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On economic design of cumulative count of conforming chart
Int. J. Production Economics 72 (2001) 89}97
On economic design of cumulative count of conforming chart M. Xie*, X.Y. Tang, T.N. Goh Department of Industrial and Systems Engineering, National University of Singapore, Kent Ridge, Singapore 119 260 Received 6 May 2000; accepted 10 August 2000
Abstract Control charts based on Cumulative Count of Conforming (CCC) have been shown to be useful in high-quality processes. In the design stage of any chart in manufacturing industry, economic factors should be taken into account. In this paper, an economic model for CCC chart design is developed. A simpli"ed algorithm is used to search the optimal setting of the sampling and control parameters. Sensitivity analysis is presented and numerical examples are used to illustrate the procedure. Compared with related studies, the model presented here could produce lower cost and smaller Type II error than the existing approach and the procedure is easy to implement. 2001 Elsevier Science B.V. All rights reserved. Keywords: Statistical process control; Economic design; CCC charts; Average run length
1. Introduction Statistical Process Control (SPC) concepts and methods have been successfully implemented in manufacturing industries for decades. As one of the primary SPC tools, control chart plays a very important role in attaining process stability. However, traditional control charts face a number of problems in high-quality processes, which are very common in the modern manufacturing environments. Cumulative Count of Conforming (CCC) control charts, which are based on the plotting of the cumulative count of conforming items between successive nonconforming ones, have been shown to be very useful. The idea of tracking cumulative
* Corresponding author. Tel.: #65 8746536; fax: #657771434. E-mail address: [email protected] (M. Xie).
counts to monitor assignable causes in high-yield process was "rst used by Calvin [1], and it was further studied by Goh [2], Xie and Goh [3,4] and Glushkovsky [5], among others. Similar to other control charts, an important stage in CCC chart implementation is its design. A common practice is to design the control chart with primarily statistical considerations. However, the design of a control chart has economic consequences as it involves various expenses, such as the costs of sampling and testing, costs associated with investigating out-of-control signals, correcting assignable causes and costs of allowing nonconforming units to reach the customer. Hence, it is more reasonable to consider the design of a control chart from an economic viewpoint. In the literature, many models have been developed to serve di!erent purposes for economic design of control charts. Considerable e!ort has been devoted to this area since Duncan's [6] pioneering work. Lorenzen and Vance [7] proposed
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 9 3 - 1
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M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
a general process model for determining the economic design of control charts. This model allows production to be continued or stopped during search and/or repair. As a result, this model has been widely used because of its uni"cation. Related reference can be found in McWilliams [8], and Surtihadi and Rachavachari [9] and they studied economic design of XM chart with LV model. Montgomery et al. [10] used it to design EWMA chart, and Simpson and Keats [11] applied it to the CUSUM chart. The study of optimal procedure has received a lot of attention since the economic models are quite complex. A simpli"ed procedure was derived and applied for di!erent economic models, e.g., the optimal design parameters of the economically based XM chart [12] and economic design of np chart with a multiplicity of assignable causes [13]. Collani et al. [14] also considered a simpli"ed economic design of control charts for monitoring the nonconforming probability. In this paper, we extend the economic design of CCC chart based on the LV model. Chung's [15] simpli"ed procedure is used to derive the optimum solution. Numerical examples are shown to demonstrate the properties of the economic design of CCC charts. Sensitivity analysis is also presented to show the e!ect of input parameters on the optimum economic design of CCC chart. Finally, the numerical comparison shows that the model presented here could produce lower cost and smaller Type II error than the existing approach.
2. The cost model for CCC chart 2.1. Notations Lorenzen and Vance [7] provided the uni"ed approach to economic design of control charts and a uni"cation of notation. The following notations will be used in the formulation of the cost function for the CCC chart in this paper. The parameters can be classi"ed into four categories. Design parameters l control limit for the CCC control chart h sampling interval
Response parameters C expected cost per hour ARL1 average run length when the process is in control ARL2 average run length when process is out of control s expected number of samples while in control expected time of occurrence of the assignable cause Cost and time parameters C quality cost/hour while the production pro cess is in control C quality cost/hour while the production pro cess is out of control ('C ) Y cost per false alarm W cost to located and repair the assignable cause a "xed cost per sample b cost per unit sampled E time to sample and chart one item ¹ expected search time when false alarm ¹ expected time to discover the assignable cause ¹ expected time to repair the process Process parameters p expected fraction defective produced when the process is in control p expected fraction defective produced when the process is out of control 1/mean time process is in control
1 if production continues during searches " 0 if production ceases during searches
1 if production continues during repair " 0 if production ceases during repair 2.2. The process model A production cycle here is de"ned as the time period from the beginning of the production or after an adjustment to the detection and elimination of an assignable cause. The cycle time consists of the following "ve parts: the time until the assignable cause occurs, the time until the next sample is taken, the time to analyze and chart the sample, the
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
91
Fig. 1. Diagram of in-control and out-of-control states of a process.
time until the chart gives an out-of-control signal and the time to discover the assignable cause and repair the process. Let A , A ,2, A denote the expected values of these above "ve segments respectively, see Fig. 1. If the production continues during the search stage, it is clear to know that A , the expected time while process is in control equals 1/. However, when the production ceases during the search stage, the process is stopped to investigate the false alarms. In this case, the time spent on investigating false alarms should also be included into the expected time while the process is in control. Suppose the total items produced in the in-control stage are s, and the time spent on investigating false alarms is s¹ /ARL1. Combine these two situations, we have 1 (1! )s¹ . A " # ARL1
(1)
Other time elements are straightforward, and they can be easily derived through A "h!, (2) A "E, (3) A "h(ARL2!1), (4) A "¹ #¹ . (5) Note that E is the expected time to sample and chart one item and it is often taken as zero. For CCC chart, A "E because the sampling is based on one-item-a-time measurement and this type of measurement is more and more widely used as automation spreads over the manufacturing.
