A study on directional returns to scale

Journal of Informetrics 8 (2014) 628–641

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Journal of Informetrics journal homepage: www.elsevier.com/locate/joi

A study on directional returns to scale Guo-liang Yang a,∗ , Ronald Rousseau b,c , Li-ying Yang d , Wen-bin Liu e a b c d e

Institute of Policy and Management, Chinese Academy of Sciences, Beijing 100190, PR China Institute for Education and Information Sciences, IBW, University of Antwerp (UA), Antwerp B-2000, Belgium KU Leuven, Department of Mathematics, Leuven B-3000, Belgium National Science Library, Chinese Academy of Sciences, Beijing 100190, PR China Kent Business School, University of Kent, Canterbury CT2 7PE, UK

a r t i c l e

i n f o

Article history: Received 31 December 2013 Received in revised form 5 May 2014 Accepted 5 May 2014

Keywords: Returns to scale (RTS) Directional returns to scale Congestion Directional congestion Evaluation of research institutes

a b s t r a c t This paper investigates directional returns to scale (RTS) and illustrates this approach by studying biological institutes of the Chinese Academy of Sciences (CAS). Using the following input–output indicators are proposed: senior professional and technical staffs, middle level and junior professional and technical staffs, research expenditure on personnel salaries and other expenditures, SCI papers, high-quality papers, graduates training and intellectual properties, the paper uses the methods recently proposed by Yang to analyze the directional returns to scale and the effect of directional congestion of biological institutes in Chinese Academy of Sciences. Based on our analysis we come to the following findings: (1) we detect the regime of directional returns to scale (increasing, constant, decreasing) for each biological institute. This information can be used as the basis for decision-making about organizational adjustment; (2) congestion and directional congestion occurs in several biological institutes. In such cases the outputs of these institutes decrease when the inputs increase. Such institutes should analyze the underlying reason for the occurrence of congestion so that S&T resources can be used more efficiently. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Efficiency in the use of scientific and technological resources is one of the important issues of concern for S&T management. During the period 2007–2011, for instance, the national financial allocation on science and technology (S&T) in China reached 1340.8 billion Yuan (RMB) which is 2.7 times more than in the period 2001–2005. Moreover, according to the “National Scientific and Technological Development Program in the 12th Five Year Plan” published by the Ministry of Science and Technology of China (MOST) in 2011, China’s R&D staff reached nearly 2.6 million people in 2010 and ranks first in the world (in absolute numbers). Yet, it is a fact of life that S&T resources are always scarce with respect to aspirations. For this reason any S&T management department is faced with the problem of rational allocation of resources, aiming at the most efficient and effective use possible. Addressing these issues we explain the method of returns to scale (RTS) in relation to efficiencies (and deficiencies) in utilization of S&T resources. Our method is illustrated by the case of the biological institutes of the Chinese Academy of Sciences (CAS). The Chinese Academy of Sciences (CAS) is the leading academic institution in China performing comprehensive research and development in the natural sciences, technology and innovation. Nowadays CAS has 12 branch offices, 117 institutes

∗ Corresponding author. Tel.: +86 10 59358816. E-mail address: [email protected] (G.-l. Yang). http://dx.doi.org/10.1016/j.joi.2014.05.004 1751-1577/© 2014 Elsevier Ltd. All rights reserved.

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as legal entities, more than 100 national engineering research centers and national key laboratories (see, e.g., Jin, Rousseau, & Sun, 2005, 2006 for more information on national key labs) and about 1000 field stations throughout the country. CAS employs more than 50,000 staff members. In 1998, supported by the Chinese Government, CAS launched the Pilot Project of the Knowledge Innovation Program (PPKIP) in an effort to build a national innovation system. During the application period of this program inputs and outputs of CAS grew significantly. For example: the total income increased from 493.598 million Yuan (RMB) in 1998 to 1703.971 million Yuan (RMB) in 2007. Moreover, the number of full time equivalent (FTE) personnel increased from 30,611 in 1998 to 44,307 in 2007; the number of SCI papers increased more than threefold from 5474 in 1998 to 23,674 in 2010. CAS’ ability to gain extra external funding improved continually. Data show an exponential increase in research funding and in high-quality papers (Zhang, Banker, Li, & Liu, 2011). Under these circumstances developing methods for analyzing the performance of these institutes became an important issue. Evaluation of CAS began in 1993. Over the past decade, models and methodologies for institutional evaluation were continuously adjusted according to different stages in experience and development. Examples of such evaluation methodologies are: “Blue Book” evaluation, “PPKIP” evaluation, “Innovation Capability Index” evaluation, “Comprehensive Quality” evaluation and “Major R&D Outcome-oriented System” evaluation (Bai, 2012). Quantitative monitoring is an important component of the current evaluation system. It makes use of indicators (such as the amount of research funding and the number of high-quality papers) for multiple inputs and outputs of each institute in CAS. The results of quantitative data monitoring show the relation of the rate of increase in output/production to the corresponding increase in inputs. In the long run all production factors are variable changing according to an increase in size and/or scale. In this context we recall the Law of Returns to Scale. This is a set of three inter-related and chronological laws (stages): the Law of Increasing Returns to Scale, the Law of Constant Returns to Scale, and the Law of Decreasing Returns to Scale. If output increases by the same proportional change then there are constant returns to scale (CRS). If output increases by less than proportional change, there are decreasing returns to scale (DRS). If output increases by more than proportional change, there are increasing returns to scale (IRS) We claim that the RTS method is important for the analysis of organizational performance. It can help decision-makers (DMs) decide if the size of the organization should be expanded or reduced. RTS is a classic economic concept tied to the relationship between production factors and variation of outputs (Pindyck & Rubinfeld, 2000). The traditional definition of RTS in economics (Banker, 1984; Pindyck & Rubinfeld, 2000) is based on the idea to measure radial changes in outputs (that is, all output components change in the same proportion) caused by those of all inputs. In some real applications, however, the increase in scale is often caused by inputs changing in unequal proportions. Based on the above thinking, Yang (2012) introduced directional RTS from a global and local (directional scale elasticity) perspective under Pareto preference, and gives specific formulations of directional RTS. Illustrating this method this paper estimates directional returns to scale of the biological institutes in CAS. Is the massive financial investment in S&T resources by the government used efficiently? Is the scale on which these institutes run optimal? All these questions relate to the rational allocation of limited S&T resources and are basic information for S&T policy. The paper is a totally reworked and greatly expanded version of (Yang, Yang, Liu, Li, & Fan, 2013). It is organized as follows: Section 2 presents the methodology used in this paper to analyze directional returns to scale (RTS). It is based on the methods proposed by Yang (2012). In Section 3, we will show the results of our analysis, including the directional RTS and the directional congestion effect. The conclusion follows in Section 4. 2. Methodology 2.1. Input–output Indicators In the latest years performance management has become more and more common in government managed organizations. This evolution is due to the following main factors: (a) increased demand for accountability by governing bodies, press, and the general public, and (b) a growing commitment by organizations to focus on improved performance (Poister, 2003). The development of performance management during the past decade indicates a change from the “output (result)” model to the “objective–process–result” model (Zhang, Yang, & Li, 2011; Zhang, Banker, et al., 2011). Geisler (2000) defined metrics for evaluating scientific work as “a system of measurement that includes: (1) the objective being measured; (2) the units to be measured; (3) the values of these units.” Keeney and Gregory (2005) studied how to effectively select measures, i.e., assessment indicators, to determine if bodies, such as government-managed organizations, meet their targets. Roper, HewittDundas, & Love (2004) discussed indicators for the pre-assessment of public R&D funding, based on the idea that increased knowledge provides beneficial outcomes to society. Moreover, Soft Systems Methodology (SSM) is another approach to systematically analyze the operation of research institutions and to build a more complete and reasonable set of evaluation indicators based on the “3E” theory (Efficacy, Efficiency, Effectiveness) (Mingers, Liu, & Meng, 2009). Zhang, Yang, et al. (2011) proposed strategy maps for National Research Institutes based on discussions of general rules of research activities so that managers can describe organizational strategies more clearly, accurately and logically. How to improve national research institutes’ efficiency in S&T resource utilization is an important management issue. Indeed, if research institutes make more efficient use of their resources they become able to play a more important role in economic development, social progress and national defence. In practical evaluation of CAS institutes, dozens of

