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Building a model for expected cost function to obtain double

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This document presents a model for determining the parameters of a double Bayesian sampling inspection plan to minimize total expected quality control costs. The model considers costs associated with inspecting samples, accepting or rejecting lots, and continuing to a second sample. It defines variables such as sample sizes, acceptance/rejection numbers, and costs. Equations are provided for the average inspection cost of the first and second samples considering the sampling distribution and conditional probability of defective items. The model aims to obtain optimal parameters (n1, a1, r1, n2, a2, r2) of the double sampling plan by minimizing the total expected regret function according to producer and consumer risk levels.

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 282 BUILDING A MODEL FOR EXPECTED COST FUNCTION TO OBTAIN DOUBLE BAYESIAN SAMPLING INSPECTION 1 Dr. Dhwyia Salman Hassan, 2 Nashaat Jaisam Al-Anber, 3 Dr.Suaad Khalaf Salman 1 Department of Statistics, College of Economic and Administration, University of Baghdad, Baghdad 2 Department of Information Technology, Technical College of Management-Baghdad, Foundation of Technical Education, Iraq 3 Department of Mathematics, College of Education for Pure Science, University of Karbala, Iraq ABSTRACT This paper deals with building a model for total expected quality control cost for double Bayesian sampling inspection plan, which consist of drawing a sample (݊ଵ) from a lot (ܰ), then checking the number of defective (‫ݔ‬ଵ) in (݊ଵ), if (‫ݔ‬ଵ ൑ ܽଵ), were (ܽଵ) is acceptance number, then accept and stop, if (‫ݔ‬ଵ ൒ ‫ݎ‬ଵ), were (‫ݎ‬ଵ) is rejection number, then take a decision to reject and stop sampling, if (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), draw a second sample (݊ଶ) and check the number of defective (‫ݔ‬ଶ) in (݊ଶ), if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൑ ܽଶ, accept and stop, if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൒ ‫ݎ‬ଶ reject and stop. The model derived assuming that the sampling process is Binomial with (݊, ‫)݌‬ and ( ‫)݌‬ is random variable, varied from lot to lot and have prior distribution ݂ሺ‫݌‬ሻ, here it is found Beta prior with parameters (ܵ, ܶ).The model is applied on the set of real data obtained from some industrial companies in Iraq, were the percentage of defective of (140) lots are studied and tested, ݂ሺ‫݌‬ሻ is found Beta prior (ܵ, ܶ). These parameters are estimated by moment estimator. After checking the distribution ݂ሺ‫݌‬ሻ and all the parameters of quality control model, we work first on obtaining single Bayesian sampling inspection plan (݊ଵ, ܿଵ) from minimizing the standard regret function (ܴଵ), then we find the parameters of double Bayesian sampling inspection plan from minimizing (ܴଶ), for various values of process average of quality (‫,)݌‬ with different lot size N. Keywords: Bayesian sampling Inspection, prior distribution, average of quality, standard regret function for single sampling plan. standard regret function for double Bayesian sampling plan. INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 2 March – April 2013, pp. 282-294 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 283 1- AIM OF RESEARCH The aim of this research is to build a model for total cost of double Bayesian sampling plans, then determine the parameters of this double Bayesian sampling plans (݊ଵ, ܽଵ, ‫ݎ‬ଵ, ݊ଶ, ܽଶ, ‫ݎ‬ଶ) from minimizing the total expected cost function. 2- NOTATIONS Here we introduce all necessary notations: ݊ଵ : First sample size drawn from lot N. ‫ݔ‬ଵ : Number of defective units in sample (݊ଵ). ܽଵ : Number of accepted defective units in (݊ଵ). ‫ݎ‬ଵ : Number of rejected units in (݊ଵ). ݊ଶ : Size of second sample taken from (ܰ െ ݊ଵ). ‫ݔ‬ଶ : Number of defective units in (݊ଶ). ܺ : Number of defective units in lot (N). ܽଶ : Number of accepted defectives in (݊ଶ). ‫ݎ‬ଶ : Number of rejected units second sample (݊ଶ). ܲ : Percentage of defectives in production. ‫݌‬ : percentage of defectives in sample. ܲ௔ ሺଵሻ ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from first sample (݊ଵ). ܲ௥ ሺଵሻ ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting from first sample (݊ଵ). ܲ௔ ሺଶሻ ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from second sample (݊ଶ). ܲ௥ ሺଶሻ ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting from first sample (݊ଶ). ܲ௦ ሺଶሻ ሺ‫݌‬ሻ : Probability of continue sampling after the first sample is chosen and the value of (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), so we cannot take a decision to accept or reject according to first sample. ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample, where; ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ ܵଵ : Cost of inspecting item from first sample. ܵଶ : Cost of repairing or replacement of item in sample (݊ଵ). ܴଵ : Cost of sampling and testing unit in rejected quantity (ܰ െ ݊). ܴଶ : Cost of repairing or replacement unit in rejected quantity (ܰ െ ݊). ‫ܣ‬ଵ : Cost of accepting good units which not represents penalty cost. ‫ܣ‬ଶ : Cost of accepting defective units. 3. INTRODUCTION In rectifying inspection plans, each item produced by a process is inspected according to some plan and then according to the number of defectives units in the sample, the lot is either accepted or the lot is rejected. Gunether [1977] explained how to obtain the parameters (݊, ܿ) for single sampling plan for ordinary sampling and for Bayesian sampling, using the model of expected total quality control cost depends on cost of inspection and sampling and
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 284 cost due to wrong decision, which is accepting bad lot and rejecting good one. Another researchers gives model for obtaining parameters of single sampling inspection plan (݊, ܿ) under truncated time to failure, see Epstein, B. [1962]. Goode and Kao [1960]. Gupta, S. S. [1962] who designed sampling inspection plans for normal and lognormal distribution according to pre – determined level of percentage of defectives. Also Gupta and Kunda [1999] discuss the properties of generalized exponential distributions and how to use it is constructing Bayesian sampling. Baklizi, A. [2003] introduced how to obtain acceptance sampling plans based on truncated life tests in the Pareto distribution of second kind. The research in this subject were developed when Balakrishnan, N. [2007] designed a group of inspection sampling for generalized Birnbaum – Saundres distribution. In [2012], Dhwyia S. Hassun, et, built a model for total expected quality control cost, and compared it with the model of Schmidt – Taylor [1973]. Also Hald, A. [1968] gives a model for Bayesian single sampling attribute plans for continuous prior distribution. The parameters of single sampling inspection plans (݊, ܿ) are obtained from minimizing total expected quality control cost. Now, in this paper we introduce ( a mixed – Binomial) quality control expected cost function, to obtain the parameters of double Bayesian sampling inspection plans, from minimizing total expected regret function, according to conditions of producers and consumers risk. Building Model of Total Expected Cost Function The model comprises: a- ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample; ݇‫ݏ‬ሺ‫݌‬ሻ ൌ ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ …………………………………………….. (1) ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ : Average cost of inspecting units of second sample, under sampling distribution ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ for number of defectives units in first sample (݊ଵ). ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൌ ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ……………………………………………… (2) And; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ‫ݎ݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൌ ‫ݕ‬ሻ ‫׊‬௬ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ …………………………………………….. (3) or; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ‫׊‬௫మ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ …………………………………………………. (4) If we assume that [‫ݔ‬ଷ ൌ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ], then the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is; ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ ൌ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ|‫ݔ‬ሻ‫݌‬ሺ‫ݔ‬ሻ ………………………………………………. (5) ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ∑ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ‫׊‬௫య ………………………………………………...(6) And; ‫݌‬ሺ ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ݃௡మ ሺ ‫ݔ‬ଶ|݊ଵ, ‫ݔ‬ଵሻ ……………………………………………….. (7) The average inspection cost for second sample is equal to:
