Robust worst-practice interval DEA with non-discretionary factors

Highlights

• A new robust worst-practice model is formulated.

• A numerical example and a case study are provided to validate the suggested model.

• Monte-Carlo simulation is used to select parameters for the suggested robust model.

Abstract

Traditionally, data envelopment analysis (DEA) evaluates the performance of decision-making units (DMUs) with the most favorable weights on the best practice frontier. In this regard, less emphasis is placed on non-performing or distressed DMUs. To identify the worst performers in risk-taking industries, the worst-practice frontier (WPF) DEA model has been proposed. However, the model does not assume evaluation in the condition that the environment is uncertain. In this paper, we examine the WPF-DEA from basics and further propose novel robust WPF-DEA models in the presence of interval data uncertainty and non-discretionary factors. The proposed approach is based on robust optimization where uncertain input and output data are constrained in an uncertainty set. We first discuss the applicability of worst-practice DEA models to a broad range of application domains and then consider the selection of worst-performing suppliers in supply chain decision analysis where some factors are unknown and not under varied discretion of management. Using the Monte-Carlo simulation, we compute the conformity of rankings in the interval efficiency as well as determine the price of robustness for selecting the worst-performing suppliers.

Introduction

In the current era of the dynamic and competitive uncertain business environment, the tendency for many businesses to underperform is quite common. Corporate bankruptcy, financial distress, and adverse market conditions such as the Covid-19 are among few risk factors that can threaten the survival of businesses. Often, the failure of a business can be avoided by early detection of certain risk factors. This sometimes requires risk assessment including the identification and quantification of risk factors to mitigate the potential loss of asset or investment and improving efficiency and productivity growth. In the financial sector, for example, banks often use the non-performing loan (NPL) ratio as a substitute variable for risk and as a result, there is an apparent attempt to quantify credit risk associated with lending activities. To this end, scholarly research on business risk has concentrated on underperforming businesses that seemingly have risk factors or have a high potential for failure. In standard productivity analysis, identifying non-performing businesses or units is very much rewarding since it is where the largest potential savings and gains can be made, particularly through a reasonable target setting (Paradi, Asmild, & Simak, 2004).

Data envelopment analysis (DEA) is an excellent data-oriented performance approach that can be used in this direction; specifically evaluating performance when multiple inputs and outputs are present in decision-making units (DMUs). The classical DEA models evaluate DMUs in pairwise comparison by maximizing the efficiency ratio scale and selecting the most favorable optimal weights for ${\mathrm{D}\mathrm{M}\mathrm{U}}_o$ (DMU under evaluation). In these pairwise comparisons, DMUs are classified into efficient and inefficient units with an optimistic view in the most favorable scenario. As Cooper et al. (2007) pointed out, the benchmarking of DMUs in this scenario is done with the best-performing frontier (BPF) and so non-performing or distressed DMUs can only be measured by how inefficient they are at being good. Although the BPF-DEA models can present an improvement path for these inefficient DMUs, the evaluation technique is not focused on identifying bad performers in the most unfavorable (worst-case) scenario. Hence, they are not an ideal fit for real-world applications where the worst business need to be clearly identified (Paradi et al., 2004, Liu and Chen, 2009).

Placing emphasis on the non-performing units requires performance models that result in putting worst-performing units on the “efficient” frontier, otherwise known as the worst-performing frontier (WPF). The WPF-DEA efficiency approach is based on this concept of identifying the bad performers in the worst-case scenario. The worst-practice DEA model was given in Shuai & Li (2005) in conjunction with rough set theory to deal with imprecise data in business failure prediction after the concept was initially introduced in Paradi et al. (2004) for credit risk evaluation. Originally, Paradi et al. (2004) demonstrated the use of a layered worst-practice DEA technique that gives much higher classification accuracies and is less sample-specific than the traditional fixed cut-off point approach. The sequential layers of poor performance incorporate risk attitudes and risk-based pricing and are found with decreasing risk rating. On the other hand, Jahanshahloo and Afzalinejad (2006) suggested a different method that ranks DMUs based on their distance from a frontier by definition of a full-inefficient frontier. Liu and Chen (2009) proposed a new model for evaluation of the worst efficiency together with the slack values in the worst-case scenario by incorporating slack-based measure (SBM) into the WPF–DEA. Using a similar frontier for the evaluation of DMUs, Azizi & Ajirlu (2011) proposed a pair of interval DEA models that consider crisp, ordinal, and interval data, as well as non-discretionary factors, simultaneously.

