An economic comparison of CUSUM and Shewhart charts

Figures & data

Table 1 Nomenclature

Notation	Description
λ	Average rate of occurrence of the assignable causes
λ j	Average rate of occurrence of assignable cause j (j = 1, 2)
τ	Expected time of occurrence of the assignable cause within an interval of length h
γ	Probability that an assignable cause will occur in an interval of h time units
γ j	Probability that assignable cause j will occur in an interval of h time units
n	Sample size
h	Sampling interval
k s	Control limit for the Shewhart chart
k c	Reference value for the CUSUM chart
H	Control limit for the CUSUM chart
σ	Standard deviation of the quality characteristic
μ0	Target mean of the quality characteristic
δ	Magnitude of the shift in the mean, measured in standard deviations
μ1	Mean of the quality characteristic when the first assignable cause has occurred
μ2	Mean of the quality characteristic when the second assignable cause has occurred
δ1	Index variable indicating whether production continues during searches (1-yes, 0-no)
δ2	Index variable indicating whether production continues during repairs (1-yes, 0-no)
L 1	Cost of restoring the process after a true alarm
L 0	Cost of a false alarm
c	Sampling and inspection cost per unit
b	Fixed cost per sample
M	Additional expected cost per time unit of operation when the process is out of control
C ij	Expected cost between two successive samples associated with departure from state (Y t = i, a t = j)
T ij	Expected time between two successive samples associated with departure from state (Y t = i, a t = j)
ARL 0	ARL while the process is in control
ARL δ	ARL while the process operates out of control with mean μ1 or μ2
α	Probability of a type I error at each sample when using a Shewhart chart
β	Probability of a type II error at each sample when using a Shewhart chart
T 0	Expected search time when a false alarm is issued
T 1	Expected search time to uncover an existing assignable cause
T 2	Expected time to repair the process
g	Time to sample and chart one item
Y t	Actual state of the process when sample t is taken
C t ^h	CUSUM statistic for detecting upward shifts
C t ^l	CUSUM statistic for detecting downward shifts
C t	Unique CUSUM statistic for use when following the approach of Crosier (1986)
m	Number of subintervals (from zero to H and from −H to zero)
w	Width of subintervals (from −(m − 1) to m − 1)
p ij ^kl	Transition probabilities of CUSUM scheme during a sampling interval where i, j (= − m, …, m) represent the value of C t and k, l (= 0, 1, 2) represent the state of the process
π ij	Steady-state probabilities where i (= 0,1,2) represents the state of the process and j (= 0, 1) represents the decision whether to overhaul (1) or not (0) (for Shewhart charts) or j (= –m, …, m) represents the value of C t (for CUSUM charts)
z t	Standardized observation at sample t
φ (z)	Density function of the standard normal distribution

Fig. 1 Time between two successive sampling instances associated with the departure from each of the six states of the Markov chain.

Table 2 Parameter sets of the 48 numerical examples (c = 1 or 4, L ₁ = 200, g = 0, T ₀ = T ₁ = T ₂ = 0, δ₁ = δ₂ = 0)

Case	b	M	L 0	λ	δ
1	0	100	100	0.01	0.5
2	0	100	200	0.01	0.5
3	0	1000	100	0.01	0.5
4	0	1000	200	0.01	0.5
5	5	100	100	0.01	0.5
6	5	100	200	0.01	0.5
7	5	1000	100	0.01	0.5
8	5	1000	200	0.01	0.5
9	0	100	100	0.1	0.5
10	0	100	200	0.1	0.5
11	0	1000	100	0.1	0.5
12	0	1000	200	0.1	0.5
13	5	100	100	0.1	0.5
14	5	100	200	0.1	0.5
15	5	1000	100	0.1	0.5
16	5	1000	200	0.1	0.5
17	0	100	100	0.01	1
18	0	100	200	0.01	1
19	0	1000	100	0.01	1
20	0	1000	200	0.01	1
21	5	100	100	0.01	1
22	5	100	200	0.01	1
23	5	1000	100	0.01	1
24	5	1000	200	0.01	1
25	0	100	100	0.1	1
26	0	100	200	0.1	1
27	0	1000	100	0.1	1
28	0	1000	200	0.1	1
29	5	100	100	0.1	1
30	5	100	200	0.1	1
31	5	1000	100	0.1	1
32	5	1000	200	0.1	1
33	0	100	100	0.01	2
34	0	100	200	0.01	2
35	0	1000	100	0.01	2
36	0	1000	200	0.01	2
37	5	100	100	0.01	2
38	5	100	200	0.01	2
39	5	1000	100	0.01	2
40	5	1000	200	0.01	2
41	0	100	100	0.1	2
42	0	100	200	0.1	2
43	0	1000	100	0.1	2
44	0	1000	200	0.1	2
45	5	100	100	0.1	2
46	5	100	200	0.1	2
47	5	1000	100	0.1	2
48	5	1000	200	0.1	2

