Pekka Korhonen Mikulas Luptacik ECO-EFFICIENCY ANALYSIS OF POWER PLANTS: AN EXTENSION OF DATA ENVELOPMENT ANALYSIS

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1 Pekka Korhonen Mikulas Luptacik ECO-EFFICIENCY ANALYSIS OF POWER PLANTS: AN EXTENSION OF DATA ENVELOPMENT ANALYSIS HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION WORKING PAPERS W-241

2 Pekka Korhonen* Mikulas Luptacik** ECO-EFFICIENCY ANALYSIS OF POWER PLANTS: AN EXTENSION OF DATA ENVELOPMENT ANALYSIS *Helsinki School of Economics and Business Administration **Vienna University of Economics and Business Administration, April 2000 HELSINGIN KAUPPAKORKEAKOULU HELSINKI SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION WORKING PAPERS W-241

3 Pekka Korhonen Helsinki School of Economics and Business Administration P.O. Box 1210, Helsinki, FINLAND Tel Fax Mikulas Luptacik Vienna University of Economics and Business Administration, Department of Economics Augasse 2-6, A-1090 Vienna, AUSTRIA Tel.: Fax: mikulas.luptacik@wu-wien.ac.at The research was supported, in part, by grants from the Foundation of the Helsinki School of Economics and the Academy of Finland. All rights reserved. This study may not be reproduced in whole or in part without the authors permission. The topic of the paper was one of the activities of the DAS-project (DAS = Decision Analysis and Support) at IIASA (IIASA = the International Institute for Applied Systems Analysis, Laxenburg, AUSTRIA), and it was started while the first author worked as the project leader of the DAS-project. Pekka Korhonen, Mikulas Luptacik and Helsinki School of Economics and Business Administration ISSN ISBN Helsinki School of Economics and Business Administration - HeSE print 2000

4 2 ABSTRACT In public discussion on environmental policy, the notion of eco-efficiency is often used. The joint production of goods and undesirable outputs, such as pollutants which may not be freely disposable without costs, cause difficulties to measure the overall performance of the firm. Because of the absence of market prices for the undesirable outputs, we are not able to estimate the environmental costs. Some of the measurement and evaluation difficulties can be overcome when Data Envelopment Analysis (DEA) is employed as the efficiency measurement vehicle. We may use two different approaches. In the first approach, we start by decomposing the problem into two parts: 1) the problem of measuring technical efficiency (as the relation of the desirable outputs to the inputs) and 2) the problem of measuring so-called ecological efficiency (as the relation of desirable outputs to the undesirable outputs) separately. Then we combine those both indicators. In the second approach, we treat the pollutants as the inputs in the sense that we wish to increase desirable outputs and reduce pollutants and inputs. The approaches are applied to the problem of measuring the efficiency of 24 power plants in an European country. Keywords: Technical Efficiency, Ecological Efficiency, Eco-Efficiency, Data Envelopment Analysis. 1. Introduction One of the most intensively discussed concepts in the international political debate nowadays is the concept sustainability. According to the Brundtland Commission, sustainability means "that current generations should meet their needs without compromising the ability of future generations to meet their own needs". In this concept, economic, social and environmental aspects are closely tied to each other. The high complexity of the notion of sustainable development requires new methodology for economic analysis and measurement of economic activities. A major issue concerns the question of how we get our economic accounting systems into a form where economic and ecological considerations are better taken care of than today. What destroys social and ecological capital, today often increases the Gross National Product and thus often gives us a wrong indication of where we stand and should go. We need new indicators to measure the economic performance of the firm and the national economy, which takes into environmental aspects as well. At the occasion of the founder-meeting of the Austrian Business Council for Sustainable Development in July 1997 in Vienna, the Swiss entrepreneur Stephan Schmidheiny said in the newspaper DER STANDARD (July 4, 1997): There is no trade-off between economy and ecology. It must be a common denominator, which he calls "ecoefficiency". How to define "eco-efficiency" in an operational way? The chairman of the board of trustees of the company Landes & Gyr, Heinz Felsner formulated this notion in the following way: "We are looking for eco-efficient solutions such that the goods and services can be produced with less energy and resources and with less waste and emissions." (DIE PRESSE, December 31,1997).

