Modeling, Analysis, and Design of Responsive Manufacturing Systems Using Classical Control Theory by Nga Hin Benjamin Fong Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University In partial fulfillment of the requirements leading to the degree of Doctor of Philosophy (Ph.D.) in Industrial and Systems Engineering (Manufacturing Systems Option) _____________________________ ____________________________ Co- Chair: Dr. John P. Shewchuk Co-Chair: Dr. Robert H. Sturges _____________________ _____________________ Dr. F. Frank Chen Dr. Ting-Chung Poon ______________________ Dr. Harry H. Robertshaw April 15, 2005 Blacksburg, Virginia Keywords: Classical Control Theory, System Responsiveness, Responsive Manufacturing Systems
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Modeling, Analysis, and Design of Responsive Manufacturing
Systems Using Classical Control Theory
by
Nga Hin Benjamin Fong
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
In partial fulfillment of the requirements leading to the degree of
Doctor of Philosophy (Ph.D.)
in
Industrial and Systems Engineering (Manufacturing Systems Option)
_____________________________ ____________________________ Co- Chair: Dr. John P. Shewchuk Co-Chair: Dr. Robert H. Sturges
_____________________ _____________________ Dr. F. Frank Chen Dr. Ting-Chung Poon
______________________ Dr. Harry H. Robertshaw
April 15, 2005
Blacksburg, Virginia
Keywords: Classical Control Theory, System Responsiveness, Responsive Manufacturing Systems
Modeling, Analysis, and Design of Responsive Manufacturing Systems
Using Classical Control Theory
by
Nga Hin Benjamin Fong
(ABSTRACT)
The manufacturing systems operating within today’s global enterprises are invariably dynamic and complicated. Lean manufacturing works well where demand is relatively stable and predictable where product diversity is low. However, we need a much higher agility where customer demand is volatile with high product variety. Frequent changes of product designs need quicker response times in ramp-up to volume. To stay competitive in this 21st century global industrialization, companies must posses a new operation design strategy for responsive manufacturing systems that react to unpredictable market changes as well as to launch new products in a cost-effective and efficient way. The objective of this research is to develop an alternative method to model, analyze, and design responsive manufacturing systems using classical control theory. This new approach permits industrial engineers to study and better predict the transient behavior of responsive manufacturing systems in terms of production lead time, WIP overshoot, system responsiveness, and lean finished inventory. We provide a one-to-one correspondence to translate manufacturing terminologies from the System Dynamics (SD) models into the block diagram representation and transfer functions. We can analytically determine the transient characteristics of responsive manufacturing systems. This analytical formulation is not offered in discrete event simulation or system dynamics approach. We further introduce the Root Locus design technique that investigates the sensitivity of the closed-loop poles location as they relate to the manufacturing world on a complex s-plane. This subsequent complex plane analysis offers new management strategies to better predict and control the dynamic responses of responsive manufacturing systems in terms of inventory build-up (i.e., leanness) and lead time. We define classical control theory terms and interpret their meanings according to the closed-loop poles locations to assist production management in utilizing the Root Locus design tool. Again, by applying this completely graphic view approach, we give a new design approach that determine the responsive manufacturing parametric set of values without iterative trial-and-error simulation replications as found in discrete event simulation or system dynamics approach.
Acknowledgements
It has been a challenging and fruitful journey to return to Virginia Tech to study my Ph.D. program in Industrial and Systems Engineering. I am very thankful to have such a unique and multi-disciplinary research committee which they have taught me how to think, learn, and communicate. I would like to take this opportunity to thank them for their guidance and support which greatly enhanced the value of my eight year journey. Dr. John P. Shewchuk, my Co-Chairman of the Committee, has spent tremendous amount of time and effort to guide me through this research journey. From initial research ideas to models implementation, from model validation to dissertation writing, he has provided a lot of good advice and help to make my Ph.D. commencement happening. In particular, his input and concern on validating CCT models to discrete manufacturing world is a crucial part to implement our new methodology from the industrial engineering point of view. Lastly, his high standard writing style has made my dissertation so completed. My heartfelt thanks to him! Dr. Robert H. Sturges, my Co-Chairman of the Committee, is truly a role model in both teaching and research excellence. His innovative ideas, kindness, courage and motivation have made me to realize how fortunate I am to have him to be my co-advisor. From academic research to industrial projects, from departmental politics to my personal issues, we can spend numerous hours to chat and discuss without feeling the time has gone so fast. He has transformed me from a below average graduate student to become awards winning research scholar. In Chinese term, he is my “Inspirational Master”!! Truly, I may never be able to complete translating CCT terminologies to manufacturing world without him. That “special Friday talk” outside Durham in September 2003 has significantly changed my career life. Thanks Dr. Bob! ☺ Dr. Harry H. Robertshaw, my former advisor for my MS in Mechanical Engineering, has guided me through many challenges throughout the past 13 years. From my MS research work to Labor Certification supporting letter, from preliminary research to latest Intel Case Study, it is unquestionable that his willingness to support is vital. I especially appreciate his extra time and effort to assist me to formulate the block diagrams and algebraic expressions for my dissertation work. Definitely, he has spent at least three times amount of time to assist my Ph.D. work than my MS program. Big thanks to him! Dr. F. Frank Chen, my most respectful industrial-managerial, research professor, has taught me one should have long term visions and plans to be success in both academia and industrial world. As the founder of the Center for High Performance Manufacturing (CHPM), Dr. Chen has provided me the complete financial support through my three years of full-time studies. Besides, he gave tremendous advice and help to make my dissertation completed. I look forward to follow his successful foot steps to work for Caterpillar at Peoria, IL. Many thanks to him!! Dr. T.C. Poon, my most admirable professor and long-time friend, has been advising me since I came to Virginia Tech in Fall 1992. Throughout these 13 years in Blacksburg, there are uncountable incidents that required his help and advice to get over the challenges. I really
iii
appreciated to have him to serve in my dissertation committee. Although the work of modeling manufacturing systems via electrical circuit network has not completed, it is noteworthy to continue these ideas for future work. Hopefully, my upcoming design engineering position will enhance my skills to work with network modeling and frequency response analysis. Thanks very much Dr. Poon! Special thanks to my former boss, Graham Swinfen, an engineering manager at BBA Friction, Inc. in Dublin, VA. Without his persistent help, I would never work at BBA Friction to get my green card and have the opportunity to begin my PhD study at Virginia Tech. From Fall 1997 to Spring 2000, I was able to drive back-and-forth for two hours to go to work-school-work-home regularly. I still remembered how much heat Graham had to take to support of the justification of my continuous education at Virginia Tech giving the time and the financial support from BBA Friction, Inc. Big thanks to Graham! Throughout my eight years of study (3.5 years part-time, 1 year off, 3.5 years full-time), I met so many helpful and interesting colleagues among the student group. Special thanks go to Nathan Ivey and Hitesh Attri in assisting me for the ARENA programming. I wish the best luck to Nate, Hitesh and Radu Babiceanu for their job hunting. It is very thankful to have known Dr. Y.A. Liu and Dr. Hing-Har Lo (Mrs. Liu) through the VT Chinese Bible Study since Fall 1992. They have been acting like my guardian throughout the years – give me the spiritual support and advice while keeping “little Ben” behaves well. ☺ Again, many thanks to Dr. T.C. Poon and Eliza Poon (Mrs. Poon), they are like my older brother and sister via Hong Kong Club and VT Chinese Bible Study. They gave me so many advices and ideas for my daily life, such as school work, buying house, raising kids, career development, retirement plan, etc. Give thanks to the Lord, I have learned so much from these two lovely families! It has been a blessing to get the role to lead and care many young Hong Kong students through the VT Chinese Bible Study and HK Club. Best wishes to my spiritual brothers, Carlos Siu, Henry Yuen, and Winston Ma for their first career challenge as engineers and/or statistical analyst. I enjoyed those uncountable hours to share our joy and sadness while we were in Blacksburg. I especially missed our weekly soccer games at Tech! No doubt in my mind, my life will ever reach to this stage without the support from my family. Thanks be to my Lord to provide me such a lovely and heart-bonded family. I would like to express my heartfelt appreciation to my parents, Hoo-Shin Fong and Shau-Shan Lai, for their unyielding love and support since I was born. My bond with my parents grew even stronger when I left home to England since 1985. They provided me with tremendous mental and financial support through these years. Particularly in the past three years, after their retirement from Hong Kong, they even came to stay with my family to help baby-sit their lovely grandchildren. For sure, their physical support has allowed me to concentrate on my research while my daughters are often crying for milk. I also give thanks to the Lord to giving me such a wonderful and supportive elder brothers, Ricky and Joe. We have grown up together back in Hong Kong, then we all went to Rishworth for high
