Efficiency Decomposition in Two-Stage Network in Data Envelopment Analysis with Undesirable Intermediate Measures and Fuzzy Input and Output

This paper presents a model for the efficiency evaluation of decision-making units comprising a network structure with undesirable intermediate measures. This model is then generalised to evaluate the efficiency of network decision-making units with triangular fuzzy data and undesirable intermediate measures. In this study, data envelopment analysis (DEA) models with fuzzy network structures and undesirable intermediate measures are considered and solved as linear triangular fuzzy planning problems. With numerical results, the application of the proposed model in the chipboard industry of wood lumber has been shown.


Introduction
Data envelopment analysis (DEA) includes techniques for calculating the efficiency or the productivity of decision-making units (DMUs) which has been proposed by Cooper et al. [1]. Their approach evaluates the efficiency of decision-making units (DMUs) in the presence of multiple inputs and outputs.
In some cases, decision-making units include subunits that are linked together, called two or multi-stage processes (series or parallel), operating with a structure called a network DEA. Fare and Grosskopf [2] for the first time delineated the importance of this topic. Chen et al. [3] reconsidered the overall efficiency calculation by common DEA models and showed their inability to evaluate the overall efficiency of these units. They developed an efficient model capable of determining the efficient frontier of a two-stage production process with intermediate measures. Liang et al. [4] proposed a DEA model for a two-stage process based on the game theory and decomposition approach. Kao and Hwang [5] presented a two-stage DEA for a Taiwanese insurance company and developed a new method to decompose the overall efficiency of such processes, which makes it possible to compare the first and second stages. Jianfeng [6] suggested a two-stage model with shared inputs and free intermediate measures. Li et al. [7] presented DEA models for extended two-stage network structures.
In real-world problems, the data are often presented imprecisely. Additionally, the intermediate measures under real-world conditions include desirable and undesirable data. In recent years, the interest in employing efficiency and productivity management has grown due to undesirable data. Jahanshahloo et al. [8] proposed a model for the development of the Topsis model for decision-making problems with fuzzy data. Jahanshahloo et al. [9] suggested a method for ranking DMUs with L 1 -norm with fuzzy data in DEA. Ebrahimnejad et al. [10] presented a primal-dual method for linear programming problems with fuzzy variables. Ebrahimnejad et al. [11] proposed a new approach for solving bounded linear programmes with trapezoidal fuzzy numbers. Khalili et al. [12] suggested a hybrid fuzzy multi-objective framework for sustainable project portfolio selection. Ebrahimnejad and Tavana [13] suggested a novel method for solving Linear programming problems with symmetric trapezoidal fuzzy numbers. Hatami-Marbini et al. [14] proposed a model that uses a lexicographic multi-objective approach to calculate fuzzy efficiency measures in data envelopment analysis. Hosseinzadeh Lotfi et al. [15] presented fuzzy data envelopment analysis models with R codes. Olfati et al. [16] established a new approach for solving fuzzy data envelopment analysis model based on uncertainty. Ebrahimnejad et al. [13] presented a new method for solving dual DEA problems with fuzzy stochastic data. Ebrahimnejad et al. [17] also analysed NATO enlargement problem in fuzzy stochastic data envelopment analysis framework. Inuiguchi et al. [18] established data envelopment analysis with fuzzy input/output data. Kao and Liu [19] used the two-stage fuzzy planning, for the evaluation of the efficiency in two-stage units with fuzzy data. Moreover, Lozano [20] presented an article that explains the process efficiency of two-stage systems with fuzzy data.
In the presence of undesirable outputs, a DMU is efficient when it has more desirable outputs and less undesirable outputs and less inputs. A network with undesirable data is a network with intermediate measures or output containing desirable and undesirable data. Jahanshahloo et al. [21] suggested a model with undesirable data in DEA. Podinovski et al. [22] presented two technologies for modelling a weak disposability in a paper entitled modelling weak disposability in data envelopment analysis under relaxed convexity assumptions. Lozano et al. [23] applied the network DEA approach to airports performance assessment considering undesirable outputs. Maghbooli et al. [24] showed the application of weak disposability in network DEA with undesirable intermediate measures. Yu et al. [25] suggested a network DEA model to combine operational and environmental performance in an integrated approach considering desirable and undesirable outputs. Nasseri and Khatir [26] presented fuzzy stochastic undesirable two-stage data envelopment analysis models and applied them to the banking industry.
So far, most papers have been presented with undesirable factor in network structure, input and output data were precise and crisp. In this paper, a new model was designed for the evaluation of a two stage with desirable and undesirable intermediate measures. The model was then generalised to a DEA model with a fuzzy network structure and desirable and undesirable outputs. For efficiency evaluation of the decision-making units with a fuzzy network structure, the input and output variables as well as the intermediate outputs are in the form of fuzzy data, eventually which yields the efficiency in the fuzzy form. The efficiency of this model can be precisely obtained by using linear fuzzy programming problem-solving methods.
In this research, the innovations carried out, are as follows.
-Development of network models in DEA with undesirable intermediate products.
-Development of network models with undesirable indicators in a fuzzy environment.
-Applying the new model for wood and chipboard industries.
In the second section, we present a new kind of network DEA model to evaluate the efficiency of decision-making units with undesirable intermediate measures. This model calculates the efficiency of the first and the second stage processes under the constraints in which overall efficiency is maintained at the same level. In the third section, we develop the model presented in the previous section of the network with undesirable intermediate actions and triangular fuzzy inputs and outputs. The model is solved using triangular fuzzy data boundaries. In the fourth section, we illustrate the application of the model with a numerical example, in the wood and chipboard industries. The numerical examples are then solved and the results are compared and analysed.

