# An optimal design of joint X and S control charts using quadratic loss function.

Introduction

Control charts are important tools of statistical quality control. These charts are used to decide whether a process has achieved a state of statistical control and to maintain current control of a process. To use any of these charts, three design parameters must be specified: the sample size n, the sampling period h, and the width of control chart k (the number of standard deviation above or below the centre line). For these charts, the design parameters are still mostly determined from experience, but general guidelines simply do not exist. How should a user determine the design parameters of control chart?

This question was first answered by Duncan (1956). He recommended the use of a concept which he called an economic design; to obtain the optimal design parameters of [Mathematical Expression Omitted] chart by minimizing the net sum of all quality cost involved. Since then,variations of Duncan's approach have appeared. For a review of the literature on [Mathematical Expression Omitted]-charts, see Montgomery (1980) and Vance (1983). Duncan (1974) noted that the joint employment of an [Mathematical Expression Omitted] chart to control process mean and R chart to control process variability will give reasonably good control of the whole process.

Recently, joint economic design of [Mathematical Expression Omitted] and R charts has been treated by Saniga (1977, 1979, 1989), Saniga and Montgomery (1981), Jones and Case (1981), Rahim et al. (1988), Rahim (1989), and Yang (1993). Nevertheless there is a drawback in using joint [Mathematical Expression Omitted] and R charts. Duncan (1974) and Besterfield (1979) report that when sample sizes are moderately large, the range statistic loses its efficiency rapidly. Alt (1981) suggests to use [Mathematical Expression Omitted] and [S.sup.2] charts for process control simultaneously. Alt's study is based on the power function approach which is a statistical criterion not an economic criterion. Rahim et al. (1988) present the economic design of joint [Mathematical Expression Omitted] and [S.sup.2] and charts when a single assignable cause process model is considered. Collani and Sheil (1989) propose an economically optimal S chart design. They note that the occurrence of assignable cause may lead to an increase in process variability without necessarily influencing the level of the process mean. Despite the extensive research work, Saniga and Shirland (1977) report the applications of the economic designs for various control charts are very limited. Elsayed and Chen (1994) indicate that the lack of practical application is due to the complexities of the mathematical models and the search for optimal solutions, and the difficulties in estimating the costs. Recently, the performance of computers has much improved, and the search for the optimal solutions has become feasible. However, the estimation of costs is still an obstacle for applying the economic control charts.

Conventionally, quality loss is considered as the cost incurred when the quality characteristic is not within the specification limits. Taguchi (1984) defines quality loss as "the loss to society caused by the product after it is shipped out". Taguchi et al. (1989) indicate that a quadratic approximation function sufficiently represents economic losses due to the deviation of quality characteristic from its target. Kacker (1986) indicates that the concept of quadratic loss emphasizes the importance of continuously reducing performance variation. Various quality evaluation systems using the loss function approach were presented by Chen and Kapur (1989).

Taguchi (1984) and Taguchi et al. (1989) provide an economic design to determine the diagnosis interval and control limits for on-line production process applying the loss function. The loss function as a rational approach for the minimization of the process variation has been widely accepted. Koo and Lin (1992) include such an approach in the economic design of [Mathematical Expression Omitted] chart. They modify Duncan's (1956) cost model with Taguchi's loss function. Elsayed and Chen (1994) present a new economic design of [Mathematical Expression Omitted] charts based on Taguchi's quality loss function for processes with continuous operations. However, no research work has been performed to include such an approach in the economic design of joint [Mathematical Expression Omitted] and S charts for the double assignable-muse process model. In this paper, we present an economic design of joint [Mathematical Expression Omitted] and S and control charts based on quadratic loss function for process with double assignable causes.

We first present assumptions and definitions of in-control and out-of-control periods in a production cycle. We then derive a double assignable-cause cost model by renewal theory approach. Finally, performance analyses of the cost model are carried out by using a fractional factorial design of experiment.

Assumptions

Assumptions 1-4 used in our cost model are similar to those given in Ho and Case (1994). Assumptions 5-10 are related to the expected time of occurrence of assignable causes and shifts in the process mean and variance for the interested quality characteristic. The assumptions are:

(1) The production process starts with a statistical in-control state with mean [Mu] and standard deviation [Sigma].

(2) There are two assignable causes, say assignable cause 1 and assignable cause 2, which influence the process parameters. One of them shifts the process mean away from the target value; the other increases the process variability. When one of the assignable causes has occurred, it can be followed by the occurrence of another assignable cause. When an assignable cause occurs, the magnitude of the shift in the process mean is [+ or -] [[Delta].sub.1]([[Delta].sub.1] [greater than] 0) and the magnitude of the increase in the process standard deviation is [[Delta].sub.2] [Sigma]([[Delta].sub.2] [greater than] 1). We assume when the shift in the process occurs, there is a probability of 0.5 that the mean shifts from [Mu] to [Mu] + [[Delta].sub.1][Sigma] and a probability of 0.5 that it shifts from [Mu] to [Mu] - [[Delta].sub.1][Sigma].

