DEA models for non-homogeneous DMUs with different input configurations

Accepted Manuscript

DEA Models for Non-Homogeneous DMUs with Different Input Configurations WangHong Li , Liang Liang , Wade D. Cook , Joe Zhu PII: DOI: Reference:

S0377-2217(16)30300-9 10.1016/j.ejor.2016.04.063 EOR 13684

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

1 December 2015 14 April 2016 28 April 2016

Please cite this article as: WangHong Li , Liang Liang , Wade D. Cook , Joe Zhu , DEA Models for Non-Homogeneous DMUs with Different Input Configurations, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.04.063

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Highlights  Conventional DEA model assumes all inputs impact all outputs  This paper considers efficiency where DMUs are non-homogeneous on the input side  We examine the case where different DMUs have different natural resource configurations  Model is applied to a set of 31 provinces in China

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DEA Models for Non-Homogeneous DMUs with Different Input Configurations by WangHong Li PhD Program, School of Management University of Science and Technology of China He Fei, An Hui Province, PR China 230026

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Liang Liang School of Management University of Science and Technology of China He Fei, An Hui Province, PR China 230026

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Wade D. Cook1 Schulich School of Business York University, 4700 Keele Street Toronto, Ontario, Canada M3J 1P3 and

November, 2015

Revised April, 2016

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Joe Zhu International Center for Auditing and Evaluation, Nanjing Audit University, Nanjing, P.R. China; and School of Business, Worcester Polytechnic Institute, Worcester, USA

Acknowledgments: Wade Cook was supported under NSERC Grant A8966 The authors are grateful to the three anonymous reviewers for their constructive comments

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Corresponding author Wade D. Cook [email protected]

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Abstract The data envelopment analysis (DEA) methodology is a benchmarking tool where it is generally assumed that decision making units (DMUs) constitute a homogeneous set; specifically, it is assumed that all DMUs have a common (input, output) bundle. In earlier work by the authors the issue of non-homogeneity on the output side was investigated. There we

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examined a set of steel fabrication plants where not all plants produced the same set of products/outputs. In the current research we investigate non-homogeneity on the input side. Such can occur in manufacturing plants, for example, when the output bundle can be produced using different mixes of machines, robots and laborers. Thus, we can have an input configuration existing in a DMU that is different from the configuration in another DMU. As a practical

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application of this phenomenon, we examine the measurement of efficiencies of a set of provinces in China. There, all provinces have the same common set of outputs in the form of GDP, supported population, and an undesirable output, nitrogen dioxide. On the input side, however, this commonality is missing. While all provinces have water, capital investment and natural resources, the latter of these (natural resources) takes several different forms, namely

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coal, natural gas and petroleum. However, not all provinces have the same mix of these resources, nor are there clear exchange rates among these very different, albeit substitutable

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inputs. This means that that one cannot directly apply the conventional DEA methodology. This then raises the question as to how to fairly evaluate efficiency when the configuration or mix of

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inputs can differ from one DMU to another. To address this, we view the generation of outputs for a province as a set of processes created by the different configurations of natural resources

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available. We develop a DEA type of methodology to evaluate these processes. This evaluation provides important insights into not only the overall performance of each province, but as well

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provides measures of the efficiency of the various configurations of the three natural resources.

Keywords: Data envelopment analysis, Non-homogeneous DMUs, Missing inputs,

Multiple processes, Input Configurations

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1. Introduction Data Envelopment Analysis (DEA), first introduced by Charnes, Cooper and Rhodes (1978), is a methodology for evaluating the relative efficiencies of a set of decision making units (DMUs). In the nearly four decades since this seminal work, literally thousands of articles and books on DEA have appeared. Useful surveys include Cook and Seiford (2009) and Paradi and

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Zhu (2013). The conventional DEA model is based on the assumption that in a multiple-input multiple-output setting, all inputs impact all outputs, or that detailed input to output relations are not closely examined. Furthermore, it is assumed in the conventional DEA model that the set of DMUs under investigation constitute a homogeneous set. This means that all DMUs have the

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same inputs and produce the same outputs.

There are many situations in which the above assumptions are violated. Regarding the assumption that all inputs affect all outputs, consider the situation in a manufacturing setting where one of the inputs is packaging resources. Clearly, this input only impacts outputs that require packaging. This gives rise to what we refer to as partial input to output impacts,

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investigated in Cook et al. (2013a). On the matter of the DMUs constituting a homogeneous set, consider the case in manufacturing where some DMUs may produce a different output mix than

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is true of other DMUs. See Cook et al. (2012), (2013b), where the efficiency of a set of steel fabrication plants is examined. This latter is a form of non-homogeneity on the output side. It

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was assumed there that the input set was common across all DMUs. In another setting, where one is comparing a set of universities, for example, it can occur

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that not all institutions have the same departments, or the same financial resources to attract students. This might be deemed a case where there is lack of homogeneity on the input side. In

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the current paper we extend that earlier research of Cook et al. (2012), (2013b) to encompass the case where the input mix can be different for some DMUs as compared to others.

In section 2 we present a problem setting in which a set of macro-level inputs are used to

generate a set of macro-level outputs; one of those inputs is the quantity of natural resources available to the decision making unit. In the case where the DMUs are a set of regions in China, those resources can take different forms in some regions versus other regions. For example, some 4

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regions have natural gas while others do not. At the same time, all regions possess natural resources in the form of coal. Hence, the conventional DEA model cannot handle this particular situation where multiple forms of an input occur. In section 3 we view the multiple forms of an input from the perspective of a set of parallel processes. Section 4 develops a type of DEAbased model to capture this situation, and section 5 applies this new methodology to the setting

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discussed in section 2. Discussion and recommendations appear in Section 6. 2. On Regional Resources in China

With three decades of rapid economic development in China, people more and more realize that developments in science and technology have promoted productivity and enlarged the

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Chinese economy. While that is important, we need to realize that ecological preservation is an important core value in Chinese culture. On June 27, 2015, at an important forum, China's Ecological Civilization Guiyang International Forum, the Chinese President set important guidelines regarding the ecology of China. At the same time, the Dean of the Chinese Academy of Sciences, in a speech at the meeting, stressed the importance of scientific research focused on

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the "optimization of national spatial development patterns, comprehensively promoting resource conservation, the natural ecosystem and environmental protection, and strengthening the

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construction of the ecological civilization system." A number of studies have examined various aspects of energy economics in the presence of

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pollution and environmental concerns. See for example Cooper et el. (1996), Lee (2005), Soyas et al. (2009), Zhou, Ang and Poh (2008), and Zhou and Ang (2008). In the current paper we

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evaluate the efficiency of a set of regions at a macro-level in terms of the economy and the environment in China; thirty-one regions are examined. Data was collected on five input factors,

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namely local water resources, capital investment, and natural resources in the form of coal, natural gas and petroleum. On the output side, two important factors to consider for use in evaluating regional economic efficiency are regional GDP and Population served. While one might postulate utilizing these economic factors as two separate outputs in a DEA analysis, a more viable choice is to combine them into a single factor, namely GDP per capita (i.e., GDP/Population). In so doing, proportional increases in this combined factor in inefficient DMUs would represent improvements in the wellbeing of society. If one were, instead, to model 5

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the two factors as separate outputs, say in an output oriented setting, an inefficient DMU could become efficient only by proportionally increasing each factor so as to project them onto the efficient frontier. There are two problems with this line of reasoning. First, logic would normally dictate that Population be viewed as nondiscretionary, since one cannot simply increase population in a given region. Second, if hypothetically Population were viewed as discretionary,

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then the same proportional increase in the two variables to get to the frontier would mean that the frontier projection would see GDP per capita at the same level after the projection as before. Such a conclusion would appear to undermine the intended purpose of efficiency analysis.

