Innovations

• Ranking of DMUs while input/output data assumes positive/negative values.
• SBM development by utilizing the PCA technique to improve the ranking power of DEA by increasing the differentiation power among DMUs
• Utilization of PCA in super-efficiency SBM model to complete ranking of DMUs with low complexity.
• Combing PCA with DEA in order to reduce evaluation indices with the least loss of information.

This paper is structured as follows: Literature Review which begins with the definitions of two methods (DEA, PCA) and follows with three parts to review previous studies: The complete ranking of DMUs with negative input/output (as this paper focuses on negative data), DEA combined with multivariate techniques except PCA (as this paper focuses on the integrated method), and DEA with PCA (as this paper utilizes these two methods). Materials and Methods, in this section a combination of steps and models that are in both methods, as well as the proper order of the steps, has been mentioned. Moreover, it has been tried to make the presented algorithm understandable for the readers with sufficient explanations of the steps. A flowchart for the ranking algorithm. In this part, the steps of the algorithm are described in the form of a flowchart that can be seen at a glance, and in fact, the roadmap of the method is presented. Results and Discussions: the efficiency and ranking of bank branches and pharmaceutical companies which are calculated with the algorithm, are presented in this section. To express the superiority of the algorithm, the efficiency scores for two case studies also were calculated with SBM (Tone (2001)) [25] (without the PCA), and super-efficiency SBM (Tone2002) [23] (without the PCA). A comparison of the results and implementation times (between our models and SBM and SSBM models) is presented at the end of this section. Conclusions and Suggestions consist of the advantages and limitations of the selected approach and recommendations for future research.

Studies on the complete ranking of DMUs with negative input/output

Studies in this field can generate complete ranking in the presence of negative data by using the super-efficiency approach or other approaches. Hadi-Vencheh et al. [28] defined a new super-efficiency model in the presence of negative indices (input/output). This model ranked all DMUs using the Range Directional Model (RDM) and determined the efficiency type of the DMUs, but in this model, sometimes infeasibilities exist which is the weakness of the model. Lin et al. proposed [29] a DDF-based (Dynamic Data Flow) super-efficiency model to complete rank of units with positive and/or negative data by applying the directional vectors. This model is sensitive to the value of parameter K which influences the power of ranking of the method. Babazadeh et al. [30] proposed a novel radial (DDF-based) super-efficiency model to distinguish between efficient and inefficient units. Furthermore, this model could generate a complete ranking order for all DMUs using a new super-efficiency measure. Additionally, it ensured feasibility regardless of whether the input/output data were positive or negative. Moreover, the given model possessed advantages such as monotonicity, unit invariance, and translation invariance properties. Some studies based on other approaches can be stated as follows: Wei et al. [31] introduced a novel modified SBM (Slacks Based Measure) that allowed for negative input/output values and ranked all the DMUs. This novel model was defined as a unit- and translation-invariant, two critical properties of DEA models. Additionally, they proposed this method to assist decision-makers in evaluating the performance of all units and assigning them a complete ranking in the presence of negative data. Lin [29] defined the VRS cross-efficiency evaluation for DMUs as a ratio, capable of handling positive and negative data in the input and output. For all the DMUs, the resulting cross-efficiencies and scores were between zero and one, which ensued in their ranking. Of course, one of the major drawbacks in complete ranking with the cross-efficiency method is that the cross-efficiency matrix is not unique. Soltanifar et al. [32] have solved this problem by combining this method with the fuzzy VIKOR method.

DEA with multivariate techniques (except PCA)