(3) The occurrence times([T.sub.1] and [T.sub.2]) of the assignable causes are independently exponentially distributed with mean 1/[Lambda], i = 1,2. They are assumed to be independent because those factors that would bring about a change in the mean are unlikely to also affect the inherent short-term variation of the process, and vice versa. Both are assumed exponential since complex processes, regardless of their failure distribution, often asymptotically approach an exponential failure distribution.

(4) The process output of quality characteristic is assumed to follow normal distribution.

(5) Samples of size n are taken every h hours under a production, rate U and the control limits of the [Mathematical Expression Omitted] chart and S chart are set at, [Mu] [+ or -] [k.sub.1][Sigma]/[square of n] and [k.sub.2][Sigma] respectively, where [k.sub.1] and [k.sub.2] are constants. If at least one data point falls outside the control limits of the [Mathematical Expression Omitted] and S charts, the process is deemed to be out of control and needs to be searched for assignable causes and an appropriate corrective action is taken. If the plotted points fall inside the control limits of these charts then the process continues and the next sample is taken after h hours.

(6) The expected time ([[Tau].sub.i]) of occurrence of the shift between the process starts and the first sampling and testing given the assignable cause i occurs is estimated as follows:

[Mathematical Expression Omitted]. (1)

Duncan (1956) proved that this approximation is quite robust for h [less than or equal to] 20.

(7) The expected time ([Tau](i) i = 1,2) of the ith occurrence of the shift between the process starts and the first sampling and testing given both assignable causes occur is estimated as follows Yang (1993):

[Mathematical Expression Omitted] (2)

[Mathematical Expression Omitted] (3)

(8) The probability ([Alpha]) that at least one sample point falls outside the control limits of joint [Mathematical Expression Omitted] and S charts when no assignable causes are actually present is:

[Mathematical Expression Omitted], (4)

where

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

([Mathematical Expression Omitted], s) are the sample mean and sample standard deviation of process output.

(9) The probability that no sample points fall outside the control limits of joint X and S charts when a shift of the process mean (variance) truly occurs can be calculated as [[Beta].sub.1]([[Beta].sub.2]), where is the probability of type II error:

[Mathematical Expression Omitted] (5)

where [Mathematical Expression Omitted] (6)

where

[Mathematical Expression Omitted]

(10) The probability that no sample points fall outside the control limits of joint [Mathematical Expression Omitted] and S charts when the shifts of process mean and variance truly occur can be calculated as [[Beta].sub.3], where [[Beta].sub.3] is also the probability of type II error:

[[Beta].sub.3] = [[Beta].sub.[x.sub.3]] - [[Beta].sub.[s.sub.2]], (7)

where

[Mathematical Expression Omitted],

[Mathematical Expression Omitted], and [Mathematical Expression Omitted]

(11) Process is discontinuous. That is, the process ceases during the search state.

The exacted production cycle time

A production cycle is defined as the time between the start of successive in-control period. The production cycle consists of four components:

(1) in-control period;

(2) time to signal an out-of-control state;

(3) time to sample and test;

(4) time needed to find and repair the assignable causes.

The last three periods form the out-of-control period.

To derive the expected cycle time applying the renewal theory approach, we have to study the process state of the system at the end of the first sampling and testing. Based on these states we may compute the expected residual cycle length. These values, together with the associated probabilities, lead us to formulate the renewal equation. We denote the times as follows:

[W.sub.0] = expected search time when false alarm;

[W.sub.1] = expected time to search and repair assignable cause 1;

[W.sub.2] = expected time to search and repair assignable cause 2;

[W.sub.3] = expected time to search and repair assignable cause 1 and assignable cause 2;

E(T) = expected cycle time.

Thus, eight states are defined as follows:

State 1 in-control and no alarm;

State 2 in-control and at least one false alarm indicted by [Mathematical Expression Omitted] chart or S chart or both;

State 3 out-of-control (assignable cause 1 occurs) and no alarm;

State 4 out-of-control (assignable cause 1 occurs) and at least one alarm indicated by one of the charts or both;

State 5 out-of-control (assignable cause 2 occurs) and no alarm;

State 6 out-of-control (assignable cause 2 occurs) and at least one alarm;

State 7 out-of-control (both assignable causes occur) and no alarm;

State 8 out-of-control (both assignable causes occur) and at least one alarm.

Table I displays the possible states of the system, the expected residual cycle time, and associated probability of being in each respective state at the end of the first sampling and testing interval.

Consequently, the renewal equation is

E(T) = h + [summation of] [P.sub.i][l.sub.i]] where i = 1 to 8 = h + E(T)[P.sub.1] + E(T)[P.sub.2] + [W.sub.0][P.sub.2] where i = 1 to 8 + [summation of] [l.sub.i][P.sub.i]] where i = 3 to 8.

