TX_1~ABS:AT/ADD:TX_2~ABS:AT 65 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 ReseaRch aRticle Determination of Single Sampling Attribute Plans Based Upon Dodge-Romig Model with Application Dler H. Kadir1, Azhin M. Khudhur1, Rebaz O. Yahya2, AbdulRahim K. Rahi3 1Department of Statistics and Informatics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq, 2Department of Business Administration, Cihan University-Erbil, Kurdistan Region, Iraq, 3Department of Business Administration, Dijlah University College, Iraq ABSTRACT Bayesian sampling plans for production inspection involve using a sampling method to assess the plan’s features, assuming that defect rates fluctuate randomly among different production batches. This study uses Bayesian sampling plans, specifically the beta distribution, to determine a single sampling plan’s (SSP) parameter (n, c). These parameters were then compared to those from other SSP. The study was conducted at the Ala corporation for soft drinks, where 120 batches were selected to calculate the defect rate. The results showed that using Bayesian and decision-making models can lead to developing a single sampling inspection procedure that closely approximates the quality level. In addition, the decision-making model resulted in a smaller sample size and lower inspection costs than other inspection plans. Keywords: Acceptance quality level, Bayesian sampling plans, operating characteristics, statistical quality control, average sample size INTRODUCTION Pepsi Company (Ala) is one of the private sector companies belonging to the food industries in the Bazian area of Sulaymaniyah Governorate. This company was established in 2006–2005 and its first production was in 2006, and the company consists of machinery and equipment (Italian and Swiss) made with high specifications as it can manufacture two types of soft drink bottles with a capacity of (1.5 L) and (330 mL), and that the allowable percentage of production damage according to international specifications is 3%, meaning that any increase in this percentage indicates a defect in the company’s production stages. Quality control is a critical aspect of production, ensuring that products meet specified standards and minimizing waste and costs. Statistical tools, such as control charts and sampling plans, play a crucial role in managing and monitoring production quality. Sampling plans, in particular, offer a precise and efficient way to assess the presence of specific characteristics in produced units by analyzing a small, randomly selected portion of the output.[1] This study focuses on the application of Bayesian sampling plans for production inspection, a method that incorporates prior knowledge and experience to make informed decisions about product acceptance or rejection. Bayesian sampling plans have been shown to be effective in various industries, and their potential benefits in the soft drink industry are worth exploring. The research aims to develop a Bayesian discriminant sampling plan for the Ala Pepsi Soft Drinks Company, a major producer in the Bazian area of Sulaymaniyah Governorate. By applying Bayesian theory and decision theory, this study seeks to establish a sampling procedure that closely approximates the actual quality level of the company’s products. The findings of this research will contribute to the optimization of quality control practices at Ala Pepsi and potentially in other similar production settings.[2,3] Hald[4] introduced a novel sampling methodology that uses cumulative conforming control chart count and compared it to conventional sampling techniques, illustrating its power for higher lot sizes and process averages. The mathematical frameworks for the suggested technique are built utilizing the count of cumulative conforming items and Markov chain Corresponding Author: Azhin M. Khudhur, Department of Statistics and Informatics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq. E-mail: azhin.khudhur@su.edu.krd Received: June 27, 2024 Accepted: August 09, 2024 Published: August 25, 2024 DOI: 10.24086/cuesj.v8n2y2024.pp65-75 Copyright © 2024 Dler H. Kadir, Azhin M. Khudhur, Rebaz O. Yahya, AbdulRahim K. Rahi. This is an open-access article distributed under the CreativeCommons Attribution License. Cihan University-Erbil Scientific Journal (CUESJ) Kadir, et al.: Sampling attribute plans based upon dodge-romig model 66 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 modeling, which both depend on Markov modeling and negative binomial distribution. This suggested method is shown to be beneficial compared to the tables of Dodge-Romig when the findings are compared to those of the single sampling plan (SSP) based on AOQL and Lot Tolerance Percent Defective (LTPD) developed by Dodge-Romig. The optimization model makes sure that the AOQL stays within a set upper limit and that the minimal ATI is attained. Ahmadi Yazdi and Fallahnezhad[5] focused on the creation of Dodge-Romig AOQL SSP using variables and specification limitations. The research suggests a generalization of Kapur and Wang’s paradigm and is to extend their earlier work. Based on the anticipated total cost per