How to deal with S-shaped curve in DEA

GRIPS Discussion Paper 13-10 How to deal with S-shaped curve in DEA Kaoru Tone Miki Tsutsui June 2013 National Graduate Institute for Policy Studies 7-22-1 Roppongi, Minato-ku, Tokyo, Japan 106-8677 How to deal with S-shaped curve in DEA Kaoru Tonea, Miki Tsutsuib a National Graduate Institute for Policy Studies, Tokyo, Japan Central Research Institute of Electric Power Industry, Tokyo, Japan [email protected], [email protected] b Abstract In DEA we are often puzzled by the big difference in CRS and VRS scores, and by the convex production possibility set syndrome in spite of the S-shaped curve often observed in many real data. In this paper we perform a challenge to these subjects. Keywords: Data envelopment analysis, S-shaped curve, CRS, VRS, scale elasticity, SAS 1. Motivation In DEA (Data Envelopment Analysis), we are often puzzled by the big difference between the constant returns-to-scale score (CRS) and the variable returns-to-scale score (VRS). Several authors (Avkiran (2001), Avkiran et al. (2008), Bogetoft and Otto (2010) among others) proposed solutions for this problem. In this paper we propose a different approach and results. Another problem is the conventional convex production possibility set assumption which is closely related to the first problem. In this paper, we discuss these two basic subjects of DEA. Several researchers have discussed non-convex production possibility set issues, see Dekker and Post (2001), Kousmanen (2001), Podinovski (2004), Olsen and Petersen (2013), among others. However, we believe there is room for further research on this subject. Another objective of this paper is the measurement of scale elasticity of production. Most of researches on this subject are based on the convex production possibility set assumption. We propose a new scheme for evaluation of scale elasticity within the cluster each DMU belongs to. This paper unfolds as follows. In Section 2, we describe a decomposition of the CRS slacks after introducing basic notations, and define the scale-independent data set. In Section 3, we introduce clusters and define the scale&cluster-adjusted score (SAS). In Section 4 we explain our scheme using a tiny example. Two illustrative examples are presented in Section 5. In Section 6, we define the scale elasticity based on the scale-dependent data set. An empirical study on Japanese universities follows in Section 7. Extensions to the radial DEA models are presented in Section 8. The last section concludes this paper. 2. Global issue 1 In this section we introduce notation and basic tools, and discuss a decomposition of slacks. 2.1. Notation and basic tools Let the input and output data matrices be respectively mn X  R (  ( xij ) (i  1, , m; j  1, , n)) and sn Y  R (  ( yrj ) ( r  1, , s; j  1, , n)), (1) where m, s and n are the number of inputs, outputs and decision making units (DMUs). Then, the production possibility set for the constant returns-to-scale (CRS) and variable returns-to-scale (VRS) models are defined respectively by P  CRS (x, y) x  Xλ, y  Yλ, λ  0 , (2) P  VRS (x, y) x  Xλ, y  Yλ, eλ  1, λ  0 , (3) where x  Rm , y  Rs and λ ( 0)  Rn are input, output, and intensity vectors, and e  R n is the row vector with all elements equal to 1. Throughout this section, we utilize the input-oriented slacks-based measure (SBM) (Tone (2001)) for the efficiency evaluation of each DMU ( xo , yo ) (o  1, , n) regarding the CRS and VRS models as follows: [CRS]  oCRS  min1  1 m si  m i 1 xio subject to Xλ  s   xo (4) Yλ  s   y o λ  0, s   0, s   0. [VRS]  oVRS  min1  1 m si  m i 1 xio subject to Xλ  s   x o (5)  Yλ  s  y o eλ  1 λ  0, s   0, s   0, 2   where λ  R n is the intensity vector and s , s are respectively input- and output-slacks. Although we present our model in the input-oriented SBM model, we can develop the model to the output-oriented and non-oriented SBM models as well as to the radial models (Section 8). We define the scale-efficiency (  o ) of DMUo by oCRS  o  VRS . o (6) We denote optimal slacks of the CRS model by (so* , so* ) . (7) Although we utilize the scale-efficiency CRS/VRS as an index of scale merits and demerits, we can make use of other indexes appropriate for discriminating handicaps due to scale. However, the index must be normalized between 0 and 1, and the larger indicates the better scale condition. 