- Email: [email protected]
Radial DEA models without inputs or without outputs
European Journal of Operational Research 118 (1999) 46±51
www.elsevier.com/locate/orms
Theory and Methodology
Radial DEA models without inputs or without outputs C.A. Knox Lovell b
a,*
, Jes us T. Pastor
b,1
a Department of Economics, University of Georgia, Athens, GA 30602, USA Vicerector de Investigaci on, Universidad Miguel Hernandez, Paseo Melchor Botella s/n, E-03206 Elche (Alicante), Spain
Received 25 August 1997; accepted 28 August 1998
Abstract In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial eciency score) and one less constraint (the convexity constraint). Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: DEA; Models without inputs (outputs) or with a single input (output)
1. Introduction DEA was initially developed by Charnes et al. (1978) (CCR) and by Banker et al. (1984) (BCC) for the purpose of evaluating the relative eciency of similar economic production systems. However Adolphson et al. (1991) noted that it is possible to adopt a broader perspective, in which DEA is also appropriate for comparing any set of homogeneous units on multiple dimensions. This broader perspective was perhaps ®rst adopted by Thompson et al. (1986), in an attempt to determine the optimal location of a superconducting supercol-
* 1
Corresponding author. E-mail: [email protected] E-mail: [email protected]
lider in the state of Texas. The authors considered six potential sites, and speci®ed three inputs (facility cost, user cost and environmental damage). They used an input-oriented CCR model, and set the output of each site equal to unity, since the output of each site is one superconducting supercollider. Adolphson et al. claim that it is possible, but not helpful, to interpret this problem as an economic production system. We show in this paper that when using the CCR model, as Thompson et al. did, such an interpretation is really necessary. Adolphson et al., went on to propose solving the superconducting supercollider problem as a pure input model (with no outputs). However instead of using an input-oriented CCR model, they proposed to use an input-oriented BCC model.
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 3 8 - 5
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
They justi®ed the model change on technical grounds: the presence of the convexity constraint in the BCC model ensures that the optimal eciency scores are positive. But they were unable to explain why they reached the same results as Thompson et al. In this paper we show why a CCR input-oriented (output-oriented) model without outputs (inputs) makes no sense. We also show that a pure input (output) DEA model is possible only if we resort to the corresponding BCC model. This apparently broader point of view matches with the classical economic point of view, since we also prove that a BCC model without inputs (outputs) is equivalent to the corresponding CCR model with a single constant input (output). This explains why Thompson et al. and Adolphson et al. reached the same result when solving the superconducting supercollider problem with ostensibly dierent DEA models: an input-oriented CCR model with a single constant output is equivalent to the corresponding BCC model with no outputs. We also provide a reformulation of all BCC models, including the simpli®ed versions mentioned above. The new formulation has one less variable (the radial eciency score) and one less constraint (the convexity constraint). Application of this new formulation to an output-oriented BCC model with a single constant input shows clearly that all that matters is the output values, i.e., the model can be considered as a pure output model. We restrict our attention to the one-stage version of the envelopment form of each radial DEA model, i.e., the version without the non-Archimedean element and without slacks in the objective function. In other words, if we solve the envelopment problem of a radial DEA model by means of a two-stage procedure (see, for instance, Lovell and Pastor, 1995), we are considering here only the ®rst stage, the one which gives us the value of the radial eciency score. The second stage is not considered because we are not primarily interested in the values of the slacks. This approach has numerous precedents in the literature (see, e.g., Seiford and Thrall, 1990). The paper unfolds as follows. In Section 2 we show that a CCR model without inputs (or out-
47
puts) is meaningless. In Section 3 we prove that a CCR model with a single constant input (output) collapses to the corresponding BCC model. In Section 4 a BCC model with a single constant input (output) is shown to coincide with a BCC model without inputs (outputs). In Section 5 we oer a new formulation of BCC models, having one less variable (the radial eciency score) and one less constraint (the convexity constraint), and we show how this reformulation applies to a BCC model without inputs (outputs). Section 6 concludes.