unit, the best inspection strategy is either approval without control or 100 observations. The sample plans are designed using the quadratic quality loss function, which has proven effective in many quality control uses. The findings paper recommends additional investigation into an integrated Bayesian SSP model for variables and boundaries. Chen[6] provided clear asymptotic formulae for the Dodge-Romig LTPD single sampling inspection plans’ sample size and approval number. Numerical experiments indicate that a simple finite population adjustment of the asymptotic formulae results in a very accurate approach to the Dodge- Romig solution. The paper’s major findings are that the sample size is asymptotically proportionate to the logarithm of the lot size and that the maximum permitted percent faulty in the sample merges to the tolerance fraction defective, with a difference of order l/V%. Klůfa[7] compared LTPD single sample plans for examination by characteristics to the matching Dodge-Romig LTPD plan from an economic standpoint. The economic effectiveness of LTPD plans for variable and property inspection is investigated, and it is discovered that they are more inexpensive than the matching Dodge-Romig attribute sample plans in several cases, with a cost savings of 80%. The research also investigates the relationship between economic efficiency and lot size N, given specified characteristics. Whenever the quantity of objects in the plenty is large, the system’s average fraction of defective is low, and while the cost of inspection by variables does not appear to be significantly higher as compared to the cost of inspection by characteristics, the economic efficiency of the LTPD plans for inspection by variables and attributes is higher. Kadir and Rahi[3] utilized the beta-binomial distribution to determine the parameters for a Bayesian sampling plan and then compared it to alternative SSP. The Bayesian and decision- making models can create a sample evaluation process that closely mimics the actual level of quality. When applying the decision-making model, the sample size was smaller than other inspection plans, leading to lower inspection costs. It is critical to completely develop quality control criteria, especially in prioritizing standard and production standards. DODGE-ROMIG MODELS In 1994 both researchers Harold and Harry used sampling plans and these plans apply in case of filtered rectifying inspection it’s an examination that examines a sample (n) that is taken from a production batch (N), the number of specified units (x) in a sample (n). If it is less or equal to several approvals (X ≤ c) then c will decide to accept the sample and the rest quantity (N – n) without, and defective units in the units will be replaced with better ones. A decision is taken to refuse the sample when the number of defective units is (x) in a sample (n) is bigger than the number of acceptances, which means when (X > c) refuses the remaining quantity (N – n) and all its units are submitted to general examination for isolating purposes of effective units and exchange them with other non- defective units. When finding the SSP for filtered examination, both researchers depended on (LTPD, and AOQL) systems, where the (LTPD) term refers to the proportion of defects allowed in the production batch, ass for the (AOQL) term refers to the highest rate of defective units in the transmitted batch and that is after implementing a filtered examination on it.[8] Single Sampling LTPD Plans LTPD SSPs are based on several assumptions: Defective units occur randomly, and the production process is under the statistical control of a binomial distribution with a constant defect rate of P1 Choose the value of LTPD to be P2 to protect the producer from delivering unsatisfactory batches, as the probability of accepting batches with quality level (P2 > P1) P2 is small and this probability is usually called Consumer’s Risk, which is the probability of accepting a batch with a quality level worse than the LTPD, and we will symbolize it as P (P2), it can be said that the producer aims to use sampling plans where the value of P (P2) is small. Rejected batches are completely re-inspected based on the decision to reject the sample, after which all defective units are replaced with non-defective (good) units or repaired if repair is possible for these units. The cost of inspecting one unit in the sample is equal to the cost of inspecting one unit in the rejected lot (N-n) and is equal to one as an economic unit. The inspection plans determined by the model aim to minimize the total inspection rate I(P1) for a product of quality (P1). This rate is based on inspecting a quantity n in the case of acceptance and n in the case of rejection, so I(P1). I(P1) = nP(P1) + NQ(P1) (1) Where: P(P1): Represents the probability of accepting N batches of the product of quality P1. The value of P(P1) depends on the type of sampling distribution, and according to the first assumption above, we obtain the value of P(P1). P P P X c b x n P B c n P x c 1 0 1 1� � � �� � � � � � � � , , ( , , ) (2) Q(P1) represents the probability of batch rejection, where: Q(P1) = 1 – P(P1) (3) A binomial distribution is used when a random sample n is drawn from batch N or from the output of a production process Kadir, et al.: Sampling attribute plans based upon dodge-romig model 67 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 whose quality rate is equal to P1. In addition, the probability of obtaining x defective units in a sample of size n taken from a production batch of size N containing x defective units is determined from the hypergeometric distribution, since P n x X N a x bx X n x N X n N, , ,� � � � �� �C C C (4) a = Max [0, n–(N – X)], b = Min [x, n] Thus, the acceptance probability of batch N containing x defective units is P(n, c, X, N) since P n x X N P n x X N x c , , , , , ,� � � � � � � 0 (5) The cumulative probability values obtained in terms of binomial sums are a good and reasonable approximation of the probability values in terms of hypergeometric sums in the case of X ≥ n, N ≥ 50 n/N ≤ 0.1 Therefore, equation (5) Can be written as follows: P n x X N P n x X N x c , , , , , /� � � � � � � 0 (6) In addition to the previous distribution, the Poisson distribution can be used to calculate the acceptance probability when the focus is on the number of defects per unit of production rather than the number of defective units. If we have a production process in which the average number of defects per unit of production is equal to μ (and these defects occur randomly so that the number of defects per unit is independent of the number of defects for any other unit of production), the acceptance probability of the quality product is μ equal to: P P x c e n xr x c n x � �� � � � �� � � � � � 0 ( ) ! P(μ) = G(c, nμ) (7) The Poisson distribution is a good approximation to a binomial distribution when P is small and n is large, so equation (2) converges to: P(P1) = B(c, n, P1) = G(c, nP1) When (P1 < 0.10) This approximation simplifies the solution for parameterizing the sampling plan because the Poisson results depend on the product (nP1) rather than the two separate parameters P1, n as in binomial. Using Dodge-Romig tables The researchers, Dodge-Romig brought out the results of the SSPs using the Hypergeometric allocation and for (LTPD) system, and the plans were set in particular tables that comprised the SSPs for eight values (LTPD ≤ 0.10) and the values are: 0.5%, 1.0%, 2.0%, 3.0%, 4.0%, 5.0%, 7.0%, and 10.0%. These techniques have improvements over the classic Dodge-Romig table, such as greater flexibility in selecting LTPD values and attaining reduced average total inspection by adapting the plan to individual process averages and lot sizes.[9] Utilizing Dodge-Romig tables The singular Dodge-Romig plans for the AOQL system rely on the following assumptions[10]: 1. The production process operates under binomial control, where the constant defective rate equals P1. 2. The examination used is of the filtering type. 3. For the producer to ensure that the quality of his product is convenient, he must pick out a value for the maximum percentage of defective units in the batch sent for the inspection (i.e., the AOQL value), Subsequently, consideration is given to the sampling plan that achieves the smallest value for the overall inspection rate I(P1) among all conjoining plans to this plan the ones that have the same (PL) value. 4. Dodge-Romig plans for the AOQL system were based on the use of a Poisson distribution rather than a binomial when determining acceptance probabilities P(P1). After mentioning the specific assumptions for the AOQL system, we proceed to reformulate equation (8) and make it more suitable for deriving the sampling plan parameters for the AQL system. First, let’s define PA: P P N n N P PA � �� � 1 1( ) (8) P P N n N e nP xA x C nP x � �� � � � � � � � �1 0 1 1( ) ! (9) If we assume that Pm represents the value of P1 that maximizes the equation (9), which equals PL, and that m = nPm, we find that: P N n N m n e m xL x c m x � �� � � � � � � � � 0 ( ) ! (10) P n N m e m xL x c m x � �� � � � � � � � �1 1 0 ( ) ! (11) Y n Nc � �� � � � � � 1 1 Whereas: Y m e m xc x c m x � � � � 0 ( ) ! In additional, Y nP n N c L� �� � � � � �1 (12) The researchers Dodge-Romig calculated the values of (m = nPm), Yc values in equation (12), for different values of c, specifically for c = 0(1)40, as they cleared that ion the table.[8] It is noteworthy that equation (12) represents the relationship between N, n, c, and when solved for n, we obtain: Kadir, et al.: Sampling attribute plans based upon dodge-romig model 68 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 n Y P Y N c L c � � �( )1 (13) This equation targets the necessary size of the sample to examine the N batch of the product after figuring the values of PL and c. The researchers Dodge-Romig extracted the result of sampling plans of the AOQL system and put it in a particular table, these tables obtain inspection plans for thirteen values of the value AOQL≤0.10 and the values are: 0.1%, 0.25%, 0.5%, 0.75%, 1.0%, 1.5%, 2.0%, 2.5%, 3.0%, 4.0%, 5.0%, 7.0%, 10.0% Each table contains six plans consistent with the worth levels, and for batch sizes groups, if N batches were distributed from