By combining (1)}(5), we can obtain the expected cycle time for CCC chart as 1 (1! )s¹ !#E E(¹)" # ARL1 #h(ARL2)#¹ #¹ . (6) Inserting the cost parameters in Eq. (6), the expected cost per cycle for CCC chart can be formulated as C E(P)" #C (!#E#h(ARL2) s> #= # ¹ # ¹ )# ARL1 (1/!#E#h(ARL2)# ¹ # ¹ ) . #(a#b) h (7) Note that since this is a renewal reward process, the expected cost per hour can be computed as the ratio of expected cost per cycle to the expected cycle time in hours. After some calculus, the expected cost per hour C for CCC chart is given as follows: S a#b S C"C # # ; D h D
(8)
where 1 s" , eHF!1 h 1 , " ! eHF!1
(9) (10)
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M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
S "(C !C )(!#E#h(ARL2) #¹ #¹ )!C [(1! )¹ #(1! )¹ ] C (1! )s¹ s> #=, ! (11) # ARL1 ARL1 (1! )s¹ D"1# ARL1 #(!#E#h(ARL2)#¹ #¹ )
(12)
and S "1#(!#E#h(ARL2)# ¹ # ¹ ). (13) The process is assumed to start in a state of statistical control with a known fraction nonconforming p . There is a single assignable cause and its occur rence results in the fraction nonconforming of process changing to p . The time between occurrences of the assignable cause is exponentially distributed with a mean of occurrences per hour. These assumptions are commonly used in the economic design, see, e.g., [7,16,17]. We assume that sampling is based on one-at-atime measurement. That is, the sample size is equal to one. This assumption greatly simpli"es the economic design of the CCC chart and it is usually the case in automated manufacturing environment. Since the sample size is always `onea, the cost of taking a sample and inspecting it is actually a known constant, a#b. Note that the sample size equals one is not a serious assumption and in fact, the idea of CCC chart has the advantage that there is no need for rational sample size. Only low-sided CCC chart studied in this article because of its simplicity and practicality. When the CCC data is out of upper control limit, the process is better and the fraction of nonconforming is smaller, which causes no actual problem. Therefore, we are more concerned with the case of out of lower control limit (LCL). Actions should be taken in this situation. In the course of inspection, one item is checked every h hours of production. The number of consecutive conforming items is counted until a nonconforming one is detected. If the count is larger than or equal to LCL, no action is taken. Otherwise, assignable causes should be searched and eliminated.
3. The optimization procedure The algorithm by Lorenzen and Vance [7] to "nd the most economical design is somewhat complicated as it consists of Newton's Method, golden section search and Fibonacci search. This could be the main reason that limits the application of their method. Furthermore, to solve formula (8), they expand the exponential in , /h, s and s/h, and ignore all terms containing powers of greater than 1. Hence, the solution is highly approximate despite its complexity. In Chung [16], a simpli"ed procedure is developed to determine the optimum parameter values based on Lorenzen and Vance's model. The procedure consists of solving an explicit formula for h in terms of n and l and yields lower loss costs than those of Lorenzen and Vance's model. Moreover, this simpli"ed procedure does not use those assumptions and approximations made in Lorenzen and Vance's model. Furthermore, no terms are neglected for "nding the optimum solution. Consequently, it was demonstrated that the procedure gives more accurate, more reliable and more feasible optimum values. Hence, it is more appropriate for economic design of CCC chart. After replacing 1/(eHF!1) in formulas (8)}(12) and (13) by 1/h!0.5, the explicit formula of h for CCC chart is derived !r #(r !4r r ) h" 2r
(14)
where
(1! )¹ r "(ARL2!0.5) (C !C ) 1! 2ARL1
> # #C [(1! )¹ 2ARL1 #(1! )¹ ]
C (1! )¹ != ! 2ARL1
!(a#b)[(ARL2!0.5)],
(15)
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
r "2(ARL2!0.5)
C (1! )¹ !> ARL1
!(a#b)(1#E# ¹ # ¹ ) ,
(16)
C (1! )¹ !> r " ARL1
2. calculate the ARL1 and ARL2 according to the value of l from Eqs. (18) and (19), respectively, 3. calculate h from Eq. (14), 4. calculate C(l , h ) from Eq. (8), 5. repeat 1}4 and "nd CH" min [C(l , h )]"C(lH, hH). ?WJ W@
(1! )¹ #E#¹ #¹ 1! ARL1
The values lH and hH which give the overall approximate minimum loss cost CH are the optimal economic design for the CCC control chart.
(1! )¹ (C !C )(E#¹ #¹ ) # ARL1 !C [(1! )¹ #(1! )¹ ] C (1! )¹ > #= # ! 2ARL1 2ARL1
93
(a#b)(ARL2!0.5)(1! )¹ # ARL1 !(a#b)(1#E# ¹ # ¹ ) (1! )¹ #E#¹ #¹ . ; 1! 2ARL1
(17)
As the above formulas show, the loss function depends on ARL1, ARL2 and h. h can be solved in terms of ARL1 and ARL2, which are functions of control limit l, the relationships between average run length and control limit are given by Xie et al. [18]. 1 1!p # ARL1"l# p p [1!(1!p )J\]
(18)
and 1!p 1 # ARL2"l# . p p [1!(1!p )J\]
(19)
In the process of obtaining the approximate optimum values lH and hH, l can be considered as discrete variables. The complete procedure is as follows: 1. given a value of l which belongs to a reasonable searching range (a)l )b),
4. Numerical illustration Lorenzen and Vance [7] presented an application of their economic model of a foundry operation which produces 84 casting per hour on one of its lines. In castings, a standard is set to prevent high carbon-silicate content as it will result in low tensile strength. The sampling process costs $4.22 per sample and takes about "ve minutes. The process stays in control for 50 hours. When the process is out of control, the system must be #ushed and restarted, and this should take about 45 minutes with a repair crew cost of $22.80 per hour and a downtime cost of $21.34 per minute. It takes about "ve minutes to get the repair crew assembled and no time to search for the cause. Here we consider the case when it is a highquality process; that is, the fraction nonconforming is very low. In such a situation, traditional p chart becomes useless because even for a moderate sample size there will seldom be any nonconforming item and thus CCC chart should be used. Suppose that historical data indicate that the process produces about 0.05% nonconforming items when in control and about 5% nonconforming items when out of control. The cost per hour while in control, C "100;84;0.0005"$4.2. Likewise, C is equal to $420. Other speci"c input parameter values are "1/50 h, E"¹ "¹ "5/60 hours, ¹ "45/60 hours, "1, "0, >"="$977.40, a"0, b"4.22. Thus, from Formula (8), the loss function approaches its optimal value CH"77.4763 when the sampling interval is hH"0.3015 and the control limit is lH"96.97.
