MECH$\mathrm{.}5710$ Quality Engineering

Homework Assignment week #$4$

Elements of $6$ Sigma

$1.$ For the following processes, complete the given table:

Specification Process $\%$ above $\%$ below

Nominal Tol Mean StdD$.\%$OK USL LSL Cp Cpk

$10\mathrm{.}00+0\mathrm{.}0410\mathrm{.}000\mathrm{.}0295\mathrm{.}45=(1002*2\mathrm{.}275)2\mathrm{.}275\%2\mathrm{.}275\%0\mathrm{.}670\mathrm{.}67$

$10\mathrm{.}00+0\mathrm{.}049\mathrm{.}990\mathrm{.}02$

$10\mathrm{.}00+0\mathrm{.}0410\mathrm{.}010\mathrm{.}02$

$10\mathrm{.}00+0\mathrm{.}0510\mathrm{.}000\mathrm{.}02$

$10\mathrm{.}00+0\mathrm{.}0610\mathrm{.}010\mathrm{.}02$

C$\_$p$=($USL$-$LSL$)/6\backslash$sigma

C$\_$pk$=$min$[($USL$-\backslash$mu $)/3\backslash$sigma $,(\backslash$mu $-$LSL$)/3\backslash$sigma $]$

$$Z$(\backslash$pm SL$)=3\backslash$times C$\_$pk $(\backslash$pm SL$)$

Row a$.$

C$\_$p$=(+0\mathrm{.}04-(-0\mathrm{.}04))/(6\backslash$times $0\mathrm{.}02)=0\mathrm{.}67$

C$\_$pk$=$min$[(10\mathrm{.}04-10\mathrm{.}00)/(3\backslash$times $0\mathrm{.}02),(10\mathrm{.}00-9\mathrm{.}96)/(3\backslash$times $0\mathrm{.}02)]=$min$[0\mathrm{.}67,0\mathrm{.}67]=0\mathrm{.}67$

$$Z$=3\backslash$times C$\_$pk$=2$

$\%$above USL,f$(2)=0\mathrm{.}00275$; $\%$below LSL$,$f$(2)=0\mathrm{.}00275$

$\%$OK$=1-0\mathrm{.}00275-0\mathrm{.}00275=0\mathrm{.}9545$

$2.$ The Lowell MFG Co$.$ has a policy of Cp $=1.$XY $($For X$,$ Y use your last two$-$digit UML ID$)$

for all new products. All sheet metal specifications are designed with a tolerance of $+4$ mils. The control charts of the sheet metal process in manufacturing are maintained with a daily monitoring of $16$ samples. The cost of each reject was calculated at $ $5\mathrm{.}00.$ Assume a manufacturing process average or X double bar of $100$ mils.

a$.$ What should the X bar and R chart control limits be to meet the CP target?

X double bar $\_\_\_\_$ UCLx $\_\_\_\_\_\_$ LCLx $\_\_\_\_\_$ Rbar$\_\_\_\_$ QLF $\_\_\_\_\_\_$

b$.$ What is the reject rate for new parts $\_\_\_\_\_\_\_\_,\_\_\_\_\_\_\_$ppm

c$.$ If the sheet metal punch press average was shifted by $1$ mil, what will the new Reject Rate be:

PPM$\_\_\_,$ Cpk $\_\_\_\_\_\_$

d$.$ If the Motorola definition of Cp was used $(1\mathrm{.}5$ sigma shift$),$ what is the reject rate?

$\_\_\_\_\_\_\_\_\%,\_\_\_\_\_\_\_$PPM$,$ Cpk $\_\_\_\_\_\_\_$

MECH$\mathrm{.}5710$ Quality Engineering

Homework Assignment week #$4$

Elements of $6$ Sigma

$3.$ The Lawrence MFG Co$.$ has a policy of Cp $=1.$XY $($For X$,$ Y use your last two$-$digit UML ID$)$

for all new products. The sheet metal department has an X Bar control chart with X double bar at $100$ mils, Upper Control Limit of $109$ mils and Lower Control Limit of $91$ mils, based on taking $9$ daily samples. A new housing is being designed with a critical parameter of $100$ mils. The cost of each reject was calculated at $$1\mathrm{.}00.$

What should the product specifications be to meet the Cp target?

USL$\_\_\_\_\_\_$ LSL$\_\_\_\_\_\_\_\_\_$

b$.$ What do expect the reject rate $\_\_\_\_\%,\_\_\_\_$PPM

c$.$ If you were using the MOTOROLA definition of Cp$=1\mathrm{.}2($all processes have an average shift of $1\mathrm{.}5$ sigma$).$ What is the reject rate in ppm$?$ QLF using the actual process as indicated by the Xbar chart?

RR $\_\_\_\_\_\_$ppm$,$ QLF $\_\_\_\_\_$