Two-Stage in DEA
In this section, first of all, the network structure in DEA will be presented as follows: If the DMU involves two or more processes, it will be called network structure decisionmaking. If the network outputs or intermediate actions are undesirable, the network is called undesirable data.

Efficiency Valuation of Two Stage
The efficiency evaluation of DMU o in the first and second stages with SBM model (presented by Tone [27] in the year 2001), are e 1 and e 2 , respectively, as follows: Figure 1. Two-stage process.

S.t.
n j=1 The optimal solutions are obtained by solving models (1) and (2) and we always have 0 < e 1 ≤ 1 and 0 < e 2 ≤ 1. According to some articles, such as Chen and Zhu [3], applying models (1) and (2) separately does not correctly model the intermediate measures because they do not address the potential conflicts between the two stages arising from the intermediate measures. It is recommended to combine these two stages with the mean weight of efficiency scores from the first and second stages, as follows: where w 1 and w 2 are the weights specified by the user, such that 0 ≤ w 1 , w 2 ≤ 1. This definition ensures that DMU is efficient if and only if it is efficient in both stages, w 1 and w 2 are, respectively, the significance or relative efficiency ratio of the first and second stages to the overall efficiency of the DMU during the entire process. Therefore, w 1 and w 2 are defined as follows.
As a result, the overall efficiency score of the two-stage process obtained with SBM is: DMU o is overall efficient if and only if e o = 1.
A decision-making unit is overall efficient if and only if the DMU is efficient at both the first and second stages, If the DMU is only efficient in the first or second stage, then the decision-making unit is not overall efficient.

Efficiency Decomposition
The optimal solution obtained from Model (5) can be used to calculate the efficiency scores in the first and second stages. Nevertheless, Model (5) can have alternative optimal solutions, the overall efficiency decomposition in Equation (3) may not be unique. As a result, the approach proposed by Kao and Hwang [5] is used in this study to produce a set of coefficients with the highest efficiency in both stages while maintaining the overall efficiency of the process. In this way, the efficiency decomposition of Equation (3) is unique. Assuming that the overall efficiency of DMU o obtained from model (5) is e * o , and the overall score of the first and second stages is maximised by maintaining the overall efficiency e * o , then we have: To describe the mentioned models with fuzzy data, in the next subsection fuzzy numbers will be presented.

