IMPROVED EFFICIENCY ASSESSMENT IN NETWORK DEA THROUGH INTERVAL DATA ANALYSIS: AN EMPIRICAL STUDY IN AGRICULTURE

. Conventional Network Data Envelopment Analysis (NDEA) models often make an assumption of data precision, which frequently does not align with the realities of many real-world scenarios. When dealing with ambiguous data, whether it involves input, output, or intermediate products represented as bounded or ordinal data, the accurate assessment of efficiency scores poses a significant challenge. This study addresses the crucial issue of handling interval data within NDEA structures by introducing an innovative methodology that integrates both optimistic and pessimistic strategies. Our proposed methodology goes beyond the mere determination of upper and lower bounds for efficiency scores; it also incorporates target-setting and improvement approaches. Through the calculation of interval efficiency for each decision-making unit (DMU), our approach offers a comprehensive framework for efficiency classification. To underscore the effectiveness of this methodology, the study presents empirical evidence through a case study in the agriculture industry. The results not only showcase the advantages of our proposed methodology but also emphasize its potential for practical application in diverse and complex real-world contexts.


Introduction
Efficiency analysis is a widely employed technique in the performance evaluation domain that aims to assess the relative performance of decision-making units (DMUs) across diverse industries.One prominent non-parametric method used for measuring the relative efficiency of DMUs by comparing their input-output relationships is Data Envelopment Analysis (DEA), which was introduced by Charnes et al. [1].DEA has found applications in diverse sectors, including healthcare, education, finance, agriculture and manufacturing.DEA models are typically based on the assumption that input and output data are precise and deterministic.However, in practical scenarios, input and output data may be imprecise or uncertain due to measurement errors, rounding errors, or subjective judgments.Interval data analysis has emerged as a promising technique to address this uncertainty by incorporating interval-valued data into DEA models.
The presence of significant variation ranges in datasets often leads to an increase in error levels during efficiency evaluation.Furthermore, data can also be uncertain, missing, or sequential, among other possibilities.To address these scenarios, an evaluation process can be employed that considers data within an interval.Interval data analysis deals with uncertain input and output data by representing them as intervals instead of exact values.By adopting this approach, interval data analysis provides a more realistic representation of the true values of input and output data and generates more accurate results in DEA models.Interval data analysis has been widely studied for the last twenty years.Cooper et al. [2] pioneered the concept of imprecise data envelopment analysis (IDEA).Imprecise data refers to a combination of interval and ordinal data.Traditional DEA models become nonlinear when imprecise data is present.To deal with this issue, evaluation procedures utilize upper and lower boundaries for each imprecise input and output, and this approach results in one upper and one lower boundary for the efficiency score of each DMU.
A literature review reveals that most IDEA studies introduce models and employ transformation and variable alteration techniques to convert nonlinear models into linear ones.Table 1 summarizes the progress of the most influential IDEA models from Cooper et al. [2] to the present day.
Traditional DEA models, including those mentioned earlier, treat DMUs as "black boxes" where only input and output values are used to assess their performance, and the internal operations and structure of the DMUs do not affect the performance evaluation.However, Färe and Grosskopf [42] introduced a new perspective that considers the impact of a DMU's internal operations and structure on performance evaluation in DEA.This approach is called network data envelopment analysis (NDEA) and has been widely used by researchers in various fields over the past two decades.Kao [43] compiled and categorized almost all publications related to this topic.Kao [44] provides a comprehensive explanation of NDEA, including its model development and practical applications.
NDEAs can be classified into three modes based on the internal structure of the DMUs being evaluated: parallel, series, or mixed (general).In the parallel mode, intermediate products are not exchanged between processes, and each process operates independently.Färe et al. [45] were among the early researchers in this field and developed the parallel mode.They used a nonparametric approach to assess the efficiency of 57 farms in Southern Illinois that were producing four types of crops simultaneously in 1994.Cook et al. [46] introduced a model to evaluate systems where multiple functions are performed, and Cook and Hababou [47] studied the performance of banks whose primary functions were sales and services.Jahanshahloo et al. [48] evaluated the performance of 39 branches of Iranian commercial banks that shared inputs and performed functions related to deposit, sales, and services.Yu and Fan [49] studied the performance of 24 bus companies that provided different urban and highway services.Kao [50] developed a parallel DEA model to measure both DMU efficiency and their corresponding process efficiency.Kao [51] later used this approach to evaluate the efficiency of 52 chemistry departments in England.Kremantzis et al. [52] discussed the incorporation of hierarchical network structures in a multi-function parallel system.
The series state is characterized by processes that can take intermediate products or exogenous input as input to produce final output or other intermediate products.Troutt et al. [53] presented a value-based model that exemplifies a serial process where multiple outputs from a process can be the input of the next process.Amirteimoori and Kordrostami [54] presented aggregation efficiency models that assess the performance of a series system, where the objective function is defined as the weighted mean of the efficiencies of  processes.Park and Park [55] suggested a method to evaluate aggregative efficiency in multi-period production systems.Tsutsui and Goto [56] used a weighted slack-based measure (SBM) to examine the performance of electric power companies in the US.Kao [57] proposed a relational model to measure the efficiency of DMUs and their processes simultaneously.Wei and Chang [58] introduced the Optimal System Design (OSD) Data Envelopment Analysis (DEA) model to optimally design series-network systems of DMUs in terms of profit maximization given the DMUs' total budget.Fang [59] discovered an error in the Wei and Chang model and suggested a new approach to derive the DMUs' optimal budgets and to verify the existence of budget congestion for the OSD network DEA models.Lin and Liu [60] extended the multiplier series DEA model to work with the directional distance function, allowing it to handle negative data and be non-oriented.Table 1.A review of studies on imprecise data envelopment analysis.