2.2. Decomposition of CRS slacks We decompose CRS slacks into scale-independent and –dependent parts as follows: so*   oso*  (1   o )so* (8) so*   oso*  (1   o )so* If DMUo satisfies  o  1 (so called in the most productive scale size), its slacks are all attributed to the scale-independent slacks. However, if  o  1 , its slacks are decomposed into the scale-independent part ( oso* ,  oso* ) and the scale-dependent part ((1   o )so* ,(1   o )so* ) . * * Scale-independent slacks = ( oso ,  oso ) (9) * * Scale-dependent slacks = ((1   o )so ,(1   o )so ). (10) 2.3. Scale-independent data set We define the scale-independent data (xo , y o ) (o  1, scale-depending slacks as: 3 , n) by deleting and adding the xo  xo  (1   o )so* Scale-independent Input Scale-independent Output y o  y o  (1   o )so* (11) See Figure 1 for an illustration. y Scale-independent slacks Scale-dependent slacks x Figure 1: Scale-independent input 3. In-cluster issue: Scale&Cluster-adjusted DEA score (SAS) In this section we introduce the cluster of DMUs and define the scale&cluster-adjusted score (SAS). 3.1. Cluster We classify DMUs into several clusters depending on their characteristics. They can be supplied exogenously (see Section 6 for an example), or determined posteriori depending on the degree of scale-efficiency. A sample of the latter case may go as follows. We already know returns-to-scale (RTS) characteristics of each DMU, i.e. IRS , CRS or DRS, from the VRS solution. We first classify CRS DMUs as Cluster C. Then we classify IRS DMUs depending on the degree of scale-efficiency σ. For example, for IRS DMUs with 1 > σ  0.8 we classify them as I1, with 0.8 > σ  0.6 as I2, and so on. For DRS DMUs with 1 > σ  0.8 we classify them as D1, with 0.8 > σ  0.6 as D2, and so on. We must decide the number of clusters and bandwidth considering the number of DMUs. We denote the name of cluster DMUj by Cluster(j) ( j  1, , n) . 3.2. Solving the CRS model in the same cluster We solve the CRS model for each DMU (xo , y o ) (o  1, same Cluster (o) which can be formulated as follows: 4 , n) referring to the ( X, Y) in the 1 m sicl   m i 1 xio subject to min1  Xμ  s cl   xo Yμ  s cl  (12)  yo  j  0 (j : Cluster( j )  Cluster(o)) μ  0, s cl   0, s cl   0. cl * cl * We denote an optimal in-cluster slacks by (so , so ) . By adding the scale-dependent slacks and in-cluster slacks, we define the total slacks as Total input slacks  so  (1   o )so*  socl * (13)  Total output slacks so  (1   o )so*  socl * Scale&cluster-adjusted data (projection) (xo , y o ) is defined by: Scale&cluster-adjusted input (Projected Input)  xo  xo  so  xo  (1   o )s o*  s ocl * Scale&cluster-adjusted output (Projected Output)  y o  y o  s o  y o  (1   o )s o*  s ocl * See Figure 2 for an illustration. y In-cluster slacks Scale-dependent slacks x Figure 2: Scale&cluster-adjusted input 5 (14) Up to this point, we deleted scale demerits and in-cluster slacks from the data set. Thus, we have obtained a scale free and in-cluster slacks free (projected) data set ( X, Y). 3.3. Scale&Cluster-adjusted score (SAS) In the input-oriented case, the scale&cluster-adjusted score (SAS) is defined by  Scale&cluster-adjusted score (SAS)  SAS o 1 m sio 1 m s cl *  sio*  1   i 1 1   i 1 io m xio m xio . (15) The reason why we utilize the above scheme is as follows. First, we wish to eliminate scale demerits from the CRS slacks. For this purpose, we decompose the CRS slacks into scaledependent and –independent parts, in the recognition of scale demerits as represented by 1-  o . If  o =1, the DMU has no scale demerits and its slacks are attributed to itself. If  o =0.25, then 75% of the slacks are attributed to its scale demerits. After deleting the scaledependent slacks, we evaluate the DMU within the cluster it belongs to and find in-cluster slacks. If the DMU is efficient among its cluster, its in-cluster slacks are zero, while, if inefficient, the DMU has in-cluster slacks against the efficient DMU. Lastly, we add the incluster and scale-dependent slacks to obtain the total slacks. Using the total slacks, we define the scale&cluster-adjusted score (SAS). [Proposition 1] The scale&cluster-adjusted score (SAS) is not less than the CRS score. oSAS  oCRS . (16) [Proposition 2] If oCRS  1 then it holds oSAS  oCRS , but not vice versa. [Proposition 3] The scale&cluster-adjusted score (SAS) is decreasing in the increase of input and in the decrease of output so long as the both DMUs remain in the same cluster. 