MINIMUM COST QUALITY CONTROL TESTS

An expected cost model of a process whose mean is con-trolled by an X chart is developed. Tow- stage numerical procedure is used to calculate the sample size, the number of units produced between sample, and the control limits of optimal control charts.


___________MINIMUM COST QUALITY CONTROL TESTS ] 2 [
The set of all possible values of the test statistic is divided in to two subsets.
One subset includes these values of the test statistics considered likely to occur if the null hypothesis is true.
The other subset includes those values considered unlikely to occur if the null hypothesis is true.
This later subset is generally called the test critical region. If a value of the test statistic falls in the critical region, the null hypothesis is rejected and the process is investigated in order to determine and correct the condition which caused the process to go out of control.
If the value of the test statistic is not in the critical region, the null hypothesis is not rejected, the process is assumed to be in control, and it is allowed to continue to operate.
As in any hypothesis testing procedure, two types of errors may be made.
One type (generally called type 1 errors) involves rejecting the null hypothesis when the process is in control.
The second type (generally called type П errors) involves the acceptance of the null hypothesis when the process is out of control.
Type П error leads to costs associated with the increase in defective products produced by an out-of-control process.
Coat of unnecessary investigation and loss of production arise from type І error.
Both of these costs can be decreased by increasing the sample size and the frequency of sampling.
This reduction in error cost is, of course, accompanied by an increase in sampling and testing costs.
Type Ι error costs can also be decreased by decreasing the size of the critical region.
Decreasing the size of the critical region increasing the probability of accepting the null hypothesis when is it true, thus decreasing the type Ι error costs.
However, if some alternative hypothesis is true (that is, if the process is out of control) the probability of accepting the null hypothesis is also increased, thus increasing type ΙΙ error costs.

2-Purpose of the article
The purpose of this article is to develop a method for choosing the test parameters (that is, the sample size, the frequency of sampling, and the critical region) in a manner that will minimize the total cost. The investigation will be limited to quality control tests involving a single process parameter.

3-The Content
The method presented in most introductory quality control texts (for example, Burr (2), Duncan (5), and Grant (8)) involves the selection of the simple size and the critical region such that the power of the test to detect some specified shift in the process parameter (that is, the probability of rejection the null hypothesis given some specific alternative hypothesis) and the type І error level are some arbitrarily selected values.
With approach, the problem of how frequently a sample should be taken is ignored.
It has also been suggested Wetler (13) that the sample size should be selected to minimize the total amount of sampling required to detect a process parameter shift of some specified size with some specific probability.
Several investigations on minimum cost selection of quality control test parameters have been reported.
Cowden (3) carried out some numerical experiments based on the assumption that the process mean is out of control at the beginning of the day.
Duncan (4) assumed that the process may shift from the incontrol state to a single out-of-control state any time during the day.
He further assumed that the process remains in the in-control state before going out-of-control is an exponential random variable with mean ) ( 1 − λ hours. Girshick and Rubin (7) considered the problem of minimizing the running and repair costs for a machine which could be considered to be in one of four states. The first two states represent levels of performance while the last two represent overhaul states.
Bather (1) developed a stopping rule which indicates when production on a machine should be halted and the machine overhauled. Tayler (11) shows that inspection of a fixed number of items at a fixed interval of time is non-optimal.
Instead, sampling should be determined at each stage by the current posterior probabilities.
Using this approach, Taylor (12) develops an optimal control procedure based on the assumption that the process has only two states, in-control and out-of-control.
Despite the non-optimal nature of fixed sample size, fixed time increment sampling plans, such plans are still widely used because of their ease of administration.
In light of the widespread use, it is desirable to obtain an optimal sampling plan within the class of fixed size -fixed time sampling plans. It is the purpose of this article to develop a method for selecting the optimal sampling plan from this class of plans.
In place of the assumption of one in-control state and one outof-control state made by other authors DUNCAN (4), TAYLOR (12), it is assumed that the process parameter, µ, is a continuous random variable which can be satisfactorily approximated by a discrete random variable.
One value, µ 0 , of the discrete random variable is associated with the in-control value of the process parameter and the remaining values,

