Operations and Supply Chain Management The Core

QUALITY MANAGEMENT AND SIX SIGMA chapter 10 335
m
​
​∑​ ​ ​R​ j ​
R ¯
j=1
​ = ______ ​ ​ [10.14]
m
​Upper control limit for X ¯
​ = ​ X ​ + ​A​ 2 ​R ¯ ​ [10.15]
​Lower control limit for X ¯
​ = ​ X ​ − ​A​ 2 ​R ¯ ​ [10.16]
​Upper control limit for R = ​D​ 4 ​R ¯ ​ [10.17]
​Lower control limit for R = ​D​ 3 ​R ¯ ​ [10.18]
LO10–6 Analyze the quality of batches of items using statistics.
∙ Acceptance sampling is used to evaluate if a batch of parts, as received in an order from a
supplier for example, conforms to specification limits.
∙ An acceptance sampling plan is defined by a sample size and the number of acceptable
defects in the sample.
∙ Since the plan is defined using statistics, there is the possibility that a bad lot will be
accepted. This is called the consumer’s risk.
∙ There is also the possibility that a good lot will be rejected. This is called the producer’s risk.
SOLVED PROBLEMS
LO10–4 SOLVED PROBLEM 1
HVAC Manufacturing produces parts and materials for the heating, ventilation, and air conditioning
industry. One of its facilities produces metal ductwork in various sizes for the home construction
market. One particular product is 6-inch-diameter round metal ducting. It is a simple product,
but the diameter of the finished ducting is critical. If it is too small or large, contractors will have
difficulty fitting the ducting into other parts of the system. The target diameter is 6 inches exactly,
with an acceptable tolerance of ±0.03 inch. Anything produced outside specifications is considered
defective. The line supervisor for this product has data showing that the actual diameter of finished
product is 5.99 inches with a standard deviation of 0.01 inch.
a. What is the current capability index of this process? What is the probability of producing a
defective unit in this process?
b. The line supervisor thinks he will be able to adjust the process so that the mean diameter of
output is the same as the target diameter, without any change in the process variation. What
would the capability index be if he is successful? What would be the probability of producing
a defective unit in this adjusted process?
c. Through better training of employees and investment in equipment upgrades, the company
could produce output with a mean diameter equal to the target and a standard deviation of
0.005 inch. What would the capability index be if this were to happen? What would be the
probability of producing a defective unit in this case?
Solution
a. ​ X ​ = 5.99 LSL = 6.00 − 0.03 = 5.97 USL = 6.00 + 0.03 = 6.03 σ = 0.01​
​C​ pk ​ = min ​ _ 5.99 − 5.97
​ ​ or ​ _ 6.03 − 5.99
​ = min [0.667 or 1.333] = 0.667​
[ 0.03
0.03
​]
This process is not what would be considered capable. The capability index is based on the
LSL, showing that the process mean is lower than the target.