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An algorithm for computing the economically optimal X -control chart for a process with multiple assignable causes
350
European Journal of Operational Research 72 (1994) 350-363 North-Holland
Theory and Methodology
An algorithm for computing the economically optimal X-control chart for a process with multiple assignable causes Kun-Jen Chung
*
Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, ROC Received May 1991; revised March 1992
Abstract: The main difficulties in the use of economic designs are the computations involved, the difficulty in specifying process parameters, and the fact that the sampling interval h rarely is a natural quantity of time. The economic design of X-charts involves the determination of the optimum values of the three control parameters: the sample size n, the control limit coefficient k and the sampling interval h. This paper deals with the economic design of X-charts for multiple assignable causes and develops an explicit equation for h in terms of n and k. With the explicit equation for h, we can give an optimization procedure, which is implementable in real time on a personal computer, for obtaining the optimal solutions of these parameters n, k and h for such charts. The results and the execution times of all numerical examples show that our optimization procedure is accurate and efficient. Keywords: Economic design; X-control chart; Process control; Assignable cause 1. Introduction
Duncan (1956) and Cowden (1957) independently pioneered the study of the design of Xcharts by an economic approach. Gibra (1975) provided a reference list of 68 periodicals and books dealing with quality control techniques. Vance (1983) complemented and updated this list by considering references form 1970 to 1980. Along with developments in control chart techniques, developments and improvements on the economic design have continued. Montgomery (1980) listed 51 references on the topic. The Correspondence to: Dr. Kun-Jen Chung, Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, ROC. * This research was supported by the National Science Council of the Republic of China (NSC 81-0415-E011-08).
above references reveal that Duncan's model has received particular attention and a considerable amount of work has been developed from it. However, despite the significant progress in the development of analytical models for the optimal economic determination of control chart parameters under different assumptions, those models have not gained analogous popularity in industry. Saniga and Shirland (1977) and Chiu and Wetherill (1975) report that very few practitioners have implemented economic models for the design of control charts. This is somewhat surprising since most practitioners claim that a major objective in the use of statistical process control procedures is to reduce costs. There are at least three reasons for the lack of practical implementation of this methodology. The first problem is the difficulty in estimating costs and other parameters. The difficulty is dis-
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 2 ) 0 0 1 5 2 - 9
K.-J. Chung / Algorithmfor computing optimalX-control charts cussed in Chiu (1976, 1977), Montgomery (1980) and Pignatiello and Tsai (1988), who indicate that in designing an X-chart control plan by an economic approach it is critical to be able to set the in-control process mean u 0 on target and to accurately estimate the process standard deviation ~r and the coefficient 6 of the effect of the assignable cause. Accurate estimates of other data p a r a m e t e r s are relatively less important. Therefore, costs do not have to be estimated with high precision although other model components such as the magnitude of the shift require relatively accurate determination. This is important, as some of the cost coefficients are difficult to determine precisely, such as the cost of producing defective units. The cost surfaces studied are generally flat in the vicinity of the optimum. Secondly, the mathematical models and their associated optimization schemes are relatively complex and are often presented in a m a n n e r that is difficult for the practitioner to understand and use. So, Lorenzen and Vance (1986) point out the need for optimization techniques implementable in real time on a personal computer. The third problem is the fact that the sampling interval h rarely is a natural quantity of time. For the X-chart, the following designs have been proposed: n = 5, h = 8, k = 3 (Ishikawa, 1976); n = 5, h = 1, k = 3 (Feigenbaum, 1983); n = 4 or 5, h = ?, k = 3 (Burr, 1953); n = 4, h = ?, k = 3 (Juran et al., 1974). For the p-chart, the following have been proposed: n > 5 0 with 3 < n / ~ < 4 , h = 8 , k = 3 (Ishikawa, 1976); n = 25, h = 1 or 8, k = 3 (Feigenbaum, 1983); n = (9 - 9 ~ ) / p , h = ?, k = 3 (Juran et al., 1974); n / 5 > 2 5 , h = ?, k = 3 (Cowden, 1957); n/5 > 1, h = ?, k = 3 (Burr, 1953); 100% inspection, h = 8, k = 3 (Grant and Leavenworth, 1980); n > 50 with @ > 4, h = ?, k = 3 (Juran et al., 1974). With the exception of k = 3, general guidelines on h simply do not exist. So, this p a p e r concerns the second and third problems and tries to solve them. A fundamental assumption of the process model studied is that there exists a single assignable cause which shifts the process mean by an amount 6o-. In practice, this assumption may not be satisfied, as it often occurs that a multiplicity of assignable causes may exist to operate on the process. So, Duncan (1971) extends his study on the economic design of X-charts to the situation of multiple assignable cause models.
351
The economic design of X-charts involves the determination of the optimum values of the three control parameters: the sample size n, the control limit coefficient k and the sampling interval h. Two models are considered in Duncan (1971). Model I assumes that when the process has been disturbed by a given assignable cause, it is free from the occurrence of other assignable causes. Model II will not be referred to here. D u n c a n (1971) finds that for multiple assignable cause model his approximate solution procedure to get the optimal economic design leads to values of the sampling interval h that are significantly different from the optimum values. So, Duncan's (1971) approximate solution procedure to get the optimal economic design does not always perform well. Tagaras (1989) proposes the power approximate method for the economic design of control charts based on an estimate of the power of the control chart at optimality. Tagaras' approximation method for multiple assignable causes essentially consists of two steps. First, multiple linear regression is employed for the derivation of the approximate formula expressing the power of the control chart for a matched single cause model as a function of the model parameters. Secondly, an approximate optimization procedure is then used to determine the economic design of the control chart for the predicted value of the chart's detection power. Tagaras' method for multiple assignable causes is academically interesting, but is practically very difficult. It may not appear concise to a practitioner with limited mathematical and computer programming knowledge. In this paper, we develop a simple and approximate algorithm which is implementable in real time on a personal computer, to get the optimal economic design. Results and execution times of all numerical examples show that our approximate solution algorithm is very accurate and efficient.
2. The m o d e l
The process is assumed to start in a state of statistical control with mean u 0 and standard deviation o.. T h e r e are s assignable causes that result in shifts in the process m e a n from u o to u 0 + 6jo. for the assignable cause A j where all 6 i are known. The time before the assignable cause
352
K.-J.
