AN INTEGRATED APPROACH FOR THE MERGER OF SMALL AND MEDIUM-SIZED INDUSTRIAL UNITS

. Considering the importance of small and medium-sized industrial units for economic growth, social cohesion, regional and local development, this study presents a model for the merger so that they can make use of each other’s capacities and facilities to achieve higher efficiency levels. The involved criteria have been chosen using the SCOR model with the consideration of sustainability, resilience and agility criteria in each part of the supply chain network. PCA has been used to reduce the dimensionality and the efficiency of units has been determined by network DEA. Next, a mathematical model has been used to determine the best combination for merger. The model chosen for the finalization of the merger process is inverse network DEA, which tries to determine the final inputs of the merged units for a specific target. In addition to theoretical benefits, the results have practical applications. The results can give supply chain partners a common language for better communication and help them settle on standardized definitions. The model has been implemented using real-world data gathered from other articles, which pertain to 26 stone industries of Iran. The DEA model and the mathematical model have been solved through GAMS and the PCA approach through MATLAB.

It can also serve as a diagnostic tool for supply chain management, helping users to understand the processes involved in a business organization and identify the critical features that lead to customer satisfaction [33].In this study, the SCOR model is used for the performance evaluation of units.After identifying the criteria for measuring performance, resilience, and agility, Principal Component Analysis (PCA) is used to reduce the dimensionality of the problem, and a version of Network Data Envelopment Analysis (NDEA) is used to determine the efficiency of units.While most articles in the literature concerning this subject have used the traditional Data Envelopment Analysis (DEA) models, in order to succeed in international markets, even the smallest sub-processes of industrial units must be taken into account during performance evaluations.This can be done by the use of inverse NDEA to analyze the changes in not only inputs and outputs but also intermediates.In the next stage of the study, the developed mathematical model is used to determine the best combination of units for merger subject to constraints such as budget, distance, and the number of employees, and with objectives such as sales maximization.For the finalization of the merger process, the inverse NDEA is utilized to determine the final inputs and outputs of the merged units for a specific performance target.Contrary to traditional DEA, which attempts to measure the efficiency of units in terms of an objective function, inverse DEA tries to determine the best possible output (input) for a given input (output) for a constant objective function value.In other words, it determines how much input should be consumed to maintain or improve efficiency at a certain output level.
A variety of different strategies can be adopted for improving profitability through mergers, which depend on whether the merger is vertical or horizontal.The present study is mainly focused on horizontal mergers.In this type of merger, one company merges with or takes control of another company that offers similar products or operates in the same industry and at the same stage of production.In this type of merger, merging companies tend to be direct competitors.This type of merger has several benefits, the most important of which is the elimination of competition, which helps increase market share, revenue, and profits.Furthermore, the increased firm size tends to result in a reduction in average costs because of higher production volume and economies of scale.This type of merger also leads to higher cost-effectiveness by reducing redundant activities and operational processes (e.g. in management, advertising, procurement, and marketing domains).Horizontal integrations tend to have an impact not only on consumer welfare through price changes, but also on consumer behavior through changes in non-price characteristics such as product quality.
In addition to their theoretical benefits, the results of this study also have some practical applications.Most importantly, the results can help various industries including the stone industry to communicate better with supply chain partners and settle on standardized definitions for processes, process elements, and criteria.From the perspective of the research domain, the present study explores not only the subject of the merger of industrial units but also the selection of the optimal combination of units for the merger.In fact, in this study, the units that are to be merged are determined with the consideration of all applying limitations and conditions as well as the purpose of the merger.
The second section of this article provides a review of the literature on the subject.The third section examines the theoretical foundations and the fourth section presents the proposed model.The fifth section presents the conclusions and provides a number of suggestions for future research.

