1. Introduction

The term sampling inspection plan, is used when the quality of product is evaluated by inspecting samples rather than by total inspects, which required cost and time. Here we introduce a model for total expected cost function by[7 & 9] whom gives simple information about the elements of this cost function, while[10] introduce the cost model when the percentage of defective is random variable follows Gamma distribution, and suggested to apply another distribution as we done in our research. The build model differs from them by including decision rule[] for acceptance and[] for rejection, the posterior distribution and finding the optimal value of acceptance number () is a closed form, also the sample size necessary to take a decision, in a closed form also. The programs required to obtain (), and total cost, arte executed.

The aim of this paper is to build the total cost of quality control model, used to find the Bayesian sampling plan () for system, the method considered the percentage

of defectives in lots is random variable having prior distribution f(p). So the method derived used to find Bayesian sampling plan.

2. Methodology of Research

Since the aim of research insist on finding Bayesian sampling plans for system , where the percentage of defective in production represents random variable varied from lot to lot and have some prior distribution[], which determined from past data and experience, and it may be Beta distribution or Gamma, or Log-Normal or any other distributions. It is necessary to determine the parameters of Bayesian sampling plan (), taken from lot by minimizing either (total inspection cost) or minimizing total expected risk, which is the risk due to taking the wrong decision.

To satisfy, the aim, first of all we define all notations necessary to build the model, these are;

: Lot size of product.

Number of defective units in the lot .

: Percentage of defective in lot .

Percentage of defective in the sample .

Number of defective in sample.

Number of defective in Lot.

Acceptance number

Cost of sampling and testing unit in the sample.

Cost of repairing or replacement of defective units in the sample.

Cost of sampling and testing units in the remaining quantity () after rejecting sample.

Cost of repairing or replacement of defective unit in quantity ().

Cost of accepting good unit and it is not a penalty cost.

Cost of accepting defective unit.

Probability of accepting the production with quality and it is;

Probability of rejecting the production with quality .

Average percentage of defectives in the lot after doing rectifying inspection on it.

Decision rule for explaining the probability of acceptance the lot (), after knowing the sample contains defective.

Decision rule for rejection.

Mean of the probability distribution of defective.

Mean of prior distribution

Mean of posterior distribution

Break even quality point;

Prior distribution of percentage of defectives in the lot of product.

Posterior distribution of percentage of defectives.

3. Construction of the Model

Suppose that

(1)

And the percentage of defectives in process is

(2)

Therefore

(3)

Is used to find the posterior distribution which is:

(4)

Therefore, the posterior distribution of percentage ofdefectives is also Beta, but with parameters

and with average;

Which can be written in terms of the mean of prior i.e.

as:

Now; let

(5)

(6)

And let be break even quality level, which is the point of percentage of defective, at which we cannot distinguish between acceptance decision and rejection, and is defined b , and it is also defined in terms of quality control cost parameters.

Therefore;

(7)

We know that the loss equal zero, when the decision is correct, and the loss is for wrong decision, i.e.

(8)

According to that, the equation of expected risk is defined;

(9)

where

(10)

Using the , can found the value of expected risk under Binomial process and continues prior distribution. See[9].

But here the proposed model depend on definition in terms of and decision rule, and it is defined as follows;

(11)

And since is random variable has then the expected value of risk in terms of is denoted by and it is defined as:

(12)

After some steps we have;

(13)

The minimum value for formula (13) can be verified by;

where;

Since equation (13), define the formula of expected risk function in terms of expectation, we want to find it in terms of sampling distribution, this required to take the distribution of defectives in the lot , and distribution of in sample (n), in consideration, and also taking the prior distribution of quality of a process, and also define the loss this tend us to find the expected value of the loss due to the decision of accepting and rejecting, which is now defined by the following equation;

(14)

(15)

Putting the following simplification in equation (15), we obtain equation (16);

Also substituting;

Where;

Where, is the posterior mean.

By applying these expectations in equation (15), we get;

(16)

Which can be simplified to;

(17)

Where;

The function in equation (17) considered to be a function of Bayesian single plan to test the product and searching for the two values which gives minimum value for function (17), can be done by applying the first partial derivatives for the function equal to zero, then solving the two equations, since the distribution of is a discrete distribution, so we cannot apply differentiation method, we can apply forward differences method for which required writing in a fitness way;

Where;

(18)

Which represents the weighted mean for according to the rule of[10], we can summarize the following rule;

(19)

From equation (19), we can get;

According to Beta – Binomial distribution when

And,

We find , so the posterior mean is;

Hence,

From applying , we find the optimal value of acceptance number, which is the value that satisfy;

(20)

And it depend on the estimated values of parameters and also on the value of sample size which is obtained from minimizing;

(21)

With respect to , where;

The result of now is;

(22)

Where depend on parameters of quality control cost and on , so for Beta – Binomial distribution, the values of is simplified to:

(23)

Where;

(24)

The estimated values of using moment's method are;

(25)

Where,

4. Application

The following data represents the distribution of percentage of defectives for (100) lots taken from Iraqi general company for producing liquid oils, after tests it found to be Beta distribution

The last three classes were grouped since its observed frequencies is less than 5, also we compute the mean of percentage of defectives and average units, the variance of percentage of defectives .

From equation (25), the estimated values of parameters by method of moments are:

The value of calculated is compared at and degree of freedom , we find for this reason the hypothesis:

is accepted

Therefore, we depend on prior Beta distribution at estimated parameters;

In finding the parameter of single Bayesian sampling plan for the system where;

Also, we have obtained the estimated parameter of total cost function, for the lots of product ( from state company vegetable oil ), these are;

Also, we find;

, is the estimated value of percentage of defective in normal condition which is also denoted by is the percentage of defective when the conditions of production is not normal.

From equation (20) and (24) by using the estimated values of also for different values of process average , the results for different size of lot are tabulated in table (2).

After testing the distribution of quality in table (1), and estimating parameters, and also computing different parameters of total cost model, we find different values of Bayesian single sampling plan and the results are explained in table (2).

The above table represents the results of sampling plan according to quality level and lot size , for equation (24) and (20), while for given data in research, we find that the proposed sampling plan when units, and and also we compute from its results found to be equal to

Table 1. Distribution of quality process for 100 lots

Table 2. Bayesian sampling plans for       under Beta prior and due to decision making proposed model

5. Conclusions

(1) The distribution of quality of 100 lots, found Beta prior with

(2) The calculated value of compared with degree of freedom since the last class of distribution of quality where grouped, it is found that for this reason, the hypothesis

(3) The value is producer's risk and is consumer risk, where considered in designing the various sampling plans, and we can use any set of .

(4) From table (2) we find the sample size is increasing when the percentage of defective is increased, which is logical result.

(5) The proposed model can be applied to another kind of Bayesian sampling which have Gamma prior, and log – Normal, since it is a general model.

(6) From the data, we found that , which is the estimated value of percentage of defectives in the normal condition of production process, but it's true value found which is greater than proposed for the factory, this results is logical also because of complex and bad conditions of process of production, we look at the percentage is very good now.