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Please cite this article as: M.C.A. Silva, Output-specific inputs in DEA: An application to courts of justice in Portugal, Omega (2017), http://dx.doi.org/10.1016/j.omega.2017.07.006

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production of a subset of outputs, but not all of them. In this paper we address only output-specific and joint inputs and will not address sub-joint inputs. In particular, we explore various possibilities to reflect the linkage between inputs and outputs. We term the alternative approaches: the separation, the ratio and the difference approaches. The separation approach is related to recent work by Cherchye et al. [4] and is also inspired in the work of Tsai and Molinero [30]. The ratio approach is linked to previous attempts by Salerian and Chan [24] and Despic et al. [8], but the final model used is an adaptation of Olesen et al. [17] to handle ratio data in DEA. The difference approach is, to our knowledge, new in the literature, but we show that it is equivalent to the use of a specific type of weight restrictions in DEA models. The various approaches are exposed and used in an empirical application to courts efficiency in Portugal (to the authors knowledge, there is only one published study on courts’ efficiency in this country). The advantages and disadvantages of each approach are highlighted through the empirical illustration. The approaches are not analytically compared given their different modelling assumptions. Promoting courts’ efficiency is part of the European Commission concerns, which has resulted in the creation of ‘The European Commission for the Efficiency of Justice’ (CEPEJ) in September 2002. CEPEJ undertakes regular processes for evaluating judicial systems (in terms of efficiency, effectiveness and quality) of the Council of Europe’s member states. In a recent survey regarding the determinants of judicial efficiency Voigt [31] distinguishes two sources of judicial efficiency: Those related with the supply side of justice (e.g. quality of the law, judicial organization, judges’ individual incentives, etc.) and those related with the demand side of justice (influenced by the regulation of lawyers, costs incurred by judicial parties, propensity to litigate, court delay, etc.). Due to the lack of other types of data, courts’ performance is mainly measured by indicators such as: cases resolved per judge, clearance rates (percentage of filed cases that are resolved), pending cases, etc. Differences between courts on their ‘operational’ performance ascertained through variables like the above, can be explained by several factors like the stability and dimension of courts’ staff, the quality of the judges’ decisions, and the complexity of the cases handled. In this paper we will be concerned just with operational performance and will not attempt to explain differences in performance between courts, nor enter into account with quality variables (due to the difficulty in obtaining such data). As a result the application to courts should be regarded as illustrative of the methods proposed to link inputs and outputs, rather than as a thorough attempt to evaluate courts in Portugal. We note however, that due to recent reforms in the organization of courts in Portugal such exhaustive analysis is a necessity. The paper contributes to the literature in several ways. It puts forth several methods to link inputs and outputs, adapting some existing models in the literature and proposing some others for the first time: It also empirically compares and highlights the advantages and disadvantages of each alternative. 2. Previous literature on courts’ efficiency There are not many empirical applications on the analysis of courts’ efficiency through frontier methods worldwide. Santos and Amado [25] reviewed the literature on courts’ efficiency and found just 24 studies applying the non-parametric Data Envelopment Analysis (DEA) technique to this context. The level of analysis can be varied, from the micro level of the court (or even the benches, as in this paper) to judicial districts (e.g. [16] and [13]), regions or even countries (e.g. [9]). The first reported study on judicial performance is that of Lewin et al. [16] who evaluated the efficiency of 30 judicial districts in the North Carolina US state. Districts were compared based on 5 inputs (size of caseload, court’s

working days, number of attorneys and other workers, number of misdemeanors in the caseload, and size of white population) and 2 outputs (total number of dispositions and the number cases pending with less than 90 days). The number of personnel at the court (aggregated or disaggregated in judges, assistants and other personnel) has been used in most efficiency applications to courts. The second most used input regards caseload, which in most cases includes pending cases and new cases, but sometimes the two are considered disaggregated. Peyrache and Zago [19] provide some arguments regarding the inclusion or exclusion of pending cases on the inputs side, exploring the arguments put forth for each option. In their empirical application to Italian courts they used two model specifications, one including and another excluding pending cases, but reached similar results for both alternatives. Regarding outputs it is clear that one output dominates the courts’ efficiency literature: The number of finished or resolved cases. This output appears in various forms, sometimes aggregated into a single figure, while other times disaggregated over different types of cases (e.g. Kittelsen [15] used 7 types of decisions as outputs, and Santos and Amado [25] used 43 types of cases). Rarely quality variables have been included in the analysis. In spite of that, some attempts have been made to explain the efficiency of courts through contextual variables (more common on macro studies and including variable like GDP per capita, percentage of population belonging to ethnic minorities, and percentage of population with higher education [13], or judges’ salaries, academic degree, and number of courts [9]), or judges’ related variables (e.g. academic qualifications of judges, and their career perspectives [26]). A quality variable has been used recently in Falavigna et al. [11], related to court delay (measured as an undesirable output in a directional distance function through the variable ‘number of days required to finish a process). Note that quality variables like those advocated within judicial literature (see e.g. Posner [23]), as the number of citations of the decision from other courts (applicable only to Anglo-Saxonic law), the number of times that the direction of the sentence is reversed by superior instances, or the distinction between the cases that finish with complete appreciation from those that did not, have been rarely used within the efficiency literature. Exceptions can be found in Pedraja-Chaparro and Salinas-Jimenez [18] and Gorman and Ruggiero [13] who distinguish between jury trial cases and non-trial cases (those that are finished without a complete process because a settlement was reached, withdrawal, etc.). In Portugal, DEA was applied for the first time to courts by Santos and Amado [25]. The authors assessed 213 first instance courts from 2007 to 2011, excluding the administrative courts. They used a single input (staff: divided into judges and administrative workers) and did not consider caseload as an input variable. As outputs, Santos and Amado [25] used finished cases of 43 types. In order to increase the discrimination of the model, weight restrictions were imposed in accordance with the complexity of cases (proxied by duration). They concluded that only 16% of courts were efficient and that inefficiency was higher in smaller courts (with less than 500 cases). Efficiency was also higher for courts with a higher percentage of workers per judge. 3. Courts input and output data In Portugal, courts can be classified into various types: judicial courts, administrative and fiscal courts, the constitutional court, accounting court, military courts, and justice of peace courts (used to solve very small conflicts). Judicial courts are the aim of our analysis, in particular 1st instance judicial courts (there are also 2nd instance judicial courts, or appeal courts, and the 3rd instance judicial court, corresponding to the Supreme Court of justice – the

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M.C.A. Silva / Omega 000 (2017) 1–11 Table 2 Descriptive statistics.