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 285 ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ………………………………………………. (8) In case of accepting produced lot from first sample, the average cost is; ‫ܧ‬ሼሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵሻ‫ܣ‬ଶ|‫ݔ‬ଵሽ ൌ ሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ‫ܧ‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫ܣ‬ଶ ൌ ሺܰ െ ݊ଵሻሼ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሺ‫ݔ‬ଵሻሽ ൌ ሺܰ െ ݊ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ The average of acceptance cost for second sample is; ‫ܧ‬ሼሺܰ െ ݊ଵ െ ݊ଶሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶሻ‫ܣ‬ଶ|‫ݔ‬ଵ, ‫ݔ‬ଶሽ ሺܰ െ ݊ଵ െ ݊ଶሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሿ …………………………………. (9) After explaining all notations, we can show that the lot size N, in case of double sampling can divide into; ܰ ൌ ݊ଵ ൅ ሺܰ െ ݊ଵሻቀ‫݌‬௔ ሺଵሻ ൅ ‫݌‬௥ ሺଵሻ ቁ ൅ ݊ଶ‫݌‬ௌ ሺଵሻ ൅ሺܰ െ ݊ଵ െ ݊ଶሻቀ‫݌‬௔ ሺଶሻ ൅ ‫݌‬௥ ሺଶሻ ቁ … (10) Assume (݇݉), is the smallest average cost, happened according to results of decisions of accepting and rejecting, where acceptance happened at (‫݌‬ ൑ ‫,)ݎ݌‬ (i.e ‫,ݎ݌‬ break even quality level) and rejection when (‫݌‬ ൐ ‫,)ݎ݌‬ (‫)݌‬ is random variable have [݂ሺ‫݌‬ሻ] then (݇݉) is equal to; ݇݉ ൌ ‫׬‬ ሺ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ௣௥ ଴ ‫׬‬ ሺܴଵ ൅ ܴଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ………………..... (11) Now the construction of total expected quality control cost [݇ଶሺ‫݌‬ሻ] is given by; ݇ଶሺ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔ ሺଵሻ ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ ሺଵሻ ሺ‫݌‬ሻቅ ൅݊ଶ݇‫ݏ‬ሺ‫݌‬ሻ‫݌‬ௌ ሺଶሻ ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔ ሺଶሻ ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻቅ ……….…(12) When ሺ‫݌‬ሻ is random variable have [݂ሺ‫݌‬ሻ], then the average cost is; ݇ଶ ൌ ‫׬‬ ݇ଶሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ݇ଶ ൌ ݊ଵ݇ܵ ൅ ሺܰ െ ݊ଵሻ ቐ ෍ ݃ሺ‫ݔ‬ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ௔భ ௫భୀ଴ ቑ ൅ ෍ ݃ሺ‫ݔ‬ଵሻ݇‫ݎ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ௡భ ௫భୀ௥భ ൅ ∑ ݃ሺ‫ݔ‬ଵሻൣ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ൧ ௥భିଵ ௫భୀ௔భାଵ ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቐ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ݇ܽሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ ௔మି௫భ ௫మୀ଴ ቑ ൅ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻሾ݇‫ݎ‬ሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ ௡మ ௫మୀ௔మି௫భାଵ ………………………..(13) Multiplying equation (10) by (݇݉) where ሾܰ݇݉ ൌ ݇݉ሿ and apply [ሺ݇2 െ ݇݉ሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ], we have obtained the equation of standard cost function (ܴଶ). ܴଶ ൌ ݊ଵ݀‫ݏ‬ ൅ ሺܰ െ ݊ଵሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ ቋ ൅ ቊන ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ቋ ൅݊ଶ ቄ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬ௌ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ‫׬‬ ߜሺ‫݌‬ሻ‫݌‬ௌ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ଴ ௣௥ ଴ ቅ ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ න ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ ሺଶሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ௣௥ ଴ ቋ … (14)
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 286 ‫ݎ݌‬ ൌ break even quality level; ‫ݎ݌‬ ൌ ஺మିோమ ஺భିோభ Equation (14), now transformed to standard cost function in terms of conditional distribution of percentage of defectives, which is [݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ] especially these parts obtained from second sampling, where; ݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ ൌ ௣௥ሺ௔భழ௫భழ௥భ|௣ሻ௙ሺ௣ሻ ∑ ௕௪ሺ௫భ,௡భሻ ೝభషభ ೌభశభ ൌ ቄ௣ೌ ሺమሻ ሺ௣ሻା௣ೝ ሺమሻ ሺ௣ሻቅ௙ሺ௣ሻ ‫׬‬ ቄ௣ೌ ሺమሻ ሺ௣ሻା௣ೝ ሺమሻ ሺ௣ሻቅ௙ሺ௣ሻௗ௣ భ బ ……… (15) This because when [ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ] a second sample (݊ଶ) is drawn, and according to this sample either we accept or reject. The final formula for standard regret function (ܴଶ) used to obtain the parameters of second sampling plan is; ܴଶ ‫כ‬ ൌ ܴଵ ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ െ ሺܰ െ ݊ଵሻ ෍ ݃ሺ‫ݔ‬ଵሻ݀‫ݎ‬ሺ‫ݔ‬ଵሻ ௥భିଵ ௫భୀ௔భାଵ ൅ ∑ ݃ሺ‫ݔ‬ଵሻܴଵሺܽଶ െ ‫ݔ‬ଵ, ݊ଶ, ܰ െ ݊ଵ|‫ݔ‬ଵ ሻ௥భିଵ ௫భୀ௔భାଵ ……………………………….