Design of Experiments
Experimental Design
By Bill Lucas
Executive Summary
Due to the
competitive market today in the semiconductor industry, ABC Co. wants to investigate
the factors, which affect the average cycle time and throughput. The objective is to minimize the average
cycle time and maximize throughput. Since the model has eight factors and two
levels each, we want to identify the factors that have large effect. By doing
this, 28-4 fractional factorial design is demonstrated and single
replication with 6 runs at the center is also used in this experiment. We have
emphasized the use of these designs in screening experiments to quickly and
efficiently identify the subset of factors that are active and to provide some
information on interaction. Half-Normal
plot is used in the ANOVA, residual analysis and model adequacy checking,
regression analysis and contour plots to help the engineer to have the better interpretation
of the experiment as well to examine the active factors in more details.
The results from
the experiment suggest that only two out of eight factors were significant, which
are release rate and dispatching rule. The model passed the tests for normality
and independence assumptions. In additions, the validity of the model was
performed based on the regression models to verify the two responses, average
cycle time and throughput. The model was verified using the confirmation run
and the error was less than one percent.
The predicted values were very close to the actual values and thus
supporting the design.
Based on the
results, we recommend that SSU dispatching rule should be used at release rate
of 19.5K wafers per month is the best combination to yield a higher throughput
and lower average cycle time.
TABLE OF CONTENTS
1. Experimental
Design in Simulation of Semiconductor Mfg. 4
1.1 Problem Statement
1.2 Description of the Model
2. Choice of
Factors Levels and Range 6
3. Selection
Response Variable 8
4. Choice of
Experimental Design 9
5. Performing the
Experiment 10
6. Statistical
Analysis of the data 11
6.1 Analysis of Variances (ANOVA)
6.2 Model Adequacy Checking
6.2.1
Normality Assumption
6.2.2
Residual Analysis
6.2.3
Box-Cox Transformation
6.3 Regression Analysis
6.3.1
Average
Cycle Time
6.3.2
Throughput
6.4 Interaction Graph of Factors A and G
6.5 Optimal
Designs
7. Conclusions 24
7.1 Confirmation Testing
7.2 Recommendations
APPENDIX 27
1. Experimental
Design in Simulation of Semiconductor Manufacturing
1.1 Problem Statement
ABC Co. is a leading semiconductor manufacturing company.
Lately they have discovered that modeling the semiconductor manufacturing and
simulating it for various conditions would save lot of time and resources. The manager of the ABC Co. wants to
investigate the factors, which affect the average cycle time and
throughput. The objective is to minimize
the average cycle time and maximize throughput.
The lesser the cycle time, the lesser the work-in-process, which means
lesser investment in inventory. The
shorter cycle time also provides market responsiveness. With this goal in mind
he wants to plan an experiment or sequence of experiments designed to take him
in the direction of that goal.
1.2 Description of the
Model
The model
represents a 300mm DRAM facility with approximately 450 process steps and 398 process tools providing 1709 total
tool ports, WIP positions, handlers, etc. that are grouped into 80 tool
groups. There are 15 operators in 8
different types and the maximum designed capacity was 20,000 wafers/month. Only one type of DRAM part, which processes
through one routing, is released into the system. The flow is a highly re-entrant, i.e. jobs
feedback through sequences of the tool-groups many times. A lot of 25 parts is
released at a fixed interval depending upon the maximum designed capacity. Twenty-one types of reticles, generic
resources, with a capacity of two each, are used. Process tool downtimes for both preventative
and unexpected maintenance are incorporated, along with employee lunches and
breaks. AutoSched AP, a commercial
simulation software package was used to model this system.
This model
simulates the manual material handling system and the various assumptions for
this system are listed below:
·
There is no operator’s traveling time to the
front of stocker when an inter-bay movement was requested.
·
Gaining access to stockers in a bay is
considered as resource contingent.
·
Load and unload times are 1 minute each.
·
The average operator’s traveling speed is
assumed to be 2 miles/hr, which is a reasonably slow walking speed, considering
the weight of the AGV (Automate Guide Vehicle).
·
To compensate for safety precautions and other
human factors in the Fab, travel times used are equal to [distance/speed]*a,
where a
is equal to 1.5.
2. Choice of
Factors Levels and Range
From the
previous experiment and experience, the Potential Design Factors and the
Nuisance factors can be identified. The potential design factors are number of
operators, release rate, dispatching rule, stocker quantity, and number of
reticles, of which number of reticle is held-constant factor and the design
factors are:
1)
For operators, there are 5 factors and two levels each.