In developing the regional economy, pollution is a major issue as well; Carbon Dioxide,

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Nitrogen Dioxide, Sulfur Dioxide, Ammonia, Nitrogen, Smoke and Dust are a natural consequence when a region sets out to improve GDP and the feeding population. Those undesirable outputs pollute the environment; in some areas, the ecological environment can be destroyed. For example, since 2013, “fog” and “haze” have become the annual keywords; Nitrogen Dioxide and Sulfur Dioxide at a level of PM2.5 were major contributors to the hazy

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atmosphere. In recognition of the importance of pollution and its impacts, Nitrogen Dioxide was

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selected as one of the most undesirable outputs impacting efficiency in terms of the environment. Not all regions have the same natural resources. The reality is that some are missing certain forms of this macro input, or more generally, the configuration of inputs differs among the

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regions. Specifically, the 31 regions fall into 4 groups according to their natural resource “holdings”: Regions in the first group have only coal, the second group, coal and petroleum, the

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third, coal and natural gas, and finally regions in the fourth group have all three forms. All regions hold water resources and investments. This is illustrated by the data in Table 1. This data

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is published by the China Statistical Yearbook (2013) and China Statistical Yearbook on the Environment (2013). Utilizing this backdrop, we set out to extend earlier research by Cook et al (2012, 2013b),

by developing a model structure to handle this missing input situation. 3. The Problem of Multiple Configurations of Inputs in DEA 6

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The conventional Data Envelopment Analysis model (output oriented CCR) takes the form:

 x min u y

i ij o

i

r

rjo

r

 x u y

(3.1)

i ij

i

r

 1,

j  1,...n

rj

r

vi , ur  0

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subject to

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Here, the constraints require that multipliers vi, ur be chosen such as to insure that the efficiency ratio of inputs to outputs for each DMUj be at or greater than unity. In the conventional DEA model it is assumed that a common set of inputs and outputs characterize all decision-making units (DMUs).

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Consider the problem discussed in section 2 where some DMUs (provinces, cities or regions) have only coal as a natural resource, while others have that resource as well as natural

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gas, and still others have petroleum as well. Referring to that particular problem setting, let j=1, 2,..., n denote the DMUs Let x1 j , x2 j denote the quantities of two inputs, water and capital

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investment, respectively, available to DMUj. These two inputs are held (in different quantities) by all DMUs. Three additional inputs in the form of natural resources are x3 j (coal), x4 j (natural gas)

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and x5 j (petroleum), are held in varying amounts by different DMUs. Let y1 j , y2 j denote outputs GDP per capita, and an undesirable output, N02. We assume each DMU possesses both of these

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outputs.

What is different about this problem setting, as indicated above, is that while coal x3 j is

common to all DMUs, this is not case for x4 j and x5 j . Figure 1 shows the DMUs split into 4 groups according to the combination of the three natural resources held by those DMUs.

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Figure1: DMU Profiles Regarding Natural Resources Inputs DMU

X1

X2

X3

X4

N1

√

√

√

_

N2

√

√

√

√

N3

√

√

√

_

N4

√

√

√

√

X5

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Group _ _

√

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√

So, regions in N1 , for example, have inputs x1 , x2 , x3 , but do not have x4 and x5 . DMUs in

N 2 have both coal and natural gas, those in N 3 have coal and petroleum, and those in N 4 have all three resources. Note as well that the 3 natural resources x3 , x4 , x5 all perform the “same

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function”, and can be considered as “substitutes” for one another. The difficulty is that we do not know the rates of substitution. In other words, if v3 , v4 denote the multipliers or prices for coal

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( x3 ) and natural gas ( x4 ) respectively, we do not know the relationship between these two multipliers. This is partially because the units of measurement for two resources (say tons and

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cubic metres respectively), are different, but as well v3 , v4 should reflect processing costs, environment impacts, etc., which are difficult to specify.

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We wish to develop a methodology for evaluating the efficiencies of DMUs in such settings; because not all inputs and outputs are common to all DMUs, the conventional model

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above would not appear to be applicable. One might conceivably treat this as simply a case of missing data as discussed by Thompson et al. (1993). In the present setting, however, the issue isn’t that the data is missing, but rather that the DMU does not have that input. Furthermore, substituting a value of zero where no value is present will produce distorted results. Thus, to address the problem of missing inputs we propose to proceed as follows.

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For a DMU jo  N2 , for example, we argue that a portion  of x1 jo and x2 jo is linked with

x3 jo to produce a portion  of that DMUs outputs y1 jo , y2 jo . The remaining amounts 1   of ( x1 jo , x2 jo ) are linked with x4 j o to produce the remainder 1   of the outputs for that DMU. (We use very simplistic notation here, namely (  ,  ) to illustrate the point regarding the splitting of

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inputs and outputs. Below, a more general and precise notation is defined in regard to these proportions of inputs and outputs). Continuing with this line of reasoning for DMU jo  N2 , we can view the process of inputs generating outputs as being divided into two separate “processes”. This situation has a minor connection to the concept of parallel processes in two-stage DEA, as discussed in Kao (2009a), but is not directly related to the multiple input configuration problem.

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See also Kao (2009b) and Kao and Hwang (2008).

Formalizing the above concept we use the following notation: Let the decision variable

iN  denote the proportion of input xij (i=1,2) for a DMU j  N p that will be linked with p

natural resource  . In case N p  N2 , for example,  takes index values 3 (associated with natural

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resource x3 or coal), and 4 (associated with natural resource x4 or natural gas). Similarly, let

rN  denote the proportion of output yrj (r=1,2) for a DMU j  N p linked with natural resource

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p

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 . We refer to  iN p and  rN p as splitting variables.

So, we view this as a situation where there are 2 production processes for a DMU j  N 2 :

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Process 1: Here, inputs [ 1N2 3 x1 j , 2 N2 3 x2 j , x3 j ] generate a portion of each of outputs y1 j , y2 j in amounts [1N2 3 y1 j ,  2 N2 3 y2 j ] . Note that we split or divide the outputs in a manner similar to that

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for the inputs.

Process 2: Here, inputs [1 N 4 x1 j ,  2 N 4 x2 j , x4 j ] generate the two outputs in the amounts 2

2

[1N2 4 y1 j ,  2 N2 4 y2 j ] .

Constraints on Splitting variables  and  For DMUs in N1 , constraints on multipliers can be expressed in the same form as that 9

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given in the original CCR model, since with only one natural resource, no splitting of inputs and outputs occurs. For DMUs in N 2 , however, we have split the “production function” into two separate processes as per Figure 2. In the model presented below, it would appear to be appropriate to impose constraints on those separate processes. At the same time we connect the processes through the splitting variables, specifically requiring the imposition of restrictions 2

2

2

2

4

 1 , for outputs r  1,2 . The same ideas hold

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 iN 3   iN 4  1 for inputs i  1,2 and  rN 3   rN for N3 , N 4 Figure 2: Two Production Processes in N2

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x1 j , x2 j , x3 j , x4 j

Process 2

Process 1

1N 3 x1 j , 2 N 3 x2 j , x3 j 2

1N 4 x1 j , 2 N 4 x2 j , x4 j

2

2

2

1N 4 y1 j ,  2 N 4 y2 j 2

2

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2

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1N 3 y1 j , 2 N 3 y2 j

2

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y1 j , y2 j

Modelling Efficiency of DMUs in N 2

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In deriving the efficiency score for any DMU j  N 2 we propose that the multipliers be

chosen in a manner that requires the ratio of inputs to outputs for each of the processes to be at or

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above unity. By doing this, our model attempts to capture both the overall efficiency of the DMU, while at the same time deriving the efficiency of the two processes for those DMUs. Given the two processes in N 2 , we suggest that the multipliers vi , ur and the splitting variables  and  satisfy

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v11N2 3 x1 j  v2 2 N2 3 x2 j  v3 x3 j

 1 (Pr ocess 1)

v11N2 4 x1 j  v2 2 N2 4 x2 j  v4 x4 j

 1 (Pr ocess 2)

u11N2 3 y1 j  u2  2 N2 3 y2 j

u11N2 4 y1 j  u2  2 N2 4 y2 j

(3.2)

 iN 3   iN 4  1, i  1,2 2

2

 rN 3   rN 4  1, r  1,2 2

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2

Modelling Efficiency of DMUs in N 3

There is a similar 2-process structure in N 3 with the difference being that instead of the second process involving x4 or natural gas (Process 2), it involves x5 or petroleum (Process 3). Figure 2 suffices to describe the setting for N 3 by substituting x5 in place of x4 (we do not

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replicate this here). Hence, iN2 is replaced by iN3 , and   5 denotes x5 not x4 . Thus, the constraints are:

v11N3 3 x1 j  v2 2 N3 3 x2 j  v3 x3 j

 1 (Pr ocess 1)

v11N2 5 x1 j  v2 2 N2 5 x2 j  v5 x5 j

 1 (Pr ocess 3)

u11N3 3 y1 j  u2  2 N3 3 y2 j

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u11N3 5 y1 j  u2  2 N3 5 y2 j

