4.2.1 Input‐Oriented RM(v) under Variable RTS

By duplicating Model (3.15) and adding an additional constraint of into the formulation, the radial model⁷, or input‐oriented RM(v), has the following mathematical structure to measure an efficiency score (θ) of the specific k‐th DMU:

(4.1)

where the side constraint is additionally incorporated into Model (4.1). Thus, the surface of a production possibility set for Model (4.1) is shaped by a convex polyhedral cone under variable RTS. It is possible for us to drop the side constraint from Model (4.1), as formulated in Model (3.15). In such a case, a production possibility set is characterized by constant RTS. Thus, Model (4.1) becomes input‐oriented RM(c) where (c) indicates the status of constant RTS.

The term becomes zero or negative on the optimality of Model (4.1). The objective value consists of an efficiency score⁸ and a total sum of slacks, where the latter is used to adjust the level of efficiency, or OE. Here, it is important to note that, if the slack adjustment term represents a level of inefficiency, then the sign before the term should be negative in the objective function in order to maintain the efficiency range between zero and unity. In contrast, Model (4.9), along with Equation (4.10), indicates a desirable output‐oriented model for RM(v) that is used for measuring a level of inefficiency in the objective function. In the case, the sign becomes positive. See Chapter 17, as well.

To describe the methodological strengths and drawbacks of Model (4.1), this chapter prepares the dual formulation of Model (4.1) that is mathematically specified as follows:

(4.2)

Here, V = (v₁,…,v_m) and U = (u₁,…,u_s) are two row vectors of dual variables related to the first and second sets of constraints in Model (4.1). An unrestricted dual variable (σ) is derived from the third constraint of Model (4.1). In transforming from Model (4.1) to Model (4.2), this chapter implicitly treats that and are changed to for all i and for all r in Model (4.2).

It is important to note that the objective function of Model (4.1) equals that of Model (4.2) on optimality. Thus, the level of OE is determined by

(4.3)

on the optimality of Models (4.1) and (4.2).

A methodological benefit of Model (4.1) is that all dual variables ( for all i and for all r) are strictly positive in their signs, so being able to function as multipliers on production factors. As a result, their information is fully utilized in performance evaluation although some dual variables may become very small as multipliers in such a manner that for some i and for some r. This analytical feature is very important in measuring a type of RTS, a marginal rate of transformation (MRT), a rate of substitution (RSU) between production factors and other scale related measures, all of which will be discussed in Chapters 8, 20 and 22.

4.2.2 Underlying Concept

This book conveys a message that DEA can be used for the measurement of not only an efficiency score but also other important measures (e.g., RTS and other scale measures). Thus, DEA can be considered as a holistic approach for performance assessment. Consequently, it is necessary for us to satisfy various conditions on DEA solutions. One of such conditions, for example, is that all dual variables should be strictly positive. Of course, the requirement may drop in many decisional cases, depending upon the type of these applications. See Chapters 9, 21, 22 and 23 in which some dual variables are unrestricted in their signs.

A drawback of Model (4.1) is that ε_n is a prescribed small number, often referred to as a non‐Archimedean small number. The specification⁹ of ε_n incorporates a subjective decision by a DEA user(s). In reality, none knows what the small number is. As a result, many users often drop the non‐Archimedean small number in their efficiency assessments. Model (4.1) without the very small number becomes

(4.4)

As mentioned previously, when we measure the level of OE concerning the k'th DMU, Models (4.1) and (4.4) may not produce any major difference between them because ε_n is the very small number. The elimination is seemingly acceptable. However, a serious problem may occur in the dual model whose multipliers, obtained from dual variables, may be zero. Such a result is unacceptable in this book on environmental assessment because it is impossible to measure RTS and the other type of scale measures (e.g., rate of substitution). See, for example, Chapters 22 and 23.

The dual formulation of Model (4.4) has the following structure:

(4.5)

Here, all mathematical descriptions in Model (4.2) are applicable to Model (4.5).