Researchers have used the combination of DEA with different techniques to improve the performance of the DEA method. Combining the DEA method with multivariate techniques (PCA, Factor Analysis, and Clustering,) is usually done for reducing input/output indicators or clustering input/output indicators. Clustering in DEA can help to solve the problem of heterogeneity in input and output data. Among the most important studies for integrating FA and DEA, the following can be stated: Nadimi and Jolai [33] combined DEA and FA for data reduction in DMUs. Their numerical results indicated that the new approach was highly consistent with DEA in terms of ranking. Vargas and Bricker [34] combined the CCR (Charnes, Cooper, and Rhodes) output-oriented model of DEA and FA to evaluate the academic units of several graduate programs with their national counterparts. They acknowledged that many academic program outputs are not immediately visible. Thus, the FA separated these outputs from the visible output and then evaluated the units using these outputs by DEA. Huang and Chen [35] developed a novel FA-DEA method based on classification and weighting, resulting in an improved model based on the classification of the original indices. Gong, Shao, and Zhu [36] proposed a new method for evaluating the energy efficiency of ethylene production by incorporating DEA and FA into the operation classification. Mengdie Huang et al. [37] conducted their evaluation employing a multi-attribute efficiency evaluation model, a multivariate statistical method, and grey theory, a so-called DEA integrated grey FA (GFA-DEA) approach, to support efficiency evaluation and ranking in uncertain systems. In this study, an efficiency measures generation technique is integrated into the grey factor analysis method for the benchmarking of DMUs. Among the most important studies for integrating Clustering and DEA, the following can be stated: Azadeh et al. [38] demonstrated how to optimize operator allocation in Cellular Manufacturing Systems using integrated fuzzy DEA (FDEA), fuzzy clustering C-means (FCM), and a computer-based simulation (CMSs). This study included integrated simulation, FDEA, FCM, and fuzzy indices, Mike G. Tsionas [39] combined Clustering and meta-frontier in DEA for solving the problem of heterogeneity of DMUs. He provides a prior design to minimize variation within groups and maximize variation across groups. The techniques are applied to a data set of large U.S. banks.

Of course, multivariate techniques are not the only way to reduce indicators in DEA, for example, Xi Bao et al [40] focused on a combination of fuzzy (QFD) and interval DEA to provide a new two-phase methodology for DEA indicator screening. The study is the first attempt to overcome the curse of the dimensionality problem and screen the evaluation indicators by combining fuzzy QFD and interval DEA.

In our paper, the enhanced models which combine PCA and SBM, PCA and Super SBM to improve the ranking power of the DEA method are proposed.

DEA with PCA

The studies that combined both methods (namely, PCA and DEA) are divided into two categories: Those that employ one method and use its output as input for another method (i) (which cannot provide any benchmark for inefficient units), and those that construct a model by combining both methods and insert data into the integrated model (ii) (which can provide benchmarking). It should be noted that this paper is classified as the second category.

1. Azadeh et al. [4] used a combination of two PCA-DEA methods to compare the wireless communication sectors of forty-two countries. The first step was to use the PCA method to convert seventeen indices into three input and five output indices. The second step was to rank countries according to DEA. Azadeh et al. [5] developed an integrated approach for assessing and optimizing total energy efficiency in energy-intensive manufacturing sectors using DEA, PCA, and Numerical Taxonomy (NT). Additionally, they presented a comprehensive approach that included structural and conventional energy efficiency indices, verification and validation mechanisms for DEA via PCA and NT; and the use of DEA for assessing total energy efficiency and optimizing consumption in energy-intensive manufacturing sectors. Wu et al. [6] connected the PCA and DEA methods to estimate online banking performance (giant banks in the USA and UK, including financial and nonfinancial variables). Amirteimoori et al. [7] proposed a method for estimating the efficiency of DMUs through PCA. This method allows for handling undesirable output while reducing the dimension of the data sets. They accomplished this by first reversing the undesirable output. Then used PCA to calculate the ratio of a single desired output to a single input. Finally, the selected PCs were imported into DEA as virtual data sets. The proposed approach was validated using a data set of the Iranian bank branches. Chunyan Yao [8] used a PCA-DEA model to determine the level of regional innovation in thirty Chinese provinces and cities and converted the correlated input/output variables to uncorrelated ones. Simultaneously, a smaller number of primary components was identified to express the original data. Finally, the DEA model was constructed using uncorrelated variables. Azadeh et al. [9] used DEA and PCA to evaluate the current maintenance and Health, Safety, and Environment (HSE) management systems of a gas transmission unit. Furthermore, they presented an integrated HSE and a performance-optimization system maintenance strategy. Moreover, PCA was used to verify the results. Poldaru and Roots [10] assessed the Quality Of Life (QOL) scores in Estonian counties using a PCA-DEA model and analyzed the model’s results. The procedure involved a two-stage analysis that began with a PCA. The standard DEA was used in the second stage. Zha et al. [11] applied PCA to analyze the original value judgment information, and the key indices in the production process are extracted. The modified DEA models are proposed. DEA efficiencies and their projections are calculated. Faed et al. [12] presented an intelligent system for handling customer complaints via PCA and DEA. They used PCA to analyze the data because it was more effective than other tools such as Exploratory Factor Analysis (EFA) in compressing data and producing more accurate results. The DEA was then used to differentiate critical customers from others and ascertain the most influential perspectives on policymaking. Omrani et al. [13] calculated the development degrees of provinces in various classes using the Common-weight DEA (CWDEA) model, the scores of which were used as PCA model indices. Additionally, they ranked the provinces using the PCA model. Xiao Shi et al. [14] evaluated the technical efficiencies of risk management of Chinese commercial banks by the DEA-BCC model. The PCA was applied to delete redundant input indicators. Rahimpour et al. [15] proposed a model to evaluate the performance of organizational units considering intellectual capital (IC) and the employee loyalty approach by applying the PCA-DEA method.
2. Some previous studies that developed a model by combining both methods (PCA and DEA) and entering data into the integrated model are summarized in Table 1. In this Table the previously proposed models are compared with the model proposed in this paper in several ways: (I) supporting both positive and negative data; (II) using CRS/VRS models; (III) utilizing Radial/Nonradial models.