iv
school in England, and we all came to States for gaining our higher education. Throughout these years, we have shared and supported each other via visit, phone calls, and prayers. It is so important to maintain this high degree of brother-hood to be able to face all those up-and-down in life. As Ricky said, I shall be thankful to have completed my Ph.D. as a by-product due to my long-waiting green card application process. While Joe always reminds me that getting a Ph.D. is just a beginning for the next chapter of life. Thank you my dearest brothers!! I also give thanks to my two lovely sisters-in-law, Susanna and Florence, for their prayers, supports, and sharing in all these years. Lastly, I would dedicate this dissertation to my beloved wife, Iris (Ching) for her unyielding love, support and care to make me become Dr. Fong! I give thanks to my Lord to give me such an understandable and dedicated wife and mother. Iris and I had gone through so many challenges since we were together in Spring 1995. Our life faced a lot of challenge in the early stage, such as, I worked over 70+ hours weekly in Hazard, KY, then she worked over 70+ hours in Hotel Roanoke, VA. Sure, we were just a cheap-labor whom decided to live in US. By summer 1998, we got married and began our next challenge for the school work at Tech. I began my part-time PhD study while I was working full-time at BBA Friction and she returned to Virginia Tech to study her MS in Accounting and Information Systems in Spring 1998. The most difficult challenge was to study together almost every night at Durham Hall until 4 am while I still had to return to work by 8 am. It is so thankful to have her support and patience to host numerous Friday night gathering with those HK students right after the bible study. We both learned so much and became more mature to take care this young students group. Without a doubt, Iris has sacrificed her career twice to choose a less competitive and lower-paid job to stay in Blacksburg to give me time to finish my PhD program. Her unique encouraging style by keep reminding me “not to waste time and move on” has made me even stronger and more self-confidence to continue to pursue my PhD program. ☺ I give thanks to her to be such a caring mother and daughter-in-law to help taking care our two lovely daughters, Vera and Audrey, and my parents while I was busy preparing my dissertation work. Iris will probably give up her career for the third time when we move to Peoria, IL. But I am sure that the kids must love to see Mommy spend more time at home. ☺ Thank you Lord for giving me such a loving family, I would never complete my PhD study without their support. My final sharing for those whom love to get a PhD degree, you should equip the following elements to be succeeded, such as willingness, hard-work, discipline, persistence, research topic, supportive professor committee and most importantly, communication skills. God Bless America!! ☺
Figure 4.8: Discrete vs. Continuous in modeling single-stage production system
Figure 4.8 shows that the state variables of the continuous model like production rate, shipment
rate, and inventory change continuously with respect to time, it yields a greater rate of change of
the state variables. For the discrete model, we have selected to update the production work force
and the shipment every 4 hours, the rate of change of the curves is much slower than the
continuous case. However, as the system reaches to its steady-state, about 12-14 days, the final
steady-state values of inventory, production rate, and shipment rate are insignificantly different
from the CCT continuous system model.
If the shipment rate and the production work force adjust at a shorter interval, say 30 minutes,
the slope of the finished inventory should increase. As the production work force and the
shipment continue to adjust at smaller time intervals, the rising slope of the finished inventory
behaves more like the continuous differential model. In other words, the CCT approach is
optimistic with respect to real discrete manufacturing systems and it always provides a lower
bound on the rise time of the system response. Additionally, the less rapid the management
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Chapter 4 Model Validation N.H. Ben Fong
changes the schedule of production work force and shipment, the more of an optimistic CCT
approximation yields (i.e., gives more error to real industrial scenario).
In real life, customer demands often come randomly and unexpected. The CCT approach is
mathematically limited in that it cannot include any stochastic elements. In addition, production
management schedule their amount of work force according to the maximum capacity of the
overall machine throughput rate. The CCT approach cannot model any non-linear elements that
contain saturation due to maximum capacity. For unreliable manufacturing processes and
machine breakdowns, we can still apply CCT approach to model the machine failure as a system
disturbance input. In order to overcome these limitations of the CCT approach, we can apply
Non-Linear Control Theory and Stochastic Linear Control Theory to further enhance model
validity.
In this chapter, we have demonstrated that the classical control theory (CCT) modeling approach
appears to provide valid approximations of real discrete manufacturing systems. To further
validate the CCT approach, we will use it to model, analyze, and design a real three-stage
semiconductor manufacturing system in Chapter 6.
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
Chapter 5 Design of Responsive Manufacturing
Systems
In this chapter, we introduce a new design methodology to assist management in identifying
operation strategies to better predict the dynamic responses of responsive manufacturing systems
subjected to rapidly changing market demands. Section 5.1 introduces the Root Locus design
technique, originating from classical control theory. In that section, we apply the Root Locus
design tool to improve the overall system responsiveness of a two-stage production control
system. By varying the system loop gain K, the location of the closed-loop poles moves
predictably on the complex s-plane. Hence, management can predict and design the particular
responsive manufacturing systems in terms of production lead time, degree of WIP overshoot
(i.e., damping ratio) and lean finished inventory. In Section 5.2, we further define and interpret
the meanings of those classical control terms, such as damping ratio, closed-loop poles location,
and settling time as they relate to the manufacturing world on a complex s-plane representation.
We give examples of various step response dynamic behaviors according to the different
locations of the closed-loop poles. This complex plane analysis offers a graphical view to
evaluate and understand the overall dynamics and the corresponding parametric sensitivity of
any responsive manufacturing systems. Finally, section 5.3 applies the Root Locus to design a
two-stage production control system with 3rd-order time delay. From section 3.2, we extended
the two-stage production control system with a 3rd-order time delay inclusion and it turned into a
fourth-order differential equation model. The corresponding root locus and dynamic behavior of
this higher-order system behaves very differently. We introduce the Dominant Roots or
72
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
dominant closed-loop poles concept to design higher-order differential system models. The
resulting root locus reveals the potential for system instability due to poor management policies
through the location of the closed-loop poles.
5.1 Root Locus Analysis and Design of a Two-Stage Production System
In this section, we introduce the Root-Locus technique from classical control theory [40,41,42]
to improve the overall system responsiveness and lead time for the two-stage production control
model. Phillips and Harbor [42] defined: “A Root Locus of a system is a plot of the roots of the
system characteristic equation (the poles of the closed-loop transfer function) as some parameter
of the system is varied”. The location of the system closed-loop poles, as given by Eq. (5.1),
determines the transient characteristics of the two-stage production system.
Closed-Loop Poles: -1 - s 2nn1,2 ζωζω j±= (5.1)
As mentioned, the closed-loop poles of a system are the roots of the characteristic equation. It is
instructive to see how the closed-loop poles move in the complex s-plane as the loop gain of the
system is varied. From a design point of view, an adjustment of the gain value(s) may bring the
closed-loop poles to certain desired locations. We apply the Root-Locus technique to plot the
roots of the characteristic equation for different values of gain K. The root locus is the locus of
roots of the characteristic equation of the closed-loop system as the gain K is varied from zero to
infinity. Such a plot clearly indicates how to modify the open-loop poles such that the response
of the manufacturing system can meet the specific customer requirements. We generally consider
the system of Fig. 5.1 in describing the concept of root locus as stated in Phillips and Harbor [42],
73
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
with 0 K< ∞. In Figure 5.1, we assume that G(s) comprises both the compensator transfer
function and the plant transfer function. The characteristic equation for this system is given in Eq.
(5.2). A value s
≤
1 is a point on the root locus if and only if s1 satisfies Eq.(5.2) for a real value of
K, with 0 ≤ K< ∞.
1 + KG(s) H(s) = 0 (5.2)
K G(s)
H(s)
+_
Figure 5.1: A system for Root Locus
For more detail on Root-Locus methodology, see Ogata [40] and Phillips and Harbor [42].