The objective value of Model (4.4) equals that of Model (4.5) on optimality. Thus, the level of OE is determined by

(4.6)

on the optimality of Models (4.4) and (4.5).

In addition to the above description, Models (4.4) and (4.5) have the other two important features. One of the two features is that Model (4.5) has because θ is unrestricted (URS) in Model (4.4). In other words, the upper bound of θ* is unity in Model (4.4), indicating the status of full efficiency, because the first constraints indicate in Model (4.5) for the k‐th DMU. The left hand side of the equation is the objective value of Model (4.5). Hence, both Models (4.4) and (4.5) indicate on the optimality of the k‐th DMU. There is no lower bound on θ* because the efficiency score is expected to be unrestricted in Model (4.4). However, a conventional use of the efficiency score, applied to a cross‐sectional data set, becomes non‐negative¹⁰ because Model (4.5) is structured by a maximization problem within an efficiency frontier.

The other concern is that (i = 1,…, m) and (r = 1,…, s) in Model (4.5) may have a possible occurrence of for some i and/or for some r. This clearly indicates that some production factors are not utilized in the efficiency assessment, so being problematic as mentioned previously. Therefore, the original DEA study (i.e., Charnes et al., 1978) incorporates into the objective function of Model (4.1). As a result of such incorporation, Model (4.2) has (i = 1,…, m) and (r = 1,…, s) so that all dual variables, or multipliers, are strictly positive in their signs. However, as mentioned previously, a problem of the non‐Archimedean small number (ε_n) is that none knows what it is in reality. Consequently, a subjective decision is included in the selection process of ε_n. Chapter 7 discusses how to handle the difficulty by incorporating strong complementary slackness conditions (SCSCs) into Models (4.4) and (4.5). The approach can overcome the difficulty related to ε_n so that it can avoid our selection subjectivity.

4.2.3 Input‐Oriented RM(c) under Constant RTS

Models (4.1) and (4.2) are formulated under variable RTS. If Model (4.1) drops , then the dual variable (σ) becomes zero in Model (4.2). In the case, the primal and dual models imply the status of constant RTS on their production possibility sets. The input‐oriented RM(c) under constant RTS has the following primal and dual formulations:

(4.7)

and:

(4.8)

Here, all mathematical descriptions on dual variables in Model (4.2) are applicable to Model (4.8).

As mentioned previously, it is possible to drop in the objective function of Model (4.7) as found in many research efforts on production economics. In the case, it is necessary for us to examine whether all dual variables are positive or not on the optimality of Model (4.8). The objective value of Model (4.7) equals that of Model (4.8) on optimality.

4.3.1 Desirable Output‐oriented RM(v) under Variable RTS

As a methodological alternative of the input‐oriented RM(v) and RM(c), previous DEA studies have paid attention to the following desirable output‐oriented model, or RM(v), to estimate an efficiency score (θ) regarding the k‐th DMU:

(4.9)

The level of OE (θ*) is measured by an unrestricted measure (τ *) and slacks on the optimality of Model (4.9) as follows:

(4.10)

There are two important concerns associated with Model (4.9). One of these concerns is that the objective value of Model (4.9) becomes more than unity () on optimality. Therefore, the OE measure of desirable output‐oriented RM(v) is determined by Equation (4.10). The other concern is that the objective function of Model (4.9) incorporates a positive sign before the term regarding the total sum of slacks like , not as found in Model (4.7), because τ* does not directly indicate the level of OE. Rather, the level of OE (θ*) is measured by the reciprocal of the inefficiency score via Equation (4.10).

Model (4.9) has the following dual formulation:

(4.11)

Here, all mathematical descriptions on dual variables in Model (4.2) are applicable to Model (4.11) by changing inputs to desirable outputs.

