J00780
J Intell Manuf
DOI 10.1007/s10845-013-0826-y
Optimization of injection-molded light guide plate with
microstructures by using reciprocal comparisons
Chung-Feng Jeffrey Kuo · Gunawan Dewantoro ·
Chung-Ching Huang
Received: 30 April 2013 / Accepted: 6 August 2013
© Springer Science+Business Media New York 2013
Abstract Injection molding is considered as an effective
way to manufacture light guide plate (LGP) with microstructures. However, the determination of processing parameters has always been difficult pertaining to shrinkage and
warpage, bringing about variations in quality of the injection molded product. This study proposed a procedure for
solving the optimization problem utilizing reciprocal comparisons in data envelopment analysis. This method attempts
to improve the comparisons of efficiency between different systems (CEBDS) which has been done in other works.
The objectives are to make the depth and angle of the
V-cut microstructures as close to the target values as possible, and make the residual stresses of light guide plate
minimum. First, Taguchi method with orthogonal array L18
was applied to reduce the number of experiments. Then, the
significant factors which have profound effect to the quality characteristics were confirmed using the ANOVA and
main effect analysis. Next, CEBDS and reciprocal comparisons were conducted to optimize the multi-parameter combination. The results were compared and investigated both
C.-F. J. Kuo (B)
Graduate Institute of Automation and Control, National Taiwan
University of Science and Technology, No. 43, Section 4,
Keelung Road, Taipei 106, Taiwan, Republic of China
e-mail: jeffreykuo@mail.ntust.edu.tw
G. Dewantoro
Department of Electronic and Computer Engineering, Satya
Wacana Christian University, 52-60 Diponegoro Street,
Salatiga 50711, Indonesia
e-mail: gunawan.dewantoro@staff.uksw.edu
C.-C. Huang
Department of Mold and Die Engineering, National Kaohsiung
University of Applied Science, 415 Chien Kung Road,
Kaohsiung 807, Taiwan, Republic of China
e-mail: cchuang@cc.kuas.edu.tw
theoretically and experimentally. The reproducibility of the
experiment was verified by confirming a confidence interval of 95 %. It is inferred that the reciprocal comparisons
approach is far superior to the CEBDS approach. The results
also demonstrated improvements of LGP qualities compared
to the best achieved in Taguchi orthogonal array experiment.
Keywords Injection molding · Taguchi method ·
Reciprocal comparisons · Optimization
Introduction
Liquid crystal display (LCD) is regarded as the most prevalent utilization among photo-electronic display devices. As
LCDs are not self-emissive display devices, they adopt backlight unit modules which provide uniform and bright light
source to display. The basic components of a typical backlight
unit module include the light guide plate, reflector, diffuser,
prism, and lamp. As shown in Fig. 1, the aim of the light guide
plate is to control the direction of the scattering of light, guiding the edge light to turn it into the entire area light. Simultaneously, the V-cut microstructures design on its surface is
used to balance out the distribution of light which increases
the luminance of the panel and ensures uniform brightness
(Jui and Ya 2012). Therefore, light guide plate is the key
element of the backlight module whose design and manufacture play significant role in determining the performance of
the backlight module’s optical properties. Injection molding
is one of the best methods for producing a complex-shaped
three dimensional product, and is thus strongly preferred for
the production of light guide plates. The key to the quality
of the injection molding process is in issues such as shrinkage and warpage. Shrinkage leads to inaccuracies in the light
guide plate dimensions, while warpage causes the shape of
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J Intell Manuf
Fig. 1 Backlight unit module
Prism sheet
Lamp
Diffuser
Light Guide Plate
Reflector
Lamp reflector
the light guide plate to become distorted (Kazmer 2007;
Umbrello et al. 2010; Bociaga et al. 2010). However, the specification and setting of processing parameters has always possessed difficulties related to shrinkage and warpage, causing
variations in quality of the injection molded product. Therefore, processing parameters that minimize thermal-induced
residual stress are to be sought, thereby deformed wedgeshaped light guide plate can be avoided. In addition to that,
V-cut microstructures, with the target depth of 2.887 µm and
target angle of 120◦ , are kept to be as close to the targets
as possible. Thus, the V-cut depth, angle, and residual stress
were chosen to be the objectives in this study. Faced with so
many processing parameters, relevant research in the past had
been trying to discover the relationship between the parameters and mold product quality. Choi and Im (1999) analyzed
the shrinkage and warpage of the final injection product using
numerical analysis method, taking the residual stress issues
in the packing pressure and cooling stages into consideration, and proving through the residual stress curve that the
packing pressure stage had the most profound effect on the
results of injection molding. Using Taguchi method, together
with nominally-the-best (type II) quality characteristic, Dragan et al. (2013) also showed that packing pressure was the
most influential process parameters affecting post-molding
shrinkage.
L9 orthogonal array from the Taguchi method can be carried out to investigate the sintering characteristics of Ti-6Al4V/HA tensile bars (Thian et al. 2002). Its final product was
produced using powder injection molding. The processing
parameters that were looked into include sintering temperature, heating rate, packing time, and cooling rate. From
the experimental result, sintering temperature had the most
significant effect on sintering characteristics, which is also
confirmed by Shye et al. (2013). Meanwhile, some research
observed that mold temperature is the most important parameter in injection molding process (Murakami et al. 2008;
Huang et al. 2009; Yun et al. 2013). They showed that high
mold temperature had been proven to significantly improve
the level of filling with melt plastics in microinjection molding and conducive to the molding of microstructures. According to Sanchez et al. (2012), cooling time was found to be
the most significant parameter to reduce warpage, as shown
in cases of the box and plastic glass. Guilong et al. (2013)
123
Microstructure
studied the effects of cavity surface temperature in rapid heat
cycle molding on mechanical strength of the molded specimens with and without a weld line. From the point of view
tensile strength and impact strength, cavity surface temperature was found to be the utmost important parameters.
While the Taguchi method is mainly used for optimizing single quality property (Nik and Shahrul 2011; Nor et
al. 2010; Hasan et al. 2007), optimal processing parameters
that determine the single quality usually do not represent the
optimization of process parameters for the overall quality.
For this reason, Taguchi method can also be combined with
other methods, such as principal component analysis as in
Anthony (2000), Saurav et al. (2009), Humberstone et al.
(2012). Nevertheless, when more than one principal component, how to trade off to select the solution is unknown.
Analytic Hierarchical Process (AHP) with Taguchi method
were combined to obtain the optimal multi-parameter of laser
scribing system for micro crystalline silicon thin film solar
cell isolation and plastic injection molding (Kuo et al. 2010;
Faisal et al. 2013). However, AHP incorporates a priori information about the response weight that increases uncertainty
in the decision-making process. Grey relational analysis was
employed in multi-objective optimization by using Taguchi
design of experiments (Yang et al. 2012; Kuo et al. 2011; Lin
2012), where the relative significance of processing parameters were still required to be known so that the optimal combinations of the processing parameters levels can be sought.
Response Surface Methodology was also used to determine
interactions among the control factors (Cicek et al. 2013;
Muhammad et al. 2012). The predictive quadratic models
were derived by the RSM to obtain the optimal responses as
a function of the parameters. Abbas and Mohammad (2011),
Abbas (2009) studied, respectively, the benevolent formula
and comparisons of efficiency between different systems
(CEBDS) technique in Data Envelopment Analysis (DEA).
The CEBDS technique turns out to be an efficient approach
for optimizing process performance with categorical data.
However, there was no confirmation experiment to prove the
reliability of the results.
This paper proposed reciprocal comparisons approach as
opposed to the CEBDS method (Abbas 2009) for optimizing processing parameters of injection molding machine. By
this way, the qualities of molded product were improved
J Intell Manuf
Table 1 Eight control factors and their levels for injection molding
machine
Start
Control factors
experimental
planning
proceed with
experiment
Main effect
analysis
ANOVA
Data analysis
CEBDS
optimal processing
parameters determination
Reciprocal
comparisons
Levels
1
2
3
A. Cooling time (s)
15
30
B. Mold temperature (◦ C)
75
80
85
(◦ C)
270
250
260
D. Injection speed (mm/s)
165
180
195
E. Injection pressure (MPa)
220
240
260
F. Packing pressure (MPa)
90
100
110
G. Packing switching (mm)
5
10
15
H. Packing time (s)
1
2
3
C. Melt temperature
verification
no
satisfactory ?
yes
order to comprehend the variance of the molded light guide
plates.
Material and equipments
End
Fig. 2 Flowchart of experimental procedure
and the computational time was reduced. First, Taguchi
method with orthogonal array L18 was applied to reduce
the number of experiments. Then, the significant factors
which have profound effect to the quality characteristics
were confirmed using the ANOVA and main effect analysis.
Next, CEBDS and reciprocal comparisons were conducted to
optimize the multi-parameter combination. The results were
compared and investigated both theoretically and experimentally. That is, the S/N ratios were predicted under the optimal conditions by addition to investigate the total anticipated
improvement and the confirmation experiments were carried
to verify reproducibility and feasibility through the proposed
approach.
The mold tooling was provided by Plastic Precision Molding Laboratory, Department of Mold and Die Engineering,
National Kaohsiung University of Applied Science, as shown
in Fig. 3. The resin material used for injection molding
was optical grade polymethyl methaacrylate (PMMA) with
refractive index of 1.49, produced by Japan’s KURARAY
Co., Ltd. The injection molding machine was manufactured
by Sodick Plustech Co., Ltd. (Model Sodick TR30EH) as
shown in Fig. 4a. The light guide plate produced by the injection molding process is shown in Fig. 4b. The surface profile
measuring system used for measuring the light guide plate
V-cut measurements is the Form Talysurf PGI 635, made by
Taylor Hobson Co., Ltd, as shown in Fig. 5a, whilst the residual stress measuring instrument is VML 250P, made by 3D
Family Tech. Co., Ltd. as shown in Fig. 5b.