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quantitative indicators (e.g., publications, awards, patents, staff, talents, funds, graduate training) are used to monitor their annual development. In this paper, we mainly refer to the succinct indicators applied in Liu, Zhang, Meng, Li, and Xu (2011) and Yang et al. (2013) to analyze the directional RTS for the biological institutes in CAS. Furthermore, we expanded the input indicators from Staff and Research Expenditure (Res. Expen.) used in Liu et al. (2011) and Yang et al. (2013) to four detailed ones to provide more information. These indicators are: (1) Four input indicators: Senior professional and technical staffs (Senior), Middle level and junior professional and technical staffs (Middle & Junior), Research expenditure on personnel salaries (Personnel Salaries) and Other expenditures (Other Expen.); (2) Four output indicators: SCI & SSCI publications (SCI Pub.), Publications in high-impact journals (High Pub.), Graduate Enrolment (Grad. Enrol.) and Intellectual Properties (IPs). As input indicators, professional and technical staffs and Res. Expen. reflect the total number (as FTE) of those who have acquired professional or technical titles or who are in the probation period of those titles in each biological institute and the total research expenditure of each institute. Professional and technical staffs can be divided into Senior and Middle & Junior. Res. Expen. can be divided into Personnel Salaries and Other Expen. Explanatory notes on these input indicators can be found in Appendix A. As output indicators, SCI Pub. and High Pub. denote the number of published papers indexed in the Science Citation Index (SCI) and Social Science Citation Index (SSCI) and the number of published papers in journals ranking among the top 15% with highest impact factor in a JCR field. Grad. Enrol. denotes the number of master and Ph.D. students currently studying in the institute. These output indicators are standard indicators to evaluate the performance of basic and applied basic research. However, these institutes are also active in technological development. As such, we added an output indicator named Intellectual Properties (IPs) to reflect the technology development of these institutes. The IPs indicator is obtained as an aggregate of number of patents (applied and granted), new varieties of plants and animals, software copyrights, new medicine, standards and similar activities. The institute’s IPs score is obtained as a weighted sum of scores determined by experts (different ones for each aspect). The process is as follows: Step 1: Experts determine the criteria for scoring different IPs for all institutes. For example, one applied patent receives a score of 3 and one granted patent a score of 5; Step 2: CAS collects the data of each institute and each IP and allocates scores for each of them; Step 3: CAS sums up the scores of all kinds of IPs to obtain the composite scores of each institute. Based on these indicators, we investigate the directional RTS and directional congestion of these biological institutes in CAS. 2.2. Data In order to obtain reliable analytical results, we use the input–output data of these 15 institutes in 2010 and 2011 respectively to form a dataset of 30 decision making units (DMUs) with 4 input indicators and 4 output indicators. The input–output data of the 30 DMUs is shown in Table 1. 2.3. Analysis methods Yang (2012) proposed the definition of directional scale elasticity in economics in the case of multiple inputs and multiple outputs using a multi-dimensional production function F. Yet, most of the time such a production function cannot be formulated in the public sector. Therefore, in practice, the DEA (data envelopment analysis) method is one of the most commonly used approaches for the analysis of RTS in the public sector (Fox, 2002).1 The estimation of RTS of DMUs using the DEA method was investigated first by Banker (1984) and Banker, Charnes, and Cooper (1984). Banker (1984) translated the definition of the RTS from classical economics into the framework of the DEA method, using the CCR-DEA model with radial measure to estimate the RTS of evaluated DMUs. Soon after that, Banker et al. (1984) proposed the BCC-DEA model under the assumption of variable RTS, and investigated how to apply the BCC-DEA model to estimate the RTS of DMUs. Existing RTS measurements in DEA models are all based on the definition of RTS in the DEA framework made by Banker (1984), who introduced the RTS in economics into the DEA framework and proposed a method to determine the RTS of DMUs in DEA models. This extended the area of application of DEA from relative efficiency evaluation to RTS measurement. RTS measurement is also conducted in the field of information science e.g., Haynes, Pollack, and Nordhaus (1983) conducted an empirical study on RTS for the Association of Research Libraries (ARL) applying input/output analysis based on a Cobb-Douglas production function. They argued that the applicability of this function to research libraries has been proved. Schubert (2014) used non-parametric techniques of multidimensional efficiency measurement (e.g., DEA) to analyze the RTS in scientific production based on survey data for German research groups from three scientific fields. DEA can also be used as a performance analysis tool to evaluate the efficiency of S&T organizations as shown by Johnes and Johnes (1992). Rousseau and Rousseau (1997, 1998) assessed the efficiency of countries using GDP, active population and R& D expenditure

1 Note: DEA is a nonparametric tool and the efficient frontier of Production Possibility Set (PPS) generated from DEA models can simulate the production function with multiple inputs and outputs. Banker (1993) argued that the efficient frontier is biased below the true efficient frontier for a finite sample size. Smith (1997) reported that in the deterministic setting assumed, using a convex production function, a well-specified DEA model will always overestimate efficiency. In effect, a larger sample size increases the possibility of encountering DMUs close to the production frontier, and therefore the DEA frontier approaches the true frontier asymptotically as sample size increases.