Table 1 Inputs and outputs used in the assessment. Inputs Workers Common cases received (Common-E) Enforcement cases received (Enforcement-E) Other cases received (Others-E)

3

Outputs Common cases solved (Common-F) Enforcement cases solved (Enforcement-F) Other cases solved (Others-F)

superior hierarchical body of judicial courts). First instance judicial courts are organized within judicial districts, which are themselves organized in circles, which in themselves are organized in judicial counties (comarcas). The word ‘court’ is associated with ‘comarcas’ where a number of judges, may specialise in certain types of cases or may have generic competences if the size of the judicial county is small and does not justify specialization. Within the courts (or judicial counties) judges are usually organized in benches (constituted mainly by a single judge and some administrative staff). The Portuguese judicial system was restructured in 2014 and 2016. The 2014 reform (Law 62/2013 regulated by Decree law, 49/2014) had two main purposes:, to implement specialized jurisdictions at the national level and to implement a new model of management of judicial counties. As a result of the reform, most generic courts and benches disappeared and work was centralized on specialized courts. Decree-Law 86/2016 undid several changes introduced in 2014, in particular because it was understood by the new central government that specialization resulted in an excessive distance between citizens and the judicial structures. As a result, several courts that were closed in 2014 (those with a procedural volume below 250 cases per year) were reopened in 2016. Our analysis focus on generic competence benches (the most disaggregate unit of analysis). On the contrary Santos and Amado [25] focused their analysis on the efficiency of judicial counties (comarcas), analysing 223 of these during the period of 2007 to 2011. These judicial counties include generic competence and specialized courts that are not necessarily comparable. The data used in the present study (from 2010 to 2012) is prior to the restructuring of 2014 and 2016. However, the recent reforms justify the need for further analysis on the efficiency of judicial courts in Portugal, and in particular on the impacts of the reforms on courts’ efficiency. Our sample comprises 267 benches of generic competence. Most of these benches are constituted by a single judge and they perform similar activities, being therefore comparable. We used data from 2010 to 2012 but these data were averaged and a single period analysis was performed (similarly to Schneider [26]). The reason for this averaging relies on the picks of cases that some benches face in some years due to mega-cases that consume most of the resources of a bench. Following the literature we chose as inputs and outputs of our analysis the variables in Table 1. Outputs considered in our assessments are the average number of cases resolved in the period under analysis. In order to reflect the mix of caseload of benches, cases were partitioned into 3 categories, from an initial number of 27 types of cases identified as part of the work of the benches (see e.g. Kittelsen [15] who used a similar approach). The construction of the 3 categories took into account the similarity of competences required to solve the cases in each category and also the percentage of total cases in the category (where we avoided considering categories where the percentage of cases handled was very small). As a result the first set of cases includes common cases, provisional orders, embargoes and divorces, the second set includes enforcement cases and service judicial notice, which are cases where debtors are forced to pay their debts against their will; The third set includes corporate reorganization/bankruptcy, guardianship cases (where the court es-

Workers Common-E Enforcement-E Others-E Common-F Enforcement-F Others-F

Average

St DEv

Max

Min

Av. CR

Max CR

Min CR

6.84 213.92 340.20 148.6 223.22 236.74 140.2

2.55 103.51 194.07 91.45 108.83 128.91 81.93

12.3 467.33 1217.33 443.2 593 672 372.2

1 28 33.33 0 27 14 0

1.05 0.72 0.97

1.37 2.58 1.52

0.8 0.17 0.63

tablishes a relationship in which a person is given legal authority over another person, e.g. a child), credit claims and workplace accidents. Note that cases within each category were aggregated through simple sums. For each of these categories the number of cases entered was considered an input. In addition to these 3 inputs we also considered the input ‘number of administrative workers’ in the court. The number of judges was not included in the input set as benches are divisions of a court represented by a single judge. As a result our assessment in fact compares the performance of judges in handling the cases that were assigned to them. Note that aggregation at the court level could be possible since a court is composed of various judges or benches (we however, do not follow that avenue in this paper). Pending cases were not considered as they were not available. As a result, the input set does not reflect the entire workload of the court but it reflects the demand for courts’ services in a given period, and the ability of the court to satisfy that demand. The comparison between supply (cases resolved) and demand (incoming cases) leads to what is usually known as clearance rate. Only when clearance rates are higher than 100% courts are able to catch up with backlog cases. This set of inputs and outputs raised the question that is addressed in this paper: What possible ways can be used to reflect the intrinsic link that exists between each input and each output? That is, the number of common cases that were resolved by the court is mainly linked to the number of cases of this type that the court received and not (directly) with the other types of incoming cases. Indirectly there could be a link, but this link is reflected in the global case load and case mix of the court (Some courts may receive more cases of a certain type and therefore devoting less time to other types of cases). The next section will describe the models put forth to resolve the problem of the intrinsic links between the inputs and the outputs in our courts example. Note however, that this problem is likely to appear in other settings too as discussed in the introduction. Descriptive statistics for the input/output variables, in Table 2, show that on average the number of cases of various types entering the benches (-E) are higher than the number of cases resolved (-F), meaning that indeed on average backlog is accumulating in Portuguese general competence benches. Average clearance rates (CR - the ratio between cases resolved and cases entered), also shown, are below 100%, except in the case of Common cases where clearance rates (CR) are on average close to 1 (1.05). Through this set of inputs and outputs, the objective of the analysis is to identify best performing benches - those that resolve the highest number of cases given the cases received and the workers of the bench. Worst performers will be those showing potential to increase the number of cases resolved given the number of cases received and the number of workers. Note that this potential to increase cannot in reality go beyond the caseload (including pending cases) of the bench (imagining that a bench resolved all the cases received, the potential to increase performance would be solely revealed by the number of pending cases). In our model

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Table 3 Some descriptive statistics for courts.

