Introduction

Data Envelopment Analysis (DEA) was originally proposed by Charnes et al (1978) as a method for evaluating the relative efficiency of DMUs, which use multiple inputs to produce multiple outputs. The different models for DEA seek to determine the DMUs that form the efficient frontier (or envelopment surface). Such DMUs are named efficient. DEA also provides measures for the relative efficiency of the remaining DMUs (the inefficient ones), which depend on the DEA model used.

There are three main DEA models: the CCR (Charnes et al, 1978), BCC (Banker et al, 1984) and additive (ADD) models (Charnes et al, 1985). This paper focuses on the additive model, whose underlying idea is the maximization of the L₁ distance of the analysed DMU to the efficient frontier. This model assumes variable returns to scale (VRS).

Three problems can be enumerated regarding the DEA additive model. There exists a scale problem, because all DEA models are projection mechanisms and the projections of the inefficient DMUs on the efficient frontier depend on the scales used to measure each input or output. A second problem is that the efficient measurement is very pessimistic, since the L₁ distance is being maximized. Finally, this efficiency measure does not have an intuitive interpretation.

This work will consider a variant of the additive DEA model, Ali et al (1995), with oriented projections, which we name 'weighted additive model'. This variant uses weights to differentiate the multiple factors (inputs and/or outputs). Setting the values of these weights is often considered a difficult task, but there exists much research on this topic in the multiple criteria decision analysis (MCDA) literature.

Several authors refer to links between DEA and MCDA. For instance, Joro et al (1998) and Halme et al (1999) relate DEA with multiple objective linear programming. Bouyssou (1999), Doyle and Green (1993) and Stewart (1996) relate DEA and discrete multiple criteria problems. Athanassopoulos and Podinovski (1997) find relationships between DEA and MCDA with imprecise information.

We will consider that the DMUs constitute the set of alternatives of a multiple criteria evaluation model, each alternative being evaluated in a number of distinct criteria. Each criterion corresponds to an input or an output factor in DEA models. A direction of preference is associated with each criterion: increasing for outputs and decreasing for inputs.

This work intends to overcome the previously mentioned disadvantages of the additive model. It proposes an approach where additive utility functions are used to aggregate the utilities associated with each alternative, based on the multiattribute utility theory (MAUT) (Keeney and Raiffa, 1976). This overcomes the problem of the scales, since all the input and output measures are translated into utility units. Moreover, the weights used in the aggregation gain a specific meaning: they are the scale coefficients of the utility functions and determine the direction of projection. Weights are chosen to benefit each DMU as much as possible, in the optimistic spirit of BCC models. Finally, the efficiency measure of each DMU will have an intuitive meaning: the 'min–max regret'.

This paper is structured as follows. In the next section, we make a brief description of DEA models, focusing on the additive DEA model and the 'weighted additive model'. A simple example with two outputs is presented, where the additive and 'weighted additive' projections are shown. Then, we review some concepts and the notation used in utility theory that is used by the method. Another section introduces a two-phase method and its application to the example introduced previously. Subsequently, we provide another example with real world data using the additive model and the two-phase method. Concluding remarks are presented in the final section.

Data Envelopment Analysis

Let us consider n DMUs to be evaluated, each of them consuming m inputs to produce p outputs. The input data are denoted by the matrix X_{m n} and the output data are denoted by the matrix Y_{p n}. We denote by x_j (the jth column of X), the vector of inputs consumed by DMU j, and by x_ij>0, the quantity of input i consumed by DMU j. A similar notation is used for outputs y_ij>0. We denote by 1 the summation vector (1,...,1)^T.

