Seminar on Basics of Taguchi Methods

Genichi Taguchi TAGUCHI’S METHODS

INTRODUCTION Professor Genichi Taguchi was the director of the Japanese Academy of quality and four times receipt of the Deming Prize.

He starts where SPC (temporarily) finishes. He can help with the identification of common cause of variation, the most difficult to determine and eliminate in process.

He attempts to go even further: he tries to make the process and the product robust against their effect (eliminate of the effect rather then the cause) at the design stage.

Even if the removal of the effect is impossible, he provides a systematic procedure for controlling the noise (through tolerance design) at the minimum cost.

When Dr. Taguchi was first brought his ideas to America in 1980, he was already well known in Japan for his contribution to quality engineering.Traditional and Taguchi’s Defination of Quality

Seminar on Basics of Taguchi Methods

Objective of Taguchi MethodsMinimize the variation in product response while keeping the mean response on target.The product can be made robust to changes in operating and environmental conditions. Since the method is applied in a systematic way at a pre-production stage (off-line), it can greatly reduce the number of time-consuming tests, thus saving in costs and wasted products.

Taguchi Loss Function : Definition “Quality is the loss a product causes to society after being shipped, other then any losses caused by its intrinsic functions.” By “loss” Taguchi refers to the following two categories:Loss caused by variability of function.

Loss caused by harmful side effects.An example of loss caused by variability if function would be an automobile that does not start in cold weather. The car’s owner would suffer a loss if he or she had to pay some to start a car. The car owner’s employer losses the services of the employee who is now late for work.

Comparing the Quality Levels of SONY TV Sets Made in JAPAN and in SAN DIEGOThe front page of the Ashi News on April 17,1979 compared the quality levels of Sony color TV sets made in Japanese plants and those made in San Diego, California, plant. The quality characteristic used to compare these sets was the color density distribution, which affect color balance. Although all the color TV sets had the same design, most American customers thought that the color TV sets made in San Diego plant were of lower quality than those made in Japan.

Distribution of the Quality characteristicColour density of the TV sets from Japanese Sony plants are normally disturbed around the target value m, and S.D is 10/6.

Color density of San Diego TV sets is uniformly distributed rather than normally distributed. Therefore, the S.D of these uniformly distributed objective characteristics is about 10/√12.Process Capability study The process capability index(Cp) is defined as the tolerance specification divided by 6 times the standard deviation of the objective characteristic: Cp=Tolerance/6*Standard deviationThe Cp of the of Japanese Sony TV sets is about 1. In addition, the mean value of the distribution of these objective characteristics is very close to the target value of m.

The process Cp of the San Diego Sony plant is calculated as follows: Cp=Tolerance/6(Tolerance/√12) = 0.577 It is obvious that the process capability index of San Diego Sony is much lower than that of Japanese Sony.

Tolerance specifications are very similar to tests in schools, where 60% is usually the dividing line between passing and failing, and 100% is ideal score.the grades between 60% and 100% in evaluating quality can be classified as follows: 60%-69% Passing (D) 70%-79% Fair (C) 80%-89% Good (B) 90%-100% Excellent (A)We can apply this scheme to the classification of the objective characteristics (color density) of these color TV sets as shown in Figure. One can see that a very high percentage of Japanese Sony TV sets are within grade B, and a very low percentage are within or below grade D. In comparison, the color TV sets from San Diego SONY have about the same percentage in grades A, B and C.

To reduce the difference in process capability indices between Japanese SONY and San Diego SONY, (and thus seemingly increase the quality level of the San Diego sets) the letter tried to tighten the tolerance specification to extend only to grade C rather than grade D. Therefore, Only the products within grades A,B and C were treated as passing.But this approach is faulty, Tightening the tolerance specifications because of a low process capability in a production plant is meaningless as increasing the passing score of school tests from 60% to 70% just because students do not learn well. On the contrary, such a school should consider asking the teachers to lower the passing score for student who do not test as well instead of rating it.Now we section will discuess how to evaluate the functional quality of products meaningfully and correctly.