Reciprocal comparisons in data envelopment analysis
Experimentation
The whole experimental procedure of this study is described
in Fig. 2:
Experimentation plan
The final selection of control factors and their levels is shown
in Table 1. Eighteen sets of experiments were then conducted
as planned by the orthogonal array, with each set of experiments repeated four times which means there would be a total
of 72 data for each quality characteristic. Four dashed lines
were followed by using surface-profile measuring system to
measure the depth and angle of molded light guide plates in
Data Envelopment Analysis (DEA) is linear programming
for measuring the productive efficiency of Decision-Making
Units (DMUs) without requiring prescribed weights attached
to the multiple inputs and multiple outputs, as described in
Cooper et al. (2006). This proposed reciprocal comparisons
method seeks the opportunity to improve CEBDS in DEA
which was introduced by Abbas (2009). Here, each DMU
is compared in reciprocal manner. That is, for each pair of
group, each DMU of one group is evaluated with respect to
the DMUs of the opposite group and vice versa. Therefore,
for a total of K groups, there will be K!/(2!x(K−2)!) reciprocal comparisons. This inter-comparison, as opposed to inter
and within-comparison in CEBDS, will result in sharper dis-
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Fig. 3 Moving mold-half (left) and fixed mold-half (right)
Fig. 4 a Sodick Plustech injection molding machine. b Light guide plate
Fig. 5 Measuring systems. a Taylor Hobson surface profilemeter. b 3D Family Tech stress measuring instrument
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crimination between two groups. For example, let DMUa
be a random DMU of group 1 to be evaluated, and E a,2
denotes the efficiency of DMUa with respect to the DMUs
of group 2. Then E a,2 is obtained by solving the following
model:
E a,2 = min θ
subject to
θ xa ≥
xjλj
yjλj
j∈gr oup2
ya ≤
j∈gr oup2
λ j ≥ 0, ∀ j ∈ gr oup2
(1)
where xa and ya are the input and output vectors for
DMUa , x j and y j are the input and output vectors for DMUs
of group 2. The objective of the linear programming is to
adjust the value of λ j such that θ is minimum. The efficiency
of each DMU of group 2 with respect to DMUs in group 1
can be easily obtained in a similar manner.
Results and discussion
Taguchi experimental method (Ross 1996)
The experimental data of the V-cut depth, V-cut angle, and the
residual stress are shown in Table 2. On the basis of the S/N
ratio available in Table 2, main effect analysis was adopted in
this study to figure out the main effect of each factor level and
the response graphs could be prepared, as shown in Fig. 6.
From the response graphs, the optimal factor combination
for single quality characteristic could be obtained.
In order to identify the control factors that have significant effects, analysis of variance (ANOVA) for S/N ratio
is performed at a certain confidence level. The results of
ANOVA for V-cut depth, angle, and residual stress are shown
in Table 3. From ANOVA of V-cut depth as shown in Table 3a,
control factors B, G, and H all possess F-ratio greater than
that obtained from table of F value with 95 % confidence
level. Therefore, all of these factors are significant factors
whose effects are strongly believed to be profound. From
ANOVA of angle as shown in Table 3b, control factors G,
and H all possess F-ratio greater than that obtained from
table of F value with 95 % confidence level. Therefore, all of
these factors are significant factors whose effects are strongly
believed to be profound. From ANOVA of residual stress as
shown in Table 3c, control factors B, G, and H all possess
Table 2 The orthogonal array with the averages and SN ratios of all qualities
Trial
A
B
C
D
E
F
G
H
Mean
of depth
(µm)
Mean
of angle
(degree)
Mean of
residual
stress
(MPa)
SNR of
depth
(dB)
SNR of
angle
(dB)
SNR of
residual
stress
(dB)
1
15
75
250
165
220
90
5
1
1.7513
140.690
0.311
−1.109
−26.334
10.149
2
15
75
260
180
240
100
10
2
2.1986
131.689
0.286
3.088
−21.795
10.887
3
15
75
270
195
260
110
15
3
2.0247
130.648
0.280
0.992
−20.723
11.054
4
15
80
250
165
240
100
15
3
2.1476
131.240
0.279
2.308
−21.234
11.094
5
15
80
260
180
260
110
5
1
1.9650
140.065
0.243
0.639
−26.128
12.273
6
15
80
270
195
220
90
10
2
2.0228
135.543
0.259
1.167
−23.835
11.732
7
15
85
250
180
220
110
10
3
2.5274
128.418
0.215
8.310
−18.658
12.832
8
15
85
260
195
240
90
15
1
1.3743
165.750
0.286
−3.620
−33.308
10.852
9
15
85
270
165
260
100
5
2
2.4465
132.615
0.263
6.727
−22.096
11.589
10
30
75
250
195
260
100
10
1
1.2800
150.708
0.289
−4.237
−29.825
10.787
11
30
75
260
165
220
110
15
2
1.5188
144.197
0.309
−2.838
−27.696
10.211
12
30
75
270
180
240
90
5
3
2.3235
131.727
0.257
4.634
−21.473
11.811
13
30
80
250
180
260
90
15
2
1.7001
138.826
0.315
−1.546
−25.548
10.041
14
30
80
260
195
220
100
5
3
2.5166
128.047
0.250
8.406
−18.274
12.039
15
30
80
270
165
240
110
10
1
1.8936
136.949
0.281
−0.167
−24.652
11.025
16
30
85
250
195
240
110
5
2
2.7172
127.588
0.195
15.299
−17.798
14.204
17
30
85
260
165
260
90
10
3
2.4924
127.954
0.231
7.310
−18.655
12.729
18
30
85
270
180
220
100
15
1
1.0134
165.857
0.311
−5.492
−33.255
10.146
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Fig. 6 The response graph for a V-cut depth b V-cut angle c residual stress
Table 3 ANOVA results for (a)
V-cut depth (b) V-cut angle (c)
residual stress
Factor
(a)
A*
SS
0.456199
d.f
1
V
F-ratio
66.89341
2
C*
10.40742
2
5.203709
2
3.063327
6.126654
33.44670
5.537093++
55.812445
11.04699
E*
17.19271
2
F*
21.31247
2
10.65624
2
84.19398
13.93829++
156.30700
31.50237
97.22433
16.09546++
182.36769
36.75469
G
H
Error
Pooled
Total
168.3880
194.4487
2
8.596357
10.94982
2
5.474911
66.44528
11
6.040480
496.1753
17
102.68817
496.17531
(b)
123
%P
0.456199
B
D*
SS′
A*
0.521994
1
0.521994
B*
5.570479
2
2.785239
C*
4.765560
2
2.382780
D*
3.192966
2
1.596483
20.69594
100
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Table 3 continued
Factor
E*
SS
V
F-ratio
2
2.607714
F*
17.02818
2
8.514089
G
83.37654
2
H
5.215428
d.f
253.6219
2
Error
21.01560
2
Pooled
57.31020
13
Total
394.3086
9.456388++
41.68827
28.76525++
126.8109
4.408477
SS′
74.559587
244.80493
74.944113
17
394.30863
%P
18.90894
62.08460
19.00646
100
(c)
∗ Pool-up
terms
++ Significant factor at 95 % CI
+ Significant factor at 90 % CI
A*
0.01555
1
B
4.64842
2
C*
0.31918
2
D*
1.31109
2
E*
0.63673
2
F
2.47414
G
2.32421
5.87759++
2
1.23707
3.12837+
1.68327
6.82858
2
3.41429
8.63424++
6.03771
28.95173
H
3.34434
2
1.67217
4.22867++
2.55346
12.24425
Error
1.27636
2
Pooled
Total
3.55892
20.8544
9
0.39544
6.72241
17
F-ratio greater than that obtained from table of F value with
95 % confidence level, whereas control factor F possesses Fratio greater than that obtained from table of F value with
90 % confidence level. Therefore, all of these factors are significant factors whose effects are strongly believed to be profound. By combining ANOVA and main effect analysis, one
can achieve the optimal parameters combination for the Vcut depth that is B3 , G1 , and H3 ; where mold temperature
is 85 ◦ C, packing switching is 5 mm, and the packing time
is 3 s. Meanwhile, the optimal parameters combination for
the V-cut angle is G1 and H3 . Lastly, the optimal parameters
combination for the residual stress is B3 , F3 , G1 , and H3 .
Optimization in Taguchi orthogonal array
3.85755
20.85440
18.49752
8.071516
32.23499
100
decreased. Since all of S/N ratios are larger-the-better
type, in consequence, set all of the responses as outputs,
which are preferred at maximum, and set a unit (one) as
the input.
Step 2. Normalize the S/N ratio, X i j , for the jth response
at the ith trial. X i j is now normalized as Z i j , which lies
between 0 and 1, by the following formula, to avoid the
effect of adopting different units. The result is shown in
Table 4. It is obvious that trial 16 is the most superior
among all trials in Taguchi orthogonal array experimentation set.
Zi j =
X i j − min X i j , i = 1, . . . , m
max X i j , i = 1, . . . , m − min X i j , i = 1, . . . , m
for i = 1, . . . , m,
j = 1, . . . , n
(2)
All trials in Taguchi’s orthogonal array turn out to be
DMU with the multi-objective as the inputs and outputs
for all DMUs. In this research, there are 18 experiments in
Taguchi’s orthogonal array of 8 factors with 3 responses.
