91影视

The lower control limit is a line below the centerline that indicates the number below which any individual data point would be considered out of statistical control due to special cause variation.

Upper Control Limit (UCL) means a value greater than the maximum value of a chemical or physical parameter that can be attributed to natural fluctuations and sampling and agree upon by the Administrator and the operator prior to initiation of mining.

Process capability index is

\({C_p} = \frac{{USL - LSL}}{{6\sigma }}\)

Were

\({C_p} = 1\)

Indicates a that a process is only marginally capable of meeting specifications. Ideally,

\({C_p} > 1.33,\)

(a).

In Exercise 4 (a), we have that\(\sigma \)= 0.02 from which we obtain

\({C_p} = \frac{{USL - LSL}}{{6\sigma }} = \frac{{3.1 - 2.9}}{{6 \times 0.02}} = 1.67 > 1.33\)

\({C_p} = \frac{{USL - LSL}}{{6\sigma }} = \frac{{3.1 - 2.9}}{{6 \times 0.05}} = 0.67 < 1\)

(b).

The capability measure that involves process mean is

\({C_{pk}} = \min \left\{ {\frac{{USL - \mu }}{{3\sigma }},\frac{{\mu - LSL}}{{3\sigma }}} \right\}\)

\(\frac{{USL - \mu }}{{3\sigma }} = \frac{{3.1 - 3.04}}{{3 \times 0.02}} = 1\)

\({C_{pk}} = \min \left\{ {1,2.33} \right\} = 1.\)

\(\frac{{USL - \mu }}{{3\sigma }} = \frac{{3.1 - 3}}{{3 \times 0.05}} = 0.67\)

\(\frac{{\mu - LSL}}{{3\sigma }} = \frac{{3 - 2.9}}{{3 \times 0.05}} = 0.67\)

\({C_{pk}} = \min \left\{ {0.67,0.67} \right\} = 0.67\)

(c).

\({C_{pk}} \le C\)

\(\frac{{USL - LSL}}{{6\sigma }} = \min \left\{ {\frac{{USL - \mu }}{{3\sigma }},\frac{{\mu - LSL}}{{3\sigma }}} \right\}\)

\(\frac{{USL - LSL}}{{6\sigma }} = \frac{{USL - \mu }}{{3\sigma }}\)

\(\mu = USL - \frac{{USL - LSL}}{2}\)

\(\mu = \frac{{USL + LSL}}{2}\)

\(\frac{{USL - \mu }}{{3\sigma }} = \frac{{USL - {{(USL + LSL)} \mathord{\left/

{\vphantom {{(USL + LSL)} 2}} \right.

\kern-\nulldelimiterspace} 2}}}{{3\sigma }} = \frac{{2USL - LSL - USL}}{{6\sigma }} = \frac{{USL - LSL}}{{6\sigma }} = {C_P}\)

\(\begin{array}{l}\frac{{\mu - LSL}}{{3\sigma }} = {C_P}\\{C_{pk}} = \min \left\{ {{C_P},{C_P}} \right\} = {C_P}\end{array}\)

We also claimed that,

\(\begin{array}{l}{C_{pk}} \le {C_P}\\\mu - LSL < USL - \mu \\{C_{pk}} = \frac{{\mu - LSL}}{{3\sigma }}\\\mu < \frac{{LSL + USL}}{2}\\{C_{pk}} = \frac{{\mu - LSL}}{{3\sigma }} < \frac{{{{(LSL + USL)} \mathord{\left/

{\vphantom {{(LSL + USL)} {2 - LSL}}} \right.

\kern-\nulldelimiterspace} {2 - LSL}}}}{{3\sigma }} = \frac{{LSL - LSL}}{{6\sigma }} = {C_P}\end{array}\)

Hence, the final answer is

a).\({C_P} = 1.67,{C_P} = 0.67\)

b).\({C_{pk}} = 1,{C_{pk}} = 0.67\)

c).\(c.\mu = \frac{{USL + LSL}}{2},{C_{pk}} \le {C_P}\)