6.1 INTRODUCTION

This chapter¹ compares radial and non‐radial models, discussed in Chapters 4 and 5, using nine desirable properties to measure the level of operational efficiency (OE). As discussed in those two chapters, RM(v) and RM(c) belong to the Debreu–Farrell measures (e.g., Farrell, 1957; Farrell and Fieldhouse, 1962) and non‐radial models belong to the Pareto–Koopmans measures (e.g., Koopmans, 1951; Russell, 1985) in OE‐based performance evaluation. As discussed in Chapter 5, Charnes et al. (1985) first proposed an additive model as a non‐radial model. Then, Charnes et al. (1982, 1983) linked the additive model to a multiplicative model by changing a data set by natural logarithm. Cooper et al. (1999) also proposed a non‐radial DEA model, referred to as “a range‐adjusted measure (RAM),” that was an extension of the additive model (Cooper et al., 2000, 2001). Aida et al. (1998) extended RAM by reorganizing it as a radial measure, so‐called “slack‐adjusted radial measure (SARM).” Meanwhile, Tone (2001) proposed another type of non‐radial model referred to as the “slacks‐based measure (SBM).” In a similar manner, Pastor et al. (1999) proposed the “enhanced Russell graph measure (ERGM),” as a non‐radial measure, that incorporated the analytical feature of the Russell measure into a framework of SBM. As discussed in Chapter 5, this chapter considers SBM = ERGM because both have the same analytical structure. In such a research trend for computational tractability, Bardhan et al. (1996b) first formulated an approximation model for the Russell measure by linear programming. The measure provided an analytical linkage between the radial and non‐radial models so that it could eliminate a need for a separate treatment of input‐oriented and output‐oriented efficiency measures in the Russell measure².

Acknowledging the contributions of these previous studies as summarized in Chapters 4 and 5, we have been long wondering which model should be used for DEA‐based performance analyses. Can we apply any DEA model to any performance evaluation? The inquiry indicates a research necessity to investigate nine criteria, based upon which we can determine the selection issue of an appropriate DEA model. Of course, such desirable criteria, discussed in this chapter, may not cover all necessities for a conventional use of DEA, but they are useful in examining the applicability of each model.

Here, this chapter clearly acknowledges the existence of other previous research efforts (e.g., Cooper et al., 1999) that have investigated such desirable properties and have compared several DEA models. However, these previous comparisons covered part of important radial and non‐radial models from a partial perspective of OE. Furthermore, these previous studies did not provide mathematical proofs on the desirable criteria in a unified manner, as found in this chapter. Therefore, following the previous contribution of Sueyoshi and Sekitani (2009), this chapter analytically investigates both what criteria (mathematical properties) are important for the OE measurement and how to select an appropriate DEA model under different OE criteria. All the criteria need to be mathematically proved by a unified analytical synthesis within a conventional framework of DEA, but not environmental assessment, which is a main research objective of this book. Such a new research extension will be found in chapters of Section II.

At the end of this section, it is necessary for us to mention that the nine desirable properties discussed here are mathematically important and analytically useful from the perspective of a conventional use of DEA. However, we clearly understand that each DEA model should have not only intuitive and visual capabilities to explain its mathematical structure but also fundamental rationales on why they are useful as a methodology for performance analysis. For example, dual variables obtained from each DEA model should be positive in these signs. Such is a general rule of DEA. However, it has several exceptions (e.g., congestion measurements in Chapters 9, 21, 22 and 23) in extending DEA for performance analysis and environmental assessment. Furthermore, in the other case where some dual variables are zero in these signs, the result indicates that production factors, corresponding to the dual variables, are not fully utilized for DEA assessment. It may be true that such a concern does not produce any major difference if we are interested in only efficiency assessment. However, it can be easily imagined that the practicality of DEA is not limited to such efficiency measurement, rather having more wide applicability such as innovation identification on production technology and eco‐technology to reduce an amount of various industrial pollutions. Thus, the performance of each radial or non‐radial model should be discussed based on their applicability and practicality, although this chapter does not discuss all of them here, rather focusing on only their analytical structures and features from optimization. Thus, the purpose of this chapter is to provide us with an analytical basis on a conventional framework of DEA‐based production analysis.

The remainder of this chapter is organized as follows: Section 6.2 summarizes desirable properties obtained from previous DEA studies. Section 6.3 reviews radial and non‐radial models from the perspective of OE criteria. Section 6.4 summarizes previous research efforts based upon the criteria on desirable properties. Section 6.5 proposes a standard model for DEA that covers the analytical features of all radial and non‐radial models. The standard model is used for mathematical proofs for examining whether each model satisfies or violates each criterion. Section 6.6 examines theoretically seven DEA models. Section 6.7 summarizes this chapter.

