A common weight MCDA–DEA approach to construct composite indicators

Ecological Economics 70 (2010) 114–120

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Ecological Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e c o n

Analysis

A common weight MCDA–DEA approach to construct composite indicators S.M. Hatefi, S.A. Torabi ⁎ Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 13 March 2010 Received in revised form 16 August 2010 Accepted 27 August 2010 Available online 20 September 2010 Keywords: Composite indicators Data envelopment analysis Common weights Multi criteria decision analysis

a b s t r a c t A common weight multi criteria decision analysis (MCDA)–data envelopment analysis (DEA) approach is proposed to construct composite indicators (CIs). The proposed MCDA–DEA model enables the construction of CIs among all entities via a set of common weights. The model is capable to discriminate entities which receive CI score of 1, i.e., the efficient entities leading to determination of a single best entity. Common weights structure of the proposed model has more discriminating power when compared to those obtained by previous DEA-like models. In order to validate the proposed MCDA–DEA model, it is applied to two case studies taken from the literature to construct two well-known CIs, i.e., Sustainable Energy Index (SEI) and Human Development Index (HDI). Both the robustness and discriminating power of the proposed method are studied through these case studies and tested by Spearman's rank correlation coefficient. The results reveal several merits of the proposed method in constructing CIs. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Composite indicator (CI) is a suitable tool for performance measurement and monitoring, benchmarking, policy analysis and public communication via providing an aggregated performance index in various fields such as sustainable energy index, Human Development Index, Environmental Performance Index, etc. (OECD, 2008; Sagar and Najam, 1998; Esty et al., 2006). The main steps in constructing CIs are weighting and aggregating of a set of given sub-indicators which directly affect the quality and reliability of the calculated CIs (Saisana et al., 2005). In this regard, multi criteria decision analysis (MCDA) and data envelopment analysis (DEA) have been accepted as two popular methods for weighting and aggregating sub-indicators to construct CIs. The implementation of MCDA methods to construct CIs can be found in some studies (e.g., Diaz-Balteiro and Romero, 2004; Ebert and Welsch, 2004; Krajnc and Glavič, 2005; Munda, 2005; Nardo et al., 2005; Hajkowicz, 2006; Zhou et al., 2006). However, determination of appropriate weights for sub-indicators is a major problem in applying MCDA methods. Nardo et al. (2005) have presented various weighting methods to derive the weights of underlying sub-indicators. However, expert opinion and analyst judgment may play an important role to derive these weights and consequently affect the quality of calculated CIs (Hope et al., 1992). Fortunately, data envelopment analysis is an effective and applicable tool to derive weights from the view point of operation research. DEA is capable to perform weighting and aggregating steps simultaneously and does not need to expert opinion and analyst judgment to do so. DEA, introduced by Charnes et al.

⁎ Corresponding author. Tel.: + 98 21 88021067; fax: +98 21 88013102. E-mail addresses: smhatefi@ut.ac.ir (S.M. Hatefi), [email protected] (S.A. Torabi). 0921-8009/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2010.08.014

(1978), has been applied for performance measurement in many fields such as economic, environmental and social sectors. Recent application of DEA for evaluating decision making units (DMUs) in these fields can be found in Kao and Hung (2008); Ramanathan and Yunfeng (2009) and Ulucan and Baris Atici (2010). Recently some studies have developed standard DEA and DEA-like models to construct CIs (Zhou et al., 2007a; Cherchye et al., 2007, Cherchye et al., 2008a;2008b). The main problem of using DEA-like models is that several DMUs or entities may receive CI score of 1 which means that all of these entities are efficient DMUs. To overcome this deficiency, Despotis (2005a;2005b) proposes a two-phase method in which the CI values of entities are first measured by the standard DEAlike model and then a parametric goal programming model is introduced to discriminate entities which receive performance score of 1. Zhou et al. (2007b) propose two classical DEA-like models to determine the best and worst set of weights for underlying subindicators and a simple additive weighting formula for data aggregation. As the best practice model may provide several efficient entities; they aggregate the best and worst efficiency scores obtained by the best and worst practice models, using an adjusting parameter called λ to obtain final CIs. Also, Zhou et al. (2010) propose similar DEA-like models but considering the weighted product (WP) method instead of weighted additive one for data aggregation. However, in final CIs construction, variation of adjusting parameter λ may strongly affect the resulting CIs. Also, the obtained weights provided by these classical DEA-like models are often unrealistic. That is, for each entity, the sub-indicators which have good performance may receive extremely high weights and those having bad performance may receive extremely small weights leading to extreme weights which are often unrealistic and impractical because of ignoring the impact of sub-indicators with extremely small weight values in CI calculation.

