THE QUALITY LOSS FUNCTION - CECS - ANUcourses.cecs.anu.edu.au/courses/ENGN8101/Loss functions-lecture 5… · THE QUALITY LOSS FUNCTION ... its standard deviation is about ... A summary

10/08/200710/08/2007 ENGN8101 Modelling and OptimizationENGN8101 Modelling and Optimization 11

THE QUALITY LOSS FUNCTION

Quality – ultimate system performance measure

Variability relates to qualityVariability increases – quality lost

Can this loss be measured?

The concept % defective has been widely used as a measure of quality level

When defective product units are not shipped it should not be considered a quality problem but a cost problem

How to evaluate the quality level of products shipped to customers is the problem of concern.


Loss Function and Quality LevelLoss Function and Quality Level

In the past, % defective, process capability In the past, % defective, process capability index and warranty cost have been used as index and warranty cost have been used as measures of quality level for shipped measures of quality level for shipped products.products.One major weakness of the process capability One major weakness of the process capability index is that there is no apparent immediate index is that there is no apparent immediate basis for specifying the optimal value of Cbasis for specifying the optimal value of Cpp..% defective or warranty costs are % defective or warranty costs are understandable because they are monetary understandable because they are monetary related measures related measures


How far from target can a system be before it should be rejected and changed?

Different to customer tolerance Δo – point at which customers have to take economic action because of off-target performance

Often Δo incurs greater costs than manufacturing limit


Quality Loss ConceptQuality Loss Concept

Deviation from Deviation from target results in loss.target results in loss.

Lower than targetLower than target

Greater than targetGreater than target

Both loseBoth lose


How can we use Quality Loss How can we use Quality Loss Concept in real life?Concept in real life?

Let’s assume that you are an Operations Manager at a company that produces custom made doors and door frames. During the winter, due to cold weather, doors tend to shrink which lets in cold air through the cracks between the door and the door frame. During the summer doors tend to expand beyond it’s normal size due to hot weather, which makes to door hard to open because it rubs against the door frame. Your job is to produce a door, where dimensions (length and width of the door) are set to a specific target level. In other words in the Winter the door cannot let any cold air in the house and it should open properly in the Summer.


How can we use Quality Loss concept in real life?


Traditional Quality MetricTraditional Quality Metric

All products within specifications equality good.All products within specifications equality good.All products outside specifications equally bad.All products outside specifications equally bad.

USL

All products equally good

Equallyunacceptable

LSL


Loss Function (Cont.)Loss Function (Cont.)Unfortunately, this definition has led to a mindset which becomeUnfortunately, this definition has led to a mindset which becomes a barrier to s a barrier to improvement in our industry. improvement in our industry. We have come to view all products which fall within specificatioWe have come to view all products which fall within specification limits as being of n limits as being of equal quality. Consider the following:equal quality. Consider the following:


Loss Function (Cont.)Loss Function (Cont.)


TV ExampleTV ExampleConsider a comparison between the quality of color Consider a comparison between the quality of color television sets produced by two factories belonging to the television sets produced by two factories belonging to the same manufacturing company. One factory (A) is located in same manufacturing company. One factory (A) is located in Japan, and the other factory (B) in America. Suppose the Japan, and the other factory (B) in America. Suppose the comparison was based on color concentration, which comparison was based on color concentration, which relates to the color balance of the television sets. Although relates to the color balance of the television sets. Although both factories used the same design, the television sets both factories used the same design, the television sets produced in the American factory had lower quality, and produced in the American factory had lower quality, and consumers consequently preferred products made in consumers consequently preferred products made in Japan. Japan.


TV Example (Cont.)TV Example (Cont.)The figure given in the next slide shows the The figure given in the next slide shows the differences in quality characteristic (i.e., color differences in quality characteristic (i.e., color concentration) distributions. The figure shows that concentration) distributions. The figure shows that the quality distribution of the Japanesethe quality distribution of the Japanese--made made television sets (shown by the solid curve) is television sets (shown by the solid curve) is approximately a normal distribution with a target approximately a normal distribution with a target value at the center; its standard deviation is about value at the center; its standard deviation is about 1/6 of the tolerance, which in this case equals 10.1/6 of the tolerance, which in this case equals 10.In quality control, the index of tolerance divided by 6 In quality control, the index of tolerance divided by 6 standard deviations is called the process capability standard deviations is called the process capability index, denoted by Cindex, denoted by Cpp..

CCpp=tolerance/(6*standard deviation)=tolerance/(6*standard deviation)


TV Example (Cont.)TV Example (Cont.)

Distribution of color concentrationDistribution of color concentration


TV Example (Cont.)TV Example (Cont.)The process capability of the JapaneseThe process capability of the Japanese--made television sets made television sets is 1.is 1.On the other hand, the quality distribution of the AmericanOn the other hand, the quality distribution of the American--made television sets (shown in the figure by a dotted curve) made television sets (shown in the figure by a dotted curve) has less outhas less out--ofof--specification products than the Japanese specification products than the Japanese ––made products and is quite similar to the uniform made products and is quite similar to the uniform distribution for those products that are within the tolerance distribution for those products that are within the tolerance limits. Since the standard deviation of the uniform limits. Since the standard deviation of the uniform distribution is given by 1/ of the tolerance, the process distribution is given by 1/ of the tolerance, the process capability index for these sets is given bycapability index for these sets is given by

CCpp=tolerance/(6*(tolerance/ ))=0.577=tolerance/(6*(tolerance/ ))=0.577

12

12


QUALITY LOSS FUNCTION

Quadratic quality loss function – relates quality loss in dollars L(y) to the deviation away from a targeted value (m) of a measured response value (y)

i.e. |y-m|such that

L(y) = k(y-m)2

If m is achieved….