Simplifying we have

[Mathematical Expression Omitted] (8)

Here, we assume the time to sample and test one item is zero. However, it should be noted that the renewal equation approach can be extended to include cases involving non-zero time to sample and test one item (Banerjee and Rahim, 1987).

Cost model

In our cost model, we first calculate the expected cost per production cycle and then divide it by the expected cycle time to obtain the expected cost per unit time. The objective is to determine four decision variables: sample size (n), sampling interval (h), control limit constant ([k.sub.1]) of [Mathematical Expression Omitted] chart and control limit constant ([k.sub.2]) of s chart.

[TABULAR DATA FOR TABLE I OMITTED]

The process costs such as cost of sampling and testing, cost of false alarm, and costs of finding and fixing the shifts in the process model. We denote the costs as follows:

[b.sub.0] fixed cost of sampling and testing;

[b.sub.1] variable cost of sampling and testing a unit of production;

[a.sub.0] cost of investigating a false alarm;

[a.sub.i] cost of finding and fixing assignable cause i,i = 1,2;

[a.sub.3] cost of finding and fixing assignable cause 1 and assignable cause 2 simultaneously;

E(C) the expected cycle cost.

The expected quality loss per unit of production can be easily estimated by calculating the process variance and the deviation of process mean from the target. Let the target value of the quality characteristic be [Mu]. An asymmetric quadratic loss function would be appropriate if the loss differs for values of a quality characteristic that are equidistant from the target. Let the asymmetric loss function be the form (9) (or Figure 1).

[Mathematical Expression Omitted] (9)

where

X [similar to] N([Mu], [[Sigma].sup.2]) during the in-control period,

X [similar to] N([Mu] [+ or -] [[Delta].sub.1] [Sigma], [[Sigma].sup.2]) if only assignable cause 1 occurs in the process,

[Mathematical Expression Omitted] if only assignable cause 2 occurs in the process,

[Mathematical Expression Omitted] if both the assignable causes occur in the process,

[A.sub.0] is the cost of repair or replacement of the product, [Mu] - [r.sub.0][Sigma] and [Mu] + [r.sub.0][Sigma] are functional limits.

According to the asymmetric quadratic loss function, the expected loss ([C.sub.0]) per unit of production during the in-control period is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted].

Once again, we assume when the shift in the process occurs, there is a probability of 0.5 that the mean shifts from [Mu] to [Mu] + [[Delta].sub.1] [Sigma] and a probability of 0.5 that it shifts from [Mu] to [Mu] - [[Delta].sub.1][Sigma].

The expected loss ([C.sub.1]) per unit of production during the period that process is only influenced by the assignable cause 1 is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted],

[Mathematical Expression Omitted].

The expected loss ([C.sub.1]) per unit of production during the period that process is only influenced by the assignable cause 2 is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted].

The expected loss ([C.sub.3]) per unit of production during the period that process is influenced by both the assignable causes is:

[C.sub.3] = 1/2 {E[L([Mu] + [[Delta].sub.1] [Sigma], [[Delta].sub.2] [Sigma])] + E[L([Mu] - [[Delta].sub.1][Sigma], [[Delta].sub.2][Sigma])]},

where [Mathematical Expression Omitted]

and [Mathematical Expression Omitted], and

[Mathematical Expression Omitted].

where [Mathematical Expression Omitted].

Therefore, the expected cycle cost can be derived by the renewal equation approach. Table II presents the total quality cost which is the sum of the expected residual cost and the cost incurred in the first sampling and testing for each possible process state of the system.

Consequently, the renewal equation is

E(C) = [P.sub.1][([b.sub.0] + [b.sub.1]n) + [C.sub.0]Uh + E(C)] + (E(C) + [a.sub.0] + ([b.sub.0] + [b.sub.1]n) + [C.sub.0]Uh)[P.sub.2] + [summation of] [P.sub.i][R.sub.i]] where i = 3 to 8

Simplifying:

[Mathematical Expression Omitted] (10)

[TABULAR DATA FOR TABLE II OMITTED]

Applying the property of renewal reward process (Ross, 1989), the expected cost per unit time (E([V.sub.[infinity]])) is derived by taking the ratio of the expected cycle cost (E(C)) and the expected cycle time (E(T)); E([V.sub.[infinity]]) = E(C)/E(T). The cost function is the function of design parameters. Hence, the optimal design parameters can be determined by minimizing the cost function.

Performance and sensitivity analysis

Sensitivity analysis on the numerical results provides some useful managerial insights that are valid at least over the wide range of the values of the parameters tested. In this section, we perform a fractional factorial experiment to study the effects and sensitivities of the input parameters on the design parameters. The algorithm used in conducting this experiment is the direct search method for a specified integer n, and specified values of parameters in the estimated range. For simplicity, we treat n, h, [k.sub.1] and [k.sub.2] as discrete variables (Chung, 1990), and assume that the values of n, h, [k.sub.1] and [k.sub.2] are within the ranges between 1 and 50 (1 [less than] n [less than] = 50, and the unit length of n is 1), 0 and 8 (0 [less than] h [less than] = 8.0, and the unit length of h is 0.5), 0 and 6 (0 [less than] k, [less than] = 6, and the unit lengths of [k.sub.1] and [k.sub.2] are 0.1) respectively. For minimizing the cost function which is subject to constraints, the method starts by searching a coarse grid to find an approximate minimum cost E([V.sub.[infinity]]). The objective function and constraints expressed mathematically are

Min. E([V.sub.[infinity]])

s.t. 0 [less than] h [less than] = 8

0 [less than] [k.sub.1] [less than] = 6 0 [less than] [k.sub.2] [less than] = 6 1 [less than] n [less than] = 50.

