Piecewise loglinear frontier and log efficiency measures

Computers Ops Res. Vol. 22, No. 10, pp. 1031-1037, 1995

Pergamon

0305-0548(94)00091-3

PIECEWISE LOGLINEAR

FRONTIER MEASURES

Copyright © 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0305-0548/95 $9.50+ 0.00

AND LOG EFFICIENCY

Kuo-Ping Chang? and Yeah-Yuh Guh:~§ Department of Economics, National Tsing Hua University,Hsinchu 300, Taiwan, R.O.C. (Received February 1994; in revisedform November 1994) Scope and Purpose--In the Data Envelopment Analysis literature several methods use loglinear production functions for measuring decision-making-units' (DMUs)efficiency. It is found, however, that some of the estimated efficiency scores are not invariant under the changes in units of outputs and inputs, i.e. the scores can not be uniquely determined. To circumvent this problem, this paper proposes a distance-oriented efficiency measure, which implicitly assumes that the way of making an inefficient DMU efficient is to move the DMU vertically to its nearest loglinear production frontier. The distance efficiency measure is easy to apply and can serve as an alternative efficiency measure in addition to the traditional radial measure. Abstract--This paper shows that the Sueyoshi and Chang [Ops. Res. Lett. 8, 205-213 (1989)] remedy for Charnes et al. [Socio-Econ. Planning Sci. 16, 223-224 (1982); Ops Res. Lett. 2, 101-103 (1983)] multiplicative (or log) efficiency measure is not unit invariant, and Banker and Maindiratta's [Mgmt Sci. 32, 126-135 (1986)] approach should use positive-multiplier loglinear production frontiers to evaluate decision-making-units (DMUs). The paper also proposes a distance-oriented efficiency measure: the distance efficiency measure, which for the case of constant output and the case of linearly homogeneous production technology is unit invariant.

1. I N T R O D U C T I O N

In evaluating technical efficiency of productive units or firms from observed data, Charnes et al. [1] introduced a fractional programming approach: Data Envelopment Analysis (DEA). The DEA approach, based on estimated piecewise linear frontier, gives ratio measure of efficiency (i.e. CCR ratios). Charnes et al. [2] extended the DEA approach by developing a multiplicative (or log) efficiency measure. This model was modified by adding constant multipliers for outputs and inputs so that the efficiency measure will be invariant under the changes in units of outputs and inputs [3-5]. Banker and Maindiratta [6] also developed an alternative treatment for the piecewise loglinear frontier. This paper shows that the Sueyoshi and Chang remedy is not unit invariant, and the Banker and Maindiratta approach should use positive-multiplier loglinear production frontiers to evaluate DMUs efficiency. The paper also suggests taking logarithms on inputs and outputs of DMUs, employing the DEA approach to estimate loglinear frontiers; then using the frontiers to construct the distance efficiency measure. The distance efficiency measure implicitly assumes that the way of making an inefficient DMU efficient is to move the DMU vertically to its nearest loglinear frontier rather than radially to the piecewise loglinear frontier. The distance efficiency measure is unit ?Kuo.Ping Chang is a professor in the Department of Economics at the National Tsing Hua University. He obtained his Ph.D. from the University of Pennsylvania. His research interests are Productivity Analysis and Industrial Organization. He has published papers in Management Science, European Journal of Operational Research, Journal of Regulatory Economics and Journal of Productivity Analysis among others. :[Present address: Department of Industrial Engineering, Yuan Z¢ Institute of Technology, Neili, Taouyan 320, Taiwan R.O.C. §Yeah-Yuh Guh is an associate professor of the Industrial Engineering Department at the Yuan Ze Institute of Technology. He received a Ph.D. degree in industrial engineering from the National Tsing Hua University in 1989. He has published papers in European Journal of Operational Research, Journal of Operations Research Society, International Journal of Information and Management Science, Journal of Mathematics Analysis and Application and Computers and Mathematics with Applications. He is currently involving in design of service systems, performance evaluation and productivity management.