s 

N. benches

N. courts

Avg wrk

Avg CR common

Avg CR enforcement

Avg CR other

1 2 3 4 5 Grand total

121 35 21 2 1 180

5.13 15.6 26.2 32 45 10.146

1.056 1.046 1.061 0.983 0.991 1.054

0.718 0.684 0.751 0.872 0.6741 0.716

0.997 0.960 0.948 0.922 0.969 0.983

we will identify potential that may go beyond backlog cases. This means that in practice we are identifying not only inefficiencies relating to the inability of the bench to resolve as many cases as similar benches, but also inefficiencies related to bad allocation of workers and cases (e.g. taking two benches with similar number of workers and number of cases received, the bench that solved more cases will be considered more efficient (i.e. it cleaned some backlog) even if the other bench did not have any backlog to clean). So our specification of inputs and outputs implies that our efficiency measure considers not only the ability of courts to process current demand, but also to reduce the backlog. The above 267 benches correspond to 180 generic competences courts that can have 1–5 benches. Some statistics of the courts according to the number of benches that they have are shown in Table 3 About 67% of our courts have a single bench, and the percentage of courts with 4 or 5 benches is very small. This is because generic competences courts are not the main courts in Portugal since in large cities courts are specialised. Note that clearance rates are relatively similar across dimensions of the courts with a tendency for smaller courts exhibiting higher clearance rates. In addition larger courts (which more benches) appear to be overstaffed, since the numbers of workers does not vary proportionally to the number of benches. 4. Output specific inputs – alternative approaches An output oriented directional distance function (see e.g. Chung et al. [6]) under variable returns to scale (VRS) will be used as a basis for efficiency computations. A VRS model is chosen because we do not expect a proportional increase in outputs generated by an increase in inputs. Since all benches have a single judge (a fixed production factor), we expect that an increase in inputs generates a diminishing marginal increase in outputs. Generically we consider a production technology with n DMUs, consuming m inputs (xi j , i = 1, . . . , m) and producing s outputs (yr j , r = 1, . . . , s). The directional vector (gyr , gxi ) specifies the direction for improvement, but in our case we set gxi to zero. The directional distance model, for DMU o, is given by (1).



β|

max λ j ,β

n 

λ j yr j ≥ yro + β gyr , r = 1, . . . , s,

j=1 n 

λ j xi j ≤ xio, i = 1, . . . , m,

j=1

n 

 λ j = 1, λ j ≥ 0 ∀ j

(1)

j=1

The multiplier directional distance model, the dual of (1), is given by (2).



min ho = ur ,v i

m 

vi xio −

i=1

−

s  r=1

s 

ur yro + γo

|

r=1

ur yr j +

m 

vi xi j + γo ≥ 0, j = 1, . . . , n,

i=1

 u r gyr = 1 , u r , vi ≥ 0

∀r, ∀i, γo ∈ R

(2)

r=1

Three approaches are going to be compared as a way of dealing with the output specific inputs in our courts example. The first approach will be called ‘Separation’ approach and corresponds to the analysis of the production process of each output separately and independently from the others. The second approach will be called the ‘Ratio’ approach and will replace outputs by clearance rates for each type of case. The third approach will be called ‘Difference’ approach and will replace outputs by the difference between cases resolved and cases entered. Such a difference actually corresponds to the variation in the pending cases (i.e. when the difference is positive pending cases reduced, when the difference is negative the pending cases increased by the amount of the difference). In the next sections each of the approaches will be detailed and results obtained for each will be shown. Note that a distinctive feature between the three approaches lies on the assumptions regarding the production technology. The separation approach assumes that the production of each output can be modelled through an individual production process, while the other approaches assume that there is a single production process for the production of all outputs, and linkages between inputs and outputs are modelled through manipulation of the variables rather than changing the production possibility set. 4.1. Separation approach The separation approach assumes separable production functions for each output as first introduced in Banker [1]. A separation model implies that the efficiency in the production of each output r can be computed independently in three distinct models or through a single model, by considering different intensity variables for each output production process λrj . The resulting model is shown in (3), where xij is now replaced by xrj , since for each output r there is a corresponding input i and therefore the same index is used, and Wj is the notation used for the joint input, number of workers.



max r λ j ,βr

s 

βr |

r=1 n 

n 

λrj yr j ≥ yro + βr gyr , r = 1, . . . , s,

j=1

λrj xr j ≤ xro, r = 1, . . . , s,

j=1 n 

n 

λrjW j ≤ Wo, r = 1, . . . , s,

j=1



λ = 1, ∀r, λ ≥ 0, ∀ j, ∀r r j

r j

(3)

j=1

This model is similar to that proposed by Cherchye et al. [4] called also the decentralized efficiency score in Cherchye et al. [3], but we consider that the convexity assumption holds for outputs. Note that we also consider an expansion factor β r associated to each output to allow for as little dependence across the three categories of outputs as possible, and also because of technological sets having zero cases entered and zero cases finished, which created problems when a single expansion factor for outputs was adopted. The above model replicates, for each output production possibilities set the constraint regarding the joint input. In practice this means that we assume that the full amount of input is available for the production of each output. Since this is not consistent with the fact that the input is shared in the production of the various outputs, we modify the above model to take this into account. As a result, a single constraint for workers replaces the r = 3 constraints for workers in model (3), assuming that a proportion α r of workers is used for the production of each output r :

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M.C.A. Silva / Omega 000 (2017) 1–11 Table 4 Results for each output inefficiency using separation approach.

Score N. Eff

Common

Enforcement

Other

25.30% 19

125.00% 5

30.80% 19

5

that benches can increase cases solved by this percentage of total observed cases. Only one court is considered 100% efficient, since it is efficient in the production of all outputs.