Publication date
Author(s) Main contribution to the literature Cooper et al. [2] Introduces Imprecise DEA, a novel approach that combines interval and ordinal data, and demonstrates the transformation of nonlinear models into linear ones using appropriate mathematical techniques.Kim et al. [3] Converts nonlinear problems into linear ones through scale transformation and variable alteration.Cooper et al. [4] Transforms nonlinear models into linear ones without requiring access to maximum data values, contrasting with Cooper et al. [2].Despotis and Smirlis [5] Introduces a novel transformation method, utilizing convex combinations of interval endpoints and variable alterations to convert nonlinear models into linear ones.Also introduces upper and lower bounds of efficiency.Entani et al. [6] Offers both an optimistic (best-case) and pessimistic (worst-case) perspective for evaluating efficiency values when dealing with interval data.Lee et al. [7] Introduces an additive model for imprecise data and converts it into a linear model through transformative techniques.Zhu [8] Introduces a method to reduce computational burden when converting nonlinear IDEA models to linear ones, building on prior approaches.Park [9] Introduces a simplified method for converting nonlinear models into linear models solely through variable alternation.Zhu [10] Demonstrates the straightforward transformation of interval and ordinal data into exact data, enabling the application of conventional DEA models to analyze them.Jahanshahloo et al. [11] Suggests an adapted model, the FDH model for interval data, and proposes upper and lower bounds for efficiency levels within this model.Jahanshahloo et al. [12] Evaluates the return to scale for DMUs with interval data, specifically those that are always efficient.Jahanshahloo et al. [13] Calculates the radius of stability for all DMUs with interval data.Amirteimoori and Kordrostami [14] Introduces a model for assessing the efficiency of multi-component structures using imprecise data.
Introduces models for upper and lower bounds of efficiency with interval data, emphasizing that different constraint sets lead to distinct efficiency frontiers, limiting direct DMU efficiency score comparisons.Haghighat and Khorram [16] In the presence of interval data in DEA, the minimum and maximum number of efficient units are determined by the most pessimistic (maximum) and optimistic (minimum) values within the interval data, respectively.Kao [17] Suggests two-level mathematical programming models for computing upper and lower bounds of efficiency when dealing with imprecise data.Smirlis et al. [18] Utilizes an interval DEA-based method to handle situations with missing data.This approach involves substituting intervals for missing data and subsequently calculating interval efficiency scores for DMUs with incomplete information.Farzipoor Saen [19] Proposes procedures for selecting the optimal technology when dealing with a mix of ordinal and cardinal data.Farzipoor Saen [20] Offers procedures for ranking the best technology when both ordinal and cardinal data are present.