6 [Proposition 4] The projected DMU (xo , y o ) is efficient under the SAS model among the DMUs in the cluster it belongs to. It is also CRS and VRS efficient among the DMUs in its cluster. All proofs are in Appendix A. 4. How does it work We demonstrate the above procedure using a tiny example. Table 1 exhibits 5 DMUs with a single input x and a single output y. Figure 3 display them where the CRS efficient frontier is the line OA while the VRS efficient lines are AB and BC. We assume DMUs B and D belong to the same cluster b while others belong to themselves. Table 1: Five DMUs DMU (I)x (O)y Cluster A 9 9 a B 6 4 b C 5 1 c D 9 4 b E 8 5 e 10 y 9 A 8 7 6 5 E 4 3 P R Q D B S 2 C 1 0O 0 x 1 2 3 4 5 Figure 3: DMUs For DMU B, we have 7 6 7 8 9 10 sB  QB  2,  B  PQ/PB  0.6667 Scale-dependent slack  RB  (1   B ) sB  0.6667 In-cluster slack  0 Total slack  0.6667. Hence  BSAS  1  RB 0.6667  1  0.8889 PB 6 Scale&cluster-adjusted input x B  xB  Total slack  5.3333. For DMU D, we have sD  QD  5,  D  PQ/PB  0.6667 Scale-dependent slack  SD  (1   D ) sD  1.6667 In-cluster slack  RS  2 Total slack  RD  RS+SD  3.6667. In-Cluster slack occurs against DMU B, because B and D belong to the same cluster b. Hence  DSAS  1  RD 3.666  1  0.5926 PD 9 Scale&cluster-adjusted input x D  xD  Total slack  5.3333. The situation of DMU E differs from other DMUs. This DMU belong to the cluster consisting of itself and is inefficient regarding to both CRS and VRS models. See Figure 4. 8 10 9 8 7 6 5 4 3 2 1 0 A Q P O0 S R E B C x 1 2 3 4 5 6 7 8 9 10 Figure 4: DMU E DMU E has the following elements:  ECRS  0.625 :  EVRS  0.825 PQ 5   0.7576 PR 6.6 sE  QE  3 E  Scale-dependent slack  SE  (1   E ) sE  0.7272 In-cluster slack  0 Total slack  SE  0.7272 Scale&cluster-adjusted score  ESAS  PS SE Total slack  1  1  0.9091 PE PE xE Scale&cluster-adjusted input x E  xe  Total slack  7.2728. DMU E has no In-cluster slack, because it is isolated in cluster. Its Scale&cluster-adjusted score SAS is larger than the VRS score. Table 2 exhibits results of computation and Figure 5 displays Scale&cluster-adjusted projections. Frontiers are non-convex. The non-convexity is caused by the recognition of scale demerits and clusters. Even when  o =1 for all DMUs, clustering may bring non-convex frontiers. 9 Table 2: Comparisons of three scores with projected input and output SAS Projection DMU A B C D E CRS-I 1 0.6667 0.2 0.4444 0.625 VRS-I 1 1 1 0.6667 0.825 SAS-I 1 0.8889 0.36 0.5926 0.9091 Input 9 5.3333 1.8 5.3333 7.2727 Output 9 4 1 4 5 10 9 A 8 7 6 E 5 B, D 4 3 2 C 1 x 0 0 1 2 3 4 5 6 7 8 9 10 Figure 5: Projected x and y (frontiers) 5. Illustrative examples In this section we present two artificial examples with a single input and a single output. The first one is totally non-convex, and the second one is a mixture of non-convex and convex frontiers. We demonstrate the above procedures using them. 5.1. Example 1 Table 3 shows 19 DMUs with input x and output y, while Figure 6 exhibits them graphically. We assume that DMUs A, B and C belong to Cluster a, and DMUs K and L to Cluster k, while other DMUs belong to themselves. 10 Table 3: Example 1 DMU (I)x (O)y Cluster DMU (I)x (O)y Cluster A B C D E F G H I 2 3 3.5 4 4.25 4.5 4.6 4.7 4.8 0.5 0.5 0.6 1 1.5 2 2.5 3 3.5 a a a d e f g h i K L M N O P Q R S 5 6 7 7.5 8 8.5 9 9.5 10 5 5 5.2 5.3 5.5 5.8 6.2 6.7 7.3 k k m n o p q r s J 4.9 4 j y 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 x Figure 6: Data plot of Example 1 First, we solved the input-oriented CRS and VRS models, and obtained the scale-efficiency and CRS slacks which were decomposed into the scale-independent and –dependent parts. Table 4 exhibits them. Since the output y has no slacks in this example, we do not display them. 11 Table 4: CRS, VRS, Scale-efficiency and Slacks CRS Slacks Scale-Independent Scale-Dependent Slacks Slacks (1   )s  DMU CRS-I VRS-I Scale-Eff. s  s A 0.25 1 0.25 1.5 0.375 1.125 B 0.1667 0.6667 0.25 2.5 0.625 1.875 C 0.1714 0.5905 0.2903 2.9 0.8419 2.0581 D 0.25 0.5833 0.4286 3 1.2857 1.7143 E 0.3529 0.6275 0.5625 2.75 1.5469 1.2031 F 0.4444 0.6667 0.6667 2.5 1.6667 0.8333 G 0.5435 0.7246 0.75 2.1 1.575 0.525 H 0.6383 0.7801 0.8182 1.7 1.3909 0.3091 I 0.7292 0.8333 0.875 1.3 1.1375 0.1625 J 0.8163 0.8844 0.9231 0.9 0.8308 0.0692 K 1 1 1 0 0 0 L 0.8333 0.8333 1 1 1 0 M 0.7429 0.7764 0.9568 1.8 1.7222 0.0778 N 0.7067 0.7536 0.9377 2.2 2.0629 0.1371 O 0.6875 0.7609 0.9036 2.5 2.2589 0.2411 P 0.6824 0.7928 0.8606 2.7 2.3237 0.3763 Q 0.6889 0.8454 0.8149 2.8 2.2816 0.5184 R 0.7053 0.9153 0.7705 2.8 2.1574 0.6426 S 0.73 1 0.73 2.7 1.971 0.729 Second, we deleted the scale-dependent slacks from the data and obtained the data set ( X,Y) . We solved the CRS model within the same cluster and found the in-cluster slacks. By adding the scale-dependent slacks and in-cluster slacks we obtained the total slacks. Table 5 records them. 12 Table 5: ( X,Y) , In-cluster slacks and Total slacks DMU Cluster A B C D E F G H I J K L M N O P Q R S a a a d e f g h i j k k m n o p q r s x 0.875 1.125 1.4419 2.2857 3.0469 3.6667 4.075 4.3909 4.6375 4.8308 5 6 6.9222 7.3629 7.7589 8.1237 8.4816 8.8574 9.271 y In-cluster slacks 