…. loss is zero


Derivation of the Loss FunctionDerivation of the Loss Function

Assume the loss due to a defective part (because Assume the loss due to a defective part (because of discarding, repairing, or downgrading) is A. of discarding, repairing, or downgrading) is A. then denote the loss function by then denote the loss function by L(yL(y) and expand ) and expand it in a Taylor series about the target value m:it in a Taylor series about the target value m:

L(yL(y)= )= L(m+yL(m+y--mm))

or or L(yL(y)=)=L(mL(m)+ ()+ (yy--mm)+ (y)+ (y--m)m)2 2 ++……!1)(mL′

!2)(mL ′′


Derivation of the Loss Function (Cont.)Derivation of the Loss Function (Cont.)

Because Because L(yL(y)=0 when y=m (by definition, quality loss is zero )=0 when y=m (by definition, quality loss is zero when y=m), and the minimum value of the function is when y=m), and the minimum value of the function is attained at this point, its first derivative with respect to m iattained at this point, its first derivative with respect to m is s zero. The first two terms of the equation then, are equal to zero. The first two terms of the equation then, are equal to zero. When we neglect terms with powers higher than 2, the zero. When we neglect terms with powers higher than 2, the equation reduces toequation reduces to

L(yL(y)= (y )= (y -- m)m)22

oror L(yL(y)= k (y )= k (y -- m)m)22

where k is a proportionality constant.where k is a proportionality constant.

!2)(mL ′′



Relationship between quality loss and deviation from the target Relationship between quality loss and deviation from the target value (m)value (m)



When the deviation productWhen the deviation product’’s functional s functional characteristic is an amount characteristic is an amount ΔΔoo from the from the target value m, the loss equals target value m, the loss equals AAoo. Then,. Then,

AAoo=k=kΔΔoo22

k=k= 020

AΔ


- unifying concept of quality and cost- relates engineering and economic terms in one model- allows for easy cost optimization strategies

k = quality loss coefficient

m+Δo = functional limitsbeyond which – 50% of system product needs customer maintenance

i.e. average customer tolerance

∴ L(y) = Ao at y = m±ΔoAo = cost to replace/repair product





CASE STUDY T1CASE STUDY T1A spring is used in the operation of a camera shutter. The manufacturing process suffers from a degree of variability, in terms of the spring constant (measured in oz/in), which significantly effects the accuracy of the shutter times. The functional limits for this spring constant are m±0.3oz/in (m=0.5oz/in), and the average cost for repairing or replacing a camera with a defective spring is $20. What is the loss function? Hence, what is the loss associated with producing a spring of constant 0.25oz/in versus the loss associated with one at 0.435oz/in.


Uses of the Loss FunctionUses of the Loss FunctionThe loss function approach can be used in evaluating the The loss function approach can be used in evaluating the effect of quality improvement. For example, assume that effect of quality improvement. For example, assume that Factory A has improved the process so that a new Factory A has improved the process so that a new standard deviation from target of 10/8 (the previous one standard deviation from target of 10/8 (the previous one is 10/6) is attained. What would be the losses caused by is 10/6) is attained. What would be the losses caused by deviations from the target value?deviations from the target value?

L=L=

L=0.08(10/8)L=0.08(10/8)22=$0.125=$0.125The loss per unit of production would decrease from The loss per unit of production would decrease from $0.222 (current process) to $0.125, resulting in $0.097 $0.222 (current process) to $0.125, resulting in $0.097 savings per unit. savings per unit.

2 2( )k y m k− = υ


Loss Function/Process Capability IndexLoss Function/Process Capability Index

Let us see how the loss function is related to the process capabLet us see how the loss function is related to the process capability index Cility index Cpp. For the . For the current process and the improved process discussed above,current process and the improved process discussed above,

LL11=k (loss with current process)=k (loss with current process)LL22=k (loss with improved process)=k (loss with improved process)

Divide the first equation by the second to obtainDivide the first equation by the second to obtain

But But and and

Then Then

This equation implies that the losses caused by deviation are reThis equation implies that the losses caused by deviation are reciprocally proportional to ciprocally proportional to the squares of the Cthe squares of the Cpp indices. indices.

22

21

2

1

υυ

=LL

161 υtoleranceC p =

22 6υ

toleranceC p =

2

2

2

1

1

2

P

p

CC

LL

=

21υ22υ


Economic Consequences of Tightening Economic Consequences of Tightening Tolerances as a Means to Improve QualityTolerances as a Means to Improve Quality

As illustrated in the following example, the loss function As illustrated in the following example, the loss function approach can be used to determine the economic impact approach can be used to determine the economic impact of tightening the tolerance to improve product quality. In of tightening the tolerance to improve product quality. In order to reduce the difference in quality and process order to reduce the difference in quality and process capability indices between television sets produced in capability indices between television sets produced in Factories A and B, the management of Factory B tightened Factories A and B, the management of Factory B tightened the tolerance from mthe tolerance from m±±5 to m5 to m±±5*(2/3). The cost of 5*(2/3). The cost of repairing an outrepairing an out--ofof--specification unit is still specification unit is still $2$2. What is the . What is the economic impact of tightening the tolerance? economic impact of tightening the tolerance?


Economic Consequences of Tightening Tolerances Economic Consequences of Tightening Tolerances as a Means to Improve Quality (Cont.)as a Means to Improve Quality (Cont.)