Fuzzy Data in DEA
A fuzzy number is a convex and normal fuzzy subset of real numbers with a membership function within the range 0 and 1. One of the most common representations of fuzzy numbers is the triangular fuzzy number with the following membership function: In its abbreviated form, the triangular fuzzy number is represented byã = (a L , a M , a U )

Definition 2.3:
A triangular fuzzy numbersã = (a L , a M , a U ) is said to be a non-negative trapezoidal fuzzy number if and only if a L ≥ 0.
be Two triangular fuzzy numbers. Then basic fuzzy arithmetic operations on these fuzzy numbers are defined as:

Two-Stage with Fuzzy Data in DEA
In this section, Model (6) and Model (7) will be presented with fuzzy numbers as follows: Assume thatx kj , k = 1, 2, . . . , K,ỹ sj , s = 1, 2, . . . , S andz pj , p = 1, 2, . . . , P are desirable and undesirable inputs and outputs of the first stage. The undesirable outputs of the first stage are used as the inputs of the second stage. In addition,x qj , q = 1, 2, . . . , Q,ỹ rj , r = 1, 2, . . . , R are the inputs and outputs of the second stage, respectively. Moreover, all the data are represented in the form of triangular fuzzy numbers.
According to models (1) and (2), suppose 1 =1 = [1, 1, 1], the efficiency evaluation in the first stage by using SBM models are, as follows: the efficiency evaluation in the first stage by using SBM models are, as follows: DMU o is efficient in the first and second stages, respectively, whenẽ 1 = 1 andẽ 2 = 1. As mentioned earlier, the intermediate measures are precisely modelled by separately applying Models (1) and (2). Similar to Section 2.1., the mean weight of the efficiency scores from the first and second stages is calculated as follows.
wherew 1 andw 2 are the weights specified by the user, such that 0 ≤w 1 ,w 2 ≤ 1. This definition ensures that DMU is efficient if and only if it is efficient in both stagesw 1 andw 2 are, respectively, the significance or relative efficiency ratio of the first and second stages to the overall efficiency of DMU during the entire process. Therefore,w 1 andw 2 are defined as follows: Wherew 1 andw 2 are fuzzy numbers, as they are obtained by applying mathematical operators of fuzzy numbers. As a result, the overall efficiency of is evaluated by the SBM model as follows:ẽ In Models (8), (9) and (12), all data are represented in the form of triangular fuzzy numbers and thus the efficiency scores from these equations are also triangular ones.  (12) is a linear fuzzy programming problem with triangular fuzzy data. By substituting Model (12) can be written as follows: According to fuzzy calculations and 1 =1 = [1, 1, 1], Model (13) can be expressed as.
For Model (14), as long as e U 0 ≤ 1 then e L 0 ≤ 1, e M 0 ≤ 1, will be automatically satisfied. The best possible values of e L 0 , e M 0 and e U 0 can be captured by the following models: The values of the optimal objective function from the three LP Equations, namely (15), (16) and (17), include the best fuzzy DMU o efficiency. As a result,ẽ = [e L 0 , e M 0 , e U 0 ] is a triangular fuzzy number.
are feasible solutions of Model (15) Therefore, Finally, it can be concluded from Proofs A and B that e L 0 ≤ e M 0 ≤ e U 0 .

Decomposition of Fuzzy Efficiency
The optimal solution obtained from Model (12) can be used to calculate the efficiency scores in the first and second stages. Since Model (12) can have alternative optimal solutions, the overall efficiency decomposition defined in Equation (10) may not be unique.
According to Section 2.2, it is assumed that the overall efficiency of DMU o gained from Model (12) isẽ * 0 , and the overall score of the first and second stages is maximised by maintaining the overall efficiencyẽ * 0 . In this case, the overall efficiency decomposition defined in Equation (10) will be unique.
In Models (18) and (19), all data are represented in the form of triangular fuzzy data and thus the efficiency scores from these equations are also triangular fuzzy numbers. Models (18) and (19) are fuzzy programming problems with triangular data, therefore, to obtain the best e t = (e L