(3.3)

 iN 3   iN 5  1, i  1,2 3

3

3

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 rN 3   rN 5  1, r  1,2 3

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Modeling Efficiency of DMUs in N 4

Note that in N 4 we have 3 not 2 sources of natural resources x3 , x4 , x5 . The appropriate

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characterization is displayed as Figure 3. In the same spirit as above, the requisite constraints are: v11N4 3 x1 j  v2 2 N4 3 x2 j  v3 x3 j

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u11N4 3 y1 j  u2  2 N4 3 y2 j

 1 (Pr ocess 1)

v11N4 4 x1 j  v2 2 N4 4 x2 j  v4 x4 j

 1 (Pr ocess 2)

v11N4 5 x1 j  v2 2 N4 5 x2 j  v5 x5 j

 1 (Pr ocess 3)

u11N4 4 y1 j  u2  2 N4 4 y2 j

u11N4 5 y1 j  u2  2 N4 5 y2 j

(3.4)

 iN 3   iN 4   iN 5  1, i  1,2 4

4

4

 rN 3   rN 4   rN 5  1, r  1,2 4

4

4

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Figure 3: Three Production Processes in N4 x1 j , x2 j , x3 j , x4 j

1N 4 x1 j , 2 N 4 x2 j , x4 j

1N 3 x1 j , 2 N 3 x2 j , x3 j 4

4

4

4

1N 4 y1 j , 2 N 4 y2 j

1N 3 y1 j , 2 N 3 y2 j

4

4

4

1N 5 x1 j , 2 N 5 x2 j , x5 j 4

4

1N 5 y1 j ,  2 N 5 y2 j 4

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4

Process 2

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y1 j , y2 j

4. Modeling Efficiency in the Presence of Multiple Processes

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To evaluate efficiencies of DMUs in the presence of non-homogeneous input sets, as in the case of the various provinces in China, we propose a three-step procedure. In the first step we determine for each DMU in each process (in which that DMU is involved), an appropriate split

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of the inputs and outputs in the form of the variables iNp  and rN p  . From these proportions, one

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derives the scaled down inputs and outputs for each DMU jo for each process in the DMU group to which jo belongs. In step 2 we use the scaled down inputs and outputs for a DMU jo (from step 1) in each of the processes involving jo , and compute the process score for that DMU. This

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,

Process 3

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Process 1

is repeated for each of the processes containing jo . In step 3, for any given DMU, we take a weighted average of the process scores for that DMU, as derived in step 2. This yields the aggregate score for that DMU. We discuss the three steps in detail.

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4.1 Step1: Deriving the Split of Inputs and Outputs In this step we wish to determine, for any given DMU, an appropriate split of inputs and outputs, across the processes to which that DMU belongs. We argue that the best way to divide up the inputs and outputs, hence determine the most appropriate alpha and beta variables, is to do so in a manner that results in the best overall score for the DMU, across all of its processes.

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Further, we argue that the overall efficiency of a DMU jo  N po can reasonably be represented as a weighted average (convex combination) of the process efficiencies. We point out that this argument is essentially that the DMU is the sum of its parts, and therefore assumes there are no economies or dis-economies of scope (see Pulley et al. (1992)). In cases where it is believed that such economies (dis-economies) of scope do exist, our approach may not accurately capture

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efficiency at the aggregate level.

Given that it is the overall efficiency of the DMU that we wish to derive, and that this overall efficiency will be represented as a convex combination of the process efficiencies, we set out to determine the  -split of inputs and  -split of outputs, with the objective of maximizing

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the overall score. In general terms, and using N 2 as an example, one can represent the overall efficiency of a DMUj in terms of its processes as

v11N2 3 x1 j  v2 2 N2 3 x2 j  v3 x3 j

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eo  min WN21[

v11N2 4 x1 j  v2 2 N2 4 x2 j  v4 x4 j u11N2 4 y1 j  u2  2 N2 4 y2 j

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WN2 2 [

u11N2 3 y1 j  u2  2 N2 3 y2 j

]

]

(4.1)

where the process weights WN 2 1 and WN2 2 are such that W N 1  W N 2

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 1, and should reflect the

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relative importance of the two processes. These weights can, of course, be chosen in any number of ways, depending on the wishes of the decision maker. One logical choice would appear to be to select them according to the contributions the respective processes make to overall production. Since the output oriented model we are using herein is directed toward output enhancement, an appropriate choice for the weight W assigned to any process, viewed from an accounting perspective, would be the proportion of weighted total output generated by the processes in

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question.

Hence, WN21 and WN2 2

for

any

given

DMUj,

WN21  [u11N2 3 y1 j  u2 2 N2 3 y2 j ] /[u1 y1 j u 2 y2 j ]

(4.2a)

WN2 2  [u11N2 4 y1 j  u2 2 N2 4 y2 j ] /[u1 y1 j u 2 y2 j ]

(4.2b) ,

can

be

expressed

as

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respectively. Following this logic (in slightly simplified notation), one arrives at similar expressions for process weights for all groups of DMUs N1 , N2 , N3 , N4 , namely: N1

Wo  1

N2

W1 

N4

W1  W2 

u11N3 3 y1 j  u 2  2 N3 3 y2 j

(Pr ocess 1)

u1 y1 j  u 2 y2 j u11N3 5 y1 j  u 2  2 N3 5 y2 j

(Pr ocess 2)

u1 y1 j  u 2 y2 j u11N 4 3 y1 j  u 2  2 N 4 3 y2 j u1 y1 j  u 2 y2 j

(Pr ocess 1)

u11N 4 4 y1 j  u 2  2 N 4 4 y2 j u1 y1 j  u 2 y2 j

u11N 4 5 y1 j  u 2  2 N 4 5 y2 j u1 y1 j  u 2 y2 j

(Pr ocess 2) (Pr ocess 3)

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W3 

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W2 

(Pr ocess 2)

u1 y1 j  u 2 y2 j

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W1 

u11N 2 4 y1 j  u 2  2 N 2 4 y2 j

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N3

(Pr ocess 1)

u1 y1 j  u 2 y2 j

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W2 

u11N 2 3 y1 j  u 2  2 N 2 3 y2 j

Clearly, defining the weights on the processes according to (4.2a) and (4.2b), (4.1)

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becomes

eo  min [

v11N2 3 x1 j  v2 2 N2 3 x2 j  v3 x3 j [u1 y1 j u 2 y2 j ]

][

v11N2 4 x1 j  v2 2 N2 4 x2 j  v4 x4 j [u1 y1 j u 2 y2 j ]

which, by definition of the alpha and beta variables in (3.2), yields

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]

(4.3)

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eo  min

v1 x1 j  v2 x2 j  v3 x3 j  v4 x4 j u1 y1 j  u2 y2 j

(4.4)

Below we present the overall efficiency model, where we denote the objective function by e o . In case there are different sets of DMUs, as is the situation herein, and applying the logic

involved:

N1

eo 

v1 x1 jo  v2 x2 jo  v3 x3 jo

N2

eo 

v1 x1 jo  v2 x2 jo  v3 x3 jo  v4 x4 jo

N3

eo 

v1 x1 jo  v2 x2 jo  v3 x3 jo  v5 x5 jo

N4

eo 

v1 x1 jo  v2 x2 jo  v3 x3 jo  v4 x4 jo  v5 x5 jo

u1 y1 jo  u2 y2 jo

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u1 y1 jo  u2 y2 jo

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that leads to (4.4), it follows that e o takes the following forms depending on the DMU group

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u1 y1 jo  u2 y2 jo

(4.5b)

(4.5c)

(4.5d)

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u1 y1 jo  u2 y2 jo

(4.5a)

The Overall Efficiency Model

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Model (4.6) below is designed to derive an overall efficiency score for any given DMU

jo in any one of the DMU sets N1 , N2 , N3 , N4 . We impose constraints on all processes in all

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DMU sets, and note specifically that DMUs in any group N p have a different set of alpha and beta splitting variables, as compared to those of the other sets. For DMU set N1 , no splitting of