5 3 The main problem in developing the eco-efficiency indicators is the lack of evaluations like market prices for the waste and emissions (or for undesirable outputs). Some of these difficulties can be overcome when data envelopment analysis is used for efficiency measurement. To our knowledge, the first paper using a non-parametric approach for multilateral productivity comparisons when some outputs are undesirable, is by Färe, Grosskopf, Lovell, and Pasurka (1989). For treating desirable and undesirable outputs asymmetrically, they use the enhanced hyperbolic output efficiency measure. This can be computed by solving a nonlinear programming problem by taking a linear approximation of the nonlinear constraint. The methodology was applied to a sample of mills producing paper and pollutants. Other related papers are Färe, Grosskopf, and Tyteca (1996) and Tyteca (1997). A comprehensive survey on the measurement of the environmental performance of firms is provided by Tyteca (1996). Golany, Roll, and Rybak (1994) have considered the problem of measuring efficiency of power plants using Data Envelopment Analysis (DEA) originally proposed by Charnes, Cooper and Rhodes [1978 and 1979] as a method for evaluating the Relative (Technical) Efficiency of Decision Making Units (DMUs) essentially performing the same task. The DEA also play a key role in our approach. The rest of this paper is organized as follows. In Section 2 we present the different variants of DEA-models which can be used for the estimation of eco-efficiency. It can be shown that the set of (strongly) efficient DMUs is the same for all models. In Section 3 we illustrate our methodology using data for a sample of power plants in one European country. Concluding remarks are given in Section Theoretical Considerations Assume we have n (homogeneous) decision making units (DMUs) each consuming m inputs and producing p outputs. The outputs corresponding to indices 1, 2,, k are desirable and the outputs corresponding to indices k+1, k+2,, p are undesirable outputs. We like to produce desirable outputs as much as possible and not to produce undesirable outputs. Let X R m n and Y Rp n be the matrices, consisting of + + nonnegative elements, containing the observed input and output measures for the DMUs. We decompose matrix Y into to parts: Y = Y g Y b, where a k n - matrix Yg is standing for desirable outputs ("goods") and a (p-k) n matrix Y b is standing for undesirable outputs ("bads"). We further assume that there are no duplicated units in the data set. We denote by x j (the j th column of X) the vector of inputs consumed by DMU j, and by x ij the quantity of input i consumed by DMU j. A similar notation is used for outputs. Occasionally, we decompose the vector y j into two parts: y j = y g j y b, where j the vectors y g j and y b j refers to the desirable and undesirable output-values of unit j. When it is not necessary to emphasize the different roles of inputs and

6 4 y g Y g (desirable/undesirable) outputs, we denote u = -y b and U = -Y b 1). Furthermore, -x -X we denote 1 = [1,..., 1] T and refer by e i to the i th unit vector in R n. We consider set T = { u u = Uλ, λ Λ}, where Λ = { λ λ R n + and Aλ b}, e i Λ, i =1,, n. Further consider matrix A R k n and vector b R k which are used to specify the feasible values of λ-variables. In classical DEA, the measure of efficiency of a DMU is defined as a ratio of a weighted sum of (desirable) outputs to a weighted sum of inputs subject to the condition that corresponding ratios for each DMU be less than or equal to one. The model chooses nonnegative weights for a DMU (whose performance is being evaluated) in a way which is most favorable for it. The original model proposed by Charnes et al. [1978, 1979] for measuring the technical efficiency of unit 0, was as follows: max h 0 = k µ r y r0 r=1 m ν i x i0 i=1 subject to: (2.1) k µ r y rj r=1 1, j = 1,2,..., n m ν i x ij i=1 µ r, ν i ε, r = 1, 2,..., k; i = 1, 2,..., m, ε > 0 ("Non-Archimedean"). We refer to the unit under consideration by subscript 0 in the functional, but preserve its original subscript in the constraints. In (2.1), only desirable outputs are used. The problem, we will study in the following is how to incorporate undesirable outputs into the model? There are at least two ways to approach the problem: 1) Because the results concerning u and U are valid for -y b and -Y b -x as well, for simplicity, we -X often refer to u and U, although we are factually interested in results concerning y x and Y X. y g Y g