Chung / Algortihm for computing optimal X-control charts
Aj occurs has an exponential distribution with parameter Aj. Samples of size n are taken every h hours and the sample mean is plotted on an X-control chart with center line u 0 and control limits u o +_ktr/v'-n. If one point falls outside the control limits, a search for an assignable cause is made. The process, however, is allowed to continue in operation during the search until the assignable cause is discovered. It is assumed that the repair cost will not be charged against the net income from the process. Economic control chart models typically assume that the time to the occurrence of an assignable cause folows an exponential distribution. If the probability of a process shift within any small interval of time is directly proportional to the length of the interval, then this assumption is probably appropriate. However, if assignable causes occur as a result of the cumulative effects of heat, vibration, shock and other similar phenomena, or as a result of improper setup or excessive stress during process start up, then use of the exponential distribution to model the duration of in-control may not be appropriate. However, McWilliams (1989) extends the exponential distribution to the Weibull distribution in order to investigate in general the impact, on economic control chart parameters and hourly costs, of the distributional assumption. It is found that the economic design is quite insensitive to the assumed distribution. The Weibull distributions can have increasing, constant, or decreasing hazard functions. Shapes closely resembling normal and lognormal distributions are possible. The results obtained for the Weibull distributions are therefore expected for other commonly used distributions. Consequently, there is no loss of generality when we assume the time to the occurrence of an assignable cause follows an exponential distribution in economic control models. When the process is in control, the probability of a sampling point falling outside the control limits is
For simplicity we shall assume that only positive shifts (Sj > O) actually occur. It has also been assumed that the cost consists of five parts: T: the cost of looking for trouble when none exists; Wj: the cost of discovering cause Aj when it has occurred; Mr: the net increase in the loss per hour resulting from a large percentage of items outside the specification limits due to the occurrence of c a u s e Aj; b: the fixed cost of taking a sample that is independent of sample size; and c: the variable cost per item of sampling, testing and plotting. There are also two time parameters: g: the time for testing and analyzing a sample item; and D / the average time it takes to discover cause A t after it has occurred. Based on the model described in this section and the parameters just defined, Duncan (1971) has shown that the expected loss-cost per hour of operation is
a = 2qb(-k)
We shall not repeat the mathematical argument here.
(1)
where q~(z) denotes the distribution function of a standard normal variable z. When the assignable cause A i has occurred, the probability of a sampling point falling outside the control limit is
=
k) +
k).
(2)
~_#AjMj( h / Pj - ~'j + gn + Dj) + o
1 + ZAj(h/Pj
- "ri + gn + Di)
b + cn
(3)
where 1 - (1 + Ajh) e x p ( - A j h ) =
,,{1- exp(-a h)} 1
h
Aj
exp(Ajh) - 1
(4)
and A
=
exp(hZa,) l-exp(-hZ ,)
=
1 exp(hEa,)- 1" (5)
3. The explicit equation for h It can be shown that Theorems 1 and 2 hold.
K.-J. Chung / Algorithm for computing optimal ,Y-control charts Theorem
1
1 - -
hh
1
e ~h -
Theorem
lira
Theorem 1 tells us that the difference between 1 / ( h h ) and 1/(e ~ h - 1) is within 0 and 1. In Duncan (1956), it is shown the expected number of samples taken during the in-control period is given by 1/(e xh - 1). So, from the point of view of the expected number of samples taken during the in-control period, the number 1/(Ah) in general is a good approximation to 1/(e xh - 1). Many researchers, e.g. Duncan (1956), Goel, Jain and
1.
1>
>0
for a l l A > O and h > O .
2. 1 -
~-,o~ x
1 e~
-1
)
1 -
Z"
(
START
353
)
L* = L(0,0,h(0,0)), n0=l, n*=0, nstop = 0, k* = 0.
Lno = L(0,0,h(0,0)), k o = 1.00, kstop=0.
ho = h(n0,k o) and L(no,k0,h o) from (12) and (3), respectively,
~, r
Compute
*=
o,
*
=ho,
. P= ,
[.no =L(no,k ,h ).
k0=k0+u
kstop=kstop+l I N
[
(where u = 0.01 or 0.1. )
I n = n o, nstop = 0, 1"
=L(n * ,k * ,h * ) L,
N
i" O0 = n 0 + l
L l
L L(n * (
Figure 1. The flowchart for the procedure
*
•
,
,k ,h )
END
)
354
K.-J. Chung / Algortihm for computing optimal X-control charts
Table 1 Comparisons among 1/(e ~h - 1), 1/(Ah) and 1/(Ah)-
Wu (1968), Goel and Wu (1973), Montgomery (1982), Panagos, Heikes and Montgomery (1985), Lorenzen and Vance (1986), Tagaras and Lee (1989), Tagaras (1989) and others adopt 1 / ( e ah 1 ) - l / ( A h ) . However, examination of the numerical model literature shows that, with very few exceptions, cost/risk parameters leading to small Ah (0.4 or less) are of most practical interest. Table 1 reveals that 1/(Ah) - 51 is better approximation to 1 / ( e *h - 1) than 1/(Ah). By Theorems 1, 2 and Table 1, we take 0.5 as a correction number. Replacing 1/(exp(ajh)1) and 1 / ( e x p ( h E a j ) - 1) in (3)-(5) by 1 / ( a j h ) - 1 and 1 / ( h E a j ) - ½ respectively, we get equations (6)(9):
-- ~2hjh 2 for rj is. The author found that for the multiple assignable cause model, use of the approximation led to values of h that were significantly different from the optimum values" (p. 110). Chiu (1973) answers the above question by comparing the loss-cost values using (4) and (10) and further confirms (10). Although (10) has been shown to be satisfactory for practical purposes and was extensively used in subsequent research (see, for example, Goel, .lain and Wu, 1968; Chiu, 1974, 1975, 1977; Chiu and Wetherill, 1974; Montgomery, 1982; Panagos, Heikes and Montgomery, 1985; Lorenzen and Vance, 1986; and others), results of all numerical examples considered in this paper show that the approximation Tj -~ l h is better than the approximation vj 1h - ]~..j.o i ~ / , 2 . Approximating r~ with ~h 1 makes =• ~.. intuitive sense as it states, on average, the assignable cause will occur in the middle of the sampling interval. Due to the use of the approximation r j - ½h and Theorem 2, we can get an equation for the optimal sampling interval h as well. The detail is discussd as follows. For a given (n, k), setting the partial derivative of L in (6) with respect to h equal to zero yields
~=
rl h2 + r2h + r 3 = 0,
1
Ah
e ;~h --
0.01 0.05 0.20 0.40 0.60 0£0 1.00
1
1
99.501 19.504 4.517 2.033 1.216 0.816 0.582
1
1
Ah
ah
2
100 20 5 2.5 1.667 1.25 1.00
99.500 19.500 4.500 2.000 1.167 0.750 0.500
EajBjMj +a-.4TEa~ + EAjWj 1 + EajB,
÷j=
1
b +cn + - h '
(6) (7)
1
A=a
hEAj
2 '
h Bi = - - - ½h + gn + Dj.
(11)
((11))
where
r, =
E a , F, -
Mj (1 + Eaj(gn + D;))
(8) (9)
×
+,,,)M; +
So (6)-(8), in general, should be reasonably good approximations for (3)-(5). Regarding (4), Duncan (1956) has shown that rj is very well approximated by the equation r j - ½h - ~1 /~ j h 2 ,
(10)
which is the result of the series expansion of the right-hand side of (4) when terms of the order a~h 4 and higher are omitted. However, Duncan (1971) remarks: "Furthermore, there is a question about how good the approximation l h
r3= -[1+
£aj(gn
+&)]
+
+¢)]}
K.-J. Chung / Algortihm for computing optimal ,Y-control charts
So / 2
h =
- r 2 + Vr2 - 4rlr 3 2r~
=h(n,
k).