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inputs and outputs is required; thus, it is necessary to impose, as per (4.6b), only the constraint requiring that the ratio of weighted inputs to outputs for each of that group’s members be at or above unity. For each DMU in any of the other three sets, however, it is necessary to impose this ratio restriction on each process for that DMU. In the case of N 2 , for example, process constraints (4.6c) and (4.6d) are required. As well, we impose on DMUs in each of N2 , N3 , N4 the requirement that the slitting variables alpha for any input for the various processes in that DMU set, sum to unity. The same applies to the output splitting variables beta. See, for example, 15

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constraints (4.6e) and (4.6f) in the case of N 2 . Finally, we place upper and lower limit constraints ((4.6p), (4.6q)) on the alpha and beta variables to control against extreme values being assigned.

u1 y1 j  u2 y2 j

(4.6a)

 1,

j  N1

(4.6b)

v11N2 3 x1 j  v2 2 N2 3 x2 j  v3 x3 j

 1,

j  N2

(Pr ocess 1)

v11N2 4 x1 j  v2 2 N2 4 x2 j  v4 x4 j

 1,

j  N2

(Pr ocess 2)

u11N2 3 y1 j  u2  2 N2 3 y2 j

u11N2 4 y1 j  u2  2 N2 4 y2 j

 iN 3   iN 4  1, i  1,2  rN 3   rN 4  1, r  1,2 2

2

 1,

j  N3

v11N3 5 x1 j  v2 2 N3 5 x2 j  v5 x5 j

 1,

j  N3

u11N3 3 y1 j  u2  2 N3 3 y2 j

u11N3 5 y1 j  u2  2 N3 5 y2 j

 iN 3   iN 5  1, i  1,2 3

 rN 3   rN 5  1, r  1,2 3

3

v11N4 4 x1 j  v2 2 N4 4 x2 j  v4 x4 j v11N4 5 x1 j  v2 2 N4 5 x2 j  v5 x5 j

(Pr ocess 2)

(4.6h) (4.6i ) (4.6 j )

(Pr ocess 1)

( 4 .6 k )

 1,

j  N4

(Pr ocess 2)

(4.6l )

j  N4

(Pr ocess 3)

(4.6m)

 1,

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u11N4 5 y1 j  u2  2 N4 5 y2 j

(4.6 g )

j  N4

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u11N4 4 y1 j  u2  2 N4 4 y2 j

(4.6 f )

 1,

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v11N4 3 x1 j  v2 2 N4 3 x2 j  v3 x3 j u11N4 3 y1 j  u2  2 N4 3 y2 j

( 4 .6 d )

(Pr ocess 1)

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v11N3 3 x1 j  v2 2 N3 3 x2 j  v3 x3 j

3

(4.6c)

(4.6e)

2

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2

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Min e o Subject to v1 x1 j  v2 x2 j  v3 x3 j

 iN 3   iN 4   iN 5  1, i  1,2

( 4 .6 n )

 rN 3   rN 4   rN 5  1, r  1,2

(4.6o)

a   iNk   b

(4.6 p)

c   rNk   d

( 4 .6 q )

ur , vi  

(4.6r )

4

4

4

4

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4

4

r , i

Again, as discussed above, the purpose of model (4.6) is to derive an appropriate set of splitting variables alpha and beta. It must be pointed out that the upper and lower bounds on these variables, as expressed in (4.6p) and (4.6q), can be difficult to estimate, depending on the 16

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application, whether highly micro or highly macro. Consider, for example, a micro problem setting involving a manufacturing plant where different mixes of machines and labor are used to make a set of products. In this relatively controlled environment, reasonably accurate estimates of the bounds could be determined regarding the portions of the time that any given resource mix (alphas) is used, and the portions of products produced under a given resource mix (betas). In the

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current highly macro level application, however, it is far more difficult to obtain estimates of these bounds. Moreover, in the controlled manufacturing environment, production relationships (which input mixes produce which portions of the outputs) are rather easily observable. More to the point, in such a micro situation where efficiencies of different processes (one machine/labor mix versus another mix) are observable and can be derived using the models here, the

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methodology developed herein would be a valuable tool for benchmarking those different processes. In the highly macro setting of the Chinese provinces, production relationships are clearly less tangible and less observable. The Linear Formulation

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The above model is nonlinear, but can be converted to linear form along the lines of the original Charnes and Cooper (1962) transformation of the fractional programming problem into a linear problem. Specifically, on the input side make the following changes of variables:

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Define the variables ziNpl  viiNpl . So, for example, ziN2 3  vi iN2 3 . Now define t  1 /[  u r yrj o ] r

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and let  iN pl  tz iN pl . Note that iN pl  1 implies that i iN pl  vi , meaning that   iN p l  vi . l

l

l

On the output side define rN p l  ur  rN p l , and then let rN pl  trN pl . As above, r Npl  r .

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l

Now replace vi , u r by i , r . The linear version of the above model is then given by (4.7). We use

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 the notation e o rather than e o to denote that it is only the numerator of e o that appears in (4.7)

(since from (4.7b) the denominator is 1). Hence, (4.7) is a linear model. We point out that the solution of (4.7), for any given DMU, yields a set of optimal

variables ˆi , ˆ r , ˆiNp , ˆrN p which are specific to that DMU being evaluated. From the above variable transformations, the values for the splitting variables (alphas and betas) are immediately available as well, since ˆiNp  ˆiNp / ˆi and ˆrN p  ˆrN p / ˆ r . Given these variables, the scaled 17

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down inputs and outputs, (denoted by xˆij , yˆ rj , respectively), are available for each process  in the DMU under analysis; specifically, xˆij  ˆ iN p xij and , yˆ rj  ˆrN p yrj .

Min e o Subject to 1 y1 jo   2 y2 jo  1

(4.7a)

1 x1 j   2 x2 j  3 x3 j  [ 1 y1 j   2 y2 j ]  0, j  N1

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(4.7b)

(4.7c)

 1N 3 x1 j   2 N 3 x2 j  3 x3 j  [1N 3 y1 j   2 N 3 y2 j ]  0, j  N 2

(Pr ocess 1)

(4.7 d )

 1N 4 x1 j   2 N 4 x2 j   4 x4 j  [1N 4 y1 j   2 N 4 y2 j  0, j  N 2

(Pr ocess 2)

(4.7e)

2

2

2

2

2

2

2

2

 iN 3   iN 4  i , i  1,2 2

(4.7 f )

2

 rN 3   rN 4   r , r  1,2 2

(4.7 g )

2

(4.7 h)

 1N 5 x1 j   2 N 5 x2 j  5 x5 j  [1N 5 y1 j   2 N 5 y2 j ]  0, j  N 3 (Pr ocess 2)

(4.7i )

 iN 3   iN 5  i , i  1,2

(4.7 j )

3

3

3

3

3

3

3

3

3

3

 rN 3   rN 5   r , r  1,2 3

3

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 1N 3 x1 j   2 N 3 x2 j  3 x3 j  [1N 3 y1 j   2 N 3 y2 j ]  0, j  N 3 (Pr ocess 1)

(4.7 k )

 1N 3 x1 j   2 N 3 x2 j  3 x3 j  [1N 3 y1 j   2 N 3 y2 j ]  0, j  N 4 (Pr ocess 1)

(4.7l )

 1N 4 x1 j   2 N 4 x2 j   4 x4 j  [1N 4 y1 j   2 N 4 y2 j ]  0, j  N 4 (Pr ocess 2)

(4.7 m)

4

4

4

4

4

4

4

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4

 1N 5 x1 j   2 N 5 x2 j  5 x5 j  [1N 5 y1 j   2 N 5 y2 j ]  0, j  N 4 (Pr ocess 3)

(4.7n)

 iN 3   iN 4   iN 5  i , i  1,2

(4.7o)

4

4

4

4

4

4

4

 rN 3   rN 4   rN 5   r , r  1,2 4

4

ai   iN pl , i, l , N p

i ,  r   , i, r

(4.7q ) (4.7r )

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a r   rN p kl , r , l , N p

(4.7 p)

ED

4

(4.7 s )

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4.2 Step 2: Deriving the Process Efficiency Scores The previous step determined a recommended set of scaled down inputs and outputs for

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each of the processes for a DMU under analysis. All DMUs have a Process 1 component. For DMUs in N1 the inputs and outputs take on their original values, while those DMUs in

N2 , N3 , N4 have scaled down values for the inputs and outputs for Process 1, as determined in

