A response to the critiques of DEA by Dmitruk and Koshevoy, and Bol

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1 J Prod Anal (2008) 29:15 21 DOI /s A response to the critiques of DEA by Dmitruk and Koshevoy, and Bol W. W. Cooper Æ Z. Huang Æ S. Li Æ J. Zhu Published online: 1 August 2007 Ó Springer Science+Business Media, LLC 2007 Abstract We here critique the articles by Dmitruk & Koshevoy (1991, J Econ Theory 55: ) and by Bol (1986, J Econ Theory 38: ) by showing how to solve the examples they erected to show the non-existence of functions for evaluating performance efficiencies in DEA. We also show that functions satisfying these criteria and other important criteria as well were already available prior to the publications of D&K and by Bol and have since been greatly extended to increase the power and scope of DEA. Keywords Performance evaluations Efficiency measures Effectiveness measures 1 Introduction In the course of reviewing one of our recent papers, a referee called our attention to the article in the Journal of W. W. Cooper (&) Department of Information, Risk, and Operations Management (IROM), The University of Texas at Austin, 1 University Station B6500, Austin, TX , USA cooperw@mail.utexas.edu Z. Huang S. Li School of Business, Adelphi University, Garden City, Long Island, NY 11530, USA Z. Huang huang@adelphi.edu S. Li susan@adelphi.edu J. Zhu Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA jzhu@wpi.edu Economic Theory by Dmitruk and Koshevoy (1991), the reading of which called our attention to the article by G. Bol (1986). As the referee indicated, in the economics literature these papers are apparently regarded as landmark contributions dealing with limitations of Data Envelopment Analysis (DEA). These articles by D&K and Bol are now outdated and, in fact, were already outdated at time of publication. We here counter the examples of non-existence that they supply and locate a major source of the difficulties that D&K and Bol identified in the fact that the technical evaluation measures considered by D&K and by Bol, as well as others in the literature they cite, are not complete. 1 That is, in the sense of Cooper et al. (1999), the measures are incomplete because they fail to reflect all of the inefficiencies that a model can identify. 2 Debreu-Farrell models and measures We start with the following technical efficiency criteria as specified in D&K which a function, F(x), is to satisfy if it is to provide a satisfactory measure, (F1) for all x 2 L, F(x) 1; (F2) if x 2 Eff L, then F(x) ¼ 1; (F3) for all x 2 L andk > 0 then FðkxÞ ¼ð1=kÞF(x); (F4) if x 0 x 00 ; x 0 6¼ x 00 ; then Fðx 0 Þ > Fðx 0 Þ; ðaþ 1 D&K (but not Bol), following Färe and Lovell (1978), explicitly assume free disposal, which means they disregard non-zero slacks as sources of inefficiency and this makes the resulting measures incomplete.

2 16 J Prod Anal (2008) 29:15 21 where x is a vector of inputs in L, the production possibility set. 2 Partly because F(2) and F(4) imply F(1), D&K replace (F2) with, ðf2 0 Þ for any x 2 L, F(x) ¼ 1, x 2 Eff L, where, we might note, the above criteria are a modification of the requirements specified for efficiency by Färe and Lovell (1978, p. 157), to which D&K are responding. The following model from Färe and Lovell (1978) see also Färe et al. (1985) can help us to understand the concerns of F&L as well as D&K, hx io ¼ Pn minh x ij k i þ s i i ¼ 1;...; m y ro ¼ Pn y rj k i s þ r r ¼ 1;...; s 0 k j ; s i ; s þ r 8i,j,r. ð1þ Here x ij, y rj respectively refer to the input amounts utilized and the output amounts produced, as determined from observed data for DMU j (the j th Decision Making Unit), j = 1,..., n. The x io and y ro refer to the inputs and outputs of DMU o, the DMU j that is to be evaluated relative to the entire collection of DMU j, j = 1,..., n, including DMU o. For the present we assume that these values are all positive, and thus follow D&K and Bol who assume this implicitly with the examples they use to show non-existence. This positivity assumption implies that solutions always exist to (1) with values of min h = h * 1 as prescribed in (F1) of A. We have here used the notation commonly employed in the DEA literature where (1) is referred to as the Farrell model (or measure). We shall, however, refer to it as the Debreu-Farrell model to reflect the fact, as cited by Farrell (1957), that Debreu (1951) is the source for his developments. This leads to the following. Definition 1 Debreu-Farrell Efficiency: DMU o is efficient if min h = h * = 1 and it is inefficient if h * < 1 with (1 h * )x io representing the amount of inefficiency in each of the i = 1,..., m inputs. As we shall see, this falls short of what is required to fulfill the following, Definition 2 Pareto-Koopmans Efficiency is attained by DMU o iff the evidence shows that it is not possible to improve any input or output without worsening some other input or output. The latter is the definition generally used in the DEA literature. We refer to it as P-K efficiency and note that it is the ordinary definition of technical efficiency that is implicitly used by D&K and reflected in (F2 ). We first turn to FðxÞ ¼ DF hðxþ ¼h, where h * is optimal for (1), as a choice of the measure of efficiency that is optimized. Commonly referred to as a radial measure, it is the value of the ratio of two measures of distance: (a) the distance on a ray from the origin to the point with coordinates that represent the performance of the DMU o being evaluated and (b) the distance from the origin to the point where this ray intersects the frontier. See Cooper et al. (2006a, Submitted) for a detailed development showing that any p measure may be used to form this ratio. 3 FDH and CCR (Charnes et al. 1978) models and measures Figure 1 is used by D&K to show that the desired function does not always exist and hence exemplifies their theorems. In this figure the production possibility set defined by L consists of all points on and to the northeast of the isoquant (e.g., the unit isoquant in which output is equal to unity) represented by the solid lines. Here Eff L [in the D&K notation] consists [only] of points a, d and the [open] interval (b, c) so that all other points evaluated by this section of the isoquant are not efficient. This figure illustrates the non existence arguments by D&K (p. 124). To start our counter analysis we note that a straightforward use of (1) in the manner usually used in DEA will result in a showing that only a and d are efficient so all other points, including the interval (b to c), have values of h * < 1 and hence are inefficient. Measured in this manner, however, evaluations can be effected on a line connecting a and d and hence are outside the production possibility set. x 2 a=(1,3) b=(2,3) p=(3,3) c=(3,2) d=(3,1) 2 See Charnes et al. (1985) for a simple axiomatic formulation that lies in the intersection of all of the commonly used axioms to characterize such production possibility sets. Fig. 1 Dmitruk and Koshevoy example x 1

3 J Prod Anal (2008) 29: To counter the D&K argument we adjoin the following bivalency condition to (1), k j 2 f0; 1g; Xn k j ¼ 1: ð2þ We are here using concepts from the FDH (Free Disposal Hull) method as introduced into DEA by H. Tulkens and his associates at CORE (Center for Operations Research and Econometrics) at the University of Louvain. See e.g., Tulkens (1993) and Deprins et al. (1984). For illustrative clarity we now use the numerical coordinates we have introduced into the portrayal of Fig. 1 and employ these values to represent the following function and its domain, 8 ðiþ For 1x 1 2;x 2 3useð1Þand ð2þ: >< ðiiþ For x 1 2;x 2 2 use ð1þ: Fx ðþ¼ Only use isoquant from b to c. >: ðiiiþ For x 1 3;x 2 1 use ð1þ and ð2þ; ð3þ where the range is 0 h * 1 for optimal h, as in (1), when the data are positive. We then define our measure of efficiency with h ¼ minff i ðxþ; F ii ðxþ; F iii ðxþg: ð4þ This will resolve ambiguities that can arise when the point being evaluated lies in an intersection or a boundary associated with the different regions delineated in (3). Cf. p for an example. We start our analysis by evaluating the point p = (3,3) (which we have added to D&K in Fig. 1) for which we use (ii) to obtain the following adaptation from (1), min h 2k b þ 3k c þ 3k p þ s 1 ¼ 3h 3k b þ 2k c þ 3k p þ s 2 ¼ 3h k b þ k c þ k p s þ ¼ 1 k b ; k c ; k p ; s 1 ; s 2 ; sþ 0: Here the third constraint reflects the unit output that is associated with the isoquant represented by the solid lines in Fig. 1. This problem has as its solution h ¼ 0:833þ; k b ¼ k c ¼ 0:5: Hence, as required, h * = < 1 shows that the DMU associated with p is not efficient. An immediate consequence of this solution is that the point represented by 2k b þ 3k c ¼ 3h ¼ 2:5 3k b þ 2k c ¼ 3h ¼ 2:5 is efficient with value ^h ¼ 1 and it satisfies P-K efficiency with both slacks zero. The same is true for all points in this interval from b to c. Also h * < 1 for points in the interior of the