Further discussion on linear production functions and DEA

Transcription

1 European Journal of Operational Research 127 (2000) 611±618 Theory and Methodology Further discussion on linear production functions and DEA Joe Zhu * Department of Management, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA , USA Received 8 December 1998; accepted 4 July 1999 Abstract The purpose of the current paper is to clarify the misunderstanding in using the constant returns to scale (CRS) model in data envelopment analysis (DEA) to estimate returns to scale (RTS) classi cation. By illustrating the following two di erent assumptions: (a) all decision making units (DMUs) are assumed to be compared to a CRS frontier in the CCR model, i.e., all DMUs are assumed to be capable of operating under CRS and (b) all DMUs exhibit CRS, several incorrect points about RTS estimation raised by Chang, K.P. and Guh, Y.Y. (1991) (European Journal of Operational Research 52, 215±223) and Chang. K.P. (1997) (European Journal of Operational Research 97, 597±599) are examined. It is shown that the assumption (a) does not mean that the RTS classi cation cannot be estimated. The CRS frontier assumption cannot change the RTS nature of each DMU. Since the CCR model mixes a technical e ciency and a scale e ciency, it is possible and meaningful to use the CCR results to determine the RTS classi cation. Finally, the infeasibility of Chang and Guh's (1991) treatment of non-archimedean in nitesimal is discussed. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Data envelopment analysis (DEA); Returns to scale (RTS) 1. Introduction Chang and Guh (1991) claim a number of conclusions regarding the estimation of returns to scale (RTS) and the treatment of non-archimedean in nitesimal e in data envelopment analysis (DEA). For the RTS issue, they argue that ``because of assuming linear production function, both BankerÕs and Banker±Charnes±Cooper methods cannot be employed to test for DMUsÕ returns to scale''. * Tel.: ; fax: address: jzhu@wpi.edu (J. Zhu). However, it has been recognized that Banker's (1984) or BCC's (Banker et al., 1984) methods may misclassify the RTS classi cation only because of multiple optimal solutions in DEA models (see Banker and Thrall, 1992). Furthermore, Zhu and Shen (1995) (i) show that a certain type of linear dependency among a set of e cient DMUs, i.e., the existence of non-extreme-e cient DMUs, is a cause of multiple optimal lambda solutions in Banker's (1984) method and therefore Chang and Guh's (1991) conclusions on the RTS estimation are incorrect; (ii) show that alternate optimal lambda values only a ect the estimation of RTS on those DMUs that truly exhibit constant returns to /00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 )

2 612 J. Zhu / European Journal of Operational Research 127 (2000) 611±618 scale (CRS), and have nothing to do with the RTS estimation on those DMUs that truly exhibit increasing or decreasing returns to scale (IRS or DRS) (also see Seiford and Zhu, 1999); and (iii) provide a remedy for Banker's (1984) method to avoid the possible e ect of multiple optimal DEA solutions on the determination of RTS. By showing that the assumption that all DMUs are capable of operating under CRS is not equivalent to the assumption that each current DMU exhibits CRS, the current paper again shows that Banker's (1984) and Banker et al.'s, (1984) methods can be employed to determine the RTS classi cation, and that it is possible and meaningful to use the magnitude of the P k j from the CCR model (CRS model) (Charnes et al., 1978) to test for DMUsÕ RTS. The current paper also shows that the Theorem 1 in Zhu and Shen (1995) is misread