Smith Manufacturing is currently considering using one of three suppliers. For a specific part, Smith's upper specification limit (USL) is 13.

Smith Manufacturing is currently considering using one of three suppliers.  For a specific part, Smith's upper specification limit (USL) is 13.8 centimeters (cm) and its lower specification limit (LSL) is 10.8 centimeters (cm). 

a) The first supplier, ABC, can adjust its mean but cannot reduce its standard deviation. Its standard deviation is 0.3 cm.  What is the range (lower and upper limits) for the mean of the process if ABC wants its process capability index (Cpk) to satisfy at least Smith's minimum acceptable value of 1.4?

b) The second supplier, DEF, cannot adjust the mean of its process which is currently 11.8 cm.  However, it can improve its standard deviation if necessary.  What is the maximum standard deviation (σ) allowed if DEF wants its process capability index (Cpk) to be at least the minimum acceptable value of 1.4?

c) The third supplier, GEF has a process whose standard deviation is 0.6 cm and the mean of its process is 12.0.  Smith is willing to adjust its specification limits if necessary to achieve a Cpk of 1.4?  What are the values of the specification limits that would be required to achieve the minimum acceptable Cpk?