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Table 1 Input–output data of the 30 DMUs. DMUs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30

Inputs

Outputs

Professional and technical staffs (FTE)

Res. Expen. (RMB million)

Senior

Middle & junior

Personnel salaries

Other Expen.

190 140 76 146 160 179 40 488 172 62 118 118 77 117 75 204 152 81 164 170 203 45 523 181 70 126 119 77 139 78

347 174 71 233 233 257 163 1090 351 98 103 145 204 229 70 363 189 82 251 267 221 196 1090 388 131 154 149 231 246 78

104.64 73.13 35.67 76.93 78.81 100.24 27.99 454.12 141.52 28.79 47.04 40.52 36.93 57.85 21.48 144.38 106.23 42.51 96.68 92.31 124.69 29.42 495.68 122.11 37.71 54.21 54.98 49.82 72.89 26.53

164.19 186.71 63.65 130 266.96 401.38 63.98 541 340.98 85.93 140.97 54.57 83.01 94.73 36.67 241.79 204.79 64.32 137.89 270.07 377.24 85.19 687.13 330.03 59.27 126.62 78.55 106.47 138.96 52.68

SCI Pub. (number)

High Pub. (number)

Grad. Enrol. (number)

IPs (score)

325 368 207 256 259 216 112 785 385 118 216 125 189 313 83 357 419 210 276 296 221 129 702 390 104 231 185 187 343 70

105 109 66 62 96 93 39 323 125 36 63 37 66 64 17 137 142 91 79 138 122 46 359 133 31 62 58 71 77 20

604 477 241 388 500 553 190 1488 417 235 481 267 232 302 126 612 482 242 396 518 555 202 1510 424 237 484 277 237 318 138

23.6 10.4 0.45 27.6 17.2 30.9 0.2 63.4 124.05 11.3 5.6 15.7 8.65 32.2 9.4 26.5 8.8 0 36.8 7.8 38.8 3.2 84.3 184.8 12.2 11.6 22 11.2 42.4 7.4

Data source: (1) Data of SCI Pub., High Pub. and IPs are from Monitoring data of institutes in CAS in 2011 and 2012; (2) Data of four inputs and Grad. Enrol. are from the Statistical yearbooks of CAS in 2011 and 2012. Note: Data were obtained from these institutes for the period of Jan. 01, 2010–Dec. 31, 2010 and Jan. 01, 2011–Dec. 31, 2011, respectively.

as inputs, and publications and patents as outputs. They showed that DEA can be used in scientometrics as a tool to measure the efficiency of decision making units (DMUs) (e.g., countries) by gauging closeness to the efficiency frontier. Similar techniques were demonstrated by (Kao & Lin, 1999; Roy & Nagpaul, 2001; Shim & Kantor, 1998). Yang and Chang (2009) used DEA under constant and variable RTS to measure firms’ efficiencies while Guan and Chen (2010) applied non-radial DEA to benchmark various R&D efficiency indexes, including scale efficiency. When studying the standard university model Brandt and Schubert (2013) observed that universities are large agglomerations of many (often loosely affiliated) small research groups. They explained this observation by typical features of the scientific production process. In particular, they argued that there are decreasing RTS on the level of the individual research groups. Somewhat similar observations (decreasing RTS) were already published by Bonaccorsi and Daraio (2005). RTS is a classic economic concept describing the relationship between changes in the scale of production and output. The traditional definition of RTS in economics is based on the idea to measure radial changes in outputs caused by changes in all inputs. Yang (2012) argued that due to the complexity of research activities in research institutions, it is often the case that production factors are not necessarily tied together proportionally, and inputs change non-proportionally. Based on this observation, he introduced directional RTS from a global and local (directional scale elasticity) perspective under Pareto preference, and gave specific formulations of directional RTS and of corresponding models. In addition, he demonstrated that traditional RTS is a special case of directional RTS in the radial direction (that is, all components of inputs or outputs change in the same proportion), so that directional RTS can provide a basis for decision-making about further development of such production processes. He gave the definition of directional RTS in a DEA framework based on the production possibility set (PPS) as follows. We consider a set of n observations of actual production possibilities (Xj , Yj ), j = 1, . . ., n, where the output vector Yj can be produced from the input vector Xj . Then the following definitions were introduced: Definition 1.

The PPSs under the assumption of variable RTS and congestion are defined as follows:

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(1) PPS under the assumption of variable RTS (Banker, 1984): PBCC (X, Y ) =

⎧ ⎨

(X, Y )|X ≥

n 

⎩

j Xj , Y ≤

n 

j=1

j Yj ,

n 

j=1

⎫ ⎬

j = 1, j ≥ 0, j = 1, . . ., n

(2) PPS under the assumption of congestion (Wei & Yan, 2004): PConvex (X, Y ) =

Definition 2.

⎧ ⎨

(X, Y )|X =

⎩

n 

j Xj , Y ≤

j=1

n 

j Yj ,

j=1

(1)

⎭

j=1

n 

⎫ ⎬

j = 1, j ≥ 0, j = 1, . . ., n

(2)

⎭

j=1

The weakly and strongly efficient frontiers of PPS are defined as follows.

