Mathematical Rationale on DEA/SCSCs:

Model (25.1) has methodological strengths and drawbacks. Table 25.1 compares DEA with DEA/SCSCs. Chapter 7 did not summarize such differences. As mentioned in Chapter 7, DEA often suffers from an occurrence of multiple reference sets and multiple projections. Model (25.1) can deal with such difficulties (i.e., multiple solutions) on efficient DMUs, but it does not directly solve the occurrence of multiple reference sets and multiple projections on inefficient DMUs. An important feature of Model (25.1) is that it can identify a unique reference set, often mathematically referred to as “a minimum face,” that can cover all possible reference sets on an efficiency frontier onto which each inefficient DMU projects. Thus, Model (25.1) can identify a unique reference set on the minimum face for efficient DMUs. Moreover, even if multiple projections occur on an inefficient DMU, DEA/SCSCs can specify their projection ranges onto the minimum face. Thus, DEA/SCSCs can solve the uniqueness issue on inefficient DMUs by a projection onto the minimum face.

Besides the description in Table 25.1, this chapter considers DEA as an identification process of an efficiency frontier because the efficiency measure of all DMUs is determined by comparing their performance with DMUs on the frontier. The efficiency frontier consists of only DMUs whose efficiency scores are unity and all slacks are zero in radial and non‐radial measurements. The requirement (i.e., zero in all the slacks) is very helpful in understanding the importance of SCSCs. To explain it in detail, let us consider Equations (7.7) and (7.8), which are for all i = 1,…, m and for all r = 1,…, s. Since all the efficient DMUs have zero in their slacks, the two groups of equations immediately indicate for all i and for all r on optimality. Thus, the incorporation of SCSCs implies that multipliers become positive on efficient DMUs.

The incorporation of SCSCs does not guarantee whether inefficient DMUs have positive multipliers because they have at least one positive slack variable(s). Such a result may not be important in Model (25.1) because inefficient DMUs do not consist of an efficiency frontier. Even if we drop an inefficient DMU(s) from the computation of DEA, such elimination does not influence the whole DEA evaluation of the other DMUs. Furthermore, when DEA projects an inefficient DMU onto an efficiency frontier, the SCSCs restrict its projection in such a manner that all multipliers of the inefficient DMU become positive on the efficiency frontier. Thus, the projection of the inefficient DMU is always restricted on a reduced range, or the minimum face, shaped by the SCSCs.

After classifying all DMUs into efficient and inefficient groups by Model (25.1), the proposed approach identifies a supporting hyperplane between the two groups. Returning to Chapter 8, we can specify the supporting hyperplane by where and are parameters for indicating the direction of a supporting hyperplane and σ indicates the intercept of the supporting hyperplane. The parameters are all unknown and need to be measured by , , where RS_k stands for a reference set of the k‐th DMU.

The above concern implies how the reference set of the k‐th DMU characterizes the location of a supporting hyperplane(s). In the case, the uniqueness of a reference set is important in identifying the supporting hyperplane. Hence, this chapter fully utilizes SCSCs for DEA assessment because of the above mathematical rationale. Moreover, it is possible for us to change the location of the supporting hyperplane to separate efficient and inefficient DMUs as a discriminant hyperplane. Therefore, this chapter needs to describe the locational change of the discriminant hyperplane by using DEA‐DA.

Acknowledging the importance of SCSCs in terms of identifying the supporting hyperplane, this chapter mentions that the proposed use of DEA/SCSCs has still another type of difficulty in reducing the number of efficient DMUs. To deal with the difficulty, this chapter proposes to use the following new approach that is an extended version of previous radial and non‐radial approaches:

• Step 1: Using results from Model (25.1), we classify all DMUs into efficient (E) and inefficient (IE) groups.
• Step 2: Apply the following DEA‐DA model to the two groups (E and IE):
  (25.2)