6.2 CRITERIA FOR OE

The OE measurement has the following desirable properties on radial and non‐radial models within a conventional framework of DEA:

1. Homogeneity (Banker et al., 1984; Russell, 1985; Dmitruk and Koshevoy, 1991; Blackorby and Russell, 1999): An output‐based OE measure should be homogeneous of degree one in desirable outputs, while an input‐based OE measure should be homogeneous of degree minus one in the inputs. For example, if we double all the inputs, then the input‐based OE measure should be cut in half.
2. Strict Monotonicity (Banker et al., 1984; Russell, 1985; Dmitruk and Koshevoy, 1991; Blackorby and Russell, 1999; Cooper et al., 1999): An OE measure should be non‐decreasing in desirable outputs and non‐increasing in inputs, along with an efficient desirable output (input) vector.
3. Efficiency Requirement (Russell, 1985; Banker et al., 1984; Cooper et al., 1999): An OE measure should be between zero and one, where “zero” implies full inefficiency and “one” implies full efficiency.
4. Unique Projection for Efficiency Comparison (Russell, 1985; Banker et al., 1984): An OE measure compares each observed input (or desirable output) vector with an efficient input (desirable output) one. This property³ was carefully discussed by Russell (1985, p. 110). The uniqueness of vector comparison is so desirable that this chapter adds the uniqueness as a desirable property. For example, a radial OE measure minimizes an input amount that can be shrunk along a projection ray, while holding the desirable output quantities constant. Meanwhile, the non‐radial OE measures do not have such a ray expansion, so that these measures minimize the total amount of slacks for determining a level of OE. Although they are different each other concerning how to adjust the observed amounts of production factors to their optimal amounts, both radial and non‐radial measures may suffer from a difficulty of multiple projection vectors and slacks for efficiency comparison. See Chapter 7 for a discussion on how to handle this type of problem related to DEA.
5. Aggregation (Pastor et al., 1999; Blackorby and Russell, 1999; Cooper et al., 2007a): A jointly measured OE among all inputs and all desirable outputs is attractive for performance analysis. The property indicates that the aggregation of inputs and desirable outputs should not influence the OE measurement of all DMUs.
6. Unit Invariance (Cooper et al., 1999; Lovell and Pastor, 1995): The unit of inputs and outputs should not influence an OE measure.
7. Invariance to Alternate Optima (Cooper et al., 1999): An occurrence of multiple solutions on DEA should not influence an OE measure.
8. Translation Invariance (Ali and Seiford, 1990; Lovell and Pastor, 1995; Pastor, 1996; Cooper et al., 1999; Pastor and Ruiz, 2007): An OE measure should be not influenced even if inputs and/or desirable outputs are shifted toward the same direction by adding or subtracting a specific real number. This property makes it possible to handle zero and negative values in a data set. See Chapters 26 and 27.
9. Frontier Shift Measurability (Färe et al., 1994b): It is desirable that an OE measure can measure a frontier shift among different time periods (or different locations). For example, DEA can link the radial‐based OE measures to the Malmquist index. The combination measures a frontier shift among different periods (Bjurek, 1996; Lambert, 1999). Consequently, the OE measure can analyze not only cross‐sections but also time‐series and panel data sets. See Sueyoshi and Aoki (2001) where the Malmquist index was combined with DEA window analysis, referred to as “Malmquist window analysis,” to avoid a retreat and/or a cross‐over among efficiency frontiers in different periods. See Chapter 19 that discusses a use of Malmquist index for environmental assessment in a time horizon.

Here, it is important to add the following two concerns on desirable properties. One of these two concerns is that the property of “unit invariance” will be discussed from an analytical perspective of DEA models in this chapter, but such a mathematical exploration is different from a computational aspect to solve these models. In short, there is a difference between analytical and computational aspects on DEA solutions. The other concern is that the property of “translation invariance” will be fully utilized in Chapter 26 that discusses how to handle an occurrence of many zeros in production factors of energy sectors. This property is very important in applying DEA for energy and environmental protection by non‐radial measurement. However, the scope of the property is limited only within the non‐radial measurement, not being extendable to the radial measurement. See Chapter 27 that discusses how to handle an occurrence of zero and negative values within the radial measurement for environmental assessment.