by Chang (1997), and as a result the counter-example used in Chang (1997) is invalid. Chang and Guh (1991) also propose a modi cation to the CCR model by replacing the e by a particular positive number. However, as pointed out by Green et al. (1996), Chang and Guh's (1991) proposal on e is found not to work in practice. The current paper further discusses this issue and shows that the failure results form an incorrect de nition of a positive-multiplier-dependent e. The structure of the paper is as follows. Section 2 discusses Chang and Guh's (1991) and Chang's (1997) incorrect statements on the CCR and BCC RTS methods, and shows Chang's (1997) misinterpretation of Zhu and Shen's (1995) Theorem 1. It is again emphasized that the possible failure of the CCR and BCC RTS methods is only due to multiple optimal solutions. Section 3 discusses the cause of infeasibility of Chang and Guh's (1991) implementation of the CCR e ciency bounds. Section 4 concludes. 2. Returns to scale 2.1. RTS related DEA models In order to develop the discussion of RTS, related DEA models are provided. Suppose we have n DMUs. Each DMU j ; j ˆ 1; 2;...; n, produces s di erent outputs, y rj r ˆ 1; 2;...; s, using m different inputs, x ij i ˆ 1; 2;...; m. Then the primal linear program for the (input-oriented) CCR model can be written as h ˆ min h k j x ij 6 hx io ; i ˆ 1; 2;...; m; k j y rj P y ro ; r ˆ 1; 2;...; s; k j P 0; j ˆ 1; 2;...;;n; 1 where x io and y ro are respectively the ith input and rth output for DMU o under evaluation. Another DEA model, which is usually referred to the BCC model, can be expressed as b ˆ min b k j x ij 6 bx io ; i ˆ 1; 2;...; m; k j y rj P y ro ; r ˆ 1; 2;...; s; k j ˆ 1; k j P 0; j ˆ 1; 2;...; n: The dual to the above linear program is max X s X s X m iˆ1 u r y ro u o u r y rj Xs iˆ1 v i x io ˆ 1; 2 v i x ij u o 6 0; j ˆ 1;...; n; u r ; v i P 0 and u o is free: 3

3 J. Zhu / European Journal of Operational Research 127 (2000) 611± If we impose P n k j 6 1 in the CCR model, then we obtain the following DEA model: f ˆ min f k j x ij 6 fx io ; i ˆ 1;...; m; k j y rj P y ro ; r ˆ 1;...; s; k j 6 1; k j P 0; j ˆ 1;...; n: 4 On the basis of Seiford and Zhu (1999), there are at least three di erent basic methods of testing a DMU's RTS nature which have appeared in the DEA literature. Banker (1984) shows that the CCR model (1) can be employed to test for DMU's RTS using the concept of most productive scale size (mpss). That is, the sum of optimal CCR optimal lambda values (k j ) can determine the RTS classi cation. We call this method CCR RTS Method. Banker et al. (1984) report that a new free BCC dual variable u o (in Eq. (3)) estimates RTS via allowing variable returns to scale (VRS) in the CCR model. That is, the sign of optimal u o determines the RTS. We call this method BCC RTS Method. Finally, Fare et al. (1985) provide the scale e ciency index method for the determination of RTS using models (1), (2) and (3) Linear production/crs assumption Chang and Guh (1991, p. 222) conclude : ``because of assuming linear production functions, both BankerÕs and the Banker±Charnes±Cooper methods cannot be employed to test for DMUsÕ returns to scale''. (In Chang (1997, p. 597), this conclusion is stated in ``Charnes, Cooper and Rhodes's (CCR) (1978) data envelopment analysis (DEA) method cannot be used to test for decisionmaking-unitsõ (DMUs) returns to scale, and Banker, Charnes and Cooper's (BCC) (1984) method for determining DMUsÕ returns to scale can fail in some cases''.) Their main argument is that because the CCR model assumes CRS, i.e., all DMUs are assumed to be capable of operating under CRS, it is impossible and meaningless to talk