7 5 1) to decompose the problem into two parts and measure efficiency in two steps: first to measure a technical efficiency and then to measure another efficiency as a ratio of a weighted sum of (desirable) outputs to the weighted sum of (undesirable) outputs, called ecological efficiency. This leads to the following two models: The first, denoted Model I (Frontier Economics) is the standard DEA model (2.1). The second, for measuring the ecological efficiency (Model II - Deep Ecology) takes the form max g 0 = k µ r y r0 r=1 p µ s y s0 s=k+1 subject to: (2.2) k µ r y rj r=1 p µ s y sj s=k+1 1, j = 1,2,..., n µ r ε, r = 1, p. ε > 0 ("Non-Archimedean") The efficiency indicators of both models, in other words technical efficiency and ecological efficiency are now the output variables for the new DEA model (with the inputs equal 1) which yields the indicator for eco-efficiency. 2) to build up the ratio, which simultaneously takes into account the (desirable) and (undesirable) outputs. We will carry out the considerations by using a CCR-model (Λ = R n + ), but the results can be generalized to other DEA-models as well. We will review some approaches and show that the seemingly different models leads to similar results. The first proposal is based on the idea to present all outputs as a weighted sum, but use negative weights for undesirable outputs. We call this model "Model A" and give it as follows:

8 6 Model A: max h A = k p µ r y r0 - µ s y s0 r=1 s=k+1 m ν i x i0 i=1 subject to: (2.3) k p µ r y rj - µ s y sj r=1 s=k+1 1, j = 1,2,..., n m ν i x ij i=1 µ r, ν i ε, r = 1, 2,..., p; i = 1, 2,..., m ε > 0 ("Non-Archimedean"). Using a standard technique (see, e.g. Charnes et al. [1978, 1979]) to transform the above fractional model (2.3) into a linear mode, we will get the following primal-dual LP-model pair. Note that the original primal formulation in Charnes et al. [1978]) is currently in the DEA-literature (see, e.g. Charnes et al. [1994]) called the dual and vice versa. Input-Oriented CCR Primal (CCR P - I) min g A = θ - ε1 T (s b + s g + s - ) s.t. (2.4a) Y g λ - s g = y g 0 Y b λ + s b = y b 0 Xλ - θx 0 + s - = 0 λ, s -, s g, s b 0 ε > 0 ("Non-Archimedean") Input-Oriented CCR Dual (CCR D - I) max h A = µ T g yg 0 - µt b yb 0 s.t. ν T x 0 = 1 µ T g Yg - µ T b Yb - ν T X 0 (2.4b) µ g, µ b,ν ε1 ε > 0 ( Non-Archimedean ) The primal model (2.4a) is almost a standard input-oriented primal CCR-model. The only difference is that the values of the undesirable outputs of the unit under consideration are considered upper bounds for a linear combination of the other undesirable outputs. Another possibility is to consider the undesirable outputs as inputs. This idea leads to the following approach, which is called Model B:

9 7 Model B: max h B = k µ r y r0 r=1 m p ν i x i0 + µ s y s0 i=1 s=k+1 subject to: (2.5) k µ r y rj r=1 1, j = 1,2,..., n m p ν i x ij + µ s y sj i=1 s=k+1 µ r, ν i ε, r = 1, 2,..., p; i = 1, 2,..., m ε > 0 ("Non-Archimedean"). This model can be presented in the following mode: Input/Undesirable Output-Oriented CCR Primal (CCR P - I/UO) min g B = θ - ε1 T (s b + s g + s - ) Input/Undesirable Output-Oriented CCR Dual (CCR D - I/UO) max h B = µ T g yg 0 s.t. (2.6a) Y g λ - s g = y g 0 Y b λ - θy b 0 + sb = 0 Xλ - θx 0 + s - = 0 λ, s -, s g, s b 0 ε > 0 ("Non-Archimedean") s.t. ν T x 0 + µ T b yb 0 = 1 µ T g Yg - µ T b Yb - ν T X 0 µ g, µ b,ν ε1 ε > 0 ( Non-Archimedean ) (2.6b) The primal model (2.6a) corresponds to a standard input-oriented primal CCR-model provided that undesirable outputs behave in the model like inputs. In this model, the DMU reduces simultaneously the inputs and undesirable outputs in order to increase eco-efficiency. Thus the eco-efficiency indicatorθ B from model B cannot be smaller than the eco-efficiency indicator from model A. The third possibility is to consider the ratio of the weighted sum of the desirable outputs minus that of the inputs to that of the undesirable outputs. This idea leads to the following approach:

10 8 Model C: max h C = k m µ r y r0 - ν i x i0 r=1 i=1 p µ s y s0 s=k+1 subject to: (2.7) k m µ r y rj - ν i x ij r=1 i=1 1, j = 1,2,..., n p µ s y sj s=k+1 µ r, ν i ε, r = 1, 2,..., p; i = 1, 2,..., m ε > 0 ("Non-Archimedean"). This approach will lead to the following LP-formulation: Undesirable Output-Oriented CCR Primal (CCR P - UO) min g C = θ - ε1 T (s b + s g + s - ) Undesirable Output-Oriented CCR Dual (CCR D - UO) max h C = µ T g yg 0 - νt x 0 s.t. (2.8a) Y g λ - s g = y g 0 Y b λ - θy b 0 + sb = 0 Xλ + s - = x 0 λ, s -, s g, s b 0 ε > 0 ("Non-Archimedean") s.t. µ T b yb 0 = 1 µ T g Yg - µ T b Yb - ν T X 0 µ g, µ b,ν ε1 ε > 0 ( Non-Archimedean ) (2.8b) This model is the same as Model A, when the roles of inputs and undesirable outputs are changed in interpretation. In order to increase the eco-efficiency, the DMUs will reduce proportionally the undesirable outputs only. Therefore, the eco-efficiency indicator of model C cannot be greater than the eco-efficiency indicator of model B. We may also consider the reciprocal models of the above models. The approach will lead so-called "output"-oriented models, where the desirable outputs are controlled. As an example, we present Model D, which is a reciprocal model of Model B:

11 9 Model D: min h D = p m µ s y s0 + ν i x i0 s=k+1 i=1 k µ r y r0 r=1 subject to: (2.9) p m µ s y sj + ν i x ij s=k+1 i=1 1, j = 1,2,..., n k µ r y rj r=1 µ r, ν i ε, r = 1, 2,..., p; i = 1, 2,..., m ε > 0 ("Non-Archimedean"). Desirable Output-Oriented CCR Primal (CCR P - DO) max g D = θ + ε1 T (s b + s g + s - ) Desirable Output-Oriented CCR Dual (CCR D - DO) min h D = µ T b yb 0 +νt x 0 s.t. (2.10a) Y g λ - θy g 0 - sg = 0 Y b λ + s b = y b 0 Xλ + s - = x 0 λ, s -, s g, s b 0 ε > 0 ("Non-Archimedean") s.t. µ T g yg 0 = 1 µ T g Yg - µ T b Yb - ν T X 0 µ g, µ b,ν ε1 ε > 0 ( Non-Archimedean ) (2.10b) When considering the primal presentations of models A - D, we see that we can use a unified presentation for all models. The presentation is called "Model G":

12 10 Model G: General Model CCR Primal (CCR P - G) max g G = σ + ε1 T (s b + s g + s - ) General Model CCR Dual (CCR D - I) min h G = - µ T g yg 0 + µt b yb 0 +νt x 0 s.t. (2.11a) Y g λ - σw g - s g = y g 0 Y b λ + σw b + s b = y b 0 Xλ + σw x + s - = x 0 λ, s -, s g, s b 0 ε > 0 ("Non-Archimedean") s.t. µ T g wg + µ T b wb + ν T w x = 1 -µ T g Yg + µ T b Yb + ν T X 0 µ g, µ b,ν ε1 ε > 0 ( Non-Archimedean ) (2.11b) w g By choosing the components of vector w = w b in a suitable way and modifying an w x objective function accordingly, we may introduce one of models A - D as follows: Table 1: Required Modifications of a General Model Model Type w g w b w x σ A 0 0 x 0 1-θ B 0 y b 0 x 0 1-θ C 0 y b θ D y g θ Note that in case of model A-C, the value of the objective function g I = 1 - g G, I = A, B, C, and in case D: g D = g G - 1. In data envelopment analysis, we are interested in the efficiency of the decision making units. Efficiency is defined as follows: Definition 1. A point u* = Uλ T is efficient iff (if and only if) there does not exist another u T such that u u*, and u u*. The unit, which is not efficient is called inefficient. However, if an inefficient unit is not an inferior point of T, we may call it weakly efficient. It is defined as follows: Definition 2. A point u* = Uλ T is weakly efficient iff there does not exist another u T such that u > u*.