0.5 0.5 0.6 1 1.5 2 2.5 3 3.5 4 5 5 5.2 5.3 5.5 5.8 6.2 6.7 7.3 0 0.25 0.3919 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Scaledependent slacks 1.125 1.875 2.0581 1.7143 1.2031 0.8333 0.525 0.3091 0.1625 0.0692 0 0 0.0778 0.1371 0.2411 0.3763 0.5184 0.6426 0.729 Total slacks 1.125 2.125 2.45 1.7143 1.2031 0.8333 0.525 0.3091 0.1625 0.0692 0 1 0.0778 0.1371 0.2411 0.3763 0.5184 0.6426 0.729 Finally we computed the adjusted score  and the projected input and output as exhibited in Table 6 while Figure 7 displays them graphically. SAS 13 Table 6: Scale&cluster-adjusted score and projected input and output Adjusted-Score Projected x Projected y DMU  SAS ( x) ( y) A 0.4375 0.875 0.5 a B 0.2917 0.875 0.5 a C 0.3 1.05 0.6 a D 0.5714 2.2857 1 d E 0.7169 3.0469 1.5 e F 0.8148 3.6667 2 f G 0.8859 4.075 2.5 g H 0.9342 4.3909 3 h I 0.9661 4.6375 3.5 i J 0.9859 4.8308 4 j K 1 5 5 k L 0.8333 6 5 k M 0.9889 6.9222 5.2 m N 0.9817 7.3629 5.3 n O 0.9699 7.7589 5.5 o P 0.9557 8.1237 5.8 p Q 0.9424 8.4816 6.2 q R 0.9324 8.8574 6.7 r S 0.9271 9.271 7.3 s Cluster y 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 Figure 7: Projection (▲) and data (○) 14 10 11 x We compare input-oriented CRS, VRS and SAS scores in Table 7 and Figure 8. Adjusted scores (SAS) of DMUs E to J and M to Q have larger than those of VRS model. This reflects non-convex characteristics of data set. Table 7: Comparison of three scores DMU CRS-I 0.25 0.1667 0.1714 0.25 0.3529 0.4444 0.5435 0.6383 0.7292 0.8163 A B C D E F G H I J VRS-I 1 0.6667 0.5905 0.5833 0.6275 0.6667 0.7246 0.7801 0.8333 0.8844 DMU SAS-I 0.4375 0.2917 0.3 0.5714 0.7169 0.8148 0.8859 0.9342 0.9661 0.9859 CRS-I 1 0.8333 0.7429 0.7067 0.6875 0.6824 0.6889 0.7053 0.73 K L M N O P Q R S CRS-I VRS-I VRS-I 1 0.8333 0.7764 0.7536 0.7609 0.7928 0.8454 0.9153 1 SAS-I 1 0.8333 0.9889 0.9817 0.9699 0.9557 0.9424 0.9324 0.9271 SAS-I 1.2 1 0.8 0.6 0.4 0.2 0 A B C D E F G H I J K L M N O P Q R S Figure 8: Comparison of three scores 5.2. Example 2 Table 8 and Figure 9 exhibit data for Example 2. These DMUs display a typical S-shaped curve. 15 Table 8: Example 2 DMU A B C D E F G H I J (I)x 2 3 4 4.5 5 6 7 8 9 10 (O)y 1 1.2 2 3 5 5.8 6.3 6.7 6.9 7 Cluster a a c d e e g h i j y 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 x Figure 9: Plot of Example 2 Table 9 and Figure 10 summarize the results of the above procedures. The projected frontiers are a mixture of non-convex and convex parts. 16 Table 9: Results of Example 2 ScaleTotal slacks dependent slacks In-cluster slacks DMU CRS-I VRS-I SAS-I ( x) ( y) A 0.5 1 0.75 1.5 1 0.5 0.5 0 B 0.4 0.7167 0.6 1.8 1.2 1.2 0.7953 0.4047 C 0.5 0.6875 0.8636 3.4545 2 0.5455 0.5455 0 D 0.6667 0.7778 0.9524 4.2857 3 0.2143 0.2143 0 E 1 1 1 5 5 0 0 0 F 0.9667 1 0.9989 5.9933 5.8 0.0067 0.0067 0 G 0.9 1 0.99 6.93 6.3 0.07 0.07 0 H 0.8375 1 0.9736 7.7888 6.7 0.2112 0.2112 0 I 0.7667 1 0.9456 8.51 6.9 0.49 0.49 0 J 0.7 1 0.91 9.1 7 0.9 0.9 0 y 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 x Figure 10: Projection (▲) and data (○) Figure 11 displays comparison of three scores. At DMUs C and D, Adjusted-scores are larger than VRS scores. This reflects non-convex characteristics of the data set. 17 CRS-I VRS-I SAS-I 1.2 1 0.8 0.6 0.4 0.2 0 A B C D E F G H I J Figure 11: Comparison of three scores 6. Scale-dependent data set and scale elasticity So far we have discussed the efficiency score issue of our proposed scheme. In this section we deal with the scale elasticity issue. Many papers have discussed this subject under the globally convex frontier assumption. See Banker and Thrall (1992), Banker et al. (2004), Färe and Primond (1995), Førsund and Hjalmarsson (2004a, 2004b), Olsen and Petersen (2013), Podinovski (2004), Kousmanen (2001) among others. However, in case of non-convex frontiers, we believe there is room for further research on this subject. Based on the decomposition of CRS slacks mentioned in Section 2, we develop a new scale elasticity which can cope with non-convex frontiers. 6.1. Scale-dependent data set We delete or add scale-independent slacks from the data, and thus define the scale-dependent data set (xˆ o , yˆ o ) . Scale-dependent input xˆo  xo   o so* Scale-dependent output yˆ o  yo   o so* Figure 12 illustrates an example. 