DEA environmental assessment: Measurement of damages to scale with unified efficiency under managerial disposability or environmental efficiency

Applied Mathematical Modelling 37 (2013) 7300–7314

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

DEA environmental assessment: Measurement of damages to scale with unified efficiency under managerial disposability or environmental efficiency Toshiyuki Sueyoshi a,⇑, Mika Goto b,1, Margaret A. Snell c,2 a

New Mexico Institute of Mining & Technology, Department of Management, 801 Leroy Place, Socorro, NM 87801, USA Central Research Institute of Electric Power Industry, 1-6-1, Otemachi, Chiyodaku, Tokyo 100-8126, Japan c New Mexico Institute of Mining & Technology, Department of Mathematics, 801 Leroy Place, Socorro, NM 87801, USA b

a r t i c l e

i n f o

Article history: Received 18 January 2012 Received in revised form 9 December 2012 Accepted 26 February 2013 Available online 7 March 2013 Keywords: Environmental assessment Data envelopment analysis Damages to scale

a b s t r a c t This study proposes a use of Data Envelopment Analysis (DEA) for environmental assessment. Firms usually produce not only desirable but also undesirable outputs as a result of their economic activities. The concept of disposability on undesirable outputs is separated into natural and managerial disposability. Natural disposability is an environmental strategy in which firms decrease their inputs to reduce a vector of undesirable outputs. Given the reduced input vector, they attempt to increase desirable outputs as much as possible. Managerial disposability involves the opposite strategy of increasing an input vector. The concept of disposability expresses an environmental strategy that considers a regulation change on undesirable outputs as a new business opportunity. Firms attempt to improve their unified (operational and environmental) performance by utilizing new technology and/or new management. Considering the two disposability concepts, this study discusses how to measure unified efficiency under managerial disposability and then discusses how to measure environmental efficiency. The proposed uses of DEA can serve as an empirical basis for measuring new economic concepts such as ‘‘Scale Damages (SD)’’, corresponding to scale economies for undesirable outputs, and ‘‘Damages to Scale (DTS)’’, corresponding to returns to scale for undesirable outputs. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Climate change and global warming have become major policy issues, internationally. Both climate change and global warming refer to an increase in average global temperature of the air, sea and land on the earth. Natural events and human activities, including all industrial and economic activities, contribute to an increase in average global temperature. Climate change is primarily caused by an increase in ‘‘greenhouse’’ gases such as carbon dioxide (CO2). In response to the issue of climate change, many researchers have applied Data Envelopment Analysis (DEA)3 to conduct environmental assessment. For example, Zhou et al. [5] summarized more than 100 DEA applications in environment and en⇑ Corresponding author. Tel.: +1 575 835 6452; fax: +1 575 835 5498. E-mail addresses: [email protected] (T. Sueyoshi), [email protected] (M. Goto), [email protected] (M.A. Snell). Tel./fax: +81 3 3201 6601. 2 Tel.: +1 575 835 5393; fax: +1 575 835 5366. 3 DEA has been long serving as a methodology to evaluate the performance of various organizations in public and private sectors. A major contributor of DEA is Professor William W. Cooper (University of Texas at Austin). Glover and Sueyoshi [1] discussed DEA theories, models and algorithms from the contributions of Professor Cooper. Ijiri and Sueyoshi [2] discussed his contributions in accounting and economics. These contributions later became the conceptual backbone of DEA development. See Dyson et al. [3]. See also Kao [4] and others for DEA applications in the area of applied mathematical modeling. 1

0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.02.027

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

7301

ergy studies. Cooper et al. [6] provided a summary of more than 100 previous studies on how OR/MS (Operations Research/Management Science) were useful in reducing air pollution. A contribution of the previous DEA studies was the separation of outputs into desirable and undesirable outputs. That was a contribution, indeed. Previous DEA research efforts in the past decade include Bevilacqua and Braglia [7], Korhonen and Luptacik [8], Kumar [9], Pasurka [10], Picazo-Tadeo et al. [11], Yang and Pollitt [12], Zaim [13], Zhou and Ang [14] and many other articles. An important feature of these previous DEA studies was that they mainly used radial models for DEA environmental assessment. In addition to these previous studies, Sueyoshi and Goto [15–20] and Sueyoshi et al. [21] applied non-radial models for DEA environmental assessment. They clearly acknowledged the contribution of the previous studies that used radial models for environmental assessment. However, they simultaneously found a difficulty in unifying desirable and undesirable outputs by using the radial models. They also found that non-radial models could unify desirable and undesirable outputs more easily than radial models because the former models do not have an efficiency score in their formulations. The level of efficiency is determined only by slacks in the non-radial measurement. Such unique features make it possible for the non-radial models to easily combine desirable and undesirable outputs into a unified treatment. After conducting a series of studies on the non-radial measures, these researchers extended their works to a new type of radial models which can measure the level of unified inefficiency and then transform it into an efficiency measure (Sueyoshi and Goto [22–29]). An important feature of the new radial models is that they can be used to identify the occurrence of desirable and undesirable congestions [24]. Furthermore, the radial models make it possible to measure the marginal rate of transformation and rate of substitution between desirable and undesirable outputs [26]. Moreover, it is possible to use them for DEA environmental assessment in a time horizon (e.g. [30]). As an extension of these previous studies, this study needs to explore two research agendas. One of the two agendas is that this study needs to review DEA environmental assessment from economic and strategic perspectives. Most of previous DEA studies on environmental assessment have used the concept of weak and strong disposability as a conceptual framework, originated from production economics. Meanwhile, the disposability is considered as a strategic concept in this study and it is separated into natural and managerial disposability by incorporating strategic aspects of environmental protection. These two disposability concepts have a close linkage with adaptive strategies for regulation change on undesirable outputs. Such a strategic concern regarding managerial disposability provides a new conceptual basis for DEA environmental assessment. The other research agenda is that this study discusses how to measure SD (Scale Damages) and DTS (Damages to Scale),4 where SD corresponds to scale economies for undesirable outputs and DTS corresponds to returns to scale for undesirable outputs. The type of DTS is determined by the upper and lower bounds of SD. The measurement of DTS has been never explored by previous DEA studies in production economics. Since the exact measurement of SD is computationally complicated with a limited practical use, this study focuses upon determining the type of DTS, not the degree of SD. The remainder of this study is organized as follows: The next section summarizes previous DEA research efforts that are closely related to this study. This section specifies the position of this research by comparing it with previous studies. Section 3 discusses why the proposed approach can measure unified (operational and environmental) efficiency of firms that produce desirable and undesirable outputs. An important aspect of this section is that this study discusses the measurement of unified efficiency from both economic and strategic viewpoints. Section 4 describes how to measure unified efficiency under managerial disposability. Section 5 discusses how to measure environmental efficiency as a simple version of unified efficiency under managerial disposability. Section 6 discusses how to measure both SD and DTS. The type of DTS is determined by measuring the upper and lower bound of SD. The last section (7) summarizes this study along with presenting future extensions. 2. Previous DEA Studies on Environmental Assessment To specify the position of this study more clearly in terms of the current DEA studies on environmental assessment, Fig. 1 visually summarizes the previous studies closely related to this study. 2.1. Weak or strong disposability s h Let us consider X 2 Rm þ as an input vector, G 2 Rþ as a desirable output vector and B 2 Rþ as an undesirable output vector. Considering both undesirable and desirable output vectors, Färe et al. [31] has specified an output vector as (G, B). Weak disposability is specified by the following vector notation:

( Pw ðXÞ ¼

ðG; BÞ : G 6

) n n n n X X X X Gj kj ; B ¼ B j kj ; X P X j kj ; kj ¼ 1; kj P 0 ðj ¼ 1; . . . ; nÞ ; j¼1

j¼1

j¼1

j¼1

4 The discussion of SD and DTS can be applied to SE (Scale Economies) and RTS (Returns to Scale) along with a mathematical modification. Because of the page limits of this journal, this study focuses solely upon the description of SD and DTS and does not discuss SE and RTS. See [20,28,29] for a detailed description of SE and RTS.