value (1) to value 100000 in 14 categories, and the value of LTPD was also determined at each level of the quality levels, with the help of these tables the parameters of the sampling plan (n, c) can be read after determining the value of PL, the quality level, and the category that includes the batch size that produces N. The use of Poisson approximation The Poisson approximation is counted as a consistent and great approximation for the accurate solution extracted from the binomial, and that’s when: 1 2 2 ( 0.10, 0.5, 0.10) P nP P N ≤ ≤ ≤ To use the Poisson approximation, we have to rewrite the equation (1) as follows: I(P1) = nP (P1) + NQ (P1) =nP(P1) + NQ (P1) + nQ(P1)– nQ(P1) I(P1) = n + (N – n) Q (P1) (14) � � �� � � � �� � � � � � � �� � �n N n e np X x C np x 1 0 1 1 ! (15) From equation (15), the parameters of the sampling plan (n, c) are determined which minimizes the value of I(P1) as claimed to the specific conditions of the consumer risk P(P2), that equals P(P2),) = G(c, nP2). Extracting the SSP for the dodge-romig models using Hald tables Here is the clarification of the method that Hald suggested to determine the parameters of the sampling plan (n, c) for the filtering inspection of the LTPD system, which relies on the direct tables that Hald found, where the values of (n, c) are determined, and for each N as an alternative of specifying the value of (n, c) for the category that contains the batch size N. This method relies on the following equation: I(P1) = n + (N–n)[1–G(c, np1)] (16) Where we multiply both sides of equation (16) by P2 to get the following equation: I(P1) P2 = nP2 + (N–n) P2 [1–G(c, np1)] (17) And it’s written briefly as follows: R(c, m) = m + (M–m)[1–G(c, np1)] (18) According to the fixed value of P2, the parameters of the SSP (n, c) are determined by minimization of the equation (17) and that under the particular condition of the consumer risk as known as: G(c, m) = 0.10, m = np2 SSPs for AOQL System The AOQL term refers to shortened for Average Outgoing Quality, where 1941 the researchers Dodge-Romig brought out simple sampling plans for the AOQL system, it expresses the average percentage of defective units in the transmitted batch, after doing filtered sampling on them, and the AOQL worth is defined by dividing the predicted value number of defective units in transmitted batch over the total number of units. If the production procedure for the product of type (P) goes under the control of a binomial the number of defective units in the produced batch will trace the binomial distribution with () parameters, and the number of defective units x in the haphazard sample n will follow the binomial distribution with (n, p). Therefore, when the batch is accepted, the number of defective units (Y = X – x) remaining quantity after pulling out the sample of size n then the binomial will follow with (N–n, p) parameters and the predicted value of Y equals P(N–n),[10,11] In case of the batch rejection, all the batch components will be re-examined, and the defective units will be replaced with better ones, that’s why Y of 0 possibly 1 in case of rejection, and the predicted value of Y can be expressed by the following equation: E(Y) = E(Y|x ≤ c)P(X ≤ c) + E(Y|X > c) Pr (X > c) =(N–n)PP(P) + (0)Q(P) E(Y) = (N–n)PP(p) (19) We use the code (PA) to express the value of AOQ, as follows: P AOQ E Y NA = = ( ) P N n N P P PA � �� � � � � � � �) . (20) According to the following rule, we can derive the value of PA where it is known that the average of the defective units in the product submitted for inspection equals Np, the number of defective units that are found during the inspection will lower this average, therefore the average number of defective units in the product emerging from inspection (NPA) equals: NPA = NP–PI(P) P NP PI P N P N I P NA � � � � � � � �[ ] [ ] (21) Kadir, et al.: Sampling attribute plans based upon dodge-romig model 69 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Therefore: I(P) = n + (N–n)Q(P) ( ) ( )( )[ ] ( ) . ( )A NP P n N n Q P N nP P P p N N − + − − = = As appeared in equation (20), but in the case of only eliminating defective units as an alternative to replacing them, the value PA is changed to * AP where: * ( ) ( )A N I PP P N PI P  − =  −  (22) The Characteristics of AOQ: If the value of the type P level was small the PA will be small as well, then it will be preferred to accept the batches without inspection. If the P was large all the batches must be thoroughly examined and as a result to repair defective units with good ones, the value of PA will also be small. PA equal Zero when P = 0, and PA equals Zero when P = 1, (when the large P = 0 the entire batch is examined PA = 0). When P has a value that means PA is the Greatest possible, which PL equals: PL = MaxP AOQ = AOQL This means: that the maximum average of the defective units in transmitted batches after the filtered inspection procedure is expressed as AOQL. Determination of the Parameters of SSPs for the AOQL System First: Usage of (Dodge-Romig) tables Dodge-Romig single plans rely on the AOQL system on the following assumption[10]: 1. The production procedure falls under the control of a binomial and the defective average is fixed it equals P1. 