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Table 1. Comparison of the studies with a model combining (PCA, DEA).

https://doi.org/10.1371/journal.pone.0290610.t001

The primary advantage of the model of this research is that it is nonradial and defined on VRS, which means that the efficiency criterion is unrelated to the data signature. This advantage has not been documented in previous research to the best of the authors’ knowledge.

Ranking algorithm steps (explanation)

Steps of the algorithm include two parts, steps 1 to 5 are the explanations of the PCA technique, and steps 6 and 7 are the explanations of the DEA models.

6- Implementing the PCA-SBM model.

The following steps are used to create the model presented in this study:

6-1- The SBM model [25].

(16)

1. x_ij: Value of i-th input variable (index) for j-th DMU
2. y_rj: Value of r-th output variable (index) for j-th DMU
3. x_ip: Value of i-th input variable (index) for the desired (p-th) DMU
4. y_rp: Value of r-th output variable (index) for the desired (p-th) DMU
5. r (1… s), s: Number of output variables
6. j (1… n), n: Number of DMUs
7. i (1…m), m: Number of input variables

6-2- The RBM model. As the x_ip and y_rp values must be positive, they can be replaced by and , given the following values: (17)

As a result, the model is modified as follows: (18)

6-3- The RBM model converted into a linear model. Assume t equals the following fraction: (19)

So the model takes on the following form: (20)

Thus, the DEA model used in this study is obtained by substituting with with and tλ_j with λ_j and also using Eq (17) for (R+, R^-) and Eq (7) for the variables (x, y).

(21)

1. x_ij: Modified value of i-th old input variable (index) for j-th DMU
2. y_rj: Modified value of r-th old output variable (index) for j-th DMU
3. x_ip: Modified value of i-th input variable (index) for the desired (p-th) DMU
4. y_rp: Modified value of r-th output variable (index) for the desired (p-th) DMU
5. q ₍1… a), a: Number of new output variables (PCs)
6. r (1 … s), s: Number of old output variables
7. j (1 … n), n: Number of DMUs
8. h (1 … f), f: Number of new input variables (PCs)
9. i (1 … m), m: Number of old input variables
10. a≤s,f≤m
11. e eigenvector (y)
12. e՛ eigenvector (x)

Ranking algorithm (flowchart)

The steps necessary to implement the algorithm which are the roadmap of the method are depicted in Fig 1.

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Fig 1. Ranking algorithm flowchart.