According to the closed-loop transfer function of the two-stage production system as derived in
Eq.(3.19), the system equation KG(s) for the root locus analysis as shown in Fig. 5.1 will
become:
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎥
⎦
⎤⎢⎣
⎡+++
⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
=
WAT1
LT1
ST1
ST1
ST1
WAT1
LT1 ss
WATLT1
LT1
FAT1
KG(s)
211
2
(5.3)
Given that the loop gain, ST1=ST2, Eq.(5.3) reduces to:
ST
1WAT
1LT1 ss
WATLT1
LT1
FAT1
KG(s)
1
2⎥⎦
⎤⎢⎣
⎡+++
⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
= (5.4)
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
For illustration, we arbitrarily set the system parameters as follows: LT=1 day, ST=5 days,
WAT=5 days, and FAT=5 days. By applying the Matlab [49] rlocus command, we find and plot
the root loci of the two-stage production control system as shown in Fig. 5.2.
-2 -1.5 -1 -0.5 0 0.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag
inar
y A
xis
Constant ζ Lines and Constant ωn Circles
ζ=0.9
0.80.7
0.6 0.5
0.4 0.3
0.20.1
ωn=1ωn=2
Figure 5.2: Root-Locus Plot of a Two-Stage Production Control System
The root-locus branches (i.e., marked as X crosses) starts from the open-loop poles from Eq.(5.4)
at s1= -1.4 and s2= 0 (K=0) at the real axis of the complex s-plane as shown in Fig. 5.2. The gain
K value increases incrementally from 1, at the closed-loop poles derived from Eq.(5.1) at s1= -
1.2 and s2= -0.2, until the increased K value is 18, with a pair of complex-conjugate roots at –0.7
± 1.96i. The arrows indicated in Fig. 5.2 provide the direction of the increasing K values from
the real-axis to the imaginary-axis. As K is varied, the location of the closed-loop poles changes.
The intersection of the horizontal K values and the vertical K values is called the breakaway
point (ζ=1) of the root-locus with s1,2= -0.7 and K value of 2.04. In a second-order dynamic
system with no additional dynamics in the numerator of the transfer function, the breakaway
75
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
point is simply (s1+s2)/2. The damping ratio ζ of the system is always equal or greater than one
as the closed-loop poles are found along the real-axis on the s-plane. Once the closed-loop poles
have passed the breakaway point, ζ begins to reduce further with pairs of complex-conjugate
roots. Figure 5.2 also plots the grid of constant damping ratio ζ lines and constant undamped
natural frequency ωn lines. The constant ζ lines are radial lines passing through the origin with a
decrement of 0.1 from ζ=1 to 0.1, whereas the constant ωn loci are circles.
In the complex s-plane, we can express the damping ratio ζ of a pair of complex-conjugate poles
in terms of the angle φ which is measured from the negative real-axis with ζ= cosφ, and
determine the distance of the pole from the origin by the undamped natural frequency ωn. As
discussed in the previous section, we quantify the transient dynamic responses of the two-stage
production system by the key parameters ζ and ωn. The importance of introducing Root-Locus
method here is to provide a predictive design technique to improve the overall dynamic behavior
of this manufacturing system. As per classical control theory, we can design a desirable transient
response of a second-order system with a damping ratio around 0.707. It is justified with results
in the next phase. Small values of ζ<0.4 yield excessive overshoot where a system with ζ>0.8
responds sluggishly as shown in Fig. 3.17 previously. The further the closed-loop pole is away
from the origin of the s-plane, the faster the response of the system behaves. However, the
system will become unstable if there is any closed-loop pole found on the right-hand side of the
s-plane.
For our example, in order to bring the system model to ζ=0.707, the K value from the root-locus
analysis is increased to 4.08 with ωn=0.99 at s1,2= -0.7 ± 0.70i as shown in Figure 5.2. Given this
76
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
particular parametric set of values, the system equation has an initial non-unity loop gain K of
0.24 as computed from Eq.(5.3). As a result, we have to multiply every root-locus gain K from
Matlab by a factor of 0.24. Although K is a function of Finished Inventory Adjustment Time
(FAT), Lead Time (LT), and WIP Adjustment Time (WAT) as given by Eq.(3.29), in order to
keep the same breakaway point while changing the K value, we can only vary K as a function of
FAT.
2n
t
WATLT1
LT1
FAT1
K y(t) limω
⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
==∞→
(3.29)
By setting FAT=1.224, we make the system respond with ζ=0.707 at K=4.08*0.24=0.98. In
designing a fast-response manufacturing system, setting ζ to 0.707 is crucial; however, moving
the breakaway point further away from the origin is another vital step. According to Eq.(3.23),
the time response of the system is a function of LT, WAT and ST.
Time constant:
⎥⎦
⎤⎢⎣
⎡++
==
1
n
ST1
WAT1
LT1
2 1 ζω
τ (3.23)
Since we have assumed the shipment time to be 5 days for a transportation schedule, we will
further study the dynamic characteristics of this system model as a function of LT and WAT.
Figures 5.3 and 5.4 show, respectively, contour plots of ζ values and breakaway points against
WAT and LT while keeping FAT=1 day.
77
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
0.5 1 1.5 2 2.5
0.5
1
1.5
2
2.5
Le ad Tim e (LT) in day s
WIP
Adj
ustm
ent T
ime
(WA
T) in
day
s
Constant ST=5 day s and Constant FAT=1 day
0.57
0.585
0.6
0.625
0.650.675
0.707
0.750.8
0.850.90.951
Figure 5.3: Contour Plot of ζ values as a function of LT and WAT
Figure 5.3 shows the contour of ζ values as a function of WAT and LT with a constant FAT=1.
A combination choice of LT and WAT along a particular contour line gives the desired damping
ratio. For example, if we pick WAT=1.5, by taking LT=1.1, it results in ζ≈ 0.707 and a
breakaway point of –0.89 (i.e., an improvement from –0.7). Figure 5.3 shows that ζ values
decrease from 1 to 0.57 with increasing values of both LT and WAT. As stated earlier, FAT is
the most significant factor that affects ζ values. With a FAT value higher than 2.0 (not shown
here), changing any value in LT and WAT will not make any impact to reduce ζ to less than
unity (i.e., the system is always overdamped or critically damped). For the contour of breakaway
points as shown in Figure 5.4, the breakaway point moves towards more negative from –0.53
to –4.0 as LT and WAT are reduced, hence it improves the system response as the breakaway
point moves further away from the origin at the complex plane. FAT does not play a role in the
location of the breakaway point. If we arbitrarily pick LT=1 and WAT=1 from Figure 5.4, it
78
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
gives a breakaway point of –1.1 (i.e., an even further negative value) with a ζ value of 0.78. We
find all of these observations without any simulation and our approach is able to predict directly
the best system response.
Figure 5.4: Contour Plot of Breakaway Points as a function of LT and WAT
0.5 1 1.5 2 2.5
0.5
1
1.5
2
2.5
Le ad Tim e (LT) in days
WIP
Adj
ustm
ent T
ime
(WA
T) in
day
s
Constant ST=5 days and Constant FAT=1 day
-0.53-0.56
-0.6-0.65
-0.7-0.75
-0.81
-0.9-1
-1.2-1.4
-1.8-2.2
-3-4
In order to verify the findings and the recommended improved parametric set of values for
applying both DOE and Root-Locus, we simulate and compare the step response of the system
model subjected to varied sets of design system parameters as shown in Fig.5.5. We use Matlab
to simulate the following four different sets of system parameters with a constant ST value of 5
days:
Set A: LT=1, WAT=5, FAT=5; Set B: LT=1, WAT=5, FAT=1.224
Set C: LT=1.5, WAT=1.1, FAT=1; Set D: LT=1, WAT=1, FAT=1
79
Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
0 5 10 150
20
40
60
80
100
120
Time in days
Fini
shed
Inve
ntor
y Le
vel i
n st
ocks
Set A: ζ=1, Brkaway Pt=-0.7
Set B: ζ=0.707, Brkaway Pt=-0.7
Set C: ζ=0.7073, Brkaway Pt=-0.8879
Set D: ζ=0.7778, Brkaway Pt=-1.1
Figure 5.5: Step Response Comparison of a Two-Stage Production Control System
The results show that the original parametric set A responds sluggishly due to its ζ value of 1
(i.e., critically damped) with a breakaway point of –0.7. The root-locus analysis suggests to
increase the gain K value to 4.08 and with FAT=1.224. The set B curve has shown a significant
improvement in term of the overall system response. Curve B has a ζ value of 0.707 with the
same breakaway point of –0.7. We further improve the system response by changing the values
of LT, WAT and FAT according to the contour plots. The set C curve gives an even quicker rise
time as compared to curve B due to the improvement of the breakaway point (i.e., increases
from –0.7 to –0.89). Curve D appears to be the best suggested system response. Again, by
studying the dynamic characteristics at the contour plots, we further improve the breakaway
point from –0.89 to –1.1 with even smaller inventory overshoot.