94
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
Table 1 The economic design of the CCC charts for some di!erent p and p values p
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009
p
0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
p /p
100 90 80 70 60 50 40 30 20 10 5 3.33 2.5 2.0 1.67 1.3 1.25 1.11
Economically optimal design lH
hH
ARLH
ARLH
CH
H
H
96.96 107.64 121.00 138.18 161.07 193.13 241.20 321.31 481.50 962.08 960.01 958.08 956.78 956.3 956.5 957.5 958.9 960.9
0.0333 0.0321 0.0306 0.0292 0.0277 0.0261 0.0244 0.226 0.0210 0.0214 0.0215 0.0216 0.0217 0.0219 0.0220 0.0221 0.0223 0.0224
1057200 952780 848410 744110 639860 535680 431560 327530 223630 120020 34605 17644 11323 8220 6440.9 5311 4542.5 3991.2
357.56 397.34 447.07 511.01 596.26 715.61 894.63 1193.0 1789.7 3580 3580 3580 3580 3580 3580 3580 3580 3580
413.86 427.46 443.21 461.90 484.60 513.02 550.10 601.48 679.79 820.08 820.44 820.93 821.49 822.07 822.7 823.2 823.7 824.26
0.0095 0.0106 0.0119 0.0136 0.0158 0.0190 0.0237 0.0315 0.0469 0.0916 0.1745 0.2496 0.3177 0.3798 0.4364 0.4882 0.5355 0.5786
0.3812 0.3813 0.3814 0.3815 0.3816 0.3817 0.3818 0.3820 0.3821 0.3822 0.3830 0.3838 0.3843 0.3845 0.3844 0.3841 0.3835 0.3827
Using formula (18) and (19), we have that: ARL1H"1.057;10 and ARL2H"357.56 and the probability of false alarm when the process is in control is H"0.0096 and the probability of failing to detect the shift when the process is out of control is H"0.3812. Table 1 shows the economic designs of CCC charts with varied values of p and p . The other parameters, such as ARL1 which represents average run length when process is in control and ARL2 which represents average run length when process is out of control, the probability of false alarms () and the probability of failing to detect the assignable cause are also listed in this table. From this table, it can be noted that the optimal loss CH is a decreasing function of p /p . For a given value of p , the optimal sampling interval hH is increasing with value of p , and for a given value of p , hH is increasing when the value of p increases. The situation of optimal value of lH is opposite to that of hH. That is, for a given value of p , the optimal control limit is decreasing with value of p , and for a given value of p , lH is decreasing when the value of p increases. The
above analysis is unsuitable for the very small shift, say, p /p (2. For instance, as last "ve lines in Table 1 shows, lH is increasing when the value of p increases. Both ARL1 and ARL2 are the increasing functions of the ratio of p and p . When the ratio is relatively large, the probability of false alarm is small. However, when the ratio is very small, for example, smaller than 2, the probability of false alarm will increase dramatically. Especially, when the ratio approaches to `1a, the probability of false alarm reaches 58%. Table 1 also shows that the probability of economic design failing to detect the assignable cause almost keeps constant in the whole range of ratio. 4.1. Sensitivity analysis When applying Lorenzen and Vance [7] model in the economic design of CCC chart, there are "fteen input parameters related to the loss function. The parameters, p and p are process parameters which specify the in-control and out-of-control states of the process monitored. The e!ect of them
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
on the economic design has been studied in above section. Here, we only consider the other input parameters. This paper adopts the approach used in Costa [19] and Bai et al. [20]. Each example in Table 2 represents the economic design of CCC chart with one di!erent parameter at a time. Therefore, sensitivity analysis of the proposed model with regard to these input parameters can be easily made. It can be seen that , which is the average occurrence rate of the assignable cause, has a signi"cant e!ect on the optimum sampling interval and the cost. A smaller implies a larger optimal sampling interval and smaller cost. From Table 2, we can see that when the value of decreases by 50%, the value of hH increases about 29% and the value of CH decreases about 20%. However, the optimum control limit lH is rather robust to the changes of . The time to sample and chart one item (E), the expected time to discover the assignable cause (¹ ) and expected time to repair the process (¹ ) all have very slight e!ects on the optimal values. However, there are still some trends that can be found. For example, the larger the ¹ , the larger CH. However, ¹ , the expected search time when false alarm, seems has no e!ect on the optimal values. It can also be observed that whether the production stops/continues during the search and whether the production stops/continues during repair a!ect the optimal values slightly. When the production continues during both searches and repairs, the cost will be slightly larger than other three modes. Variations in the quality cost per hour when process is in control and out of control (C and C ) can cause relatively big e!ects on optimum values of hH and CH, while hardly a!ect the optimum value of lH. On the other hand, it is somewhat a bit surprise that the cost per false alarm (>) and the cost to locate and repair the assignable cause (=) only have little e!ect on the optimum values. This example shows that the optimum control limit lH is relatively robust to all the input parameters (except for p and p ). Among all the factors, C , C and are the most sensitive factors which a!ect the optimal cost and optimal sampling interval for CCC chart.
