CHAPTER 12
Design for Quality and Product Excellence
Teaching Notes The precise manner in which a person or team approaches product design, solving problems to achieve product excellence, or developing product reliability is not as critical as doing it in a system systemati aticc fashio fashion. n. Students Students have been exposed exposed to process process manage managemen mentt and improve improvement ment in Chapter 7, but they may still have some difficulty in understanding how measurement (metrology) and Six Sigma proects can be used at the design stage to ma!e fre"uent, but gradual changes as an approach to process improvement. improvement. #ey obectives for this chapter should include$ To explore the typical structured product prod uct development development process consisting consisting of idea • generation, preliminary concept development, product%process development, full&scale production, product introduction, and mar!et mar!et evaluation.
•
To learn that concurrent, or simultaneous, engineering is an effective approach for managing the product development process by using multi&functional teams to help remove organi'ational organi'ational barriers barriers between departments and therefore reduce product development time. esign reviews help to facility product development by stimulating discussion, raising "uestions, and generating new ideas
•
To introduce introduce the concept of Design for Six Sigma (DFSS) consisting of a set of tools and methodologies used used in the product development development process to ensure that goods and services meet customer needs and achieve performance performance obectives, and that the processes proc esses used to ma!e and deliver them achieve Six Sigma capability. SS consists of four principal activities of$ Concept development, Design development, Design optimization, and activities are often incorporated into a variation of the Design verification. These activities *+C process, !nown as DMADV, which stands for Define, Measure, Analyze, Design, and Verify.
•
To define concept development as the process of applying scientific, engineering, and business !nowledge to produce a basic functional design that meets both customer custome r needs and manufacturing manufacturing or service delivery re"uirements. This involves developing creative ideas, evaluating them, and selecting the best concept. -
esign for /uality and 0roduct 1xcellence
•
To explore ualit! Function Deplo!ment (FD) && a planning planning process to guide the design, manufacturing, and mar!eting of goods by integrating the voice of the customer throughout the organi'ation. + set of matrices, matrices, often called called the "ouse of technical re"uirements, re"uirements, ualit! , is used to relate the voice of the customer to a product2s technical component re"uirements, process control c ontrol plans, and manufacturing manufacturing operations.
•
To investi investigate gate good product design, which anticipates anticipates issues issues related related to cost, manufacturability, manufacturability, and "uality. "uality. mprovements in cost and "uality "ua lity often result from simplifyi simplifying ng designs, and employing techni"ues such s uch as design for manufactura#ilit! (DFM) $ the process of designing a product for efficient production at the highest level of "uality.
•
To study social responsibilities in the design process including product safety and environmental concerns, which have made Design for %nvironment (Df%) and design for disassem#l! important features of products, because they permit easy removal of components for recycling or repair, eliminate other environmental ha'ards, and ma!es repair more affordable.
•
To explore Design for %xcellence %xcellence (DF&) , an emerging concept that includes many design&related initiatives such as concurrent engineering, design for manufacturability design for assembly, design for environment and other 3design for4 approaches. 5 obectives include higher functional performance, physical performance, user friendliness, reliability and durability, maintainability and serviceability, safety, compatibility and upgradeability, environmental friendliness, and psychological characteristics.
•
To introduce concept engineering (C%) && a focused process for discovering customer re"uirements re"uirements and using them to select superior product produc t or o r service concepts that meet those re"uirements. re"uirements.
•
To investigate manufacturing specifications, consisting of nominal dimensions and tolerances. 'ominal refers to the ideal dimension or the target value that manufacturing see!s to meet6 tolerance is the permissible variation, recogni'ing the difficulty difficulty of meeting a target consistently. olerance design involves determining the permissib permissible le variation variation in in a dimensio dimension. n.
•
esign optimi'ation includes setting proper tolerances to ensure maximum product performance performance and ma!ing ma!ing designs designs ro#ust 6 that is, insensitive to variations in manufacturing or the use environment.
•
+ scientific approach to tolerance design uses the aguci loss function. Taguchi assumes that losses can be approximated by a "uadratic function so that larger deviations from target correspond to increasingly larger losses. or the case in which a specific target value, T, is determined to produce the optimum performance, and in which "uality deteriorates as the actual value moves away from the target on either side (called 3nominal is best4), the loss function is represented by (x) 8 !(x & T).
esign for /uality and 0roduct 1xcellence
•
To explore ualit! Function Deplo!ment (FD) && a planning planning process to guide the design, manufacturing, and mar!eting of goods by integrating the voice of the customer throughout the organi'ation. + set of matrices, matrices, often called called the "ouse of technical re"uirements, re"uirements, ualit! , is used to relate the voice of the customer to a product2s technical component re"uirements, process control c ontrol plans, and manufacturing manufacturing operations.
•
To investi investigate gate good product design, which anticipates anticipates issues issues related related to cost, manufacturability, manufacturability, and "uality. "uality. mprovements in cost and "uality "ua lity often result from simplifyi simplifying ng designs, and employing techni"ues such s uch as design for manufactura#ilit! (DFM) $ the process of designing a product for efficient production at the highest level of "uality.
•
To study social responsibilities in the design process including product safety and environmental concerns, which have made Design for %nvironment (Df%) and design for disassem#l! important features of products, because they permit easy removal of components for recycling or repair, eliminate other environmental ha'ards, and ma!es repair more affordable.
•
To explore Design for %xcellence %xcellence (DF&) , an emerging concept that includes many design&related initiatives such as concurrent engineering, design for manufacturability design for assembly, design for environment and other 3design for4 approaches. 5 obectives include higher functional performance, physical performance, user friendliness, reliability and durability, maintainability and serviceability, safety, compatibility and upgradeability, environmental friendliness, and psychological characteristics.
•
To introduce concept engineering (C%) && a focused process for discovering customer re"uirements re"uirements and using them to select superior product produc t or o r service concepts that meet those re"uirements. re"uirements.
•
To investigate manufacturing specifications, consisting of nominal dimensions and tolerances. 'ominal refers to the ideal dimension or the target value that manufacturing see!s to meet6 tolerance is the permissible variation, recogni'ing the difficulty difficulty of meeting a target consistently. olerance design involves determining the permissib permissible le variation variation in in a dimensio dimension. n.
•
esign optimi'ation includes setting proper tolerances to ensure maximum product performance performance and ma!ing ma!ing designs designs ro#ust 6 that is, insensitive to variations in manufacturing or the use environment.
•
+ scientific approach to tolerance design uses the aguci loss function. Taguchi assumes that losses can be approximated by a "uadratic function so that larger deviations from target correspond to increasingly larger losses. or the case in which a specific target value, T, is determined to produce the optimum performance, and in which "uality deteriorates as the actual value moves away from the target on either side (called 3nominal is best4), the loss function is represented by (x) 8 !(x & T).
9 esign for /uality and 0roduct 1xcellence
•
To examine the characteristics of Design Design Failure Mode And %ffects Anal!sis (DFM%A) && a methodology to identify all the ways in which a failure can occur, to estimate the effect and seriousness of the failure, and to recommend corrective design actions.
•
To study the dimensions of relia#ilit! ability of a product to perform as relia#ilit! :the ability expected over time. ormally, reliability is defined as the probability that a product, piece of e"uipment, or system performs its intended function for a stated period of time under specified operating opera ting conditions. n practice, the number of failures failures per unit time determines reliability during the duration under consideration (called the failure rate), loo! at functional failure at the start of product life (The early failure period is sometimes called the infant mortalit! period ), relia#ilit! failure after some period of use.
•
To understand why reliability is often modeled using an exponential probability probability distribution distribution and use the reliabi reliabilit lity y function, function, specify specifying ing the probability probability of λ survival, which is$ ;(T) 8 - < e& T.
•
To explore systems composed of individual components with !nown reliabilities, configured in series, in parallel , or in some mixed combination, and how it ties into various aspects of design, including optimi'ation, tolerance design, and design verification.
•
To learn that design optimi'ation includes setting proper tolerances to ensure maximum product performance and ma!ing ma!ing designs designs ro#ust6 a scientific approach to tolerance design uses the aguci aguci loss function. functi on. Techni"ues for design verification include formal reliability evaluation, using techni"ues such as accelerated life testing and #urn*in .
•
To appreciate that the purpose of a design revie+ is to stimulate discussion, raise "uestions, and generate new ideas and solutions to help designers anticipate problems before they occur. occur.
•
To understand techni"ues for design verification including formal reliability evaluation. These include accelerated life testing, which involves overstressing components to reduce the time to failure and find wea!nesses6 and #urn*in , or component stress stress testing, which involves exposing integrated circuits to elevated temperatures in order to force latent defects to occur.
•
To appreciate that Six Sigma performance depends on reliable measurement systems. Common types of measuring instruments used in manufacturing today fall into two categories$ 3low&technology4 and 3high&technology.4 ow&technology instruments are primarily primarily manual devices that have been availabl availablee for many many years6 years6 high&techno high&technology logy describes those those that depend on modern electronics, electronics, microprocessors, lasers, lasers, or advanced optics.
•
To define metrolog!&&the science of measurement < broadly as the collection of people, e"uipment, faciliti facilities, es, methods, and procedures used to assure the correctness or
= esign for /uality and 0roduct 1xcellence ade"uacy of measurements, and is a vital part of global competitiveness, including characteristics such as$ accurac!, precision, repeata#ilit! or euipment variation , reproduci#ilit! or operator variation , cali#ration and tracea#ilit! .
•
To appreciate that process capa#ilit! is the range over which the natural variation of a process occurs as determined by the system of common causes6 that is, what the process can achieve under stable conditions. The relationship between the natural variation and specifications is often "uantified by a measure !nown as the process capa#ilit! index, C p.
•
To learn that a process capa#ilit! stud! is a carefully planned study designed to yield specific information about the performance of a process under specified operating conditions. Three types of studies are a peak performance study, process characterization study, and component variability study.
A'S-%S / 0A123 2' 4AC2C% 5%3 2SS0%S
Testing +udio Components at Shure, nc. -.
The general definition of reliability as$ the probability that a product, piece of equipment, or system performs its intended function for a stated period of time under specified operating conditions, is thoroughly tested by Shure. Tests are tailored to various mar!et segments, according to the type of use (or abuse) the e"uipment is li!ely to incur. or the consumer mar!et, Shure uses the cartridge drop and scrape test, which is particularly important to test for, in the light of how 3scratch4 >2s use the e"uipment. or presentation and installation audio systems, they use the microphone drop test and perspiration test. or mobile communications, the two above tests, temperature, and cable and cable assembly flex tests are applicable. or the performance audio, the microphone drop test, perspiration test, se"uential shipping, cable and cable assembly flex, and temperature storage would all be appropriate. The purpose of the tests is to simulate actual operating conditions so that the products can sustain accidents and rough handling and perform effectively over a useful life. /uality characteristics that are studied are achieved reliability and performance.
.
or the microphone drop test, the measures are probably variable measures of sound and response levels, within an acceptable range. Thus, standard variables control charts may be used. or the perspiration test, it may be that a p&chart or u&chart is used for attribute measures. The cable and cable assembly flex test might use a p&chart to measure the percentage of cables tested that failed due to roc!ing motions or twisting motions. The se"uential shipping tests would probably show varying proportions of failures due to dropping, vibration, and rough handling. These might be sorted out using a 0areto chart. Then efforts could be made to improve the most fre"uently occurring causes. The cartridge drop and scrape test could also use p& or np&charts (see Chapter -9) to show results per sample of -?? repetitions of the test. The temperature tests would most li!ely use standard variables charts to measure whether test performance was within control limits, or not.
@ esign for /uality and 0roduct 1xcellence +pplying / in a *anaged Care Argani'ation -.
+lthough this example of / involved the design of a tangible items, it is more difficult to implement in a service context, as opposed to a pure manufacturing context, because both customer re"uirements and technical re"uirements are harder to "uantify and assess that with tangible products.
.
The detailed calculations in the mportance of the ho!s row and "ercentage of importance of the ho!s row used to arrive at these figures can be shown and verified on a spreadsheet. Bote that some discrepancies involving incorrect multiplication, were found in part of the / 3ouse of /uality.4
Direction of Improvement )ase*+se Acc+rac( Timeliness Clarit( Conciseness Import. of o!s % of Import. of o!s
Rate of Import . ,.-.3 ./ .5 /.-
Co. o ! ./ .0 .5 /.1 ,.0
"la n ,.,.1 .5 .2 ,.0
Rate of Impro v. 0., 0.0.3 0.0.3
Abso l. #gt. 1. 4., ./ -.4 /.-
% Improv e /-./% /2.-% 0/.4% //.4% 2.2%
Font si$e
0 0 035.0 -.1%
Upat e 0 2 2 0 ,/4.2 //.%
"oto s
0 0-., 5.30 %
Use of color s
Gloss.
