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Methodology Description

Slacks-Based Measure of Efficiency

DEA is a set of models and methods of mathematical programming, which enables comparisons of the relative efficiency of the Decision-Making Units (DMUs). DEA is a specific part of OR which focuses on ranking the DMUs based on a set of criteria, where it is assumed that each DMU produces outputs based on production inputs. The terminology used within this area comes from DEA being developed for production firms. The term inputs refers to the variables which a DMU is aiming to reduce, while the term outputs refers to the variables which should be greatest possible. Furthermore, the term relative efficiency denotes the efficiency of one DMU compared to others. Thus, this methodology compares a set of DMUs one to another, not to the best possible solution determined by the researcher, industry, etc. For a basic introduction to the DEA terminology, ideas, and models, interested readers are referred to Cooper et al. (2011), Sexton (1986), Doyle and Green (1994), or Sickles and Zelenyuk (2019).

The basic model notation is as follows. The DMU uses m inputs (x{px2p.. .,х„ф to produce s outputs (УрУгр-■ .,yjy), where у denotes the DMU,у e {1, 2,..., n}. Inputs and outputs in this study refer to the financial ratios which are used for comparison purposes. If the data are put in matrix form, matrices X and Y consist of all inputs and outputs, respectively:

Each column in matrices X and Y consists of data for the y'-th DMU, *o =(*io,*2o,: and = (у2о,: where *0>0. дгпф0 andya> 0.y0#0

holds. The two most basic models which were developed (and mostly used today) are the Charnes-Cooper-Rhodes and the Banker-Charnes-Cooper model. The main difference between them is the assumption of returns to scale in the production

(constant versus variable returns to scale). However, these two models suffer from several disadvantages. They are sensitive to data translation. As the data have to be nonnegative, depending on whether this refers to inputs or outputs, some variants of these models cannot be used. Furthermore, both models assume an equal proportion of input decrease and/or output increase of a selected DMU in order to get to the efficient frontier. That is why many different models and extensions have been developed over the years.

The SBM (slacks-based measure) model (Tone 1997, 2001) has a certain advantage compared to the two mentioned models, which is being an additive model, with units-invariance property, alongside the possibility of being nonoriented and nonra- dial. This means that the optimization process consists of simultaneously decreasing inputs and increasing outputs, without forcing the same rate of improvements for inputs and outputs. This is especially useful when ratios are used as inputs and/or inputs, which is often the case in analyzing the financial performance of companies. The model is dimensionally free in that way (Thrall 1996).

In order to measure the (in)efficiency of each DMU, the following model is optimized:

where s and s+ are vectors of input and output slacks, respectively. A DMU is called SBM-efficient if and only if P*= 1. This is equivalent to s =0 and s+ = 0, which means that no input excess and no output shortfalls are present in the optimal solution. The SBM projection to the efficient frontier for a DMU under consideration is

Thus, the main idea in this research is that when comparing the performances of companies one to another, the DEA methodology can be very useful due to it comparing the efficiency in “producing” outputs by using the smallest amount of inputs. Financial ratios that should be the smallest possible will be observed as inputs, while the opposite is true for the output variables.

Often the problem with real data is missing data. The values in input and output vectors should not be missing in order for the DEA model to be optimized.

One approach is to delete those DMUs from the analysis which have missing data. However, this could result in a small sample in the end which cannot be used for the analysis. Another approach is to add penalties to missing data. Kuosmanen (2009) advises this. Missing input values should be set to a large value, greater than the maximal input value for the existing data. The missing values for output values should be given such that these values are smaller compared to the lowest output value which is available in the sample. As Kuosmanen (2009) showed, the optimal values in such an approach are as good as the approach of deleting the DMUs with missing data.

Finally, correlations between inputs and outputs are important in the analysis as well. The main idea is that the correlation between inputs and outputs is as greatest possible, while the correlations between inputs (outputs) themselves should be lowest possible (due to containing the same information if the correlation is great). Thus, in order to reduce the unnecessary number of inputs and/or outputs in the analysis, the correlation matrices will be observed. For more information on the correlations within DEA analysis, please see Lopez et al. (2016).

Multiple Criteria Decision-Making

In order to test the robustness of the results, another approach will be made in order to obtain rankings of the companies in the empirical part of the analysis. MCDM is also a part of the OR, in which optimization of mathematical models is done on a set of different alternatives based on different criteria. As many economic and financial decisions are often based on conflicting criteria, the MCDM modelling aids in constructing the ranking system of the alternatives which are taken into consideration. Here, the alternatives are the DMUs from the previous subsection terminology and approach, and the criteria are the inputs and outputs from the DEA terminology. Details on the MCDM methodology with applications in finance can be found in Hurson and Zopounidis (1995).

The main flow of this approach consists of the following. The problem of the analysis, with objectives and decisions, has to be made, by identifying the alternatives which are compared one to another. Then, the mathematical model is formulated and needs to be solved. The solution of the model has to be observed by the decision-maker, and (s)he needs to make the final decision, based on the topic observed, or redefine some of the steps of the whole process. For more details, please see Bigaret et al. (2017). As a simple model, this research utilizes the MOORA model (multiobjective optimization by ratio analysis). As Brauers and Zavadskas (2010) showed, this approach is robust with respect to seven criteria analyzed. First, this approach is more robust when more stakeholders are involved compared to one decision-maker (Brauers, 2007). Second, the noncorrelated multiobjective-based method is more robust compared to the limited number of objectives. Third, it is nonsubjective; fourth, if it is based on cardinal numbers, it is more robust when compared to ordinal numbers. Fifth, the method is robust in cases when all interrelations between objectives and alternatives are taken into consideration when compared to examination two-by-two (Brauers, 2004). Sixth, this method is more robust when using the newest data possible, and finally, when all of the previously mentioned conditions are met, the optimization which utilizes two methods compared to one is more robust, three methods are more robust compared to two, etc. For more details, please see Brauers and Ginevicius (2009).

Every alternative j is observed with its criteria i. All of the data are put into matrix Xjj= |x,;], where i e {1,2,..., n),j e {1, 2,..., m). Every value xtj is put into the following ratio, in order to obtain the following normalization:

so that values in (6.4) fall in the range [0,1]. These normalized values are summed up if the criterion should be greatest possible or detracted if the main goal is to obtain minimized values of any criterion. In that way, the normalized assessment value is calculated as:

Each value y* in (6.5) can be finally ranked. Each criterion in (6.5) has equal weights in calculating y). The researcher can give different weights to each criterion, based on previous knowledge, experience, etc. More details on this modelling approach in finance can be found in BaleZentis et al. (2012) or Xidonas et al. (2009). Finally, in terms of the financial ratios observed in this study, the criteria which should be greatest possible (addition in (6.5)) are those ratios which should be greatest possible, while the opposite is true for those ratios which should be minimal possible.

Empirical Results

 
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