Data envelopment analysis and principal component analysis
Adler and Golany have used a combination of data envelopment analysis and principal component analysis to reduce dimensions when the number of inputs and outputs is much larger than the decision units [1].Azadeh examined an integrated framework for evaluating and ranking manufacturing systems -based on management and organizational performance indicators -for Iran's telecommunications sector [9].The first major component has the highest variance in the sample data.The second component has the second level of variance and so the principal components are calculated [3].
Principal components can be used to replace all inputs or outputs simultaneously or to replace specific groups of variables [2].Tavakoli and Shirouyehzad used PCA-DEA to evaluate the performance of the steel company based on its human capital management [35].Fu et al., in their research, concluded that by combining PCA and DEA in evaluating the performance of energy projects, the results improved in comparison to those who used only the simple DEA method [20].
PCA is a multivariate technique used to analyze the relationships of variables and explains the variables based on their components [24].To avoid subjectivity in selecting the index, PCA (principal component analysis) was used to reduce the dimensions and obtained comprehensive and objective indicators, followed by performance evaluation and analysis [34].Jothimani et al. used financial ratios as input and output performance evaluation parameters by PCA-DEA in an Indian stock market study [27].Jakaitiené et al. proposed the PCA-DEA evaluation method to evaluate the performance of the European education system [26].In a paper developed in 2020, a combination of data envelopment analysis and principal component analysis was used to evaluate the performance of 100 companies based on their 2015 financial statements [11].
In 2020, an article was developed with the aim of considering carbon emission factors in the evaluation of logistics performance.In this paper, two approaches of principal component analysis and DEA cover approach were used [17].In another paper, the financial performance of 46 financial institutions was calculated using a combination of two approaches: data envelopment analysis and principal component analysis [13].In 2021, the confidence area method was used in data envelopment analysis and combined with the PCA approach to evaluate the energy security performance of 125 countries for 21 consecutive years [37].In addition, in 2022, with the aim of evaluating the efficiency of stone industrial units, a combination of two methods, DEA and PCA has been used [31].In another research, in order to analyze export conditions in the stone industry, in one of the stages, a combination of PCA and DEA approaches was used [32].

Inverse data envelopment analysis
Unlike standard data envelopment analysis, which aims to find performance scores, inverse data envelopment analysis, once performance is known, aims to determine the levels of inputs and outputs needed to achieve the desired performance score.The idea of inverse data envelopment analysis was first introduced by Wei to evaluate inputs and outputs for a resource allocation problem [36].
The source of inverse data envelopment analysis is reverse optimization.Unlike normal optimization, in which the goal is to find the optimal solution, in a reverse optimization, a practical solution is given that is not necessarily desirable [4].
Then, Jahanshahloo et al. developed an inverse data envelopment analysis model.The purpose of their proposed model was to estimate DMU0 output with an increase in input.In this study, a sensitivity analysis was proposed in the inverse data envelopment analysis model on inputs and outputs [25].Alinejad examined how to modify the number of outputs and inputs to keep the relative efficiency score constant and used the multiobjective interactive linear programming method [5].Using reverse data envelopment analysis, Lin measured the performance and revenue scores of a number of chain stores in Taiwan [29].
Lertworasirikul et al. discussed the possibility of simultaneously increasing some outputs and decreasing other outputs using inverse data envelopment analysis [28].Hosseinzadeh Lotfi et al. developed an inverse data envelopment analysis model to estimate output if some inputs increase [30].In 2017, Ghiyasi also addressed the issue of inverse data envelopment analysis when price data is known.In fact, the proposed models guarantee not only the technical efficiency but also the labor cost (revenue) of all DMUs.In this paper, the theoretical foundations of this problem are developed and a MOLP structure is developed [22].Chen et al. proposed an inverse data envelopment analysis model to deal with adverse outputs [15].
In 2018, Emrouznejad et al. examined the solution to the problem of allocating greenhouse gas quotas set by the government in the Chinese manufacturing industry to different regions of China with the inverse data envelopment analysis approach [18].In 2021, an article developed a new model of inverse data envelopment analysis with undesirable outputs to ensure the achievement of the safety goal at the current technical level and reduce the number of fatalities in road accidents.Two safety objectives and two additional objectives are defined and then their realization paths are determined by using the new inverse DEA model, respectively.The model calculated the optimal adjustments of inputs, desirable and undesirable outputs, and the corresponding realization paths are thus obtained to achieve these objectives [16].
Because of limitation of the change range of outputs/inputs on the assumption of VRS, Chen et al. provided the specific optimization scheme for the given efficiency level; then the practicability and effectiveness of inverse DEA method were improved [14].