2. A CCR model without inputs (or without outputs) is meaningless We begin by considering CCR models, both output-oriented and input-oriented. We prove that an output-oriented (input-oriented) CCR model without inputs (without outputs) is unable to distinguish between ecient and inecient units. It is in this sense that we refer to such a model as being meaningless. Proposition 1. An output-oriented (input-oriented) CCR model without inputs (without outputs) rates all DMUs as in®nitely inecient. Proof. An output-oriented CCR envelopment problem without inputs is formulated as max / /;k
s:t: Y k P /Yo ; k P 0n ; where Yo (Yo1 ,. . .,Yom ) is an m ´ 1 output vector of the unit being evaluated, Y is an m ´ n matrix of output vectors of the n units in the sample, k (k1 ,. . .,kn ) is an n ´ 1 vector of intensity variables, and n is the number of units in the sample. Since {k0 1; kj 0, 8j 6 0} is a feasible solution with / 1, it is clear that we can consider a sequence of feasible solutions {k0 k; kj 0, 8j 6 0} with / k, and consequently max / +1. This means that all units are assigned the worst possible eciency score (if we admit +1 as an eciency
48
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
score), and all units are in®nitely inecient. As is well known in linear programming theory, the corresponding dual multiplier problem is infeasible and, consequently, no shadow prices are available in this case. For the case of an inputoriented CCR model without outputs, a similar proof assigns to each unit the worst possible eciency score (h 0), and therefore this model is also meaningless. (An alternative and more direct proof is obtained by considering the dual multiplier problem in each case.)
where e is an n ´ 1 vector of 1s and all other variables are as previously de®ned. Assume that (/ , k ) is an optimal solution to the problem. It suces to show that eT k 1. This is accomplished by proving that the strict inequality eT k < 1 leads to a contradiction. In fact, if the strict inequality holds, and knowing that eT k is positive because k P 0n , we can build a new feasible solution kf k /eT k satisfying Y kf Y k =eT k P / Yo =eT k / =eT k Yo > / Yo :
3. An output-oriented (input-oriented) CCR model with a single constant input (output) Having considered and rejected as meaningless CCR models having no inputs or no outputs, we now consider CCR models having a single constant input or a single constant output. Such a model has been considered by Thompson et al. (1986). We prove that such models are indistinguishable from the corresponding BCC models. This explains why a graphical display of a CCR output-oriented model with one constant input and two outputs is exactly the same as a graphical display of the corresponding BCC model. Proposition 2. An output-oriented CCR model with a single constant input and an input-oriented CCR model with a single constant output coincide with the corresponding BCC models. Proof. Let us ®rst consider an output-oriented CCR model with a single constant input. We can always assume that the constant input is at level 1 by changing its scale of measurement. This change can only in¯uence the value of the slacks, unless a normalized model is considered as proposed by Lovell and Pastor (1995). The envelopment form of the model is max / /;k
s:t:
Y k P /Yo ; eT k 6 1; k P 0n ;
Consequently, /f / /eT k > / shows that (/f , kf ) is a strictly better solution than (/ , k ), i.e., (/ , k ) is not optimal. For the input-oriented CCR model with a single constant output, the proof is completely similar. The same proof goes through for a CCR model with multiple constant inputs (or multiple constant outputs). We always end up with the corresponding BCC model. 4. A BCC model without inputs (or without outputs) It makes no sense to speak of an input-oriented model without inputs. Therefore, an output orientation is adopted when we consider a BCC model without inputs. Due to the presence of the convexity constraint, the presence or absence of a single constant input in an output-oriented BCC model is irrelevant, as the following proposition shows. Proposition 3. An output-oriented (input-oriented) BCC model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Proof. We can always assume that the constant input is at level 1, since a rescaling of any variable does not aect the optimal eciency score obtained by means of any radial DEA model. Therefore the restriction associated with a single constant input is eT k 6 1. The presence of the convexity constraint eT k 1 converts the preced-
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