  Here, M is a prescribed large number and ε is a prescribed small number. It is necessary for us to prescribe the two numbers before solving Model (25.2). The objective function minimizes the total number of incorrectly classified DMUs by counting a binary variable (μ_j). In the classification, the efficient DEA group (E) has more priority than the inefficient group (IE). Therefore, Model (25.2) adds M to the efficient group in the objective of Model (25.2). The discriminant score is expressed by () and (), respectively, after changing their locations from the left hand side to the right hand side of Model (25.2). The sign of σ is not important because it is unrestricted. The small number (ε) is incorporated into the right hand side of Model (25.2) in order to avoid a case where an observation(s) exists on the estimated discriminant function. All the DMUs on desirable outputs and inputs connect to each other by a discriminant function, which we consider as the supporting hyperplane in DEA. Unknown weights v_i for i = 1, …, m and u_r for r = 1, …, s are determined by the minimization of incorrect classifications. The last two constraints indicate that all the variables (v_i for i = 1, …, m, and u_r for r = 1, …, s) are positive so that the discriminant function is a full model. Thus, Model (25.2) incorporates the constraints to “normalize” weight estimates in the manner that all weights are expressed by a positive percentage and the sum of all weights is unity. Such a change depends upon the degree of freedom between the number of observations and that of weights.

• Step 3: After applying Model (25.2) to the data set, this chapter obtains an optimal solution and compute the following score for the j‐th DMU:
  (25.3)
• Step 4: Using the ρ_j scores, this chapter computes their adjusted efficiency scores by the following procedure:
  1. Find the maximum and minimum values of ρ by and .
  2. Find the range between them by
     • (b-1) Range (A) = if is non‐negative or
     • (b-2) Range (B) = if is negative.
  3. The adjusted efficiency score for the j‐th DMU is determined by
     • (c-1) Efficiency = []/[range (A)] if is non‐negative or
     • (c-2) Efficiency = /[range(B)] if is negative.

The following five concerns are important in describing the combined use between DEA‐DA and DEA/SCSCs. First, Figure 25.2 visually describes the relationship between a supporting line obtained from Model (25.1) and a discriminant line determined by Model (25.2). The figure contains three efficient DMUs and the remaining inefficient DMUs. The symbol ○ stands for an efficient DMU and the symbol × stands for an inefficient DMU. For our visual convenience, the figure consists of a single input (x) on a horizontal axis and a single desirable output (g) on a vertical axis. The line s–s′ indicates a supporting line passing on DMU{B} and the line is expressed by . Meanwhile, the line d–d′ indicates a discriminant line between efficient and inefficient DMUs. Both the supporting line and the discriminant line are different in their analytical features. In the proposed approach, we consider that the supporting hyperplane determined by DEA can serve as a discriminant function for DEA‐DA by shifting an intercept of the supporting line, as depicted in Figure 25.2. This is just an idea without any mathematical rationale. Second, Model (25.2) incorporates a unique feature originated from DEA. That is, DEA/SCSCs, structured by Model (25.1), provides information for distinguishing between the efficient (E) and inefficient (IE) groups. The proposed approach classifies all DMUs into two groups under the condition that Model (25.2) minimizes the number of misclassification. Third, the research of Sueyoshi and Goto (2009a, b)^7,8 has used DEA‐DA as a methodology for bankruptcy and financial assessments by applying the two models. See Chapter 11. Their studies have used one of the two models for identifying an overlap between default and non‐default groups. The other model classifies all observations belonging to the overlap. This chapter does not follow their research direction, rather making a direct linkage between DEA/SCSCs and DEA‐DA. That is, this chapter uses a “standard” model, or Model (25.2), in order to classify DMUs in efficient and inefficient groups based upon their operational performance, or efficiency scores. See Chapter 11 for a description on the standard model. Thus, the proposed combination can measure the operational performance of DMUs. Fourth, since the proposed approach combines DEA/SCSCs with DEA‐DA, it can provide a single criterion (i.e., an industry‐wide evaluation by common multipliers) to assess the performance of all DMUs. Such a unique feature can drastically reduce the number of efficient DMUs so that we can identify a single best DMU and the remaining others with some level of inefficiency. Finally, the adjusted efficiency score of all DMUs locates between zero and unity so that it satisfies the efficiency requirement as discussed in Chapter 6.

Averch–Johnson effect:

Their study discussed the economic implication of regulation, often referred to as the “Averch–Johnson effect”. The concept implies that regulated companies tend to engage in an excessive amount of capital accumulation to expand the volume of their profits under no competition. As a result, there is a strong incentive to over‐invest in their facilities in order to increase their profits overall. Their investments often go beyond any optimal efficiency point for capital, so producing a high level of inefficiency within their firms because of large excess facilities (i.e., assets). The concept has served as a policy basis to promote the liberalization of the electric power industry in many industrial nations.

Data Accessibility:

The data source is Handbook of Electric Power Industry (2010).