The primal and dual (envelopment and multiplier forms) linear programming formulations, to evaluate the efficiency of DMU k, for the (input-oriented) CCR model are the following:

The primal and dual (envelopment and multiplier forms) linear programming formulations, to evaluate the efficiency of DMU k, for the (input-oriented) BCC model are the following:

The primal and dual (envelopment and multiplier forms) linear programming formulations, to evaluate the efficiency of DMU k, for the additive (VRS) model are:

Each of these models seeks to determine the efficiency of a particular DMU (X_k,Y_k) regarding the efficient frontier determined by the efficient DMUs. The solution of a DEA model for DMU (X_k,Y_k) results in a measure of efficiency and an efficient projection point, (_k,_k), on the efficient frontier. The efficiency evaluation and the location of this efficient projected point depend on both the envelopment surface and the evaluation system of the chosen DEA model.

The primal formulations show that CCR and BCC models exhibit only one difference: the latter includes a 'convexity' constraint (1=1). Because of that, if a DMU is characterized as efficient in the CCR model, it will also be characterized as efficient with the BCC model; the contrary does not necessarily hold.

We can also observe that placing =1 (and =1) in the BCC model yields the additive model (ADD). The distinction between the ADD model and the oriented models (in this case input oriented) lies in the fact that the latter have a two-stage process: the maximal reduction of inputs is achieved first, via the optimal ^*; then, in the second stage, the movement towards the efficient frontier is achieved via the slack variables (e and s). The ADD model uses the second stage only. The solution in the first stage of the oriented models allows obtaining an inefficiency measure, between 0 and 1, for the inefficient DMUs with respect to the orientation used. The ADD model does not return a one-dimensional inefficiency measure for the inefficient DMUs, but it returns a non-positive value, which allows checking of the relative efficiency of the DMU under analysis. If the obtained value is negative, then the DMU under analysis is operating inefficiently in some factors. This value is symmetric to the sum of the distances of each dimension up to the envelopment surface (L₁ distance).

The type of envelopment surface, commonly referred to as constant returns-to-scale (CRS) or VRS, results from the particular DEA model chosen. The CCR model produces a CRS envelopment surface, while the BCC model and the additive model produce a VRS envelopment surface. The envelopment surfaces for the BCC and additive DEA models are identical, whereas the objective function values (efficiency scores) and, more importantly, the projections on the efficient frontier differ. Thus, the sets of DMUs determined to be efficient are exactly the same for both models. Differences in the actual efficiency scores or projections simply reflect the metrics used in the two models.

Considering the primal problem for DMU k, the optimal vector _j^*0, j where _j=1ⁿ_j^*=1 defines a projected point (_k,_k)=(_j=1ⁿ_j^*X_j,_j=1ⁿ_j^*Y_j) on the efficient frontier. If the projected point coincides with the point under analysis, then the DMU is efficient and belongs to the efficient frontier. Otherwise, DMU k is inefficient and the corresponding projected point, which is on the efficient frontier, can be obtained as a linear combination for the CRS model, or a convex combination for the VRS model, of DMUs that lie on the efficient frontier.

The DEA additive model

The additive model (ADD) maximizes the L₁ distance of the DMU under analysis to the projected point on the efficient frontier.

The projected point (_k,_k) and the point (X_k,Y_k) under analysis coincide if and only if the optimal value of the primal ADD model is null and the slack values are also null. Thus, the DMU k is efficient if and only if the optimal value of the primal ADD model is null. The DMU k is inefficient if it does not lie on the frontier, that is, if any component of the slack variables, e or s is not zero; the values of these nonzero components identify the sources and amounts of inefficiency in the corresponding outputs and inputs.

This process is repeated n times, once for each DMU k to be evaluated. The objective function values obtained effectively partition the set of DMUs in two subsets: the DMUs for which the optimal value is zero are efficient and determine the envelopment surface, while the DMUs for which the optimal value is negative are inefficient and lie beneath the surface.

An optimal comparison point (_k,_k) on the envelopment surface is associated with each inefficient DMU k, (X_k,Y_k), which usually may be expressed as a convex combination of the DMUs, that is, (_k,_k)=(_j=1ⁿ_j^*X_j,_j=1ⁿ_j^*Y_j)=(X_k-e,Y_k+s), with _j=1ⁿ_j^*=1. Optimal values for the slacks variables obtained from solving the primal problem ADD_P measure the L₁ distance from (X_k,Y_k) to this projected point (_k,_k), on the frontier. These slacks measure the surplus of inputs, e, and the deficit of outputs, s, with respect to an efficient point.