There are 18 input parameters for the derived cost function. We assign two levels to each of 12 input parameters in the factorial experiment design. The remaining six input parameters are assigned one level, because they are always not sensitive on the design parameters (Panagos et al., 1985). Table III presents the artificial values of input parameters and their assigned levels.

For convenience, the levels are basically the same as those used in Panagos's experiment(1985) except that the quality cost is calculated using the asymmetric quadratic loss function that contains cost of repair or replacement ([A.sub.0]), customer tolerance ([r.sub.0], [r[prime].sub.0] and production rate (U) instead of the in-control and out-of-control hourly costs. We should note that if the levels of input parameters are varied over different ranges, the experimental results may be different. However, the levels used for study here are not unrealistic.

The orthogonal array [L.sub.32]([2.sup.31]) used for the performance and sensitivity study is resolution IV. The experimental design of resolution IV means that no main effect is confounded with any other main effect or with any two-factor interaction, but two-factor interactions are confounded with one another. A total of 32 runs are required to conduct the experiment. The experimental results are shown in the Appendix.
```Table III. Input parameters and levels

Levels
No.         Factors                  Low         High

01         [A.sub.0]                  1          50
02         [r.sub.0]                  1           3
03      [r[prime].sub.0]              1           3
04       [[Delta].sub.1]              1           2
05       [[Delta].sub.2]              1.5         2.5
06      [[Lambda].sub.1]              0.05        0.01
07      [[Lambda].sub.2]              0.04        0.0083
08         [b.sub.0]                  0.5         5
09         [b.sub.1]                  0.1         1.0
10             U                    100        1000
11         [a.sub.0]                 35         250
12         [W.sub.0]                  0.4         1.0
13         [a.sub.1]                 50
14         [a.sub.2]                 60
15         [a.sub.3]                100
16         [W.sub.1]                  2
17         [W.sub.2]                  3
18         [W.sub.3]                  5

Note: [Mu] = 0, [Sigma] = 1
```

We consider these seven two-factor interactions ([A.sub.0][r.sub.0], [A.sub.0][r.sub.0], [A.sub.0][[Delta].sub.2], [A.sub.0]U, U[[Delta].sub.1], U[[Delta].sub.2]) since they are multiplied together to estimate the out-or-control expected loss. The remaining two-factor interactions are considered relatively insignificant. Therefore, we calculate the F-ratio values for 12 main effects and seven two-factor interactions and consider the effects of the remaining two-factor and higher order interactions as residual with 12 degrees of freedom in the ANOVA table. The significant levels of effects determined by the analysis of variance indicate the relative importance of effects rather than the evidence of statistical hypothesis.

We consider the input factors which have significant effects on the expected total cost. Table IV shows the analysis of variance for the response of optimized expected cost.

Table IV shows that the optimized expected cost is significantly affected by two main effects: [A.sub.0] and U and one two-factor interaction: [A.sub.0]U. We take 5 per cent significant level. The significance levels indicate that the cost of repair or replacement ([A.sub.0]) and production rate per hour (U) have a dominate effect on the optimized expected cost. The effect of the interaction shows that the optimized expected cost is very sensitive to U when the [A.sub.0] is high (see Figure 2). Table V shows the analysis of variance for the response of optimized sample size.

As shown in Table V, the sample size is significantly affected by two main effects: [A.sub.0] and [b.sub.1]. Their significance levels show that the cost of repair or replacement ([A.sub.0]) has a dominate effect on the optimized sample size. The variable sampling and testing cost ([b.sub.1]) is also important. Table VI shows the analysis of variance for the response of optimized sampling interval.

As shown in Table VI, the optimized sampling interval is significantly affected by three main effects' [A.sub.0], [r.sub.0] and U, and two two-factor interactions' A, [r.sub.0] and [A.sub.0]U. The effect of interaction between [A.sub.0] and [r.sub.0] suggests that the sampling interval is more sensitive to U when [A.sub.0] is low (Figure 3). The effect of interaction between [A.sub.0] and U also suggests that the sampling interval is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 4 OMITTED]. Table VII shows the analysis of variance for the response of optimized control limit coefficient of [Mathematical Expression Omitted] chart.