I031

1032

Kuo-Ping Chang and Yeah-Yuh Guh

invariant for the case of constant output and the case of linearly homogeneous production technology. Unlike the Banker and Maindiratta, and All and Seiford approaches, the distance efficiency measure does not need the addition of a constant term in the loglinear frontier, so it will rate fewer D M U s as being efficient. This paper is organized as follows. The following section examines the problems of employing the Charnes et al. [2,3], Sueyoshi and Chang [4] and Banker and Maindiratta [6] approaches in measuring technical inefficiency. The distance efficiency measure is discussed in Section 3. Concluding remarks appear in Section 4. 2. A R E V I E W O F T H E A L T E R N A T I V E

APPROACHES

The Charnes et al. [3] multiplicative model can be stated as follows (assuming that there are n DMUs, and each D M U uses m inputs to produce s outputs): For the k-th DMU" Max m k

s.t. e 0 \

=

e

~"

e

ei

vi

,

v~ j _< 1 , j = 1,...,n,

r=l

~) > 0 , ~ > 0,#r_> 6 > 0, and vi > 6 > 0.

(1)

By taking logarithms on inputs, Xij's, and outputs, Yrj's, (1) can be rewritten as: s

m

Max Mk ---- (~-- ~) + E

#r Yrk -- E

r=l m

$

s.t. ( ~ - ~) + E # r Y ~ j - E v i f ( i j r=l

vi)(ik

i=l

_ O, Vr, i,

(2)

where Xij = log Xij and I~ = log rrj. The dual of (2)is Min - 6

s

+

s

,

I.r=l ) t j ~rrj - - Sr+ ----

s.t.

]rrk, r =

1,..

.

,s,

j=l

- ~

AjXij - si- = -f(ik, i = 1,... ,m,

j=l n

EAj

= 1,Aj, S+r,S; > o, Vy, r,s.

(3)

j=l

The Charnes et al. [2] model has the same formulation as (1) except that the model requires = ~=

0 a n d 6 = 1.

The problem of the Charnes et al. [2] model is that the ranking of D M U s is dependent on the units of measurement employed. For example, assume that a firm uses two inputs (X 1 and X2) to produce a single output (Y), and there are four observations (DMUs): Pl(10 1 , 104 , 10 l ), P2(10 4 , 10 l, 101), P3(103, l04, 101), P4(107, 104, 102). Taking logarithms ((lOgloX1, logloX2, loglo Y)), and substituting them into (2) (where ~ = ~ = 0 and 6 = 1). The results show that v~ = t,~ = 1, #* = 5, P1 and P2 are termed efficient (MI* = M2* = 0), and /'3 and P4 are termed inefficient (M3* = - 2 , M ~ = - 1 ) . However, when the unit of the first input, ,I"1, is changed from one to ten (e.g. Pi changes from (101 , 104, 101) to (102, 104, 101), etc.),/'4 will be termed efficient (i.e. solving for P4 we get: v~ = v~ = 1, #* = 6 and M4* = 0).

L o g l i n e a r f r o n t i e r a n d log efficiency

1033

Charnes et al. [3] claimed the objective value of (2) is unit invariant, and (2) can give a piecewise Cobb-Douglas production frontier. However, (2) will not give meaningful results if 6 in (2) is an infinitesimal. For example, with DMUs P1, P2, P 3 and P4, say 6 = 10-1'°°°'°°°, all the DMUs will be termed efficient (the solution is obtained for ~ ) * = ~ * = 0 , ~ , ~ = ~ , ~ = # * = 1 0 -l'°°°'°°°, MI* ----M2* -- M3* -- M4* ~ 0). To circumvent the above problem two remedies have been suggested: Sueyoshi and Chang [4], and Ali and Seiford [5]. Sueyoshi and Chang [4] adopted Charnes et al. [7] suggestion by replacing the objective function in (3) with

hk = - 6

s

k) +

S

,

(4)

where 6 = 1/(m + s). With (4) the dual of (3) is identical to (2), except that the third constraint of (2) becomes #r > 1/[(m + s)(l?rk)], I:/> 1/[(m + S)(f(ik)], Vr and i. Sueyoshi et al. [4] claimed that hk in (4) is unit invariant. Unfortunately, this is not true. For example, with DMUs P1, P2, P 3 and P4, solving (2) for P 3 : (~* - ~*) = ~'~ = 1/~ = 1/9, #* = 6/9, h; = -2/9. When the unit of the first input is changed from one to 10, then h~ = - 1 / 6 , (~* - ~*) = 0, v~ = ~,~ = 1/12,/z* = 6/12. Ali et al. [5] argued that because of the convexity constraint, EjAj = 1, of (3), the value of the objective function is unit invariant. They suggested setting 6 of (3) equal to 1, i.e., in (2)/z, and v; > 1. In fact, 6 can be any real positive number as long as it is not very close to zero.i" It will be shown that because 6 of the Banker and Maindiratta [6] approach is an infinitesimal, the approach cannot be employed in the constant output case, and may fail to give meaningful efficiency scores in the varying output case. The Banker and Maindiratta approach can be stated as follows: Max ~k = gk + 6