The dual problem ADD_D yields an alternative geometric interpretation. Here one seeks the closest supporting (facet-defining) hyperplane, that is, Y_k-X_k+_k=w_k, with maximal value. An efficient point (X_k,Y_k) will lie on the facet-defining hyperplane with equation ^*Y_k-^*X_k+^*=0, where the superscript (*) denotes an optimum value.

The envelopment surface, determined by the dual ADD problem, is defined by the supporting hyperplanes in R^m+s that form the facets of the convex hull associated with the points representing the set of DMUs (X_j,Y_j) j=1,...,n. The supporting hyperplane Y-X+=0 intersects the convex hull in a facet of the surface if and only if all the points (X_j,Y_j) belong or are involved by it and, besides, at least one of these points belongs to it.

The first constraints of the dual ADD problem guarantee that all the points belong to the envelopment surface or are part of its interior. The maximization of the objective function seeks to find the set of (_k,_k,_k) that minimizes the distance of the DMU k under analysis to the envelopment surface. Due to these constraints the objective function value is non positive, and a null value indicates that the DMU k belong to a hyperplane, which characterizes one facet of the envelopment surface. The inefficient DMUs belongs to the closest supporting (facet-defining) hyperplane and, because of that, the objective function returns a negative value.

The distance between the hyperplane of DMU k and the frontier supporting hyperplane is given by _k=-_ks-_ke, hence the points (_k,_k) and (X_k,Y_k) belong to parallel hyperplanes which are _k distant.

Let us consider an illustrative example with eight DMUs evaluated considering two outputs (see Table 1).

Table 1 - Data for the illustrative example.

Full table

The results displayed in Figure 1 are obtained by solving the ADD problem with the data in Table 1. In Figure 1 it is possible to recognize that DMUs 1, 2, 3 and 4 are efficient. The envelopment surface is given by , and . DMUs 5, 6, 7 and 8 are inefficient and are enveloped by that surface.

Figure 1.

Example of the ADD model.

Full figure and legend (8K)

The additive model returns the projected points based on the maximum L₁ distance between the DMU under analysis and the efficient frontier. For example, DMUs 5 and 6 have their maximum L₁ distance to the envelopment surface in DMU 2.

It is important to emphasize that the projections depend on the scales. For instance, if the scale of Y₂ was multiplied by a factor of 2, then the results would be completely different (see Figure 2).

Figure 2.

Example of the ADD model with the scale of Y₂ multiplied by a factor of 2.

Full figure and legend (11K)

Ali et al (1995) presented an additive model with oriented projections. The primal and dual (envelopment and multipliers forms) linear programming formulations, to evaluate the efficiency of DMU k, for this 'weighted additive model' are:

The projection mechanism is similar to the ADD model, but new parameters u^k,v^k are introduced. The vectors (u^k,v^k) are the coefficients of the objective function, in the primal formulation, and thus are a component of the evaluation mechanism. They provide and determine the directions of the projection. Setting equal weights (eg, u^k=1 and v^k=1) in the primal 'weighted additive' problem yields the additive model. In Ali et al (1995), u^k and v^k are considered as given constants. (Note that if the vectors (u^k,v^k) were also a variable of the problem, then the model would become non-linear.) Considering the data in Table 1 and solving the primal 'weighted additive' problem, with the vector (u₁^k,u₂^k)=(1,0), yields the results displayed in Figure 3.

Figure 3.

Example of the 'weighted additive model' using (u₁^k,u₂^k)=(1,0).