6.3 SUPPLEMENTARY DISCUSSION

To initiate our description on desirable OE properties, this chapter first needs to describe supplementary comments on additive model (AM) and slack‐adjusted radial measure (SARM).

6.4 PREVIOUS STUDIES ON DESIRABLE PROPERTIES

Following Sueyoshi and Sekitani (2009), Table 6.1 summarizes the relationship between seven DEA models and nine desirable properties in terms of the OE measurement. In the table, the symbol “○” indicates that each model satisfies a desirable property specified in each cell of the table. The symbol “×” indicates the opposite case.

Table 6.1 also lists previous research on the examination on OE properties at the bottom of each cell. If we cannot identify the previous research, there is no specification on previous research source in the cell. In such cases, this chapter depends upon the effort of Sueyoshi and Sekitani (2009). Based upon our best knowledge, Table 6.1 covers most of the previous studies related to desirable OE properties. Of course, this chapter knows that there may be other important studies on OE‐based desirable properties in previous literature. However, to avoid a descriptive duplication, we select a few example studies in each cell of Table 6.1.

Table 6.1 has two theoretical inconsistencies between previous research and this chapter. One of the two inconsistencies is that, as summarized in Table 6.1, RM(v) and RM(c) are unit invariant. Meanwhile, the research of Lovell and Pastor (1995) have discussed that they are unit invariant with respect to the radial measures, but did not pay attention to the existence of slack components. Furthermore, in terms of the translation invariance, their study (Lovell and Pastor, 1995) has discussed that these models have the property of partial invariance (i.e., invariance with respect to inputs or desirable outputs, but not both, depending upon model orientation). A difference between this chapter and their work is that we are interested in whether the OE measure is influenced by a unit change. Meanwhile, in their study (Lovell and Pastor, 1995), all the components of an optimal solution are required to be the same under the property of unit invariance. Partial invariance is attained if part of these optimal components is not influenced by the unit change. From their perspective, the claim of this chapter is looking for partial unit invariance in the OE measure. The other inconsistency is that, in this chapter, the additive model is neither unit invariant nor translation invariant. However, Lovell and Pastor (1995) have claimed that a normalized weighted additive model satisfies the two properties. Their normalized (weighted) additive model corresponds to RAM in this chapter. As indicated in Table 6.1, RAM is unit invariant and translation invariant. Thus, our claim is consistent with theirs. Chapter 16 extends RAM for environmental assessment because it has the property of translation invariance. See Chapter 26, as well.

In reviewing the previous studies, the following three concerns are important in examining strict monotonicity. First, the strict monotonicity has been often discussed as a desirable OE property in the literature on production economics. See, for example, Dmitruk and Koshevoy (1991). Unfortunately, most of the previous discussions assume a single desirable output in the OE measurement. In contrast, this chapter examines the property of strict monotonicity in a framework of multiple desirable outputs and inputs. Our discussion on the desirable property does not have such an assumption on a single desirable output. Second, RAM’s strict monotonicity examined in this chapter is different from Cooper et al. (1999, p. 17). Later, this chapter will explain why RAM violates the property of strict monotonicity. Finally, in discussing the strict monotonicity, previous studies did not provide a unified mathematical definition on the property. For example, the numerator in SBM is decreasing in an input vector and the denominator is increasing in a desirable output vector, so that the ratio is monotonically decreasing in the two production factors. Meanwhile, the Russell measure, as discussed in Färe et al. (1994a), is decreasing in the input vector and the desirable output vector. See Bardhan et al. (1996b) and Cooper et al. (2006). Thus, it is necessary for us to examine the property of strict monotonicity in a unified analytical synthesis with multiple inputs and multiple desirable outputs.

6.5 STANDARD FORMULATION FOR RADIAL AND NON‐RADIAL MODELS

To prove the claims summarized in Table 6.1, this chapter considers a standard DEA model that serves as a mathematical basis for our examination. Given unknown matrixes (X > 0 and G > 0) and observed column vectors (X_k > 0 and G_k > 0) regarding a specific k‐th DMU to be examined, this chapter formulates the standard model by the following vector and matrix expressions for an observed data set (A) regarding inputs and desirable outputs:

(6.1)

The important feature of the standard model, or Model (6.1), is that it is structured by a vector of λ which expresses a set of intensity variables on production factors. R_h stands for the right hand side to express a vector related to inputs and desirable outputs on the k‐th DMU. f is a function for production activities. To condense our description, this chapter uses the vector and matrix expressions, not the scalar expression found in most of the other chapters.