,k +

~'k ,

n

s.t. ~k + Yrk "+"~3rk = E

Aj)'~j, r = 1,...,S,

j=l ?1

f(ik + ~ik= E Ajflij, i : l , . . . , m , j=l

~'~ Aj = 1,~k, ~/k, ~,k, )~j _>0,

(5)

j=l

where 6 is an infinitesimal. The dual of (5) is Min

Zk = ~ i=1

b'iY/k-

#r ~k

" ~ O~00

r=l

s.t.

1,...,n, i=1

r=l

E#

r = 1,ui,/z r _> 6 > 0 , V i , r.

(6)

r=l

In the constant output case (e.g. with P1, P2 and P 3 ) L,~ and u~ can be set equal to the infinitesimal, 6, and #* = a~o = 1; then Z~ = 6(f~k + X2k) -- 1 + 1,Vk. With a very small positive 6, all efficiency measures (2~'s) will be indistinguishable from one another. In the varying output case, for example, assume four DMUs: P~, P2,/'4 and P5 ( 108, 104, 102)• P5 is less efficient than/'4 because it uses more unit of X~ t h a n / 4 . Solving (6) for P1 and P2: #* = 1, t~o = 0, t,~ = u~ = 1/5, Z~ = Z~ = 0; for P4 ?Whe s6 is an infinitesimal, we will have the same problem as Chames et al. [3]. For example, substitute P1, P2, P3, P4 and P5 (1.0, 1 0 , 10") into (2) where 6 - - 1 0 -I'°eo'°e°, all the D M U s are termed etficient (i.e. ~ * = ~ * - - 0 , * * I 000 000 * * * * , , vl = ~ = # = 10- ' ' , M1 = M~ = . M~ = M~ = M~^ ~ 0). When 6 is not , an infinitesimal, the magnitude of M~ • ^, , • , • , • Is dependent on that of & For example, ff 6 = 1, ~7 -- 0, ~* = 1, ul = v2 = 1, # = 6, M1 = M~ = M~ = 0, M~ = - 2 , M ; = - 1 . I f 6 - - 0.5,~* --0,~* -- 0.5, v~ = u ~ = 0.5,/~* = 3, Ml* =M2* =M4* = 0, M3* = - 1 , Ms* -- -0.5. I f 6 = 0.01, O* ----O, ~ = 0.01, u~ = u~ = 0.01, #* = 0.06, MI* = 342* = M,~ -----O, M3* = -0.02, Ms* --- -0.01.

1034

Kuo-Ping Chang and Yeah-Yuh Guh

and Ps: #* = 1, a~0 = 2, 1/~ = v~ = 6, Z~ = 6 ( 7 + 4),and Z~ = 6(8 + 4). Again, since 6 is an extremely small positive number, Z1, ^* Z~, ^* Z~ and 2~5 are indistinguishable from one another. [Note that (6) will always term the D M U having the highest output level efficient. For example, with three D M U s P1,P2 and P ' (10 ;~, 10~, 102), solving (6) for P': #* = 1, a~0 = 2, v~ = v~ =/~, P ' is always termed efficient as long as/31 and/32 are real numbers.] A remedy for this problem is to follow Chang and Guh's [8] suggestion: set 6 of (6) equal to zero, from (6) find all positive-multiplier loglinear frontiers [i.e. #r and vi in (6) must be positive], and then assign each D M U to the frontier which gives the highest efficiency score (2~k). 3. T H E D I S T A N C E

EFFICIENCY

MEASURE

The Ali and Seiford [5], and Banker and Maindiratta [6] approaches need to add a constant term in the loglinear frontier, i.e. r) and ~ in (2), and a00 in (6). In this section we suggest an alternative measure: the distance efficiency measure, which does not need adding a constant term. Chang and Guh [8] have shown that the reason why the DEA approach gives a piecewise linear frontier: because the approach linearly aggregates inputs and outputs. Thus, to estimate a piecewise loglinear frontier it is necessary to replace linear aggregations in the DEA method with loglinear aggregations, i.e. the "multiplicative" DEA method can be stated as:

MaxIYlk=(r=~lUr~"k)/(i=~lViXik)

'

U,,Vi >_ 0, I7"0 = log Yrj, f(ij = logXij, Vr, i, and/.

