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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
Spring 2008
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6.780 Assignment 4, 2008 -- solutions
Problem 8-1.
µ0 = 1050; σ = 25; δ = 1σ; K = (δ/2)σ = (1/2)25 = 12.5; H = 5σ = 5(25) = 125
(a)
MTB > Stat > Control Charts > Time-Weighted Charts > CUSUM
CUSUM Chart of Molecular Weight ( Ex8 -1 mole)
target = 1050, std dev = 25, k = 0.5, h = 5
1250
Cumulat ive Sum
1000
750
500
250
UCL= 125
0
0
LCL= -125
2
4
6
8
10
12
Sample
14
16
18
20
The process signals out of control at observation 10. The point at which the assignable
cause occurred can be determined by counting the number of increasing plot points. The
assignable cause occurred after observation 10 – 3 = 7.
(b)
σˆ = MR2 d 2 = 38.8421/1.128 = 34.4345 equation 5-6. No. The estimate used for σ is much smaller than that from the data.
Note: can equivalently use Sbar/c4, where Sbar is the mean of
the standard deviations of all pairs of consecutive samples
(a moving standard deviation, cf. a moving range)
8-2
Problem 8-2.
MTB > Stat > Control Charts > Time-Weighted Charts > CUSUM
CUSUM Chart of Standardized Molecular Weight ( Ex8 -2 std)
target = 1050, std dev = 25, k = 0.5, h = 5
50
Cumulat ive Sum
40
30
20
10
UCL= 5
0
0
LCL= -5
2
4
6
8
10
12
Sample
14
16
18
20
The process signals out of control at observation 10. The assignable cause occurred after
observation 10 – 3 = 7.
8-3
Problem 8-21 (8-19, 4th ed.).
λ = 0.1, L = 2.7, σ̂ = 12.16 , CL = µ0 = 950, UCL = 957.53, LCL = 942.47.
MTB > Stat > Control Charts > Time-Weighted Charts > EWMA
EWMA Chart of Temperature Readings ( Ex8 -7 temp)
lambda = 0.1, L = 2.7
960
+2.7SL= 957.53
EWMA
955
_
_
X= 950
950
945
-2.7SL= 942.47
1
8
16
24
32
40
48
Sample
56
64
72
80
Test Results for EWMA Chart of Ex8-7temp
TEST. One point beyond control limits.
Test Failed at points: 12, 13
Process is out of control at samples 8 (beyond upper limit, but not flagged on chart), 12 and
13.
Note that the estimate of sigma used here is again calculated
as MR2bar/d2 (the mean of a moving average of 2 samples, divided
by d2)
8-5
8-22 (8-20 4th ed.).
λ = 0.4, L = 3, σ̂ = 12.16 , CL = µ0 = 950, UCL = 968.24, LCL = 931.76.
MTB > Stat > Control Charts > Time-Weighted Charts > EWMA
EWMA Chart of Temperature Readings ( Ex8 -7 temp)
lambda = 0.4, L = 3
970
UCL= 968.24
EWMA
960
_
_
X= 950
950
940
LCL= 931.76
930
1
8
16
24
32
40
48
Sample
56
64
72
80
Test Results for EWMA Chart of Ex8-7temp
TEST. One point beyond control limits.
Test Failed at points: 70
With the larger λ, the process is out of control at observation 70, as compared to the chart
in the Exercise 21 (with the smaller λ) which signaled out of control at earlier samples.
8-6