Step1. More explicitly, in Process 1(   3 ), for DMUs j  N1 , xˆij  xˆij 3  xij , yˆ rj  yˆ rj3  yrj (the original values), while for j  N p , p  2,3,4 , xˆij 3  ˆiN p 3 xij , yˆ rj3  ˆrN p 3 yrj . In Process 2 (   4) we

18

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consider only DMUs j  N p , p  2,4 , and xˆij 4  ˆiN p 4 xij , yˆ rj 4  ˆrN p 4 yrj . In Process 3 (   5) , we have DMUs j  N p , p  3,4 , and xˆij 5  ˆ iN p 5 xij , yˆ rj5  ˆrN p 5 yrj . Using these definitions of xˆij , yˆ rj we implement model (4.8) for each of the three processes:

 xˆ min  u yˆ

i ijo 3

i ijo 5

i

rjo 3

r

r

(4.8 P1)

 xˆ  u yˆ

i

rjo 4

r

r

subject to i ij 3

i

r

 xˆ min  u yˆ

i ijo 4

i

r

Process 3 Model

subject to

( 4 .8 P 2 )

 xˆ  u yˆ

i ij 4

 1, j  1,...n

i

rj 3

r

r

rj 4

r

vi , ur  0

rjo 5

r

vi , ur  0

 1, j  N 2  N 4

subject to

(4.8 P3)

 xˆ  u yˆ

i ij 5

i

 1, j  N 3  N 4

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 xˆ min  u yˆ

Process 2 Model

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Process 1 Model

r

rj 5

r

vi , ur  0

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4.3 Step 3: Deriving the Aggregate Efficiency Scores

In this final step, to get the aggregate score for any DMU, we take the weighted average

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of its process scores. Recalling the definition of the process weights W above, they are immediately available from model (4.7). In other words, for DMUs in N 2 , the process 1 and process 2 weights are simply the output components in constraints (4.7d) and (4.7e),

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respectively. Hence,

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W1  1N2 3 y1 j  2 N2 3 y2 j and W2  1N3 5 y1 j  2 N3 5 y2 j (4.9) The same idea applies to DMUs in sets N3 , N 4 . In the following section we apply the models developed herein to evaluate the

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efficiencies of the 31 regions in China. 5. Efficiency Measurement of Provinces in China Table1 displays performance data on 31 provinces/regions in China. This data is published by the China Statistical Yearbook (2013) and China Statistical Yearbook on Environment (2013). We evaluate the regions, taking as inputs water, capital investment and natural resources; the latter inputs fall into three categories, namely coal, natural gas and 19

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petroleum. On the output side, the regions are viewed in terms of GDP per capita and an undesirable factor, nitrogen dioxide. For analysis purposes, the latter output has undergone translation by subtracting its values from a number larger than the maximum of all the observed values. In this case we used 60 as the translation constant. See Seiford and Zhu (2002). We can interpret the translated values as the amounts of nitrogen dioxide avoided or not experienced,

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which can be seen as having desirable status. We have not shown the translated values. The regions fall into 4 groups as shown in Figure 1. Eight of the provinces have only coal as a natural resource, constituting group N1 . DMU groups N 2 (the coal and natural gas set) and N 3 (the coal and petroleum set), contain only one member each. The 21 members of group N 4

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have all three natural resources. We remind the reader again that there are three processes involved in generating the outputs for the DMUs , namely process 1 which uses a portion of inputs 1 and 2 together with coal, process 2 using portions of inputs 1 and 2 together with natural gas, and finally process 3 using portions of inputs 1 and 2 together with petroleum.

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We note that in solving the overall model (4.7) we have imposed lower and upper limits of 10% and 90%, respectively, on the alpha and beta variables (as per (4.7q) and (4.7r)). We

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point out that these limits were set rather arbitrarily, however, some sensitivity tests on these limits did not result in significant changes in the scores. Tables 2 and 3 display the alpha and beta

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splitting variables for N 2 and N 3 respectively. The alpha and beta values for N 4 are presented in Tables 4 and 5, respectively. Recall again that the purpose of model (4.7) is to split the inputs

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and outputs such as to transform the DMU into what we might view as a set of parallel processes. The alpha variables derived in (4.7) describe the portions of each of inputs x1 , x2 in a DMU group that are paired up with an energy variable (e.g. petroleum). Consider, for example,

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the first row (Tianjin) of Table 4:

Region

1N 3

2N 3

1N

Tianjin

0.56

0.64

0.28

4

4

44

2N

44

0.20

1N 5

2N 5

0.16

0.17

4

4

Thus, for this region (in N 4 ) 56% of input x1 is linked with coal (Process 1), 28% is linked with natural gas, and the remaining 16% is linked with petroleum. Similarly the other three elements 20

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in this row are the allocations for input x2 . Tables 6, 7 and 8 contain the scaled data for the three processes as described in step 1 above. Table 9 contains the final results from the analysis. It is noted that we have displayed the

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inverse of the output oriented scores that actually arise from the models (3.1) and (4.7). It is more appropriate to declare the efficiencies in this manner. Column e0 is the (inverse of the) score arising from the solution of the step 1 model (4.7). Recall that this model yields the alpha and beta values given in Tables 2, 3, 4 and 5. Columns e1, e2 and e3 contain the scores for processes 1, 2 and 3, respectively, from step 2 described above. Displayed in columns W1 ,W2 and W3 are

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the process weights arising from the solution of (4.7). These are used to construct the aggregate scores eT.

We note that the aggregate efficiency eT of a DMU is at the full 100% efficiency level only when all processes for that DMU are efficient as well. Full efficiency occurs for 6 of the

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provinces, while another 17 provinces are inefficient in all processes. (Out of this set of 17 regions, however, 6 of those are in N1 , where there is only one process (coal)). Each of the

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remaining 8 provinces shows a mix of efficient and inefficient processes.

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We note that the aggregate efficiency eT of a DMU is at the full 100% efficiency level only when all processes for that DMU are efficient as well. The 6 fully efficient provinces are

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Tibet, Shanghai, Hainan, Guangdong, Ningxia and Tianjin. The first two of these have only process 1 (in N1 ), the third has 2 processes (in N 3 ), and the latter three have all three processes

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(in N 4 ). These six provinces have a superior environmental and economic reputation. Guangdong province is, for example, highly developed in economic terms; it is a recognized leader among 31 provinces, hence it seems appropriate that it is efficient in all three processes. Similar cases can be made for some of the other efficient DMU’s. In the case of Tibet, for example, there is a flourishing tourism economy, and as well has little pollution. These two factors would appear to contribute to its full efficient status.

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In N 4 , 11 provinces/regions are inefficient in all processes, with Heilongjiang, Jilin and Liaoning being some of the worse. Heilongjiang (.2430) has process efficiency scores of (.2367), (.2820), and (2541) for processes 1, 2 and 3, respectively; Jilin (.4043) has two process efficiency scores less than 30%, namely process 2 (.1185) and process 3 (.2424); Liaoning (.5797) has two process efficiency scores less than 20%, that is process 1 (.1556) and process 3

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(.1403), respectively. The results for these three cases are as one might expect, in that are old industrial provinces with poor reputations in an environmental and economic sense.