(12)
The above h is the unique, real positive solution of (11), and in general, this is the h that minimizes the cost function L(n, k, h) when n, k and h are fixed.
4. The optimization algorithm In the process of obtaining the approximate optimum values n*, k * and h*, we treat n and k as dicrete variables and assume that the unit length of k is u ( = 0.1 or 0.01). The numbers n and k have no u p p e r bounds. The stopping rules for n and k are described in the flowchart. The search algorithm is as follows. In each step of the outer loop, an n-value is given and the value of k is nested for the value of
355
n. Referring to Collani (1986), for n = 0, we set k = 0. Values of n = 0, k = 0 and h = h 0 > 0 m e a n that instead of sampling the product, we just examine the machine every h 0 hours for evidence of any changes in the machine and repair it if necessary. (i) Given a pair (n o, k o) where 0 _< n o, and 0 < k o. (ii) Calculate a and ~ from (1) and (2). (iii) Calculate h o from (12). (iv) Calculate L(n o, k o, h o) from (3). (v) Find L'n,, = minimumo<_k~(L(no, ko, ho)). (vi) Calculate L* = m i n i m u m 0 _
Table 2 Cost and risk factors and parameters for Duncan's model Example No.
~
a
M
e
D
T
W
b
c
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0,5 0.5 0.5
0.01 0.02 0.03 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
100.00 100.00 50.00 1000.00 10000.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 1000.00 12.87 128.70 12.87 12.87 12.87 2.25 225.00 2.25 2.25 2.25
0.05 0.05 0.05 0.05 0.05 0.05 0.50 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
50 50 50 50 50 50 50 50 5 500 5000 50 50 50 50 50 50 500 50 50 50 50 500 50 50
25,0 25,0 25.0 25.0 25.0 25.0 25.0 25.0 2.5 250.0 2500.0 25.0 25.0 25.0 25.0 25.0 25.0 250.0 25.0 25.0 25.0 25.0 250.0 25.0 25.0
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 5.00 0.50 0.50 0.50 0.50 0.50 0.50 5.00 0.50 0.50 0.50 0.50 5.00 0.50
0.10 0.10 0.10 0.10 0,10 0.10 0,10 0,10 0,10 0,10 0.10 0,10 1.00 I 0.00 1.00 0.10 0.10 0.10 0.10 1.00 0.10 0.10 0.10 0.10 1.00
356
/~-J.
Chung / Algortihm for computing optimal.Y-control charts
?.o ,--, II II . *
"~
7~
~..-.
e~
o i.
_=
e.
e.~
[. i
Q
u
c5 II
ta
K.-J. Chung / Algorithm for computing optimal X-control charts
357
t"q
[-
'
0
Z z -8 0
0
0 o
0 0
$
~ II
H ©
"-8
358
K.-J. Chung / Algortihm for computing optimal X-control charts
less than 1.00 and 0.001, respectively. Duncan (1986) has observed that in the economic design of X-charts the average run length in the out-ofcontrol state often has a value between 1 and 2. Duncan's observation supports that the optimal value of Pj is rarely less than 0.001. Hence, in the search algorithm we take n = 1 and k = 1 as the starting values of n and k and require Pj not less than 0.001 to avoid the overflow problem from 1/Pj. In general, M / A j is rather large and hence r I is greater than 0. Although this is so, if Pj is sufficiently small, r 1 may be nonpositive. To keep the sampling interval h in (12) positive, we should avoid the occurrence that r~ is nonpositive. Hence, we have to put the restriction on the positivity of r I in the flowchart. To accelerate the process of getting the optimal design, the search algorithm requires the stopping rules for n and k. Let nstop and kstop be two positive integers. The stopping rules for n and k are described as follows. The stopping rule for the sample size n: Suppose that L*n o = L ( n o, k o, h(n o, ko)) is the current optimum value and L*n o -< L*no+ j for all j = 1, 2, 3 . . . . , nstop. The search procedure will terminate. The optimal solution will be n * - - - n o, k * = k 0 and h* = h ( n 0, k0). The stopping rule for the control limit coefficient k: Suppose that L(no, ko, h(no, ko)) is the current optimum value for the fixed n o and
L ( n o , k o + uj, h(no, k o + uj))
<<-L(,,o,
h( no, ko) )
Table 5 The prior distributions used in the study Shift in m e a n , t~jO"
0.75~ 1.25~ 1.75~ 2.25g 2.75~ 3.25g 3.75g 4.25~ 4.75g 5.25g 5.75~ 6.25g
;~j Negative exponential
Uniform
Half Normal
0.001098 0.000855 0.000666 0.000519 0.000404 0.000314 0.000245 0.000191 0.000148 0.000115 0.000090 0.000070
0.000988 0.000988 0.000988 0.000988 0.000988 0.000988
0.001429 0.001261 0.001045 0.000814 0.000596 0.000409 0.000264 0.000160 0.000091 0.000048 0.000025 0.000012
A = 0.0047
A = 0.0059
A = 0.0062
Leavenworth, 1980); Bowker and Lieberman, 1959; Ishikawa, 1976; Feigenbaum, 1983; Burr, 1953; Juran et al., 1974; Shewhart, 1939; or others), the X- and R-charts are based upon the m e a s u r e m e n t of a single measurable quality characteristic (such as a dimension, weight, tensile strength) of sample items drawn from the production process. The observations are taken periodically in small samples (called subgroups), commonly of size 4 or 5, where each subgroup is as homogeneous as possible. According to the above arguments, we take nstop = kstop = 5. On the other hand, Montgomery (1982) present a computer program for the optimal economic design
for all j = 1, 2, 2 . . . . . kstop. The search algorithm will execute the next n o (that is, n o + 1) and terminate on k 0. For the fixed n 0, the optimal solution will be k * = k 0 and
h* = h(n o, ko). The Shewhart control charts are still the basic tools for a state of statistical control. Shewhart developed the use of 3-sigma control limits as action limits. The justification of 3-sigma limits is based on empirical-economic considerations rather than on a formal statistical basis. Shewhart settled on subgroup sizes of 4 or 5 and left the intersample interval to be determined by the quality control engineer or other personnel. Duncan (1986) wrote that samples consist of 4 or 5 items taken fairly frequently. Furthermore, as is described in various books (e.g., see Grant and
Table 6 The reference set for cost and probability. Parameters of D u n c a n ' s model
(4j)
g~
~ "
Dj
1 2 3 4 5 6 7 8 9 10 11 12
S 7.22 27.60 76.14 165.69 302.36 433.64 570.32 659.86 708.40 728.97 735.78 737.56
$19.63 14.57 11.81 9.84 9.06 8.66 8.37 8.17 8.05 7.93 7.83 7.73
4.17 3.08 2.50 2.08 1.92 1.84 1.77 1.72 1.70 1.68 1.66 1.64
a These values Wj are changed proportionately as T is changed.