… (16) Where; ݀‫ݎ‬ሺ‫ݔ‬ଵሻ ൌ න ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ௣௥ ଴ ܴଵ ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ ൌ ௄ሺே,௡,௖ሻ ି ே௞௠ ௞௦ ି ௞௠ ………………………………… (17) Where; ‫ܭ‬ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ቂሺ‫ܣ‬ଶ െ ܴଶሻ ௌ ௌା் ‫ܨ‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻቃ ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ቂሺܴଵ ൅ ܴଶሻ ௌ ௌା் ቃ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ ቀ ௌ ௌା் ቁ , ‫݌‬ ൌ ் ௡ା் ݇ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ሺܰ െ ݊ሻሺ‫ܣ‬ଵ ൅ ܴଶ‫݌‬ሻ ൅ ሺܰ െ ݊ሻሺܴଵ െ ‫ܣ‬ଵሻ Which can be simplified to; ݇ሺܰ, ݊, ܿሻ ൌ ݊ሺ݇‫ݏ‬ െ ݇݉ሻ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ ൅ሾሺܴଵ െ ‫ܣ‬ଵሻሾ1 െ ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿሿ ൅ ܰ݇݉ ……………... (18) Therefore the equation of standard expected cost (Regret function ܴଵ) is; ܴଵሺܰ, ݊, ܿሻ ൌ ௞ሺே,௡,௖ሻ ି ே௞௠ ௞௦ ି ௞௠ Which is the first part of equation (ܴଶ ‫כ‬ ) defined in equation (16), so first of all and after determining the parameters of single Bayesian sampling plan [given in table (1)] we work on minimizing (16) to find the parameters of double Bayesian sampling inspection plan given in table (4). Equation (16) shows that the average of standard cost function for double sampling plan, is represented by standard cost function for single sampling plan and also standard conditional cost function weighted by the probability distribution of number of defective units [݃ሺ‫ݔ‬ଵሻ], from first sample, it is obtained from; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ‫׊‬௫మ
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 287 To satisfy the aim of research which is to find the parameters ሺܽଵ, ‫ݎ‬ଵ, ܽଶሻ from minimizing (ܴଶ), we apply difference equation since the distribution of number of defectives in production is of discrete type, and also we consider; ∆ ܴଶ ൌ ∆ ௞మ ஺మିோమ The differences equations according to ሺ‫ݎ‬ଵ, ܽଵ, ܽଶሻ, required to rewrite (݇ଶ) as in equation (19) follows; Since; ݇ଶ ൌ ݇ଵሺܽଵ, ݊ଵ, ܰሻሺ‫ܣ‬ଶܴଶሻ ∑ ݃ሺ‫ݔ‬ଵሻ൛݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ௥భିଵ ௫భୀ௔భାଵ ݊ଶሻ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ൟ …………………………………….. … (19) Then using; ∆ ܴଶ ൌ ∆ ௞మ ஺మିோమ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ … (20) We can find [ ∆௥భ ܴଶሺ‫ݎ‬ଵሻ] as; ∆ ௥భ ܴଶሺ‫ݎ‬ଵሻ ൌ ݃௡భ ሺ‫ݎ‬ଵሻቐ݊ଶߜቂ‫݌‬௡భ ሺ௥భሻ ቃ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃௡మ ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻቂ‫݌‬௡భା௡మ ሺ௥భ, ௫మሻ െ ‫ݎ݌‬ቃ ௔మି௥భ ௫మୀ଴ ቑ … (21) And also; ∆ ௔భ ܴଶሺܽଵሻ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଶ െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ … (22) And also the difference of [ܴଶሺܽଵሻ] with respect to ሺܽଵሻ; ∆௔భ ܴଶሺܽଵሻ ൌ ݃௡భ ሺܽଵ ൅ 1ሻ ቄ݊ଶሺ‫ܣ‬ଶ െ ܵଵሻቂ‫݌‬௡భ ሺ௔భାଵሻ ቃ െ ሺܵଵ െ ‫ܣ‬ଵሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ ൅ ݃௡భ ሺܽଵ ൅ 1ሻሺܰ െ ݊ଵ െ ݊ଶሻ ∑ ݃௡మ ሺ‫ݔ‬ଶ|ܽଵ ൅ 1ሻቂ‫݌‬௡భା௡మ ሺ௔భାଵା ௫మሻ െ ‫ݎ݌‬ቃ ௡మ ௫మୀ௔మି௔భ ቅ …………. … (23) From ∆௔మ ܴଶሺܽଶሻ we can find the value of parameter (ܽଶ) for given (݊ଵ, ݊ଶ ) which satisfy the inequality (24) follows. ‫̂݌‬௡భା௡మ ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫̂݌‬௡భା௡మ ሺܽොଶ ൅ 1ሻ …………………………………….. (24) Then; ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 ‫݄݊݁ݓ‬ ݇‫ݏ‬ ൌ ݇‫ݎ‬ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 ݂‫ݎ݋‬ ݇‫ݏ‬ ൐ ݇‫ݎ‬
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 288 And when [ܽଶ ൌ ܽଵ], the second term of ∆௔భ ܴଶሺܽଵሻ is positive, so we assume that [ܽଵ ൌ ܽଵ ‫כ‬ ] is the value at which the first term of ∆௔మ ܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ ‫כ‬ሻ ൐ 0 and when ݇‫ݏ‬ ൌ ݇‫ݎ‬ ฺ ሺ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1ሻ, and we see that the average cost of inspection equal to average cost of reject, therefore we need only to find the parameters (ܽଵ, ܽଶ ) and from ∆௔భ ܴଶሺܽଵሻ we obtain ‫݌‬௡భ ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to ‫ݎ‬ଵ ൑ ݊ଵ ൅ 1. After completing model for double Bayesian sampling inspection plan, this model is applied to distribution of Beta – Binomial. From ∆௔మ ܴଶሺܽଶሻ we can find (ܽଶ) that satisfy the inequality (24). ‫݌‬௡భା௡మ ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫݌‬௡భା௡మ ሺܽොଶ ൅ 1ሻ ………………………………………. (25) When (ܽଵ ൌ ܽଶ), the second term of ∆௔భ ܴଶሺܽଵሻ is positive, so we assume that (ܽଵ ൌ ܽଵ ‫כ‬ ) is the value of which the first term of ∆௔మ ܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ ‫כ‬ ሻ ൐ 0. When; ݇‫ݏ‬ ൌ ݇‫ݎ‬ Then; ∆ ܴଶሺ‫ݎ‬ଵሻ ൑ 0 ฺ ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 And for ݇‫ݏ‬ ൐ ݇‫ݎ‬ ฺ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 And when ݇‫ݏ‬ ൌ ݇‫,ݎ‬ we saw that average cost of inspection equal average cost of reject, therefore we need only to find the parameters (ܽଵ, ܽଶ), and from equation ∆௔భ ܴଶሺܽଵሻ, we obtain ‫݌‬௡భ ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to (‫ݎ‬ଵ ൑ ݊ଵ ൅ 1). 