Publication date
Author(s) Main contribution to the literature Farzipoor Saen [21] Proposes procedures for selecting the best supplier when dealing with a combination of ordinal and cardinal data.Kazemi Matin et al. [22] Introduces an additive model for evaluating the technical efficiency of DMUs in the context of interval data.Utilizes the translation invariant property of the additive model to transform the nonlinear model into a linear one.Park [23] Examines imprecise data in DEA and presents two distinct programs for computing the upper and lower bounds of efficiency.Jahanshahloo et al. [24] Introduces a generalized model for interval data utilizing a domination structure.Farzipoor Saen [25] Introduces a novel pair of DEA models for selecting the best suppliers when dealing with imprecise data and nondiscretionary factors concurrently.Park [26] Analyzes the relationship between envelopment and multiplier forms of DEA models and their respective solutions in the context of Interval Data Envelopment Analysis (IDEA).Azizi and Ganjeh-Ajirlu [27] Suggests a pair of DEA models for assessing relative efficiency in scenarios involving nondiscretionary and imprecise data.Farzipoor Saen [28] Develops modifications to traditional DEA models to address media selection problems when dealing with both imprecise data and flexible factors.Farzipoor Saen [29] Introduces a model for selecting optimal markets that simultaneously accounts for weight restrictions, dual-role factors, and imprecise data.Kao and Liu [30] Presents an approach for evaluating interval efficiency within two-stage network systems when dealing with fuzzy data.Emrouznejad et al. [31] Introduces nonparametric multiplicative and general models designed to handle interval inputs and outputs.Zhu and Zhou [32] Examines the impact of imprecise data within a two-stage network structure.Bouzembrak et al. [33] Extends a comprehensive possibilistic linear model for supply chain systems with imprecise inputs and presents a solution approach for a possibilistic mixed-integer program.Hadi Vencheh et al. [34] Suggests an inverse DEA model that replaces fluctuating DMUs with modified counterparts.Additionally, introduces an imprecise inverse DEA model designed for assessing relative efficiency when dealing with interval data.Azizi et al. [35] Presents two approaches for obtaining efficiency scores in the presence of imprecise data using the SBM model: one optimistic (best-case) perspective and one pessimistic (worst-case) perspective.He et al. [36] Develops interval efficiencies for inefficient DMUs through the utilization of interval ideal points, allowing for an increase in the lower bound of efficiency while enabling the upper bound of efficiency to reach unity (one).Shirdel et al. [37] Proposes a model for assessing DMUs using interval data and applies its outcomes to rank the DMUs.Creates a robust efficiency analysis procedure for cases with incomplete data, and introduces methods for perfect and potential efficiency analysis to establish lower and upper bounds for robust efficiency ratings.2018 Toloo et al. [39] Suggests a pair of interval DEA models for calculating interval efficiencies, considering inputs, outputs, and dual-role factors as intervals.These models utilize optimal multipliers for each dual-role factor to determine whether it functions as an input, output, or remains in equilibrium.2020 Mo et al. [40] Introduces an Interval Modified Slack-Based Measure (SBM) model designed for situations involving undesirable outputs.2021 Ghobadi [41] Suggests solutions to address challenges in merging DMUs when dealing with interval data.
A significant amount of literature has been dedicated to the study of two-stage DEA models, which can be considered a specialized type of the series mode.Sexton and Lewis [61] developed a model and used it to evaluate the efficiency of Major League Baseball.Kao and Hwang [62] proposed a method to decompose the efficiency of two-stage structures.Chen et al. [63] developed a similar efficiency decomposition method for variable returns to scale (VRS).Liang et al. [64] introduced efficiency measures for two-stage structures, including feedback loops where the output of the second stage may be used as input for the first stage.Liu and Lu [65] used the concept of eigenvector centrality in social network analysis to identify the efficient DMUs in two-stage systems.Halkos et al. [66] used an additive efficiency decomposition method and proposed a weight assurance region model to restrict the weights of the relative importance of the first and second stage performance.Aviles-Sacoto et al. [67] considered a situation where some outputs from the first stage may also be considered final outputs and developed a methodology to handle this situation.Tavana et al. [68] presented a two-stage DEA method for evaluating the performance of multi-level supply chains.Azizi and Kazemi-Matin [69] proposed a method for ranking two-stage production systems based on the efficiency score of each stage.Halkos et al. [70] provided a classification of the literature on two-stage DEA structures.Finally, Shahbazifar et al. [71] introduced a new model for evaluating the group efficiency of two-stage production systems.
In NDEA, the mixed or general state combines the series and parallel modes, where a process can use input from other processes and produce intermediate or final outputs.Limited research has been conducted on modeling the general structure of NDEA.Färe and Grosskopf [42] coined the term NDEA and developed an envelopment model for certain forms of network mixed states.Lewis and Sexton [72] extended the definition of NDEA and applied it to evaluate the American Baseball League.Kao [57] proposed a multiplier model for a mixed structure, and Kao and Hwang [73] explored further forms of mixed state and their impact on performance in the banking industry.Lozano [74] studied general network structure by defining a production possibility set (PPS) for each process and the entire system, and introduced a radial model for technical efficiency and another for cost efficiency.Kazemi Matin and Azizi [75] presented a model for network general state in both multiplier and envelopment forms, where a section of the process output could be used as input to other processes, and the remaining part of the output is the final output.Boloori et al. [76] proposed dual multiplier and envelopment DEA models for a more general state where a factor could be both an intermediate product and the main input-output simultaneously.Kalhor and Kazemi-Matin [77] introduced a new PPS for general network DEA in the presence of undesirable outputs and proposed network DEA models for performance evaluation of DMUs, emphasizing the use of distinct abatement factors and selecting an appropriate balance constraint.Yang et al. [78] proposed a two-level maximin NDEA model, considering the efficiency of all divisions and DMUs simultaneously.