K.-J. Chung / Algortihmfor computing optimalX-control charts of an X-control chart. Montgomery's search procedure essentially consists of two phases. About the second phase, Montgomery (1982, p. 41) remarks: " T h e second phase of the optimization finds the optimal k and h for each value of n in the interval max{l, n*-10} < n _
5. Comparisons with published results (I) Single assignable cause (A) Duncan's and Tagaras' algorithms. Optimum designs for 25 examples taken from Table 2 in Duncan (1956) were obtained by using the new algorithm and Tagaras' (1989) algorithm. Table 3 compares them with Duncan's solutions and shows that in all the cases considered the new algorithm yields lower loss-costs than those of Duncan (1956) and Tagaras (1989). The gains from using the new algorithm are between 0% and 24% for Duncan's (1956) algorithm and between 0% and 22% for Tagaras' (1989) algorithm. (B) Goel, Jain and Wu's algorithm. Optimum designs for 15 representative examples taken from Table 2 in Duncan (1956), were obtained by using the new algorithm. Table 4 compares them with the results of Goel et al. (1968) and shows that in all the cases considered the new algorithm yields lower loss-costs than those of Goel et al. (1968). The gains from using the new algorithm are between 0% and 12%. Goel et al.'s (1968) algorithm essentially consists of solving an implicit equation in the variables n and k and an
359
explicit equation for h in terms of n and k. Table 4 shows that Goel et al.'s algorithm is rather accurate. Comparing Goel et al.'s algorithm with the new algorithm, the new algorithm is simpler to solve than Goel et al.'s algorithm.
(II) Multiple assignable causes Optimum designs for 48 representative examples taken from Table 4 in Duncan (1971) were obtained by using the new algorithm and Tagaras' (1989) algorithm. Table 7 compares them with Duncan's solutions and shows that both our solutions and Tagaras' solutions are all better than Duncan's (1971) solutions and that our solutions are, except for Example 45 in Table 7, all better than Tagaras' solutions. The gains from using the new algorithm are between 6% and 42% for Duncan's (1971) algorithm and between - 3 % and 17% for Tagaras' (1989) algorithm. Although Tagaras' search algorithm is reasonably good in general, it has at least the following drawbacks. (1) Tagaras (1989) uses the multiple linear regression to get an estimate of the power of the control chart at optimality. Basically, Tagaras' approach may be unpredictable in some cases because it contains parameters derived empirically from regression models. Consequently, Tagaras' search algorithm will bear some risks because linear regression analysis is a statistical technique and open to chances of errors. (2) Recall values of n = 0, k = 0 and h = h 0 > 0 to mean that instead of sampling the product, we just examine the machine every h 0 hours for evidence of any changes in the machine and repair it if necessary. Sometimes, such a sampling plan is optimal. In this case, the power P is 1.00. The expression for P in Tagaras (1989) never equals 1.00. So, Tagaras' search algorithm never reaches such a solution. Sometimes, it involves significant cost penalties. The optimal loss-cost of Example 25 in Table 3 is 22% higher than the true optimum. (3) Usually the efficiency of Tagaras' search algorithm is better than ours. However, its accuracy does not need to be so. Many examples in Tables 3 and 7 support our observations.
360
~-J. Chung / Algorithm for computing optimal .~-control charts
M~M~NMMNM~M~M~MMMMMNMMM~
e~
d d d d d d d d d d d d ~ d d d d d d d d d ~ d d
o
Z ~ Z Z ~ Z Z ~ Z Z ~ Z Z ~ Z Z ~ Z Z
o
t~
N~N~~NN~NNNNNNNNNNNN~
K.-J. Chung / Algortihm for computing optimal X-control charts
e-
~
M
~
N
~
N
~
'
~
M
~
M
~
.~8
~o M
M
d
d
~
4
~
~
M
d
4
~
4 ~ M d d ~ ¢-,
M M M M ~ M M M M N M ~ ~
~
~ M 4 ~
u
8 I
M M M N M M M M M M M d d d d ~
~'i ~=
d ~ d d d d ~
II
I
~4 d d d ~ d d d d d d d d d d d d
M M M M M M M M M M M ~ ~
i i
d d d d d d d
i 8.~p
.
•
~ = z z ~ z z ~ = z z ~ z z ~ z z ~ = z
7£i ~ ~
~
1 ~
~ 1
~ ~
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~
............
~
8 I
~ M ~ d @ M
~ i
ii N z
~
II
361
362
K.-J. Chung /Algorithm for computing optimal X-control charts
Our optimization algorithm is free from the drawbacks mentioned above. Basically, our optimization algorithm is a reasonably good approximation method and will usually terminate at the optimal solution. Tables 3 and 7 demonstrate this fact. From computational experiences, it is antipated that our optimization algorithm will indeed be more efficient than those employed by Duncan (1956), Goel et al. (1968), and the direct method employed by Duncan (1971). However, Tagaras' search algorithm seems to be more efficient than ours. In fact, this is not important since the reported execution times are low anyway. Calculations of all examples in Tagaras (1989) were executed on a VAX 8600 mainframe computer. A VAX 8600 is not designed for personal use. Lorenzen and Vance (1986) point out the need for optimization techniques implementable in real time on a personal computer. Calculations of all examples in this paper were executed on a personal computer E N S O N T E C H (PC-386). We think that it is a big advantage from the point of view of applications. The results and the execution times of all numerical examples show that our optimization algorithm is quite accurate and efficient indeed. To sum up, we conclude that our optimization algorithm is not only quite accurate and efficient but also simpler to solve than all algorithms of the above-mentioned models.