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
5.2 Interpretation of CCT Terms to the Manufacturing World
As stated previously, the root locus is the locus of roots of the characteristic equation of the
closed-loop system as a specific parameter (usually, gain K) varies from zero to infinity. This
technique is very useful since it provides guidelines in which the open-loop poles and/or zeros
(definition of these CCT terms will be described later) should be modified such that the system
responses meet the desired performance specifications. By using the root-locus method, we can
determine the value of the loop gain K that makes the damping ratio, ζ of the dominant closed-
loop poles as prescribed. In the manufacturing world, industrial engineers and managers prefer to
predict and control the critical system variables, like lead time, inventory level, settling time for
production control planning purposes. In this section, we define and interpret some key
manufacturing system variables as they relate to the classical control theory terms described in
the complex s-plane. The aim of this interpretation is to offer a new operation planning strategy
for manufacturing managers to better predict and study the transient behavior of responsive
manufacturing systems. This approach offers a systematic and graphical view to evaluate and
understand the overall dynamics and its corresponding parametric sensitivity of any responsive
manufacturing system model.
Responsiveness is an overall strategy focused on thriving in an unpredictable and dynamic
environment. Lean is a philosophy that seeks to minimize all waste that includes long lead time,
excess WIP inventory, non-value added activities. Responsiveness refers to the dynamics of
manufacturing system behavior in term of damping ratio ζ value. Fowler [31] described an MRP
system as a feedforward system where the production is pulled through the system by a
feedforward scheduling system, referenced to the particular customer demand. He further stated
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
that the Kanban control system may be seen as the production pulled through the system by a
sequence of cascaded feedback control loops. However, there has never been an example to
describe whether these loops are stable or tuned properly. We will now do this with a new
systematic method to study and predict production operation strategies based upon the key
manufacturing variables as they relate to the closed-loop poles location in the complex s-plane.
Again, we pick the Root Locus plot of the two-stage production control system as shown in
Figure 5.2 to perform the investigation. As stated previously, the root-locus branches of this
model (marked as X crosses) starts from the open-loop poles at s1= -1.4 and s2=0 (K=0) at the
real axis of the complex s-plane as shown in Figure 5.6. The gain K value increases
incrementally from 1, at the closed-loop poles at s1= -1.2 and s2= -0.2, until the increased K
value is 18, with a pair of complex-conjugate roots at -0.7 ± 1.96i. We added nine “+ crosses” on
the s-plane to examine different dynamic characteristics according to the location of the closed-
loop poles as shown in Figure 5.6 (next page).
In this section, by varying the location of different closed-loop poles or roots on the complex s-
plane as shown in Figure 5.6, we specifically define and interpret the significance of each
location as it relates to the dynamic behavior found in the manufacturing environment. The x-
axis and the y-axis on Figure 5.6 represent the real axis and the imaginary axis, respectively. The
closed-loop poles can be interpreted as the production buffers (i.e., accumulates products). The
pole located at point A indicates a negative real root of the characteristic equation, Eq. (5.3). This
pole location gives the production inventory level to decay exponentially until it reaches steady
stable condition. Any manufacturing system that contains only real roots, closed-loop poles on
the x-axis will not obtain any inventory overshoot and dynamic oscillation (i.e., ζ ≥1.0).
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
-2 -1.5 -1 -0.5 0 0.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag
inar
y A
xis
X X
+
+
+
+
+
+
+
A B
C
D
E
F
G
H
J
+
+
K
L
Figure 5.6: Complex s-plane interpretation of varying location of closed-loop poles The pole located at point B is special case when production maintains at a certain inventory level
steadily (i.e., no rise or fall), thus it is considered as a limited stable condition due to the potential
of inventory growing exponentially. The pair of complex-conjugate roots with negative real parts
located at points C and D give dynamics that makes production inventory oscillate and dies
down to stable condition within its envelope. The points C and D are found at the contour line of
damping ratio, ζ = 0.8. In terms of the speed of the production lead time, points A, C, and D shall
behave similarly except there is no oscillation at point A. As the closed-loop poles get closer to
the right, the speed of the system responsiveness shall decrease. The pair of complex-conjugate
roots with negative real parts located at points E and F is expected to respond slower than the
pair located at points C and D. In addition, the degree of oscillation shall magnify as the damping
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
ratio decreases from ζ =0.8 to 0.5. Their rate of exponential decay will be less than the pair from
C and D but it still leads to a steady stable condition. The pair of complex-conjugate roots on the
imaginary axis located at points G and H yields a limit cycle for the manufacturing system.
Given ζ =0, the production inventory will behave periodically within a particular set range of
inventory level. The production attempts to respond due to the seasonal customer demand,
however given the fixed maximum customer demand, the corresponding production inventory
oscillate within the same envelope. Obviously, operation management does not want to keep a
fixed amount of annual inventory costs to forecast seasonal customer requirements. Rather,
management shall determine a strategy to make shipments as needed. In CCT terms, this type of
manufacturing system is considered to be marginally stable due to the potential to grow
exponentially positive. In practice, of course, saturation of some limited resource will eventually
occur to keep the results finite.
Once the closed-loop poles go beyond the imaginary axis to the right-hand-side of the s-plane,
the manufacturing system will behave unstable without bound or until the plant capacity
saturates. For instance, the pole located at point L has a positive real root. Thus, the production
inventory will grow positively without bound. This is the typical scenario when the management
decided to build as much inventory as production can provide to prevent the unexpected sudden
request by their customers. Truly, this kind of operation policy should not continue to run, given
the concept of Kanban from the Lean manufacturing system. Lastly, the pair of complex-
conjugate roots with positive real parts located at points J and K indicates that production
inventory continues to increase with dynamics and oscillation. This system grows positive and
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
sinusoidal with an increasing size of envelope with no bound. Obviously, it turns into a poor
management policy to let inventory costs add accumulatively in seasons without bound.
We graphically display the dynamic behavior of those six special cases as discussed as shown in
Figure 5.7. Again, the two stable cases, include, a single negative real root (A) and a pair of
complex conjugate roots with negative real parts (C and D). The two marginally stable cases,
contain, a single root at the origin (B), and a single pair of complex conjugate roots on imaginary
axis (G and H). Finally, the two unstable cases comprise, a positive real root (L) and a complex
conjugate roots with positive real parts (J and K).
