1. Introduction
Industries across the globe rely on the efficiency evaluation of manufacturing processes to create quality products and save on expensive manufacturing and quality costs [
1,
2]. In practice, information asymmetries arise in the only collected data due to subjective judgments based on experience and historical data or variations, which may significantly affect the accuracy of the efficiency assessment results and the effectiveness of the improvement decisions [
3]. Consequently, developing effective approaches for efficiency evaluation under asymmetry input and output has become a real challenge. This research, therefore, develops a framework for window analysis in data envelopment analysis (DEA) to evaluate efficiency from asymmetry input and output data for manufacturing processes.
Data envelopment analysis (DEA) is a mathematical programming-based non-parametric approach that is widely used for assessing the relative efficiency of homogeneous decision-making units (DMUs) [
4]. In DEA, process engineers usually rely on multiple crisp inputs and crisp outputs for assessing process efficiency. In DEA models, a DMUs’ efficiency is defined by its relative distance from the production frontier [
5,
6]. Usually, two DEA techniques are used to evaluate the DMUs’ efficiency, including the Charnes–Cooper–Rhodes (CCR) [
7] and Banker, Charnes, and Cooper (BCC) [
8]. The CCR model measures technical efficiency (TE) by maximizing the output from a given set of inputs at an optimal scale of operation, or constant returns to scale (CRTS). To gain valuable information about the source of inefficiencies, TE is composed of two efficiencies: pure technical efficiency (PTE) and scale efficiency (SE). The BCC model assumes that a DMU is operating under variable returns to scale (VRS) and measures PTE by only comparing a DMU to a unit of a similar scale. PTE assesses the extent to which a DMU utilizes its sources in exogenous environments and evaluates managerial performance [
9,
10]. Finally, the scale efficiency (SE), calculated as TE divided by PTE, is used to evaluate the effect of the scale size on efficiency and enables management to select the optimal resource size to obtain the target production level. Inappropriate scale size causes technical inefficiency [
11,
12]. Scale inefficiency (SIE) is due to increasing returns-to-scale (IRTS) when the manufacturing process is too small for its scale of operations. SIE is due to decreasing returns-to-scale (DRTS) when the process is too large for its scale of operations. To reduce costs and maximize revenues, the process has to operate at the most productive scale, which is CRTS [
13,
14,
15]. The traditional DEA models for evaluating DMUs relative efficiency in various business applications [
16,
17,
18,
19,
20].
Nevertheless, for DEA models to avoid producing multiple efficient DMUs, the number of DMUs should be at least two times the sum of the number of inputs and outputs [
6]. Fortunately, DEA window analysis was introduced to improve discriminating power by increasing the number of DMUs when using a limited number of DMUs [
2]. DEA window analysis regards the same DMU in distinct periods, which are then treated as entirely different DMUs. The TE, PTE, and SE of a DMU in any period can then be estimated using the inputs and outputs of the same DMU in other periods as well as those of other DMUs. The window analysis was applied to evaluate process efficiency under crisp input and output data in a wide range of manufacturing and service applications [
21,
22,
23,
24,
25]. In manufacturing systems, however, variations in process and measurement result in asymmetry data. As a result, the data available for efficiency analysis cannot be presented as crisp data. Consequently, window analysis should be developed to deal with asymmetry data, represent real-world problems more realistically, and obtain a reliable assessment of manufacturing processes.
In this context, this research proposes a DEA window analysis to assess the DMUs relative efficiency using asymmetry or fuzzy dynamic data. In this research, the relative efficiency is calculated for each element of the triangular fuzzy number of the input and output data. Then, the fuzzy efficiency is transformed into a crisp value. The efficiency evaluation for a blowing machine used to manufacture plastic products during the year 2020 is utilized to illustrate the developed fuzzy window analysis. The remainder of this paper, including the introduction section, is organized in the following sequence:
Section 2 reviews the relevant background on fuzzy DEA techniques and applications.
Section 3 presents this research methodology.
Section 4 presents an application of the developed window analysis and research results.
Section 5 summarizes the research conclusions.