https://doi.org/10.1371/journal.pone.0290610.g001

Ranking the branches of a bank

In many papers, the DEA models are used to calculate the efficiency of banks. For example, Hatami-Marbini et al. [42] presented a method to assess the relative efficiencies of bank branches when interval and negative data were present. They classified bank branches as either strictly efficient, marginally efficient, or inefficient. Li et al. [43] estimated the efficiency of Chinese commercial banks by employing the super-efficiency DEA, Zhujia Yin et al [44] evaluated the banking efficiency by using the proposed DEA model and also Hssan Shah et al [45] investigated the efficiency, production technology gap, and productivity growth among 147 Central Banks (CBs) of South Asian countries from 2013–2018. Starting with financial indicators of CBs, various techniques such as DEA (CCR, BCC), DEA meta-frontier and Malmquist productivity index (MPI) are employed for the empirical analysis. Saif Ullah et al. [46] used the Data Envelopment Analysis (DEA) approach to measure the efficiency of the banks of Pakistan based on output and input variables. The study used data from seventeen commercial banks from 2011 to 2020 and also Hassan Shah et al. [47] employed the DEA Super-SBM with the undesirable output for the efficiency evaluation of 24 Central Banks of Pakistan. They were evaluated for 12 years, ranging from 2006 to 2017.

In our paper, the presented algorithm was applied to the data of forty bank branches in 2022. The data in this case study were initially positive, but some of them became negative by applying the PCA technique. The algorithm was used to estimate the efficiency scores of the branches, and then they were ranked.

Three input and eight output indices were used in this case study. To select indicators, first the available data were examined. Empirically, this fact can be used to distinguish between input and output indicators. The lower the input index value, the higher the efficiency of the banking system and the higher the output index value, the higher the efficiency of the banking system. The input indices include personnel score (privilege of staff), interests paid, and overdue receivables. The output indices comprise facilities, four deposits, interests received, fees, and other resources.

The indices are described as follows (except for personnel score, the units of measurement for the given indices are in million Rials (Iranian currency)):

Personnel score (privilege of staff): Refers to an index obtained from the number of personnel, their experiences, and their level of education and training. The index is calculated from the total weighted sub-index.

Interest paid: Banks pay interest on customer deposits. The total interest paid per month on all short- and long-term deposits is referred to as interest paid.

Overdue receivables: Lending money to customers is one of the bank’s activities. After receiving these loans, customers are required to repay them in monthly installments. Unfortunately, some customers do not pay them on time or at all. Overdue receivables refer to the amount owed by customers in installments.

Facilities: An index representing the value of the bank loans paid to customers. Markedly, this is the total amount of all facilities paid in one month.

Four deposits: Include interest-free, current, short- and long-term savings accounts held by customers. Notably, this is the balance of these deposits in a single month distinctively.

Interests received: Banks are always paid interest on facilities. The interests received are a percentage of loans. The sum of all interest received by banks for all previous loans, in a month, is referred to as the interest received.

Fees: Banks provide various services to their customers, including money transfers and guarantees, for which they charge a fee.

Other sources: The value of all other customer deposits, such as the government, is referred to as other sources.

Note 1: The number of new input indices/output indices to be extracted from the existing three inputs/ eight outputs is an important issue that depends on the opinion of the DM. Owing to the fewer the number of inputs/outputs, the lesser the number of efficient units, the higher the loss of information, so it seems to be a trade-off.

Note 2: Given that relation (1) is a research hypothesis, the number of inputs/outputs and the number of DMUs should apply in relation (1). When all three inputs and all eight outputs are considered, relation (1) holds: 40≥max{(3*(3+8)), (3*8)} [1]

The number of efficient units and the percentage of the variance of data used in the model were calculated for various inputs/outputs quantities, as expressed in Table 2.

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Table 2. Number of efficient DMUs in bank branches, in different conditions.

https://doi.org/10.1371/journal.pone.0290610.t002

• While the total number of primary indices (three inputs and eight outputs) are included in the model (the last row in the Table), thirty-one out of forty units become efficient, posing a significant challenge in ranking units, even though 100% of the variance of the input/output data is used.
• When the algorithm reduces the number of inputs and outputs to one (the first row in the Table), the number of efficient units decreases to four, which is significantly better than the previous condition for ranking units. The model retains 99.424% and 80.333% of the total variance of the input/output data, respectively. It is desirable if the DM is of the opinion that 80% of the variance of the input/output data is sufficient and it seems to be the best option for DM to accept because the number of efficient units is significantly less compared to other modes.
• When the number of inputs is reduced to two, 100% of the variance of the input data is utilized, indicating that, the three input indices are unnecessary. Thus, the model’s two independent inputs provide sufficient information.

As shown in Fig 2, the more the number of inputs and outputs, the more the number of efficient units, however the less the loss of information.