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
0 5 10 15 20 25 300
20
40
60
80
100Negative Real Root (A)
Inve
ntor
y (U
nit)
0 5 10 15 20 25 30-60
-40
-20
0
20
40
60
80Complex Roots w/ Negative Real Parts (C&D)
Inve
ntor
y (U
nit)
0 5 10 15 20 25 3099
99.5
100
100.5
101Single Root at Origin (B)
Inve
ntor
y (U
nit)
Time (Day)0 5 10 15 20 25 30
-100
-50
0
50
100Complex Roots on Imaginary Axis (G&H)
Inve
ntor
y (U
nit)
Time (Day)
0 5 10 15 20 25 300
2000
4000
6000
8000
10000Positive Real Root (L)
Inve
ntor
y (U
nit)
Time (Day)0 5 10 15 20 25 30
-1000
-500
0
500
1000Complex Roots w/ Positive Real Parts (J&K)
Inve
ntor
y (U
nit)
Time (Day)
Figure 5.7: Transient mode shapes associated with locations of roots in the complex s-plane
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
5.3 Root Locus Design of a Two-Stage Production System with Time Delay
Delays are inherent in many physical and engineering systems. There are different kinds of
delays found as described by Forrester and Sterman [5,16]. They are such as material delays,
pipeline delay or transportation lag, and information delays. For the manufacturing sector,
material delay is a kind that captures the physical flow of material through a delay process. For
example, this delay often happens in a supply chain, distribution business, and construction
management. For multi-stage manufacturing processes, between each station, there is a delay
caused by transportation and order handling. Each station operates individually based on demand
information provided from upstream. As the number of stations increase, the demand signal
amplifies from station to station as orders go through the chain of supply. Forrester described
those oscillations in demand along the chain as the bullwhip effect [2]. In the previous sections,
we stated that the higher the order of the time delay goes, the better the production system can be
characterized. Wikner [27] has described that a third-order delay has proved to be an appropriate
compromise between model complexity and model accuracy for most dynamic modeling of
production-inventory systems. In this section, we investigate the dynamic behavior of the two-
stage production control system with a 3rd-order time delay from Figure 3.7 on the complex s-
plane environment. As recall from Eq.(3.33), the 4th-order differential equation formulated for
the two-stage with a 3rd-order time delay model is as:
( )( )
[ ] [ ] [ ] [ ]D s C s B s A sWATLT
1LT
1FAT
27
DI(s)FI(s)
234
23
++++
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛
= (5.5)
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
where,
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )( )
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟
⎠⎞
⎜⎝⎛ +=
+++=
+++=
++=
212
12
123
112
1
ST1
ST1
FAT1
WAT1
LT1
LT27 D
WATSTLT9
WATLT27
STLT27
LT27 C
WATST1
WATLT9
STLT9
LT27 B
WAT1
ST1
LT9 A
Given the set of selected system parameters: LT=4; ST=8; WAT=1; FAT=1, we observe a
significant dynamic oscillation difference between the third-order delay model and the first-order
exponential smoothing model for the two-stage production system from Figure 3.20.
0 5 10 15 20 25 300
20
40
60
80
100
120
140
160
180
Time in days
Fini
shed
Inve
ntor
y Le
vel (
Uni
t)
1st-Order Delay
3rd-Order Delay
Figure 3.21: Dynamic Responses between 1st-order time delay and 3rd-order time delay
Again, we apply the Root Locus technique to investigate the effect of the closed-loop poles
location for the dynamic production control system. Since it is a fourth-order differential
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
equation model, there are four closed-loop poles found on the complex s-plane as shown in
Figure 5.8
-2.5 -2 -1.5 -1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5Im
ag A
xes
Real Axis
LT=4, ST=8, WAT=1, FAT=1
ζ=0.9 ζ=0.8
ζ=0.7ζ=0.6
X X
X
X
+
+
+
+
Figure 5.8: Root-Locus Plot of a two-stage production control system with 3rd-order time delay
The root-locus branches starts from the open-loop poles location where s1= 0, s2= -1.74, and s3,s4
= -0.819±0.909i, (K=0) on the complex s-plane as shown in Fig. 5.8. The gain K value increases
in the direction of the arrows from the starting poles location (K=0) to the stopping poles
location (K=8). As observed in Figure 5.8, the complex conjugate pair of poles enters the right-
hand side of the s-plane as K increases over 1.305 (not explicitly shown here). As the gain K
continues to increase to infinity (K→∞), the production system grows oscillatory with no bound
(i.e., unstable). Whereas, the two real roots (poles) continue to go towards to the further left of
the s-plane (i.e., stable) as the gain K value increases to infinity. The major challenge here is to
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
design and improve this 3rd-order time delay, two-stage production control system to give quick-
responsive production lead time, less oscillation, and lean final inventory level. Although we
have derived the fourth-order differential equation for this particular production system, there is
no standard formulation that describes damping ratio, time constant, rise time and settling time
for higher-order control systems. In MATLAB programming software [49], we can apply a
function command, rltool, from the Control System Toolbox, to determine the corresponding
gain K value of the corresponding higher-order manufacturing system that yields fast production
lead time and less inventory overshoot on the rlocus plot as shown in Figure 5.8. For higher-
order dynamic systems, it is a good rule of thumb to set the damping ratio between 0.7 and 0.9 to
obtain a fast responsive system behavior [40,41,42,43]. As the gain K value is varied, the
location of the closed-loop poles changes. However, there are altogether four closed-loop poles
or roots to be adjusted in this fourth-order system model. The challenge is to select the particular
K gain value such that the overall production control system yields the best system performance.
Fortunately, there is a concept called Dominant Roots or dominant closed-loop poles from
classical control theory [40,41,42] to help in determining the most influential pole(s) for the
system. We have seen that a time constant τ is a measure of the decay rate of an exponential e-at,
where τ=1/a. The time constant corresponds to the characteristic roots, s= -a, or a complex pair,
s= -a ± ib. If a stable dynamic system model has several roots with different real parts, the root
having the largest time constant (i.e., the one lying the farthest to the right on the complex s-
plane) is the root whose exponential term dominates the overall system response. This particular
root is called the dominant root. There can be two dominant roots (complex pair) existing in a
system because they both have the same real part and same time constant. We illustrate different
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
step input responses of the two-stage production system with a 3rd-order time delay under various
gain K values as shown in Figure 5.9.
0 10 20 300
50
100Gain K = 0.1
Inve
ntor
y (U
nit)
0 10 20 300
50
100
150Gain K = 0.35
Inve
ntor
y (U
nit)
0 10 20 300
50
100
150Gain K = 0.6
Time (Day)
Inve
ntor
y (U
nit)
0 10 20 300
50
100
150
200Gain K = 1.0
Time (Day)
Inve
ntor
y (U
nit)
Figure 5.9: Step Responses of a two-stage system w/ 3rd-order time delay under various K values
Given the set of selected system parameters: LT=4; ST=8; WAT=1; FAT=1, we show four
different step response curves by varying K values as shown in Figure 5.9, As the gain K values
varies from the root locus, the location of the closed-loop poles (roots) changes on the complex
s-plane. As described earlier, the location of the closed-loop poles determines the overall system
responsiveness for the two-stage production control system. The multiple oscillated and dynamic
step response as shown in Figure 3.20 is set with K=1.0. By varying K values via rltool from
MATLAB, we are able to improve the system response by minimizing the inventory overshoot
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
and reducing the production lead time as shown in Figure 5.9. In this particular case, by reducing
the K value, we decrease the amount of oscillation and improve the system settling time.
However, if we reduce K value too low, like K=0.1, the higher-order system behaves sluggishly
like a first-order dynamic system as shown on the top left hand corner of Figure 5.9. Obviously,
we would like the system respond similar to the case when K=0.35. As recall from the previous
section, the gain K is a function of Finished Inventory Adjustment Time (FAT), Lead Time (LT),
and WIP Adjustment Time (WAT). Plus, this fourth-order system equation has an initial non-
unity loop gain K of 2.109. Hence, in order to set K=0.35 from the rltool in MATLAB, we have
to multiply 0.35 by a factor of 2.109 to give 0.7383. The dynamic structure of this fourth-order
differential model is very similar to its previous 2nd-order system except for the third-order delay
term of 1/LT3. Again, in order to keep the same breakaway point while changing the K value, we
only vary K as a function of FAT. By setting FAT=2.8571, we make the system respond with a
settling time of 12 days at ζ = 0.67 and Kactual = 0.7383. Its resulting closed-loop poles are s1,2 =
-0.4521±0.4884i, s3,4= -1.2354±0.3747i. Given the same breakaway point, we can improve the
system response by changing the K value from 0.35 to 0.300676, thus FAT becomes 3.3258. The
production lead time reduces to 10 days with no oscillation with ζ = 0.83 and Kactual=0.6342. Its
corresponding closed-loop poles are s1,2= -0.6203±0.4148i, s3,4= -1.067.