2. Literature Review
The traditional DEA models were reported as powerful techniques for efficient evaluation of homogeneous DMUs from crisp input and output process data. In reality, however, production processes are usually volatile and complex, which makes it difficult to obtain accurate or precise input and output data. Therefore, significant research efforts were directed at developing DEA models that can evaluate the relative efficiency of DMUs for a manufacturing process from fuzzy input and output data. For example, Guo and Tanaka [
26] proposed two fuzzy DEA models for evaluating the efficiencies of DMUs from fuzzy input and output data. The efficiencies were represented by fuzzy numbers to reflect the inherent fuzziness of evaluation problems. The fuzzy DEA models extended the CCR model to more general forms that can handle crisp, fuzzy, and hybrid data. Lertworasirikul et al. [
27] transformed fuzzy DEA models into possibility DEA models by using possibility measures of fuzzy constraints. The fuzzy membership functions of fuzzy data were of the trapezoidal type. Liu and Chuang [
28] proposed a DEA approach to determine the fuzzy efficiency measures embedded with the assurance region (AR) concept. The fuzzy DEA/AR model was transformed into a family of crisp DEA/AR models by calculating the lower and upper bounds of efficiency scores at a specific level. A study of twenty-four university libraries in Taiwan was employed to illustrate their model. Wen and Li [
29] proposed a credibility measure in the fuzzy DEA model, followed by DMUs ranking. A hybrid algorithm combined with the fuzzy simulation and genetic algorithmgenetic algorithm was used to solve the model for trapezoidal or triangular fuzzy inputs and outputs. Puri and Yadav [
30] proposed the concept of fuzzy input mix-efficiency and evaluated the fuzzy input mix-efficiency using the α-cut approach. A real case study from the State Bank of Patiala in the Punjab state of India, with districts, was provided for illustration. Wanke et al. [
31] employed bootstrapped regressions and fuzzy-DEA (FDEA) models to capture vagueness in the input and output measurements obtained from Nigerian airports. Barak and Dahooei [
32] proposed FDEA and fuzzy multi-attribute decision-making (F-MADM) for ranking the airlines’ safety. The FDEA was adopted to estimate criteria weights, which were then employed to rank each airline using MADM methods. Arana-Jiménez et al. [
33] considered a slacks-based additive inefficiency measure and compared it with the existing fuzzy DEA methods. However, their model could not discriminate between efficient and weakly efficient DMUs. Khoshandam and Nematizadeh [
34] proposed an inverse network DEA model for two-stage processes to evaluate the amount of change in one or more indicators of one stage of a process by changing indicators of another stage to preserve the level of efficiency in the presence of undesirable factors. The model was implemented in poultry farming. Mohanta et al. [
35] developed an intuitionistic fuzzy DEA (IFDEA) model based on triangular intuitionistic fuzzy numbers (TIFNs). The weighted possibility means for TIFNs were then utilized to compare and rank the TIFNs.
Little research has been reported on using window analysis to assess efficiency when the input and output data are asymmetry and dynamic. For example, Wang et al. [
36] combined DEA window analysis and fuzzy techniques for order of preference by similarity to the ideal solution to assess the capabilities of 42 countries in terms of renewable energy production potential. Three inputs (population, total energy consumption, and total renewable energy capacity) and two outputs (gross domestic product and total energy production) Peykani et al. [
37] developed credibility-based fuzzy window analysis to evaluate the dynamic performance of hospitals during different periods of data ambiguity. The proposed approach was implemented on a real data set to evaluate the performance of hospitals in the USA. Al-Refaie [
2] proposed a DEA window analysis and Malmquist productivity index under fuzzy data. The proposed window analysis was based on providing crisp efficiency scores by solving a single model.
This research provides an extension to ongoing research [
38,
39,
40] by developing a framework for window analysis with asymmetry data. The collected input and output data are represented by triangular fuzzy numbers. Process efficiency is then calculated at three levels and transformed into a single, crisp optimal efficiency. The proposed DEA window analysis contributes to literature and practice by: (1) providing reliable assessments of process efficiency under asymmetry input and output data; (2) determining effective improvement actions based on slack analysis of inputs and outputs that lead to enhanced process performance; and (3) transforming the fuzzy efficiency into a crisp score that facilitates understanding and interpreting process efficiency.