References Bowker, A.H., and Lieberman, G.J. (1959), Engineering Statistics, Prentice-Hall, Englewood Cliffs, NJ. Burr, I.W. (1953), Engineering Statistics and Quality Control, McGraw-Hill, New York. Chiu, W.K. (1973), "Comments on the economic design of ,~-charts", Journal of the American Statistical Association 68, 919-921. Chiu, W.K. (1974), "The economic design of Cusum charts for controlling normal means", Applied Statistics 23, 420-433. Chiu, W.K. (1975), "Minimum cost control schemes using np-Charts", International Journal of Production Research 13, 341-349. Chiu, W.K. (1976), "On the estimation of data parameters for economic optimum ,X-charts", Metrika 23, 135-147. Chiu, W.K. (1977), " A sensitivity study for minimum cost np-charts", International Journal of Production Research 15, 237-242. Chiu, W.K., and Wetherill, G.B. (1974), "A simplified scheme
for the economic design of X-charts", Journal of Quality Technology 6, 63-69. Chiu, W.K., and Wetherill, G.B. (1975), "Quality control practices", International Journal of Production Research 13, 175-182. Collani, E.v. (1986), "A simple procedure to determine the economic design of an ,~-control chart", Journal of Quality Technology 18, 145-151. Cowden, D.J. (1957), Statistical Methods in Quality Control, Prentice-Hall, Englewood Cliffs, NJ. Duncan, A.J. (1956), "The economic design of ,X-charts used to maintain current conrol of a process", Journal of the American StatisticalAssociation 51,228-242. Duncan, A.J. (1971), "The economic design of X-charts when there is a multiplicity of assignable causes", Journal of the American Statistical Association 66, 107-121. Duncan, A.J. (1986), Quality Control and Industrial Statistics, 5th. ed., Irwin, Homewood, I1. Feigenbaum, A.V. (1983), Total Quality Control, 3rd. ed., McGraw-Hill, New York. Gibra, I.N. (1975), "Recent developments in control chart techniques", Journal of Quality Technology 7, 183-192. Goel, A.L., Jain, S.C., and Wu, S.M. (1968), "An algorithm for the determination of the economic design of ,~-charts based on Duncan's model", Journal of the American Statistical Association 63, 304-320. Goel, A.L., and Wu, S.M. (1973), "Economically optimum design of Cusum charts", Management Science 19, 12711282. Grant, E.L., and Leavenworth, R.S. (1980), Statistical Quality Control, McGraw-Hill, New York. IMSL (1982), "IMSL library reference manual", IMSL Inc., Houston, TX. Ishikawa, K. (1976), "Guide to quality control", Asian Productivity Organization, Tokyo. Juran, J.M., Gryna, Jr., F.M., and Bingham, Jr., R.S., Quality Control Handbook, McGraw-Hill, New York. Lorenzen, TJ., and Vance, L.C. (1986), "The economic design of control charts: A unified approach", Technometrics 28, 3-10. McWilliams, T.P. (1989), "Economic control chart designs and in-control time distribution: A senstivity study", Journal of Quality technology 21, 103-110. Montgomery, D.C. (1980), "The economic design of control charts: A review and literature survey", Journal of Quality Technology 12, 75-87. Montgomery, D.C. (1982), "Economic design of an X-control chart", Journal of Quality Technology 14, 40-43. Panagos, M.R., Heikes, R.G., and Montgomery, D.C. (1985), "Economic design of ,X-control charts for two manufacturing process models", Naval Research Logistics Quarterly 32, 631-646. Pignatiello, Jr., J.J., and Tsai, A. (1988), "Optimal economic design of "X-control charts when cost model parameters are not precisely known", liE Transactions 20, 103-110. Saniga, E.M., and Shirland, L.E. (1977), "Quality control in practice - A survey", Quality Progress 10, 30-33. Shewhart, W.A., with the editorial assistance of Deming, W. Edwards (1939), "Statistical method from the viewpoint of quality control", Graduate School, US Department of Agriculture, Washington, DC.
K.-J. Chung / Algorithm for computing optimal X,-control charts Tagaras, G. (1989), "Power approximations in the economic design of control chart", Naval Research Logistics 36, 639-654. Tagaras, G., and Lee, H.L. (1989), "Approximate semi-economic design of control charts with multiple control limits", Naval Research Logistics 36, 337-353.
363
Vance, L.C. (1983), "A bibliography of statistical quality control chart techniques, 1970-1980", Journal of Quality Technology 15, 59-62.
European Journal of Operational Research 72 (1994) 350-363 North-Holland
Theory and Methodology
An algorithm for computing the economically optimal X-control chart for a process with multiple assignable causes Kun-Jen Chung
*
Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, ROC Received May 1991; revised March 1992
Abstract: The main difficulties in the use of economic designs are the computations involved, the difficulty in specifying process parameters, and the fact that the sampling interval h rarely is a natural quantity of time. The economic design of X-charts involves the determination of the optimum values of the three control parameters: the sample size n, the control limit coefficient k and the sampling interval h. This paper deals with the economic design of X-charts for multiple assignable causes and develops an explicit equation for h in terms of n and k. With the explicit equation for h, we can give an optimization procedure, which is implementable in real time on a personal computer, for obtaining the optimal solutions of these parameters n, k and h for such charts. The results and the execution times of all numerical examples show that our optimization procedure is accurate and efficient. Keywords: Economic design; X-control chart; Process control; Assignable cause 1. Introduction
Duncan (1956) and Cowden (1957) independently pioneered the study of the design of Xcharts by an economic approach. Gibra (1975) provided a reference list of 68 periodicals and books dealing with quality control techniques. Vance (1983) complemented and updated this list by considering references form 1970 to 1980. Along with developments in control chart techniques, developments and improvements on the economic design have continued. Montgomery (1980) listed 51 references on the topic. The Correspondence to: Dr. Kun-Jen Chung, Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, ROC. * This research was supported by the National Science Council of the Republic of China (NSC 81-0415-E011-08).
above references reveal that Duncan's model has received particular attention and a considerable amount of work has been developed from it. However, despite the significant progress in the development of analytical models for the optimal economic determination of control chart parameters under different assumptions, those models have not gained analogous popularity in industry. Saniga and Shirland (1977) and Chiu and Wetherill (1975) report that very few practitioners have implemented economic models for the design of control charts. This is somewhat surprising since most practitioners claim that a major objective in the use of statistical process control procedures is to reduce costs. There are at least three reasons for the lack of practical implementation of this methodology. The first problem is the difficulty in estimating costs and other parameters. The difficulty is dis-
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 2 ) 0 0 1 5 2 - 9
K.-J. Chung / Algorithmfor computing optimalX-control charts cussed in Chiu (1976, 1977), Montgomery (1980) and Pignatiello and Tsai (1988), who indicate that in designing an X-chart control plan by an economic approach it is critical to be able to set the in-control process mean u 0 on target and to accurately estimate the process standard deviation ~r and the coefficient 6 of the effect of the assignable cause. Accurate estimates of other data p a r a m e t e r s are relatively less important. Therefore, costs do not have to be estimated with high precision although other model components such as the magnitude of the shift require relatively accurate determination. This is important, as some of the cost coefficients are difficult to determine precisely, such as the cost of producing defective units. The cost surfaces studied are generally flat in the vicinity of the optimum. Secondly, the mathematical models and their associated optimization schemes are relatively complex and are often presented in a m a n n e r that is difficult for the practitioner to understand and use. So, Lorenzen and Vance (1986) point out the need for optimization techniques implementable in real time on a personal computer. The third problem is the fact that the sampling interval h rarely is a natural quantity of time. For the X-chart, the following designs have been proposed: n = 5, h = 8, k = 3 (Ishikawa, 1976); n = 5, h = 1, k = 3 (Feigenbaum, 1983); n = 4 or 5, h = ?, k = 3 (Burr, 1953); n = 4, h = ?, k = 3 (Juran et al., 1974). For the p-chart, the following have been proposed: n > 5 0 with 3 < n / ~ < 4 , h = 8 , k = 3 (Ishikawa, 1976); n = 25, h = 1 or 8, k = 3 (Feigenbaum, 1983); n = (9 - 9 ~ ) / p , h = ?, k = 3 (Juran et al., 1974); n / 5 > 2 5 , h = ?, k = 3 (Cowden, 1957); n/5 > 1, h = ?, k = 3 (Burr, 1953); 100% inspection, h = 8, k = 3 (Grant and Leavenworth, 1980); n > 50 with @ > 4, h = ?, k = 3 (Juran et al., 1974). With the exception of k = 3, general guidelines on h simply do not exist. So, this p a p e r concerns the second and third problems and tries to solve them. A fundamental assumption of the process model studied is that there exists a single assignable cause which shifts the process mean by an amount 6o-. In practice, this assumption may not be satisfied, as it often occurs that a multiplicity of assignable causes may exist to operate on the process. So, Duncan (1971) extends his study on the economic design of X-charts to the situation of multiple assignable cause models.