18 (17) y Scale-independent slacks x Figure 12: Scale-dependent input ˆ ,Y ˆ ) in the same cluster. Thus, we We first project (xˆ o , yˆ o ) onto the VRS frontier of ( X Proj Proj denote them (xˆ o , yˆ o ) : ( xˆo , yˆo )  ( xˆoProj , yˆoProj ) . (18) 6.2. Scale elasticity The scale elasticity or degree of scale economies is defined as the ratio of marginal product to average product. In a single input/output case, if the output y is produced by the input x, we define the scale elasticity by  dy dx y . x (19) In the multiple input-output environments, it is determined by solving linear programs related to the supporting hyperplane at the respective efficient point. See Cooper et al. (2007, pp. 147-149) for details. ˆ The production set ( X Proj ˆ Proj ) defined above has convex frontiers at least within each ,Y Proj Proj cluster, we can find a supporting hyperplane at (xˆ o , yˆ o ) that supports all projected DMUs in the cluster and has the minimum deviation t from them. This scheme can be formulated as follows: 19 min t subject to vxˆ oProj  1 uyˆ oProj  u0  1 (20)  vxˆ Proj  uyˆ Proj  u0  w j  0 (j : Cluster( j )  Cluster(o)) j j  w j  t  0 (j : Cluster( j )  Cluster(o)) v  0, u  0, w j  0(j ), t  0 : u0 free in sign. * Let the optimal u0 be u0 . We define the scale elasticity of DMU (xo,yo) by: Scale Elasticity  o  1 . 1  u0* (21) * If uo is not uniquely determined, we check its min and max while keeping t at the optimum. The reason why we apply the above scheme is as follows. (1) Conventional methods assume a global convex production possibility set for identifying RTS characteristics of each DMU. However, as we observed, the data set not always exhibits convexity. Moreover, the RTS property is a local one, but not global, as the formula (19) indicates. Hence, we discuss this issue within the cluster the DMU belongs to, after deleting the scale-independent slacks. * (2) Conventional methods usually find multiple optimal values of u0 and there is a big gap between its min and max. The scale elasticity  o defined above remains between the min and max, but has much small allowance. 7. An empirical study In this section we apply our scheme to a data set comprising 37 Japanese National Universities with the faculty of medicine. 7.1. Data Table 10 exhibits the data set of Japanese National Universities with the faculty of medicine at the year 2008 (Report by Council for Science and Technology Policy, Japanese Government, 2009). We chose two inputs: (I) Subsidy (unit: one million Japanese yen) and (I) No. of faculty, and three outputs: (O) No. of publication, (O) No. of JSPS (Japan Society for Promotion of Sciences) fund and (O) No. of funded research. We classified them into four clusters: A, B, C and D depending on the sum of No. of JSPS fund and No. of funded research. Cluster A is defined as the set of universities with the sum larger than 2000, Cluster B between 2000 and 1000, Cluster C between 1000 and 500, and Cluster D less than 500. 20 Table 10: Data set 6359 4776 3786 4009 2605 2560 2443 (O) JSPS fund 2896 2304 1952 1941 1396 1310 1351 (O) No. of funded res. 2280 1504 1382 1357 1186 922 796 1667 1814 1567 1303 1505 1549 1362 1089 1143 1264 911 811 751 606 606 507 543 401 453 430 B B B B B 19200 17569 20467 16124 14515 17154 13196 12357 14850 13138 16884 14589 14436 1129 1010 1224 1151 867 1084 898 830 799 855 1121 970 976 803 722 706 582 643 685 481 446 628 576 531 562 550 537 446 428 309 351 378 325 242 266 353 311 277 311 314 302 317 418 321 284 329 357 319 228 265 274 229 C C C C C C C C C C C C C 10631 11319 10202 10953 13017 11355 11522 10637 8936 11054 10888 10686 629 795 657 668 859 775 779 785 656 692 749 645 293 465 300 311 382 339 391 287 267 343 323 254 199 190 170 184 201 191 162 174 157 158 157 152 231 233 240 191 159 156 171 142 153 134 132 135 D D D D D D D D D D D D University (I)Subsidy (I)Faculty (O)Publication A1 A2 A3 A4 A5 A6 A7 96174 60868 50717 50615 42398 41014 35985 4549 3562 2619 2877 2207 2086 1792 B1 B2 B3 B4 B5 48106 28896 22898 18245 18255 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 Cluster A A A A A A A Figure 13 plots 47 universities regarding no. of faculty (input) and no. of publication (output). Globally non-convex characteristics are observed. Especially between big seven universities (A) and other universities (B, C and D), there is a gap. We can see similar gaps among other inputs vs. outputs. 