(1) Weakly efficient frontier: EFweak =





¯ Y¯ ) ∈ PPS such that (−X, ¯ Y¯ ) > (−X, Y ) (X, Y ) ∈ PPS|there is no (X,

(2) Strongly efficient frontier: EFstrong =



(3)



¯ Y¯ ) ∈ PPS such that (−X, ¯ Y¯ ) > (−X, Y ) and (X, ¯ Y¯ ) = (X, Y ) ∈ PPS|there is no (X, / (X, Y )

(4)

m , Y ∈ Rs , we let Definition 3 (directional RTS). Assuming DMU(Y0 , X0 ) ∈ PPS and X0 ∈ R+ 0 +





ˇ(t) = max ˇ|(˝t X0 , ˚ˇ Y0 ) ∈ PPS, t = / 0

T

m and (ı , . . ., ı ) ∈ Rs represent where ˝t = diag {1 + ω1 t, . . ., 1 + ωm t} and ˚ˇ = diag {1 + ı1 ˇ, . . ., 1 + ıs ˇ}; (ω1 , . . ., ωm )T ∈ R+ s 1 +

input and output directions respectively and satisfy We let

m

i=1

ωi = m;

s

ı r=1 r

= s where t, ˇ are input and output scaling factors.

− = lim

ˇ(t) t

(5)

+ = lim

ˇ(t) t

(6)

t→0−

t→0+

Then we have (a) if − > 1 (or + > 1) holds, then increasing directional RTS takes place on the left-hand (or right-hand) side of this point (Y0 , X0 ) in the direction of (ω1 , ω2 , . . ., ωm ) and (ı1 , ı2 , . . ., ıs ); (b) if − = 1 (or + = 1) holds, then constant directional RTS takes place on the left-hand (or right-hand) side of this point (Y0 , X0 ) in the direction of (ω1 , ω2 , . . ., ωm ) and (ı1 , ı2 , . . ., ıs ); (c) if − < 1 (or + < 1) holds, then decreasing directional RTS takes place on the left-hand (or right-hand) side of this point (Y0 , X0 ) in the direction of (ω1 , ω2 , . . ., ωm ) and (ı1 , ı2 , . . ., ıs ). Without going into details the method (Upper and Lower Bound Method, ULBM) proposed in (Yang, 2012) is as follows (but more details are available from the authors upon request). The directional scale elasticity of a strongly efficient DMU ω1 = ω2 on the strongly efficient frontier in the BCC-DEA model (denoted as PBCC (X, Y )2 ), can be determined through the following Model (7): V T WX0 ( ¯ ) = max(min) T U Y0

⎧ T U Yj − V T Xj + 0 ≤ 0, j = 1, . . ., n ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T T

s.t.

(7)

U Y0 − V X0 + 0 = 0

⎪ ⎪ V T X0 = 1 ⎪ ⎪ ⎪ ⎩

U ≥ 0, V ≥ 0, 0 free

2

PBCC (X, Y ) =

n 

(X, Y )|

j=1

j Xj ≤ X,

n 

j=1

j Yj ≥ Y, j ≥ 0,

n 

j=1

 j = 1, j = 1, . . ., n

.

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where U = (u1 , u2 , . . ., us )T and V = (v1 , v2 , . . ., vm )T are vectors of multipliers, and  = diag {ı1 , ı2 , . . ., ıs } and W = diag {ω1 , ω2 , . . ., ωm } are matrixes of input and output directions, respectively. Based on the optimal solutions of Model (7), we have the following procedure for determining the directional RTS of DMU (X0 , Y0 ) in the direction of (ω1 , . . ., ωm )T and (ı1 , . . ., ıs )T .

(1) The directional RTS to the “right” of DMU (X0 , Y0 ): (a) (X0 , Y0 ) > 1, increasing directional RTS holds; (b) (X0 , Y0 ) = 1, constant directional RTS holds; (c) (X0 , Y0 ) < 1, decreasing directional RTS holds; (2) The directional RTS to the “left” of DMU (X0 , Y0 ): (a) (X ¯ 0 , Y0 ) > 1, increasing directional RTS holds; (b) (X ¯ 0 , Y0 ) = 1, ¯ 0 , Y0 ) < 1, decreasing directional RTS holds; (d) if the upper bound of Model (3) constant directional RTS holds; (c) (X is unbounded, there is no data to determine the directional RTS to the “left” of DMU (X0 , Y0 ).

Inefficient or weakly efficient DMUs can be projected onto the strongly efficient frontier using DEA models so that we can estimate the directional RTS to the “right” and “left” of them according to the directional RTS of these projections. Model (7) is a fractional program and difficult to solve. For this reason we transform it into an equivalent mathematical program (Model (8)) through a Charnes–Cooper transformation (Charnes & Cooper, 1962). Then Model (7) becomes: ( ¯ ) = max(min) T X0 -⎧ ⎪ T −1 Yj −  T W −1 Xj + 0 ≤ 0, j = 1, . . ., n s.t.

⎪ ⎪ ⎪ ⎨ T −1 Y0 −  T W −1 X0 +  = 0 0

(8)

⎪  T W −1 X0 = t ⎪ ⎪ ⎪ ⎩

 ≥ 0,  ≥ 0, t ≥ 0, 0 free

Solving Model (8), we can obtain the directional scale elasticity and directional RTS of DMU (X0 , Y0 ). RTS often involves a congestion effect. A congestion effect means that the reduction of one (or some) input(s) results in the increase of the maximum possible value of one (or some) output(s) under the premise that other inputs or outputs do not become deteriorated (Cooper, Seiford, & Zhu, 2004). Essentially, the congestion effect describes the issue of excessive inputs (Wei & Yan, 2004). Färe and Grosskopf (1983, 1985) investigated congestion effects using quantitative methods and proposed corresponding DEA models to deal with this issue. Soon after that, Cooper, Huang, and Li (1996) proposed another model to study the congestion effect. Cooper, Gu, and Li (2001) compared the similarities and differences of the above two models. To detect the congestion effect of DMUs Wei and Yan (2004) and Tone and Sahoo (2004) built a new DEA model based on a new production possibility set under the assumption of weak disposal. The above methods are all based on the idea of radial changes in all inputs (i.e., all components of inputs change in the same proportion). Based on the idea of nonproportional changes, as explained above, Yang (2012) determined the directional congestion for a strongly efficient DMU (X0 , Y0 ) on the strongly efficient frontier of the production possibility set determined in Model (9) (denoted by Pconvex (X, Y )3 ), through the following Model (10):

max 0

⎧ ⎪ j xij = xi0 , i = 1, . . ., m ⎪ ⎪ ⎪ ⎪ j ⎪  ⎪ ⎪ ⎨ j yrj − sr+ = 0 yr0 , r = 1, . . ., s

s.t.