13 11 We may prove that eco-efficiency of a unit can be analyzed with Model G, and the w g result does not depend on vector w = w b provided w 0, w 0. For simplicity, we w x consider the Model G in the form: max g G = σ + ε1 T s s.t. (2.12) Uλ - σw - s = u 0 λ, s 0 ε > 0 ("Non-Archimedean") Theorem 1. The following results hold: 1. u 0 is efficient iff the value of σ at the optimum σ* = 0 for all w 0, w u 0 is inefficient, but not weakly efficient iff the value of σ at the optimum σ* > 0 for all w 0, w 0. Proof. 1.a u 0 is efficient the value of σ at the optimum σ* = 0 for all w 0, w 0. Assume σ* > 0 for some w 0, w 0. Hence it follows that Uλ* - σ*w u 0 Uλ* u 0 Uλ* u 0, because σ* > 0 and w 0, w 0 u 0 is not efficient, which contradicts with the assumption. 1.b u 0 is efficient the value of σ at the optimum σ* = 0 for all w 0, w 0. Assume u 0 is inefficient u T such that u u 0 u u 0. Assume u j > u 0j, j {1, 2,, p}. Choose w j > 1 w i = 0, i = 1,2,, p, i j u = Uλ - σw u 0, when σ = u j - u 0j > 0, which contradicts with the assumption that σ* = 0. 2.a u 0 is inefficient, but not weakly efficient the value of σ at the optimum σ* > 0 for all w 0, w 0. Assume u 0 is inefficient, but not weakly efficient u such that u > u 0 for all w 0, w 0 σ > 0 such that u = Uλ - σw > u 0 σ* > 0. 2.b u 0 is inefficient, but not weakly efficient the value of σ at the optimum σ* > 0 for all w 0, w 0. Because σ* > 0 for all w 0, w 0 we choose w = [1, 1,, 1]. Hence it follows that at the optimum u* = Uλ* - σ*w u 0 u* > u 0, because σ* >0 and w = [1, 1,, 1]. Remark 1. A DMU is eco-efficient iff σ* = 0 and all slack variables s b, s g, and s - equal zero for all w 0, w 0; otherwise it is eco-inefficient.

14 12 Remark 2. If a DMU is eco-inefficient, but weakly eco-efficient, then for some w 0, w 0, σ* = 0, but in this case at least one of the components of the slack variable vectors s b, s g, and s - is positive. Theorem 1 and Remarks 1 and 2 shows that to analyze the eco-efficiency, we may use any of the models A-G. The eco-efficient units are eco-efficient, no matter which model is used, and eco-inefficient - but not weakly eco-efficient - units can be diagnosed inefficient by using the value of σ at the optimum. For the weakly ecoefficient solution, not only the value of σ is not sufficient. 3. Eco-Efficiency of Power Plants In this section, we describe how we used our approach to evaluate the eco-efficiency and the emission reduction programme of 24 power plants in an European country. The desirable output is electricity generation with a minimum of MW and a maximum of MW. The total costs are considered as an input (min US$, max US$). The undesirable outputs or the pollutants are DUST, NOx and SO2. By the emission reduction programme (with total investment between TUS$ and TUS$) the power plants reduced the emission quantities considerably. The emission levels are available before and after the emission reduction. The emission quantities of DUST (t/a) were before reduction within the range [574, 14097], and after [175, 1418]. The corresponding ranges for NOx (t/a) were: before [1926, 5509] and after [963, 2754], and for SO2 (t/a) before: [1401, 24459] and after [1401, 12230]. All data are per year. Solving the Models I (2.1) and II (2.2) we obtained the measures of the technical and ecological efficiency respectively. Those measures provide the first indicators for the performance of power plants from the eco-efficiency point of view. The results are given in Table 2. The first column denoted by "Tech. Eff." contains the technical efficiency, which is the results of Model I using total costs as an input and the electricity generation as an output. It is a very simple CCR-model with only one efficient unit namely power plant 1 which is a small one with the lowest output level and the lowest total costs. At column 3 denoted by "Ecol. Eff. (before)", the ecological efficiency before emission reduction programme is presented. The results are obtained solving Model II with electricity generation as the desirable output and with DUST, NOx and SO2 as pollutants or undesirable outputs. The ecological efficient power plants are 1, 2, 4, 8, 13, and 14. The fourth column is standing for the ecological efficiency after emission reduction. Only the power plant 1 is technical and ecological efficient - before emission reduction.

