1 Introduction

The constant and variable returns-to-scale (CRS and VRS) models of Charnes et al. (1978) and Banker et al. (1984) have long remained the two basic convex models in data envelopment analysis (DEA). The former assumes that the technology exhibits full proportionality between all inputs and outputs called “ray unboundness” in Banker et al. (1984). The latter does not incorporate this assumption. Consequently the CRS model generally discriminates better between the efficiency of decision making units (DMUs) than the VRS model, but its underlying assumption of full proportionality is often considered too demanding to be acceptable (Cooper et al. 2000; Thanassoulis 2001).

Recently a spectrum of new technologies was introduced based on the assumption of selective proportionality (Podinovski 2004). In such technologies only a subset of outputs is assumed proportional to a subset of inputs. In other words, such technologies exhibit CRS with respect to some inputs and outputs, and VRS with respect to the remaining ones. For this reason such technologies were called hybrid returns-to-scale (HRS) technologies, and DEA models based on them HRS models.

The CRS and VRS models are special extreme cases in the spectrum of HRS models. Thus, in the case of CRS the subsets of inputs and outputs assumed proportional include all inputs and outputs. In the case of VRS both sets are empty.

If the subset of outputs deemed proportional to the inputs is expanded, the discrimination of the HRS model improves. This is not, however, so if the subset of selected inputs is expanded. This point will be further illustrated below. The result of this observation is that neither the HRS nor CRS technology is a subset of the other. Consequently, neither the HRS nor CRS model can be said to discriminate better than the other.

Interestingly, some technologies may exhibit both the full and selective proportionality. Assume, for example, that the performance of university departments is assessed using the two inputs: teaching and research staff, and two outputs: students and publications. Note that it is common for universities to maintain a certain student to staff ratio. In addition, a certain number of publications is usually required from every member of staff. Based on these practices, the use of the CRS technology with respect to university departments may be justified.

Further, based on the same practice, it may also be reasonable to assume that the number of students is proportional to the teaching staff. For example, an increase of, say, 10% of the teaching staff should be sufficient to teach 10% more students. Therefore, this technology exhibits not only the full but also selective proportionality. The latter involves only teaching staff and students and excludes research staff and publications.

A similar property may be observed in many other technologies which exhibit CRS and where a certain range of activities is exclusively performed by a subset of inputs. In health services, units may specialise in the provision of treatment for a certain condition while also providing general care. If we split general doctors from specialists, then it might be reasonable to assume proportionality between the specialists and patients who need special treatment.

In all such cases we might use either the CRS or HRS models but either model would use only one part of available information about the underlying technology, that is either full or selective proportionality, while completely ignoring the other part. This of course would generally result in different efficiencies of the same DMUs in the two models.

A better approach would be to use a combination of both types of proportionality in the same model. In this paper we consider a technology which is based on two simultaneous assumptions of full and selective proportionality. More precisely, we complement the axioms of the CRS technology (Banker et al. 1984) by the axiom of selective proportionality (Podinovski 2004). The resulting C-HRS technology incorporates the CRS and HRS technologies as subsets, and the corresponding C-HRS DEA models provide a generally better discrimination than both CRS and HRS models.

We proceed as follows. In Sect. 2 we give a brief overview of the notion of selective proportionality and HRS technologies based on it. In Sect. 3 we obtain technology C-HRS that satisfies both assumptions of full and selective proportionality. In Sect. 4 we illustrate the use of C-HRS models by a computational example. In Sect. 5 we comment on the use of the C-HRS technology for testing scale efficiency in the HRS technology.

2 Selective proportionality

Below we outline the idea of selective proportionality and based on it hybrid returns-to-scale production technology. Its full development can be found in Podinovski (2004).

Let \( T \subset R_{ + }^{m + s}\) be a production technology with m inputs and s outputs. The set of all inputs is denoted I and the set of all outputs is denoted O. Each DMU in T is represented by the pair (X,Y) where X is the vector of inputs and Y the vector of outputs. Let J = {1, 2,…, n} be the set of observed DMUs. These are denoted (X j ,Y j), and their individual inputs and outputs are, respectively, \( X_{i}^{j}\) and \( Y_{r}^{j}\), where i ∈ I and r ∈ O. Each observed DMU is assumed to have at least one strictly positive input and output.