Taguchi’s Quadratic Quality Loss FunctionQuality Loss Occurs when a product’s deviates from target or nominal value.

Deviation Grows, then Loss increases.

Taguchi’s U-shaped loss Function Curve.Taguchi’s U-shaped loss Function CurveTaguchi loss FnScrap or Rework Cost. LossMeasuredcharacteristicLTLNominalUTL

Taguchi’s method: Loss function..Loss = L(y) = L( m + (y-m)) = L(m) + (y-m) L’(m)/ 1! + (y – m)2 L”(m)/ 2! + …Ideally:(a) L(m) = 0 [if actual size = target size, Loss = 0], and(b) When y = m, the loss is at its minimum, therefore L'(m) = 0Taguchi’s Approximation: L(y) ≈ k( y – m)2

Formula to find Taguchi’s Loss FnTaguchi uses Quadratic Equation to determine loss CurveL (x) = k (x-N)² Where L (x) = Loss Function,k = C/d² = Constant of proportionality, where C – Loss associated with sp limitd - Deviation of specification from target valuex = Quality Features of selected product,N = Nominal Value of the product and (x-N)= Tolerance

In the case of the SONY colour TV sets, let the adjustment cost be A= 300 Rs, when the colour density is out of the tolerance specifications. Therefore, the value of k can be calculated by the following equation: k = 300/5² = 12 RsTherefore, the loss function is L(y) = 12(y – m)²This equation is still valid even when only one unit of product is made.The mean square deviation of objective characteristics from their target values can be applied to estimate the mean value of quality loss Equation. One can calculate the mean square deviation from target σ² (σ² in this equation is not variance) by the following equation (the term σ² is also called the mean square erroror the variationof products): σ² =mean value of (y-m)² So L(y) = k(y-m)² becomes L = kσ²

PlantLocationMean Value of Objective FunctionQuality level of Sony TV setStandard DeviationVariationLoss L(in Rs)Defective RatioWe can evaluate the differences of average quality levels between the TV sets from Japanese Sony and those from San Diego Sony.

it is clear that although the defective ratio of the Japanese Sony is higher than that of the San Diego Sony, the quality level of the former is 3 times higher than the latter.If Tighten the toleranceAssume that one can tighten the tolerance specifications of the TV sets of San Diego Sony to be m ± 10/3. Also assume that these TV sets remain uniformly distributed after the tolerance specifications are tightened. The average quality level of San Diego Sony TV sets would be improved to the following quality level: L = 12[(1/ √12) (10) (2/3)] ² = 45Rs

Where the value of the loss function is considered the relative quality level of the product. This average quality level of the TV sets of San Diego Sony is 56Rs lower than the original quality level but still 11Rs higher than that of Japanese Sony TV sets. In addition, in this type of quality improvement, one must adjust the products that are between the two tolerance limits,m± 10/3 and m ± 5, to be within m ± 10/3. In the uniform distribution shown in Figure, 33.3% would need adjustment, which would cost 300Rs per unit. Therefore each TV set from San Diego Sony would cost an additional 100Rs on average for the adjustment: (300)(0.333) = 100RsConsequently, it is not really a good idea to spend 100Rs more to adjust each product, which is worth only 56Rs.A better way is to apply quality management methods to improve the quality level of products.

Example of loss functionSuppose that the specification on a part is 0.500 ± 0.020 cm. A detailed analysis of product returns and repairs has discovered that many failures occur when the actual dimension is near the extreme of the tolerance range (that is, when the dimensions are approximately 0.48 or 0.52) and costs $50 for repair. Thus, in Equation, the deviation from the target, x – T , is 0.02 and L(x) = $50. Substituting these values, we have: 50 = k(0.02)2 or k = 50/0.0004 = 125,000Therefore, the loss function for a single part is L(x) = 125000(x – T)2.

