Chapter 5 – Part 1

Solutions to SHW

1. What do we mean by a process that is consistently on target?

Amount of Toner

LSL

Target

USL

Process is consistently on target if the distribution is tightly clustered around the target, or, equivalently, if the variance around the target is small.

Amount of Toner

2. What do we mean by a process that is consistently on target?

LSL USL

Target Mean

Process is consistently off target if the distribution is off target but tightly clustered—or has a small variance—around its mean.

3. What do we mean by a process that is haphazardly on target?

USL

Target

LSL

Process is haphazardly on target if the distribution is on target but exhibits a great deal of variation aroundThe target.

4. Refer to Jane and Sam in the notes to Chapter 5, Part 1. Draw the distribution of two machines, one behaving like Jane and the other like Sam. Which machine is easier to fix? How would you fix it?

Target

Sam(Machine A)

Jane(Machine B)

Problem 4 - Continued

• Easier to adjust mean to target than to reduce variance.

• Machine A: reduce variance

• Machine B: adjust mean to target

5. How does better quality increase productivity?

As quality improves, rework and scrap decrease

This results in fewer inputs being used to produce a units of output.

Also, more good units are produced the first time.

Since better quality means high output of good units and fewer inputs, productivity—output divided by inputs—increases.

Problem 6

a) Find the loss function

LSL = 0.6

USL = 1.6

Target = (USL+ LSL)/2 =(1.6+.6)/2 = 1.1

USL = Target + a

1.6 = 1.1 + a

a=0.5

48$)5.0(

12$22

a

Rk

2)( TXkL

2)1.1(48$ XL

Problem 6

Problem 6

b) If X = 1.3,

c) If the company ships a brake pad with a thickness of 1.3 inches, the company will impose a loss of $1.92 on the customer. The loss is due to the thickness being 0.2 inches off target.

92.1)1.13.1(48$ 2 L

36$)75.0)(48($

)(

kLossE

Problem 6d) Expected loss

The company imposes, on average, a loss of $36 on its customers from shipping off target units. (Note that some units will have a loss Greater than $36 but other units will a loss of less than $36. The average of the losses of all units shipped will be $36.)

Problem 7

• Ashi Newspapers on April 17, 1979 reported that: Identical sets were assembled by Sony in a plant in Japan and in a plant in the U.S. with same design, and the same parts.

• U.S. customers preferred TV’s assembled in Japan to those assembled in U.S., because of better color.

• The U. S. plant performed 100% inspection. As a result, none of the sets assembled in U.S. were out of specs.

• However, Japan shipped all set “as is” without inspection.

Problem 7

• The result was that 0.0027% of sets assembled in Japan were out-of-spec, and thus defective.

• The color density (the quality characteristic of interest, X) of sets produced in Japan was normally distributed, while the color density of the sets produced in the U.S. had a uniform distribution. The specification limits are 10 and 20.

• The distributions at each plant are shown below.

Color DensityDistribution

y

10 15 20

Problem 7

US-built sets:100% within limit

Japan-built sets:.3% out of limits

The cost of repairing a TV set that is at the specification limits to the target value of 15 was $6 at both plants.

a. Find the expected loss at each plant and explain the meaning of your answer. (Ans. U.S. plant, $2.00; Tokyo plant, $0.69)

b. Why is the expected loss less at the plant that has a higher percentage of out-of-spec TV sets?

c. Is your result consistent with the response of consumers? Hint: If the random variable X has a uniform distribution over the range a to b, variance of the distribution is

12

)( 2ab

For the Tokyo plant, you need to use the appropriate table to find the z value (standard normal random variable) that corresponds to the desired area under the normal distribution. Once you find the z value, you should be able to obtain the standard deviation.

Problem 7

Solution to 7a –Tokyo Defect rate = .003. Split between two tails

beyond spec limits = .0015 in each tail.

LSL USL

.0015 .0015

1510 20

LSL USL

Use Appendix B, p. 652 to find z. Since the tail area

above USL is .0015 and Appendix B gives the area

between 0 and z, we Look up the area between 0

and z, which is .5000 -.0015 =.4985

.0015 .0015

z = 2.9650

Table area =.4985

From Appendix B, the z value is z = 2.96 or 2.97,

so split z to get 2.965. See next slide.

z Table (Text, p. 652)

z .00 .01 .02 . . . .06 .07

0.0

0.1

0.2

.

.

.

2.9 .4985 .4985

Solution to 7a -Tokyo

Use formula for z to solve for standarddeviation:

69.11520

965.2

meanUSL

z

Solution to 5a - Tokyo

24.0$5

6$22

a

Rk

Expected loss

69.0$)69.1)(24.0($

)(2

kLossE

Solution to 5a-Tokyo

• On average, each TV set shipped from the Tokyo plant imposes a loss $0.69 on the customer.

Solution to 5a – U.S. Plant

33.812

)1020(

12

)( 222

ab

00.2$)33.8)(24.0($

)(

kLossE

Solution to 5b• Why is the expected loss less at the plant

that is producing a higher percentage of out of spec sets—the Tokyo plant?

Solution to 5b

• E(Loss -Tokyo) =$.69 • E(Loss - U.S.) =$2.00 • The reason why the expected loss is less at

the Tokyo plant than at the U.S. plant is because the variance is smaller at the Tokyo plant, so there is less variability around the target.

• The sets are therefore more consistently on target, even if 3 out of 1000 sets are out-of-spec.

Solution to 5b

• At the U.S. plant, all sets are in spec. but they are haphazardly on target.

• Since the U.S. distribution is uniform, the percentage of sets on target is the same as the percentage of set near either on of the spec. limits.

• This means that the customer is just as likely to get a set that is on target as one that it on the lower or upper spec. limit.

Solution to 5b• Note that, although the U.S. plant is

performing 100% inspection and is not producing any out-of-spec sets, it still has a higher expected loss. Why?

• The reason is that mass inspection will not reduce the expected loss because it does not reduce the variance of color density.

• To reduce variance, we must improve the process by, for example, using better parts and/or materials, better maintenance of tools and equipment, etc.