Merging of industrial units
Gattoufi et al. developed the concept of inverse data envelopment analysis in the concept of unit merger analysis to find the optimal levels of inputs and outputs required from entity mergers.Using inverse data envelopment analysis, the authors enabled the integrated entity to achieve an efficient goal [21].
In a 2016 article, the merging of decision-making units despite negative variables was discussed.In this model, the integrated entity can adjust its inputs and outputs to achieve a specific efficiency goal.The proposed method examines the maximum improvement in negative output of DMUs after merging [7].Amin et al. also developed two-stage inverse data envelopment analysis (DEA) models for estimating potential gains from bank mergers for the top US commercial banks.The results showed additional intermediate and final outputs at different predefined target levels of technical efficiencies.The paper confirms that there are financial gains to improving technical efficiency as the merged bank improves its optimal mix of inputs at higher efficiency levels [8].Estimation of operating efficiency gains from possible M&A between banks is another topic that has been developed by Al Tamimi and et al.Their empirical analysis was based on conventional and Islamic banks located in the MENA region and Turkey [6].
Ghobadi studied the problem of merging units in the presence of interval data by considering inverse data envelopment analysis.An insightful method was provided to obtain the minimum and maximum achievable lower and upper bounds of the efficiency interval of the merged unit [23].

Problem definition and modeling
One of the methods widely used in the performance evaluation of supply chains is the Network Data Envelopment Analysis (NDEA).NDEA measures the performance and efficiency of a unit whilst considering all the components involved based on the most detailed criteria.In this study, this method is used to determine the efficiency of industrial units.Given the large number of criteria considered in this study, which greatly undermine the ability of NDEA models to provide acceptable results, Principal Component Analysis (PCA) is used for dimensionality reduction in order to increase the method's differentiation power by significantly reducing the amount of data fed to the model.The CRS version of NDEA is then used to determine the efficiency of sub-networks and ultimately the efficiency of the entire supply chain network for the units.Next, a mathematical model is used to determine the optimal conditions for the merger of industrial units.After identifying the units that have to merge to achieve efficiency improvement, inverse DEA is used to determine the merged units' optimal inputs levels for achieving the target efficiency level.The proposed structure of the problem is shown in Figure 1.

DEA and Network DEA
Data Envelopment Analysis (DEA) is a mathematical programming model for evaluating the efficiency of decision-making units (DMUs) with multiple inputs and multiple outputs.This method was originally developed by Farrell for measuring the efficiency of production units [19].Later, Charnes, Cooper, and Rhodes developed the DEA model for measuring the efficiency of units with multiple inputs and outputs under the assumption of a constant returns-to-scale (RTS).This model became known as the CCR model [12].In 1984, Banker, Charnes, and Cooper generalized the CCR method for variable RTS, developing the model known as the BCC model [10].DEA is one of the most widely used mathematical programming methods for evaluating and comparing the performance of a set of decision-making units (DMUs) in applications such as supplier selection.This model is known to maintain a good level of performance even when the analysis involves a large number of DMUs.For this reason, this study uses the DEA method for its evaluations.
In this study, the criteria for the evaluation of industrial units and supply chain components and subnetwork are chosen from among the list of criteria previously used in other articles.These criteria, which also include sustainability criteria, are selected using the SCOR model.As mentioned, this study makes use of the network DEA approach for its evaluations.The general network layout considered in this study is illustrated in Figure 2. The mathematical model of the network DEA is presented in other sections.

Principal component analysis
In DEA, for efficiency evaluations to be reliable, the number of DMUs under evaluation must be at least three times the number of input and output criteria [10].Considering the large number of criteria considered in this study for supply chain network evaluations, which also requires examining the conditions of a large number of industrial units, Principal Component Analysis (PCA) is used to reduce the dimensionality of the data while preserving the majority of the information contained in the data in order to make sure that the proposed model can be properly implemented with fewer DMUs.For a problem with  variables, the principal component  is expressed as: In the above equation,  = 1, . . .,  denotes the principal components and  = 1, . . .,  denotes the main problem variables (one can define as many principal components as there are problem variables).Also,   is the -th component of the linear transformation vector or eigenvector derived from the correlation matrix, and   is the -th main variable.In this study, the PCA module provided in MATLAB was used to analyze the input and output criteria and choose the principal components that can preserve a high percentage of the total variance in the data for use in the NDEA model.

