PERFORMANCE EVALUATION OF TWO-STAGE PRODUCTION SYSTEMS WITH TIME-LAG EFFECTS: AN APPLICATION IN THE HORTICULTURE INDUSTRY

. In standard data envelopment analysis (DEA), it is assumed that inputs of a specific production period are used to generate outputs of the same period. However, in some practical examples, time-lag effects exist between inputs and outputs. The inputs of one period are used to generate outputs for several periods, or inputs of several periods are used to create outputs for one period. In this paper, we present some new DEA models for performance assessment of network production systems with time-lag effects. An empirical application in the horticulture sector in Iran shows the usefulness and capabilities of our proposed approach


Introduction
Data envelopment analysis (DEA), initially introduced by Farrell [14], is a non-parametric mathematicalprogramming method for evaluating the efficiency of a set of decision-making units (DMUs) with multiple inputs and outputs.Charnes et al. [4] presented the CCR model by extending Farrell's work to evaluate DMUs considering multiple inputs and multiple outputs.Later on, Banker et al. [2] extended the CCR model to the BCC model by assuming variable returns to scale into the evaluation.In addition to these basic DEA models, other evaluation approaches, such as Additive and Slack Base Measure (SBM) models, have also been proposed in the DEA literature [40].In the last four decades, an impressive number of methods and applications have been reported in the DEA framework.For further information, the interested readers could refer to Emrouznejad et al. [10] and Kaffash et al. [19] for comprehensive surveys and analysis of related studies in DEA theory and applications.
In classic DEA models, DMUs are considered black-boxes; the intrinsic activity and internal structures of sub-processes do not account for the unit's assessment (see [16,17,20,26,30] for more details).In a production unit, the inputs may pass through multiple processes to produce outputs.As a result of applying traditional DEA models, a black-box DMU may be seen efficient while its subunits are performing inefficiently [12,24].
In recent DEA literature, considerable efforts have been devoted to developing new models for analyzing multi-stage production units.Seiford and Zhu [39] attempted to abandon the traditional perspective in the performance evaluation of bank branches and consider the internal structure by separating the bank's operations into two successive stages of profiting and marketing.However, they used two DEA models to assess the two stages and ignored the transfer of information between the two stages.
According to Cook et al. [7], in two-stage models where the first stage outputs are used as the second stage inputs, if the second stage becomes inefficient, we have to reduce the input to make it efficient.However, reducing the second stage input results in decreasing the output of the first stage, which yields to the inefficiency of the first stage.Although they looked at the internal structures in these types of research, they did an incomplete evaluation due to overlooking subunit communications and how to transfer information from one stage to the next.To troubleshoot the issues raised in independent and classic DEA models, Färe and Groskopf [13] introduced network data envelopment analysis (NDEA) models to evaluate the processes operations in assessing the efficiency of the DMUs with multiple-stage structure.Unlike the classic models, the NDEA models of Kao [20] depend on the structure of the DMU, relations of its subdivisions, and the type of inputs and outputs.Despotis et al. [9] used a weak-link approach to provide a simple two-stage model for evaluating the efficiency of DMUs.Khoveyni et al. [28] examined the concept of variations effect in a two-stage NDEA to see how output products would change if the intermediate products rise due to the increasing inputs in the first stage.Research in this area is still of interest to many researchers [5,6,21,23,31,32].
In most DEA models, the efficiency of the companies and organizations is usually evaluated for a specific time, which is not an accurate assessment of the performance of the whole system.It is more realistic to assess and compare the efficiency of these kinds of operations over several periods.In this regard, there are many extensions of the original DEA models that take more than a one-time period to evaluate the efficiency.Some examples of these models are window analysis and Malmquist productivity evaluation.Regardless of minor differences between these models, they all have the common purpose of evaluating DMUs over multiple periods.Nemoto and Goto [34] proposed a dynamic DEA model for measuring the performance of multiple-time production systems.Golany et al. [15] presented an efficiency measurement framework for systems composed of two subsystems arranged in series that simultaneously computes the efficiency of the aggregate system and each subsystem.Park and Park [36] introduced a two-stage approach to measure aggregative efficiency over several periods.Amirteimoori and Kordrostami [1] presented a model to measure cumulative efficiency across all periods and showed that cumulative efficiency is a convex combination of periods' efficiency