S.M. Hatefi, S.A. Torabi / Ecological Economics 70 (2010) 114–120

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Relying on recent DEA literature, this paper proposes a MCDA–DEA model using common weights approach to construct CIs which have more discriminating power than the best weights. The model is capable to discriminate efficient entities which may receive same CI value of 1 by using exiting methods and therefore specifies a single best entity, and at the same time, it does not require the adjusting parameter λ, i.e., removing the Zhou et al. (2007b) method deficiencies. The rest of the paper is organized as follows. In Section 2, a brief discussion is provided about the problem definition and the Zhou et al. (2007b) which is the most recent and efficient approach in the CI literature. In Section 3, the best practice DEA-like model of Zhou's et al. method is modified to introduce the proposed common weight MCDA– DEA model. The application of the proposed model to construct CIs is shown by two case studies in Section 4. The numerical results regarding the robustness and discriminating power of the proposed model is also presented in this section. Finally, the concluding remarks are reported in Section 5.

Consequently, a set of indices gI1, gI2, …, gIm for all entities are provided by solving model (1) repeatedly for each entity. Model (1) is actually equivalent to an input-oriented CRS DEA model with n outputs and one dummy input of 1 for all the entities. Furthermore, the objective function of model (1) is similar to simple additive weighting (SAW) aggregation method, while the weights of subindicators are changeable for each entity calculated endogenously by the model. The second DEA-like model that produces the worst favorable weights for each entity is as follow:

2. Problem Description

Model (2) is equivalent to an output-oriented CRS DEA model with n inputs and one dummy output of 1 for all the entities. In a similar way, model (2) produces bI1, bI2, …, bIm values by solving this model repeatedly for each entity. In this manner, models (1) and (2) seek for the “best” and “worst” weights which are then used to aggregate subindicators into a composite performance score. It is noteworthy to mention that the appropriate weights for the sub-indicators should reflect their preferences. However, deriving the preferential weights is often problematic due to inherent subjectivity for deriving them. Fortunately, the weights derived from the DEA-like models do not require any subjective information but at the same time, they have not a preferential interpretation. In fact, the main purpose of a DEA-like model is to derive an efficient frontier in a production context by which we can identify the efficient and nonefficient DMUs. After calculating the gIi and bIi values for each entity, Zhou et al. (2007b) propose the following equation to combine them to construct the final CI value of respective entity:

Suppose there are m entities or decision making units (DMUs) with same inputs and outputs like a number of universities or oil refineries whose aggregated performances are to be measured based on n sub-indicators with different measurement units. Let Iij denotes the value of sub-indicator j with respect to entity i. In CIs construction, the benefit type is considered for those of sub-indicators that satisfy the property of “the larger the better”. Noteworthy, for the cost type sub-indicators (i.e., those of sub-indicators with the property of “the smaller the better”), they can be easily converted to benefit type ones by considering their reciprocal values (Zhou et al., 2007b). The problem of aggregating these sub-indicators into a composite index CIi (i = 1, 2,..., m) has been depicted in Fig. 1. Notably, the value of CIi indicates the aggregated performance of entity i with respect to all of underlying sub-indicators. In constructing CIs, a weight is first assigned to each sub-indicator, and then a certain aggregation function is applied to calculate CIs. All sub-indicators are often normalized before aggregation because they usually have incommensurable measurement units. Recently, Zhou et al. (2007b) propose a mathematical programming approach to construct CIs. In their approach, two sets of weights are first calculated by using two slightly different DEA-like models and then they are combined to construct the final CI values. The first DEAlike model that produces the most favorable weights for each entity is as follow: n

g

gIi = max ∑ wij Iij

ð1Þ

j=1

n

g

s:t: ∑ wij Ikj ≤1 k = 1; 2; …; m j=1

g

wij ≥0;

j = 1; 2; …; n

Model (1) provides the most favorable aggregated performance score for entity i in terms of all the underlying sub-indicators.