The investigation of interval data in NDEA has been relatively limited.One of the few studies in this area was conducted by Amirteimoori and Kordrostami [14], who proposed a model for measuring the efficiency of a multi-component structure in a network parallel state with imprecise data.Kao and Liu [30] developed an approach for calculating interval efficiency in a two-stage network system with fuzzy data.They used a multiplier model to determine the upper bound of efficiency and a development model to calculate the lower bound of efficiency.Building on the work of Kao and Liu [30], Zhu and Zhou [32] explored the presence of imprecise data in a two-stage network structure.
While NDEA has seen limited exploration in the context of interval data, it is essential to examine its applications in various domains, including agriculture.The previous sections have laid the foundation by discussing the significance of DEA models in performance evaluation, the emergence of interval data analysis as a promising technique, and the categorization of NDEA models based on the internal structure of DMUs.In this section, we delve deeper into the intersection of interval data and NDEA, aiming to bridge the gap between these domains and extend the practical utility of NDEA models.Our investigation not only contributes to the theoretical understanding of interval data in network DEA but also sets the stage for its real-world application, as demonstrated in subsequent sections.
The field of agricultural economics has a profound influence on food security and economic stability, making it a prime arena for the application of DEA models.As modern agriculture becomes increasingly intricate, the necessity for advanced DEA models becomes apparent.These models are instrumental in addressing complex challenges such as scale inefficiency, environmental sustainability, and technological progress.Recent systematic reviews in the agricultural DEA domain have shed light on the shortcomings of traditional approaches like Constant Returns to Scale (CRS) and VRS, underscoring the demand for more sophisticated techniques.Remarkably, despite its paramount significance, NDEA models remain relatively underexplored in the agricultural sector, as noted by Kyrgiakos et al. [79].
Furthermore, Network DEA has been deployed in several studies to uncover inefficiencies within various subsystems of the agricultural process.Saputri et al. [80] employed this methodology to evaluate the efficiency of three distinct stages in the agri-food supply chain for Indonesian rice producers.Kord et al. [81] adopted a similar approach, presenting agricultural activity as two separate stages (environmental and economic) and conducting a sustainability assessment for different Iranian regions by leveraging shared inputs between these stages.Lu et al. [82] introduced a three-stage Network model to assess agricultural food production systems in European Union countries under the principles of a circular economy, where the final output carries over into the next period.
Additionally, Kord et al. [83] incorporated sensitivity analysis into their approach to evaluate the allocation of human resources in a two-stage Network DEA model.Specifically, they examined the optimal allocation of human resources in the initial cultivation and maintenance stage, as well as in the harvesting stage.Abbas et al. [84] also utilized this analysis to investigate changes in crop output under varying input conditions.Furthermore, Grey Relational Analysis was applied by Yang et al. [85] to assess the impact of included variables on the environmental performance of Chinese households.
In light of these developments, it is worth noting that there is a noticeable gap in research that combines NDEA and agriculture, especially when dealing with interval data.Recognizing this research vacuum, our study aims to fill this void by presenting a practical application that focuses on evaluating wheat farming practices across 30 Iranian provinces.In doing so, our objective is to showcase the potential of NDEA models in enhancing the efficiency and sustainability of agricultural operations, thus pushing the boundaries of knowledge in the assessment of agricultural efficiency.
This study addresses the research gap in the limited number of papers that examine the presence of interval data in network DEA.The contribution of this study is twofold: firstly, it provides a new tool for evaluating network production processes with imprecise data, and secondly, it demonstrates the practical application of this tool in the agriculture sector.
The subsequent section of this paper examines the impact of interval data on general network structures of DEA and proposes novel models for assessing the upper and lower bounds of efficiency.In Section 3, an illustrative example is formulated to demonstrate the effectiveness of the proposed models.Additionally, Section 4 showcases a real-world application of the models in evaluating wheat production in Iran.Finally, the study concludes with remarks presented in Section 5.In the following sections, we delve into the impact of interval data on general network structures, provide novel models for assessing efficiency bounds, present an illustrative example, discuss a real-world application in wheat production, and conclude with our findings and remarks.