The HRS technology is an extension to the VRS technology in that it allows for proportional scaling of some inputs and outputs. Examples of such technologies were discussed in the Introduction. Central to the definition of the HRS technology is the notion of selective proportionality. It is defined separately in the two possible scenarios.

In the expansion scenario we assume that the outputs from the subset \( {\rm O}_{ + }^{P} \subseteq {\rm O}\) will increase in proportion to the increase of the inputs from the subset \( {\rm I}_{ + }^{P} \subseteq {\rm I}\). More precisely, by applying an expansion factor α > 1 to these inputs and outputs of any DMU (X,Y), and leaving the remaining inputs and outputs unchanged, we obtain the following DMU (X α ,Y α):

$$ X_{i}^{\alpha } = \left\{ \begin{gathered} \alpha X_{i} , {\text{ if }}i \in {\rm I}_{ + }^{P} , \hfill \\ X_{i} , {\text{ if }}i \in {\rm I}\backslash {\rm I}_{ + }^{P} , \hfill \\ \end{gathered} \right.\quad Y_{r}^{\alpha } = \left\{ \begin{gathered} \alpha Y_{r} , {\text{ if }}r \in {\rm O}_{ + }^{P} , \hfill \\ Y_{r} , {\text{ if }}r \in {\rm O}\backslash {\rm O}_{ + }^{P} . \hfill \\ \end{gathered} \right.$$
(1)

In the contraction scenario we assume that the outputs from the subset \( {\rm O}_{ - }^{P} \subseteq {\rm O}\) will decrease in proportion to the reduction of the inputs from the subset \( {\rm I}_{ - }^{P} \subseteq {\rm I}\). (These subsets are generally different from \( {\rm I}_{ + }^{P}\) and \( {\rm O}_{ + }^{P}\).) Since this is also likely to affect the remaining outputs from \( {\rm O}\backslash {\rm O}_{ - }^{P}\) but in an unspecified proportion, we have to assume that these will be reduced to zero (see Podinovski (2004) for a discussion of this assumption). Specifically, by applying a contraction factor α, such that 0 ≤ α < 1, to the selected subsets of inputs and outputs of DMU (X,Y) we obtain the following DMU (X α ,Y α):

$$ X_{i}^{\alpha } = \left\{ \begin{gathered} \alpha X_{i} , {\text{ if }}i \in {\rm I}_{ - }^{P} , \hfill \\ X_{i} , {\text{ if }}i \in {\rm I}\backslash {\rm I}_{ - }^{P} , \hfill \\ \end{gathered} \right. \quad Y_{r}^{\alpha } = \left\{ \begin{gathered} \alpha Y_{r} , {\text{ if }}r \in {\rm O}_{ - }^{P} , \hfill \\ 0 , {\text{ if }}r \in {\rm O}\backslash {\rm O}_{ - }^{P} . \hfill \\ \end{gathered} \right.$$
(2)

The HRS technology T HRS is defined as the intersection of all sets T that satisfy the following five axioms.Footnote 1

  1. (A1)

    Feasibility of observed data. (X j, Y j) ∈ T for any j ∈ J.

  2. (A2)

    Convexity. The set T is convex.

  3. (A3)

    Free disposability. (X,Y) ∈ T, Y ≥ Y′ ≥ 0 and X ≤ X′ implies (X′,Y′) ∈ T.

  4. (A4)

    Selective proportionality. Let (X, Y) ∈ T. For any α > 1, define unit (X α ,Y α) as in (1). For any α such that 0 ≤ α < 1, define (X α ,Y α) as in (2). Then (X α ,Y α) ∈ T.