scores.Kao and Liu [25] proposed a method to measure the cumulative efficiency values of several periods.In other words, they used a network approach to assess the efficiency of periods and the overall efficiency of DMUs.Jablonsky [18] analyzed the performance of multiple-time systems and presented the efficiency and hyper-efficiency concepts in multipletime DEA models.Razavi et al. [37] introduced a two-stage approach based on Chebyshev inequality bounds related to multiple-time production systems.Kordrostami and Jahani [29] presented a method for evaluating the efficiency of multi-time production systems with negative data.Recently, Esmaeilzadeh and Kazemi Matin [11] expanded the concept of multiple-time production in the NDEA.Their models were based on series and parallel approaches in network data envelopment analysis.
An essential issue in traditional DEA is that the inputs for one period are used for generating outputs of the same period, but in practice, the input of one period may be used for the output generation of several periods, or the input of several periods is used to generate the output of one period.This condition is called production processes with time-lag effects.For example, in the horticultural industry, the costs incurred for a garden as input in one year are presented as output in subsequent years.For the first time, Özpeynici and Köksalan [35] introduced DEA models with time-lag effects.They presented two multiple models with input and output lag times.On the other hand, there are evaluation models of stocks portfolio performance in which variables such as returns of funds are considered time-lag components.In this field, the black-box models have also been proposed to evaluate the performance of stocks portfolio (see, e.g., [3,27,33,38]).
In some real-world applications, we may encounter a network process that one of its stages has a time delay in transforming inputs to outputs.For example, in the horticultural industry, we have a delay of a few years to produce fruits, and if we want to compote the product, we will have a network whose first stage has a time delay.Given that no previous research has evaluated the performance of DMUs in a network structure with a time lag, the question is how to deal with a time lag in a network process.
This paper aims to answer the research question and provide models for NDEA with a time lag.For this purpose, we consider a simple two-stage network whose first-stage inputs have a time lag.We first evaluate the efficiency of the stages independently, and then we provide a multiplier DEA model for the overall performance assessment of the two-stage production units.We will also present an envelopment DEA model for evaluating the efficiency of production processes in both black-box and network cases, by considering time-lag effects.Besides, the applicability of the proposed models is demonstrated by applying them in a real-world case in the horticulture industry in Iran.
The rest of the paper is organized as follows.Section 1 introduces basic multiple period models of DEA with time lag effects in the black-box case.Section 2 presents the new approach in modeling the time lag effect in a two-stage network DEA framework.The new suggested models are presented in both multiple and envelopment forms.An empirical application of performance assessments of horticulture sectors in Iran is presented in Section 3, as well as data analysis and discussions.Section 4 contains conclusions and suggestions.
The CCR multiplier model to evaluate the efficiency of DMU  in period  is stated as follows: where   ( = 1, . . ., ) and   ( = 1, . . ., ) are the output and input weights for evaluating DMU  , respectively.Also, at optimality, 0 <  *  ≤ 1 is the efficiency score of DMU  .Note that in most traditional DEA models, it is assumed that the inputs in a specific production period are used to generate outputs of the same period.
The time lag may occur at inputs or outputs.For example, Figure 1a displays the time lag in the inputs.In this figure, a time lag of three years ( = 3) has been considered for 6 years ( = 6).As seen in Figure 1a, the outputs start from the third year, and we can use the inputs of the first to the third year for the third year output.For the fourth year's output, we can use the inputs of the second to the fourth year.Similarly, for the fifth year's output, the inputs of the third to the fifth year and, for the sixth year's outputs, the inputs of the fourth to the sixth year can be used.However, we do not have outputs for the first two years.
A similar analysis can be performed for output time lags, as shown in Figure 1b.Here the first-year input is only used for the outputs of the first three years.Similarly, the fourth year's inputs will generate outputs of the fourth to the sixth year.Also, we will not have input in the fifth and sixth years.The following multiplier DEA model (MPI) is proposed by Özpeynici and Köksalan [35] for evaluating the efficiency of the DMU  with the time lag in inputs (shown typically in Fig. 1a).Here,  (−) shows the th input of DMU  in the period ( − ).Also,  is the time lag duration.For the case of the output delay, shown in Figure 1b, the multiplier model for efficiency evaluation in the presence of output time lag (MPO) is represented as follows:

𝐸
As explained in the previous section, the traditional DEA models consider a production unit a black box, and the internal processes of the units are ignored.Inputs may go through several stages to produce outputs.The overall efficiency depends on the efficiency of these stages.For accurate performance evaluation, the efficiency values of these stages and the system as a whole must be calculated, and their relationships should be determined.As a result, the efficiency value obtained is more reliable, and the sources of inefficiency are better realized [22].To resolve this issue, we will present a new two-stage DEA model with time lag effects in the next section.

New two-stage multi-time DEA modeling with time lag effects
In this section, we present some novel two-stage DEA models with time lag effects in both multiplier and envelopment forms.

A two-stage multiplier network DEA model with a time lag
For the simplicity of presentation, the network is considered a two-stage process in a series case as follows.The research question is, when there is a time lag in the first stage of a two-stage network production system, how the evaluation process is done.The main purpose of this section is to answer this question in detail.In other words, we aim to present new network DEA models that evaluate the efficiency with a time lag in the first stage; the extension to the case of time lag in both production stages would be straightforward.
Suppose that all DMUs use  different inputs to generate  intermediate products in the first stage, and the second stage consumes the intermediate products as inputs to generate  final outputs in the second stage.
We suggest the following multiplier DEA models for evaluating the efficiency values of individual stages of DMU  : (Stage 1) Here, (   ), (  ) and (  ) are input, intermediate, and output weights, respectively.Also, in this Model, the time lag is applied to inputs of the first stage, and  denotes the index of the unit under evaluation.
The following multiplier model is also suggested for calculating system efficiency by considering constraints for both stages.)︀ is an optimal solution of this model, then the system (overall) efficiency and the efficiency of the stages in the presence of time lag for inputs of the first stage are calculated as follows: Note that the efficiency decomposition is provided as  *  =  * 1 ×  * 2 for system efficiency evaluation.Using the following theorem, we note that system efficiency evaluation in Model (2.3) provides a more accurate assessment than the traditional black-box approach.
Theorem 2.1.The optimal value of Model (2.3) is always less than or equal to the optimal value of Model (1.2).
By integrating these two constraints, we will obtain the following constraint: Because the other constraints of the two models are the same, then we infer that (︀  *  ,  *

𝑟𝑡
)︀ is a feasible solution for Model (1.2).In other words, we showed that the optimal solution of Model (2.3) is a feasible solution for Model (1.2).Since both objective functions of Model (1.2) and Model (2.3) are the same and of the maximization form, it can be concluded that the optimal value of Model (2.3) is less than or equal to the optimal value of Model (1.2).