about determining RTS by the CCR RTS method (see e.g., Chang, 1997, p. 598). This argument seems rational. However, it confuses the following two di erent assumptions: (a) all DMUs are evaluated with respect to a CRS frontier in the CCR model (i.e., all DMUs are assumed to be capable of operating under CRS) 2 and (b) all DMUs exhibit CRS. It is true that the CCR model assumes CRS and the e cient DMUs must exhibit CRS. It is also true that the CCR model (1) improves the e ciency of each DMU by assuming that each DMU should operate on a CRS frontier. However, assumption (a) is not equivalent to assumption (b). That is, evaluating DMUs in the CCR model (1) does not necessarily mean that each current DMU exhibits CRS. Each DMU has its own RTS nature no matter which DEA model is employed. Thus the real RTS classi cation should be estimated and uncovered. In fact, when a particular CRS frontier is determined by the CCR model (1), some CCR ine cient DMUs may be operating in a scale ine cient manner (either in IRS or DRS regions), and the violation of the CRS assumption can be examined by a scale e ciency index proposed by Fare et al. (1985). Furthermore, note that the CCR model (1) is a mixture of a technical e ciency measure and a scale e ciency measure. Therefore, the CCR model (1) does contain the RTS and scale e ciency information, and the RTS classi cation or scale e ciency can be derived from the CCR 1 RTS generally has an unambiguous meaning only if a DMU is on the VRS (BCC) e ciency frontier. If a DMU is not BCC e cient, we use the information from (2) to project this DMU onto the BCC frontier (see Banker et al., 1995). 2 In Chang (1997, p. 597), this is stated as ``all DMUs are assumed under constant returns to scale''.

4 614 J. Zhu / European Journal of Operational Research 127 (2000) 611±618 model when a DMU is CCR-ine cient. In the CCR model (1), a DMU does not achieve 100% technical and scale e ciency unless it is on (or moved onto) the CRS frontier. The scale ine ciency in the CCR model is caused by IRS or DRS. Consider a set of the ve hypothetical DMUs, (A, B, C, D, andg) in Fig. 1. Points B and C are CCR-e cient. The remaining three DMUs (A, D, and G) can be projected onto the CRS frontier by an input reduction in order to achieve CRS as assumed in the CCR model (1). Points A and D are technically and scale ine cient. Point G is technically ine cient but scale e cient, since it is in a CRS region. (If an output-oriented CCR model is employed, point G is also scale ine cient and in a DRS region.) Obviously, the assumption that all DMUs are assumed to be capable of operating under CRS cannot change the IRS nature of A and the DRS nature of D. Finally, Chang and Guh (1991) suggest using the scale e ciency index method to estimate the RTS classi cation. However, as shown by Fare and Grosskopf (1994), Banker et al. (1996) and Seiford and Zhu (1999), the three basic RTS methods are equivalent. This again indicates that (i) the failure of Banker's (1984) and Banker et al. (1984) methods, i.e., the CCR and BCC RTS methods, has nothing to do with the (piecewise) CRS linear production frontier assumption in DEA and (ii) Chang's (1997, p. 598) claim: ``since the CCR method assumes constant-returns-to-scale...; it will be impossible and meaningless to use any information from the method... to test for DMUsÕ returns to scale'', is incorrect and misleading Most productive scale size concept Banker's (1984) mpss concept is misinterpreted in Chang (1997, p. 597) by stating ``since all the points on the constant-returns-to-scale frontier are mpss, it is not meaningful to nd which is mpss''. Fig. 1. The CCR model, CRS frontier and RTS.