4-GENERAL COST MODEL
The expected total cost ) (c E , per unit of product, associated with a quality control test procedure can be written as is the expected cost per unit associated with the production of defective product. Both Cowden (3) and Duncan (4) consider the cost of sampling and testing to consist of a constant a mount independent of the number of units sampled plus a constant a mount for each unit sampled.
In view of the difficulty of obtaining accurate cost estimates, more complex cost functions appear to be unwarranted.
Thus, the expected sampling and testing cost per unit is µ There may be some reason to suppose that the costs of determining the cause of a shift will depend upon the true value n ___________MINIMUM COST QUALITY CONTROL TESTS ] 6 [ of the parameter µ , since it is likely that the cause of small shifts will be more difficult to find than the cause of large shift.
However, the cost of correcting process after the cause has been determined is often larger for large shifts than for small shifts.
It is also difficult to conceive of a situation prior information will be available concerning the cost of correcting a process as a function of the true value of the parameter µ .
Prior information is generally available concerning how often the process goes out of control, how long the process is inoperative, and the cost per hour of an inoperative process.
From this information, the average cost of getting the process back into operation can be determined with reasonable accuracy.
Thus, it will be assumed that the cost of investigating and correcting a process that is apparently out of control is a random variable, (ν ), with mean a 3 whose distribution does not depend on the parameter µ .
If(u) is a random variable which takes on the value one if the null hypothesis is rejected and zero otherwise, and if (ν ) equals zero when (u) equals zero (that is, investigation and correction costs are not incurred unless the null hypothesis is rejected ), then the expected cost per unit for rejecting the null hypothesis is  There is some intuitive appeal to the argument that the relationship between the number of defectives produced and the cost of producing defectives is nonlinear, since a small number of defectives may go unnoticed by the custom while a large number of defective may cause loss of future business.
However, in view of the inherent difficulties involved in determining the nature of this relationship, a simple linear relationship is assumed.
If a 4 is the cost associated with producing a defective unit of In the above function, the a's are cost coefficients which are assumed to be functionally independent of the test parameter.
The vector f depends only on the nature of the process parameter and the definition of defective unit and, thus, does not functionally depend on the test parameters.
The vector q ,α and γ are, however, functionally dependent on the test parameters.
The form of this dependency is developed in later sections.
Thus far, only two (that is, n and k ) of the three test parameters have been defined. In order to express the third test parameter (that is, the test critical rejoin) as a single parameter, some restrictions must be placed on the nature of the test. n ___________MINIMUM COST QUALITY CONTROL TESTS ] 8 [ It will be assumed that the test statistic, (T ), is normally distributed with mean ( µ ) and variance ( N / 2 σ ), and that the critical rejoin is symmetric and defined by the critical rejoin

4-1 PROBABILITY VECTOR q
On the basis of the assumption that (T ) is normally distributed with mean ( µ ) and variance (

4-2 PROBABILITY VECTOR α
The elements, i α , of the probability vector α represent the steady-state probability that the process is in state i (that is, i µ µ = ) at the time a sample is selected.
To obtain these steady-state probabilities, the transition probability matrix, ( p represent the probability that the process will shift from the in-control state (that is, If it is assumed that the time the process remains in the incontrol state before going out-of-control is an exponential random variable with mean ) ( 1 − λ hours, then the probability of remaining in state If a production rate of ) (R units per hour is assumed and the production of a fraction of a unit is allowed, is the average number of units produced before an out-of-control shift occurs).
Same method is needed to assign the remaining probability    Several additional assumptions must be made before the transition probabilities are defined.
First it is assumed that when the process goes out-of-control (that is, µ shifts from 0 µ to i µ ) it stays out-of-control until detected (that is, until 0 H is rejected). In practice this means that the process will not correct itself. It will be further assumed that when the process goes out-ofcontrol it will not improve, but it may get worse. This One way to do this, while staying within the restriction that the process cannot get better by itself (that is, the process cannot shift directly from

5-SOULUTION METHOD
Tabulation of optimal quality central test parameter for the model as shown in equation (5)   A ), and for the desired values of the a priori distribution parameters (that is, π and s ).
From these results, the general behavior of the model can be studies and preliminary estimates of the optimal values of (n, k and l ) can be obtained.
In the second stage, the preliminary estimates obtained from the first stage are used as the starting point for a search method designed to locate the optimal values of the test parameters within any desired accuracy.