95
4.2. Comparisons The economic design of CCC chart has been studied in this paper. The cost model derived for CCC chart in this paper is in the spirit of Lorenzen and Vance's [7] general model, and a combination of properties of CCC chart is used here. In addition, the use of a simpli"ed procedure greatly reduces the di$culty in getting the optimal parameters of CCC chart. The model used in this article allows production to be continued or stopped during search and/or repair. Xie et al. [18] presented a number of examples where the CCC chart is used to monitor a highquality process. Speci"c input parameter values used in their paper are given by p "500 ppm, p "0.005, "0.01, < "150, < "150, A "10, A "150, t "0.1, t "0.3 and c"0.5. The 18 examples presented in Section 5 of Xie et al. [18] are used for comparison. Table 3 shows the comparison of the results, `Aa and `Ba respectively represent the solutions in this paper and the earlier paper. The approximate optimum loss-cost value in Xie et al. [18] is always higher than the optimum value obtained here and on the average, the percentage is about 2%. In addition, both the traditional and economic design presented in Xie et al. [18] can hardly detect the small shifts since their Type II error is extremely big in those situations. Especially, as the last line in Table 3 shown, the ability of detecting the special shift presented in their paper is only 33%, while our design can still keep ability at a moderate level.
5. Discussions Economic considerations should be incorporated in control chart design in manufacturing industry. However, due to the existence of di!erent process models and the number of parameters involved, such a study is not straightforward, and it may add to the complexity to the implementation
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
1 2 3
E1 E2 E3
T01 T02 T03
T11 T12 T13
T21 T22 T23
1 2 3 4
C01 C02 C03
C11 C12 C13
Y1 Y2 Y3
W1 W2 W3
A1 A2 A3
B1 B2 B3
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
Example Parameter values no. p p E
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
5/60 5/60 5/60
¹
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
¹
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
0.005 5/60 0.01 5/60 0.02 5/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60 45/60
45/120 45/60 45/30
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
¹
Table 2 Sensitivity analysis of economic design for CCC control charts
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 0 0 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 1 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
14.24 114.24 214.24
114.24 114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
C
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
849.2 949.2 1049.2
949.2 949.2 949.2
949.2 949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
C
0 0 0
0 0.5 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
a
2.22 3.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
b
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
777.40 877.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
>
977.40 977.40 977.40
977.40 977.40 977.40
777.40 877.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
=
189.2 190.5 191.2
191.2 191.5 191.7
191.2 191.2 191.2
191.7 191.5 191.2
191.3 191.2 191.2
191.2 191.2 191.3
191.2 191.4 191.3 191.3
191.2 191.2 191.2
191.2 191.2 191.2
191.2 191.2 191.2
191.2 191.2 191.2
190.9 191.1 191.2
lH
0.0173 0.0219 0.0262
0.0262 0.0283 0.0304
0.0261 0.0262 0.0262
0.0262 0.0262 0.0262
0.0287 0.0262 0.0242
0.0243 0.0262 0.0283
0.0262 0.0267 0.0261 0.0268
0.0263 0.0262 0.0260
0.0262 0.0262 0.0262
0.0262 0.0262 0.0262
0.0262 0.0262 0.0262
0.0438 0.0332 0.0262
hH
422.4 472.9 514.1
514.1 532.9 549.1
511.2 512.6 514.1
513.9 514.0 514.1
486.1 514.1 540.5
441.6 514.1 585.0
514.1 524.6 512.7 526.0
516.9 514.1 508.6
513.5 513.7 514.1
514.1 514.1 514.1
513.5 513.7 514.1
328.1 408.9 514.1
CH
Optimal economic design
96 M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
97
Table 3 Comparison results Example no.
1 2 3 4 5 6 7 8 9
A
Example no.
B
CH
H
CH
H
25.26 26.51 27.96 29.70 31.83 34.52 38.08 43.11 51.08
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
25.53 26.76 28.22 29.96 32.08 34.77 38.32 43.34 51.29
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
of control charting technique, which is supposed to be a simple approach for process monitoring. The focus here is on the use of CCC chart which is based on cumulative count of conforming items, and this is a better alternative than the traditional attribute chart which requires a rational sample size. A different model than a previous paper is used and a simple optimization procedure is adopted in this paper. Sensitivity analysis is presented and numerical examples are used to illustrate the procedure.
References [1] T.W. Calvin, Quality control techniques for &zero-defects'. IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-6 (1983) 323}328. [2] T.N. Goh, Acontrol chart for very high yield processes, Quality Assurance 13 (1987) 18}22. [3] M. Xie, T.N. Goh, Some procedures for decision making in controlling high yield processes, Quality and Reliability Engineering International 8 (1992) 355}360. [4] M. Xie, T.N. Goh, The use of probability limits for process control based on geometric distribution, International Journal of Quality & Reliability Management 14 (1997) 64}73. [5] E.A. Glushkovsky, &On-line' G-control chart for attribute data, Quality and Reliability Engineering International 10 (1994) 217}227. [6] A.J. Duncan, The economic design of XM charts used to maintain current control of a process, Journal of the American Statistical Association 51 (1956) 228}242. [7] T.J. Lorenzen, L.C. Vance, The economic design of control charts: A uni"ed approach, Technometrics 28 (1986) 3}10.
