Non-Parametric (Programming) Approach

Unlike econometric models, non-parametric approaches do not require a pre-defined functional formulation but use linear programming techniques to determine rather than estimate the efficiency frontier. Much of the research using linear programming techniques involves the application of data envelopment analysis (DEA) and the free disposal hull (FDH). FDH is a non-parametric technique but differ from DEA by excluding linear combinations of production units from the analysis.

Primarily, DEA seeks to measure technical efficiency (TE) without using price and cost data or specifying a functional formulation. However, when information about costs and prices is available, DEA allows for the calculation of allocative efficiency (AE).

Assuming a set of N (n=1,2,...,N ) DMUs (Decision Making Units)³ in the sample, each observation, DMU (_j j =1,2,...,n), uses m inputs x (_ij i =1,2,...,m) to produce s outputs y (_rj r =1,2,...,s). The efficiency ratio of DMU_j can be defined as the ratio of its weighted sum of outputs over its weighted sum of inputs:

=

3: We use the phrase Decision-Making Units (DMUs) throughout this thesis to refer to benchmarked units or firms under study. The phrase was first used by Charles (1978) to include non-market units such as schools and hospitals.

In an output orientation, we seek to find the maximum output that can be produced while holding the input at its current level. This is a maximisation problem, which can be solved using linear programming with the following objective function:

n

In equation 10, each DMU selects input and output weights that maximize its efficiency score and the problem is run N times to identify the relative efficiency scores of all DMUs. Input-oriented models can be formulated in the same way by minimising the input while holding the output constant. Equation (11) shows the CCR formulation for the input oriented model. and Rhodes) for CRS but can also be expressed as a DEA-BCC model (due to Banker, Charnes and Cooper) to account for VRS by adding the extra constraint

∑

= choice of orientation depends on the objective of benchmarking (input conservation versus output augmentation), and on the extent to which inputs and outputs are controllable. Both models should estimate exactly the same frontier, with the same set of DMUs being identified as efficient under either model. However, efficiency scores of inefficient DMUs may differ under VRS.

Figure 5: DEA production frontier under the single input and single output scenario (Adapted from De Borger et. al, 2002)

In the simple scenario of a single-input and a single-output, Figure 5 illustrates DEA models and efficiencies under different orientations and scale technologies. The DEA frontier consists of a convex hull of intersecting planes that envelops the efficient data points A, B, C, D, E and F. Note that only units B and C are efficient under both CRS and VRS, which confirms that DEA-CRS is more restrictive than DEA-VRS. For the inefficientDMU , the projection towards the CRS frontier (the straight line) makes _j point j the new target, while _c j_i j_oand j are the VRS targets for the input, output and _a additive orientations, respectively. Unlike for CCR and BCC, the additive model is un-oriented and combines simultaneous input reduction and output increase.

In Figure 5, both DMUs E and F are on the frontier indicating that they have an efficiency score of 1. However, DMU F can still reduce its inputs by some units to reach DMU E. This individual input reduction is called input slack. Input and output slack formation is the product of the convex structure of the DEA frontier. The revised input-oriented VRS model from equation 11 can write as in equation 12 where ε _{is an} infinitesimally small positive number, while is s and _i⁻ s are the input and the output _r⁺ slacks, respectively.

Input

Output

O

A B

C

D E

I

J_o

J_a

J_c J_i

F

J

VRS convex frontier

0 model. Figure 6 depicts TE and AE measures in both orientations. When cost and price information is available, one can draw the iso-cost line (CC’ combination of x₁ and x₂ giving rise to the same level of cost expenditure) for the input-oriented model and the iso-revenue line DD’ (combination of y₁ and y₂ giving rise to the same level of revenue) for the output-oriented model. Allocative efficiencies for input (AEi) and output (AEo) orientations can therefore be calculated, corresponding in our example to the ratios OJb/OJ and OJ/OJb, respectively. Finally, note that the reference set or peers for the inefficient DMU are E and F in the input-oriented model, and F and G in the _j output-oriented model.