7302

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

Production Factors

Unified Efficiency Measurement (Category II) Managerial Disposability

Natural Disposability

RTS

Unified Efficiency Measurement Under Natural Disposability

Unified Efficiency Measurement Under Managerial Disposability

RTS

Operational Efficiency Measurement

Environmental Efficiency Measurement

DTS

DTS

Fig. 1. Structure of Unified Efficiency in Previous Studies. Note: (a) RTS stands for Returns to Scale and DTS stands for Damages to Scale. (b) Category I (for output unification) does not have any direct linkage to this study. Therefore, Fig. 1 does not provide the conceptual structure of Category I. See [17,18] for a description on Category I. (c) Fig. 1 depicts two groups of efficiency measurements. One of the two groups (listed at the left hand side) focuses upon the unified (operational and environmental) performance of firms that decrease an input vector until a vector of undesirable outputs can satisfy a governmental standard on regulation. They attempt to increase a vector of desirable outputs as much as possible by using a reduced input vector. In contrast, the other group (listed at the right hand side) focuses upon the unified performance of firms that increase an input vector while decreasing a vector of undesirable outputs by utilizing new technology and/or new management. Considering that the regulation on undesirable outputs provides firms with a business opportunity, they attempt to increase a vector of desirable outputs along with increasing an input vector. (d) The concept of strong and weak disposability considers only a decrease in the input vector, but does not consider an increase in the input vector. In contrast, the concept of natural and managerial disposability considers both directions of the input vector from the perspective of corporate strategies for environmental protection. See [24] for a comparative description of strong & weak disposability versus natural & managerial disposability.

where the subscript (j) stands for the jth DMU (Decision Making Unit, corresponding to an organization in the private or public sector) and kj indicates the jth intensity variable (j = 1, . . . , n). In the specification, desirable outputs are strongly dispos  P able, but undesirable outputs are not. The inequality constraints G 6 nj¼1 Gj kj allow a vertical extension, reflecting the   P strong disposability of desirable outputs. Meanwhile, the equality constraints B ¼ nj¼1 Bj kj indicate a possible occurrence of congestion by weak disposability on undesirable outputs. Strong disposability is specified by [31] as the following vector notation on the two output vectors:

( Ps ðXÞ ¼

ðG; BÞ : G 6

) n n n n X X X X Gj kj ; B 6 Bj kj ; X P X j kj ; kj ¼ 1; kj P 0 ðj ¼ 1; . . . ; nÞ : j¼1



j¼1

Pn



j¼1

j¼1

allow for strong disposability of undesirable outputs. j¼1 Bj kj P P It is important to note that previous research [31] has used the constraints B ¼ nj¼1 Bj kj or B 6 nj¼1 Bj kj in order to identify P an occurrence of congestion on undesirable outputs. However, this study uses the constraints B P nj¼1 Bj kj because we are not interested in congestion. See [24] for a detailed description on differences between [31] and the approach proposed in this study. The economic concept of weak and strong disposability has attracted considerable attention from production economists. For example, Korhonen and Luptacik [8] proposed seven radial models, derived from the original ratio form, to examine the efficiency of an emission reduction program used by twenty-four power plants. In addition to weak and strong disposability, the research [31] discussed a null-joint relationship between desirable and undesirable outputs. In our interpretation, this relationship indicates that undesirable outputs can be produced only if desirable outputs are produced. Acknowledging the contribution of such previous research efforts, this study needs to investigate three research agendas regarding DEA environmental assessment. First of all, DEA models proposed in the previous studies belonged to radial measurement. They have dual formulations. Their formulations indicate that some multipliers (i.e., dual variables) may become zero. This indicates that the corresponding production factors are not fully utilized in terms of their environmental assessments. This type of methodological difficulty has been long discussed by many DEA researchers (e.g., Thompson et al. [32]). However, no previous research has addressed this methodological issue. Furthermore, the zero-value multipliers in strong disposability indicate the occurrence of zero RTS and/or zero DTS, both of which suggest that they should belong to weak disposability. Thus, there is a conceptual problem in distinguishing between strong and weak disposability. See the study of [24] for a detailed description on the problem in distinguishing between them. This problem occurs because production economists do not examine the dual formulation of radial models proposed in their studies. Second, weak and strong disposability are proposed for environmental assessment that utilizes radial models. However, the use of weak disposability in this setting is problematic, as is expressed in [33]. Finally, an input vector is assumed to be oriented toward a decreasing direction in the previous studies. However, this assumption is inconsistent with the reality of environmental protection. For

The inequality constraints B 6

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

7303

example, let us consider a coal-fired power plant. The power plant may increase the amount of coal combustion if the increase can reduce the amount of CO2 emission. Such a reduction may result from either a managerial effort to use high quality coal with less CO2 emission and/or an engineering effort to utilize new generation technology (e.g., clean coal technology). Such action is considered as a corporate effort for environmental protection, or managerial disposability, in this study. No previous DEA study in production economics has addressed strategic effort related to managerial disposability.

2.2. Natural disposability and managerial disposability To overcome these difficulties in DEA environmental assessment, the recent research in this area has analytically shifted to non-radial measurement. Furthermore, using a non-radial model, previous research [15,21] has investigated the operational, environmental and unified (both operational and environmental) performance measures of coal-fired power plants in the United States. Previous research [16] also investigated the unified efficiency of Japanese fossil fuel power generation. They proposed the use of a non-radial model which measures the unified (operational and environmental) efficiency of these firms. The non-radial model is referred to as ‘‘Category II’’ by Sueyoshi and Goto [17], where the two types (i.e., Categories I and II) of unification were proposed for desirable and undesirable outputs. An important feature of both Categories I and II is that the directional vector of inputs can increase or decrease along with an increase in desirable outputs and a decrease in undesirable outputs. Since the proposed models use the total amount of slacks for measuring unified efficiency, they can avoid the problem associated with weak disposability discussed in preceding research [24]. Earlier research [18] also mathematically explored how to measure the type (increasing, constant or decreasing) of RTS (Returns to Scale) and the type of DTS from the dual formulation of Category I. The contribution of Category I is not listed in Fig. 1 because it is not directly related to this study. As depicted at the center of Fig. 1, the research [19] has discussed that, in the unified measurement of Category II, a computational difficulty may arise in formulating the dual model because the problem was solved using non-linear or mixed integer programming. To resolve the computational difficulty of Category II, [19] has proposed the following two approaches that originate from corporate strategy for environmental protection: The first strategy, referred to as ‘‘natural disposability’’, indicates that a firm may decrease a directional vector of inputs to decrease a directional vector of undesirable outputs. Given the reduced vector of inputs, the firm increases a directional vector of desirable outputs as much as possible. This study considers natural disposability as a negative adaptation to changes in environmental regulation. A cost concern associated with natural disposability is operational cost. It is easily expected that the total operation cost will decrease but the average operation cost will increase due to natural disposability as a result of decreased production. This type of strategy has been long supported by many economists (e.g., Palmer et al. [34]). The second strategy, referred to as ‘‘managerial disposability’’, involves the opposite approach to natural disposability. In managerial disposability, a firm increases a directional vector of inputs to decrease a directional vector of undesirable outputs by utilizing technology innovation. Given the increased input vector, the firm increases a directional vector of desirable outputs as much as possible. This study considers managerial disposability as a positive adaptation to changes in environmental regulation. A cost concern associated with managerial disposability is opportunity cost. This is the cost that occurs when consumers do not purchase products from a dirty-imaged company that does not pay attention to environmental protection. The opportunity cost poses a great risk to management. It is easily imaginable that the total operation cost may increase because of technology investment but the average operational cost may decrease when managerial disposability is used because of increased production. Furthermore, the opportunity cost associated with various environmental problems is much larger than the operation cost in modern businesses. Thus, it is necessary for corporate leaders to pay serious attention to various environmental protections. This type of strategy has been long supported by corporate strategists in U.S. business schools (e.g. Porter and van der Linde [35] in Harvard Business School). Finally, unified efficiency can be measured by using the two types of non-radial models, conceptually defined by natural and managerial disposability (depicted in the middle of Fig. 1). As discussed at the bottom of Fig. 1, [36] simplified the two types of disposability by dropping either undesirable outputs from natural disposability or desirable outputs from managerial disposability. They measured the level of operational efficiency by inputs and desirable outputs. The level of environmental efficiency is measured by inputs and undesirable outputs. These two efficiency measures are treated separately so that there is no interaction between them. This approach, which is discussed in [36], has provided a quick-and-easy method for measuring both RTS and DTS. As summarized above, all the previous studies did not discuss adaptive behaviors of firms when they faced a regulation change on undesirable outputs. Therefore, this study first reviews DEA environmental assessment from economic and strategic perspectives. No previous DEA study has examined the strategic aspect of firms from the perspective of environmental protection. In this study, the direction of an input vector is examined under natural and managerial disposability, both of which have a close relationship to corporate strategies which are meant to allow adaptation to regulation changes on undesirable outputs. This strategic concern provides a new conceptual basis for preparing DEA environmental assessment. Furthermore, this study discusses how to classify the type of DTS measurement. The type classification guides firms in identifying their appropriate operation sizes for environmental protection. The type of DTS is determined by the upper and lower bounds of SD in this study. No previous study has made a comparison between DTS under managerial disposability and environmental efficiency.