Fig. 1. Graphical representation of CI construction.

n

b

ð2Þ

bIi = min ∑ wij Iij j=1

n

b

s:t: ∑ wij Ikj ≥1 k = 1; 2; …; m j=1

b

wij ≥0;

CIi ðλÞ = λ

j = 1; 2; …; n

gIi −gI − bI −bI− + ð1−λÞ i : gI⁎−gI − bI⁎−bI −

ð3Þ

Where: −

gI⁎ = f max gIi ; i = 1; 2; …; mg; gI = f min gIi ; i = 1; 2; …; mg; − bI⁎ = fmax bIi ; i = 1; 2; …; mg ; bI = fmin bIi ; i = 1; 2; …; m g and 0≤λ≤1:

In formula (3), λ is an adjusting parameter which is determined by the decision maker. Choosing appropriate value for this parameter is crucial and depends on DM's preferences on two aforementioned extreme cases. However, it should be noted that such flexibility in the selection of λ value may also cause difficulty for DM to make his/her final subjective choice since there may not exist enough objective evidences to support any choices of λ values. Moreover, different values of λ may lead to distinct and misleading results for CIs. As mentioned earlier, although the above model can be considered as one of the interesting and applicable tools in constructing CIs, but it may leads to extreme weighting of underlying sub-indicators which is unrealistic in practice. In the other words, when specific entity is under evaluation, for example, model (1) has the flexibility to provide the weights in its own favor for maximizing its own efficiency score. This weight flexibility may identify an entity to be efficient by giving an extremely high weight to a sub-indicator which has extremely good performance and an extremely small weight to that which has extremely bad performance. However, such an extreme weighting is unrealistic and causes models (1) and (2) to have a poor discriminating power. To alleviate this deficiency, some authors have recently studied the problem of finding a common set of weights in the DEA literature (Golany, 1988; Roll et al., 1991; Roll and Golany, 1993; Li

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and Reeves, 1999; Karsak and Ahiska, 2005; Amin et al., 2006; Ertay et al., 2006; Kao, 2010). In this regard, Golany (1988) proposed an interactive multi objective procedure to determine efficient output levels, based on a multi goal programming model which can be considered as a first work integrating DEA and MCDA for finding common weights. This paper employs efficiency measures that are not specific to a particular entity, but common to all entities by using common weights approach. For doing so, the model (1) is converted into a common weight MCDA–DEA model with an improved discriminating power among entities which receive composite index of 1. Notably, the model (1) is the main model of Zhou et al. (2007b) in CIs construction, but it has some difficulties (as the model (2) does). For example, it may determine several efficient DMUs and so fails in identifying the best entity based on CI values (i.e., full ranking of DMUs). To remove this deficiency, Zhou et al. (2007b) propose the above formula (3) which can be considered as an extension of model (1). Other extension of model (1) for constructing CIs is proposed by Despotis (2005a,2005b). Nevertheless, the extensions of classical DEA models to common weights ones are usually derived from the respective best practice models to improve discriminatory power (e.g., Karsak and Ahiska, 2005, 2007; Amin et al., 2006). Accordingly, by relying on the common weight DEA literature, the proposed model is also derived from the best practice model (1), therefore, it can also be considered as an extension of model (1) via a set of common weights. As discussed earlier, a similar extension could also be employed on the worst practice model (2), but it is not the case as it is usual in the literature. More details of the proposed method are provided at the next section. 3. Proposed Common Weight MCDA–DEA Method A common weight MCDA–DEA model is derived from model (1). Let di denotes the deviation of the efficiency of entity i from unity when it is under evaluation. The following model is equal to model (1) which should be solved m times to minimize the efficiency deviation of each entity. Note that in model (4), we have: gIi = 1 − di. ð4Þ

min di n

wij ≥0; dk ≥0;