Full figure and legend (7K)

Solving the same problem for each DMU in Table 1, but now considering the vector (u₁^k,u₂^k)=(0,1), yields the results depicted in Figure 4. As can be seen in Figures 3 and 4, the directions of projection of inefficient units are different in both cases. We could also obtain other projected points if other vectors (u₁^k,u₂^k) were used, that is, intermediate points between the two situations depicted in Figures 3 and 4. The 'weighted additive model', when compared to the ADD model, has the advantage of allowing the flexibility of changing the projection direction, which may be less pessimistic than the one proposed by the ADD model. However, a question raised by this approach is: 'how to choose the vectors (u^k,v^k)?' This question will be addressed with the presentation of a two-phase method.

Figure 4.

Example of 'weighted additive model' using (u₁^k,u₂^k)=(0,1).

Full figure and legend (8K)

MCDA with utility functions

MCDA is a field where concepts of efficiency have also been developed, although with a different meaning from DEA (Bouyssou, 1999). In MCDA, an alternative is efficient or non-dominated if there is no other alternative that is (strictly) better at some criteria without being worse at some other criteria. An alternative/DMU may be efficient according to MCDA but inefficient according to DEA, in case it is dominated by a convex combination of two other alternatives/DMUs.

The method we will present appeals to concepts from the MAUT within MCDA that we briefly review in this section. MAUT has been developed as a normative theory about the behaviour of a rational decision maker (DM) in a decision situation under uncertainty with multiple criteria (see Keeney and Raiffa, 1976; von Winterfeldt and Edwards, 1986; Goodwin and Wright, 1998). In MAUT, the concept of attribute is equivalent to our concept of criterion.

MAUT is used to evaluate the utility of each alternative, assuming that each one is as better as greater as is its utility. MAUT aims to simplify the formulation of multiattribute utility functions by allowing the DM to focus his/her attention on one attribute at a time, and subsequently on the aggregation of attributes, instead of evaluating directly the global utility. Note that, although we only mention utility functions, we may as well work with value functions, see von Winterfeldt and Edwards (1986).

Let A={a₁,a₂,...,a_n} denote a set of alternatives to be evaluated according to q criteria. Let u_j(a_i) denote the utility of an alternative a_i according to the jth criterion (j=1,...,q). The additive utility function aggregates these functions through a weighted sum where q scale coefficients w₁,...,w_q are used: u(a_i)=_j=1^qw_ju_j(a_i), where w_j0, j=1,...,q and _j=1^qw_j=1 (by convention).

Obtaining precise values from the DM for these scale coefficients is often considered very difficult; hence there are many studies on additive utility functions that accept imprecise information from the DM, for example, Athanassopoulos and Podinovski (1997), Miettinen and Salminen (1999), Salo and Hämäläinen (2001), Dias and Clímaco (2000). In most of these studies, the imprecise information is related to the scale coefficients, which are generally the most difficult parameters to elicit. Such approaches consider a set of acceptable vectors for the scale coefficients, rather than a single precise vector. Hence, the global utility of each alternative is no longer precisely determined.

In this context of additive aggregation with imprecise weights, the ideas of using the well-known max–min and min–max regret rules have been proposed as a way to compare the alternatives (eg Dias and Clímaco, 2000; Salo and Hämäläinen, 2001;). The max–min rule compares the alternatives based on the worst utility they may attain given the acceptable values for the scaling coefficients. Alternatives with higher worst-case utility are preferred. The min–max regret rule compares the alternatives based on the greater difference, taking into account all acceptable vectors of the scaling coefficients, to the optimal alternative for that vector. The maximum regret associated with a given alternative a_k, denoted R_max(a_k), is the maximum loss of opportunity associated with choosing that alternative:

Alternatives with the minimum R_max are preferred.

The aim of the approach presented in the next section is to find the scale coefficients w that, for each alternative minimize the utility difference to the best alternative, according to the min–max regret rule.

A two-phase method

This section presents a two-phase method that overcomes some of the disadvantages of the additive model. Let us return to the 'weighted additive model'. In the primal problem, the vectors (u^k,v^k) are the coefficients of the objective function, which provide the direction of the projections. Observing this model, one may ask the question of how to choose the vectors (u^k,v^k). To answer this question, we are going to consider the set of DMUs as being a set of alternatives to be evaluated according to an additive MAUT model. The utility of each DMU is

where the scale coefficients w₁,...,w_q are the weights of the utility functions.