Here, both CEBDS and reciprocal comparisons are investigated to demonstrate the improvement of the latter method.
The procedure is outlined in following steps:
Step 3. For each factor in Table 2, the 18 DMUs are then
separated into groups, each at the same factor level, as
shown in Table 5. In this case, factor A is assigned at two
levels; A1 and A2. As a result, the 18 DMUs for factor
A are divided into two groups each of 9 DMUs.
Step 1. Trial j ( j = 1, . . ., 18) shown in Table 2
are considered as DMU j. The relative efficiency of
DMU j, which is calculated as the ratio of the sum of
the weighted outputs relative to the sum of weighted
inputs, is improved when the sum of weighted outputs
is increased and/or the sum of the weighted inputs is
Step 4. For each of the factors A to H, the efficiencies
of the 18 DMUs are separately measured by solving linear programming (1). Appendix summarizes the obtained
DMU’s efficiencies for all factors. Let y j1 , y j2 , and y j3
be the outputs, and all inputs are set to be one. For factor B in Appendix, DMU1 is evaluated as follows: The
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Table 4 Normalized S/N ratio
(dB)
Trial no.
Depth
Angle
Stress
1
0.2108
0.4496
0.3353
2
0.4127
0.7423
3
0.3119
0.8114
4
0.3752
5
Trial no.
Depth
Angle
Stress
10
0.0604
0.2245
0.2876
0.0223
11
0.1277
0.3618
0.1177
0.0717
12
0.4870
0.7631
0.0000
0.7784
0.0834
13
0.1898
0.5003
0.0675
0.2949
0.4629
0.7257
14
0.6685
0.9693
0.0673
6
0.3203
0.6108
0.2715
15
0.2561
0.5581
0.0631
7
0.6639
0.9445
0.3009
16
1.0000
1.0000
1.0000
8
0.0900
0.0000
0.0121
17
0.6158
0.9447
0.8600
9
0.5877
0.7229
0.2293
18
0.0000
0.0034
0.1574
Table 5 The index of DMU that belong to groups of each factor
Group (level)
Factor
A
B
C
D
E
F
G
H
1
1–9
2
10–18
3
–
1, 2, 3, 10,
11, 12
4, 5, 6, 13,
14, 15
7, 8, 9, 16,
17, 18
1, 4, 7,10,
13, 16
2, 5, 8, 11,
14, 17
3, 6, 9, 12,
15, 18
1, 4, 9,11,
15, 17
2, 5, 7, 12,
13, 18
3, 6, 8, 10,
14, 16
1, 6, 7, 11,
14, 18
2, 4, 8, 12,
15, 16
3, 5, 9, 10,
13, 17
1, 6, 8,12,
13, 17
2, 4, 9, 10,
14, 18
3, 5, 7, 11,
15, 16
1, 5, 9,12,
14, 16
2, 6, 7, 10,
15, 17
3, 4, 8, 11,
13, 18
1, 5, 8, 10,
15, 18
2, 6, 9, 11,
13, 16
3, 4, 7, 12,
14, 17
efficiency, E 1,1 of DMU1 with respect to the 6 DMUS of
group 1 is obtained by solving the following model:
subject to
θ≥
6
λj +
j=4
E 1,1 = min θ
subject to
θ≥
λj
j=13
0.2108 ≤
3
15
6
y j1 λ j +
6
y j2 λ j +
6
y j3 λ j +
j=4
λj +
12
0.4496 ≤
λj
y j1 λ j
j=1
12
j=10
0.4496 ≤
3
y j2 λ j +
12
y j2 λ j
0.3353 ≤
3
y j3 λ j +
12
y j3 λ j
0.2108 ≤
3
y j1 λ j +
15
y j2 λ j
15
y j3 λ j
j=13
j=13
j=4
λ4 , λ5 , λ6 , λ13 , λ14 , λ15 ≥ 0
j=10
λ1 , λ2 , λ3 , λ10 , λ11 , λ12 ≥ 0
(3)
E 1,3 = min θ
subject to
The E 1,1 is measured as 1.000.
Similarly, the efficiency of DMU1 with respect to the six
DMUs of group 2, E 1,2 , is obtained by solving the following
model:
θ≥
9
λj +
123
18
λj
j=16
j=7
0.2108 ≤
E 1,2 = min θ
(4)
The E 1,2 is evaluated as 0.693.
Finally, the efficiency of DMU1 with respect to the six
DMUs of group 3, E 1,3 , is obtained by setting z 3 = 1 and
z 1 = z 2 = 0 and solving the following model:
j=10
j=1
j=1
0.3353 ≤
y j1 λ j
j=13
j=4
j=10
j=1
15
9
j=7
y j1 λ j +
18
j=16
y j1 λ j
J Intell Manuf
0.4496 ≤
y j2 λ j
j=7
18
j=16
9
18
y j3 λ j
9
y j2 λ j +
B2 . In a similar manner, the optimal levels for the other
factors are decided at A1 , C1 , D2 , E3 , F3 , G1 , H3 .
The E 1,3 is evaluated as 0.45.
The optimal efficiency of DMU1, E 1 , is then decided as
0.45; which is the minimum of {1.000, 0.693, 0.45}. The
E 2 to E 18 for DMU2 to DMU18 are evaluated in a similar manner. The E j values for the 18 DMUs using the
CEBDS approach for the rest of the factors are obtained
similarly.
Step 6. Reciprocal comparisons solve the problem in the
same way as that of CEBDS. However, unlike CEBDS,
each DMU is only evaluated based on intergroup comparison rather than both inter- and within- group. Therefore,
comparison between group 2 and group 2 (within-group
comparison) is not required. For example, one does not
need to solve E 4,2 , E 5,2 , E 6,2 , E 13,2 , E 14,2, E 15,2 for factor B because DMU4 , DMU5 , DMU6 , DMU13 , DMU14 ,
and DMU15 belong to group 2 in factor B. This step is
repeated until the reciprocal comparison efficiencies of
18 DMUs for all factors are obtained.
Step 5. The level efficiency is calculated as the average
of the E j values measured using CEBDS approach for
the DMUs at that factor level. Table 6 displays the calculated efficiencies of all factor levels. For illustration,
the efficiency of B1 , 0.56929, is calculated as the sum
of the efficiencies for the six DMUs at level 1 divided
by six. Similarly, the efficiency for B2 and B3 are calculated as 0.690419 and 0.64327, respectively. Obviously,
the largest level efficiency for factor B corresponds to
B2 , and hence, the optimal level of factor B is selected at
Step 7. The efficiency is calculated as the average of
the E j,k values evaluated by the reciprocal comparisons
approach at that factor level, as shown in Table 7. For
illustration, the efficiency of B1 , 1.485, is calculated as
the average of E j,1 values. Similarly, the efficiency for
B2 and B3 are calculated as 0.86 and 0.63, respectively.
The minimum of {1.485, 0.86, and 0.63} corresponds to
B3 , and hence, the optimal level of factor B is decided
at B3 . In a similar way, the optimal levels for factors A
to H are decided at A2 , B3 , C1 , D3 , E2 , F3 , G1 , and H3 ,
respectively.
0.3353 ≤
y j3 λ j +
j=16
j=7
λ7 , λ8 , λ9 , λ16 , λ17 , λ18 ≥ 0
(5)
Table 6 The level of efficiency using the CEBDS approach
Factor
Confirmation experiments
Level
1
2
3
Optimal level
A
0.652839
0.615814
B
0.569290
0.690419
0.643270
A1
B2
C
0.660084
0.638963
0.603932
C1
D
0.635920
0.638877
0.628182
D2
E
0.582243
0.655311
0.665426
E3
F
0.559751
0.609653
0.733575
F3
G
0.771748
0.681333
0.449898
G1
H
0.378082
0.656330
0.868567
H3
After the optimum light guide plate processing parameters was obtained through both approaches, the confirmation
experiments were conducted.
Verification of CEBDS approach
Take the V-cut depth for instance, the S/N ratio of optimal
processing parameters is:
Sˆ N = T + (B2 − T ) + (G 1 − T ) + (H3 − T )
= B2 + G 1 + H3 − 2T
Table 7 The level of efficiency using the reciprocal comparisons
DMU j
Factor
A
B
C
D
E j,1
E j,2
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
Average
0.833
0.65
1.485
0.86
0.63
0.62
0.721
1.471
0.744
0.834
0.64
Optimal level
A2
B3
C1
D3
DMU j
E
F
G
H
Average
1.165
0.623
0.61
Optimal level
E2
0.62
0.719
0.795
F3
1.384
0.585
0.57
G1
0.711
2.93
1.729
H3
123
J Intell Manuf
= 1.801199 + 5.765991 + 5.326707 − 2 × 2.215056
= 8.463784 dB
(6)
where T is the total average value of S/N ratio; B2 , G1 and
H3 are the average of the S/N ratio for those factor level.
Confidence Interval (CI) of theoretically-predicted value is
calculated:
1
C I1 = Fα;1,v2 × Verr or ×
ne f f
7
= 3.371962 d B
(7)
= 4.84 × 6.04048 ×
18
The 95 % confidence interval of S/N ratio for V-cut depth,
V-cut angle, and residual stress are 5.091822 < µconfirmation
< 11.83575; −20.2842 < µconfirmation < −15.5014; and
11.93522 < µconfirmation < 14.05634, respectively. The
surface profile of V-cut microstructures and the residual
stress under the optimal combination is shown in Fig. 7. The
S/N ratio (dB) derived from four verification data for the
V-cut depth, V-cut angle, and residual stress are 10.91939,
−14.4397, and 14.9668, respectively. Compared to trial no.