The operator in this model is responsible for loading and unloading the wafers
on the machines and they are also responsible for transportation of wafers
within the Fab. Varying the number of operators would possibly affect the
performance of the system.
2)
Release Rate, i.e. the rate at which the wafers are
released into the factory, has two levels. The release rate is measured by the
number of wafers scheduled to release into the Fab per month. The release rate
affects the machine utilization, specially the batching machine that in turn
affects the system performance.
3)
The dispatching rule for the bottleneck workstations
has two levels. The bottleneck machines were identified from the previous
experiments. According to the theory of constraints, the bottleneck machine
determines the capacity of the Fab that determines the throughput.
4)
Stocker Quantity, which has two levels. In this model
the stockers are treated as stations and there is one stocker at each bay. The
shortage of stockers can cause blocking which can severely delay the
manufacturing processes.
The details of
the factors, level and range are given in the table below
Table 1 Design Factors and their Levels
Factor
|
Levels
|
Range
|
|
Operator
|
|||
OP_DIFF
|
2
|
2
|
4
|
OP_PHOTO
|
2
|
2
|
4
|
OP_ETCH
|
2
|
3
|
5
|
OP_WET
|
2
|
2
|
4
|
OP_MOVE
|
2
|
35
|
55
|
Release Rate
|
2
|
18K
|
19K
|
Dispatching Rules
|
2
|
FIFO
|
Same
Setup
|
Stockers Qty.
|
2
|
2
|
4
|
The Nuisance
Factors are the various distributions for the processing time and the down time,
which are uncontrollable or the noise factors. The controllable nuisance factor
that can be identified is the random number stream that is to be used for the
simulation. We intend to keep the random number stream constant throughout the
experiment.
3. Selection
Response Variable
The purpose of
the study is to determine the time taken for a lot of wafers to be produced.
This factor is best represented by the average cycle time and thus the average
cycle time happens to be one of our response variables.
Various other
parameters are necessary to determine the proper running of the factory, one of
which is the throughput. Thus the two-response variables for our project are:
1)
Average cycle time and
2)
Throughput
Cycle time is
defined as total elapsed time from lot creation to lot completion that include
process time, move time, queue time, and hold time.
The average
output of a production process (machine, workstation, line, and plant) per unit
time is defined as the system’s throughput.
These response
variables can be obtained from the simulation output report. The simulation
would run for a period of time at steady state. The steady state would be
determined by a long initial run and the statistics collected during this
warm-up period would be eliminated from the simulation output.
4. Choice of
Experimental Design
Since the model
has eight factors and two levels each, we need to identify the factors that
have large effects. To do so, screening
experiments will be used at the initial stage of the experiment. We will use the 2k Fractional
Factorial Design for this screening experiment.
We choose 28-4 Fractional Factorial Design, single
replication with 6 runs at the center point as shown in Table 2. We determine
the number of runs from the results of the Design Expert software.
The alias for
this design is a bit different for factor A. The possible reason for this could
be the use of center points and also that the factor A is a categorical factor.
The defining relation and the aliases is shown in Appendix 1.
Table 2:
Design Matrix
Std
|
Run
|
Block
|
Factor1: DISPATCHING RULE
|
Factor2: OP_DIFF
|
Factor3: OP_PHOTO
|
Factor4: OP_ETCH
|
Factor5: OP_WET
|
Factor6: OP_MOVE
|
Factor7: RELEASE RATE
|
Factor8: STOCKERS QTY.