351
The economic design of X-charts involves the determination of the optimum values of the three control parameters: the sample size n, the control limit coefficient k and the sampling interval h. Two models are considered in Duncan (1971). Model I assumes that when the process has been disturbed by a given assignable cause, it is free from the occurrence of other assignable causes. Model II will not be referred to here. D u n c a n (1971) finds that for multiple assignable cause model his approximate solution procedure to get the optimal economic design leads to values of the sampling interval h that are significantly different from the optimum values. So, Duncan's (1971) approximate solution procedure to get the optimal economic design does not always perform well. Tagaras (1989) proposes the power approximate method for the economic design of control charts based on an estimate of the power of the control chart at optimality. Tagaras' approximation method for multiple assignable causes essentially consists of two steps. First, multiple linear regression is employed for the derivation of the approximate formula expressing the power of the control chart for a matched single cause model as a function of the model parameters. Secondly, an approximate optimization procedure is then used to determine the economic design of the control chart for the predicted value of the chart's detection power. Tagaras' method for multiple assignable causes is academically interesting, but is practically very difficult. It may not appear concise to a practitioner with limited mathematical and computer programming knowledge. In this paper, we develop a simple and approximate algorithm which is implementable in real time on a personal computer, to get the optimal economic design. Results and execution times of all numerical examples show that our approximate solution algorithm is very accurate and efficient.
2. The m o d e l
The process is assumed to start in a state of statistical control with mean u 0 and standard deviation o.. T h e r e are s assignable causes that result in shifts in the process m e a n from u o to u 0 + 6jo. for the assignable cause A j where all 6 i are known. The time before the assignable cause
352
K.-J.
Chung / Algortihm for computing optimal X-control charts
Aj occurs has an exponential distribution with parameter Aj. Samples of size n are taken every h hours and the sample mean is plotted on an X-control chart with center line u 0 and control limits u o +_ktr/v'-n. If one point falls outside the control limits, a search for an assignable cause is made. The process, however, is allowed to continue in operation during the search until the assignable cause is discovered. It is assumed that the repair cost will not be charged against the net income from the process. Economic control chart models typically assume that the time to the occurrence of an assignable cause folows an exponential distribution. If the probability of a process shift within any small interval of time is directly proportional to the length of the interval, then this assumption is probably appropriate. However, if assignable causes occur as a result of the cumulative effects of heat, vibration, shock and other similar phenomena, or as a result of improper setup or excessive stress during process start up, then use of the exponential distribution to model the duration of in-control may not be appropriate. However, McWilliams (1989) extends the exponential distribution to the Weibull distribution in order to investigate in general the impact, on economic control chart parameters and hourly costs, of the distributional assumption. It is found that the economic design is quite insensitive to the assumed distribution. The Weibull distributions can have increasing, constant, or decreasing hazard functions. Shapes closely resembling normal and lognormal distributions are possible. The results obtained for the Weibull distributions are therefore expected for other commonly used distributions. Consequently, there is no loss of generality when we assume the time to the occurrence of an assignable cause follows an exponential distribution in economic control models. When the process is in control, the probability of a sampling point falling outside the control limits is
For simplicity we shall assume that only positive shifts (Sj > O) actually occur. It has also been assumed that the cost consists of five parts: T: the cost of looking for trouble when none exists; Wj: the cost of discovering cause Aj when it has occurred; Mr: the net increase in the loss per hour resulting from a large percentage of items outside the specification limits due to the occurrence of c a u s e Aj; b: the fixed cost of taking a sample that is independent of sample size; and c: the variable cost per item of sampling, testing and plotting. There are also two time parameters: g: the time for testing and analyzing a sample item; and D / the average time it takes to discover cause A t after it has occurred. Based on the model described in this section and the parameters just defined, Duncan (1971) has shown that the expected loss-cost per hour of operation is
a = 2qb(-k)
We shall not repeat the mathematical argument here.
(1)
where q~(z) denotes the distribution function of a standard normal variable z. When the assignable cause A i has occurred, the probability of a sampling point falling outside the control limit is
=
k) +
k).
(2)
~_#AjMj( h / Pj - ~'j + gn + Dj) + o
1 + ZAj(h/Pj
- "ri + gn + Di)
b + cn
(3)
where 1 - (1 + Ajh) e x p ( - A j h ) =
,,{1- exp(-a h)} 1
h
Aj
exp(Ajh) - 1
(4)
and A
=
exp(hZa,) l-exp(-hZ ,)
=
1 exp(hEa,)- 1" (5)
3. The explicit equation for h It can be shown that Theorems 1 and 2 hold.
K.-J. Chung / Algorithm for computing optimal ,Y-control charts Theorem
1
1 - -
hh
1
e ~h -
Theorem
lira
Theorem 1 tells us that the difference between 1 / ( h h ) and 1/(e ~ h - 1) is within 0 and 1. In Duncan (1956), it is shown the expected number of samples taken during the in-control period is given by 1/(e xh - 1). So, from the point of view of the expected number of samples taken during the in-control period, the number 1/(Ah) in general is a good approximation to 1/(e xh - 1). Many researchers, e.g. Duncan (1956), Goel, Jain and
1.
1>
>0
for a l l A > O and h > O .
2. 1 -
~-,o~ x
1 e~
-1
)
1 -
Z"
(
START
353
)
L* = L(0,0,h(0,0)), n0=l, n*=0, nstop = 0, k* = 0.
Lno = L(0,0,h(0,0)), k o = 1.00, kstop=0.
ho = h(n0,k o) and L(no,k0,h o) from (12) and (3), respectively,
~, r
Compute
*=
o,
*
=ho,
. P= ,
[.no =L(no,k ,h ).