16 in L18 orthogonal array, which shows the best S/N ratio
among all trials, the result of CEBDS is able to improve the
quality of V-cut angle and residual stress. However, the V-cut
depth quality was deteriorated with the decrease of −4.38 dB.
The 95 % Confidence Interval of experimental V-cut depth
value is calculated as:
C I2 =
Fα;1,v2 × Verr or ×
=
4.84x6.04048 ×
1
ne f f
1
7
+
18 4
+
1
r
= 4.321978 d B (8)
The 95 % confidence interval of S/N ratio for V-cut depth,
V-cut angle, and residual stress are 6.597413 < µconfirmation
< 15.24137; −17.7361 < µconfirmation < −11.1434; and
13.6897 < µconfirmation < 16.2438, respectively. The diagram for the CI of the V-cut depth, angle, and residual stress
verification experiment values and theoretical predicted values are shown in Fig. 8. From the diagrams, the CI from the
verification experiment and the theoretical predictions did
indeed coincide; therefore the results from the experiment
using CEBDS approach are indeed reliable had been proven.
Fig. 7 a V-cut microstructure and b residual stress under optimal combination with CEBDS approach
123
J Intell Manuf
Verification of reciprocal comparisons approach
Using Eq. 6, the S/N ratio of optimal processing parameters
for V-cut depth is 11.41823 dB. Confidence Interval (CI) of
theoretically-predicted value is calculated using Eq. 7, yieldFig. 8 Diagram for the CI of
the a V-cut depth b V-cut angle
and c residual stress with
CEBDS approach
ing 3.60468 dB. The 95 % confidence interval of S/N ratio for
V-cut depth, V-cut angle, and residual stress are 7.81355 <
µconfirmation < 15.02291; −20.2842 < µconfirmation <
−15.5014; and12.57461 < µconfirmation < 14.79927,
respectively. The surface profile of V-cut microstructures and
(a)
(b)
confirmation experiment value
confirmation experiment value
theoretical predicted value
theoretical predicted value
(c)
confirmation experiment value
theoretical predicted value
Fig. 9 a V-cut microstructure and b residual stress under optimal combination with reciprocal comparisons
123
J Intell Manuf
F i g . 1 0 Diagram for the CI of
the a V-cut depth b V-cut angle
and c residual stress, with
reciprocal comparisons
approach
(a)
(b)
(c)
residual stress under the optimal combination is shown in
Fig. 9. The S/N ratio (dB) derived from four verification
data for the V-cut depth, V-cut angle, and residual stress are
14.6251, −13.174, and 15.5299, respectively. Compared to
trial no. 16 in L18 orthogonal array, which shows the best
S/N ratio among all trials, the result of reciprocal comparisons is able to improve the quality of V-cut angle and residual
stress. The V-cut quality is a bit worse with the decrease of
−0.674 dB. However, the total improvement for all qualities
is +5.276 dB with more balanced manner without sacrificing one of the qualities. Obviously, the total improvements
of all qualities in reciprocal comparisons obviously outperform that in CEBDS method. The 95 % Confidence Interval of experimental V-cut depth value is calculated using
Eq. 8, yielding 4.505978 dB. The 95 % confidence interval
of S/N ratio for V-cut depth, V-cut angle, and residual stress
are 10.11911 < µconfirmation < 19.13107; −16.4701 <
µconfirmation < −9.87743; and 14.20946 < µconfirmation <
16.85025, respectively. The diagram for the CI of the V-cut
depth, angle, and residual stress verification experiment values and theoretical predicted values are shown in Fig. 10.
From the diagrams, the CI from the verification experiment
and the theoretical predictions did indeed coincide; therefore the results from the experiment using reciprocal comparisons approach are indeed reliable had been proven. It
can easily be observed that using the reciprocal comparisons
approach to measure DMU’s efficiency in the proposed procedure results in better outcome than the CEBDS approach
(Abbas 2009). Thus, reciprocal comparisons are promising
to be used in optimization problems for its simplicity and
powerfulness.
123
Conclusion
The reciprocal comparisons approaches in DEA were proposed to find out the optimal processing parameters of injection molding machine. The optimal processing parameters in
reciprocal comparisons approach are A2 , B3 , C1 , D3 , E2 , F3 ,
G1 , H3 , that is: the cooling time of 30 s, mold temperature of 85 ◦ C, melt temperature of 250 ◦ C, injection speed
of 195 mm/s, injection pressure of 240 MPa, packing pressure of 110 MPa, packing switching of 5 mm, and the packing time of 3 s. The reciprocal comparisons methods were
able to search optimal parameters combinations beyond the
orthogonal array experimentation set. Confirmation experiments showed that the optimal parameters combination
demonstrated qualities improvements in a more balanced
manner. Compared with CEBDS approach, the reciprocal
comparisons approach shows a better effectiveness because
not only provides larger anticipated improvements in the
responses, but also requires less computations since withincomparisons is not required. The reliability and reproducibility of the results were verified using 95 % confidence
interval.
Acknowledgments This research was financially supported by the
National Science Council of the Republic of China under Grant No.
NSC 97-2622-E-011-001-CC3.
Appendix
See Table 8.
J Intell Manuf
Table 8 The efficiencies for all DMUs
DMU j
Factor
A
B
C
D
E j,1
E j,2
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
DMU1
0.645
0.45
1
0.693
0.45
0.45
0.474
1.235
0.476
0.645
0.45
DMU2
0.786
0.742
0.956
0.766
0.742
0.742
0.766
0.954
0.786
0.786
0.742
DMU3
0.859
0.811
1
0.849
0.811
0.811
0.838
1
0.859
0.859
0.811
DMU4
0.824
0.778
1.04
0.825
0.778
0.778
0.804
0.985
0.824
0.824
0.778
DMU5
1
0.726
2.164
1
0.726
0.726
0.844
2.673
0.844
1
0.726
DMU6
0.715
0.611
1.127
0.803
0.611
0.611
0.637
1
0.647
0.715
0.611
DMU7
1
0.945
1.872
1.181
0.945
0.945
1.016
1.309
1.078
1
0.945
DMU8
0.136
0.09
0.205
0.137
0.09
0.09
0.135
0.153
0.146
0.136
0.09
DMU9
0.885
0.723
1.595
1.016
0.723
0.723
0.896
1
0.955
0.885
0.723
DMU10
0.428
0.288
0.858
0.437
0.288
0.288
0.334
1.059
0.334
0.428
0.288
DMU11
0.385
0.362
0.575
0.443
0.362
0.362
0.376
0.506
0.383
0.385
0.362
DMU12
0.808
0.763
1
0.787
0.763
0.763
0.787
1
0.808
0.808
0.763
DMU13
0.53
0.5
0.652
0.541
0.5
0.5
0.517
0.632
0.53
0.53
0.5
DMU14
1.026
0.969
1.487
1
0.969
0.969
1
1.291
1.086
1.026
0.969
DMU15
0.591
0.558
0.74
0.594
0.558
0.558
0.577
0.702
0.591
0.591
0.558
DMU16
2.02
1
3.745
2.218
1
1
1.585
3.683
1.624
2.02
1
DMU17
1.493
0.945
2.719
1.573
0.945
0.945
1
3.168
1
1.493
0.945
DMU18
0.217
0.158
0.47
0.217
0.158
0.158
0.183
0.58
0.183
0.217
0.158
DMU j
Factor
E
F
G
H
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
DMU1
1
0.45
0.476
0.476
1.236
0.45
0.45
0.476
2.398
0.878
0.45
0.474
DMU2
0.766
0.742
0.786
0.786
0.766
0.742
0.742
0.786
1.1
1.479
0.742
0.766
DMU3
0.839
0.811
0.859
0.859
0.859
0.811
0.811
0.859
1
1.454
0.811
0.838
DMU4
0.806
0.778
0.824
0.824
0.845
0.778
0.778
0.824
1
1.432
0.778
0.804
DMU5
2.164
0.726
0.844
0.844
2.577
0.726
0.726
0.844
4.979
1
0.726
0.844
DMU6
0.857
0.611
0.647
0.647
1.093
0.611
0.611
0.647
2.126
1.177
0.611
0.637
DMU7
1
0.945
1.078
1.078
1.311
0.945
0.945
1
2.744
2.251
0.945
1
DMU8
0.135
0.09
0.146
0.146
0.137
0.09
0.09
0.136
0.24
0.305
0.09
0.135
DMU9
0.884
0.723
0.955
0.955
1
0.723
0.723
0.885
2.193
1.993
0.723
0.884
DMU10
0.858
0.288
0.334
0.334
1
0.288
0.288
0.334
1.959
0.469
0.288
0.334
DMU11
0.39
0.362
0.383
0.383
0.51
0.362
0.362
0.383
0.965
0.668
0.362
0.376
DMU12
0.787
0.763
0.808
0.808
0.787
0.763
0.763
0.808
1.298
1.652
0.763
0.787
DMU13
0.52
0.5
0.53
0.53
0.563
0.5
0.5
0.53
0.731
0.899
0.5
0.517
DMU14
1
0.969
1.086
1.086
1
0.969
0.969
1.026
1.782
2.267
0.969
1
DMU15
0.578
0.558
0.591
0.591
0.61
0.558
0.558
0.591
0.738
1
0.558
0.577
DMU16
3.063
1
1.624
1.624
3.774
1
1
1.575
7.605
3.391
1
1.575
DMU17
2.581
0.945
1
1
3.154
0.945
0.945
1
6.235
2.088
0.945
1
DMU18
0.47
0.158
0.183
0.183
0.547
0.158
0.158
0.183
1
0.217
0.158
0.183
123
J Intell Manuf
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DOI 10.1007/s10845-013-0826-y
Optimization of injection-molded light guide plate with
microstructures by using reciprocal comparisons
Chung-Feng Jeffrey Kuo · Gunawan Dewantoro ·
Chung-Ching Huang
Received: 30 April 2013 / Accepted: 6 August 2013
© Springer Science+Business Media New York 2013
Abstract Injection molding is considered as an effective
way to manufacture light guide plate (LGP) with microstructures. However, the determination of processing parameters has always been difficult pertaining to shrinkage and
warpage, bringing about variations in quality of the injection molded product. This study proposed a procedure for
solving the optimization problem utilizing reciprocal comparisons in data envelopment analysis. This method attempts
to improve the comparisons of efficiency between different systems (CEBDS) which has been done in other works.