With the original tolerance, the expected loss is With the original tolerance, the expected loss is L=k =L=k =$$0.667. The expected loss after tightening 0.667. The expected loss after tightening the tolerance is the tolerance is L=k = 0.08L=k = 0.08[[(2/3)*(10/ )(2/3)*(10/ )]]22= = $$0.296/unit0.296/unitIf improvement of the process was obtained by If improvement of the process was obtained by repairing the failed units (units outside the new repairing the failed units (units outside the new tolerance mtolerance m±±5*(2/3)) at a cost of $2 per unit, then 5*(2/3)) at a cost of $2 per unit, then the average cost of repair is as follows: the average cost of repair is as follows:

Average cost of repair per unit = percent of Average cost of repair per unit = percent of production that needs repair to meet the tightened production that needs repair to meet the tightened tolerancetolerance * * repair cost per unit = 0.333*2 = $0.667repair cost per unit = 0.333*2 = $0.667

12

2υ

2υ


Economic Consequences of Tightening Tolerances Economic Consequences of Tightening Tolerances as a Means to Improve Quality (Cont.)as a Means to Improve Quality (Cont.)

A summary of the results of this example is shown in the table gA summary of the results of this example is shown in the table given iven below. In this case, tightening tolerance is an uneconomical altbelow. In this case, tightening tolerance is an uneconomical alternative ernative because the expected total loss of tightening tolerance and repabecause the expected total loss of tightening tolerance and repair ir (0.667+0.296=$0.963) is greater than the expected loss using the(0.667+0.296=$0.963) is greater than the expected loss using the

original tolerance ($0.667).original tolerance ($0.667).


The Loss Function and The Loss Function and Justification of ImprovementsJustification of ImprovementsThe loss function can also be used to justify The loss function can also be used to justify improvements of the process, as illustrated in the improvements of the process, as illustrated in the following example.following example.Assume that Factory A wishes to improve the Assume that Factory A wishes to improve the quality of its television sets by reducing deviations quality of its television sets by reducing deviations from the target value so that the new standard from the target value so that the new standard deviation will be 10/8. This improvement can be deviation will be 10/8. This improvement can be technologically achieved at an additional cost of technologically achieved at an additional cost of $0.05 per unit of production. Should the factory $0.05 per unit of production. Should the factory improve its process? (Assume that no inspection is improve its process? (Assume that no inspection is performed.)performed.)


The Loss Function and Justification of The Loss Function and Justification of Improvements (Cont.)Improvements (Cont.)

Total loss per unit of the current process:Total loss per unit of the current process:L= k = 0.08(10/6)L= k = 0.08(10/6)22 = $0.222= $0.222

Total loss per unit after improving the process:Total loss per unit after improving the process:L=0.08(10/8)L=0.08(10/8)2 2 = $0.125= $0.125Additional cost of improvement = $0.05/unitAdditional cost of improvement = $0.05/unitAdditional cost plus loss per unit = 0.05+0.125=$0.175Additional cost plus loss per unit = 0.05+0.125=$0.175

The net gain resulting from improvement in the process The net gain resulting from improvement in the process capability is 0.222capability is 0.222--0.175= $0.047 per unit of production. If 0.175= $0.047 per unit of production. If the production rate of this factory is 100,000 units per the production rate of this factory is 100,000 units per month, then the expected savings will be $4700 per month, month, then the expected savings will be $4700 per month, or $56,400 annually.or $56,400 annually.

2υ


The Loss Function and InspectionThe Loss Function and Inspection

The loss function approach can be used effectively to The loss function approach can be used effectively to determine whether 100determine whether 100--percent inspection can be percent inspection can be justified or not. It should be noted that the objective justified or not. It should be noted that the objective of inspection is to screen or repair defective products of inspection is to screen or repair defective products that cannot meet the given specifications. Therefore, that cannot meet the given specifications. Therefore, inspection cannot be used to improve the quality of inspection cannot be used to improve the quality of items within the specifications. The improvement of items within the specifications. The improvement of the process can only be accomplished through the process can only be accomplished through improved manufacturing techniques or product improved manufacturing techniques or product design, not through screening or 100design, not through screening or 100--percent percent inspection. inspection.


The Loss Function and Inspection The Loss Function and Inspection (Cont.)(Cont.)

Consider the case where the diameter of a stainlessConsider the case where the diameter of a stainless--steel bar is msteel bar is m±±55μμm. The cost of repairing a defective m. The cost of repairing a defective bar is $6, and the cost of inspection is $0.03 per unit. bar is $6, and the cost of inspection is $0.03 per unit. Would a 100Would a 100--percent inspection of items be justified? percent inspection of items be justified? The estimated standard deviation of the process is The estimated standard deviation of the process is 10/6. 10/6. The expected loss without inspection is The expected loss without inspection is

L=kL=kwhere k = A/where k = A/ΔΔ2 2 = $6.00/5= $6.00/52 2 = $0.24= $0.24therefore L = 0.24(10/6)therefore L = 0.24(10/6)2 2 = $0.667/unit = $0.667/unit

2υ



Assuming that the characteristic of the product follows a Assuming that the characteristic of the product follows a

normal distribution, the proportion of the products falling normal distribution, the proportion of the products falling

outside the specification moutside the specification m±±5 is 0.27 percent. The variance 5 is 0.27 percent. The variance

after screening defective products by using 100after screening defective products by using 100--percent percent

inspection ( ) is obtained using the procedure shown inspection ( ) is obtained using the procedure shown

below.below.