Table 3 Shewhart charts compared with CUSUM charts (c = 1)

	Optimal Shewhart				Optimal CUSUM
Case	h	n	k s	ECT 1	h	n	k c	H	ECT 2	Percentage cost improvement (%)
1	7.2	24	1.6	11.76	6.9	23	1.1	0.6	11.72	0.4
2	8.3	32	1.9	12.64	7.8	30	1.3	0.7	12.59	0.4
3	2.2	24	1.6	33.92	2.1	23	1.1	0.6	33.77	0.4
4	2.5	34	2.0	36.81	2.4	31	1.3	0.7	36.64	0.5
5	8.3	27	1.6	12.41	8.0	26	1.1	0.6	12.39	0.2
6	9.1	34	1.9	13.22	9.0	34	1.4	0.6	13.19	0.2
7	2.6	28	1.6	36.01	2.5	27	1.2	0.5	35.94	0.2
8	2.8	35	1.9	38.69	2.7	34	1.4	0.6	38.58	0.3
9	2.8	21	1.5	45.46	2.6	20	0.9	0.7	45.32	0.3
10	3.2	28	1.8	47.70	3.0	26	1.1	0.8	47.53	0.4
11	0.7	23	1.6	117.66	0.7	23	1.1	0.6	117.18	0.4
12	0.8	31	1.9	126.46	0.8	31	1.3	0.7	125.90	0.4
13	3.3	23	1.4	47.13	3.2	22	0.9	0.6	47.05	0.2
14	3.6	31	1.8	49.17	3.5	30	1.2	0.7	49.08	0.2
15	0.8	26	1.6	124.09	0.8	26	1.1	0.6	123.86	0.2
16	0.9	34	1.9	132.18	0.9	34	1.4	0.6	131.86	0.2
17	4.4	10	2.2	7.82	4.4	10	1.5	0.8	7.79	0.4
18	4.7	12	2.5	8.20	4.7	12	1.7	0.9	8.16	0.5
19	1.4	10	2.2	20.82	1.3	10	1.6	0.7	20.72	0.5
20	1.6	13	2.5	22.03	1.4	12	1.7	0.9	21.90	0.6
21	5.9	12	2.2	8.78	5.8	12	1.7	0.6	8.77	0.1
22	6.3	14	2.4	9.09	6.2	14	1.9	0.6	9.08	0.2
23	1.8	12	2.2	23.91	1.8	12	1.7	0.6	23.89	0.1
24	1.9	14	2.5	24.93	1.9	14	1.9	0.6	24.88	0.2
25	1.7	10	2.2	35.90	1.5	9	1.4	0.9	35.80	0.3
26	1.9	12	2.4	36.90	1.7	11	1.6	0.9	36.78	0.3
27	0.5	11	2.2	78.39	0.4	9	1.5	0.8	77.95	0.6
28	0.5	13	2.5	81.97	0.5	12	1.7	0.8	81.66	0.4
29	2.2	11	2.1	38.48	2.2	11	1.6	0.6	38.45	0.1
30	2.3	13	2.4	39.30	2.3	13	1.8	0.6	39.27	0.1
31	0.6	12	2.2	87.81	0.6	12	1.7	0.5	87.75	0.1
32	0.6	14	2.5	90.96	0.6	13	1.8	0.7	90.82	0.2
33	2.7	4	2.8	5.31	2.2	3	1.7	1.1	5.29	0.4
34	2.8	4	2.9	5.46	2.7	4	2.0	1.0	5.44	0.5
35	0.9	4	2.7	12.64	0.7	3	1.7	1.1	12.57	0.6
36	0.9	4	2.9	13.14	0.8	4	2.0	1.1	13.05	0.7
37	4.6	5	2.7	6.71	4.5	5	2.2	0.6	6.71	0.0
38	4.5	5	2.9	6.81	4.6	5	2.3	0.6	6.81	0.0
39	1.4	5	2.7	17.13	1.4	5	2.2	0.6	17.12	0.1
40	1.4	5	2.9	17.47	1.4	5	2.3	0.6	17.46	0.0
41	1.0	4	2.8	29.26	0.8	3	1.7	1.1	29.16	0.3
42	1.0	4	2.9	29.66	1.0	4	2.0	1.0	29.59	0.2
43	0.3	4	2.7	53.17	0.2	3	1.7	1.2	53.02	0.3
44	0.3	4	2.9	54.75	0.3	4	2.0	1.0	54.52	0.4
45	1.7	5	2.7	33.11	1.6	4	2.0	0.6	33.11	0.0
46	1.7	5	2.9	33.38	1.7	5	2.2	0.7	33.36	0.0
47	0.5	5	2.7	67.26	0.5	5	2.2	0.5	67.26	0.0
48	0.5	5	2.8	68.34	0.5	5	2.2	0.7	68.29	0.1