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Fig 2. Comparing the number of inputs and outputs with the number of efficient units (bank branches).

https://doi.org/10.1371/journal.pone.0290610.g002

In Table 3, the bank branches are ranked according to their efficiency scores determined by the proposed algorithm for one input index and one output index. To express the superiority of this algorithm, the efficiency scores of the bank branches were calculated with SBM (Tone (2001)) [25] (without the PCA), and the score rank of the efficient units was estimated, using super-efficiency SBM (Tone2002) [23] (without the PCA).

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Table 3. Ranking of bank branches (2022).

https://doi.org/10.1371/journal.pone.0290610.t003

We implemented our proposed method with GAMS software. In all cases, it was successful in calculating efficiency scores.

In comparing the two methods (PCA-SBM, SBM), it is clear that the PCA-SBM model works better than the SBM model (four efficient DMUs versus twenty-nine). If a method can assign the efficiency score of one to a smaller number of units, the ranking power of that method is higher. Because it is impossible to distinguish between units that have an efficiency of 1. the SBM method cannot capable to determine the rank for 29 DMUs: DMU2, DMU3, DMU4, DMU5, DMU7, DMU8, DMU9, DMU11, DMU12, DMU13, DMU14, DMU15, DMU16, DMU18, DMU20, DMU23, DMU27, DMU28, DMU29, DMU30, DMU31, DMU32, DMU33, DMU34, DMU36, DMU37, DMU38, DMU39 and DMU40. But the method PCA-SBM is not capable to determine the rank for only 4 DMUs: DMU5, DMU18, DMU31and DMU34.

When the two methods (PCA-SSBM, SSBM) are compared, it is distinct that the PCA-SSBM model works in a more enhanced manner than the SSBM model. Since the SSBM model cannot rank non-extreme units (DMU7, DMU29, and DMU40, as their efficiency scores remain one and it is impossible to distinguish between their ranks). Though, the PCA-SSBM is capable of ranking DMU7, DMU29, and DMU40 by assigning an efficiency score not equal to one to these units.

In S1 and S2 Tables in the Supporting Information section, raw data of input and output indices for the bank branches are presented (2022) [48].

Ranking the pharmaceutical companies

Using the data available in the stock market to rank companies and select a portfolio via DEA has always been interesting for researchers. For example, Peykani et al. [49] proposed a robust two-phase approach for portfolio construction problem by using DEA and robust optimization approaches. The efficiency of all stocks that can be invested, was evaluated and measured using real-world data from Tehran Stock Exchange (March 2013 to March 2014).

Also using the DEA method for evaluating the efficiency of Health industry sectors such as hospitals and pharmaceutical factories is common. Anteneh Lamesgen et al [50]. was employed input-oriented DEA with a variable return to scale assumption to estimate the efficiency scores of neonatal health service and its associated factors among primary hospitals in three zones of Northwest Ethiopia. Wanhui Zheng et al. [51] utilized Four-Stage DEA to measure the relative efficiencies of Chinese public hospitals from 2010 to 2016. Rodrigo Moreno-Serra [52] applied DEA to analyze efficiency in the achievement of health system objectives and its relationship with certain system characteristics at the country level across Latin American and Caribbean (LAC) countries.

In this paper, the presented algorithm (PCA-DEA) was used to calculate the financial efficiency scores and rank twenty-seven pharmaceutical companies listed in the Iranian stock market (real-world data 2021). Ranking these companies can help stock market investors to select stock portfolios.

Some of the data in this case study was initially negative for example, GPM (Gross Profit Margin) and ROA (Return On Assets), unlike the previous case study that there is no negative data between inputs and outputs of bank branches and after implementing the PCA technique, negative data is generated.

In this case study, three input indices, including total investments, current assets, and TFA (Tangible Fixed Assets), and eight output indices including GPM, OPM (Operating Profit Margin), NPM (Net Profit Margin), ROA, ROE (Return On Equity), reverse P/E (Price/Earning), reverse P/B (Price/Book Value), and reverse P/S (Price/Sales) were used. To select indicators, we have first examined the available data, the ones we need to perform manufacturing operations are the input indicators and the ones obtained after performing the manufacturing operations are the output indicators.

Note 1: For the last three indices (namely, P/E, P/B, P/S), the lower the values, demonstrate, the better organizational performance, hence, the inverse of these values are utilized for the output indices.