Interval data in NDEA
As previously stated, NDEA can be classified into three main states based on the internal structure of DMUs, namely parallel, series, and mixed.Since the overall network structure encompasses all three states, this section focuses solely on the general states in relation to interval data.

General network structure with interval data
In a general state, each process has the ability to consume exogenous inputs and intermediate products produced by other processes as inputs, and produce final outputs and other intermediate products as outputs.Let us consider  observed DMUs. Figure 1 illustrates the internal relationships among processes for the production units in the general mode.Specifically, the figure represents DMU  , where   ;  = 1, . . ., , refers to the observed amount of the -th exogenous input to that DMU.Each DMU is divided into  processes, labelled as ;  = 1, . . ., .The amount of the -th input consumed by the -th process in DMU  is indicated by    , representing the total amount of the -th input consumed by all processes in DMU  (  = ∑︀  =1    ).Similarly,   ;  = 1, . . ., , denotes the amount of the -th final output of the DMU, while    represents the amount of the -th final output produced by the -th process in DMU  .The flow of outputs among the processes is represented by   = ∑︀  =1    .Additionally, intermediate products are generated and consumed by the various processes within the DMU.
Let    denote the amount of the -th intermediate product that is produced by the -th process and consumed by the -th process in DMU  .We define ∑︀  =1    as the total amount of the -th intermediate product produced by the -th process in DMU  , and ∑︀  =1    as the total amount of the -th intermediate product consumed by the -th process in DMU  .
The Model (1) proposed by Kazemi-Matin and Azizi [75] is used to evaluate the efficiency score of DMU  for relations and internal structures similar to those depicted in Figure 1:  [5].Specifically, a convex combination of the lower and upper bounds of each input, output, and intermediate product interval is used.To illustrate this approach, let's consider these equations: To The above variable substitutions are made to transform the original nonlinear model into a linear model that can be easily solved using linear programming techniques.In addition, the new decision variables are introduced to represent the deviation of the actual input/output values from the lower bound of the corresponding intervals.By using a convex combination of the lower and upper bounds of each input, output, and intermediate product interval, we ensure that the resulting linear model is equivalent form of the original nonlinear model.
Solving the linear model (Model (2)) yields efficiency scores for each DMU  , which indicate how well the DMU  utilizes its inputs to generate outputs relative to the other DMUs in the sample.These efficiency scores can be used to identify best practices and benchmarking targets for improving the efficiency of the DMUs under evaluation.
To reduce the number of variables and computation cost while obtaining upper and lower bounds of efficiency scores in DEA with interval data, the strategy is to use the principle of convexity.For a specific DMU, the inputs are assumed to be at the lower bound and the outputs at the upper bound within the intervals to determine the highest efficiency score.The worst-case scenario within the intervals is selected for both inputs and outputs of the other units in this case.Conversely, for the lowest efficiency score of the same DMU, the inputs are assumed to be at the upper bound and the outputs at the lower bound, while the best-case scenario within the intervals is selected for both inputs and outputs of the other units.This approach allows for the computation of the upper and lower bounds of the efficiency score without calculating the actual efficiency score for each DMU, which is challenging when dealing with large datasets.It should be noted that since intermediate products serve as both the outputs of the previous stage and the inputs of the next stage, the model can select the best values for them.
In Model (2), the DMU  achieves its highest efficiency levels when its outputs reach their maximum values ∀ : ŷ =   and its inputs reach their minimum values ∀ : x =   , at the same time.However, the outputs of the remaining DMUs reach their minimum values while their inputs reach their maximum levels, i.e. ∀∀ ̸ =  : ŷ =   and ∀∀ ̸ =  : x =   ; indicating that these DMUs achieve their highest values of inputs and lowest values of outputs.Conversely, DMU  reaches its lowest efficiency level when its outputs reach their minimum values and inputs reach their maximum values, while the remaining DMUs achieve their highest efficiency levels with their outputs reaching their maximum values and inputs reaching their minimum values.This implies that the remaining DMUs achieve their lowest values of inputs and highest values of outputs.
Intermediate products generated by a process must be entirely utilized by other processes, meaning that altering the value of an intermediate product from one process has a ripple effect on the other processes.If the value of an intermediate product produced by process  rises, the input values of other processes will also rise, leading to a decrease in the efficiencies of those processes.Therefore, intermediate products cannot be treated as independent variables since changes in their values have an impact on the efficiencies of the other processes.
Similarly, the same principle can be applied to intermediate products that are consumed by process .These intermediate products must have been influenced by other processes as well.Thus, if we reduce the value of an intermediate product produced by process  to improve its efficiency, it will also decrease the output values of other processes.Consequently, the efficiencies of these processes will decrease as well.The optimal value of Model (2)'s objective function increases consistently as  decreases and as  increases.This means that the optimal value of  should be the highest possible, denoted as  * =  * , while the optimal value of  should be the lowest possible, represented as  * = 0.As a result, the th constraints of Model (2) can be modified accordingly: Therefore, to achieve the highest total efficiency, the optimal values of intermediate products should be treated as variables within the interval of [   ,    ].Thus, a convex combination of intervals should be utilized when modelling to obtain the upper bound of efficiency of intermediate products.
Note that Model (2) is designed to incorporate all data levels within the intervals, whereas Model (3) acts as its linear counterpart tailored for precise data.Notably, we demonstrate that Model (2) can be seamlessly converted into Model (3), enabling the computation of upper bounds on efficiency scores in scenarios with imprecise data.This transformation significantly bolsters practicality and computational efficiency, rendering Model (3) particularly well-suited for large-scale applications and real-time decision-making.
Consequently, DMU  achieves the lowest efficiency level in the most unfavorable state, where the outputs of each process of DMU  reach their minimum values while the inputs of each process of DMU  reach their maximum values simultaneously.Meanwhile, the remaining DMUs' processes reach their maximum output values and minimum input values simultaneously.As we discussed earlier, to determine the optimal values of intermediate products that lead to the lowest efficiency, they should be considered as variables within their intervals.Thus, an optimal solution for the following Model ( 4 The optimal value obtained through Model ( 4) can be regarded as the lowest boundary for efficiency scores of DMU  .