(9)

j  ⎪ ⎪ ⎪ j = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j +

j , sr ≥ 0, r = 1, . . ., s; i = 1, . . ., m; j = 1, . . ., n

3

PConvex (X, Y ) =

n 

(X, Y )|

j=1

j Xj = X,

n 

j=1

j Yj ≥ Y, j ≥ 0,

n 

j=1

 j = 1, j = 1, . . ., n

.

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V T WX0 ( ¯ ) = max(min) T U Y0

⎧ T U Yj − V T Xj + 0 ≤ 0, j = 1, . . ., n ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T T

s.t.

U Y0 − V X0 + 0 = 0

(10)

⎪ ⎪ V T X0 = 1 ⎪ ⎪ ⎪ ⎩

U ≥ 0, V, 0 free

Similarly as before we transform Model (10) into an equivalent mathematical program (Model (11)) through a Charnes–Cooper transformation (Charnes & Cooper, 1962) as follows: ( ¯ ) = max(min) T X0 ⎧ T −1  T −1 ⎪   Yj −  W Xj + 0 ≤ 0, j = 1, . . ., n

⎪ ⎪ ⎪  T −1 T −1 ⎪ ⎨   Y0 −  W X0 + 0 = 0

s.t.

(11)

 T W −1 X0 = t

⎪ ⎪ ⎪ ⎪ T ⎪ ⎩  Y0 = 1

 ≥ 0, t ≥ 0, , 0 free

Based on the results of Model (11), we have the following procedure for determining the directional congestion effect of strongly efficient DMU (X0 , Y0 ) on the strongly efficient frontier of Pconvex (X, Y ) in the direction of (ω1 , . . ., ωm )T and (ı1 , . . ., ıs )T . (1) If the optimal objective value (X0 , Y0 ) in Model (11) is bounded and satisfies (X0 , Y0 ) < 0, then a directional congestion effect occurs to the “right” of the DMU (X0 , Y0 ). If the lower bound (X0 , Y0 ) of Model (11) is unbounded, there is no data to determine the directional congestion effect to the “right” of DMU (X0 , Y0 ). (2) If the optimal objective value (X ¯ 0 , Y0 ) in Model (11) is bounded and satisfies (X ¯ 0 , Y0 ) < 0, then a directional congestion ¯ 0 , Y0 ) of Model (11) is unbounded, then there is no effect occurs to the “left” of the DMU (X0 , Y0 ). If the upper bound (X data to determine the directional congestion effect to the “left” of DMU (X0 , Y0 ). Inefficient or weakly efficient DMUs can be projected onto the strongly efficient frontier using DEA models so that we can detect the directional congestion effect to the “right” and “left” of the inefficient or weakly efficient DMUs through the directional congestion of these projections. Further mathematical details are available from the authors upon request. 3. Analysis of directional RTS and directional congestion effect 3.1. Directional RTS Firstly, we determine the strongly efficient frontier using an input-based BCC-DEA model (Model (12)) with radial measurement. min 0 − ε

 

sr+

+





si−

r

s.t.

i ⎧ −  x + s =

x ⎪ 0 i0 , i = 1, . . ., m j ij i ⎪ ⎪ ⎪ j ⎪ ⎪ ⎨ +

(12)

j yrj − sr = yr0 , r = 1, . . ., s

⎪ j ⎪  ⎪ ⎪ ⎪ j = 1, j , si− , sr+ ≥ 0, j = 1, . . ., n ⎪ ⎩ j

where ε > 0 is a non-Archimedean infinitesimal. That is, ε > 0 is smaller than any positive real number. Variables si− , sr+ ≥ 0 are slack variables. According to Model (12), we obtain the projections of the 30 DMUs on the strongly efficient frontier (see Table 2) through the following formulae (13).

x˜ i0 ← 0∗ xi0 − si−∗ , i = 1, . . ., m y˜ r0 ← yr0 + sr+∗ , r = 1, . . ., s

(13)

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Table 2 The projections of the 30 DMUs on the strongly efficient frontier. DMU

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30

Inputs

Outputs

Senior (FTE)

Middle & junior (FTE)

Personnel salaries (RMB million)

Other Expen. (RMB million)

SCI Pub. (number)

High Pub. (number)

Grad.Enrol. (number)

IPs (score)

190 140 76 128.0338 144.3393 157.1333 40 488 165.287 62 118 118 77 117 75 189.6825 152 81 137.5981 170 203 45 523 181 70 126 110.4001 70.3153 139 73.4286

347 174 71 202.3525 191.0286 226.1761 163 1090 321.2641 98 103 145 204 229 70 312.1786 189 82 235.0296 267 221 196 1090 388 131 154 147.8356 167.0720 246 73.4286

104.64 73.13 35.67 59.4705 71.0961 88.2175 27.99 454.12 109.3081 28.79 47.04 40.52 36.93 57.85 21.48 134.2469 106.23 42.51 70.4273 92.31 124.69 29.42 495.68 122.11 37.71 54.21 46.8044 43.0068 72.89 22.5692

164.19 186.71 63.65 114.0027 198.9935 191.8121 63.98 541 268.5406 85.93 140.97 54.57 83.01 94.73 36.67 224.8203 204.79 64.32 129.1165 270.07 377.24 85.19 687.13 330.03 59.27 126.62 77.9361 97.2269 138.96 43.0533

325 368 207 256 260.6405 284.4449 112 785 385 118 216 125 189 313 83 385.0547 419 210 276 296 221 129 702 390 104 231 185 187 343 70

105 109 66 64.4126 96 93 39 323 125 36 63 37 66 64 17 137 142 91 79 138 122 46 359 133 31 62 58 71 77 20

604 477 241 388 500 553 190 1488 419.0087 235 481 267 232 302 126 612 482 242 396 518 555 202 1510 424 237 484 277 243.4994 318 140.8955

23.6 10.4 0.45 27.6 17.2 30.9 0.2 63.4 124.05 11.3 5.6 15.7 8.65 32.2 9.4 26.5 8.8 0 36.8 7.8 38.8 3.2 84.3 184.8 12.2 11.6 22 11.2 42.4 9.5070