Table 4 Shewhart charts compared with CUSUM charts (c = 4)

	Optimal Shewhart				Optimal CUSUM
Case	h	n	k s	ECT 1	h	n	k c	H	ECT 2	Percentage cost improvement (%)
1	14.9	0	*	14.73	14.9	0	*	*	14.73	0.0
2	12.6	15	1.2	18.94	11.9	14	0.7	0.6	18.86	0.4
3	4.5	0	*	45.37	4.5	0	*	*	45.37	0.0
4	3.8	17	1.3	58.68	3.6	16	0.8	0.6	58.42	0.4
5	14.9	0	*	14.73	14.9	0	*	*	14.73	0.0
6	13.4	16	1.2	19.32	12.7	15	0.7	0.6	19.26	0.3
7	4.5	0	*	45.37	4.5	0	*	*	45.37	0.0
8	4.0	18	1.3	59.97	3.8	17	0.8	0.6	59.79	0.3
9	5.7	0	*	48.94	5.7	0	*	*	48.94	0.0
10	8.2	0	*	56.15	8.2	0	*	*	56.15	0.0
11	1.5	0	*	147.34	1.5	0	*	*	147.34	0.0
12	1.3	16	1.2	189.41	1.2	14	0.7	0.6	188.57	0.4
13	5.7	0	*	48.94	5.7	0	*	*	48.94	0.0
14	8.2	0	*	56.15	8.2	0	*	*	56.15	0.0
15	1.5	0	*	147.34	1.5	0	*	*	147.34	0.0
16	1.4	17	1.2	193.23	1.3	16	0.7	0.6	192.60	0.3
17	7.2	6	1.6	11.76	7.1	6	1.1	0.6	11.72	0.4
18	8.3	8	1.9	12.64	8.1	8	1.3	0.7	12.59	0.4
19	2.2	6	1.6	33.92	2.2	6	1.1	0.6	33.78	0.4
20	2.5	8	1.9	36.82	2.5	8	1.4	0.6	36.65	0.5
21	8.5	7	1.6	12.41	8.4	7	1.2	0.5	12.39	0.1
22	9.5	9	1.9	13.22	8.6	8	1.3	0.7	13.19	0.2
23	2.6	7	1.6	36.01	2.5	7	1.2	0.5	35.95	0.2
24	2.9	9	1.9	38.69	2.8	9	1.4	0.6	38.59	0.3
25	2.7	5	1.5	45.47	2.6	5	0.9	0.7	45.32	0.3
26	3.2	7	1.8	47.70	3.1	7	1.2	0.7	47.54	0.3
27	0.7	6	1.6	117.68	0.7	6	1.1	0.6	117.21	0.4
28	0.8	8	1.9	126.49	0.8	8	1.3	0.7	125.96	0.4
29	3.2	6	1.5	47.13	3.3	6	1.0	0.5	47.07	0.1
30	3.7	8	1.8	49.18	3.5	7	1.1	0.7	49.09	0.2
31	0.9	7	1.6	124.23	0.8	6	1.1	0.5	123.95	0.2
32	0.9	8	1.9	132.36	0.9	8	1.3	0.7	132.00	0.3
33	4.9	3	2.3	7.86	3.7	2	1.4	0.9	7.82	0.4
34	4.7	3	2.5	8.20	4.7	3	1.7	0.9	8.16	0.5
35	1.5	3	2.3	20.90	1.1	2	1.4	0.9	20.85	0.3
36	1.5	3	2.5	22.03	1.4	3	1.7	0.9	21.90	0.6
37	5.9	3	2.2	8.78	5.8	3	1.7	0.6	8.77	0.1
38	6.7	4	2.5	9.13	5.7	3	1.7	0.8	9.11	0.2
39	1.8	3	2.2	23.91	1.8	3	1.7	0.6	23.89	0.1
40	2.0	4	2.5	25.03	2.0	4	2.0	0.6	25.00	0.1
41	1.5	2	2.1	36.03	1.4	2	1.3	1.0	35.84	0.5
42	1.9	3	2.4	36.90	1.8	3	1.7	0.8	36.81	0.2
43	0.5	3	2.3	78.60	0.4	2	1.3	1.0	78.44	0.2
44	0.5	3	2.4	82.04	0.5	3	1.7	0.8	81.66	0.5
45	2.3	3	2.1	38.50	2.3	3	1.6	0.6	38.47	0.1
46	2.3	3	2.3	39.35	2.2	3	1.7	0.7	39.29	0.2
47	0.6	3	2.2	87.81	0.6	3	1.7	0.5	87.75	0.1
48	0.7	4	2.5	91.36	0.6	3	1.7	0.7	91.17	0.2