We note that 7 of the provinces in N 4 show a mix of efficient and inefficient processes. There are few conclusive observations one can make here, although there is again some evidence

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of coal being a disadvantageous factor. For example, in the case of Anhui (eT=.5259) and Jiangsu (eT=.7267), process 2 is 100% efficient in both, process 3 is rated at .5189 in Anhui, and at .4694 in Jiangsu, while process 1 rates only at .1074 and .1336 in Anhui and Jiangsu, respectively. Given the low values of the process 1 efficiency scores, it might be argued again

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that reduced reliance on coal could benefit the efficiency status of these provinces. In summary, it is worth commenting on the capabilities of the various regions in terms of

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the processing of the three natural resources. If one examines the 21 regions in N 4 where all three resources exist, we note some useful statistics. First, only 4 of the 21 regions here are efficient in

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their consumption of coal (process 1); compare this to 7 and 5 efficient regions in terms of the consumption of natural gas and petroleum (processes 2 and 3), respectively. Going further, 15 of

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the 21 regions in N 4 have process 1 efficiency scores less than 50%, while the corresponding frequencies for each of processes 2 and 3 are 9 and 12, respectively. Ten of the 21 DMUs in N 4

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have process 1 efficiency scores under 20%. The corresponding scores for processes 2 and 3 are 5 and 5 DMUs, respectively. Finally, the average process 1 efficiency among the DMUs in N 4 is 0.337, as compared to 0.599 and 0.508 for processes 2 and 3. These statistics would appear to point to process 1 as being significantly less efficient than the other two processes. The reader is advised, of course, to view these statistics with some caution, given that different frontiers exist across the three processes. 22

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6. Discussion and Further Directions In the usual DEA setting it is assumed that a set of DMUs is to be evaluated in terms of an input/output bundle that is common to all members of that set. In some situations, however, this commonality assumption does not hold. This paper examines the problem of measuring relative efficiency in the presence of different input configurations across a set of DMUs. Such multiple configurations appear rather regularly in manufacturing environments. We develop a DEA-like

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methodology to address this non-conventional situation, and apply it to 31 provinces in China where a set of macro level inputs is assumed to contribute to the generation of a set of macro level outputs. The DMUs (the provinces) are, however, not homogeneous on the input side. Specifically, an important input to the process is natural resources, which, in the current setting takes three different forms, namely coal, natural gas and petroleum. The issue is that some of

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these resources are held by a subset, but not all, of the provinces. To evaluate the DMUs in a fair manner, a multiple-process type of model is developed which allows the overall efficiency of a DMU to be expressed as a weighted average of its process efficiencies. The approach is intended to provide a fair evaluation of efficiency for each DMU, meaning that it is neither penalized nor rewarded for having or not having a given input.

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To be able to interpret the performance of each province, it is important to view efficiency at the level of the DMU, and at the level of the sub-DMU (process). This provides

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important insights into how efficiently a province utilizes each of its natural resources. As discussed in the previous section, the efficiency with which coal is utilized is substantially worse than is the case for the other two natural resources. In the provinces/cities/regions in where all

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three of the natural resources are available, coal is the least efficient in 11 of the 21 DMUs. Corresponding figures for processes 2 and 3 are 5 and 2, respectively. As well we point out, as

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discussed above, that the averages of the efficiency scores for the three processes in N4 are 0.337, 0.599 and 0.508 for processes 1, 2 and 3 respectively. This evidence would appears to point to the need for less reliance on coal and more on cleaner resources in the form of natural

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gas and petroleum.

Limitations: Our methodology has limitations. It is important to emphasize again the key assumption we are making herein, namely that the three natural resource inputs perform the same function in their roles as aids in the generation of outputs. Specifically, coal, natural gas and petroleum are substitutes for one another, although as indicated earlier, the rates of substitution may be difficult 23

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to specify. Hence, one may not be able to convert the different resources into a single input. It might be argued that the models developed herein are more suited to macro settings at regional or provincial levels, and may not necessarily be completely adaptable to micro-level applications such as a set of firms, industries or different markets. Hence, generalizability of this concept to micro economic situations may not necessarily be achievable. On the other hand, in the

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hypothetical manufacturing setting discussed in section 4, the methodology would appear to be entirely appropriate. If the application (whether micro or macro) is one where the available natural resources are generally aimed, for example, at generating electricity, it might be contended that the multiple natural resources are, in a sense, convertible to a single resource (electricity), hence the inherent complication created by non-homogeneity is removed from the

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picture. This, however, would appear to call for a two-stage efficiency evaluation process, rather than a single stage setting, as discussed in this paper. This approach has its own set of challenges, however, in that we still have some DMUs possessing certain resources that other DMUs do not have. So, regardless of the modelling approach, one must still confront some form of the non-

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homogeneity issue.

Arguably, the application discussed here is not only at a macro level, but as well, it is one

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where strict government control is in place, hence it is important to provide for the inclusion of undesirable outputs such as Nitrogen Dioxide, a major contributor to poor air and water quality in China. Centralized control would seem to imply, as well, that provinces or regions have less

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autonomy as to how, and in what combinations, natural resources are deployed. At this level, we are not concerned about issues such as usage of the different resources (e.g. electricity for

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factories, petroleum for transportation needs (e.g. automobiles), natural gas for home heating, etc.). In a firm-level or industry-level environment, where a more micro view can be taken, and

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where management is at liberty to choose whether or not to be concerned about environmental factors, a different approach may be warranted. Presumably at this level, management might choose to exclude such environmental factors, and as well, individual managers may choose to not deploy certain resources, but rather to adopt their own policies. This can have implications about the splitting variables (the alphas and betas). In some industries management can, for example, choose to implement technologies aimed at controlling pollution. The result of this can be the phasing out of, or reducing the use of certain natural resources (e.g. coal). In this situation, 24

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the implication may be that as a resource is phased out, the alphas (proportions of other inputs linked to that “declining” natural resource) should, over time, be required to move to zero. Specifically, less and less of the other (linked) inputs should be deployed to output production. This would mean, of course, that the associated proportions of outputs (defined by the betas) generated by that declining bundle of inputs, should be forced to zero as well. This being said, it

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would appear that the application of our methodology would need to be carefully monitored to accommodate the changing environment in which DMU management finds itself. Methodologically, this implies that the upper and lower bounds on the alphas and betas need to be carefully tied to changing conditions. While such changing conditions may appear to point to a shortcoming of the methodology developed here, it is worth remembering that the DEA model

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arguably suffers from a similar lack of provision for dealing with such changes. Thus, to some extent, the DEA model and the frontier involved are rather moving targets. Another limitation is, of course, the small samples of DMUs (regions) in certain groups (the second and third groups have only 1 DMU each). Clearly, it is difficult to make any

M

meaningful inferential claims regarding those two groups. Future applications of this

ED

methodology should seek to have statistically valid sampled sizes in each of any groups defined.

In the current paper, only multiple configurations on the input side are considered, as compared to the earlier work by Cook et al (2012, 2013), where non homogeneity on the output

PT

side was examined. An important direction for future research is the investigation into the

CE

general case dealing with non-homogeneity on both input and output sides. Another important area for future research is the problem of economies of scope. Specifically, we have assumed herein that multiple processes can be aggregated via a weighted

AC

average (arithmetic mean), to arrive at an overall DMU score. This aggregation approach presumes that no economies or diseconomies of scope are present. When scope considerations are necessary, the weighted average may not provide an adequate measure of efficiency.

25

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References Charnes, A., Cooper, W.W., 1962. Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9(3-4), 181-186.

China Statistical Yearbook. 2013 China Statistical Yearbook and the Environment. 2013

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Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444.

Cook W. D., Harrison, J, Rouse, P, Zhu, J., 2012. Relative efficiency measurement: The problem of a missing output in a subset of decision making units. European Journal of Operational Research, 220(1), 79–84.

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Cook, W. D., Imanirad, R., Zhu, J., 2013a. Partial input to output impacts in DEA: Production considerations and resource sharing among business sub-units. Naval Research Logistics, 60(3), 190-207. Cook W. D., Harrison, J., Imanirad, R, Rouse, P, Zhu, J, 2013b. Data envelopment analysis with nonhomogeneous decision making units. Operations Research, 61(3), 666-676.

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Cook, W. D., Seiford, L., 2009. Data envelopment analysis (DEA)-Thirty years on, European Journal of Operational Research, 192 (1),1-17

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Cooper, W.W., Huang, Z., Li, S., Lelas, V., Sullivan, D.W., 1996. Survey of mathematical programming models in air pollution management. European Journal of Operational Research, 96, 1–35.

PT

Kao, C., 2009a. Efficiency measurement for parallel production systems. European Journal of Operational Research, 196(3), 1107–1112.

CE

Kao, C., 2009b. Efficiency decomposition in network DEA: A relational model. European Journal of Operational Research, 192, 949-962.

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Kao, C., Hwang, S.N., 2008, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European Journal of Operational Research, 185(1), 418-429. Lee, C.-C., 2005. Energy consumption and GDP in developing countries: A co-integrated panel analysis. Energy Economics, 27, 415–427. Paradi, J.C., Zhu, H., 2013. A survey of bank branch efficiency and performance research with data envelopment analysis. Omega, 41(1), 61-79.

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Pulley, L. B., Braunstein, Y. M., 1992. A composite cost function for multiproduct firms with an application to economies of scope in banking. Review of Economics and Statistics, 74, 221–230. Seiford, L., Zhu, J., 2002. Models for undesirable factors in efficiency evaluation. European Journal of Operational Research, 142, 16-20.