A DEA two-stage envelopment model with a time lag
In DEA, there are two different views to evaluate efficiency.One is the multiplier form, and the other is the envelopment form.Each one has unique features that can complement each other.Although these forms are generated with different views, it is possible to express the relationships between these models using the properties of primal-dual models in linear programming [8].For example, the envelopment CCR form of the dual model is the multiplier CCR form.However, the envelopment form can present the vector point for an inefficient DMU while the multiplier form cannot.
Here, we present models to evaluate the efficiency with a time lag in envelopment form.The envelopment form makes it possible to assess the system and individual efficiencies, as well as benchmark targets.To do so, we will present the following DEA model, which is the extension of the dual form for Model (1.2) to evaluate the efficiency of observed units with inputs time lag: In this model,    is the intensity weights for constructing a non-negative combination of the observed DMUs, and   is the ratio of the proportional decrease in inputs of DMU  for time period .
To present the efficiency scores, we suggest taking an average of the input efficiency scores of the periods.So,  *  as the optimal efficiency score for the DMU  , is calculated as This Model calculates the system efficiency of the network for the DMU under evaluation. *  , the overall efficiency score for DMU  , is suggested as , where  * is the optimal value of Model (2.5).Need to be noted that the efficiency obtained from Model (2.5) does not suffer from existence of multiple optimal solutions.Although, it is possible that each   in the objective function has multiple optimal solutions, but the optimal value of the objective function,  * is always unique.So, the aggregate efficiency score  *  for DMU  is well-defined and unique.
Besides the system efficiency score, a key feature of the new proposed envelopment model is to provide an ideal or benchmark point for the unit under evaluation.So, an ideal or benchmark point for evaluating DMU  )︀ is the optimal solution of Model (2.5) for estimating DMU  .Similar to the multiplier form, the following proposition can be stated for the envelopment models.
Theorem 2.2.The optimal value of Model (2.5) is always smaller than or equal to the optimal value of Model (2.4).
Proof.The proof of this theorem is similar to that of Theorem 2.1.
Here, a simple numerical example is presented to illustrate the discussed approach.
Example 2.3.Suppose four DMUs with one input, one intermediate product, and one output for 3 time periods.Also, suppose that  = 2, i.e., the output is produced after two years.The information on these indices is given in Table 1 below.
The multiplier DEA models for black-box efficiency evaluation of DMU  is as follows: By solving this linear programming (LP) optimization model, the efficiency score is obtained as , where  * is optimal value of the above model.The proposed network model for performance evaluation of DMU  in the presence of output time lag could be stated as follows: By solving this LP model, the network efficiency score is calculated as  *  = 0.76.Similarly, one can obtain the black-box and network delayed efficiency and the stages efficiency scores.The results are presented in Table 2.
As you can see in Table 2, the black-box efficiency for DMU  is 1, and the other units are classified as inefficient.But it cannot be guaranteed that the network efficiency will reach 1.It means that in the network mode, the inefficiency sources are better recognized, and the efficiency values of the DMUs are more accurate than the black-box model.Also, the traditional and network efficiency of DMUs, as well as the efficiency values of the first and second stages of DMUs, have been presented.As columns 1 and 2 show, the network efficiency value is equal to the product of the efficiency values of the stages.For example, for DMU  , the efficiency value of stage 1 equals 1, and the efficiency value of stage 2 is 0.76, and their product is 0.76.
In the next section, we present a practical application of the new proposed approach.

A practical example
This section will provide an empirical example of applying the models presented in this article in the horticulture industry to illustrate the importance and applicability of the proposed method.