Mathematical model of the merger of industrial units
After determining the efficiency of industrial units, a mathematical model was developed for the merger of these units with the purpose of increasing their interactions, enabling them to utilize each other's capacities to increase efficiency, and achieving cost reduction (e.g. by reducing redundant labor costs).In this model, the goal is to find the best combinations for the binary integration of units with the objective of cost reduction considering the units' capacities and limitations.This model also assumes that the units to be merged must be in the same location.The parameters and variables of this mathematical model are as follows: The mathematical model for the merger of units is given in equations ( 6) to (23).
+ = min(  ,   ) ∀, ( In the above mathematical model, equation (6) computes the profit to be earned from the reduction of labor costs after the merger.In this equation, it is assumed that after the merger, all the capacities of the industrial units are used.So, the number of labor units after the merger is obtained by dividing the total capacity of two units by the average production capacity of each person.Since the goal is to maximize the difference between costs before and after the merger and the max-min composition is commonly used when a system requires conservative solutions, we have: Since using the above model makes the problem nonlinear, the parameter   is defined as follows: By substituting this parameter into the first term of the objective function and adding the constraint to equation ( 25), the problem becomes linearized as shown below: This constraint only applies if there are fewer labor units after the merger than before the merger.In other words, Industrial units will get this profit only if the number of labor units decreases after the merger.Otherwise, this profit amount will be zero.With the increase in the number of human resources, costs such as salary increases will be added, which is considered in equation (7).Therefore, the right side of constraint ( 27) is multiplied by the variable   so that it gets activated whenever the mentioned condition applies.Otherwise,   = 0 makes sure that the right side of constraint ( 27) remains zero, which results in   becoming zero as well.Thus, the constraint added to the model is in the form of equation (11).As mentioned earlier, equation (7) in the objective function is the profit to be earned from the reduction of procurement costs after the merger.Since the volume of procurements is expected to increase after the merger (assuming that more procurement will be made in the unit with lower procurement cost, as expressed in Eq. ( 13)), a volume discount is considered in the model.Equation ( 8) expresses the recruitment costs after the merger.In some mergers, units need to hire more people in order to make full use of their new capacities, in which case the cost gets a negative sign in the model.Equation ( 9) computes the increase in taxes because of the increase in sales after the merger.In the case of the stone industry, this tax cost only applies to domestic sales because the export of building stones is exempt from tax.
Equations ( 10) and (11) determine the value of variable   .If the number of labor units before the merger is greater than after the merger, then the parameter   must become non-zero and the recruitment costs should not be applied, and therefore the variable   will be equal to one.Conversely, if the merger causes an increase in labor, the recruitment costs should be applied and the parameter   should be zero, and therefore   will be zero.Equation ( 14) determines the maximum inefficiency required for the merger.In this equation, the average level of inefficiency for each sub-network and each decision unit has been compared with the maximum acceptable level for total inefficiency.This constraint only applies if the two units are merged (  = 1) and is otherwise removed from the model.Equations ( 15) and ( 16) consider the reasonability of the merger.Only the units that are located in the same city can be merged, and each unit can only be merged with one other unit.Equation (17) indicates that the units located in different cities cannot be merged (  = 0).Equation (18) indicates that a unit cannot be merged with itself.Equation (19) states that if unit  is merged with unit , neither of them can then be merged with another unit like  or .Thus, we have: These two constraints can be combined together and placed in the model as one constraint.Equations ( 20) and ( 21) of the model make sure that in the matrix of binary mergers, the variables   can be non-zero at most once in each row and each column (meaning that each unit can only be merged with one unit in the same city).Equations ( 22) and ( 23) define the type of variables used in the model, which are all binary.