3. Research Methodology
The developed framework for window analysis with asymmetry data are depicted in
Figure 1.
The steps of the developed window analysis with asymmetry data are presented as follows:
Step 1: Specify the fuzzy input variables (planned production quantity, number of defectives, and idle time) and a single fuzzy output variable (actual production quantity) for the manufacturing process under consideration. Classify the variables into inputs and outputs for DEA window analysis, where input (output) variables are these variables to be minimized (maximized) [
7]. Then, collect asymmetry data for those variables over a time horizon (T). It is assumed that the input and output data follow triangular fuzzy numbers (TFNs).
Step 2: Let the collected fuzzy input and output data of DEA variables at time t; t = 1, …, T, be denoted as ; i = 1,…, m, and , r = 1, …, s, respectively. The time horizon, T, is then divided into n windows. Let z denote window width. Let wj denote the jth window, which can be determined as follows: w1: , w2: , …, and so on. Finally, treat each window as a Decision-Making Unit (DMU).
Step 3.1: Evaluate the technical efficiency (TE) of each DMU
j (
j = 1, …,
n). Let
be the efficiency of DMU
k (
). Then, the objective of the optimization model aims to maximize
[2, 9]. Generally,
where
ur (
r = 1,…,
s) and
vm (
i = 1,…,
m) are the input and output weights. The ratio of the virtual output versus the virtual input of DMU
j cannot exceed one. Then, the following constraints are formulated [3, 8]:
Let
and
denote the negative input and positive output slacks, respectively. The equivalent dual problem of the input-oriented fractional model (Formulas (1)–(4)) is formulated as [
7]:
Let denote the optimal efficiency of DMUk. Then, DMUk is identified as CCR-efficient if is equal to one and all slacks are zeros. Otherwise, DMUk is identified as CCR-inefficient.
Because the input and output
and
are fuzzy numbers, then
is a triangular fuzzy numbers [
28]. Let
,
, and
, denote the low, middle, and high levels of the optimal TE values, respectively. Let the fuzzy input
, where
aitk,
bitk and
citk denote the low (L), middle (M), and high (H) elements for fuzzy
ith input;
i = 1,…,
m, of DMU
k;
. In addition, the fuzzy output
, where
drtk,
ertk, and
frtk represent the L, M, and H levels of fuzzy
rth output;
r = 1,…,
s, of DMU
k;
. For illustration, the representation of the collected data for a window of six periods of DMU
1 is displayed in
Table 1.
The optimal low TE,
, of DMU
k at a specific time
q;
, in window
k is estimated using the dual formulation of the CCR model as follows:
Table 2 displays the optimal TE and PTE efficiencies from low-level input and output data for DMU
1.
The middle and high optimal efficiencies for the remaining periods in window k; and t ≠ q, are estimated similarly. Repeat this step to obtain the optimal efficiencies and for all periods in window k; , of DMUk. In a similar manner, obtain the optimal efficiencies and from the middle and high levels of input and output data.
Step 3.2: Evaluate the pure technical efficiency (PTE) of DMU
j. Let
,
, and
, denote the low, middle, and high optimal
PTE efficiencies, respectively. For illustration, the optimal PTE,
, at the low data level DMU
k, is estimated using an input-oriented BCC model without slacks [
2]. Mathematically,
The obtained optimal values of
are shown in
Table 2. Similarly, the
values are calculated at the remaining periods (
t ≠
q) of DMU
k. In a similar manner, the
and
, are obtained at window periods of DMU
j (
j ≠
k).
Table 3 summarizes the calculated
,
,
, and
.
Step 3.3: Calculate the elements of the TE and PTE averages
and
, respectively, of DMU
j;
j = 1, …,
n. For example, in
Table 2, the low-level optimal efficiency of DMU
1,
, is the average of the period efficiencies
,
, …, and
. Similarly, the low-level optimal PTE of DMU
1,
.