Q&A
Terms 2 0
0
2
'ect. 0
25./ -.0 %
,13.3 /,.3 %
/,,.4 0/.45 %
Tbl. of Contn t. 2
0
/,2.0 0.30 %
04.3 2.3, %
The numbers in the original table were verified by the calculations shown above (some columns of the original table were rearranged for convenience of calculation). The rates of improvement, absolute weights, and percent improvements, based on the given values for 3rate of importance4 and 3company now4 and 3plan4 were validated. +s in the original table, the 3importance of hows4 and 3percent of importance of hows4 turned out to be accurately calculated. Specific factors shown as the most important were 3glossary terms4 and 3updates.4
9.
The lessons that can be learned and applied to other service organi'ations that see! to design or redesign their products and services include the facts that / provides for a systematic approach to lin!ing the 3voice of the customer4 to operational re"uirements. Dy doing so, operating efficiencies can be reali'ed and customer satisfaction can be enhanced. n addition, employee satisfaction often can be improved, as well, as found in the case. t must be recogni'ed that time and effort is involved in gathering, sorting, and analy'ing the characteristics and factors. +lso, there is subectivity in applying ratings and weights to variables. ence, the results are not easy to predict and guarantees are limited.
ANSERS T! RE"#E Q$EST#!NS -.
0roduct design and development consists of six steps$
• dea Eeneration. Bew or redesigned product ideas should incorporate customer needs and expectations.
• 0reliminary Concept evelopment. n this phase, new ideas are studied for feasibility. 0roduct%0rocess evelopment. f an idea survives the concept stage, the actual
Lang. Frinl (. 0
9.
The lessons that can be learned and applied to other service organi'ations that see! to design or redesign their products and services include the facts that / provides for a systematic approach to lin!ing the 3voice of the customer4 to operational re"uirements. Dy doing so, operating efficiencies can be reali'ed and customer satisfaction can be enhanced. n addition, employee satisfaction often can be improved, as well, as found in the case. t must be recogni'ed that time and effort is involved in gathering, sorting, and analy'ing the characteristics and factors. +lso, there is subectivity in applying ratings and weights to variables. ence, the results are not easy to predict and guarantees are limited.
ANSERS T! RE"#E Q$EST#!NS -.
0roduct design and development consists of six steps$
• dea Eeneration. Bew or redesigned product ideas should incorporate customer needs and expectations.
• 0reliminary Concept evelopment. n this phase, new ideas are studied for feasibility.
• 0roduct%0rocess evelopment. f an idea survives the concept stage, the actual design process begins by evaluating design alternatives and determining engineering specifications for all materials, components, and parts. This phase usually includes prototype testing, design reviews, and development, testing, and standardi'ation of the manufacturing processes
• ull&Scale 0roduction. f no serious problems are found, the company releases the product to manufacturing or service delivery teams.
• *ar!et ntroduction. The product is distributed to customers. • *ar!et 1valuation. +n ongoing product development process that relies on mar!et evaluation and customer feedbac! to initiate continuous improvements. .
Competitive pressures are forcing companies to reduce time to mar!et, which means that the time for product development is also s"uee'ed. The problems incurred in speeding up the process are well !nown. f done too hastily, the result will be the need to revise or scrap the design, cost increases or proect over&runs, difficulty in manufacturing the product, early product failure in the field, customer dissatisfaction, and%or lawsuits due to product liability. Ane of them most significant impediments to rapid design is poor intra& organi'ational coordination. ;educing time to mar!et can only be accomplished by process simplification, eliminating design changes, and improving product manufacturability. This re"uires involvement and cooperation of many functional groups to identify and solve design problems in order to reduce product development and introduction time.
.
Design for Six Sigma (DFSS) uses a set of tools and methodologies in the product development process to ensure that goods and services will meet customer needs and
7
Design for Six Sigma
6
achieve performance obectives, and that the processes used to ma!e and deliver them achieve Six Sigma capability. SS consists of four principal activities$ • Concept development , in which product functionality is determined based upon customer re"uirements, technological capabilities, and economic realities6 • Design development , which focuses on product and process performance issues necessary to fulfill the product and service re"uirements in manufacturing or delivery6 • Design optimization, which see!s to minimi'e the impact of variation in production and use, creating a 3robust4 design6 and • Design verification, which ensures that the capability of the production system meets the appropriate sigma level =.
Concept engineering (C%) emerged from a consortium of companies that included 0olaroid and Dose along with researchers at *T. C1 is a focused process for discovering customer re"uirements and using them to select superior product or service concepts that meet those re"uirements, and it puts the voice of the customer into a broader context and employees numerous other techni"ues to ensure effective processing of "ualitative data. ive maor steps comprise the process$ • #nderstanding the customer$s environment . This step involves first proect planning activities such as team selection, identifying fit with business strategy, and gaining team consensus on the proect focus. t also includes collecting the voice of the customer to understand the customer2s environment < physical, psychological, competitive, and so on. • Converting understanding into requirements. n this step, teams analy'e the customer transcripts to translate the voice of the customer into more specific re"uirements using the #> method. This step focuses on identifying the technical re"uirements we discussed in the context of /, selecting the most significant re"uirements, and 3scrubbing4 the re"uirements to refine them into clear and insightful statements. • %perationalizing !hat has been learned . nvolves determining how to measure how well a customer re"uirement is met. The principal re"uirement is to focus on throughput time, so the concept of 3"uic!ly4 needs to be operationali'ed and measured. Ance potential metrics are defined, they are evaluated to reduce the number of metrics that need to be used while ensuring that they cover all !ey re"uirements. This usually re"uires some sort of customer "uestionnaire to identify the importance of the re"uirements and prioriti'ed them. • Concept generation. This step generates ideas for solutions that will potentially meet customers2 needs. The approach re"uires brainstorming ideas that might resolve each individual customer re"uirement, selecting the best ones, and then classifying them under the traditional functional product characteristics. This helps to develop a 3mar!et in4 rather than a 3product out4 orientation. Creative thin!ing techni"ues are applied here to increase the number and diversity of potential ideas. • Concept selection. The potential ideas are evaluated for their capability to meet re"uirements, tradeoffs are assessed, and prototyping may begin. The process ends with reflection on the final concept to test whether the decision 3feels right4 based on all the !nowledge that has been ac"uired.
Design for Six Sigma
7
Concept engineering is an important tool for assuring "uality because it provides a systematic process that leaves a strong audit trail bac! to the voice of the customer. This ma!es it difficult to challenge the results of s!eptics and convert them. The process also helps to build consensus and gives design teams confidence in selling their concept to management. owever, it ta!es a lot of discipline and patience. @.
/ benefits companies through improved communication and teamwor! between all constituencies in the production process, such as between mar!eting and design, between design and manufacturing, and between purchasing and suppliers. 0roduct obectives are better understood and interpreted during the production process. Fse of / determines the causes of customer dissatisfaction, ma!ing it a useful tool for competitive analysis of product "uality by top management. 0roductivity as well as "uality improvements generally follow /. / reduces the time for new product development. / allows companies to simulate the effects of new design ideas and concepts. Companies can reduce product development time and bring new products into the mar!et sooner, thus gaining competitive advantage.
G.
n the / development process, a set of matrices is used to relate the voice of the customer to a product2s technical re"uirements, component re"uirements, process control plans, and manufacturing operations. The first matrix, called the "ouse of ualit! , provides the basis for the / concept. Duilding the ouse of /uality consists of six basic steps$ H? dentify customer re"uirements. H- dentify technical re"uirements. H ;elate the customer re"uirements to the technical re"uirements. H9 Conduct an evaluation of competing products or services H= 1valuate technical re"uirements and develop targets. H@ etermine which technical re"uirements to deploy in the remainder of the production%delivery process. The first ouse of /uality in the / process provides mar!eting with an important tool to understand customer needs and gives top management strategic direction. Three other 3houses of "uality4 are used to deploy the voice of the customer to (in a manufacturing setting) component parts characteristics, process plans, and "uality control. The second house applies to subsystems and components. +t this stage, target values representing the best values for fit, function, and appearance are determined. n manufacturing, most of the / activities represented by the first two houses of "uality are performed by product development and engineering functions. n the last two stages, the planning activities involve supervisors and production line operators. n the third house, the process plan relates the component characteristics to !ey process operations, the transition from planning to execution. #ey process operations are the basis for a control point . + control point forms the basis for a "uality control plan delivering those critical characteristics that are crucial to achieving customer satisfaction. This is specified in the last house of "uality. These are the things that must be measured
Design for Six Sigma
89
and evaluated on a continuous basis to ensure that processes continue to meet the important customer re"uirements defined in the first ouse of /uality. 7.
0roduct design can have a maor impact on manufacturability. f careful thought and planning is not done by the designer (or design team), the end product can end up being difficult or impossible to build due to placement of components, methods for attachments, 3impossible4 tolerances, difficulties in attaching or fastening components and%or difficulties in getting the whole assembled 3system4 to wor! smoothly, even with the highest "uality components. n addition time, materials, and other resources may be wasted unnecessarily due to a poor manufacturing design. The concept of esign for *anufacturability (*) is the process of designing a product so that it can be produced efficiently at the highest level of "uality. ts goal is to improve "uality, increase productivity, reduce lead time (time to mar!et, as well as manufacturing time) and maintain flexibility to adapt to future mar!et conditions.
I.
#ey design practices for high "uality in manufacturing and assembly include$ -) analy'e all design re"uirements to assess proper dimensions and tolerances, ) determine process capability, 9) identify and evaluate possible manufacturing "uality problems, =) select manufacturing processes that minimi'e technical ris!s, and @) evaluate processes under actual manufacturing conditions.
J.
Social responsibilities in the design process include safety and environmental concerns, which have made Design for &nvironment (1) and Design for Disassembly important features of products. egal and environmental issues are becoming critical in designing products and services, today. 0roduct safety and its conse"uences, product liability, should be of primary concern because of the damage that ha'ardous designs can do to consumers of the product. +lso, liability lawsuits can do maor damage to the financial health of an organi'ation, as well as its image and reputation in the mar!etplace. ;ecords and documentation relating to the design process are the best defense against liability lawsuits. These would include records on prototype development, testing, and inspection results. 1nvironmental issues involve "uestions of whether 3environmentally friendly4 designs (those that minimi'e damage to the environment in manufacture and product use) are being developed, what impacts will the design of the product have on the environment when it is scrapped, and how can consumers be given the most value for their money, while balancing the other two issuesK The above "uestions can often be addressed by considering it as a 3design for environment4 concept (often combined with and 3design for disassembly4). Lhat is the best design for repairability%recylabilityK
-?.
Design for %xcellence (DF&) is an emerging concept that includes many design&related initiatives such as concurrent engineering, design for manufacturability design for assembly, design for environment and other 3design for4 approaches. 5 obectives include higher functional performance, physical performance, user friendliness, reliability and durability, maintainability and serviceability, safety, compatibility and upgradeability, environmental friendliness, and psychological characteristics. 5 represents a total approach to product development and design involves the following activities$
Design for Six Sigma
88
• Constantly thin!ing in terms of how one can design or manufacture products better, • • • •
not ust solving or preventing problems ocusing on 3things done right4 rather than 3things gone wrong4 efining customer expectations and going beyond them, not ust barely meeting them or ust matching the competition Aptimi'ing desirable features or results, not ust incorporating them *inimi'ing the overall cost without compromising "uality of function
--.
*anufacturing specifications consist of nominal dimensions and tolerances. 'ominal refers to the ideal dimension or the target value that manufacturing see!s to meet6 tolerance is the permissible variation, recogni'ing the difficulty of meeting a target consistently. Traditionally, tolerances are set by convention rather than scientifically. + designer might use the tolerances specified on previous designs or base a design decision on udgment from past experience. Setting inappropriate tolerances can be costly, since tolerance settings often fail to account for the impact of variation on product functionality, manufacturability, or economic conse"uences. The Taguchi loss function is a scientific approach to tolerance design. Taguchi assumed that losses can be approximated by a "uadratic function so that larger deviations from target cause increasingly larger losses.
-.
The Taguchi loss function is a useful concept for process design. Taguchi suggests that there is not strict cut&off point that divides good "uality from poor "uality. ;ather, he assumed that losses can be approximated by a "uadratic function so that larger deviations from target correspond to increasingly larger losses. or the case in which a specific target value, T, is determined to produce the optimum performance, and in which "uality deteriorates as the actual value moves away from the target on either side (called 3nominal is best4), the loss function is represented by (x) 8 !(x & T) where x is any actual value of the "uality characteristic and ! is some constant. Thus, (x < T) represents the deviation from the target, and the loss increases by the s"uare of the deviation.
-9.