15 13 Table 2: Technical and Efficiency Analysis (CCR-model) Units Tech. Eff. Ecol. Eff. (before) Ecol. Eff. (after) Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit In order to get an indicator of the eco-efficiency we took technical and ecological efficiency as output variables for the new DEA model with input equal 1. In this way the eco-efficiency is decomposed into technical and ecological efficiency. The ecoefficiency frontier before and after the emission reduction is illustrated in Fig. 1 and Fig. 2. As we can see Figs. 1 and 2, only the power plant 1 in case of before emission reduction is and power plants 1 and 2 in case of after emission reduction are ecoefficient. The units 2, 4, 8, 13, and 14 (in case of before emission reduction) and the units 4, 5, 8, 13, and 14 (in case of after emission reduction) are only weakly efficient, because they are technical inefficient. Because all eco-inefficient units lie outside the eco-efficiency cone in both cases, the indicator of the eco-efficiency is simple the better value of efficiency scores obtained from the Models I and II. Thus the eco-efficient scores are the same as ecological efficiency scores in Table 2 except that Unit 1 is also eco-efficient in case of after emission reduction. In order to show the importance of the technical (ecological) efficiency in determining the eco-efficiency of the units 1 and 2 we computed the ratio of weighted technical (ecological) efficiency to the virtual output (the weighted sum of technical and ecological efficiency). This is useful indication of the importance of technical (ecological) efficiency in determining the eco-efficiency. In both power plants the

16 14 technical efficiency was given an importance of approximately 60 % and the ecological efficiency of approx. 40 % in determining of the eco-efficiency. The strength of both units lie more in the technical efficiency, while the eco-inefficient units have a weakness primarily in the technical inefficiency. From the corresponding slack variables of the new DEA model, the potential eco-efficiency improvement with respect to technical and ecological efficiency respectively can be seen. It is obvious that the eco-efficiency after the reduction programme is in average higher than before. Unit 14 Unit 13 Unit 8 Unit 4 Unit 2 Unit 1 Ecological efficiency Technical efficiency Figure 1: Eco-Efficiency Frontier before Emission Reduction Unit 14 Unit 13 Unit 8 Units 5, 4 Unit 2 Unit 1 Ecological efficiency Technical efficiency Figure 1: Eco-Efficiency Frontier after Emission Reduction

17 15 An alternative approach to analyse eco-efficiency is to use models A, B, and C. Table 3 shows the results of model B before and after emission reduction, both under the assumption of constant returns to scale. We will discuss in more detail the results of model B after emission reduction. A similar analysis can be done for the models A and C (before and after emission reduction) and for variable returns to scale. Table 3: Eco-Efficiency Scores Using the Combined Model B Units Before Emission After Emission Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit Unit The input variables in model B are total costs, the investment for emission reduction, the emission of DUST, emission of NOx and emission of SO2, all after the reduction programme. The only output variable is electricity generation. Comparing the results of model B (Table 3) with the eco-efficiency obtained as a composition of technical and ecological efficiency (Table 2 and Fig. 2) the tendency of the same results can be observed. Because of the property of DEA models yielding the best possible results for every decision making unit, the eco-efficiency defined by model B cannot be lower than the eco-efficiency in Table 2. For instance, the weakly eco-efficient power plants 4, 5, 8, and 13 from Table 2 (after) are eco-efficient according to model B. The units 1 and 2 are efficient in both cases. But model B provides a deeper insight in the causes of eco-inefficiency and shows the potential improvement to particular inputs and outputs. Nevertheless, the decomposition of the eco-efficiency into technical and ecological efficiency can be useful.

