Manufacturing System Research from a Design Point of View:
Optimization vs. System Design
by
ZHENWEI ZHAO
B.S., Automotive Engineering, 1998
Tsinghua University
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
June 2002
0 2002 Massachusetts Institute of Technology
All rights reserved
A uth o r ...................................................................................................
Department of Mechanical Engineering
May 10, 2002
C ertifie d by .................................................................
.
.
....................
David S. Cochran
Associate Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by .................................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
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Manufacturing System Research from a Design Point of View:
Optimization vs. System Design
by
ZHENWEI ZHAO
Submitted to the Department of Mechanical Engineering
on May 10, 2002 in partial fulfillment of the requirement for
the degree of Master of Science in Mechanical Engineering
ABSTRACT
Research methodologies in manufacturing system can be generally divided into two
groups: optimization methodologies and system design methodologies. Optimization
methodologies study the abstracted mathematical models of real systems and aim to find
the optimal solutions; while system design methodologies study system requirements and
aim to design solutions to meet these requirements.
Industrial practice has shown that manufacturing system design based on optimal
solutions often lead to poor overall system performance, however, the reason for this is to
a large extent unclear. This thesis analyzes typical optimization models in a system
design point of view. It shows that since these models apply insufficient number of DPs
to satisfy system FRs, the FRs cannot be fully fulfilled. Some of the system FRs are
hence compromised and overall system performance sacrifices. Modified designs are
presented based on axiomatic design methodology and MSDD to fully achieve all system
FRs.
A comparison of mathematical model based manufacturing system analysis
methodologies and system design methodologies is conducted. The result shows that
mathematical models analysis are consistent with manufacturing system design
framework MSDD, therefore the DPs provided by MSDD are supported by mathematical
analysis. It is also pointed out that the analysis models strongly rely on their assumptions
so that the analysis results may become inapplicable when system changes. MSDD is
based on decoupled decomposition from general system requirements; therefore it is
robust and applicable to a wide range of manufacturing systems.
Thesis Supervisor: David. S. Cochran
Title: Associate Professor of Mechanical Engineering
3
4
Acknowledgements
Looking back to the nearly two years that I have spent in MIT, I feel I was so lucky to be
part of the Production System Design Lab of MIT. The lab has provided everybody a
warm, friendly and happy atmosphere. Everybody I met here became my good friend and
each of them had given me invaluable help to learn from and involve in this different
country and culture.
First of all, I want to specially thank my thesis supervisor, Prof. David Cochran. It has
been the most rewarding experience in my life to learn from him, the insights on both
academia and life. Without his consistent trust and support, I would have collapsed many
times when I was feeling desperate, let alone to finish this thesis.
I would also want to thank all friends that I have been working with in PSD lab, Jochen,
Jongyoon, Partic, Jey, Kola, Keith, Quinton, Yongsuk, Steve, Carlos, Memo, Jose, Cesa,
Abhinav, Martin and Henning. I also want to show my appreciation to Pat for her
kindness help on so many things.
5
6
Table of Contents
ABSTRACT ......................................................................................................
3
ACKNOW LEDGEMENTS.................................................................................
5
TABLE OF CONTENTS...................................................................................
7
LIST OF FIGURES............................................................................................
9
LIST OF TABLES ...........................................................................................
12
CHAPTER 1: INTRODUCTION .....................................................................
13
1.1 M otivation..............................................................................................................
13
1.2 Thesis Outline........................................................................................................
14
CHAPTER 2: EVOLUTION OF MANUFACTURING SYSTEM AND RESEARCH
17
METHODOLOGY............................................................................................
2.1 The History of M anufacturing System .................................................................
17
2.2 M anufacturing System Design Framework..........................................................
2.2.1 Systematic Approach for Manufacturing System Design................
2 .2 .2 A x iomatic D esign ............................................................................................
2.2.3 Manufacturing System Design Decomposition..................................................
26
26
27
31
CHAPTER 3: INVENTORY AND PRODUCTION CONTROL MODELS FROM
43
SYSTEM DESIGN POINT OF VIEW .............................................................
3.1 General Introduction of Optimization M ethodologies .......................................
43
3.2 Inventory and Production Control M odels .......................................................
3.2.2 Economical Order Quantity Model ...................................................................
3.2.3 The Newspaper Vender Model .......................................................................
3 .2 .4 (Q , r) M o del ....................................................................................................
3 .2 .5 (s ,S ) M o d e l .......................................................................................................
46
46
49
53
59
3.3 Analysis of Inventory and Production Control M odels .....................................
3.3.1 General Analysis on Inventory and Production Control Models .....................
3.3.2 Analysis of EOQ Model ..................................................................................
3.3.3 Analysis of Newspaper Vender Model............................................................
63
63
64
67
7
3.3.4 A naly sis of (Q ,r) M odel....................................................................................
3.3 .5 A naly sis of (s, S) M odel ...................................................................................
3 .3 .6 C on clu sion .......................................................................................................
3.4 Applying System Design Methodology to Solve Optimizing Problem........
3 .4 .1 P rob lem D escription .........................................................................................
3.4.2 Solution based on Mathematical Optimization Methodology .........................
3.4.3 Solution based on System Design Methodology................................................
70
75
78
79
79
82
84
CHAPTER 4: MANUFACTURING SYSTEM ANALYSIS BASED ON
STOCHASTIC MODELS FROM SYSTEM DESIGN POINT OF VIEW........... 97
4.1 Introduction of stochastic models for manufacturing system analysis ........
4.1.1 Introduction of M arkov process ........................................................................
4.1.2 Discrete Time Discrete State Markov Process .................................................
4.1.3 Continuous Time Discrete State Markov Process ............................................
97
97
99
102
4.2 Transfer line analysis based on stochastic models .............................................
4.2.1 Introduction of Transfer Line Analysis ...........................................................
4.2.2 Zero Buffer and Infinite Buffer Model ............................................................
4.2.3 General Assumptions of Finite Buffer Transfer Line Analysis ........................
4.2.4 Deterministic Two-Machine Line ...................................................................
4.2.5 Exponential Two-Machine Line......................................................................
107
107
108
111
112
116
4.3 Stochastic Model Analysis in a System Design Point of View .............
4 .3 .1 Lin e B alan cin g ...............................................................................................
4 .3 .2 H igh R ep airing R ate .......................................................................................
4 .3 .3 Low F ailure Rate ............................................................................................
4 .3 .4 Role of A uton om ou s.......................................................................................
4.4.4 Summary on Manufacturing System Analysis.................................................
117
1 18
122
12 5
12 6
129
CHAPTER 5: CONCLUSION...........................................................................
131
REFERENCES.................................................................................................
133
8
List of Figures
FIGURE 2-
1: SCHEMATIC OF A TYPICAL HIGH-SPEED LINE LAYOUT OF ASSEMBLY-TYPE
2
M AN UFACTU RING SY STEM ......................................................................................
1
FIGURE 2- 2: SCHEMATIC OF ATYPICAL DEPARTMENTAL LAYOUT OF MACHINING-TYPE
22
M AN UFACTU RING SY STEM ......................................................................................
FIGURE 2- 3: EVOLUTION OF PRODUCTION VOLUME AND PRODUCT VARIETY OF
AUTOMOTIVE INDUSTRY [WOMACK 1990].............................................................
23
FIGURE 2- 4: COMPARISON OF US AND JAPANESE AUTOMOTIVE PRODUCTION BETWEEN
24
1947 AND 1989 [W OM ACK 1990] ..........................................................................
FIGURE 2- 5: MAPPING BETWEEN CUSTOMER DOMAIN, FUNCTIONAL DOMAIN AND PHYSICAL
DOM AIN [M ODIFIED FROM SUH 1990] ...................................................................
28
FIGURE 2- 6: ZIGZAGGING PROCESS OF MULTI-LEVEL DESIGN DECOMPOSITION [MODIFIED
..
FRO M S U H 19 9 0 ] .................................................................................................
29
FIGURE 2- 7: SIx M SDD BRANCHES [LINCK 2001] .......................................................
32
FIGURE 2- 8: M SDD STRUCTURE [LINCK 2001]............................................................
33
FIGURE 2- 9: QUALITY BRANCH OF M SDD ...................................................................
34
FIGURE 2- 10: PROBLEM IDENTIFYING AND RESOLVING BRANCH OF MSDD..................
35
FIGURE 2- 11: PREDICTABLE OUTPUT BRANCH OF MSDD ............................................
37
FIGURE 2- 12: DELAY REDUCTION BRANCH OF MSDD .................................................
39
FIGURE 2- 13: THE OPERATION COST BRANCH OF MSDD ..............................................
41
FIGURE 2- 14: THE INVESTMENT BRANCH OF M SDD......................................................
42
FIGURE 3- 1: RELATIONSHIP BETWEEN ORDER QUANTITY AND AVERAGE COST IN EOQ
49
MO D E L ...................................................................................................................
FIGURE 3- 2: THE EFFECT OF DIFFERENT PRODUCTION COST ON OPTIMAL PRODUCTION
VO LUM E ...........................................................................................................
.....
52
FIGURE 3- 3: THE EFFECT OF DIFFERENT SALVAGE VALUE AND OPTIMAL PRODUCTION
V O LUM E .................................................................................................................
FIGURE 3- 4: THE COST STRUCTURE OF EOQ MODEL.......................................................
9
53
64
FIGURE 3- 5: DECOMPOSITION ANALYSIS OF EOQ MODEL .............................................
65
FIGURE 3- 6: M ODIFIED DESIGN FOR EOQ MODEL .........................................................
67
FIGURE 3- 7: COST STRUCTURE OF NEWSPAPER VENDER MODEL .....................................
67
FIGURE 3- 8: DECOMPOSITION ANALYSIS OF NEWSPAPER VENDER MODEL ......................
68
FIGURE 3- 9: MODIFIED DESIGN FOR NEWSPAPER VENDER MODEL .................................
70
FIGURE 3-
10: COST STRUCTURE OF (Q,R) MODEL ..........................................................
71
FIGURE 3- 11: DECOMPOSITION ANALYSIS FOR (Q,R) MODEL ..........................................
72
FIGURE 3- 12: Two STEPS CHANGEOVER TIME REDUCTION [COCHRAN 2002] ................
74
FIGURE 3-
13: M ODIFIED DESIGN FOR (QR) MODEL ......................................................
75
FIGURE 3-
14: THE COST STRUCTURE OF (S,S) MODEL ....................................................
76
FIGURE 3- 15: DECOMPOSITION ANALYSIS OF (S ,S) MODEL ............................................
FIGURE 3-
16: M ODIFIED DESIGN FOR (S,S) MODEL .........................................................
FIGURE 3- 17: SUPPLY CHAIN FOR APPAREL PRODUCTION................................................
77
78
80
81
FIGURE 3-
18: SUPPLY CHAIN PRODUCTION TIMELINE.....................................................
FIGURE 3-
19: INTERACTIVE SOLVING RESULT OF OPTIMAL COST SEARCHING [CARO ET AL,
2 0 0 1 ] .....................................................................................................................
FIGURE 3- 20: RETAILER-MANUFACTURER INTEGRATION ..............................................
84
87
FIGURE 3- 21: COMPARISON OF PRODUCTION TIMELINE BETWEEN OLD AND NEW DESIGN . 88
FIGURE 3- 22: HIGH-LEVEL DECOMPOSITION OF NEW DESIGN .........................................
89
FIGURE 3- 23: PRODUCTION LEADTIME REDUCTION DESIGN .............................................
89
FIGURE 3- 24: SUPPLIER LEADTIME REDUCTION DESIGN ................................................
90
FIGURE 3- 25: SCHEMATIC OF TRANSPORTATION WASTE IN OLD SUPPLY CHAIN ................
92
FIGURE 3- 26: MATERIAL FLOW ORIENTED TRANSPORTATION DESIGN .............................
92
FIGURE 3- 27: MANUFACTURER LEADTIME REDUCTION DESIGN .....................................
93
FIGURE 3- 28: DESIGN DECOMPOSITION OF SUPPLY CHAIN OPTIMIZATION PROBLEM ......... 95
FIGURE 4-
1: TRANSITION PROBABILITY OF DISCRETE TIME TWO-STATE MARKOV PROCESS
............................................................................................................................
FIGURE 4- 2: ASYMPTOTIC BEHAVIOR OF MACHINE STATUS PROBABILITY DISTRIBUTION
1 00
101
FIGURE 4- 3: TRANSITION PROBABILITY OF CONTINUOUS TIME TWO STATE MARKOV
PR O C E S S .............................................................................................
10
. ...............
10 4
FIGURE 4- 4: RELATIONSHIP BETWEEN THE DELAY OF A M/MI QUEUE WITH ARRIVAL RATE
............................................................................................................................
1 07
FIGURE 4- 5: SCHEMATIC OF A TRANSFER LINE .............................................................
107
FIGURE
4- 6: RELATIONSHIP BETWEEN LINE EFFICIENCY AND BUFFER SIZE [GERSHWIN
1 9 94 ] ...................................................................................................................
FIGURE 4- 7: WORK LOOPS IN A CELLULAR LAYOUT MANUFACTURING SYSTEM..............
115
119
FIGURE 4- 8: MMC OF WORK LOOP DESIGN WITH
10 OPERATORS [OROPEZA 2001] ........ 120
FIGURE 4- 9: MMC OF WORK LOOP DESIGN WITH
14 OPERATORS [OROPEZA 2001] ........ 121
FIGURE 4-
10: ACTION CHAIN WITH MULTIPLE CONNECTIONS ........................................
122
FIGURE 4-
11: EXAMPLE OF ANDON BOARD ..................................................................
123
FIGURE 4-
12: SHORTENED ACTION CHAIN WITH ONE CONNECTION ................................
124
FIGURE 4- 13: EXAMPLE OF STANDARDIZED WORK SHEET [COCHRAN 2002] ..................
FIGURE 4-
14: EXAMPLE OF STANDARDIZED PREVENTIVE MAINTENANCE SHEET [COCHRAN
2 0 0 2 ] ...................................................................................................................
FIGURE 4-
12 6
15: TOYOTA PRODUCTION SYSTEM DESIGN MODEL [COCHRAN 1999] ............ 127
FIGURE 4- 16: EXAMPLE OF POKE-YOKE [LOW, 2001] ..................................................
11
125
129
List of Tables
TABLE 2-
1: SUMMARY OF MAIN INNOVATIONS IN THE HISTORY OF MANUFACTURING
SY STEM S [C OCHRAN , 1994]....................................................................................
26
TABLE 2- 2: REPRESENTATIONS OF DIFFERENT TYPE OF DESIGN [LINCK 2001] .............
30
TABLE 3-
1: COMPARISON OF INVENTORY AND PRODUCTION CONTROL MODELS ............... 79
TABLE 3- 2: M ANUFACTURER' S FORECAST DATA ..........................................................
86
TABLE 3- 3: ESTIMATED IMPLEMENTING COSTS OF LEAF-LEVEL DPS .............................
93
1: TRANSIENT STATS IN A TWO-MACHINE TRANSFER LINE MODEL ...................
112
TABLE 4-
12
Chapter 1: Introduction
1.1 Motivation
Methodologies that have been used in manufacturing system analysis and design areas
can be generally categorized into to groups: optimization and system design. Many
different optimization methodologies have been used to analyze manufacturing system
performance and design optimal system control policies, including linear programming,
non-linear programming, dynamic programming, variation analysis, network analysis,
etc. Mathematical models are established based on deterministic, statistical or stochastic
analysis according to different views of system and modeling assumptions. Constraints
and decision variables are then identified; and finally optimization algorithms are applied
to solve the optimal solutions.
System design methodology, on the other hand, approaches the manufacturing system
problems in a design point of view. It starts from the customer needs, which are the origin
of system design functional requirements (FRs). Design process proceeds to find out
design parameters (DPs) to satisfy FRs. Decomposition approach may be applied to break
down high-level design intents into implementable design parameters. If there is a
constraint that refrains the customer needs from being realized, the system will be
modified to eliminate it to ensure the fulfillment of customer demand.
Therefore, optimization approach and system design approach are logically different in
that the former admits the constraints and optimizes the system output, while the latter
tackles the constraints to ensure the system output can meet customers' needs.
Optimization methodology based research in manufacturing system usually addresses
specific problems by studying mathematical models. This incurs two possible problems:
First, manufacturing system is a complex system that includes many hierarchies and
relationships. Looking at a local problem without considering its relationship with other
elements in the system will result in local optimization instead of superior performance of
the overall system. Second, most models that have been used rely heavily on some strong
assumptions, which are not generally valid in real situations. This causes most of the
optimal solutions lose their optimality when systems change.
13
In spite of the shortcomings they have, optimization approach is still important for
research in manufacturing system. The optimization results usually can provide valuable
insights for system design. It also quantitatively validates the ideas and intuitions that
being used in conceptual design.
This thesis attempts to review these optimization-based methodologies, analyze them in a
system design point of view, find out their inferiorities and limitations in a design point
of view, and develop ways to interface between two approaches to achieve better system
design.
1.2 Thesis Outline
The thesis includes three major chapters. Chapter 2 is the first part that serves as a
general review of the evolution of manufacturing system and research methodologies.
The chapter begins with a review of manufacturing system evolution since the first
industrial revolution till later 20th century. The review shows that manufacturing system
has evolved through a history from simple to complex, form process oriented to system
oriented. A discussion of manufacturing system evolution explains the reason of the
emergence and prosperity of optimization methodologies in old style production
environment and their declination during the postwar period. The changes in technology
development and international markets of manufacturing industry put much higher
requirements onto manufacturing system than any other time in the history. Modem
manufacturing systems' complexity and dynamic characters decide the optimization
methods, which view the system statically and base themselves on many oversimplified
assumptions, will never work. Manufacturing system has to be designed and operated in a
systematic manner.
Axiomatic design approach is then introduced as a fundamental methodology that is
going to be frequently applied into the later analysis of this thesis. Manufacturing System
Design Decomposition (MSDD) is discussed with considerable detail. MSDD will serve
as a knowledge base for optimization model analysis in the future chapters.
Chapter 3 discusses in depth the optimization methodologies that have been widely
applied in manufacturing system research. The scope of this chapter focuses on the
14
inventory and production control models The reason is that these models have been most
commonly applied in guiding manufacturing system design, and also they are good
representatives of optimization methodologies in that these models can cover most of the
optimization modeling and solution solving techniques. Four models ranging from
deterministic to stochastic, from one-parameter to multiple-parameter, are discussed and
analyzed in detail. The analysis shows the in a design point view, all these models are
coupled design in that they are trying to satisfy FRs with insufficient number of DPs.
Therefore the optimal solution is a compromising of system FRs and the result is far from
optimal seen from the system level. Modified designs are suggested for each model based
on axiomatic design methodology and MSDD to convert coupled unacceptable designs to
decoupled designs. A case study is presented at the end of this chapter. The case is a
classical supply chain management problem, which has been used as a practice of
applying optimization methodologies to solve supply chain conflicts. Solutions based
system design methodology as well as optimization methodology are derived. The
comparison of the two shows that the system design can reach a much better solution
than optimization methods.
Chapter 4 studies a widely used methodology for manufacturing system analysis transfer line models based on stochastic process. Two fundamental types of stochastic
models, discrete time discrete state and continuous time discrete state, are discussed in
detail, and the analysis results derived from these models are studied.
The study of these analysis results and comparing them with MSDD design framework
shows that the mathematical analysis is perfectly consistent with system design
methodology. While the model analysis shows "what does an good system need", MSDD
provides the solution of "how to achieve a good system." The limitation of stochastic
models is also discussed. Since the models strongly rely some of their assumptions, their
analysis may not apply when system changes. MSDD on the other hand, is based on
decoupled decomposition from high-level system requirements. Therefore it is robust and
generally applicable in a wide range of manufacturing systems
15
16
Chapter 2: Evolution of Manufacturing System and
Research Methodology
2.1 The History of Manufacturing System
The modem industry started in England when James Watt invented and sold his first
steam engine during the mid- 18th century. Prior to that, manufacturing was small scale
and for local and very limited market. Manufacturing work was normally carried out in
two systems, the domestic system and craft guilds. In domestic system, no professional
facility existed at all. Jobs were distributed to people's home where they finished them
and then "sell back" to the merchant. In the craft guilds system, specialists with
professional skills passed jobs sequentially among different shops. For example, in order
to make leather products, it could first be sent to a tanner to get tanned, then to curriers
and finally to suitcase maker or shoemaker.
The first industrial revolution dramatically changed the manufacturing processes of
human being. Numerous machines and manufacturing methods has been invented and
developed in that period, which greatly improved both manufacturing productivity and
variety of goods that people could make [Hopp, Spearman 2001]. These prominent
technological advances including the flying shuttle developed by John Kay in 1733, the
spinning jenny invented by James Hargreaves in 1765, and the water frame developed by
Richard Arkwright in 1769. By facilitating capital for labor, these innovations firstly
brought the manufacturing industry the economies of scale that greatly promoted
centralized production.
The industrial revolution in America was a little later than that of in European. Due to
England's the technology protection to keep its competitive advantages over most of the
other countries, it was not until in 1790s that the first advanced textile machine appeared
in America. Moses Brown established the first textile mill in 1793 at Pawtucket, Rhode
Island, which was memorized as the famous "Rhode Island System". The system, which
was originally an exact mimic of their English predecessor however, evolved in a
different way that the English system did. By the 1820s, the American system
distinguished itself from the English system by having consolidated and integrated many
17
different production processes in the same manufacturing facility, which was latterly
referred as vertical integration.
Vertical integration became popular in American manufacturing plants due to two
reasons:
1. Unlike England, American had no strong tradition of craft guilds. Therefore the
American manufacturing production relied primarily on the domestic system,
which required no specific skills for production people. This resulted that the
American system didn't have barriers among people with different crafts that the
English system had, therefore it was much easier to realized vertical integration.
2. America started its production industry based on waterpower in 18th and
19 th
centuries. The steam engine, which had become popular in England at that period
of time, did not replace the wide-use waterpower until the Civil War. The
manufacturing plants were usually built close to the waterpower wheel, which
sent the power to the plant by a spinning shaft. This power input configuration
essentially generated a layout constraint of the plants. It is desired for the plants to
put all their machines as close to the wheel draft as possible, which necessarily
facilitated the integration of manufacturing processes.
The second fundamental step in the American manufacturing system evolution after
vertical integration is the production of interchangeable parts. This concept was
developed and had been widely used in American manufacturing industries during the
mid and late
19 th
century. The 1851 Crystal Palace Exhibition in London witnessed a
display of American products such as locks, repeating pistol and mechanical reaper, all
produced with interchangeable parts [Hopp, Spearman 2001].
Eli Whitney and Simeon North first proved the feasibility of the concept of
interchangeable parts. They contracted to produce 10000 muskets for the American
government in 1801. Although it took them 9 years to finish the production, Whitney and
North showed indisputably that the interchangeable parts, which they called "uniform
system", worked.