4.2. Ratio approach n j=1

n

λ1j α1W j +

j=1

λ2j α2W j +

n j=1

λ3j α3W j ≤ Wo. This constraint

is in the spirit of Tsai and Molinero [30], where a single constraint was used to model joint inputs in teaching and research in an university example, as pioneered by Beasley [2] in a assessment of university departments. Note that similar models have been proposed by Cook and Hababou [7] in the context of assessing service and sales performance in bank branches, where some inputs and outputs were specific to each of these functions but some inputs were shared by both. Both Tsai and Molinero [30] and Beasley [2] have let the model determine the optimal shares of joint resource allocated to each process. In our case, since the percentage of cases in each category is similar, we assumed that the joint resource was equally allocated to each process (and so αr = 1/s ). As a result, the separation approach model that we applied to the courts example is shown in (4).



max r λ j ,βr

s 

βr |

r=1 n 

n 

λrj yr j ≥ yro + βr gyr , r = 1, . . . , s,

j=1

λrj xr j ≤ xro, r = 1, . . . , s,

j=1 s  n 

λ

r jW j

≤ sWo,

r=1 j=1

n 

 λ = 1 ∀r, λ ≥ 0, ∀ j, ∀r r j

r j

(4)

j=1

The optimal solution to model (3) is also a solution to model (4), which corresponds to a relaxation of model (3) consistent with the fact that the full amount of human resources is not available in each output production process. Applying model (4) to our data, with a directional vector equal to the output level of the assessed unit yro , resulted in the outputs inefficiency scores shown in Table 4. The choice of a directional vector equal to observed output levels, implies that β r associated to each output equals the ratio between the output augmentation and the observed output. As a result the β r reported above should be interpreted as the estimated required increase in each output expressed as a percentage of observed outputs. It is therefore an inefficiency measure obtained for each output. The overall measure of inefficiency can be obtained through a consistent aggregation of the β r s (see e.g. [12]). Note that, defining y∗ro as the target output level of DMU y∗ −y

o then βr∗ = royro ro . In order to assure aggregation consistency in the overall efficiency score computed we used a weighted average of the individual efficiency scores, where weights for each output r were the proportion of output r of the unit being assessed in its total output (wr = syro y ). When this aggregation weight r=1 ro  is used, the resulting overall aggregate score ( sr=1 wr βr ) is equal to γ =

 y∗ro − sr=1 yro r=1 s r=1 yro

s

. Note that inefficiencies can be expressed as

efficiency scores through the transformation 1+1γ . The output showing the lowest potential for augmentation is common cases, which can increase by 25.3% of observed levels. Enforcement cases show an abnormal high potential for improvement (above 100%) meaning that benches are particularly ineffective in dealing with this type of cases, or that there is a huge potential for dealing with more of these cases. In terms of aggregate inefficiency the mean for the total sample of courts is 61.75%, meaning

The second approach that we consider for dealing with outputspecific inputs is based on the fact that the ratios between the outputs and the output-specific inputs entail a meaning: clearance rates, used commonly as a KPI in assessing the performance of courts. Benches with a clearance rate above 1 are clearing pending cases, as they resolve more cases than the ones that entered the bench in the period considered. On the contrary, clearance rates below 1 mean that pending cases are accumulating. The use of ratios to reflect direct linkages between volume inputs and outputs can be seen as the opposite approach to that taken in Thanassoulis et al. [27], where ratios were replaced by volume measures. The replacement of volume measures by ratios has been suggested before, although not necessarily with the purpose addressed in this paper. For example, Despic et al. [8] propose solving DEA models through a replacement of all variables in the model by all possible ratios between outputs and inputs. Ratios were considered then the outputs of a DEA model with a single unitary input. Results from this modified model and the traditional DEA model are not coincident, but are somehow related, as shown by Despic et al. [8]. Clearly, only the ratios that correspond to links of interest between inputs and outputs may be considered under this approach, and as a result the linkages can be reflected in the chosen ratios. In our case we did not follow this approach given the existence of the joint input relative to workers, and the fact that its use would result in multiple targets for workers, since there would be two ratios for each output: the clearance rate and the ratio of output per number of workers. As a result, we decided to keep on the input side the number of workers and use the clearance rates on the output side. This raised another problem. That of mixing a volume measure with ratio data, which by definition are adimensional. The problems arising from mixing ratio data with volume measures are relatively well known and have been addressed before (e.g. Hollingsworth and Smith [14] or Emrouznejad and Amin [10]). There are two known problems related to DEA models applied to ratio data: (1) the violation of the proportionality assumption implicit in constant returns to scale (this violation is not a problem under variable returns to scale), and (2) the convexity problem analysed in Emrouznejad and Amin [10], where convex targets formed from ratios may lie on the interior or outside of the production possibility set formed by volume measures. Recently Olesen et al. [17] addressed extensively the issue of ratio and volume variables in DEA models and proposed models that can address the above problems. An interesting remark made by the authors is that under CRS ratios must be distinguished as far as the way they respond to the proportional changes in volume inputs. As a result, ratios are classified into: proportional ratios (those that increase proportionality with the increase in volume measures), fixed ratios (those that do not vary when volume measures change), downward proportional ratios (are proportional when volume measures decrease but fixed when they increase), and upward proportional ratios (are proportional when volume measures increase, but fixed when volume measures decrease). In our case we will use a VRS model, and as a result proportionality is not assumed. Regarding the convexity assumption, that ratios also do not satisfy, this problem is solved adapting the linear model proposed by Olesen et al. [17]. In this model, (5), all outputs are ratios (Rrj ) and a single input measure (Wj ) is used.

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 max λ j ,βr

s 

βr |

r=1

n 

4.3. Difference approach

λ jW j ≤ Wo,

j=1

λ j (Rr j − (Rro + βr gRr )) ≥ 0, r = 1, . . . , s, j = 1, . . . , n  n 

λ j = 1, λ j ≥ 0 ∀ j

(5)

j=1

The first constraint relates to the volume input, and the second set of constraints assure that only when the ratio r of the peer unit j is greater than the ratio for the observed unit (Rro ), the intensity variable for peer j (λj ) can be greater than zero. Note that we use a non-radial model with a β r for each output for consistency with the previous approach (a radial directional model with a single expansion factor could also be used). Note that model (5) is in fact a FDH model as shown in Podinovski [20]. This fact implies that the reference technologies for this ratio model and the previous model are distinct, with the FDH model providing more conservative estimates of efficiency. However, model (5) will favour units with a low number of workers and high clearance rates, and smaller benches handling a low volume of cases have in fact the tendency to exhibit higher clearance rates. As a result, for comparability reasons and since knowledge exists on the numerator and denominator of the used ratios, we added the volume outputs ‘number of cases solved’ to reflect the fact that an increase in the volume input (number of workers) may have an impact on the volume of outputs produced (but not necessarily on the clearance rates.) The above non-linear model (5) was therefore added with this set of constraints and was modified into a mixed integer linear model (6) for computation purposes.