The WPF-DEA models unlike the BPF-DEA models assign the most-unfavorable weights to each DMU. The strategy selects the input and output weights that reflect the underutilization of resources or failures to perform, as particularly prevalent with vulnerable companies in a competitive business climate or financial crisis. A key advantage of this approach over the traditional DEA models is that, by focusing on identifying the worst-performing units, the WPF-DEA approach is able to identify the largest potential improvement that can be made in the most unlikely worst-case scenario. The overarching benefit of applying the worst-practice DEA models in practice is evident; the models are fit for purpose in several applications where potentially distressed and non-performing businesses must be identified. For example, in the financial sector, the worst-practice DEA has been successfully applied to business failure prediction (Shuai and Li, 2005), credit risk evaluation (Paradi, Asmild, & Simak, 2004), and banks bankruptcy evaluation (Liu and Chen, 2009). The approach can also be used in risk-taking sectors such as insurance companies and investment firms. Moreover, in supply chain analysis, using the WPF-DEA to select suppliers with the worst performance presents a potential advantage of blacklisting suppliers that are least competitive and will potentially go out of business in comparison with others in a similar unfavorable scenario (Azadi and Saen, 2011, Azadi and Saen, 2012). Generally, and in all applications, where identifying the least performing DMUs is paramount, the worst-practice DEA is practically more useful than the traditional DEA models that focus on best-performing DMUs. The concern in this paper is to present novel models that achieve this goal, particularly, when the decision environment is uncertain, i.e., input and outputs factors are imprecisely unknown, and when such factors could be non-discretionary. It should be noted that both traditionally proposed BPF-DEA and WPF-DEA models assume precise data in measurement which may not be the case in real-world situations.

Uncertainty and imprecision are pervasive in DEA models. They can be addressed with several approaches such as the stochastic DEA (Olesen and Petersen, 2016, Charles and Cornillier, 2017), Imprecise DEA (IDEA) (Cooper, Park, & Yu, 1999), fuzzy DEA (Hatami-Marbini et al., 2011, Saati et al., 2011), interval DEA (Despotis and Smirlis, 2002, Entani et al., 2002, Zhu, 2003) and robust DEA (RDEA) (Sadjadi and Omrani, 2008, Shokouhi et al., 2010, Toloo and Mensah, 2019, Hatami-marbini and Arabmaldar, 2021) and including some probabilistic approaches (Landete et al.,2017). See Mensah (2019) for a general overview. Cooper, Park, & Yu (1999) addressed imprecision in data in its general form by proposing an IDEA model. Bounded/interval data, ordinal data, and ratio-bounded data including a mix of imprecise and exact data were the imprecisions this model was designed to address, however, its nonlinearity makes it difficult to be used. Zhu (2003) addressed this problem by transforming the nonlinear IDEA to a standard linear DEA through scale transformation and variable alternations. By similar procedure, Despotis & Smirlis (2002) proposed an approach that treats interval DEA as a peculiar case of DEA with exact. They define lower and upper bound for interval efficiency scores and further discriminate DMUs into fully efficient, efficient, and inefficient units. Wang et al. (2005) proposed a modified interval DEA model of Despotis & Smirlis (2002) suggesting a unified production frontier and correcting the distance measures efficiency scores of the former. Entani et al. (2002) adopt the interval DEA to determine the optimistic and pessimistic ranking of DMUs with a fuzzy approach. Toloo et al. (2008) presented a framework where DEA is used to measure overall profit efficiency with interval data. It is shown that when the inputs, outputs, and price vectors each vary in intervals, the DMUs cannot be easily evaluated and the interval DEA provides the efficiency of DMUs with their lower and upper bound values which in turn leads to the optimistic and pessimistic standpoints from the bounds of the interval. Khalili-Damghani et al. (2015) developed a customized DEA method for elucidating the returns to scale issue when interval data and undesirable outputs exist. The applicability of the developed approach is investigated using a six-year study of 17 combined cycle power plants in Iran.

Recently, the robust DEA based on the robust optimization introduced in Soyster (1973) has been heavily employed to deal with uncertainty in DEA. We refer the reader to Bertsimas et al., 2011, Gabrel et al., 2014, and Mensah & Rocca (2019) for extensive reading on robust optimization techniques. Robust DEA models are formulated as a conservative approach based on a specific robust optimization concept, notably the robust concept of Ben-Tal and Nemirovski, 2000, Bertsimas and Sim, 2004. As clearly stated in Mensah (2020), the approach offers to immunize the uncertain inputs and outputs data of DMUs in a user-defined uncertainty set and provides a probability guarantee for reliable efficiency scores, robust discrimination, and ranking of DMUs. The robust DEA was first introduced in Sadjadi and Omrani (2008) for output data uncertainty and performance evaluation of electricity distribution companies. Since then, the approach has found several real-world applications whereas research on its theoretical extension in different DEA domains continues to increase. Within the traditional DEA context of constant returns to scale (CRS) and variable returns to scale (VRS) technology, Lu (2015) first developed two robust DEA models using both the Ben-Tal and Nemirovski, 2000, Bertsimas and Sim, 2004 robust optimization approaches to evaluate algorithm performance. Arabmaldar et al. (2017) proposed a new RDEA model and two linear robust super-efficiency DEA measures under CRS technology based on Bertsimas & Sim (2004) approach. Toloo & Mensah (2019) studied the non-negativity conditions in a robust optimization framework and extended it to RDEA models and developed a new RDEA model under VRS technology with an application made to the efficiency of the largest 250 banks in Europe. See also Sadjadi et al. (2011) where ellipsoidal uncertainty induced robust framework of Ben-Tal & Nemirovski (2000) was utilized to propose a robust super-efficiency DEA model.