The purpose of Design 'ailure Mode and &ffects Analysis (*1+) is to identify all the ways in which a failure can occur, to estimate the effect and seriousness of the failure, and to recommend corrective design actions. + *1+ usually consists of specifying the following information for each design element or function$ ailure modes6 effect of the failure on the customer6 severity, li!elihood of occurrence, and detection rating6 potential causes of failure, and corrective actions or controls. + simple example of a *1+ for an ordinary household light soc!et is provided in the chapter.
-=.
;eliability has grown increasingly important among the "uality disciplines due to safety needs of consumers, the search for competitive advantage by companies, growing consumer awareness, and rising expectations and the difficulty of achieving high reliability in more sophisticated and complex modern products.
-@.
(eliability is the probability that a product, piece of e"uipment, or system performs its intended function for a stated period of time under specified operating conditions. There are four !ey components of this definition, including probability, time, performance, and operating conditions. +ll of these have to be considered in a comprehensive definition of reliability. 0robability allows comparison of different products and systems, time allows us to
Design for Six Sigma
8:
measure the length of life of the product, performance relates to the ability of the product to do what it was designed to do, and operating conditions specify to amount of usage and the environment in which the product is used. -G.
+ functional failure is one incurred at the start of the productMs life due to defective materials, components, or wor! on the product. + reliability failure is one that is incurred after some period of use. or example, if a new TN set suffers a blown picture tube during the first wee!, itMs a functional failure. There was obviously a defect in the manufacture of the tube. f the vertical hold feature of the set goes out (perhaps 9 days after the - year warranty is up), that is a reliability failure. t should reasonably be expected to last much longer than one year, but it didnMt.
-7.
'ailure rate is defined as the number of failures per unit of time during a specified time period being considered. or example, if -@ *0&9 players were tested for @?? hours and there were two failures of the units, the failure rate would be$ % (-@ x @??) 8 - % 97@? or ?.???G7.
-I.
The cumulative failure rate curve plots the cumulative percent of failures against time on the hori'ontal axis. The failure rate curve is obtained by determining the slope of the failure rate curve at a number of points to obtain the instantaneous failure rate (failures per unit time) at that point. + plot of these values yields the failure rate curve.
-J.
The average failure rate over any interval of time is the slope of the line between the two endpoints of the interval on the failure rate curve.
?.
The product life characteristics curve, is the so&called Obath&tub curveO because of its shape. t is actually the failure rate curve, described above. Such curves can be used to understand the distinctive failure rate patterns of various designs and products, over time.
-.
The reliability function represents the probability that an item will not fail within a certain period of time, T. t is directly related to the cumulative distribution function$ (T) 8 - & e&λT, that yields the probability of failures. Since (T) is the probability of failure, the reliability function, ;(T) can be defined as the complement, e.g. probability of not failing$ ;(T) 8 - & (- & e &λT) 8 e&λT t can also be expressed using the mean time to failure (*TT) value θ as$ ;(T) 8 e&T%θ
.
The reliability of series, parallel, and series parallel is relatively easy to compute, given the reliability of components in each system. or the series system, ; S 8 ; -; ; 9. Thus reliabilities are multiplicative. or a parallel system, the relationships are a little more complex, since the units are designed to use redundant components, so that if one unit fails the system can continue to operate. The system reliability is computed as$ ; S 8 - & P(- & ; -)(- & ; )(- & ; n)Q
Design for Six Sigma
8;
or series¶llel systems, the e"uivalent reliabilities of each parallel sub&system are calculated, successively, until there are no more parallel sub&systems. The system is then reduced to a serially e"uivalent system in which all component reliabilities can be multiplied to get the final reliability value. 9.
The purpose of a design review is to stimulate discussion, raise "uestions, and generate new ideas and solutions to help designers anticipate problems before they occur. To facilitate product development, a design review is generally conducted in three maor stages of the product development process$ preliminary, intermediate, and final. The preliminary design review establishes early communication between mar!eting, engineering, manufacturing, and purchasing personnel and provides better coordination of their activities. t usually involves higher levels of management and concentrates on strategic issues in design that relate to customer re"uirements and thus the ultimate "uality of the product. The preliminary design review evaluates such issues as the function of the product, conformance to customer2s needs, completeness of specifications, manufacturing costs, and liability issues. +fter the design is well established, an intermediate review ta!es place to study the design in greater detail to identify potential problems and suggest corrective action. 0ersonnel at lower levels of the organi'ation are more heavily involved at this stage. inally, ust before release to production, a final review is held. *aterials lists, drawings, and other detailed design information are studied with the purpose of preventing costly changes after production setup.
=.
*ethods of product testing for reliability include$ life testing, accelerated life testing, environmental testing and vibration and shoc! testing. n life and accelerated life testing the product is tested until it fails. The latter speeds up the process by overstressing the item to hasten its eventual failure. 1nvironmental and shoc! tests are performed to determine the productMs ability to survive and operate under adverse conditions of heat, cold, or shoc!.
@.
)atent defects are fre"uently found in electronic devices, such as semi&conductors. The term refers to the fact that a certain small proportion of the units will have defects which show up during the early life of the product, perhaps the first -,??? hours of operation. Then the remaining components, after the Oinfant mortalityO period has passed, the remaining components may operate for years without many failures.
G.
o#ust designs are those that are insensitive to variations in manufacturing or in the use environment.
7.
Common types of measuring instruments (see Donus *aterials folder on the 0remier website) used in manufacturing today fall into two categories$ 3low&technology4 and 3high&technology.4 ow&technology instruments are primarily manual devices that have been available for many years and include rulers, calipers, mechanical micrometers, go&no go gauges, etc.6 high&technology describes those that depend on modern electronics, microprocessors, lasers, or advanced optics, such as micrometers with digital readouts, electronic optical comparators, and computeri'ed coordinate measuring machines.
Design for Six Sigma
8<
I.
*etrology is the science of measurement. t formerly included only the measurement processes involved in gauging the physical attributes of obects. Today, metrology is much more broadly defined as$ the collection of people, e"uipment, facilities, methods, and procedures used to assure correctness or ade"uacy of measurements. t is vital to "uality control because of the increasing complexity of modern manufacturing and service operations. n particular, the increasing emphasis and oversight of government agencies, the implications of measurement errors on safety and product liability, and the need for reliance on improved "uality control methods, such as S0C, ma!e metrology an important branch of science.
J.
Accuracy is defined as the closeness of agreement between an observed value and an accepted reference value or standard. +ccuracy is measured as the amount of error in a measurement in proportion to the total si'e of the measurement. Ane measurement is more accurate than another if it has a smaller relative error. "recision is defined as the closeness of agreement between randomly selected individual measurements or results. 0recision, therefore, relates to the variance of repeated measurements. + measuring instrument having a low variance is said to be more precise than another having a higher variance. (eproducibility is the variation in the same measuring instrument when it is used by different individuals to measure the same parts. Causes of poor reproducibility include poor training of the operators in the use of the instrument or unclear calibrations on the gauge dial.
9?.
Calibration is the comparison of a measurement device or system having a !nown relationship to national standards to another device or system whose relationship to national standards is un!nown. Calibration is necessary to ensure the accuracy of measurement and hence to have confidence in the ability to distinguish between conforming and nonconforming production. *easurements made with uncalibrated or inade"uately calibrated e"uipment can lead to erroneous and costly decisions.
9-.
;epeatability and reproducibility (;R;) re"uire a study of variation and can be addressed through statistical analysis. ;R; studies must be done systematically, and re"uire "uite a number of steps. + repeatability and reproducibility study is conducted in the following manner (Bote$ formulas are omitted for the sa!e of brevity).
-. . 9.
=.
Select m operators and n parts. Typically at least operators and -? parts are chosen. Bumber the parts so that the numbers are not visible to the operators. Calibrate the measuring instrument. et each operator measure each part in a random order and record the results. ;epeat this for a total of r trials. +t least two trials must be used. et * i! represent the !th measurement of operator i on part . Compute the average measurement for each operator and the difference between the largest and smallest average.
Design for Six Sigma
@.
G.
8=
Compute the range for each part and each operator (these values show the variability of repeated measurements of the same part by the same operator)6 compute the average range for each operator6 compute the overall average range. Calculate 3control limits4 on the individual ranges ; i , using a constant (=) that depends on the sample si'e (number of trials, r) and can be found in a table for control charts. +ny range value beyond the control limits might result from some assignable cause, not random error. 0ossible causes should be investigated and, if found, corrected. The operator should repeat these measurements using the same part. f no assignable cause is found, these values should be discarded and all statistics in step @ as well as the control limit should be recalculated. Ance these basic calculations are made, an analysis of repeatability and reproducibility can be performed, e"uipment variation (1N) is computed as reproducibility, and operator variation as appraisal variation (+N). Constants # - and # are chosen and depend on the number of trials and number of operators, respectively. These constants provide a JJ percent confidence interval on these statistics. +n overall measure of repeatability and reproducibility (;R;) is given by$
;epeatability and reproducibility are often expressed as a percentage of the tolerance of the "uality characteristic being measured. The +merican Society for /uality suggests the following guidelines for evaluating these measures of repeatability and reproducibility$ • Fnder -? error$ This rate is acceptable. • -? to 9? error$ This rate may be acceptable based on the importance of the application, cost of the instrument, cost of repair, and so on. • Aver 9? error$ Eenerally, this rate is not acceptable. 1very effort should be made to identify the problem and correct it. 9.
"rocess capability is the range over which the natural variation of a process occurs as determined by the system of common causes. t is the ability of the combination of people, machines, methods, materials, and measurements to produce a product or service that will consistently meet design specifications. 0rocess capability is measured by the proportion of output that can be produced within design specifications6 in other words, it is a measurement of the uniformity of the product.
99.
+ process capability study is a carefully planned study designed to yield specific information about the performance of a process under specified operating conditions. Three types of studies are often conducted. + pea! performance study is focused on determining how a process performs under actual operating conditions. + process characteri'ation study is designed to determine how a process performs under actual operating conditions. + component variability study has the goal of determining the relative contribution of different sources of total variation. The six steps involved in ma!ing a process capability study are listed in the chapter.
9=.
The following are brief definitions of the various process capability indexes$
Design for Six Sigma
8>
C p is the ratio of the specification width to the natural tolerance of the process C pl is the lower one&sided index that relates the distance from the process mean to the lower tolerance limit to its 9 σ natural spread C pu is the upper one&sided index that relates the distance from the process mean to the upper tolerance limit to its 9 σ natural spread These indexes are calculated to determine the ability of a process to meet or exceed design specifications and are only meaningful when a process is !nown to be under control. Eeneral a process is considered to be capable if its index is -.? or above. These indexes may be used to establish "uality policy in operating areas or with a supplier by stating an acceptable standard, such as$ all capability indexes must be at .? (called G σ "uality) or above if the process is to be considered acceptable for elimination of inspection processes by customers.
S!%$T#!NS T! PR!&%E'S Bote$ ata sets for several problems in this chapter are available in the 1xcel wor!boo! C*+Data on the 0remium website for this chapter accompanying this text. Clic! on the appropriate wor!sheet tab as noted in the problem (e.g., "rob. *+-) to access the data. -.
Tonia2s Tasty Tacos conducted consumer surveys and focus groups and identified the most important customer expectations as
Tasty, moderately healthy food Speedy service +n easy&to&read menu board +ccurate order filling 0erceived value
evelop a set of technical re"uirements to incorporate into the design of a new facility and a ouse of /uality relationship matrix to assess how well your re"uirements address these expectations. ;efine your design as necessary, based upon the initial assessment.
Ans!er 0.
Anal(sis of c+stomer responses for Tonia2s Tasty Tacos inicates tat tere are li6el( to be several strong relationsips bet!een c+stomer re7+irements an associate tecnical re7+irements of te pro+ct tat Tonia esigns 8for e9ample. a b+rrito:; s+c as val+e vs. price< n+trition vs. calories 8an oter n+tritional content val+es; s+c as soi+m; an percent trans*fat:. ote te tree c+stomer response categories tat are +nrelate to te esign of te b+rritos ** orer acc+rac(; spee( service; an men+
Design for Six Sigma
8?
boar. boar. Tese Tese factor factors s !o+l !o+l re7+i re7+ire re a separat separate e anal(si anal(sis s as part part of a facilit( an process esign.
PART#A% H!$SE !( Q$A%#T) 'ATR#* 'ATR#* F/ /'2A@S AS3 AC/S
"ric e Taste Taste
>al+ al+e
Calo ries
'oi+ m
% t*Fat
Imprtnc e 0/ ,-
∆
=oistness
Compet. )val. 0/ , -
'elling "ts. 0 / , -
⊕ ∆
>is+all( Appealing +tritio+s o+s
⊕ ⊕ ∆
•
Goo >al+e al+e
•
Flavor >is+a l ?ealt alt
'i$e
∆
•
• •
⊕
Co+,etiti-e E-aluation.