|
R1: AVG. CYCLE TIME
|
R2: THROUGHPUT
|
Hours
|
Lots
|
|||||||||||
1
|
21
|
Block 1
|
{-1}
|
-1
|
-1
|
-1
|
-1
|
-1
|
-1
|
-1
|
||
2
|
14
|
Block 1
|
{1}
|
-1
|
-1
|
-1
|
-1
|
1
|
1
|
1
|
||
3
|
17
|
Block 1
|
{-1}
|
1
|
-1
|
-1
|
1
|
-1
|
1
|
1
|
||
4
|
22
|
Block 1
|
{1}
|
1
|
-1
|
-1
|
1
|
1
|
-1
|
-1
|
||
5
|
3
|
Block 1
|
{-1}
|
-1
|
1
|
-1
|
1
|
1
|
1
|
-1
|
||
6
|
12
|
Block 1
|
{1}
|
-1
|
1
|
-1
|
1
|
-1
|
-1
|
1
|
||
7
|
9
|
Block 1
|
{-1}
|
1
|
1
|
-1
|
-1
|
1
|
-1
|
1
|
||
8
|
20
|
Block 1
|
{1}
|
1
|
1
|
-1
|
-1
|
-1
|
1
|
-1
|
||
9
|
10
|
Block 1
|
{-1}
|
-1
|
-1
|
1
|
1
|
1
|
-1
|
1
|
||
10
|
2
|
Block 1
|
{1}
|
-1
|
-1
|
1
|
1
|
-1
|
1
|
-1
|
||
11
|
8
|
Block 1
|
{-1}
|
1
|
-1
|
1
|
-1
|
1
|
1
|
-1
|
||
12
|
13
|
Block 1
|
{1}
|
1
|
-1
|
1
|
-1
|
-1
|
-1
|
1
|
||
13
|
11
|
Block 1
|
{-1}
|
-1
|
1
|
1
|
-1
|
-1
|
1
|
1
|
||
14
|
19
|
Block 1
|
{1}
|
-1
|
1
|
1
|
-1
|
1
|
-1
|
-1
|
||
15
|
6
|
Block 1
|
{-1}
|
1
|
1
|
1
|
1
|
-1
|
-1
|
-1
|
||
16
|
18
|
Block 1
|
{1}
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
||
17
|
16
|
Block 1
|
{-1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
||
18
|
1
|
Block 1
|
{1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
||
19
|
7
|
Block 1
|
{-1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
||
20
|
15
|
Block 1
|
{1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
||
21
|
4
|
Block 1
|
{-1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
||
22
|
5
|
Block 1
|
{1}
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
5. Performing
the Experiment
The experiments were performed using the AutoSched AP simulation software package. In the initial stage, one long run was made to determine the warm-up period. The warm-up period is the time taken by the simulation model to reach a steady state, where no statistics is collected. Cycle time was plotted against time (in days) and the period was determined to be 94 days as shown in Figure 1.
Fig.1
Warm-up Period Determination
The run length
was determined to be 3 years and only single replication was made at each run
due to the limited resources. A single run took about two and half hours on a
fast machine (PIII, 800Mhz). Refers to Appendix 2 and see the Result
Matrix.
6. Statistical
Analysis of the data
Upon completion of the runs, the results were fed to the Design Expert
software and the results were analyzed. As mentioned earlier, the design chosen
was a resolution IV, 28-4 fraction factorial design. The analysis
includes ANOVA, residual analysis and model adequacy checking, regression
analysis, and contour plots. These analyses are discussed in detail below.
6.1 Analysis of
Variances (ANOVA)
Figure 2 below shows the half-normal plot, which shows the effects
of various factors. Based on this graph, where the response variable is average
cycle time, the factors that lie along the line are negligible and three
factors seem to be significant. The two main effects from this analysis are A
and G and a two-factor interaction AG.
 |
Fig.2 Half-Normal Plot of Average Cycle Time
The similar analysis was performed for response variable, throughput, as
shown in the Figure 3 below.
 |
Fig.3 Half-Normal Plot of Throughput
Table 3 shows the results of analysis of variance.
Based on the response variable, average cycle time, it shows that the factors
that we chose are significant and their interaction is significant, and that
there is no evidence of second-order curvature in the response.
Table 3
ANOVA for Avg. Cycle Time
 |
The Table 4 supplements ANOVA table. The high value of R-Squared indicates that
the major proportion of variability is included in the model.
Table 4
Supplementary Data of Avg. Cycle Time
The similar
analysis was performed for response variable, throughput, as shown in the Table5&6
below.
Table 5
ANOVA for Throughput
 |
Table 6 Supplementary Data of Avg. Throughput
6.2 Model Adequacy
Checking
6.2.1 Normality
Assumption
The adequacy of
the underlying model should be checked before the conclusions from the analysis
of variance are adopted. Violation of the basic assumptions and model adequacy
can be easily investigated by the examination of residuals. For example, if the
model is adequate, the residuals should be structure less and that is, they
should contain no obvious patterns. In Figure 4, presents a normal
probability plot of the residuals for average cycle time. There is no severe
indication of non-normality, nor is there any evidence pointing to possible
outliers and the equality of variance assumption does not seem to be
violated. Figure 5, Normal Plot
of Residuals for Throughput shown below is also normally distributed and it
resembles a straight line.
 |
Fig 4 Normal Plot of Residuals for Avg. Cycle Time
 |
Fig 5 Normal Plot of Residuals for Throughput
6.2.2 Residual
Analysis
 |
Various residual plots are shown in this section below. Figure 6-10 show diagnostic plots of the model. The residuals are normally distributed and the equality of variance does not seem to be violated.