Classifying DMUs
After solving the models for the upper and lower boundaries of efficiency, DMUs can be categorized into one of the following four groups: After solving the DEA models for the upper and lower boundaries of efficiency, DMUs can be classified into one of four groups:  ++ ,  + ,  − , and  −− .These groups are defined based on the upper and lower bounds of efficiency scores of the DMUs relative to the other DMUs in the network.
The first group,  ++ , consists of DMUs whose worst efficiency score is greater than or equal to the best observed efficiency score of all other DMUs in the network.These DMUs are fully efficient in both the upper and lower bounds and represent the upper boundary of the production frontier.They are the most efficient units in the network.
The second group,  + , includes DMUs that have an efficiency score equal to the maximum observed efficiency score on at least one of the efficiency bounds when compared to other DMUs in the network.These DMUs are efficient, but they may not be the most efficient in the network.This group encompasses DMUs that are on the frontier as well as those that are not but are still relatively efficient.
The third group,  − , comprises inefficient DMUs that are not the best in either the upper or lower bounds.Specifically, a DMU is in this group if its efficiency score in both the upper and lower bounds is less than the efficiency scores of all other DMUs.
The fourth group,  −− , includes DMUs that are fully inefficient in both upper and lower bounds and are dominated by other DMUs in the dataset.In other words, their efficiency scores in both bounds are less than or equal to the efficiency scores of all other DMUs.
In summary, categorizing DMUs into  ++ ,  + ,  − , and  −− groups provides a way to identify the most efficient, efficient, inefficient, and fully inefficient units in a network.This information can be used to develop strategies for improving the efficiency of DMUs in the network.For example, DMUs in the  ++ group may be considered as benchmarks for other DMUs to strive towards, while DMUs in the  + group may require further investigation to determine why they are only efficient in one bound.On the other hand, DMUs in the  − and  −− groups may need improvement to increase their efficiency scores.

Efficiency improvement and target setting
To improve the efficiency of DMUs in the  − or  + groups, we need to identify the sources of inefficiency by examining the lower boundary of their efficiency scores.The lower boundary indicates the minimum efficiency level that the DMUs can achieve, and the difference between the lower boundary and the highest efficiency score shows the amount of inefficiency.For these units We need to optimize the use of inputs by identifying ways to reduce input usage without affecting the quantity of the produced outputs.This can be achieved through the following target settings.
Let's DMU  : (  ,   ,   ) belongs to  −− .So, in evaluating this unit with Model (3) we get   < min ̸ =   < 1, then the new activity (︀     ,   ,   )︀ is considered as a benchmark for this unit.A similar improvement scenario can also be implemented for units in the  − and  + group.
Proof.It is enough to show that for the improved activity is efficient in Model (3).If we assume that the data are exact, by taking dual for the linear Model (1) and by removing the slack variables from the objective function and the constraints we have the following envelopment model which is equivalent to the multiplier Model (1) when the variables are non-negative: It is easy to see that this model is always feasible and in optimality 0 < ℎ *  ≤ 1.Note that for calculating upper bound of the efficiency for DMU  , this unit is set at its best level while the other units are at their worst level and the computed score is   < max    ≤ 1.Now consider the interval data when the new activity (  x , ẑ , ŷ ) is under evaluation for its upper bound of the efficiency score.We show that the new upper bound computed by the following model is equal to max