• @ >er( strong relationsip ⊕ @ 'trong relationsip ∆ @ #ea6 relationsip .
Bewfo Bewfoni nia, a, nc., nc., is wor!in wor!ing g on a desi design gn for a new new sma smartp rtphone hone.. *ar!e *ar!eti ting ng staff staff conduct conducted ed extensive surveys and focus groups with potential customers to determine the characteristics that the customers want and expect in a smartphone. Bewfonia2s studies have identified the most important customer expectations as
nitial cost ;eliability 1ase of use eatures Aperating cost Compactness
evelop a set of technical re"uirements re"uirements to incorporate into the design of a ouse of /uality relationship matrix to assess how well your re"uirements address these expectations. ;efine your design as necessary, based upon the initial assessment.
Design for Six Sigma
86
Ans!er /.
Bewfonia2s propose Anal(si Anal( sis s of c+st c+stom omer er resp respon onse ses s for for Bewfonia2s propose smartpone smartpone ini inica cate tes s te te li6 li6eli elioo oo of seve severa rall str strong ong relat elatio ions nsi ips ps bet! bet!ee een n c+stomer c+stomer re7+ir re7+irement ements s an associate associate tecnical tecnical re7+irement re7+irements s of te esign; s+c as val+e vs. price< feat+res vs. compactness< an ease of +se vs. feat+res. perating costs ma( possibl( be istantl( relate to initial cost an feat+res. Tecnical caracteristics re7+ire to translate te Bvoice of te c+stomer into operational or engineering terms migt be meas+res of p+rcase cost; operating programs 8e.g.; ranc'; or ote oterr sim similar lar s( s(st ste ems:; ms:; n+mb n+mber er an an t(pe t(pe of feat feat+r +re es; !ei !eigt; gt; imensions; batter( life; cost of replacement batteries; an periperals.
9.
Toni To nia2 a2ss Tasty Tacos Tacos (0rob (0roble lem m -) ac"u ac"uir ired ed some some addi additi tiona onall inf infor orma mati tion. on. t t found found that that consumers placed the highest importance on healthy food, followed by value, followed by order accuracy and service. The menu board was only casually noted as an important attribute in the surveys. Tonia faces three maor competitors in this mar!et$ Erabby2s, Taco!ing, and Sandy2s. Studies of their products yielded the information shown in the table in C*+Data file for "rob.*+ on the 0remium website for this chapter. ;esults of the consumer panel ratings for each of these competitors can also be found there (a -<@ scale, with @ being the best). Fsing this information, modify and extend your ouse of /uality from 0roblem - and develop a deployment plan for a new burrito. An what attributes should the company focus its mar!eting effortsK
Ans!er 9.
Lith Lith the new data given for ToniaMs customers, custome rs, a partial ouse of /uality for the design of the burritos can be built, as shown below. Bote that the relationships between customer re"uirements (flavor, health, value) and associated technical re"uirements ( fat, calories, sodium, price) of the burrito design are strong.
Te inter inter*re *relation lationsips sips of te te roof roof are are not so!n so!n 8limitat 8limitations ions of ='# ='#or� soft!are:; tese ma( be s6etce in. For e9ample; te( !o+l so! a strong inter*relationsip inter*relationsip bet!een fat an calories.
PART#A% H!$SE !( Q$A%#T) 'ATR#* 'ATR#* F/ /'2A@S AS3 AC/S
"ric e Taste Taste
=oistness
'i$e
∆
Calo ries
'oi+ m
% t*Fat
⊕
Imprtnc e 0/ ,9
Compet. )val. 0/ , G
Q'
'elling "ts. 0 / , -
Design for Six Sigma
∆
>is+all( Appealing +tritio+s o+s
⊕ ⊕ ∆
•
Goo >al+e al+e
•
⊕
-
, H3. /1 o$. E
, 4.3 o$.
Flavor >is+a l ?ealt alt >al+ al+e
87
Co+,etiti-e E-aluation. Grabb(s Taco6ing Taco6ing 'an(s Targets Targets
Deplo(ment
∆
•
9 9
E
•
•
-
-
/ 53 o$.
/
/ , 0%
5mg.
E
G' Q
9
Q' G 9
-
'G Q
Q
E
'G
E
E
• @ >er( strong relationsip ⊕ @ 'trong relationsip ∆ @ #ea6 relationsip Tonia2s Tasty Tacos tecnical re7+irements re7+irements m+st be place on a more e7+al basis; !ic !o+l best be so!n as +nitso+nce; e9cept for te percent fat val+e. Tese are so!n belo!.
Compan(
"riceo$.
Calorieso$.
'oi+mo$. % Fat
Grabb(s
H 3./5/
53
0.1
H 3.33 3.33
5-
0/.14 0/.14
H 3./2/
23
0.
0 Taco6i Taco6ing ng / 'an(s 01 Alt Alto+ o+g g Tonia onias s is lo! lo! in pric price e per per o+nc o+nce; e; as !ell !ell as calo calori ries es;; an percent fat; tis anal(sis s+ggests tat Tonias so+l tr( to increase its si$e an vis+al appeal; !ile contin+ing to re+ce te cost per o+nce. At te same time; it so+l b+il on te strengt of te n+trition tren b( 6eeping te soi+m an percent fat lo!; as i Grabb(s; an sligtl( re+cing te n+mber of calories per o+nce to be even more competitive.
Design for Six Sigma
:9
f Tonia2s can design a flavorful, healthy, 7 o'. taco and sell it at an attractive price (say, U-.I@ or less), it should be a very profitable underta!ing. =.
Bewfonia, nc. (0roblem ), faces three maor competitors in this mar!et$ Aldphonia, Simphonia, and Colliefonia. t found that potential consumers placed the highest importance on reliability (measured by such things as freedom from operating system crashes and battery life), followed by compactness (weight%bul!iness), followed by flexibility (features, ease of use, and types of program modules available). The operating cost was only occasionally noted as an important attribute in the surveys. Studies of their products yielded the information shown in the table in C*+Data file for "rob.*+/ on the 0remium website for this chapter. ;esults of the consumer panel ratings for these competitors are also shown in that spreadsheet. Fsing this information, modify and extend your ouse of /uality from 0roblem and develop a deployment plan for the new smartphone. An what attributes should the company focus its mar!eting effortsK
Ans!er ,.
Lith the new data given for e!fonias potential customers, a partial ouse of /uality for the design of the smartphone can be built, as shown below. Bote the strong relationships between customer re"uirements and associated technical re"uirements of the smartphone design.
Te inter*relationsips of te roof are not so!n 8limitations of ='#or� soft!are:; b+t tese ma( be s6etce in. For e9ample; te( !o+l so! a strong inter*relationsip bet!een si$e an !eigt.
PART#A% H!$SE !( Q$A%#T) 'ATR#* (!R NEPH!N#A/S S'ARTPH!NE CASE Cost 'i$e 8in. :
#t. 8o$.:
Featr. 8n+m. :
pr. "rog .
at . Life
pr. Cos t
Import ance
Compet )val. 0/ ,-
'elling "ts. 0/ ,-
G' ?
E
0/ ,Reliabl e Compa ct
Jeeps operati ng Fits poc6et ot
∆
∆
⊕
• 9
• ∆
⊕
9
•
∆
9
G'? 'G Q
Design for Six Sigma
:8
eav( Feat+r es
Calena r; contact mgt.; etc. )ase of Int+itive +se operati ons >al+e Goo val+e Co+,etiti-e E-aluation. lponia 'imfonia Colliefonia Targets Deplo(ment
⊕
•
∆
•
•
∆
∆
•
9
G' ?
9
•
Q' G
9
Q
'G
,
-
,
-
-
-
-
,
/
/
/
• @ >er( strong
,
,
,
,
relationsip ⊕ @ 'trong relationsip
H/3 E
- 9 ./ E
1 o$. E
03
#in. C)
-
=o .
E
E
E
∆ @ #ea6 relationsip
Tis anal(sis s+ggests tat e!fonia so+l tr( to position itself bet!een 'imfonia an Colliefonia in price an feat+res. It so+l b+il on te strengt of te c+stomers reliabilit( concern; 6eeping batter( life near - o+rs an +se a proven operating program; s+c as DranchAS. )no+g feat+res 803: so+l be oKere to be competitive. f e!fonia can design a high&value smartphone and sell it at an attractive price (say, U@? or less), it should be a very profitable underta!ing. @.
+ genetic researcher at Eenab, td. is trying to test two laboratory thermometers (that can be read to -%-??,???th of a degree Celsius) for accuracy and precision. She measured @ samples with each and obtained the results found in the C*+Data file for "rob.*+- on the 0remium website for this chapter. The true temperature being measured is ? degrees C. Lhich instrument is more accurateK Lhich is more preciseK Lhich is the better instrumentK
Ans!er -.
Acc+rac( of Termometer A Termometer Abs M?.???9-& ?N
Abs M*.3333- * 3N
Design for Six Sigma
::
033 9 ************************ @ 3.30 % 3.33-% 0 eg.
033 9
*********************** @ 0 eg.
Termometer is more acc+rate. Te )9cel*calc+late 8see spreaseets "rob0/*-a.9ls an "rob0/* -b.9ls on te "remi+m !ebsite for etails: statistics an fre7+enc( istrib+tion; so!s tat Termometer is also more precise tan Termometer A; as inicate b( a smaller stanar eviation. Termometer is a better instr+ment; beca+se it is li6el( tat it can be aO+ste to center on te nominal val+e of 3.
Freuenc! a#le * 4ro#lem 8:*=a 0pper Cell oundaries Freuencies
Cell Cell Cell 9 Cell = Cell @ Cell G Cell 7
&?.??@&?.??-GJ &?.???IG &?.????9 ?.???I? ?.??-G9 ?.??=G
Standard Statistical Measures
*ean *edian *ode Standard deviation Nariance *ax *in ;ange
?.???9- ?.???=G VB%+ ?.??-9=9 ?.????? ?.??=@G &?.??@-= ?.??=J7?
9 @ @ G =
Design for Six Sigma
:;
Freuenc! a#le 4ro#lem 8:*=# 0pper Cell oundaries Freuencies
Cell Cell Cell 9 Cell = Cell @
&?.??&?.???7? ?.????@ ?.??-@G ?.??9
Standard Statistical Measures
*ean *edian *ode Standard deviation Nariance *ax *in ;ange
&?.????=G &?.???-9 VB%+ ?.??-?= ?.??????.??9-G &?.???J ?.??=@@
7 7 7 9
Design for Six Sigma
G.
:<
Two scales were at +ussieburgers, td. used to weigh the same @ samples of hamburger patties for a fast&food restaurant in +ustralia. ;esults are shown in C*+Data file for "rob.*+0 on the 0remium website for this chapter. The samples were weighed in grams,
and the supplier has ensured that each patty weighs --= grams. Lhich scale is more accurateK Lhich is more preciseK Lhich is the better scaleK
Ans!er 1.
'ee spreaseets "rob0/*1a.9ls an "rob0/*1b.9ls for etails. Acc+rac( of
'cale A
AbsM00.21 *00,N 033 9 ************************ @ 3.3- % 0.15-% 00,
'cale AbsM00-.2/ * 00,N 033 9 *********************** @ 00,
'cale A is more acc+rate. Te fre7+enc( istrib+tion; ta6en from te )9cel printo+t; so!s tat 'cale is more precise tan 'cale A.
Design for Six Sigma
:=
'cale is a better instr+ment; beca+se it is li6el( tat it can be aO+ste to center on te nominal val+e of 3. 'cale A Freuenc! a#le * 4ro#lem 8:*>a
Cell Cell Cell 9 Cell = Cell @ Cell G Cell 7
0pper Cell oundaries
Freuencies
--.?? --.G7 --9.99 --=.?? --=.G7
[email protected] --G.??
9 ? @ J ? G
Standard Statistical Measures
*ean *edian *ode Standard deviation Nariance *ax *in ;ange
--9.JG --=.?? --=.?? -.-= -.J --G.?? --.?? =.??
Design for Six Sigma
:>
'cale Freuenc! a#le 4ro#lem 8:*># 0pper Cell oundaries Freuencies
Cell Cell Cell 9 Cell = Cell @
--=.??
[email protected] --G.?? --7.99 --I.??
Standard Statistical Measures
*ean *edian *ode Standard deviation Nariance *ax *in ;ange
[email protected] --G.?? --G.?? -.- -.= --I --= =.??
9 @ -? @
Design for Six Sigma
7.
:?
+ blueprint specification for the thic!ness of a dishwasher part at 0lataimpia, nc. is ?.9@ W ?.?@ centimeters (cm). t costs U-@ to scrap a part that is outside the specifications. etermine the Taguchi loss function for this situation.
Ans!er 4.