ing restriction into a redundant restriction and so it can be deleted. The following result is a direct consequence of Propositions 2 and 3. Corollary 3.1. An output-oriented (input-oriented) CCR model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Hence the ®nal formulation of the envelopment problem for an output-oriented BCC model without inputs is max /;k
s:t:
/ Y k P /Yo ; eT k 1; k P 0n :
From an economic point of view it is dicult to accept a DEA model without inputs. However Lovell and Pastor (1997) have recently considered such a model for evaluating the performance of branch oces in relation to the target setting procedure established by a large Spanish savings bank. The outputs were the ``percentage of target coverage'' by each branch oce for a set of 17 dierent targets, and we considered no resource as an input. Hence we are facing a pure output model. Nevertheless, from a production perspective we can argue that each branch oce is by itself ``the input'' and, therefore, a single constant input was at hand. As a consequence of the theory developed above, we know that only one radial DEA model can be used with this data set: an output-oriented BCC model without inputs. This is exactly the model we used. In these special circumstances scale eciency is not a major issue. In fact, in this particular case and as a consequence of Corollary 3.1, we know that scale eciency is equal to 1 for every branch oce. The fact that an output-oriented BCC model without inputs coincides with an output-oriented CCR model with a single constant input implies that returns to scale are globally constant, not in branch oce service provision, but in branch of-
49
®ce target coverage. Size is irrelevant in this model. One may think that we may lose some dual information when considering a BCC model without inputs instead of a BCC model with a constant input, since in the former case we are unable to obtain a shadow price for the constant input. Fortunately, if we resort to any DEA software package, we always work with the BCC model with a constant input. Moreover, since we have shown that the latter model is just the same as the corresponding CCR model, the ratio form of the CCR model tells us that we can always assign a shadow price of 1 to the constant input and get relative prices for the outputs. Going back to our starting point, since prices are relative, we can simply consider the prices associated with outputs in the BCC model without inputs, and gather the same information with regard to outputs as with the more complex model. A ®nal point is worth mentioning. It is completely equivalent to deal with a radial DEA model with a single constant input (output) or with multiple constant inputs (outputs). By rescaling all the constant inputs (outputs) we can consider that all of them are at level 1, and, consequently, the corresponding linear programming restrictions collapse to a single restriction.
5. A new formulation of BCC models We focus our attention on an output-oriented BCC model. (A similar treatment is valid for an input-oriented BCC model.) Its envelopment problem is formulated as max /k
/
s:t: Y k P /Yo ; X k 6 Xo ; eT k 1; k P 0n : Since / P 1, we can consider a new variable h 1/ /. Substituting h for / in the BCC envelopment problem, we get
50
min h;k
s:t:
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
h Y k P 1=hYo ; X k 6 Xo ; eT k 1; k P 0n :
Multiplying the ®rst set of restrictions by h > 0, and considering a new vector of intensity variables s hk, we obtain the reformulation min h;s
s:t:
h Y s P Yo ;
mulation. In fact, if we look at the transformation, we can derive the value of the eciency score as / 1/(eT s). Since Yo is a semipositive vector, s cannot be a zero vector, and so 0 < eT s . Moreover, since {s0 1; sj 0, "j ¹ 0} is a feasible solution, the optimal value s satis®es eT s 6 1. The output-oriented BCC model which has been reformulated contains inputs as well as outputs. An output-oriented BCC model without inputs can be formulated as min s
s:t: Y s P Yo ; s P 0n :
X s 6 hXo ; eT s h; s P 0n : Comparing this model with the input-oriented BCC model, the next result holds. Proposition 4. When rating unit0 , the results of an output-oriented BCC model coincide with the results of an input-oriented BCC model if, and only if, unit0 is weakly ecient. Proof. The formulation of the input-oriented BCC model coincides with the last formulation except for the last constraint, where h has to be replaced by 1. Hence both models collapse if, and only if, h 1, i.e., when unit0 is rated as weakly ecient. Finally, in the reformulation of an output-oriented BCC model, we can dispense with h because h eT s. We then obtain min s
s:t:
eT s Y s P Yo ; X s 6 eT sXo ; s P 0n :