This means when the deviation is 0.10, the firm can still expect an average loss per unit of: L(0.10) = 125,000(0.10)2 = $12.50 per partKnowing the Taguchi loss function helps designers to determine appropriate tolerances economically. For example, suppose that a simple adjustment can be made at the factory for only $2 to get this dimension very close to the target. If we set L(x) = $2 and solve for x – T, we get: 2 = 125000(x – T)2 x – T = 0.004Therefore, if the dimension is more than 0.004 away from the target, it is more economical to adjust it at the factory and the specifications should be set as 0.500 ± 0.004.

Variation of the Quadratic Loss FunctionNominal the besttype: Whenever the quality characteristic y has a finite target value, usually nonzero, and the quality loss is symmetric on the either side of the target, such quality characteristic called nominal-the-best type. This is given by equation L(y) =k(y-m)² Example: Color density of a television set and the out put voltage of a power supply circuit.

2)Smaller-the-better type: Some characteristic, such as radiation leakage from a microwave oven, can never take negative values. Also, their ideal value is equal to zero, and as their value increases, the performance becomes progressively worse. Such characteristic are called smaller-the-better type quality characteristics. Examples: The response time of a computer, leakage current in electronic circuits, and pollution from an automobile.In this case m = 0 L(y) =ky² This is one side loss function because y cannot take negative values.

3)Larger-the-better type: Some characteristics do not take negative values. But, zero is there worst value, and as their value becomes larger, the performance becomes progressively better-that is, the quality loss becomes progressively smaller. ,, also Their ideal value is infinity and at that point the loss is zero. Such characteristics are called larger-the-better type characteristics.Example: Such as the bond strength of adhesives.Thus we approximate the loss function for a larger-the-better type characteristic by substituting 1/y for y in L(y) = k [1/y²]

4)Asymmetric loss function: In certain situations, deviation of the quality characteristic in one direction is much more harmful than in the other direction. In such cases, one can use a different coefficient k for the two directions. Thus, the quality loss would be approximated by the following asymmetric loss function: k(y-m) ²,y>m L(y) = k(y-m) ², y≤m

Introduction to Analysis of variation(ANOVA)What is ANOVA ANOVA is a statistically based decision tool for detecting any differences in average performance of groups of items tested. ANOVA is a mathematical technique which breaks total variation down into accountable sources; total variation is decomposed into its appropriate components.Degrees of Freedom (dof)Degree of freedom are the number of observations that can be varied independently of each other.

Two –Way ANOVAThere are two controlled parameters in this experimental situation Let us consider an experimental situation. A student worked at an aluminum casting foundry which manufactured pistons for reciprocating engines. The problem with the process was how to attain the proper hardness (Rockwell B) of the casting for a particular product. Engineers were interested in the effect of copper and magnesium content on casting hardness.According to specifications the copper content could be 3.5 to 4.5% and the magnesium content could be 1.2 to 1.8%.

Two –Way ANOVA continue…The student runs an experiment to evaluate these factors and conditions simultaneously. If A = % Copper Content = 3.5 = 4.5If B = % Magnesium Content = 1.2 = 1.8 The experimental conditions for a two level factors is given by = 4 which are Imagine, four different mixes of metal constituents are prepared, casting poured and hardness measured. Two samples are measured from each mix for hardness. The result will look like:

Two –Way ANOVA continue…To simplify discussion 70 points from each value are subtracted in the above observations from each of the four mixes. Transformed results can be shown as:Two way ANOVA calculations: The variation may be decomposed into more components:Variation due to factor AVariation due to factor BVariation due to interaction of factors A and BVariation due to error

Two –Way ANOVA continue…An equation for total variation can be written asA x B represents the interaction of factor A and B. The interaction is the mutual effect of Cu and Mg in affecting hardness. Some preliminary calculations will speed up ANOVAGrand Total =29 =26 =21 =34 T = 55, N = 8

Two –Way ANOVA continue…Total Variation = 6² + 8² + 3² + ----------- + 10² - = 40.85Variation due to factor A = 1.125=Mathematical check : Numerator 29 + 26 = 55 and Denominator 4 + 4 = 8

Two –Way ANOVA continue…For a two level experiment, when the sample sizes are equal, the equation above can be simplified to this special formula: = = 1.125Similarly the variation due to factor BTo calculate the variation due to interaction of factors A and BLet represent the sum of data under the ith condition of the combination of factor A and B. Also let c represent the number of possible combinations of the interacting factors and the number of data under this condition.