2. The used inspection is filtered type. 3. The manufacturer must be pleased with the products so that he confirms it. After the inspection, he must choose a value for the highest percentage of defective units in the transmitted batch (i.e., the AOQL value). He considers the sampling plan that fulfills the smallest value for the entire rate (P1) between all the plans close to it that have the same value of (P1). 4. Dodge-Romig plans were based on using a Poisson distribution rather than a binomial for the AOQL system when determining acceptance probabilities P(P1). After mentioning propositions AOQL own system plans, we reframe equation (20) and make it more acceptable for the reason of deriving the sampling plan parameters for AQL system. Define PA: P P N n N P PA � �� � 1 1( ) 1 1 1 0 ( ) ! C np x A x N n e nPP P N X − = − =   ∑ (23) If we presume that (Pm) represents the value of P1 that makes the equation (23) as maximum as possible, that is, we find that nPm = m is equal to PL: P N n N m n e m XL x c m x � �� � � � � � � � � 0 ( ) ! (24) P n N m e m XL x c m x � �� � � � � � � � �1 1 0 ( ) ! (25) Y n Nc � �� � � � � � 1 1 While, Y m e m Xc x c m x � � � � 0 ( ) ! Meaning that: Y nP n N c L� �� � � � � �1 (26) The researchers, Dodge-Romig measured the (m = npm) and Yc values from the equation (26) for the other c value, in particular, c = 0(1)40 values, they clarified that in the table.[12] We notice that equation (26) stands for the relationship between N, n, c, and when we solve the (n) we get: n Y P Y N c L c � � �( )1 (27) This equation (27) defines the needed sample size for batch N of product knowing the c and PL values. Dodge-Romig brought out the outcome of the sampling plan for the AOQL system and set them in its table, these tables involve sampling plans for thirteen values AOQL ≤ 0.10. Moreover, the values are: 0.1%, 0.25%, 0.5%, 0.75%, 1.0%, 1.5%, 2.0%, 2.5%, 3.0%, 4.0%, 5.0%, 7.0%, and 10.0% Every table consists of six plans corresponding to the quality levels and category of batch sizes. If N batches were distributed from value (1) to (100,000) in 14 categories, the value of LTPD was selected at every quality level, and through these tables, the sampling plan parameters can be read (n, c) after selecting PL value and level of quality and the category that involves the batch size producer N. Second: Using hald table [28] The inspections of SSPs for the (AOQL) system, which was brought out by Hald depend on minimizing the total inspection and the results of multiplying sides of equation (15) by the PL value, which means: I(P1) PL = nPL + (NPL–nPL)Q(P1) (28) Kadir, et al.: Sampling attribute plans based upon dodge-romig model 70 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Which can be written in this form: I(P1) PL = mL + (ML–mL)(1–G(c, PmL) (29) The results of the SSPs were set which replaced the equation (29) in the tables Hald,[11,12] as the value of P, YC, c IMPLEMENTING DODGE-ROMIG MODEL The process of (Ala Pepsi 1.5 L bottle) production reveled to us that it falls under the control of the beta binomial and because of that beta distribution is considered X p as an evaluate for the type of the product; therefore, the value of beta distribution (P = 0.005349) which is the value that is pointed in the Dodge-Romig model, and from the public facility for the fizzy drinks (Ala Pepsi) it appeared to us that the authorized value of the defective percentage (LTPD) is (P = 1%) whereas the parameters of the SSP (n, c) are extracted, which is mandatory for inspection batch N = 39388 which it represent the everyday production ratio and that’s through depending on Table 1[13] that knows the value that is comparable to the size of the batch, which falls under the category (20000–50000) and quality level (0.4–0.5%) and it was found that the parameter sampling plan is (2570.19), and regarding the average total inspection I(P) which is formalized to apply to this plan which is known in equation (14). I(P) = n + (N–n)Q(P) It was extracted after determining the possibility of approval of the product from the following equation: I p P x Cr x x n x n x� � � �� � � � � � � � ��19 0 005349 1 0 005349 0 93389 0 19 . ( . ) . Q(Pi) = 1–0.93389 = 0.06611 I(Pi) = 2570 + (39388–2570) 0.611 = 5004 Sampling Units Poisson Approximation was implemented after finding the sampling plan for the (LTPD) system through using Dodge-Romig tables to determine the Sampling plan with the help of Hald tables, where it relies on locating the value M = NP2 = 39388 (0.01) = 39388 which falls within the column P = 0.5349 in the [Table 2],[11] then knowing the values of (c, mc) that is comparable to the value of M from the first and second columns, and it was discovered that (c = 24, mc = 31.58) and that’s because the parameters of the SSP are (n = mc, P2 = 3158, c = 24), and to find the total inspection ratio which is connected to the SSP (3158,24), the value of Q(P1) was found in the Table 3[11] where Q(P1) = 0.0195 and therefore the total average is (The average of the accepted and rejected inspection quantity) is: I(P1) = 3158 + (39388–3158) (0.0195) = 3864 It appeared to us previously that the total average inspection in using the Poisson approximation is smaller than in the case of using the binomial. IMPLEMENTING THE DECISION-MAKING MODEL Based on the decision-making model, a set of Bayesian strategies was selected to assess the product by using past defective proportion data to calculate sample size (n) and acceptance criteria (c). The sampling plans for the product were listed in [Table 2] based on quality levels and batch sizes produced. Regarding the necessary parameters for the sampling plan to analyze the daily production of the (Ala Pepsi 1.5L) product with a quality of ( X p = 0 005349. ), quantity of (0.00617), and extracted based on the sample size (n, c) = (1495,14). The expected dangerous value for the sampling plan was extracted (1495, 14) and it was found that the value R{f(p) n, c} equals (43.210$). IMPLEMENTING [HALD] MODEL The defective percentages distribution p(f) is a continuous distribution that can be altered at a specific point P = Pr Bayesian plans for product examination will be derived from direct formulas by [Hald], with the required groups of Bayesian plans needing to be calculated beforehand: Average cost of inspection per unit 1. k S S Ps � �1 2 =0.0011 + (0.03) (0.005349) = 0.00126$ Average cost of rejection per unit 2. k R R P kr s� � �1 2 Average cost of acceptance per unit 3. k A A Ps � �1 2 =0 + 0.208 (0.005349) = 0.0011$ Also the evaluation quantities (4, 5, 6, 7) 4. ( ) 0 ( ) ( ) , ( 1, ) rp r r Pr prP P f p dp P IB PIBα β α β− = − +∫ =(0.006179) (0.789972)–(0.005349) (0.734239) = 0.000954 5. 2 2 0 ( ) ( ) ( ) rp m r rk k A A P P f P dp= − − −∫ =0.00126–(0.178) (0.0009544) = 0.0010901 6. ( )2 22 1 ) 2 ( , )( r r s m p q A R B k k α β λ α β − = − �1 2 148 25059� . Whereas: 2 20.0061 , 0.208 , 0.03 , ˆˆ 25 , 4648rP A R α β= = = = = 7. � � � � � � � � 2 23 11 2 3 1 3 1 1 1 � � � �� � � � � � �� � � � ( ) ( ) p q p qr r r r The value of λ1, λ2 is extracted, therefore the value on (n) needed for the inspection batch will be N = 39388 and it’s the particular value of the upcoming context: n N* � � �� �1 2 n Unit* . .� � � � �12 17582 39388 55 4084 2361 Kadir, et al.: Sampling attribute plans based upon dodge-romig model 71 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Table 1: Bayesian plans to test the product against the decision-making model and the Hald model [Hald] model Decision-making model Batch size Pn(c) c n Pn(c) c n 0.005998286 10 1162 0.00580307 6 669 10000 0.00610687 11 1222 0.005768515 6 701 11000 0.006049403 11 1278 0.005919349 7 733 12000 0.005994006 11 1333 0.005886681 7 763 13000 0.006107626 12 1385 0.005856515 7 791 14000 0.006056638 12 1436 0.00600874 8 819 15000 0.006008444 12 1485 0.005979344 8 846 16000 0.005962933 12 1532 0.005951307 8 872 17000 0.006079027 13 1578 0.005924596 8 897 18000 0.006035578 13 1623 0.006076854 9 922 19000 0.005993691 13 1667 0.006050899 9 946 20000 0.006110937 14 1709 0.006026232 9 969 21000 0.006070984 14 1751 0.006001765 9 992 22000 0.006033416 14 1791 0.005978548 9 1014 23000 0.00599631 14 1831 0.006130671 10 1036 24000 0.006113404 15 1870 0.006107137 10 1058 25000 0.006078104 15 1908 0.00608484 10 1079 26000 0.006044122 15 1945 0.006063756 10 1099 27000 0.006010518 15 1982 0.00621547 11 1119 28000 0.00597818 15 2018 0.006194081 11 1139 29000 0.006094842 16 2054 0.00617284 11 1159 30000 0.006064192 16 2088 0.006152794 11 1178 31000 0.006032961 16 2123 0.006132879 11 1197 32000 0.006003807 16 2156 0.00611413 11 1215 33000 0.006119773 17 2190 0.006264815 12 1233 34000 0.006091371 17 2222 0.00624578 12 1251 35000 0.006062356 17 2255 0.00622686 12 1269 36000 0.006034483 17 2287 0.006208054 12 1287 37000 0.006007724 17 2318 0.006190397 12 1304 38000 0.006123612 18 2349 0.006339673 13 1321 39000 0.006096696 18 2380 0.006321743 13 1338 40000 0.006070874 18 2410 0.006303915 13 1355 41000 0.006045269 18 2440 0.006287227 13 1371 42000 0.006020722 18 2469 0.006270627 13 1387 43000 0.005995538 18 2499 0.006254115 13 1403 44000 0.006111111 19 2527 0.006401838 14 1419 45000 0.006086596 19 2556 0.006385069 14 1435 46000 0.006063111 19 2584 0.006369427 14 1450 47000 0.006039808 19 2612 0.006352826 14 1466 48000 0.006016683 19 2640 0.006337342 14 1481 49000 0.00599455 19 2667 0.006321932 14 1496 50000 Kadir, et al.: Sampling attribute plans based upon dodge-romig model 72 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Table 2: Bayesian plans for product inspection relative to the Beta-Prior distribution extracted from the decision-making model Quality level Quality level Quality level Quality level Quality level Batch size 0.005 0.004 0.003 0.002 0.001 Accepted number Sample size Accepted number Sample size Accepted number Accepted number Sample size Accepted number Sample size Accepted number C n c n c n c N c n 6 669 5 527 4 449 3 398 7 753 10000 6 701 5 553 4 471 4 418 7 790 11000 7 733 5 578 4 492 4 436 8 825 12000 7 763 5 602 4 512 4 454 8 859 13000 7 791 6 624 5 532 4 471 8 891 14000 8 819 6 646 5 550 4 488 9 923 15000 8 846 6 667 5 569 4 504 9 953 16000 8 872 6 688 5 586 5 519 9 982 17000 8 897 6 708 5 603 5 534 9 1011 18000 9 