This is our new formulation of an output-oriented BCC model which, when compared to the original formulation, has one less variable (the radial eciency score) and one less constraint (the convexity constraint). We stress the important role of the (new) vector of intensity variables in this refor-
eT s
This reformulation shows clearly that, from a mathematical programming point of view, we are not considering any input at all. In the absence of slacks we can give a nice geometrical interpretation of this pure output model. Assume that s is the optimal value with, let us say, s1 , s2 and s3 all positive. Then Yo s1 Y1 s2 Y2 s3 Y3 and, consequently, / Yo 1=eT s Yo s1 =eT s Y1 s2 =eT s Y2 s3 =eT s Y3 : Hence the projection of Yo onto the ecient frontier, / Yo , is a convex combination of Y1 , Y2 and Y3 . The minimization of the sum of the ss corresponds to the selection of the rays de®ned by ecient units which are jointly as close as possible to the ray de®ned by Yo , i.e., we are searching for the smallest cone de®ned by rays associated with ecient units which is able to generate the ray associated with unit0 . The generating rays of the cone correspond to the peer group for unit0 . A similar chain of transformations gives rise to a new formulation for an input-oriented BCC model without outputs: max s
eT s
s:t: Y s P eT sYo ; X s P Xo ; s P 0n :
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
As before, to obtain an input-oriented BCC model without outputs, all we have to do is delete the ®rst set of constraints from the model above. 6. Conclusions In this paper we have considered radial DEA models without inputs (or without outputs). We have proved that it is not possible to consider constant returns to scale models without inputs (or without outputs) because they are not able to distinguish between ecient and inecient units. In fact, all units are classi®ed as in®nitely inecient. We have also considered an output-oriented (input-oriented) CCR model with a single constant input, and we have found that it coincides with the corresponding BCC model if only radial eciency scores are of interest. We have further proved that an output-oriented (input-oriented) BCC model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Moreover, considering a single constant input (output) is equivalent to considering multiple constant inputs (outputs). It is easy to verify that the two last statements are also valid for the FDH model (Deprins et al., 1984). Finally we have proposed a new formulation for BCC models, with one less constraint (the convexity constraint) and one less variable (the radial eciency score). It is easy to incorporate the non-archimedean component into the objective function of our new formulation, if desired. Our future research will be devoted to the study of the family of additive DEA models (see Charnes et al., 1985; Pastor, 1994). This has not only interest by
51
itself, but also as a tool for analyzing the two-stage version of the radial model, i.e., we would like to determine not only the behavior of the radial eciency score, but also the behavior of the slacks, in DEA models without inputs or without outputs.
References Adolphson, D.L., Cornia, G.C., Walters, L.C., 1991. A uni®ed framework for classifying DEA models. In: Operational Research'90. Pergamon Press, New York, pp. 647±657. Banker, R.D., Charnes, A., Cooper, W.W., 1984. Some models for estimating technical and scale ineciencies in data envelopment analysis. Management Science 30 (9), 1078± 1092. Charnes, A., Cooper, W.W., Golany, B., Seiford, L.M., Stutz, J., 1985. Foundations of data envelopment analysis for Pareto±Koopmans ecient empirical production functions. Journal of Econometrics 30, 91±107. Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the eciency of decision making units. European Journal of Operational Research 2 (November), 429±444. Deprins, D., Simar, L., Tulkens, H., 1984. Measuring laboreciency in post oces. In: Marchand, M., Pestieau, P., Tulkens, H. (Eds.), The Performance of Public Enterprises: Concepts and Measurements. North-Holland, Amsterdam. Lovell, C.A.K., Pastor, J.T., 1995. Units invariant and translation invariant DEA models. Operations Research Letters 18 (3), 147±151. Lovell, C.A.K., Pastor, J.T., 1997. Target setting: An application to a bank branch network. European Journal of Operational Research 98 (2) (April 16), 290±299. Pastor, J.T., 1994. New additive DEA models for handling zero and negative data. Paper presented at the Alaska TIMS/ ORSA Meetings, Anchorage, AK. Seiford, L.M., Thrall, R.M., 1990. Recent developments in DEA: The mathematical programming approach to frontier analysis. Journal of Econometrics 46, 7±38. Thompson, R.G., Singleton, Jr., F.D., Thrall, R.M., Smith, B.A., 1986. Comparative site evaluation for locating a highenergy physics lab in Texas. Interfaces 16, 35±49.