Two –Way ANOVA continue…Note that when the various combinations are summed, squared, and divided by the number of data points for that combination, the subsequent value also includes the factor main effects which must be subtracted. While using this formula, all lower order interactions and factor effects are to be subtracted. For the example problem: = 14, = 15, = 7 = 19And no. of possible combinations c = 4And since there are two observations under each combination = 2

Two –Way ANOVA continue…= 15.125SinceDegrees of Freedom (Dof) – Two way ANOVA= N – 1 = 8 – 1 = 7==- 1 = 1- 1 = 1

Two –Way ANOVA continue…ANOVA summary Table (Two-way)* at 95% confidence** at 90% confidenceThe ANOVA results indicate that Cu by itself has no effect on the resultant hardness, magnesium has a large effect (largest SS) on hardness and the interaction of Cu and Mg plays a substantial part in determining hardness.

Two –Way ANOVA continue…Geometrically, there is some information available from the graph that may be useful. The relative magnitudes of the various effects can be seen graphically. The B effect is the largest, A x B effect next largest and the A effect is very small. See

Hard-ness10B2B effectA effect86Mid pt. A2B1 & A2B2A x B effectMid pt. on line B1B142A2A1Two –Way ANOVA continue…

Introduction To Orthogonal ArrayEngineers and Scientists are often faced with two product or process improvement situations. 1st development situation is to find a parameter that will improve some performance characteristic to an acceptable and optimum value. This is the most typical situation in most organizations.

2nd situation is to find a less expensive, alternate design, material, or method which will provide equivalent performance Orthogonal ArrayWHAT IS ARRAYAn Array’s name indicates the number of rows and column it has , and also the number of levels in each of the column. Thus the array L4 has four rows and three 2-level column.WHAT IS ORTHOGONALITYOne main requirement of orthogonality is a balanced experiment which means equal number of samples under various test conditions.

Most common test plan is to evaluate the effect of one parameter on product performance. This is what is typically called as one factor experiment.This experiment evaluates the effect of one parameter while holding everything else constant. The simplest case of testing the effect of one parameter on performance would be to run a test at two different conditions of that parameter.

Several Factors One at a Time(OFAT)1-LOW LEVEL2-HIGH LEVEL

OFATIsolate what is believed to be the most important factor

Investigate this factor by itself, ignoring all others

Make recommendations on changes to this crucial factor

Move on to the next factor and repeatExampleA process producing steel springs is generating considerable scrap due to cracking after heat treatment. A study is planned to determine better operating conditions to reduce the cracking problem. There are several ways to measure crackingSize of the crackPresence or absence of cracks The response selected was Y: the percentage without cracks in a batch of 100 springs Three major factors were believed to affect the responseT: Steel temperature before quenchingC: carbon content (percent)O: Oil quenching temperature

Traditional ApproachFour runs at each level of T with C and O at their low levels Carrying out similar OFAT experiments for C and O would require a total of 24 observations and provide limited knowledge.Conclusion: Increased T reduces cracks by 10%

Factorial ApproachInclude all factors in a balanced design:

To increase the generality of the conclusions, use a design that involves all eight combinations of the three factors.The above eight runs constitute a FULL FACTORIAL DESIGN.The design is balanced for every factor. This means 4 runs have T at 1450 and 4 have T at 1600. Same is true for C and O.

Advantages Of Factorial approach over OFATThe effect of each factor can be assessed by comparing the responses from the appropriate sets of four runs.More general conclusions8 runs rather than 24 runs

A Full Factorial Experiment If a full factorial is to be used in this situation will have to be conducted. (As shown in figure below)

This is 1/16th FFE which has only 8 of the possible 128 combinations represented

A Case Study Of Orthogonal ArrayDuring the late 1980s, Modi Xerox had a large base of customers (50 thousand) for this model spread over the entire country. Many buyers of these machines earn their livelihood by running copying services. Each of these buyers ultimately serves a very large number of customers (end user). When copy quality is either poor or inconsistent, customers earn a bad name and their image and business gets affected.