Practical Example
In this section, we consider the chipboard industry of wood lumber as a two-stage process.
In the first process, the lumber wood, a number of nails and glue are introduced into the system to produce tables and chairs. Some wood chips are produced, which is an undesirable output data and enters the second process as intermediate measures. By adding additives (Special glue, Urea formaldehyde glue, Solid paraffin and chemicals) as the second stage entrance the chipboard will be produced ( Figure 2). It is expected to produce 15 workshops in January and July, so each workshop is a decision-making unit(DMU) and j represents the number of workshops. The lumberx 1j , the nailsx 2j , the gluesx 3j are inputs, the wood chipz 1j , the tablesỹ 1j and the chairsỹ 2j are the undesirable and desirable outputs in first stage, respectively, the wood chipz 1j as intermediate measures, the additivesx 1j are inputs, the chipboardỹ 1j is output in second stage.
We develop Models (18), (19) and (12) in order to calculate the efficiency of the first and second stages, and the whole system for five manufacturing plants.
The data used in this paper are the average figures for these items from June to July in 2019. Using averages to calculate efficiencies will obtain results that do not reveal the real situation because the data fluctuates from month to month. In contrast, fuzzy numbers are more appropriate to display inaccurate data.
Using the data from June to July to represent the domain of fuzzy numbers and their averages to show the vertex, triangular fuzzy numbers are constructed byx 1j ,x 2j ,x 3j ,z 1j , y 1j ,ỹ 2j ,x 1j andx 2j ,ỹ 1j j = 1, . . . , 15, as shown in Tables 1 and 2, respectively. By Applying Models (15), (16) and (17), the upper and middle and lower bound of theẽ o ,ẽ 1 andẽ 2 . The system efficiency and the efficiency of the first and the second stages are obtained. Table 3 shows the results forẽ * o ,ẽ * 1 andẽ * 1 refer to system, Process 1, and Process 2, respectively.
According to Table 3, the overall efficiency scores and the efficiency scores in the first and second stages of the first and third workshop are equal to1 = (1, 1, 1), so the first and the third workshops are strongly efficient. Also the efficiency score of the first stage of the 4th, 8th, 11th and 14th workshops is equal to (1, 1, 1) but the efficiency score of stage 2 and the overall efficiency score of these workshops are less (1, 1, 1) so these workshops are strongly efficient in the first stage. But in the second stage and the overall workshops are inefficient. The efficiency scores of workshops 2 and 9 equal to are (0.956, 1, 1) and (0.999, 1, 1) in the first stage, respectively, so workshops 2 and 9 are efficiency in the first stage but they are inefficient in the second stage and the overall. Also, they are not strongly efficient. Workshops 10 and 13 are strong efficient in the second stage, but they are inefficient because the overall efficiency score and the first stage are less than one. The efficiency scores of workshops 6 equal to are (0.810, 0.860, 1) in the second stage, so workshops 6 are weakly efficiency in the first stage but they are inefficient in the first stage and the overall. Other workshops are inefficient in the first, the second stages and the overall.

Conclusions
In this paper, we presented a new kind of network DEA model to evaluate the efficiency of decision-making units with undesirable intermediate measures. This model calculated the efficiency of the first and the second stage processes under the constraint, which overall efficiency are maintained at the same level. This model is generalised to evaluate the efficiency of decision units with network structure with triangular fuzzy data and undesirable intermediate measures.
Then we solve the model as a linear triangular fuzzy programming problem. Often in the real world, the data is fuzzy and, in some cases, outputs are undesirable. Using these models in the industry, the desired outputs can be achieved by applying undesirable outputs.
In future research, we will show that, firstly these models can be extended to networks with more than two stages with undesirable and desirable intermediate measures and outputs, then extend the models to a state of fuzzy data. Secondly, the model will be solved utilising triangular fuzzy data boundaries and will be calculated by considering αcut.