7304

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

3. Unified (operational and environmental) efficiency 3.1. Formulation In DEA environmental assessment, each DMU is characterized by its production activity that uses inputs to produce not only desirable but also undesirable outputs. An important feature of DEA performance assessment is that the achievement of each DMU is determined by being compared with those of the remaining others. The performance level is referred to as ‘‘an efficiency score’’ or ‘‘an efficiency measure’’. To describe DEA environmental assessment, this study considers n DMUs. The jth DMU (j = 1, . . . , n) uses a column vector of inputs (Xj) in order to yield not only a column vector of desirable (good) outputs (Gj) but also a column vector of undesirable (bad) outputs (Bj), where Xj = (x1j, x2j, . . . , xmj)T, Gj = (g1j, g2j, . . . , gsj)T and Bj = (b1j, b2j, . . . , bhj)T. The superscript ‘‘T’’ indicates a vector transpose. These vectors are referred to as production factors in this study. It is assumed that Xj > 0, Gj > 0 and Bj > 0 for all j = 1, . . . ,n, for our mathematical convenience. The research [19,20] has proposed a non-radial model for the measurement of unified (operational and environmental) efficiency. The following non-radial model of the kth DMU is reformulated under natural and managerial disposability:

Maximize

m s h X X X  xþ x  g b Rxi di þ di þ Rgr dr þ Rbf df ;

s:t:

n X xþ x xij kj  di þ di ¼ xik

r¼1

i¼1

f ¼1

ði ¼ 1; . . . ; mÞ;

j¼1 n X g g rj kj  dr ¼ g rk

ðr ¼ 1; . . . ; sÞ

j¼1

ð1Þ

n X b bfj kj þ df ¼ bfk

ðf ¼ 1; . . . ; hÞ;

j¼1 n X kj ¼ 1;

kj P 0

ðj ¼ 1; . . . ; nÞ;

j¼1 xþ

di P 0 ði ¼ 1; . . . ; mÞ; g

b

dr P 0 ðr ¼ 1; . . . ; sÞ; x

g

x

di P 0 ði ¼ 1; . . . ; mÞ; and df P 0 ðf ¼ 1; . . . ; hÞ:

b

Here, di (i = 1, . . . , m), dr (r = 1, . . . , s) and df (f = 1, . . . , h) are all slack variables related to inputs, desirable outputs, and undex xþ sirable outputs, respectively. The slack variable di related to the ith input is separated into positive and negative parts (di x and di Þ to express managerial or natural disposability, respectively. These parts are incorporated together in the first group of constraints of Model (1). The column vector k = (k1, . . . , kn)T stands for unknown variables (often referred to as ‘‘structural’’ or ‘‘intensity’’ variables) involved in connecting the input and output vectors by a convex combination. Since the sum of the structural variables is restricted to be unity in Model (1), the production possibility set for Model (1) is structured under variable RTS and DTS. An important feature of Model (1) is that ranges are incorporated in the formulation. They are determined by the upper and lower bounds on inputs and those of desirable and undesirable outputs. The upper and lower bounds of inputs are specified by  xi ¼ maxj fxij g and xi = minj{xij} for all i. The range for inputs becomes Rxi ¼ 1=½ðm þ s þ hÞð xi  xi Þ for all i. The upper and lower bounds of desirable outputs are gr ¼ maxj fg rj g and gr = minj {grj} for all r. The range for desirable outputs becomes Rgr ¼ 1=½ðm þ s þ hÞðgr  g r Þ for all r. The upper and lower bounds of undesirable  ¼ maxj fb g and b = min {b } for all f. Then, Rb ¼ 1=½ðm þ s þ hÞðb   b Þ for all f indicates outputs are expressed by b f fj f f f j fj f the range for undesirable outputs. The existence of these ranges indicates that Model (1) can always produce positive multipliers (i.e., dual variables) so that information about all production factors is fully utilized in the environmental assessment.  x   x  xþ x x x Mathematically, the two slacks related to the ith input are defined as di ¼ di  þ di =2 and di ¼ di   di =2. The xþ x x parentheses in the objective function of Model (1) indicate that di þ di ¼ jdi j. The input slacks in the first group of conxþ x x straints in Model (1) indicate that di  di ¼ di . The variable transformation of input slacks needs the nonlinear conditions xþ x di di ¼ 0 (i = 1, . . . , m), implying that the two slack variables are mutually exclusive. Consequently, the simultaneous occurxþ x rence of both di > 0 and di > 0 (i = 1, . . . , m) is excluded from the optimal solution of Model (1). When Model (1) results in the simultaneous occurrence discussed above, a computer code may produce ‘‘an unbounded solution’’ because of the violation of the nonlinear conditions. In order to make Model (1) satisfy the nonlinear conditions, the previous studies (e.g. [19]) have suggested the following two computational alternatives: (a) Model (1) can incorporate the nonlinear conditions into the formulation as side constraints. Then, Model (1) can be xþ x solved with di di ¼ 0 (i = 1, . . . , m) as a nonlinear programming problem.

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314 xþ

7305

x

þ þ   (b) Model (1) can incorporate the following side constraints: di 6 Mzþ 6 Mz i ; di i ; zi þ zi 6 1; zi and zi : binary (i = 1, . . . , m) into the formulation. Then, Model (1) can be solved with the side constraints as a mixed integer programming problem. Here, M stands for a very large number that we need to prescribe before computation.

It is important to note that Model (1) often has a feasible solution. However, it may also produce an unbounded solution because the dual formulation is infeasible. In this case, we must depend upon one of the above two alternatives. Both usually produce similar optimal solutions [17]. Here, the unbounded solution implies the violation of the nonlinear conditions, which is different from a conventional unbounded solution (where an optimal objective value becomes infinite) in linear xþ programming. The violation indicated by the unbounded solution implies a simultaneous occurrence of di > 0 and x di > 0 for some i in Model (1). Such a computational problem indicates a methodological drawback of Model (1). However, Model (1) also has methodological strengths. For example, it can express the two efficiency frontiers by a single set of intensity variables. Furthermore, the direction of possible projections in Model (1) includes both natural and managerial disposability in environmental assessment. This study reorganizes Model (1) into two non-radial models based upon natural and managerial disposability so that the proposed approach can overcome the computational difficulties of Model (1). It is important to note that dual feasibility can be easily attained by slightly modifying Model (1). See the Appendix A of this study for a description of how to obtain dual feasibility by modifying Model (1). The formulation discussed in the Appendix A is almost the same as the one under managerial disposability. Therefore, this study does not discuss it further here. After obtaining an optimal solution from Model (1), this study determines a unified efficiency score (h) as follows:

h¼1

" # m s h    X X X xþ x g b Rxi di þ di Rgr dr þ Rbf df : þ r¼1

i¼1

ð2Þ

f ¼1

Here, all slack variables represent the level of inefficiency in Eq. (2) and they are determined on optimality of Model (1). The unified efficiency is determined by subtracting the level of inefficiency from unity.