4 - APPLICATION OF PROPOSED MODEL After the total cost function are built, it is necessary to apply this model for sampling distribution to show how the parameters of double sampling plans are obtained, to take a decision for accept or reject, assume (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) are three independent random variables follow Binomial distribution as: ‫ݔ‬ଵ~ܾሺ݊ଵ, ‫݌‬ሻ ‫ݔ‬ଶ~ܾሺ݊ଶ, ‫݌‬ሻ ‫ݔ‬ଷ~ܾሺ݊ଷ, ‫݌‬ሻ And (‫)݌‬ is random variable varied from lot to lot, and it's prior distribution [݂ሺ‫݌‬ሻ] is assumed here ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ, (Beta prior), so the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is; ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ሻ ൌ න ‫ܥ‬௫భ ௡భ ଵ ଴ ‫݌‬௫భ‫ݍ‬௡భ ି ௫భ‫ܥ‬௫మ ௡మ ‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫ܥ‬௫య ௡య ‫݌‬௫య‫ݍ‬௡య ି ௫య 1 ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ି ଵ ݀‫݌‬ ൌ ஼ೣభ ೙భ஼ೣమ ೙మ஼ೣయ ೙య ஻௘௧௔ ሺ௦,௧ሻ ‫׬‬ ‫݌‬௫భା௫మା௫యାௌିଵଵ ଴ ‫ݍ‬௡భ ା௡మ ା௡య ି ௫భି௫మି௫యା௧ିଵ ݀‫݌‬ ൌ ஼ೣభ ೙భ஼ೣమ ೙మ஼ೣయ ೙య ஻௘௧௔ ሺ௦,௧ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ሻ …. (26) Also we can find the posterior distribution ݂ሺ‫ݔ|݌‬ଵሻ and ݂ሺ‫ݔ|݌‬ଵ, ‫ݔ‬ଶሻ ൌ ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ, and; ‫݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ ݀‫݌‬ ଵ ଴ Since; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ݆‫ݐ݊݅݋‬ ሺ‫,݌‬ ‫ݔ‬ଵሻ ݃ሺ‫ݔ‬ଵሻ ݃ሺ‫ݔ‬ଵሻ ൌ න ‫ܥ‬௫భ ௡భ ଵ ଴ ‫݌‬௫భ‫ݍ‬௡భ ି ௫భ 1 ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ି ଵ ݀‫݌‬
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 289 ݃ሺ‫ݔ‬ଵሻ ൌ ‫ܥ‬௫భ ௡భ ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ܵ, ݊ଵ െ ‫ݔ‬ଵ ൅ ‫ݐ‬ሻ And from equation below which is Beta distribution; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ௚ሺ௣,௫భ ሻ ௚ሺ௫భሻ The conditional distribution of (‫)݌‬ given (‫ݔ‬ଵ) is; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ଵ ஻௘௧௔ሺ௫భାௌ,௡భ ି ௫భା௧ሻ ‫݌‬௫భାௌିଵ ‫ݍ‬௡భ ି ௫భା௧ିଵ …………………….. (27) And; ‫ܧ‬ሺ‫ݔ|݌‬ଵሻ ൌ ‫ݔ‬ଵ ൅ ܵ ݊ଵ ൅ ܵ ൅ ‫ݐ‬ Also we can find ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ; ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ ሻ ଵ ଴ ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ………………………………….. (28) ൌ ଵ ௞ ‫׬‬ ‫ܥ‬௫మ ௡మଵ ଴ ‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫݌‬௫భାௌିଵ ‫ݍ‬௡భ ି ௫భା௧ିଵ ݀‫݌‬ ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ଵ ௞ ‫ܥ‬௫మ ௡మ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ൅ ‫ݐ‬ሻ ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ௌା௫భ ௌା௫భା௫మ ஼ೣమ ೙మ஼ೄశೣభ ೙భశ೟శೄషభ ஼ೄశೣభశೣమ ೙భశ೙మశ೟శೄషభ ………………………………………. (29) Which is [݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ] Assume ‫ݔ‬ଷ ൌ ܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ‫ݔ‬ଷ~ ‫݈ܽ݅݉݋݊݅ܤ‬ ሺ݊ଷ, ‫݌‬ሻ Then the conditional distribution of (‫ݔ‬ଷ) given (‫ݔ‬ଵ, ‫ݔ‬ଶ) is; ‫ݎ݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ ଵ ଴ ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ݀‫݌‬ ……………………….. (30) To solve equation (30), we need to find ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ; Let ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ‫׶‬ ‫ݔ‬ଵ~ ܾሺ݊ଵ, ‫݌‬ሻ ‫ݔ‬ଶ~ ܾሺ݊ଶ, ‫݌‬ሻ And are independent, then; ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ~ ‫ܤ‬ሺ݊ଵ ൅ ݊ଶ, ‫݌‬ሻ ݃ሺ‫,݌‬ ‫ݕ‬ሻ ൌ ‫ܥ‬௬ ௡భା௡మ ‫݌‬௬ ‫ݍ‬௡భା௡మି௬ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ିଵ ‫׵‬ ݃ሺ‫݌|ݕ‬ሻ ൌ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ න ‫ܥ‬௬ ௡భା௡మ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మି௬ା௧ିଵ ݀‫݌‬ ଵ ଴ ݃ሺ‫݌|ݕ‬ሻ ൌ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ ‫ܥ‬௬ ௡భା௡మ ‫ܽݐ݁ܤ‬ ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ ‫׵‬ ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൌ ݆‫ݐ݊݅݋‬ ݉ܽ‫݈ܽ݊݅݃ݎ‬ ൌ భ ಳሺೄ,೟ሻ ஼೤ ೙భశ೙మ௣೤శೄషభ௤೙భశ೙మష೤శ೟షభ భ ಳሺೄ,೟ሻ ஼೤ ೙భశ೙మ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ ൌ ଵ ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మି௬ା௧ିଵ 0 ൑ ‫݌‬ ൑ 1 …………. (31) Then ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ is distributed as Beta distribution with parameters (‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫.)ݐ‬ This distribution is necessary to find the distribution of number of defectives ሺ‫ݔ‬ଷ) after drawing the first and second sample, i.e to find the distribution of ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 290 ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ න ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶ ൌ ‫ݕ‬ሻ݀‫݌‬ ଵ ଴ ൌ න ‫ܥ‬௫య ௡య ଵ ଴ ‫݌‬௫య‫ݍ‬௡యି ௫య 1 ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మା௧ି௬ିଵ ݀‫݌‬ ൌ ‫ܥ‬௫య ௡య ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ න ‫݌‬௫యା௬ାௌିଵ ଵ ଴ ‫ݍ‬௡భା௡మା௡యି௫యା௧ି௬ିଵ ݀‫݌‬ ൌ ‫ܥ‬௫య ௡య ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ ൌ ‫ܥ‬௫య ௡య Γሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ Γ ሺ‫ݕ‬ ൅ ܵሻΓ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ ൅ ܵሻ Γ ሺܵ ൅ ‫ݐ‬ ൅ ݊ଵ ൅ ݊ଶ ൅ ݊ଷሻ ൌ ஼ೣయ ೙యሺ௡భା௡మା௧ାୗିଵሻ! ሺ௬ାௌିଵሻ!