The objectives are to make the depth and angle of the
V-cut microstructures as close to the target values as possible, and make the residual stresses of light guide plate
minimum. First, Taguchi method with orthogonal array L18
was applied to reduce the number of experiments. Then, the
significant factors which have profound effect to the quality characteristics were confirmed using the ANOVA and
main effect analysis. Next, CEBDS and reciprocal comparisons were conducted to optimize the multi-parameter combination. The results were compared and investigated both
C.-F. J. Kuo (B)
Graduate Institute of Automation and Control, National Taiwan
University of Science and Technology, No. 43, Section 4,
Keelung Road, Taipei 106, Taiwan, Republic of China
e-mail: jeffreykuo@mail.ntust.edu.tw
G. Dewantoro
Department of Electronic and Computer Engineering, Satya
Wacana Christian University, 52-60 Diponegoro Street,
Salatiga 50711, Indonesia
e-mail: gunawan.dewantoro@staff.uksw.edu
C.-C. Huang
Department of Mold and Die Engineering, National Kaohsiung
University of Applied Science, 415 Chien Kung Road,
Kaohsiung 807, Taiwan, Republic of China
e-mail: cchuang@cc.kuas.edu.tw
theoretically and experimentally. The reproducibility of the
experiment was verified by confirming a confidence interval of 95 %. It is inferred that the reciprocal comparisons
approach is far superior to the CEBDS approach. The results
also demonstrated improvements of LGP qualities compared
to the best achieved in Taguchi orthogonal array experiment.
Keywords Injection molding · Taguchi method ·
Reciprocal comparisons · Optimization
Introduction
Liquid crystal display (LCD) is regarded as the most prevalent utilization among photo-electronic display devices. As
LCDs are not self-emissive display devices, they adopt backlight unit modules which provide uniform and bright light
source to display. The basic components of a typical backlight
unit module include the light guide plate, reflector, diffuser,
prism, and lamp. As shown in Fig. 1, the aim of the light guide
plate is to control the direction of the scattering of light, guiding the edge light to turn it into the entire area light. Simultaneously, the V-cut microstructures design on its surface is
used to balance out the distribution of light which increases
the luminance of the panel and ensures uniform brightness
(Jui and Ya 2012). Therefore, light guide plate is the key
element of the backlight module whose design and manufacture play significant role in determining the performance of
the backlight module’s optical properties. Injection molding
is one of the best methods for producing a complex-shaped
three dimensional product, and is thus strongly preferred for
the production of light guide plates. The key to the quality
of the injection molding process is in issues such as shrinkage and warpage. Shrinkage leads to inaccuracies in the light
guide plate dimensions, while warpage causes the shape of
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J Intell Manuf
Fig. 1 Backlight unit module
Prism sheet
Lamp
Diffuser
Light Guide Plate
Reflector
Lamp reflector
the light guide plate to become distorted (Kazmer 2007;
Umbrello et al. 2010; Bociaga et al. 2010). However, the specification and setting of processing parameters has always possessed difficulties related to shrinkage and warpage, causing
variations in quality of the injection molded product. Therefore, processing parameters that minimize thermal-induced
residual stress are to be sought, thereby deformed wedgeshaped light guide plate can be avoided. In addition to that,
V-cut microstructures, with the target depth of 2.887 µm and
target angle of 120◦ , are kept to be as close to the targets
as possible. Thus, the V-cut depth, angle, and residual stress
were chosen to be the objectives in this study. Faced with so
many processing parameters, relevant research in the past had
been trying to discover the relationship between the parameters and mold product quality. Choi and Im (1999) analyzed
the shrinkage and warpage of the final injection product using
numerical analysis method, taking the residual stress issues
in the packing pressure and cooling stages into consideration, and proving through the residual stress curve that the
packing pressure stage had the most profound effect on the
results of injection molding. Using Taguchi method, together
with nominally-the-best (type II) quality characteristic, Dragan et al. (2013) also showed that packing pressure was the
most influential process parameters affecting post-molding
shrinkage.
L9 orthogonal array from the Taguchi method can be carried out to investigate the sintering characteristics of Ti-6Al4V/HA tensile bars (Thian et al. 2002). Its final product was
produced using powder injection molding. The processing
parameters that were looked into include sintering temperature, heating rate, packing time, and cooling rate. From
the experimental result, sintering temperature had the most
significant effect on sintering characteristics, which is also
confirmed by Shye et al. (2013). Meanwhile, some research
observed that mold temperature is the most important parameter in injection molding process (Murakami et al. 2008;
Huang et al. 2009; Yun et al. 2013). They showed that high
mold temperature had been proven to significantly improve
the level of filling with melt plastics in microinjection molding and conducive to the molding of microstructures. According to Sanchez et al. (2012), cooling time was found to be
the most significant parameter to reduce warpage, as shown
in cases of the box and plastic glass. Guilong et al. (2013)
123
Microstructure
studied the effects of cavity surface temperature in rapid heat
cycle molding on mechanical strength of the molded specimens with and without a weld line. From the point of view
tensile strength and impact strength, cavity surface temperature was found to be the utmost important parameters.
While the Taguchi method is mainly used for optimizing single quality property (Nik and Shahrul 2011; Nor et
al. 2010; Hasan et al. 2007), optimal processing parameters
that determine the single quality usually do not represent the
optimization of process parameters for the overall quality.
For this reason, Taguchi method can also be combined with
other methods, such as principal component analysis as in
Anthony (2000), Saurav et al. (2009), Humberstone et al.
(2012). Nevertheless, when more than one principal component, how to trade off to select the solution is unknown.
Analytic Hierarchical Process (AHP) with Taguchi method
were combined to obtain the optimal multi-parameter of laser
scribing system for micro crystalline silicon thin film solar
cell isolation and plastic injection molding (Kuo et al. 2010;
Faisal et al. 2013). However, AHP incorporates a priori information about the response weight that increases uncertainty
in the decision-making process. Grey relational analysis was
employed in multi-objective optimization by using Taguchi
design of experiments (Yang et al. 2012; Kuo et al. 2011; Lin
2012), where the relative significance of processing parameters were still required to be known so that the optimal combinations of the processing parameters levels can be sought.
Response Surface Methodology was also used to determine
interactions among the control factors (Cicek et al. 2013;
Muhammad et al. 2012). The predictive quadratic models
were derived by the RSM to obtain the optimal responses as
a function of the parameters. Abbas and Mohammad (2011),
Abbas (2009) studied, respectively, the benevolent formula
and comparisons of efficiency between different systems
(CEBDS) technique in Data Envelopment Analysis (DEA).
The CEBDS technique turns out to be an efficient approach
for optimizing process performance with categorical data.
However, there was no confirmation experiment to prove the
reliability of the results.
This paper proposed reciprocal comparisons approach as
opposed to the CEBDS method (Abbas 2009) for optimizing processing parameters of injection molding machine. By
this way, the qualities of molded product were improved
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Table 1 Eight control factors and their levels for injection molding
machine
Start
Control factors
experimental
planning
proceed with
experiment
Main effect
analysis
ANOVA
Data analysis
CEBDS
optimal processing
parameters determination
Reciprocal
comparisons
Levels
1
2
3
A. Cooling time (s)
15
30
B. Mold temperature (◦ C)
75
80
85
(◦ C)
270
250
260
D. Injection speed (mm/s)
165
180
195
E. Injection pressure (MPa)
220
240
260
F. Packing pressure (MPa)
90
100
110
G. Packing switching (mm)
5
10
15
H. Packing time (s)
1
2
3
C. Melt temperature
verification
no
satisfactory ?
yes
order to comprehend the variance of the molded light guide
plates.
Material and equipments
End
Fig. 2 Flowchart of experimental procedure
and the computational time was reduced. First, Taguchi
method with orthogonal array L18 was applied to reduce
the number of experiments. Then, the significant factors
which have profound effect to the quality characteristics
were confirmed using the ANOVA and main effect analysis.
Next, CEBDS and reciprocal comparisons were conducted to
optimize the multi-parameter combination. The results were
compared and investigated both theoretically and experimentally. That is, the S/N ratios were predicted under the optimal conditions by addition to investigate the total anticipated
improvement and the confirmation experiments were carried
to verify reproducibility and feasibility through the proposed
approach.
The mold tooling was provided by Plastic Precision Molding Laboratory, Department of Mold and Die Engineering,
National Kaohsiung University of Applied Science, as shown
in Fig. 3. The resin material used for injection molding
was optical grade polymethyl methaacrylate (PMMA) with
refractive index of 1.49, produced by Japan’s KURARAY
Co., Ltd. The injection molding machine was manufactured
by Sodick Plustech Co., Ltd. (Model Sodick TR30EH) as
shown in Fig. 4a. The light guide plate produced by the injection molding process is shown in Fig. 4b. The surface profile
measuring system used for measuring the light guide plate
V-cut measurements is the Form Talysurf PGI 635, made by
Taylor Hobson Co., Ltd, as shown in Fig. 5a, whilst the residual stress measuring instrument is VML 250P, made by 3D
Family Tech. Co., Ltd. as shown in Fig. 5b.