Te Tag+ci Loss F+nction for 0lataimpia, nc. part is L89: @ 6 89 * T: / H0- @ 6 83.3/-: / 6 @ /,333
∴ L89: @ 6 89 * T: / @ /,333 89 * T: / I.
+ team was formed to study the dishwasher part at 0lataimpia, nc. described in 0roblem 7. Lhile continuing to wor! to find the root cause of scrap, they found a way to reduce the scrap cost to U-? per part. a. etermine the Taguchi loss function for this situation. b. f the process deviation from target can be held at ?.?-@ cm, what is the Taguchi lossK
Ans!er 5.
Te Tag+ci Loss F+nction is L89: @ 6 89 * T:/ a: H03 @ 6 83.3/-: / 6 @ 01333
∴ L89: @ 6 89 * T: / @ 01333 89 * T: / b: L89: @ 01333 89 * T: / ∴ L83.30-: @ 01333 83.30-: / @ H.13
J.
+ specification for the length of an auto part at 0artsimensions, nc. is @.? W ?.-? centimeters (cm). t costs U@? to scrap a part that is outside the specifications. etermine the Taguchi loss function for this situation.
Ans!er 2.
Te Tag+ci Loss F+nction is L89: @ 6 89 * T:/ H-3 @ 6 83.03: / 6 @ -333
Design for Six Sigma
:6
∴ L89: @ 6 89 * T: / @ -333 89 * T:/ -?.
+ team was formed to study the auto part at 0artsimensions described in 0roblem J. Lhile continuing to wor! to find the root cause of scrap, the team found a way to reduce the scrap cost to U9? per part. a. etermine the Taguchi loss function for this situation. b. f the process deviation from target can be held at ?.?? cm, what is the Taguchi lossK
Ans!er 03.
Te Tag+ci Loss F+nction is L89: @ 6 89 * T:/ a: H3 @ 6 83.03:/ 6 @ 333
∴ L89: @ 6 89 * T: / @ 333 89 * T: / b: L89: @ 333 89 * T: / ∴ L83.3/3: @ 333 83.3/3: / @ H 0./3 --.
;uido Fnlimited ma!es electronic soundboards for car stereos. Autput voltage to a certain component on the board must be - W ?. volts. 1xceeding the limits results in an estimated loss of U@?. etermine the Taguchi loss function.
Ans!er 00.
Te Tag+ci Loss F+nction is L89: @ 6 89 * T:/ H-3 @ 6 83./: / 6 @ 0/-3
∴ L89: @ 6 89 * T: / @ 0/-3 89 * T:/ -.
+n electronic component has a specification of -?? W 9 ohms. Scrapping the component results in a UI- loss. a. Lhat is the value of ! in the Taguchi loss functionK b. f the process is centered on the target specification with a standard deviation of - ohm, what is the expected loss per unitK
Ans!er
Design for Six Sigma
0/.
:7
For a speciPcation of 033 oms a: L89: @ 6 89 * T:/ H50 @ 6 8 :/ 6@2 b: )L89: @ 6 8σ
-9.
/
D/: @ 2 8 0/ 3/ : @ H2
+n automatic coo!ie machine must deposit a specified amount of @ W ?. grams (g) of dough for each coo!ie on a conveyor belt. f the machine either over& or underdeposits the mixture, it costs U?.? to scrap the defective coo!ie. a. Lhat is the value of ! in the Taguchi loss functionK b. f the process is centered on the target specification with a standard deviation of ?.?G g, what is the expected loss per unitK
Ans!er 0.
For a speciPcation of /- 3./ grams a: L89: @ 6 89 * T:/ H3.3/ @ 6 8 3./ : / 6 @ 3.b: For σ )L89: @ 6 8σ
-=.
@ 3.31 /
D/: @ 3.- 8 3.31 / 3/ : @ H3.3305
+ computer chip is designed so that the distance between two adacent pins has a specification of .??? W ?.?? millimeters (mm). The loss due to a defective chip is U. + sample of @ chips was drawn from the production process and the results, in mm, can be found in the C*+Data file for "rob.*+*/ on the 0remium website for this chapter. a. Compute the value of ! in the Taguchi loss function. b. Lhat is the expected loss from this process based on the sample dataK
Ans!er 0,.
For a speciPcation of /.333 .33/ mm an a H/ scrap cost Anal(sis of te ataset for problem 0/*0, provies te follo!ing statistics
Design for Six Sigma
;9
@ /.33335< D @ /.33335 * /.33 @ 3.33335 @ 3.3303, σ x
L89: @ 6 89 * T:/ H/ @ 6 83.33/: /
a:
b: H3.-,,
-@.
∴ 6 @ -33;333
)L89: @ 6 8σ
/
D/: @ -33;333 8 3.3303, / 3.33335 / : @
n the production of transformers, any output voltage that exceeds -? W -@ volts is unacceptable to the customer. 1xceeding these limits results in an estimated loss of U=@?. owever, the manufacturer can adust the voltage in the plant by changing a resistor that costs U.@. a.
etermine the Taguchi loss function.
b. Suppose the nominal specification is -? volts. +t what tolerance should the transformer be manufactured, assuming that the amount of loss is represented by the cost of the resistorK
Ans!er 0-.
a:
Te Tag+ci Loss f+nction is L89: @ 6 89 * T:/ ,-3 @ 6 80-:/ 6 @ 3.'o; L89: @ 3.- 89*T:/
b:
H/./- @ 3.- 89*0/3: / ,.-3 @ 89 * 0/3: / 89 * T: Tolerance @
=.@?
@ /.0/ volts
/.0/ @ 9 * 0/3
∴ 9 @ 0//.0/ -G.
+t 1le!troparts *anufacturers2 integrated circuit business, managers gathered data from a customer focus group and found that any output voltage that exceeds -? W @ volts was
Design for Six Sigma
;8
unacceptable to the customer. 1xceeding these limits results in an estimated loss of U??. owever, the manufacturer can still adust the voltage in the plant by changing a resistor that costs U.??. a. etermine the Taguchi loss function. b. Suppose the nominal specification remains at -? volts. +t what tolerance should the integrated circuit be manufactured, assuming that the amount of loss is represented by the cost of the resistorK
Ans!er 01.
a: Te Tag+ci Loss f+nction is L89: @ 6 89 * T: / /33 @ 6 8-:/ 6 @5 'o; L89: @ 5 89*T:/
b: Te Tag+ci Loss f+nction is L89: @ 6 89 * T:
/
H/.33 @ 5 89*0/3: / 3./- @ 89 * 0/3: / 89 * T: Tolerance @
?.@ @
3.- volts
3.- @ 9 * 0/3
∴ 9 @ 0/3.-7.
Two processes, 0 and /, are used by a supplier to produce the same component, X, which is a critical part in the engine of the +ir0ort 77I airplane. The specification for X calls for a dimension of ?.= mm W ?.?9. The probabilities of achieving the dimensions for each process based on their inherent variability are shown in the table found in the C*+Data file for "rob.*+*1 on the 0remium website for this chapter. f ! 8 G?,???, what is the expected loss for each processK Lhich would be the best process to use, based on minimi'ing the expected lossK
Ans!er 04.
or the +ir0ort 77I plane parts (see spreadsheets 0rob-&-7.xls for detailed calculations)$
Design for Six Sigma
;:
Specifications are = Y%& 9 mm (x) 8 G???? (x & T) or a typical calculation$
∴ (?.-) 8 G???? (?.- & ?.=) 8 U @=.?? Leighted loss 8 ?.- 5 U@=.?? 8 U G.=I
Air2Port Airplane Co. Calculation of Taguchi Loss Values Va lue
0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28
Loss ($)
Process P Probabilit y
Weighted Loss ($)
Process Q Probability
Weighted Loss ($)
96.00 54.00 24.00 6.00 0.00 6.00 24.00 54.00 96.00
0 0.12 0.12 0.12 0.28 0.12 0.12 0.12 0
0.00 6.48 2.88 0.72 0.00 0.72 2.88 6.48 0.00
0.02 0.03 0.15 0.15 0.30 0.15 0.15 0.03 0.02
1.92 1.62 3.60 0.90 0.00 0.90 3.60 1.62 1.92
Execte! Loss
20.16
16.08
Therefore, 0rocess / incurs a smaller loss than 0rocess 0, even though some output of / falls outside specifications. -I.
The average time to handle a call in a the Call&Bowait call processing center has a specification of G W-.@ minutes. The loss due to a mishandled call is U-G. + sample of @ calls was drawn from the process and the results, in minutes, can be found in the C*+Data file for "rob.*+*2 on the 0remium website for this chapter. a. Compute the value of ! in the Taguchi loss function. b. Lhat is the expected loss from this process based on the sample dataK
Design for Six Sigma
;;
Ans!er 05.
For a speciPcation of G ± -.@ min+tes an a H01 call misanling cost x@ σ
a:
1.301< D @ 1.301 * 1.33 @ 3.301 @ 3.52-4 L89: @ 6 89 * T:/ H01 @ 6 80./- : / < ∴ 6 @ 03./,
b: ) ML89: @ 6 8σ H5./05
-J.
/
D/:N @ 03./, 8 3.52-4 / 3.301 / : @
Compute the average failure rate during the intervals ? to =?, =? to 7?, and 7? to -??, and ? to -??, based on the information in igure -.I.
Ans!er 02.
ase on te C+m+lative Fail+re Rate c+rve From 3 * ,3; slope @ /2.- ,3 @ 3.45 From ,3 * 43; slope @ 8,3 S /2.-: 843 * ,3: @ 3.-3 From 43 * 033; slope @ 823 * ,3: 8033 * 43: @ 0.145 From 3 * 033; slope @ 23 033 @ 3.2 'ee te spreaseet "rob0/*02 for more etails; incl+ing a iagram so!ing te fail+re rate c+rve.
Design for Six Sigma
?.
;<
The life of a cell phone battery is normally distributed with a mean of J?? days and standard deviation of @? days. a. Lhat fraction of batteries is expected to survive beyond J7@ daysK b. Lhat fraction will survive fewer than I?? daysK c. S!etch the reliability function. d. Lhat length of warranty is needed so that no more than -? percent of the batteries will be expected to fail during the warranty periodK
Ans!er /3.
a: "89 24-: @ 3.- * "8233 9 24-: 24-*233 "8233 9 24-:@ " 8$ ****************: @ " 83 $ 0.-: @ 3.,/ -3
Terefore; "89 54-: @ 3.- * 3.,/ @ 3.3115 or 1.15% so+l s+rvive be(on 24- a(s. 533 *233 b: " 89 533: @ " 8$ *************: @ " 8$ */.3 : -3 Terefore; " 89 553: @ 3.- * 3.,44/ @ 3.3//5 or /./5% so+l s+rvive less tan 533 a(s.
c: Te reliabilit( f+nction loo6s appro9imatel( spreaseet "rb0/*/3.9ls for etails:
as
follo!s
8see
Design for Six Sigma
;=
: Let 9! be te limit of te !arrant( perio. " 89 9!: @ 3.03< $ @ *0./5; for $ @ 9*233 @ *0./5 ; 9! @ 51 o+rs for te -3
-.
!arrant( limit.
ifetred, nc., ma!es automobile tires that have a mean life of 7@,??? miles with a standard deviation of ,@?? miles. a. Lhat fraction of tires is expected to survive beyond 77,@? milesK b. Lhat fraction will survive fewer than GI,7@? milesK c. S!etch the reliability function. d. Lhat length of warranty is needed so that no more than -? percent of the tires will be expected to fail during the warranty periodK
Ans!er /0.
a: "89 44/-3: @ 3.- * "84-333 9 44/-3:
44/-3 * 4-333 "84-333 9 44/-3:@ "8$ *******************: @ " 83 $ 3.2: @ 3.0-2 /-33
Terefore; "89 44/-3: @ 3.- * 3.0-2 @ 3.05,0 or 05.,0% so+l s+rvive be(on 44/-3 miles.
Design for Six Sigma
;>
154-3* 4-333 * 1/-3 b: " 89 154-3: @ "8$ ******************** : @ "8$ ********** : @ " 8$ */.-3: @ /-33 /-33 3.- * "8154-3 9 4-333: @ 3.- * 3.,25 @ 3.331/ Terefore; " 89 154-3: @ 3.331/ or 3.1/% so+l s+rvive less tan 154-3 miles. c:
Te reliabilit( f+nction loo6s appro9imatel( as follo!s
: Let 9! be te limit of te !arrant( perio. " 89 9!: @ 3.03< $ @ *0./5; for $ @ 9*4-333 @ *0./5 ; 9 ! @ 40;533 miles for te /-33 !arrant( limit. .
*assive Corporation2s tested five motors in an I??&hour test. Compute the failure rate if, three failed after ??, 97@, and =@? hours and the other two ran for the full I?? hours each.
Ans!er
Design for Six Sigma
//.
;?
=assive Corporations motors ave a fail+re rate of
λ@
@
o+r M8/ 9 533: /33 4- ,-3N 9.