 max

λ j ,βr ,k j

s  r=1

βr |

n 

λ j yr j ≥ yro, r = 1, . . . , s

j=1

λ j ≥ 0, k j ∈ {0, 1} ∀ j

ho = −

min

ur ,vr ,v,γo

s 

ur yro +

r=1

−

s 

ur yr j +

r=1

s 

vr xro + vWo + γo|

r=1 s 

vr xr j + v W j + γo ≥ 0, j = 1, . . . , n,

r=1



ur gyr = 1, ∀ r, ur = vr , ur , ≥ 0, vr ≥ 0

∀r, v ≥ 0, γo ∈ R (7)

Setting ur = vr = wr results in the equivalent model (8):



n 



ho = −

min

wr ,v,γo

s 

wr (yro − xro ) + vWo + γo|

r=1

λ j = 1, −

j=1

(6)

Targets from model (6) are obtained directly form the ratio targets: R∗ro, which can be converted to the target finished cases that the court in question should attain (y∗ro = R∗ro× cases type r entered), when it is compared to other benches with higher clearance rates and simultaneously lower number of workers. Results from applying the ratio model (6) to our sample, using the same directional vector (i.e. finished cases yro ) revealed an average aggregate inefficiency of 26.7% with 51 benches appearing with 100% efficiency (see Table 5). The inefficiency value is oby∗ −y tained consistently with the previous approach as royro ro . Aggregation procedures were also the same. Enforcement cases are the ones where the benches reveal higher inefficiency with a potential to increase the number of cases dealt with by about 51%. This values is much lower than the potential identified under the separation approach.

s 

wr (yr j − xr j ) + vW j + wo ≥ 0, j = 1, . . . , n,

r=1



wr gyr = 1, ∀ r, wr ≥ 0,

∀r, v ≥ 0, γo ∈ R

(8)

The dual is shown in (9).



max λ j ,βr

s  r=1

n 

βr |

n 

λ j (yr j − xr j ) ≥ (yro − xro ) + βr gyr , r = 1, . . . , s

j=1

λ j W j ≤ Wo,

j=1

n 

 λ j = 1, λ j ≥ 0

(9)

j=1

Running model (9) or (8) using a directional vector consistent with that used in the previous approaches (equal to the output observed finished cases) yields the results shown in Table 6 for each output average inefficiency. Overall 12 units have been identified as 100% efficient and the aggregate inefficiency is 59.50%.

Table 6 Results for each output inefficiency using difference approach.

Table 5 Results for each output inefficiency using the ratio approach.

Score N. Eff



λ jW j ≤ Wo,

j=1

(Rro + βr gRr ) − Rr j ≤ M (1 − k j ), r = 1, . . . , s, j = 1, . . . , n, λ j ≤ k j , j = 1, . . . , n n 

The third and last approach that we consider for dealing with output-specific inputs is based on the fact that the differences between the outputs and the input-specific outputs entail a meaning related to pending cases. That is, pending casest = pending casest−1 + incoming casest – resolved casest , and therefore the difference between resolved cases and incoming cases yields the variation in pending cases (pending casest−1 – pending casest ). A positive value in the difference means that there was an effective reduction in pending cases while a negative value means that pending cases built up. Taking differences between outputs and output-specific inputs is equivalent to the consideration of weight restrictions where one imposes the weights on inputs and corresponding outputs to be the same. Weight restrictions that link inputs and outputs are called ARII constraints (see Thompson et al. [28,29]). Note that the weights determined in a DEA model are units dependent meaning that the equality between weights can be done directly when the unit of measurement is the same, or should take the units of measurement into account when that is not the case. Model (1) added of these weight restrictions (WR), is shown in (7), for our specific setting.

Common

Enforcement

Others

16.32% 56

51.13% 52

21.06% 52

Score N. Eff

Common

Enforcement

Others

11.80% 18

140.10% 11

11.50% 21

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Table 7 Results for each output inefficiency and overall inefficiency in three approaches. Approach

Common (%)

Enforcement (%)

Others (%)

Overall (%)

N. frontier

Separation Ratio Difference Standard

25.30 16.32 11.8 11.03

125.00 51.13 140.10 47.43

30.80 21.06 11.50 12.06

61.75 26.70 59.50 24.11

1 51 12 58

Enforcement cases are again those where courts reveal higher average inefficiency, with the remaining type of cases revealing similar average levels of inefficiency. 5. Comparative analysis of the approaches Efficiency results obtained from the various approaches can be compared empirically because the inefficiency scores computed are comparable. As mentioned before, we do not attempt an analytical comparison of the approaches given their different modelling assumptions. The average inefficiencies obtained from the 3 approaches in the production of the 3 outputs are shown in Table 7. In this table we also show results from a standard approach where links between inputs and outputs are not taken into account. In this standard approach results were obtained through an output oriented directional distance model, under VRS and using different expansion factors for each output for comparability reasons. Inputs were the number of workers and the cases entered disaggregated in the three types considered. Average results are similar for the separation and differences approaches. The ratio approach is the most dissimilar to the other two, showing lower inefficiencies for benches but still agreeing with the other approaches on the identification of enforcement cases as the ones where more potential for improvement exists. The ratio approach is also the most similar with the standard approach where links between inputs and outputs are ignored, both in terms of overall average inefficiency, in terms of the number of frontier units, and in terms of output specific inefficiencies. Regarding the cases that are dealt with more efficiently, these are common cases under the ratio and separation approaches, and other cases in the difference approach. Note that the approach that is more stringent in terms of identifying inefficiencies is the separation approach, and the approaches that identify lower overall inefficiencies are the ratio and the standard approaches. Correlations between the ratio and the separation approaches are the lowest (0.38 between ranks and 0.39 between aggregate efficiency scores), meaning that these approaches show important differences in ranking. The separation and the difference approaches show much higher correlation values (0.67 between ranks and the same value between efficiency scores) implying some concordance between the two approaches in terms of the ranking of benches and their inefficiencies. The correlations between the ratio and difference approaches are also high (0.74 for the correlation between ranks and 0.67 for correlation between aggregate efficiencies). The standard approach has the highest correlation with the ratio approach (0.73 between efficiency scores and 0.71 between ranks) and the lowest correlation with the separation approach (0.47 between efficiency scores and 0.43 between ranks). Computing differences between the aggregate efficiencies (recall that inefficiencies in the previous tables are easily converted into efficiency values) of the various methods results in the summary values shown in Table 8. Results in Table 8 show that the ratio approach tends to result in efficiency scores that are higher than those yielded by the separation and difference approaches. On the contrary, comparing the difference and separation approaches results in an average difference on efficiencies close to zero with almost as much benches