922 7 727 6 620 5 549 10 1038 19000 9 946 7 746 6 636 5 563 10 1065 20000 9 969 7 765 6 651 5 577 10 1092 21000 9 992 7 783 6 667 5 591 11 1117 22000 9 1014 7 800 6 682 5 604 11 1143 23000 10 1036 8 817 6 696 6 617 11 1167 24000 10 1058 8 834 6 711 6 630 11 1191 25000 10 1079 8 851 7 725 6 642 11 1215 26000 10 1099 8 867 7 739 6 654 12 1238 27000 11 1119 8 883 7 752 6 666 12 1261 28000 11 1139 8 899 7 766 6 678 12 1283 29000 11 1159 8 914 7 779 6 690 12 1305 30000 11 1178 9 929 7 792 6 701 13 1327 31000 11 1197 9 944 7 804 6 712 13 1348 32000 11 1215 9 959 8 817 7 724 13 1369 33000 12 1233 9 973 8 829 7 734 13 1389 34000 12 1251 9 987 8 841 7 745 13 1410 35000 12 1269 9 1001 8 853 7 756 14 1430 36000 12 1287 9 1015 8 865 7 766 14 1449 37000 12 1304 10 1029 8 876 7 776 14 1469 38000 13 1321 10 1042 8 888 7 787 14 1488 39000 13 1338 10 1055 8 899 7 797 14 1507 40000 13 1355 10 1069 8 910 7 807 15 1526 41000 13 1371 10 1082 9 921 7 816 15 1544 42000 13 1387 10 1094 9 932 8 826 15 1562 43000 13 1403 10 1107 9 943 8 836 15 1581 44000 14 1419 11 1120 9 954 8 845 15 1598 45000 14 1435 11 1132 9 964 8 854 15 1616 46000 14 1450 11 1144 9 975 8 864 16 1634 47000 14 1466 11 1156 9 985 8 873 16 1651 48000 14 1481 11 1168 9 995 8 882 16 1668 49000 14 1496 11 1180 9 1005 8 891 16 1685 50000 Kadir, et al.: Sampling attribute plans based upon dodge-romig model 73 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Table 3: Bayesian plans for the inspection of the Ala Pepsi 1.5L product that’s extracted from [Hald]’s model Quality level Quality level Quality level Quality level Quality level Batch size 0.005 0.004 0.003 0.002 0.001 Number of acceptances Sample size Number of acceptances Sample size Number of acceptances Sample size Number of acceptances Sample size Number of acceptances Sample size c n c n C n c n c n 10 1162 8 798 7 671 3 588 6 529 10000 11 1222 8 839 7 707 3 620 6 557 11000 11 1278 8 879 8 741 3 650 7 584 12000 11 1333 9 917 8 773 3 679 7 611 13000 12 1385 9 954 8 804 3 706 7 636 14000 12 1436 9 989 8 835 3 733 7 660 15000 12 1485 9 1023 8 864 3 759 7 683 16000 12 1532 10 1057 9 892 3 784 7 706 17000 13 1578 10 1089 9 920 3 808 8 728 18000 13 1623 10 1120 9 946 3 832 8 750 19000 13 1667 10 1151 9 972 3 855 8 771 20000 14 1709 10 1181 9 998 3 878 8 791 21000 14 1751 10 1210 9 1022 3 900 8 811 22000 14 1791 11 1238 9 1047 3 921 8 830 23000 14 1831 11 1266 10 1070 3 942 8 849 24000 15 1870 11 1293 10 1094 3 963 8 868 25000 15 1908 11 1320 10 1116 3 983 8 886 26000 15 1945 11 1346 10 1139 3 1003 9 904 27000 15 1982 11 1372 10 1161 3 1022 9 922 28000 15 2018 12 1397 10 1182 3 1041 9 939 29000 16 2054 12 1422 10 1203 3 1060 9 956 30000 16 2088 12 1446 11 1224 3 1078 9 973 31000 16 2123 12 1470 11 1245 3 1096 9 989 32000 16 2156 12 1494 11 1265 3 1114 9 1006 33000 17 2190 12 1517 11 1285 3 1132 9 1022 34000 17 2222 13 1540 11 1304 3 1149 9 1037 35000 17 2255 13 1563 11 1323 3 1166 10 1053 36000 17 2287 13 1585 11 1342 3 1183 10 1068 37000 17 2318 13 1607 11 1361 3 1200 10 1083 38000 18 2349 13 1629 12 1380 3 1216 10 1098 39000 18 2380 13 1650 12 1398 3 1232 10 1113 40000 18 2410 13 1672 12 1416 3 1248 10 1127 41000 18 2440 13 1693 12 1434 3 1264 10 1142 42000 18 2469 14 1713 12 1452 3 1280 10 1156 43000 18 2499 14 1734 12 1469 3 1295 10 1170 44000 19 2527 14 1754 12 1486 3 1310 10 1184 45000 19 2556 14 1774 12 1503 3 1325 10 1197 46000 19 2584 14 1794 12 1520 3 1340 10 1211 47000 19 2612 14 1813 13 1537 3 1355 11 1224 48000 19 2640 14 1833 13 1553 3 1370 11 1237 49000 19 2667 14 1852 13 1570 3 1384 11 1251 50000 Kadir, et al.: Sampling attribute plans based upon dodge-romig model 74 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 However, the number of is extracted from the c n pr * *� � �1 context which means: 1 1 2 ˆ ˆr rP qβ β α= − − Therefore, the value of the accepted unit (n = 2361) equals: β1 = (4648) (0.0061) – (25) (0.9939) – 0.5 β1 = 3.0053 And added: c* = 2361 (0.061) + 3.0053) = 17 Unit As a result, a SSP is necessary to assess the production mean of N=39388, as specified in [Hald]’s model (2361, 17). This plan entails inspecting a random sample size of (2361) bottles, and if the number of defective units in the sample is (17) or fewer, all units are accepted; otherwise, the batch is rejected. The cost of quality control, determined by the sampling plan, will be derived from the smallest standard cost R0 (N), resulting in the best sampling plan (2361.17) being implemented: ( )* 2 0 1 2 2 2 2 , ( ) s mk kR n ds ds A R λ λ − = − − = − ds = 0.000954 R0 (N) = [2(2361)–148.2506 + 55.4084] (0.000954) R0 (N) = 4.4162 Therefore, the total value cost k(p) is expected to control quality type which equals to: k(p) = R0 ((A2–R2) + NKm =4.4162 (0.178) + 39388 (0.001091) =42.9731 $ Table 3 displays the outcomes of Bayesian strategies for assessing the Ala Pepsi 1.5L item derived from [Hald]’s model, based on the Beta distribution, categorized into quality levels X � � �0 001 0 001 0 005. . . and production batch sizes N = 10000 (1000) 50000. CALCULATING THE VALUE OF THE DEFECTIVE FRACTION IN UNEXAMINED QUANTITIES Based on decision-making models and the Hald model, a set of Bayesian strategies