The indices are described as follows:

Total investments: Includes the total amount of initial capital as well as short- and long-term investments.

Current assets: Refers to cash and other assets that have been converted to cash or that have been sold and consumed during normal business operations within one year.

TFA: Generally refers to assets that have a physical value and are used by business units for their primary activities and are not excluded from any activity in the following fiscal year.

GPM: The company’s sales revenue after deducting direct production costs. GPM measures the gross profit-to-sales ratio; the higher this ratio, the more successful the business.

OPM: The company’s revenue after deducting expenses associated with its primary operations. The distinction between GPM and OPM is that operating costs are deducted in addition to production costs when estimating operating profit. Similarly, OPM calculates the operating profit-to-sales ratio; the higher this ratio, the more successful the business.

NPM: This type of profit is the residual income after all company expenses are deducted. The distinction between NPM and OPM is in nonoperational revenues and expenses. The most important items in this section are investment income, income from asset sales, financial expenses, and tax expenses. NPM calculates the net profit-to-sales ratio; the higher this ratio, the more successful the business.

ROA: Demonstrates a company’s management’s efficiency in utilizing its assets to generate earnings. ROA is also calculated by dividing the net profit by total assets; the higher this ratio, the more successful the business is.

ROE: Calculated by dividing net profit by total shareholders’ equity. Thus, ROE is regarded as a measure of a company’s profitability in terms of stockholders’ equity.

P/E ratio: This index is critical for stock valuation. It is calculated by dividing the market value per share (VPS) by the earnings per share (EPS). For calculating this ratio for a given year, the average stock price for the year is used as P, and the net profit for the year is divided by the number of common shares in the company as EPS; thus, the lower the index, the higher the value of the company’s shares; hence, the lower the index, the higher the value of the company’s shares.

P/B ratio: This index is calculated by dividing the company’s market value by its shareholders’ equity book value. This ratio is calculated in the desired year using the average stock price and total equity in the balance sheet of that year; thus, the lower the index, the higher the value of the company’s shares.

P/S ratio: This index is calculated by dividing the market value of the company by its sales. To calculate this ratio in a particular year, the average share price and the company’s sales from that year’s balance sheet are used; thus, the lower the index, the higher the value of the company’s shares.

Note 2: The number of new input/output indices to be extracted from the existing three inputs/ eight outputs is an important issue that depends on the opinion of DM. Because the fewer the number of inputs/outputs, the fewer the number of efficient units but the higher the loss of information, so it seems to be a trade-off.

Note 3: Given that relation (1) is a research hypothesis, the number of inputs/outputs and the number of DMUs should apply in relation (1). When all three inputs and all eight outputs are considered, relation (1) does not hold: 27≤max{(3*(3+8)), (3*8)}

[1] therefore, it is necessary to reduce the number of inputs and outputs.

The number of efficient units and the percentage of the variance of data used in the model were calculated for various inputs/outputs quantities, as expressed in Table 4.

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Table 4. Number of efficient DMUs in companies, in different conditions.

https://doi.org/10.1371/journal.pone.0290610.t004

• While the total number of primary indices (three inputs and eight outputs) is included in the model (the last row in the Table), nineteen out of the twenty-seven units became efficient, posing a significant challenge in ranking the units, even though 100% of the variance of the input/output data is used. On the other hand, since the relation (1) is not established, this state is not accepted at all.
• When the algorithm reduces the number of inputs and outputs to one (the first row of the Table), the number of efficient units decreases to four and is significantly better than the prior situation for ranking units. The model maintains 81.963% and 82.886% of the total variance of the input/output data, respectively. It is desirable if the DM is certain that 80% of the variance of the input/output data is sufficient.
• While the number of inputs is reduced from three to one and the number of outputs is reduced from eight to two, the model retains 81.963% and 92.801% of the total variance of the input/output data, (the second row of the Table). As evident, the number of efficient units reached 5. As the number of efficient units has not much changed compared to the previous state, it seems to be the best option for DM to accept.
• While the number of inputs is reduced from three to two and the number of outputs is reduced from eight to two, the model retains 97.053% and 92.801% of the total variance of the input/output data, (the fourth row of the Table). As evident, the number of efficient units reached 14, reducing the ranking power of the model.

As shown in Fig 3, the more the number of inputs and outputs, the more the number of efficient units, however the less the loss of information.