18
It is difficult to overstate the importance the role of interchangeable parts in the history of
America. Boorstein [1956] called it " the greatest skill-saving innovation in human
history." The concept of interchangeable parts essentially decoupled the processes from
operator skills, therefore greatly reduced the need for experienced worker with special
skills, which made the large-scale mass-production possible. Under the American
manufacturing system, workers without special skills can make very complex parts by
producing interchangeable products in numerous consecutive processes, each of which
requires simple operational skills. This early rise of undifferentiated worker directly led
to the history of labor relations in America. It also paved the way for the separation
between management and execution in the early
2 0 th
century.
In spite of the great achievements in the textile industry in 18th and
1 9 th
century, most
industry before 1840s was in small scale. One of the important reasons was the
waterpower supply that most industry used. Since there was huge seasonal variations in
the power supply itself, the workers were mostly part time and the class of permanent
works was very small and the class of professional management hardly existed. A survey
on American manufacturing system conducted by the Secretary of Treasure in 1832
pointed out that in 10 states that the survey had covered, only 36 enterprises with 250 or
more workers, of which 31 were textile factories. The majority of enterprise had only a
few thousand dollars of assets and a dozen employees. This situation was finally broken
by the second industrial revolution in American, which started with using new industrial
energy and development of mass transportation means.
Railroad were the spark for the second industrial revolution. Colonel John Stevens
received the first railroad charter from the government in 1815. By 1890, the total
railroad in America has reached 199,876 miles, 72,473 of while were west of Mississippi.
Unlike in the eastern, the western railroads were general built in sparsely populated states
and tried to connect to the anticipated places for future development [Hopp, Spearman
2001].
Railroad building had led to great changes in American production industry. Since the
capital needed to build railroads was far greater than that required to build a textile
factory, also, because of the complexity and the distributed nature of its operations, many
19
stakeholders of the railroad companies were not directly managing the operations.
Therefore for the first time in the history, a new class of salaried employees - middle
management - emerged in American industries. Also because of the complexity of the
railroad operational system, large amount of data needed to be collected and analyzed.
This caused the emergence of technical analysis and accounting agents. In the large scale
production as railroad industry and later the mass retailers, cost was viewed as the
extreme important factor. While the railroad industry focused mainly on ton -mile cost
ratios, the mass retailers used gross margins. Examples of these early accounting
practices include: Marshall Filed was tracking inventory turns as early as in 1870
[Johnson and Kaplan 1987], and maintained an average of between five and six turns a
year during 1870s and 1880s [Chandler 1977].
Large-scale production in American began from steel industry and introduced by Andrew
Carnegie, who started his career in steel industry in 1872. He combined the new process
technologies and management methods together and brought the steel industry to an
unprecedented level of integration and efficiency. He named his first integrated plant the
Edgar Thompson Works, whose goal was "a large and regular output". By relentlessly
exploiting his scale advantages and increasing the speed of production, Carnegie soon
became the most efficiency steel producer in the world. By 1879, American steel
production volume was close to the Britain; but by 1902, America produced 9,138,000
tons compared with 1,826,000 tons in Britain.
If Andrew Carnegie were viewed as the inventor of large-scale production, Henry Ford
would be the inventor of fast-speed mass production. Like Carnegie, Ford recognized the
importance the fast speed production to increase throughput. He innovatively abandoned
the old style assemble methods that were dominating most assembly industry by that
time. Instead of having skilled workers assemble complex sub-assemblies and then gather
around a static chassis to complete the final assembly, Ford introduced the moving
assembly line. Products were traveling on the moving assembly line in a continuous, non stop manner, workers stand aside of the line and carry out simple operations. In this way,
complex sub-assembling skills became unnecessary, production speed had been increased
and unit cost was dramatically reduced. In 1906 the Model N was introduced with a price
of $600, which was far less expensive than the price of $1000 of normal four -cylinder
20
automobiles at that time. In 1908 Ford started producing the legendary Model T with an
original price of $850. By continuously improving the production speed and reducing
cost, he brought the price down to $360 by 1916 and $290 in the 1920s. Ford sold
730,041 Model T's in the fiscal year 1916/17, which was roughly one-third of the
American automobile market.
Ford had started a general production management style that has been followed by most
American industries in the
2 0 th
century, which is commonly referred as "mass
production". The basic spirits included in mass production is to reduce unit production
cost by product variability reduction, standardization and simplifying operations. Less
product variability required less changeover operations therefore the system can keep the
same production pattern for a long period of time with high production speed. Also the
operations that each worker needs to perform are very simple therefore the workers can
keep a very fast production pace. The mass-type production normally led the
departmental layout of machining department and the high-speed transfer line layout for
assembly department. Figure 2- 1 and Figure 2- 2 demonstrate the typical high-speed
assembly department (line) and departmental machining department.
Cycle time for each operation (seconds)
2.4 3.8 7.4 4.2 3.7 3.7 3.1 7.4 5.8 5.1 4.4 6.2 4.3 4.7 5.5 3.4 5.7 5.7 3.8 6.6 6.6 2.2 4.6 3.5
From inventory
Figure 2- 1: Schematic of a typical high-speed line layout of assembly-type
manufacturing system
21
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&m
.Y YTESTE
-0-0-
12
LAPPERS
Q&
24
36
L-APPER;
APPBFR
$
9WilE
LAPPER
ELABRATOR
O6THEEN (7)
PACKOUT
Figure 2- 2: Schematic of a typical departmental layout of machining-type manufacturing
system
Another person that had important contribution of manufacturing system evolution is
Alfred Sloan, who successfully directed GM to take over Ford to become the biggest
automaker in US. Contrary to Ford's getting lowest unit cost by lowest product variety
philosophy, Sloan developed various divisions that were targeting different market
sections. Under Sloan's management, GM also adopted sophisticated mew procedures for
demand forecasting, inventory tracking and market share estimation. Under this system
GM achieved more flexibility and customer satisfaction by regularly introducing new
models to the market. In 1929 GM increased its market share to 32.3 percent and took
over the first place in the American automotive industry. Figure 2- 3 shows the evolution
of production volume and product variety in different manufacturing systems.
22
Mass Production (Ford), 1914
Mass Production (Sloan), 1920s
Lean Production, 1970s
2000s
Craft Production, 1900
Number of Products on Sale
Figure 2- 3: Evolution of production volume and product variety of automotive industry
[Womack 1990]
Having mastered the techniques of mass production and distribution and management of
large-scale enterprises, American manufacturing won the undisputable world-leading
position after World War II. In 1945 the American market was eight times the size of the
next-largest market in the world, which offered American manufacturing companies vast
opportunity to reap the economies of scale advantage.
The American manufacturing industry experienced an exhilarating postwar boom. The
per capita income (in contrast to the 1958's) rose from $1 to $3 in 1970 [U.S. Department
of Commerce 1972]. In 1947, the 200 largest industrial firms in America covered 30
percent of the world's value added in manufacturing and 47.2 percent of total corporate
manufacturing assets. By 1969 the top 200 American industrials accounted for 60.9
percent of the world's manufacturing assets [Chandler 1977].
Along with the huge profit and wealthy life that the golden era had brought to American
people, the seeds of future bitters were also buried. Since the postwar American
industries were facing almost non-competing marks all over the world, they did not even
worry about the details in their manufacturing system. As long as the products can be
made, they will be sold and bring back profit. This attitude however, soon changed the
American manufacturing industry from a golden boom to a miserable bust in the 1970s
and 1980s. Because of the American technological advantage and lacking of competition
23
from out side, the manufacturing companied lacked incentives to refine and improve their
manufacturing system to achieve higher quality, better customer service and lower
production cost. On the other hand, manufacturing industries in other countries did not
have any competitive advantages and had to compete with the powerful America. The
only way left for them was to relentlessly improve and hope one day they can recover and
challenge the America.
As the result, from early 1970s, the manufacturing industry in some postwar recovered
countries such as Japan and Korea had gathered enough strength and achieved
considerable competitive advantage over US in product quality, on-time delivery, product
variety and customer service. Manufacturing companies from those countries
successfully won big share of the international as well as the American domestic markets
that used to be controlled by American companies (Figure 2- 4). American industry was
facing a deep trouble.
US and Japanese Motor Vehicle Production
14
12
7
0
\
10
8
---------
.2
Japan
U
4
0
.
140
1950
1960
1970
1980
1990
20
0
Year
Figure 2- 4: Comparison of US and Japanese automotive production between 1947 and
1989 [Womack 1990]
The competitors' competitive advantage came from their manufacturing system. The new
system, which was usually referred "lean" system in contrast with the American
traditional "mass" system, was originated form Toyota. A fundamental character of
Toyota Production System (TPS) is that it focuses on continuous improvement and
24
elimination of waste [Monden 1998]. The manufacturing plant is considered as an
integrated system other than just an assembly of departments. Production is information
and material flow oriented and anything that was not value adding would be viewed as
waste and would be eliminated. Guiding by this philosophy and years of continuous
improvement, TPS achieved better quality, higher product variety, shorter production
leadtime and much lower production cost than the Big Three in US.
To end this section, a milestone list of the history of manufacturing system evolution is
shown as in Table 2- 1.
1785
1792
1798
1801
1809
1811
1812
1815
1818
1819
1819
1822
1825
1834
1839
1845
1860s
1894
1896
1896
1898
1898
1899
1900
1903
1905
1906
1907
1908
1909
1913
1922
1923
1928
1945
1948
1949
1950
25
Thomas Jefferson proposes that Congress mandate interchangeable parts for all musket contracts.
Eli Whitney invents cotton gin.
Eli Whitney contract for 4000 muskets in 1.5 years.
Eli Whitney demonstrates interchangeability to Congress.
Eli Whitney delivers, 8.5 years late, non-interchangeable parts.
John Hall patents breech-loading rifle.
Roswell Lee becomes superintendent of Springfield Armory.
Congress orders Ordnance Dept. to require interchangeable parts.
Blanchard invents trip hammer for making gun barrels.
Blanchard invents lathe for making gunstocks.
Lee introduces inspection gauges; Springfield Armory.
John Hall announces success at Harpers Ferry using system of gauges to measure parts.
Eli Whitney dies.
Simeon North at Middletown CT, adopts Hall's gauges, delivers rifles (parts) interchangeable with
Harper's Ferry production.
Samuel Colt and Eli Whitney Jr. revolver contract.
The Armory Practice spreads to private contractors.
Steam-powered cars multiply, but do not reach public acceptance.
Charles King in Detroit invents four-cylinder engine.
King's 4 cylinder attains top speed of 5 mph in March, weight 1500 lbs.
Ford's 4 cylinder attains top speed of 20 mph in June, weight 500 lbs.
Stanley Steamer won hill-climbing test, order for 200 resulted for $600.
Percy Maxin creates range of designs for electrics. Range 35 miles at 12 mph between charges.
Electrics out sell all others.
1500 Electrics sold, twice the number of steamers.
Ford Model A, twin horizontally opposed engine, $750 ea., 1708 in 1904.
25 made/day, Ford Mfg. Co. formed to produce engines/transmissions.
Model N outsells Oldsmobile with 8,729, a 4 cylinder at $500.
Models N, R ($750) and S ($700) sold 14,887 and 10,202 in 1908.
Model T introduced, single cast 4 cylinder, 5 body styles: $825 - $1000.
100 produced per day. 17,771 Model T's sold.
Moving assembly line at Highland Park. 308K-1914, 501K-1915 at $440.
Over 1 Million model T's sold yearly to 1926.
1.82 million produced at average of $300 with more options standard.
Chevrolet out-sells Ford and Produces 1.2 million vehicles.
Need to Re-build wide variety of products in low volume after World War II. Only had six
presses, requiring frequent and fast changeover.
Withdrawal by subsequent processes.
Intermediate warehouses abolished.
"In-line cells". Horseshoe or U-shaped machine layout.
1950
1953
1955
1955
load.
1958
1961
1962
1962
1965
1966
1971
1971
1981
1990
Machining and assembly lines balanced.
Supermarket system in machine shop.
Assembly and body plants linked.
Main plant assembly line production system adopts visual control (andon), line stop and mixed
Automation to autonomation.
Warehouse withdrawal slips abolished.
Andon installed, Motomachi assembly plant.
15-minute main plant setups.
Kanban adopted company-wide. Full work control of machines baka-yoke.
Kanban adopted for ordering outside parts for 100% of supply system; began teaching affiliates.
First autonomated line Kamigo plant.
Main office and Motomachi setups reach 3 minutes.
Body indication system at Motomachi Crown line.
Publication of Toyota Production System in English & infusion in U.S.
Publication of the "Machine that Changed the World"
Table 2- 1: Summary of main innovations in the history of manufacturing systems
[Cochran, 1994]
2.2 Manufacturing System Design Framework
2.2.1 Systematic Approach for Manufacturing System Design
Traditional American approach to study systems is first trying to break it down to many
simple modules, and then rigors scientific analysis tools are applied to each of the module
to achieve superior performance of each module. Applying this approach to
manufacturing system analysis and design can be traced back to Frederick Taylor's
motion study. This approach might work for simple systems since that the overall
performance of these systems mainly results from the performance of their components.
However, when system is getting more and more complex, optimizing components does
not necessarily result in better system performance, or in many cases, worse performance.
The reason for this is that complex system includes huge amount of relationships among
system elements. Every element affects many other elements and the system performance
is the aggregation of them all. Optimizing some of the elements without considering the
relationships may cause severe damage on other elements therefore result even worse
overall system performance.
To overcome the shortcoming of the traditional methods, systematic methodologies need
to be established for manufacturing system design. A systematic design approach is topbottom type. It starts from designing the system to achieve high-level system
requirements. The design will then be decomposed into detail from the high-level design
26
framework. In this way, all design detail will be consistent with high-level system
requirements and desired system performance is ensured.
2.2.2 Axiomatic Design
Axiomatic design is a methodology that guides the design process through a scientific
and controllable path. For a long time people have been thinking that the design process
is something relative to arts rather than rigorous science or engineering. The quality or
performance of a design work mainly depends on the inspiration and talent of the
designer. However, this art-type design can hardly be incorporated into modem scientific
or engineering practice due to the following two main reasons:
1.
The design is unexplainable. Most of design work is based on the designer's
personal perceptions and judgments; therefore it is almost impossible to explain
the exact reasons that lead to the design result.
2. The design is unpredictable. Since the design process cannot be explicitly listed
out, it is impossible to control the design time schedule.
To overcome these shortcomings, explicit rules need to be established that can guide the
design process in the right direction and lead to predictable result. This way of thinking
resulted in the development of axiomatic design.
Axiomatic design defines the design as "an interplay between what we want to achieve
and how we want to achieve it." [Suh, 2001] "What we want to achieve" will come from
the customer needs. Axiomatic assumes that all design work must begin with customer
needs; if there is no customer needs, there is no design. Once the customer needs are
identified, they will be further transformed into a minimum set of specifications, or
design functional requirements (FRs). According to the FRs, design parameters DPs will
be designed to realize "how we want to do that."
27
What
Customer
Wants
(Internal &
External)
Customer Domain
- Customer needs
- Expectations
- Specifications
- Constraints, etc.
FR'
How
DP's
0064-
Functional Domain
* Design Objectives
Physical Domain
* Physical
Implementation
Figure 2- 5: Mapping between customer domain, functional domain and physical domain
[Modified from Suh 1990]
The axiomatic design mapping process is shown in Figure 2- 5. It is noted that the system
functional requirements are not exactly equivalent to customer needs. Customer needs are
usually phrased in a non-scientific way with ambiguity and overlapping. The designer
should define a set of unambiguous and independent specifications to be design FRs.
In most cases, the high level DPs designed for system FRs are not physically
implementable. These DPs could either be subsystems that need to be designed in detail
or just general design directions that need to be further materialized. In either case these
high level DPs need to be decomposed until physically implementable DPs have been
achieved.
28
"Zig"
FRI
FRIl
FR12
'_
"Zag"
FRl3
Functional Requirements
Functional Domain
DP I
DPl1
DP12
DP13
Design Parameters
Physical Domain
Figure 2- 6: Zigzagging process of multi-level design decomposition [Modified from Suh
1990]
To decompose the high level FRs and DPs, it is need to zigzagging the design process
between the functional domain and the physical domain. As shown in Figure 2- 6, the
design starts from highest-level functional requirement FRI. To satisfy this FR, the
designer needs to go to the physical domain to find out the appropriate DP 1. Since DP 1 is
not physically implementable, the design process will come back to the functional
domain to decompose FRI to low-level requirements FRI 1, FR12 and FR13. This
decomposition step would base on both the high-level FR and DP, because different DP
could result in different FR decomposition. After the low-level FRs are decomposed, the
design process will proceed to physical domain and design parameters will be selected to
meet those FRs. Keep conducting this process until all DPs are physically implementable,
which are called leaf-level DPs. Only if all lowest-level DPs are implementable should
we call a design process finished and terminated.
Axiomatic design assumes there are two fundamental rules (axioms) that lead to a
successful design [Suh, 2001]:
Axiom 1: The Independence Axiom. Maintain the independence of the functional
requirements (FRs)
Axiom 2: The Information Axiom: Minimize the information content of the
design.
29
The Independence Axiom defines the relationship between FRs and DPs in an acceptable
design. The relationship can be expressed in the forms of design matrix, graphical
representations or path illustrations. According to Axiom 1, there are three different types
of designs: uncoupled design, decoupled (partially coupled) design and couple design,
which are shown in Table 2- 2.
Mathematical
FR,
representation
FR2
X
0.DR
FR1
1
FR2
XDDP2
= 0
Coupled
design
Partially coupled
design
Uncoupled
design
X
DPfD
XXHDP2 J
_R
FR2
[
X.D
DP2
X X
FR,
FR2
FR 1
FR
2
FR 1
FR
2
DP,
DP 2
DP 1
DP 2
DP 1
DP
2
Graphical
representation
DPP DC
Illustration of
DP2
FR2
FR2
FR2
DPI
DPI
DP1
path dependency
going from A to
B
FR1
FR2(B)
FR2(B)
FR2(A)
FR1
FR2(A)
A
FR1(A)
FR1(B)
FR1
FR2(B)
FR2(A)
A
FR1(A)
FR1(B)
A
FR1(A)
FR1(B)
Table 2- 2: Representations of different type of design [Linck 2001]
In an uncoupled design, the DPs and FRs are independent in the sense that one DP only
affects one FR and the design matrix is diagonal. Therefore the FRs can be met by
implementing each DP independently. In a decoupled design, the design matrix is
triangular. Although the DPs and FRs are not uncoupled, a specially path can be found so
that the FRs can be met one by one by implementing DPs following this path. For a lower
triangular design matrix, this design path is implementing DPs from up to bottom in the
DP vector. The third type of design is called coupled design, which has a full design
matrix. In this design the FRs cannot be directly met without iteration. The coupled
30
design is unacceptable and designers should always ensure their designs are uncoupled or
decoupled.
Table 2- 2 shows different cases where the numbers of DPs and FRs are equal, which
represents most situations. However, it is worthwhile to address the situations when
number of DPs is not equal to number of FRs. It has been proved [Sun, 2001] that, when
the number of DPs is less than the number of FRs, the design is always coupled. When
the number of DPs is larger than the number of FRs, it is a redundant design. Whether or
not it can be simplified to an uncoupled or decoupled design depends on if a diagonal or
triangular design matrix can be resulted by DP elimination.
2.2.3 Manufacturing System Design Decomposition
Manufacturing system design decomposition (MSDD) is a manufacturing system design
framework developed in Production System Design Lab (PSDL) at MIT. MSDD attempts
to show a general logic map of achieving a manufacturing system that can meet its
requirements. MSDD is an axiomatic design based framework that clearly separates the
system FR and design DPs, which differentiates itself from the traditional manufacturing
system design methodologies that focused on applying "lean tools". Starting from
highest-level system FR/DPs, MSDD decomposes them into multiple levels of FR/DP
pairs until all DPs become implementable. The decomposition therefore ensures all detail
DPs are consistent with high-level system level FRs. MSDD presents a decoupled design
and provides an unambiguous path to achieve system FRs in an non-iterative way.
The highest-level FR of manufacturing system design should represent the general goal
that a manufacturing system aims to achieve. Hopp and Spearman [1996] defined the
goal as "the fundamental objective of a manufacturing firm is to increase the well-being
of its stakeholders by making a good return on investment over the long term". The
highest-level FR of MSDD is defined as FR- 1 "Maximize long-term return on
investment" and the design parameter is selected as DP-I "Manufacturing system
design".
ROI = Revenue - Cost ... (2.1)
Investment
31
The definition of return on investment is shown in the formula (2.1). It is straight forward
that in order to increase ROI, a manufacturing system needs to increase its revenue and
reduce its cost and investment. These arguments compose the second level decomposition
of MSD, which include three FRs: FR 1I "Maximize sales revenue;" FR12 "Minimize
manufacturing cost" and FRI3 "Minimize investment over production system life cycle",
and their corresponding DPs: DP 1 "Production to maximize customer satisfaction;"
DP12 "Elimination of non-value adding sources of cost;" DP13 "Investment based on a
long term strategy." These three second level of FR/DP pairs compose a decoupled
design with design matrix as the following:
FR -Ill
X
X
FR -12=
FR -13
_X
0
0
DP -II
X
0
X
X_
DP -12
IDP- 13
... (2.2)
The decomposition under the second level can be divided into six branches, namely
quality, identifying and resolving problems, predictable outputs, delay reduction,
operational cost and investment, as shown in Figure 2- 7.
FR
DP
Quality
Identifying Predictand resolving able
problems
output
Delay reduction
Operational
costs
Invest-
ment
FR: Functional Requirement
DP: Design Parameter
Figure 2- 7: Six MSDD branches [Linck 2001]
Figure 2- 8 shows the relationship between MSDD branches and high-level FR/DP pairs.
The first four branches are under FR/DP 11; the fifth branch is under FR/DP 12 and the
sixth branch is under FR/DP 13. The following discussion would be based on each of the
six branches.
32
FR-1
Maximize long-term
return on Investment
DP-1
Manufacturdng syste m
design
Maiize
FR-12
Minimize manufacturing
costs
sales reven
FR-13
Minimize investment
over production system
life
DP-12
Elimination of non-value
adding sources of cost
DP-11
Production to
maximize customer
satisfaction
FR-111
Manufacture products to
target design
specifications
DP-111
Production processes
with minimal variation
from the target
FR-112
Deliver products on time
FR113
Meet customer
expected lead time
DP-112
Throughput time
variation reduction
D P113
FR-R1
Respond rapidly to
production disruptions
FR-P1
Minimize production
disruptions
DP-R1
Procedure for detecbion
&response to
production disruptions
DP-P1
Predictable production
resources (information,
equpment people,
Identifying
Quality
and
resolving
problems
Predictable
Output
dce
DP-13
Investment based on a
long term strategy
Mean throughput time
reduction
Delay
Reduction
Operational
Costs
Investment
Figure 2- 8: MSDD structure [Linck 2001]
Quality Branch
The quality branch of MSDD begins with FR 11 "Manufacture products to target design
specifications." DP 11 "Production processes with minimal variation from the target" is
selected to satisfy it. FR/DP 111 is further decomposed into three low-level FR/DP pairs.