Reciprocal comparisons in data envelopment analysis
Experimentation
The whole experimental procedure of this study is described
in Fig. 2:
Experimentation plan
The final selection of control factors and their levels is shown
in Table 1. Eighteen sets of experiments were then conducted
as planned by the orthogonal array, with each set of experiments repeated four times which means there would be a total
of 72 data for each quality characteristic. Four dashed lines
were followed by using surface-profile measuring system to
measure the depth and angle of molded light guide plates in
Data Envelopment Analysis (DEA) is linear programming
for measuring the productive efficiency of Decision-Making
Units (DMUs) without requiring prescribed weights attached
to the multiple inputs and multiple outputs, as described in
Cooper et al. (2006). This proposed reciprocal comparisons
method seeks the opportunity to improve CEBDS in DEA
which was introduced by Abbas (2009). Here, each DMU
is compared in reciprocal manner. That is, for each pair of
group, each DMU of one group is evaluated with respect to
the DMUs of the opposite group and vice versa. Therefore,
for a total of K groups, there will be K!/(2!x(K−2)!) reciprocal comparisons. This inter-comparison, as opposed to inter
and within-comparison in CEBDS, will result in sharper dis-
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Fig. 3 Moving mold-half (left) and fixed mold-half (right)
Fig. 4 a Sodick Plustech injection molding machine. b Light guide plate
Fig. 5 Measuring systems. a Taylor Hobson surface profilemeter. b 3D Family Tech stress measuring instrument
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crimination between two groups. For example, let DMUa
be a random DMU of group 1 to be evaluated, and E a,2
denotes the efficiency of DMUa with respect to the DMUs
of group 2. Then E a,2 is obtained by solving the following
model:
E a,2 = min θ
subject to
θ xa ≥
xjλj
yjλj
j∈gr oup2
ya ≤
j∈gr oup2
λ j ≥ 0, ∀ j ∈ gr oup2
(1)
where xa and ya are the input and output vectors for
DMUa , x j and y j are the input and output vectors for DMUs
of group 2. The objective of the linear programming is to
adjust the value of λ j such that θ is minimum. The efficiency
of each DMU of group 2 with respect to DMUs in group 1
can be easily obtained in a similar manner.
Results and discussion
Taguchi experimental method (Ross 1996)
The experimental data of the V-cut depth, V-cut angle, and the
residual stress are shown in Table 2. On the basis of the S/N
ratio available in Table 2, main effect analysis was adopted in
this study to figure out the main effect of each factor level and
the response graphs could be prepared, as shown in Fig. 6.
From the response graphs, the optimal factor combination
for single quality characteristic could be obtained.
In order to identify the control factors that have significant effects, analysis of variance (ANOVA) for S/N ratio
is performed at a certain confidence level. The results of
ANOVA for V-cut depth, angle, and residual stress are shown
in Table 3. From ANOVA of V-cut depth as shown in Table 3a,
control factors B, G, and H all possess F-ratio greater than
that obtained from table of F value with 95 % confidence
level. Therefore, all of these factors are significant factors
whose effects are strongly believed to be profound. From
ANOVA of angle as shown in Table 3b, control factors G,
and H all possess F-ratio greater than that obtained from
table of F value with 95 % confidence level. Therefore, all of
these factors are significant factors whose effects are strongly
believed to be profound. From ANOVA of residual stress as
shown in Table 3c, control factors B, G, and H all possess
Table 2 The orthogonal array with the averages and SN ratios of all qualities
Trial
A
B
C
D
E
F
G
H
Mean
of depth
(µm)
Mean
of angle
(degree)
Mean of
residual
stress
(MPa)
SNR of
depth
(dB)
SNR of
angle
(dB)
SNR of
residual
stress
(dB)
1
15
75
250
165
220
90
5
1
1.7513
140.690
0.311
−1.109
−26.334
10.149
2
15
75
260
180
240
100
10
2
2.1986
131.689
0.286
3.088
−21.795
10.887
3
15
75
270
195
260
110
15
3
2.0247
130.648
0.280
0.992
−20.723
11.054
4
15
80
250
165
240
100
15
3
2.1476
131.240
0.279
2.308
−21.234
11.094
5
15
80
260
180
260
110
5
1
1.9650
140.065
0.243
0.639
−26.128
12.273
6
15
80
270
195
220
90
10
2
2.0228
135.543
0.259
1.167
−23.835
11.732
7
15
85
250
180
220
110
10
3
2.5274
128.418
0.215
8.310
−18.658
12.832
8
15
85
260
195
240
90
15
1
1.3743
165.750
0.286
−3.620
−33.308
10.852
9
15
85
270
165
260
100
5
2
2.4465
132.615
0.263
6.727
−22.096
11.589
10
30
75
250
195
260
100
10
1
1.2800
150.708
0.289
−4.237
−29.825
10.787
11
30
75
260
165
220
110
15
2
1.5188
144.197
0.309
−2.838
−27.696
10.211
12
30
75
270
180
240
90
5
3
2.3235
131.727
0.257
4.634
−21.473
11.811
13
30
80
250
180
260
90
15
2
1.7001
138.826
0.315
−1.546
−25.548
10.041
14
30
80
260
195
220
100
5
3
2.5166
128.047
0.250
8.406
−18.274
12.039
15
30
80
270
165
240
110
10
1
1.8936
136.949
0.281
−0.167
−24.652
11.025
16
30
85
250
195
240
110
5
2
2.7172
127.588
0.195
15.299
−17.798
14.204
17
30
85
260
165
260
90
10
3
2.4924
127.954
0.231
7.310
−18.655
12.729
18
30
85
270
180
220
100
15
1
1.0134
165.857
0.311
−5.492
−33.255
10.146
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Fig. 6 The response graph for a V-cut depth b V-cut angle c residual stress
Table 3 ANOVA results for (a)
V-cut depth (b) V-cut angle (c)
residual stress
Factor
(a)
A*
SS
0.456199
d.f
1
V
F-ratio
66.89341
2
C*
10.40742
2
5.203709
2
3.063327
6.126654
33.44670
5.537093++
55.812445
11.04699
E*
17.19271
2
F*
21.31247
2
10.65624
2
84.19398
13.93829++
156.30700
31.50237
97.22433
16.09546++
182.36769
36.75469
G
H
Error
Pooled
Total
168.3880
194.4487
2
8.596357
10.94982
2
5.474911
66.44528
11
6.040480
496.1753
17
102.68817
496.17531
(b)
123
%P
0.456199
B
D*
SS′
A*
0.521994
1
0.521994
B*
5.570479
2
2.785239
C*
4.765560
2
2.382780
D*
3.192966
2
1.596483
20.69594
100
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Table 3 continued
Factor
E*
SS
V
F-ratio
2
2.607714
F*
17.02818
2
8.514089
G
83.37654
2
H
5.215428
d.f
253.6219
2
Error
21.01560
2
Pooled
57.31020
13
Total
394.3086
9.456388++
41.68827
28.76525++
126.8109
4.408477
SS′
74.559587
244.80493
74.944113
17
394.30863
%P
18.90894
62.08460
19.00646
100
(c)
∗ Pool-up
terms
++ Significant factor at 95 % CI
+ Significant factor at 90 % CI
A*
0.01555
1
B
4.64842
2
C*
0.31918
2
D*
1.31109
2
E*
0.63673
2
F
2.47414
G
2.32421
5.87759++
2
1.23707
3.12837+
1.68327
6.82858
2
3.41429
8.63424++
6.03771
28.95173
H
3.34434
2
1.67217
4.22867++
2.55346
12.24425
Error
1.27636
2
Pooled
Total
3.55892
20.8544
9
0.39544
6.72241
17
F-ratio greater than that obtained from table of F value with
95 % confidence level, whereas control factor F possesses Fratio greater than that obtained from table of F value with
90 % confidence level. Therefore, all of these factors are significant factors whose effects are strongly believed to be profound. By combining ANOVA and main effect analysis, one
can achieve the optimal parameters combination for the Vcut depth that is B3 , G1 , and H3 ; where mold temperature
is 85 ◦ C, packing switching is 5 mm, and the packing time
is 3 s. Meanwhile, the optimal parameters combination for
the V-cut angle is G1 and H3 . Lastly, the optimal parameters
combination for the residual stress is B3 , F3 , G1 , and H3 .
Optimization in Taguchi orthogonal array
3.85755
20.85440
18.49752
8.071516
32.23499
100
decreased. Since all of S/N ratios are larger-the-better
type, in consequence, set all of the responses as outputs,
which are preferred at maximum, and set a unit (one) as
the input.
Step 2. Normalize the S/N ratio, X i j , for the jth response
at the ith trial. X i j is now normalized as Z i j , which lies
between 0 and 1, by the following formula, to avoid the
effect of adopting different units. The result is shown in
Table 4. It is obvious that trial 16 is the most superior
among all trials in Taguchi orthogonal array experimentation set.
Zi j =
X i j − min X i j , i = 1, . . . , m
max X i j , i = 1, . . . , m − min X i j , i = 1, . . . , m
for i = 1, . . . , m,
j = 1, . . . , n
(2)
All trials in Taguchi’s orthogonal array turn out to be
DMU with the multi-objective as the inputs and outputs
for all DMUs. In this research, there are 18 experiments in
Taguchi’s orthogonal array of 8 factors with 3 responses.