6-SOME NUMERICAL RESULTS
To illustrate the design of an optimal sampling plan, consider the following example: a1=10 Dr. per sample. a2= 1 Dr. per unit sampled. a3= 100 Dr. per investigation. a4= 10 Dr. per defective unit produced, ‫ג‬  = 0.001 (that is, on the average, the process shifts out of control every 1000 units),and π =0.376 (that is ,on the average shift is 2.  Table 1, the optimal sampling plan is shown as E * ( c )=0.0737 , n=3 K=0.046, and L=2.75.
Since K=‫ג‬  k, the optimal value of k is k= In the sample problem, it was assumed that, on the average, 1000 units are produced before the process goes out of control ( that is, ‫ג‬  =0.001 ). It was also assumed that when the process goes out of control the mean µ shifts on the average 2.4 σ. using Equations 8 and 9, and the fact that i i + = 0 µ µ σ , it can be shown that π must equal 0.376 in order for the expected value of µ , given that µ is out of control (that is , given µ ≠ 0 µ ), to equal 0 µ +2.4 σ . If a sample is taken every 40 units ( that is, k=40 ), then K=‫ג‬  k=(0.001)(40)=0.04 and, thus, the a priori distribution ___________MINIMUM COST QUALITY CONTROL TESTS ] 20 [ of µ from Equations 8 and 9 is ( p 0 ,p 1 ,p 2 ,p 3 , p 4 ,p 5 ,p 6 )=(0.961, 0.009, 0.013, 0.011, 0.005, 0.001, 0.000 ). If the number of units between samples should be increased ( say, from 40 to 80 ) it would be exacted that the probability of finding the process in control ( that is , p 0 ) should decrease and the other values of p i should increase. Using K=(0.001)(80)=0.08 in equations 8 and 9, the following a priori distribution is obtained: (p 0 , p 1 , p 2 , p 3 , p 4 , p 5 , p 6 )=(0.923, 0.018, 0.026, 0.021, 0.010, 0.002, 0.000). in order to obtain the transition probability matrix B, the probability of rejecting H 0 ,given that µ = i µ (that is, q i ), is required for all values of i. these probabilities are a function of the control limits L and the sample size n. For example, when n=4 and L=3, the vector q is calculated, using Equations ( 6 ) and ( 7 ), as (q 0 ,q 1 ,q 2 ,q 3 ,q 4 ,q 5 ,q 6 )=(0.003,0.159,0.841,0.998,1.000,1.000 1.000).
Using the above values for the vectors p and q for ( k=40, n=2, and L=3), the modified transition probability matrix B * defined in Equation ( 16 ) is calculated as The steady-state vector (α ) of probabilities that the process mean is i µ at the time a test is conducted is obtained by finding the last row of the inverse of the above matrix.
The results for (k=40, n=2 and L=3 ) are ( α 0 ,α 1 ,α 2 ,α 3 ,α 4 ,α 5 ,α 6 ) =(0.949, 0.011, 0.018,0.014, 0.006, 0.002, 0.000 ). The steady-stute vector ( γ ) of probabilities that the process mean is i µ at any point in time is calculated using Equation (19) and ( 20 ). The result for ( k=40, n=2, and L=3 ) are (γ 0 ,γ 1 ,γ 2 ,γ 3 ,γ 4 ,γ 5 ,γ 6 ) =(0.931, 0.011, 0.022, 0.022, 0.011, 0.003, 0.000). The only additiond information required in order to calculate the expected cost is the vector (f ) of probabilities of producing a defective given that the process mean is i µ . Since it has been assumed that a defective is any unit whose measurement falls out side the range( i To illustrate the behavior of the three cost components as the number of units produced between samples is increased. As would be expected, using a fixed sample size, (n=4 ) and fixed control limits ( L=3 ), the expected sampling and inspection costs per unit E(c 1 ) and the expected cost per unit associated with rejecting the null hypothesis E(c 2 ) both decrease as the number of units produced between samples increases. surprising that the total expected cost achieves a minimum at a value of k close to the optimal value of 46. To illustrate the effect of changing k, n, and L, the total expected cost was calculated using the five values of k ( 20, 40, 60, 80, 100 ), three values of n ( 2,3,and 4), and two values of L ( 2 and 3) The minimum total expected cost was obtained with a sample size of three ( n=3 ), three sigma control limits ( L=3), and a frequency of sampling of 40 units between samples (k=40).The total expected cost associated with this scheme is 0.745 Dr. per unit.