According to statistic quality control, all operation outs puts need to be inside the control
limits. In addition to this minimum requirement, if the manufacturer wants to achieve
higher and assured high quality production, the process mean needs to be adjusted to be
on its target (desired) value and process variation should be as small as possible. The
former statement requires the process be essentially "right" in a statistical point view and
the second statement requires the process be good in a sense that most of the process
outputs will be very close to its desired value. These requirements are formally expressed
33
-
as FR-Q 1 "Operate processes within control limits;" FR-Q2 "Center process mean on the
target;" and FR-Q3 "Reduce variation in process output." Three DPs are chosen to
address these FRs, they are DP-Q1 "Elimination of assignable causes of variation;" DPQ2 "Process parameter adjustment;" and DP-Q3 "Reduction of process noise."
FRIl1
Manufackre products
totargetdesgn
speclicaltons
P1111
Process capablty
DP-111
Producion process
wsit minirel naon
from toe target
limit
FR-Q2
Centerprocess
meanon fie trget
PM-Qi
Numberofdefbct
pern parS with an
assignable cause
PM-Q2
Diference between
process mean and
FR-Q1
Operate processes
withincontrol
FR-Q3
Reducevariatonin
process output
PM-03
Variance
ofprocess
output
target
DP-Q1
ElIrination of
assignable causes of
DP-Q2
Process pararer
adjlusenent
DP-Q3
Reducton of
process nose
varialion
FR-Q11
FR-Q12
FR-Q13
FR-Q14
FR-Q31
FR-Q32
Elminate
operatr
assignable
Elminate
machine
assignable
Eliminate
method
Eliminate
metanal
Redice noise
in process
assignaba
assignable
Reduce mpact
ofinputnoise on
process output
Causes
causes
causes
causes
inputs
PM-QiI
NjTer of
ddts per
PM-Q12
Nreter of
dfte
rn
PM-Q13
Nerrrof
defeots Wrn
PM-Q14
frr
of defwts
par
br5
h
PM-Q31
Variance of
process irPus
to
operators
Stable ouput
fromoperatrs
Output
variance /
irnput Neriance
Ithe procOSS
.i ty of
DP-Q12
Failuremode
DP-Q13
Pr es plan
DP-Q14
DP-Q31
DP-Q32
design
Supplierquality
program
Convrsion of
mon
mo
RobuSpr
endefcf
to
DP-Q11
P-3
ecprl
to
assignable
cause
operatorhasof
knowledge
operator
comsistenly
FR-M1
Ensure fiat
FR-Q1 13
Ensure tiat
required tasks
perforrs tasks
translate to
FR-Q111
Ensure fiat
operatrhuman
errors do not
defects
correctly
PM-Q113
PM-Q111
nrrter of dfecs per
dtby an
npa Se,operator's lark of
understadarg
methods
abou
PM-Q112
tterofrdefws
rnparrscaused CV
ter of defes per
rtsca edby
rhuman sor
N
per
n
nestarard
methodis
DP-Q111
DP-Q112
DP-Q113
Trenongprogrrm
Standardwork
Mistakeproof
operatos (Poke-
mehocs
Figure 2- 9: Quality branch of MSDD
In a manufacturing system, assignable variations can come from all factors that involved:
operators, machines, operations and material. Therefore in order to eliminate assignable
causes of variation, all these factors have to be considered. To eliminate operator related
variations, stable output needs to be achieved to ensure production output will not vary
with different operators (FR/DP-Q11). Means to achieve operator output stabilization
include operator training program, standard work methods and application of mistake
34
~~~~1
proof devices (Poka-Yoke). To eliminate machine assignable causes, failure mode and
effects analysis need to be conducted to find out the root causes of these variations and
apply procedures to prevent them from happening again (FR/DP-Q 12). A carefully
designed process plan will be helpful to eliminate method assignable causes (FR/DPQ13). And supplier quality program will be selected to eliminate material assignable
causes (FR/DP-Q 14). The full decomposition of quality branch is shown in Figure 2- 9.
Problem Identifying and Resolving
FR-Ri
Respond
repidyie
production
dsruplors
PM-R1
Time between
occurrence and
resoluton of
dirupions
DP-R1
Procedure for
deteoione&
resposee to
prodbcion
disruplors
FR-R12
Communicate
probkems I ie
FR-R11
Rapidly
recognize
p dton
PM-R12
PMRI11
ocurerce o
desoticand
occur
they
PM-R111
Time beween
occurrenceand
recogrin that
desrupior
occurred
DP-RiII
Increased
operator
sampltg
teof
udZtarce
ti
deslto is and
torotreeme
uretandirg
wha l
deeruptons
probee,
resolution
DP-R11
Coniguraonto
DP-R12
Specified
communicatlon
paths and
procedures
DP-R13
Stend rd
meioodt
FR-R121
Identycorrect
support
FR-R122
Minimizedelay
resources
correctstpport
resources
ot4oerrest
to ndbstand
to
tie
FR-R113
Identilywhat
hedsrupionIs
whet
dteeqoptiorni and
idenIyand
elIminteroot
inconetcing
FR-R123
nimizelimefor
e
3
PM-R112
Trebeween
PM-R113
PM-R121
PM-R122
PM
Tmebewnee
iderifiioneol
Timebtween
iderificaimof
Time beWeen
iderlificaoniof
Cortwt
sopqotresouce
dcoretAeand
itterlifioabon of
whre It.
dp
equipment
slats
FR-RI12
IdenIfy
disruptore
where they
occur
Tirebeleen
eup Ore-e
sa
je~tlifceemoel
disrupteos
derupions
When
f the
what ft
dsrptin s
enable
de tectonof
Identfy
PM-R13
Timsbewen
ic~fitmofi~
T me esa
FR-R1II
FR-R13
SolW problerre
immedately
rightpeople
wha edsption
-dere
and
iertifiationof
ds ruti
nis
DP-R113
DP-R112
Simplified
matenalflow
Feedbk
pal
ste
suetem
of
nht
nise
dsrqotii.
iderfpcato
s
r
ad
or
identiction
aedcotactof
eoreetstitppo?
resource
cucep"
DP-R121
Specitfied
DP-R122
Rapidsupport
stpport
contact
resourcesfor
each failure
m ode
procedure
ofCcrct
anderes
=
urderst
DP-R123
System Fat
conreyswhat
hedisrupIonis
Figure 2- 10: Problem identifying and resolving branch of MSDD
To be able to respond rapidly to production disruptions, procedures for detection and
response to production disruptions need to be established. This is shown as FR/DP-R1 in
MSDD as the root of the problem identifying and resolving branch.
35
Problem identifying and resolving procedures should be able to cover three basic steps:
rapidly recognize production disruptions when they occur (FR-Rl 1); communication the
problems to the right people (FR-R12) and apply measures to solve the problems
immediately (FR-R13). To facilitate the problem identification, the manufacturing system
configuration needs to enable the detection of the disruptions (DP-R1 1); specified
communication paths and procedures should be established (DP-R12) to ensure the
information communication channels are clear and effective when problems happen.
Standard problem solving procedures (DP-R13) need to be defined to ensure the problem
can be solved in the shortest possible time.
To detect an occurred problem, the information of when the problem occurred, where it
occurred and the nature of the problem needs to be collected (FR-Ri 11-3). Increasing
operators' sampling rate of equipment status (DP-Rl 11) will help detecting the problem
in a timely manner. Simplified material flow (DP -R 112) is an effective way to quickly
identify where the problem happened. Feedback of sub-system state (DP- 113) can tell
operators what type the problem just happened is.
Fast communication procedures (DP-R12) required identifying correct support resources
(FR-R121) when disruptions occur and minimizing the time to contact the support
resources (FR-122). Specified support resources for different failure modes (DP-R121)
are designed to satisfy FR-R121 and rapid support contact procedure (DP -R122) is
designed to achieve FR-R 122. The full decomposition of problem identifying and
resolving branch is shown in Figure 2- 10.
Predictable Output
The third branch of MSDD begins with FR-P 1 "Minimize production disruptions" and its
corresponding DP-P1 "Predictable production resources". To achieve predictable
production, the system needs to ensure the availability of relevant product information
(FR-P 11); ensure predictable worker output (FR-P12); ensure predictable equipment
output (FR-P 13) and ensure material availability even though fallout exists (FR-P 14).
Motivated workforce performing standard work (DP-P 12) will lead to predictable worker
output. It is critical that the operators can complete the operations in standard times (FR121). Standard work methods (DP -P121) need to be established to ensure the operations
36
are conducted in a standardized and predictable manner. Perfect attendance program (DPP 122) is designed to ensure the availability of worker for the system (FR-P 122). Mutual
relief system with cross-trained worker (DP-P123) aims to eliminate the interruptions due
to worker allowance (FR-P 123).
FR-PI
Mnimize
production
dsruplors
PM-P1
de Arvoe o
&
Amount of tire
kost to ds.pti-r
dlsrupliorm
Predomble
producton
resources
equiPment, nfo)
4
F _
FR-P11
Ensure
evenlablityof
FR-P12
Ensure
n
releon
IFR-P1
predicable
workeroutput
rrneartof
intee
Nirrter
td
m-
disrup
of
PM-P13
Nu-rber
of
eo
WCirrmt of
to
dteions
DP-P12
Motetedwork
and
MaintRnance
equipment
re
perrming
nomptefntn
of
task
compleFon
ima
slandardwork
laretm.
FR-P122
Ensu re
evailablityof
workers
PM-P122
Numberof
eqet
upnt
FR-P123
Do not
interrupt
production
worker
for
allowances
OI
task operalbr
to p
lon tim
Stardardwork
methodh to
required to
latenes
intemruption time
[)P-P122
IOP-P123
DP-P131
attendance
Mulial
Perfect
program
s"Gtem relief
wit)
cross-raned
workers
32
regularly
PM-P131
Amountoftme
Machn-s
desiged
for
seviceablity
Inment
h
enn
eserviceable
2-ly
operaor
TWle
n
rabelity
r
FIR-PI
Service
equipmentis
serAce
equipment
tmonof
of
FR-P131
Ensure hiat
PM-P123
occurrences of Nurrter of
dsruplior die to
pr.
completionTm e laiiness,
PM-P121
Variance in
repaal
SC
DP-P13
e
force
reliable
Informatn
Ss lam
io"-Ce
4
of
r ue
e
oWn tme
Capable
DP-P121
P
e
ent
e
p
fa operatens
DP-P11
ariabity
r
tO
disrupors
FR-P121
Reduce
eityenen
faiout
output
PM-P12
0n
o
of
material
predictle
equipment
production
informeton
PM-P121
Nurrtw
Verene
Ensure
equpment
32
equpment
PM-PI
Frequencyof
I
flat
o
I1
of
nices
semcing
DP-P1 32
Regular
preventalwe
maintenance
program
FIR-P142
Ensure proper
tm ing ofpart
arcfls tD
PM-P142
Pars
dema nded-
'l09
delivered
dI
DPP142
oparaiors at
sub-paeo
derrard
Figure 2- 11: Predict
output branch of MSDD
Maintenance of equipment reliability (DP-P13) is critical to eliminating equipment
disruptions. The equipment in system should be designed in a way that easy to service
(FR-P 131) and regular machine service should be performed to maintain that equipment
constantly be in perfect condition (FR-P 132). These two FRs are achieved by DP-P 131
"Machines designed for serviceability" and DP-P132 "Regular preventive maintenance
program", respectively.
37
Standard work in process (DP-P141) and parts moved to downstream at customer
consumption rate (DP-P 142) are helpful to ensure parts are always available to material
handlers (FR-P141) and ensure proper timing of parts arrival (FR-P142), both of which
compose the standard material replenishment approach (DP-P 14). The full decomposition
of predictable output branch is shown in Figure 2- 11.
Delay Reduction
Five types of delays are involved in manufacturing system: lot delay, process delay, run
size delay, transportation delay and systematic operational delay. Eliminating these five
types of delays (FR-T 1-3) will lead to mean throughput time reduction (DP 113).
Lot delay occurs when products are transferred between processes with big batch size.
Each part has to wait each other part both before and after operations. Transfer batch
reduction (single piece flow) (DP-T1) will reduce the lot delay time.
Process delay occurs when parts arrival interval is shorter than machine processing
interval; therefore products would be accumulating in front of the machines. Producing at
customer takt time will eliminate the time difference between part arrival and process
cycling. Production at takt time requires defining takt time (FR-T2 1), ensure production
cycle time equals to takt time (FR-T22) and parts arrive at service rate (FR-T23). DP-T21
"Definition or grouping of customer to achieve takt times with an ideal range", DP- T22
"Subsystem enabled to meet the desired takt time (design and operation)" and DP-T23
"Arrival of parts at downstream operations according to pace of customer demand" are
designed to achieve their corresponding FRs.
Run size delay is caused by the manufacturing system not being able to produce customer
required product mix. Products have to wait in the inventory area until all customer
required product types have been produced. In order to produce customer required mix
and quantity during each demand interval, customer demand information needs to be
transferred to each process in the system (FR-T3 1) and production run size should be
sufficient small (FR-T32). Information flow design (DP- T3 1) and changeover time
reduction (DP-T32) are essential to meet those two FRs.
38
To avoid production interruptions, the system should ensure support resources do not
interfere with production resources (FR-T5 1); ensure production resources do not
interfere with each other and ensure support resources do not interfere with one another.
FR113
Meetcustomerexpeced
lead tne
PM113
Dffwerne beWeen mean
throufghpu tie od
customer's ewprsted lead
DPI 13
Mean throughpout
reducton
FR-T1
Reduce
del) y
lot
Inentorydue
FR-T3
Reduce
processdelay
sizedelay
(casedblyr.,>r,)
PM-T2
n
PM-T
FR-T2
Reduce
t
Reduce
transportalon
delay
PM-T4
In-e
ntorydue
transportaon
torunsize
delay
delay
FR-T5
Reduce
s sIBM etc
operatonal
delaw
PM-Ts
FR-T4
Pn
PM-T3
H Innntory due
Inbe
ntorydue
to process
tolotsizdelay
Ime
tot
Pfodtion time
delay
arrng resouces
DP-21
D euorno
co nstrt
d zhetaktme
DP-2
snpieced
o)
FR-T21
Da
ine
takttimes)
FR-T22
Ensure
t at
production
Hastakttme
been defined?
(Yes/No)
DP-T21
Definiom
grotpiro of
or
Custornes to
an
rwit
idsal
Ensure Onhat part
FR-T1
Promide
arriigl ran is
knowledge
PM-723
between arrival
Difference
rates
DP-T22
Subsystema
ermbled
to
ft deeired tkt
tim (egard
dOrrt been
accun
c
FR-T222
PM-T221
been
Has
d? (Yes
PM-T222
been
Has
achieved? (Yes
/No)
Is average
cwcle Wme less
than tak, imemi
DP- T22 2
DP-T223
Stagger
this
achiew
/No)
DP- T22 1
Design of
appropniate
automtic wo rk
conten'tat each
statin
Ensure that
manual
tim a -
FR-T52
Ersure
suppot
produtn
that
reS-
iont
size
PM-T51
Prodoti-
lot due to
prodotion
resowrcm
DP-T32
Design quick
changeoer
material
d
for
handin
tires
esodile
1
ter
equiment to
configured
seprta m
DP-T5
Subsw r nd
a~orw-pmn
t
oaO
FR-T53
Ers ue that
support
resara
i
on) do'rt interfere
<brit nterfse
on)
Wih ore a-#-e
with one arolher
roua
suppot
downs ream
ustomer
FR-T221
Ensure Ilat
autom abc cWle
fimne 4
m him urn tak t
ime
at Irun sizes
-targetrun
2PT3
eo
FR-T51
Ensure
pnmnded?9
=n.raIon
flow -om
ptoe
sm
PM-T32
derrand
rarg.
tern
design avoid
producton
ineouptons
that
Actual run size
P11-11131
ths
informaton
DP-T2 3
Arrival of pars a[
dw r-a
operatiOns
rreet
Suts
FR-T32
Ha
andserice
e and
cP-T5
flow
oentednade
ut
des ign
Produce in
sufgicignty
of
demanded
docrtdn productmo
of
(partwes and
between
production
each[
DP-T4
Material
dem andindarel
FR-T23
rate (rp=r,)
Difference
r
rdurng
er
d
PM-T22
t tee
thedesiedmix
and quant
ep
equal to service
ime
equals apktime
PM-T21
Productonof
e
hvd
)
s
cyle
DP-T3
Pro ducton
des Igned for
tn (d
PM-T52
Protion
due to
prodution
loet
inme
-s
istefereseth
one anolher
DP-T52
Ensure
coordnaion
andseparaton
ofproducon
work paterns
PM-T53
drotion
lot du
support
to
resore.
inoesfere
ime
esofl
one another
DP-T53
Ensure
coordinaton
andsepareton
ofsupportwork
pamrs
FR-T223
Ensure level
cycle
cWle Ome mix
takt Orm e
PM-T223
this
Das ign of
appropnale
operatorwork
conlantAcoos
des red tm a
interval?
producon
parts wilh
of
diflbrentcycle
lim e
OE
Figure 2- 12: Delay reduction branch of MSDD
These three requirements are achieved by DP-T51 "Subsystems and equipment
configured to separate support and production access requirements", DP-T52 "Ensure
coordination and separation of production work patterns" and DP-T53 "Ensure
39
coordination and separation of support work patterns", respectively. The full
decomposition of delay reduction branch is shown in Figure 2- 12.
Operational Cost
Total operational cost is divided into direct labor cost and indirect labor cost. Therefore
cost reduction requires both direct labor (FR- 121) cost reduction and indirect labor cost
reduction (FR-122). Non-value adding activities of direct labor include operators' waiting
on machines (FR-D 1), waste motion of operators (FR-D2) and operators' waiting on
other operators (FR-D3). Operators waiting for machines can be eliminated by human machine separation. Machines are designed to be able to operate without operators'
constant attendance (DP-D 1). Operators work tasks and work loops should be studied and
design to eliminate any non-value adding motions (DP-D2). Balanced work loop design
that ensures all operators have same cycle time (DP-D3) will eliminate operators waiting
time on each other.
Reducing indirect labor cost requires improving effectiveness of production managers
(FR-Il) and eliminating information disruptions (FR-12). Self directed work teams
(horizontal organization) (DP-Il) could effectively reduce the amount management that
the system needs; seamless information flow (visual factory) (DP-12) is a key factor to
reduce the amount of indirect labor required to schedule the system. The full
decomposition of operation cost branch is shown in Figure 2- 13.
40
FRI2
Mnimize
manufacumig
costs
PM12
Manufac
costs
tring
DP12
Elminaon of
non-\alue
addnigsources
ofcost
FR121
Reduce waste
FR122
Reduceowast
FR123
Mnimiz
indirctiaboor
inindrectlabor
PM121
Percentage of
operaetrs' time
spenton
wta.dmotons
andwaitng
PM122
Amountof
required
inditctlabor
PM123
Facite scost
DP121
Elminaton of
DP122
Reduction of
DP123
Reducl
non-relue
maral
tasks
facilites
cost
-------------
indirectlabor
addng
tasks
FR-Di
Eliminate
operators'
waiing on
FR-D2
Elminate
FR-D3
Elmnate
operats'
waitingo oher
operteos
FR-l
PM-D3
Percentageof
operators'time
spentwalig on
PM-lI
Amountof
onequipment
PM-D2
Percentageof
operaors'lame
spenton
wastdmoors
DP-DI
DP-D2
DP-D3
HumanMachine
sepaation
Deigo
of
works ators/
workfoopsto
facilitate
operatrtasks
Bderoed
Selfdirected
koo
workleams
(honzortal
organizaton)
wrasedmoon
ofoperats
machies
PM-Dl
Percentageof
operators'lame
spentwaiig
FR-Di1
adice tir
operators
spendn nonas.aadded
station
PM-DIl
Prcentage of
operators'tir
spent on non
valuo-addnig
tasks while
wating at a
FR-D12
Enable
workerto
operalsmore
thanone
madlinel
satblon
PM-D12
Percentage of
satons ina
sslam
eachworker
can opeae
that
Indirectlabor
requiradt.
ranagesystem
otheroperators
DP-l
FR-D21
Minimize
wasladmotion
ofoperatrs
between
slateos
FR-D22
PM-21
Percentageof
operatrs'e
spentwalking
PM-22
Percentage of
operats'tine
spent onwasted
rotlons during
between
sate
improve
effecteness of
producion
manageis
Mnimize
wasladmoton
in operators'
work
prepralion
workpreparaton
constm
space
onof
ad floor
FR-2
Eliinate
inormaton
disruptorns
PM-12
Amountof
indrectlabor
requitedto
schedue
st
m
DP-12
Seamless
informaeontow
(usualfactory)
FR-D23
fotnimize
woreledmoton
in opetas'
worktasks
PM-023
Percentage of
operalors'ime
spenton
wasetdmoors
dr
dri or
8H
station
DP-D1l
Machinest&
smtors
desigedt
run
autonomously
DP-D21
Machines
s tios
DP-D22
Standardtoolt/
locatedateach
mIlple
contguredt
reducewa kng
sttion
sateos
distance
(5s)
DP-D12
Workers
frainedto
operate
equiment
DP-D23
Ergonomic
Interface between
theworkar.
ahineand
Figure 2- 13: The operation cost branch of MSDD
Investment
MSDD does not give out a general decomposition of the investment branch since it is
extremely case specific and dependent on particular system circumstances. However
MSDD shows some general comments on the investment issue in a manufacturing
system: let the system drive the investment decisions, not the investment decisions drive
the system. MSDD puts the investment branch on the right most position means it should
be considered only after all requirements to its left have been satisfied. A manufacturing
41
system should never sacrifice meeting system design requirements to investment
decisions. The position of the investment branch in MSDD is shown in Figure 2- 14.
Figure 2- 14: The investment branch of MSDD
42
Chapter 3: Inventory and Production Control Models
from System Design Point of View
3.1 General Introduction of Optimization Methodologies
From the early 50's when scientific approaches began to be applied in manufacturing
research areas, different methodologies have been developed in order to analyze the
performance of manufacturing processes and systems. The most commonly used
methodologies include:
" Machine line analysis based on stochastic models for buffer/capacity
analysis/design
" Linear Programming or Non-Linear Programming based optimization methods for
resources planning
" Statistics variation analysis based logistics system design and supply chain
management
All of these three groups of methodologies are essentially mathematic optimization
approaches. They either aim to find out optimal shop floor control policies, or the best
ways to allocate the constraint resources, or minimizing some crucial system parameters
(for example, stock-out possibilities).