Figure 6: Illustration of DEA input and output orientations, excluding the effect of technological change (Adapted from De Borger et. al, 2002)

J_a

DEA applications in ports are quite recent with the first attempt being attributed to Roll and Hayuth (1993). Estache et al. (2002) provide a detailed review of the use of DEA techniques in ports although since then many studies have been published on the subject. The literature in the field may be divided into a-four categorisation criteria:

• Between CCR models (Valentine and Gray, 2001; Tongzon, 2001) and DEA-BCC models (Martinez-Budria et al., 1999; Serrrano and Castellano, 2003), although recent studies use both models;

• Between input-oriented models (Barros, 2003) and output oriented models (Wang and Cullinane, 2005);

• Between applications looking at aggregate port operations (Barros and Athanassious, 2004) and those focusing on a single port operation (Cullinane et. al, 2004);

• Between studies relying on DEA results solely and those complementing DEA with a second stage analysis such as regression or bootstrapping (Turner et. al, 2004; Bonilla et. al, 2002).

The DEA approach to efficiency analysis has many advantages over parametric approaches. The methodology accommodates multiple inputs and outputs, and provides information about the sources of their relative (factor specific) efficiency. DEA neither imposes a specification of a functional form, nor requires assumptions about the technology. In DEA, firms (or DMUs) are benchmarked against the achievable best performance rather than against a statistical measure, an average or theoretical standard.

There is also no necessity to pre-define relative weight-relationships, which should free the analysis from subjective weighting. Similarly, each input/output variable can be measured in its natural measurement units, e.g. dollar values versus physical measures.

Another useful feature of DEA is that it attempts to find one or more efficient reference point(s) (a peer or combination of peers) for each inefficient DMU, which also informs about improvement projection possibilities in terms of specific input reductions, output increases, or both. In addition and although DEA requires a dataset of at least three to four times the number of input and output parameters (Bowlin, 1998), this is still smaller than the dataset required under SFA. All such features and others make DEA particularly attractive for port-related efficiency studies; which justifies the increasing number of academic literature on the subject.

On the other hand, one could argue that the same features that make DEA a powerful tool also create major limitations. Primarily, one may question the logic behind the virtual output/input construction under DEA, especially when outputs and inputs of a different nature are considered. A major drawback of DEA stems from the sensitivity of efficiency scores to the choice of, and the weights attached to, input and output variables. This is of major concern because a DMU can appear efficient simply because of its patterns of inputs and outputs. Moreover, input (output) saving (increase)

potentials identified under DEA are not always achievable in port operational settings, particularly if this involves small amounts of indivisible input or output units.

Another problem with DEA is that while there is no prior requirement of weight selection, the technique does not investigate relationships between variables within and across the sampled DMUs. As such, the technique does not account for substitution possibilities between inputs or transformation possibilities between outputs. This is of particular importance in the context of container port benchmarking because factor endowments, utilisation and substitution vary largely between different port operating systems. A similar issue in DEA is that inefficient DMUs and their benchmarks may not be similar in their operating practices. This is largely because the composite DMU that dominates the inefficient DMUs either depicts an inherently different technology or does not exist in reality. As a solution to these problems, some authors propose to add weight multipliers to DEA models by introducing expert judgements, such as through survey or AHP-based techniques, or by incorporating prior views on efficient firms and on the relationship between inputs and outputs. Others have used performance-based clustering and other similar methods in order to discriminate between efficient firms or identify more appropriate benchmarks (Sharma, 2005; Wang et al., 2006).

Analytically, DEA does not allow for stochastic factors and measurement errors and there is no information on statistical significance or confidence intervals. For economists, the non-statistical attribute of DEA is a major impediment against its validity. Although a second-stage regression analysis is sometimes used to solve this, regression assumes data interdependency and requires the imposition of a functional form which deprives DEA from its major advantage. It is worth underlying that several recent works have tried to close the gap of statistical grounding in DEA analysis (see for instance Banker and Cooper, 1994; Simar and Wilson, 1995 Gstach, 1998; and Cooper et al., 2002). Suggested solutions that allow DEA to work in stochastic environments include chance-constrained programming and DEA bootstrapping, the latter is becoming more popular among researchers. Other solutions include the use of panel data to filter noise across time periods (Banker and Maindiratta, 1992), and the inclusion of some sort of parameterisation, for instance by constructing dummy efficiency variables from DEA to be used as additional repressors in OLS or SFA estimation (Sengupta, 1989).