with positive efficiency differences and negative efficiency differences. This also happens when one compares the standard approach with the ratio approach, where the number of units with positive and negative differences are distributed and average differences are close to zero. On the contrary, the full set of benches showed higher efficiencies under the standard approach than under the separation and difference approach. In relation to the difference approach this is an expected result since the difference approach can be seen as the standard model with restrictions on the weights, and the addition of weight restrictions cannot improve efficiency scores. A more detailed analysis was performed regarding the characteristics of the best and worst performing benches under all approaches. For that purpose we divided the sample into the top 25% performers (the 67 units with highest efficiency) and the bottom 25% performers (the 67 units with lowest efficiency), and computed the average inputs and outputs for each of these groups. Results are shown in Table 9, where average clearance rates and differences between cases are also shown. In bold we signal the highest average values and we use underscript to signal the lowest. In spite of some high correlations between efficiency scores obtained under each approach, there are large differences on the characteristics of the best and worst performing units under each approach. Top performers under the separation approach tend to be the benches with higher inputs and outputs while under the two other linking approaches top performers are remarkably similar and smaller on average. However, the smallest units are the ones identified as top performers by the standard approach. In terms of clearance rates, we see that top performers under the separation approach show the worst clearance rates, whereas the best clearance rates are identified by the ratio approach (as expected). In terms of differences, the top performers for the separation approach show amongst the lowest differences, and the highest differences of cases handled are identified by the difference approach, as expected. Note that generally top performers are contributing to clearing backlog since they exhibit on average clearance rates above 1. This is true for common cases and for other cases. Enforcement cases appear to be the main problem in the benches analysed, since even the best performing benches show clearly low clearance rates for this output – meaning that backlog is on average building up for this type of cases. For bottom performers, the opposite happens with the ratio approach identifying largest benches as the most inefficient (and with characteristics similar to those identified by the standard approach) and the separation approach identifying the smallest benches as the worst performers. In terms of clearance rates, the bottom performers for the separation approach are the ones experiencing the best clearance rates, and the worst performers for the ratio approach are indeed those that show the worst clearance rates. In terms of differences, the separation approach is also the one for which worst performers show the highest values of the differences, while the ratio approach is the one for which worst performers show the lowest values (this last result is unexpected, as the difference approach should be the one discriminating better on the differences between best and worst performers – note however that differences of worst performers are similar for all the approaches except for the separation approach). Bottom

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M.C.A. Silva / Omega 000 (2017) 1–11 Table 8 Differences between efficiency scores of the 4 approaches.

Average N. positive diferences N. negative differences N. zero differences

Difference-separation

Difference-ratio

Difference-standard

Ratio-separation

Ratio-standard

Separation-standard

0.39% 118 128 21

−16.91% 11 242 14

−20.22% 0 255 12

17.30% 232 26 9

−3.31% 85 132 50

−20.61% 0 266 1

Table 9 Characteristics of best and worst performers in the 4 approaches. Top 25%

Wrk Common-E Enforcement-E Other-E Common-F Enforcement-F Other-F Common-CR Enforcement-CR Other-CR Common-Diff Enforcement-Diff Other-Diff

Bottom 25%

Separation

Ratio

Difference

Standard

Separation

Ratio

Difference

Standard

7.83 292.05 507.91 229.95 318.21 375.36 214.90 1.10 0.80 0.96 26.17 −132.56 −15.06

5.96 213.69 338.80 153.65 239.48 265.44 154.73 1.13 0.85 1.08 25.79 −73.36 1.09

6.15 220.68 315.98 154.18 248.68 262.78 154.59 1.12 0.87 1.05 28.01 −53.20 0.41

5.5 198.66 303.18 141.75 218.04 233.11 141.56 1.10 0.81 1.06 19.38 −70.07 −0.19

5.25 129.38 188.50 75.6 128.76 117.83 68.68 1.02 0.65 0.95 −0.62 −70.67 −6.92

7.68 230.00 411.73 160.85 221.26 238.52 141.54 0.98 0.59 0.89 −8.73 −173.21 −19.31

6.46 175.75 295.27 113.70 168.68 163.41 97.97 0.98 0.60 0.90 −7.07 −131.86 −15.73

7.28 206.10 343.71 145.48 200.26 206.05 128.80 0.99 0.63 0.90 −5.85 −137.65 −16.67

performers show mostly below 1 clearance rates, and very low clearance rates for the most problematic type of cases (enforcement). In addition, bottom performers also exhibit negative differences, meaning that on average these are the courts where backlog is accumulating. As a result, in spite of backlog being not explicitly considered in our models, clearly we have been able to distinguish those benches that are clearing backlog from those that are accumulating it. Fig. 1 helps visualising these results, where we use as a basis the average of each variable of the top performers under the separation approach and compare the average performance of the various groups in relation to this base. The graph on the top left shows that best performers are dissimilar for the separation approach when compared to the other three approaches. The graph on the bottom compares worst performers between approaches, where it is clear that the bottom performers identified in the ratio approach are on average dealing with more cases than the worst performers in the other three approaches. The separation approach clearly identifies worst performers as the smallest benches on average. In summary, the four approaches show rather dissimilar average characteristics of worst performers, but similar characteristics of best performers as far as the ratio, the standard, and difference approaches are concerned. Comparing now best and worst performers within approaches (see Fig. 2) we can see that the separation approach is the one that shows more differences between worst and best performers, being these differences mostly related with size, in spite of the use of a VRS model. For the difference approach worst performers show a higher number of workers, and similar/lower number of cases entered than best performers, but a lower ability to handle these cases. For the ratio approach, a similar situation happens, but differences between worst and best performers happen more on the input side (with emphasis for the number of workers that are identified in this approach as an important source of inefficiency), since in terms of outputs both resolve a similar number of cases with a slight disadvantage for worst performing units. The ratio approach is the one that least discriminates between best and worst performing benches. Note that conclusions for the standard approach are very similar to those for the ratio approach, since the graphs are remarkably similar.