www.elsevier.com/locate/orms
Theory and Methodology
Radial DEA models without inputs or without outputs C.A. Knox Lovell b
a,*
, Jes us T. Pastor
b,1
a Department of Economics, University of Georgia, Athens, GA 30602, USA Vicerector de Investigaci on, Universidad Miguel Hernandez, Paseo Melchor Botella s/n, E-03206 Elche (Alicante), Spain
Received 25 August 1997; accepted 28 August 1998
Abstract In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial eciency score) and one less constraint (the convexity constraint). Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: DEA; Models without inputs (outputs) or with a single input (output)
1. Introduction DEA was initially developed by Charnes et al. (1978) (CCR) and by Banker et al. (1984) (BCC) for the purpose of evaluating the relative eciency of similar economic production systems. However Adolphson et al. (1991) noted that it is possible to adopt a broader perspective, in which DEA is also appropriate for comparing any set of homogeneous units on multiple dimensions. This broader perspective was perhaps ®rst adopted by Thompson et al. (1986), in an attempt to determine the optimal location of a superconducting supercol-
* 1
Corresponding author. E-mail: [email protected] E-mail: [email protected]
lider in the state of Texas. The authors considered six potential sites, and speci®ed three inputs (facility cost, user cost and environmental damage). They used an input-oriented CCR model, and set the output of each site equal to unity, since the output of each site is one superconducting supercollider. Adolphson et al. claim that it is possible, but not helpful, to interpret this problem as an economic production system. We show in this paper that when using the CCR model, as Thompson et al. did, such an interpretation is really necessary. Adolphson et al., went on to propose solving the superconducting supercollider problem as a pure input model (with no outputs). However instead of using an input-oriented CCR model, they proposed to use an input-oriented BCC model.
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 3 3 8 - 5
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
They justi®ed the model change on technical grounds: the presence of the convexity constraint in the BCC model ensures that the optimal eciency scores are positive. But they were unable to explain why they reached the same results as Thompson et al. In this paper we show why a CCR input-oriented (output-oriented) model without outputs (inputs) makes no sense. We also show that a pure input (output) DEA model is possible only if we resort to the corresponding BCC model. This apparently broader point of view matches with the classical economic point of view, since we also prove that a BCC model without inputs (outputs) is equivalent to the corresponding CCR model with a single constant input (output). This explains why Thompson et al. and Adolphson et al. reached the same result when solving the superconducting supercollider problem with ostensibly dierent DEA models: an input-oriented CCR model with a single constant output is equivalent to the corresponding BCC model with no outputs. We also provide a reformulation of all BCC models, including the simpli®ed versions mentioned above. The new formulation has one less variable (the radial eciency score) and one less constraint (the convexity constraint). Application of this new formulation to an output-oriented BCC model with a single constant input shows clearly that all that matters is the output values, i.e., the model can be considered as a pure output model. We restrict our attention to the one-stage version of the envelopment form of each radial DEA model, i.e., the version without the non-Archimedean element and without slacks in the objective function. In other words, if we solve the envelopment problem of a radial DEA model by means of a two-stage procedure (see, for instance, Lovell and Pastor, 1995), we are considering here only the ®rst stage, the one which gives us the value of the radial eciency score. The second stage is not considered because we are not primarily interested in the values of the slacks. This approach has numerous precedents in the literature (see, e.g., Seiford and Thrall, 1990). The paper unfolds as follows. In Section 2 we show that a CCR model without inputs (or out-
47
puts) is meaningless. In Section 3 we prove that a CCR model with a single constant input (output) collapses to the corresponding BCC model. In Section 4 a BCC model with a single constant input (output) is shown to coincide with a BCC model without inputs (outputs). In Section 5 we oer a new formulation of BCC models, having one less variable (the radial eciency score) and one less constraint (the convexity constraint), and we show how this reformulation applies to a BCC model without inputs (outputs). Section 6 concludes.
