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

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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.

Introduction

The free disposal hull (FDH) model originally was designed as an alternative to data envelopment 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 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 FDH’s 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 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.

Section snippets

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   K, where K is an index set, producing R outputs with I inputs. Then (xk,yk)R+R+I,k  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 Färe et al., 1994). Let a technology T be defined byT=(x,y):ycanbeproducedbyx.Throughout the paper, we use the

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

Conclusion

The use of linear programming instead of traditional enumeration algorithms (Tulkens, 1993, 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 LP’s. However, an integrated LP framework as developed in this paper does offer some advantages. First, it allows to deal with more sophisticated FDH

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 (13)

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    These models, presented by Deprins, Simar, and Tulkens (1984), work under nonconvex technologies and evaluate the DMUs considering the closest inner approximation of the true strongly disposable (but possibly nonconvex) technology (Cherchye, Kuosmanen, & Post, 2001). FDH models have been studied by many scholars, including Tulkens (1993), Kerstens and Eeckaut (1999), Cherchye et al. (2001), Podinovski (2004c), Leleu (2006), Briec and Kerstens (2006), Soleimani-damaneh and Reshadi (2007), Soleimani-damaneh and Mostafaee (2009), Soleimani-damaneh and Mostafaee (2015), Kerstens and de Woestyne (2014), Kerstens and de Woestyne (2018), Diewert and Fox (2014), Abdelsalama, Fethi, Matallínc, and Tortosa-Ausina (2014), Cesaroni and Giovannola (2015), Cesaroni, Kerstens, and de Woestyne (2017a), Cesaroni, Kerstens, and de Woestyne (2017b), Krivonozhko and Lychev (2017), Krivonozhko and Lychev (2019). Returns to Scale (RTS) is an important criterion for analyzing the performance and productivity of DMUs (Cooper, Sieford, & Tone, 2007).

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    These models, which were first presented by Deprins, Simar, and Tulkens (1984), evaluate the Decision Making Units (DMUs) considering the closest inner approximation of the true strongly disposable technology. FDH models have been studied by various scholars, including (Briec & Kerstens, 2006; Briec, Kerstens, & Vanden Eeckaut, 2004; Cherchye, Kuosmanen, & Post, 2000; 2001; Kerstens, & Vanden Eeckaut, 1999; Leleu, 2006; Podinovski, 2004; Soleimani-damaneh, Jahanshahloo, & Reshadi, 2006; Soleimani-damaneh & Mostafaee, 2009; Soleimani-damaneh & Reshadi, 2007; Tulkens, 1993) and Mostafaee (2011). See also (Kerstens & Van de Woestyne, 2014) for a recent review of the solution methods in FDH models.

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