ሺ௡భା௡మା௧ି୷ିଵሻ! ൈ ሺ௫యା௬ାௌିଵሻ!ሺ௡భା௡మା௡యି௫యି௬ା௧ିଵሻ! ሺௌା௧ା௡భା௡మା௡యିଵሻ! ൌ ஼ೣయ ೙య஼೤శೄ ೙భశ೙మశ೟శೄషభ ሺ௫యାௌା௬ሻ ஼ೣయశೄశ೤ ೙భశ೙మశ೙యశ೟శೄషభ ሺ௬ାௌሻ Which is mixed hypergeometric distribution.We can obtain the conditional mean of ሺ‫ݔ‬ଷሻ given ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ; ‫ܧ‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ݊ଷ‫݌‬௡భା௡మ ሺ௫భ,௫మሻ Where, ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫݌‬௡భା௡మ ሺ௫భା௫మሻ 5- CASE STUDY The following data represents the percentage of defectives of (140) lots from some industrial companies in Iraq, these percentages are grouped in frequency table as follows; Table (1): The frequency table for percentage of defectives of 140 lots Percentage of defectives Observed frequency ሺ݂௜ሻ Class mark ‫݌‬௜ ‫݌‬௜݂௜ ‫݌‬௜ ଶ ݂௜ 0.0 - 6 0.01 0.06 0.0116 0.02 - 13 0.03 0.39 0.0117 0.04 - 17 0.05 0.85 0.0425 0.06 - 24 0.07 1.68 0.1176 0.08 - 18 0.09 1.62 0.1456 0.10 - 16 0.11 1.76 0.1936 0.12 - 15 0.13 1.95 0.2535 0.14 - 10 0.15 1.50 0.2250 0.16 - 9 0.17 1.53 0.2601 0.18 - 7 0.19 0.33 0.2527 0.20 - 3 0.21 0.63 0.1323 0.22 – 0.24 2 0.23 0.46 0.1058 Total 140 13.699 1.7413
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 291 Then we draw the histogram and frequency curve, the curve indicate it is Beta distribution with parameters (S, T), which are estimated by moments method using the equations; ܵመ ൌ ‫ݔ‬௣ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ ଶ ሿหܵ௣ ଶ ܶ෠ ൌ ‫ݔ‬௤ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ ଶ ሿหܵ௣ ଶ Where; ‫ݔ‬௣ ൌ ∑ ௣೔௙೔ ∑ ௙೔ ൌ 0.09785 ܵ௣ ଶ ൌ ∑ ௣೔ మ ௙೔ ∑ ௙೔ െ ‫݌‬ ଶ ൌ 0.00286 Then; ܵመ௠௢௠ ൌ 2.9 ؆ 3 ܶ෠௠௢௠ ൌ 26.9 ؆ 27 Then the distribution of (‫݌‬௜ሻ is tested using ( ߯ଶ ) test, where the cumulative probabilities are computed from; ‫ܨ‬ሺ‫݌‬ሻ ൌ ‫׬‬ ଵ ஻ሺௌ,ఛሻ ௣ ଴ ‫ݑ‬ௌିଵ ሺ1 െ ‫ݑ‬ሻ்ିଵ ݀‫ݑ‬ ൌ ‫ܧ‬ሺܵ, ܵ ൅ ܶ െ 1, ‫݌‬ሻ ൌ ‫ݎ݌‬ሺ‫ݔ‬ ൐ ܵሻ ൌ ∑ ‫ܥ‬௫ ௌା்ିଵௌା்ିଵ ௫ୀௌ ‫݌‬௫ ‫ݍ‬ௌା்ି௫ିଵ Where; ܵመ ൌ 3 , ܶ෠ ൌ 27 The hypothesis tested is; ‫ܪ‬௢: ‫݌‬ ~ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ ‫ܪ‬ଵ: ‫݌‬ ‫؂‬ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ Using Chi – square test, which depend on the following formula; ߯ଶ ൌ ෍ ሺܱ௜ െ ‫ܧ‬௜ሻଶ ‫ܧ‬௜ ௡ ௜ୀଵ The following table represents the necessary computations for goodness of fit. Table (2): The necessary computations for goodness of fit Proportion of defectives Observed frequency (ܱ௜ሻ ‫ݎ݌‬ሺ‫݌‬ଵ ൏ ‫݌‬ ൏ ‫݌‬ଶሻ ‫ܧ‬௜ ሺܱ௜ െ ‫ܧ‬௜ሻଶ ‫ܧ‬௜ 0.0 - 6 0.0231 3.234 2.3657 0.02 - 13 0.0974 13.636 0.0296 0.04 - 17 0.1662 23.268 1.6384 0.06 - 24 0.1612 22.568 0.0908 0.08 - 18 0.1500 21.00 0.4285 0.10 - 16 0.1252 17.528 0.1332 0.12 - 15 0.0827 11.578 1.0114 0.14 - 10 0.0715 10.01 0.00001 0.16 - 9 0.0435 6.09 1.2904 0.18 - 7 0.0373 5.222 0.0605 0.20 - 3 0.0252 3.528 0.0790 0.22 – 0.24 2 0.0167 2.338 0.0488 Total 140 1 140 7.1931
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 292 Comparing [ ߯௖௔௟ ଶ ൌ 7.1931ሿ with ሾ߯௧௔௕ሺ଴.଴ହ,଼ሻ ଶ ൌ 15.507ሿ, we find that, ߯௖௔௟ ଶ ൏ ߯௧௔௕ ଶ , we accept the hypothesis that, ‫݌‬௜ ~ ‫ܽݐ݁ܤ‬ distribution with parameters ሺS,Tሻ. Then the prior distribution of percentage of defective is Beta with ܵመ ൌ 3 , ܶ෠ ൌ 27. After we know that the prior distribution of percentage of defective ሾ݂ሺ‫݌‬ሻሿ is Beta – prior with the estimated parameters ሺܵመ, ܶ෠ ሻ we also obtain the parameters of quality control cost for each produced units, which are; ‫ܣ‬ଵ ൌ 0, ܴଵ ൌ 10.000 ‫ܦܫ‬ ܵଵ ൌ 10,000 ‫ܦܫ‬ ܴଶ ൌ 28,000 ‫ܦܫ‬ ܵଶ ൌ 28,000 ‫ܦܫ‬ ‫ܣ‬ଶ ൌ 153.000 ‫ܦܫ‬ ‫ݎ݌‬ ൌ ோభି஺భ ஺మିோమ ൌ 0.08, which is the break even quality control point. According to the above parameters we find the parameters of single Bayesian sampling inspection plan from; ݊ଵ ‫כ‬ ൌ ߣଵ√ܰ ൅ ߣଶ ‫ܥ‬ଵ ‫כ‬ ൌ ݊ଵ ‫כ‬ ‫ݎ݌‬ ൅ ‫ܤ‬ଵ ‫ܤ‬ଵ ൌ ܶ෠‫ݎ݌‬ െ ܵመ‫ݎݍ‬ െ ଵ ଶ ‫݌‬ ൌ ௣ ௣௥ ൌ 1.2 ‫ݎ݌‬ ൌ 0.08 And; ߣଵ ଶ ൌ ௣ೝ ೄ ௤ೝ ೅ሺ஺మିோమሻ ଶ஻ሺௌ,்ሻሺ௞௦ି௞௠ሻ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬ ݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ ߣଶ is a constant obtained according to the prior distribution of percentage of defective as; ߣଶ ൌ ሾ3ሺܵ ൅ ܶሻଶ െ 11ሺܵ ൅ ܶሻ െ 2 െ ሺ3ܵ െ 1ሻܵ|‫ݎ݌‬ െ ሺ3ܶ െ 1ሻܶ|‫ݎݍ‬ െ 1| ‫ݎݍݎ݌‬ሿ 12⁄ And ݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ Since ܵመ ൌ 3 , ܶ෠ ൌ 27 ߣଵ ൌ 4.663, ߣଶ ൌ െ27, ܰ ൌ 200 The parameters of single Bayesian sampling inspection plan are; ݊‫כ‬ ൌ ߣଵ√ܰ ൅ ߣଶ ൌ 39 ‫ܥ‬‫כ‬ ൌ ݊‫כ‬ ‫ݎ݌‬ ൅ ‫ܤ‬ଵ ൌ 2 ‫׵‬ parameters of single Bayesian sampling plan is; ሺ݊‫כ‬ ൌ 39, ‫ܥ‬‫כ‬ ൌ 2 ሻ, for different values of ሾ ‫݌‬ ൌ 0.07ሺ0.05ሻ0.09ሿ for ሺܰ ൌ 200, 225, 250,275,300ሻ, the following table summarize single sampling plans.