@ 3.3300, fail+res
/1/-
ivelong, nc.2s computer monitors have a failure rate of ?.????@ units per hour. +ssuming an exponential distribution, what is the probability of failure within -?,??? hoursK Lhat is the reliability functionK
Ans!er /.
Te reliabilit( f+nction for Livelongs monitors is R8T: @ 0 * F8T: @ e*λ T
λ @ 3.3333-< Use F8T: @ "89 03333: F8T: @ "89 03333: @ 0 * e *3.3333- 803333: @ 0* 3.134 @ 3.2 or 2.% probabilit( tat a monitor !ill s+rvive less tan 03;333 o+rs.
/,.
An electronic component in a satellite raio as fail+re rate of
λ@ .
33330-. Fin te mean time to fail+re 8=TTF:. #at is te probabilit( tat te component !ill not ave faile after 0/;333 o+rs of operationV Ans!er /,.
Te =TTF is θ @
0
<
so; θ @ 11111.14
λ R 8T: @ e* Tθ @ e* 0/333 11111.14 @ e s+rviving for at least 0/;333 o+rs
@.
*3.05
@ 3.5- or 5.-% probabilit( of
The *TD of an integrated circuit made by ceeF, nc. is -I,??? hours. Calculate the failure rate.
Ans!er /-.
The failure rate (λ) for ceeF, nc.2s integrated circuits is$
λ@
0
@ 3.3333-1 fail+res r.
Design for Six Sigma
G.
;6
05333
+ manufacturer of *09 players purchases maor electronic components as modules. The reliabilities of components differ by supplier (see diagram, below). Suppose that the configuration of the maor components is given by$
The components that can be purchased from three different suppliers. The reliabilities of the components are as follows$ Component Supplier 9
Supplier -
Supplier
+ D C
.J7 .I@ .J@
.J .J? .J9
.J@ .J? .II
Transportation and purchasing considerations re"uire that only one supplier be chosen. Lhich one should be selected if the radio is to have the highest possible reliabilityK
Ans!er /1.
'+pplier 0 RaRbc @ 83.24: M0 * 80 * 3.5-:80 * 3.2-:N @ 3.21 '+pplier / RaRbc @ 83.2/: M0 * 80 * 3.23:80 * 3.2:N @ 3.20, '+pplier RaRbc @ 83.2-: M0 * 80 * 3.23:80 * 3.55:N @ 3.22 Terefore; coose '+pplier 0.
7.
+n electronic missile guidance system consists of the following components$ Components +, D, C, and have reliabilities of ?.JI, ?.J@, ?.I@, and ?.JJ, respectively (see the following diagram). Lhat is the reliability of the entire systemK
Ans!er
Design for Six Sigma
;7
/4. Te reliabilit( of te parallel R cc so!n in te iagram above te problem; is calc+late as Rcc @ 0 * 80 * 3.5-: / @ 3.25 RaRbRccR @ 83.25:83.2-: 83.25:83.22: @ 3.23 /5.
A estronics store processes c+stomers tro+g !or6 stations !en te( !is to b+( a certain pop+lar pro+ct. =o+lar components for te pro+ct m+st be cec6e electronicall( at t!o !or6 stations before Pnal cec6o+t; !ere te casier collects cas or creit cars for te sale. a: If !or6station 0 as reliabilit( of testing e7+ipment of 3.25; !or6station / as reliabilit( of testing e7+ipment of 3.2/ an te Pnal cec6o+t register as a reliabilit( of 3.23; !at is te overall cec6o+t s(stem reliabilit(V b: If te store manager !ants to ens+re at least a 23% s(stem reliabilit( can se o so b( eicating t!o Pnal cec6o+t registers to te process; in parallel; eac aving a 3.23 reliabilit(; !it te same reliabilit( at !or6stations 0 an /V Ans!er
/5.
a: RaRbRc @ 83.25:83.2/:83.23: @ 3.500 b: RaRbc @ 83.25: 83.2/: M0 * 80 * 3.23:80 * 3.23:N @ 3.52 o; tis !ill not provie te minim+m re7+ire s(stem reliabilit(. Te manager m+st Pn a !a( to improve reliabilit( of one or more !or6stations or cec6o+t registers.
29. *anuplex, nc. has a complex manufacturing process, with three operations that are performed in series. Decause of the nature of the process, machines fre"uently fall out of adustment and must be repaired. To !eep the system going, two identical machines are used at each stage6 thus, if one fails, the other can be used while the first is repaired (see accompanying figure).
Design for Six Sigma
<9
The reliabilities of the machines are as follows$ *achine
;eliability
+ D C
.7? .I? .J@
a. +naly'e the system reliability, assuming only one machine at each stage (all the bac!up machines are out of operation). b. ow much is the reliability improved by having two machines at each stageK
Ans!er /2.
a: RaRbRc @ 83.43:83.53:83.2-: @ 3.-/ b: RaaRbbRcc @ M0 * 80 * 3.43: /N M0 * 80 * 3.53: /N M0 * 80 * 3.2-: /N @ 83.20: 83.21: 83.224-: @ 3.540 Te improvement is signiPcant; rising 3.2 from 3.-/ to 3.540
9?.
+n automated production system at +utoprod, nc. consists of three operations$ turning, milling, and grinding. ndividual parts are transferred from one operation to the next by a robot. ence, if one machine or the robot fails, the process stops. a. f the reliabilities of the robot, turning center, milling machine, and grinder are ?.JI, ?.J?, ?.J9, and ?.I@, respectively, what is the reliability of the systemK b. Suppose that two grinders are available and the system does not stop if one fails. Lhat is the reliability of the systemK
Ans!er 3.
a: RtRmRg @ 83.25:83.23:83.2:83.5-: @ 3.124 b: RtRmReg @ 83.25:83.23:83.2:M0 * 80 * 3.5-: /N @ 3.53/
Design for Six Sigma
<8
E+E1 ;R; 0;AD1*S
9-.
+ gauge repeatability and reproducibility study at ran!ford Dra!e Systems collected the data found in the C*+Data file for "rob.*+* on the 0remium website for this chapter. +naly'e these data. The part specification is -.? W ?.?G mm.
Ans!er 0.
Detaile calc+lations for te Prst operator are as follo!s @ 8∑∑=iO6: nr @ /2.4/3 3 @ 3.2234
x 0
@ 8∑RiO: n @ 3./53 03 @ 3.3/5
; 0
Use tis meto to calc+late val+es for te secon operator @ /2.230 3 @ 3.2214<
x /
@ 3.53 03 @ 3.35
; /
@ ma9 W x iX * min W x iX @ 3.2214 * 3.2234 @ 3.331
x D
@ 8∑ ; i: m @ 83.3/5 3.35: / @ 3.3 D, @ /.-4, < UCL R @ D, ; @ 8/.-4,: 83.3: @ 3.35,2; all ranges belo! ;
J 0 @ .3- < J / @ .1- 8from Table 0/.: )> @ J 0 ; @ 8.3-: 83.3: @ 3.0331A> @ RR @
(# . x )
.
.
& (1N % nr)
(1N) Y (+N)
@ 3.3002
@ 3.030,
)7+ipment variation @ 033 83.0331- 3.0/: @ 5.55% perator variation @ 033 83.3002 3.0/: @ 2.2/% R & R variation @ 033 83.030, 3.0/: @ 5,.-3% For etaile spreaseet ata; see "rob0/*0RR.9ls. 'preaseet res+lts conPrm prior calc+lations; as follo!s Tolerance analysis Aerage range 0.033 !epeatability ("V) 0.101 83.88" #bar range ( x %) 0.006 !eproducibility (AV) 0.012 9.93" !epeatability and !eproducibility (!&!) 0.101 84.46" Control li'it or indiidual ranges 0.085
Design for Six Sigma
<: ote* any ranges beyond this li'it 'ay be the result o assignable causes. +dentiy and correct. %iscard alues and reco'pute statistics.
∴ Concentrate on re+cing e7+ipment variation ote tat te calc+lator val+es; so!n in te etaile calc+lations above; an comp+ter val+es o not matc precisel(; beca+se a greater n+mber of ecimal places are +se b( te comp+ter to carr( o+t calc+lations. All form+las are ientical; o!ever. 9.
+ gauge repeatability and reproducibility study was made at 0recision 0arts, nc., using three operators, ta!ing three trials each on identical parts. The data that can be found in the C*+Data file for "rob.*++ on the 0remium website for this chapter were collected. o you see any problems after analy'ing these dataK Lhat should be doneK The part specification for the collar that was measured was -.G W ?. inches.
Ans!er /.
Detaile calc+lations for te Prst operator are as follo!s @ 8∑∑=iO6: nr @ ,5.,5 3 @ 0.101
x 0
@ 8∑RiO: n @ 0. 03 @ 3.0
; 0
Use tis meto to calc+late val+es for te secon operator @ ,1.4, 3 @ 0.--5<
x /
@ 0.-5 03 @ 3.0-5
; /
Also; +se tis meto to calc+late val+es for te tir operator @ ,4.3- 3 @ 0.-15<
x
@ 3.103 03 @ 3.310
;
@ ma9 W x iX * min W x iX @ 0.101 S 0.--5 @ 3.3-5
x D
;
@ 8∑ ; i: m @ 83.0 3.0-5 3.310: @ 3.004
D, @ /.-4, < UCL R @ D,
;
@ 8/.-4,: 83.004: @ 3.30/; all ranges belo!
J 0 @ .3- < J / @ .1- 8from Table 0/.: )> @ J 0 ; @ 8.3-: 83.004: @ 3.-12 A> @
(# . x ) . & (1N . % nr)
@ 3.0,/,
Design for Six Sigma
RR @
(1N) Y (+N)
<;
@ 3.5,
)7+ipment variation @ 033 83.-12 3.,3: @ 52./% perator variation @ 033 83.0,/, 3.,3: @ -.13% R & R variation @ 033 83.5, 3.,3: @ 21.35% ote tat te range in sample 4 e9ceee te control limit of 3.30 b( for te Prst operator. Tis point co+l ave been +e to a misreaing of te ga+ge. If so; tis sample so+l be tro!n o+t; anoter one ta6en; an te val+es recomp+te. For etaile spreaseet ata; see "rob0/*/RR.9ls. 'preaseet res+lts conPrm prior calc+lations; as follo!s Aerage range #bar range ( x %)
0.117 0.058
!epeatability ("V) !eproducibility (AV) !epeatability and !eproducibility (!&!) Control li'it or indiidual ranges ote* any ranges beyond this li'it 'ay be the result o assignable causes. +dentiy and correct. %iscard alues and reco'pute statistics.
Tolerance analysis 0.358 89.47" 0.142 35.58" 0.385 96.28" 0.302
∴ Concentrate on re+cing e7+ipment variation ote also tat te calc+lator val+es; so!n in te etaile calc+lations above; an comp+ter val+es o not matc precisel(; beca+se a greater n+mber of ecimal places are +se b( te comp+ter to carr( o+t calc+lations. All form+las are ientical; o!ever.
33.A mac hi ni ngpr oces satMac h3ToolCo.hasar equi r eddi mensi ononapar tof0. 575 Da t a ±0 . 0 0 7i nc h .T we nt y fiv epa r t se a c hwe r eme a s ur e da sf o undi nt heC12 fil ef or P r o b . 1 2 3 3o nt hePr e mi um
we b s i t ef o rt hi sc h a pt e r .Wha ti si t sc a p a bi l i t yf o r
pr oduc i ngwi t hi na c c e p t a bl el i mi t s ? Ans!er .
For sample statistics at *ach9 Tool Co. of a tolerance of 3.-4- W 3.334
x
@ 3.-4-3< σ @ 3.331- an
Cp @ UTL * LTL @ 3.-5/ * 3.-15 @ 3.-2< not capable * +nsatisfactor( 1σ 1 83.331-:
Design for Six Sigma
<<
'ee spreaseet "rob0/*.9ls for more escriptive anal(sis. Bote$ There is some rounding error in the above calculations. See spreadsheet for more descriptive analysis.
'ominal specification 0pper tolerance limit 1o+er tolerance limit
?.@7@? ?.@I? ?.@GI?
?.@7@? Cp ?.??G@ Cpl
Average Std. deviation
CpB
?.9@G7 ?.9@=7 ?.9@I7 ?.9@=7
9=. +dustments were made in the process at Ma c h3ToolCo. ,discussed in 0roblem 99 and @ more samples were ta!en. The results are given in the C*+Data file for "rob.*+/ on the 0remium website for this chapter. Lhat can you observe about the processK s it now capable of producing within acceptable limitsK
Ans!er ,. For sample statistics of 3.-4- W 3.334
x
@ 3.-4--< σ @ 3.3304 an a tolerance of
The standard deviation is smaller than previously, indicating less 3spread4 within the data. See spreadsheet 0-
[email protected] for more descriptive analysis.