As mentioned previously, it is possible to drop the non‐Archimedean small number (ε_n) from Model (4.9) and Equation (4.10). In such a case, the desirable output‐oriented RM(v) becomes

(4.12)

The level of OE (θ*) is measured by an unrestricted measure (τ *) by

(4.13)

Model (4.12) has the following dual formulation:

(4.14)

Here, all the descriptions for the input‐oriented RM(v) in Model (4.5) are applicable to Model (4.14).

4.3.2 Desirable Output‐oriented RM(c) under Constant RTS

It is possible to reorganize Model (4.9) under constant RTS that measures the level of OE (θ *) as follows:

(4.15)

Model (4.15) drops from Model (4.9). The OE (θ*) is measured by an unrestricted measure (τ*) via . The elimination of the side constraint makes it possible to measure the desirable output‐oriented RM(c) under constant RTS.

The dual formulation of Model (4.15) has the following formulation:

(4.16)

Here, and are two row vectors of dual variables related to the first and second sets of constraints in Model (4.15).

As an extension of Models (4.15) and (4.16), this chapter reorganizes them as follows:

(4.17)

The efficiency score (θ*) is measured by an unrestricted measure (τ*) via .

The dual formulation of Model (4.17) has the following formulation:

(4.18)

See a description of Model (4.16) regarding and .

4.4.1 Comparison between Input‐Oriented and Desirable Output'Oriented Radial Models

There are several differences between the four radial models (RMs), two of which are summarized as follows. One of the two differences is that an efficiency score of the input‐oriented RM(v) is different from that of the desirable output‐oriented RM(v) under variable RTS. In contrast, the efficiency measure of input‐oriented RM(c) equals that of desirable output‐oriented RM(c) under constant RTS. See Proposition 4.1 about the mathematical proof. The other difference is shown in Figure 4.1 which depicts a visual difference between efficiency frontiers under constant RTS and variable RTS in the x–g domain. A straight line from the origin indicates an efficiency frontier under constant RTS, while a piece‐wise linear contour line, or {A‐B‐C‐D}, indicates the efficiency frontier under variable RTS.

As depicted in Figure 4.1, the two production possibility sets imply two areas for covering a possible combination between inputs and desirable outputs. They are axiomatically specified as follows:

(4.19)

The first production possibility set is formulated under variable RTS, or “VRTS,” while the second one is formulated by constant RTS, or “CRTS” in Equation (4.19). The difference between the two production possibility sets can be found in only the incorporation of . In Figure 4.1, the first production possibility set under variable RTS consists of the area [C] and the second one under constant RTS consists of the three areas expressed by [A]∪[B]∪[C] where the symbol ∪ indicates a union set.

In examining RM(c), it is necessary for this chapter to prove the following proposition:

4.4.2 Hybrid Radial Model: Modification

As an extension of the radial models, we can control the sum of structural (intensity) variables by adding the lower (lb) and upper (ub) bounds, or in the proposed RM formulations. Such models are often referred to as “hybrid models” in the DEA community.

To discuss a hybrid model, this chapter considers it by adding the condition on the lower and upper bounds in Model (4.1). The hybrid model for input‐oriented measurement becomes

(4.23)

where the previous side constraint for variable RTS in Model (4.1) is replaced by the new one in Model (4.23).

The dual formulation of Model (4.23) becomes as follows:

(4.24)

Using Model (4.23), the input‐oriented radial measurement is classified into the following three categories, for example: (a) (b) , and (c) under increasing RTS, respectively. This type of classification can be applied to the desirable output‐oriented radial model, as well. See Tone (1993) for the first discussion on the hybrid expression on RM. See also Sueyoshi (1997) which provided a detailed description on the hybrid expression and its applications.

4.5.1 Multiplier Restriction Methods

Assurance Region (AR) Analysis: To reduce the number of efficient DMUs by restricting the range of multipliers (i.e., dual variables), Thompson et al. (1986), Dyson and Thanassoulis (1988) and others proposed the use of AR analysis, which was discussed at the initial stage of DEA development. AR analysis incorporates the following multiplier restrictions into the dual formulations of the proposed RMs:

(4.25)

where the ratio of the i‐th input multiplier to the first one is expressed by its lower and upper bounds. The ratio combinations are applicable to the r‐th desirable output to the first one, as specified in Equation (4.25). It is trivial that multipliers are usually reqiured to be positive in many cases, whose magnitudes are expected to be much larger than ε_n.