3.2. Economic implications Fig. 2 visually describes the analytical structure for measuring unified efficiency by Model (1). For our visual convenience, Fig. 2 considers the achievement (listed as X) of each DMU that uses an input (x on the horizontal axis) to produce both a desirable output (g on the vertical axis) and an undesirable output (b on the vertical axis). Let the achievement (listed as K) of the kth DMU be (xik,grk, bfk) in Fig. 2. To unify the operational and environmental performance of firms, Model (1) needs to incorporate two efficiency frontiers into the proposed analytical structure that is visually described in Fig. 2. One of the two frontiers is an efficiency frontier for a desirable output. The frontier is depicted as the top frontier in Fig. 2. The other (a bottom frontier) is an efficiency frontier for an undesirable output. The efficiency frontier for an undesirable output is separated into the frontier under natural dispos-

A-B-C: Frontier for Desirable Outputs

g&b

C x

×

×

×

B ×

x drg

A

grk bfk

×

K K

×

d bf

x ×

×

G

×

H

× D

×F x E

D-E-F: Frontier for Undesirable Outputs (Managerial Disposability)

G-H-D: Frontier for Undesirable Outputs (Natural Disposability)

xik − dix −

xik

xik + dix+

x

Fig. 2. Natural and Managerial Disposability. Source: Sueyoshi and Goto [19,20].

7306

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

ability (the left hand side of the efficiency frontier for undesirable outputs in Fig. 2) and the frontier under managerial disposability (the right hand side of the efficiency frontier for undesirable outputs in Fig. 2). NW (North–West) projections: Let us consider the kth DMU whose performance is measured as the combination between an input and a desirable output, as depicted in Fig. 2. If the kth DMU is inefficient in operational performance, then the DMU g needs to improve its operational efficiency by increasing the amount of a desirable output, expressed by dr , and/or by x decreasing the amount of an input, expressed by di . Such a direction of possible projections can be found on B from K in Fig. 2 as an example. The contour line (A–B–C) indicates an efficiency frontier for a desirable output onto which K may be projected. The direction of such possible projections on A–B–C should be expressed by both the sign of an input-related slack and the sign of a desirable output-related slack in Model (1). From now on, this study considers that the direction of such possible projections is to the ‘‘North–West’’, which indicates the enhancement of operational efficiency. SW (South–West) projections: Fig. 2 visually describes natural disposability in an input (on the horizontal axis) and an undesirable output (on the vertical axis) that can be found in projections toward the ‘‘South–West’’; the direction from K to G in Fig. 2 is an example of a projection toward the ‘‘South–West’’. Here, K stands for an observed performance and G stands for its performance on the frontier for an undesirable output. The contour line (G–H–D) indicates the frontier for the undesirable output under natural disposability. SE (South–East) projections: The projection from K to E is an example of the opposite case, indicating managerial disposability in the South–East direction. In this case, an efficiency frontier for K can be found on the contour line (D–E–F) on which we can determine the level of environmental efficiency on the frontier for an undesirable output. Production technology depicted in Fig. 2 can axiomatically express natural and managerial disposability by the following two output vectors and an input vector:

( Pn ðXÞ ¼

ðG; BÞ : G 6 (

Pm ðXÞ ¼

) n n n n X X X X Gj kj ; B P Bj kj ; X P X j kj ; kj ¼ 1; kj P 0 ; j¼1

j¼1

j¼1

j¼1

j¼1

j¼1

j¼1

j¼1

) n n n n X X X X ðG; BÞ : G 6 Gj kj ; B P B j kj ; X 6 X j kj ; kj ¼ 1; kj P 0 :

The difference between the two concepts of disposability is that production technology under natural disposability, or Pn(X), P P has X P nj¼1 X j kj . Meanwhile, that of the managerial disposability, or Pm(X), has X 6 nj¼1 X j kj . The unification of two disposability sets produces the following output set:

Pu ðXÞ ¼ Pn ðXÞ [ Pm ðXÞ: In this unification, where [ stands for a union set, an efficiency frontier (EF) is for both desirable and undesirable outputs. The level of unified (operational and environmental) efficiency on a DMU is identified on EF in this study. 3.3. Strategic implications Natural or managerial disposability can be reexamined by considering the environmental strategies of firms that need to adapt to a regulation change on undesirable outputs. To describe the relationship between the two types of disposability and adaptive strategies, this study reviews the research of [37] that has examined the corporate strategy of 220 Japanese manufacturing firms (2004–2007) from the perspective of environmental protection. Their study identified that large firms had managerial and financial capabilities to improve their operational and environmental performance. However, the research could not find such a business linkage in the other (relatively small and medium-sized) firms, even after looking at all firms listed in the Tokyo stock exchange market. These firms improved their operational performance and then directed efforts toward the improvement of their environmental performance. Their environmental performance was the second priority for the firms even though the Japanese government’s policies were putting pressure on all Japanese manufacturing firms to pay attention to various environmental issues related to global warming and climate change. Fig. 3 visually describes the dynamic process in which firms gradually adapt to a governmental regulation change. Furthermore, their study [37] indicates that firms cannot immediately improve their environmental performance even if consumers’ environmental consciousness increases the firm’s business opportunities. The firms need to accumulate capital and then invest the capital in new production technology and management.

Regulation Change

Improve Operational Performance

Improve Environmental Performance

Improve Both Performance Measures

Fig. 3. Strategy to Adapt a Regulation Change. Note: A causality process from the regulation change to the improvement of operational and environmental performance indicates that it is necessary for us to consider a time lag in the process. Therefore, this study needs to develop DEA environmental assessment in a time horizon. The research [30] provides a detailed description on how to reorganize DEA models for measuring a frontier shift in a time horizon.

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

7307

Table 1 Strategic Differences between Natural and Managerial Disposability. Natural disposability (negative adaptation)

Managerial disposability (positive adaptation)

Adaptive strategy

This strategy reduces the amount of inputs (e.g., coal combustion) in order to decrease undesirable outputs (e.g., CO2 emission) to the level required by government regulation. Given the reduced vector of inputs, a coal-fired power plant attempts to increase the amount of desirable outputs (e.g., electricity) as much as possible

Total cost and revenue

Operation cost: Decrease in total operation cost but increase in average operation cost Opportunity cost: Increase Investment cost: No change Revenue: Decrease This is a short term strategy. A power plant can decrease its operation cost by making a decrease in an input vector. However, it still maintains the negative image of a dirty coalfired power plant so that consumers who are environmentally conscious try to avoid the use of electricity generated by the power plant. Thus, the power plant increases an opportunity cost in the long run. Since the power plant decreases the amount of inputs and desirable outputs, it decreases its revenue in the short and long term

This strategy considers a regulation change as a new business opportunity. Using high quality coal with less CO2 emission and/or incorporating engineering efforts to utilize new generation technology (e.g., clean coal technology), this strategy increases the amount of inputs but it decreases the amount of undesirable outputs. Using the increased input vector, a coal-fired power plant increases desirable outputs as much as possible by using new generation technology Operation cost: Increase in total operation cost but decrease in average operation cost Opportunity cost: Decrease Investment cost: Increase Revenue: Increase This is a long term strategy. The power plant needs to invest in new generation technology. So, the investment cost is increased by the strategy in the short run. However, the plant can decrease its operation cost due to the use of new technology. Furthermore, consumers who are environmentally conscious want to purchase clean electricity generated by a new coal fired power plant. Thus, the power plant decreases an opportunity cost in a long run. Since the power plant increases the amount of inputs and desirable outputs, it increases its revenue in the short and long term Input formulation:

Short and Long Runs

Input Formulation in DEA and Possible Projections

Input formulation: n X

xij kj 6 xik

ði ¼ 1; . . . ; mÞ;

j¼1

Projections: Combinations between NW and SW

n X xij kj P xik ði ¼ 1; . . . ; mÞ; j¼1

Projections: Combinations between NW and SE

Note: This study focuses upon unified efficiency under managerial disposability in this study because it is an important component of DEA environmental assessment.