In a manufacturing system, each process aims to achieve its preferred outputs with
limited resources. For example, in a machining department, one of the desired outputs
would be throughput rate and the limited resources would be machine hours. A
straightforward mathematical model can be established to show the basic relationships in
this problem.
Max: N
St.
Nm <M
Ni
L;
In the equation above, N is the number of products being produced during each time
interval; m and 1 are the machine hour and labor needed for each product, while M and L
are the total machine hours and labor available in the time interval. Solving this simple
43
optimization problem will show that, assuming machine capacity is less than labor
capacity, the optimal number of products that need to be produced during each time
interval will be M / m. Apparently, system design based on this model would be in favor
of maximizing machine utilization. This solution, however, has been proved to be just the
wrong way to go in modem large-scale manufacturing systems.
The example above is naive comparing the complex optimization problems that appeared
in manufacturing system research literature. However, this simple model exemplifies the
underlining shortcomings that the optimization methodology generally has, which could
be summarized as following:
1. Since the optimization methods are based on a mathematic simplification of
the real problem, they are not able to capture all relationships in complex
systems.
Optimization models are established to capture some of the properties of the
problem under study. These models have worked with satisfactory results in
many engineering design areas, such as optimal control, signal
recognition/transfer, etc. However, the accuracy of these optimization
solutions is highly dependent on how close they are to the reality. In simple
engineering analysis and design, it is generally easy to assure that math
models can capture most aspects of the problems (i.e., mechanical systems or
electrical systems). However, for system design with much more complexity,
the optimization methods generally have risk of mis-capturing some of the
important factors and therefore the optimal results based on the oversimplified
models would be seriously biased.
2. Commonly used optimization models view the system under study as static.
Therefore the solutions are not robust and may lose optimality even when
systems change slightly.
Linear programming and non-linear programming based optimization models
are based on static system relationships and constraints. For large-scale
system like manufacturing system, system elements and their interactions are
constantly changing. Therefore, even if a sophisticated optimization model
44
can be established and solved, the optimality of the solution can be severely
degraded when system configuration changes.
3. The amount of calculation work to solve large-scale optimization problems
can be huge, which makes real-time optimal control usually not applicable in
actually manufacturing system.
The example discussed before is an extremely simple problem. If we make
some changes, for example the labor and machine hours that each job
consumes are probabilistic rather than deterministic, the problem becomes
totally different and much harder to solve than the original one. Problems such
as logistics optimization or optimal control are usually NP-hard, which means
there is no known polynomial algorithm to solve them. The time needed to
solve these problems will grow exponentially with the increasing of the
problem's scale. For example, finding out the optimal dispatch sequence of 25
jobs (a classical topic in job shop planning) will take a very fast computer
about 77 years [Hopp, Spearman 2001]. Therefore, although the optimization
models might be mathematically valid, the intolerable time they need to
generate the solutions renders them useless practically.
Great amount of literature work has been done to push the optimization methodologies
into a more refined state. On one hand, the models have become more and more complex
in order to capture more reality from actually problem; On the other hand, in order to
solve these complex problems, many researchers have been exploring mathematic
methods and algorithms that can be applied with reasonable computation time. This is a
fairly tough work. Even if some algorithm could be found out, they tend to be very tricky
and imply many strong assumptions, which are not always making sense. Optimization
methodologies have entered an impasse.
While most people were focusing on refining the optimization models and attacking the
mathematical obstacles to solve them, something magic was happening in the other part
of the globe. Without the super-computer based production planning system, without the
complicated "optimal" inventory control policies, without even trying to optimize
anything, Toyota, an automotive company who resumed production after WWII with 1/3
45
labor effectiveness of its American counterparts, beat the Big Three in less than 25 years.
It kept their competitive advantages over the Big Three for almost 30 years. Toyota has
brought changes in people's mind: maybe manufacturing system design is not so
"mathematical" as people thought, it just needs a different way to look at.
This chapter will focus on discussing a very important type of problem in manufacturing
system, namely inventory and production control. Inventory and production control is the
area in which most optimization methodologies have been applied. Therefore study and
discussion on optimization and system design methodologies in this area will be able to
represent the general relationships of the two.
3.2 Inventory and Production Control Models
3.2.1 Introduction of Inventory and Production Control Models
Inventory and production control models are viewed as among the oldest and most
fundamental mathematical models that have been used in manufacturing system
management. The reason is that inventory and production control policy is one of the
most important factors in many production industries. It has significant affect the total
production cost, production leadtime and customer service. Inventory and production
control policy has been one of the characteristics to differentiate different manufacturing
systems. For example, the two most commonly used manufacturing system structures,
push-based system and pull-based system, adopt very different inventory and production
control concepts, which in turn determine their very different performance.
Inventory/production control models have been evolving for almost a hundred years. The
first and most famous model was Harris's Economical Order Quantity (EOQ) model.
Although hundreds of different models have been developed afterwards from very simple
deterministic ones to the very complex stochastic ones [Hopp, Spearman 2001], the
fundamental concepts behind haven't changed much from the EOQ model.
3.2.2 Economical Order Quantity Model
The initial problem that inspired Harris to develop this model was the following: A
factory produces more than one kind of products. Producing each product incurs a unit
46
production cost, while switching production between different product types will entail a
setup cost, which is assumed to be much higher than the unit cost. The problem assumes
that the production capacity is greater than customer demand rate, so the redundant
products will be stored as inventory, which incurs an inventory holding cost. The trade
off is if the factory wants to hold less (average) inventory to reduce inventory holding
cost, it needs to increase the number of setups, which adds more setup costs. Therefore, a
(optimal) production pattern needs to be determined to balance the two conflicting cost
and find out the minimum total cost.
Harris developed his mathematic model to solve this experience-based optimization
problem. The model was constrained by the mathematical precision of his own day and
used several strong assumptions and simplifications as the following [Hopp, Spearman
2001]:
1.
Production is instantaneous. The production capacity is infinite hence the entire
lot is produced and finished simultaneously.
2. Delivery is immediate. Products produced can be shipped to customers
immediately.
3. Demand is deterministic. The quantity and timing of customer demand is known
for sure.
4. Demand is constant over time. There is no quantity variation over time. Equal
time interval corresponds to equal demand quantity.
5. A production run incurs a fixed setup cost. The setup cost is view as fixed and
independent of product type, production volume and shop floor status.
6. Independed products. There are no relationships among the production of
different products, i.e. there is no sequential relationships or resource sharing.
Under these assumptions, the production cost (for fixed time interval T) model can be
expresses as the following:
CT
(Q) =
47
2
±+ CD ... (3.1)
Q
The notation for this model is listed below.
D
Customer demand in time T (units)
Q
Production batch size (units)
A
Fixed setup cost (dollar)
h
Inventory holding cost (dollar per unit per time interval)
c
Unit production cost (dollar per unit)
CT(Q)
Total production cost during time interval T
The decision variable for this optimization problem is the production batch size
optimal value
Q*needs
Q. The
to be found to balance the inventory holding and setup cost to
achieve minimum total production cost C* (Q). It is straightforward to verify that the
cost function is convex therefore the (global) optimal point would be the unique
stationary point, which is
.2AD
h
The optimal value shown above is the well-known economic order quantity (EOQ), also
referred to as economical lot size. Figure 3- 1 shows a numerical example of EOQ model.
As order quantity increases, the inventory holding cost curve goes up while the setup cost
goes down. Total production cost curve combines the two trends and attains a minimum
point.
48
30 25
20
oU 15
10
<
5
0
0
100
200
300
400
500
Order quantity (Q)
Figure 3- 1: Relationship between order quantity and average cost in EOQ model
The EOQ model discussed in this section is very simple. It assumes all the information
(production, demand, inventory) is known perfectly and deterministic. Although most of
these assumptions are too strong to be realistic, the model demonstrates the fundamental
tradeoff of between manufacturing and inventory management. In real world, all the
parameters in a manufacturing system are random in nature. The deterministic
assumption for EOQ model is only valid if the variation is fairly small and can be
practically ignored. However, this is not always the case. In order to analyze problems
with more complexity and develop more generic models, strong assumptions have to be
relaxed and random effect needs to be introduced. This type of model that based on
random parameters is named as stochastic inventory model.
3.2.3 The Newspaper Vender Model
The newspaper vendor model is one of the simplest stochastic inventory models. The
name of this model comes from the original problem that led to the appearance of this
model, which can be stated as the following: A newspaper vender makes his life by
selling newspaper everyday. In the morning he buys newspapers from wholesaler and
then sells them during the day with a retail price higher than the wholesale price to make
a profit. If, however, by the end of the day he hasn't sold all of the newspapers he bought,
there is no chance for him to sell them tomorrow. So he has to sell them to paper recycle
49
company with a much lower price, which causes him a lost. Since the customer demands
vary everyday, he needs to figure out how many newspapers to buy in the morning to
maximize his (expected) profit.
The same problem also exists in a manufacturing context. Consider a situation that a
manufacturer who faces a very seasonal demand. While production is being done year
round, most of the sale occurs in a specific time period (demand burst season).
Overproduction will cause the manufacturer a lost since he has to sell the redundant
products with a price lower that production cost; Underproduction is also a lost since the
manufacturer will miss the potential profit. Since the precise demand information is
unknown when the production begins, the manufacturer, in the same situation of the
newspaper vendor, needs to decide what the production volume would be based on
stochastic model, to maximize the expected profit.
The Newspaper vender model is aimed to decide the optimal production volume. It is
based on the following assumptions:
1.
Customer demand is random. Customer demand volume varies with time in a
random way, however, its probability density function (p.d.f.) is known.
2. Planning horizon is separable. The manufacturer makes production plan for each
production cycle (e.g., one year), and no product will be carried across different
cycles.
3. Production is finished before customer demand occurs. The production and
delivery stages are separated. Manufacturer will finish all planned production
volume before shipping them to customers.
Based on the assumptions above, the total expected cost for one production cycle is
[Bramel, Simchi-Levi 1997]:
z(y) = cy - r
50
D
-
min(y, D)dF(D) v
(y - D)dF(D) for yO....(3.2)
The model is using the following notation:
C
Unit production cost
R
Unit selling price
V
Unit salvage value
D
Customer demand
Y
Production volume, decision variable
By rewriting the term fDmin(y,D)dF(D) as
fDO
DdF(D)+ f_,ydF(D), equation
(3.2) becomes:
z(y)= cy - rE(D) - r
(y - D)dF(D) -
'
(y - D)dF(D) ... (3.3)
It can be easily verified that the cost function is convex, since each of the terms involved
in Z(y) calculating is convex. Therefore, the optimal solution for equation and (3.3) can
be solved by taking the derivative of z(y) with respect to y. With the aid of Leibunitz
rule, the optimality condition is the following:
c - r( - Pr{D : y}) - v Pr{D:y} = 0 ... (3.4)
which implies that the optimal production quantity S should satisfy
Pr{D ! S} = r
.. (3.5)
r -v
The equation (3.5) makes sense only if r>c >v, which is consistent with the real situation:
unit selling price is higher than production cost, while production cost is higher than
salvage cost. If either of these two inequalities doesn't hold, the optimization result is
invalid. The reason for this is straightforward: if, for example, the production cost c is
lower than the salvage value v, then there is actually no risk for the manufacturer to
overproduce since even selling as salvage price can still be profitable.
The cumulative probability function Pr{D S} is non-decreasing with S. Therefore, for a
fixed selling price, the optimal production volume increases with salvage value and
decreases with production cost. Figure 3- 2 and Figure 3- 3 show example curves of S vs.
51
c and S vs. v. The demand is assumed to be uniformly distributed with average value of
500. Other parameters are identified in the caption of each figure. The trends that are
shown in the figures are consistent with the intuition: when profit margin is high (p>>c),
the manufacturer tends to take the risk to produce more products to avoid sellout; when
the salvage value is very low, the manufacturer tends to be conservative in productions
since most value of overproduced products will be lost.
400
E 350
> 300
0
250
0
.
0
200
150
100
1
1.2
1.4
1.6
1.8
2
2.2
Production Cost c ($)
Figure 3- 2: The effect of different production cost on optimal production volume
52
350
C)
E
.
300
0
0
250
0
- 200
E
0.
150
0
0.2
0.4
0.6
0.8
1
Salvage Value v ($)
Figure 3- 3: The effect of different salvage value and optimal production volume
3.2.4 (Q, r) Model
The newspaper vender model discussed in the previous section is based on very strong
assumptions, which rarely hold in real manufacturing context. Even for production with
strong seasonality, manufacturing and shipping are still done in a multi-stage way. For
most manufacturing systems, customer demand is continuous and doesn't vary much with
time, therefore inventory control must be viewed in a more general prospective. Over
production in one time period can be carried to the next period as inventory. Also, if the
production volume and inventory cannot meet the customer demand in a time period, the
manufacturer will not only lose the potential profit, but also suffer from extra penalty cost
such as backorder handling or losing creditability from customer.
The (Q, r) model, similar to the EOQ and Newspaper Vender models, is aiming to solve
the tradeoff between setup cost (order cost) and the inventory holding cost under
stochastic customer demand. Because of the existence of the setup cost (which is usually
much higher than unit production cost), the manufacturer cannot produce corresponding
to each specific customer need. Therefore, similar as in the EOQ model, an optimal
production volume
Q needs to be found to minimize the expected cost. However, since
the customer demand is stochastic and unfilled demand will incur stockout cost, and also
53
the production requires a certain leadtime, an indicator of when to place an order needs to
be defined. This decision variable is referred as the reorder point r. Therefore the (Q, r)
model is to minimize the overall cost (production cost, inventory holding cost and
stockout cost) by setting optimal
Q and r values.
One of the major characteristics of (Q, r) model in opposed to the previous two models is
that (Q, r) model has two decision variables. In depth studies of (Q, r) model show that
the two variables actually sever for two different purposes and they are separable to some
extent. As in the EOQ model, the order quantity
Q affects
the tradeoff between the setup
cost and inventory holding cost. Large production quantity
Q will result in lower setup
cost but higher inventory holding cost. The reorder point r, however, affects the
likelihood of a stockout in inventory. A higher reorder point reduces the risk of stockout
by keeping higher inventory, and a lower reorder point will reduce the inventory level at
the cost of a higher probability of having a stockout.
It is important to recognize that the two parameters generate two different types of
inventories: cycle stock and safety stock. Production quantity
Q controls
the cycle stock,
which will circulate to fill customer demands; Reorder point r keeps the safety stock,
which aims to deal with the variation during the production leadtime.
A typical (Q, r) model would be based on the following modeling assumptions [Hopp,
Spearman 2001]:
1. Production is independent. The products can be analyzed individually and there
are no interactions among different products.
2. Single piece customer demands. Customer demands are single piece and arrives
one at a time.
3. Unfilled customer demands are backordered. If the customers' orders cannot be
filled, they will wait until being filled in a future time.
4. Replenishment time is deterministic and known. The time lag between placing a
production order and the ordered products actually arrive at inventory is fixed.
5. The order quantity for production is fixed. Each time the production order is
triggered, the order quantity is fixed.
54
The idea of the model is to find out the optimal values of Q and r which solves the
following:
min {fixed setup cost + stockout cost + inventory holding cost}... (3.6)
Q,r
A mathematical equation can be derived based on equation (3.6) by using the notation
listed below.
D
Expected customer demand during each time interval
L
Replenish (production) lead time
X
Demand occurred during lead time, an random variable
0
Expected demand during the lead time, E[X]
C-
Standard deviation of demand during the lead time
p(x)
Probability mass function of demand during lead time
G(x)
Cumulative probability function of demand during lead time
A
Setup cost
C
Unit production cost
H
Unit inventory holding cost for each time interval
B
Unit backorder cost for each time interval
Q
Replenish quantity, decision variable
R
Reorder point, decision variable
S
Safety stock implied by r, equals r-0
F(Q, r)
Order frequency as a function of Q and r
S(Q, r)
Faction of orders filled from stock (fill rate) as a function of Q and r
B(Q, r)
Average number of outstanding backorders as a function of Q and r
I(Q, r)
Average on-hand inventory level as a function of Q and r
55
Setup Cost
Since the expected customer demand for each time interval is D, and the replenish
quantity is fixed
F(Q, r) =
D
Q, the expected
order frequency can be computed as
... (3.7)
Q
Therefore the setup cost would be the expected order frequency multiplied by the fixed
setup cost A.
Coste
= F(Q, r)A =
Q
A ... (3.8)
Backorder Cost
According to the model assumption, unfilled customer orders would be backordered.
Backorder cost represents the penalty cost for stockout. The back order cost would be
unit backorder cost multiplied by the average backorder level B(Q, r).
In order to calculate the backorder level B(Q, r), it is needed to further study the
inventory position. Define B(R) as the expected backorder number when the inventory
position is R, which can be calculated by the following:
B(R) -(x
-
R)g(x)dxwhere g(x) is the continuous analog p.d.f. of p(x) ... (3.9)
The logic of (3.9) is that, the expected production orders places are equal to the customer
demands during the leadtime. Backorder happens only if the customer demand x is larger
than inventory position R.
Also, it can be proved that the inventory position is uniformly distributed in the sense that
inventory position is equally likely to take any value in its possible range [r+l, r+Q].
Combining this result and equation (3.9), the final expression of average backorder level
when production quantity
B(Q, r)
56
-
1 r+Q
ZB(x)
Q x=r
I
Q and reorder point r is shown in (3.10).
[B(r +1)+
Q
+ B(r + Q)]...(3.10)
This calculation can be easily conducted by an iterative loop. For simplification, a good
approximation can be used, namely using B(r) to approximate all B terms in the right side
of (3.10). The simplified backorder formula is shown (3.11).
B(Q, r)> B(r) ... (3.11)
Revisiting the definition of B(R) in (3.9) can show that (3.11) is an overestimation of
(3.10). The expected backorder cost is just the backorder number B(Q,r) multiplied by
unit backorder cost b.
= bB(r) ... (3.12)
Costbckord,,
Inventory Holding Cost
The inventory holding cost can be calculated by multiplying average inventory level
I(Q,r) with inventory holding cost h. By analyzing the reorder policy it can be seen that
the real inventory level varies from Q+s and s+1. Hence, in a long run, it is reasonable to
estimate the average inventory by the following expression:
I(Q, r) ~ (Q+s)+(s +1)
2
-
Q+1 +s = Q+1 +r-O ... (3.13)
2
2
Although mathematically we treat a backorder as a negative inventory level, however, in
reality the inventory can't physically drop below zero. Therefore (3.13) is an
underestimation of real inventory level, which should be modified by incorporating the
backorders, which is shown as (3.14).
I(Q,r)= Q
Q +l+r -0+ B(Q, r) ... (14)
2
And the inventory holding cost, therefore, is the following:
Cost hldking
=hI(Q,r) = h[ Q +1 + r - 0 + B(Q, r)] ... (3.15)
2
Now all there cost terms have been established based on the model assumptions. The
verbal expression (3.6) then can be converted into a mathematical expression.
D
Q
57
Plug in all the approximations that have been derived above, the approximated overall
cost is:
D
Y(Q, r) ~ Y(Q, r)
Q
A + bB(r)+ h[
Q+
2
+ r - 0 + B(Q, r)] ... (3.16)
Equation (3.16) is again a linear combination of convex functions therefore the optimal
solution can be easily found be setting the partial derivative with Q and r to zero
respectively. The optimal reorder quantity Q* and reorder point r*are given by
Q
2AD
.
=
h
*
, G(r )=
b
b+h
... (3.17)
Notice optimal reorder quantity is exactly same as in EOQ model. This is because the
(Q,r) model is based on the long run average cost calculation, and the approximation is
cut at the first derivative. Therefore the optimal solution for the deterministic model
(EOQ) also optimizes the average cost of stochastic model.
The optimal reorder point is expressed in term of its cumulative probability function.
Since G(r) is non-decreasing, so r*is increasing with b and decreasing with h, which is
also consistent with intuition. If the backorder cost is very high, the manufacturer tends to
keep high safety stock to prevent backorder from happening. On the contrary, if holding
inventory is very expensive, the manufacture might reduce the safety stock and take more
risk of stockout. If we further assume the distribution of customer demand during
leadtime is normally distributed with mean 0 and standard deviation Y, then the
expression of r*can be simplified as the following:
r* 0 + z-, where z is the normal distribution z-value such that 1D(z) =
b
... (3.18)
b+h
The (Q,r) model is a relatively complex and close-to-reality inventory model. Lots of
further research had been done based on it to incorporate more random factors. Apart
from the mathematic complexity to derive optimal solutions, the (Q,r) model carries the
most fundamental concept of stochastic inventory control, namely that
Cycle stock quantity trades off the setup cost and inventory holding cost.
and
58
Safety stock provides a buffer against stockouts.
And also, (Q,r) model offers some very valuable insight on the factors that affect the
stocking policy, which, will be shown later, the manufacturing system design. These
insights can be concluded to the following four points.
1.
Increasing customer demand will increase the optimal reorder quantity.
2. Increasing the average customer demand during the production leadtime 0 will
result in higher safety stock. An increase of 0 could due to two reasons: higher
customer average demand D or longer production leadtime 1. This implies that
either high customer demand or long production leadtime will lead to high safety
stock.
3. Increasing the standard deviation of customer demand during the production
leadtime a- will tend to increase the safety stock. If the demand during leadtime is
very unstable and varies a lot, the manufacturer needs to put more safety stock to
protect against stockouts.
4. Increasing unit inventory holding cost will reduce both optimal reorder quantity
and the reorder point r. This is very straightforward. Since both the cycle stock
and safety stock will incur inventory, with high unit inventory holding cost, the
manufacturer tends to reduce both and shift the inventory holding cost by setup
cost and stockout cost.
3.2.5 (s,S) Model
The basic (Q,r) model shown above is a very generalized model. It is also a relatively
open platform for modifications to fit into more complex situations. However, in spite of
these virtues, (Q,r) model is limited by two very strong constraints: the reorder quantity is
fixed and placing production order only when inventory position drops to the reorder
point. These are the pre-defined "rules" for (Q,r) model, and the model develops the
optimal values under these two rules. However, it is natural to probe that if there is other
rules that can result even lower overall cost that the (Q,r) rules? Or in other words, what
is the optimal value of optimal for inventory control "rules"?
59
The (s,S) model is the result of searching a "general" optimal inventory control policy
inspired by the questions asked above [Bertsekas 2000] [Bramel, Simchi-Levi 1997]. The
model needs the following two assumptions:
1. Stockout is backlogged and unit stockout cost is justifiable.
2. Production is instantaneous.
The second assumption can be relaxed to any fixed production leadtime, which, however,
will need much more calculation work.
