A linear programming framework for free disposal hull technologies and cost functions: Primal and dual models

European Journal of Operational Research 168 (2006) 340–344 www.elsevier.com/locate/ejor

Continuous Optimization

A linear programming framework for free disposal hull technologies and cost functions: Primal and dual models Herve´ Leleu

*

CRESGE/LABORES/CNRS, Catholic University of Lille, 60 Boulevard Vauban, B.P. 109, F-59016 Lille Cedex, France Received 22 October 2003; accepted 19 April 2004 Available online 28 July 2004

Abstract The free disposal hull (FDH) model, introduced by Deprins et al. [The Performance of Public Enterprises Concepts and Measurements, Elsevier, 1984], is based on a representation of the production technology given by observed production plans, imposing strong disposability of inputs and outputs but without the convexity assumption. In its traditional form, the FDH model assumes implicitly variable returns to scale (VRS) and the model was solved by a mixed integer linear program (MILP). The MILP structure is often used to compare the FDH model to data envelopment analysis (DEA) models although an equivalent FDH LP model exists (see Agrell and Tind [Journal of Productivity Analysis 16 (2) (2001) 129]). More recently, specific returns to scale (RTS) assumptions have been introduced in FDH models by Kerstens and Vanden Eeckaut [European Journal of Operational Research 113 (1999) 206], including non-increasing, non-decreasing, or constant returns to scale (NIRS, NDRS, and CRS, respectively). Podinovski [European Journal of Operational Research 152 (2004) 800] showed that the related technical efficiency measures can be computed by mixed integer linear programs. In this paper, the modeling proposed here goes one step further by introducing a complete LP framework to deal with all previous FDH models.
2004 Elsevier B.V. All rights reserved. Keywords: Free disposal hull model; Data envelopment analysis; Returns to scale; Linear programming

1. Introduction The free disposal hull (FDH) model originally was designed as an alternative to data envelop-

*

Tel.: +33 32013 4060; fax: +33 32013 4070. E-mail address: [email protected]

ment analysis (DEA) models, where only the strong (free) disposability of inputs and outputs is assumed. As introduced by Deprins et al. (1984), their main contribution was to relax the convexity assumption of DEA models. As such, the FDH model was initially presented as a variable returns to scale (VRS) DEA model in which activity variables were binary thereby excluding

0377-2217/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.04.042

H. Leleu / European Journal of Operational Research 168 (2006) 340–344

the linear combinations of observed production plans. Therefore, the FDH model was traditionally represented as a mixed integer linear programming (MILP) problem for nearly two decades. Because of the MILP approach, however, the more profound nature of the FDH model was hidden. A first step towards recognizing FDHÕs potential was made in the paper of Kerstens and Vanden Eeckaut (1999) where specific returns to scale assumptions (RTS) were introduced in the FDH approach. Further discussion can be found in a recent contribution by Briec et al. (2004). The approach presented here is mainly motivated by the use of RTS assumptions to infer the characterization of returns to scale in a FDH VRS model. As shown in Briec et al. (2000), RTS on a VRS technology are unambiguously defined by comparing optimal solutions of the NIRS and NDRS problems. Podinovski (2004b) developed further this approach by introducing the distinction between local and global RTS. There are two computational methods used to solve the FDH models. The first one is based on enumeration algorithms as proposed by Tulkens (1993), Cherchye et al. (2001) or Briec et al. (2004). The second one is the use of mathematical programming. Based on the technologies defined in Kerstens and Vanden Eeckaut (1999), the computation of the technical efficiency measures require solving non-linear mixed integer programs. Recently, Podinovski (2004a) simplified the approach by showing that there exist equivalent MILP models. These approaches, however, do not take into account that an LP model exists which can solve the FDH VRS model as demonstrated in Agrell and Tind (2001). In this paper, we rely on the Agrell and Tind (2001) approach to introduce RTS in FDH models within a LP framework. We also give the insightful economic interpretation of the dual program. Apart from the effect on the technology, the convexity assumption also impacts all other economic value functions. In fact, the non-convexity of the technology leads to non-convex cost, revenue or profit functions (Kuosmanen, 2003). While Briec et al. (2004) introduced the FDH-cost function along with enumeration algorithms, we extend our approach to estimate the FDH-cost

