Question

K-Log produces cereals that are sold in boxes labeled to contain 390 grams. If the cereal
content is below 390 grams, K-Log may invite auditor’s scrutiny. Filling much more than 390 grams costs
the company since it essentially means giving away more of the product. Accordingly, K-Log has set
specification limits at 400± 10 grams for the weight of cereal boxes. Currently a filling machine fills the
boxes. The boxes weigh on average 395 grams with a standard deviation of 20 grams.

In parts a)-d) you should work with a process mean of 395 grams. This describes an unlikely, but otherwise ordinary situation. Part e) asks you to rework the problem with a process mean of 380 grams. This situation is possible but very unusual. There is a didactic point to its solution, so I would ask you to work through both situations.

a) Determine the process capability Cp ratio and the process capability index Cpk.
b) Briefly comment on the implication of your finding in part a).
c) Calculate the probability that a randomly selected cereal box will not conform to specifications.
d) For a process capability index of 2, determine what process targets (in terms of mean and standard
deviation of the filling process) are needed? (Hint: What are the required process characteristics to
achieve 6-sigma?)
e) Repeat parts a) and b) under the assumption that the mean is 380 grams instead of 395 grams.

Answer 1

a) Upper Specification Limit = 400+10 = 410g

Lower Specification Limit = 400-10 = 390g

Mean = 395g

Standard Deviation = 20

Cp = (USL - LSL ) /

= (410 - 390) / 20

Cp = 1

Cpk = min[USL−μ/3σ,μ−LSL/3σ]

= min[ 410-395 / (3x20) , 395-390 / (3x20) ]

= min(0.25,0.083)

Cpk = 0.083

b) The value of Cpk is too low to satisfy aby customer. There is a very high chance that the product will have a defect. The value of Cp=1 says that the process is capable with tight control.

c) The Cumulative Probability Associated with USL = 0.773

The Cumulative Probability Associated with LSL = 0.401

Hence probability of conforming = 0.773-0.401 =0.372

Therefore probability of non-conforming = 1-0.372 = 0.628

d) For a value Cp=2 we have Z=6

Process is 99.99% conforming.

2= (LSL-Mean) / Std. deviation

Mean=395

Std. deviation =0.75

e)

pper Specification Limit = 400+10 = 410g

Lower Specification Limit = 400-10 = 390g

Mean = 380g

Standard Deviation = 20

Cp = (USL - LSL ) /

= (410 - 380) / 20

Cp = 1.5

Cpk = min[USL−μ/3σ,μ−LSL/3σ]

= min[ 410-380 / (3x20) , 380-390 / (3x20) ]

= min(0.5,-5.16)

Cpk = -5.16

The value of Cpk is negative which means that the manufactured product will always be defective.The value of Cp=1 says that the process is capable.