The lead time has been reduced from 12 days to 10 days because the dominant pair of roots go
further left from the imaginary axis (i.e., from -0.45 to -0.62). In addition, the damping ratio
increases from 0.67 to 0.83, especially the improved roots, s3 and s4, are located on the real axis
with real parts to give fast response time. Finally, we can further reduce the production lead time
and system settling time by changing the parametric set of system variables to LT=1, ST=5,
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
WAT=1, and FAT=1.4. For this particular parametric set of values, we find a different initial
non-unity loop gain K of 54. The root-locus branches start from the open-loop poles with arrows
at s1=0, s2= -5.1042, s3,4= -2.5479±2.345i (K=0) and go towards the poles ending at K=13 as
shown in Figure 5.10.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2-4
-3
-2
-1
0
1
2
3
4
Real Axis
Imag
Axe
s
LT=1, ST=5, WAT=1, FAT=1.4
ζ=0.9
+
+
+
+
X
X X
X ζ=0.8
ζ=0.7
Figure 5.10: Root-Locus Plot of a two-stage production control system with 3rd-order time delay
It is shown that the root-locus plot in Figure 5.10 is very different from the Figure 5.8. All the
closed-loop poles can be found on the real axis depending on their selected K values. Whereas,
the pair of dominant poles in the previous case could never reach ζ>0.9 nor the real axis as gain
K varies. As mentioned, the starting open-loop poles location and its breakaway points are
determined by the given set of design system variables in terms of LT, ST, WAT, and FAT. By
changing the K value to 0.7143 with FAT=1.4, we make the system respond with ζ=0.89 and
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
Kactual =0.7143*54=38.57. Its corresponding closed-loop poles are located at s1,2= -1.5087
±0.7675i and s3,4= -3.5913±0.7513i. The resulting production lead time has significantly
improved and reduced from 10 days to 4.3 days. In order to verify the findings and the
recommended improved parametric set of system variables applying Root Locus technique, we
simulate and compare those four particular step responses of the two-stage production control
model with a 3rd-order time delay subjected to varied sets of design system parameters as shown
in Figure 5.11.
Set A: LT=4, ST=8, WAT=1, FAT=1; (Multiple oscillations with very long settling time)
Set B: LT=4, ST=8, WAT=1, FAT=2.8571; (Single Overshoot with reduced lead time, 12 days)
Set C: LT=4, ST=8, WAT=1, FAT=3.326; (No Overshoot with improved lead time, 10 days)
Set D: LT=1, ST=5, WAT=1, FAT=1.4; (No Overshoot, quick-response lead time, 4.3 days)
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
160
180
Time (Day)
Fini
shed
Inve
ntor
y (U
nit)
Set A
Set B
Set C
Set D
Figure 5.11: Various Step Response of a two-stage production system with 3rd-order time delay
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
We have demonstrated a new operation design method that permits production management to
better predict and design manufacturing system variables that yield responsive production lead
time and minimal inventory build-up (i.e., leanness) under transient manufacturing conditions.
This Root Locus design approach gives a completely graphic view to understand the sensitivity
on the location of the closed-loop poles by varying the loop gain K values to the dynamic system
models in the complex s-plane. We are able to determine responsive production parametric
values without iterative trial-and-error simulation as found in discrete event simulation or the
system dynamics approach.
The Root Locus design method we demonstrated here is based on varying the loop gain K values,
plus changing the manufacturing system parameters to reduce the time constant of the dominant
poles. In many industrial cases, however, the adjustment of the gain K values alone may not
provide sufficient alternation of the manufacturing system structure to meet the specific customer
requirement. As we learned from this section, as gain K values increases, it improves the steady-
state behavior but it could make the system in a poor stability region (even unstable) as K goes to
infinity. If it is the case, it may be necessary to redesign the dynamic manufacturing system
structure to alter the overall transient behavior to meet the management specification. Such a
redesign approach is called System Compensation technique found in classical control theory
[40,41,42]. We can easily perform this redesign approach using rltool from MATLAB by adding
selective poles and zeros into the original differential equation system model. In general, by
adding poles (more feedback loops) to the system, we destabilize the dynamic behavior with the
addition of production buffers. Whereas, adding zeros is similar to anticipating the customer
demand (i.e., forecasting with feedforward loops) to the system, in result, it has a stabilizing
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
influence. In root locus plotting, the gain K always begins at its open-loop location as K
increases until it reaches either to a zero location or infinity. System compensation is definitely
something we shall consider for responsive manufacturing systems applications in the near future.
5.4 Guidelines to perform Root Locus Design in Responsive Manufacturing Systems In this chapter, we have demonstrated how to apply Root Locus design technique from classical
control theory to improve the overall dynamic responsiveness of responsive manufacturing
systems according to the closed-loop poles locations on the complex s-plane. We here
summarize and give some basic guidelines for those whom may interest to apply Root Locus to
design and improve their particular manufacturing system dynamic behavior.
(1) Model the responsive manufacturing system into transfer function (i.e., differential equation
format).
(2) Apply Root Locus design technique to identify the root locus of the particular dynamic
manufacturing system behavior by varying the loop gain K values on a complex s-plane (you
can use MATLAB Control Toolbox to perform this function task).
(3) Select your desired closed-loop poles location (i.e., damping ratio in the range of 0.7-0.9)
and record your corresponding system loop gain K value.
(4) Adjust your key manufacturing system variables to match the same loop gain K value.
(5) Plot your resulting dynamic step response according to your selected parametric set values.
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Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong
(6) Shorten your production lead time by moving your dominant closed-loop poles location
further to the left-hand side of the complex s-plane and adjust your key manufacturing
system variables to match the new loop gain K value.
(7) Verify your dynamic step response behavior according to your selected parametric set values.
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Chapter 6 Industrial Case Study N.H. Ben Fong
Chapter 6 Industrial Case Study
The purpose of this chapter is to enhance the validation of applying classical control theory
(CCT) methodology to responsive manufacturing systems. By using a real industrial case study,
we aim to further validate the potential of using CCT approach to model, design, and improve
the overall responsiveness of manufacturing systems. We have chosen a hybrid push-pull
production system for semiconductor manufacturing to represent a particular Intel Corporation
plant facility as extracted from the proceedings of the 22nd International Conference of the
System Dynamics Society (July 2004) written by Goncalves et.al. [50]. In their industrial case
study, Goncalves, Hines, Sterman, and Lertpattarapong undertook a year-long, in-depth research
project to develop and analyze a manufacturing model of producing semiconductors via the
System Dynamics (SD) approach. Their case study addressed the causes of oscillatory behavior
in capacity utilization at a semiconductor manufacturer and the role of endogenous customer
demand in influencing the company’s production and service level. For more a detailed
understanding of this 41-pages case study, please read Goncalves et al. [50].
We intend to make use of this particular Intel hybrid push-pull semiconductor production system
in the following way:
(1) Demonstrate the use of classical control theory approach to convert this real industrial SD-
based semiconductor manufacturing system into block diagram representation and transfer
function.
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Chapter 6 Industrial Case Study N.H. Ben Fong
(2) Apply the Root Locus design technique to study the sensitivity of the closed-loop pole
locations and the dynamic behavior of this higher-order differential equation system model.
(3) Investigate any new findings and difficulties to design this industrial higher-order
responsive manufacturing system.
We will find that the CCT approach suggests new ways to control the studied production system
and to represent its improvement potential. This hybrid push-pull semiconductor production
consists of three stages (i.e., Fabrication WIP for Wafers, Assembly WIP for dies, and Finished
Inventory for chips) as shown in Figure 6.1. The push system is found at the upstream stages and
a pull system is at the downstream stages.