Illustrative example
The present example's structure and data have been derived from Kao's [57] work.However, certain modifications have been made and the data has been assumed to be interval.It is important to note that the Kao [57] example's data has been utilized as the lower boundary for the interval data in the current instance.The current study examines five DMUs that exhibit a network structure comparable to that shown in Figure 2.
Each DMU consumes two inputs, namely  1 and  2 , which are common to all three processes.Three outputs, namely  1 (generated by process 1),  2 (generated by process 2), and  3 (generated by process 3), are produced.Additionally, intermediate product  1 is generated by process 1, intermediate product  2 is generated by process 2, and both intermediate products are consumed by process 3. The data in this example is interval in nature and is presented in Table 2.
Due to the mixed (general) internal structure of the DMUs, it was necessary to modify Model (3) to obtain the upper boundary of the efficiency score in Model (6).The modification was carried out in the following manner: The results obtained by applying Model (6) to the current scenario, while considering a value of  = 10 −5 , are presented in Table 3.
The determination of variables related to the convex combination of intermediate products can be achieved by acquiring knowledge of both   and    , where  denotes a particular product.The optimal values of intermediate products can be computed by utilizing    in equation ẑ The results of this computation are presented in two columns of Table 3.
An interesting observation is that the highest efficiency score for DMUs does not necessarily correspond to the lowest or highest values of the intermediate product interval.Additionally, it is apparent that for DMU3, the highest efficiency score is achieved when the intermediate products of both DMU3 and DMU5 are situated within the range of their lowest and highest values.Likewise, the lower limit of efficiency scores can be determined for all five DMUs by utilizing Model (4), and the resulting values are displayed in the last column of Table 3.
Following the computation of upper and lower boundaries of the efficiency score, an interval efficiency score has been calculated for each DMU.As a result, DMUs 1, 2, and 5 have been identified as members of the set E + due to their complete efficiency scores being at the upper boundary, while DMUs 3 and 4 have been identified as members of the set E − due to their efficiency scores being less than best observed boundary values.Both E ++ and E −− groups are empty.supplies for all nations has become a top priority.The criticality and urgency of monitoring the agricultural industry and assessing its production units lies in the significant impact it has on countries' economies, not only for satisfying domestic demands but also for exports.
Efficiency evaluation is an integral component of agricultural economics.As global population continues to grow and demand for food increases, it is important to maximize production while minimizing costs.Efficiency evaluation enables the identification of areas of improvement in the production process, allowing producers to optimize resources and increase yields.This evaluation also helps policymakers in formulating and implementing agricultural policies that incentivize farmers to adopt best practices and increase productivity [86].
Moreover, efficiency evaluation aids farmers to stay competitive in the global market by reducing production costs and increasing yields.It also leads to more sustainable agricultural practices, which is critical for preserving natural resources, reducing pollution, and mitigating the impacts of climate change [87].By improving efficiency, farmers can reduce the use of resources such as water, fertilizers, and pesticides, promoting a more sustainable and eco-friendly agriculture industry.
Overall, the importance of efficiency evaluation in agricultural economics is undeniable.It is an essential tool for improving productivity, reducing costs, and promoting sustainability in the agriculture industry [88].By utilizing efficiency evaluation techniques, farmers, policymakers, and other stakeholders can work together to ensure a more efficient, productive, and sustainable agriculture industry for the future.
Iran's agricultural sector has traditionally played an important role in its economy, but it has faced significant challenges such as drought, water scarcity, and outdated farming practices.In response, the Iranian government has implemented policies aimed at modernizing the sector and improving productivity, including investments in irrigation infrastructure, subsidies for agricultural inputs, and support for research and development.However, the efficiency of these policies in increasing productivity and reducing production costs remains unclear.Despite the efforts made by the government, the agricultural sector still struggles with low productivity and limited access to credit and markets.This has led to an increased reliance on imports, negatively impacting the country's food security.Therefore, evaluating the efficiency of the production process in the agricultural sector in Iran is crucial for addressing the challenges it faces and achieving sustainable growth in the sector.
Iran possesses extensive agricultural land, of which almost one-third is suitable for farming.However, due to insufficient water distribution and poor soil quality, only a small portion of this land is currently being utilized for agriculture.Rainfed and irrigated farming practices are implemented in different parts of the cultivated area, with rainfed agriculture being limited to areas with sufficient natural water resources and precipitation.Notably, extensive precipitation is only found in the Caspian lowlands, such as Guilan and Mazandaran, making them ideal for rainfed agriculture.Azerbaijan, Khorasan, and Fars also receive adequate winter rains for growing grains, which eliminates the need for additional irrigation.However, the arid plateaus of Iran, as well as its eastern and southeastern regions like Yazd and Hormozgan, are not suitable for rainfed farming and require irrigation for crop growth.
Wheat is a fundamental agricultural commodity in Iran, serving as a staple food for Iranians and playing a significant role in their livelihoods, nutrition, and employment.Iran has a per capita consumption of 160 kg of bread wheat, which exceeds that of most other countries.In order to increase wheat production and meet the population's demand for this essential crop, Iran's government encourages farmers to grow more wheat.
This section aims to analyze the efficiency of wheat farming in Iran's provinces during the 2018-2019 crop year, which lasted from 22 September 2018 to 22 September 2019.Iran had 30 provinces during this period, and to obtain a more precise evaluation of wheat production performance, we used Models (3) and (4) on the network structure shown in Figure 3.The provinces were treated as four-stage network systems, with two parallel processes, each consisting of two series stages.The agricultural operations were divided into two parallel processes, namely irrigation and rainfed, and each parallel process included sowing-growing and harvesting series stages.The data used for this analysis was obtained from the Iranian Ministry of Agricultural Jihad1 , and we illustrated the Iran wheat farming system in Figure 4.The complete dataset, which includes interval inputs, intermediate products, and outputs related to wheat farming in the 30 provinces, can be found in Tables 4 and 5.The optimization tasks in this study were carried out using Lingo 18.0 software, with computations performed on a laptop computer equipped with an 11th Gen Intel R ○ Core TM i5 processor running at 2.40 GHz.This configuration guaranteed efficient and dependable performance throughout the analysis.Notably, all process times were negligible.