The Japanese electric power industry consists of nine investor‐owned utility firms with regional monopoly, which are listed from A to I in this chapter. One electric power company (i.e., Okinawa) is much smaller than the other firms in terms of operation size and so it is excluded from our research for this chapter. The observed period is from 2005 to 2009. This chapter measures the performance of the nine electric power firms by examining two outputs and five inputs.

The two desirable outputs are the total amount of electricity sold and the total number of customers. The five inputs are total generation asset, total transmission asset, total distribution asset, total operation cost without labor cost, and number of employees. As mentioned previously, the electric power industry is a large capital‐intensive industry. Therefore, this chapter pays attention to the total assets related to generation, transmission and distribution.

Table 25.2 summarizes DEA efficiency scores and multipliers (i.e., dual variables) of the nine Japanese electric power firms from 2005 to 2009, measured by original radial formulations, or Models (7.1) and (7.2). In similar manner, Table 25.3 summarizes their efficiency scores and multipliers measured by the proposed formulation, or Model (25.1).

Comparing Table 25.2 with Table 25.3 provides us with the following two methodological implications. One of the two implications is that all efficient and inefficient firms contain zero in their multipliers in Table 25.2. However, as found in Table 25.3, the multipliers of efficient firms are all positive but those of inefficient firms contain zero. The results indicate that SCSCs change efficient DMUs to avoid their multipliers from being zero, but making them positive. Results produced by DEA/SCSCs can fully utilize information on the production factors of efficient DMUs, but not that of inefficient ones. The other implication is that even if DEA incorporates SCSCs, the combination still produces many efficient DMUs as summarized in Table 25.3. This indicates that the approach of DEA/SCSCs needs to improve the empirical effectiveness on performance assessment by combining it with DEA‐DA in order to provide common multipliers.

Table 25.4 documents the efficiency scores of Japanese electric power companies, measured by a conventional use of Model (7.1), and their adjusted efficiency scores measured by the proposed approach (DEA/SCSCs combined with DEA‐DA). The proposed approach produces their adjusted efficiency scores on [0, 1] so that they satisfy the efficiency requirement. In the proposed approach, the sixth electric power company {F} was the best performer (an adjusted efficiency score is 1 and rank is 1) in 2009. The second electric power company {B} was the worst performer (an adjusted efficiency score is 0 and rank is 45) in 2009. Thus, these results indicated that there was a single best performer and the remaining other firms were inefficient from the industry‐wide perspective.

Table 25.4 also provides an interesting difference between the conventional use of DEA and the proposed approach. To explain such a difference, let us see the efficiency and adjusted efficiency scores of company {E}, for example. The electric power company {E}, being smaller than the other firms, has produced unity (efficiency score = 1) in the conventional efficiency measures from 2005 to 2009. Therefore, the use of DEA evaluates company {E} as an efficient electric power firm. In contrast, the proposed approach has evaluated company {E} as 0.168 (2005), 0.201 (2006), 0.194 (2007), 0.188 (2008) and 0.194 (2009) on its adjusted efficiency scores. The result indicates that company {E} is inefficient. Such an evaluation difference is because the proposed approach has an industry‐wide assessment by common multipliers. In contrast, the conventional use of DEA does not have such an analytical capability for performance assessment. Thus, the proposed approach overcomes a methodological drawback of DEA, often observed in many previous studies.

Using the adjusted efficiency scores in Table 25.4, this chapter conducts the rank sum test on two null hypotheses. The first null hypothesis is whether liberalization in 2005 has not yet influenced the operational performance of Japanese electric power industry. The H statistic, measured by Equation (17.A3) in the appendix of Chapter 17, becomes 1.641 (which is less than 9.49: the χ² score at the 5% significance level under four degrees of freedom). Thus, this chapter cannot reject the null hypothesis, so confirming that liberalization has not yet produced any major change in the Japanese electric power industry after 2005. The result also supports the statistical validity of our computational procedure in which this chapter pools data sets from 2005 to 2009 and treats them as a single data set.

The second null hypothesis is whether Japanese electric power firms have shown a same result on their operational performance even after liberalization. The H statistic, measured by Equation (17.A3) in Chapter 17, becomes 38.282 (which is larger than 20.09: the χ² score at the 1% significance level under eight degrees of freedom). Thus, this chapter can reject the null hypothesis, so accepting the alternative hypothesis that Japanese electric power firms have taken different strategies in their operations so that they have produced different results in their efficiency measures.