Since this model is more generalized and takes fewer assumptions, it is more
mathematically challenging than the previous models. (s, S) model is developed by using
dynamic programming (DP) algorithm to find out the optimal rules to control inventory.
Consider a discrete time sequence k, k=0, 1.. .N-1. If we view the inventory level evolves
with the time index and at each time we call a different stage, the inventory level at each
stage is given by the following:
Xk+
= xk
+Uk
- W , k=0,1L,...N
... (3.19)
In which, xk is the inventory level at stage k, uk is the production order placed at stage k
and wk is the customer demand at stage k. Since it is assumed that production leadtime is
zero, therefore the inventory level at (the beginning of) stage k+1 equals to the inventory
level at stage k plus the ordered amount uk minus shipped amount wk. Notice that we
assumed that the stockouts are backordered, the inventory level x can be negative.
At first, we will attack an easy situation in which the setup cost is assumed to be zero.
Under this assumption, the overall cost of a stage can be represented by the following
equation:
r(x) = p max(0,-x) + h max(0, x) ... (3.20)
where p is unit backorder cost and h is unit inventory holding cost.
Therefore, the total cost during all N stages is the following:
N-
E{Z(cuk
k=C
60
+
±pmax(O,Wk
xk-uk)±h max(, xk+u-wk))V ..( 3 -2 l)
It is assumed that the purchasing cost c is positive and is strictly less than the backorder
cost p. The second assumption is necessary for the problem to be well posed. If, however,
the production cost c were greater than p, it would never be optimal to produce. This is
analogical to the assumption in newspaper vender model that the salvage value v is
strictly less than the production cost c.
By applying DP algorithm and setting terminal cost to zero, we have
JN(XN)
0
J,(x,) = min[cu,
+ H(xk
Uk 0
+Uk)+
E{Jkl(Xk +Uk -w)}]
... (3.22)
where the function H is defined by
H(y) = pE{max(0, w, - y)} + hE{max(0, y -
By introducing the variable
yk - Xk + Uk,
Wk)}
equation (3.22) can be rewritten as the
following
ik(Yk) = min[cyk+ H(yk)+ E{Jk+l (yk
Wk}}] - CXk
... (3.23)
Yk Xk
Since the function max(0, Wk - y) and max(0, y - wk) are both convex in y for each fixed
W, taking expectation preserves their convexity. Therefore the function
convex function. It can also be proved that the cost-to-go function Jk+1
H(yk)is
(Yk - w)
a
is also
convex. Hence the function in brackets of right hand side of equation (3.23) is convex.
These characteristics ensue equation (3.23) has a global minimum value, denoted by Sk.
Considering the constraint Yk
> Xk,
a minimum yk equals Skif Yk
> Xk
and xk otherwise.
The optimal control policy has the form
kk
p
Sk XkIif
0
if xk > Sk
... (3.24)
This is the simple version of (s, S) model, in which S equals S k and s equals 0. Equation
(3.24) shows that, at each stage, the inventory level is checked. If it is lower than S k, an
order is placed to bring the inventory level to Sk. Therefore the order-to point Sk behaves
61
like a inventory cap, whenever the inventory level drops below it, the amount difference
Sk-Xk
will be placed.
For more general cases, when the setup cost is not zero but fixed, more effort needs to be
put to find out the optimal solution. Unlike the zero setup cost situation, the production
cost C(u), which is shown in equation (3.25), is no longer continuous.
Ki+cu,
af u > 0
0,=fu=
C(u)
if U = 0
0,
. .. (3.25)
The DP algorithm for this problem takes the form of
JN(XN) =
0
Jk (Xk)=min[C(uk)+H(xk+uk)+E{Jk+l(Xkuk -wk)}]...(
Plug in the substitution of Yk =
Gk (y) = cy + H(y)+ E{ Jk(y
Xk +
3 26
.
)
uk and define the function
--w}
The Jk is written as
ik (xk)
=
minGk (xk),
inK + Gk (Y
]
-CXk
... (3.27)
Unfortunately, unlike the K=0 case that has been discussed before,
Gk
is not necessarily
be convex. This means that a global optimal solution cannot be resulted by deriving
equation (27). However, it can be proved that even though Gkis not convex, it satisfies
the following property:
K +Gk(z +y)
Gk(y) ±Z
Gk(y)
Gk(Yb)
b
j , for all z 0, b>0,y
This property is called K-convexity. After conducting through the mathematical
derivation, it can be shown that there exist two parameters S and that define the optimal
inventory control policy for the K-convex cost-to-go function:
p*(X)= k
62
0
i
,>S... (3.28)
S-xk ifxk<s
This means that an upper bound S and a lower bound s are controlling the inventory.
Production order is placed only when the inventory level drops below the lower bound s.
The order quantity is the difference between the current inventory level and the upper
bound S.
3.3 Analysis of Inventory and Production Control Models
3.3.1 General Analysis on Inventory and Production Control Models
The four inventory/production control models shown in the previous section can cover
the ideas behind most of the inventory control research. The models are valuable in the
sense that they show the fundamental tradeoffs in inventory and production control and
provide insights of the major reasons that lead to those tradeoffs. However, these models
have their inherent shortcomings to be applied directly for manufacturing system design:
1. The models are aiming to study local cost problems (e.g. minimizing cost in a
production department). The cost structure for a manufacturing system is very
complicated [Cochran 2002]. Many system level factors have significant affect on local
cost. Therefore optimizing local cost will not necessarily lead to superior system wide
performance.
2. As can be seen from the detail discussion in the previous section, the inventory control
models are based on strong assumptions that cannot always hold in real situations. When
the real system is far away from those assumed postulations, the "optimal" cost derived
from those models may be far from the lowest cost that the system can reach.
3. The most important downside of those inventory and production models is that they
are not customer needs oriented. Almost all stochastic models (which need to consider
non-deterministic customer needs) are trying to tradeoff customer service level with
inventory holding cost. This is unacceptable in a system design point of view. Since
revenue of a manufacturing system comes from customer satisfaction, it makes no sense
to "optimize" the customer service. Customer needs for a manufacturing system are
fixed. If the system cannot meet them, it's out of business, which mathematically means a
negative infinite cost.
63
While many people were enjoying playing math models in their fictional "manufacturing
system", they ignored one thing: the models should be able to work in the real situations.
Many successful "lean" manufacturers achieved superior system performance by using
very simple rules that are mathematically very loose, while the MRP planning (which
incorporates complex inventory and planning models) based companies messed up with
every thing from shop floor control to inventor management. To see why this is
happening, we need to jump out from math tricks and look at the problems in a system
design point of view.
3.3.2 Analysis of EOQ Model
The high-level functional requirement (FRI) of EOQ problem is to minimize overall cost.
The design parameter that the EOQ model selected to achieve this FR is cost optimization
based on model assumptions (DPTl).
Setup cost
Total cost
CT(Q)
hQ AD|
+cD
+
2Q
=
Production cost
(Constant)
Inventory holding cost
Figure 3- 4: The cost structure of EOQ model
The EOQ model identifies the three fundamental ingredients for overall cost, namely
setup cost, inventory holding cost and production cost, as shown in Figure 3- 4.
Therefore, the model decomposes the FRI into two low-level FRs given the high-level
DP: FR 11 "minimize setup cost" and FRT 12 minimize inventory holding cost". One
DP (DPTl 1) is chosen to achieve both FR T1
quantity
64
Q". The
and FRT12, namely "Optimizing reorder
decomposition analysis for EOQ model is show in Figure 3 - 5.
FRI:
Minimize Overall Cost
DPT1:
Optimization based on cost
model
FRTl 1:
FRT12:
Minimize setup cost
Minimize inventory holding Cost
DPT 1:
Optimization of order
quantity Q
Figure 3- 5: Decomposition analysis of EOQ model
The design matrix of EOQ model therefore is:
FRT1I
FRT 12
=
~XfPT
l}'1
_X
Since FRs are independent by definition, it is impossible to minimize both setup cost and
inventory holding cost by one parameter. In fact according to model assumptions,
reducing one cost will necessarily result increasing the other. This design is a coupled
design since the number of DP is less than the number of FRs. According to axiomatic
design, a coupled design is unacceptable in that it is unable to meet all FRs in practice.
Design based on EOQ model views the setup cost and inventory holding cost as
conflictions. In order to reduce inventory cost, setup cost will increase; on the other hand,
if we want to reduce setup cost, inventory holding cost will rise. This statement is based
on the following reasoning that is mathematically shown in the model: since customer
demand rate is constant, if the manufacturer produces more products each time it
produces, he/she needs produce less times. However, since the production is must faster
than customer consumption, the manufacture needs to store the products in inventory
65
until they all consumed by the customer and next production begins. Therefore, the more
products the manufacturer produces, the higher inventory level he/she is going to keep.
This seemingly plausible reasoning, however, is not true in general. FR 1I and FR12 by
nature are not dependent according to AD. What relate them together and makes them
conflict to each other are two assumptions that have been used in building EOQ model:
1.
The setup cost for each setup (unit setup cost) is constant and fixed.
2. The production capacity is infinite and fixed.
From manufacturing system design point of view, neither of these two statements is true.
Setup cost reduction and production capacity flexibility are among the most important
system design parameters. If these two unnecessary assumptions are removed, the
coupling relationship between FR 1I and FR12 is broken. By reducing unit setup cost,
total setup cost can be reduced independently. To reduce inventory level, production rate
needs to be modified to be as close to customer consumption rate as possible. The ideal
scenario would be the unit setup cost is cut to nearly zero and production rate perfectly
matches the customer consumption rate. In this case the overall cost would approach to
zero.
A new design based on system design approach is proposed in Figure 3- 6. A different
design parameter DPs1 "cost reduction based on system design" is selected to implement
system design approach to satisfy FRI. FRsi land FRs12 are decomposed as before. To
reduce total setup cost, the unit setup cost needs to be reduced instead of trying to reduce
the number of setups. To achieve inventory holding cost reduction, flexibility need to be
built in production capacity to ensure the manufacturer produces only customerconsumed amounts. DPs1 affects FRs2 since practically the production rate might not be
able to be very close to customer consumption rate, therefore setup can still affect the
inventory level.
The design matrix based on the system design decomposition is the following:
FRs11
FRs12
66
FX
0
DPS1I
LX
X
DP12
This new design has two DPs to achieve two FRs and the design matrix is triangular,
therefore it is a decoupled design. Path dependency requires DP s1 1 should be
implemented before DPs 12 in order to achieve both FRs 11 and FRs12 without iteration.
FRI:
Minimize Overall Cost
DPs1:
Cost reduction based on
system design
I
FRSl1:
FRs12:
Minimize setuLp cost
Minimize inventory holding
cost
- -------
--- --------------DPs11:
DPs12:
Setup cost redluction
Customer consumption
oriented production
Figure 3- 6: Modified design for EOQ model
3.3.3 Analysis of Newspaper Vender Model
Newspaper vender model is similar as EOQ model but is based on non-deterministic
customer demand. Therefore sale revenue needs to be considered (as negative cost) in the
model.
Overall Cost
1-r-------------------------z(y)=rcy
r
Dmin(y,
D)dF(D)
-
v
DO
(y
D)dF(D)
I --
Production Cost
Sale Revenue
Figure 3- 7: Cost structure of newspaper vender model
67
From the analysis shown in Figure 3- 7 it can be seen that the newspaper model deals the
tradeoff between sales revenue and production cost. If the vender produces less products
that the customers actually need, he/she loses the opportunity of selling more products;
however, if the vender produces more products than the customer needs, by the end of
selling season he/she has to trade the overproduced produces with salvage price, which is
assumed much lower than the production cost and therefore causes overproduction cost.
Therefore the vendor needs to find out optimal production volume based on the statistical
information of customer demand.
The production is assumed as one time production, so setup cost is not addressed in this
situation. Inventory holding cost is also neglected by assuming it is sufficiently small
comparing the overproduction cost. The idea of newspaper vender model is shown as
Figure 3- 8.
FRi:
Minimize Overall Cost
DPTl:
Optimization based on cost
model
r
FRT1 1:
FRT12:
Maximize sale revenue
Minimize overproduction cost
DPT11:
Optimization of production
quantity
Figure 3- 8: Decomposition analysis of newspaper vender model
It can be seen that this model tries to meet two system requirements (FRT11 and FRTl2)
with one design parameter (DP' 11). The design matrix for newspaper vender model is
the following:
68
FRT1I
FRT 12
XfpT11
X
Obviously this is again a coupled design with insufficient number of DPs. This design is
unacceptable since two FRs cannot be met independently by applying one DP.
Modification of this design would be based on new design parameters. Optimal
production quantity (DPTl 1) is not a correct design parameter for maximizing revenue
(FRT, 1). In the scope of manufacturing system design, sale revenue can be achieved by
producing customer-wanted quantity and variety of products with perfect quality at
customer required time. Therefore production quantity is a requirement for
manufacturing system from customer side. The system has to be designed to achieve this
requirement instead of "optimizing" it. Under the assumptions that are used for the
newspaper vender model, a new design parameter, DPs 1I "Perfect quality products with
on-time delivery", is selected to achieve the FRs1 1 "Maximize sale revenue". In order to
achieve the other FR to minimize overproduction cost, a pull based information flow
must be established. Since the customer demands are stochastic, the only way to produce
customer wanted quantity and mix is to direct the customer demand information into the
entire production system. Each process produces the quantity and mix that is required by
the downstream process, while external customer pulls from the finishing process. The
new design decomposition is shown as Figure 3- 9.
69
FRI:
Minimize Overall Cost
DPsI:
Cost reduction based on
system design
FRs11:
FRS 12:
Maximize sale revenue
Minimize overproduction cost
DPs11:
DPs11:
Perfect quality products with
on-time delivery
Information flow design &
customer integration
Figure 3- 9: Modified design for newspaper vender model
The design matrix of this new decomposition is
FRsll
FX
0
DPs11
FRsl2
0
X
DPs12
This is uncoupled design and the two FRs can be achieved independently. By applying
DPS 11 the manufacturer can over perform its competitors and win more market share
which will lead to more sale revenue. DPs12 perfectly aligns the production system with
customer demand and makes it robust with customer demand variation. In this pull-based
system overproduction cannot happen therefore the ideal overproduction cost would be
zero.
3.3.4 Analysis of (Q,r) Model
An analysis of the total cost structure of the (Q,r) model is shown in Figure 3 - 10. It
composes three parts: setup cost, stockout cost and inventory holding cost, therefore it is
much more complicated than the previous models, whose total cost only has two major
components.
70
Total cost
'
Q+1
ID
Y(Q,r) ~ Y(Q,r) =+ A+bB(r)|+h[-+r-0+B(Qr)]
IQ
i
i
I
i
2
I11
tT
Setup Stockout
cost
cost
ii
Inventory
holding cost
Figure 3- 10: Cost structure of (Q,r) model
In order to minimize the total cost, two optimization parameters are selected: production
volume Q and reorder point r. The production is assumed to use the following policy:
when inventory drops to reorder point r, the manufacturer will start the production and
produce Q products. Customer demand is assumed to be stochastic while production
leadtime is taken as constant in the typical model.
If the production Q is set at a high value, the setup cost will be small since less
production is needed for each time interval. However, large production batch will
increase average inventory level and hence increase the inventory holding cost. On the
other hand, higher re-order point will give the manufacturer more protection of stockout,
but it also increases the inventory level therefore incurs additional inventory holding cost.
The model is trying to find out the optimal Q and r combination that minimized the sum
of the three cost components. The decomposition of this idea is shown in Figure 3- 11.
71
FRi:
Minimize Overall Cost
DPTl:
Optimization based on cost
model
FRT11:
FRT12:
FRT13:
Minimize setup cost
Minimize inventory holding cost
Minimize stockout cost
DPT11:
DPT 12:
Optimization o forder
quantity Q
Optimization of
reorder point r
Figure 3- 11: Decomposition analysis for (Q,r) model
DP T "Optimization based on cost model" is selected to satisfy FRI "Minimize overall
cost". Three low level FRs are further decomposed, they are FRT 1
minimize setup
cost"; FRT 12 "minimize inventory holding cost" and FRT 13 "minimize stockout cost".
In the optimization model, two DPs are selected to achieve these three FRs, they are: DPT
11 "optimization of order quantity
Q" and DPT
12 "optimization of reorder point r".
From the cost equation it can be seen that the order quantity affects setup cost and
inventory holding cost, while reorder point affects stockout cost and inventory holding
cost. There for the design equation for (Q,r) model can be expressed as the following.
FRT I1
V0
FRT 12
IVX
0 X
FR T 13
DPTrl
DPT12
This design equation apparently represents a coupled design since there is insufficient
number of DPs to achieve the FRs. In order to achieve an acceptable design, three DPs
need to be designed and the design matrix needs to be at least triangular.
To minimize setup up cost, procedures that can reduce setup cost need to be established.
In manufacturing system design, setup is normally referred to as system changeover.
72
Minimizing changeover cost is usually realized by changeover time reduction through
externalized internal setup procedures and standardization of setup operations
There are two types of changeover activities: internal and external. Internal changeover
activity refers to the activities that have to be performed when the system is down (e.g.
changing machine tools). External changeover activity means the operations that can be
done parallel while the system is still running (e.g. preparing the new tools). Changeover
time reduction is usually conducted in two steps. First step is converting as much of
internal changeover activity as possible to external activity. By doing this, the
unnecessary system down time is eliminated. Second step is to reduce the internal
changeover time that is left after conversion. This step includes the following procedures:
1.
Standardize changeover operations to establish unambiguous and effective
procedures to perform changeover.
2. Train operators as well as management to consistently perform the standardized
changeover operations.
3. Implement devices (e.g. positioning pins for die changeover) to facilitate
changeover activities.
The two-step changeover reduction is shown in Figure 3- 12. It is important to point out
that, according to many industrial applications, 50% of internal changeover time can be
easily externalized and 50% of the rest internal changeover time can be reduced by the
three procedures listed above. Therefore, a rough estimation of 75% system down time
due to changeover can be avoided.
73
System downtime
Product A
]
tion
Actual Changeover
FCleaning up
Product B
...........
Conversion to external changeover
Preparation I
Cleaning up
-I
Product A
Actual
F
Product B
Changeover
*__--
--
__Reduction of internal changeover
System downtime
Figure 3- 12: Two steps changeover time reduction [Cochran 2002]
To satisfy the FR "minimizing inventory holding cost", production needs to be customer
consumption oriented, namely producing only the customer consumed quantity and mix.
This is realized by designing pull-based information flow system. Similar as the
discussion in newspaper vender model, customer requirements requirement information
needs to be thoroughly and consistently directed into the entire manufacturing system.
It can be seen from the derivation of (Q,r) model that the probability of stockout cost is a
function of reorder point r, customer demand distribution p and production leadtime 1.
Customer demand distribution is external and cannot be changed. Reorder point r affects
both stockout cost and inventory holding cost. Therefore selecting r as DP would result in
a coupled design. Production leadtime reduction is an appropriate DP to reduce stockout
risk. If production leadtime were reduced to zero, then the production would be stockout
free. This means production leadtime is an independent DP to solve to stockout cost FR.
74
FR1:
Minimize Overall Cost
DPsl:
Cost reduction based on
system design
FRs 11:
FRs 12:
FRs13:
Minimize setup cost
Minimize inventory holding Cost
Minimize stockout cost
DPs 11:
DPs12:
DPs13:
Setup cost reduction
Customer consumption oriented
production/information flow
Production leadtime
reduction
Figure 3- 13: Modified design for (Q,r) model
The new design decomposition is shown in Figure 3- 13. The path dependencies shown
in the decomposition indicates that it is a decoupled design. Setup cost reduction will
affect inventory holding cost in the sense that, the more setup cost can be reduced the less
inventory is needed to hold. Information flow and customer oriented production bring
real-time accurate customer demand information to all production processes. It helps to
reduce customer demand variation from inside and out side of the manufacturing system,
which reduces the risk of inventory stockout. The new design equation corresponding to
the decomposition above is:
FRsl
_X0 0
DPsll
FRs12 =
XX 0
DPs12
_0XX
DPs13
FRs13
3.3.5 Analysis of (s, S) Model
The (s, S) model is similar with the (Q,r) model except it doesn't confine itself to any
predefined policies therefore its solution would have more optimality than the one of
(Q,r) model. Mathematically speaking, the optimal cost of (s,S) would be the lower
75
bound of all possible optimal cost of (Q,r) model assuming they use the same set of
assumptions.
Analysis of the total cost structure in Figure 3- 14 shows that there are also three cost
components in (s, S) model: setup cost, stockout cost and inventory holding cost.
Total cost
JO(x 0 ) = E{j(C(uk)+pmax(,wk -xk -U )I+ih max(O,xk
k=O
I
Setup
cost
1
+Uk
W)
------------------------------
Stockout
cost
Inventory
holding cost
Figure 3- 14: The cost structure of (s,S) model
Similar to (Q,r) model, two optimization parameters are selected to minimized three cost
components. The order-to amount S affects the setup cost and inventory holding cost,
while the reorder point s affects the stockout cost and the inventory holding cost. A
decomposition analysis of (s,S) model is shown in Figure 3- 15.
76
FRI:
Minimize Overall Cost
DPTl:
Optimization based on cost
model
FRT11:
FRT12:
FRT 13:
Minimize setu pcost
Minimize stockout cost
Minimize inventory holding
DPT11:
DPT12:
Optimization of order-to
quantity S
Optimization of
reorder point s
Figure 3- 15: Decomposition analysis of (s,S) model
The design equation of (s,S) model is the same as the one of (Q,r) model, which is shown
below.
FRT'
_X 0-DFlI
IFR T12f = XX
X
FRT 13
p1
0 X_
To correct this coupled design, three DPs need to be designed to achieve a decoupled
design. Setup cost reduction is selected to minimize setup cost; customer consumption
oriented production/information flow is chosen for minimizing stockout cost; production
run size reduction is selected to minimize inventory-holding cost. The DPs for this design
is different from those for (Q,r) model. The reason is that these two models are based on
different assumptions. For example, in (Q,r) model the production leadtime is assumed to
be constant and fixed, while in (s, S) model it is assumed to be zero. Therefore leadtime
reduction is a DP for (Q,r) model but it's not applicable for (s,S) model, new DP (in this
case, batch size reduction) needs to be determined to complete the design.