are selected to assess the product. Determining the expected value of the fraction defective in the untested quantities (N–n) is crucial. and they are going to be approved based sample’s acceptance, after that we will depend on the average value of the distribution following the detective amount E(p/x) when X = c which means Pn(c) and it appeared to us in the equation (31) that the next distribution f(p/x) is also a Beta with parameters (x + α, α + β + n) therefore it is: E x P x x nn� � � � � � � � � � � � (30) And when X = c it is: P c c nn � � � � � � � � � (31) The forthcoming [Table 1] shows a comparison of Bayesian designs based on [Hald]’s model and the Decision- making model, considering the quality level ( X p = 0 005349. ), value Pn(c), and all plans from both models. Table 1 clearly shows that the expected value of the defective fraction in the quantities (N – n) that will accepted under the sample approval decision equals 0.6%, according to the Decision-Making model equals 0.6%, and according to [Hald] model that is the same value as the sells percentage that is allowed through (LTPD) that relies on the Pepsi company for fizzy drinks and the correspondences of the average value of (Pn) and (LTPD) shows the efficiency of the Bayesian plans that relies on the previous distribution of defective percentages, as it takes all available and previous information about the quality in consideration when estimating the quality in subsequent production batches, and the correspondence indicates the importance on the Bayesian plans and efficiency and the quality level of actual production X p = 0 005349. , as we find that the size of the sample for different batch sizes is small compared to other sampling plans which means reducing the cost of them and therefore the reduction of the total cost. CONCLUSION The Beta-Binomial distribution with a rate of 0.005349 was found to be a suitable probability distribution for modeling the proportion of defective products in the Ala Pepsi factory. In this specific case, the Poisson approximation resulted in a smaller total average inspection compared to the binomial distribution, suggesting potential cost savings. The study also demonstrated the effectiveness of the decision-making model in determining a sampling plan (n = 1495, c = 14) that resulted in smaller sample sizes and reduced inspection costs compared to the Dodge-Romig tables method. The expected value of the defective fraction in the untested quantities was consistent with the allowed LTPD, highlighting the efficiency of Bayesian plans in incorporating prior knowledge about quality. These findings have significant implications for quality control practices in the soft drink industry, suggesting that Bayesian sampling plans can be a valuable tool for optimizing inspection processes and reducing costs. Future research could explore the generalizability of these findings to other industries and investigate the use of different prior distributions in Bayesian sampling plans. REFERENCES [1] P. Banerjee. Sampling inspection procedures. Calcutta Statistical Association Bulletin, vol. 6, no. 3, pp. 132-148, 1955. [2] D. H. Kadir and A. R. K. Rahi Al-Harthy. Application of Bayesian Technique for Ala Pepsi Softdrink Company in Sampling Plan Design. University of Sulaimani, Iraq, 2007. [3] D. H. Kadir and A. R. K. Rahi. Applying the bayesian technique in designing a single sampling plan. Cihan University-Erbil Scientific Kadir, et al.: Sampling attribute plans based upon dodge-romig model 75 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (2): 65-75 Journal, vol. 7, no. 2, pp. 17-25, 2023. [4] A. Hald. Some limit theorems for the dodge-romig LTPD single sampling inspection plans. Technometrics, vol. 4, no. 4, pp. 497-513, 1962. [5] A. Ahmadi Yazdi and M. S. Fallahnezhad. Comparison between count of cumulative conforming sampling plans and Dodge- Romig single sampling plan. Communications in Statistics-Theory and Methods, vol. 46, no. 1, pp. 189-199, 2017. [6] C. H. Chen. Economic design of Dodge-Romig AOQL single sampling plans by variables with the quadratic loss function. Journal of Applied Science and Engineering, vol. 8, no. 4, pp. 313-318, 2005. [7] J. Klůfa. Economic aspects of the LTPD single sampling inspection plans. Agricultural Economics, vol. 61, no. 7, pp. 326-331, 2015. [8] H. F. Dodge and H. G. Roming. Sampling Inspection Tables. Wiley, United States, 1959. [9] W. C. Guenther. On the use of standard tables to obtain Dodge‐ Romig LTPD sampling inspection plans. Naval Research Logistics Quarterly, vol. 18, no. 4, pp. 531-542, 1971. [10] A. Z. Salman. Using Decision-Making Methods to Build the Best Cost Function Model in Quality Control. Ph.D Thesis. Administration and Economics University of Baghdad, Baghdad, 1991. [11] A. Hald. Statistical Theory of Sampling Inspection by Attributes (Probability and Mathematical Statistics). Academic Press, London, New York, 1981. [12] D. Harold and R. Harry. Sampling Inspection Tables. Ediciones Revolucionarias. Wiley, New Jersey, U.S, 1959. [13] D. C. Montgomery. Introduction to Statistical Quality Control. John Wiley and Sons, New Jersey, U.S, 2007.