Fig 6
Residual vs. Predicted Plot for Cycle Time
 |
Fig.7 Residual vs. Predicted Plot for Throughput
 |
Fig.8 Residual vs. Run Number for Avg. Cycle Time
Fig.9 Residual vs. Run
Number for Throughput
 |
Fig.10
Residuals vs. Significant Factor
6.2.3 Box-Cox
Transformation
 |
The model was tested for any transformations that could have been applied, but the Box-Cox Plot did not suggest any new transformations for both response variables, namely Average Cycle Time and Throughput as shown in Figure 11 and 12.
Fig.11
Box Cox Plot for Cycle Time
 |
Fig.12
Box Cox Plot for Throughput
6.3 Regression Analysis
The equations
below are fitted regression model representations of the two-factor factorial
experiments for both responses.
6.3.1 Average
Cycle Time
Final Equation
in Terms of Coded Factors:
Avg Cycle Time = 1292.14 - 333.07 * A +178.70 * G - 94.60 * A * G
Final Equation in Terms of Actual Factors:
Dispatching Rule FIFO
Avg Cycle Time = -5207.44845 + 0.36441 * RELEASE RATE
Dispatching Rule SSU
Avg Cycle Time = -1143.43634 + 0.11213 * RELEASE RATE
Avg Cycle Time = 1292.14 - 333.07 * A +178.70 * G - 94.60 * A * G
Final Equation in Terms of Actual Factors:
Dispatching Rule FIFO
Avg Cycle Time = -5207.44845 + 0.36441 * RELEASE RATE
Dispatching Rule SSU
Avg Cycle Time = -1143.43634 + 0.11213 * RELEASE RATE
6.3.2
Throughput
Final
Equation in Terms of Coded Factors:
Throughput = 8004.00 + 769.55 * A + 14.62 * G + 252.75 * A * G
Final Equation in Terms of Actual Factors:
Dispatching Rule FIFO
Throughput = 13187.57955 - 0.31750 * RELEASE RATE
Dispatching Rule SSU
Throughput = 2089.17045 + 0.35650 * RELEASE RATE
Throughput = 8004.00 + 769.55 * A + 14.62 * G + 252.75 * A * G
Final Equation in Terms of Actual Factors:
Dispatching Rule FIFO
Throughput = 13187.57955 - 0.31750 * RELEASE RATE
Dispatching Rule SSU
Throughput = 2089.17045 + 0.35650 * RELEASE RATE
6.4 Interaction Graph
of Factors A and G
Figure 13 shows the interaction effect
of factors A and G. The average cycle time decreases when changing from FIFO to
SSU and from 19.5K wafers per month to 18K wafers per month. The other factors
do not have significant effect on the responses. The contour plot was
constructed by converting the type of factors from categorical to numeric as
shown in Appendix 3.
 |
Fig.13 Interaction Graph of Avg. Cycle Time
Figure 14 below shows the interaction effect of
factors A and G. Using the dispatching rule FIFO, throughput is higher at 18K
release rate than at 19.5K release rate compared to the dispatching rule SSU
where the throughput is higher at 19.5K than 18K release rate. The other
factors do not have significant effect on the responses. The contour plot was
constructed by converting the type of factors from categorical to numeric as
shown in Appendix 3
 |
Fig.14 Interaction Graph of Throughput
6.5 Optimal Designs
 |
The Design Expert provides optimal designs with the desirability factor of 0.852, which determines the optimal level for each factor as shown in the Figure 15. The “circle” mark and two ends on the line represent the current operating condition and its ranges. The last two boxes show the ranges of two responses. Cycle time follows the hierarchical principle, while throughput follows the linear relationship. The alternative solutions are shown in the Appendix 4. The constraints used are the high and the low level of each factor, where the objective used is to maximize the Throughput and minimize the Avg. Cycle Time.
Fig. 15 Ramps for various
factors
7. Conclusions
The 28-4
fractional factorial designs were run and analyzed to determine the effect of
various factors on average cycle time and throughput. The following conclusions
were made:
- Only two factors and their interaction were
significant: release rate and dispatching rule
- The model was tested for its adequacy and found that
the assumption of normality and independency are not violated
- R2 value was very high, that suggesting
that model accounted for most of the variability
- Box-Cox Plot did not suggest any transformation
- The graphs for the significant factors were analyzed
and the best value is obtained at high value of release rate and using
dispatching rule as SSU, which is in conjunction with our intuition.