The adaptive process depicted in Fig. 3 has a close linkage with natural and managerial disposability. To describe this relationship, let us consider a coal-fired power plant as an illustrative example. Let us also consider the relationship between total cost and the amount of CO2 emission. In this study, the total cost consists of the following three cost components: (a) an operational cost to generate electricity, (b) an opportunity cost to lose a business chance to provide customers with a utility service and (c) an investment cost to purchase a new generation facility and/or to utilize new management at the coal-fired power plant. Positive or negative adaptation: When a government changes its regulation policy on CO2 emission (as an undesirable output), a coal-fired power plant may be required to change its strategy to adapt to the regulation change. Acknowledging that there are many adaption strategies, this study focuses upon two (positive and negative) strategies as summarized in Table 1. A positive adaptation implies that a firm considers the regulation change as a new business opportunity. The firm replaces old generation facility and management with new ones that use high quality coal with less CO2 emission and/or incorporate engineering efforts to utilize new generation technology (e.g., clean coal technology). In this case, a cost for technology investment occurs within the power plant. However, the new technology decreases both the operation cost and the opportunity cost. In particular, the decrease in the opportunity cost is increasingly important because modern consumers pay attention to environmental protection and they dislike purchasing electricity from a coal-fired power plant with a dirty image. In contrast, a negative adaptation implies that a coal-fired power plant does not make take major strategic action. Instead, the power plant decreases the amount of coal combustion until they reach a level of CO2 emission that satisfies the government standard. Given the reduced level of coal combustion, the plant manager then tries to enhance the amount of electricity produced. This type of negative adaptation can be found in firms that do not have enough capital to invest in new generation technology. 4. Unified efficiency under managerial disposability As discussed in Section 3.1, the computational problem of Model (1) is that it is formulated by nonlinear or mixed integer programming. Consequently, it is not easy for us to formulate the dual of Model (1). This is a difficulty of determining the type of DTS because there is no direct access to the dual formulation. To obtain the dual formulation, the studies [19,20] have replaced Model (1) by the following DEA model that combines desirable and undesirable outputs of the kth DMU under managerial disposability:

7308

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

Maximize

m s h X X X x g b Rxi di þ Rgr dr þ Rbf df ;

s:t:

n X x xij kj  di ¼ xik

r¼1

i¼1

f ¼1

ði ¼ 1; . . . ; mÞ;

j¼1 n X

g

ðr ¼ 1; . . . ; sÞ;

j¼1 n X

b

ðf ¼ 1; . . . ; hÞ;

g rj kj  dr ¼ g rk

ð3Þ

bfj kj þ df ¼ bfk

j¼1 n X kj ¼ 1; kj P 0 j¼1 x di P b df P

ðj ¼ 1; . . . ; nÞ; g

0 ði ¼ 1; . . . ; mÞ;

dr P 0 ðr ¼ 1; . . . ; sÞ;

and

0 ðf ¼ 1; . . . ; hÞ: x

Model (3) considers only deviations di (i = 1, . . . , m) of Model (1) to attain managerial disposability. The formulation for natx x ural disposability is obtained by changing di to þdi in the first constraint group of Model (3). Returning to Model (3), a unified efficiency score (h⁄) of the kth DMU under managerial disposability is measured by

h ¼ 1 

! m s h X X X x g b Rxi di þ Rgr dr þ Rbf df : r¼1

i¼1

ð4Þ

f ¼1

Here, all slack variables are determined on optimality of Model (3). The equation within the parentheses, obtained from the optimality of Model (3), indicates the level of unified inefficiency under managerial disposability. The unified efficiency is obtained by subtracting the level of inefficiency from unity. It is possible to use Eq. (4) in the case of natural disposability, as well. However, the implication of input-related slacks is different between managerial and natural disposability although both use Eq. (4). Model (3) has the following dual formulation: m s h X X X Minimize  v i xik  ur g rk þ wf bfk þ r; r¼1

i¼1

s:t:

f ¼1

m s h X X X ur g rj þ wf bfj þ r P 0  v i xij  i¼1

vi P

r¼1 Rxi

ðj ¼ 1; . . . ; nÞ; ð5Þ

f ¼1

ði ¼ 1; . . . ; mÞ; ðr ¼ 1; . . . ; sÞ;

ur P Rgr wf P Rbf

ðf ¼ 1; . . . ; hÞ;

r : URS: where vi (i = 1, . . . , m), ur (r = 1, . . . , s), wf (f = 1, . . . , h) are all dual variables related to the first, second and third groups of constraints in Model (3). The dual variable (r) is obtained from the fourth equation of Model (3). The symbol ‘‘URS’’ indicates that a variable is unrestricted. The objective value of Model (3) equals that of Model (5) on optimality. It is clear that Model P (5) can always produce positive multipliers (dual variables). In the case of natural disposability,  m i¼1 v i xik needs to be rePm placed by i¼1 v i xik in both the objective function and the first constraint of Model (5). 5. Environmental efficiency To simplify Model (3) by excluding desirable outputs, [36] has proposed the following model to measure the environmental efficiency of the kth DMU:

Maximize

m h X X x b Rxi di þ Rbf df

s:t:

n X x xij kj  di ¼ xik

i¼1

f ¼1

j¼1 n X

ði ¼ 1; . . . ; mÞ;

b

bfj kj þ df ¼ bfk

ðf ¼ 1; . . . ; hÞ;

j¼1 n X kj ¼ 1; kj P 0 j¼1 x di P

ðj ¼ 1; . . . ; nÞ; b

0 ði ¼ 1; . . . ; mÞ and df P 0 ðf ¼ 1; . . . ; hÞ:

ð6Þ

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

7309

An important feature of Model (6) is that it determines the environmental efficiency on the k-th DMU by comparing its inputs and undesirable outputs with those of the other DMUs. The efficiency score (h⁄) of the k-th DMU stands for environmental performance that is measured by

! m h X X  x x b b h ¼1 Ri di þ Rf df : 

i¼1

ð7Þ

f ¼1

The equation within the parentheses, obtained from the optimality of Model (6), indicates the level of environmental inefficiency. The environmental efficiency is obtained by subtracting the level of inefficiency from unity, as specified in Eq. (7). The dual formulation of Model (6) is as follows: m h X X Minimize  v i xik þ wf bfk þ r i¼1

s:t:

f ¼1

m h X X  v i xij þ wf bfj þ r P 0 i¼1

ðj ¼ 1; . . . ; nÞ;

ð8Þ

f ¼1

vi P

Rxi

ði ¼ 1; . . . ; mÞ; wf P

Rbf

ðf ¼ 1; . . . ; hÞ;

r : URS; where V = (v1, . . . , vm)Tand W = (w1, . . . , wh)T are two column vectors of dual variables related to the first and second sets of constraints in Model (6). The dual variable (r) is obtained from the last constraint of Model (6). The objective value of Model (6) equals that of Model (8) on optimality. It is clear that Model (8) can always produce positive multipliers (dual variables). Returning to the example of the coal-fired power plant in Section 3.3, Fig. 4 depicts economic and strategic implications of possible projections using Model (6). In Fig. 4, the performance of firms is measured by the amount of coal consumption on the horizontal axis and the amount of CO2 emission on the vertical axis. An efficiency frontier for an undesirable output is expressed by D–A–B–C. The kth power plant has four types of possible projections (e.g. K–D, K–A, K–B and K–C) toward the efficiency frontier. The first projection (from K–D) indicates a decrease in the amount of generated electricity. This decrease leads to a reduction in the amount of CO2 emission. Consequently, the power plant can satisfy the maximum level of CO2 emission required by government regulation. The coal-fired power plant, using a current generation facility, may choose this type of environmental strategy. This strategy can reduce the generation cost, but may increase an opportunity cost because consumers may change electricity suppliers if they know that their current supplier’s electricity is generated by a current generation facility with a dirty image. Moreover, the reduced amount of generation results in revenue loss to the firm that owns the coal-fired power plant. Therefore, Model (1) incorporates the projection (K–D), but Model (6) does not incorporate that projection. The second projection (from K to A) indicates a strategic effort by which a coal-fired power plant decreases the amount of CO2 emission while maintaining the current level of electricity. The third projection (from K to C) indicates another strategic effort by which the power plant increases the amount of electricity (due to an increase in the consumption of coal), while holding the current level of CO2 emission.

b (CO2)

Frontier for Undesirable Output

x ×

K

b fk

d bf

×

×

×

C x B

A

D

xik − d ix −

x ik

xik + dix +

x (coal)

Fig. 4. Projection Direction toward Frontier for Undesirable Output.