2. A CCR model without inputs (or without outputs) is meaningless We begin by considering CCR models, both output-oriented and input-oriented. We prove that an output-oriented (input-oriented) CCR model without inputs (without outputs) is unable to distinguish between ecient and inecient units. It is in this sense that we refer to such a model as being meaningless. Proposition 1. An output-oriented (input-oriented) CCR model without inputs (without outputs) rates all DMUs as in®nitely inecient. Proof. An output-oriented CCR envelopment problem without inputs is formulated as max / /;k
s:t: Y k P /Yo ; k P 0n ; where Yo (Yo1 ,. . .,Yom ) is an m ´ 1 output vector of the unit being evaluated, Y is an m ´ n matrix of output vectors of the n units in the sample, k (k1 ,. . .,kn ) is an n ´ 1 vector of intensity variables, and n is the number of units in the sample. Since {k0 1; kj 0, 8j 6 0} is a feasible solution with / 1, it is clear that we can consider a sequence of feasible solutions {k0 k; kj 0, 8j 6 0} with / k, and consequently max / +1. This means that all units are assigned the worst possible eciency score (if we admit +1 as an eciency
48
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
score), and all units are in®nitely inecient. As is well known in linear programming theory, the corresponding dual multiplier problem is infeasible and, consequently, no shadow prices are available in this case. For the case of an inputoriented CCR model without outputs, a similar proof assigns to each unit the worst possible eciency score (h 0), and therefore this model is also meaningless. (An alternative and more direct proof is obtained by considering the dual multiplier problem in each case.)
where e is an n ´ 1 vector of 1s and all other variables are as previously de®ned. Assume that (/ , k ) is an optimal solution to the problem. It suces to show that eT k 1. This is accomplished by proving that the strict inequality eT k < 1 leads to a contradiction. In fact, if the strict inequality holds, and knowing that eT k is positive because k P 0n , we can build a new feasible solution kf k /eT k satisfying Y kf Y k =eT k P / Yo =eT k / =eT k Yo > / Yo :
3. An output-oriented (input-oriented) CCR model with a single constant input (output) Having considered and rejected as meaningless CCR models having no inputs or no outputs, we now consider CCR models having a single constant input or a single constant output. Such a model has been considered by Thompson et al. (1986). We prove that such models are indistinguishable from the corresponding BCC models. This explains why a graphical display of a CCR output-oriented model with one constant input and two outputs is exactly the same as a graphical display of the corresponding BCC model. Proposition 2. An output-oriented CCR model with a single constant input and an input-oriented CCR model with a single constant output coincide with the corresponding BCC models. Proof. Let us ®rst consider an output-oriented CCR model with a single constant input. We can always assume that the constant input is at level 1 by changing its scale of measurement. This change can only in¯uence the value of the slacks, unless a normalized model is considered as proposed by Lovell and Pastor (1995). The envelopment form of the model is max / /;k
s:t:
Y k P /Yo ; eT k 6 1; k P 0n ;
Consequently, /f / /eT k > / shows that (/f , kf ) is a strictly better solution than (/ , k ), i.e., (/ , k ) is not optimal. For the input-oriented CCR model with a single constant output, the proof is completely similar. The same proof goes through for a CCR model with multiple constant inputs (or multiple constant outputs). We always end up with the corresponding BCC model. 4. A BCC model without inputs (or without outputs) It makes no sense to speak of an input-oriented model without inputs. Therefore, an output orientation is adopted when we consider a BCC model without inputs. Due to the presence of the convexity constraint, the presence or absence of a single constant input in an output-oriented BCC model is irrelevant, as the following proposition shows. Proposition 3. An output-oriented (input-oriented) BCC model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Proof. We can always assume that the constant input is at level 1, since a rescaling of any variable does not aect the optimal eciency score obtained by means of any radial DEA model. Therefore the restriction associated with a single constant input is eT k 6 1. The presence of the convexity constraint eT k 1 converts the preced-
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
ing restriction into a redundant restriction and so it can be deleted. The following result is a direct consequence of Propositions 2 and 3. Corollary 3.1. An output-oriented (input-oriented) CCR model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Hence the ®nal formulation of the envelopment problem for an output-oriented BCC model without inputs is max /;k