In MAUT, a preference direction is associated with each criterion's original scale: increasing (criterion to maximize) or decreasing (criterion to minimize). These scales have to be converted into utility scales, using an appropriate questioning procedure (see von Winterfeldt and Edwards, 1986). The utilities of the DMUs in every criterion are treated as outputs, without loss of generality. Indeed, utilities are always to maximize: for input factors, the lower the consumption, the higher the utility; for output factors, the higher the production, the higher the utility.

In what follows, let d denote the distance defined by the utility difference to the best of all alternatives. The purpose of our approach is, for DMU k, to calculate the vector w of utility function weights that minimizes the distance (the utility difference) of this unit to the best one (note that the best alternative will also depend on w). For that purpose, the following problem is solved:

which can be written as:

The variables of the problem are the weighting vector w and the efficiency measure d.

If the optimal value d^* of the objective function is null, then the DMU k under evaluation is efficient, otherwise it is inefficient and d^* is the minimum difference of utility to the best DMU (ie the DMU with higher global utility). Let us note, however, that d^* can be null for a DMU that would be deemed as inefficient in the additive model (and dominated in MCDA). For instance, in Figure 4 a DMU with utilities (2,8) would yield d^*=0 for w=(0,1), although it is dominated by DMU 4 with utilities (3,8). Such DMUs would be called 'weakly efficient' in MCDA language and can be easily detected. This effect is natural if null weights are accepted, which means accepting that some criteria might be deleted. If this is considered undesirable, then the analysts may choose to constrain the weights to positive values or to solve a second linear program to maximize the minimum weight subject to the constraints (2) with d=0.

The problem (1) is solved in phase 1 of our method. Then, the 'weighted additive' problem can be solved using the optimal weighting vector w^* obtained in phase 1 to compute the projected point:

Phase 1: Convert inputs and outputs in utility scales and compute the efficiency measure d of each DMU and the vector of weights associated with this utility difference (linear program (2)).

Phase 2: Solve the 'weighted additive' problem, using the weighting vector resulting from phase 1, and determine the corresponding projected point of the DMU under evaluation. Since utilities are seen as outputs, the primal formulation becomes (the variables are and s):

The approach will now be illustrated using the same example described before, considering the data in Table 1. To allow the results to be compared to the ones presented before, we will convert the values in Table 1 into 'utilities' by dividing them by 10. In practice, however, a proper conversion would have to be made by an elicitation process, see von Winterfeldt and Edwards (1986).

The application of phase 1 of the two-phase method leads to the results in Table 2. In phase 2, the problem to solve is the 'weighted additive', using the weighting vector (w₁^*,w₂^*) in the objective function, and the results are described graphically in Figure 5.

Figure 5.

Illustrative example.

Full figure and legend (11K)

Table 2 - Efficiency measure and weights, for each DMU.

Full table

Table 2 gives, for each of the eight DMUs, the utility difference d^* and the weights obtained in phase 1 of the method by solving problem (1) for each DMU. The projections of inefficient DMUs 5, 6, 7 and 8 on the efficient frontier using the weighting vector resulting from phase 1 are displayed in Figure 5 (but multiplied by 10).

The efficient DMUs are obviously the same as those that have already been attained using the ADD model, but now the DMUs 5 and 7 are projected onto the efficient DMU 3. The DMU 6 is projected onto the DMU 1, and DMU 8 is projected on the line that connects the DMUs 3 and 4. The dotted lines in Figure 5 are parallel to the lines that connect the DMUs where the projection is going to be made. Thus, these dotted lines are orthogonal to the weighting vector that resulted from phase 1, for each DMU.