The pattern of blurred images (skips) observed in the copy is shown in Figure above. It usually occurs between 10-60 mm from the lead edge of the paper. Sometimes, on a photocopy taken on a company letterhead paper, the company logo gets blurred, which is not appreciated by the customer. This problem was noticed in only one-third of the machines produced by the company, with the remaining two-thirds of machines in the field working well without this problem. The in-house test evaluation record also confirmed the problem in only about 15% of the machines produced. Data analysis indicated that not all the machines produced were faulty. Therefore, the focus of further investigation was to find out what went wrong in the faulty machines or whether there are basic differences between the components used in good and faulty machines.

Selection of Factor and Intersection To EvaluateA copier machine consists of more than 1000 components and assemblies. A brainstorming session by the team helped in the identification of 16 components suspected to be responsible for the problem of blurred images. Each Suspected component had at least two possible dimensional characters which could have resulted in the skip symptom. This led to more than 40 probable causes (40 dimensions arising out of 16 components) for the problem. An attempt was made by the team to identify the real causes among these 40 probable causes. Ten bad machines were stripped open and various dimensions of these 16 component were measured. It was observed that all the dimensions were well within specifications Hence, this investigation did not give any clues to the problem. Moreover, the time and effort spent in dismantling the faulty machines and checking various dimensions in 16 components was in vain. This gave rise to the thought that conforming to specification does nor always lead to perfect quality. The team needed to think beyond the specification in order to find a solution to the problem

Selection of No. of factorAn earlier brainstorming session had identified 16 components that were likely to be the cause of this problem. A study of travel documents of 300 problem machines revealed that on 88% of occasions, the problem was solved by replacing one or more of only four parts of the machine. These four parts were from the list of 16 parts identified earlier. They were considered to be Critical and it was decided to conduct an experiment on these four parts. These parts were the following:(a) Drum shaft (b) Drum gear (c) Drum flange (d) Feed shaftTwo sets of these parts Were taken for Experiment I, one from an identified problem machine and one from a problem free machine. The two levels considered in the experiment were good and bad; ‘good’ signifying parts from the problem-free machine and ‘bad’ signifying parts from the problem machine. The factors and levels thus identified are given in Table below.

Selection of no. of factor and level

Selection of appropriate Orthogonal Array

Assignment of Factor/Interaction to Columns

Types of Quality ControlOnline quality controlOff- line quality control

What is RobustnessRobustness means small variation in performance.All products look good when they are precisely made.Robust products work well . Only robust products provide consistent customer satisfaction .

Example of Robustness Suppose two shooter “X” and “Y” go to the target range, and each shoots an initial round of 10 shots. X has his shots in a tight cluster, which lies outside the bull’s-eye,but Y actually has one shot in the bull’s-eye, but his success results only from his hit-or-miss pattern. “X” “Y” In this initial round Y has one By more bull’s-eye than X , but X is the robust shooter. a simple adjustment of his sights, X will move his tight cluster into the bull’s-eye for the next round. Y faces a much more difficult task. He must improve his control altogether, systematically optimizing his arm position, the tension of his spring, and other critical parameters.

Why we use Robust DesignEvery customer wants a product with high quality and a minimum cost, this is achieved by robust designing. There are mainly four types of cost which is under the category of obvious quality costs. Internal quality cost

Prevention cost These obvious quality cost are incurred directly by the producer and then passed on to the customer through the purchase price of the product. Robust Design is a symmetric method for keeping the producer’s cost low while delivering a high quality product. This is done by ROBUSTNESS STRATEGY.

Tools for robustness strategySignal to Noise Ratio P- DiagramQuadratic loss functionOrthogonal Array

Seminar on Basics of Taguchi Methods

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Seminar on Basics of Taguchi Methods