77
FRI:
Minimize Overall Cost
DPsl:
Cost reduction based on
system design
FRs11:
FRs12:
FRs13:
Minimize setup cost
Minimize stockout cost
Minimize inventory holding cost
--------------------------------------------------DPs11:
DPs12:
DPs13:
Setup cost reduction
Customer consumption oriented
production/information flow
Production run size
reduction
Figure 3- 16: Modified design for (s,S) model
It is clear that both setup cost reduction and information flow affect inventory holding
cost. But it's not clear that whether setup cost reduction also affects the stockout cost. In
order to decide whether this path dependency exist or not, it is worthy to revisit the
derivation of the simpler version (s,S) model with zero setup cost assumption. The result
(3.24) clearly shows that if setup cost were zero, the reorder point (and therefore stockout
cost) would also zero. It means that setup cost reduction is strongly relative to stockout
cost, which can be stated as a path dependency between DPs 1 and FRs 12. The new
design decomposition is shown in figure above and the design equation is shown in the
following. Obviously it is a decoupled design therefore is theoretically superior to the
original coupled design.
FRs 1
FXO 0 ]DPs11
FRs12
XX 0
DPs12
FRs13
XXX
DPs13
3.3.6 Conclusion
This section analyzed the four inventory and production control models. A comparison of
their assumption and design type is listed in the table below.
78
EOQ
Model Assumptions
Deterministic
Customer Demand
(Qr)
(s,S)
X
X
X
X
Stochastic
Zero leadtime
NV
X
X
X
Production Leadtime
X
Non-zero leadtime
One period
X
Prouduction Horizon
Multi-period
X
X
Fixed
X
X
X
Production Runsize
Variable
X
Dependent
Dep-Independent
Production Dependency
Type of Design
---X
X
X
X
EOQ
NV
(Qr)
(s,S)
Uncoupled Design
Not-coupled Design
Coupled Design
---
---
X
-
----
---
Decoupled Design
Sufficient Design
Insufficient Design
X
X
X
X
Table 3- 1: Comparison of inventory and production control models
3.4 Applying System Design Methodology to Solve Optimizing
Problem
3.4.1 Problem Description
A world-class skiing apparel manufacturer is facing a planning problem for their
production of one of their key products - skiing parkas. Due to the long production
leadtime, the manufacturer mainly relies on forecasting based on previous years' sale
data to schedule their production. However, because the large product variety and great
variation in their forecasting, the company always ended up with overproduction of some
79
types of parkas and underproduction some of the others. As has been addressed before,
both over and under production bring the company significant financial or opportunity
loss. The supply chain of the skiing parkas manufacturing is shown in the following
figure.
ProductFlow
Material Suppliers
Retailers
-
Orderingflow
Figure 3- 17: Supply chain for apparel production
The manufacturer orders raw material from material suppliers. The most important
material for parka production is fabric and insulation material. While all parkas are using
the same insulation material, there variation in fabric material is big. Approximately 10
different types of shell fabrics are needed each year for parkas. For each type of fabric, it
will be dyed or printed into 10 different colors or prints. The production leadtime for
fabric material is roughly 5 month and for dying and printing is 6 weeks. Because both
the fabricating and dying processes of fabric production are done in huge batch,
minimum order quantities are needed for each fabric type and individual color or print.
After the company receives the material from material supplier, it distributes the parka
manufacturing between two of its major production areas: Mainland China and Hong
Kong. The hourly labor cost of Mainland China is about 1/30 of that of in Hong Kong,
however, due to lack of training and low level of skill, the workers' production rate is
lower and defect rate is much higher in Mainland China production facilities. Therefore
the production in China needs longer leadtime and requires larger batch size.
More than 80% of customer demand is received on March each year in Las Vegas trade
show, and most of the demand needs to be fulfilled around September of the same year.
80
Since there are only 6 months between customer placing their order and production, and
the total production leadtime is about 15 months, the manufacturer has to plan most of
their production based on forecast. A detailed timeline for parka production, customer
demand information and material production is shown in Figure 3- 18. For the
convenience of time reference, it is assumed the company is planning the production
schedule for the year of 1993.
Jan 94
Jan 93
Jan 92
Desion
I
eb 92
Reta ding
Production
Jan 3
Sep 93
Production
Mar 93
Demand
AAA
Information
Material
Supply
Additional Order
Las Vegas 93'
Jul 92
Fabric
Nov 92
Dying
Order
AA A
Replen'ish Order
Mar 93
Dying
Order
Figure 3- 18: Supply chain production timeline
The production can be divided into three stages: design, production and retailing. After
the parka styles are designed, all the different fabric/color combinations are produced and
tested in small quantity. This process usually takes more than a year, from February 1992
to January 1993. Full-scale production begins at January 1993 till September. After that
will be the retailing stage, in which the manufacturer ships the products to retailer and
small batch production for replenish orders will still being conducted subjected to the
availability of material and order size. Demand information shows that the majority of
customer orders are received at the Las Vegas show while small amount or additional
orders and replenish orders will arrive afterwards. In order to start the full-scale
production on January 1993, the dying order needs to be placed at November 1992.
Dying order is usually split into two orders. The reason is that the manufacturer believes
81
that the later it makes a forecast the more accurate it would be. Therefore it places first
order just to start the production and place the order later based on a more accurate
forecast. To process the dying order, however, the fabric order has to be ordered at July
1992 to allow a 5-month production leadtime. Figure 3- 18 demonstrates timeline of the
entire production process.
Therefore, in order to maximize the expected profit, the manufacturer needs to make
following two decisions:
1.
How much fabric of each style/color combination should be ordered in the first
order and how much in the second order?
2. How much production needs to be done in Mainland China facility and how much
in Hong Kong?
3.4.2 Solution based on Mathematical Optimization Methodology
The case study is a very typical one for supply chain management courses. The
traditional optimization methods to solve this problem can be summarized in the
following steps:
1.
Establishing cost (profit) calculation model by simplifying the problem.
2. Identifying the constraints and expressed them mathematically.
3. Formulate the case into an optimization form based on 1 and 2.
4. Solve the optimization problem to find the optimal solution.
Various optimization models can be applied to this problem, including non-linear
programming and dynamic programming based models. The optimality of their results
depends on the modeling assumptions and solving algorithms. This section will show a
two-stage dynamic programming algorithm based model as an example of applying
optimization methodology to solve this supply chain management problem [Caro et al,
2001].
The decision parameters that this model selects are the production volume of each
different product type at each of the two production stages. The cost-to-go function of the
82
second stage is shown below. The xi and yi are the production quantity of each product i
at the first and second stage respectively; K is the total production capacity, which equals
to 20,000; pi is the wholesale price and mi is the minimal ordering quantity of each
product type i.
J,(D, x) = max,?
{ 0.24 p, min(D,,xi + y)
-0.08 pi(xi + yj -min(Di,x + yi)}
s.t. y ? m, or yj =0
? y.?K-? x,
i
By changing notations, the cost-to-go function can be simplified into the standard form:
10
J1 (D,x) =max,? pi(0.24z +0.08w )
i=1
s.t. z, ? D
z, ? X, + yi
z, ? 0
Wi ? Di - xi - y
? y, ? K-? xi
Kb, ? y, ? m, .
hi. w,
W y,. ? 0, 6, ? {0,j}
And the first stage cost-to-go function can hence be expressed as the following:
JO = maxX ED 1 (D, x)]
s.t. x ? m1 or xi = 0
? x ? K
x.? M
M is the minimal ordering quantity in the first stage, which equals to 10,000.
83
It can be seen that the cost-to-go functions of this dynamic programming model are nonlinear which involves non-continuous identity functions and mathematical expectation.
Finding out the analytical algorithm is extremely difficult, if not impossible. Two suboptimal solving procedures were presented in the reference [Caro et al, 2001]. One is
based on rank-sorting methods and the other is based on Monte-Carlo simulation. Suboptimal solutions were resulted from both methods that including the production order
quantity vector and the optimal profit. The simulation result is shown in the following
figure. The production vectors were iterated according to their objective profit value. The
sub-optimal profit was reached when the iteration converges. Figure 3 - 19 shows that the
value of the sub-optimal profit is close to $490K.
X 10 5
Tipical run of the local search in the first stage.
4.88 -
4.86 -
4.84-
4.82.0
4.8-
4.78-
4. 1
3
4
5
6
7
8
9
10
Local search iteration
Figure 3- 19: Interactive solving result of optimal cost searching [Caro et al, 2001]
3.4.3 Solution based on System Design Methodology
The optimization model says that $490K is the highest profit that the manufacturer can
achieve. However, since the customer demand is 200,000 and the average profit for each
parka would be $50 if produced in China, the ideal profit is $1,000K. This means that no
matter how good the planning it makes, the manufacturer will lose more than half of its
potential profit. So the question is: Is that possible for the manufacturer to avoid this big
84
loss and make higher profit? Obviously the optimization planning will say no, but we will
show the system design will say yes.
No matter how sophisticated that people think manufacturing system design will be, the
beginning of it is surprisingly simple. The only thing people need to start system design is
asking a one-word question: "Why"?
"Why can'tI make $1 000K profit?"
"Because you cannot sell allyouproducts and you have to trash some of them and that brings
you a big loss."
"Why can't I sell all my products?"
"Because you produce accordingto forecast basedplanning, which is not what customers will
really buy."
"Why do I produce according to planningrather than by real customer need?"
"Because you have long production leadtime so you have to startyour production before
customers need actually happen."
"Why do I have long production leadtime?"
"Because both your materialsupplierandyourself have big production batch."
"Why do I have big production batch?"
"Because your changeovers cost your suppliers a fortune and your workers are not trainedto do
mixture in your sewing line."
"So, the problem is my suppliers' changeover and my training."
High Level Design Process:
The conversation above represents a decomposition process, which is one of the key
concepts for manufacturing system design. The first question is actually the statement of
the general system design goal: To achieve full profit that can be made by the customer.
A straightforward analysis would show that the reason the manufacturer cannot achieve
full profit is that it cannot produce only customer required quantity and mix. Under this
situation customer demand in some product types cannot be met while the manufacture
may overproducing in the other types. The former will result of losing potential profit and
85
the latter causes overproduction cost. The combination of these two effects leads to the
result that the manufacturer can only make half of its potential profit in the best case.
To allow the manufacture make more profit, the system must be redesigned to break the
constraints that prevent it from doing so. The second and the third questions showed that
the two major constraints are not knowing customers' demand and long production
leadtime. Production under forecasting data instead of real customer demand data will
necessarily result in either unmet customer demand or overproduction, both of which will
result in profit loss. Therefore providing the manufacturer as accurate as possible
customer demand information will be the first requirement for profit loss reduction. The
design parameter to meet this requirement is customer integration, which is a key concept
of supply chain management. In the original case, there is no information communication
between manufacturer and the retailers before the Las Vegas trade show, where the
orders are actually placed. The manufacture has to "guess" what the retailers' order
quantity and mix would be. The following table shows the average value and standard
deviation of customer demand forecast for each product type. It can be seen that the
forecast is very unreliable in the sense the c.v. value (the ratio of standard deviation to
average value, which shows how "concentrating" the statistical data points are) is very
high.
Style
Gail
Isis
Entice
Assault
Teri
electra
Stephnaie
Seduced
Anita
Daphne
Forecast
SAverage
1,017
1,042
1,358
2,525
1,100
2,150
1,113
4,017
3,296
2,383
Forecast
Std. Dev.
Harf
Forecast
Forecast
c.v
388
646
496
680
762
807
1,048
1,113
2,094
1,394
508
521
679
1263
550
1075
556
2008
1648
1192
38.18%
62.04%
36.49%
26.95%
69.23%
37.56%
94.18%
27.71%
63.53%
58.48%
Table 3- 2: Manufacturer's forecast data
However, if the manufacturer can cooperating with the retailer to share the demand
information, it could get more accurate information for production information, which
would greatly reduce the risk of overproduction or unmet customer demand. And also,
86
since the retailers are also collecting and forecasting information from their customers,
they compose a two-stage forecasting supply chain with the manufacturer. This will
typically result in bullwhip effect, which means the inaccuracy of forecasting increases as
the supply chain element goes away from external customer. By forecast information
sharing, the manufacturer and retailers are integrated into one element in the supply chain
and generating sharing forecasting information. One unnecessary forecasting step in the
supply chain is eliminated and more accurate demand information is resulted. The
following figure shows reducing bullwhip effect by retailer-manufacture integration.
Manufacture
Retaffer
?
Customer
?
?
?
--
r...
Figure 3- 20: Retailer-manufacturer integration
Given that the best demand information can be provided from the retailer to the
manufacturer, it still needs to be able to meet these demand information on time. The
design parameter to achieve this is leadtime reduction. As has been analyzed in the
introduction section, the major reason that the manufacturer has to begin production
before accurate information can be achieved is that both the manufacturer itself and the
supplier have a very long production leadtime. Therefore the manufacturer has to make
enough time allowance for itself as well as its supplier.
By production leadtime reduction, the production processes can be postponed
accordingly. The manufacturer therefore can take advantage of more accurate customer
demand information that would greatly helpful in reducing profit loss.
87
Old Design
Jan 94
Jan 93
Jan 92
Feb 92
Retailing
Production
,
Design
Jan 93
Sep 93
Production
Mar 93
Demand
Information
New Design
A
AAAA
Additional Order
Las Vegas 93'
A
Replenish Order
,JProduction
Design
Retailing
Production
Demand
Information
A A A A
A
A A A
Additional Order
Retailer Info
Las Vegas 93'
A A A
Replenish Order
Figure 3- 21: Comparison of production timeline between old and new design
As shown in Figure 3- 21, manufacturer/retailer integration leads to earlier and well distributed demand information to the manufacturer. And also, by leadtime reduction, the
production phase can be shifted to the right, which allows additional information to feed
in the production schedule. These two high level FR/DP pairs compose the top level of
system design decomposition, as shown in Figure 3- 22.
88
FR1:
Achieve full profit
DPI:
Customer needs
driven production
FR1:
FR12:
Acquire actual customer
demand information
Meet customer required
quantity and mix
---- -----------------------------------------------------DP11:
DP12:
Manufacturer/retailer
integration
Production leadtime
reduction
Figure 3- 22: High-level decomposition of new design
Production leadtime is a key issue in "lean" manufacturing system design. In this
particular case, there are two leadtime components involved: the supplier's leadtime and
the manufacturer's leadtime. In order to reduce the overall production leadtime, both
components need to be considered. This is shown in the FR/DP 12 decomposition.
FR12:
Meet customer required
quantity and mix
DP12:
Production leadtime
reduction
FR121:
FR122:
Reduce supplier's
leadtime
Reduce manufacturer's
leadtime
DP121:
DP122:
Supplier leadtime
reduction
Manufacturer leadtime
reduction
Figure 3- 23: Production leadtime reduction design
Supplier integration is a critical step to achieve supplier leadtime reduction. As Monden
discussed in TPS [Monden 1998], it is inefficient to apply the "lean" concepts in some of
89
the elements in the supply chain while the other elements remain "mass". Lean
production needs fast material flow with great variety. Without well-designed
information share between supplier and manufacturer, the "lean" in manufacture will be a
heavy burden for suppliers, who have to maintain an even greater amount of inventory to
satisfy the manufacturer's demand. Mutual trust and cooperation need to be established
between the supply chain elements. The manufacturer is responsible to help the suppliers
to adopt the same system design concepts. Minimizing material cost by squeezing
supplier is a widely used optimization strategy for manufacturers. However, it is just a
short-term solution because the suppliers will eventually give up or go bankruptcy.
Similar to the situation of retailer integration, supplier integration gives supplier more
real-time and accurate information for their production. To meet the manufacturer's
demand, however, the supplier also needs to follow similar design parameters to cut
down their batch size. As pointed out in the EOQ model, the changeover cost needs to be
reduced. By doing this the supplier can have more changeovers than before with the same
changeover cost, and therefore the batch size can be reduced without extra cost. The
decomposition of FR/DP 121 is shown below.
FR121:
Reduce supplier's
leadtime
DP121:
Supplier leadtime
reduction
FR-S 1:
FR-S2:
Transfer information to
supplier
Reduce supplier's batch
size
DP-S2:
P-S 1:
Supplier integration
Supplier's changeover
cost reduction
Figure 3- 24: Supplier leadtime reduction design
90
The manufacture's leadtime can be decomposed into two parts: the production leadtime
that is needed inside of production facility and the transportation leadtime that is
consumed in transportation processes. In order to shorten the manufacturer's leadtime,
design parameters need to be established to reduce both of its components
Large batch size is the reason for long leadtime in the production facilities. When
products are produced in a batch size larger than 1, the phenomena that parts waiting each
other occurs. When one product is being processed, the parts behind it have to wait. On
the other hand, when one product is finished, it has to wait to be shipped until all
products behind it have been finished. The waiting time is proportional with the batch
size. In industries with huge batch size, the waiting time can be many hours while the
actually processing cycle time is less than a minute. Production leadtime can be reduced
dramatically by batch size reduction.
The nature of production of the manufacturer is different from of the supplier. The former
is mainly human operation while the latter is mostly machine processing. Therefore the
DP to achieve batch size reduction for the manufacturer is necessarily different from the
one designed for supplier. The main constraint for the manufacturer to achieve small
batch size is the cost of human learning curve. Operators tend to be more productive and
make less defective products when they get used to an operation. When that operation
changes due to changing to a different product, the production rate drops and the defect
rate is likely to increase, which incurs cost for the system. Operator training is critical to
overcome this problem. The training program usually include two general steps:
1. Standardized operation training.
2. Cross-functional training.
The first step will eliminate the variations in system due to operators' different operation.
By establishing standardized operations and enforcing them to be strictly followed,
system performance will be stable and independent of particular operators. Crossfunctional training allows operators to change from on operation to the other without
losing productivity. If operators can smoothly transfer from one product's operation to
another product's operation, the mix-product production can be realized and therefore
changeover cost can be eliminated.
91
Aligning material transportation flow with information flow is the solution to reduce
transportation cost. The original transportation flow is shown in Figure 3- 25.
winnipeg
Thunder Bay
Seattle
Sa
Bismarck
HeI na
Mi
Pierre
Boi se
Sioux Falls@
nne
Salt LAkO Ci
wichita
d 10
p @
SLa e4itl~eldock
San Francisco
Los Angeles
Lincoln
Springfield
SaCI Denver
S a rta
Mad ison
Phoenix
Dallas . Jackson
Figure 3- 25: Schematic of transportation waste in old supply chain
The products will be shipped to Seattle central warehouse by ship. Then they are
transported by truck to Denver distribution center where all products are distributed to the
retailers. It can be seen that the transportation from Seattle to Denver is unnecessary. In
the new design that is shown in Figure 3- 26, the distribution information is sent to
Seattle from Denver and products are distributed directly from Denver to retailers.
mpLi ORa
T fi ner
isrnarck
Vi~
A
~O
Pierrek
Bo is
Seattle
kK5)
Cheyenne
Salt LAke C
Lincoln
Denver
a FgA
se Denver
ail~~~tl
t
I0
i0e
Figure 3- 26: Material flow oriented transportation design
92
The decomposition of FR/DP122 is shown in below.
FR122:
Reduce manufacturer's
leadtime
DP122:
Manufacturer leadtime
reduction
F R-M1:
FR-M2:
Reduce batch size delay]
Reduce transportation
delay
DP-M1:
DP-M2:
Operators' training
Material flow oriented
transportation
Figure 3- 27: Manufacturer leadtime reduction design
The full decomposition is shown in the following figure. It is
While no accurate monetary value can be resulted from this new design based on system
design methodology, an approximate number can be estimated looking at each leaf-level
DPs. The leaf-level DPs and their estimated cost are listed in Table 3- 3 below.
DP Number
DP 11
DP-S 1
DP-S2
DP-M1
DP-M2
DP Name
Manufacturer/retailer integration
Supplier integration
Supplier's changeover cost reduction
Operators' training
Material flow oriented transportation
Total Cost:
Estimated Cost (K)
10
10
50
10
-10
70
Table 3- 3: Estimated implementing costs of leaf-level DPs
As shown in the table, the total cost to achieve this new design is about $70K, therefore
the best-case profit would be:
$1000K-$70K=$930K
93
Clearly this maximal cost is much higher than the optimal value $490K that can be
reached by optimization methodology.
As discussed in the beginning of this chapter, system design methodology can always
achieve much better performance than optimization methodologies. A theoretical
explanation can be given by sensitivity analysis of optimization problems. If an
optimization problem has many extreme points that are very close to optimal solution, it
would very difficult to find out the real optimal solution. This process is going to take
numerous iterations with minor optimality improvement. However, if some constraint
function can be changed with even a very small value, the object value can be improved
dramatically.
A system design methodology is using exactly this property. Instead of searching for
"optimal" solutions, it aims to meet all requirements by attacking unnecessary constraints
that have been assumed by optimal methodologies. Through applying effort to move or
eliminate constraints (with some cost), great benefit can be achieved by meeting all the
system requirements in a path dependent, non-iterative way.
94
1:1'
I ii
I,
I
'FE
.I.
iI
M
'I
i R
O.
Ir
I
ii!
I
~Ii
MirM
I~J~
NU
I
IFFI
91
Figure 3- 28: Design decomposition of supply chain optimization problem
95
96
Chapter 4: Manufacturing System Analysis based on
Stochastic Models from System Design Point of View
4.1 Introduction of stochastic models for manufacturing system
analysis
The inventory/production control models that have been addressed in the previous
chapter have been dominating the manufacturing system quantitative analysis and design
for a long time. These models are based on one-stage production scenario. The
production system is view as an integrated "big machine" that has aggregated properties
such as production rate, production leadtime and production batch size. Research based
on these models can be viewed as a "macro level" study of manufacturing system, since
only the relationships among the "big machine", inventory and customers' demand are
addressed. However, as the manufacturing system getting more and more complex and
competition in product quality, production rate and overall production cost getting more
and more fierce, more precise and refined analysis methodology needs to be developed to
study the "micro level" of manufacturing system, namely the relationship between
machines and inline buffers (WIP).
Stochastic models of manufacturing systems were developed under such as situation. It is
widely recognized that applying stochastic models in manufacturing system analysis
originated from Jonh. A. Buzacott's work in 1960's [Buzacott 1993], which can be
viewed as the basis of most of the later work in this area.
4.1.1 Introduction of Markov process
Stochastic process is the study of the sequences of events governed by probabilistic laws.
A most common type of stochastic process is concerned with the investigation of the
structure of families of random variables Xt, where t is a parameter running over a
suitable index set T. The variable X could be one- dimensional or n-dimensional, or even
more general, infinite-dimensional. It could also be either discrete, continuous, or a
combination of both. The second index T is usually one dimensional, either continuous or
discrete. Stochastic processes with T=[0, ? ) are particularly important in applications, in
which t can be interpreted as time.
97
Markov process is one of the most important stochastic processes that have been widely
used in many applications such as physics, chemistry, transportation system,
manufacturing system, aero and astronaut systems, etc. A rough definition of Markov
process can be stated as the following [Karlin, Taylor 1975]: a Markov process is a
process with the property that, given the value of Xt, the values of Xs, s>t, do not depend
on the values of X., u<t; that is, the current state of the process will provide sufficient
information to predict future behavior of the process, and knowing the process
information before current state has no additional benefit. In formal terms a Markov
process can be expressed as the following:
Prfa < X, <b| X, = x,
X
=x2,---X
=x,}= Prfa < X, <b| X, =x,
whenever t, < t 2 < ... < tn < t ... (4.1)
It can be seen from its definition that a Markov chain is a probabilistic dynamic system in
which the future behavior depends on only the present situation, not the past. Specifically
the two components of a Markov process, the random variable and the index, is defined
as state and time index of the process.