Cp @ UTL * LTL @ 3.-5/ * 3.-15 @ 0.4< Te process capabilit( is no! insie 1 σ
1 83.3304:
te tolerance limits; at an
acceptable level. Bote, however, that the other process capability indexes, below, show that there are still some slight problems with process centering that must be addressed. ?.@7@@ Cp ?.??-GJ Cpl C,u CpB rom the data for #ermit Theatrical 0roducts, construct a histogram and
'ominal specification 0pper tolerance limit 1o+er tolerance limit
9@.
?.@7@? ?.@I? ?.@GI?
Average Std. deviation
-.9I9I -.=IGG -.I-? -.I-? estimate the
process capability. f the specifications are = W ?.?9, estimate the percentage of parts that will be nonconforming. inally, compute C p, C pu, and C pl. Samples for three parts were ta!en as shown in the C*+Data file for "rob*+- on the student 0remium website for this chapter.
Design for Six Sigma
<=
Ans!er -. '+mmar( statistics an te istogram from spreaseet "rob0/*-.9ls so! Column 1 #ean $tan!a%! E%%o% #e!ian #o!e $tan!a%! &e'iation $a(le Va%iance *u%tosis $+e,ness -ange #ini(u( #axi(u( Confi!ence Le'el95.0"/
24.0014 0.00097 24.001 24.000 0.00967 9.4E)05 0.53132 0.05271 0.058 23.971 24.029 0.00192
Bin 23.971 23.977 23.983 23.988 23.994 24.000 24.006 24.012 24.017 24.023 #o%e
Frequency 1 0 0 7 14 26 20 19 7 5 1
,istogra' 30 25 y c n e u / e r .
20 %euenc
15 10 5 0 1 7 9 . 3 2
7 7 9 . 3 2
3 8 9 . 3 2
8 8 9 . 3 2
4 9 9 . 3 2
0 0 0 . 4 2
-in
6 0 0 . 4 2
2 1 0 . 4 2
7 1 0 . 4 2
3 2 0 . 4 2
e % o #
Design for Six Sigma
<>
or sample statistics of$
x
8 =.??-=6 σ 8 ?.??J7
Specification limits for the process are$ 9.J7 Z µ Z =.?9
' 8 =.?9?? & =.??-= 8 .J@ ?.??J7
' 8 9.J7?? & =.??-= 8 &9.= ?.??J7
0( ' [ .J=) 8 (?.@ & ?.=JI=) 8 ?.??-G that items will exceed upper limit
0( ' Z &9.=) 8 ?.?? that items will exceed lower limit
Therefore, the percent outside is$ ?.??-G, or ?.-G C p 8 FT & T 8 =.?9? & 9.J7? 8 -.?9Gσ G (?.??J7) C pu 8 FT & x 9σ
C pl 8
x
8 =.?9? & =.??-= 8 ?.JI9 9 (?.??J7)
& T 8 =.??-= & 9.J7? 8 -.?7J 9σ 9 (?.??J7)
The process capability indexes are slightly out of tolerance for the upper index, and within minimum limits for the lower and overall index. These results indicate that the process may be minimally ade"uate if it can be centered on the nominal dimension of =. owever, the ideal situation would be to launch process improvement studies so that the capability indexes could be at least doubled.
9G.
Samples for three parts made at ;iver City 0arts Co. were ta!en as shown in the C*+Data file for "rob.*+0 on the 0remium website for this chapter. ata set - is for part -, data set is for part , and data set 9 is for part 9. a. Calculate the mean and standard deviations for each part and compare them to the following specification limits$
0art
Bominal
Tolerance
9
-.7@? .??? -.@?
W ?.?=@ W ?.?G? W ?.?9?
Design for Six Sigma
b. Lill the production process permit an acceptable fit of all parts into a slot with a specification of @ W ?.?I- at least JJ.79 percent of the timeK
Ans!er 9G.
a) Sample statistics as shown in spreadsheet 0rob.-&9G.xls are$ ata set -$
x
8 -.7==G6 s 8 ?.?-G96 9s 8 ?.?=IJ
ata set $
x
8 -.JJJJ6 s 8 ?.??7I6 9s 8 ?.?9=
ata set 9$
x
8 -.=I@6 s 8 ?.??@6 9s 8 ?.?-@G
0art - will not consistently meet the tolerance limit since its W 9s value is greater than the tolerance limit. 0arts and 9 are well within their tolerance limits, since their W 9s values are smaller than the stated tolerances. b)
x T
8 =.JJ9? 6 1stimated
σ 0rocess 8
. . . s- Y s. Y s9 8
?.?-G9 Y ?.??7I Y ?.??@ 8 ?.?-II
0rocess limits$ =.JJ9? W 9(?.?-II) or =.J9GG to @.?=J= vs. specification limits of =.J-J to @.?I- for a confidence level of ?.JJ79. The parts will fit within their combined specification limit with a ?.JJ79 confidence level. 97.
Amega Tecnology td. (AT) is a small manufacturing company that produces various parts for tool manufacturers. Ane of AT2s production processes involves producing a Teflon\ spacer plate that has a tolerance of ?.?@ to ?.-?? cm in thic!ness. An the recommendation of the "uality assurance (/+) department and over obections of the plant manager, AT ust purchased some new e"uipment to ma!e these parts. ;ecently, the production manager was receiving complaints from customers about high levels of nonconforming parts. e suspected the new e"uipment, but neither /+ nor plant management would listen. The manager discussed the issue with one of his production supervisors who mentioned that she had ust collected some process data for a study that the "uality assurance department was underta!ing. The manager decided that he would prove his point by showing that the new e"uipment was not capable of meeting the specifications. The data
Design for Six Sigma
<6
provided by the supervisor are shown in the C*+Data file for 0roblem -&97 on the 0remium website for this chapter. 0erform a process capability study on these data and interpret your results.
Ans!er 97.
Amega Technology td.2s process capability results from the 1xcel spreadsheet software are shown below. (See spreadsheet 0rob-&97.xls for details.) Aerage 0.0764 0tandard deiation 0.0104
Cp Cpl
0.8019 0.8468
Cpu Cp1
0.7569 0.7569
Tese ata so! tat te process as a rater lo! overall capabilit(; !it Cp @ 3.5302 an a total of 0.40% of te val+es falling o+tsie of te speciPcation limits of 3.3- * 3.03 "rocess statistics $ @ te part
x
@ 3.341,; σ @ 3.303,
3.03 * 3.341, @ /./4
"8 $ /./4: @ 8.-* 3.,55,: @ 3.3001 tat
3.303,
$ @ 3.3- * 3.341, @ */.-, tat te part 3.303, limit
!ill e9cee +pper limit
"8 $ */.-,: @ 8.-* 3.,2,-: @ 3.33-!ill e9cee lo!er
Terefore; te percent o+tsie is 3.3040; or 0.40 % 9I. Suppose that a refrigeration process at Coolfoods, td. as a normally distributed output with a mean of @.? and a variance of -.==. a. f the specifications are @.? W 9.@, compute C p, C p! , and C pm. s the process capable and centeredK b. Suppose the mean shifts to 9.? but the variance remains unchanged. ;ecompute and interpret these process capability indexes. c. f the variance can be reduced to =? percent of its original value, how do the process capability indices change (using the original mean of @.?)K
Design for Six Sigma
<7
Ans!er 5.
8a:
x
Cp @
@ /-.3< σ @ 0./ UTL * LTL @ /5./- S /0.4- @ 3.23 1 σ 1 80./:
Cm @ Cp
- + P( mean − target ) % σ Q
@ 3.23
- + P(@.? − @.?)
% -. Q
@
3.23 Cp+ @ UTL * σ Cpl @
x
x
@ /5./- * /-.3 @ 3. 23 80./:
* LTL @ /-.3 * /0.4- @ 3.23 σ 80./:
Concl+sion Te process is centere on te mean; b+t it oes not ave ae7+ate capabilit( at tis time. Cp6@ min 8Cpl ; Cp+ : @ 3.23 8b:
Cp @ cange.
x
@ /< σ @ 0./
UTL *LTL @ 1 σ
CmoiPe @ C p
/5./- S /0.4- @ 3.23
Tis res+lt as not
1 80./: - + P(mean − target ) % σ Q
@ 3.23
- + P(9.? − @.?) % -. Q
@
3.-5, eca+se of te sift a!a( from te target; capabilit( is lo!er. Cp+ @ UTL * σ Cpl @
x
x
* LTL @
@
/5./- * /.3 80./: /.3 * /0.4-
@ 0.,-5
@ 3.,4 < Cp6@ min 8C pl ; Cp+ : @
3.,4 σ
80./:
Concl+sion Te process is s6e!e an still oes not ave ae7+ate capabilit( at tis time. 8c:
σ /ne! @ 3., 80.,,: @ 3.-41
∴ σne! @ 3.4-2
Design for Six Sigma
Cp @
UTL *LTL 1 σ
CmoiPe @ Cp
=9
@
/5./- S /0.41 83.4-2:
- + P(mean − target ) % σ Q
@ 0.,/4
@ 0.,/4
- + P(@.? − @.?)
% ?.7@J Q
@
0.,/4 If tere is no sift a!a( from te target; capabilit( is e7+al to C p. Cp+ @ /5./- * /-.3 83.4-2:
@ 0.,/4
Cpl @ /-.3 * /0.4- @ 0.,/4 83.4-2: Cp6@ min 8Cpl ; Cp+ : @ 0.,/4 Re+cing te variance brings te C pl an Cp+ to te point of ae7+ac(; provie te process can remain centere.
9J.
+ process has upper and lower tolerance limits of @.I? and @.??, respectively. f the customer re"uires a demonstrated C p of .?, what must the standard deviation beK f both C pu and C pl must also be .?, determine the process mean, using that calculated standard deviation, assuming a normal distribution of output.
Ans!er 2.
Cp @ UTL *LTL @ /.3 @ 1σ
Cp+ @ UTL *
x
σ
Cpl @
x
* LTL
σ =?.
@
-.53 * -.33 @ 3.5 < Terefore; σ @ 3.3114 1σ 1σ
-.53 *
x
@
/.3< Terefore; !e get
@ -.,
σ
@
x
* -.33 @ /.3< Terefore; !e get
σ
Clearly demonstrate that Six Sigma re"uires C p 8 .? and C pk 8 -.@.
Ans!er
x
x
@ -.,
Design for Six Sigma
=8
=?.
+s explained in Ch. --$ The easiest way to understand this is to thin! of the distance from the target to the upper or lower specification (half the tolerance), measured in terms of standard deviations of the inherent variation, as the sigma level. + !&sigma "uality level satisfies the e"uation$ ! H process standard deviation 8 tolerance % Bote that in igure --.-, if the design specification limits were only = standard deviations away from the target, the tails of the shifted distributions begin to exceed the specification limits by a significant amount. Table --.- shows the number of defects per million for different sigma "uality levels and different amounts of off¢ering. Bote that a "uality level of 9.= defects per million can be achieved in several ways, for instance6
•
with ?.@&sigma off¢ering and @&sigma "uality
•
with -.?&sigma off¢ering and @.@&sigma "uality
•
with -.@&sigma off¢ering and G&sigma "uality
n many cases, controlling the process to the target is less expensive than reducing the process variability. This table can help assess these trade&offs. The sigma level can easily be calculated on an 1xcel spreadsheet using the formula$ 8BA;*SBN(-&Bumber of efects%Bumber of Apportunities) Y ST
Design for Six Sigma
=:
or e"uivalently, 8BA;*SBN(-&dpmo%-,???,???) Y ST ST refers to the off¢ering as used in Table --.-. Fsing the airline example discussed earlier, if we had 9 lost bags for I???(-.G) 8 -,I?? opportunities, we would find 8BA;*SBN(-&9%-I??) Y -.@ 8 =.JJII or about a @&sigma level.
Using ata from problem ,3; above; !e can so! tat Cp @ UTL *LTL @ /.3 @ 1σ
Cp+ @ UTL *
x
σ Cpl @
x
* LTL
σ
@
-.53 * -.33 @ 3.5 < Terefore; σ @ 3.3114 1σ 1σ
-.53 *
x
@
/.3< Terefore; !e get
x
@ -.,
σ @
x
* -.33 @ /.3< Terefore; !e get
x
@ -.,
σ
Tis meets te re7+irement tat ! H process standard deviation 8 tolerance %
1 E .3114 @ -.53 * -.33 @ 3.,33 / #it a mean sift of 0.- σ CmoiPe @ Cp @ 0.,0,
- + P(mean − target ) % σ Q
@ /.3
- + P(@.= − @.=GG7)
eca+se of te sift a!a( from te target; capabilit( is lo!er
S$00EST#!NS (!R PR!ECTS ETC3 0.