Discussion
In the first and third stages of this production analysis, we employ consumed seeds, measured in kilograms, and cultivated area, measured in hectares, as crucial inputs.Harvested area, measured in hectares, plays a pivotal role as an intermediate product, serving as both the output of the first and third stages and the input for the second and fourth stages of the process.Ultimately, the final output from stages 2 and 4 is the production of wheat, measured in tons.Among the provinces under evaluation, it's important to note that Sistan-Baluchestan, Hormozgan, and Yazd do not engage in rainfed farming, and, consequently, their data have been recorded as zero in this context.However, the remaining provinces engage in a combination of both irrigation and rainfed agriculture.Notably, some provinces, such as Azerbaijan-East, Azerbaijan-West, Ardabil, Ilam, Chahar Mahal, Bakhtiari, Golestan, and Khuzestan, exhibit a higher reliance on rainfed farming practices, while others primarily rely on irrigationbased farming methods.
In Table 6, we present interval efficiency data and efficiency classifications for thirty Iranian provinces, focusing primarily on their performance in wheat production.Our comprehensive analysis utilizes the Network DEA models we propose, taking into account the inherent uncertainty associated with interval data.In this table, the upper and lower bounds represent the maximum and minimum efficiency scores achieved by each province, respectively.These bounds are essential in our evaluation as they provide vital efficiency intervals, forming the basis for a holistic ranking of the production units.
Additionally, in Figure 4, we visually depict the efficiency scores of these provinces for the 2018-2019 crop year, using hollow and solid columns to represent the lower and upper efficiency scores, respectively.The chart's horizontal axis displays the provinces, offering a clear view of their respective efficiency scores.This graphical It's noteworthy that provinces 18 and 28 are the only ones with an upper bound of efficiency greater than the maximum lower bound of efficiency among all provinces.This indicates that these two provinces possess the potential to achieve higher levels of efficiency than any other province in the sample.
Overall, Table 6 serves as a valuable tool for policymakers to identify provinces performing well and those in need of improvement.It also facilitates benchmarking provincial performance and the identification of best practices that can be shared among provinces.Based on the data in Table 6 and Figure 4, it's evident that significant variation exists in the efficiency scores of different provinces, and some provinces have greater potential for improvement than others.
The efficiency intervals in Table 6 enable the classification of provinces into four categories:  ++ ,  + ,  − , and  −− .These categories are determined based on the upper and lower bounds of the efficiency scores for each province.
As the results indicate, there are no provinces in the  ++ category in the given data.Tehran is the sole province in the  + category, signifying its status as the most efficient province in the sample. − includes provinces with lower efficiency scores than Tehran.In this case, all provinces except Tehran fall into this category.Lastly,  −− includes provinces with the lowest efficiency scores, where the upper bound of the efficiency score is less than the lower bound of the efficiency score of the least efficient among the other provinces.Consequently, Bushehr falls into this category for the given data.
The incorporation of these additional details enriches the depth and clarity of our results presentation.This comprehensive approach provides valuable insights into the performance of the production units, effectively capturing the subtleties of interval data and enabling meaningful comparisons with the conventional model.

Conclusions
In summary, this research sought to tackle the challenge of performance measurement in general network production units when dealing with interval data.Our study introduced novel models for assessing the upper and lower bounds of efficiency scores and introduced the concept of interval efficiency for each Decision-Making Unit (DMU).Notably, our findings underscored the critical role played by intermediate products in network systems, emphasizing their inclusion as variables in efficiency score calculations.
To illustrate the practicality and applicability of our proposed approach and models, we conducted an empirical application within the Iranian wheat farming industry.This application demonstrated the feasibility of our methodology in real-world scenarios, offering a comprehensive evaluation of the overall efficiency of production units with general network structures.Future investigations may explore the potential for assessing the efficiency of individual sub-units within these networks.
Moreover, further research avenues could delve into the adaptability of our proposed models in different types of network production systems, such as transportation networks or supply chain networks.Additionally, the study could explore the incorporation of alternative data types, such as fuzzy or uncertain data, to enhance the precision of efficiency score calculations.
In conclusion, this research constitutes a valuable contribution to the realm of performance measurement in network production units.By presenting models that account for both intermediate products and interval data, we lay a solid foundation for future studies to build upon.These future endeavors can explore innovative approaches and models that promise more accurate and efficient performance evaluations of diverse network production systems.

Figure 1 .
Figure 1.A production series network in a general operational mode.

Figure 2 .
Figure 2. Internal structure of a DMU in illustrative example.

Figure 3 .
Figure 3. Internal structure of each province's agriculture process.
Therefore, we cannot consider the lowest values of the consumed intermediate products as independent variables.Model (3) is proposed below to calculate the highest efficiency level of DMU  : . . ., , ,  = 1, . . ., ).

Table 2 .
Inputs, intermediate products and outputs for 5 DMUs with mixed structure relations in sub-processes.

Table 4 .
Upper bound data for 30 provinces.

Table 6 .
Efficiency interval and classification of 30 provinces.