21 Faculty vs. Publication 7000 6000 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 Figure 13: Plot of no. of faculty (horizontal) vs. no. of publication (vertical) 7.2. Adjusted score Table 11 compares the three scores and Figure 14 displays them graphically. Table 11: Comparisons of CRS, VRS and SAS (Adjusted score) DMU CRS-I VRS-I SAS-I DMU CRS-I VRS-I SAS-I DMU CRS-I VRS-I SAS-I A1 0.9246 1 0.9943 C1 0.6824 0.9003 0.9232 D1 0.7301 1 0.9272 A2 0.9764 1 0.9994 C2 0.626 0.8921 0.8885 D2 0.6406 0.9857 0.8742 A3 1 1 1 C3 0.5265 0.7342 0.7287 D3 0.7604 1 0.9426 A4 1 1 1 C4 0.8013 0.8563 0.9872 D4 0.5777 0.9514 0.8033 A5 1 1 1 C5 0.7398 0.9713 0.938 D5 0.394 0.814 0.6426 A6 0.8415 0.9036 0.9891 C6 0.5478 0.8149 0.769 D6 0.4349 0.8796 0.6904 A7 1 1 1 C7 0.7865 0.9994 0.9545 D7 0.4713 0.916 0.7009 B1 0.6126 0.6776 0.9628 C8 1 1 1 D8 0.4089 0.8646 0.6481 B2 0.6645 0.7642 0.8576 C9 0.7554 1 0.9402 D9 0.523 1 0.7725 B3 0.7476 0.8759 0.963 C10 0.626 1 0.8601 D10 0.4029 0.9521 0.6556 B4 0.7794 1 0.9513 C11 0.5005 0.7255 0.6506 D11 0.3847 0.8991 0.6162 B5 0.7395 1 0.9321 C12 0.5985 0.8543 0.7641 D12 0.4206 0.9504 0.6381 C13 0.5107 0.843 0.7192 22 CRS-I VRS-I SAS-I 1.2 1 0.8 0.6 0.4 0.2 A1 A2 A3 A4 A5 A6 A7 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 0 Figure 14: Comparisons of three scores The SAS of B1, B2 and B3 are remarkably larger than those of VRS, demonstrating the nonconvex structure of the data set. Universities in Cluster A are judged almost efficient by adjusted scores. Table 12 summarizes averages of CRS, VRS and SAS for each cluster. For Cluster A universities, gaps among three scores are small and have the highest marks in each model. For Cluster B universities, the average SAS is larger than the average of VRS scores. This indicates the existence of non-convex frontiers around B sized universities. For Cluster C universities, discrepancy between CRS and VRS comes large, and the average of SAS is between them, closer to VRS. For Cluster D universities, the discrepancy comes largest indicating the smallest scale-efficiency. Adjusted scores position around the middle of CRS and VRS. Average SAS decreases monotonically from A to D. Table 12: Average scores Cluster CRS-I VRS-I SAS-I A 0.9632 0.9862 0.9975 B 0.7087 0.8635 0.9334 C 0.6693 0.8916 0.8556 D 0.5124 0.9344 0.7426 7.3. Scale elasticity Table 14 reports the scale elasticity  computed by the formula (26). 23 Table 14: Scale elasticity DMU A1 A2 A3 A4 A5 A6 A7 Scale El. 0.961 0.9954 1.0267 1.0299 1.0525 1.051 1.0669 DMU B1 B2 B3 B4 B5 Scale El. 1.1522 1.0915 1.1965 1.3262 1.2003 DMU C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 Scale El. 1.137 1.422 1.296 1.152 1.416 1.33 1.197 1.139 1.311 1.56 2.043 2.02 1.56 DMU D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 Scale El. Ave. Max Min StDev 1.0262 1.0669 0.961 0.0369 Ave. Max Min StDev 1.1933 1.3262 1.0915 0.0863 Ave. Max Min StDev 1.429 2.043 1.137 0.303 Ave. Max Min StDev 1.642 1.9736 0.6433 0.4143 1.6564 1.0532 1.7399 3.1328 1.9453 2.034 1.9234 3.5783 2.1912 2.0527 2.1179 2.1913 We observe that for Cluster A universities the scale elasticity is almost unity with the max 1.0669 and min 0.961. This cluster exhibits constant returns-to-scale. Clusters B, C and D universities have elasticity higher than unity and the averages are increasing in this order. They have increasing returns-to-scale characteristics. 8. The radial model case In this section, we apply the above approaches to the radial DEA models. 8.1. CCR and BCC models Throughout this section, we utilize the input-oriented radial measures: CCR (CharnesCooper-Rhodes (1978)) and BCC (Banker-Charnes-Cooper (1984)) models, for the efficiency evaluation of each DMU ( xo , yo ) (o  1, , n) as follows: [CCR]  oCCR  min  subject to Xλ   x o Yλ  y o λ  0,  : free. 24 (22) [BCC]  oBCC  min  subject to Xλ   x o Yλ  y o (23) eλ  1 λ  0,  : free, where λ  R n is the intensity vector. Although we present our model in the input-oriented radial model, we can develop the model in the output-oriented radial model as well. We define the scale-efficiency (  o ) of DMUo by o  oCCR . oBCC (24) 8.2. Decomposition of slacks We decompose CRS score into scale-independent and –dependent parts as follows: The radial input-slacks can be defined as so  (1  oCCR )xo  Rm . (25) We decompose the radial input-slacks into scale-dependent and scale-independent slacks as: so  (1   o )so   oso Scale-dependent input slacks soScaleDep   (1   o )so  (1   o )(1  oCCR )xo Scale-independent input slacks soScaleIndep    oso   o (1  oCCR )xo (26) (27) 8.3. Scale-adjusted input and output We define scale-adjusted input x o and output y o by xo  xo  soScaleDep   ( o  oCCR   ooCCR )xo yo  yo . [Definition 1] (Scale-adjusted score) We define scale-adjusted score by 25 (28) oscale   o  oCCR   ooCCR . (29) x o is the scale accounted (free) input. We have the following propositions. [Proposition 5] 1   o  oCCR   ooCCR  max(oCCR ,  o ) (30)  o  oCCR   ooCCR  1if and only if  o  1. (31) [Proposition 6] Proofs are in Appendix A. 8.4. In-cluster issue: Scale&cluster-adjusted score (SAS) In this section we introduce the cluster of DMUs and define the scale&cluster-adjusted score (SAS). We classify DMUs into several clusters depending on their characteristics. We denote the name of cluster DMUj by Cluster(j) ( j  1, , n) . 