After the total inspection, the outAfter the total inspection, the out--ofof--specification products specification products are removed. The probability density function of those items are removed. The probability density function of those items that have passed the screening (acceptable items) is given that have passed the screening (acceptable items) is given by dividing the probability density function of the normal by dividing the probability density function of the normal distribution by Q, the proportion of acceptable items. distribution by Q, the proportion of acceptable items.

2outυ



Let Let f(yf(y) be the density function of the normal ) be the density function of the normal distribution, which is given by distribution, which is given by


The Loss Function and Inspection (Cont.)The Loss Function and Inspection (Cont.)



One might conclude that in this case 100One might conclude that in this case 100--percent inspection percent inspection is useless in improving quality, because the fraction defective is useless in improving quality, because the fraction defective is only 0.27 percent. It is different, of course, when the is only 0.27 percent. It is different, of course, when the purpose of 100purpose of 100--percent inspection is to find serious percent inspection is to find serious defectives. defectives. In the case of a normal distribution with a standard deviation In the case of a normal distribution with a standard deviation that is that is ¼¼ of the tolerance, the loss without inspection, L, is of the tolerance, the loss without inspection, L, is

L = 0.24*(10/4)L = 0.24*(10/4)2 2 = $1.50/unit= $1.50/unitThe proportion of the product falling outside the specification The proportion of the product falling outside the specification is 4.55 percent, and the variance of the outgoing items is is 4.55 percent, and the variance of the outgoing items is (0.88)(0.88)22 times that of the original value. The total loss in the times that of the original value. The total loss in the case of 100case of 100--percent inspection when percent inspection when σσ equals 10/4 becomesequals 10/4 becomes

L = 0.03+6.00*0.0455+0.24*(0.88)L = 0.03+6.00*0.0455+0.24*(0.88)22*(10/4)*(10/4)22

= $1.465/unit= $1.465/unit



This result is an improvement of $0.035 per item. If there This result is an improvement of $0.035 per item. If there are 200,000 items produced each month, the amount of are 200,000 items produced each month, the amount of improvement is $7,000 each month. improvement is $7,000 each month. Assuming that the standard deviation is Assuming that the standard deviation is ½½ the tolerance and the tolerance and the production output still follows a normal distribution, the the production output still follows a normal distribution, the portion of the product falling outside the specification is 31.7portion of the product falling outside the specification is 31.7percent. Even if all the products are inspected and the percent. Even if all the products are inspected and the defective ones screened out, the standard deviation of the defective ones screened out, the standard deviation of the outgoing quality is reduced to only 53.9 percent of the outgoing quality is reduced to only 53.9 percent of the original value (original value (σσ=tolerance/2). Therefore, the loss caused =tolerance/2). Therefore, the loss caused by variation isby variation is

L=0.24L=0.24[[0.539*(10/2)0.539*(10/2)]]22= $1.743= $1.743



Not only is this worse than the loss of Not only is this worse than the loss of $0.667 for $0.667 for σσ=10/6 and no inspection, but =10/6 and no inspection, but it is also worse than the loss of $1.50 for it is also worse than the loss of $1.50 for σσ=10/4 and no inspection, with 4.55 =10/4 and no inspection, with 4.55 percent defective products. Thus, the percent defective products. Thus, the solution to the quality problem is, in this solution to the quality problem is, in this case, through improvement of the process case, through improvement of the process and not through 100and not through 100--percent inspection.percent inspection.


The Loss Function and Inspection (Cont.)The Loss Function and Inspection (Cont.)



The table shows a summary of the expected losses caused The table shows a summary of the expected losses caused by variation for different probability distributions. These by variation for different probability distributions. These expected losses do not include the cost of inspection or expected losses do not include the cost of inspection or loss caused by defective products found by inspection. loss caused by defective products found by inspection. Cases 1 through 6 demonstrate how screening reduces Cases 1 through 6 demonstrate how screening reduces total losses for the given parameters. A detailed analysis of total losses for the given parameters. A detailed analysis of Case 2 follows, in order to illustrate how the results of the Case 2 follows, in order to illustrate how the results of the

table are obtained.table are obtained.



Since L=kSince L=k2υ




Determinations of TolerancesDeterminations of TolerancesLoss function can also be used to determine tolerances of Loss function can also be used to determine tolerances of the quality characteristics. The determination of tolerance the quality characteristics. The determination of tolerance is illustrated in the following example.is illustrated in the following example.

Example: Consider the production of high-voltage transformers. During the life of this kind of transformer, output voltage might change because of the deterioration of transistors in the power circuit. Assume that a transformer is not suitable for its intended function when its output voltage exceeds the tolerance limits of 115±25V. Exceeding the limits results in a loss (denoted by A) of $300. Before shipping to a customer, the manufacturer can adjust the voltage in the plant by changing a resistor at a cost of $1. What should the manufacturer’s specifications be?

Solution: The loss caused by product variation from the target value, L(y), is

L(y)=k(y-m)2

where m is the target value (115V in this case) and k is the proportionality constant.


2 20

300 0.48(25)

Ak = = =Δ

The loss function2( ) 0.48( 115)L y y= −

It is assumed that the allowable varying range of the output voltage for the customer is 115±25V. The allowable varying range in the plant will be different, because it is easy to adjust the voltage to the target value by changing a resistor in the circuit. The loss or cost of adjust to the manufacturer is $1. Substitution of this value in Equation above yields

21.0 0.48( 115) 115 1/ 0.48 115 1.4Vy y= − ⇒ = ± ≈ ±

As shown above:2 2

. . 0 0 02 2

. .