Data
Qazvin Agriculture Jihad Organization3 offered 150 hectares of land to boost production and reduce youth unemployment.Cherry, sour cherry, and walnut trees will cultivated in this land.Also, the bank facilities as loans are available to 15 people chosen by lot.Because of the bank limited resources, this loan will be paid to farmers within 5 years as the work progresses.Every farmer can have a maximum of ten acres of land.We know that these trees will yield after 6 years.That is, after six years of spending money and work, the trees will bear fruit.We want to calculate and compare the efficiency of these farmers for over ten years.We consider the whole period as 6 years.Each farmer can develop the land until the fifth year and then stop the development.This action aims to create conversion industries, add value to agricultural products, and prevent waste.This example is solved in two stages.Inputs, intermediate, and output variables in this evaluation are considered as follows: -The inputs to the first stage: (  ) (1) Land area (ha) The weight of each compote or package is half a kilo, and their unit is number.The time lag diagram is also shown in Figure 2 below.
It is noteworthy that each farmer should consume the first stage inputs in 5 years and does not produce any output during these 5 years; the output is produced after 5 years.This is the reason for this time lag in the output of the first stage.In other words, this farmer uses water and fertilizer for the trees as inputs for 5 years, and after this time, he will have the fruits of walnuts, cherry, sour cherry as delayed output.As shown in Figure 3, there is a time lag in the first stage for 6 years, i.e., the trees will yield after 6 years.But in the second stage, we have no time lag.The input of the first six-year is spent and used to produce the output of the first stage in the sixth year.The intermediate output of the sixth year is used as the input for the second stage to produce the output of this stage.The inputs of the second to the seventh year are used for the output production of the first stage of the seventh year.Also, these intermediate measures are used to produce the second stage output for the seventh year and so on until the tenth year.Besides, Table 3 presents the first farmer's input, intermediate, and output data for ten years.
Data for other farmers are given in the appendix.

Results
To evaluate these 15 farmers, we used two DEA models of (2.4) and (2.5) and two multiplier models of (2.1) and (2.2) (presented in this paper).Model (2.4) is a traditional model with a black-box structure with only input and output.But Model (2.5) has a network structure and includes intermediate products.GAMS software was used to solve all linear programming models in this study.
The results of the model implementation are reported in Table 4 below.
According to the traditional model, nine farmers are classified as efficient.But in the network model, none of the farmers were fully efficient, and farmer 1 has the highest efficiency value.This farmer has the highest efficiency in both models.Consider farmer number 4. This farmer has the lowest efficiency value in the traditional model, which means he has the worst performance.But in the network model, he is higher than seven farmers because the traditional model does not consider the intermediate measures in calculating this farmer's efficiency.Besides, the efficiency values of stages 1 and 2 are presented in the fifth and sixth columns.The product of efficiency values of the stages yields the exact value of network efficiency.As you can see, the number of efficient units is very high in the first stage.But in the second stage, none have worked well.So, the second stage has a significant impact on the inefficiency of the farmers.Therefore, in the network mode, inefficiency resources are better analyzed than the traditional model, which increases the accuracy and validity of the efficiency.
As we expected, the efficiency of the network case does not exceed the efficiency of black-box modeling.As a result, the new approach provided more discriminant power in the performance evaluation of the production units.There is no efficiency score interference in the network model, and a unique ranking is provided for the farmers.But for black-box efficiency, the efficiency score interference is not negligible, especially in score one.In the sixth column of Table 4, farmers' ranking is presented based on network efficiency scores.This ranking is based on the fact that any farmer with a high network efficiency score has a better position.Farmers 1, 15, and 6 are in the first to third ranks, and farmers 9, 8, and 12 are in the last three positions, respectively.There is also no ranking interference in network efficiency.Besides, 15 farmers without time lag are evaluated and compared.In this regard, the corresponding output with time lag is assumed 0, and we use a regular two-stage DEA [20] to evaluate the farmers.The results are presented in the third column of Table 4. Using this method, farmers 1, 14, and 15 are in the first, second, and third ranks, respectively.Compared with positions resulting from network efficiency with a time lag, only farmer 1 gets the same rank, and these two methods yield different results; it shows the importance of considering time lag in evaluating the farmers.As you can see, the height of the efficiency score in the network model is always less than that in the black box.The reason for is ignoring the intermediate structure for DMUs in black-box evaluation.Therefore, the traditional model is incapable of comprehensively evaluating DMUs with time-lag.In this case, it seems essential to consider the time-lagged network models, including the model presented in this paper.