Several extensions of the robust DEA to advanced DEA models have been made in the literature. To peruse a few, Omrani (2013) provided a goal programming technique with the robust DEA to study common set of weights in DEA. Salahi et al (2020) in recent times also proposed a new envelopment-based robust DEA model to find the common set of weights in DEA. On the other hand, Sadjadi, Omrani, Makui, et al. (2011) by integrating the DEA and multi-objective linear programming proposed an interactive robust DEA model to find the input and output target values based on the expert’s preference in the presence of uncertain data. Amirkhan et al. (2018) formulated a mixed model using fuzzy and robust DEA. Salahi et al. (2019) suggested robust Russell measure and its enhanced models under interval and ellipsoidal uncertainties in their best- and worst-cases. Shirazi & Mohammadi (2019) developed a robust slacks-based measure (SBM) model with undesirable outputs to measure the efficiency of airlines in Iran. Tavana et al. (2021) developed two DEA adaptations to rank DMUs in the presence of interval data and undesirable outputs. An interval DEA model and a robust cross-efficiency DEA model are proposed to extend the traditional classification of efficient and inefficient units. More recently, Hatami-marbini and Arabmaldar (2021) estimated cost efficiency measures under polyhedral uncertainty set using robust optimization approaches. Firstly, they presented robust DEA models to cope with uncertain inputs and outputs and then developed a pair of robust cost efficiency models to attend to uncertain input prices.

In this paper, we propose a robust optimization approach for dealing with uncertainties that cover interval data for the WFP-DEA. The earlier consideration of robust optimization for interval DEA, notably of Despotis & Smirlis (2002) can be found in Shokouhi et al (2010) for the BPF-DEA. Our approach is specifically based on the adaptation of the robust optimization of Bertsimas and Sim (2004) and follows similarly, the research work of Shokouhi et al (2014) for the BPF-DEA. Shokouhi et al (2014) model is a modified robust DEA model based on a unified production frontier of Wang et al. (2005). As will be seen subsequently, the latter’s approach has some drawbacks. We aim to capture these drawbacks and propose a correct version and extension to the WPF-DEA. The proposed models in this paper are less computationally demanding and essential for adjusting the conservativeness of the model while their efficiencies vary between interval bounds. It should be mentioned here that, our robust-based approach is the first to be applied to the WPF-DEA. Furthermore, we propose a robust worst-practice model in the presence of non-discretionary factors which is practically useful in situations in which identifying the units with bad performance is indispensable. Non-discretionary factors in DEA are particularly useful where business owners cannot directly control some inputs and/or outputs. Again, the consideration of non-discretionary factors in robust DEA in this paper makes it the first of its kind and adds to the literature studies on the topic for the traditional DEA models (see for e.g., Banker and Morey, 1986, Muñiz et al., 2006, Huguenin, 2015, Taleb et al., 2018) and imprecise DEA models (e.g. Fried et al., 2002, Farzipoor Saen, 2009, Saati et al., 2011, Azizi and Ajirlu, 2011, Zerafat Angiz and Mustafa, 2013, Toloo et al., 2018, Toloo et al., 2021). Finally, our contribution provides a numerical example and an application to supplier selection to compare and illustrate the efficacy and applicability of the proposed models.

The rest of the paper is organized as follows: Section 2 provides the background of WPF-DEA and demonstrates it with interval data within the optimistic and the pessimistic scenario in the worst-practice concept. Section 3 reviews the robust BPF-DEA with interval data. A revised study of Shokouhi et al (2014)’s modeling approach is given in Section 4 and the robust WPF-DEA with interval data is then developed. Section 5 further develops a robust WFP-DEA model in the presence of non-discretionary factors. Two numerical examples are provided in Section 6 to compare and illustrate the practical application of the proposed models. Conclusions and future research directions are presented in the final section.