(7)

Using the same estimation procedures as in the DEA method, this fractional programming problem can be transformed into a linear programming problem. With the transformation Ur = fUr, v i = t V i , Vr, i, and t -1 = ~iVif(ik, ( 7 ) becomes

s Max ~k ----~

r=l u r )rrj- ~

s.t. r=l

Urgk ~3iXij ~ O,

i=1

~-~viXik = 1,u~,vi > 0,Vr, i, andj.

(8)

i=1 The loglinear production possibility set is

/ T =

n (i", Y) I log Y~ _< K ~

A}log Y~], log t"i _> K

j=l

.

.

j=l

j=l

x ~ AjlogXij, K > 0, A~. < 0,~--~ Aj = 1,Vr, iandj

}

(9)

The supporting hyperplane for this set, i.e. the loglinear frontier, is Y~.=lUr s • Yrk ^ -- ~i=l'Ui m • S ^i k = O. Dividing the frontier by Y2"r=i ~ Ur, * it becomes ~']~r=l~r s , Yrk ^ _ ~']~m=l • l/~Xik = 0, where ~r -~- Ur/~']rU; and u~ = v*/~,,u*, which is exactly the loglinear frontier (6) would give if there were no a00. The loglinear CCR ratio [:~kin (8)] is not unit invariant. However, the loglinear frontiers derived from (8) can be used to construct the unit-invariant distance efficiency measure. In the constant output case (e.g. with P1, P2 and P3), the distance efficiency measure is derived by first calculating the shortest vertical distances between each D M U and the loglinear frontiers (e.~., PI: 0, P2: 0, P3: v~, where, because all IT"k'Sare equal to one, the loglinear frontier is: (1/5X1 + (1/5),('2 = 1); secondly, by dividing each of the distances by the largest one, and multiplying by - 1 , distance

b~glinear frontier and log efficiency

1035

Table 1. Theloglinear frontier and the distance e5ciency measmt The distance cIIicicncy measure

i;

u:

u:

v;

1

l/5

l/5

I

0

1

115

l/5

I

0

517

117

l/7

517

-215

519

119

119

519

-415

5/10

l/IO

l/l0

5110

-1

5/W

Wt

Wt

516

-l/5

I

116

W

1

0

1

116

116

1

0

618

l/8

IlfJ

618

-215

6/10

l/IO

l/l0

6110

-415

6111

l/11

l/11

6/11

-1

617

117

117

617

-115

10’

P, =

0 10’

10'

10’ P3 = (1IO4 10’

tps is evaluated by the positive-multiplier loglinear frontier

(l/5)2, + (1/5),tz

= t rather than by (O)$, + L = P.

efficiency measures can be obtained (Table 1). The range of distance efficiency measures are between 0 and - 1, where 0 represents the most efficient DMU, and -1, the least efficient one. Table 1 indicates that when the unit of the first input is changed from one to 10 (e.g. Pr , P2 and P3 becomes PI, 4 and Ps), the distance efficiency measures remain unchanged. This is because (as shown in Fig. 1) a change in the unit of A’, results in a horizontal shift of locations of all the DMUs, which does not affect the relative positions of the DMUs. In the varying output case, another assumption can be added: the firm has a linearly homogeneous production technology (i.e. the firm’s production frontier is tY =f (tXi, Hz), where f is a scalar) so that all DMUs can have the same output level. For example, with PI, Pz, P3, P4 and Ps, assume linearly homogeneous production technology, to have the same output level