  5. (A5)

    Closedness. The set T is closed.Footnote 2

For the constructive description of the HRS technology, define \( \bar{X}\) and \( \bar{Y},\) the m × n and s × n matrices whose columns are respectively the input and output vectors X j and Y j of the observed DMUs. Define the matrices \( \hat{X}^{ + },\) \( \hat{X}^{ - }\) and \( \hat{Y}^{ + }\) as follows (notably, we do not need \( \hat{Y}^{ - }\)). The matrices \( \hat{X}^{ + }\) and \( \hat{X}^{ - }\) are obtained from \( \bar{X}\) by changing to zero all rows \( i \in {\rm I}\backslash {\rm I}_{ + }^{P}\) and \( i \in {\rm I}\backslash {\rm I}_{ - }^{P},\) respectively. The matrix \( \hat{Y}^{ + }\) is obtained from \( \bar{Y}\) by changing to zero every row \( r \in {\rm O}\backslash {\rm O}_{ + }^{P}.\)

Theorem 1

(Podinovski 2004) Technology T HRS is the set of all nonnegative pairs (X, Y) such that

$$ X = \bar{X}\lambda + \hat{X}^{ + } \mu - \hat{X}^{ - } \nu + d$$
(3)
$$ Y = \bar{Y}\lambda + \hat{Y}^{ + } \mu - \bar{Y}\nu - e$$
(4)
$$ \sum\limits_{j = 1}^{n} {\lambda_{j} } = 1$$
(5)
$$ \lambda_{j} \ge v_{j} \quad {\text{for }} {\text{all}}\,j \in J$$
(6)
$$ \lambda ,\mu ,v \in R_{ + }^{n} ,\;d \in R_{ + }^{m} ,\;e \in R_{ + }^{s} .$$
(7)

(The use of the term \( \bar{Y}\nu\) in (4) may seem to contradict our definition (2) but it is nevertheless correct and is a side effect of the convexity axiom (A2), see Podinovski (2004) for detail.)

Depending on the subsets of inputs and outputs assumed proportional, different special cases of the HRS technology are obtained. These include the CRS and VRS technologies, and also technologies in which selective proportionality is assumed only in the expansion or contraction scenario.Footnote 3

Finally, we note the following important observation that follows from Theorem 3 in Podinovski (2004): if \( {\text{I}}_{ + }^{P} = {\text{I}},\) the HRS technology T HRS is a subset of the CRS technology T CRS. If \( {\text{I}}_{ + }^{P} \ne {\text{I}},\) the HRS technology T HRS is not a subset of T CRS. This observation is discussed in greater detail in the next section where it is used to motivate our further extension to the HRS model.

3 Combining CRS with selective proportionality

As noted in the introduction, some technologies may exhibit both the full and selective proportionality between the inputs and outputs. It may appear that if the full proportionality is assumed, no further benefits could be obtained from the additional assumption of selective proportionality. The following is a counterexample.

Suppose that unit A shown in Table 1 is an observed DMU. The hypothetical unit B is obtained by the full proportional expansion of A. Consequently, B ∈ T CRS , where T CRS is the CRS technology induced by unit A.

Table 1 Units generated by DMU A under different assumptions
Full size table

The hypothetical unit C is obtained by the selective proportional expansion of A defined by the sets \( {\rm I}_{ + }^{P}\) = {Input 1} and \( {\rm O}_{ + }^{P}\) = {Output 1}, using the expansion factor α = 2. Unit C is a member of the corresponding HRS technology T HRS induced by A.

It is easy to see that that \( B \notin T_{\text{HRS}}\) and \( C \notin T_{\text{CRS}}.\) This implies that neither technology T HRS nor T CRS is a subset of the other. Note that in our example \( {\text{I}}_{ + }^{P} \ne {\text{I}},\) and this is entirely consistent with our final observation in the preceding section. Further consider unit D in Table 1. This unit can be obtained either by the full proportional expansion of unit C or selective expansion of unit B. It is interesting that \( D \notin T_{\text{HRS}}\) and \( D \notin T_{\text{CRS}}.\) Unit D is only feasible in the technology induced by A if both assumptions of full and selective proportionality are valid simultaneously.