Step 4: To provide a more practical interpretation of the optimal fuzzy
TE and
PTE, the
and
are transformed into single crisp values. Let
D denote the defuzzified optimal efficiency, which is calculated as follows [
25]:
where
U,
M, and
L are the high, middle, and low of efficiency TFN, respectively. Use Equation (22) to calculate the optimal
and
of DMU
j;
j = 1, …,
n, respectively, as follows:
Step 5: Calculate the optimal SE,
, as follows:
Table 3 also summarizes the values of
for all DMUs.
Step 6: Let
,
, and
denote the TE, PTE, and SE at period
t, respectively. Obtain
and
as follows. Firstly, calculate the average period TE,
, at low-level data, from the values of
for the DMUs that include the period
t in window
j, as shown in Equation (26).
where
denotes the number of DMUs that include period
t in window
j. Similarly, calculate the
and
in a similar manner. Apply Equations (23) and (24) to calculate the crisp
and
, respectively. Finally, calculate the values of
using Equation (25). The results are shown in
Table 3.
Step 7: Analyze the results for the DMUs and period fuzzy and crisp optimal efficiencies (TE, PTE, and SE). Determine and examine the input and output negative and positive slacks for all DMUs and periods. Recommend the required actions to improve process performance. Finally, validate the anticipated improvement.
4. Research Results
The efficiency evaluation of the blowing machine in the plastics industry was considered and is presented as follows. In steps 1 and 2, the relevant fuzzy input and output data for a blowing process were collected for 12 months;
t = 1, …, 12, from the production reports. The planned production quantity (
), number of defectives (
), and idle time (
) were treated as the inputs, whereas the actual production quantity in production units (
) was set as the output for all periods in the DEA model as shown in
Table 4. The length of each window consisted of six periods. Hence, seven DMUs; DMU
1 to DMU
7 for
t1–
t6,
t2–
t7, …, and
t7–
t12, respectively, were obtained.
In step 3.1, the input-oriented
CCR model was employed to estimate the optimal TE;
,
, and
, at the low, middle, and high levels respectively.
Table 5 displays the obtained optimal values of
,
,
, and
.
From
Table 5, the following remarks are obtained: Firstly, the estimated optimal TE listed at each period (column) reveals a stable performance because almost all the percentages of the coefficient of variation, CV%, are smaller than 0.05 for all columns. However, the CV percentages corresponding to the DMUs (rows) are greater than 0.05, which implies significant dispersion or trend in the TE scores of the same window. Secondly, the averages of
,
, and
are found to be smaller than one for all seven DMUs, and thereby it can be concluded that the blowing process was TE-inefficient at all data levels. Furthermore, the optimal TE values for each window period;
, and
were estimated and found to be equal to one in two, two, and three out of twelve periods, respectively. In step 3.2, the optimal values of the pure technical efficiency;
,
, and
, were calculated at low, middle, and high data levels to explain the causes of the
TE inefficiency (TIE) for all window periods.
Table 6 displays the results of the optimal
PTE, where it is found that: (1) the CV% indicates the existence of less dispersion in
PTE scores than the TE scores of the same window for all DMUs. For example, the
(=0.9785) for DMU
1 implies that the same output level could be produced by 97.85% of the recourses. In other words, about 2.15% of recourses could be saved by enhancing the machine’s performance to the highest level, and (2) the averages of
,
, and
are smaller than one for all seven DMUs, and thereby the blowing machine is judged PTE-inefficient in all DMUs. Further, the optimal PTE at each window period;
,
, and
) was calculated and found to be equal to one in six, six, and eight out of the twelve periods, respectively.
Table 7 displays the estimated optimal fuzzy values of
and
for all DMUs. It is obvious in
Table 7 that the existence of variations in the input and output data results in reasonable differences between the
,
, and
values. A similar conclusion is obtained when comparing between
,
, and
values. Such differences may lead to erroneous improvement directions and complications in the decision-making process. Consequently, in step 4, the defuzzified values of the optimal TE and PTE,
and
, respectively, were calculated using Equations (23) and (24), respectively. From
Table 7, the optimal TE is found to be smaller than one for all DMUs. Hence, all DMUs can be characterized as TE-inefficient. Moreover, the optimal PTE is smaller than one for all DMUs, and consequently, the seven DMUs can be identified as PTE inefficient (PTIE). Step 5 follows to determine the optimal SE,
, values shown in
Table 7.