As shown in Table VII, the optimized control limit coefficient ([K.sub.1]) of chart is significantly affected by two main effects: [A.sub.0] and U, and one two-factor interaction: [A.sub.0] and U. Their significance levels show that the cost of repair or replacement ([A.sub.0]) has a dominate effect on the optimized control limit coefficient ([K.sub.1]) of [Mathematical Expression Omitted] chart. The production rate (U) is also important. The effect of interaction between [A.sub.0] and U suggests that the optimized control limit coefficient ([K.sub.1]) is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 5 OMITTED]. Table VIII shows the analysis of variance for the response of optimized control limit coefficient ([K.sub.2]) of S chart. We find that the optimized control limit coefficient ([K.sub.2]) of S chart is also significantly affected by two main effects: [A.sub.0] and U, one two-factor interaction: [A.sub.0]U. Their significance levels also show that the cost of [TABULAR DATA FOR TABLE IV OMITTED] repair or replacement ([A.sub.0]) has a dominate effect on the optimized control limit coefficient ([K.sub.2]) of chart. The production rate (U) is also important. The effect of interaction between [A.sub.0] and U also suggests that the optimized control limit coefficient ([K.sub.2]) is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 6 OMITTED].

Summary and conclusions

We developed the first economic design of [Mathematical Expression Omitted] and S charts based on the quadratic loss function, a double assignable-cause cost model is [TABULAR DATA FOR TABLE V OMITTED] derived applying renewal theory approach. The expression for the expected cycle length and the expected cost per cycle is easier to obtain by the proposed approach, and the cost model, including the customers' voice, reveals the importance of quality. In practice, the economic [Mathematical Expression Omitted] and S charts can be constructed applying the proposed approach when the values of cost and process parameters are known. Economic [Mathematical Expression Omitted] and S charts should be preferable if quality engineers would like to maintain the whole process with minimum cost.

In data analysis, the effects of input factors have been investigated. The optimized expected cost is significantly affected by [A.sub.0], U and [A.sub.0]U. The sample size is significantly affected by [A.sub.0] and [b.sub.1]. The optimized sampling interval is significantly affected by [A.sub.0], [r.sub.0], U, [A.sub.0][r.sub.0] and [A.sub.0]U. The control limit coefficients of X and S and charts are all significantly affected by [A.sub.0], U and [A.sub.0]U. It is also clear that the cost of repair or replacement ([A.sub.0]) and customers' tolerance ([r.sub.0]), which are related to the loss function, are critical when designing economically based on [Mathematical Expression Omitted] and S charts. This provides a clear incentive for production [TABULAR DATA FOR TABLE VI OMITTED] [TABULAR DATA FOR TABLE VII OMITTED] engineers to place more efforts on off-line process design and consistent quality improvement in order to maintain a robust production process.

Several important extension of the present model can be studied. Generally it is straightforward to extend the model proposed to study other control charts, like the np-charts for attributes. Their differences lie in the definition of the control limits and the derivation of the probabilities of Type I and Type II errors. The method proposed can also be applied to the classic case of economic design of [Mathematical Expression Omitted]-charts or S-charts for process with a single assignable cause.

[TABULAR DATA FOR TABLE VIII OMITTED]

A particularly interesting research area revolves around the economic modelling of production processes subject to a multiplicity of assignable causes. Finally, the modelling of imperfect inspection and restoration procedures also provides additional topics of interest for research in the area.

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[TABULAR DATA FOR TABLE A1 OMITTED]

(8) The probability ([Alpha]) that at least one sample point falls outside the control limits of joint [Mathematical Expression Omitted] and S charts when no assignable causes are actually present is:

[Mathematical Expression Omitted], (4)

where

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

([Mathematical Expression Omitted], s) are the sample mean and sample standard deviation of process output.

(9) The probability that no sample points fall outside the control limits of joint X and S charts when a shift of the process mean (variance) truly occurs can be calculated as [[Beta].sub.1]([[Beta].sub.2]), where is the probability of type II error:

[Mathematical Expression Omitted] (5)

where [Mathematical Expression Omitted] (6)

where

[Mathematical Expression Omitted]

(10) The probability that no sample points fall outside the control limits of joint [Mathematical Expression Omitted] and S charts when the shifts of process mean and variance truly occur can be calculated as [[Beta].sub.3], where [[Beta].sub.3] is also the probability of type II error:

[[Beta].sub.3] = [[Beta].sub.[x.sub.3]] - [[Beta].sub.[s.sub.2]], (7)

where

[Mathematical Expression Omitted],

[Mathematical Expression Omitted], and [Mathematical Expression Omitted]

(11) Process is discontinuous. That is, the process ceases during the search state.

The exacted production cycle time

A production cycle is defined as the time between the start of successive in-control period. The production cycle consists of four components:

(1) in-control period;

(2) time to signal an out-of-control state;

(3) time to sample and test;

(4) time needed to find and repair the assignable causes.

The last three periods form the out-of-control period.