The above example with unit D shows that, if \( {\text{I}}_{ + }^{P} \ne {\text{I}},\) the simultaneous assumption of the full and selective proportionality induces hypothetical units beyond the scope of technologies T CRS and T HRS. This implies that a technology based on both assumptions is generally larger than the simple union of technologies T CRS or T HRS.

We shall refer to such a technology as the C-HRS technology T C-HRS. It is based on the set of axioms (A1)–(A5) and the additional axiom of full proportionality:

  1. (A6)

    Full proportionality. If (X,Y) ∈ T, then (αX,αY) ∈ T for any α ≥ 0.

Technology T C-HRS can be viewed as the minimal technology, in the sense of the minimal extrapolation principle, which expands T HRS by allowing full proportional scaling of the units in T HRS. In other words, technology T C-HRS is the cone technology induced by T HRS. This is similar to the fact that technology T CRS is the cone extension of the VRS technology T VRS.

The following theorem gives a constructive definition of technology T C-HRS.

Theorem 2

Technology T C-HRS is the set of all nonnegative pairs (X, Y) that satisfy conditions (3), (4), (6) and (7).

The proof follows from two separate statements: T C-HRS satisfies all axioms (A1)–(A6), and \( T_{\text{C-HRS}} \subseteq T,\) where T is any set satisfying (A1)–(A6). Both statements can be proved in the same way as Lemmas 1 and 2 in Podinovski (2004) and are not given here.

According to Theorem 2, the technology T C-HRS is described by exactly the same linear conditions as technology T HRS except for the normalising equality (5) which is removed. Note that the CRS technology T CRS as the cone extension of the VRS technology T VRS is also obtained by dropping condition (5).

It is worth emphasising that technology T C-HRS defined by Theorem 2 is only interesting if \( {\text{I}}_{ + }^{P} \ne {\text{I}}.\) Otherwise the assumption of selective proportionality becomes redundant because of the assumed full proportionality, and technology T C-HRS is the same as the standard technology T CRS. Examples of technologies where \( {\text{I}}_{ + }^{P} \ne {\text{I}}\) can be found in the Introduction.

4 Example

The following example is based on a slightly modified data set from Podinovski (2004). It demonstrates that the combination of full and selective proportionality assumptions may lead to a significant improvement of the discrimination of the resulting model.

Table 2 represents five units whose radial efficiency is assessed using two inputs and two outputs. Assume that Output 1 is proportional to Input 1 in both the expansion and contraction scenarios, that is \( {\rm I}_{ + }^{P} = {\rm I}_{ - }^{P}\) = {Input 1} and \( {\rm O}_{ + }^{P} = {\rm O}_{ - }^{P}\) = {Output 1}. This leads to the construction of the following matrices as defined before the formulation of Theorem 1.

$$ \bar{X} = \left( {\begin{array}{*{20}c} 8 \\ {20} \\ \end{array} \quad \begin{array}{*{20}c} {12} \\ {25} \\ \end{array} \quad \begin{array}{*{20}c} {20} \\ {25} \\ \end{array} \quad \begin{array}{*{20}c} 7 \\ {40} \\ \end{array} \quad \begin{array}{*{20}c} {16} \\ {24} \\ \end{array} } \right),\quad \hat{X}_{ + } = \hat{X}_{ - } = \left( {\begin{array}{*{20}c} 8 \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} {12} \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} {20} \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} 7 \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} {16} \\ 0 \\ \end{array} } \right),$$
$$ \bar{Y} = \left( {\begin{array}{*{20}c} {11} \\ 8 \\ \end{array} \quad \begin{array}{*{20}c} {15} \\ 4 \\ \end{array} \quad \begin{array}{*{20}c} {12} \\ 5 \\ \end{array} \quad \begin{array}{*{20}c} 9 \\ 8 \\ \end{array} \quad \begin{array}{*{20}c} 5 \\ 6 \\ \end{array} } \right)\quad \hat{Y}_{ + } = \left( {\begin{array}{*{20}c} {11} \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} {15} \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} {12} \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} 9 \\ 0 \\ \end{array} \quad \begin{array}{*{20}c} 5 \\ 0 \\ \end{array} } \right),$$
Table 2 The data set
Full size table