% .?GG7 Q
C+stomer attrib+tes an tecnical re7+irements migt be Attrib+tes Tecnical Re7+irements a. oo6 p+rcase ?o+rs 'ce+le of open o+rs rgani$ation ( ept.co+rseprofessor "re*processing availabilit( Reservations on Internet )ase of pa(ment Cas; cec6; creit car
Design for Six Sigma
Time >al+e )mpat( b. Registration Convenience 'pee Costs Acc+rac( )mpat( c. ?otel room * b+siness Convenience metos 'pee * cec6 ino+t 6no!lege Tecnolog( Amenities e9ercise facilities Costs Acc+rac( . ?otel room * famil( Convenience moerate
=;
'pee of cec6o+t Lo!est available price Unerstaning!illingness of personnel to solve problems Time; ates; Internet; pone "rocess stanars Fees )rror prevention Unerstaning!illingness of personnel to solve problems +siness location ; ates; "rocess stanars; s(stem FAY; Internet connection Resta+rant; in*room !or6 areas; Internet connections; Fees * relate to services )rror prevention
Location near recreation;
'pee * cec6*in
ining facilities; ates; metos "rocess stanars; s(stem
Amenities Costs Acc+rac(
"la(; famil(*relate facilities Fees * relate to famil( b+get )rror prevention
6no!lege
Construction of the matrix is left to the student.
/.
C+stomer re7+irements !o+l li6el( incl+e fresness; taste; consistenc(; appearance of te pro+ct< 6no!lege; attentiveness; frienliness of c+stomer service personnel< spee an acc+rac( of te coo6s an orer Pllers< acc+rac( an frienliness of te co+nter personnel. Tecnical re7+irements migt be e9plore to etermine !at !o+l be re7+ire to eliver te pro+ct to in*o+se vers+s eliver( c+stomers. Te former !o+l re7+ire !ait staK training in c+stomer service tecni7+es; !ile te latter !o+l re7+ire 6no!legeable rivers s+bstit+ting 8in some !a(s: for !ait staK. Tecnolog( for eliver orers !o+l involve e7+ipment to receive orers via FAY macines or over te Internet. Regaring coo6s an orer Pllers 8common to bot in*o+se an e9ternal pro+ct eliver(:; o! m+c of te assembl( of te pi$$a
Design for Six Sigma
=<
so+l be one b( an 8vers+s macine*mae or macine assiste:V Tis is merel( s+ggestive of te t(pes of 7+estions concerning c+stomer an tecnical re7+irements tat st+ents so+l consier. Constr+ction of te ?o+se of Q+alit( is left to te st+ent. .
For a glier te follo!ing re7+irements migt be Attrib+tes )ase of assembl( instr+ctions )as( to [( Fligt caracteristics D+rabilit( >al+e
=.
c+stomer
attrib+tes
an
tecnical
Tecnical Re7+irements ZDesign for assembl(Z<
'imple
ZLa+ncZ mecanism #ing; tail; bo( esign Q+alit( of !oo "rice+rabilit( ratio
The best way to prioriti'e the voice of the customer would be to have a focus group of typical customers, such as craftspeople, Odo&it&yourselferMsO, hobbyists to provide input on how they used the screwdriver and their priorities. Delow is a possible configuration of the matrix, with priorities for a 3serious4 craftsperson. Such a person would loo! for "uality and functionality over price or 3extra4 features, such as ratchets or interchangeable bits.
"/0S% /F 0A123 MA2& F/ A S2M41% SC%-D2V%
0rice
nterchg Dits
Steel Shaft
1asy to use
;atchet Capabil.
urable
Comfortable
0lastic andle
oes not rust
Nersatile
;ubber Erip
Design for Six Sigma
nexpensive 0riority
==
9
-
G
@
=
• @ >er( strong relationsip @
'trong relationsip @ #ea6 relationsip -.
Ans!ers !ill var(; epening on te service processes cosen b( te st+ents. Te case on Bappl(ing QFD to a Universit( '+pport '(stem migt be +se as a starting point to etermine !at t(pes of information to loo6 for to complete tis proOect.
1.
Tis e9ercise !ill give st+ents an appreciation of te callenges of esigning an calibrating meas+rement s(stems.
4. Ans!ers !ill var(; epening on te inivi+al !ebsites an topics cosen b( te st+ents. 5.
Tis e9ercise is esigne to f+rter st+ents a!areness of te breat of te Z7+alit( movementZ an elp tem conPrm o! an !eter te teor( of 7+alit( is being applie in practical settings in b+siness an in+str(. =eas+rement is commonl( +se in man+fact+ring an services; b+t it varies !iel( in precision an acc+rac( concepts; epening on te si$e of te Prm an te in+str(.
A'S-%S / CAS% 0%S2/'S
Case 4 A,,lying Quality (unction De,loy+ent to a $ni-ersity Su,,ort Ser-ice -.
The answer to the "uestion of whether students agree or disagree with the relative importance ran!ings obtained from the study of the ;;C at Tennessee Tech ultimately depends on students2 opinions. owever, a strong case might be made that the relative importance score would depend on the situation. or small, rush, duplicating obs, prompt service would seem to be of greatest importance. or research obs where specific information has to be found, !nowledge and courtesy of the employees would be highly desirable, as well as accuracy, which might be a close second in importance. or the inexperienced user, such as a freshman student, empathy and willingness to help would possibly be ran!ed as the two highest criteria.
.
Concentrating on the top four characteristics, the following weighted scores can be calculated$ ;esources (personnel) Customer handling nformation handling
-9@ GI I7
Design for Six Sigma
+ttitudes and morale
=>
GI
The three areas on which the analysts focused were document handling, training, and layout. The above weighted scores would seem to lend little support to the need to deploy a new document handling process (=@ point score), nor to improve the layout (G point score), which have very little impact on customer "uality criteria. Training may be re"uired, but the focus on document handling would seem to be unnecessary. 9.
Eiven the high ran!ing of resources (personnel), it appears that more attention should be paid to selection and retention issues. nformation handling, in second place, also has a maor impact, with customer handling, and attitudes and morale tied for third place. These categories could be improved by training and by process analysis to determine if the best processes were being used. +s a result, it could be predicted that morale and customer satisfaction would li!ely increase.
Case * lacB %lB Medical Center
-.
The next steps would include gathering data using the chec!list form that the committee designed. The committee might also want to develop process flow charts, while waiting for the fall data to be gathered and analy'ed.
.
The data from the chec!list should be put into a format, perhaps in a spreadsheet, where it could be analy'ed. +nalysis tools might include 0areto analysis and histograms. Segmentation should also be used to find out the incidence of falls in li!ely locations.
9.
mproved processes and systems should be developed based on analysis of the chec!list and process charting. The 3significant few4 causes, based on the 0areto analysis, should be addressed first, in order to achieve the greatest immediate impact. +nalysis of the process flow chart could reveal places where processes could be simplified, and might also identify conditions that would contribute to patient falls, so they could be eliminated.
onus Materials Case * "!draulic 1ift Compan!
-.
The !ey to the calculation of an estimated process capability for this case is to calculate an estimated standard deviation for each condition. Fsing the simplifying assumption that the sample standard deviation is a good approximation of the population standard deviation will allow us to ma!e a reasonable estimate, even though for the cases of the small sample si'es of 9? or 9@ that assumption would be open to argument by statisticians. Le will concentrate on the calculation of C p for only case (a) and (e), since it is obvious that the capability became drastically worse during the experimental stages from (b) to (d). ;eading the data from the histograms, we can use the calculation of the sample standard
Design for Six Sigma
=?
deviation with grouped data from the chapter. The fre"uency histogram for condition (a) shows$ mp, Eroup x re"uency fx fx ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] =@ 9 -9@ G?7@ @? G 9?? -@??? 9 @@ ? ? ? = G? -G JG? @7G?? @ G@ = G? -GJ?? G 7? -@=? -?7I?? 7 7@ G =@? 997@? I I? 9 -I=? -=7?? J I@ @ =@ 9G-@ -? J? -? J?? I-??? -J@ ? ? ? - -?? @ @?? @???? 79-? @@-=@?
Σfx x
s8
8
n
79-? 8 -?? 8 79.-
P Σfx % ( n − -)
− P( Σ( fx)
% n) % (n & -)Q
P@@-=@? % JJQ − P( 79-?) % -?? % JJQ
8 8 -9.-9I
∴ c p 8 FT − T 8 -?? & @? 8 ?.G96 not acceptable G σ G(-9.-9I) The fre"uency histogram for condition (e) shows$ mp, Eroup x re"uency fx fx ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] G? -? 7?? G@ ? ? ? 9 7? - I=? @II?? = 7@ 7 @@ 9J97@ @ I? -II? 7?=?? G I@ 9 @@ -G7@ G? -J7=@?
Σfx
G?
Design for Six Sigma x
8
s8
n
=6
8 9@ 8 7=.I@7
P Σfx % ( n − -)
− P( Σ( fx)
% n) % (n & -)Q
P-J7=@? % 9=Q − P( G?) % 9@ % 9=Q
8 8 G.=-
∴ c p 8 FT − T 8 -?? & @? 8 -.9=6 not outstanding, but much better Gσ G (G.=-) .
The process used here was obviously a systematic process of problem solving similar to the one suggested in this chapter. The first step was a) to understand the Omess.O + 0areto&li!e approach found that @? of the defective items were due to dimensional problems on one diameter of the valve stem. Bext, b) find facts on the process capability6 c) specific problems were identified$ over adustment by the operator, and lac! of ability of the machine to hold tolerances6 d) ideas on machine adustments and improvements were generated6 e) solutions were implemented, with the machine being adusted and later overhauled6 f) as eming said, Oo it over, again and again.O PStep 7, added to the process, is continuous improvementQ.
Case * loomfield ool Co.
-.
See spreadsheet -blomrrcase.xls for details. Bote that there are some rounding errors below that ma!e answers on the spreadsheet appear slightly different. Calculations for the repeatability and reproducibility (;R;) study are as follows$ x - 8 (∑∑*i! ) %nr 8 .997I % 9? 8 ?.?77J 8 (∑; i) % n 8 ?.-?9?% -@ 8 ?.??GJ
; -
Fse this method to calculate values for operator x 8 .J7= % 9? 8 ?.?7GG6
8 ?.?J=@% -@ 8 ?.??G9
;
x 8 max ^ x i_ & min ^ x i_ 8 ?.?77J & ?.?7G@ 8 ?.??-= ; 8 (∑ ; i ) % m 8 (?.??GJ Y ?.??G9) % 8 ?.??GG = 8 9.G76 FC; 8 =
;
8 (9.G7) (?.??GG) 8 ?.?-@, all ranges below
# - 8 =.@G 6 # 8 9.G@ (from Table --. in the text) 1N 8 # - ; 8 (=.@G) (?.??GG) 8 ?.?9?AN 8
(# . x ) . & (1N . % nr) 8 P(9.G@)(?.??-9)Q & P( ?.?9?-) % 9?Q 8
Design for Six Sigma
=7
& ?.?????77
BAT1$ +ccording to the Measurement 3ystems Analysis (eference Manual published by the +utomotive ndustry +ction Eroup (++E), Troy *, -JJ?, p. ==$ O... if a negative value is calculated under the s"uare root sign, the PANQ defaults to 'ero (?).O ;; 8
(1N)
Y (AN)
8
(?.?9?-) Y (?)
8 ?.?9?-
Therefore, in this case, ;; 8 1N 1"uipment variation (related to tolerance) 8 -?? (?.?9?- % ?.?@) Aperator variation (related to tolerance) ; R ; variation (related to tolerance)
8 G?. 8 ? 8 G?.
∴ Concentrate on reducing e"uipment variation, not operator variation. Total Nariation$
TN8
(;;) Y (0N)
, where 0N is 0art Nariation
0N 8 ; p # 9 ; p 8 ;ange of part averages for the entire sample$ ?.-?-9 to ?.?@-G 8 ?.?=J7 # 9 8 -.=@ from Table --.@ 0N 8 (?.?=J7) (-.=@) 8 ?.?7Thus TN 8
(;;) Y (0N)
8
(?.?9?-) Y (?.?7-)
8 ?.?7I-
AN 8 ? 1N 8 ;;, related to TN 8 -?? (?.?9?- % ?.?7I-) 8 9I.@ 0N, related to TN 8 -?? (?.?7- % ?.?7I-) 8 J.9 BAT1$ The sum of the above percentages will not add to -??. Dased on the OrulesO for process capability given in the text, it can be assumed that the e"uipment and the process need to be improved, since none of the percentages fall below the 9? or -? minimums. The operators are consistent in their measurements, so their methods are not in "uestion at this point. Lorn or faulty gauges should be discarded and the rest should be calibrated.
Case 4 The P#"!T #nitiati-e 5 Part ##