Inverse NDEA for the merged units
After determining the optimal combination for the merger, it is necessary to determine the units' post-merger inputs.Here, this was done using the inverse NDEA method.In the approach of inverse data envelopment analysis, an attempt is made to determine the optimal amount for inputs and outputs in order to achieve a certain efficiency.Since in this article, the aim is to determine the optimal combination for the integration of factories, it should be determined how much these units should maintain their criteria after the integration in order to achieve the desired efficiency.The best approach for this purpose is inverse data envelopment analysis.To develop the inverse NDEA model, first, the dual of DEA models had to be obtained.In this work, the mathematical model of inverse DEA was developed using the criteria considered for the case study, which are discussed in the next section, because mathematical models depend on the conditions of the case study.

Model implementation
As mentioned, in this study, evaluations with the NDEA approach are performed based on the real data reported or used in other articles.The supply chain network of the considered factories is shown in Figure 3. Considering the sequence of activities in the selected industry and their prerequisite and dependency relations, the supply chain of this industry can be divided into four sub-networks.The first sub-network is related to the procurement of raw materials such as stone blocks and consumables such as resin, the second sub-network is related to support operations such as human resource management, accounting, finance, and R&D, the third sub-network is related to the production process, and finally, the fourth sub-network is related to sales and delivery.
After determining the supply chain network of the units and the relation between sub-networks, the criteria involved in the evaluation of sub-networks (inputs and outputs for each part of the network) were classified as shown in Table 1.
The efficiency of units was determined based on the real-world data used in other articles, which were collected by observation and face-to-face interviews.These data are provided in Table 2.
The results of the application of PCA to the above data in MATLAB are presented in Table 3.In this analysis, the principal components that were able to preserve the highest percentages of the total variance in the data were chosen for use in the rest of the study.The amount of variance preserved by these principal components is shown in Table 4.
Since DEA models cannot accept negative inputs, the following equations ( 30) and (31) were used to make all the figures related to principal components non-negative: Also, since each principal component preserves one part of the total variance in the data, different components have different levels of importance for subsequent analyses.In other words, the greater the variance preserved by a component, the more valuable it will be.Therefore, the values of input and output criteria were entered into the DEA model based on the component's value.This value was calculated based on the ratio of the variance preserved by each individual component to the sum of the total variance covered by all of the selected components.First, the variance preserved by each component was used to compute a weight for that component in DEA.Then, the weight obtained for each criterion was multiplied by the non-negative values of that criterion in the respective column to obtain the weighted criteria values for use in the DEA model.The final results are presented in Table 5.
After preparing the data related to the inputs and outputs of the sub-networks of the supply chain, the CRS version of the DEA model was used to determine the efficiency of the stone processing units in all four sub-networks.The definitions of variables and parameters used in the model are provided in Table 6.
The DEA models developed for the raw material and consumable procurement department based on the CRS model are in the form of equations ( 32)- (35): The DEA models formulated for the support department based on the CRS model are provided in equations ( 36)-(42): The DEA models for the manufacturing department based on the CRS model are in the form of equations ( 43) to (49): 4 1 , The DEA models developed for the sales department based on the CRS model are given in equations ( 50) to (55): The results of the implementation of the four DEA models in GAMS software to determine the efficiency of each department are shown in Table 7.
After determining the efficiency of each unit, the mathematical model for the merger was applied to the case study data.In this model, the goal is to determine the best combinations for the binary merger of units with the objective of cost reduction considering the units' capacities and the defined constraints.The results of model implementation in GAMS showed that there should be a merger between units 1 and 5, units 10 and 26, units 12 and 15, units 13 and 22, units 14 and 19, and units 23 and 24.
Table 1.Classification of criteria for the performance evaluation of units based on the combination of BSC and SCOR.After obtaining the optimal combinations for the merger, the units' post-merger inputs were determined using the inverse NDEA method.To develop the inverse NDEA model, first, the dual of DEA models had to be obtained.The dual of the CRS-type DEA model for raw material and consumable procurement department are as follows:   The inverse DEA models developed for the procurement department in the binary merger of industrial units (units  and ) according to the dual of the model for this department are given in equations ( 60)-(65): Table 6.Variables and parameters used in the NDEA model.