5 J. Zhu / European Journal of Operational Research 127 (2000) 611± In fact, the mpss concept is about nding a way to move a CCR ine cient DMU onto a mpss point rather than nding the mpss points on the frontier. For example, in Fig. 1, all DMUs between B and C are mpss points determined by the CCR model (1). The mpss concept can move point A to B by the optimal values of h and P k j from (1), i.e., h x= P k j ; y= P k j. However, because of the multiple optimal solutions caused by the coexistence of B and C (linear dependency and multiple mpss), G can be moved onto any point on BC by h and associated P k j with di erent values. (For more discussion on mpss concept and multiple optimal solution, the reader is referred to Banker and Thrall (1992).) 2.4. Multiple optima It has been noted that the CCR RTS method (or the BCC RTS method) may fail to work only when multiple optimal solutions are present in model (1) (or (3)). That is, when multiple optimal solutions occur, RTS classi cation may be misclassi ed by only using a single set of resulting optimal solutions from (1) (or (3)). However, as shown in Zhu and Shen (1995) and Seiford and Zhu (1999), alternate optimal lambda values only a ect the estimation of RTS on those DMUs that truly exhibit CRS, and have nothing to do with the RTS estimation on those DMUs that truly exhibit IRS or DRS. This indicates that for e.g., P k j derived from the CCR model (1) can be used to estimate the RTS classi cation for a scale ine cient DMU. Therefore whether the CCR RTS method works is independent of the CRS linear production frontier assumption in the CCR model as claimed in Chang and Guh (1991) and Chang (1997). For the situation when a DMU truly exhibits CRS with multiple optimal lambda solutions, one may secure the help from the BCC model (2) (see, e.g., Theorem 5 in Zhu and Shen, 1997). We shall point out that this remedy is not ``the BCC method'' as claimed in Chang (1995, footnote 1 on p. 598). Note that the BCC RTS method uses optimal u o from model (3), while the CCR RTS method uses P k j from (1). Since Zhu and Shen's (1995) Theorem 5 still employs P k j from (1) to determine the RTS classi cation, it is a modi ed CCR RTS method. 3 In fact, it is possible that we use P k j from (1) to determine the RTS classi cation without linear dependency in e cient DMUs or (and) multiple optimal solutions. On the basis of Charnes et al. (1991), this particular linear dependency corresponds to the existence of non-extreme-e cient DMUs. Sueyoshi (1992) determined that the weakly e cient DMUs may also cause multiple optimal solutions. 4 However, multiple optimal solutions do not necessarily result in multiple values of P k j in (1). Finally note that in Chang and Guh (1991, p. 220), seven of their sample DMUs (P 1 ;...; P 7 ) exhibit CRS with the unique result of P k j ˆ 1, and P P exhibits DRS with the unique result of k j ˆ 10 > 1. By using the scale e ciency index method, we have h ˆ b (CRS) for P 1 ;...; P 7 and b ˆ f 6ˆ h (DRS) for P. Thus the CCR RTS method and the scale e ciency index method yield the consistent results as expected. The fact that the seven DMUs (P 1 ;...; P 7 ) exhibit CRS is not due to the CRS frontier assumption in the CCR model, but their RTS nature Zhu and Shen's (1995) Theorem 1 Theorem 1 in Zhu and Shen (1995) is misread and misinterpreted by Chang (1997) with an invalid counter-example. This theorem states that (Zhu and Shen, 1995, p. 591) If no linearly dependent relationship can be found in e cient DMUs, then we have the unique result of k j and further P k j can be uniquely determined. k j represent optimal values in the CCR model (1). 3 Also, a similar remedy can be developed for the BCC RTS method (see Seiford and Zhu, 1999). 4 See Seiford and Zhu (1998) for necessary and su cient conditions on the presence of multiple optimal solutions and the existence of weakly e cient DMUs.