To derive the expected cycle time applying the renewal theory approach, we have to study the process state of the system at the end of the first sampling and testing. Based on these states we may compute the expected residual cycle length. These values, together with the associated probabilities, lead us to formulate the renewal equation. We denote the times as follows:

[W.sub.0] = expected search time when false alarm;

[W.sub.1] = expected time to search and repair assignable cause 1;

[W.sub.2] = expected time to search and repair assignable cause 2;

[W.sub.3] = expected time to search and repair assignable cause 1 and assignable cause 2;

E(T) = expected cycle time.

Thus, eight states are defined as follows:

State 1 in-control and no alarm;

State 2 in-control and at least one false alarm indicted by [Mathematical Expression Omitted] chart or S chart or both;

State 3 out-of-control (assignable cause 1 occurs) and no alarm;

State 4 out-of-control (assignable cause 1 occurs) and at least one alarm indicated by one of the charts or both;

State 5 out-of-control (assignable cause 2 occurs) and no alarm;

State 6 out-of-control (assignable cause 2 occurs) and at least one alarm;

State 7 out-of-control (both assignable causes occur) and no alarm;

State 8 out-of-control (both assignable causes occur) and at least one alarm.

Table I displays the possible states of the system, the expected residual cycle time, and associated probability of being in each respective state at the end of the first sampling and testing interval.

Consequently, the renewal equation is

E(T) = h + [summation of] [P.sub.i][l.sub.i]] where i = 1 to 8 = h + E(T)[P.sub.1] + E(T)[P.sub.2] + [W.sub.0][P.sub.2] where i = 1 to 8 + [summation of] [l.sub.i][P.sub.i]] where i = 3 to 8.

Simplifying we have

[Mathematical Expression Omitted] (8)

Here, we assume the time to sample and test one item is zero. However, it should be noted that the renewal equation approach can be extended to include cases involving non-zero time to sample and test one item (Banerjee and Rahim, 1987).

Cost model

In our cost model, we first calculate the expected cost per production cycle and then divide it by the expected cycle time to obtain the expected cost per unit time. The objective is to determine four decision variables: sample size (n), sampling interval (h), control limit constant ([k.sub.1]) of [Mathematical Expression Omitted] chart and control limit constant ([k.sub.2]) of s chart.

[TABULAR DATA FOR TABLE I OMITTED]

The process costs such as cost of sampling and testing, cost of false alarm, and costs of finding and fixing the shifts in the process model. We denote the costs as follows:

[b.sub.0] fixed cost of sampling and testing;

[b.sub.1] variable cost of sampling and testing a unit of production;

[a.sub.0] cost of investigating a false alarm;

[a.sub.i] cost of finding and fixing assignable cause i,i = 1,2;

[a.sub.3] cost of finding and fixing assignable cause 1 and assignable cause 2 simultaneously;

E(C) the expected cycle cost.

The expected quality loss per unit of production can be easily estimated by calculating the process variance and the deviation of process mean from the target. Let the target value of the quality characteristic be [Mu]. An asymmetric quadratic loss function would be appropriate if the loss differs for values of a quality characteristic that are equidistant from the target. Let the asymmetric loss function be the form (9) (or Figure 1).

[Mathematical Expression Omitted] (9)

where

X [similar to] N([Mu], [[Sigma].sup.2]) during the in-control period,

X [similar to] N([Mu] [+ or -] [[Delta].sub.1] [Sigma], [[Sigma].sup.2]) if only assignable cause 1 occurs in the process,

[Mathematical Expression Omitted] if only assignable cause 2 occurs in the process,

[Mathematical Expression Omitted] if both the assignable causes occur in the process,

[A.sub.0] is the cost of repair or replacement of the product, [Mu] - [r.sub.0][Sigma] and [Mu] + [r.sub.0][Sigma] are functional limits.

According to the asymmetric quadratic loss function, the expected loss ([C.sub.0]) per unit of production during the in-control period is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted].

Once again, we assume when the shift in the process occurs, there is a probability of 0.5 that the mean shifts from [Mu] to [Mu] + [[Delta].sub.1] [Sigma] and a probability of 0.5 that it shifts from [Mu] to [Mu] - [[Delta].sub.1][Sigma].

The expected loss ([C.sub.1]) per unit of production during the period that process is only influenced by the assignable cause 1 is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted],

[Mathematical Expression Omitted].

The expected loss ([C.sub.1]) per unit of production during the period that process is only influenced by the assignable cause 2 is:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted].

The expected loss ([C.sub.3]) per unit of production during the period that process is influenced by both the assignable causes is:

[C.sub.3] = 1/2 {E[L([Mu] + [[Delta].sub.1] [Sigma], [[Delta].sub.2] [Sigma])] + E[L([Mu] - [[Delta].sub.1][Sigma], [[Delta].sub.2][Sigma])]},

where [Mathematical Expression Omitted]

and [Mathematical Expression Omitted], and

[Mathematical Expression Omitted].

where [Mathematical Expression Omitted].

Therefore, the expected cycle cost can be derived by the renewal equation approach. Table II presents the total quality cost which is the sum of the expected residual cost and the cost incurred in the first sampling and testing for each possible process state of the system.