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 293 Table (3): Single Bayesian sampling inspection plans P 0.07 0.075 0.08 0.085 0.09 N ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ 150 10 0 71.6 15 0 80.16 23 1 102.4 17 0 70.9 20 1 72.1 175 13 0 81.8 19 1 94.37 27 1 116.0 22 1 87.9 23 1 88.9 200 17 0 95.4 23 1 107.9 30 2 126.2 27 1 104.9 27 1 105.7 225 20 0 105.65 26 1 118.2 35 2 143.2 32 2 120.3 35 2 122.3 250 23 1 115.82 29 1 128.4 38 2 153.6 36 2 135.6 38 3 137.1 275 26 1 126.02 32 2 138.6 41 3 163. 6 40 2 149. 2 41 3 150.2 300 29 1 135.22 35 2 148.8 45 3 177.2 43 2 159.4 45 3 161.4 After we find the single Bayesian sampling inspection plans, from minimizing (ܴଵ), then minimizing (ܴଶሻ gives the double Bayesian sampling inspection plans, which works on minimizing total expected quality control cost, derived in equation (10) and then transformed to standard cost function (ܴ2) in equation (14). Table (4): Double Bayesian sampling inspection plans P 0.07 0.075 0.08 0.085 0.09 N ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ 150 7 1 1.00 17 1 0.996 20 2 1.008 18 2 0.996 20 2 0.94 175 13 2 0.9 21 2 0.986 25 2 1.06 21 2 0.986 22 2 0.96 200 16 1 0.88 23 2 0.97 28 2 0.99 26 2 0.977 25 3 0.97 225 19 2 0.85 25 2 0.85 32 2 0.97 30 3 0.967 30 3 0.99 250 20 2 0.80 29 2 0.83 36 2 0.88 32 3 0.933 39 3 1.04 275 23 1 0.75 31 3 0.79 40 3 0.86 36 3 0.942 40 3 1.65 300 30 1 0.74 36 3 0.75 42 3 0.85 38 4 0.931 45 4 1.85 From the data we find the mean of percentage of defective is ‫݌‬ ൌ 0.09785, ‫ݎ݌‬ ൌ 0.08, ܰ ൌ 200. The single Bayesian sampling inspection plan is found ሺ݊ଵ ‫כ‬ ൌ 39, ‫ܥ‬ଵ ‫כ‬ ൌ 2ሻ, which defined as drawing at random from lot (ܰ ൌ 200ሻ, a sample of size ሺ݊ଵ ‫כ‬ ൌ 39ሻ, and testing it's units, if the number of defective is found (‫ܥ‬ଵ ‫כ‬ ൌ 2ሻ, then accept from first sample (‫ݔ‬ଵ ൑ 2ሻ but if ሺ‫ݔ‬ଵ ൒ 2) then reject from first sample and also stop, but according to the model, we find the single Bayesian sampling inspection plans (݊ଵ, ‫ܥ‬ଵ, ܴଵሻ as tabulated in table (5), then the second Bayesian sampling inspection plans are found and put in table (4). CONCLUSION 1- Finding plan which minimize the linear combination of two expected cost function for first sampling and second sampling required a procedure carried out by means of difference equation, since the cost function is of discrete type, and the minimization is done in two stages to find (ܴଵሻ and ሺܴଶ). 2- We try also to find a family of optimum plans through maximizing the OC curve, but it depend on probability of sampling under double sampling plans, and do not consider the parameters of cost function. 3- The model was built under Beta – Binomial distribution since the distribution of process is Binomial (݊, ‫݌‬ሻ, and ሺ‫)݌‬ is random variable have [݂ሺ‫݌‬ሻሿ, and it found to be Beta (ܵ, ܶ). 4- The parameters of [݂ሺ‫݌‬ሻ], (ܵ, ܶሻ are estimated by moments method and it can be estimated also by any other statistical method.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 294 5- The decision to accept or reject obtained from second sample is more effective than the decision just obtained from first sampling especially for expensive units. 6- The producer risk for the studied company was (‫ܤ‬ ൌ 0.10), but it is found greater than this number, because of bad station of production in Iraq, but in spite of this the company work to satisfy the required quality control level. 7- The percentage of defectives in 140 lot are varied from to lot, so the distribution of these percentage is considered, and it is found to be Beta prior with estimated parameters (ܵመ, ܶ෠), therefore the model is build for Bayesian sampling plan. REFERENCES [1] Baklizi, A. (2003), "Acceptance sampling plans based on truncated life tests in the Pareto distribution of second kind", Advances and Applications in Statistics, Vol. 3, 33- 48. [2] Balakrishnan, N., Leiva, V. and Lopez, J. (2007), " acceptance sampling plans from truncated life test based on generalized Birnbaum – Saunders distribution", Communications in Statistics – Simulation and Computation, Vol. 36, 643 – 656. [3] Epstein, B. (1954), "Truncated life test in the exponential case", Annals of mathematical Statistics, Vol. 25, 555 – 564. [4] Goode, H. P. and