6 616 J. Zhu / European Journal of Operational Research 127 (2000) 611±618 It is obvious that ``no linear dependency'' is the condition of the Theorem 1 and that ``P k j can be uniquely determined'' is the conclusion. (T1.i) in the proof of Theorem 1 (p. 591) should read: ``If p < m s, (because of the linear independency assumption) these p DMUs must be linearly independent'', where p is the number of e cient DMUs. Chang's (1997) counter-example (p. 599) violates the linear independency condition, therefore it is invalid. In fact, the proof of this theorem is partitioned into two cases: (i) p < m s and (ii) p P m s. Whether p < m s or p P m s has nothing to do with the linear independency. The linear independency condition is given by Theorem 1. Therefore, there is no mistake in Zhu and Shen's (1995) Theorem 1. The theorem is misinterpreted by taking p < m s (or p P m s) as the condition. max Xs and X s X m iˆ1 u r y ro u r y rj Xs iˆ1 v i x io ˆ 1; u r ; v i P e max h o ˆ h j ˆ v i x ij 6 0; j ˆ 1;...; n; P s P l ry ro m iˆ1 w ix io P s P l ry rj m iˆ1 w 6 1; j ˆ 1; 2;...; n; ix ij l r Pm iˆ1 w ix io P e; w i P m iˆ1 w ix io P e: Non-archimedean in nitesimal e As discovered by Green et al. (1996), Chang and Guh's (1991) proposal regarding the nonarchimedean in nitesimal e cannot work in practice. However, Green et al. (1996) do not point out that Chang and Guh (1991) incorrectly de ne their positive-multiplier-dependent e. In fact, this e should be de ned in a fractional DEA formulation, rather than multiplier DEA formulation. Chang and Guh's (1991) verbal suggestion itself `` nd all positive-multiplier linear production frontiers, and then, assign each DMU to the production frontier which gives the highest e ciency score'' is correct. If we correctly replace the in- nitesimal e, then Chang and Guh's (1991) e-approach works. However, this requires the existence of full dimensional e cient facets (FDEFs) (Olesen and Petersen, 1996). Obviously, if one cannot nd the associated FDEF (sometimes the associated FDEF may not exist), then a speci c DMU is unable to be assigned as suggested by Chang and Guh (1991). To start, we write the dual to (1) and its equivalent fractional programming model. Chang and Guh (1991) propose that by selecting a multiplier-dependent e, one can directly run the DEA model and obtain a scalar e ciency score for each DMU, particularly for DMUs with nonzero slacks. They suggest (Chang and Guh, 1991, p. 219) (A) replacing e by d ˆ min r;i {u r ; v i }, where u r ; v i represent positive multipliers obtained from (5) with e ˆ 0, and rerunning (5), or (B) nding all positive-multiplier linear production frontier and then assigning each DMU to the production frontier which gives the highest e ciency score. The intent of the verbal proposal (B) is quite clear: DMUs with non-zero slacks are re-evaluated by an extrapolated e cient facet. However, Green et al. (1996) show that Chang and Guh's (1991) proposal (A) fails to work because of the infeasibility of (5) with e ˆ d. In fact, based upon (B) and (6), proposal (A) should read: (A 0 ) replacing e by min r;i fu r ; v i g= P m iˆ1 w ix io, and then rerunning the following CCR model: P s max P l ry ro m iˆ1 w ix io P s P l ry rj m iˆ1 w 6 1; j ˆ 1; 2;...; n; ix ij l r P d; w i P d: 7

7 J. Zhu / European Journal of Operational Research 127 (2000) 611± Therefore, Chang and Guh (1991) misde ne the e and proposal (A) is incorrect. Proposal (B) is not equivalent to (A) but (A 0 ). Clearly, Chang and Guh (1991) employ proposal (A 0 ) and (7) to adjust the e ciency scores for their seven-dmu example. Therefore they do not encounter the infeasibility which is discovered by Green et al. (1996) when proposal (A) is employed. For more discussion of the remedy for these proposals, the readers are referred to Green et al. (1996) and Olesen and Petersen (1996). 