ð5Þ

min M M−di ≥0;

i = 1; 2; …; m

n

s:t: ∑ wj Iij + di = 1; j=1

wj ≥0; di ≥0;

i = 1; 2; …; m

i = 1; 2; …; m; j = 1; 2; …; n

where wj denotes the common weight with respect to sub-indicator j among all entities. Notably, the constraints M − di ≥ 0 ; ∀ i assure that M = max{ di, i = 1, 2,..., m}. According to the literature of common weights (e.g., Karsak and Ahiska, 2005, 2007), all entities are evaluated using a set of common weights to enable a fair comparison among them which contrasts with the conventional DEA models in which each entity is evaluated by different sets of weights. Since the aim of model (5) is minimizing the maximum deviation from the full efficiency among all entities, the derived common weights which restrict the freedom of a specific entity to choose the sub-indicators' weights in its own favor, may not be favorable and ideal for each specific entity. However, all entities are evaluated through a common base by using these common weights. Now, to ensure having non-zeros common weights for subindicators, a non-Archimedean infinitesimal value (epsilon) is introduced as the lower bound of these common weights. Notably, Cook et al. (1996) addressed that it is unreasonable to credit the worth of any subindicator with negligible importance(ε ≈ 0). Arguably, a larger value of ε is preferable to a smaller value because of increasing the discriminating power among entities. Also, if the choice of ε is not small enough, then the DEA models such as CCR model or the proposed model (6) (at below) may generate infeasible results. In this regard, Ali and Seiford (1993) and Mehrabian et al. (1998) proposed an assurance interval on ε value for ensuring the feasibility of the multiplier side and boundedness of the envelopment side of the CCR and BCC models. However, the proposed model in this paper has enough discriminating power such that setting an infinitesimal value for ε is not a critical issue. In this manner, the model (5) can be reformulated as follows: ð6Þ

min M

s:t: ∑ wij Ikj + dk = 1; j=1

manner, the proposed MCDA–DEA model can be initially written as follows:

k = 1; 2; …; m

j = 1; 2; …; n; k = 1; 2; …; m

M−di ≥0; n

i = 1; 2; …; m

s:t: ∑ wj Iij + di = 1; j=1

Model (4) provides a set of best weights for entity i. Therefore, solving this model for each entity provides m different sets of weights for sub-indicators each of which might be unrealistic as said before. The spirit of model (4) is similar to those of proposed by Cherchye and Kuosmanen (2004) and Kuosmanen et al. (2006) in which price variations are restricted across firms in contrast to the previous price/ weight restriction tools that impose bounds on the relative prices of inputs/outputs within the evaluation of a single firm. Notably, Despotis (2005b) resorted to the CRS DEA model (1) to develop a parametric goal programming model for measuring HDI. He addressed that efficiency has no special meaning in CI construction, as no kind of transformation of inputs to outputs is assumed. Accordingly, this paper does not focus on other types of DEA modeling, e.g., variable returns to scale introduced by Banker et al. (1984), because they cannot have any specific meaning in CI construction area. The objective function of model (4) only involves the efficiency deviation of entity i. However, to maximize the efficiencies of all entities simultaneously, we need to reformulate this model via a set of common weights. For doing so, this paper uses minimax approach to minimize the maximum efficiency deviation among all entities. In this

wj ≥ε; di ≥0;

i = 1; 2; …; m

i = 1; 2; …; m; j = 1; 2; …; n

Using the model (6), the composite indicator of ith entity is computed by CIi = 1 − di ; ∀ i = 1, 2,..., m. It is worth mentioning that Zhou et al. (2007b) addressed that if an entity has a value dominating other entities in terms of a certain sub-indicator, this entity would always obtain a score of 1 even if it has severely bad values in other more important sub-indicators because of assigning negligible or zero values for their weights with respect to model (1) and/or model (2). Therefore, the best and worst weights derived from models (1) and (2) cannot be an upper and lower bounds for the common weights. Also, it should be considered that the entities are evaluated based upon 2m different weight vectors which are sequentially generated by applying the model (1) and (2) for each entity, while just one weight vector is generated by using model (6). In the DEA-like models, the weights are endogenously generated from the given data by which a number of sub-indicators (with zero weights) may be ignored in aggregation. To overcome this difficulty, various ways of exogenously restricting the flexibility of weights such as direct