Equation (4.25) is further extendable by the following input and desirable output weighted measure on the specific k‐th DMU:

(4.26)

where the ratio is determined by the i‐th weighted input divided by the sum of the total weighted inputs. In a similar manner, the ratio can be applied to the r‐th desirable output as formulated in Equation (4.26). Each equation indicates the importantce of the i‐th input and the r‐th desirable output among these production factors.

Figure 4.2 visually describes an implication of multiplier restriction in the data domain of two inputs (x1 and x2). All DMUs from {D1} to {D5}, consisting of an efficiency frontier, have the same magnitude of a desirable output so that the figure drops it. Let us consider A1 and A2 as the two vectors to restrict input multipliers. They are used as the upper and lower bounds for the multiplier restriction, expressed by Ṽ in Figure 4.2.

As a result of such multiplier restriction, the number of efficient DMUs decreases from five to one, that is, {D3} because only the DMU can exist only within the multiplier range (Ṽ) for inputs. This chapter uses Ṽ because input multipliers are applied for restriction on v_i (i = 1,…, m).

The proposed AR analysis has two difficulties in the practical uses. One of the two difficulities is that the approach depends upon prior information (e.g., expreience and previous emprical evidence) for multiplier restriction. It is often difficult for us to assess such prior information. The other difficulty is that the approach may produce an infeasible solution on optimality as a result of strict restriction. It is necessary for us to carefully think how to restrict multipliers on a use of the AR approach. This type of difficulty will be solved in DEA environmental assessment. See Chapters 22 and 23 which describe a new type of multiplier restriction, referred to as “explorative analysis,” which does not use any prior information along with a data adjustment. The AR analisis does not incoporate the data adjustment.

4.5.2 Cone Ratio Method

As mentioned previously, DEA produces many efficient DMUs. Charnes et al. (1989) and Brockett et al. (1997) proposed another type of approach for multiplier restriction, referred to as the “cone ratio (CR) method,” to reduce the number of efficient DMUs. To describe the methodology, this chapter starts from the following non‐negative input vector with m components, where the vector and matrix notations are for our visual convenience:

(4.27)

where is used for a column vector for input restriction (j = 1,…, n). The symbol (Tr) stands for a vector or matrix transpose. The restriction vector Ṽ is a (1 × m) column vector. So, Ṽ^Tr becomes a (m × 1) row vector.

In a similar manner, this chapter specifies the non‐negative directional vector for desirable outputs with s components as follows:

(4.28)

where is used for a column vector for restriction on desirable outputs (j = 1,…, n). The restriction vector Ũ is a (1 × s) column vector. So, Ũ^Tr becomes a (s × 1) row vector.

Using Equations (4.27) and (4.28), the input‐oriented RM(c), or Model (4.8), becomes the following vector notation under the proposed CR method:

(4.29)

where is a raw vector of s dual variables for desirable outputs, is a raw vector of m dual variables for inputs, is a column vector of m observed inputs of the j‐th DMU, is a column vector of m observed inputs of the specific k‐th DMU and is a column vector of s observed desirable outputs of the j‐th DMU.

Model (4.29) can be reorganized by incorporating Equations (4.27) and (4.28). The resulting formulation becomes

(4.30)

Here, in Model (4.29) are replaced by and in Model (4.30).

The dual formulation of Model (4.30) becomes as follows:

(4.31)

Like the AR analysis, the proposed CR method has also a difficulty in both accessing prior information and being often infeasible on optimality. Another difficulty of the CR method is that the AR analysis is more intutive and easily used than the CR method. Therefore, Chapters 22 and 23 extend the AR analysis, not the CR method, to prepare “explorative analysis.” A benefit of CR is that it requires less computation time than AR because CR has less constraints than AR in the formulation. See Chapter 10 on DEA network computing which takes advantage of the mathematical structure with less constraints in the formulation.