300 ( ) for 300= ( )

1 ( ) for 1= ( )cus cus

manu manu

k y m y m A k

k y m y m A k

= − ≥ ± Δ ⇒ = Δ

= − ≥ ± Δ ⇒ = Δ

002 2

0 0

A A AkA

= = ⇒ Δ = ΔΔ Δ


and the manufacturer’s tolerance is

0 00

(functional orcustomer limit) /

(safety factor)AA

ΔΔ = Δ =

φ

Δ=manufacturer’s tolerance limit

A=manufacturer’s loss when the product does not conform to the specification limits

A0=loss to the customer caused by the failure of the product



Main function of quality loss function:

= define manufacturing tolerancesor more generally…define system variability

QUALITY LOSS FUNCTION – 4 types

In the following illustrates the evaluation of the quality level of products by using the loss function approach for four types of tolerances. The three types are listed below:

1 target-is-best2 smaller-is-best3 larger-is-best4 asymmetric target-is-best


Target-is-beste.g. previous example!

-quality characteristics is usually a nominal output, for example--most parts in mechanical fittings have nominal dimensions--Ratios of chemicals or mixtures are nominally the best type. --Thickness should be uniform in deposition /growth /plating

/etching..

Average quality lossThis type of tolerance is required for many products, parts,

elements, and components when a nominal size (or characteristic) is preferred.

calculated in terms of MSD (mean square deviation)

for n observations: ∑=

−=n

in my

nMSD

1

2)(1


From this – the average loss function for multiple lots:

S2 = variance =

[ ]22 )()( mySkyL −+=

∑ =−

−n

i i yyn 1

2)(1

1

CASE STUDY T2CASE STUDY T2

Data table…

Winder S2 Ῡ (Ῡ-m)2 L(y)

New 0.0007 0.385 0.0132 3.08Old 0.0184 0.539 0.0015 4.41

Continuing from above, it was thought that if a new machine was purchased – that the losses would reduce. To test this, 8 springs were tested from each machine, as detailed below. Which machine is best, and why? New machine: 0.37, 0.41, 0.37, 0.43, 0.39, 0.35, 0.40, 0.36 Old machine: 0.55, 0.67, 0.70, 0.54, 0.41, 0.32, 0.46, 0.66


New machine: lower variance but off-targetOld machine: higher variance but on-target

Loss incurred – influenced by variability more than target value

To reduce loss further (new machine)…… use an adjustment parameter

i.e. reduce variability then adjust average response

A 2-stage optimization!


Target-is-best

Example: A manufacturer of ball bearings used in gas Example: A manufacturer of ball bearings used in gas turbines requires that tolerances of the diameter and turbines requires that tolerances of the diameter and hardness of each ball be as follows:hardness of each ball be as follows:Tolerance of diameterTolerance of diameter mm11±± 0.6 0.6 μμmmTolerance of hardnessTolerance of hardness mm22±± 2.0 (Brinell hardness)2.0 (Brinell hardness)

where mwhere m11 and mand m22 are the target values of the diameter and are the target values of the diameter and the hardness, respectively. The production rate is 80,000 the hardness, respectively. The production rate is 80,000 balls per day at a cost of 30balls per day at a cost of 30¢¢ per ball. Defective balls per ball. Defective balls cannot be reworked and are scrapped. The following cannot be reworked and are scrapped. The following deviations from the diameter and the hardness target deviations from the diameter and the hardness target values were recorded.values were recorded.


Deviations from the target diameter:Deviations from the target diameter:0.3 0.0 0.3 0.0 --0.1 0.0 0.3 0.2 0.1 0.1 0.0 0.3 0.2 0.1 --0.2 0.6 0.40.2 0.6 0.4

--0.2 0.1 0.0 0.2 0.1 0.0 --0.4 0.5 0.4 0.4 0.5 0.4 --0.2 0.0 0.2 0.0 0.00.0 0.2 0.2

Deviations from the target hardness:Deviations from the target hardness:--1.0 1.0 --1.6 1.6 --0.4 0.4 --1.0 0.6 0.4 1.0 0.6 0.4 --1.2 1.2 --1.3 1.3 --0.2 0.2 --0.40.4--0.4 0.5 0.4 0.5 --0.3 0.6 0.3 0.6 0.60.6 --0.9 0.9 --0.7 0.7 --0.9 0.9 --0.7 0.7 --1.31.3Based on the diameter and hardness measurements Based on the diameter and hardness measurements recorded above, determine the quality levels of the recorded above, determine the quality levels of the production process for the diameter and hardness production process for the diameter and hardness attributes of the balls. attributes of the balls.

Target-is-best


Target-is-best


Target-is-best





A process without adjustmentA process without adjustment


Example: By examining the diameter data we find that more positive deviations than negative ones, whereas the hardness data show more negative deviations than positive ones. Assume that the manufacturer can shift the means of the data to the target values. What are the quality levels of the diameter and the hardness after the adjustments?

Solution: The process should first be adjusted so that the value of everydiameter is adjusted by an amount e*= , the predicted deviation of the diameter from the target value. The new deviation (after adjustment) from the target value (m+ e*) is

Deviation after adjustment ( )y m y m y y= − − − = −

( )y m−










Smaller-is besthere – ideal response = zero

L(y) = k(y)2

examples?