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restrictions, adjusting the observed input–output levels to capture value judgments and restricting virtual inputs and outputs or “proportional constraints” are proposed in the literature (see e.g., Allen et al., 1997; Thanassoulis et al., 1995; Wong and Beasley, 1990). Recently, Zhou et al. (2007b) incorporate additional information on the weights by restricting their flexibility through “proportional constraints”. However, Wong and Beasley (1990) discuss that implementing this type of restriction is not straightforward, due to the fact that the implied restrictions on the DEA weights are entity specific. Consequently, since all entities are evaluated simultaneously in the proposed approach, incorporating such kind of weight restrictions to the model is problematic. Also, using this type of weight restrictions requires subjective information which must be provided by making a consensus among all decision makers or domain experts as the relative importance of each sub-indicator. Therefore, using of weight restrictions is not recommended in the proposed model. In summary, the merits of proposed model (6) when compared to Zhou et al. (2007b), can be summarized as follows: • Obtained CIs by using common weights approach, i.e., model (6), are more discriminated than those of obtained by using the model (1) of Zhou et al's. method, since the model (6) seeks for the common weights of all entities simultaneously by preventing each particular entity to choose the weights in its own favor, • The common weights are computed in just one step via solving model (6), while those of Zhou et al's. method require models (1) and (2) to be solved m times (i.e., running 2m LP models), • It enables a more fair evaluation of entities based on composite indicators calculated by common weights, while in the Zhou et al's. method; entities are actually evaluated by different weights. To sum up, in spite of the above merits, developing common weights in CI construction has other promising advantages. It removes any subjectivity which may lead to confusing interpretation of CI results. As discussed earlier, interpretation of adjusting parameter λ in Zhou et al. (2007b) is crucial and involves subjectivity in determining the appropriate value for it. Also, the proposed method has a notable saving in the number of LP models which must be solved. Therefore, the proposed method has less computational complexity than that of Zhou's et al. method. In addition to aforementioned advantages, sometimes the model (6) may result in more than one best entity, thus, fails in determining the best entity. If this is the case, the following common weight MCDA model is proposed to overcome this difficulty:

117

sustainable energy index (SEI) based on three specific sub-indicators in eighteen APEC economies at 2002. Energy supply, energy efficiency and environmental protection are the main elements for sustainable energy development (Jefferson, 2006). In this case, the proposed method is applied to construct SEI by aggregating energy efficiency indicator (EEI), renewable energy indicator (REI) and climate change indicator (CCI). Table 1 shows the required data and the calculated SEI values of eighteen APEC economies using the Zhou et al. (2007b) and proposed methods. According to Table 1, the second to forth columns show the relevant sub-indicators values for evaluating APEC economies. Also, the fifth to seventh columns show indices gIi, bIi and CIs respectively which are obtained by applying models (1) and (2) on the three subindicators and then using formula (3) with λ = 0.5. The eighth column provides CIs values obtained by proposed model (6). As the Table 1 indicates, Peru, Philippines, Papua New and New Zealand gain same performance score of “1” by model (1) while the number of efficient economies reduces from four to two namely Peru and Philippines by the proposed model (6). This reduction implies the higher discrimination power of proposed common weight model (6) compared to the model (1). Hereafter, in order to specify the most efficient country based on SEI value, we employ model (7). According to the results of model (6), EF set consists of Peru and Philippines. The step size and initial value of discrimination parameter K are set to 0.01 and 0.05, respectively. In this way, the model (7) finally converges to a single best entity namely Peru at K = 0.09 as determined by Zhou et al. model. Notably, the non- Archimedean value (ε)is also set to 0.00001when solving models (6) and (7). The seventh and ninth columns show the composite indicators (SEI values) by applying model (3) and proposed model (7). By comparing these columns, four economies which gain same performance score of 1 by model (1), receive similar rank based on their respective SEI values. The robustness and efficiency of the proposed MCDA–DEA method is evaluated in comparison with Zhou et al. (2007b) method by Spearman's rank correlation test. For more detail on Spearman's rank correlation test, the reader is referred to (Sheskin, 2000). The test indicates whether or not there is a positive correlation between the two sets of calculated SEIs by using the proposed and Zhou's et al. methods. For this, null hypothesis H0 : rs = 0 is tested against alternative hypothesis H1 : rs N 0 as follow: H0. There is no correlation between the CIs obtained by the proposed MCDA–DEA model and the Zhou et al. model.