341

function by LPs. We, therefore, obtain the static decomposition of the economic inefficiency into the technical and the allocative parts under a complete LP framework. 2. The linear FDH technology In the first step, the production technology used in the paper is defined from a set of observed production plans (xk, yk), k 2 K, where K is an index set, producing R outputs with I inputs. Then RþI ðxk ; y k Þ 2 Rþ ; "k 2 K. Note also that we assume at least one input and at least one output of each production plan is strictly positive to ensure the feasibility of all linear programs (see Fa¨re et al., 1994). Let a technology T be defined by T ¼ fðx; yÞ : y can be produced by xg:

ð2:1Þ

Throughout the paper, we use the radial input distance function to compute the technical inefficiency of observed production plans but the results may be straightforwardly extended to other types of distance functions. Therefore, the technical inefficiency of an observed production plan (xo, yo) is defined by Eðxo ; y o Þ ¼ min fho : ðho xo ; y o Þ 2 T g: o

ð2:2Þ

h

As defined here, the FDH technology exhibits a strong (free) disposability assumption on T but does not impose any convexity assumption. Without any reference to specific RTS, the traditional FDH technology is under variable returns to scale and is labeled TFDH  VRS. Traditionally, the FDH technology is represented by its production possibility set ( K K X X T FDHVRS ¼ ðx; yÞ : zk y k P y; zk xk 6 x; k¼1 K X

k¼1

)

zk ¼ 1; zk 2 f0; 1g; k 2 K :

k¼1

ð2:3Þ Obviously, this representation of the FDH technology leads to a MILP problem with binary variables when computing the technical inefficiency of an observed production plan (xo, yo). Starting with this definition, Kerstens and Vanden Eeckaut

342

H. Leleu / European Journal of Operational Research 168 (2006) 340–344

(1999) have introduced returns to scale in the FDH technology but as pointed out by Podinovski (2004a) in order to compute the technical efficiency measure, we need to solve three mixed integer nonlinear models. Podinovski (2004a) then shows that this task can be accomplished by solving equivalent mixed integer linear programs. This simplification is valuable but we can go further by using the representation of the FDH technology developed in Agrell and Tind (2001). They have shown that there exists an equivalent LP problem to the traditional MILP problem to compute the FDH efficiency measure under variable returns to scale. We propose here to extend their work to include the alternative RTS assumptions. To begin with, the following linear program is derived from the Agrell and Tind (2001) approach although they present the problem under an output orientation including the slacks. EFDHVRS ðxo ; y o Þ ¼ min o hk ;zk

s:t: zk y kr P zk y or ; zk xki 6 hok xoi ; K X zk ¼ 1;

K X

hok

ðP1Þ

k¼1

zk P 0;

k¼1

s:t: ðzk þ wk Þy kr P zk y or ; wk Þxki

ðzk þ K X zk ¼ 1;

6 hok xoi ;

r 2 R; k 2 K; i 2 I; k 2 K;

k¼1

zk P 0;

8k 2 K;

wk 2 Ck ; where

ðP2Þ

8k 2 K

Ck 2 fNIRS; NDRS; CRS g with NIRS ¼ fwk : wk 6 0g;

Proof. Following (2.4), RTS assumptions are introduced in (P1) via the scaling parameter (1 + d). This leads to EFDHRTS ¼ min o

k 2 K:

hk ;zk ;d

We introduce the non-increasing, non-decreasing and constant returns to scale assumptions (respectively, NIRS, NDRS and CRS) by reference to VRS technology. We use a slightly modified definition compared to the one given in Podinovski (2004a). T FDHRTS ¼ fðx0 ; y 0 Þ : ðx0 ; y 0 Þ ¼ ð1 þ dÞðx; yÞ; ðx; yÞ 2 T FDHVRS ; d 2 Cg; where C 2 f NIRS; NDRS; CRSg; with NIRS ¼ fd : 1 6 d 6 0g: NDRS ¼ fd : d P 0g; CRS ¼ fd : d P  1g:

hk ;zk ;wk

NDRS ¼ fwk : wk P 0g; CRS ¼ fwk : wk unconstrained g:

k¼1

r 2 R; k 2 K; i 2 I; k 2 K;