Some remarks can be made from the comparison of the ADD model with the proposed two-phase method:

• The values of the objective function given by the ADD model, displayed in Table 3, are the weighted L₁ distances considering equal weights. When the DMUs are able to choose their weights the efficiency measures become those in Table 2, which are naturally lower (at most equal), due to the pessimistic nature of L₁ distance.
• The comparison of Tables 2 and 3 shows a rank reversal concerning DMUs 7 and 8. In the two-phase method, DMU 8 appears closer to efficiency than DMU 7. As we can observe in Figure 5 (two-phase method), if DMUs 7 and 8 could choose their weights, the DMU 7 is never closer to the efficient frontier than DMU 8. Using the ADD model the projections are different, the variables are also the weights but the metric L₁ leads to the type of projection that is displayed in Figure 1.
• We can observe that if DMU 8 chooses its weights, the utility difference to the best one (the one that has the higher utility) is smaller if its projection is made onto the line instead on the line .
• The DMUs 5 and 6 are projected onto DMU 2 in Figure 1, whereas in Figure 5 the DMU 5 is projected onto DMU 3, and the DMU 6 is projected onto DMU 1.

Table 3 - Objective function and slack values for each DMU, according to the ADD model.

Full table

An example with real world data

In this section, the intention is to compare the results using the ADD model and the two-phase method, using an example with 12 DMUs and three factors in the efficiency evaluation of electricity distribution companies. The data have been collected from OFGEM's 2003 report 'Background to work on assessing efficiency for the 2005 distribution price control review', (http://www.ofgem.gov.uk/temp/ofgem/cache/cmsattach/4636_back
ground_cepa_report_and_efficiency_dpcr300903.pdf, consulted in 28/06/2006) and concern the period 2002/2003 (Table 4).

Table 4 - Utilities for the illustrative example.

Full table

For the purpose of this illustration, the factor data are already converted into utilities, using a linear transformation. In practice, a DM would have the opportunity to consider, for instance, decreasing marginal utilities. The DMUs utilities in every criterion are treated as outputs, without loss of generality.

The results in Table 5 are obtained by solving the linear program (2), referred to in the previous section. Table 5 gives, for each of the 12 DMUs, the efficiency measure d^* and the weights obtained in phase 1 of the method. The efficient DMUs are 1, 5, 6, 9, 10 and 12 (the value of the objective function is null).

Table 5 - Efficiency measure and weights, for each DMU.

Full table

The projections of inefficient DMUs, 2, 3, 4, 7, 8 and 11, on the efficient frontier using the weighting vector resulting from phase 1 are displayed in Table 6 (phase 2).

Table 6 - Objective function and slack values for each DMU.

Full table

Some remarks can be made from the comparison of the DMUs' utilities with the projection returned by the two-phase method. Observing Table 7 we conclude that:

• The projection of DMU 2 and DMU 4 has the same utility than the efficient DMU 1. Therefore, the DMUs 2 and 4 are projected onto DMU 1.
• The DMUs 3, 7 and 11 are projected near the efficient DMU 6, because the values of utilities are the closest. The efficient DMU 6 is the reference to the inefficient DMUs 3, 7 and 11.
• The efficient DMU 10 is the reference to the inefficient DMU 8, because the utility value of the projection of DMU 8 is near to the utility value of the efficient DMU 10.
• The rank of the inefficient DMUs starts with DMU 3 and continues with DMUs 11, 4, 8, 2 and finally, with the greater utility difference, DMU 7.
• Note that the utility difference is the efficiency measure d^*, that was obtained in phase 1.

Table 7 - DMUs' utilities (weights in Table 5).

Full table

The results corresponding to the traditional additive DEA model (the ADD model) are displayed in Table 8. The values of the objective function given by the ADD model are weighted L₁ distances, considering all weights equal to 1, although we replace those weights by the weighting vector . This leads to the same solutions, but the L₁ distance values appear divided by a factor of 5.

Table 8 - Objective function and slack values for each DMU, according to the ADD model using .

Full table

The efficient DMUs are obviously the same as those that have already been obtained using the two-phase method, but now it is interesting to compare the values of the objective functions in both models.