ω2

ω1=ω2 R1: Decreasing ω1

Fig. 1. The directional RTS to the right of DMU4 (ω3 = ω4 = 1)

ω2

R1: Decreasing

ω1=ω2 R2: Constant

R3: Increasing ω1 Fig. 2. The directional RTS to the left of DMU4 (ω3 = ω4 = 1)

where ( 0∗ , si−∗ , sr+∗ ) is the optimal solution of Model (12). Secondly, we can determine the directional RTS of the 30 DMUs in CAS using the methods mentioned in Section 2 (Model (8)). We take DMU4 as an example because it shows some interesting results. Without loss of generality, we set the direction of the outputs to ı1 = ı2 = ı3 = ı4 = 1. In order to show the input directions in the two-dimension coordinates, we set ω3 = ω4 = 1

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R1: Increasing ω4

R2: Constant ω3=ω4 R3: Decreasing ω3

Fig. 3. The directional RTS to the right of DMU4 (ω1 = ω2 = 1)

ω4

R1: Increasing

ω3=ω4 R2: Constant R3: Decreasing ω3

Fig. 4. The directional RTS to the left of DMU4 (ω1 = ω2 = 1)

and ω1 = ω2 = 1 respectively and let ω1 , ω2 and ω3 , ω4 vary to see the changes of directional RTS. See Figs. 1–4. The detailed data is shown in Table B.1 in Appendix B. By the above analysis, we come to the following findings. (a) The directional RTS to the “right” of DMU4 (a.1) When ω3 = ω4 = 1 (i.e., Personnel salaries and Other Expen. increase proportionally), on the basis of existing inputs, if Senior and Middle & Junior increase in any proportion (under Pareto preference), decreasing directional RTS holds for DMU4 , i.e., DMU4 is located on the region with decreasing directional RTS in any direction of inputs increase. See Fig. 1. (a.2) When ω1 = ω2 = 1 (i.e., Senior and Middle & Junior increase proportionally), on the basis of existing inputs, if the proportion of Personnel salaries and Other Expen. increase is located in area R1 in Fig. 3, increasing directional RTS holds for DMU4 . If their increase proportion is located in area R2 (solid line) in Fig. 3, constant directional RTS takes place. Finally, if the proportion is located in area R3, decreasing directional RTS shows. See Fig. 3. (b) The directional RTS to the “left” of DMU4 (b.1) Consider ω3 = ω4 = 1 (i.e., Personnel salaries and Other Expen. decrease proportionally): if, on the basis of existing inputs, Senior and Middle & Junior decrease in the same proportion (dotted line) decreasing directional RTS takes place. If the proportion of Senior and Middle & Junior decrease is located in area R1 in Fig. 2, decreasing directional RTS takes place. If the proportion of Senior and Middle & Junior decrease is located in area R3, increasing directional RTS takes place. Finally, if the proportion is located in area R2 (solid line), constant directional RTS takes place (see Fig. 2). (b.2) Consider now ω1 = ω2 = 1 (i.e., Senior and Middle & Junior decrease proportionally): if, on the basis of existing inputs, Personnel salaries and Other Expen. decrease in any proportion in area R1, increasing directional RTS takes place. Area R2 (solid line) and R3 denote constant and decreasing directional RTS, respectively. See Fig. 4. Similarly, we can have the directional RTS to the “right” and “left” of other DMUs.

3.2. Directional congestion effect As mentioned above, the congestion effect essentially describes the issue of excessive inputs (Wei & Yan, 2004). In this case, an increase (decrease) in one or more inputs causes a decrease (increase) in one or more outputs (Cooper et al., 2001). If a research institution suffers a congestion effect, its inputs should be decreased so that its outputs could increase. This information is essential for DMs to decide on the allocation of S&T resources. Therefore, in this section we will detect whether or not there exists a congestion effect. Furthermore, if a congestion effect exists, we will determine in which directions it exists so that we can obtain information about which increase (decrease) in multiple inputs of these universities and in which direction the decrease (increase) in one or more outputs is caused.

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Table 3 The congestion effect of the 30 DMUs using the WY-TS model. DMUs

Inputs

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6 DMU7 DMU8 DMU9 DMU10 DMU11 DMU12 DMU13 DMU14 DMU15 DMU16 DMU17 DMU18 DMU19 DMU20 DMU21 DMU22 DMU23 DMU24 DMU25 DMU26 DMU27 DMU28 DMU29 DMU30

Senior (FTE)

Middle & Junior (FTE)

Personnel Salaries (RMB million)

Other Expen. (RMB million)

SCI Pub. (number)

Outputs High Pub. (number)

Grad. Enrol. (number)

IPs (score)

Congestion effect ϕ = ∗ / ∗ (WY-TS model)

190 140 76 146 160 179 40 488 172 62 118 118 77 117 75 204 152 81 164 170 203 45 523 181 70 126 119 77 139 78

347 174 71 233 233 257 163 1090 351 98 103 145 204 229 70 363 189 82 251 267 221 196 1090 388 131 154 149 231 246 78

104.64 73.13 35.67 76.93 78.81 100.24 27.99 454.12 141.52 28.79 47.04 40.52 36.93 57.85 21.48 144.38 106.23 42.51 96.68 92.31 124.69 29.42 495.68 122.11 37.71 54.21 54.98 49.82 72.89 26.53

164.19 186.71 63.65 130 266.96 401.38 63.98 541 340.98 85.93 140.97 54.57 83.01 94.73 36.67 241.79 204.79 64.32 137.89 270.07 377.24 85.19 687.13 330.03 59.27 126.62 78.55 106.47 138.96 52.68

325 368 207 277.2045 259 216 112 785 385 118 216 125 189 313 83 420.582 419 210 276 296 221 129 702 390 104 231 185 187 343 111.6301

105 109 66 74.418 96 93 39 323 125 36 63 37 66 64 17 145.1545 142 91 79 138 122 46 359 133 31 62 58 71 77 28.2745

604 477 241 420.138 500 553 190 1488 417 235 481 267 232 302 126 648.4275 482 242 396 518 555 202 1510 424 237 484 277 237 318 176.1984

23.6 10.4 0.45 29.8861 17.2 30.9 0.2 63.4 124.05 11.3 5.6 15.7 8.65 32.2 9.4 28.0773 8.8 0 36.8 7.8 38.8 3.2 84.3 184.8 12.2 11.6 22 11.2 42.4 9.4483