Let Q(X_k, G_k, X, G, A, R_h) be the optimal objective value that is obtained from Model (6.1). The optimal value indicates a degree of OE measure for all DEA models except the additive model and RAM. In the two models, the optimal objective value indicates the level of inefficiency. The OE measure is determined by subtracting the degree of inefficiency from unity. The unique feature on RAM will be later incorporated into all non‐radial models used for environmental assessment.

The following three models indicate, as examples, how Model (6.1) can express each of DEA models:

RM under Variable RTS (input‐oriented):

Reformulation of RM(v) produces the following formulation:

(6.2)

Here, e is a row vector whose components are all 1. θ, an efficiency measure for OE, which is usually incorporated in the objective function of the input‐oriented RM(v), is replaced by Q(X_k, G_k, X, G, A, R_h) in the RM(v). Consequently, the standard model (6.1) can express Model (6.2) as follows:

(6.3)

Model (6.2) omits slacks, associated with a non‐Archimedean small number (ε_n), that are originally incorporated in the RM(v). The superscript (Tr) stands for a vector transpose. For our descriptive convenience, this chapter drops slacks from Model (6.2) so that Model (6.1) can express Model (6.2) in a condensed form.

Additive Model:

This chapter can reformulate the additive model as follows:

(6.4)

Hence, the standard model (6.1) can express the additive model as follows:

(6.5)

SBM (= ERGM):

The chapter reformulates SBM (= ERGM) as follows:

(6.6)

Hence, SBM becomes the following standard model:

(6.7)

The above reformulation process, discussed for the three models: RM(v), the additive model and SBM, is applicable to other DEA models. See Table 6.2 for a summary on how the standard model (6.1) can express all the DEA models examined in this chapter.

Model	f(Xk, Gk, X, G, λ)	A	Rh
RM(v): input oriented
Additive model
RAM
SARM
RM (c): input‐oriented
SBM and ERGM
Russell measure

The following three concerns are important in discussing the standard model (6.1) and Table 6.2. First, a methodological comparison on DEA may be dated back to Ahn et al. (1988). Their study combined the two RMs with conventional statistical tests and analyses. Then, they characterized these measures by comparing them with statistical tests. Second, a general DEA model was first proposed by Yu et al. (1996a, 1996b) and Wei and Yu (1997). Their studies provided a unified framework for four types of RMs under different conditions on RTS. Their general model incorporated a case (called “K‐cone” or “predilection cone”) where it restricted multipliers, measured by dual variables, by using prior information. As a result of such multiplier restriction, different DEA models had a restricted projection surface on an efficiency frontier. In contrast, the standard model (6.1) proposed in this chapter covers not only radial but also non‐radial models. In Model (6.1), six DEA models, except the Russell measure, are structurally simplified and unified into a standard linear programming model. This chapter uses the proposed standard model (6.1) only for mathematical proofs on desirable OE properties. The Russell measure is formulated by nonlinear programming. Finally, assuming that SARM has an optimal solution, Proposition 6.1 implies that it has the following objective function:

(6.8)

Thus, Table 2 has Equation (6.8) as the objective function of SARM.

Here, it is important to note that radial approaches discussed for DEA environmental assessment are based upon SARM and non‐radial approach for the assessment are based upon RAM, along with some modifications (e.g., incorporation of undesirable outputs). The two models used for conventional DEA will become an analytical basis for environmental assessment in Section II. The SARM originates from RAM. So, it is possible to consider that all the models used in chapters of Section II are originated from RAM.

We have four rationales concerning why we pay attention to RAM. First, it has the property of “translation invariance” so that it can handle an occurrence of zero and/or negative values in a data set without any difficulty. Chapter 26 describes how a non‐radial model, originated from RAM, can utilize the property of translation invariance for dealing with an occurrence of zero and/or negative values in environmental assessment. Chapter 27 discusses a radial model, originated from SARM, can handle zero and/or negative values for environmental assessment. Second, radial and non‐radial models, originated from RAM and SARM, can always produce positive multipliers (dual variables) in their signs so that all production factors will be fully utilized in environmental assessment. Third, the proposed models can easily incorporate undesirable outputs and then unify between desirable and undesirable outputs in their formulations. Finally, as documented in Chapters 20, 21 and 23, these models have structural flexibility in the manner that they can measure various scale‐related measures such as RTS and damages to scale (DTS). For these measures, some multipliers are set to be unrestricted in their signs.