4.5.3 Cross‐reference Method

Besides the multilier restriction approaches, a cross‐reference method is widely used by many DEA researchers to classify efficient DMUs into two categories. One of the two categories includes efficient ones that locate on a vertex of an efficiency frontier. The other category includes the other efficient ones that locate on the efficiency frontier, but not locating on the vertex. See the DMU classification discussed in Chapter 9 (for algorithm development for network computing). Also, see Charnes et al. (1986, 1991) for the original DMU classification, parts of which are used in this chapter. A gist of the cross‐reference method is the classification of DMUs on an efficiency frontier.

In this chapter, the three groups of DMUs are characterized by an optimal solution of radial measurement regrading the k‐th DMU, for example, by Model (4.20) with slacks. The assumption on constant RTS is necessary so that the proposed cross‐reference method, using RM(c), can produce a feasible solution on optimality. In contrast, RM(v) may produce an infeasible solution in the case of variable RTS.

The three groups of DMU classifications considered in this chapter are mathematically specfied in the following manner:

(4.32)

Here, all DMUs are expressed by a whole set (J).

The first two groups of efficient DMUs locate on an efficiency frontier. The efficient group of DMUs (E) locates on the vertexes of an efficiency frontier. The efficient DMUs in the other group (E′) locate between such efficient DMUs that belong to E. The DMUs in E′ may suffer from an occurrence of multiple solutions on a combination(s) of a reference set. The last group (IF) consists of inefficient DMUs, but they exist on the efficiency frontier. They are not efficient because of at least one positive slack.

As mentioned previously, DEA may produce many efficient DMUs in . Therefore, it is necessary for us to classify them further if we want to prepare their ranking assessment. That is the purpose of the cross‐reference method. See Chapter 25 on the ranking analysis.

To describe the proposed approach, let us select DMU {a}, belonging to and return to input‐based RM(c), or Model (4.20), to obtain the computational feasibility. The cross‐reference efficiency of the b‐th DMU in J is measured by the following radial model (e.g., Doyle and Green, 1994; Green et al., 1996):

(4.33)

An important feature of Model (4.33) is that if the efficient DMU {a} belongs to E, then the elimination changes the shape of an efficiency frontier. As a result, the degree of may be changed from the original efficiency measure. In contrast, if it belongs to E′, then the elimination does not make any change on the frontier shape because it does not locate on the vertex of an efficiency frontier. In the case, the degree of does not produce any efficiency change. Thus, the cross‐reference method can provide us with a new measure on efficient DMUs, based upon which we can classify them. The most important benefit of the proposed approach is that it is possible for us to measure a degree of cross‐reference of the a‐th DMU in by examining the degree of . In this case, may be more than unity so that becomes positive. Thus, all efficient DMUs in group E on the vertexes of an efficiency frontier are reevaluated by the degree of . Thus, the cross‐reference method can provide us with an additional measure on efficient DMUs, based upon which we can classify them for ranking analysis.

Finally, it is important to note three concerns on a use of the cross‐reference method. First, the approach is feasible under constant RTS, but it may produce an infeasible solution under variable RTS. Second, there is no theoretical rationale on the validity of the cross‐reference method although it intuitively appeals to DEA users. Finally, the assessment on efficient DMUs is based upon local evaluation by considering a limited number of efficient DMUs in a reference set. It does not cover the group‐wide performance assessment of all DMUs by using a single set of common multipliers (weights) on efficient DMUs. See, for example, Chapter 25 that discusses how to determine common multipliers for the group‐wide assessment. Thus, the cross‐reference approach has a high level of computational tractability on DMU ranking analysis, but lacking any theoretical support within the framework of RM(c).