FabricationWIP (FWIP)
FinishedInventory
(FI)Wafer Start(WS) Net Wafers
Outflow (NWO)
Shipment Rate(SR)
Desired FinishedInventory (FI*)
Desired FabWIP (FWIP*)
Adjustment forFGI (AFGI)
Adjustment forFab WIP (AFWIP)
Desired WaferStart (WS*)
ManufacturingCycle Time (MCT)
FabWIPAdjustment Time
(FWAT)
+
-
FI AdjustmentTime (FIAT)
+
-
+
- +
-
+
AssemblyWIP (AWIP)
Net AssemblyOutflow (NAO)
Desired NetAssembly Outflow
(NAO*)
DesiredAssembly WIP
(AWIP*)
Adjustment forAWIP (AAWIP)
CompleteAssembly Time
(CAT)
AWIP AdjustmentTime (AWAT)
+
+
-
-
Die Inflow (DI)+
Desired Net WaferStart (NWS*)
Minimum OrderProcessing Time
(MOPT)
ExpectedShimpent
Rate (ESR)
UpdateShipments Time
(UST)
-
+
+
+
<Total Demand(TD)>
+-
++
+
-+
+
+
+-
+
Safety StockPercentage (SSP)
+
Figure 6.1: Hybrid Push-Pull Intel Semiconductor Production System
The push system characterizes the front-end: weekly updates from the total demand and the
adjustment from Fabrication and Assembly WIP serve as the basis for the desired wafer
production rate (i.e., Wafer Start). In contrast, the back-end operates as a pull system, with
assembly/testing, packing, and distribution based on current customer demand. Production
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Chapter 6 Industrial Case Study N.H. Ben Fong
decisions often rely on customer demand. The current customer demand drives the particular
shipment and its assembly completion, whereas the demand forecasts influence production start
rate. All the incoming orders are logged by the Intel’s information system and tracked until they
are shipped to customers or cancelled. If the finished products (i.e., microprocessors) are
available in Finished Inventory (FI), orders can be filled immediately. Hence, incoming customer
orders “pull” the available microprocessors from FI. Consequently, the replenishment of FI
shipped to customers “pull” microprocessors from the Assembly WIP (AWIP).
The current customer demand drives the pull characteristic of assembly WIP and finished
inventory. The actual shipment operates in a pull mode, with shipment being determined by the
desired rate. However, if there is not enough finished inventory, the system will ship out only
what it is available. The Finished Inventory (FI, units) is the accumulation of difference between
Net Assembly Outflow and Shipment Rate. The shipment rate (SR, units/month) depends on the
stock of FI and the minimum order processing time (MOPT, month) via a simple first-order
delay process. The expected shipment rate (ESR, units/month) is computed under the feedback
of the current shipment rate with a first-order delay of Update Shipment Time (UST, month).
The desired net chips (Adjustment finished inventory, AFI (units/month)) is adjusted above or
below recent shipment to close any gap between the desired finished inventory FI* (units) and
the actual FI proportional to the Finished Inventory Adjustment Time (FIAT, month). The
desired finished inventory is calculated by the product of ESR and MOPT with a safety stock
percentage (SSP) factor. The Desired Net Assembly Outflow (NAO*, units/month) is the
summation of AFI and ESR.
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Chapter 6 Industrial Case Study N.H. Ben Fong
At the Assembly WIP process stage, the Adjustment Assembly WIP (AAWIP, units/month) is
adjusted between the desired Assembly WIP (units) and the current Assembly WIP (units)
proportional to an Assembly WIP Adjustment Time (AWAT, month). The desired Assembly
WIP (AWIP*, units) is the product of desired Net Assembly Outflow (NAO*, units/month) and
Complete Assembly Time (CAT, month). The desired Net Wafer Start (NWS*, units/month) is
the summation of AAWIP (units/month) and the total demand by the customer (TD, units/month).
The stock level of Assembly WIP (AWIP, units) is the accumulation of difference between Die
Inflow (DI, unit/month) and Net Assembly Outflow (NAO, units/month). NAO is computed as a
first-order delay between Assembly WIP proportional to Complete Assembly Time.
The wafers produced in the fabrication process stage are pushed into the Assembly WIP where
they are stored until orders for specific products pull them from AWIP into Finished Inventory
for shipment. While the Net Wafer Outflow (NOW, units/month) depletes fabrication WIP
(FWIP, units), wafer start (WS, unit/month) replenishes it. The Net Wafer Outflow is a first-
order time delay of Manufacturing Completion Time (MCT, month) from the Fab WIP (FWIP,
units). The decision on actual production rate, WS, is based directly on the desired Wafer Start
(WS*, units/month). The Fab planners determine the desired wafer start considering the desired
Net Wafer Start (NWS*, units/month) requested by the Assembly Stage and an adjustment for
fabrication WIP (AFWIP, units/month). The AFWIP is calculated as the difference between the
current Fab WIP and the desired Fab WIP (FWIP*) proportional to a FWIP Adjustment Time
(FWAT, month). The FWIP* is the product of the desired Net Wafer Start (NWS*, units/month)
and the Manufacturing Completion Time (MCT).
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Chapter 6 Industrial Case Study N.H. Ben Fong
To simplify the non-linear mathematical expression for the non-negativity constraints to prevent
negative production at Wafer Start, Desired Net Wafer Start, and Desired Net Assembly Outflow,
we assume that there is no backlog in this industrial case study. The system variables, WS,
NWS*, and NAO* always have positive productive rates. For the detail descriptions of this case
study, please refer to Goncalves et al. [50].
Following the guidelines stated in Section 3.3, we now convert the cause-and-effect expressions
from CLDs and SFDs into different sets of system equations as shown in Figures 6.2, 6.3, 6.4.
Fabrication WIP Stage:
FWIP = WS - NWO∫
FWIPNWO = MCT
WS = Max (0, WS*) WS* = AFWIP + NWS*
FWIP* - FWIPAFWIP = FWAT
⎛ ⎞⎜ ⎟⎝ ⎠
FWIP* = (NWS*) (MCT)
Figure 6.2: System Equations for 1st Stage Process – Fabrication WIP
Assembly WIP Stage:
AWIP = DI - NAO∫
AWIPNAO = CAT
NWS* = Max (0, AAWIP +TD)
AWIP* - AWIPAAWIP = AWAT
⎛ ⎞⎜ ⎟⎝ ⎠
AWIP* = (NAO*) (CAT)
Figure 6.3: System Equations for 2nd Stage Process – Assembly WIP
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Chapter 6 Industrial Case Study N.H. Ben Fong
:
Finished Inventory Stage
FI = NAO - SR∫
FISR = M OPT
NAO* = Max (0, AFI + ESR)
FI* - FIAFI = FIAT
⎛ ⎞⎜ ⎟⎝ ⎠
FI* = (ESR) (MOPT) (SSP)
ESR = Delay1(SR, UST)
Figure 6.4: System Equations for 3rd Stage Process – Finished Inventory
By constructing sets of functional blocks according to the above system equations, we integrate
and link each of those functional blocks to generate our complete block diagram representation
of a three-stage semiconductor production system as shown in Figure 6.5
1MCT
1FWAT
1MCT
1/s 1/s
1CAT
1MOPT
1AWAT
1MOPT
1FIAT
11+(UST) s
11+(UST) s
MCT
SSP
CAT
TD NWS*FWIP* FWIP NWO
WS
FWIP
AWIP
NAOAWIP*
AAWIP NAO
FISR
ESR
FI
FI*
NAO*
+
+ +
++ +
− − −
−
NWO
+
+
+
+
+
−
− AWIP
1CAT
1/s
SR+
AFI
Figure 6.5: Block Diagram Representation of a Three-Stage Semiconductor Production System
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Chapter 6 Industrial Case Study N.H. Ben Fong
To simplify the algebraic expression from Figure 6.5, we use basic capital letters (A,B,C, etc.) to
represent individual key production system variables. In addition, we apply the block diagram
reduction technique to make the mathematical relationship between Net Wafer Outflow (NOW)
and Desired Net Wafer Start (NWS*) to the following:
( )
( )A + ENWO
NWS* AEs + A + E=
where A = MCT
B = CAT
C = MOPT
D = UST
E = FWAT
F = AWAT
G = FIAT
V = A E+
H = SSP
1C
1F
1G
1Ds + 1
H
B
TD AWIP
AWIP*
AAWIP
FI
ESR
FI
FI*
NAO*
+
−
+
+
+
+
− AWIP
SR+
AFI
VAEs + V
BBs + 1
CCs + 1
1B
1Ds + 1
Figure 6.6: Simplified BD Representation of a Three-Stage Semiconductor Production System
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Chapter 6 Industrial Case Study N.H. Ben Fong
After algebraic simplification, the block diagram representation of a three-stage semiconductor
production system is shown in Fig. 6.6. We again apply the block diagram reduction technique to
reduce the algebraic expression into a single, 4th-order differential equation, transfer function as
stated as below:
( )4 3 2
V Ds + 1FI ABDETD s [I] s [II] s [III] s + [IV]
⎛ ⎞⎜ ⎟⎝ ⎠=
+ + + (6.1)
where
[ ]1[I] = ADE(B+C) + BC(AE+DV)ABCDE
( )1 V[II] = AE B+C+D DV(B+C) BCV + ABCDE F
⎡ ⎤+ +⎢ ⎥⎣ ⎦
BCD
1 VBC[III] = V(B+C+D) + AE + 1ABCDE F C G
⎡ ⎤⎛ ⎞+ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
D D
( )1 VBC[IV] = V + 1 HABCDE FG
⎡ ⎤−⎢ ⎥⎣ ⎦
We observe a new finding from equation (6.1) that there is a Laplace s term found in the
numerator of the transfer function. From the classical control theory [40,41,42], we know that a
zero exists whenever the differential equation contains numerator dynamics. This particular
numerator dynamic is caused by the first-order delay from the expected shipment rate while the
finished inventory (FI) is sending back as a velocity feedback (or tachometer feedback) through
its derivative action. In manufacturing terms, a zero is acting like a forecast for product demand.