6.6 DESIRABLE PROPERTIES FOR DEA MODELS

6.6.1 Aggregation

As mentioned by Blackorby and Russell (1999, p. 10), the previous definition of aggregation was considerably “general and too abstract.” Therefore, this chapter follows the definition on aggregation referred to as “aggregation index (AI)” in their study (Blackorby and Russell, 1999, p. 11). The AI axiom implies that “all DMUs are efficient if and only if an aggregated DMU is efficient.” This chapter redefines the AI axiom as follows: Let us consider the aggregation of input and desirable output vectors as and . Then, this chapter can specify input and desirable output matrixes as and after incorporating the aggregated input and desirable output vectors into these original matrixes. Here, the subscript “TA” stands for total aggregation. Suppose that are generated by (X′, G ') where stands for the right hand side of an aggregated matrix, or . Here, let and , then the AI axiom has the following definition:

(6.9)

Here, γ indicates a value of the functional form (f) when a DMU is efficient in terms of OE. For example, stands for the status of efficiency in the additive model and RAM. In the other DEA models, indicates the status of efficiency.

DMU	Input	Desirable output	RM(v)	Efficiency status
{A}	1	1	1	Efficient
{B}	2	3	1	Efficient
{C}	3	2	0.5	Inefficient
{TA}	6	6	1	Efficient

As listed in Example 6.1, two DMUs {A, B} are efficient, while DMU {C} is inefficient by RM(v). DMU {TA} (= DMU{A} + DMU{B} + DMU{C}) is efficient by the RM(v). The AI axiom in Equation (6.9) requests that if DMU {TA} is efficient, then all DMUs should be efficient in each RM(v)’s individual measurement. However, DMU{C} is inefficient. The result of Example 6.1 illustrates that the RM(v) does not satisfy the AI axiom. Since the status of inefficiency measured by the RM(v) is equivalent to that of other DEA models such as the additive model, RAM and SARM, this chapter confirms that these DEA models violate the property of aggregation, as well, by the following example:

DMU	Input	Desirable output 1	Desirable output 2	Russell measure	Efficiency status
{A}	1	1	2.5	1	Efficient
{B}	1	2	2	1	Efficient
{C}	1	3	0.5	1	Efficient
{TA}	3	6	5	0.944	Inefficient

Now, let us consider the case of Russell measure. As listed in Example 6.2, all three DMUs {A, B, C} are efficient by the Russell measure. However, DMU {TA} is inefficient by the Russell measure. The AI axiom indicates that if all DMUs are efficient, then the aggregated DMU {TA} (= DMU{A} + DMU{B} + DMU{C}) should be efficient. Thus, the Russell measure violates the property of aggregation. Next, SBM (= ERGM) measures the OE on DMU {TA} as 0.909 and RM(c) determines the level of OE as 0.933. Thus, the result (i.e., the Russell measure violates the property of aggregation) can be also applied to the RM(c) and SBM (= ERGM)⁴.

6.6.2 Frontier Shift Measurability

Using sensitivity analysis directly linked to the frontier shift measurability, this chapter examines a case where DEA shifts inputs and desirable outputs regarding the k‐th DMU from the current production level to another one. Let us consider the matrix (A) that consists of X and G. The data matrix is written as A(X, G). Similarly, R_h is expressed by R_h(X_k, G_k) as the right‐hand side of DEA models.

When R_h(X_k, G_k) is changed to , then the standard model (6.1) becomes

(6.10)

Here, Δ_x indicates the amount of inputs and Δ_g indicates that of desirable outputs, both are changed from these current production levels.

Model (6.10) cannot guarantee that the new production level () always belongs to an original production possibility set, shaped by the original production structure of (X, G). Consequently, Model (6.10) may produce an infeasible solution. The following proposition describes analytical relationships among DEA models due to the sensitivity analysis:

The relationships between seven DEA models due to sensitivity analysis are visually summarized in Figure 6.1. See Cooper et al. (2005) for a detailed description on DEA‐based sensitivity analysis. Figure 6.1 visually indicates that, if input‐oriented RM(v) violates the property of frontier shift measurability, then the three models (i.e., an additive model, SARM and RAM) violate the property. Similarly, if SBM violates the property, then Russell measure and SBM (= ERGM) violate the property. As indicated by Cooper et al. (1999), only input‐oriented RM(c) can satisfy the property. The property of frontier shift measurability is applicable to output‐oriented RM(c) because the radial model does not have the side constraint (i.e., the sum of λ_j for j = 1,…, n) in its formulation.

The following example illustrates why an optimal solution does not exist in sensitivity analysis of input‐oriented RM(v) and SBM:

DMU	Input	Desirable output
{A}	1	2
{B}	2	3
{C}	2 (3/2)	1/10 (4)

Let us consider the sensitivity analysis of DMU {C} where it reduces the amount of input from 2 to 3/2 and increases the amount of desirable output from 1/10 to 4.