Proposition 1. The FDH technical efficiency measure under NIRS, NDRS and CRS can be computed from the following linear program: K X hok EFDHRTS ¼ min o

K X

hok ;

k¼1

s:t: ð1 þ dÞzk y kr P zk y or ; ð1 þ K X

dÞzk xki

6 hok xoi ;

r 2 R; k 2 K; i 2 I; k 2 K;

ð2:5Þ

zk ¼ 1;

k¼1

zk P 0;

8k 2 K;

d 2 C:

ð2:4Þ

Now, by defining wk = dzk"k 2 K, we obtain (P2) where Ck can be directly deduced from the definition of the wk and the non-negativity of the zk. h

Now, by combining (P1) and (2.4), we can state the main result of the paper.

In (P2), RTS are introduced via a scaling parameter for each production plan belonging to the VRS technology allowing the constraint on the sum of activity variables to remain unchanged

H. Leleu / European Journal of Operational Research 168 (2006) 340–344

PK ð k¼1 zk ¼ 1Þ. In comparison to the DEA framework, we note that the RTS are introduced by modifying the constraint on the sum of activity variables. This is not intuitive since this constraint stands for the convex combinations of production plans and it is not related at all with RTS. In fact, this is a way to linearize the problem (see Banker et al., 1984) precluding an intuitive economic interpretation of RTS. Therefore, the approach defined here appears more appealing and leads also to a relevant economic interpretation of the dual formulation of (P2) which is defined by EFDHRTS ðxo ; y o Þ ¼ max po ; ukr ;vir ;po

s:t: R X r¼1 I X

ukr ðy kr  y or Þ  ¼ 1;

ukr y kr 

ukr y kr 

r¼1

vki xki P 0;

k 2 K; under NIRS;

I X

ukr y kr 

I X

I X

3. Extension to the FDH-cost functions We can now directly extend the model to compute allocative and economic inefficiencies under the FDH technology. While we consider the input orientation throughout the paper, the focus of the discussion is on cost function (note that similar arguments prevail for revenue or profit functions). It is worth noting that in most cases the FDH-cost function will be not convex since the related technology is not convex. The impact of convexity on economic value functions are discussed in details in Kuosmanen (2003). Cost functions for FDH technologies have been recently introduced by Briec et al. (2004) along with enumeration algorithms. The following linear program which computes the minimum cost for each evaluated production plan, can be easily derived from (P2), given a set of input prices: ! I K X X k k o e C FDHRTS ¼ e x i ; zk ; wk min pi xi i¼1

vki xki 6 0;

k 2 K; under NDRS;

ðzk þ wk Þy kr P zk y or ; wk Þxki

vki xki ¼ 0;

k 2 K; under CRS;

i¼1

ukr y kr 

firms while, in NIRS and NDRS, the shadow profit must be, respectively, non-negative and non-positive, noting that the VRS assumption does not impose any constraint on the shadow profits.

k¼1

s:t:

i¼1

r¼1

or R X

I X

ðP3Þ

i¼1

r¼1

or R X

k 2 K;

k 2 K;

r 2 R; k 2 K; i 2 I; k 2 K;

r¼1

or R X

vki xki þ po 6 0;

i¼1

vki xoi i¼1 ukr P 0; vki P 0; and R X

I X

343

ðzk þ K X zk ¼ 1;

vki xki ;

k 2 K;