4.6.1 Cost Efficiency Measures

A methodological benefit of the radial measurement is that it can conceptually connect these models to cost efficiency analysis that minimizes the amount of total production cost (Sueyoshi, 1999b). Assuming that we can access an input price vector, or , of all DMUs (j = 1,…, n), the cost efficiency of the k‐th DMU is measured by the following model:

(4.34)

Here, is an unknown input quantity vector, not an observed vector as previously discussed for radial models, or RMs. It is important to note that Model (4.34) does not incorporate slacks into the formulation because it looks for an input vector that attains cost minimum, not an efficiency score. Moreover, dual variables become positive, as discussed below, because an input‐related dual vector corresponds to an input price vector. Therefore, this chapter needs to drop a slack vector in the objective function of Model (4.34).

The cost efficiency of the k‐th DMU is determined by the following manner (Farrell, 1957):

(4.35)

The cost efficiency implies the mimimized cost divided by an observed cost regrading the k‐th DMU.

The dual formulation of Model (4.34) becomes:

(4.36)

On optimality, both primal and dual models have a cost minimum expressed by . Moreover, always exsist between the two models as part of complementary slackness conditions (CSCs) of linear programming. If for all i, then Model (4.36) becomes the following condensed formulation:

(4.37)

because is maintained on the optimality of Model (4.37). Equation (4.35) indicates that the cost minimum of the k‐th DMU is less than or equal all the cost estimates measured by its observed input price vector.

4.6.2 Type of Efficiency Measures in Production and Cost Analyses

Previous discussions used the term “efficiency.” Hereafter, the concept of efficiency is separated into more specific classifications related to the proposed input‐oriented radial measurement. Some of these efficiency meaures were first conceptually specified by Farrell (1957) in the case of a single desirable output with multiple inputs under constant and diminishing RTS. Then, Sueyoshi (1999b) elaborated them by extending these production factors under constant RTS and variable RTS.

The efficiency measures used in this chapter are specified as follows:

1. Operational efficiency (OE_v) under variable RTS: the degree of opetational efficincy that is measured by the optimal objective value of RM(v), for example, Model (4.1),
2. Operational and Scale Efficiency (OE_c) under constant RTS: the degree of operational efficiency that is measured by the optimal objective value of RM(c), for example, Model (4.7),
3. Production‐based scale efficiency (PSE): .
4. Cost efficiency (CE_v) under variable RTS: from Equation (4.35) via Model (4.34),
5. Cost and Scale Efficiency (CE_c) under constant RTS: from Equation (4.35) via Model (4.34) without .
6. Cost‐based scale efficiency (CSE): .
7. Allocative efficiency (AE_v) under variable RTS: .
8. Allocative and scale efficiency (AE_c) under constant RTS:

The desirable output‐oriented measures have another type of efficiency classifications. This chapter discribes only the input‐oriented efficiency ones because they can directly link to cost‐based efficiency concepts.

Figure 4.3 visually describes various efficiency measures, all of which are defined by the eight classifications: (a–h) as discussed above. The figure has two efficiency frontiers, depending upon the assumption on RTS (i.e., constant or variable). The efficiency frontier, consisting of a piece‐wise linear contour line, or DMUs {F‐G‐B‐H‐I‐J}, is shaped under variable RTS, while the other curve (I–I′) is shaped under constant RTS. For our descriptive convenience in Figure 4.3, this chapter starts with a visual explanation on OE_V.

The degree of OE_V is visually specified by the distance OB/OA under variable RTS, where OB stands for a distance between the origin O and DMU {B}. In a simular manner, OA indicates the distance between the origin O and DMU {A}. This chapter drops the term DMU if a specified point in the figure is used to indicate a projected point, so not indicating a DMU. Shifting from variable to constant RTS, OE_c is measured by OC/OA. The difference between the two measures is expressed by PSE = OE_c/OE_v = (OC/OA)/(OB/OA) = OC/OB. For example, DMU {A} needs to reduce the amount of an input vector from {A} to {B} under variable RTS. The level of such input reduction, indicating an amount of inefficiency, is expressed by the distance of (OA–OB)/OA. By controling the operation size, the DMU {A} can attain the input decrease until it attains a projected point, C. The inefficiency due to an inapproporiate operation size is expressed by (OB–OC)/OA. Thus, OE_C is separated into two components in the manner of .