1036

Kuo-Ping Chang and Yeah-Yuh Guh X

\

\

I I l

I I I

I \\

I

\p~11

\_\ [

P3

P3

/%

i

\

\

{I/SlXl +

P2

-

(llSlX2 = 1

-

\~/\

-

~2\,~, / \

\

0

Xl

Fig. h The piecewise loglinear frontier and the distance efficiency measure. (101) the inputs and outputs of P4 and P5 can be divided by 10:P4 becomes P~ (106, 103, 10 l) and P5 becomes P~(107, 103, 101). Table 1 shows that the distance efficiency measures for Pl, P2, P3, P~ and P~ are unit invariant. Note that with D M U s Pl, P2, P3, P4, P5 and P6(105, 101, 101), (8) will term/'6 efficient, although it uses more X1 unit than Ps. This is because P6 is evaluated by the frontier (0)~'1 + X2 = Y. A remedy for this problem is to adopt Chang and Guh's suggestion [8]: finding all the positivemultiplier loglinear frontiers, and then assigning D M U s to the frontier which gives the highest efficiency score. P6 is then evaluated by (1/5)Xl + (1/5)X2 = l~, and its distance efficiency score is -1/5 (Table 1). Note also that unlike (2) and (6), the loglinear production frontier in (8) does not have a constant term; hence the distance efficiency measure will term less D M U s efficient than the Ali and Seiford [5], or Banker and Maindiratta [6] approaches.~" The distance efficiency measure suggests the way of making an (technically) inefficient D M U efficient is to move the D M U vertically to its nearest frontier, rather than radially to the piecewise frontier (i.e., the decision-maker may not reduce all inputs equiproportionally as the CCR ratio measure suggests). Since there is an infinite number of ways to move a D M U to a given frontier, and "there is no reason to measure technical efficiency radially, even for homothetic technologies" [9: p. 151], the distance efficiency measure can serve as an alternative log efficiency measure.~ 4. C O N C L U D I N G R E M A R K S This paper has shown that the Sueyoshi and Chang remedy for Charnes et al. [2,3] is not unit invariant, and the Banker and Maindiratta [6] approach should use positive-multiplier leglinear tFor example, with D M U s PI, P2, P3,/'4 and Ps, as shown in Table I, only PI and/'2 are termed efficientby the distance efficiencymeasure method. By the All and Seiford[5]approach where ~ isnot an infinitesimal,Pt,/'2 and/'4 are termed efficient(seefootnoteon,p.3).By the B M approach (wi~ Chang and Guh's remedy [8]),P,, P2 and P4 are termed efficient: From (6),~ = 2~ = ~; = 0, ~, = I, ~i = ~'2 = I/6, ~00 = I/6.

~:Ifinput pricesdata are available,the way of making an inefficientD M U production frontierwhich has minimum costs.

emcient is to move the D M U

to the point of a

Loglinear frontier and log efficiency

1037

production frontiers to evaluate DMUs' efficiency. The paper also suggests taking logarithms on DMUs' inputs and outputs, employing the original DEA method to estimate loglinear frontiers; then using derived frontiers to construct the distance efficiency measure. The distance efficiency measure is unit invariant for the case of constant output and the case of linearly homogeneous production technology. Unlike the Ali and Seiford [5], and Banker and Maindiratta [6] approaches, the distance efficiency measure does not need the addition of a constant term, so it will rate fewer DMUs as being efficient. Acknowledgement--The authors are indebted to Prof. William T. Ziemba and two anonymous referees for their helpful comments and suggestions. Views expressed in this paper, however, are those of the authors.

REFERENCES 1. A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decisionmaking units. Eur. J. Opl Res. 2, 429-44 (1978). 2. A. Charnes, W. W. Cooper, L. Seiford and J. Stutz, A multiplicative model for efficiency analysis. Socio-Econ. Planning Sci. 16, 223-224 (1982). 3. A. Charnes, W. W. Cooper, L. Seiford and J. Stutz, Invariant multiplicative efficiency and piecewise Cobb-Douglas envelopments. Ops Res. Lett. 2, 101-103. 4. T. Sueyoshi and Y. Chang, Efficient algorithm for additive and multiplicative models in data envelopment analysis. Ops Res. Lett. 8, 205-213 (1989). 5. A. I. Ali and L. M. Seiford, Translation invariance in data envelopment analysis. Ops Res. Lett. 9, 403-405 (1990). 6. R. Banker and A. Maindiratta, Piecewise loglinear estimation of efficient production surfaces. Mgmt Sci. 32, 126-135 (1986). 7. A. Charnes, W. W. Cooper, B. Golany, L. Seiford and J. Stutz, Foundations of data envelopment analysis for ParetoKoopmans efficient empirical production functions. J. Econometrics 30, 91-107 (1985). 8. K.P. Chang and Y. Y. Guh, Linear production functions and the data envelopment analysis. Eur. J. Opl Res. 52, 215-223 (1991). 9. R. F/ire and C. A. K. Lovell, Measuring the technical efficiency in production. J. Econ. Theory 19, 150-162 (1978).