𝑥𝑖𝑗
Value of input  of the procurement department for unit    criteria weight Value of output 1 of the procurement department for unit   1 Value of input 2 of the human resources management department for unit Value of input 3 of the accounting and finance department for unit   3 Value of input 4 of the research and development department for unit   4 Value of output 2 of the human resources management department for unit   1 Value of output 3 of the accounting and finance department for unit   2 Value of output 4 of the research and development department for unit   3 Value of input 1 of the rock offloading department for unit   4 Value of output 2 of the rock offloading department (input of the stone cutting department) for unit   5 Value of input 3 of the stone cutting department for unit   6 Value of output 4 of the stone cutting department (input of the storage and packaging department) for unit Value of output 5 of the storage and packaging department for unit   1 Value of input 3 of the marketing and advertising department for unit   3 Value of output 1 of the delivery department for unit   1 Value of output 2 of the marketing and advertising department for unit Notes.All parameters and variables have been used in relations 32-55.Subscripts and superscripts are used only to name them.
The dual of the DEA models for the support department based on the CRS model are expressed in equations ( 66)-(73): The inverse DEA models obtained for the sales department in the binary merger of industrial units (units  and ) according to the dual of the model for this department are provided in equations ( 112)-(121): Min After coding the inverse DEA models in GAMS, the residuals of each input for each unit to be merged was obtained, and the percentage change in each criterion was determined accordingly.
As previously explained, the data used in the DEA model were first subjected to a transformation to make them positive and then weighted based on the percentage of variance they preserve.Therefore, in the model, PC()  was used instead of PC  , resulting in the following formulation: The results are presented in Tables 8-11.
Because of the use of PCA to reduce the dimensionality of the data to be fed into DEA, the obtained changes are related to the selected principal components and not the original problem data.Therefore, following the principles of PCA, the new inputs of the units to be merged were determined by the use of eigenvectors, which were obtained from equation (124) (where   denotes eigenvectors) The eigenvectors obtained by implementing PCA in MATLAB are given in Tables 12-19.
Knowing the percentage change in each principal component after a merger of an industrial unit, one can estimate the amount of change to be expected in each of the main criteria to help the managers of the industry make more informed decisions about the modification of their inputs in order to achieve a target performance level.For example, to estimate the amount of change in each of the three inputs of the marketing and advertisement department in the sales network, considering that some criteria are not applicable, one should use   In this paper, these changes are limited in order to minimize the change in the original problem data.This has been achieved by applying some constraints to the extent of change possible in each input.For example, for percentage-type criteria, the change interval should such that input values remain between zero and 100.
As another example, qualitative data that have been quantified based on the Likert scale, inputs should be discrete values ranging from 1 to 5.These constraints along with the objective function were analyzed in a mathematical model developed in GAMS.The results are presented in Table 20.As indicated in the previous sections, 6 optimal combinations were obtained for integration.For example, according to the objective functions and constraints of the integration model, it was concluded that units 1 and 5 meet the integration conditions.Since in this article it is assumed that after the merger of two units, none of them will be removed, the remaining amount of each of the criteria should be obtained for each of the units 1 and 5.The data of this the table shows how much of each of its inputs should be retained after unit 1 is merged with unit 5 to achieve the selected target performance in cooperation with the other unit.
The application of the proposed model to real data and the examination of its effect on the efficiency of industrial units showed an efficiency improvement compared to the average efficiency of units before the merger; a difference that is illustrated in Figure 4.

Conclusions and suggestions
Recognizing the importance of attention to SMEs operating in different industries and their growth potentials, many governments have adopted support policies and incentives to help them achieve higher efficiency levels.The efforts to enhance the efficiency of SMEs in an industry cannot succeed without a sound knowledge of the supply chain network of that industry and the criteria that influence its efficiency.Since the use of a large number of criteria tends to result in more logical conclusions, this study attempted to consider as many criteria as possible.However, since using large volumes of data greatly undermines the ability of DEA models to differentiate between DMUs, the PCA method was used to reduce the dimensionality of the problem while preserving the information contained in the data as much as possible.After developing NDEA models for evaluating the efficiency of DMUs, the paper explored the subject of the merger of industrial units with attention to efficiency criteria and certain constraints such as the location of units so that they can benefit from each other's competitive advantages, experiences, and facilities, modernize their equipment with joint capital, gain more resistant against unforeseen changes, and try to improve the quality of their products.After determining the optimal combination of units for the merger by the mathematical model, changes in the inputs of each sub-network of the merged units were estimated and then used to predict the increase in the efficiency level of the merged units relative to their premerger conditions.One of the innovations of this paper was the implementation of the model on the real data

Figure 1 .
Figure 1.The proposed structure of the problem.