- The optimal value with specified desirability was
calculated using the software.
- Since only two factors have been identified as
significant, more detailed experiment can be designed to study the effects
are various levels of these factors.
7.1 Confirmation Testing
Based on the regression models, runs were made to verify the results obtained for both responses, average cycle time and throughput. Table 7 shows the predicted and actual results.
Table 7
Confirmation Testing values
The table
indicates that the predicted values are very close to the actual values and
thus supporting the design.
7.2 Recommendations
Based on the
conclusions and the validity of the model, we recommend:
- To use SSU as a dispatching policy
- To operate at a release rate of 19.5K to yield a higher
throughput and lower average cycle time.
- To design a new experiment to take into consideration
more levels of these significant factors, such as 32 or higher
APPENDIX
1
Defining
Relation and Aliases
APPENDIX 2
Result
Matrix
Std
|
Run
|
Block
|
Factor1: OP_DIFF
|
Factor2: OP_PHOTO
|
Factor3: OP_ETCH
|
Factor4: OP_WET
|
Factor5: OP_MOVE
|
Factor6: RELEASE RATE
|
Factor7: DISPATCHING RULE
|
Factor8: STOCKERS QTY.
|
R1: AVG. CYCLE TIME
|
R2: THROUGHPUT
|
Hours
|
Lots
|
|||||||||||
1
|
16
|
Block 1
|
2
|
2
|
3
|
2
|
35
|
18000
|
FIFO
|
2
|
1358.95
|
7476
|
2
|
5
|
Block 1
|
4
|
2
|
3
|
2
|
35
|
19500
|
SSU
|
4
|
1034.83
|
9060
|
3
|
10
|
Block 1
|
2
|
4
|
3
|
2
|
55
|
18000
|
SSU
|
4
|
1894.73
|
6998
|
4
|
8
|
Block 1
|
4
|
4
|
3
|
2
|
55
|
19500
|
FIFO
|
2
|
877.549
|
8503
|
5
|
11
|
Block 1
|
2
|
2
|
5
|
2
|
55
|
19500
|
SSU
|
2
|
1895.17
|
7010
|
6
|
2
|
Block 1
|
4
|
2
|
5
|
2
|
55
|
18000
|
FIFO
|
4
|
890.351
|
8497
|
7
|
15
|
Block 1
|
2
|
4
|
5
|
2
|
35
|
19500
|
FIFO
|
4
|
1347.63
|
7453
|
8
|
18
|
Block 1
|
4
|
4
|
5
|
2
|
35
|
18000
|
SSU
|
2
|
1081.39
|
8978
|
9
|
20
|
Block 1
|
2
|
2
|
3
|
4
|
55
|
19500
|
FIFO
|
4
|
1337.59
|
7490
|
10
|
21
|
Block 1
|
4
|
2
|
3
|
4
|
55
|
18000
|
SSU
|
2
|
1068.04
|
8995
|
11
|
22
|
Block 1
|
2
|
4
|
3
|
4
|
35
|
19500
|
SSU
|
2
|
1895.25
|
6990
|
12
|
3
|
Block 1
|
4
|
4
|
3
|
4
|
35
|
18000
|
FIFO
|
4
|
888.456
|
8495
|
13
|
7
|
Block 1
|
2
|
2
|
5
|
4
|
35
|
18000
|
SSU
|
4
|
1884.61
|
7014
|
14
|
13
|
Block 1
|
4
|
2
|
5
|
4
|
35
|
19500
|
FIFO
|
2
|
867.791
|
8503
|
15
|
12
|
Block 1
|
2
|
4
|
5
|
4
|
55
|
18000
|
FIFO
|
2
|
1339.14
|
7498
|
16
|
1
|
Block 1
|
4
|
4
|
5
|
4
|
55
|
19500
|
SSU
|
4
|
1012.69
|
9104
|
17
|
6
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
FIFO
|
3
|
1642.32
|
7197
|
18
|
14
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
SSU
|
3
|
934.913
|
8789
|
19
|
9
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
FIFO
|
3
|
1591.25
|
7307
|
20
|
19
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
SSU
|
3
|
927.695
|
8807
|
21
|
17
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
FIFO
|
3
|
1648.65
|
7180
|
22
|
4
|
Block 1
|
3
|
3
|
4
|
3
|
45
|
18750
|
SSU
|
3
|
924.109
|
8812
|
APPENDIX
3
 |
Appendix 3A Contour Plot for Cycle Time
Appendix 3B
Contour Plot for Throughput
APPENDIX 4
Optimal
Solution