production possibility set that are evaluated by the isoquant from a to b. Therefore (F1) and (F2) of (A) are satisfied in this interval. The same results are obtained from (1) and (2) for every other point in Fig. 1. That is, we will have h * = 1for points on the frontier indicated by the solid line and h * < 1 for points in the interior of the production possibility set. Hence (F1) and (F2) in (A) are always satisfied. Turning to (F3) in (A) we utilize the ratio form presented in Charnes et al. (1978), min zx; ð yþ ¼ u ry P ro m v ix io u ry P ro m v ix io 1; j ¼ 1;...; n u r ; v i 0 8r; i: t Xm We then choose a variable t such that v i x io ¼ 1; ð5þ ð6þ which implies t >0. 3 Multiplying all numerators and denominators by this t does not change the value of any ratio. Hence setting l r ¼ tu r ; m ¼ tv i ; r ¼ 1;...; s i ¼ 1;...; m we replace (5) by max z(y) ¼ Ps m i x io ¼ 1 l r y ro l r y rj Pm m i x ij 0; l r ; m i 0 8r; i: j ¼ 1;...; n ð7þ 3 We are here using the development on pp in Cooper et al. (2006b) but the basic development is taken from fractional programming as first presented in Charnes and Cooper (1962). See Schaible (1994) for a survey of the 900+ papers on fractional programming and its extensions since publication of this paper by Charnes and Cooper. See Bradley and Frey (1974) for fractional programming formulations that extend the Cobb-Douglas (and related homogeneous functions) that are frequently used in economics.

4 18 J Prod Anal (2008) 29:15 21 We therefore relate (5) to (7) via u r y ro v i x ¼ Xs l r y ro ð8þ io where * represents an optimal value. However, (7) is the dual ( multiplier ) model to (1), so l r y ro m i x io ¼ h ðxþ: ð9þ Thus, F(x) is homogenous of degree minus one as stipulated in F(3), and multiplication of the x io by any constant, k >0,i = 1,...,m, gives l r y ro ð Þ ¼ 1 k h ð10þ m i kx io as in (F3). This is to say that if h * (x) is the efficiency measure associated with input vector x then altering all its component, x io, by the same k > 0 should alter the value of the efficiency measure to k 1 h * (x). See the discussion of the Russell measure on p. 157 of Färe and Lovell (1978). In any case, we have now satisfied (F3) in Satisfaction of (10) is not simply a mathematical coincidence. For a rationale embodying the result in (9) we can interpret (5) as the real problem of interest so that (7) and (1) are simply parts of the algorithm used to solve this nonlinear (and non convex) problem. (See, for instance, Cooper et al. (2007) where this approach is used to obtain an aggregate measure of efficiency from the efficiency scores of the individual DMUs). This interpretation is analogous to the manner in which simplex tableaus represent solutions to problems en route to solving the problem of interest with the simplex method. See the discussion of algorithmic completion of a model in Charnes and Cooper (1965). D&K (p. 124) refer to F(x) as having a discontinuity at point b. However, this is clearly not the case since h * =1at all points on the frontier, as has now been shown. D&K therefore probably mean that the efficiency measure should change to a value less than unity at b. This is accomplished, as in Charnes et al. (1978), 4 by replacing the objective in (1) with min zx; ð yþ ¼ h e X s s þ r þ Xm s i 4 This is the same year as Färe and Lovell (1978).! ð11þ where the s þ r ; s i 0 are the output and input slacks that are represented in (1) and e > 0 is a non-archimedean element smaller than any positive real number. Remark This non-archimedean concept is drawn from the field of non-standard mathematics. See A. Robinson (1966, 1967). See also the discussion in Arnold et al. (1998) which uses this non-standard approach to extend the usual duality relations in linear programming. Moreover, this e > 0 need not be specified explicitly. It is usually handled in a two-stage manner as follows: Stage 1: An optimal value is secured for h ¼ h with (1). Stage 2: the h in the constraints of (1) are fixed at h = h * and the sum of the slacks represented in (11) is then maximized. Use of (11) replaces Definition 1 with the following, for full P-K efficiency, Definition 3: Part 1: The performance of DMU o is evaluated by z(x,y) 1 and is efficient if and only if z * (x,y) =1. Part 2: The latter condition is achieved with the CCR (Charnes et al. 1978) measure if and only if the following two conditions are both satisfied at an optimum, (i) h * =1 (ii) All slacks are zero. This satisfies (F2 ), the condition added to (A) by D&K, as well as (F2) because this measure is complete i.e., it reflects all inefficiencies that the model can identify. We now turn to (F4) in (A) and start this discussion by using (i) to evaluate point b of Fig. 1 with the objective in (1) replaced with the objective in (11) to obtain, min h e s 1 þ s 2 þ sþ 1k a þ 2k b þ 3k c þ 3k d þ s 1 ¼ 2h 3k a þ 3k b þ 2k c þ 1k d þ s 2 ¼ 3h k b þ k b þ k c þ k d s þ ¼ 1; along with non-negativity for the variables and the integer condition in (2), as required for region (i) in (3). This problem has as its solution h * =1,k * a =1,s * 1 =1. Hence the objective has value zx,y) =h * 1e ) = (1 e )<1 so b is evaluated by point a and identified as not efficient by Definition 3. To show how (F4) of (A) is satisfied we start by noting that this slack value of unity represents the difference in the coordinates of b at x 1 = 2 and the coordinate of a at x 1 =1. That is, the slack s * 1 = 1 represents the difference in the x 1 coordinate for points a and b on this interval of the isoquant. Now as movement is effected from b to a, the value of the slack variable decreases because the associated x 1 coordinate value decreases. This conforms to the

5 J Prod Anal (2008) 29: conditions specified in (F4). That is, as the coordinate for x 1 is reduced and the value of the objective is increased. This same condition is present at all points as movement is made from b to a with a slack value of zero at the latter point which is thus characterized as efficient by Definition 3. Thus, in strictly monotonic fashion (F4) is satisfied at every point on the segment from a to b. This same situation occurs on the segment of the frontier extending upward from point a with x 2 replacing x 1 as the input of interest. Jumping to the other side of Fig. 1 this same situation also occurs with d serving as the efficient point from which all points on the vertical and horizontal segments are evaluated. Finally, the points in the open interval from b to c are all efficient as was shown in our opening example. Hence conformance to the requirements in (F4) is secured in this manner. Conversely, on the open interval between points b and c and at points a and d it is not possible to reduce either input without also increasing the other input in order to stay within the production possibility set. Hence the conditions for recourse to (F4) are not present in this interval. Clearly the Farrell-Debreu measure defined in (1) cannot satisfy (F4) since that measure is incomplete. This situation is remedied by the CCR measure defined in (11) which was introduced into the DEA literature in Charnes et al. in Therefore, we have now shown that the pertinent points in Fig. 1 all satisfy (F4). Finally, as already noted, use of (11) for the objective satisfies (F2 ). Hence all of the conditions specified by D&K are satisfied. 4 Bol and multiplicative models We now turn to Bol (1986) and utilize the multiplicative model introduced in Charnes et al. (1982) to find that Bol s nonexistence example was also already outdated at time of publication. For this purpose we take only the first of the nonexistence examples in Bol since his second example was designed only to show that non-convexity was not the source of his nonexistence demonstration. The example supplied by Bol is represented in Fig. 2, below, where the level line L(u) is given by: x 1 x 2 u 2 for 0 x 1 u x 2 u for u x 1 2u x 1 x 2 2u 2 for 2u x 1 : ð12þ For our counter to this example we will simply show how Fig. 2, as defined in (12), can be reduced to a variant of Fig. 1. For this purpose we turn to the class of multiplicative models, introduced into the DEA literature in the following form by Charnes et al. (1982) max Qs Q s rj =Qm ro =Qm x m i io x m i ij 1; l r ; m i 1; 8r; i: j ¼ 1;...; n ð13þ Noting that Qs y l i ro =Qm x m i io 1 is included as one of the constraints we have Definition 4 only if max Ys The performance of DMU o is efficient if and ro =Ym x m i io ¼ Ys ro =Ym x m i io ¼ 1: To convert (13) to a linear programming problem we use max Ps l r^y ro Pm m i^x io l r^y rj Pm m i^x ij 0; l r ; m i 1 8i; r: j ¼ 1;...; n ð14þ where the caret, ^, denotes logarithm. We now relate this to the additive model. Because (14) is an ordinary linear programming problem it has a dual that can be represented by max Ps s þ r þ Pm s i P n P n x 2 ^y rj k j s þ r ¼ ^y ro ; r ¼ 1;...; s ^x ij k j þ s i ¼ ^x io ; i ¼ 1;...; m k j ; s þ r ; s i 0 8i; j; r; L(u) Fig. 2 Example from Bol (1986) 2u x 1 ð15þ