ð7Þ

min M−K ∑ de e∈EF

M−di ≥0; n

i = 1; 2; …; m

s:t: ∑ wj Iij + di = 1; j=1

wj ≥ε; di ≥0;

i = 1; 2; …; m

Table 1 Data and the SEI values of eighteen APEC economies. Economies

Sub-indicators

Zhou et al. model

The proposed model

EEI

REI

CCI

Model Model CIs 1 2

CIs

CIs (k=0.09)

13.825 17.758 12.381 5.473 10.79 4.286 6.95 8.647 8.424 8.516 8.204 8.178 5.901 6.208 5.767 5.539 4.683 2.453

53.6 44.6 23.5 56.9 30 46.8 32.2 8.2 9.5 7.8 4.8 11.1 6 5.6 4 2.6 0.6 11.5

4.51 4.136 5.039 2.281 2.478 1.608 2.542 2.522 2.059 1.784 1.891 1.372 1.614 1.425 1.442 1.391 1.437 0.652

1.000 1.000 1.000 1.000 0.642 0.822 0.597 0.558 0.488 0.480 0.462 0.461 0.366 0.350 0.339 0.326 0.312 0.211

1 1 0.73865 0.71966 0.63025 0.57368 0.56024 0.39824 0.38124 0.35573 0.32832 0.35526 0.26994 0.26468 0.24155 0.22187 0.1869 0.1869

1 0.87839 0.80394 0.76191 0.55449 0.5947 0.58066 0.37197 0.32796 0.28057 0.26748 0.25982 0.24456 0.21862 0.20684 0.1887 0.17689 0.17689

i = 1; 2; …; m; j = 1; 2; …; n

where EF denotes the set of entities that are currently received composite indicator value of 1 and K ∈ [0, 1] is a discriminating parameter. In this way, the model (7) finally converges to a single best entity with CI value of 1 by augmenting the value of K from zero to one with a predetermined step size like 0.1. In the next section the proposed CI construction method including the models (6) and (7) are applied for two case studies taken from the CI literature (Zhou et al., 2007b; Despotis, 2005b). 4. Case Studies: Sustainable Energy Index and Human Development Index The proposed MCDA–DEA model is applied for two case studies taken from the literature. The first case study is about the calculation of

Peru Philippines Papua New New Zealand Vietnam Canada Chile Japan Mexico Indonesia Thailand China United States Australia Malaysia Taiwan, China Korea Russia

5.159 4.964 3.577 2.231 3.134 1.747 2.817 2.103 1.853 1.585 1.511 1.467 1.382 1.235 1.169 1.066 1.000 1.000

1 0.977 0.81 0.648 0.529 0.477 0.463 0.353 0.278 0.24 0.22 0.214 0.144 0.116 0.101 0.081 0.064 0

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H1. There is a positive correlation between the CIs obtained by the proposed MCDA–DEA model and the Zhou et al's. model. According to the results, the Spearman's rank correlation coefficient is 0.993. Also, the P-value of the test is zero and hence H0 is rejected at α = 0.00001. In the other hand, there is a high positive correlation between the two sets of SEI values obtained by the proposed and Zhou et al's. methods. In the second case study we compute Human development index (HDI) based on three socioeconomic indicators which reflect three dimensions of human development: longevity, educational attainment and standard of living. Longevity is measured by life expectancy at birth (LE). Educational attainment is measured by a weighted average of the adult literacy rate (ALR) and the combined-primary, secondary and tertiary-gross educational enrollment ratios (GEER). An adjusted GDP per capita (PPP USD) is used as a measure of standard of living. Further details and definitions of these indicators can be found in (United Nations Development Programme (UNDP), 2008). In this manner, the HDI is calculated by aggregating three socioeconomic indicators, namely the life expectancy index (LEI), education index (EDI) and GDP index (GDPI). The original data for 27 economies in the Asia and Pacific region are shown in Table 2 as reported in (Despotis, 2005b). According to Table 2, the sixth and seventh columns show the gIi and bIi values computed by solving models (1) and (2). Three countries gain same performance score of “1”, namely Hong KongChina (SAR), Republic of Korea and Singapore, by model (1), while the number of efficient countries reduces to one, namely Hong KongChina (SAR), by solving the proposed model (6). Two final columns of Table 2 show final HDI values calculated by Zhou's et al. and the proposed methods. According to the results, two models introduce Hong Kong-China (SAR) as the best entity. It should be noted that in this case, the Zhou et al's. method requires solving 54 linear programming models while the proposed method requires solving only model (6), even any need to applying model (7). By a similar way, the robustness of the proposed MCDA–DEA method is tested by Spearman's rank correlation test in this case