k0=k0+u
kstop=kstop+l I N
[
(where u = 0.01 or 0.1. )
I n = n o, nstop = 0, 1"
=L(n * ,k * ,h * ) L,
N
i" O0 = n 0 + l
L l
L L(n * (
Figure 1. The flowchart for the procedure
*
•
,
,k ,h )
END
)
354
K.-J. Chung / Algortihm for computing optimal X-control charts
Table 1 Comparisons among 1/(e ~h - 1), 1/(Ah) and 1/(Ah)-
Wu (1968), Goel and Wu (1973), Montgomery (1982), Panagos, Heikes and Montgomery (1985), Lorenzen and Vance (1986), Tagaras and Lee (1989), Tagaras (1989) and others adopt 1 / ( e ah 1 ) - l / ( A h ) . However, examination of the numerical model literature shows that, with very few exceptions, cost/risk parameters leading to small Ah (0.4 or less) are of most practical interest. Table 1 reveals that 1/(Ah) - 51 is better approximation to 1 / ( e *h - 1) than 1/(Ah). By Theorems 1, 2 and Table 1, we take 0.5 as a correction number. Replacing 1/(exp(ajh)1) and 1 / ( e x p ( h E a j ) - 1) in (3)-(5) by 1 / ( a j h ) - 1 and 1 / ( h E a j ) - ½ respectively, we get equations (6)(9):
-- ~2hjh 2 for rj is. The author found that for the multiple assignable cause model, use of the approximation led to values of h that were significantly different from the optimum values" (p. 110). Chiu (1973) answers the above question by comparing the loss-cost values using (4) and (10) and further confirms (10). Although (10) has been shown to be satisfactory for practical purposes and was extensively used in subsequent research (see, for example, Goel, .lain and Wu, 1968; Chiu, 1974, 1975, 1977; Chiu and Wetherill, 1974; Montgomery, 1982; Panagos, Heikes and Montgomery, 1985; Lorenzen and Vance, 1986; and others), results of all numerical examples considered in this paper show that the approximation Tj -~ l h is better than the approximation vj 1h - ]~..j.o i ~ / , 2 . Approximating r~ with ~h 1 makes =• ~.. intuitive sense as it states, on average, the assignable cause will occur in the middle of the sampling interval. Due to the use of the approximation r j - ½h and Theorem 2, we can get an equation for the optimal sampling interval h as well. The detail is discussd as follows. For a given (n, k), setting the partial derivative of L in (6) with respect to h equal to zero yields
~=
rl h2 + r2h + r 3 = 0,
1
Ah
e ;~h --
0.01 0.05 0.20 0.40 0.60 0£0 1.00
1
1
99.501 19.504 4.517 2.033 1.216 0.816 0.582
1
1
Ah
ah
2
100 20 5 2.5 1.667 1.25 1.00
99.500 19.500 4.500 2.000 1.167 0.750 0.500
EajBjMj +a-.4TEa~ + EAjWj 1 + EajB,
÷j=
1
b +cn + - h '
(6) (7)
1
A=a
hEAj
2 '
h Bi = - - - ½h + gn + Dj.
(11)
((11))
where
r, =
E a , F, -
Mj (1 + Eaj(gn + D;))
(8) (9)
×
+,,,)M; +
So (6)-(8), in general, should be reasonably good approximations for (3)-(5). Regarding (4), Duncan (1956) has shown that rj is very well approximated by the equation r j - ½h - ~1 /~ j h 2 ,
(10)
which is the result of the series expansion of the right-hand side of (4) when terms of the order a~h 4 and higher are omitted. However, Duncan (1971) remarks: "Furthermore, there is a question about how good the approximation l h
r3= -[1+
£aj(gn
+&)]
+
+¢)]}
K.-J. Chung / Algortihm for computing optimal ,Y-control charts
So / 2
h =
- r 2 + Vr2 - 4rlr 3 2r~
=h(n,
k).
(12)
The above h is the unique, real positive solution of (11), and in general, this is the h that minimizes the cost function L(n, k, h) when n, k and h are fixed.
4. The optimization algorithm In the process of obtaining the approximate optimum values n*, k * and h*, we treat n and k as dicrete variables and assume that the unit length of k is u ( = 0.1 or 0.01). The numbers n and k have no u p p e r bounds. The stopping rules for n and k are described in the flowchart. The search algorithm is as follows. In each step of the outer loop, an n-value is given and the value of k is nested for the value of
355
n. Referring to Collani (1986), for n = 0, we set k = 0. Values of n = 0, k = 0 and h = h 0 > 0 m e a n that instead of sampling the product, we just examine the machine every h 0 hours for evidence of any changes in the machine and repair it if necessary. (i) Given a pair (n o, k o) where 0 _< n o, and 0 < k o. (ii) Calculate a and ~ from (1) and (2). (iii) Calculate h o from (12). (iv) Calculate L(n o, k o, h o) from (3). (v) Find L'n,, = minimumo<_k~(L(no, ko, ho)). (vi) Calculate L* = m i n i m u m 0 _
Table 2 Cost and risk factors and parameters for Duncan's model Example No.
~
a
M
e
D
T
W
b
c
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 1.0 1.0 0.5 0.5 0,5 0.5 0.5
0.01 0.02 0.03 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
100.00 100.00 50.00 1000.00 10000.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 1000.00 12.87 128.70 12.87 12.87 12.87 2.25 225.00 2.25 2.25 2.25
0.05 0.05 0.05 0.05 0.05 0.05 0.50 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
50 50 50 50 50 50 50 50 5 500 5000 50 50 50 50 50 50 500 50 50 50 50 500 50 50
25,0 25,0 25.0 25.0 25.0 25.0 25.0 25.0 2.5 250.0 2500.0 25.0 25.0 25.0 25.0 25.0 25.0 250.0 25.0 25.0 25.0 25.0 250.0 25.0 25.0
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 5.00 0.50 0.50 0.50 0.50 0.50 0.50 5.00 0.50 0.50 0.50 0.50 5.00 0.50
0.10 0.10 0.10 0.10 0,10 0.10 0,10 0,10 0,10 0,10 0.10 0,10 1.00 I 0.00 1.00 0.10 0.10 0.10 0.10 1.00 0.10 0.10 0.10 0.10 1.00
356
/~-J.
Chung / Algortihm for computing optimal.Y-control charts
?.o ,--, II II . *
"~
7~
~..-.
e~
o i.
_=
e.