8.5. Solving the CCR model in the same cluster We solve the input oriented CCR model for each DMU (xo , y o ) (o  1, , n) referring to the ( X, Y) in the same Cluster (o) which can be formulated as follows:  ocl*  min  ocl subject to Xμ   ocl xo  0 Yμ  y o  j  0 (j : Cluster( j )  Cluster(o)) μ  0,  ocl : free. 26 (32) Scale&cluster adjusted data (projection) (xo , y o ) is defined by: Scale&cluster-adjusted input (Projected Input) xo   ocl* xo   ocl* ( o   oCCR   o oCCR )xo Output (33) yo  yo. [Definition 2] (In-cluster score) We define  ocl* as in-cluster score. Up to this point, we deleted scale demerits and in-cluster slacks from the data set. Thus, we have obtained a scale free and in-cluster slacks free (projected) data set ( X, Y). 8.6. Scale&cluster-adjusted Score (SAS) [Definition 3] (Scale&cluster-adjusted score) In the input-oriented case, the scale&cluster-adjusted score (SAS) is defined by Scale&cluster-adjusted score (SAS) oSAS  ocl* ( o  oCCR   ooCCR ) (34) . Similarly to Propositions 1 to 4, we have the followings. [Proposition 7] The scale-cluster adjusted score (SAS) is not less than the CCR score. oSAS  oCCR . (35) [Proposition 8] If oCCR  1 then it holds oSAS  oCCR , but not vice versa. [Proposition 9] The scale-cluster adjusted score (SAS) is decreasing in the increase of input and in the decrease of output so long as the both DMUs remain in the same cluster. [Proposition 10] The SAS-projected DMU (xo , y o ) is radially efficient under the SAS model among the DMUs in the cluster it belongs to. It is also CCR and BCC efficient among the DMUs in its cluster. 9. Concluding remarks 27 We have developed a scale&cluster-adjusted DEA model assuming scale-efficiency and cluster of DMUs. This model can deal with S-shaped frontiers smoothly. The adjusted score (SAS) reflects the inefficiency of DMUs after deleting the inefficiency caused by scale demerits and accounting in-cluster inefficiency. We also propose a new scheme for evaluation of scale elasticity. We applied this model to a data set comprising Japanese universities. The managerial implications of this study are as follows. (1) We are free from the big difference in CRS and VRS scores. Hence, use of DEA becomes more convenient and simple. (2) We need not any statistical tests on the range of the intensity vector λ. (3) We can cope with the non-convex frontiers, e.g. S-shaped curve. In such cases, VRS scores are too stringent to the DMUs. The optimal slacks are not necessarily determined uniquely. In such a case, we can set the “importance level” of input (output) items and can solve the associated linear programs recursively. Although we presented the scheme in input-oriented form, we can extend it to output-oriented and non-oriented (both-oriented) model. Future research subjects include studies in alternative scale-efficiency measures other than the CRS/VRS ratio and clustering methods. Extensions to cost, revenue and profit models are also our future research subjects. References Avkiran, N.K. (2001) Investigating technical and scale efficiencies of Australian universities through data envelopment analysis. Socio-Economic Planning Science, 35, 57-80. Avkiran, N.K, Tone, K. and Tsutsui, M. (2008). Bridging radial and non-radial measures of efficiency in DEA. Annals of Operations Research, 164, 127-138. Banker, R. D., Charnes, A. and Cooper, W. W. (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30, 10781092. Banker, R.D. and Thrall, R. M. (1992) Estimation of returns to scale using data envelopment analysis. European Journal of Operational Research, 62, 74-84. Banker, R. D., Cooper, W. W., Seiford, L. M., Thrall, R. M. and Zhu, J. (2004) Returns to scale in different DEA models. European Journal of Operational Research, 154, 345362. Bogetoft, P. and Otto, L. (2010) Benchmarking with DEA, SFA, and R. Springer. 28 Charnes A., Cooper W. W. and Rhodes, E. (1978) Measuring the efficiency of decision – making units. European Journal of Operational Research, 2, 429-444. Cooper, W. W., Seiford, L. M. and Tone, K. (2007) Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software. Spriger. Dekker, D. and Post, T. (2001) A quasi-concave DEA model with an application for bank branch performance evaluation. European Journal of Operational Research, 132, 296311. Färe, R. and Primond, D. (1995) Multi-output Production and Duality: Theory and Application. Kluwer Academic Press. Førsund, F.R. and Hjalmarsson, L. (2004) Are all scales optimal in DEA? theory and empirical evidence. Journal of Productivity Analysis, 21, 25-48. Førsund, F.R. and Hjalmarsson, L. (2004) Calculating scale elasticity in DEA models. Journal of the Operational Research