study. The Spearman's rank correlation coefficient regarding the HDI values obtained by the proposed MCDA–DEA and Zhou et al's. methods is 0.968. Also, the P-value of the test is zero and hence H0 is rejected at α = 0.00001. Therefore there is a high rank correlation between these two sets of HDI values. Although the robustness of proposed method is verified when compared to Zhou et al. (2007b) method by Spearman's rank correlation test, but it should be noted that the Zhou's et al. method requires 2 × m LP models to be solved, which limits its desirability when compared with the proposed method where only one LP model has to be solved for each value of k until a single entity remains the best. For example, the proposed method requires the resolution of only six and one linear programs to calculate the SEI and HDI indices, respectively (K = 0, 0.05, 0.06, 0.07, 0.08, 0.09 for calculating SEI and K = 0 for HDI) while 36 and 54 linear programs were respectively solved by using Zhou et al. method. Hence, the proposed method, which is based on the successive resolutions of a common weight MCDA–DEA model with an improved discriminating power and notable saving in the number of LP models to be solved, can be considered as a convenient decision tool and a sound alternative to the DEA-like models in CI construction.

5. Concluding Remarks This paper proposes a common weight MCDA–DEA method with a more discriminating power over the existing ones that enable us to construct CIs using a set of common weights. The model discriminates entities which gain same CI score of 1 by using previously developed models in the literature. (See for example, Zhou et al., 2006, 2007b, 2010). It should be noted that there are several works in the literature proposing input–output-oriented common weight DEA models for decision problems with multiple inputs and outputs (like those of Karsak and Ahiska, 2005;2007). But, we believe that our common weight model is the first one proposed in the CI construction literature.

Table 2 Data and the HDI values for the countries of Asia and Pacific. Country

Bangladesh Bhutan Brunei Darussalam Cambodia China Fiji Hong Kong-China (SAR) India Indonesia Islamic Republic of Iran Republic of Korea Lao People's Democratic Rep. Malaysia Maldives Mongolia Myanmar Nepal Pakistan Papua New Guinea Philippines Samoa (Western) Singapore Solomon Islands Sri Lanka Thailand Vanuatu Viet Nam

LE

58.6 61.2 75.7 53.5 70.1 72.9 78.6 62.9 65.6 69.5 72.6 53.7 72.2 65 66.2 60.6 57.8 64.4 58.3 68.6 71.7 77.3 71.9 73.3 68.9 67.7 67.8

ALR

40.1 42 90.7 65 82.8 92.2 92.9 55.7 85.7 74.6 97.5 46.1 86.4 96 83 84.1 39.2 44 63.2 94.8 79.7 91.8 62 91.1 95 64 92.9

GEER

36 33 72 61 72 81 64 54 65 69 90 57 65 75 57 56 61 43 37 83 65 73 46 66 61 47 63

GDP

1361 1536 16,765 1257 3105 4231 20,763 2077 2651 5121 13,478 1734 8137 4083 1541 1199 1157 1715 2359 3555 3832 24,210 1940 2979 5456 3120 1689

Zhou et al. model

The proposed model

Model 1

Model 2

CIs

CIs

0.62687 0.67537 0.9764 0.67018 0.89 0.96838 1 0.70709 0.84109 0.84938 1 0.56282 0.91219 0.93684 0.82358 0.78667 0.61194 0.73507 0.63581 0.95649 0.88324 1 0.875 0.94206 0.90127 0.79664 0.88594