e.~
[. i
Q
u
c5 II
ta
K.-J. Chung / Algorithm for computing optimal X-control charts
357
t"q
[-
'
0
Z z -8 0
0
0 o
0 0
$
~ II
H ©
"-8
358
K.-J. Chung / Algortihm for computing optimal X-control charts
less than 1.00 and 0.001, respectively. Duncan (1986) has observed that in the economic design of X-charts the average run length in the out-ofcontrol state often has a value between 1 and 2. Duncan's observation supports that the optimal value of Pj is rarely less than 0.001. Hence, in the search algorithm we take n = 1 and k = 1 as the starting values of n and k and require Pj not less than 0.001 to avoid the overflow problem from 1/Pj. In general, M / A j is rather large and hence r I is greater than 0. Although this is so, if Pj is sufficiently small, r 1 may be nonpositive. To keep the sampling interval h in (12) positive, we should avoid the occurrence that r~ is nonpositive. Hence, we have to put the restriction on the positivity of r I in the flowchart. To accelerate the process of getting the optimal design, the search algorithm requires the stopping rules for n and k. Let nstop and kstop be two positive integers. The stopping rules for n and k are described as follows. The stopping rule for the sample size n: Suppose that L*n o = L ( n o, k o, h(n o, ko)) is the current optimum value and L*n o -< L*no+ j for all j = 1, 2, 3 . . . . , nstop. The search procedure will terminate. The optimal solution will be n * - - - n o, k * = k 0 and h* = h ( n 0, k0). The stopping rule for the control limit coefficient k: Suppose that L(no, ko, h(no, ko)) is the current optimum value for the fixed n o and
L ( n o , k o + uj, h(no, k o + uj))
<<-L(,,o,
h( no, ko) )
Table 5 The prior distributions used in the study Shift in m e a n , t~jO"
0.75~ 1.25~ 1.75~ 2.25g 2.75~ 3.25g 3.75g 4.25~ 4.75g 5.25g 5.75~ 6.25g
;~j Negative exponential
Uniform
Half Normal
0.001098 0.000855 0.000666 0.000519 0.000404 0.000314 0.000245 0.000191 0.000148 0.000115 0.000090 0.000070
0.000988 0.000988 0.000988 0.000988 0.000988 0.000988
0.001429 0.001261 0.001045 0.000814 0.000596 0.000409 0.000264 0.000160 0.000091 0.000048 0.000025 0.000012
A = 0.0047
A = 0.0059
A = 0.0062
Leavenworth, 1980); Bowker and Lieberman, 1959; Ishikawa, 1976; Feigenbaum, 1983; Burr, 1953; Juran et al., 1974; Shewhart, 1939; or others), the X- and R-charts are based upon the m e a s u r e m e n t of a single measurable quality characteristic (such as a dimension, weight, tensile strength) of sample items drawn from the production process. The observations are taken periodically in small samples (called subgroups), commonly of size 4 or 5, where each subgroup is as homogeneous as possible. According to the above arguments, we take nstop = kstop = 5. On the other hand, Montgomery (1982) present a computer program for the optimal economic design
for all j = 1, 2, 2 . . . . . kstop. The search algorithm will execute the next n o (that is, n o + 1) and terminate on k 0. For the fixed n 0, the optimal solution will be k * = k 0 and
h* = h(n o, ko). The Shewhart control charts are still the basic tools for a state of statistical control. Shewhart developed the use of 3-sigma control limits as action limits. The justification of 3-sigma limits is based on empirical-economic considerations rather than on a formal statistical basis. Shewhart settled on subgroup sizes of 4 or 5 and left the intersample interval to be determined by the quality control engineer or other personnel. Duncan (1986) wrote that samples consist of 4 or 5 items taken fairly frequently. Furthermore, as is described in various books (e.g., see Grant and
Table 6 The reference set for cost and probability. Parameters of D u n c a n ' s model
(4j)
g~
~ "
Dj
1 2 3 4 5 6 7 8 9 10 11 12
S 7.22 27.60 76.14 165.69 302.36 433.64 570.32 659.86 708.40 728.97 735.78 737.56
$19.63 14.57 11.81 9.84 9.06 8.66 8.37 8.17 8.05 7.93 7.83 7.73
4.17 3.08 2.50 2.08 1.92 1.84 1.77 1.72 1.70 1.68 1.66 1.64
a These values Wj are changed proportionately as T is changed.
K.-J. Chung / Algortihmfor computing optimalX-control charts of an X-control chart. Montgomery's search procedure essentially consists of two phases. About the second phase, Montgomery (1982, p. 41) remarks: " T h e second phase of the optimization finds the optimal k and h for each value of n in the interval max{l, n*-10} < n _
5. Comparisons with published results (I) Single assignable cause (A) Duncan's and Tagaras' algorithms. Optimum designs for 25 examples taken from Table 2 in Duncan (1956) were obtained by using the new algorithm and Tagaras' (1989) algorithm. Table 3 compares them with Duncan's solutions and shows that in all the cases considered the new algorithm yields lower loss-costs than those of Duncan (1956) and Tagaras (1989). The gains from using the new algorithm are between 0% and 24% for Duncan's (1956) algorithm and between 0% and 22% for Tagaras' (1989) algorithm. (B) Goel, Jain and Wu's algorithm. Optimum designs for 15 representative examples taken from Table 2 in Duncan (1956), were obtained by using the new algorithm. Table 4 compares them with the results of Goel et al. (1968) and shows that in all the cases considered the new algorithm yields lower loss-costs than those of Goel et al. (1968). The gains from using the new algorithm are between 0% and 12%. Goel et al.'s (1968) algorithm essentially consists of solving an implicit equation in the variables n and k and an
359
explicit equation for h in terms of n and k. Table 4 shows that Goel et al.'s algorithm is rather accurate. Comparing Goel et al.'s algorithm with the new algorithm, the new algorithm is simpler to solve than Goel et al.'s algorithm.
(II) Multiple assignable causes Optimum designs for 48 representative examples taken from Table 4 in Duncan (1971) were obtained by using the new algorithm and Tagaras' (1989) algorithm. Table 7 compares them with Duncan's solutions and shows that both our solutions and Tagaras' solutions are all better than Duncan's (1971) solutions and that our solutions are, except for Example 45 in Table 7, all better than Tagaras' solutions. The gains from using the new algorithm are between 6% and 42% for Duncan's (1971) algorithm and between - 3 % and 17% for Tagaras' (1989) algorithm. Although Tagaras' search algorithm is reasonably good in general, it has at least the following drawbacks. (1) Tagaras (1989) uses the multiple linear regression to get an estimate of the power of the control chart at optimality. Basically, Tagaras' approach may be unpredictable in some cases because it contains parameters derived empirically from regression models. Consequently, Tagaras' search algorithm will bear some risks because linear regression analysis is a statistical technique and open to chances of errors. (2) Recall values of n = 0, k = 0 and h = h 0 > 0 to mean that instead of sampling the product, we just examine the machine every h 0 hours for evidence of any changes in the machine and repair it if necessary. Sometimes, such a sampling plan is optimal. In this case, the power P is 1.00. The expression for P in Tagaras (1989) never equals 1.00. So, Tagaras' search algorithm never reaches such a solution. Sometimes, it involves significant cost penalties. The optimal loss-cost of Example 25 in Table 3 is 22% higher than the true optimum. (3) Usually the efficiency of Tagaras' search algorithm is better than ours. However, its accuracy does not need to be so. Many examples in Tables 3 and 7 support our observations.
360
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Our optimization algorithm is free from the drawbacks mentioned above. Basically, our optimization algorithm is a reasonably good approximation method and will usually terminate at the optimal solution. Tables 3 and 7 demonstrate this fact. From computational experiences, it is antipated that our optimization algorithm will indeed be more efficient than those employed by Duncan (1956), Goel et al. (1968), and the direct method employed by Duncan (1971). However, Tagaras' search algorithm seems to be more efficient than ours. In fact, this is not important since the reported execution times are low anyway. Calculations of all examples in Tagaras (1989) were executed on a VAX 8600 mainframe computer. A VAX 8600 is not designed for personal use. Lorenzen and Vance (1986) point out the need for optimization techniques implementable in real time on a personal computer. Calculations of all examples in this paper were executed on a personal computer E N S O N T E C H (PC-386). We think that it is a big advantage from the point of view of applications. The results and the execution times of all numerical examples show that our optimization algorithm is quite accurate and efficient indeed. To sum up, we conclude that our optimization algorithm is not only quite accurate and efficient but also simpler to solve than all algorithms of the above-mentioned models.
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K.-J. Chung / Algorithm for computing optimal X,-control charts Tagaras, G. (1989), "Power approximations in the economic design of control chart", Naval Research Logistics 36, 639-654. Tagaras, G., and Lee, H.L. (1989), "Approximate semi-economic design of control charts with multiple control limits", Naval Research Logistics 36, 337-353.
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Vance, L.C. (1983), "A bibliography of statistical quality control chart techniques, 1970-1980", Journal of Quality Technology 15, 59-62.