18 16 Computing the ratio of a weighted inputs to the weighted sum of inputs we obtain an indication for the importance of particular inputs. For example, in units 2 and 5, an importance of 82 % was given to abatement investment in determining their ecoefficiency. The investment in emission reduction were highly efficient. The strengths of unit 1 lie in the abatement investment (48 %) and in the lower level of DUST emission (47 %). The abatement activity was oriented to reduction of NOx only. An interesting result can be found for power plant 8. The most important factor for the eco-efficiency of this unit is the low level of SO2 emission (the lowest level under all power plants). To this input was given an importance ranking of 99 % in determining the eco-efficiency. The most important factors for the power plan 13 are the abatement investment and the relative low level of NOx emission with respect to the high level of output. Power plant 13 is the plant with the highest electricity generation. Power plant 14 is a weak eco-efficient because of input inefficiency. The potential improvements lie in reducing of abatement investment by 52 % and in reducing of total costs by 24 %. Similar results can be found for the inefficient power plants They have a weakness in technical efficiency and they should primarily reduce the inputs. The last Table 4 contains the results of models A, B, and C under the assumption of variable returns to scale. As shown by Theorem 1 and Remarks 1 and 2 in Section 2, the efficient units are efficient, no matter which model is used (the power plant 14 is weak eco-efficient). Furthermore, it can be seen that the efficiency indicator θ B from model B is greater or equal than be efficiency indicator θ A from model A and greater than or equal to the efficiency indicator θ C from model C. 4. Concluding Remarks In this paper we presented two approaches which can be used for the estimation of eco-efficiency. In the first approach, we measured the eco-efficiency in two steps: We estimated the technical efficiency and the so-called ecological efficiency separately. Then we took the results of both models as the output variables for the new DEA model (with the inputs equal 1) which provides the indicator for eco-efficiency. In the second approach, we formulated the different variants of DEA models, which simultaneously take into account the inputs, the pollutants or undesirable outputs and the desirable outputs. It was shown that the efficient units are efficient, no matter which model variant is used. However, the efficiency scores may differ. Comparing the above mentioned two approaches, in tendency both lead to the same results. However, the second approach provides a more deeply insight into the causes of the eco-inefficiency and shows the potential improvement with respect to the particular inputs and outputs. The first approach yields the decomposition of the ecoefficiency into the technical efficiency and ecological efficiency respectively.

19 17 As a topic for further research we intend to introduce into our models the environmental standards. In this way the impact of environmental policy for the efficiency measure and for the multilateral productivity comparisons across the firms or particular industries in different countries may be evaluated. Table 4: Results of Models A, B, C Before emission reduction After emission reduction DMUs Model A Model B Model C Model A Model B Model C DMU DMU DMU03 0, , , , , ,98739 DMU DMU DMU , , ,94549 DMU07 0, , , , , ,98754 DMU DMU09 0,8152 0, , , , ,94342 DMU10 0, , , , , ,91388 DMU11 0, , , , , ,96286 DMU , , ,99969 DMU DMU14 0, DMU15 0, , , , , ,83986 DMU16 0, , , , , ,83765 DMU17 0, , , , , ,82439 DMU18 0, , , , , ,96724 DMU19 0, , , , , ,99577 DMU20 0, , , , , ,99531 DMU21 0, , , , , ,98684 DMU22 0, , , DMU23 0, , , , , ,99319 DMU24 0, , , , , , References Banker, R.D., Charnes, A. and Cooper, W.W. (1984), Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science 30, Charnes, A., Cooper, W.W. and Rhodes, E. (1978), Measuring Efficiency of Decision Making Units, European Journal of Operational Research 2, Charnes, A., Cooper, W.W. and Rhodes, E. (1979), Short Communication: Measuring Efficiency of Decision Making Units, European J. Operational Res. 3, 339.

20 18 Färe, R., Grosskopf, S., Lovell, K., and Pasurka, C.: (1989), "Multilateral Productivity Comparisons When Some Outputs Are Undesirable: A Nonparametric Approach", The Review of Economics and Statistics, 71/1. Färe, R., Grosskopf, S., and Tyteca, D.: (1996), "An Activity Analysis Model of the Environmental Performance of Firms - Application to Fossil-Fuel-Fired Electric Utilities", Ecological Economics, 18, pp Golany, B., Roll, Y., and Rybak, D.: (1994), "Measuring Efficiency of Power Plants in Israel by Data Envelopment Analysis", IEEE Transactions on Engineering Management, Vol. 41, No. 3. Tyteca, D. (1996), "On the Measurement of the Environmental Performance of Firms - A Literature Review and a Productive Efficiency Perspective", Journal of Environmental Management 46, Tyteca, D. (1997), "Linear Programming Models for the Measurement of Environmental Performance of Firms - Concepts and Empirical Results", Journal of Productivity Analysis, 8,