7310

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

The fourth projection (from K to B) indicates a strategic effort that lies in between the two projections from K to A and from K to C. This projection indicates that the power plant increases the amount of electricity (due to an increase in the consumption of coal) and decreases the amount of CO2 emission. Finally, it is important to note that coal combustion may decrease the amount of CO2 emission and increase the amount of electricity under managerial disposability as depicted by the projection from K to B. That is due to a strategic effort to decrease the amount of CO2 emission (e.g., an effort to use high quality coal with less CO2 emission) and/or an engineering effort to utilize new generation technology that can reduce the amount of CO2 emission (e.g., clean coal technology). This strategic effort involves an investment cost to acquire new technology, but decreases average operation and opportunity costs in the short and long term. The total cost decreases in the long run because the opportunity cost becomes much larger than before. As mentioned previously, modern consumers are environmentally conscious and a majority will prefer a green generator which produces electricity with less CO2 emission. If an electric power company cannot provide green electricity, consumers may choose another utility firm and/or an alternative energy supplier. Furthermore, many governments, including Japan and EU nations, recently indicated targets for CO2 emission reductions by a certain period in future. If firms do not comply with these targets, they must pay a financial penalty. In that case, the opportunity cost includes the penalty charge. 6. Scale damage and damages to scale 6.1. Measurement of DTS under managerial disposability To determine the degree and type of DTS by Model (5), this study describes the concept of SD (Scale Damages). The degree of SD depends upon the efficiency/inefficiency status of the kth DMU. If the DMU is efficient, then the degree of SD is determined as follows:

SD ¼

,

m X

v i xik

i¼1

h m X X wf bfk ¼ v i xik i¼1

f ¼1

,

m X

s X ur g rk  r

i¼1

r¼1

v i xik þ

!

, ¼1

1



r  fgk



m .X

v i xik

!! :

ð9Þ

i¼1

P P Ps    Here, hf¼1 wf bfk ¼ m value becomes zero i¼1 v i xik þ r¼1 ur g rk  r is incorporated into Eq. (9) because the optimal objective P  on an efficient DMU. All optimal dual variables are obtained from Model (5). The new variable fgk ¼ sr¼1 ur g rk is incorporated in Eq. (9) and it indicates the level of influence from desirable outputs on SD. If the kth DMU is inefficient, then the DMU needs to be projected on an efficiency frontier. The degree of SD can be determined on its projected point. That is,

SD ¼

m X

v

 i



xik þ

x  di

!,

! h   X b  wf bfk  df :

i¼1

ð10Þ

f ¼1

The degree of SD, measured by Eq. (10), is further reformulated as follows:

SD ¼

m X

v

i¼1

¼

 i



x  xik þ di

!,

h X

wf

! ! !, m h     X X  b x     b ¼ bfk  df v i xik þ di wf bfk  wf df i¼1

f ¼1

!, ! m s X      X   g  v i xik þ dxi v i xik þ Rxi dxi þ ur g rk þ Rgr dr  r

f ¼1

m X i¼1

¼1 1

i¼1



g

r  fk

m .X

v

 

x  i xik þ di

r¼1

!! :

ð11Þ

i¼1

P Ps Ph Pm x x Ps Ph b b g g     Here,  m i¼1 v i xik  r¼1 ur g rk þ f ¼1 wf bfk þ r ¼ i¼1 Ri di þ r¼1 Rr dr þ f ¼1 Rf df is obtained from the optimal objective value of Model (3) and that of Model (5). This equation is incorporated in the reformulation from the first to the second equation in (11). x g The status of inefficiency is due to the existence of positive slack (s) such that di > 0 for some i, dr > 0 for some r and/or   x b df > 0 for some f. The Complementary Slackness Conditions (CSCs) of linear programming indicate that v i  Rxi di ¼ 0 for     g b all i, ur  Rgr dr ¼ 0 for all r and wf  Rbf df ¼ 0 for all f on optimality. Consequently, this study has v i ¼ Rxi for some i, ur ¼ Rgr for some r, and wf ¼ Rbf for some f in the case of an inefficient DMU with positive slacks. In reformulating Eq. (11), this study uses two assumptions. One of the two assumptions is that v i ¼ Rxi for all i, ur ¼ Rgr for all r, and wf ¼ Rbf for all f under CSCs. This assumption makes it possible to reformulate the second equation into the third  P P  g  equation in (11). The other assumption is fgk ¼ sr¼1 ur g rk þ dr ffi sr¼1 ur g rk because this study is interested in the type of g DTS that is mainly determined by inputs and undesirable outputs, as formulated in (10). Hence, dr ffi 0 is assumed for all r. The assumption indicates that all slacks related to desirable outputs are positive but very close to zero. In addition to the two assumptions, this study needs to mention that Eq. (11) for an efficient DMU equals Eq. (9) because x di ¼ 0 for all inputs. The comparison between Eqs. (9) and (11) indicates that both the sign of a dual variable (r⁄) and the   influence from desirable outputs fg determine the degree of SD and the type of DTS on all DMUs. k

7311

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

Next, assuming both a unique projection of an inefficient DMU onto an efficiency frontier and a unique reference set for   the projected DMU, this study can determine the upper and lower bounds of the dual variable (r) and the influence fgk from desirable outputs using the following model:

Max=Min

r  fgk

s:t: all constraints in both ð3Þ and ð5Þ; s X ur g rk ; fgk ¼

ð12Þ

r¼1 m s h m s h X X X X X X x g b Rxi di þ Rgr dr þ Rbf df ¼  v i xik  ur g rk þ wf bfk þ r; r¼1

i¼1

fgk

r¼1

i¼1

f ¼1

f ¼1

P 0:

The first additional constraint indicates the influence from desirable outputs. The second additional constraint is due to the condition that the objective function of Model (3) equals that of Model (5) on optimality.  Þ and lower bound (s⁄) of the objective value are obtained from the maximization and minimization The upper bound ðs of Model (12), respectively. An optimal solution of (3) and (5) is feasible in Model (12) and vice versa. Therefore, the optimal  and s⁄ corresponds to the upper and lower bounds of the objective value of Model (12). solution pair of s Based upon the upper and lower bounds, the type of DTS of the kth DMU is determined by the following classification:

 P s > 0; ðaÞ Increasing DTS $ s   P 0 P s and ðbÞ Constant DTS $ s   P s : ðcÞ Decreasing DTS $ 0 > s

ð13Þ

6.2. Measurement of DTS under environmental efficiency To determine the type of DTS under environmental efficiency, the concept of SD needs to be determined in terms of efficiency/ inefficiency status of the kth DMU. That is, if the DMU is environmentally efficient, then the degree of SD is determined by

SD ¼

,

m X

v i xik

i¼1

h m X X wf bfk ¼ v i xik

,

i¼1

f ¼1

m X

!

,

v i xik  r

¼1

!!

m X

v i xik

r =

1

i¼1

ð14Þ

:

i¼1

P Ph    Here,  m i¼1 v i xik þ f ¼1 wf bfk þ r ¼ 0 obtained from Model (8) is incorporated into Eq. (14) because environmental efficiency implies an optimal objective value of zero in Model (8). In contrast, if the kth DMU is inefficient in environmental performance, then its projected point on an efficiency frontier is     x Xk þ d x b of Model (6). The degree of SD of the inefficient DMU can be found on from an optimal solution k ; d ; d b Bk  d determined on its projected point. That is,

SD ¼

m X

v

 i



xik þ

x  di

!,

i¼1

! h   X b  wf bfk  df :

ð15Þ

f ¼1

Eq. (15) is further reformulated in the following manner:

SD ¼

m X

v

 i



xik þ

x  di

i¼1

¼

m X

v i



x 

!, !,

xik þ di

i¼1

! h   X b  wf bfk  df ¼

m X

f ¼1

i¼1

m X 

v i xik þ Rxi dxi

i¼1



v

!  r

 i



!,

xik þ

x  di

f ¼1

,

¼1

! h   X   b wf bfk  wf df

1

r =

m X

v i



x 

xik þ di

!! :

ð16Þ

i¼1

P Ph Pm x x Ph b b    Here,  m i¼1 v i xik þ f ¼1 wf bfk þ r ¼ i¼1 Ri di þ f ¼1 Rf df is obtained from the objective function of Model (6) and that of Model (8) on optimality and this condition is incorporated into the reformulation of the first equation into the second equax tion in (16). Environmental inefficiency implies the existence of positive slack (s) in such a manner that di > 0 for some i and      x b b x b df > 0 for some f. The CSCs of linear programming indicate that v i  Ri di ¼ 0 for all i and wf  Rf df ¼ 0 for all f on optimality. Consequently, we have

v i ¼ Rxi

for some i and wf ¼ Rbf for some f. Here, assuming that the environmental inef-

ficiency of the kth DMU is due to all inputs and all undesirable outputs, we have

v i ¼ Rxi for all i and wf ¼ Rbf for all f. Thus,

we reformulate the second equation into the third equation of (16). x Here, it is important to note that Eq. (16) for an environmentally efficient DMU becomes Eq. (14) because di ¼ 0 for all inputs. In examining both (14) and (16), this study finds that the sign of the dual variable (r) determines the degree of SD and the type of DTS on all efficient and inefficient DMUs in terms of their environmental performance.