Figure 2 depicts the optimal TE, PTE, and SE for all DMUs, where the optimal PTE is found to be larger than the corresponding SE in four DMUs: DMU
1 to DMU
3 and DMU
7. Consequently, the reason behind the TIE for these DMUs was scale inefficiency. whereas the TIE was caused by managerial inefficiency for the remaining three DMUs, DMU
4 to DMU
6. The largest differences (
> 0.05) between PTE and SE correspond to DMU
1, DMU
2, and DMU
4. Further, the averages of optimal
,
, and
were calculated from all DMUs and found 0.9169, 09672, and 0.9501, respectively. Consequently, it is concluded that TIE was caused by SIE.
The period averages of TE, PTE, and SE were analyzed in step 6 and then shown in
Figure 3. It is noted that the TIE in months 1 to 3, 5, 8, 10, and 12 were caused by SIE, whereas the TIE was caused by PTE in months 4, 7, and 11. Finally, the process operated at an optimal TE in months 6 and 9.
Furthermore, the data projections resulting from the CCR and BCC models were calculated and then displayed in
Table 8 for all DMUs. It is found that the largest input excesses correspond to IT, followed by DQ for all DMUs. Thus, reductions in excess inputs should be made to become technically efficient. To illustrate, for DMU
1 to become CCR-efficient at the same PQ, the PP, DQ, and IT have to be reduced by 9.77%, 24.93%, and 57.67%, respectively. On the other hand, for DMU
1 to become BCC-efficient, the PP, DQ, and IT have to be reduced by 1.54%, 3.82%, and 27.72%, respectively, while the PQ should be increased by 3.11%. On average, for the blowing process to become CCR-efficient, the inputs PP, DQ, and IT should be reduced by 8.10, 23.3, and 69.21%, respectively. On the other hand, to become BCC-efficient, the inputs PP, DQ, and IT should be reduced by 3.83, 14.04, and 34.55%, respectively, while increasing the output by 1.01%.
In practice, the optimal SE and PTE obtained enable the identification of the causes of DMUs technical inefficiency: managerial or scale inefficiencies. Management actions, such as controlling resources to improve PTE, expanding/decreasing operational scale to boost overall technical efficiency, implementing effective quality control procedures to reduce defectives, increasing the sampling and testing scale to avoid producing defective outputs, and adopting efficient production scheduling and sequencing to reduce idle time (IT) and better utilize input resources and available production capacities, are needed.
Implementing these actions, the optimal TE, PTE, and SE,
, and
, respectively, were estimated and then displayed in
Table 9. It is noted after improvement that all the optimal TE,
, values are less than one, which implies that the blowing process is still technically inefficient. Comparing the
values in
Table 6 with the corresponding values in
Table 9, it is obvious that the TE efficiency has improved for all DMUs. Moreover, the TIE is still caused by SIE in three DMUs (DMU
1 to DMU
3), while it is caused by PTIE in the remaining four DMUs. Nevertheless, all the differences are less than 0.05. The averages of
,
, and
after improvement are 0.9481, 0.9692, and 0.9784, respectively. On average, the cause of the TIE is managerial inefficiency because improvement actions take longer to take effect. Despite that, these
,
, and
averages are larger than their corresponding values in
Table 4.
Figure 4 compares the optimal TE, PTE, and SE before and after improvement actions.
From
Figure 4, the following remarks are obtained: (a) the optimal TE after improvement is larger than its corresponding before-improvement TE for all DMUs; (b) the optimal PTE after improvement is larger than its corresponding PTE before improvement for almost all DMUs. The optimal PTE after improvement started to become larger than the corresponding before-improvement optimal PTE from DMU
3. The reason behind such a delay may be attributed to improving quality and production procedures that require time to take effect, and (c) the SE after improvement is larger than its corresponding before-improvement SE due to the improvement of the optimal technical efficiency for all DMUs.
The improvement analysis shows that the proposed DEA window analysis is effective in evaluating and enhancing the performance of the blowing process under asymmetry input and output data. Further, the improvement actions conducted based on the proposed efficiency analyses have resulted in significant improvements in the process’s technical efficiency and, thereby, savings in costly production and quality resources.