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Soytas, L., Sari, R., 2009. Ecological economics, energy consumption, economic growth and carbon emissions: Challenges faced by an EU candidate member. 68(6), 1667-1675. Thompson, R.G., Dharmapala, P. S.,Thrall, R. M., 1993. Importance for DEA of zeros in the data, multipliers, and solutions. Journal of Productivity Analysis, 4, 379-390. Zhou, P., Ang, B.W., Poh, K.L., 2008. A survey of data envelopment analysis in energy and environmental studies. European Journal of Operational Research, 189, 1–18.

Table 1: Data on Regions in China Variables

AN US

Zhou, P., Ang, B.W., 2008. Linear programming models for measuring economy-wide energy efficiency performance. Energy Policy. 36, 2911–2916.

Inputs

Coal Resource 100 million tons

Natural Gas Resource 100 million cu. m

8870.8

3.73

8311.7 19243.6

1511.4 2174.4 1988.9

Petroleum Resource

Per Capita

Nitrogen

10000 tons

10000 Yuan

tons

0

0

8.64

3.36

908.42 0.43

0 0

0 0

3.52 6.33

8.45 9.23

13850.7 12103.1 15898.5

4.44 4.11 6.61

0 0 0

0 0 0

5.26 2.88 3.34

9.57 11.38 22.05

4196.4 33.9 974 364.3 32.9

696.7 6961.2 5949.1 2755.8 8853.6

0.12 0.1 69.39 1.19 2.97

0 0 5.44 0 278.78

0 0 0 297.5 3034.52

2.28 8.48 1.97 3.22 9.13

0.57 1.55 4.68 4.12 3.29

Hebei Inner Mongolia Liaoning Jilin

235.9 510.3 547.3 460.5

20106 12174.6 24225.6 9694.4

39.51 401.66 31.92 9.82

315.37 8344.3 178.54 776.22

26934.54 8517.07 16946.82 18304.08

3.65 6.38 5.66 4.61

36.04 26.7 20.53 12.9

Heilongjiang

841.4

10400.5

61.64

1381.51

50137.48

3.57

24.75

Beijing

39.5

AC

Tibet Shanghai Guizhou Hainan Tianjin

CE

Fujian Jiangxi Hunan

106.2 1444.8

100 million Yuan

PT

Shanxi Zhejiang

M

Units

Investment

ED

Water Resource 100 million cu. m

Outputs

27

ACCEPTED MANUSCRIPT

36552.9 16587.8 33538.2 21710.1 16884.5 22005.9 10506.8 10312 18204 8047.5 13222.3 5365.8 1982.7 1998.3 6572.9

10.82 80.38 79.73 99.09 3.25 0.23 2.08 19.85 54.53 59.09 108.99 34.08 15.97 32.34 152.47

Table2: Alpha and Beta Values for N2 Region Guizhou

1 N 3

2N

0.52

0.65

2

1 N

23

2N

24

0.48

1N 3

2N 3

1N

0.45

0.56

0.55

Hainan

3

Table4: Alpha Values for N4

1N 3

35

1N 3

2 N 3

1N

0.35

0.49

0.35

0.51

24

2N

1N 5

2 N

0.44

0.47

0.59

0.53

0.41

3

24

35

1N 5

2N 5

0.20

0.16

0.17

0.39

0.42

0.22

0.27

0.33

0.34

0.34

0.33

0.33

0.39

0.26

0.25

0.45

0.36

0.29

0.64

0.28

0.39

0.32

0.33

Liaoning Jilin

PT

CE

4

44

2N

44

4

4

0.80

0.65

0.10

0.17

0.10

0.17

Heilongjiang

0.80

0.70

0.10

0.15

0.10

0.15

Jiangsu

0.33

0.30

0.27

0.42

0.40

0.28

Anhui

0.10

0.34

0.10

0.31

0.80

0.35

Shandong

0.36

0.28

0.29

0.42

0.36

0.30

Henan

0.36

0.27

0.30

0.43

0.34

0.30

Hubei

0.80

0.65

0.10

0.17

0.10

0.17

Guangdong

0.33

0.10

0.33

0.76

0.34

0.14

Guangxi

0.10

0.34

0.10

0.31

0.80

0.35

AC

17.47 18.12 56.27 41.86 19.46 19.46 11.59 5.41 22.14 7.61 8.58 4.75 0.67 2.73 16.21

0.65

33

0.56

Hebei Inner Mongolia

2

2 N

3

1N

4

Tianjin

2

1N 3

3

2N 3

Region

6.82 2.87 5.16 3.15 3.92 5.39 2.68 3.87 2.96 2.21 3.85 2.19 3.30 3.62 3.36

2N 5

ED

3

3061.03 260.06 34302.35 5160.24 1328.7 7.9 139 158.63 804.63 12.21 31397.94 19184.32 6499.44 2299.47 56464.74

24

M

Table3: Alpha and Beta Values for N3 Region

24.35 0.3 345.9 75.08 49.68 0.3 1.24 1928.31 9351.09 2.24 6376.2 224.58 1281.6 294.96 9324.37

CR IP T

373.3 701 274.3 265.5 813.9 2026.5 2087.4 476.9 2892.4 1689.8 390.5 267 895.2 10.8 900.6

AN US

Jiangsu Anhui Shandong Henan Hubei Guangdong Guangxi Chongqing Sichuan Yunnan Shanxi Gansu Qinghai Ningxia Xinjiang

28

ACCEPTED MANUSCRIPT

0.10

0.31

0.10

0.34

0.80

0.35

Sichuan

0.10

0.33

0.10

0.33

0.80

0.34

Yunnan

0.10

0.34

0.10

0.31

0.80

0.35

Shanxi

0.80

0.65

0.10

0.17

0.10

0.17

Gansu

0.34

0.34

0.35

0.27

0.32

0.39

Qinghai

0.80

0.10

0.10

0.44

0.10

0.46

Ningxia

0.29

0.27

0.29

0.25

0.42

0.48

Xinjiang

0.40

0.34

0.32

0.34

0.27

0.32

2N

1N 5

2N 5

0.19

0.13

0.18

0.33

Region

1N 3

2 N 3

1N

Tianjin

0.52

0.71

0.29

0.15

Hebei Inner Mongolia Liaoning

0.21

0.31

0.61

0.36

0.22

0.31

0.17

0.33

Jilin

0.80

0.79

Heilongjiang

0.80

0.31

Jiangsu

0.20

0.32

Anhui

0.33

0.10

Shandong

0.19

0.33

Henan

0.18

0.33

Hubei

0.80

0.79

Guangdong

0.65

Guangxi

0.33

Chongqing

0.29

ED

Table5: Beta Values for N4

Sichuan

44

44

AN US

4

4

4

0.32

0.55

0.37

0.63

0.33

0.20

0.34

0.10

0.11

0.10

0.10

0.10

0.36

0.10

0.32

0.61

0.35

0.19

0.33

0.30

0.10

0.37

0.80

0.61

0.33

0.20

0.33

0.62

0.34

0.20

0.33

0.10

0.11

0.10

0.10

0.80

0.19

0.10

0.17

0.10

0.10

0.33

0.10

0.34

0.80

0.10

0.28

0.10

0.43

0.80

M

0.23

PT

4

CR IP T

Chongqing

0.10

0.10

0.10

0.10

0.80

0.80

0.33

0.10

0.33

0.10

0.34

0.80

0.80

0.79

0.10

0.10

0.10

0.11

0.33

0.10

0.34

0.79

0.33

0.11

Qinghai

0.80

0.31

0.10

0.34

0.10

0.35

Ningxia

0.38

0.23

0.43

0.21

0.19

0.55

Xinjiang

0.34

0.31

0.34

0.31

0.32

0.38

Shanxi

AC

Gansu

CE

Yunnan

29

ACCEPTED MANUSCRIPT

Table6: Scaled Data for Process One Inputs

Jiangxi Hunan Tibet Shanghai Guizhou Hainan Tianjin Hebei Inner Mongolia Liaoning Jilin Heilongjiang Jiangsu Anhui Shandong Henan Hubei Guangdong Guangxi Chongqing Yunnan Shanxi