1 1 1 0.9768 0.9592 0.9461 1 1 0.9637 1 1 1 1 1 1 0.9978 1 1 0.9475 1 1 1 1 1 1 1 0.9935 0.9164 1 1

We detect the congestion effect of these DMUs using the WY-TS model (Färe, Grosskopf, & Lovell, 1985; Kao, 2010; Tone & Sahoo, 2004; Wei & Yan, 2004) based on the input–output data of these DMUs. We draw up the output-based BCC-DEA model with radial measurement as follows: max 0 − ε

 

sr+

+





si−

r

s.t.

i ⎧ −  x + s = x ⎪ j ij i0 , i = 1, . . ., m i ⎪ ⎪ ⎪ j ⎪ ⎪ ⎨ +

(14)

j yrj − sr = 0 yr0 , r = 1, . . ., s

⎪ j ⎪  ⎪ ⎪ ⎪ j = 1, j , si− , sr+ ≥ 0, j = 1, . . ., n ⎪ ⎩ j

where ε > 0 is a non-Archimedean construct, i.e., an arbitrary small positive entity. We can use Model (9) to measure the pure technical efficiency (Kao, 2010; Tone & Sahoo, 2004). As proposed by Färe et al. (1985) and Kao (2010), the congestion effect ϕ is also measured as the ratio of the objective value in Model  (9) to that of Model

j xij + si− = xi0 , i =

(14): ϕ = ∗0 / 0∗ . Note that Model (9) differs from Model (14) only in the first constraint set where 1, . . ., m is replaced by



j

j xij = xi0 , i = 1, . . ., m. The equality constraints in Model (9) are stronger than the inequality

j

constraints of Model (14). This assures the congestion measure ϕ to be bounded by 1. If ϕ = 1, then there is no congestion effect. If ϕ < 1, a congestion effect does exist. We observe congestion effects for DMU4 , DMU5 , DMU6 , DMU9 , DMU16 , DMU19 , DMU27 and DMU28 . See Table 3 for detailed information about the projections of these DMUs on Pconvex (X, Y ) through Model (9) and congestion effect in WY-TS model. Let us mention that the results on this table only detect output-based congestion when the outputs increase proportionally. Secondly, we can analyze the directional congestion effect of the above DMUs using the methods mentioned in Section 2.3 (Model (11)). We also consider DMU4 as the example to see the results. Without loss of generality, we set the outputs

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Table 4 The directional congestion effect of DMU4 in different inputs directions. ω3 = ω4 = 1

ω1 = ω2 = 1

ω1

ω2

 -

¯

0.3 0.5 0.7 0.9 1 1.1 1.3 1.5 1.7

1.7 1.5 1.3 1.1 1 0.9 0.7 0.5 0.3

1.17 1.14 0.81 0.48 0.31 0.15 −0.19 −0.52 −0.85

1.47 1.16 0.84 0.53 0.37 0.21 −0.10 −0.42 −0.73

Directional congestion effect (right)

Directional congestion effect (left)

ω3

ω4

 -

¯

No No No No No No Yes Yes Yes

No No No No No No Yes Yes Yes

0.3 0.5 0.7 0.9 1 1.1 1.3 1.5 1.7

1.7 1.5 1.3 1.1 1 0.9 0.7 0.5 0.3

0.48 0.43 0.39 0.34 0.31 0.29 0.24 0.19 0.14

0.55 0.50 0.45 0.40 0.37 0.35 0.30 0.24 0.19

Directional congestion effect (right)

Directional congestion effect (left)

No No No No No No No No No

No No No No No No No No No

direction to ı1 = ı2 = ı3 = ı4 = 1. Moreover, we set ω3 = ω4 = 1 and ω1 = ω2 = 1 respectively and let ω1 , ω2 and ω3 , ω4 vary to see the changes of directional congestion effect of DMU4 in different inputs directions. See Table 4 for details. Based on the above analysis, we find that congestion effect occurs on DMU4 when using the WY-TS model. However, for DMU4 , the directional congestion effect occurs in certain directions (e.g., ω1 = 1.3, ω2 = 0.7, ω3 = 1, ω4 = 1; ı1 = ı2 = ı3 = ı4 = 1) and does not occur in other directions (e.g., ω1 = 0.7, ω2 = 1.3, ω3 = 1, ω4 = 1; ı1 = ı2 = ı3 = ı4 = 1). Trying to determine the reason behind this phenomenon would lead us too far and is beyond the scope of this paper. We can analyze the directional congestion effect for other DMUs in a similar way. 3.3. Policy implications From the above analysis and findings, we see that: (1) the regions of directional RTS (increasing, constant, decreasing) can be detected through the methods in Section 2 so that DMs can refer to this information when making decisions or formulate S&T policies with certain prior information. Taking DMU4 in Section 3 as example, when Personnel salaries and Other Expen. increase proportionally, we found that if Senior and Middle & Junior increase in any proportion, decreasing directional RTS holds on DMU4 so its professional and technical staff should be reduced to improve its scale efficiencies. When Senior and Middle & Junior increase proportionally, DMU4 ’s Personnel salaries and Other Expen. should be increased with the proportion in area R1 in Fig. 3 so that the scale efficiency of resource utilization could be improved further. On the contrary, from the left side directional RTS, when Senior and Middle & Junior decrease proportionally, DMU4 should choose the suitable proportion of Personnel salaries and Other Expen. decrease in area R1 in Fig. 2 to improve its scale efficiency. When Personnel salaries and Other Expen. decrease proportionally, it’s better for DMU4 to choose the proportion of Senior and Middle & Junior decrease in area R3 in Fig. 4. (2) A congestion effect occurs for DMU4 using traditional WY-TS model. That fact normally indicates that the inputs of this DMU should be reduced to achieve more outputs. However, based on our analysis of directional congestion of DMU4 , the directional congestion effect does not occur for the same DMU in certain directions (e.g., ω1 = 0.7, ω2 = 1.3, ω3 = 1, ω4 = 1; ı1 = ı2 = ı3 = ı4 = 1), which means if the inputs increase in these directions with no directional congestion, the outputs also could be expanded. Therefore, these institutes should analyze their own strengths carefully and identify their own development path for resources so that their scale efficiencies can be improved. What is interesting is that the area R3 in Fig. 2 is the same to the directions in which directional congestion occurs when ω3 = ω4 = 1 in Table 4. Combing the results on directional RTS and directional congestion, we can see that the inputs of DMU4 should be reduced to achieve more outputs. In the process of reducing its inputs, we should choose the direction of inputs decrease in area R1 instead of area R3 in Fig. 2 or the direction in area R3 in Fig. 4 to improve scale efficiency and avoid directional congestion effect. 4. Conclusions and discussion This paper investigated the directional returns to scale of biological institutes in CAS. Firstly, input–output indicators are proposed, including Staff, Research funding, SCI papers, High-quality papers and Graduates training. Secondly, this paper used the methods proposed by Yang (2012) to analyze the directional returns to scale and the effect of directional congestion of biological institutes in CAS. Based on the analytical results, we have the following findings: (1) for each biological institute we can detect the regions of increasing (constant, decreasing) directional returns to scale. This information is essential for decision-making on organizational adjustment; (2) we find that congestion and directional congestion occurs in several biological institutes. The outputs of these institutes will decrease with an increase of inputs. For this reason these institutes should analyze the underlying reason for the occurrence of this congestion effect so that science and technology (S&T) resources can be used more efficiently.