10 11 12 13 14 15 16 17 18
A
B
CH
H
CH
H
66.86 66.87 66.89 66.90 66.92 66.94 66.95 66.97 66.99
0.382 0.3822 0.3823 0.3825 0.3825 0.3825 0.3824 0.3823 0.3821
67.03 67.04 67.06 67.12 67.72 68.73 69.94 71.23 72.53
0.3819 0.3823 0.3823 0.4109 0.491 0.5531 0.6021 0.642 0.6742
[8] T.P. McWilliams, Economic, statistical, and economicstatistical X chart designs, Journal of Quality Technology 26 (1994) 227}238. [9] J. Surtihadi, M. Rachavachari, Exact economic design of XM charts for general time in-control distributions, International Journal of Production Research 32 (1994) 2287}2302. [10] D.C. Montgomery, J.C. Torng, J.K. Cochran, F.P. Lawrence, Statistically constrained economic design of the EWMA control chart, Journal of Quality Technology 27 (1995) 250}256. [11] J.R. Simpson, J.B. Keats, Sensitivity study of the CUSUM control chart with an economic model, International Journal of Production Economics 40 (1) (1995) 1}19. [12] K.J. Chung, Determination of optimal design parameters of an XM -chart, Journal of the Operational Research Society 43 (1992) 1151}1157. [13] K.J. Chung, An algorithm for the economic design of np charts for a multiplicity of assignable causes, Journal of the Operational Research Society 46 (1995) 1374}1385. [14] E.V. Collani, K. Drager, Asimpli"ed economic design of control charts for monitoring the nonconforming probability, Economic Quality Control 10 (1995) 231}247. [15] K.J. Chung, A simpli"ed procedure for the economic design of control charts: A uni"ed approach, Engineering Optimization 17 (1991) 313}320. [16] W.K. Chiu, Economic design of attribute control charts, Technometrics 17 (1975) 81}87. [17] D.C. Montgomery, Economic design of a XM control chart, Journal of Quality Technology 14 (1982) 1272}1284. [18] M. Xie, T.N. Goh, W. Xie, Astudy of economic design of control charts for cumulative count of conforming items, Communications in Statistics } Simulation and Computation 26 (1997) 1009}1027. [19] A.F.B. Costa, Joint economic design of XM and R control charts for processes subject to independent assignable causes, IIE Transactions 25 (1993) 27}33. [20] D.S. Bai, K.T. Lee, An economic design of variable sampling interval XM control charts, International Journal of Production Economics 54 (1) (1998) 57}64.
On economic design of cumulative count of conforming chart M. Xie*, X.Y. Tang, T.N. Goh Department of Industrial and Systems Engineering, National University of Singapore, Kent Ridge, Singapore 119 260 Received 6 May 2000; accepted 10 August 2000
Abstract Control charts based on Cumulative Count of Conforming (CCC) have been shown to be useful in high-quality processes. In the design stage of any chart in manufacturing industry, economic factors should be taken into account. In this paper, an economic model for CCC chart design is developed. A simpli"ed algorithm is used to search the optimal setting of the sampling and control parameters. Sensitivity analysis is presented and numerical examples are used to illustrate the procedure. Compared with related studies, the model presented here could produce lower cost and smaller Type II error than the existing approach and the procedure is easy to implement. 2001 Elsevier Science B.V. All rights reserved. Keywords: Statistical process control; Economic design; CCC charts; Average run length
1. Introduction Statistical Process Control (SPC) concepts and methods have been successfully implemented in manufacturing industries for decades. As one of the primary SPC tools, control chart plays a very important role in attaining process stability. However, traditional control charts face a number of problems in high-quality processes, which are very common in the modern manufacturing environments. Cumulative Count of Conforming (CCC) control charts, which are based on the plotting of the cumulative count of conforming items between successive nonconforming ones, have been shown to be very useful. The idea of tracking cumulative
* Corresponding author. Tel.: #65 8746536; fax: #657771434. E-mail address: [email protected] (M. Xie).
counts to monitor assignable causes in high-yield process was "rst used by Calvin [1], and it was further studied by Goh [2], Xie and Goh [3,4] and Glushkovsky [5], among others. Similar to other control charts, an important stage in CCC chart implementation is its design. A common practice is to design the control chart with primarily statistical considerations. However, the design of a control chart has economic consequences as it involves various expenses, such as the costs of sampling and testing, costs associated with investigating out-of-control signals, correcting assignable causes and costs of allowing nonconforming units to reach the customer. Hence, it is more reasonable to consider the design of a control chart from an economic viewpoint. In the literature, many models have been developed to serve di!erent purposes for economic design of control charts. Considerable e!ort has been devoted to this area since Duncan's [6] pioneering work. Lorenzen and Vance [7] proposed
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 9 3 - 1
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a general process model for determining the economic design of control charts. This model allows production to be continued or stopped during search and/or repair. As a result, this model has been widely used because of its uni"cation. Related reference can be found in McWilliams [8], and Surtihadi and Rachavachari [9] and they studied economic design of XM chart with LV model. Montgomery et al. [10] used it to design EWMA chart, and Simpson and Keats [11] applied it to the CUSUM chart. The study of optimal procedure has received a lot of attention since the economic models are quite complex. A simpli"ed procedure was derived and applied for di!erent economic models, e.g., the optimal design parameters of the economically based XM chart [12] and economic design of np chart with a multiplicity of assignable causes [13]. Collani et al. [14] also considered a simpli"ed economic design of control charts for monitoring the nonconforming probability. In this paper, we extend the economic design of CCC chart based on the LV model. Chung's [15] simpli"ed procedure is used to derive the optimum solution. Numerical examples are shown to demonstrate the properties of the economic design of CCC charts. Sensitivity analysis is also presented to show the e!ect of input parameters on the optimum economic design of CCC chart. Finally, the numerical comparison shows that the model presented here could produce lower cost and smaller Type II error than the existing approach.
2. The cost model for CCC chart 2.1. Notations Lorenzen and Vance [7] provided the uni"ed approach to economic design of control charts and a uni"cation of notation. The following notations will be used in the formulation of the cost function for the CCC chart in this paper. The parameters can be classi"ed into four categories. Design parameters l control limit for the CCC control chart h sampling interval
Response parameters C expected cost per hour ARL1 average run length when the process is in control ARL2 average run length when process is out of control s expected number of samples while in control expected time of occurrence of the assignable cause Cost and time parameters C quality cost/hour while the production pro cess is in control C quality cost/hour while the production pro cess is out of control ('C ) Y cost per false alarm W cost to located and repair the assignable cause a "xed cost per sample b cost per unit sampled E time to sample and chart one item ¹ expected search time when false alarm ¹ expected time to discover the assignable cause ¹ expected time to repair the process Process parameters p expected fraction defective produced when the process is in control p expected fraction defective produced when the process is out of control 1/mean time process is in control
1 if production continues during searches " 0 if production ceases during searches
1 if production continues during repair " 0 if production ceases during repair 2.2. The process model A production cycle here is de"ned as the time period from the beginning of the production or after an adjustment to the detection and elimination of an assignable cause. The cycle time consists of the following "ve parts: the time until the assignable cause occurs, the time until the next sample is taken, the time to analyze and chart the sample, the
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
91
Fig. 1. Diagram of in-control and out-of-control states of a process.