4. Conclusion The paper points out that RTS misclassi cation is not due to the (piecewise) linear production function (or CRS) assumption in the CCR model, but a result of multiple optimal solutions in DEA model. Chang and Guh (1991) and Chang (1997) conclusions on whether Banker (1984) and Banker et al. (1984) RTS methods can be employed to estimate the RTS classi cation are incorrect. It is shown that Chang and Guh's (1991) and Chang's (1997) incorrect conclusions on RTS estimation result from their indi erence to the following two di erent assumptions: (a) all DMUs are assumed to be capable of operating under a CRS technology via the CCR model (1) and (b) all DMUs exhibit CRS. The current paper clari es the misunderstanding and further emphasizes that both CCR and BCC RTS methods can be employed to determine the RTS classi cation. Although the CCR model assumes that all DMUs are capable of operating under CRS, the RTS nature of each DMU is independent of this assumption. We should note that a number of modi cations to the CCR and BCC RTS methods have been developed for the situation when multiple optimal solutions are present (see e.g., Banker et al., 1996; Seiford and Zhu, 1999) and that numerous realworld applications show that both CCR and BCC RTS methods can be employed to determine the RTS classi cation (see Seiford, 1996). We also should note that RTS conditions and evaluations have been con ned to models using ``radial measures'' of relative e ciency. See Cooper et al. (1999) for the situation where RTS is discussed in the form of ``additive'' and ``multiplicative'' models. Finally, the current paper also clari es another problem in Chang and Guh's (1991) proposal of replacing the non-archimedean in nitesimal e by a particular positive number. References Banker, R.D., Estimating most productive scale size using data envelopment analysis. European Journal of Operational Research 17, 35±44. Banker, R.D., Bardhan, I., Cooper, W.W., A note on returns to scale in DEA. European Journal of Operational Research 88, 583±585. Banker, R.D., Chang, H., Cooper, W.W., Equivalence and implementation of alternative methods for determining returns to scale in data envelopment analysis. European Journal of Operational Research 89, 473±481. Banker, R.D., Charnes, A., Cooper, W.W., Some models for the estimation of technical and scale ine ciencies in data envelopment analysis. Management Science 30, 1078± Banker, R.D., Thrall, R.M., Estimation of returns to scale using data envelopment analysis. European Journal of Operational Research 62, 74±84. Chang, K.P., A note on a discussion of testing DMUs returns to scale. European Journal of Operational Research 97, 597±599. Chang, K.P., Guh, Y.Y., Linear production functions and the data envelopment analysis. European Journal of Operational Research 52, 215±223. Charnes, A., Cooper, W.W., Rhodes, E., Measuring the e ciency of decision making units. European Journal of Operational Research 2, 429±441. Charnes, A., Cooper, W.W., Thrall, R.M., A structure for classifying and characterizing e ciency and ine ciency in data envelopment analysis. Journal of Productivity Analysis 2, 197±237. Cooper, W.W., Seiford, L.M., Thrall, R.M., Zhu, J., Returns to scale in di erent DEA models. Working paper. Fare, R., Grosskopf, S., Estimation of returns to scale using data envelopment analysis: A comment. European Journal of Operational Research 79, 379±382. Fare, R., Grosskopf, S., Lovell, C.A.K., The Measurement of E ciency of Production. Kluwer Nijho, Boston. Green, R.H., Doyle, J.R., Cook, W.D., E ciency bounds in data envelopment analysis. European Journal of Operational Research 89, 482±490. Olesen, O.B., Petersen, N.C., Indicators of ill-conditioned data sets and model misspeci cation in data envelopment analysis: An extended facet approach. Management Science 42, 205±219.

8 618 J. Zhu / European Journal of Operational Research 127 (2000) 611±618 Seiford, L.M., Data envelopment analysis: The evolution of the state of the art (1978±95). Journal of Productivity Analysis 7, 99±137. Seiford, L.M., Zhu, J., On alternative optimal solutions in the estimation of returns to scale in DEA. European Journal of Operational Research 108 (1), 149±152. Seiford, L.M., Zhu, J., An investigation of returns to scale in data envelopment analysis. OMEGA 27 (1), 1±11. Sueyoshi, T., Measuring technical allocative and overall e ciencies by data envelopment analysis. Journal of Operational Research Society 43, 141±155. Zhu, J., Shen, Z.H., A discussion of testing DMUs returns to scale. European Journal of Operational Research 81, 590±596.