Consequently, the renewal equation is

E(C) = [P.sub.1][([b.sub.0] + [b.sub.1]n) + [C.sub.0]Uh + E(C)] + (E(C) + [a.sub.0] + ([b.sub.0] + [b.sub.1]n) + [C.sub.0]Uh)[P.sub.2] + [summation of] [P.sub.i][R.sub.i]] where i = 3 to 8

Simplifying:

[Mathematical Expression Omitted] (10)

[TABULAR DATA FOR TABLE II OMITTED]

Applying the property of renewal reward process (Ross, 1989), the expected cost per unit time (E([V.sub.[infinity]])) is derived by taking the ratio of the expected cycle cost (E(C)) and the expected cycle time (E(T)); E([V.sub.[infinity]]) = E(C)/E(T). The cost function is the function of design parameters. Hence, the optimal design parameters can be determined by minimizing the cost function.

Performance and sensitivity analysis

Sensitivity analysis on the numerical results provides some useful managerial insights that are valid at least over the wide range of the values of the parameters tested. In this section, we perform a fractional factorial experiment to study the effects and sensitivities of the input parameters on the design parameters. The algorithm used in conducting this experiment is the direct search method for a specified integer n, and specified values of parameters in the estimated range. For simplicity, we treat n, h, [k.sub.1] and [k.sub.2] as discrete variables (Chung, 1990), and assume that the values of n, h, [k.sub.1] and [k.sub.2] are within the ranges between 1 and 50 (1 [less than] n [less than] = 50, and the unit length of n is 1), 0 and 8 (0 [less than] h [less than] = 8.0, and the unit length of h is 0.5), 0 and 6 (0 [less than] k, [less than] = 6, and the unit lengths of [k.sub.1] and [k.sub.2] are 0.1) respectively. For minimizing the cost function which is subject to constraints, the method starts by searching a coarse grid to find an approximate minimum cost E([V.sub.[infinity]]). The objective function and constraints expressed mathematically are

Min. E([V.sub.[infinity]])

s.t. 0 [less than] h [less than] = 8

0 [less than] [k.sub.1] [less than] = 6 0 [less than] [k.sub.2] [less than] = 6 1 [less than] n [less than] = 50.

There are 18 input parameters for the derived cost function. We assign two levels to each of 12 input parameters in the factorial experiment design. The remaining six input parameters are assigned one level, because they are always not sensitive on the design parameters (Panagos et al., 1985). Table III presents the artificial values of input parameters and their assigned levels.

For convenience, the levels are basically the same as those used in Panagos's experiment(1985) except that the quality cost is calculated using the asymmetric quadratic loss function that contains cost of repair or replacement ([A.sub.0]), customer tolerance ([r.sub.0], [r[prime].sub.0] and production rate (U) instead of the in-control and out-of-control hourly costs. We should note that if the levels of input parameters are varied over different ranges, the experimental results may be different. However, the levels used for study here are not unrealistic.

The orthogonal array [L.sub.32]([2.sup.31]) used for the performance and sensitivity study is resolution IV. The experimental design of resolution IV means that no main effect is confounded with any other main effect or with any two-factor interaction, but two-factor interactions are confounded with one another. A total of 32 runs are required to conduct the experiment. The experimental results are shown in the Appendix.
```Table III. Input parameters and levels

Levels
No.         Factors                  Low         High

01         [A.sub.0]                  1          50
02         [r.sub.0]                  1           3
03      [r[prime].sub.0]              1           3
04       [[Delta].sub.1]              1           2
05       [[Delta].sub.2]              1.5         2.5
06      [[Lambda].sub.1]              0.05        0.01
07      [[Lambda].sub.2]              0.04        0.0083
08         [b.sub.0]                  0.5         5
09         [b.sub.1]                  0.1         1.0
10             U                    100        1000
11         [a.sub.0]                 35         250
12         [W.sub.0]                  0.4         1.0
13         [a.sub.1]                 50
14         [a.sub.2]                 60
15         [a.sub.3]                100
16         [W.sub.1]                  2
17         [W.sub.2]                  3
18         [W.sub.3]                  5

Note: [Mu] = 0, [Sigma] = 1
```

We consider these seven two-factor interactions ([A.sub.0][r.sub.0], [A.sub.0][r.sub.0], [A.sub.0][[Delta].sub.2], [A.sub.0]U, U[[Delta].sub.1], U[[Delta].sub.2]) since they are multiplied together to estimate the out-or-control expected loss. The remaining two-factor interactions are considered relatively insignificant. Therefore, we calculate the F-ratio values for 12 main effects and seven two-factor interactions and consider the effects of the remaining two-factor and higher order interactions as residual with 12 degrees of freedom in the ANOVA table. The significant levels of effects determined by the analysis of variance indicate the relative importance of effects rather than the evidence of statistical hypothesis.

We consider the input factors which have significant effects on the expected total cost. Table IV shows the analysis of variance for the response of optimized expected cost.

