Question

Control Chart Chocolate Viscosity 570 540 530 520 10 Subgroup # Control Chart-Chocolate Viscosity 50 s45 40 35 25 20 15 10 10 15 20 25 Subgroup #

Answer 1

Solution

Back-up Theory

Let Xdouble bar = (1/k)∑(i = 1, k)x_ibar, and sbar = (1/k)∑(i = 1, k)s_i where x_ibar = mean and s_i = standard deviation of the ith sub-group

The Xbar part of Xbar-s Chart has:

Central line: CL = Xdouble bar

Lower Control Limit: LCL = Xdouble bar – A₃sbar

Upper Control Limit: UCL = Xdouble bar + A₃sbar;

The s part of Xbar-s Chart has:

Central line: CL = sbar

Lower Control Limit: LCL = B₃sbar

Upper Control Limit: UCL = B₄sbar, where

A₃, B₃, B₄ are constants which can be directly obtained from standard Control Chart Constants Table.

Now to work out the solution,

Only final results are given below. Detailed working follow at the end.

Part (A)

Final CL and Control Limits are:

Xbar Chart
CL	554.1
LCL	528.7988
UCL	579.4013
sChart
CL	25.95
LCL	7.3698
UCL	44.5302
ANSWER 1

At the first stage 5 sub-groups were found to be out of control limits. Reworking the limits after eliminating these sub-groups, saw all remaining sub-groups within limits.

Since only 20% of the sub-groups were out of limits, the process is considered ‘in control’. ANSWER 2

Part (B)

Given 530 ± 90, tolerance band is 180.

Estimate of process standard deviation = sbar = 25.95 and (6 x sbar) = 155.7.

Since (6 x sbar) < tolerance band, the process is considered ‘capable of meeting the specification’.

However, since process centering being at 554, which is much above the center of the specification band, despite being capable process, rejections are likely. ANSWER

Part (C)

Cpk compares well with the cp indicating in-built process capability. Nevertheless, the actual performance would depend on the process centering. ANSWER

Part (D)

With Xdoublebar at 554.1, sbar at 25.95 and USL at 620, probability of rework

= P[Z > {(620 – 554.1)/25.95}], where Z ~ N(0, 1)

= P(Z > 2.539)

= 0.0056.[using Excel Function: Statistical NORMSDIST]

So, number of units requiring rework in a lot of 1000 = 1000 x 0.0056

= 5.6 ~ 6 ANSWER

DONE

Details of working

k	25
n	10
A3	0.975
B3	0.284
B4	1.716
∑xibar	13798
∑si	670
Xdouble bar	551.92
sbar	26.8
Xbar Chart
CL	551.92
LCL	525.79
UCL	578.05
sChart
CL	26.8
LCL	7.6112
UCL	45.9888
Subgroups 6,9,15,16, 18 are out of control limits. Deleting these sub- groups
k	20
n	10
A3	0.975
B3	0.284
B4	1.716
∑xibar	11082
∑si	519
Xdouble bar	554.1
sbar	25.95
Xbar Chart
CL	554.1
LCL	528.7988
UCL	579.4013
sChart
CL	25.95
LCL	7.3698
UCL	44.5302
Now all sub-groups are within limits.