한 제조공정의 표준편차는 0.005이고, 규격은 0.99± 0.02로 주어져 있다.(1) 공정평균이 0.99로 목표값(Target)과 일치할 때, 공정능력지수 와 및 시그마 수준을 구하라.(2) 공정평균이 1.00으로 이동한 경우, 와 및 시그마 수준을 구하라.

1. Break Down the Problem

We need to calculate the following:

1. Process Capability Index $C_p$
2. Process Capability Index $C_{pk}$
3. Sigma level for two scenarios:
   • When the process average is 0.99
   • When the process average is 1.00

2. Relevant Concepts

1. The formulas for process capability indices are: $C_p = \frac{USL - LSL}{6\sigma}$ $C_{pk} = \min{(C_{pu}, C_{pl})}$ where:
   • $USL$ = Upper Specification Limit
   • $LSL$ = Lower Specification Limit
   • $\sigma$ = Standard deviation
   • $C_{pu} = \frac{USL - \mu}{3\sigma}$
   • $C_{pl} = \frac{\mu - LSL}{3\sigma}$

3. Analysis and Detail

Given Data:

• Standard deviation $\sigma = 0.005$
• Target (process mean) when it matches = 0.99
• Upper Specification Limit (USL) = 0.99 + 0.02 = 1.01
• Lower Specification Limit (LSL) = 0.99 - 0.02 = 0.97

(1) When Process Mean is 0.99

1. Calculate $C_p$: $C_p = \frac{1.01 - 0.97}{6 \times 0.005} = \frac{0.04}{0.03} = 1.33$

2. Calculate $C_{pu}$ and $C_{pl}$:

   • $C_{pu}$: $C_{pu} = \frac{1.01 - 0.99}{3 \times 0.005} = \frac{0.02}{0.015} = 1.33$
   • $C_{pl}$: $C_{pl} = \frac{0.99 - 0.97}{3 \times 0.005} = \frac{0.02}{0.015} = 1.33$

3. Thus, $C_{pk}$: $C_{pk} = \min{(C_{pu}, C_{pl})} = \min{(1.33, 1.33)} = 1.33$

4. Sigma Level: $\text{Sigma Level} = C_p \times 3 = 1.33 \times 3 = 4.00$

(2) When Process Mean is 1.00

1. Calculate $C_p$ (remains the same): $C_p = \frac{1.01 - 0.97}{6 \times 0.005} = \frac{0.04}{0.03} = 1.33$

2. Calculate $C_{pu}$ and $C_{pl}$:

   • $C_{pu}$: $C_{pu} = \frac{1.01 - 1.00}{3 \times 0.005} = \frac{0.01}{0.015} = 0.67$
   • $C_{pl}$: $C_{pl} = \frac{1.00 - 0.97}{3 \times 0.005} = \frac{0.03}{0.015} = 2.00$

3. Thus, $C_{pk}$: $C_{pk} = \min{(C_{pu}, C_{pl})} = \min{(0.67, 2.00)} = 0.67$

4. Sigma Level: $\text{Sigma Level} = C_p \times 3 = 1.33 \times 3 = 4.00$

4. Verify and Summarize

• For the process mean of 0.99:
  • $C_p = 1.33$, $C_{pk} = 1.33$, Sigma Level = 4.00
• For the process mean of 1.00:
  • $C_p = 1.33$, $C_{pk} = 0.67$, Sigma Level = 4.00

Final Answer

1. When Process Mean is 0.99:

   • $C_p = 1.33$
   • $C_{pk} = 1.33$
   • Sigma Level = 4.00

2. When Process Mean is 1.00:

   • $C_p = 1.33$
   • $C_{pk} = 0.67$
   • Sigma Level = 4.00