s:t:
/ Y k P /Yo ; eT k 1; k P 0n :
From an economic point of view it is dicult to accept a DEA model without inputs. However Lovell and Pastor (1997) have recently considered such a model for evaluating the performance of branch oces in relation to the target setting procedure established by a large Spanish savings bank. The outputs were the ``percentage of target coverage'' by each branch oce for a set of 17 dierent targets, and we considered no resource as an input. Hence we are facing a pure output model. Nevertheless, from a production perspective we can argue that each branch oce is by itself ``the input'' and, therefore, a single constant input was at hand. As a consequence of the theory developed above, we know that only one radial DEA model can be used with this data set: an output-oriented BCC model without inputs. This is exactly the model we used. In these special circumstances scale eciency is not a major issue. In fact, in this particular case and as a consequence of Corollary 3.1, we know that scale eciency is equal to 1 for every branch oce. The fact that an output-oriented BCC model without inputs coincides with an output-oriented CCR model with a single constant input implies that returns to scale are globally constant, not in branch oce service provision, but in branch of-
49
®ce target coverage. Size is irrelevant in this model. One may think that we may lose some dual information when considering a BCC model without inputs instead of a BCC model with a constant input, since in the former case we are unable to obtain a shadow price for the constant input. Fortunately, if we resort to any DEA software package, we always work with the BCC model with a constant input. Moreover, since we have shown that the latter model is just the same as the corresponding CCR model, the ratio form of the CCR model tells us that we can always assign a shadow price of 1 to the constant input and get relative prices for the outputs. Going back to our starting point, since prices are relative, we can simply consider the prices associated with outputs in the BCC model without inputs, and gather the same information with regard to outputs as with the more complex model. A ®nal point is worth mentioning. It is completely equivalent to deal with a radial DEA model with a single constant input (output) or with multiple constant inputs (outputs). By rescaling all the constant inputs (outputs) we can consider that all of them are at level 1, and, consequently, the corresponding linear programming restrictions collapse to a single restriction.
5. A new formulation of BCC models We focus our attention on an output-oriented BCC model. (A similar treatment is valid for an input-oriented BCC model.) Its envelopment problem is formulated as max /k
/
s:t: Y k P /Yo ; X k 6 Xo ; eT k 1; k P 0n : Since / P 1, we can consider a new variable h 1/ /. Substituting h for / in the BCC envelopment problem, we get
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min h;k
s:t:
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
h Y k P 1=hYo ; X k 6 Xo ; eT k 1; k P 0n :
Multiplying the ®rst set of restrictions by h > 0, and considering a new vector of intensity variables s hk, we obtain the reformulation min h;s
s:t:
h Y s P Yo ;
mulation. In fact, if we look at the transformation, we can derive the value of the eciency score as / 1/(eT s). Since Yo is a semipositive vector, s cannot be a zero vector, and so 0 < eT s . Moreover, since {s0 1; sj 0, "j ¹ 0} is a feasible solution, the optimal value s satis®es eT s 6 1. The output-oriented BCC model which has been reformulated contains inputs as well as outputs. An output-oriented BCC model without inputs can be formulated as min s
s:t: Y s P Yo ; s P 0n :
X s 6 hXo ; eT s h; s P 0n : Comparing this model with the input-oriented BCC model, the next result holds. Proposition 4. When rating unit0 , the results of an output-oriented BCC model coincide with the results of an input-oriented BCC model if, and only if, unit0 is weakly ecient. Proof. The formulation of the input-oriented BCC model coincides with the last formulation except for the last constraint, where h has to be replaced by 1. Hence both models collapse if, and only if, h 1, i.e., when unit0 is rated as weakly ecient. Finally, in the reformulation of an output-oriented BCC model, we can dispense with h because h eT s. We then obtain min s
s:t:
eT s Y s P Yo ; X s 6 eT sXo ; s P 0n :
This is our new formulation of an output-oriented BCC model which, when compared to the original formulation, has one less variable (the radial eciency score) and one less constraint (the convexity constraint). We stress the important role of the (new) vector of intensity variables in this refor-