6 20 J Prod Anal (2008) 29:15 21 which is a version of the additive model introduced in Charnes et al. (1985). This measure is complete so Definition 4 yields the following condition for efficiency:dmu o has performed efficiently if and only if an optimum is achieved with all slacks equal to zero in (15). Proceeding on this route we can now replace (12) with ^x 1 þ ^x 2 2^u for 1^x 1 ^u ^x 2 ^u for ^u ^x 1 ln 2 þ u ^x 1 þ ^x 2 ln 2 þ 2^u for 2^u ^x 1 : ð16þ Figure 3 provides a graphic portrayal of this transformation from Fig. 2. Figure 3 is very much like Fig. 1 so, again, we find that the inefficiencies fail to be identified by (1) when the data stated in logarithmic units lie in the interval ^u ^x 1 ln 2 þ ^u. Because non-zero slack is present, however, this difficulty is avoided by using (15) because, by Definition 3, this non-zero slack designates DMU o as inefficient. Hence, again, the difficulty is confined to cases in which the measure is not complete. There are deficiencies in the use of this measure because (14), like the additive model, need not equal unity when DMU o is efficient. Turning to the multiplier (dual) model in (13), however, and taking anti-logs we return to Definition 4 and the unity criterion is satisfied. n Another deficiency is identifiable because (13) is not units invariant a property not specified by Bol or by D&K. However, this is remedied in Charnes et al. (1983) where a units invariant version of the multiplicative model results in a generalization of Cobb-Douglas functions to multiples of such functions, with each involving multiple outputs and multiple inputs. A great variety of standard formulations in economics, finance and other fields, are also opened for possible use and further research along these lines. Banker and Maindiratta (1986), for example, use piecewise loglinear extensions of (13) to comprehend production possibility sets which are quasi-concave as well as quasi-convex. Hence the usual sequence of increasing, constant and decreasing returns to scale is thereby relaxed. In addition Banker et al. (2004b) show how exact (numerically valued) elasticities of scale can be obtained from these multiple output-multiple input formulations. Moreover, these elasticity values do not depend on the existence of derivatives so the findings by Forsund (1996, p. 290) 5 that exact elasticities cannot be obtained in DEA is not correct because, once again, the findings of nonexistence result from confining attention to models with radial measures that are not complete. 5 Summary and conclusions We have here shown that D&K and Bol both have shortcomings because the measures they consider are incomplete. We have also shown how to remove this deficiency by replacing the Debreu-Farrell measure in (1) with the CCR measure in (11). We then showed how multiplicative models can be used to relate the supposed counter example in Bol to an example like the one used by D&K. As noted earlier, all of these models and measures were published prior to the publication by D&K and by Bol. Moreover, the development of DEA has proceeded at accelerating rates to introduce new models and methods which have many more attractive features. An example is the RAM model of Cooper et al. (1999) which does not use non-archimedean concepts but provides a measure that is complete and is also both strongly monotonic and affine invariant so that it is (1) units invariant and (2) translation invariant. Hence, the assumption of data positivity is not necessary when RAM is used. Acknowldgements We are grateful to a referee whose comments led to a great improvement over the original version of this paper. Support from the IC2 Institute for this research is also gratefully acknowledged by W.W. Cooper ˆx 2 References 2û û ˆx 1 + ˆx 2 2û ˆx 2 û ˆx 1 + x ln2 ˆ2 + 2uˆ Arnold V, Bardhan I, Cooper WW, Gallegos A (1998) Primal and dual optimality in computer codes using two-stage solution procedure in DEA. In: Aranson J, Zionts S (eds) Operations research: methods, models and applications. Westport, Conn. Quorum Books Banker RD, Cooper WW, Seiford LM, Zhu J (2004b) Returns to scale in DEA. In: Cooper WW, Seiford LM, Zhu J (eds) Handbook on data envelopment analysis. Norwell, Mass, Kluwer Academic Publishers. Banker RD, Maindiratta A (1986) Piecewise loglinear estimation of efficient production surfaces. Manage Sci 32: û ln 2 + û ˆx 1 Fig. 3 Bol example in logarithmic units 5 See also Fukuyama (2000, p. 105).

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