Here, both CEBDS and reciprocal comparisons are investigated to demonstrate the improvement of the latter method.
The procedure is outlined in following steps:
Step 3. For each factor in Table 2, the 18 DMUs are then
separated into groups, each at the same factor level, as
shown in Table 5. In this case, factor A is assigned at two
levels; A1 and A2. As a result, the 18 DMUs for factor
A are divided into two groups each of 9 DMUs.
Step 1. Trial j ( j = 1, . . ., 18) shown in Table 2
are considered as DMU j. The relative efficiency of
DMU j, which is calculated as the ratio of the sum of
the weighted outputs relative to the sum of weighted
inputs, is improved when the sum of weighted outputs
is increased and/or the sum of the weighted inputs is
Step 4. For each of the factors A to H, the efficiencies
of the 18 DMUs are separately measured by solving linear programming (1). Appendix summarizes the obtained
DMU’s efficiencies for all factors. Let y j1 , y j2 , and y j3
be the outputs, and all inputs are set to be one. For factor B in Appendix, DMU1 is evaluated as follows: The
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Table 4 Normalized S/N ratio
(dB)
Trial no.
Depth
Angle
Stress
1
0.2108
0.4496
0.3353
2
0.4127
0.7423
3
0.3119
0.8114
4
0.3752
5
Trial no.
Depth
Angle
Stress
10
0.0604
0.2245
0.2876
0.0223
11
0.1277
0.3618
0.1177
0.0717
12
0.4870
0.7631
0.0000
0.7784
0.0834
13
0.1898
0.5003
0.0675
0.2949
0.4629
0.7257
14
0.6685
0.9693
0.0673
6
0.3203
0.6108
0.2715
15
0.2561
0.5581
0.0631
7
0.6639
0.9445
0.3009
16
1.0000
1.0000
1.0000
8
0.0900
0.0000
0.0121
17
0.6158
0.9447
0.8600
9
0.5877
0.7229
0.2293
18
0.0000
0.0034
0.1574
Table 5 The index of DMU that belong to groups of each factor
Group (level)
Factor
A
B
C
D
E
F
G
H
1
1–9
2
10–18
3
–
1, 2, 3, 10,
11, 12
4, 5, 6, 13,
14, 15
7, 8, 9, 16,
17, 18
1, 4, 7,10,
13, 16
2, 5, 8, 11,
14, 17
3, 6, 9, 12,
15, 18
1, 4, 9,11,
15, 17
2, 5, 7, 12,
13, 18
3, 6, 8, 10,
14, 16
1, 6, 7, 11,
14, 18
2, 4, 8, 12,
15, 16
3, 5, 9, 10,
13, 17
1, 6, 8,12,
13, 17
2, 4, 9, 10,
14, 18
3, 5, 7, 11,
15, 16
1, 5, 9,12,
14, 16
2, 6, 7, 10,
15, 17
3, 4, 8, 11,
13, 18
1, 5, 8, 10,
15, 18
2, 6, 9, 11,
13, 16
3, 4, 7, 12,
14, 17
efficiency, E 1,1 of DMU1 with respect to the 6 DMUS of
group 1 is obtained by solving the following model:
subject to
θ≥
6
λj +
j=4
E 1,1 = min θ
subject to
θ≥
λj
j=13
0.2108 ≤
3
15
6
y j1 λ j +
6
y j2 λ j +
6
y j3 λ j +
j=4
λj +
12
0.4496 ≤
λj
y j1 λ j
j=1
12
j=10
0.4496 ≤
3
y j2 λ j +
12
y j2 λ j
0.3353 ≤
3
y j3 λ j +
12
y j3 λ j
0.2108 ≤
3
y j1 λ j +
15
y j2 λ j
15
y j3 λ j
j=13
j=13
j=4
λ4 , λ5 , λ6 , λ13 , λ14 , λ15 ≥ 0
j=10
λ1 , λ2 , λ3 , λ10 , λ11 , λ12 ≥ 0
(3)
E 1,3 = min θ
subject to
The E 1,1 is measured as 1.000.
Similarly, the efficiency of DMU1 with respect to the six
DMUs of group 2, E 1,2 , is obtained by solving the following
model:
θ≥
9
λj +
123
18
λj
j=16
j=7
0.2108 ≤
E 1,2 = min θ
(4)
The E 1,2 is evaluated as 0.693.
Finally, the efficiency of DMU1 with respect to the six
DMUs of group 3, E 1,3 , is obtained by setting z 3 = 1 and
z 1 = z 2 = 0 and solving the following model:
j=10
j=1
j=1
0.3353 ≤
y j1 λ j
j=13
j=4
j=10
j=1
15
9
j=7
y j1 λ j +
18
j=16
y j1 λ j
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0.4496 ≤
y j2 λ j
j=7
18
j=16
9
18
y j3 λ j
9
y j2 λ j +
B2 . In a similar manner, the optimal levels for the other
factors are decided at A1 , C1 , D2 , E3 , F3 , G1 , H3 .
The E 1,3 is evaluated as 0.45.
The optimal efficiency of DMU1, E 1 , is then decided as
0.45; which is the minimum of {1.000, 0.693, 0.45}. The
E 2 to E 18 for DMU2 to DMU18 are evaluated in a similar manner. The E j values for the 18 DMUs using the
CEBDS approach for the rest of the factors are obtained
similarly.
Step 6. Reciprocal comparisons solve the problem in the
same way as that of CEBDS. However, unlike CEBDS,
each DMU is only evaluated based on intergroup comparison rather than both inter- and within- group. Therefore,
comparison between group 2 and group 2 (within-group
comparison) is not required. For example, one does not
need to solve E 4,2 , E 5,2 , E 6,2 , E 13,2 , E 14,2, E 15,2 for factor B because DMU4 , DMU5 , DMU6 , DMU13 , DMU14 ,
and DMU15 belong to group 2 in factor B. This step is
repeated until the reciprocal comparison efficiencies of
18 DMUs for all factors are obtained.
Step 5. The level efficiency is calculated as the average
of the E j values measured using CEBDS approach for
the DMUs at that factor level. Table 6 displays the calculated efficiencies of all factor levels. For illustration,
the efficiency of B1 , 0.56929, is calculated as the sum
of the efficiencies for the six DMUs at level 1 divided
by six. Similarly, the efficiency for B2 and B3 are calculated as 0.690419 and 0.64327, respectively. Obviously,
the largest level efficiency for factor B corresponds to
B2 , and hence, the optimal level of factor B is selected at
Step 7. The efficiency is calculated as the average of
the E j,k values evaluated by the reciprocal comparisons
approach at that factor level, as shown in Table 7. For
illustration, the efficiency of B1 , 1.485, is calculated as
the average of E j,1 values. Similarly, the efficiency for
B2 and B3 are calculated as 0.86 and 0.63, respectively.
The minimum of {1.485, 0.86, and 0.63} corresponds to
B3 , and hence, the optimal level of factor B is decided
at B3 . In a similar way, the optimal levels for factors A
to H are decided at A2 , B3 , C1 , D3 , E2 , F3 , G1 , and H3 ,
respectively.
0.3353 ≤
y j3 λ j +
j=16
j=7
λ7 , λ8 , λ9 , λ16 , λ17 , λ18 ≥ 0
(5)
Table 6 The level of efficiency using the CEBDS approach
Factor
Confirmation experiments
Level
1
2
3
Optimal level
A
0.652839
0.615814
B
0.569290
0.690419
0.643270
A1
B2
C
0.660084
0.638963
0.603932
C1
D
0.635920
0.638877
0.628182
D2
E
0.582243
0.655311
0.665426
E3
F
0.559751
0.609653
0.733575
F3
G
0.771748
0.681333
0.449898
G1
H
0.378082
0.656330
0.868567
H3
After the optimum light guide plate processing parameters was obtained through both approaches, the confirmation
experiments were conducted.
Verification of CEBDS approach
Take the V-cut depth for instance, the S/N ratio of optimal
processing parameters is:
Sˆ N = T + (B2 − T ) + (G 1 − T ) + (H3 − T )
= B2 + G 1 + H3 − 2T
Table 7 The level of efficiency using the reciprocal comparisons
DMU j
Factor
A
B
C
D
E j,1
E j,2
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
Average
0.833
0.65
1.485
0.86
0.63
0.62
0.721
1.471
0.744
0.834
0.64
Optimal level
A2
B3
C1
D3
DMU j
E
F
G
H
Average
1.165
0.623
0.61
Optimal level
E2
0.62
0.719
0.795
F3
1.384
0.585
0.57
G1
0.711
2.93
1.729
H3
123
J Intell Manuf
= 1.801199 + 5.765991 + 5.326707 − 2 × 2.215056
= 8.463784 dB
(6)
where T is the total average value of S/N ratio; B2 , G1 and
H3 are the average of the S/N ratio for those factor level.
Confidence Interval (CI) of theoretically-predicted value is
calculated:
1
C I1 = Fα;1,v2 × Verr or ×
ne f f
7
= 3.371962 d B
(7)
= 4.84 × 6.04048 ×
18
The 95 % confidence interval of S/N ratio for V-cut depth,
V-cut angle, and residual stress are 5.091822 < µconfirmation
< 11.83575; −20.2842 < µconfirmation < −15.5014; and
11.93522 < µconfirmation < 14.05634, respectively. The
surface profile of V-cut microstructures and the residual
stress under the optimal combination is shown in Fig. 7. The
S/N ratio (dB) derived from four verification data for the
V-cut depth, V-cut angle, and residual stress are 10.91939,
−14.4397, and 14.9668, respectively. Compared to trial no.