RM(v):

If the sensitivity analysis of input‐oriented RM(v) is applied to DMU {C}, then the constraints on λ in Model (6.2) become , and . Since the radial model has λ_B + (39/10)λ_C. From the non‐negativity of λ_A, λ_B and λ_C, we have Hence, we find an infeasible solution in the input‐oriented RM(v) if the data is shifted from the current production level of DMU {C} to the other level as listed in Example 6.3.

SBM:

If we apply SBM to DMU{C}, then the constraints regarding λ become , , and . Since , the equation becomes . From the non‐negativity of λ_A, λ_B and λ_C, we have . Hence, an infeasible solution occurs in SBM if the data changes the current production level of DMU {C} to the other level as listed in Example 6.3.

Here, it is important to note the following concern on the property of frontier shift measurability. That is, an important feature of RM(c) is that it does not have in the formulation so that it can maintain the property. The radial formulation for environmental assessment discussed in Chapter 19, originated from SARM, drops the side constraint so that it can measure the frontier shift in a time horizon. Thus, the radial model for environmental assessment discussed in Chapter 19 does not have in the formulation in order to measure a shift of Malmquist index and its related subcomponents in a time horizon.

6.6.3 Invariance to Alternate Optima

The remaining five properties (homogeneity, strict monotonicity, efficiency requirement, unit invariance and translation invariance) depend upon the property of invariance to alternate optima. Therefore, this chapter starts examining the property of invariance to alternate optima by the following proposition:

As documented in Table 6.2, the additive model measures the degree of OE by

(6.11)

where λ* is an optimal solution of Model (6.4). The additive model does not use the optimal objective value of Model (6.5) as the degree of OE measure. Hence, let Equation (6.11) be Q(X_k, G_k, X, G, A(X, G), R_h(X_k, G_k)) for the additive model. Then, it is sufficient to show that Equation (6.11) suffers from an occurrence of multiple optimal solutions in Model (6.4).

DMU	Input	Desirable output
{A}	1	2
{B}	2	3
{C}	2	1/10

The optimal objective value of the additive model (6.4), applied to DMU {C}, becomes –2.9. The optimal solution needs to satisfy .

Hence, Equation (6.11) becomes

(6.12)

The above inequalities indicate that the additive model applied to DMU {C} suffers from an occurrence of multiple OE measures. The upper and lower bounds are specified by Equation (6.12).

6.6.6 Homogeneity

The property of homogeneity should be discussed only on an inefficient DMU (so, ). The status change of an efficient DMU due to μ does not have any theoretical implication because the change reorganizes the structure of an efficiency frontier.

The non‐radial models violate the property of homogeneity. For instance, this chapter can confirm the violation of RAM by Example 6.5.

DMU	Input	Desirable output
{A}	1	2
{B}	μ	1/2
{C}	2	1/2

In Example 6.5, μ is an arbitrary positive number. If , then DMU {A} is fully efficient. If , then DMU {C} is fully inefficient. If , then DMU {B} is fully inefficient. The objective of RAM becomes 1–0.5 μ on DMU {B} if 2 1 and it is 0 if 2. When μ = 1,the OE measure of DMU {B} is 0.5. The homogeneity of RAM for (1.5, 0.5) has the following relationship:

for all μ such that and

for all . Hence, RAM does not satisfy the property of homogeneity on (1.5, 0.5).

The following example indicates that SBM and Russell measure violate the property of homogeneity:

DMU	Input 1	Input 2	Output
{A}	1	2	1
{B}	5/4	1	1
{C}	μ	3 μ	1

In Example 6.6, μ is an arbitrary positive number. If μ1, then DMUs {A, C} are efficient. If , then the OE measure of DMU {B} is . If μ = 1, then the inputs and output of DMU {C} become and . In this case, DMU {B} becomes efficient and the OE measure of DMU {C} is 5/6. Hence, a DMU {k} with and has the following OE measure under :

Thus, the SBM violates the property of homogeneity.

The Russell measure of DMU {B} is 8/9. Under , the OE measure of DMU {B} is . Thus, the DMU with and has the following OE measure under :

Thus, the Russell measure violates the property of homogeneity.

Note that this chapter does not prove that the additive model and SARM violate the property of homogeneity, but the above proof is applicable to the two DEA models in a similar manner.