Background density on a text imageRadiation leakageCorrosion of metalsSignal to noise ratios!Defective components

-quality characteristics is usually an undesired output, for example

--Defects like pin holes, particulates in deposition processes

--Unwanted by-product or side effect


Features:Features:TheThe--SmallerSmaller--TheThe--Better type tolerance involves a nonnegative Better type tolerance involves a nonnegative characteristic, whose ideal value is zero. A typical example of characteristic, whose ideal value is zero. A typical example of such such a characteristic is impurity. Wear, shrinkage, deterioration, ana characteristic is impurity. Wear, shrinkage, deterioration, and d noise level are also examples of this type. noise level are also examples of this type. Under TheUnder The--SmallerSmaller--TheThe--Better (SBetter (S--type) tolerance, the type) tolerance, the characteristic value is ycharacteristic value is y≥≥0, the target value is m=0, and the upper 0, the target value is m=0, and the upper tolerance limit is tolerance limit is ΔΔ..



In the copier industry, one measure of the acceptability of a copy is the amount of background toner particles that adhere to the portion of the copy that is intended to be white. Minimizing the residual toner in white areas is a smaller-is-best objective. It has been determined that approximately half of the customers will not tolerate a background level beyond 1.2 background units. Beyond that, a service call is placed at a cost of $200 plus the cost of the down time of the copier - $150 per hour. If the average copier down time is 2.5 hours, what is the associated loss function? As time progresses, it is apparent that one copier is not enough, so a second one is introduced. 8 sample background measurements are taken from each machine to compare and contrast their performances. From the data below – what conclusions can you draw about the cost of this facility? Machine 1: 0.64, 0.56, 0.71, 0.55, 0.59, 0.75, 0.64, 0.76 Machine 2: 0.55, 0.67, 0.70, 0.94, 0.71, 0.82, 0.86, 0.96


Here: Δo=2 = 1.2, Ao = 200 + 150(2.5) = $575

So k = Ao/(Δo)2 = 575/1.22 = $399.30

∴ L(y) = 399.30(y2)

Average quality loss…

Data table =

Machine S2 Ῡ S2+Ῡ2 L(y)

1 0.0068 0.65 0.4293 171.412 0.0203 0.77 0.6229 248.70

Note: S2+Ῡ2 not (Ῡ-m)2

Clearly – machine 1 is best – but is probably sub-optimal?


TheThe--LargerLarger--TheThe--BetterBetter (L Type)(L Type)


Larger-is-better

here – ideal response = max

L(y) = k(1/y2)

and k = AoΔo2

examples?..etc…!

--quality characteristics is usually a desired output, for example

--Bond strength--Critical Current



The seal strength of a vacuum blower housing in an office copier is an example of a larger-the-better case. The better it can run under widely varying use environments, the better it is for minimizing loss. When the blower seal fails to operate, it costs $40 to replace: $20 in part costs and another $20 in installation labour costs. While the device that uses the vacuum blower sits idle, the cost to the customer is $340/hour. On average it takes 30 minutes to replace the blower. Seal integrity is measured by testing the seal adhesion strength. The seal level at which the vacuum loss becomes objectionable is 20 psi. What is the loss function and what is the loss incurred for a machine whose seal strength is measured as 13 psi?


Ao = 340/2 + 40 = $210Δo = 20 psi

so k = 210 x 202

giving L(y) = 84000/y2

When y = 13 psi:L(y) = 84000/132 = $497

BUT/ the actual cost of repair is only $210!

This illustrates that costs continue to increase beyond the acceptable value – due to other consequences on the system…

Average quality loss…⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛== ∑

=

n

i iynkMSDkyL

1

211)()(


Example: Consider two types of cables, T1 and T2. The price and strength for either type are proportional to the cable’s cross sectional area. The prices are P1=$1750/mm2 and P2=$2250/mm2, and the strengths are S1=220kgf/mm2 and S2=265kgf/mm2 for types T1 and T2, respectively. The lower tolerance limit of the cables breaking strength is 20000kgf, and the loss caused by falling below the lower tolerance limit is $58 million. Perform tolerance design and determine the tolerance limits for the better cable. Solution: We first calculate the total cost for each cable (price +quality loss). Let x be the cross-sectional area of the cable, which is the parameter being sought. Cable T1. The total cost C is obtained as the sum of the price and the quality loss.

20 0

1 21

2

2

( )

58,000,000 (20,000) 1750(220 )

AC P x

S x

xx

Δ= +

×= +

(3.13)


The total cost is minimized by taking the derivative of Eq (1) with respect to x and equating it to zero:

20 0

1 2 31

20

AdC Pdx S x

Δ= − =

or

1/ 3220 0

21 1

2818mm

Ax

S P⎛ ⎞Δ

= =⎜ ⎟⎝ ⎠

The price of this cable is

1750×818=$1.43 million Cable type T2: The corss-sectional area is

1/ 322

2

2 58,000,000 (20,000) 665mm2250 (265)

x⎛ ⎞× ×

= =⎜ ⎟×⎝ ⎠


The price of cable T2 is

2250×665=$1.50 million Cable T1 is selected, since the price of T1 is less than T2. The tolerance of this cable is obtained using the formulation presented before:

02 2

0(1/ ) (1/ )AAk = =

Δ Δ

It turns out

00

58,000,000 20 metric tons force1,430,000

=127.4 metric tons force

AA

Δ = Δ = ×


Asymmetric target-is-best

i.e. when it is more harmful for the variable to be off-target in one direction over the other

here – 2 lossfunctions required:

i.e.L+(y) = k+(y-m)2, y>mL-(y) = k-(y-m)2, y≤m








Consider the temperature drift in a refrigerator. The standard target for most refrigerators is 40°F. Consider the consequences of being above and below this targeted temperature. When the temperature gets above 50°F, several things can happen that annoy the consumer. These include tepid food and drink that is not pleasant to the taste, and spoilage due to accelerated bacterial growth. Each of these can cause economic loss, losses due to discarding and replacement of food, and losses due to illness from ingestion of tainted food. When the temperature gets below 30°F, there may be some damage due to ice crystals, but there should be little food lost. When too hot – the losses incurred include $50 for lost food replacement and $100 for a service call. When too low, the losses incurred include $10 for lost food replacement and $100 for a service call. What are the loss functions in this case?


here, k- < k+

k+ = Ao/ΔO2 = $150/(10°F)2 = $1.50/°F2

k- = Ao/ΔO2 = $110/(-10°F)2 = $1.10/°F2

∴

L+(y) = 1.5(y-40)2 L-(y) = 1.1(y-40)2



TAGUCHI S/N RATIOS

Taguchi idea – use signal/noise ratios as performance measures

signal = target valuenoise = scatter around target value

Performance measure – when maximized…variation is minimized ∴ loss is minimized


Advantages of S/N MethodAdvantages of S/N Method


When to Use the S/N Ratio for When to Use the S/N Ratio for AnalysisAnalysis

Whenever an experiment involves repeated Whenever an experiment involves repeated observations at each of the trial conditions, the observations at each of the trial conditions, the S/N ratio has been found to provide a practical S/N ratio has been found to provide a practical way to measure and control the combined way to measure and control the combined influence of deviation of the population mean from influence of deviation of the population mean from the target and the variation around the mean. In the target and the variation around the mean. In standard ANOVA they are treated separately. standard ANOVA they are treated separately.


When to Use the S/N Ratio for When to Use the S/N Ratio for Analysis (Cont.)Analysis (Cont.)

S/N offers the following two main advantages:S/N offers the following two main advantages:1.1. It provides a guidance to a selection of the It provides a guidance to a selection of the

optimum level based on least variation around optimum level based on least variation around the target and also the average value closest to the target and also the average value closest to the target. the target.

2.2. It offers objective comparison of two set of It offers objective comparison of two set of experimental data with respect to variation experimental data with respect to variation around the target and the deviation of the around the target and the deviation of the average from the target value.average from the target value.


Signal to Noise RatioSignal to Noise Ratio

The relevance of the S/N ratio equation is The relevance of the S/N ratio equation is tied to interpreting the signal or numerator of tied to interpreting the signal or numerator of the ratio as the ability of the process to build the ratio as the ability of the process to build good product, or of the product to perform good product, or of the product to perform correctly. By including the impact of the noise correctly. By including the impact of the noise factors on the process or product as the factors on the process or product as the denominator, we can then adapt the S/N denominator, we can then adapt the S/N ratio as the barometer of the ability of the ratio as the barometer of the ability of the system (product or process) to perform well system (product or process) to perform well in relation to the effect of noise. in relation to the effect of noise.


Signal to Noise Ratio (Cont.)Signal to Noise Ratio (Cont.)

By successfully applying this concept to By successfully applying this concept to experimentation, we can determine the experimentation, we can determine the control factor settings that can produce control factor settings that can produce the best performance (high signal) in a the best performance (high signal) in a process or product while minimizing the process or product while minimizing the effect of those influences we can not effect of those influences we can not control (low noise).control (low noise).



To obtain a better understanding of how this approach To obtain a better understanding of how this approach works and what it means, letworks and what it means, let’’s discuss a practical s discuss a practical example (car radio) illustrated below:example (car radio) illustrated below:





For improved For improved additivityadditivity of the control factor effects, it of the control factor effects, it is common practice to take log transformation of is common practice to take log transformation of μμ22//σσ2 2 express the S/N ratio in decibels.express the S/N ratio in decibels.

The range of values of The range of values of μμ22//σσ2 2 is (0,is (0,∞∞), while the range ), while the range of values of Z is (of values of Z is (--∞∞, , ∞∞). Thus, in the log domain, we ). Thus, in the log domain, we have better have better additivityadditivity of the effects of two or more of the effects of two or more control factors. Since log is a monotone function control factors. Since log is a monotone function maximizing maximizing μμ22//σσ2 2 is equivalent to maximizing Z.is equivalent to maximizing Z.

2

10 210logZ μ=

σ


very important reasonvery important reason

Taking Taking ““logarithmlogarithm”” improves improves ‘‘additivityadditivity’’. . Why? Why?

Most functions in Most functions in ‘‘naturenature’’ follow the powerfollow the power--lawslawsY = A exp (Y = A exp (--Ea/Ea/kTkT) ) (EXPONENTIAL)(EXPONENTIAL)Y = B AY = B Ax . . . . x . . . . ( a ( a –– RAISED RAISED –– POWER X)POWER X)Y = B Y = B XXaa . . . . . . . . ( X ( X –– RAISED RAISED –– POWER a)POWER a)

On taking logarithm On taking logarithm these become these become ‘‘additiveadditive’’For example For example LogLog10 10 ( B( B XXaa . . expexp((--Ea/Ea/kkTT)) ) )

= = LogLog1010(B) + a. Log(B) + a. Log1010 ((XX) + () + (--Ea/Ea/kkTT)) . . LogLog1010 (e) (e) These are These are ‘‘additiveadditive’’ for variables (Logfor variables (Log1010 XX) and ) and (1/(1/T)T)



Consider the following two sets of observations around Consider the following two sets of observations around the target and the deviation of the average from the the target and the deviation of the average from the target value. target value. Let m=75Let m=75

Observation A: 55 58 60 63 65Observation A: 55 58 60 63 65 = 60.2= 60.2Dev. Of mean from target = 75 Dev. Of mean from target = 75 -- 60.2 = 14.860.2 = 14.8

Observation B: 50 60 76 90 100 Observation B: 50 60 76 90 100 = 75= 75Dev. Of mean from target = 75 Dev. Of mean from target = 75 -- 75 = 075 = 0

y

y




Conversion of Results into S/N Conversion of Results into S/N Ratios (Cont.)Ratios (Cont.)