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Fig 3. Comparing the number of inputs and outputs with the number of efficient units (pharmaceutical companies).

https://doi.org/10.1371/journal.pone.0290610.g003

In Table 5, the Pharmaceutical companies are ranked according to their efficiency scores determined by the proposed algorithm for one input index and two output indices. To express the superiority of this algorithm, the efficiency scores of the companies were calculated with SBM (without the PCA) [25], and the score rank of the efficient units was estimated using the super-efficiency SBM (without the PCA) [23].

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Table 5. Ranking of the pharmaceutical companies (2021).

https://doi.org/10.1371/journal.pone.0290610.t005

Whilst comparing the two methods (PCA-SBM, SBM), it is evident that the PCA-SBM model works in a superior way than the SBM model (5 efficient DMUs versus 19). If a method can assign the efficiency score of one to a smaller number of units, the ranking power of that method is higher. Because the ranking of the units that get an efficiency score of one requires another complementary method. In this case study, the SBM method cannot capable to determine the rank for 18 DMUs: DMU3, DMU5, DMU6, DMU8, DMU9, DMU10, DMU11, DMU12, DMU14, DMU15, DMU16, DMU17, DMU18, DMU19, DMU20, DMU22, DMU24, DMU25 and DMU27. But the method PCA-SBM is not capable to determine the rank for only 4 DMUs: DMU5, DMU18, DMU31and DMU34.

When the two methods (PCA-SSBM, SSBM) are put to comparison, it is clear that the PCA-SSBM model works better than the SSBM model, because the SSBM cannot rank non-extreme units (DMU10, DMU16, and DMU18, as their efficiency scores remain one and it is impossible to distinguish between their ranks). Though, the PCA-SSBM is capable of ranking DMU10, DMU16, and DMU18 by assigning an efficiency score not equal to one to these units.

In S3 and S4 Tables in Supporting Information Section, raw data of input and output indices for the pharmaceutical companies are presented (2021) [53].

Comparison of the complexity of two methods (SBM, Super SBM) with (PCA-SBM, PCA-Super SBM)

To compare the complexity of the proposed method with the complexity of the prior ranking method (SBM, Super SBM), the time of implementation (execution) can be applied as a complexity index. The less the time of implementation, the less the complexity.

The data set was different divisions of branches of one of the Iranian commercial banks. The maximum number of branches of this bank is around 2000. We have increased the number of DMUs exponentially in order to make the differences in execution times clear.

We implemented two methods with GAMS software, and applied several data sets from 50 to 2000 DMUs. In all cases, the software was successful in calculating efficiency scores. In SBM, Super SBM method, all indices (3 inputs and 8 outputs) are considered. The first step is to calculate the efficiency score of 50 DMUs with SBM and note down the time of its execution, the second step is to calculate the score rank of efficient units out of 50 with Super SBM and also note down the time, and finally add up these times. In PCA-SBM, PCA-Super SBM method only one input and one output are considered (which are actually a linear combination of the previous 3 inputs and 8 outputs). The first step is to calculate the efficiency score of 50 DMUs with PCA-SBM and note down the time of its execution, second step is to calculate the score rank of efficient units out of 50 with PCA-Super SBM and also note down the time, and finally add up these times. This time calculation method is continued for 100, 500, 1000, and 2000 DMUs.

Table 6 shows the implementation time of two methods for various numbers of DMUs (50, 100, 500, 1000, and 2000).

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Table 6. Comparison of two methods.

https://doi.org/10.1371/journal.pone.0290610.t006

The results in the table show that in all cases, especially in the high number of DMUs, the complexity of the presented method (PCA-SBM, PCA-Super SBM) is far less than the previous method (SBM, Super SBM). The reason for this issue can be expressed from two perspectives: First, when the number of indices is reduced, the complexity of the model is less and it takes less time to solve the model. Second, as in the presented method, fewer units are selected as efficient units, so less time is required to calculate rank scores and perform their ranking.

In Fig 4 it is obvious that the complexity of the proposed method (PCA-SBM, PCA-Super SBM) is less than the complexity of the (SBM, Super SBM) method, especially in excessive amounts of DMUs.

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Fig 4. Complexity comparing.

https://doi.org/10.1371/journal.pone.0290610.g004