6.6.8 Unique Projection for Efficiency Comparison

The following examples confirm the violation of unique projection for efficiency comparison:

DMU	Input 1	Input 2	Desirable output
{A}	1	2	2
{B}	2	1	2
{C}	2	2	2
{D}	6	6	1

When RAM, SBM and Russell measure are applied to the data set of Example 6.8, the three models have multiple optimal solutions on DMU {C} such as . Multiple projections occur on the following line segment:

DMU	Input	Desirable output 1	Desirable output 2
{A}	1	2	1
{B}	1	3	1
{C}	1	1	1

When input‐oriented RM(c) is applied to DMU{C} of Example 6.9, the DMU has unity in the degree of OE. However, the DMU suffers from an occurrence of multiple solutions on the following line segment:

Thus, the RM(c) suffers from an occurrence of multiple projections. The assertion is applicable to input‐oriented RM(v), as well. Therefore, the radial measures violate the property of unique projection.

DMU	Input	Desirable output 1	Desirable output 2
{A}	1	1	2
{B}	1	2	1
{C}	2	1	1

The optimal solution of SARM applied to DMU {C} of Example 6.10 is on Hence, becomes a projection point set for DMU {C}. In other words, a projection point(s) of the DMU is not unique and the projection(s) may occur on a line segment: . The discussion on the above three examples can be applied to other DEA models, including the additive model.

6.7 SUMMARY

This chapter reviewed desirable properties for OE in a unified analytical synthesis. Then, we theoretically compared seven radial and non‐radial DEA models based upon the OE criteria. The findings on OE criteria were summarized in Table 6.1. This table provided us with the following guidance in selecting an appropriate DEA model. First, there is no perfect DEA model that can satisfy all desirable OE properties. It is necessary for us to develop the perfect DEA model that can satisfy all the desirable OE properties. That will be important but being a difficult future research task. Second, all DEA models violate both the property of unique projection for efficiency comparison and the property of aggregation. Cooper et al. (2007a) discussed how to deal with the aggregation issue. Third, the additive model violates all desirable OE properties. This chapter recommends the use of RAM as an extended formulation of the additive model because RAM satisfies several desirable OE properties. Fourth, both the Russell measure and SBM (= ERGM) perform as well as RAM as a non‐radial measure. If we are interested in the property of strict monotonicity, the two models (SBM and Russell measure) outperform the other models including RAM. In contrast, if we are interested in the property of translation invariance, RAM outperforms the Russell measure and SBM (= ERGM). In Section II (DEA environmental assessment), the proposed DEA model for non‐radial measurement originates from the RAM because it needs to utilize the property of translation invariance. See Chapter 26 which discusses how to handle an occurrence of many zeros in a data set. Fifth, if we are interested in the property of homogeneity, the two RMs are useful for such a purpose. All the non‐radial models do not satisfy the desirable property⁹. Sixth, RM(c) is useful in measuring a frontier shift among different periods¹⁰. See Chapter 19 which describes how to measure a frontier shift among different periods by using the radial measurement for environmental assessment, under constant RTS. Finally, if a data set contains zero and/or negative values, then we need to depend upon the property of “disaggregation” although this chapter has not discussed it. See Chapter 27 for a detailed description on the property of disaggregation. In the case of non‐radial measurement, RAM becomes an appropriate model for the data set¹¹ that contains zero and/or negative values. Meanwhile, in the case of both radial and non‐radial measurements, the property of disaggregation is useful in handling such zero and/or negative values.

In addition to the above implications for DEA users, Table 6.1 indicates that all DEA models examined in this chapter suffer from an occurrence of multiple projections. As a result of such a difficulty, all the DEA models cannot uniquely determine the OE measure for efficiency comparison. The difficulty is often associated with an occurrence of multiple reference sets and multiple projections. Unfortunately, the research issue has been insufficiently explored in previous DEA studies. To overcome the difficulty, we need to develop a new approach that can identify the existence of unique projection for efficiency comparison. The proposed approach combines a DEA model (primal formulation) with its dual model and then adds SCSCs in a combined model. The incorporation of SCSCs restricts projections of an inefficient DMU in a “minimum face” on an efficiency frontier when it changes its status from inefficiency to efficiency. Consequently, the combined primal/dual model with SCSCs can identify a projection set as the minimum face on an efficiency frontier. See Chapter 7 that discusses how to incorporate the SCSCs into DEA assessment.