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Chapter 6 Industrial Case Study N.H. Ben Fong
It usually improves the stability of the manufacturing system as the varying loop gain K
increases until it reaches the location of a zero.
Next, we apply the Root Locus design technique per Section 5.4 guidelines, to study the
sensitivity of the closed-loop poles location of this three-stage production control system. Given
the complexity of this, multiple feedback and forward loops, system transfer function as stated in
eq. (6.1), it is not an easy task to derive its corresponding open-loop system transfer function for
Root Locus analysis. Instead, we can add a gain block K in the simplified block diagram from
Fig. 6.6, such that we can adjust the gain K value to bring the closed-loop poles to certain desired
locations. The simplified system block diagram with an additional block K is displayed in Figure
6.7. Referring to eq. (5.2) from Section 5.1, we assume that G(s) comprises both the compensator
transfer function and the plant transfer function. The characteristic equation for the system is
computed as follows, where H(S) =1:
1+KG(s) H(s) = 0 (6.2)
We can algebraically derive the new system transfer function from the block diagram in Figure
6.7. Its corresponding characteristic equation, G(s) for this new system is computed as in eq.
(6.3).
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Chapter 6 Industrial Case Study N.H. Ben Fong
1C
1F
1G
1Ds + 1
H
B
TD AWIP
AWIP*
AAWIP
FI
ESR
FI
FI*
NAO*
+
−
+
+
+
+
− AWIP
SR+
AFI
VAEs + V
BBs + 1
CCs + 1
1B
1Ds + 1
K
Figure 6.7: Simplified BD Representation with block K of the Three-Stage Semiconductor System
( )
[ ] [ ] [ ] [ ]
2
4 3 2
1- HV 1 1 1s + s AEF C D G DG
G(s) = s + s + s + s +δ ε λ σ
⎡ ⎤⎛ ⎞+ + +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
(6.3)
where
( ) (1 = ADE B+C + BC AE+DVABCDE
δ ⎡ ⎤⎣ ⎦) ;
( ) ( )1 = AE B+C+D DV B+C BCVABCDE
ε ⎡ ⎤+ +⎣ ⎦ ;
( )1 = V B+C+D AEABCDE
λ ⎡ ⎤+⎣ ⎦ ;
V = ABCDE
σ
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Chapter 6 Industrial Case Study N.H. Ben Fong
There are total of four closed-loop poles (roots) found in this particular system. In addition, there
are two numerator dynamics (i.e., zeros) found according to where we added the gain block K.
Assume that production management wants to investigate the overall production lead time and
its dynamic behavior to manufacture a target of 5000 chips. Given the four different sets of
system parameters shown in Table 6.1, we use MATLAB [49] to investigate each individual step
response dynamic behavior and its corresponding root loci. These system parameters are
disguised to maintain confidentiality for Intel Corporation. Figure 6.8 shows the step response of
the three-stage semiconductor production system with four different sets of system parameters,
labeled as Set A, Set B, Set C, and Set D.
Table 6.1: Parametric Values for a three-stage semiconductor production system
SET A SET B SET C SET DA = MCT (month) 1 1 1 1B = CAT (month) 0.1 0.4 0.1 0.1C = MOPT (month) 7 0.7 1.3 0.94D = UST (month) 0.2 0.5 0.2 0.1E = FWAT (month) 1 0.2 1 1F = AWAT (month) 0.1 0.2 1 0.4G = FIAT (month) 0.3 0.3 0.3 0.4H = SSP 1.1 1.1 1.1 1.1TD = 5000 Chips
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Chapter 6 Industrial Case Study N.H. Ben Fong
0 5 10 15 20 25 30 35 40 45 500
1000
2000
3000
4000
5000
6000
7000N
umbe
r of C
hips
Time (month)
SET A
SET B
SET C SET D
Figure 6.8: Step Response of a three-stage production system with different parametric set values
As shown in Figure 6.8, Set A curve yields a sluggish dynamic response (i.e., overdamped with
ζ>1) with an offset of 400 chips less than the target 5000 units as it reaches the steady-state at
time 20 months. Set B curve responds with a less rising slope but gives a surplus of 1400 chips at
time 50 months. Set C curve gives a much steeper rising slope and it reaches its steady-state at
time 10 months. Unfortunately, the system passes the target value with a surplus of 1800 chips.
Set D curve yields the fastest rising slope with no surplus made. It reaches the target value of
5000 chips at time 8 months. Among all four different parametric sets values, Set D gives the
best overall dynamic response that leads to shorter production lead time, minimal WIP overshoot,
fastest rising slope at the transient period. Figures 6.9-6.10 show the corresponding root loci
from the parametric set values of Set B and Set D.
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Chapter 6 Industrial Case Study N.H. Ben Fong
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2-8
-6
-4
-2
0
2
4
6
8
Real Axis
Imag
Axe
s
ζ=0.9
X X X X
ζ=0.8 ζ=0.7
Figure 6.9: Root Locus of a three-stage semiconductor production system (Set B)
-15 -10 -5 0 5-15
-10
-5
0
5
10
15
Real Axis
Imag
Axe
s
ζ=0.9
X X X
ζ=0.8
ζ=0.7
Figure 6.10: Root Locus of a three-stage semiconductor production system (Set D)
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Chapter 6 Industrial Case Study N.H. Ben Fong
Figure 6.9 shows that there are two zeros found on the root locus plot, located at -6.7 and +0.15
respectively. The location of the four closed-loop poles with K=0 begins at s1 = -6.0, s2 = -2.5, s3
= -2.0, s4 = -1.4286 (i.e., all real roots on the real-axis). The arrows indicate the direction of the
four closed-loop poles movement as the gain K values varies from 0 to infinity. The breakaway
points of the pair of complex conjugate roots are located at -2.273. The major difference between
this three-stage semiconductor production system from the previous two-stage production system
with a 3rd-order delay is that the two dominant closed-loops (i.e., closest to the imaginary axis)
will terminate their movement once they reach the zeros at -6.86 and +0.097 respectively. By
applying rltool from Matlab, we determine that as K>2.142, the dominant closed-loop pole will
reach a positive value on the real axis. Hence, the manufacturing system could lead unstable or
marginally stable response.
We can improve the overall dynamic response by moving the breakaway point and the dominant
poles further away from the imaginary axis as shown in Figure 6.10. The location of the four
closed-loop poles are found at s1 = -11, s2,s3 = -10, and s4 = -1.0638 when K=0. As gain K
increases, the repeated roots, s2 and s3 breakaway from each other and stay with the same real
value of -9.1345 with increasing complex conjugate values as K continues to increase. The two
zeros are located at -13.7 and +0.182. By comparing Figure 6.9 to Figure 6.10, the breakaway
point of the complex conjugate pair has moved from -2.273 to -10. Hence, Set D responds much
faster than Set B with a shorter production lead time. Again, the arrows indicate the direction of
the four closed-loop poles movement as the gain K values varies from 0 to infinity Although all
the closed-loop poles are located at the right-half plane, if gain K goes beyond 17, the
manufacturing system will behave unstable or marginally stable. Obviously, it takes much higher
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Chapter 6 Industrial Case Study N.H. Ben Fong
gain K value to bring the manufacturing system to be unstable, hence Set D is the best parametric
set values among the four choices.
We address that the root locus of any dynamic system could behave differently according to the
system parametric set values chosen. For instance, if we chose the following parametric set