Table 5 Shewhart charts compared with CUSUM charts for n = 1 (c = 1)

	Optimal Shewhart			Optimal CUSUM
Case	h	k s	ECT 1	h	k c	H	ECT 2	Percentage cost improvement (%)
1	14.9	*	14.73	0.3	0.2	6.6	12.57	14.7
2	21.5	*	19.32	0.3	0.2	8.0	13.77	28.7
3	4.5	*	45.37	0.09	0.2	6.7	36.72	19.1
4	6.5	*	62.58	0.07	0.2	9.1	40.46	35.3
5	14.9	*	14.73	14.9	*	*	14.73	0.0
6	21.5	*	19.32	21.5	*	*	19.32	0.0
7	4.5	*	45.37	4.5	*	*	45.37	0.0
8	6.5	*	62.58	6.5	*	*	62.58	0.0
9	5.7	*	48.94	0.1	0.2	7.1	47.44	3.1
10	8.2	*	56.15	0.1	0.2	8.5	49.75	11.4
11	1.5	*	147.34	0.03	0.2	6.6	125.68	14.7
12	2.1	*	193.26	0.03	0.2	8.0	137.72	28.7
13	5.7	*	48.94	5.7	*	*	48.94	0.0
14	8.2	*	56.15	8.2	*	*	56.15	0.0
15	1.5	*	147.34	1.5	*	*	147.34	0.0
16	2.1	*	193.26	2.1	*	*	193.26	0.0
17	0.7	2.2	12.46	0.4	0.5	4.5	8.43	32.3
18	0.5	2.5	15.21	0.4	0.5	5.1	8.91	41.4
19	0.2	2.2	36.65	0.1	0.5	4.9	23.14	36.9
20	0.2	2.4	46.07	0.1	0.5	5.5	24.46	46.9
21	14.9	*	14.73	1.8	0.4	2.3	13.78	6.5
22	2.9	1.7	18.77	1.4	0.4	3.3	15.23	18.9
23	4.5	*	45.37	0.5	0.4	2.4	40.89	9.9
24	0.7	1.8	58.72	0.4	0.4	3.5	45.80	22.0
25	0.3	2.2	46.38	0.2	0.4	4.5	37.45	19.2
26	0.3	2.4	52.24	0.2	0.4	5.3	38.94	25.5
27	0.07	2.2	124.56	0.04	0.5	4.5	84.34	32.3
28	0.05	2.5	152.11	0.04	0.5	5.1	89.14	41.4
29	5.7	*	48.94	5.7	*	*	48.94	0.0
30	8.2	*	56.15	0.7	0.4	2.8	52.88	5.8
31	1.5	*	147.34	0.2	0.4	2.1	138.05	6.3
32	0.3	1.7	187.83	0.2	0.4	2.7	155.09	17.4
33	0.8	2.6	6.79	0.7	1.0	2.6	5.61	17.4
34	0.7	2.8	7.65	0.6	1.0	3.1	5.82	24.0
35	0.3	2.5	17.57	0.2	1.0	2.7	13.61	22.6
36	0.2	2.8	20.42	0.2	1.0	3.0	14.28	30.0
37	2.8	2.0	9.64	2.4	0.9	1.6	9.12	5.3
38	2.6	2.2	10.89	2.0	0.9	2.1	9.78	10.2
39	0.9	2.0	26.93	0.7	0.9	1.6	25.16	6.6
40	0.8	2.2	31.09	0.6	0.9	2.1	27.31	12.1
41	0.3	2.6	33.04	0.3	0.9	2.6	30.06	9.0
42	0.3	2.7	35.31	0.2	1.0	3.1	30.65	13.2
43	0.08	2.6	67.89	0.07	1.0	2.6	56.05	17.4
44	0.07	2.8	76.54	0.06	1.0	3.1	58.18	24.0
45	1.2	1.9	40.11	1.0	0.8	1.6	39.02	2.7
46	1.2	2.1	43.11	0.9	0.9	1.9	40.67	5.7
47	0.3	1.9	96.40	0.2	0.9	1.8	91.88	4.7
48	0.3	2.1	109.24	0.2	0.9	2.1	97.77	10.5

Crosier , R. B. 1986 . A new two-sided cumulative sum quality control scheme . Technometrics , 28 : 187 – 194 .

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