Based on Theorems 1 and 2, we can now construct the models for the assessment of radial efficiency of DMUs in both technologies T HRS and T C-HRS, and in both scenarios involving input minimisation and output maximisation. For example, the output radial efficiency of DMU 3 in technology T C-HRS, which will be of the most interest to us in the discussion below, is obtained by solving the following linear program and subsequently inverting the optimal value of its objective function.

$$ \begin{aligned} {\text{Max}}\quad \theta \hfill \\ {\text{Subject}}\;{\text{to}}\quad \,\,\,8\lambda_{1} + 12\lambda_{2} + 20\lambda_{3} + 7\lambda_{4} + 16\lambda_{5} \hfill \\ + 8\mu_{1} + 12\mu_{2} + 20\mu_{3} + 7\mu_{4} + 16\mu_{5} \hfill \\ -8{{\upnu}}_{1} - 12{{\upnu}}_{2} - 20{{\upnu}}_{3} - 7{{\upnu}}_{4} - 16{{\upnu}}_{5} \le 20 \hfill \\ 20{{\uplambda}}_{ 1} + 2 5 {{\uplambda}}_{ 2} + 2 5 {{\uplambda}}_{ 3} + 40{{\uplambda}}_{ 4} + 2 4 {{\uplambda}}_{ 5} \le 25 \hfill \\ 11 {{\uplambda}}_{ 1} + 1 5 {{\uplambda}}_{ 2} + 1 2 {{\uplambda}}_{ 3} + 9 {{\uplambda}}_{ 4} + 5 {{\uplambda}}_{ 5} \hfill \\ + 11\mu_{1} + 15\mu_{2} + 12\mu_{3} + 9\mu_{4} + 5\mu_{5} \hfill \\-11{{\upnu}}_{1} - 15{{\upnu}}_{2} - 12{{\upnu}}_{3} - 9{{\upnu}}_{4} - 5{{\upnu}}_{5} \ge 12\theta \hfill \\ 8\lambda_{1} + 4\lambda_{2} + 5\lambda_{3} + 8\lambda_{4} + 6\lambda_{5} \hfill \\ -8{{\upnu}}_{1} - 4{{\upnu}}_{2} - 5{{\upnu}}_{3} - 8{{\upnu}}_{4} - 6{{\upnu}}_{5} \ge 5\theta \hfill \\ \lambda_{j} - \nu_{j} \ge 0\quad {\text{for}}\;j = 1, \ldots ,5 \hfill \\ \lambda_{j} , \mu_{j} , \nu_{j} \ge 0\quad {\text{for}}\;j = 1, \ldots ,5 \hfill \\ \theta \;{\text{sign}}\;{\text{free}}. \hfill \\ \end{aligned}$$

Table 3 shows the input radial efficiency of all units in the four technologies: VRS, CRS, HRS and C-HRS. Table 4 shows the output radial efficiencies in the same technologies.

Table 3 Input efficiency in four different models
Full size table
Table 4 Output efficiency in four different models
Full size table

The following observations can now be made. Firstly, as already discussed, if \( {\text{I}}_{ + }^{P} \ne {\text{I}}\) then neither the HRS nor CRS technology is a subset of the other. This can be seen by referring to Unit 3 whose efficiency in the HRS technology is lower than in the CRS technology, in both input and output orientations. At the same time we see the opposite in the case of Unit 5 whose efficiency is lower in the CRS technology.

Secondly, the efficiency of Unit 3 in technology C-HRS is 0.5 which is significantly lower than in either technology T HRS or T CRS, in both Tables 3 and 4.

Thirdly, the input and output radial efficiencies of each DMU are identical not only in technology T CRS, which is a well-known fact, but also in technology T C-HRS. This is not surprising because both are cone technologies. Of course, the radial projections of inefficient units on the boundary depend on the orientation of such models.