Society, 55, 1012-1038. Kousmanen, T. (2001) DEA with efficiency classification preserving conditional convexity. European Journal of Operational Research, 132, 326-342. Olesen, O. B. and Petersen, N. C. (2013) Imposing the Regular Ultra Passum law in DEA models. Omega: The International Journal of Management Science, 41, 16–27. Podinovski, V.V. (2004) Local and global returns to scale in performance measurement. Journal of the Operational Research Society, 55, 170-178. Tone, K. (2001) A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research,130, 498-509. Appendix A Proof of Propositions Let us define the production possibility sets P( X, Y) and P( X, Y) for (x j , y j ) and (x j , y j ) ( j  1, , n) , respectively by  P( X, Y)  (x, y ) x    y  , λ  0 . P( X, Y)  (x, y ) x   j 1 x j  j , 0  y   j 1 y j  j , λ  0 n n n x j  j , 0  y   j 1 n j 1 j j [Lemma 1] P(X, Y) = P( X, Y) . Proof . We define the scale&cluster-adjusted DMU (x j , y j ) ( j  1, 29 , n) by (A1) x j  x j  (1   j )s j * y j  y j  (1   j )s j * . (A2) If  j  1 (DMUj is efficient), then we have x j  x j and y j  y j . If  j  1 (DMUj is inefficient), then x j  x j  (1   j )s j *  x j  s j * y j  y j  (1   j )s j *  y j  s j * , where (x j  sj * , y j  sj * ) ( x j , y j ) ( j  1, (A3) is the projection of (x j , y j ) onto the P(X,Y) frontiers. Thus, , n) belongs to P(X,Y). Hence, efficient frontiers are common to P(X,Y) and P( X, Y) . Q.E.D. [Proposition 1] oSAS  oCRS (o  1, , n). Proof. The CRS scores for (xo , y o ) and ( xo , y o ) are, respectively, defined by [CRS]  oCRS  min1  1 m si  m i 1 xio subject to Xλ  s   xo (A4)  Yλ  s  y o λ  0, s   0, s   0. and [SAS]  oSAS  min1  1 m sicl   (1   o ) si*  m i 1 xio subject to Xμ  s cl   xo Yμ  s cl   yo  j  0 (j : Cluster( j )  Cluster(o)) μ  0, s cl   0, s cl   0. We prove this proposition in two cases. (Case 1) All DMUs belong to the same cluster. In this case (A5) comes to: 30 (A5) [SAS]  SAS o 1 m ti  (1   o ) so*  min1   i 1 m xio subject to Xλ  t   xo (A6)  Yλ  t  y o λ  0, t   0, t   0. * * * Let (λ , t , t ) be an optimal solution for (A5). Since P( X, Y) = P( X, Y) and both sets have the same efficient DMUs which span (xo , y o ) , we have Xλ *  t *  xo  xo  (1   o )so* (A7) Yλ *  t *  y o  y o  (1   o )so* Hence, we have Xλ *  t *  (1   o )so*  xo (A8) Yλ *  t *  (1   o )so*  y o . * * * * * This indicates that (λ , t  (1   o )so , t  (1   o )so ) is feasible for (A4) and hence CRS its objective function value is not less than the optimal value  o . oSAS  1  1 m ti*  (1   o ) sio*  oCRS .  i 1 m xio (A9) (Case 2) Multiple clusters exist. In this case, we have additional constraints to (A6) for the cluster restriction as follows. [SAS]  oSAS  min1  1 m ti  (1   o ) si*  m i 1 xio subject to Xλ  t   xo (A10) Yλ  t   y o  j  0 (j : Cluster( j )  Cluster(o)) λ  0, t   0, t   0. Since adding constrains result in an increase in the objective value, it holds that oSAs  oCRS . (A11) Q.E.D. [Proposition 2] If oCRS  1 then it holds oSAS  1 , but not vice versa. * * CRS Proof. If o  1 then, we have so  0 and so  0 . Hence we have Total slacks = 0 and oSAS  1 . The converse is not always true as demonstrated by the example below where all DMUs belong to an independent cluster. 31 DMU A B C (I)x 2 4 6 (O)y 2 2 2 Cluster a b c DMU A B C CRS-I 1 0.5 0.3333 SAS-I 1 1 1 Cluster a b c Q.E.D. [Proposition 3] The scale&cluster-adjusted score (SAS) is decreasing in the increase of input and in the decrease of output so long as the both DMUs remain in the same cluster.     Proof. Let x p , y p and xq , y q with x p  xq and y p  y q be respectively the original and varied DMUS in the same cluster. Since the projected point of     x p , y p  on the SAS frontiers is feasible for x q , y q and slacks between x q , y q and the frontier point are larger than the slacks between x p , y p  and the frontier point. We have this proposition. Q.E.D. [Proposition 4] The projected DMU (xo , y o ) is efficient under the SAS model among the DMUs in the cluster it belongs to. It is also CRS and VRS efficient among the DMUs in its cluster. Proof. From the definition of (xo , y o ) it is SAS efficient. It is also CRS (VRS) efficient in its cluster. Q.E.D. [Proposition 5] 1   o  oCCR   ooCCR  max(oCCR ,  o ) (A12)   CCR CCR CCR CCR CCR CCR . Proof.  o  o   oo   o (1  o )  o  o (1   o )   o  max  o ,o This term is increasing in  o and is equal to 1 when  o =1. Q.E.D. [Proposition 6]  o  oCCR   ooCCR  1if and only if  o  1. CCR CCR Proof. If  o  1 , it holds  o  o   oo  1. 32 (A13) Conversely, if  CCR o  o  oCCR   ooCCR  1 , we have  o (1  oCCR )  1  oCCR  1   o  1, else if  CCR o 1 BCC o  1and  o  1. 33 Hence, if Q.E.D.