The time index of Markov process may be continuous or discrete. When it is discrete, it is
usually only allow the time value to take integer values or some other countable number
set. Such Markov processes are characterized by difference equations. When the time
index is continuous, it usually takes all real number or positive real number. This type of
Markov process is usually characterized by a set of differential equations. The state of a
Markov process can be discrete, continuous or a combination of both. In a discrete state
Markov process, the random variable can be of one of the value of a discrete set at each
time interval, the state "jumps" from one value to another along time index. In a
continuous state Markov process, however, the state changes continuously along the time
index.
Two types of Markov processes are widely used in manufacturing system analysis:
discrete time, discrete state and continuous time, discrete state. The following two
sections will explain them in detail.
98
4.1.2 Discrete Time Discrete State Markov Process
The basic assumption of a discrete time discrete state Markov process is the following:
Pr{X(t + 1) = x(t + 1)1 X(t) = x(t), X(t -1)
= x(t - 1), X(t - 2) = x(t - 2), ... X(0) = x(0)}
= Pr{X(t + 1) = x(t + 1)1 X(t) = x(t)} ...(4.2)
where x(r) ? S, r = 0,1,-- -t +1,
S is the state space.
Define the transition probability from state i to state j as
Pr{X(t +1) = i I X(t) = j} = i
... (4.3)
and define
Pr{X(t) = i} = p,(t) ... (4.4)
The stochastic process evolving equation can be written as the following:
pan(t +1)
=gee.p (t) ... (4.5)
and more generally, in a matrix form:
{p(t + 1)} = [P]{p(t)} ... (4.6)
? p 1 (t + 1)?
in which {p(t + W)=
P2(t +1)?
[P]l
,[P]=
p1
P12
{,
I=
By definition, p(t) is the transition probability vector, therefore it satisfies the
normalization equation:
? p,(t)=1, and ?
p (t) =1I... (4.7)
The formal equation mean that the process must be somewhere in the state space, while
the second equation states that the process keeps evolving along time index.
The future state distribution can be calculated by the following transition equation:
{p(t + n)} = [P]" {p(t)} ... (4.8)
99
A classical example of applying discrete state discrete time Markov process is machine
reliability. Consider the following situation: A machine can be operated in two states,
operational (up) and under repair (down). When the machine is operational in one time
interval, it has certain probability p going down in the next time interval. On the other
hand, if the machine is down in one time interval, it also has certain probability r being
fixed in the next time interval. The question is to determine the long -term average
production rate of this machine.
p
1-r
I -P
0
r
Figure 4- 1: Transition probability of discrete time two-state Markov process
If we define 1 for up state and 0 for down state, the system probability transition
equations can be written as the following:
? p(O, t + 1) = p(O, t)(1 - r) + p(l, t)p
p(1, t + 1) = p(O, t)r + p(l, t)(1 - p)
...
(4.9)
Given the initial conditions p(0,0) and p(O,1), the probability distribution along time t can
be solved [Gershwin 1994]:
[1-(l-p-r)t ]
,p(0,t) = p(0,0)(1-p-r)t +
r+p
?p(1, t) = p(1,0)(1 - p - r)' +
r
r[1
r+p
... (4.10)
-(1 -p -r)']
Assume the breaking probability p equals 0.02 and fixing probability r equals 0.08, and
the probabilities that the machine is operational and under repair at the beginning are both
0.5, the probability distribution as a function of time is show in Figure 4- 1.
100
Unreliable Machine Productivity
0.8 0.6 -
LO 0.4
a.
0.2 0-
0
20
60
40
80
100
t
Figure 4- 2: Asymptotic behavior of machine status probability distribution
The limit of the two probabilities can be calculated by the following steady state
probability equation:
p(O) =r+p
r~p
which is the solution of
? p(O) = p(O)(l - r) + p(l)p
p(l) = p(l)(l - p) + p(O)r
.(4.12)
A steady state probability is defined as the probability that satisfies the following:
p 1 = pj (t) = p,.(t +1)
= limp(t) ... (4.13)
t?
?
Therefore, it also can be derived from (4.9) by assuming
?p(O, t +1)
= p(O, t)
? p(, t + 1) = p(, t)
101
4.1.3 Continuous Time Discrete State Markov Process
In continuous time discrete state Markov process, the system evolves along continuous
time. The basic assumption of this type of Markov process is the following:
Pr{X(t) = x(t) X(s) = x(s),s < r} = Pr{X(t) = x(t) IX(r) = x(r) ... (4.14)
For all t > - and x(t) ? E where S is the discrete state space.
The system transition probability can be defined as
Pr{X(t)= i} =? Pr{X(t) = IiX()= j}Pr{X(r) = i} ... (4.15)
To derive the differential transition probability, replace t by t + St and r by t, (4.15)
becomes:
Pr{X(+St) =i} =? Pr{X(t+St) =ijX(t) j}Pr{X(t)= j} ... (4.16)
I
Assume St small and , exists for all i ? j such that
Pr {X(t + St) = i I X(t) = j} = A, St + o(St) ... (4.17)
plug (4.17) into (4.16), after simplification the transition probability equation can be
written as the following form:
Ali p (t)St + Pr {x(t + St) = i IX(t) = i}p,(t)+o(St) ... (4.18)
j?i
Considering the normalization equation
? Pr{X(t +St)
j IX(t) = i} = l and defining A2i = -? A,. , it can be further simplified
]
j?i
as the following:
(4.19)
pi(t+t)=? 1'jpj(t)+o(tt) ...
By applying first order Taylor expansion to the left side term of the equation above, it
becomes
102
dp
dt
therefore
di= ? 11,jpj (t), for all i ...
(4.21)
dt
*i
If the ergodic distribution exists, it will satisfy the above equation with the left hand term
equals zero, which is the following:
0 = ?
j?i
+ Aiip, ... (4.22)
plugging in the definition of A)y , this can be written as:
Pi? AJ.i =? A pj ... (4.23)
j?i
pi
This equation is called the "balance equation". It is the most important equation in
continuous time discrete state Markov process. Most of further analysis would be based
on this basic relationship. The left hand side of this equation can be views as the rate that
the process leaves state i, where the right hand side is the rate that the process enters the
state i. In the steady state, it is intuitively right that these two rates should be equal to
each other to achieve a "balance" status. The balance equation provides a mathematical
proof to support this conclusion.
The unreliable machine example discussed in the previous section can also be modeled as
continuous time discrete state Markov process. If the machine operation time is
exponential distributed with the parameter p rather than deterministic, which is assumed
in the previous example, the model essentially becomes continuous time since state
changes can occur at any (real number) time index. The probability that an operation is
completed during the time interval [t, t + 5t] while the machine is up is pt .
Accordingly, machine state change is adjusted to fit in the continuous time situation. The
probability that a failure occurs during an interval [t, t + 5t] while the machine is up is
114. The probability that a repair is completed during an interval [t, t + t] while the
machine is down is rt . It is also assumed by convention that the machine can only
103
break when it is operational and can only be repaired when it is down. The graph of this
continuous time Markov process is shown as the following.
p
I1P
0
1-r
r
Figure 4- 3: Transition probability of continuous time two state Markov process
Similar to the discrete time case, the probability distribution time function of this
continuous time problem is the following:
?p(Ot + t) = p(0,t)(l - r t) + p(l,t)pt + o(t)
p(l,t + t) = p(O, t)rdt + p(l,t)(l - p~t) = ot)
... (4.24)
after applying Taylor expansion and some simplification, it becomes
?dp(0,.t) = -p(O, t)r +
p(l, t)p
dt
dp(l, t)
... (4.25)
p(0,t)r - p(l,t)p
dt
?
Given the initial condition of p(0,0) and p(1,0), the equation can be solved and the
solution is the following
p(0,t) =
*r~p
? p(l, t) =
+ [p( 0 ,0 ) r -[p(0,0)r+p
]e-(p)t
r+p
... (4.26)
]e-(
p''
r+p
As t ? ? , the steady state solution can be expressed as the following:
104
P(O) =.(4.27)
r+
r+p
r~p
Therefore the average production rate is p(1)p or
rp
r+p
The other example of continuous time discrete state Markov process is known as the
M/M/1 queue. Consider a situation that a queuing system with infinite amount of storage
space. Products arrive according to a Poisson process, which means the time interval
between two consecutive parts is exponential distributed. The arrival rate of products is
A, which means if a product arrives at time s, the probability that a product arrives at
during the time interval [s + t,.s + t + t] is e-&2A&t. The system service rate is p , which
means that if a product leaves at time s and the buffer is not empty, the probability that
another product leaves the system during time interval [s + t,.s + t + t] is e-"put .
The system evolving equation is the following
p(n, t + 3t) = p(n - 1,t)A&t+ p(n + 1,t)put + p(n,t)(1 - (Aitt + p&)) + o(St),
forn > 0 ... (4.28)
And the boundary condition is
p(O,t + t) = p(l, t)p(t + p(O,t)(1
-
A&) + o(St) ... (4.29)
The system differential equation and boundary condition then can be derived as the
following:
?p(n, t)
p(n -,t)A + p(n +1, t)p - p(n, t)(A + p), for n > 0 ... (4.30)
?t
and
?t
=P
-A,
p(1t t)p - p(0, t)A ...
(4.31)
If a steady state distribution exists, it should satisfy the balance equation and boundary
condition, which are
105
0 =
(4.32)
p(n -1I)A + p(n + 1)p - p(n)(A + p) , n > 0 ...
and
0 = p(l)p + p(O)A ... (4.33)
the solution for the above equation considering the boundary condition is
p(n)=(
)()
p p
=(I- p)P", n > 0, p = -..
P
(4.34)
the expected storage level be calculated according to different p values:
if p <1:
n=? np(n)
i-p
P-A
if p?1
n
=
By applying Little's law to the expected queuing product number, the average delay
experienced by each product is
T=
1
Figure 4- 4 shows the relationship between average waiting time and the arrival rate A,
where service rate p equals 1.
106
Delay in M/M/1 Queue
120
------
100E
80 60
40
20 0
0.2
0
0.4
0.6
1
0.8
Arrival Rate
Figure 4- 4: Relationship between the delay of a M/M/1 queue with arrival rate
It can be seen from Figure 4- 4 that when the arrival rate approaches to service rate, the
average storage level will go infinite. A mathematical interpretation of this phenomenon
is that the M/M/l queue is a special case of random walk. When the arrival rate is equal
to the service rate, the random walk becomes symmetric and hence recurrent in any state.
Under this situation the process has equal probability to be anywhere therefore the
expected storage value is infinite. The M/M/I model shows that, to avoid the storage
going to very large, the machine service rate should always be larger than the production
arrival rate.
4.2 Transfer line analysis based on stochastic models
4.2.1 Introduction of Transfer Line Analysis
Transfer line is a simplification of a manufacturing system that composes linear
connections of machines and buffers, as show in Figure 4- 5.
Mu
F
:
4-M2Set
of M3
Figure 4- 5: Schematic of a transfer line
107
M4f-+
If the machines in a transfer line are perfectly reliable, they can be designed to operate at
the same speed therefore the whole line is synchronized and buffers are not necessary.
However, since all machines will eventually fail and a failed machine need to be repaired,
if there is no buffer between machines, failures of any machine will stop the entire
transfer line. The introduction of buffers would dampen this effect. When upstream
machine is broken down, the downstream machine can still work by pulling products
from the buffer in between; on the other hand, if d own stream machine is down, the
upstream machine also can keep working by storing products in the buffer. The buffers
between machines act as decouplers that greatly reduce the dependency between adjacent
machines.
Intuitively, the bigger the buffers are, the less chance that the machines will be starved or
blocked due to the failure of other machines. Therefore big buffers tend to increase the
production rate of the line since the machines have more working time. However, as
shown in the previous analysis of M/M/1 queue, the higher the buffer level, the longer the
average waiting time would be. Therefore the major goal for the study of transfer line
stochastic model is to show quantitatively how buffer size will affect the production rate
and leadtime (summation of all waiting time through the entire line).
4.2.2 Zero Buffer and Infinite Buffer Model
Zero and infinite buffer transfer lines are two extreme cases for transfer line study.
Despite of their lacking of realisticity, the analysis of these models is valuable because
they essentially determine the upper and lower bounds of production rate that a finitebuffer transfer line can reach.
The machine operating times of zero and infinite buffer are assumed to be constant and
equal to unit time. Machines can only fail when they are working and can be repaired
only when they are down. Both the up time and down time are assumed to be geometric
distributed. The probability that machine M fails during a unit time interval when it is
operating is pi, and the probability of machine M, is repaired when it is down is r.. By
convention another important assumption is made: the first machine in the transfer line is
108
pulling products from an infinite source therefore it is never starved; the last machine is
sending products to an infinite sink therefore it is never blocked. It is also defined that
r.
ei =
... (4.35)
r + pi
to be the isolated efficiency of machine Mi.
Zero buffer transfer line
The analysis of zero buffer transfer line is relatively simple. Since there are no buffers
between machines, the production of all machines in the line are totally coupled. When
one machine fails the entire line is blocked. It is assumed that no two machines may fail
simultaneously, therefore by the operational failure assumption only one machine may be
down at any time.
Assume machine M, has been down m times during long time interval T. Therefore the
total down time for the transfer line is the summation of each individual machine's down
time:
k
D=
' , k is the number of machines in the transfer line.
i=1 r,
So the total up time for the line is approximately
k
m.
U=T- ?
-' ... (4.3 6)
r.
i=1
Consider the operational failure assumption, if the time interval T is long enough, the
following relationship between m and p, holds:
k
... (4.37)
U= T - U
1 r,
or,
U
1
= EODF
T
.
k
1
kP
1+ 7
j=1
109
r
(4.38)
EODF
is the efficiency of the transfer line since it measure the number of products that
can be produced in a unit time interval. It can be seen that it is the ratio
'-of each
ri
machine that affects the overall efficiency of the whole line rather than the individual
pi or r.. If a machine is replaced with a new one with higher p, and r but fixed ratio of
the two, the line efficiency is not changed. This conclusion, however, is generally not true
for finite buffer transfer line.
Infinite buffer transfer line
Consider a two-machine line with an infinite buffer in between. Suppose the average
production rate of the first machine is u, and for the second machine is u2 In the case
that ul < u 2 , which means the second machine consumes faster than the first machine
produces, it is expected that the buffer will be frequently empty, the production rate of
the two machine line will depend on the how many products can the first machine
produce in a long run. Consequently, in the case that u1
>
u2 , which means the second
machine is slower than the first machine, the buffer will go infinite in a long run since the
second machine cannot consume as many products as the first machine can produce. The
production rate of the line therefore depends on how many products can the second
machine produce in a long run. As a summary, the production rate of a two-machine line
with unequal isolated production rate machines (unbalanced two-machine line) equals to
the production rate of the slower one.
Complexity arises when the isolated production rate of the two machines are equal (a
balanced two-machine line). In this case the two machines have exactly the same average
performance, and the scenarios that the first machine is faster than the second and the
second machine is faster than the first is equally likely to happen. Intuitively it can be
imagined that the inventory behaves like neither of the unbalance line cases, it will
fluctuate widely.
A rigorous mathematical proof can show that the expected buffer level of a balanced twomachine line is infinite. And the production rate of the line is exactly equal to the
individual isolated production rate.
110
The two-machine line analysis can be applied to analyze more complicated long transfer
lines with infinite buffers. Consider a long transfer line with more than two machines. It
can always be broken down with two parts with different production rate. Since the
product rate of the line is determined by the slower part, the fast part can therefore be
discarded. By keep breaking the long line into smaller parts, the slowest two-machine
line can be finally reached, and that would be the product rate of the entire line. If, in a
special case where all machines in the line are operating at the same speed, the buffer
level would fluctuate widely with infinite expectation value and the line production rate
will equal to the production rate of each individual machine [Gershwin 1994].
4.2.3 General Assumptions of Finite Buffer Transfer Line Analysis
It is worthwhile to have some discussion about the general assumptions that adopted by
finite buffer transfer line analysis. These assumptions essentially define the context where
the analysis can be applied. When the actual system is not close the situation postulated
in these assumptions, caution needs to be taken since the validity of the analysis result
may deteriorate.
1.
Line balancing. The transfer line stochastic models assume no yield loss, which
means that the machines are not producing any defective products. Under this
situation, there is no reason to have machines with different production rate in the
transfer line, otherwise the faster machines will always need to wait for the slower
machines. Therefore all machines should be designed to have equal isolated
production rate, which is usually referred as line balancing.
2. Infinite repair resources. By assuming the repair processes of each individual
broken machine are independent, the transfer line models essentially assume
infinite repair resources. The repair process therefore can be viewed as purely
machine character that has nothing to do with system-wide properties.
3.
Independent machine failures. This assumption is analogous to the previous one.
The failure of one machine has no effect on the status of the other machines. This
assumption also excludes the issues such as power failure that could have effect
on all machines.
111
4. Infinite input and output into and from the transfer line. As discussed before, the
transfer line models assume that there is an infinite material source before the first
machine therefore it is never starved and there is an infinite sink after the last
machine therefore it is never blocked.
4.2.4 Deterministic Two-Machine Line
The two-machine line is the simplest but non-trivial case of transfer line. Long transfer
line analysis is based on the two-machine line analysis by conducting line decomposition
that has been mentioned is section 4.2.2. Since long transfer line is actually a simulationbased approximation of two-machine line analysis, it will not be addressed in this
chapter.
In the two-machine deterministic transfer line model, the process state is defined as
s
(n, a, a 2 ) , where n is the buffer level (0 ? n ? N) and a, is the status of machine i
(i =1,2, ai = 0,1) .
It can be found that some of the states are transient in a sense that they cannot be reached
from any other state except themselves or other transient states. Table 4- 1 lists all of the
transient states in a deterministic transfer line model.
(0,1,0)
(N,0,0)
(0,1,1)
(N,0,1)
(0,0,0)
(N,1,1)
(1,1,0)
(N-1,0,l)
Table 4- 1: Transient stats in a two-machine transfer line model
By convention, it is also assumed that the repair rates for the first and the second
machines are r and r2 , respectively and the failure rates for the first and the second
machines are p, and P2 The lower boundary (n ? 1) equations of system evolvement are the following:
p(0,0,1) = (1- r)p(0,01) + (1 - r)r 2 p(1,0,0) + (1 - r,)(1 - P2 )p(,O,) + p 1 (1- P 2 )P(1,1,1)
112
p(1,0,0) = (1 - r)(l - r2 )p(1,0,0) + (1 - i)p 2 p(,0,1)+ p+p 2 pl,,l)
(1 - r)r2 p(2,0,0) (1- r )(l
p(1,0,1)
-
p 2 )p(2,0,1) + plr 2p(2,1,0) + p, (1 - p 2 )p(2,1,1)
p(1,1,1) = rip(0,0,1)+ r r2 p(1,0,0) + r (1 - p 2 )p(1,0,1) + (I - p)(
p(2,1,0) = r (1 - r2 )p(1,,0) + rp 2 p(l,0,) + (1-
-
p 2 )p(lll)
p1 )p 2 pl1,11)
... (4.39)
The internal (2 ? n ? N - 2) system evolving equations are the following:
p(n,0,0) = (1- r)(1 - r2 )p(n,0,0) + (1 - ri)P2(p(n,0,1) + p, (1 - r2 )p(n,1,0) + pp 2p(n,1,l)
p(n,0,1) = (1 - r)r 2p(n +1,0,0) + (1- r0)(1 - p 2 )p(n +1,0,1) + p, r2p(n +1,1,0) + p, (1 - p 2 )p(n + 1,1,1)
p(n,10) = r (1 - r 2 )p(n - 1,0,0) + r p 2 p(n - 1,0,1) + (1 - p)(1 - r2 )p(n - 1,1,0) + (1 - p1 )p 2 p(n - 1,1,1)
p(n,1,1) = rr2 p(n,0,0) + r,(1- p 2 )p(n,0,1) + (1 - p)r 2 p(n,1,0)+ (1- P
1
-
p 2 )p(n,1,1)
... (4.40)
The upper boundary equations are
p(N - 2,0,1)
(1 - ri)r
2 p(N -
1,0,0) + plr2 p(N - 1,1,0) + p (1- p 2 )p * N - 1,1,1)
p(N - 1,0,0) = (1 - ri)(1 - r2 )p(N - 1,0,0) + p, (1 - r 2 )p(N - 1,1,0) + pp
p(N - 1,1,0) = r(1
p(N - 1,1,1)
-
2 p(N
- 1,1,1)
r2 )p(N - 1,0,0) + p (1 - r2 )p(N - 1,1,0)+ p+p 2 p(N - 1,1,1)
r r 2p(N - 1,0,0) + (1 - p)r 2p(N - 1,1,0) + (1 - P1)(1 - p 2 )p(N - 1,1,1) + r2p(N,1,0)
p(N,1,0) = r,(1 - r2 )p(N - 1,0,0) + (1 - p)(l - r2 )p(N - 1,1,0 + (1 - p1 )p 2 p(N - 1,1,1) + (1 - r2 )p(N,1,0)
... (4.41)
and finally the normalization is the following:
N
1
1
?
n
113
1 =Oa2Op2=0
p(n,a9,a2)
...
(4.42)
The efficiency E,of machine M, is defined as the probability that it can do production at
any time. Therefore the efficiency El is defined as the probability that machine M, is up
and not blocked, which is:
El =9 p(n,al,a2) ...-(4.43)
n<N
a1 =1
E2
is therefore the probability that machine M 2 is up and not starved:
E 2 = ? p(n, a, a 2 ) ...
(4.44)
n>0
a2 =1
Since the buffers are infinite, the efficiency of the two machines must be equal to each
other, otherwise the buffer level would eventually goes to zero or infinite. A rigorous
mathematical proof can also support this conclusion. Therefore efficiency of the transfer
line is hence equal to the efficiency of the machines.
E=EI= E ... (4.45)
This property is called conservation of flow, which means that the same amount of
products will flow through every machine in the line.