The effect of the combined assumption of full and selective proportionality is not always as pronounced as in the case of DMU 3 above. The other four units in our example were less interesting, as their efficiency in technology T C-HRS was equal to the minimum of their efficiencies in technologies T CRS and T HRS.

5 Technology T C-HRS and analysis of scale efficiency

Technology T C-HRS can be viewed as the cone technology induced by the convex technology T HRS. Following the general framework of Färe et al. (1983), we can define the scale efficiency of any DMU0 operating in technology T HRS as the ratio E C-HRS/E HRS of its efficiencies in the C-HRS and HRS technologies, respectively.

If DMU0 exhibits scale inefficiency, its nature can be investigated by the use of the corresponding non-increasing and non-decreasing returns-to-scale (NIRS and NDRS) technologies induced by T HRS. If the radial efficiency of DMU0 in the NIRS technology is greater than in the CRS technology, DMU0 exhibits decreasing returns to scale, otherwise DMU0 exhibits increasing returns to scale. A modification of this method, which makes use of both NIRS and NDRS technologies, was proposed by Kerstens and Vanden Eeckaut (1999).

In order to define the NIRS and NDRS reference technologies induced by T HRS, we replace the last axiom (A6) by one of the following reduced axioms, respectively:

  1. (A7)

    Full proportional contraction. If (X,Y) ∈ T, then (αX,αY) ∈ T for any 0 ≤ α ≤ 1.

  2. (A8)

    Full proportional expansion. If (X,Y) ∈ T, then (αX,αY) ∈ T for any α ≥ 1.

The NIRS technology T NI-HRS is the intersection of all technologies that satisfy axioms (A1)–(A5) and (A7). Similarly, the NDRS technology T ND-HRS is intersection of all technologies that satisfy axioms (A1)–(A5) and (A8).

Theorem 3

Technology T NI-HRS is the set of all nonnegative pairs (X,Y) that satisfy conditions (3) to (7), where the equality sign in equation (5) is changed to “≤”. Similarly, for the technology T ND-HRS , the equality sign in (5) is changed to “≥”.

The proof of Theorem 3 is similar to the proofs of Theorems 1 and 2 and is not given.

Using the reference technologies T C-HRS, T NI-HRS and T ND-HRS in the general framework of Färe et al. (1983) and its later modifications, the analysis of scale efficiency and the nature of returns to scale of units in T HRS becomes straightforward.

6 Conclusion

This paper takes further the recent development of production technologies based on the notion of selective proportionality between inputs and outputs. Such technologies are referred to as hybrid returns-to-scale technologies.

In this new development, we first observed that some technologies might exhibit both types of proportionality: full, as in CRS, and selective, as in HRS technologies. Examples of such technologies were discussed and included the functioning of university departments with the teaching and research staff on the input side, and students and publications on the output side.

Such examples motivated this paper to explore the advantages of combining both types of proportionality in one model. Although intuitively the assumption of selective proportionality appears weaker than the assumption of full proportionality, there is a special case where this is not so. More specifically, the HRS technology is not a subset of the CRS technology if at least one input is excluded from the assumption of selective proportionality in the expansion scenario. The example with university departments belongs to this special case.

Based on the simultaneous assumptions of full and selective proportionality, we constructed a new C-HRS technology and DEA models based on it. In the special case that motivated our development in the first place, that is when some inputs are excluded from selective proportionality, the new C-HRS technology is larger than the simple union of CRS and HRS technologies. Outside this special case the C-HRS technology coincides with the standard CRS technology and is not interesting.

To see the difference to the efficiency scores that the use of the C-HRS technology might bring to DEA compared to the CRS and HRS technologies in the above special case, we considered a numerical example. The input radial efficiency of one of the units, being equal to about 80% in the HRS and CRS models, dropped to 50% in the C-HRS model. Note that this drop was caused by simply incorporating two assumptions of proportionality in one model.

Finally, we observed that the new C-HRS technology could be regarded as the cone extension of the HRS technology. We further developed the two corresponding non-increasing and non-decreasing returns-to-scale technologies. These reference technologies can be used to investigate the type of returns to scale in the HRS technology using standard procedures.