time until the chart gives an out-of-control signal and the time to discover the assignable cause and repair the process. Let A , A ,2, A denote the expected values of these above "ve segments respectively, see Fig. 1. If the production continues during the search stage, it is clear to know that A , the expected time while process is in control equals 1/. However, when the production ceases during the search stage, the process is stopped to investigate the false alarms. In this case, the time spent on investigating false alarms should also be included into the expected time while the process is in control. Suppose the total items produced in the in-control stage are s, and the time spent on investigating false alarms is s¹ /ARL1. Combine these two situations, we have 1 (1! )s¹ . A " # ARL1
(1)
Other time elements are straightforward, and they can be easily derived through A "h!, (2) A "E, (3) A "h(ARL2!1), (4) A "¹ #¹ . (5) Note that E is the expected time to sample and chart one item and it is often taken as zero. For CCC chart, A "E because the sampling is based on one-item-a-time measurement and this type of measurement is more and more widely used as automation spreads over the manufacturing.
By combining (1)}(5), we can obtain the expected cycle time for CCC chart as 1 (1! )s¹ !#E E(¹)" # ARL1 #h(ARL2)#¹ #¹ . (6) Inserting the cost parameters in Eq. (6), the expected cost per cycle for CCC chart can be formulated as C E(P)" #C (!#E#h(ARL2) s> #= # ¹ # ¹ )# ARL1 (1/!#E#h(ARL2)# ¹ # ¹ ) . #(a#b) h (7) Note that since this is a renewal reward process, the expected cost per hour can be computed as the ratio of expected cost per cycle to the expected cycle time in hours. After some calculus, the expected cost per hour C for CCC chart is given as follows: S a#b S C"C # # ; D h D
(8)
where 1 s" , eHF!1 h 1 , " ! eHF!1
(9) (10)
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M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
S "(C !C )(!#E#h(ARL2) #¹ #¹ )!C [(1! )¹ #(1! )¹ ] C (1! )s¹ s> #=, ! (11) # ARL1 ARL1 (1! )s¹ D"1# ARL1 #(!#E#h(ARL2)#¹ #¹ )
(12)
and S "1#(!#E#h(ARL2)# ¹ # ¹ ). (13) The process is assumed to start in a state of statistical control with a known fraction nonconforming p . There is a single assignable cause and its occur rence results in the fraction nonconforming of process changing to p . The time between occurrences of the assignable cause is exponentially distributed with a mean of occurrences per hour. These assumptions are commonly used in the economic design, see, e.g., [7,16,17]. We assume that sampling is based on one-at-atime measurement. That is, the sample size is equal to one. This assumption greatly simpli"es the economic design of the CCC chart and it is usually the case in automated manufacturing environment. Since the sample size is always `onea, the cost of taking a sample and inspecting it is actually a known constant, a#b. Note that the sample size equals one is not a serious assumption and in fact, the idea of CCC chart has the advantage that there is no need for rational sample size. Only low-sided CCC chart studied in this article because of its simplicity and practicality. When the CCC data is out of upper control limit, the process is better and the fraction of nonconforming is smaller, which causes no actual problem. Therefore, we are more concerned with the case of out of lower control limit (LCL). Actions should be taken in this situation. In the course of inspection, one item is checked every h hours of production. The number of consecutive conforming items is counted until a nonconforming one is detected. If the count is larger than or equal to LCL, no action is taken. Otherwise, assignable causes should be searched and eliminated.
3. The optimization procedure The algorithm by Lorenzen and Vance [7] to "nd the most economical design is somewhat complicated as it consists of Newton's Method, golden section search and Fibonacci search. This could be the main reason that limits the application of their method. Furthermore, to solve formula (8), they expand the exponential in , /h, s and s/h, and ignore all terms containing powers of greater than 1. Hence, the solution is highly approximate despite its complexity. In Chung [16], a simpli"ed procedure is developed to determine the optimum parameter values based on Lorenzen and Vance's model. The procedure consists of solving an explicit formula for h in terms of n and l and yields lower loss costs than those of Lorenzen and Vance's model. Moreover, this simpli"ed procedure does not use those assumptions and approximations made in Lorenzen and Vance's model. Furthermore, no terms are neglected for "nding the optimum solution. Consequently, it was demonstrated that the procedure gives more accurate, more reliable and more feasible optimum values. Hence, it is more appropriate for economic design of CCC chart. After replacing 1/(eHF!1) in formulas (8)}(12) and (13) by 1/h!0.5, the explicit formula of h for CCC chart is derived !r #(r !4r r ) h" 2r
(14)
where
(1! )¹ r "(ARL2!0.5) (C !C ) 1! 2ARL1
> # #C [(1! )¹ 2ARL1 #(1! )¹ ]
C (1! )¹ != ! 2ARL1
!(a#b)[(ARL2!0.5)],
(15)
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
r "2(ARL2!0.5)
C (1! )¹ !> ARL1
!(a#b)(1#E# ¹ # ¹ ) ,
(16)
C (1! )¹ !> r " ARL1
2. calculate the ARL1 and ARL2 according to the value of l from Eqs. (18) and (19), respectively, 3. calculate h from Eq. (14), 4. calculate C(l , h ) from Eq. (8), 5. repeat 1}4 and "nd CH" min [C(l , h )]"C(lH, hH). ?WJ W@
(1! )¹ #E#¹ #¹ 1! ARL1
The values lH and hH which give the overall approximate minimum loss cost CH are the optimal economic design for the CCC control chart.