AC

Gansu

13850.7 12103.1 15898.5

4.44 4.11 6.61

4196.4 33.9 506.48 163.94 18.31 91.14 168.52 214.38 368.40 673.12 121.35 70.10 98.60 96.10 651.12 662.21 208.74 47.69 289.24 168.98 312.40 90.09 716.16 3.12 364.18

696.7 6961.2 3866.92 1543.25 5652.05 6390.36 4037.87 6269.28 6349.39 7304.28 10831.13 5690.33 9454.69 5910.92 11058.67 2200.59 3604.29 3164.38 6033.20 2760.64 8660.01 1821.67 198.27 540.06 2247.56

0.12 0.1 69.39 1.19 2.97 39.51 401.66 31.92 9.82 61.64 10.82 80.38 79.73 99.09 3.25 0.23 2.08 19.85 54.53 59.09 108.99 34.08 15.97 32.34 152.47

CE

Sichuan

1511.4 2174.4 1988.9

Qinghai Ningxia

Xinjiang

Per Capita

8.64 3.52 6.33

30

Nitrogen

56.64 51.55 50.77

5.26 2.88 3.34

50.43 48.62 37.95

2.28 8.48 0.97 1.51 4.72 0.78 1.42 0.99 3.69 2.86 1.36 0.95 0.98 0.58 3.14 3.48 0.89 1.12 0.30 0.73 3.08 0.73 2.63 1.38 1.15

59.43 58.45 19.36 32.97 40.45 7.41 10.46 13.14 36.99 11.10 13.69 4.19 1.25 5.96 31.84 32.43 4.84 5.46 3.79 5.24 40.39 5.53 18.47 13.41 13.56

AN US

Fujian

3.73 908.42 0.43

M

Zhejiang

Coal Resource

8870.8 8311.7 19243.6

ED

Shanxi

investment

39.5 106.2 1444.8

PT

Beijing

Water Resource

CR IP T

Variables

Outputs

ACCEPTED MANUSCRIPT

Table7: Scaled Data for Process Two Outputs

Inputs

Liaoning Jilin Heilongjiang Jiangsu Anhui Shandong Henan Hubei Guangdong Guangxi Chongqing Sichuan Yunnan Shanxi Gansu Qinghai Ningxia

171.55 134.58 46.05

4109.37 10902.32 1685.67

8344.3

84.14 100.99 70.10 78.32 80.43 81.39 672.03 208.74 47.69 289.24 168.98 39.05 92.40 89.52 3.16 290.88

1548.11 15422.68 5090.94 14061.55 9391.31 2937.12 16725.85 3224.63 3537.57 6033.20 2469.85 2266.78 1440.85 881.00 497.74 2223.48

1381.51

AC

CE

Xinjiang

5.44 278.78 315.37 178.54 776.22 24.35 0.3 345.9

1.00 2.65 2.22

31

75.08 49.68 0.3

1.24

1928.31 9351.09 2.24 6376.2 224.58 1281.6 294.96 9324.37

Nitrogen

35.96 8.72 8.59

CR IP T

Inner Mongolia

2082.19 1731.47 8361.69

Per Capita

1.45 3.57 0.46

10.68 13.00 5.23

0.36 4.17 0.87 3.14 1.95 0.39 1.00 0.89 1.08 0.30 0.73 0.39 0.74 0.33 1.55 1.14

12.73 14.92 4.19 1.24 6.25 4.51 4.05 4.84 5.46 3.79 5.24 5.37 43.41 20.17 12.15 13.59

AN US

Hebei

Natural Gas Resource

467.52 9.18 92.66

M

Tianjin

investment

ED

Guizhou

Water Resource

PT

Variables

ACCEPTED MANUSCRIPT

Table8: Scaled Data for Process Three Inputs

Heilongjiang Jiangsu Anhui Shandong Henan Hubei Guangdong Guangxi Chongqing Sichuan Yunnan Shanxi Gansu Qinghai Ningxia

1212.55 1470.08 5353.95

297.5 3034.52 26934.54

1.71 1.76 0.65

170.24 198.34 46.05

4027.36 7054.01 1659.34

8517.07 16946.82 18304.08

3.51 1.11 0.46

84.14 150.96 560.80 97.38 88.96 81.39 692.25 1669.92 381.52 2313.92 1351.84 39.05 84.50 89.52 4.52 245.54

1548.11 10299.09 5806.53 10021.96 6407.88 2888.71 3079.46 3677.89 3610.05 6137.60 2817.01 2295.51 2103.28 903.43 960.50 2101.86

50137.48 3061.03 260.06 34302.35 5160.24 1328.7 7.9 139 158.63 804.63 12.21 31397.94 19184.32 6499.44 2299.47 56464.74

0.36 1.29 1.05 1.04 0.63 0.39 0.91 0.90 1.67 2.37 0.74 0.39 0.72 0.33 0.69 1.07

AC

CE

Xinjiang

200.37 5.42 52.10

32

Nitrogen

22.91 7.54 7.95

CR IP T

Jilin

Per Capita

AN US

Hebei Inner Mongolia Liaoning

Petroleum Resource

M

Tianjin

investment

ED

Hainan

Water Resource

PT

Variables

Outputs

12.16 13.33 4.88 11.41 13.91 33.50 1.24 5.94 4.19 4.05 38.73 43.67 30.29 41.91 5.67 6.31 20.69 31.71 16.63

ACCEPTED MANUSCRIPT

Table 9: Overall, Aggregate and Process Efficiency Measures

Jiangxi Hunan Tibet Shanghai Guizhou Hainan Tianjin Hebei Inner Mongolia Liaoning Jilin Heilongjiang Jiangsu Anhui Shandong Henan Hubei Guangdong Guangxi Sichuan

Gansu Qinghai

AC

Ningxia

CE

Yunnan Shannxi

Xinjiang

1.0000 0.4737 0.1133 0.8336 0.1185 0.2820 1.0000 1.0000 0.5305 0.6428 0.3747 1.0000 1.0000 0.1038 0.0236 0.9956 0.0929 1.0000 0.7599 1.0000 0.2316

PT

Chongqing

1.0000 1.0000 1.0000 0.1129 0.6983 0.1403 0.2424 0.2541 0.4694 0.5189 0.0861 0.1876 0.2944 1.0000 0.6822 1.0000 0.4370 1.0000 0.1607 0.2840 0.6937 1.0000 0.4031

e0 0.8744 0.7299 0.2839 0.3743 0.3779 0.2403 1.0000 1.0000 1.0000 0.8919 1.0000 0.1414 0.3293 0.1863 0.3817 0.2330 0.1526 0.2436 0.1220 0.1175 0.1929 0.2897 0.3798 0.5080 0.1424 0.5832 0.1918 0.5621 0.9978 1.0000 0.2773

eT 0.8655 0.2484 0.2782 0.2922 0.2482 0.1661 1.0000 1.0000 0.7250 1.0000 1.0000 0.3298 0.4392 0.5797 0.4043 0.2430 0.7267 0.5259 0.3544 0.4475 0.2559 1.0000 0.6643 0.8359 0.3558 0.9138 0.2618 0.8410 0.8116 1.0000 0.2815

33

W1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3500 0.5500 0.5227 0.2138 0.2231 0.1742 0.7978 0.8000 0.1993 0.1000 0.1897 0.1832 0.7978 0.8000 0.1000 0.1000 0.1000 0.1000 0.7972 0.1000 0.3113 0.2856 0.3420

W2

CR IP T

Fujian

e3

AN US

Zhejiang

e2

M

Shanxi

e1 0.8655 0.2484 0.2782 0.2922 0.2482 0.1661 1.0000 1.0000 0.2143 1.0000 1.0000 0.1030 0.1321 0.1556 0.4612 0.2367 0.1336 0.1074 0.0729 0.0700 0.2359 1.0000 0.1849 0.2552 0.0386 0.1425 0.2961 0.2275 1.0000 1.0000 0.2171

ED

Region Beijing

W3

0.6500 0.2866 0.6071 0.2270 0.6299 0.1017 0.1000 0.6111 0.1000 0.6093 0.6182 0.1017 0.1000 0.1000 0.1000 0.1000 0.1000 0.1008 0.7858 0.3400 0.2877 0.3383

0.4500 0.1907 0.1790 0.5499 0.1959 0.1005 0.1000 0.1896 0.8000 0.2010 0.1986 0.1005 0.1000 0.8000 0.8000 0.8000 0.8000 0.1019 0.1142 0.3487 0.4267 0.3197