Analysing specifically the cases that showed the highest differences in ranking between approaches, we show in Table 10 the observed values of the volume measures, the clearance ratios, and the differences between finished cases and entered cases for these units. We also show the total average of the sample as a means of comparison. Bench T16 is one of the worst performers under the differences approach, but it is considered a top performer under the separation approach. It is clear from Table 10 that this bench shows the worst values in terms of differences in performance for all outputs – values that are clearly below average differences observed and values that are too high, like 481 enforcement cases that were not handled. The separation approach could not identify this unit as low performer and instead classified it under the top performers – a result of a high number of workers and high volume of outputs, given the (also high) input levels. Bench T179 is one of the worst performers under the ratio approach, but a top performer under the separation approach. In Table 10 we can indeed see that the clearance rates for this bench are very low and smaller than average. When compared to T16 the clearance rates are not lower, but T179 shows simultaneously higher number of workers and lower volume of finished cases, which contributed to its inefficiency under the ratio approach. T146 is one of the worst performers under the separation approach mainly due to its reduced inputs and respectively low volume of outputs. So, the model could identify peers with similar or lower inputs and much higher output levels, which turned T146 inefficient under this approach. Under the ratio approach, this bench is considered a top performer since its clearance rates are good and mostly above average. 5.1. Discussion The results above show that the three approaches to link inputs and outputs capture different aspects of performance and each of them may lead to the identification of best performance under one perspective, but low performance under another. For example, under the ratio approach an excessive importance may be placed in very good clearance rates, in spite of these representing a very small amount of cases resolved (see e.g. bench T146, where a clearance rate of 1.178 represented just 3 additional cases solved); under the separation approach the volume of cases

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Fig. 1. Characteristics of top performers and bottom performers under each approach.

handled is clearly influencing performance measures and is leading to higher volume and sized benches to outperform smaller benches (see e.g. bench T16). The comparison with the standard approach shows that efficiency is overestimated when the links between inputs and outputs are not accounted for, particularly when the links are taken into consideration through the differences and separation approaches that always result in lower efficiency scores. As for the ratio approach, there are not many differences between the standard and this approach, especially as far as good performers are considered. However, the ratio approach appears to identify better than the standard approach the worst performers: those benches that are on average larger than the others but exhibit the lowest clearance rates, meaning that backlog is accumulating in these benches. As a result, we tend to favor the use of the difference approach as a way to link inputs and outputs that relate directly. The results from the courts example has shown that this approach provides less extreme results than the remaining approaches, being able to account for the volume of work at the court without penalising excessively smaller units or without benefiting excessively larger units. The use of differences is equivalent to the use of weight restrictions that impose equal weighting to the linked inputs and outputs. It may however be a less general approach in the sense that the computation of differences needs to yield an observable and

meaningful measure (which in our case was a measure of reduction in backlog). In case the differences cannot be computed the ratio approach appears as a close alternative, where rank correlations with the difference approach are close to 70% and the absolute differences in rankings is the lowest on average between these two approaches. When using the ratio approach, care is needed, however, with the assumptions implicit in ratio measures that do not vary proportionally to input volume measures. We handled this issue in this paper by adding also absolute values on the output side, and this prevented the model from identifying very small units as the most efficient, in spite of the most inefficient being on average the largest. Note that the ratio approach and the standard approach reveal an additional source of inefficiency related to the possible excessive number of workers in some benches. This is an important finding, and one that deserves further political considerations since some benches may be overstaffed while others may be understaffed. As a result, an avenue for future research could be to analyse the extent to which some branches are overstaffed and move the staff from these branches to understaffed ones. Note that the comparison between benches T16 and T179 in Table 10 showed that bench T16 receives almost the double of cases than bench T179, but has one less worker and the same number of judges. Although most of the work is made by the judge it is true

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Fig. 2. Characteristics of top performers and bottom performers within each approach.

Table 10 Characteristics of some units with large differences in ranking between approaches. Eff Eff Eff Eff

Separation Ratio Diff Std

Wkr Commum-E Enforcement-E Others-E Commum-F Enforcement-F Other-F Common-CR Enforcement-CR Other-CR Common-Diff Enforcement-DiffEnf Other-Diff

0.791 0.885 0.559 0.911

0.409 0.936 0.521 1

0.750 0.480 0.616 0.841

T16

T146

T179

Average sample

8 416.33 909 443.167 409 428 331.67 0.982 0.471 0.748 −7.33 −481 −111.5

4 53 126 17.83 51.33 95.67 21 0.969 0.759 1.178 −1.67 −30.33 3.167

9 226.67 582 182.33 234 403.67 172.33 1.032 0.694 0.945 7.33 −178.33 −10

6.84 213.92 340.199 148.60 223.22 236.74 140.22 1.051 0.723 0.973 9.31 −103.46 −8.38

that workers have an impact on the workload that is left for the judge to perform. This is consistent with the study of Santos and Amado [25] that concluded that most efficient courts were those where the proportion of judges on overall staff was below the median, implying that administrative workers have a role to play on the efficiency of Portuguese courts. In addition, re-allocation can be investigated not only regarding workers but also regarding cases allocated to each bench. Other policy implications relate to the high congestion that happens in Portuguese benches for enforcement cases, that are handled much more ineffectively than

the two other types of cases considered in this assessment. Clearly benches that are considered efficient on handling these cases could be investigated and actions taken to analyse the reasons for this. Under the difference approach 12 units are efficient in handling enforcement cases. These benches have on average less staff and receive more cases than the inefficient ones. Still they manage to show clearance rates of enforcement cases of 1.05 against clearance rates of 0.7 for inefficient benches on the production of this output (details of computations not shown for brevity). 6. Conclusion This paper has addressed the topic of establishing links between specific inputs and outputs in efficiency assessments. This is a topic that has deserved recent attention in the literature, with some new developed models advocating the use of output specific production functions and thus separate efficiency assessments. Through an illustrative example on Portuguese courts’ efficiency we have compared separate assessments with two other approaches that could be used in establishing the links between inputs and outputs: One is the ratio approach, where output specific inputs are linked to outputs through a ratio, and the other is the differences approach, where linkages are established through differences. From this comparative study we conclude that the approach used to reflect the linkages between inputs and outputs is not indifferent in terms of results produced. The various approaches revealed some strengths and fragilities. Average characteristics of low performers are very different between approaches, but the ratio and difference approaches identified best performers with very close characteristics. The separation approach was the