Figure 2 .
Figure 2. General network layout of the supply chain of industrial units.

Figure 3 .
Figure 3. Network structure of the supply chain of the stone industry.

S14- 1 2
Ability to meet the diverse needs of customers Purchase outputs & Finance inputs S3-1 Share of raw materials and consumables in the total product cost S14-2 Share of marketing costs in the total product cost S3-2 Cost of raw materials (e.g.stone blocks and consumables) Avs.Outputs & R&D inputs S15-1 Feedback from customers S3-3 Transport cost per unit volume of stone per kilometer S15-2 Knowledge of the target markets S3-4 Financial stability of stone quarries affiliated with the processing unit Unloading inputs S16-1 Maintenance and repair cost per period Finance inputs S4-1 Percentage of checks cleared and proceeds received S16-2 Amount of electricity consumed per period in normal time Human resource management inputs S5-1 Number of employees in the nonproduction parts of the business Unloading outputs & Cutting inputs S17-1 Precautionary storage of raw materials and consumables (e.g.resin) S5-2 Number of employees in the production part of the business Cutting inputs S18-1 Average machinery repair time S5-3 Number of hours of formal training S18-2 Use of machinery after the end of their life S5-4 Number of hours of informal and on-thejob training S18-3 Amount of water consumed by each machinery per period S5-5 Management's training level Cutting outputs & Warehousing inputs S19-1 Speed of cutting and processing of stone into the final product S5-6 Employees' safety training level S19-2 Difference between the unit's nominal capacity and Cost of transporting one unit volume of product from the unit to the customer per kilometer Human resource management outputs & Unloading inputs S7-1 Number of shifts and working hours S20-3 Ratio of exports to total production S7-2 Ratio of native workers to non-native workers S20-4 Share of production costs in the total product cost S7-3 Ratio of non-native workers without work license Warehousing outputs S21-1 Amount of product stored at the end of each period S7-4 Fines paid due to unlicensed non-native workers Delivery outputs S22-1 Time required to meet customer needs S7-5 Safe environment for employees S22-2 Direct sales to consumers S7-6 Employees' exposure to environmental hazards and pollution S22-3 Sales to brokers and intermediaries Human resource management outputs & Cutting inputs S8-1 Level of expertise of human resources based on their degree Adv.outputs S23-1 Percentage change in sales due to marketing and advertising S8-2 Level of work experience of human resources

Figure 4 .
Figure 4. Percentage increase in the efficiency of units after the merger.

Table 2 .
Data used in the NDEA model.

Table 4 .
Amount of variance preserved by the selected principal components.

Table 5 .
Non-negative and weighted values of the selected principal components.

Table 7 .
Efficiency of each sub-network according to the NDEA model.The inverse DEA models formulated for the support department in the binary merger of industrial units (units  and ) according to the dual of the model for this department are as follows: The inverse DEA models developed for the manufacturing department in the binary merger of industrial units (units  and ) according to the dual of the model for this department are: 2, . . .,  2 }

Table 8 .
Percentage change in each criterion in the raw material and consumable procurement department.

Table 9 .
Percentage change in each criterion in the manufacturing department.

Table 10 .
Percentage change in each criterion in the support department.

Table 11 .
Percentage change in each criterion in the sales department.

Table 12 .
Eigenvectors related to principal components-inputs of the procurement department.

Table 13 .
Eigenvectors related to principal components-inputs of the human resources management department.

Table 14 .
Eigenvectors related to principal components-inputs the accounting and finance department.

Table 15 .
Eigenvectors related to principal components-inputs of the research and development department.

Table 16 .
Eigenvectors related to principal components-inputs of the stone offloading department.

Table 17 .
Eigenvectors related to principal components-inputs of the stone cutting department.

Table 18 .
Eigenvectors related to principal components-inputs of the delivery department.

Table 19 .
Eigenvectors related to principal components-inputs of the advertisement department.