eT s
This reformulation shows clearly that, from a mathematical programming point of view, we are not considering any input at all. In the absence of slacks we can give a nice geometrical interpretation of this pure output model. Assume that s is the optimal value with, let us say, s1 , s2 and s3 all positive. Then Yo s1 Y1 s2 Y2 s3 Y3 and, consequently, / Yo 1=eT s Yo s1 =eT s Y1 s2 =eT s Y2 s3 =eT s Y3 : Hence the projection of Yo onto the ecient frontier, / Yo , is a convex combination of Y1 , Y2 and Y3 . The minimization of the sum of the ss corresponds to the selection of the rays de®ned by ecient units which are jointly as close as possible to the ray de®ned by Yo , i.e., we are searching for the smallest cone de®ned by rays associated with ecient units which is able to generate the ray associated with unit0 . The generating rays of the cone correspond to the peer group for unit0 . A similar chain of transformations gives rise to a new formulation for an input-oriented BCC model without outputs: max s
eT s
s:t: Y s P eT sYo ; X s P Xo ; s P 0n :
C.A.K. Lovell, J.T. Pastor / European Journal of Operational Research 118 (1999) 46±51
As before, to obtain an input-oriented BCC model without outputs, all we have to do is delete the ®rst set of constraints from the model above. 6. Conclusions In this paper we have considered radial DEA models without inputs (or without outputs). We have proved that it is not possible to consider constant returns to scale models without inputs (or without outputs) because they are not able to distinguish between ecient and inecient units. In fact, all units are classi®ed as in®nitely inecient. We have also considered an output-oriented (input-oriented) CCR model with a single constant input, and we have found that it coincides with the corresponding BCC model if only radial eciency scores are of interest. We have further proved that an output-oriented (input-oriented) BCC model with a single constant input (output) is equivalent to an output-oriented (input-oriented) BCC model without inputs (outputs). Moreover, considering a single constant input (output) is equivalent to considering multiple constant inputs (outputs). It is easy to verify that the two last statements are also valid for the FDH model (Deprins et al., 1984). Finally we have proposed a new formulation for BCC models, with one less constraint (the convexity constraint) and one less variable (the radial eciency score). It is easy to incorporate the non-archimedean component into the objective function of our new formulation, if desired. Our future research will be devoted to the study of the family of additive DEA models (see Charnes et al., 1985; Pastor, 1994). This has not only interest by
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itself, but also as a tool for analyzing the two-stage version of the radial model, i.e., we would like to determine not only the behavior of the radial eciency score, but also the behavior of the slacks, in DEA models without inputs or without outputs.
References Adolphson, D.L., Cornia, G.C., Walters, L.C., 1991. A uni®ed framework for classifying DEA models. In: Operational Research'90. Pergamon Press, New York, pp. 647±657. Banker, R.D., Charnes, A., Cooper, W.W., 1984. Some models for estimating technical and scale ineciencies in data envelopment analysis. Management Science 30 (9), 1078± 1092. Charnes, A., Cooper, W.W., Golany, B., Seiford, L.M., Stutz, J., 1985. Foundations of data envelopment analysis for Pareto±Koopmans ecient empirical production functions. Journal of Econometrics 30, 91±107. Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the eciency of decision making units. European Journal of Operational Research 2 (November), 429±444. Deprins, D., Simar, L., Tulkens, H., 1984. Measuring laboreciency in post oces. In: Marchand, M., Pestieau, P., Tulkens, H. (Eds.), The Performance of Public Enterprises: Concepts and Measurements. North-Holland, Amsterdam. Lovell, C.A.K., Pastor, J.T., 1995. Units invariant and translation invariant DEA models. Operations Research Letters 18 (3), 147±151. Lovell, C.A.K., Pastor, J.T., 1997. Target setting: An application to a bank branch network. European Journal of Operational Research 98 (2) (April 16), 290±299. Pastor, J.T., 1994. New additive DEA models for handling zero and negative data. Paper presented at the Alaska TIMS/ ORSA Meetings, Anchorage, AK. Seiford, L.M., Thrall, R.M., 1990. Recent developments in DEA: The mathematical programming approach to frontier analysis. Journal of Econometrics 46, 7±38. Thompson, R.G., Singleton, Jr., F.D., Thrall, R.M., Smith, B.A., 1986. Comparative site evaluation for locating a highenergy physics lab in Texas. Interfaces 16, 35±49.