16 in L18 orthogonal array, which shows the best S/N ratio
among all trials, the result of CEBDS is able to improve the
quality of V-cut angle and residual stress. However, the V-cut
depth quality was deteriorated with the decrease of −4.38 dB.
The 95 % Confidence Interval of experimental V-cut depth
value is calculated as:
C I2 =
Fα;1,v2 × Verr or ×
=
4.84x6.04048 ×
1
ne f f
1
7
+
18 4
+
1
r
= 4.321978 d B (8)
The 95 % confidence interval of S/N ratio for V-cut depth,
V-cut angle, and residual stress are 6.597413 < µconfirmation
< 15.24137; −17.7361 < µconfirmation < −11.1434; and
13.6897 < µconfirmation < 16.2438, respectively. The diagram for the CI of the V-cut depth, angle, and residual stress
verification experiment values and theoretical predicted values are shown in Fig. 8. From the diagrams, the CI from the
verification experiment and the theoretical predictions did
indeed coincide; therefore the results from the experiment
using CEBDS approach are indeed reliable had been proven.
Fig. 7 a V-cut microstructure and b residual stress under optimal combination with CEBDS approach
123
J Intell Manuf
Verification of reciprocal comparisons approach
Using Eq. 6, the S/N ratio of optimal processing parameters
for V-cut depth is 11.41823 dB. Confidence Interval (CI) of
theoretically-predicted value is calculated using Eq. 7, yieldFig. 8 Diagram for the CI of
the a V-cut depth b V-cut angle
and c residual stress with
CEBDS approach
ing 3.60468 dB. The 95 % confidence interval of S/N ratio for
V-cut depth, V-cut angle, and residual stress are 7.81355 <
µconfirmation < 15.02291; −20.2842 < µconfirmation <
−15.5014; and12.57461 < µconfirmation < 14.79927,
respectively. The surface profile of V-cut microstructures and
(a)
(b)
confirmation experiment value
confirmation experiment value
theoretical predicted value
theoretical predicted value
(c)
confirmation experiment value
theoretical predicted value
Fig. 9 a V-cut microstructure and b residual stress under optimal combination with reciprocal comparisons
123
J Intell Manuf
F i g . 1 0 Diagram for the CI of
the a V-cut depth b V-cut angle
and c residual stress, with
reciprocal comparisons
approach
(a)
(b)
(c)
residual stress under the optimal combination is shown in
Fig. 9. The S/N ratio (dB) derived from four verification
data for the V-cut depth, V-cut angle, and residual stress are
14.6251, −13.174, and 15.5299, respectively. Compared to
trial no. 16 in L18 orthogonal array, which shows the best
S/N ratio among all trials, the result of reciprocal comparisons is able to improve the quality of V-cut angle and residual
stress. The V-cut quality is a bit worse with the decrease of
−0.674 dB. However, the total improvement for all qualities
is +5.276 dB with more balanced manner without sacrificing one of the qualities. Obviously, the total improvements
of all qualities in reciprocal comparisons obviously outperform that in CEBDS method. The 95 % Confidence Interval of experimental V-cut depth value is calculated using
Eq. 8, yielding 4.505978 dB. The 95 % confidence interval
of S/N ratio for V-cut depth, V-cut angle, and residual stress
are 10.11911 < µconfirmation < 19.13107; −16.4701 <
µconfirmation < −9.87743; and 14.20946 < µconfirmation <
16.85025, respectively. The diagram for the CI of the V-cut
depth, angle, and residual stress verification experiment values and theoretical predicted values are shown in Fig. 10.
From the diagrams, the CI from the verification experiment
and the theoretical predictions did indeed coincide; therefore the results from the experiment using reciprocal comparisons approach are indeed reliable had been proven. It
can easily be observed that using the reciprocal comparisons
approach to measure DMU’s efficiency in the proposed procedure results in better outcome than the CEBDS approach
(Abbas 2009). Thus, reciprocal comparisons are promising
to be used in optimization problems for its simplicity and
powerfulness.
123
Conclusion
The reciprocal comparisons approaches in DEA were proposed to find out the optimal processing parameters of injection molding machine. The optimal processing parameters in
reciprocal comparisons approach are A2 , B3 , C1 , D3 , E2 , F3 ,
G1 , H3 , that is: the cooling time of 30 s, mold temperature of 85 ◦ C, melt temperature of 250 ◦ C, injection speed
of 195 mm/s, injection pressure of 240 MPa, packing pressure of 110 MPa, packing switching of 5 mm, and the packing time of 3 s. The reciprocal comparisons methods were
able to search optimal parameters combinations beyond the
orthogonal array experimentation set. Confirmation experiments showed that the optimal parameters combination
demonstrated qualities improvements in a more balanced
manner. Compared with CEBDS approach, the reciprocal
comparisons approach shows a better effectiveness because
not only provides larger anticipated improvements in the
responses, but also requires less computations since withincomparisons is not required. The reliability and reproducibility of the results were verified using 95 % confidence
interval.
Acknowledgments This research was financially supported by the
National Science Council of the Republic of China under Grant No.
NSC 97-2622-E-011-001-CC3.
Appendix
See Table 8.
J Intell Manuf
Table 8 The efficiencies for all DMUs
DMU j
Factor
A
B
C
D
E j,1
E j,2
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
DMU1
0.645
0.45
1
0.693
0.45
0.45
0.474
1.235
0.476
0.645
0.45
DMU2
0.786
0.742
0.956
0.766
0.742
0.742
0.766
0.954
0.786
0.786
0.742
DMU3
0.859
0.811
1
0.849
0.811
0.811
0.838
1
0.859
0.859
0.811
DMU4
0.824
0.778
1.04
0.825
0.778
0.778
0.804
0.985
0.824
0.824
0.778
DMU5
1
0.726
2.164
1
0.726
0.726
0.844
2.673
0.844
1
0.726
DMU6
0.715
0.611
1.127
0.803
0.611
0.611
0.637
1
0.647
0.715
0.611
DMU7
1
0.945
1.872
1.181
0.945
0.945
1.016
1.309
1.078
1
0.945
DMU8
0.136
0.09
0.205
0.137
0.09
0.09
0.135
0.153
0.146
0.136
0.09
DMU9
0.885
0.723
1.595
1.016
0.723
0.723
0.896
1
0.955
0.885
0.723
DMU10
0.428
0.288
0.858
0.437
0.288
0.288
0.334
1.059
0.334
0.428
0.288
DMU11
0.385
0.362
0.575
0.443
0.362
0.362
0.376
0.506
0.383
0.385
0.362
DMU12
0.808
0.763
1
0.787
0.763
0.763
0.787
1
0.808
0.808
0.763
DMU13
0.53
0.5
0.652
0.541
0.5
0.5
0.517
0.632
0.53
0.53
0.5
DMU14
1.026
0.969
1.487
1
0.969
0.969
1
1.291
1.086
1.026
0.969
DMU15
0.591
0.558
0.74
0.594
0.558
0.558
0.577
0.702
0.591
0.591
0.558
DMU16
2.02
1
3.745
2.218
1
1
1.585
3.683
1.624
2.02
1
DMU17
1.493
0.945
2.719
1.573
0.945
0.945
1
3.168
1
1.493
0.945
DMU18
0.217
0.158
0.47
0.217
0.158
0.158
0.183
0.58
0.183
0.217
0.158
DMU j
Factor
E
F
G
H
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
E j,1
E j,2
E j,3
DMU1
1
0.45
0.476
0.476
1.236
0.45
0.45
0.476
2.398
0.878
0.45
0.474
DMU2
0.766
0.742
0.786
0.786
0.766
0.742
0.742
0.786
1.1
1.479
0.742
0.766
DMU3
0.839
0.811
0.859
0.859
0.859
0.811
0.811
0.859
1
1.454
0.811
0.838
DMU4
0.806
0.778
0.824
0.824
0.845
0.778
0.778
0.824
1
1.432
0.778
0.804
DMU5
2.164
0.726
0.844
0.844
2.577
0.726
0.726
0.844
4.979
1
0.726
0.844
DMU6
0.857
0.611
0.647
0.647
1.093
0.611
0.611
0.647
2.126
1.177
0.611
0.637
DMU7
1
0.945
1.078
1.078
1.311
0.945
0.945
1
2.744
2.251
0.945
1
DMU8
0.135
0.09
0.146
0.146
0.137
0.09
0.09
0.136
0.24
0.305
0.09
0.135
DMU9
0.884
0.723
0.955
0.955
1
0.723
0.723
0.885
2.193
1.993
0.723
0.884
DMU10
0.858
0.288
0.334
0.334
1
0.288
0.288
0.334
1.959
0.469
0.288
0.334
DMU11
0.39
0.362
0.383
0.383
0.51
0.362
0.362
0.383
0.965
0.668
0.362
0.376
DMU12
0.787
0.763
0.808
0.808
0.787
0.763
0.763
0.808
1.298
1.652
0.763
0.787
DMU13
0.52
0.5
0.53
0.53
0.563
0.5
0.5
0.53
0.731
0.899
0.5
0.517
DMU14
1
0.969
1.086
1.086
1
0.969
0.969
1.026
1.782
2.267
0.969
1
DMU15
0.578
0.558
0.591
0.591
0.61
0.558
0.558
0.591
0.738
1
0.558
0.577
DMU16
3.063
1
1.624
1.624
3.774
1
1
1.575
7.605
3.391
1
1.575
DMU17
2.581
0.945
1
1
3.154
0.945
0.945
1
6.235
2.088
0.945
1
DMU18
0.47
0.158
0.183
0.183
0.547
0.158
0.158
0.183
1
0.217
0.158
0.183
123
J Intell Manuf
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