Managerial implications
As mentioned in the practical example, a time delay always happens in the production process.For instance, trees lack any output at first, and after a few years, they start to bear fruits.So if a time delay is missed in the evaluation, we may incorrectly evaluate the performance, leading to faulty management decisions by the decision-maker.The next thing seen in the numerical example is the network structure of performance appraisal.According to the practical example, in the black-box structure, most farmers were efficient; that renders an incorrect evaluation of them.However, this shortcoming was eliminated in the envelopment data analysis model of the presented network data, and the farmers were ranked using the network model.Even in the network model, the efficiency of each step is given for further analysis of management results.In general, the proposed model can be used as the main model for evaluating performance in manufacturing industries with a time delay factor, in which other models of data envelopment analysis do not adequately assess them.

Conclusion and suggestions
In this paper, we present a network DEA model with a time lag.Based on a real-world example and a twostage production system with a time lag for stage 1, some new models for efficiency evaluation were presented in both multiplier and envelopment forms.Finally, to illustrate the importance of the issue, we evaluated the efficiency in the Iran horticulture industry using the models presented.It is shown that the new approach provides successful modeling of time lag effects in the performance evaluation of two-stage production systems.
Some industries may have time delay indicators, which leads to incorrect results in performance evaluation using traditional approaches.Therefore, developing performance appraisal models is essential in dealing with these conditions.But the type of model expansion should always match the needs of the real world.A real-world example discussed in this article has a two-stage network structure with a time delay at the input of the first stage.For this reason, the extended model of the input version had a two-stage with a time delay in the first stage.The results showed that a complete evaluation of decision-making units could be done using the proposed model.So, DMUs were ranked using the results of the proposed model.In contrast, the traditional model was not able to do this.
The model presented in this paper was proposed based on the type of time delay in the practical example.But in real-world applications of evaluation, other structures of time delay may be seen.The different appearances of these structures are one of the limitations of this research.However, researchers can introduce new timedelay structures in future studies based on practical examples in the real world and offer other new models for evaluations.In addition to these unfavorable output indicators, uncontrollable inputs can be considered performance evaluation indicators in dealing with time delays.Even we may need to present multi-period models.All of these issues can be viewed by researchers in future studies.
Appendix A. Data for farmers in the horticulture industry

Figure 1 .
Figure 1.(a) The time lag effects for inputs.(b) The time lag effects for outputs.

Figure 2 .
Figure 2. A simple two-stage process.

Figure 3 .
Figure 3.The diagram of the network production for the practical example.

Figure 4 .
Figure 4. Network efficiency scores with time lag for farmers.

Figure 4
Figure 4 the network efficiency values of the farmers with a time lag.As you can see, the height of the efficiency score in the network model is always less than that in the black box.The reason for is ignoring the intermediate structure for DMUs in black-box evaluation.Therefore, the traditional model is incapable of comprehensively evaluating DMUs with time-lag.In this case, it seems essential to consider the time-lagged network models, including the model presented in this paper.

Table 1 .
Data for the four DMUs over 3 periods.

Table 4 .
The efficiency values of 15 farmers with a time lag.

Table A . 1 .
Farmer No. 2's input, intermediate, and output data for 10 years.Table A.2. Farmer No. 3's input, intermediate, and output data for 10 years.Table A.3.Farmer No. 4's input, intermediate, and output data for 10 years.Table A.5. Farmer No. 6's input, intermediate, and output data for 10 years.Table A.6.Farmer No. 7's input, intermediate, and output data for 10 years.Table A.7. Farmer No. 8's input, intermediate, and output data for 10 years.Table A.8. Farmer No. 9's input, intermediate, and output data for 10 years.Table A.9. Farmer No. 10's input, intermediate, and output data for 10 years.

Table A .
10. Farmer No. 11's input, intermediate, and output data for 10 years.Table A.11. Farmer No. 12's input, intermediate, and output data for 10 years.Table A.12. Farmer No. 13's input, intermediate, and output data for 10 years.

Table A .
13. Farmer No. 14's input, intermediate, and output data for 10 years.Table A.14. Farmer No. 15's input, intermediate, and output data for 10 years.