These two sets of observations may have These two sets of observations may have come from the two distributions shown in the come from the two distributions shown in the figure above.figure above.Observe that the set B has an average value Observe that the set B has an average value which equals to target value, but has a wide which equals to target value, but has a wide spread around it. For the set A, the spread is spread around it. For the set A, the spread is smaller, but the average itself is quite far smaller, but the average itself is quite far from the target. Which of the two is better? from the target. Which of the two is better?


Conversion of Results into S/N Conversion of Results into S/N Ratios (Cont.)Ratios (Cont.)

Based on the average value the product Based on the average value the product shown by shown by obsobs. B appears to be better. Based . B appears to be better. Based on consistency, product A is better. How an on consistency, product A is better. How an one credit A for less variation? How does one one credit A for less variation? How does one compare the distances of the averages from compare the distances of the averages from the target? Surely comparing the averages is the target? Surely comparing the averages is one method. Use of S/N ratio offers an one method. Use of S/N ratio offers an objective way to look at the two objective way to look at the two characteristics together.characteristics together.


Computation of S/N RatioComputation of S/N Ratio


Computation of S/N Ratio Computation of S/N Ratio (Cont.)(Cont.)


Most Common S/NMost Common S/N’’s for the Static s for the Static CaseCase

NominalNominal--isis--Best (N.B.)Best (N.B.)


Most Common S/NMost Common S/N’’s for the Static Case (Cont.)s for the Static Case (Cont.)

2

210 log yZs

⎛ ⎞= ⎜ ⎟

⎝ ⎠


Most Common S/NMost Common S/N’’s for the Static s for the Static Case (Cont.)Case (Cont.)

SmallerSmaller--isis--Better (S.B.)Better (S.B.)



2

1

110logn

ii

Z yn =

⎛ ⎞= − ⎜ ⎟⎝ ⎠∑


Most Common S/NMost Common S/N’’s for the Static s for the Static Case (Cont.)Case (Cont.)

LargerLarger--isis--Better (L.B.)Better (L.B.)



⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=∑=

nyZ

n

i 12

1

log10


Target-the best

Ῡ = sample mean s = sample standard deviationaim

to maximize Z through parameter design

Smaller-is-best

here, needs to be maximized

Larger-is-best

here, needs to be maximized

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛= 2

2

log10syZ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= ∑ =

ny

Zn

i i12

log10

⎟⎠

⎜⎝

⎟⎟⎟⎞

⎜⎜⎜⎛

−=∑=

nyZ

n

i 12

1

log10


But what about quality characteristics that approach But what about quality characteristics that approach

an ideal value?an ideal value?Examples areExamples are

Efficiency : all efficiencies approach the ideal value of Efficiency : all efficiencies approach the ideal value of 100%100%

Weld strength : approaches the ideal strength of the Weld strength : approaches the ideal strength of the materialmaterial

Critical temperature or Critical current density for High Critical temperature or Critical current density for High Temperature superconductors (YBCO) : Temperature superconductors (YBCO) :

These approach ideal values, say 92K and 10These approach ideal values, say 92K and 1088 A/cmA/cm22

Which SNWhich SN--Ratio is most suitable among the following Ratio is most suitable among the following ??

smallersmaller--thethe--betterbetterLARGERLARGER--THETHE--BETTERBETTER

NOMINALNOMINAL--thethe--BEST BEST


Taguchi Method is most effective when Taguchi Method is most effective when there is at least one quality characteristic there is at least one quality characteristic that is sensitive to variations or NoIsE that is sensitive to variations or NoIsE

Desirable QualitiesDesirable Qualitiese.g. e.g. ““NominalNominal--thethe--BestBest”” type are sensitive to type are sensitive to NoIsENoIsE

S/N RatioS/N Ratio ZZ = 10 Log= 10 Log1010 ( ( meanmean22 / / VarianceVariance ))

Question: the large of mean, the better the quality is?Question: the large of mean, the better the quality is?Undesirable propertiesUndesirable propertiese.g. e.g. ““SmallerSmaller--thethe--better better ”” type are type are alsoalso sensitive to sensitive to NoIsENoIsE

S/N Ratio ZS/N Ratio Z = = –– 10 Log10 Log1010 ( 1/n ( 1/n ΣΣ YYii22 ))

≅≅ –– 10 Log10 Log1010 ((VarianceVariance) . . . ) . . . if ideal value if ideal value = zero= zero


Best approach for experimental design

Maximize these ratios to minimize variabilityto maximize system robustness

thenadjust target value to desired value

2-stage optimization!

Some drawbacks, however, to Taguchi approach…

e.g. assumes Ῡ and s are of equal importance/influence+ noticeable confounding issues

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THE QUALITY LOSS FUNCTION - CECS - ANUcourses.cecs.anu.edu.au/courses/ENGN8101/Loss functions-lecture 5… · THE QUALITY LOSS FUNCTION ... its standard deviation is about ... A summary