ðP3Þ

unconstrained under VRS: In (P3), the shadow profit of the observed production plan is compared to the shadow profit of all other production plans by using a specific set of shadow prices for each comparison. Clearly, the RTS assumptions are directly linked to some constraints on the value of the shadow profits. As a classical result in economics, the CRS assumption imposes a zero shadow profit for all

r 2 R; k 2 K; i 2 I; k 2 K; ðP4Þ

k¼1

zk P 0;

i¼1

6e x ki ;

k 2 K;

e x ki P 0; i 2 I; k 2 K; wk 2 Ck ; k 2 K: The economic inefficiency can now be computed as the ratio of the observed to the minimum cost and, as usual, the allocative inefficiency is computed as the ratio between the economic and the technical inefficiencies. 4. Conclusion The use of linear programming instead of traditional enumeration algorithms (Tulkens, 1993;

344

H. Leleu / European Journal of Operational Research 168 (2006) 340–344

Cherchye et al., 2001; Briec et al., 2004) to compute FDH technologies deserves some comments. It is worthy to note that the approach used here does not afford any computational advantage over enumeration algorithms which are computationally superior to LPÕs. However, an integrated LP framework as developed in this paper does offer some advantages. First, it allows to deal with more sophisticated FDH models without high additional costs in terms of program development. For instance, program (P2) can be readily extended to introduce constraints on input or output substitution, regulatory constraints or environmental variables. Second, the duality in linear programming is also a major benefit. As presented here, the dual formulation of the FDH model with various RTS assumptions enhances the economic interpretation of the FDH technology in terms of shadow prices. Duality also offers a way to include economic information in the evaluation process. Along with the dual program (P3), one can easily include weight restrictions or constraints on the prices (see for example, Kuosmanen and Post, 2001). Finally, from a practical point of view, the use of LPÕs can be justified by the lack of proper enumeration software for these relatively new FDH models. While most people use specialized LP software to solve the family of DEA models, it could be useful to have a similar framework to solve the family of FDH models without having to implement a specific enumeration algorithm for each new model. Acknowledgement We thank three anonymous referees for helpful comments which greatly improved the exposition of the paper. The paper benefited from comments by L. Coudeville and B. Dervaux. The usual disclaimer applies.

References Agrell, P.J., Tind, J., 2001. A dual approach to nonconvex frontier models. Journal of Productivity Analysis 16 (2), 129–147. Banker, R.D., Charnes, A., Cooper, W.W., 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 1078–1092. Briec, W., Kerstens, K., Vanden Eeckaut, P., 2004. Non-convex technologies and cost functions: Definitions, duality and nonparametric tests of convexity. Journal of Economics 81 (2), 155–192. Briec, W., Kerstens, K., Leleu, H., Vanden Eeckaut, P., 2000. Returns to scale information on nonparametric deterministic technologies: A simplification of goodness-of-fit methods. Journal of Productivity Analysis 14 (3), 267–274. Cherchye, L., Kuosmanen, T., Post, G.T., 2001. FDH directional distance functions: With an application to European commercial banks. Journal of Productivity Analysis 15, 201–215. Deprins, D., Simar, L., Tulkens, H., 1984. Measuring labor efficiency in post offices. In: Marchand, M., Pestieau, P., Tulkens, H. (Eds.), The Performance of Public Enterprises Concepts and Measurements. Elsevier, Amsterdam, pp. 247–263. Fa¨re, R., Grosskopf, S., Lovell, C.A.K., 1994. Production Frontiers. Cambridge University Press, Cambridge. Kerstens, K., Vanden Eeckaut, P., 1999. Estimating returns to scale using non-parametric deterministic technologies: A new method based on goodness-of-fit. European Journal of Operational Research 113, 206–214. Kuosmanen, T., 2003. Duality theory of non-convex technologies. Journal of Productivity Analysis 20 (3), 273–304. Kuosmanen, T., Post, G.T., 2001. Measuring economic efficiency with incomplete price information: With an application to European commercial banks. European Journal of Operational Research 134 (1), 43–58. Podinovski, V.V., 2004a. On the linearisation of reference technologies for testing returns to scale in FDH models. European Journal of Operational Research 152, 800–802. Podinovski, V.V., 2004b. Local and global returns to scale in performance measurement. Journal of Operational Research Society 55, 170–178. Tulkens, H., 1993. On FDH efficiency: Some methodological issues and application to retail banking, courts, and urban transit. Journal of Productivity Analysis 4 (1), 183– 210.