7312

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

Assuming a unique projection and a unique reference set, this study can measure the upper and lower bounds of the dual variable (r) by using the following model:

Max=Min

r

s:t: all constraints in both ð6Þ and ð8Þ; m h m h X X X X x b Rxi di þ Rbf df ¼  v i xik þ wf bfk þ r: i¼1

f ¼1

i¼1

ð17Þ

f ¼1

  Þ and lower bound (r⁄) from the maximization and minimization of the model, Model (17) produces the upper bound ðr respectively. An optimal solution of Models (6) and (8) is feasible in Model (17) and vice versa. Therefore, the optimal solu  and r⁄ corresponds to the upper and lower bounds of r⁄ in Model (17). tion pair of r The classification of DTS is determined by the upper and lower bounds of r⁄ in the following manner:

  P r > 0; ðaÞ Increasing DTS $ r   P 0 P r and ðbÞ Constant DTS $ r   P r : ðcÞ Decreasing DTS $ 0 > r

ð18Þ

6.3. Comments on DTS classifications This study brings up two concerns in terms of DTS classification. One of the two concerns is that the difference between the DTS classification of (13) and (18) is that the classification in (13) depends upon both the dual variable (r⁄) and the influ    ence fg from desirable outputs. Meanwhile, the classification in (18) excludes the influence fg and depends only upon k k ⁄ the dual variable (r ). The other concern is that in the proposed DTS classifications in (13) and (18), ‘‘increasing DTS’’ implies that the kth DMU needs to ‘‘decrease’’ the current operation size. ‘‘Constant DTS’’ implies that it is not recommended but acceptable for the kth DMU to ‘‘maintain’’ the current operation size. ‘‘Decreasing DTS’’ implies that it is not recommended but acceptable for the kth DMU to ‘‘increase’’ the current operation size. Thus, the connotations of DTS classifications are the opposite to those of RTS classifications in terms of their economic implications. 6.4. Computational problem of the proposed approach The proposed approach is not perfect. For example, as discussed in Section 6.1, the proposed SD and DTS have several assumptions to obtain Model (12). Some DMUs may not satisfy such assumptions. Furthermore, multipliers (dual variables) are not restricted by prior information. As a result, it is possible to measure the type of DTS, but it is impossible to measure the degree of SD. To deal with such difficulties, this study proposes the approach that can incorporate SCSCs (Strong Complementary Slackness Conditions) into Model (12), as discussed in [29], although the research is concerned with determining the type of DTS, not the degree of SD. The SCSCs can function as multiplier restrictions without any prior information. It is true that the determination of DTS is sufficient enough from a practical perspective. However, when we extend the proposed approach into a time horizon, the measurement on SD will be essential in examining ‘‘inter-temporal DTS’’ among different multiple periods. The computational difficulty associated with SD will be explored in a future research extension. 7. Conclusion and future extensions This study discussed DEA environmental assessment for organizations which produced not only desirable but also undesirable outputs as a result of their economic activities. An important feature of this study was that the concept of weak and strong disposability, widely used in production economics, was replaced by the concept of natural and managerial disposability. In this study, we explored how to measure economic concepts such as SD and DTS. The type of DTS discussed in this study indicated opposite implications to the type of RTS, widely used as an economic measure among DEA researchers. In addition to SD and DTS, this study considered another type of computational feasibility of the proposed DEA approach, which changed original non-linear or mixed integer programming formulations by reformulating them by linear programming. Theoretical extensions on SD and DTS as well as the new unified efficiency measurement under managerial disposability were explored from the perspective of environmental protection. In addition to the future research extensions discussed previously, this study has the following four research tasks that need to be overcome in future: First, this study does not document an illustrative example or any real application of the proposed approach. This is a drawback of this study. This study does not provide the illustrative example or real application because [29] has documented an application of SD/DTS along with unified efficiency under managerial disposability and environmental efficiency. Second, this study does not discuss how to deal with an occurrence of multiple projections and multiple references sets in the proposed SD/DTS measurement. It is necessary for us to explore SD/DTS measurement from the mathematical perspective of SCSCs. See [25,27–29] for a detailed description of SCSCs. That is another important future extension of this study. Third, this study needs to further explore economic and managerial features of DTS measurement.

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

7313

For example, we need to investigate the relationship between the amount of investment in new environmental technology for renewable energies and the type of DTS in terms of electricity generation under various cases of fuel mixtures. No study has discussed such a research issue. Finally, it is necessary to consider the use of a stochastic DEA model for future planning and assessment. Future planning is as important as the current performance assessment. However, this study did not involve such a research agenda. In conclusion, it is hoped that this study can make a contribution to DEA environmental assessment. We look forward to seeing future research extensions, as discussed in this study. Acknowledgements This work is supported by JSPS Grant-in-Aid for Scientific Research (C) 24530287. The authors thank three reviewers whose constructive comments have improved the quality of this study. Appendix A. Computational feasibility for Model (1) It is possible to solve Model (1) by non-linear or mixed integer programming, as discussed in this study. However, it is more practical to solve Model (1) using linear programming. To attain such a computational practicality, this study slightly modifies Model (1):

Maximize

m s h X X X  xþ x  g b Rxi di  di þ Rgr dr þ Rbf df

s:t:

n X xþ x xij kj  di þ di ¼ xik

r¼1

i¼1

f ¼1

ði ¼ 1; . . . ; mÞ;

j¼1 n X g g rj kj  dr ¼ g rk

ðr ¼ 1; . . . ; sÞ

j¼1

ðA-1Þ

n X b bfj kj þ df ¼ bfk

ðf ¼ 1; . . . ; hÞ;

j¼1 n X kj ¼ 1;

kj P 0

ðj ¼ 1; . . . ; nÞ;

j¼1 xþ

x

di P 0 ði ¼ 1; . . . ; mÞ; di P 0 ði ¼ 1; . . . ; mÞ; g dr

P 0 ðr ¼ 1; . . . ; sÞ; and

b

df P 0 ðf ¼ 1; . . . ; hÞ:

 x  The difference between changes from  x  Models (1) and (A-1) is that in Model (A-1), the sign of the positive slack þdi positive to negative di . This is the only difference between the two models. Paying attention to only the slacks on inputs,  xþ   x  this study can assert that Model (A-1) maximizes a positive slack di , simultaneously minimizing a negative slack di . xþ x Consequently, it is expected that Model (A-1) has di > 0 and di ¼ 0 on optimality. Thus, the model puts more weight on environmental performance than operational performance of DMUs, as found in managerial disposability. hP i  xþ P P x  g b m x The efficiency score is measured by h ¼ 1  þ di þ sr¼1 Rgr dr þ hf¼1 Rbf df . All optimal slack variables for i¼1 Ri di  xþ x  determining the efficiency score are measured by Model (A-1). Although di  di is used in the objective function of Mod xþ x  is used for determining the efficiency measure because such an objective term indicates el (A-1), the slack sum di þ di an absolute value of input-related slacks. The dual formulation of Model (A-1) becomes the following: m s h X X X Minimize  v i xik  ur g rk þ wf bfk þ r i¼1

s:t:

r¼1

f ¼1

m s h X X X  v i xij  ur g rj þ wf bfj þ r P 0 i¼1

r¼1

ðj ¼ 1; . . . ; nÞ;

f ¼1

ðA-2Þ

v i ¼ Rxi

ði ¼ 1; . . . ; mÞ; ur P Rgr

ðr ¼ 1; . . . ; sÞ; wf P

Rbf

ðf ¼ 1; . . . ; hÞ;

r : URS: where vi (i = 1, . . . , m), ur (r = 1, . . . , s), wf (f = 1, . . . , h) are all dual variables related to the first, second and third groups of constraints in Model (A-1). The dual variable (r) is obtained from the fourth equation of Model (A-1).