1 1.00688 1.73686 1 1.39741 1.51537 1.78337 1.18657 1.31794 1.47086 1.66464 1 1.59708 1.40333 1.11706 1.01456 1 1.09777 1.12844 1.43195 1.47377 1.77766 1.2111 1.38627 1.50604 1.32385 1.15451

0.07325 0.13311 0.94332 0.12279 0.62785 0.79278 1 0.28408 0.52119 0.62827 0.92422 0 0.78067 0.68520 0.37295 0.26531 0.05618 0.25940 0.16546 0.72594 0.66885 0.99636 0.49178 0.68028 0.71007 0.47412 0.46817

0.56074 0.59071 0.97392 0.62697 0.8863 0.96269 1 0.6887 0.83472 0.84792 0.99147 0.56074 0.90978 0.8782 0.81939 0.75961 0.5899 0.64892 0.63471 0.92707 0.88233 0.9968 0.79547 0.93894 0.89501 0.75735 0.87928

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In summary, the proposed common weight MCDA–DEA method has the following merits: 1. As all common weights must be greater than zero (i.e.,wj ≥ ε ; ∀ j), therefore, the proposed models (6) and (7) consider the impact of all sub-indicators to construct composite indicators. 2. The recent DEA-like models especially Zhou et al's. method provide two sets of weights for underlying sub-indicators namely the best and worst weights when evaluating each entity. It was discussed earlier that such weighting values are unrealistic. Instead, the proposed method obtains a set of common weights for evaluating all entities which leads to CIs calculated by similar weights which is very essential for fair comparison of entities. 3. In Zhou et al. (2007b), choosing an appropriate value for adjusting parameter λ subjectively may cause difficulty for DM since there may not exist enough objective evidences to support any subjective choices of λ values. Moreover, different values of λ may lead to distinct and misleading results for CIs, making difficulty in reaching to a final decision. While in the proposed method there is no adjusting parameter needing the DM subjective opinion. 4. The proposed model (6) has more discriminating power than the model (1) reducing the number of efficient entities which receive CI value of 1. In the cases in which model (6) provides more than one efficient entity, by assigning an appropriate value to the discriminating parameter K, model (7) converges to a single efficient entity. 5. The proposed method does not require solving 2 × m models as it is the case in the Zhou et al. (2007b). That is, by a single formulation (6), the composite indicator of each entity can be computed. In the situation which there exist several best (efficient) entities, the proposed model (7) finally converges to a single best entity by setting appropriate value for discriminating parameter. However, in the worst case of applying proposed method, the number of models required to be solved, were less than 2m in our numerical tests. The proposed MCDA–DEA method is applied for two case studies to construct sustainable energy index and Human Development Index. The robustness and discriminating power of the proposed approach are tested by using these case studies. It is illustrated that number of best or efficient entities reduces by considering the common weight approach rather than those of DEA-like models. Also, the proposed method and Zhou et al's. model introduces a same entity as a best entity in two case studies which indicate that the proposed method can yield CIs at least as good as those of obtained by Zhou et al's. method with less computation efforts as well as considering common weights in constructing the more fair CI values for all entities. Finally, Spearman's rank correlation test reveals that there is high rank correlation between CIs obtained by the proposed and Zhou et al's. methods. Acknowledgement This study was supported by the University of Tehran under the research grant no. 8109920/1/07. The authors are grateful for this financial support. We are also grateful to the anonymous reviewers for their valuable comments and constructive criticism. References Ali, A.I., Seiford, L.M., 1993. Computational accuracy and infinitesimals in Data Envelopment Analysis. INFOR 31, 290–297. Allen, P., Athanassopoulos, A., Dyson, R.G., Thanassoulis, E., 1997. Weights restrictions and value judgements in Data Envelopment Analysis: evolution, development and future directions. Annals of Operations Research 73, 13–34. Amin, G.R., Toloo, M., Sohrabi, B., 2006. An improved MCDM DEA model for technology selection. International Journal of Production Research 44 (13), 2681–2686. Banker, R.D., Charnes, A., Cooper, W.W., 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30, 1078–1092. Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the efficiency of decisionmaking units. European Journal of Operational Research 2, 429–444.

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