At the end of this chapter, it is necessary to describe the following four concerns for research extension from conventional DEA to its environmental assessment. First, chapters in Section II need to incorporate undesirable outputs in their computational frameworks. Unification between desirable and undesirable outputs will be very important in terms of DEA environmental assessment. See Chapters 15, 16 and 17 on output unification. Second, a unified efficiency measure for environmental assessment has the assumption that undesirable outputs are “by‐products” of desirable outputs. As a result of the assumption, a desirable output vector should have an increasing trend, and an undesirable output vector should have an increasing and decreasing trend along with an increase in an input vector. To identify an occurrence of such an increasing and decreasing trend, DEA environmental assessment needs to incorporate the concept of “congestion.” See Chapters 9, 20, 21, 22 and 23. Thus, “strict monotonicity” and “liner homogeneity,” for example, are desirable properties for conventional DEA as discussed in this chapter, but are not necessary for environmental assessment. Third, this chapter did not discuss that dual variables, or multipliers, obtained from a radial or non‐radial model should be positive in their signs. Otherwise, production factors, corresponding to dual variables with zero, will be not utilized in DEA assessment. RAM and SARM can satisfy the property. Therefore, the two models will be utilized for non‐radial and radial models for environmental assessment, respectively. Finally, all models used for environmental assessment should be intuitively and visually discussed. As explored in this chapter, DEA environmental assessment will consider a balance between theory and practicality. These concerns will be explored in chapters in Section II.

APPENDIX

Proof of Proposition 6.1

Any combination between , and becomes a feasible solution. The objective function is bounded, because and (j = 1, 2,…, n). If , then SARM does not have an optimal solution. In contrast, if , then the objective function of SARM has a lower bound because

Let (θ*, λ*) be an optimal solution of SARM, then , and . It follows from that ∀ = 1,⋯,m. This indicates that is a feasible solution of SARM and . Hence, the following relationship is maintained on optimality:

The above relationship implies that is on the optimality of SARM.

Q.E.D.

Proof of Proposition 6.6

For any , is maintained in two RMs (input‐oriented), SBM and the Russell measure. For a feasible solution of λ, RAM has , (i = 1,…, m) and (r = 1,…, s). Hence, The solution of and is feasible in two RMs (input‐oriented), SBM and the Russell measure. For a feasible solution of λ, two RMs (input‐oriented), SBM and the Russell measure have . Hence, RAM has and two RMs, SBM and Russell measure have . In other words, two RMs (input‐oriented), RAM, SBM and Russell measure have .

Next, we prove that if is identified by SARM, RAM, SBM and Russell measure, then DMU {k} is fully efficient by using contradiction. We assume that there exists a feasible solution of Model (6.1)for DMU {k} such that , and . Since X > 0 and G > 0, we have

and

Let and for all , then we have

for SBM and Russell measure. This is in contradiction to .

Similarly, we have

and

Therefore, SARM has

and RAM has

This is in contradiction to .

Q.E.D.

Proof of Proposition 6.7

Both of the two input‐oriented RMs have A(X,G) = A(X′,G′) and R_h(X_k, G_k) = R_h(μX_k, G_k). Let λ* be an optimal solution in Model (6.1). Then we have and . It follows from and that

Since is maintained in the above equation, we have

Q.E.D.

Proof of Proposition 6.8

Let λ* be an optimal solution of Model (6.1) for SBM, then we have . It follows from the definitions of and , and that . Therefore, λ* is a feasible solution of

where is an (i,j) component of X^' and g' _rj is an (r,j) component of G^'. Since , we have

Since , we have either

or

Therefore, it follows that

The satisfaction of the Russell measure can be proved in a similar manner. Q.E.D.

Proof of Proposition 6.10

is equivalent to . The objective function of two input‐oriented RMs, SBM and the Russell measure consists of (i = 1,…, m) and (r =1,…, s). Let be an (i,i) component of D^x and let an (r,r) component of D^g, then we have (i = 1,…, m) and (r = 1,…, s). Both and are unit invariant. Therefore, two input‐based RMs, SBM and the Russell measure are unit invariant.

The objective function of RAM and SARM consists of and . Since for D^xX is (i =1,…, m) and for D^gG is , we have and . Therefore, both RAM and SARM are unit invariant.

Q.E.D.

Proof of Proposition 6.11

For RAM, is equivalent to . The objective function of RAM consists of

Since , we have

for i = 1,…, m, where π_i is the i‐th component of a m‐dimensional non‐negative vector (π).

In the same manner, we can prove (g_rk + ψ_r) for all r = 1,…, s, where ψ_r is the r‐th component of an s‐dimensional non‐negative vector ψ. From the definitions, the range of and the range of are equal to those of and , respectively. Therefore, we have

for every k = 1,…, n.

Q.E.D.