(1! )¹ (C !C )(E#¹ #¹ ) # ARL1 !C [(1! )¹ #(1! )¹ ] C (1! )¹ > #= # ! 2ARL1 2ARL1
93
(a#b)(ARL2!0.5)(1! )¹ # ARL1 !(a#b)(1#E# ¹ # ¹ ) (1! )¹ #E#¹ #¹ . ; 1! 2ARL1
(17)
As the above formulas show, the loss function depends on ARL1, ARL2 and h. h can be solved in terms of ARL1 and ARL2, which are functions of control limit l, the relationships between average run length and control limit are given by Xie et al. [18]. 1 1!p # ARL1"l# p p [1!(1!p )J\]
(18)
and 1!p 1 # ARL2"l# . p p [1!(1!p )J\]
(19)
In the process of obtaining the approximate optimum values lH and hH, l can be considered as discrete variables. The complete procedure is as follows: 1. given a value of l which belongs to a reasonable searching range (a)l )b),
4. Numerical illustration Lorenzen and Vance [7] presented an application of their economic model of a foundry operation which produces 84 casting per hour on one of its lines. In castings, a standard is set to prevent high carbon-silicate content as it will result in low tensile strength. The sampling process costs $4.22 per sample and takes about "ve minutes. The process stays in control for 50 hours. When the process is out of control, the system must be #ushed and restarted, and this should take about 45 minutes with a repair crew cost of $22.80 per hour and a downtime cost of $21.34 per minute. It takes about "ve minutes to get the repair crew assembled and no time to search for the cause. Here we consider the case when it is a highquality process; that is, the fraction nonconforming is very low. In such a situation, traditional p chart becomes useless because even for a moderate sample size there will seldom be any nonconforming item and thus CCC chart should be used. Suppose that historical data indicate that the process produces about 0.05% nonconforming items when in control and about 5% nonconforming items when out of control. The cost per hour while in control, C "100;84;0.0005"$4.2. Likewise, C is equal to $420. Other speci"c input parameter values are "1/50 h, E"¹ "¹ "5/60 hours, ¹ "45/60 hours, "1, "0, >"="$977.40, a"0, b"4.22. Thus, from Formula (8), the loss function approaches its optimal value CH"77.4763 when the sampling interval is hH"0.3015 and the control limit is lH"96.97.
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M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
Table 1 The economic design of the CCC charts for some di!erent p and p values p
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009
p
0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
p /p
100 90 80 70 60 50 40 30 20 10 5 3.33 2.5 2.0 1.67 1.3 1.25 1.11
Economically optimal design lH
hH
ARLH
ARLH
CH
H
H
96.96 107.64 121.00 138.18 161.07 193.13 241.20 321.31 481.50 962.08 960.01 958.08 956.78 956.3 956.5 957.5 958.9 960.9
0.0333 0.0321 0.0306 0.0292 0.0277 0.0261 0.0244 0.226 0.0210 0.0214 0.0215 0.0216 0.0217 0.0219 0.0220 0.0221 0.0223 0.0224
1057200 952780 848410 744110 639860 535680 431560 327530 223630 120020 34605 17644 11323 8220 6440.9 5311 4542.5 3991.2
357.56 397.34 447.07 511.01 596.26 715.61 894.63 1193.0 1789.7 3580 3580 3580 3580 3580 3580 3580 3580 3580
413.86 427.46 443.21 461.90 484.60 513.02 550.10 601.48 679.79 820.08 820.44 820.93 821.49 822.07 822.7 823.2 823.7 824.26
0.0095 0.0106 0.0119 0.0136 0.0158 0.0190 0.0237 0.0315 0.0469 0.0916 0.1745 0.2496 0.3177 0.3798 0.4364 0.4882 0.5355 0.5786
0.3812 0.3813 0.3814 0.3815 0.3816 0.3817 0.3818 0.3820 0.3821 0.3822 0.3830 0.3838 0.3843 0.3845 0.3844 0.3841 0.3835 0.3827
Using formula (18) and (19), we have that: ARL1H"1.057;10 and ARL2H"357.56 and the probability of false alarm when the process is in control is H"0.0096 and the probability of failing to detect the shift when the process is out of control is H"0.3812. Table 1 shows the economic designs of CCC charts with varied values of p and p . The other parameters, such as ARL1 which represents average run length when process is in control and ARL2 which represents average run length when process is out of control, the probability of false alarms () and the probability of failing to detect the assignable cause are also listed in this table. From this table, it can be noted that the optimal loss CH is a decreasing function of p /p . For a given value of p , the optimal sampling interval hH is increasing with value of p , and for a given value of p , hH is increasing when the value of p increases. The situation of optimal value of lH is opposite to that of hH. That is, for a given value of p , the optimal control limit is decreasing with value of p , and for a given value of p , lH is decreasing when the value of p increases. The
above analysis is unsuitable for the very small shift, say, p /p (2. For instance, as last "ve lines in Table 1 shows, lH is increasing when the value of p increases. Both ARL1 and ARL2 are the increasing functions of the ratio of p and p . When the ratio is relatively large, the probability of false alarm is small. However, when the ratio is very small, for example, smaller than 2, the probability of false alarm will increase dramatically. Especially, when the ratio approaches to `1a, the probability of false alarm reaches 58%. Table 1 also shows that the probability of economic design failing to detect the assignable cause almost keeps constant in the whole range of ratio. 4.1. Sensitivity analysis When applying Lorenzen and Vance [7] model in the economic design of CCC chart, there are "fteen input parameters related to the loss function. The parameters, p and p are process parameters which specify the in-control and out-of-control states of the process monitored. The e!ect of them
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
on the economic design has been studied in above section. Here, we only consider the other input parameters. This paper adopts the approach used in Costa [19] and Bai et al. [20]. Each example in Table 2 represents the economic design of CCC chart with one di!erent parameter at a time. Therefore, sensitivity analysis of the proposed model with regard to these input parameters can be easily made. It can be seen that , which is the average occurrence rate of the assignable cause, has a signi"cant e!ect on the optimum sampling interval and the cost. A smaller implies a larger optimal sampling interval and smaller cost. From Table 2, we can see that when the value of decreases by 50%, the value of hH increases about 29% and the value of CH decreases about 20%. However, the optimum control limit lH is rather robust to the changes of . The time to sample and chart one item (E), the expected time to discover the assignable cause (¹ ) and expected time to repair the process (¹ ) all have very slight e!ects on the optimal values. However, there are still some trends that can be found. For example, the larger the ¹ , the larger CH. However, ¹ , the expected search time when false alarm, seems has no e!ect on the optimal values. It can also be observed that whether the production stops/continues during the search and whether the production stops/continues during repair a!ect the optimal values slightly. When the production continues during both searches and repairs, the cost will be slightly larger than other three modes. Variations in the quality cost per hour when process is in control and out of control (C and C ) can cause relatively big e!ects on optimum values of hH and CH, while hardly a!ect the optimum value of lH. On the other hand, it is somewhat a bit surprise that the cost per false alarm (>) and the cost to locate and repair the assignable cause (=) only have little e!ect on the optimum values. This example shows that the optimum control limit lH is relatively robust to all the input parameters (except for p and p ). Among all the factors, C , C and are the most sensitive factors which a!ect the optimal cost and optimal sampling interval for CCC chart.