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Acknowledgements We would like to acknowledge the support of the National Natural Science Foundation of China (Grant No. 71201158) and the German Academic Exchange Service (DAAD, No. A1394033). We thank anonymous referees for valuable comments which helped us to improve the manuscript. We also thank Prof. Li, X.X. and Prof. Fan, C.L. for their help in providing data and materials for the research explained in this manuscript.

Appendix A. Explanatory notes on input indicators4 1 Regular staff Regular staff refers to persons who are on the year-end payroll, working directly under the management of an institution or unit and receiving remuneration for their work. The retired are not included. Professional and technical staff is the main part of regular staff and refers to those who have acquired professional or technical titles or who are in the probation period of those titles. (1) Senior: research fellow and associate research fellow, professor and associate professor, senior engineer, senior agronomist, chief (and associate chief) physician (pharmacist, nurse and technician), senior laboratorian, senior statistician, senior economist, senior accountant, senior editor and associate senior editor, translation editor and associate translation editor, senior journalist, senior librarian and associate librarian, and so on. (2) Middle level: research associate, lecturer, engineer, agronomist, physician (pharmacist, nurse and technician) in charge, laboratorian, statistician, economist, accountant, editor, translator, journalist, librarian, and so on. (3) Junior: research assistant, teaching assistant, assistant engineer, technician, assistant agronomist, agrotechnician, physician (pharmacist, nurse and technician), assistant laboratorian, laboratory technician, assistant physician (pharmacist, nurse and technician), assistant statistician, statistical clerk, assistant economist, assistant accountant, accounting clerk, assistant editor, internship, assistant translator, assistant journalist, library assistant, library clerk, and so on. 2 Total expenditure Personnel expenditure: spending from various kinds of funds, such as basic salary, subsidiary salary, other salaries, and social security expenditure. (1) Basic salary: basic salary and proportioned allowance paid to the staff according to relevant state policies. (2) Subsidiary salary: allowances and subsidies paid to the staff according to relevant state policies, including working post allowance, inflation subsidies, local allowance, allowance for winter heating, night snack allowance, traffic allowance for the staff to and back from work, and overtime pay, etc. (3) Other salaries: allowances and subsidies paid to the on-job staff, which is the component of the total salary set according to the relevant state policies. This item is not included in the basic salary and subsidiary salary. (4) Social security expenditure: the pension, allowances or subsidies paid to retired personnel according to the relevant state policy, and various basic social insurance premium paid by the units according to the state policy. Public expenditure: spending from various kinds of funds, such as official business spending, expenditure for professional activities, expenditure for purchasing equipment, renovation fees and others. (1) Expenditure for public operation: the expenses for organizing and managing professional work and auxiliary activities, mainly including expenses on administration, postage and communications, water and electricity, repair maintenance, property costs, winter heating, vehicles and fuel, etc. (2) Expenditure for professional activity: expenses occurred in the process of carrying out professional work and auxiliary activities, mainly including raw and processed materials consumed, computation and testing, fuel and power, expenses for attending conferences and making business trips, etc. (3) Expenditure for purchasing equipment: expenses spent on purchasing equipment for scientific research, production, development, business operation and office instruments, which does not fall into the category of capital construction expenditure and is managed as fixed assets. It mainly includes expenses for purchasing various kinds of instruments and equipment, vehicles, books, etc., according to the relevant policies. Balanced and transferred self-raised capital construction funds: expenditure from non financial subsidiary funds for capital construction projects which are approved by the state departments concerned and which are included in the capital construction plan of the year. Business expenditure: expenses occurred in the process of carrying out non-independent accounting business activities other than professional and auxiliary work, as well as the actual cost of products sold by those internal cost-accounting units. Designated expenditure: expenses paid from above-mentioned “Designated funds”.

4

Explanatory notes are from Statistical yearbooks of CAS in 2012.

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Table B.1 The directional RTS of DMU4 in different input directions. Input directions

ω3 = ω4 = 1

ω1

ω2

 (lower bound)

¯ (upper bound)

Directional RTS (right)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.64 0.67 0.70 0.72 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.73 0.72 0.70 0.69 0.67 0.66 0.64 0.62

0.97 0.95 0.93 0.91 0.89 0.87 0.86 0.88 0.90 0.92 0.95 0.98 1.00 1.04 1.07 1.10 1.13 1.16 1.19

Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing

Input directions

ω1 = ω2 = 1

Directional RTS (left)

ω3

ω4

 (lower bound)

¯ (upper bound)

Directional RTS (right)

Directional RTS (left)

Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Constant Increasing Increasing Increasing Increasing Increasing Increasing

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1.02 1.00 0.97 0.95 0.93 0.90 0.87 0.83 0.79 0.75 0.70 0.64 0.58 0.52 0.45 0.39 0.33 0.27 0.21

1.47 1.41 1.35 1.29 1.23 1.16 1.10 1.04 0.98 0.92 0.86 0.80 0.75 0.71 0.68 0.66 0.63 0.61 0.59

Increasing Constant Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing

Increasing Increasing Increasing Increasing Increasing Increasing Increasing Increasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing

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