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one producing the most dissimilar results being particularly influenced by the size of the benches in spite of the use of a variable returns to scale model. In spite of that, this approach is the one that provides a better discrimination between best and worst performers – something that the ratio approach does not do when we look at average characteristics of these two groups of units. The analysis of Portuguese courts has been illustrative in this paper. In spite of that, there appears to be huge inefficiencies in courts in Portugal, with large discrepancies between the number of cases received and solved and the staff employed. Future work should pass through an in-depth analysis of these inefficiencies (particularly after the reforms in 2014 and 2016) and an investigation of how resources could be better re-allocated between courts. This could be a valuable input for the executive government, which could use this type of analysis to better allocate its scarce resources. Acknowledgements The author is grateful to Raquel Prata who collected and performed a first analysis of the data used in this study. This analysis is published in her Master thesis entitled “Contributo para a avaliação da eficiência dos tribunais”, which is available at ISCTE Business School, Lisbon. The author also benefited from some comments and remarks received when discussing the paper with Victor Podinovski, Ole Olesen, Veerle Hennebel and Laurens Cherchye. All contents are the sole responsibility of the author. References [1] Banker R. Selection of efficiency evaluation models. Contemp Account Res 1992;9(1):343–55. [2] Beasley J. Determining teaching and research efficiencies. J Oper Res Soc 1995;46:441–52. [3] Cherchye L, De Rock B, Hennebel V. Coordination efficiency in multi-output settings: a dea approach. Ann Oper Res 2017;250(1):205–33. [4] Cherchye L, Rock BD, Dierynck B, Roodhooft F, Sabbe J. Opening the black box of efficiency measurement: input allocation in multi-output settings. Oper Res 2013;61(5):1148–65. [5] Cherchye L, Rock BD, Walheer B. Multi-output efficiency with good and bad outputs. Eur J Oper Res 2015;240:872–81. [6] Chung Y, Färe R, Grosskopf S. Productivity and undesirable outputs: a directional distance function approach. J Environ Manag 1997;51(3):229–40. [7] Cook WD, Hababou M. Sales performance measurement in bank branches. Omega Int J Manag Sci 2001;29:299–307. [8] Despic O, Despic M, Paradi J. DEA-R: ratio-based comparative efficiency model, its mathematical relation to DEA and its use in applications. J Product Anal 2007;28:33–44.

[m5G;August 31, 2017;21:53] 11

[9] Deyneli F. Analysis of relationship between efficiency of justice services and salaries of judges with two stage DEA method. Eur J Law Econ 2012;34(3):477–93. [10] Emrouznejad A, Amin GR. Dea models for ratio data: convexity consideration. Appl Math Model 2009;33(1):486–98. [11] Falavigna G, Ippoliti R, Manello A, Ramello G. Judicial productivity, delay and efficiency: a directional distance function (ddf) approach. Eur J Oper Res 2015;240:592–601. [12] Färe R, Karagiannis G. A postcript on aggregate farrell efficiencies. Eur J Oper Res 2014;233:784–6. [13] Gorman MF, Ruggiero J. Evaluating U.S. judicial district prosecutor performance using dea: are disadvantaged counties more inefficient? Eur J Law Econ 2009;27(3):275–83. [14] Hollingsworth B, Smith P. Use of ratios in data envelopment analysis. Appl Econ Lett 2003;10:733–5. [15] Kittelsen SAC, Førsund FR. Efficiency analysis of Norwegian district courts. J Product Anal 1992;3(3):277–306. [16] Lewin AY, Morey RC, Cook TJ. Evaluating the administrative efficiency of courts. Omega Int J Manag Sci 1982;10(4):401–11. [17] Olesen OB, Petersen NC, Podinovski VV. Efficiency analysis with ratio measures. Eur J Oper Res 2015;245:446–62. [18] Pedraja-Chaparro F, Salinas-Jimenez J. An assessment of the efficiency of spanish courts using DEA. Appl Econ 1996;28(11):1391–403. [19] Peyrache A, Zago A. Large courts, small justice! the inefficiency and the optimal structure of the italian justice sector. Omega Int J Manag Sci 2016;64:42–56. [20] Podinovski V. Selective convexity in DEA models. Eur J Oper Res 2005;161:552–63. [21] Portela M, Camanho A. Analysis of complementary methodologies for the estimation of school value-added. J Oper Res Soc 2010;61:1122–32. [22] Portela M, Camanho A, Borges D. Performance assessment of secondary schools: the snapshot of a country taken by dea. J Oper Res Soc 2012;63:1098–115. [23] Posner R. Is the ninth circuit too large? a statistical study of judicial quality. J Legal Stud 20 0 0;29(2):711–19. [24] Salerian J, Chan C. Restricting multiple-output multiple-input DEA models by disagregating the output-input vector. J Product Anal 2005;24:5–29. [25] Santos S, Amado C. On the need for reform of the portuguese judicial system - does data envelopment analysis assessment support it? Omega Int J Manag Sci 2014;47:1–16. [26] Schneider M. Judicial career incentives and court performance: an empirical study of the german labour courts of appeal. Eur J Law Econ 2005;20(2):127–44. [27] Thanassoulis E, Boussofiane A, Dyson RG. Exploring output quality targets in the provision of perinatal care in England using data envelopment analysis. Eur J Oper Res 1995;80:588–607. [28] Thompson R, Langemeier L, Lee C, Lee E, Thrall RM. The role of multiplier boundas in efficiency analysis with application to kansas farming. J Econ 1990;46:93–108. [29] Thompson RG, Lee E, Thrall RM. DEA/AR-efficiency of U.S. independent oil/gas producers over time. Comput Oper Res 1992;19(5):377–91. [30] Tsai P, Molinero CM. A variable returns to scale data envelopment analysis model for the joint determination of efficiencies with an example of the UK health service. Eur J Oper Res 2002;141(1):2138. [31] Voigt S. Determinants of judicial efficiency: a survey. Eur J Law Econ 2016;42(2):183–208.

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