95
4.2. Comparisons The economic design of CCC chart has been studied in this paper. The cost model derived for CCC chart in this paper is in the spirit of Lorenzen and Vance's [7] general model, and a combination of properties of CCC chart is used here. In addition, the use of a simpli"ed procedure greatly reduces the di$culty in getting the optimal parameters of CCC chart. The model used in this article allows production to be continued or stopped during search and/or repair. Xie et al. [18] presented a number of examples where the CCC chart is used to monitor a highquality process. Speci"c input parameter values used in their paper are given by p "500 ppm, p "0.005, "0.01, < "150, < "150, A "10, A "150, t "0.1, t "0.3 and c"0.5. The 18 examples presented in Section 5 of Xie et al. [18] are used for comparison. Table 3 shows the comparison of the results, `Aa and `Ba respectively represent the solutions in this paper and the earlier paper. The approximate optimum loss-cost value in Xie et al. [18] is always higher than the optimum value obtained here and on the average, the percentage is about 2%. In addition, both the traditional and economic design presented in Xie et al. [18] can hardly detect the small shifts since their Type II error is extremely big in those situations. Especially, as the last line in Table 3 shown, the ability of detecting the special shift presented in their paper is only 33%, while our design can still keep ability at a moderate level.
5. Discussions Economic considerations should be incorporated in control chart design in manufacturing industry. However, due to the existence of di!erent process models and the number of parameters involved, such a study is not straightforward, and it may add to the complexity to the implementation
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
0.0005 0.0005 0.0005
1 2 3
E1 E2 E3
T01 T02 T03
T11 T12 T13
T21 T22 T23
1 2 3 4
C01 C02 C03
C11 C12 C13
Y1 Y2 Y3
W1 W2 W3
A1 A2 A3
B1 B2 B3
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
0.005 0.005 0.005
Example Parameter values no. p p E
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
0.02 0.02 0.02
5/60 5/60 5/60
¹
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
¹
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60
5/60 5/60 5/60 5/60
5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
5/240 5/60 5/120 5/60 5/60 5/60
0.005 5/60 0.01 5/60 0.02 5/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60 45/60
45/120 45/60 45/30
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
45/60 45/60 45/60
¹
Table 2 Sensitivity analysis of economic design for CCC control charts
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 0 0 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 1 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
14.24 114.24 214.24
114.24 114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
114.24 114.24 114.24
C
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
849.2 949.2 1049.2
949.2 949.2 949.2
949.2 949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
949.2 949.2 949.2
C
0 0 0
0 0.5 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
a
2.22 3.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
4.22 4.22 4.22
b
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
777.40 877.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
>
977.40 977.40 977.40
977.40 977.40 977.40
777.40 877.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
977.40 977.40 977.40
=
189.2 190.5 191.2
191.2 191.5 191.7
191.2 191.2 191.2
191.7 191.5 191.2
191.3 191.2 191.2
191.2 191.2 191.3
191.2 191.4 191.3 191.3
191.2 191.2 191.2
191.2 191.2 191.2
191.2 191.2 191.2
191.2 191.2 191.2
190.9 191.1 191.2
lH
0.0173 0.0219 0.0262
0.0262 0.0283 0.0304
0.0261 0.0262 0.0262
0.0262 0.0262 0.0262
0.0287 0.0262 0.0242
0.0243 0.0262 0.0283
0.0262 0.0267 0.0261 0.0268
0.0263 0.0262 0.0260
0.0262 0.0262 0.0262
0.0262 0.0262 0.0262
0.0262 0.0262 0.0262
0.0438 0.0332 0.0262
hH
422.4 472.9 514.1
514.1 532.9 549.1
511.2 512.6 514.1
513.9 514.0 514.1
486.1 514.1 540.5
441.6 514.1 585.0
514.1 524.6 512.7 526.0
516.9 514.1 508.6
513.5 513.7 514.1
514.1 514.1 514.1
513.5 513.7 514.1
328.1 408.9 514.1
CH
Optimal economic design
96 M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
M. Xie et al. / Int. J. Production Economics 72 (2001) 89}97
97
Table 3 Comparison results Example no.
1 2 3 4 5 6 7 8 9
A
Example no.
B
CH
H
CH
H
25.26 26.51 27.96 29.70 31.83 34.52 38.08 43.11 51.08
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
25.53 26.76 28.22 29.96 32.08 34.77 38.32 43.34 51.29
0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
of control charting technique, which is supposed to be a simple approach for process monitoring. The focus here is on the use of CCC chart which is based on cumulative count of conforming items, and this is a better alternative than the traditional attribute chart which requires a rational sample size. A different model than a previous paper is used and a simple optimization procedure is adopted in this paper. Sensitivity analysis is presented and numerical examples are used to illustrate the procedure.
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10 11 12 13 14 15 16 17 18
A
B
CH
H
CH
H
66.86 66.87 66.89 66.90 66.92 66.94 66.95 66.97 66.99
0.382 0.3822 0.3823 0.3825 0.3825 0.3825 0.3824 0.3823 0.3821
67.03 67.04 67.06 67.12 67.72 68.73 69.94 71.23 72.53
0.3819 0.3823 0.3823 0.4109 0.491 0.5531 0.6021 0.642 0.6742
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