114
0.4'
0.4
0.3
.06-
,
---- r1 = 0.14-
0--
- 3
-
50
r1 = 0.12
= 0 .1 1
=0.08~rl=00
200
150
100
N
Figure 4- 6: Relationship between line efficiency and buffer size [Gershwin 1994]
Line efficiency E can be calculated by solving the system evolving equations. A
numerical result is shown in the following figure, which demonstrates the effects of
buffer size N and the repair rate r, of machine M 1 . The system parameters are p, = 0.1,
r2 =
0.1, and p 2 = 0.1. Figure 4- 6 shows the line efficiency-buffer size relationships
under different r, values 0.14,0.12,0.10,0.08 and 0.06 from the top curve to the bottom
curve. Note that the line is balanced when r equals 0.10. When r < 0.10 the first
machine is less efficient than the second machine therefore the line efficiency is
approaches the Mi 's efficiency when N gets large, which is
= 0.5. When
ri +p 1
r > 0.10, machine M 2 is the slower among the two and therefore the line's asymptotic
efficiency is determined by M 2 's efficiency
2
r2 + p 2
value.
115
,
which varies with different r2
4.2.5 Exponential Two-Machine Line
In the previous discussion on deterministic transfer line analysis, the machine cycle time
is deterministic and constant. However, in real situation this assumption might not hold
and the cycle time may vary according to some probability distribution each time the
machine processes a product. Exponential two-machine model is established on the
assumption that the machine cycle time is exponential distributed and therefore the
stochastic process that governs the system evolving becomes continuous time, discrete
state type.
Introducing the new parameter p,, i = 1,2, which is the exponential distribution parameter
for each machine, the system evolving equations can be derived as the following:
p(n,0,0)(r1 + r2 ) = p(n,,O)p 1 + p(n,O,1)p 2
p(0,0,0)(r1 + r2 )
=
p(N,0,0)(r + r2 )
p(O,1,O)p 1
=
p(N,Ol)p
2
p(n,0,1)(r, + P2 + P 2 ) = p(n,O,O)r 2 +P(n,1,)p + p(n 1,,1)p
p(0,0,1)r = p(OOO)r
2
2
+ p(0,1,1)p + p(Ol)p 2
p(N,0,1)(r, +P 2 + p 2 ) = p(N,O,O)p
2
p(n,1,0)(p, + p, + r2 ) = p(n - 1,1,0)pl + p(n,O,O)r + p(nll)p2
p(O,1,0)(p1 + P1 + r2 )
p(0,,0)r
p(N,1,0)r2 = p(N - 1,1,O),u + p(N,0,0)r + p(N,1,l)p
p(n,1,1)(p, + p
2
p 2 + P2)
=
p(n - 11,l)pl + p(n ±,111"P2 + p(n,1,O)r 2 + p(n,0,1)r,
p(O,1,1)(p1 + PI) = p(1,1,1)p
2
+ p(0,1,O)r 2 + p(0,,1)r
p(N,1,1)(p
... (4.46)
116
2
+ P2) =
p(N - 1,1,1)pl + p(N,1,0)r2 + p(N,0,1)r
Similar to the deterministic case, the efficiency of machine is defined as the probability
that the machine can produce a product at any time:
1
N-1
El =
?9 p(n,l, a 2 )
n=O a 2 =0
N
I
? p(n, al,1)
E 2 =?
n=1 a 1 =O
... (4.47)
The normalization is equation is
N
1
1
? ? ? p(n,aj,a2)=
n=O a1 =0a 2=0
...
(4.48)
and the conservation of flow is expressed as
pIEl
=
p12E2
. (4.49)
Line efficiency is again defined as the flow rate of the both machines. By solving the
system evolving equations and plugging the probabilities into the flow rate expression,
the relationships between line efficiency, buffer size and machine parameter can be
determined.
4.3 Stochastic Model Analysis in a System Design Point of View
The transfer line stochastic model analysis demonstrates many valuable insights for
manufacturing system design, which are summarized as the following:
1.
A manufacturing machine line should be designed as balance as possible to avoid
faster machines waiting for slower machine.
2. The production rate of the line (efficiency) increases as the repairing rate
increases and the failure rate decreases.
3.
The variation in machine cycle time and unreliability is the root cause for the high
buffer level between machines.
117
The analysis insights actually present the functional requirements to design a satisfactory
manufacturing system. System design parameters (methodologies) to satisfy these
functional requirements are addressed in detail in the rest of this section.
4.3.1 Line Balancing
From the line balancing point of view, it will be ideal that all production processes have
very close operation cycle time, therefore the line is naturally balanced. However this is
rarely the case in real situation. Due to the machine design constraints or special process
requirements, some processes need longer or shorter cycle time than other processes, in
which case the line is not naturally balanced and design work needs to be done to achieve
line balancing.
A commonly used methodology for line balancing is called work loop design. The idea
for this methodology is that although the machine cycle times may differ from each other,
there are always ways that the combinations of some of these cycle times be close to each
other. Figure 4- 7 demonstrates an example of work loop design.
118
Assembly Cell
Storage/container
Operation
ell
Conveyor
Figure 4- 7: Work loops in a cellular layout manufacturing system
As shown in Figure 4- 7, there are I1I machines in the manufacturing cell. In order to
balance the line, 5 work loops are designed to operate the I1I operations. Therefore even
the I1I operations may have different cycle times, by designing work loops and machine
layouts, balanced work loop cycle times can be achieved. Another benefit of operation
the cell in a work loop manner is that not only the line can be balanced, but also the loop
cycle time can be changed by adding or retrieving operators from the cell. For example,
new work loops can be designed with 4 operators in operating the cell with a longer loop
cycle time than before. The following two figures show MMCs of two different work
loop designs for one manufacturing cell. The first design has 10 operators with a
balanced loop cycle time of 70 seconds and the second design has 14 operators with a
balanced loop cycle time of 40 seconds.
119
Operalur:
PART: U24 Gear
PROCESS
TIME (sec
Operator
1(-1) Start shaline
1 (2) Load housing
1(0)Install check valne
1 (1) Load pinion bearingand seal (auto cycle)
2 (A Grab rack and assemble ret ring
2() Change racks and press PS to swage racks
2 (4)Push pallet into rack insertion station, load rack into station. prose P S
2 (2) Tq. shrt tumline Q1t), St. long turntine (both) 19. shrt turnline (rght), tg. Long tureline (both)
5.333
.41
3.
1b.5
~Preto
P.6, to lift pateS, 1q. Bashing, ins. Busht. trnext op. press P.28
3
s
3 (.) Raise pallet lift &notate 90 dg
S 2)Retrisee teols 8.insert rakve, replace tools, neat clip. P.8 to tower lift &ret.
4
4
A
4
4
4
55
8.333
19 83
32
"
28 ...
.
1425
37
(1) Press palm buttons to raise pallet
.
.
5.5
6 (11) Toruapinion cap_
5 )l1)
89
1
leak test (including transfer time)
S(11) Load yoke components
5 (11) Rotate 90 dag
5
.
...
118
.
Pres palm beuttons to seat cip
(6) Palm buttons to raise pallet
(6) Torque nut.
(28) Apply grease
(8) Press palm batton to lower pallet
(28) Rotate & release pallet
(7) Load nput earing. sa and enap ring
Auto
6
2.05
18
31
42)
(
80
80
50
45 17
3.3
37803
41J11
30
20
10
Man Walk Auto
1.5 1
10.5
6 12
3
14 2.13
(St. No.) OPERATION - Sequence as described
Matk goar wth pin,
-3.
13,33
2
4.333
5 (11) Install yoke
5 (11) Press palm buttons to lower pallet
8 (11) Rotate back 80d
8 (11) Press palm button to realease
5 9J) Insert tie rods into machine
2.6E7
1.18
7..1.1.
..
(12) Auto bumish rack teeth
(14) Mash laud and final set
.1.
A and 18 combined Qncludiog tranfer
17) Functional Testa
yoke.
.Stake
.
.
.
.
.
.
time)
(21 Retrieve and grease boot 6 P1) Grease one boot grocve
8 (21) Rotatsl7ldeg
8(21)install boot
6 (21) Insert tinnerman clip, start jam nut, place tinn. Clip with tool, torque jam nut with
pallet bocki 70 dg....
(21) Mn ore
b5S(21Rotate
6
(21) Press palm button
8
to
lower
& rel.
.
..
.
......
.....
Ill
...........
Ih
1.5
tt
8
Place travel restrictor and press palm buttons
7 (22) Retrieve boot and place t on greaser (can be pertonned as part appnsaches)
7 (22) Grease one bost groves
7(22) Place plastic clip and cut with tool
7 (22) Rotate pallet 70 deg.
7 (22) Install bool
7 (2) Insert tinnerman clip, start jam nut, p lace ti nn Clip with tool, torque jam nut with tool
7 22) Rotate patet buck 7dgg
.
.
.
.
.
..
.
7 (22) Press palm buttons to lower & ret pallet
.....
.....
28
1.5
05
pallet
......... ...
...........
.
2
...........
......
8
17
B
1:5
20
.
5
.
8 (23) Retrieve breather tube & dip
8 23) Install breather tube
...
.5
P)2 Crimp tie rod boots clips
Grab tie rod ends
8 3) Install rght tee red end
8 (23) Install left tie rod and
8 (23) Press palm buttons to ral. pallet
8 Rotate pallet 10 deg. atead of atest
8 23)
1
105
15
9.333
2:13
2.13
9 (24)Place centenog too & raise
9 (24) Press palm buttons to lift pallet
9 4) Torque right tie rod and
9 24) Torque lol tie rod end
9 (24) Press palm buttons to lower pallet
95)Install plugs
9 25Mark gearwith pin
.
2.
.
77
.13
13
1
5
2.11
1
2.11
.
10 (28 Bushings in
Load gear from holder
1i (2)
1 @ Press palm buttons
10 (25) Unclamp gear from previous station
15 (26) Lease gear at holder
10 R) Unload finished gear to current unload area and mark gear w dh p in.
15 Pack gear to dunnage
9
1.
15.5
1
13.25
11.5
66
-
1...
2
4
Figure 4- 8: MMC of work loop design with 10 operators [Oropeza 2001]
120
PART: U2014 Gear
PROCESS
Operator
141
Operaors:
TIME J
Man Walk Auto
(St. No.) OPERATION - Sequence as described
1 (-l) Start shortline (both ends)
1 Load housin
1 (t) Install check calve
15.5
16 5
2(1) Load pinion beanng and seal (auto
2(2) St long turmline (both ends)
end
3 (3)st ohrt lurnline (rglht
Long
,q
(both
ends)
to lift
pallet tq
Bushing,
6
1 3+
1 16
1 16
.1- iH II 1' F
92
B
r
III H Il-4l- il-
PB.to
lower lift&
i
raise pallet
S(7)Lsad iput bearigseal and snap rng, press PB to initiate auto cycle
P (B)
Torque nut and apply grease (simultaneously with both hands)
PEI) Press palm buttons to lower pallet
6 PB) Rotate & release pallet
i
"-,i t r-i i
Ti
i1
2
105
19.63
2
325
Il
-t-t
t[44-
4
(11)
Press
palm
. ....-.....
button
2
to
-
ii
i's 2
(21)
9(21) Mark gear with
.
jam
70 deg,
nut, place
tinn
Clip with tool,
tq
22
1
1l
6
nut with
jam
1010111
.44
4
.
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14 Pack gearto dunnage
i
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1"1
Figure 4- 9: MMC of work loop design with 14 operators [Oropeza 2001]
121
!!11!
8'5
leave itat holder at next station
gear fros
11
"
4.
lower pallet
14(25) Press palm buttons
i-
1
I
Tirtrit-i
8H
12 (24) Place cente ng tool & raise
12 (24)Preos palrn buttaons alit pallet
12 (24) Torque rght lie rod and
12 (24) Torque left tie rod end
Unload finished
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11 (23) Retieve breather tube & dip (can be done pror to pallet arrval4
11 (2) Install breather tube
11 (23) C mp tie rod boots clips
11
b3) Kit tie rod ends to pallet
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13 (25) Mark gear with pin
13 (25)Unlarnpgear and
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12 (24) Press palm buttons to
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4
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20
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10
1.3
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30
40
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4.3.2 High Repairing Rate
In stochastic models discussed previously, the repairing rate is referred as the probability
that a failed machine can be fixed in a unit time interval. In the actual system, this rate is
determined by how fast the maintenance personnel can identify and response a problem
that has occurred in the system.
In most of the mass production manufacturing systems, problem identification and
resolution is conducted in a very complex hierarchical system. For example, when a
machine is broken, the operator is not authorized to do anything but file either a paper or
electrical report to his/her line supervisor. The line supervisor then sends a maintenance
requirement to the maintenance supervisor, who usually belongs to an independed
department. The maintenance supervisor will usually check their reports every day or
every shift and then plan their maintenance schedule, which includes which maintenance
person to be sent to fix the problem and what material should be prepared. The
maintenance person will go to the broken machine at his/her scheduled time and get it
fixed. The entire repairing process can easily take a day or more. This process can be
viewed as an action chain that is shown below:
Solution
Problem
Line
Maintenance
Maintenance
Supervisor
Supervisor
Personnel
Operator
Connection 1
Connection 2
Connection 3
Figure 4- 10: Action chain with multiple connections
There are three connections between the problem being identified by the operator and the
personnel solve it. Each connection will take a certain amount of time. The more
connections between problem identification and resolution, the lower the repairing rate
would be, and hence the lower the efficiency of the line will be.
122
Clearly, an effective way to increase repairing rate is to shortening the action chain
therefore to shorten the time needed to fix a problem. In Toyota Production System,
information board (Andon board) is used to achieve fast problem identification and
resolution.
Figure 4- 11: Example of Andon board
The figure above shows a very simple model of information board. All machine status is
shown in the bottom row of the board. A green light means the process is in problem-free
operation while a red light shows there is a problem in a specific process. More complex
information board may be used in actual production that uses different color lights to
show different nature of the problem that has occurred. An alarm will be triggered when
a red light is going on. The information boards will be installed in many placed in the
plant all people can be notified when problems occur.
During production, when the operator identifies a problem he/she will turn the red light
on in the information board. The maintenance person, who also has information board
around, will be immediately notified the place where the problem occurs and what is the
123
nature of the problem. He/she therefore will immediately go to the broken process and
solve the problem. The action chain for this repairing process is shown in the following
figure.
Solution
Problem
Operator
k
Maintenance
Personnel
Connection 1
Figure 4- 12: Shortened action chain with one connection
Fast problem resolution also requires that the maintenance personnel can solve the
problem in an effective and predictable way, which requires the non -ambiguity of how to
solve a specific problem and time variation between different maintenance people to be
eliminated. Standardized working instruction sheet is the design solution to meet this
requirement. Instead of leaving the problem solution to people, standardized work sheet
lists out instructions and sequence of operations that the operator should follow. In this
way, the best way to do work can be predefined and formulated as "rules". By following
standardized work instructions, the work will be done in the most effective way and
human based variation can be eliminated. The following figure shows an example of
standardized working instruction sheet.
124
Standardized Work
Department: 308 Lamination
Activity
Safety Equipment
Check knob location
I Glasses
Ear Plugs
I Gloves
1.
On the control panel, move from AUTO to Manual
2.
Take the template for the part you are checking from the hanger
3.
Lay the template flush with the side of the windshield
4.
Check to see if the knob is within the window of thetemplet
5.
Check to see if the knob is "square" (not skewed/crooked)
6.
Adjust the setup as necessary
7.
Fill out check sheet 09 85 00 07
Figure 4- 13: Example of standardized work sheet [Cochran 2002]
4.3.3 Low Failure Rate
The failure rate is defined as the possibility of a machine can break down in a unit time
interval. Machine failure is a major source of production interruptions. In addition to
machine down times, machine failures some times are difficult to notice and an out-ofcontrol machine can produce large amount of defective products before its problem being
identified.
In a real manufacturing system, machine failures usually due to lack of maintenance.
Therefore a well-designed maintenance program is vital to maintaining a predictable
production out. Standardized maintenance operations should be established to define
effective and unambiguous steps the maintenance operators should follow. The following
figure shows an example of standardized maintenance sheet.
125
Standardized Preventive Maintenance
Department: 308 Pre-Lamination
Equipment
Requirement
Check every 2 hours:
not worn
-
V-belt conveyors
-
parating unit vacuum cups
-
auxiliary air knives
on, positioned to wings
-
arm brushes & felt
free of loose glass, not worn
-
templet position in arms
-
washer water temperature
-
washer water
-
conveyor
centered, air on, not worn
adequate distance from air base
100-140 F
both valves on / adequate flow
hain aligned, no missing roller/rubber
Figure 4- 14: Example of standardized preventive maintenance sheet [Cochran 2002]
4.3.4 Role of Autonomous
The stochastic models, similar to most of the mathematical models have been used in
manufacturing system research, view the events in the system as independent and
probabilistically distributed. The most fundamental assumption of stochastic process, as
discussed in the beginning of this chapter, is that the occurrence of one event doesn't
depend on the events that happened before the one just before the current event. And also
the major parameter, such as the machine cycle time and the failure and repairing rates
are probabilistic events whose probability distribution can be mathematically expressed.
However these assumptions ignored a very important factor in a real manufacturing
system: human autonomous. Human beings, not like machines, are autonomous and can
have judgment and communications. Therefore it is questionable to use simple
mathematical models to describe systems that involve human beings.
For example, the queue theory that has been discussed in previous chapters shows that in
a symmetric two-machine transfer line that has exactly the same machine parameters, the
buffer level between them will eventually go infinity. However, this problem can be
126
easily solved by well-designed information and material flow in a manufacturing system.
As discussed in Mondon's book Jidoka (autonomous), which means communication and
help between upstream and down stream operators are one of two pillars of Toyota
Production System.
Voume and Mix Flexibil~ity
Level and~ B alaiwe Production.
Figure 4- 15: Toyota production system design model [Cochran 1999]
In TPS operators' working loops, instead of machines, determine the cycle time of the
production. And standard work in process (WIP) is established between to operator loops.
When the downstream operator notices that he/she is faster than the upstream operator by
observing that the SWIP level is dropping, he/she would either slow down (if it is
because he/she works too fast) or go to help the upstream operator to speed up (if it is
because the upstream operator has some problem). Since the two operators have equal
average capacities, in this way they can absorbing their capacity discrepancies and keep
the line balanced all the time. Therefore the SWIP between them can be kept constant and
the production will be stable.
A mathematical explanation of autonomous is that if considering human factors, the
assumptions about independed events do not hold. Specifically, the discussion of some of
the important assumptions are listed as the following:
127
1.
The incoming and out coming products cannot be viewed as independent of
machine status.
As discussed just now, because of the communications between operators, the
production output and input are decided by the operators which obviously
dependent on machine status. Also, since the human judgment and mutual help is
involved in the system, a fixed mathematical description of system parameters
would not be appropriate.
2. The repairing rate and failure rate is not probabilistic. Because of fast problem
identification procedures, standardized maintenance programs and operators'
authorization on handling machine problems, machines can hardly fail during
operation and repairing can be done in a fixed, standardized way. Therefore these
parameters also hardly obey any mathematical distribution, especially exponential
distribution, which assumes historical independent event.
3. Production cycle time is not probabilistic. Since it is the operators working loop
rather than machine automatic cycle time determines the production cycle time,
autonomousity can eliminate the production cycle time variation. In fact, each
operator has his/her own pre-defined working loops and standardized operation
sequence. If the operator finds that he/she is slower than the standardized pace,
he/she will speed up to catch up; on the other hand, if he/she finds that he/she is
faster that the schedule, he/she will slow down to meet the pace. Therefore the
operator can keep a constant production cycle time rather than a probabilistic
distributed one.
4. Defect rate is not probabilistic
It has been widely believed that a manufacturing system cannot achieve perfect
product quality. This statement is plausible in a mathematical point of view since
defective product is assumed to be probabilistic distributed therefore no matter
how good the system would be there is always positive probability that a defect
can happen. TPS successfully achieved defect free production by combining
autonomous and Poka-Yoke. The later term refers to defect-proof devices that
installed in the manufacturing system. In addition to human based quality
128
checking system, the poke-yoke device physically prevents defective products
from being produced.
One example of a poka-yoke is a physical feature designed into the pallets, as
shown in Figure 4- 16. For the housing part, the nest on the pallet has two small
features that protrude into slots in the housing when it is seated. At one point in
assembly, vanes are inserted into the slots. The vanes are purposely designed
asymmetric therefore if the vanes are inserted upside-down, the pallet features
prevent complete insertion. The subsequent assembly task will fail if the vane
orientation is not corrected. This insertion makes it impossible for the operator to
continue the process without noticing a defective product has been produced.
VANE
HOUSING
PALLET NEST
Figure 4- 16: Example of Poke-Yoke [Low, 2001]
4.4.4 Summary on Manufacturing System Analysis
Manufacturing system analysis based on stochastic models is important in that they can
show many quantitative insights among different system parameters such as repairing
rate, failure rate and production cycle time. However, it is not enough to stop at the
analysis level, as most research work did. It is more important to use these insights as the
guidance to design a better manufacturing system. This is where the manufacturing
system analysis and manufacturing system design meet. Manufacturing analysis shows
how the system parameters affect the system performance, either in desired direction or
undesired direction. System design therefore needs to find out appropriate design
parameters that can facilitate the system to move towards the desired direction.
129
Manufacturing system analysis can, however, be part of the foundation of manufacturing
system design. As discussed previously, some of the assumptions that the mathematical
models adopted are too strong to be realistic in most of the situations, which could
severely affect the validity the analysis result from those models.
An appropriate conclusion about manufacturing system analysis and manufacturing
system design would be: Manufacturing system analysis results can serve as a general
guidance of the fundamental relationships of a manufacturing system. Manufacturing
system design has to be customer requirement oriented and case specific. Trying to
blindly apply the general quantitative result to guide a manufacturing system design
would bound to be a miserable failure.
130
Chapter 5: Conclusion
This thesis reviewed typical optimization-based methodologies that have been applied in
guiding manufacturing system design. The models are studied in a system design point
view by analyzing the system requirements they aim to achieve and design parameters
they applied. Analysis result showed that the models are insufficient according to
axiomatic design and hence lead to compromising system requirements and sacrificing
overall system performance. Modifications were recommended to achieve decoupled
design that ensures all system FRs can be fulfilled. A case study example was presented
to show the effectiveness of system design methodology comparing to optimization
result.
The thesis also studied a most widely used manufacturing system analysis methodology
based on stochastic process models. The model analysis results were compared with
manufacturing design framework MSDD. The comparison showed MSDD FRs are
consistent with the insights provided by stochastic process models. The limitation of
manufacturing system analysis models was also discussed. These models strongly rely on
assumptions therefore their analysis result may lose validity when system changes. The
system design framework, however, is robust and generally applicable.
The contribution of this thesis can be viewed in the following three aspects:
1. This thesis reviewed the evolution of manufacturing system and proposed the reason of
why manufacturing system designed with system design methodologies rather than
optimization models.
2. This thesis analyzed typical optimization methodologies from a system design point of
view and concluded the reasons that lead the models to local rather than system wide
optimization.
3. The thesis studied the manufacturing system analysis models and concluded the
analysis results were consistent with the FRs of system design framework MSDD.
131
132
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134