The production‐based description can be applied to the cost‐based one such as CE_v, CE_c and CSE. Given an input price vector (P), H indicates the cost‐minimum for DMU {A} in the case of variable RTS. Here, lines p₁–p₁′ and p₂–p₂′ indicate a level of cost under a fixed ratio of input prices with p₁/p₂. D and H locate on the line (p₂ and p₂′), with both having the same value of cost‐minimum. Therefore, CE_v = OD/OA. If the line is closer to the origin, the level of cost is lower. Meanwhile, under constant RTS, K indicates such a cost‐minimum. The total production cost becomes same on K and E because both points locate on the line (p₁ and p₁′). Therefore, CE_c= OE/OA. The degree of CSE is determined by the ratio between CE_v and CE_c. Thus, this chapter has CSE = CE_c/CE_v = (OE/OA)/(OD/OA) = OE/OD. The decomposition related to the cost minimization process indicates that DMU {A} needs to attain H by an effort for cost mimimization and further attain K by controlling its operational size.

In addtion to these efficiency concepts, AE_v and AE_c have unique features that cannot be found in the other efficiency measures. For example, the degree of AE_v is determined by and AE_V = (OD/OA)/(OB/OA) = OD/OB in Figure 4.3. Given an input price vector (P), the efficiency measure indicates how each DMU controls the ratio of input allocation by itself for cost‐mimimization under variable RTS. The degree of AE_C has the same implication under constant RTS.

4.6.3 Illustrative Example

Table 4.1 lists a data set to discuss three efficiency measures related to two radial models. The table contains eight DMUs {A–H}, each of which uses one input to produce one desiable output. Table 4.2 summarizes computational results on OE_v, OE_c and PSE. Figure 4.4 visually describes OE_v measures under variable RTS and OE_c measures under constant RTS, both are associated with the two efficiency frontiers.

The formulation to measure OE_V of DMU {A}, for example, is formulated as follows:

It is necessary to repeat the computation process on DMU {A} to measure the other DMUs {B–H}.

As listed in Table 4.2, along with Figure 4.4, only DMU {B} attains the status of efficiency in the three measures. The four DMUs {A, B, C and D} attain the status of efficiency on OE_v under variable RTS, but only DMU {B} become efficient under constant RTS. The other DMUs {E, F, G and H} have exhibited some level of inefficiency on OE_v. To describe the implication of PSE, measured by OE_c/OE_v, let us pay attention to the inefficient DMU {F}. Since the computational results of Table 4.2 are determined by the input‐oriented measurement, the ideal performance of DMU {F} can be found in a projected point R on the efficiency frontier under variable RTS and another projected point Q under constant RTS.

The degree of OE_v on DMU {F} is measured by PR/PF = 0.810. The degree of OE_c is measured by PQ/PF = 0.643. Based upon the two efficiency measures, the degree of PSE on the DMU is determined by OE_c/OE_v = (PQ/PF)/(PR/PF) = PQ/PR = 0.643/0.810 = 0.794. The description of DMU {F} in terms of the three efficiency measures is applicable regarding the other DMUs.

Table 4.3 lists a data set to discuss eight efficiency measures related to production and cost‐based efficiency measures. First, let us describe the formulation to measure the level of CE_v on DMU {F}, for example, which is structured as follows:

Table 4.4 summarizes eight efficiency measures related to production and cost analyses. As summarized in the table, DMU {C} is efficient in terms of the eight measures. The DMUs {A, B, D} have mixed efficiency measures in the eight measures and the remaining DMUs {E, F, G, H} are inefficient in these measures.