7314

T. Sueyoshi et al. / Applied Mathematical Modelling 37 (2013) 7300–7314

The original model (A-1) shows primal feasibility and the dual model (A-2) shows dual feasibility. Consequently, we can solve the two models using linear programming. Furthermore, if we change v i ¼ Rxi to v i P Rxi in (A-2), then it becomes Model (5). Thus, Models (A-1) and (A-2) provide a mathematical linkage between Models (1), (3) and (5). References [1] F. Glover, T. Sueyoshi, Contributions of Professor William W. Cooper in operations research and management science, Eur. J. Oper. Res. 197 (2009) 1– 16. [2] Y. Ijiri, T. Sueyoshi, Revisiting accounting essays by William W. Cooper: In commemorating his 95th birthday, ABACUS: J. Account. Finan. Business Stud. 46 (2010) 464–505. [3] R. Dyson, F. Glover, Y. Ijiri, A. Whinston, T. Sueyoshi, New concepts, methodology and algorithm for business education and research in the 21st century, Decis. Support Syst. 48 (2010) 427–429. [4] C. Kao, Weight determination for consistently ranking alternatives in multiple criteria decision analysis, Appl. Math. Model. 34 (2010) 1779–1787. [5] P. Zhou, B.W. Ang, K.L. Poh, A survey of data envelopment analysis in energy and environmental studies, Eur. J. Oper. Res. 189 (2008) 1–18. [6] W.W. Cooper, Z. Huang, S. Li, V. Lelas, D.W. Sullivan, Survey of mathematical programming models in air pollution management, Eur. J. Oper. Res. 96 (1996) 1–35. [7] M. Bevilacqua, M. Braglia, Environmental efficiency analysis for ENI oil refineries, J. Cleaner Product. 2002 (2002) 85–92. [8] P.J. Korhonen, M. Luptacik, Eco-efficiency analysis of power plants: an extension of data envelopment analysis, Eur. J. Oper. Res. 204 (2004) 437–446. [9] S. Kumar, Environmentally sensitive productivity growth: a global analysis using Malmquist–Luenberger index, Ecol. Econom. 56 (2006) 280–293. [10] C.A. Pasurka Jr, Decomposing electric power plant emissions within a joint production framework, Energ. Econom. 28 (2006) 26–43. [11] A.J. Picazo-Tadeo, E. Reig-Martı´nez, F. Hernández-Sancho, Directional distance functions and environmental regulation, Resour. Energ. Econom. 27 (2005) 131–142. [12] H. Yang, M. Pollitt, The necessity of distinguishing weak and strong disposability among undesirable outputs in DEA: environmental performance of Chinese coal-fired power plants, Energ. Policy 38 (2010) 4440–4444. [13] O. Zaim, Measuring environmental performance of state manufacturing through changes in pollution intensities: a DEA framework, Ecolog. Econom. 48 (2004) 37–47. [14] P. Zhou, B.W. Ang, Linear programming models for measuring economy-wide energy efficiency performance, Energ. Policy 36 (2008) 2911–2916. [15] T. Sueyoshi, M. Goto, Should the US clean air act include CO2 emission control?: Examination by data envelopment analysis, Energ. Policy 38 (2010) 5902–5911. [16] T. Sueyoshi, M. Goto, DEA approach for unified efficiency measurement: assessment of Japanese fossil fuel power generation, Energ. Econom. 33 (2011) 95–208. [17] T. Sueyoshi, M. Goto, Methodological comparison between two unified (operational and environmental) efficiency measurements for environmental assessment, Eur. J. Oper. Res. 210 (2011) 684–693. [18] T. Sueyoshi, M. Goto, Measurement of returns to scale and damages to scale for DEA-based operational and environmental assessment: how to manage desirable (good) and undesirable (bad) outputs?, Eur J. Oper. Res. 211 (2011) 76–89. [19] T. Sueyoshi, M. Goto, Data environment analysis for environmental assessment: comparison between public and private ownership in petroleum industry, Eur. J. Oper. Res. 216 (2012) 668–678. [20] T. Sueyoshi, M. Goto, Returns to scale and damages to scale under natural and managerial disposability: strategy, efficiency and competitiveness of petroleum industry, Energ. Econom. 34 (2012) 645–662. [21] T. Sueyoshi, M. Goto, T. Ueno, Performance analysis of U.S. coal-fired power plants by measuring three DEA efficiencies, Energ. Policy 38 (2010) 1675– 1688. [22] T. Sueyoshi, M. Goto, DEA radial measurement for environmental assessment and planning: desirable procedures to evaluate fossil fuel power plants, Energ. Policy 41 (2012) 422–432. [23] T. Sueyoshi, M. Goto, Environmental assessment by DEA radial measurement: U.S. coal-fired power plants in ISO (independent system operator) and RTO (regional transmission organization), Energ. Econom. 34 (2012) 663–676. [24] T. Sueyoshi, M. Goto, Weak and strong disposability vs. natural and managerial disposability in DEA environmental assessment: comparison between Japanese electric power industry and manufacturing industries, Energ. Econom. 34 (2012) 686–699. [25] T. Sueyoshi, M. Goto, DEA radial and non-radial models for unified efficiency under natural and managerial disposability: theoretical extension by strong complementary slackness conditions, Energ. Econom. 34 (2012) 700–713. [26] T. Sueyoshi, M. Goto, Returns to scale, damages of scale, marginal rate of transformation and rate of substitution in DEA environmental assessment, Energ. Econom. 34 (2012) 905–917. [27] T. Sueyoshi, M. Goto, Returns to scale and damages to scale with strong complementary slackness conditions in DEA environmental assessment: Japanese corporate effort on environmental protection, Energ. Econom. 34 (2012) 1422–1434. [28] T. Sueyoshi, M. Goto, DEA environmental assessment of coal fired power plants: methodological comparison between radial and non-radial models, Energ. Econom. 34 (2012) 854–1863. [29] T. Sueyoshi, M. Goto, Returns to scale and damages to scale on U.S. fossil fuel power plants: radial and non-radial approaches for DEA environment assessment, Energ. Econom. 34 (2012) 2240–2259. [30] T. Sueyoshi, M. Goto, Window analysis for DEA environmental assessment in a time horizon: an application to U.S. coal-fired power plants. Working Paper #2012-20. Department of Management, New Mexico Tech. Energy Policy, submitted for publication. [31] R. Färe, S. Grosskopf, C.A.K. Lovel, C. Pasurka, Multilateral productivity comparison when some outputs are undesirable: a nonparametric approach, Rev. Econom. Statist. 71 (1989) 90–98. [32] R.G. Thompson, F.D. Singleton, R.M. Thrall, B.A. Smith, Comparative site evaluation for locating a high-energy physics lab in Texas, Interface 16 (1986) 35–49. [33] T. Kousmane, Weak disposability in nonparametric production analysis with undesirable outputs, Amer. J. Agricul. Econom. 87 (2005) 1077–1082. [34] K.W. Palmer, W.E. Oates, P.R. Portney, Tightening environmental standards – the benefit-cost or the no-cost paradigm, J. Econom. Perspect. 9 (1995) 19–132. [35] M.E. Porter, C. van der Linde, Toward a new conception of the environment competitiveness relationship, J. Econom. Perspect. 9 (1995) 97–118. [36] T. Sueyoshi, M. Goto, Returns to scale vs. damages to scale in data envelopment analysis: an impact of U.S. clean air act on coal-fired power plants, OMEGA: Int. J. Manag. Sci. 41 (2013) 164–175. [37] T. Sueyoshi, M. Goto, Measurement of a linkage among environmental, operational and financial performance in Japanese manufacturing firms: a use of data envelopment analysis with strong complementary slackness condition, Eur. J. Oper. Res. 207 (2010) 1742–1753.