If we compare the ADD model with the two-phase method, we have lower objective function values attained in phase 1 of the two-phase method, compared with the L₁ distances returned with the ADD model. Therefore, when the DMUs are able to choose their weights the efficiency measures become those in Table 5, and similarly to the example presented in previous section, the values of the objective functions are lower (at most equal) than in the ADD model. This only emphasizes the pessimism of the L₁ metric used in the ADD model.

The comparison of Tables 5 and 8 also shows a rank reversal concerning DMUs 4 and 8 (third and fourth positions). In the two-phase method, the DMU 4 is closer to efficiency than DMU 8; this is because the DMUs are free to choose their weights and they do not have to use a specific metric to project themselves onto the efficient frontier in a way that minimizes the distance to the best one (the DMU with higher global utility).

Table 9 shows the utilities of DMUs and the utilities of the projected DMUs obtained using the ADD model with all weights equal to . Observing Table 9 we conclude that:

• All DMUs have utilities different from the ones observed in Table 7, because in this case we use the weighting vector .
• The utility difference is higher in Table 9, comparing with the values in Table 7. That fact only reinforces the idea that the L₁ metric used in the ADD model is very pessimistic.
• The projections of inefficient DMUs are different, that is, the inefficient DMUs are projected in other directions, when compared with the ones that we observe in Table 7. For example, the projection of DMU 8 has the same utility than the efficient DMU 1, that is, the DMU 8 is projected onto DMU 1; The DMU 3 is projected near the efficient DMU 10, because the values of utilities are the closest.
• The rank reversal concerning DMUs 4 and 8 is also visible when we compare the utility difference in Tables 7 and 9.

Table 9 - DMUs' utilities (using weights .

Full table

Concluding remarks

This work proposes a two-phase method to overcome some problems usually found in the ADD DEA model: the problem of scales, the pessimistic nature of its efficiency measure, and the lack of an intuitive interpretation for this measure.

The proposed method exploits links between DEA and MCDA (namely multiattribute utility models with imprecise information), where DMUs play the role of decision alternatives. The consumption of inputs and the production of outputs are converted into multiple utility functions that are aggregated using a weighed sum (MAUT's additive model). The main idea of this approach is to let each DMU choose the weights associated with these utility functions in a way that minimizes the difference of utility to the best DMU (min–max regret rule).

In a second phase, the 'weighted additive' DEA model of Ali et al (1995) can then be solved to obtain a projection onto the efficient frontier. This phase uses the weights obtained in the previous one, rather than being arbitrarily set.

The two-phase approach overcomes the scales problem, since all the inputs and outputs are converted into utilities. For instance, in the ADD model, if an input expressed in Tonnes has its scale changed to kg, then the results are likely to change. However, when these units are converted into utilities in the first phase of the method, then the utility of 1 Tonnes is equal to the utility of 1000 kg. The conversion to utilities, although may represent an increase in the effort required for the analysis, introduces a new dimension of flexibility to the model, allowing to incorporate subjective assessments of how much satisfaction corresponds to increases in outputs and decreases in inputs (eg, stating that an increase from 1 to 2 Tonnes brings higher satisfaction than an increase from 1000 to 1001 Tonnes). The conversion of the original scales to utility functions also presents the advantage of giving a clear meaning to the weights. Such weights represent scaling factors or 'price coefficients' related to preference trade-offs among single-criterion utility functions. Therefore, a DM may meaningfully indicate some constraints on such weights.

The two-phase approach also overcomes the pessimistic character of the DEA ADD model's efficiency measure, and the lack of an intuitive interpretation for this measure. Indeed, the DMUs choose their weights in a way that minimizes the distance to the best alternative, and such a distance has a natural 'maximum regret' interpretation.

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Acknowledgements

The authors wish to thank the remarks of an anonymous referee. This research was partially supported by FCT/FEDER Grants POSI/SRI/37346/2001 and POCI/EGE/58371/2004.