Table IV shows that the optimized expected cost is significantly affected by two main effects: [A.sub.0] and U and one two-factor interaction: [A.sub.0]U. We take 5 per cent significant level. The significance levels indicate that the cost of repair or replacement ([A.sub.0]) and production rate per hour (U) have a dominate effect on the optimized expected cost. The effect of the interaction shows that the optimized expected cost is very sensitive to U when the [A.sub.0] is high (see Figure 2). Table V shows the analysis of variance for the response of optimized sample size.

As shown in Table V, the sample size is significantly affected by two main effects: [A.sub.0] and [b.sub.1]. Their significance levels show that the cost of repair or replacement ([A.sub.0]) has a dominate effect on the optimized sample size. The variable sampling and testing cost ([b.sub.1]) is also important. Table VI shows the analysis of variance for the response of optimized sampling interval.

As shown in Table VI, the optimized sampling interval is significantly affected by three main effects' [A.sub.0], [r.sub.0] and U, and two two-factor interactions' A, [r.sub.0] and [A.sub.0]U. The effect of interaction between [A.sub.0] and [r.sub.0] suggests that the sampling interval is more sensitive to U when [A.sub.0] is low (Figure 3). The effect of interaction between [A.sub.0] and U also suggests that the sampling interval is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 4 OMITTED]. Table VII shows the analysis of variance for the response of optimized control limit coefficient of [Mathematical Expression Omitted] chart.

As shown in Table VII, the optimized control limit coefficient ([K.sub.1]) of chart is significantly affected by two main effects: [A.sub.0] and U, and one two-factor interaction: [A.sub.0] and U. Their significance levels show that the cost of repair or replacement ([A.sub.0]) has a dominate effect on the optimized control limit coefficient ([K.sub.1]) of [Mathematical Expression Omitted] chart. The production rate (U) is also important. The effect of interaction between [A.sub.0] and U suggests that the optimized control limit coefficient ([K.sub.1]) is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 5 OMITTED]. Table VIII shows the analysis of variance for the response of optimized control limit coefficient ([K.sub.2]) of S chart. We find that the optimized control limit coefficient ([K.sub.2]) of S chart is also significantly affected by two main effects: [A.sub.0] and U, one two-factor interaction: [A.sub.0]U. Their significance levels also show that the cost of [TABULAR DATA FOR TABLE IV OMITTED] repair or replacement ([A.sub.0]) has a dominate effect on the optimized control limit coefficient ([K.sub.2]) of chart. The production rate (U) is also important. The effect of interaction between [A.sub.0] and U also suggests that the optimized control limit coefficient ([K.sub.2]) is more sensitive to U when [A.sub.0] is low [ILLUSTRATION FOR FIGURE 6 OMITTED].

Summary and conclusions

We developed the first economic design of [Mathematical Expression Omitted] and S charts based on the quadratic loss function, a double assignable-cause cost model is [TABULAR DATA FOR TABLE V OMITTED] derived applying renewal theory approach. The expression for the expected cycle length and the expected cost per cycle is easier to obtain by the proposed approach, and the cost model, including the customers' voice, reveals the importance of quality. In practice, the economic [Mathematical Expression Omitted] and S charts can be constructed applying the proposed approach when the values of cost and process parameters are known. Economic [Mathematical Expression Omitted] and S charts should be preferable if quality engineers would like to maintain the whole process with minimum cost.

In data analysis, the effects of input factors have been investigated. The optimized expected cost is significantly affected by [A.sub.0], U and [A.sub.0]U. The sample size is significantly affected by [A.sub.0] and [b.sub.1]. The optimized sampling interval is significantly affected by [A.sub.0], [r.sub.0], U, [A.sub.0][r.sub.0] and [A.sub.0]U. The control limit coefficients of X and S and charts are all significantly affected by [A.sub.0], U and [A.sub.0]U. It is also clear that the cost of repair or replacement ([A.sub.0]) and customers' tolerance ([r.sub.0]), which are related to the loss function, are critical when designing economically based on [Mathematical Expression Omitted] and S charts. This provides a clear incentive for production [TABULAR DATA FOR TABLE VI OMITTED] [TABULAR DATA FOR TABLE VII OMITTED] engineers to place more efforts on off-line process design and consistent quality improvement in order to maintain a robust production process.

Several important extension of the present model can be studied. Generally it is straightforward to extend the model proposed to study other control charts, like the np-charts for attributes. Their differences lie in the definition of the control limits and the derivation of the probabilities of Type I and Type II errors. The method proposed can also be applied to the classic case of economic design of [Mathematical Expression Omitted]-charts or S-charts for process with a single assignable cause.

[TABULAR DATA FOR TABLE VIII OMITTED]

A particularly interesting research area revolves around the economic modelling of production processes subject to a multiplicity of assignable causes. Finally, the modelling of imperfect inspection and restoration procedures also provides additional topics of interest for research in the area.

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[TABULAR DATA FOR TABLE A1 OMITTED]
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