Kao, J.H.K. (1961), "Sampling plans based on the Weibull distribution", Proceeding of the seventh national symposium on reliability and quality control, Philadelphia, 24 – 40. [5] Guenther, W.C. (1977), "Sampling inspection in statistical quality control", Charles Grifin and company LTD. [6] Gupta, S. S. and Groll, P. A. (1961), "Gamma distribution in acceptance sampling based on life tests", Journal of the American statistical Association, Vol. 56, 942 – 970. [7] Gupta, S. S. (1962), "life test plans for normal and log – normal distributions", Technometrics, Vol. 4, 151 – 160. [8] Hald, A. (1962), "some limit theorems for the Dodge – Romige LTPD single sampling inspection plans", Technometrics, Vol. 4, 497 – 513. [9]Hald, A. (1968), "Bayesian single sampling attribute plans for continuous prior distributions", Technometrics, Vol. 10, 667 – 683. [10] Nilesh Parihar and Dr. V. S. Chouhan, “Extraction of Qrs Complexes using Automated Bayesian Regularization Neural Network”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 37 - 42, ISSN Print: 0976-6480, ISSN Online: 0976-6499.

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Building a model for expected cost function to obtain double

  • 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 282 BUILDING A MODEL FOR EXPECTED COST FUNCTION TO OBTAIN DOUBLE BAYESIAN SAMPLING INSPECTION 1 Dr. Dhwyia Salman Hassan, 2 Nashaat Jaisam Al-Anber, 3 Dr.Suaad Khalaf Salman 1 Department of Statistics, College of Economic and Administration, University of Baghdad, Baghdad 2 Department of Information Technology, Technical College of Management-Baghdad, Foundation of Technical Education, Iraq 3 Department of Mathematics, College of Education for Pure Science, University of Karbala, Iraq ABSTRACT This paper deals with building a model for total expected quality control cost for double Bayesian sampling inspection plan, which consist of drawing a sample (݊ଵ) from a lot (ܰ), then checking the number of defective (‫ݔ‬ଵ) in (݊ଵ), if (‫ݔ‬ଵ ൑ ܽଵ), were (ܽଵ) is acceptance number, then accept and stop, if (‫ݔ‬ଵ ൒ ‫ݎ‬ଵ), were (‫ݎ‬ଵ) is rejection number, then take a decision to reject and stop sampling, if (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), draw a second sample (݊ଶ) and check the number of defective (‫ݔ‬ଶ) in (݊ଶ), if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൑ ܽଶ, accept and stop, if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൒ ‫ݎ‬ଶ reject and stop. The model derived assuming that the sampling process is Binomial with (݊, ‫)݌‬ and ( ‫)݌‬ is random variable, varied from lot to lot and have prior distribution ݂ሺ‫݌‬ሻ, here it is found Beta prior with parameters (ܵ, ܶ).The model is applied on the set of real data obtained from some industrial companies in Iraq, were the percentage of defective of (140) lots are studied and tested, ݂ሺ‫݌‬ሻ is found Beta prior (ܵ, ܶ). These parameters are estimated by moment estimator. After checking the distribution ݂ሺ‫݌‬ሻ and all the parameters of quality control model, we work first on obtaining single Bayesian sampling inspection plan (݊ଵ, ܿଵ) from minimizing the standard regret function (ܴଵ), then we find the parameters of double Bayesian sampling inspection plan from minimizing (ܴଶ), for various values of process average of quality (‫,)݌‬ with different lot size N. Keywords: Bayesian sampling Inspection, prior distribution, average of quality, standard regret function for single sampling plan. standard regret function for double Bayesian sampling plan. INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 2 March – April 2013, pp. 282-294 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  • 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 283 1- AIM OF RESEARCH The aim of this research is to build a model for total cost of double Bayesian sampling plans, then determine the parameters of this double Bayesian sampling plans (݊ଵ, ܽଵ, ‫ݎ‬ଵ, ݊ଶ, ܽଶ, ‫ݎ‬ଶ) from minimizing the total expected cost function. 2- NOTATIONS Here we introduce all necessary notations: ݊ଵ : First sample size drawn from lot N. ‫ݔ‬ଵ : Number of defective units in sample (݊ଵ). ܽଵ : Number of accepted defective units in (݊ଵ). ‫ݎ‬ଵ : Number of rejected units in (݊ଵ). ݊ଶ : Size of second sample taken from (ܰ െ ݊ଵ). ‫ݔ‬ଶ : Number of defective units in (݊ଶ). ܺ : Number of defective units in lot (N). ܽଶ : Number of accepted defectives in (݊ଶ). ‫ݎ‬ଶ : Number of rejected units second sample (݊ଶ). ܲ : Percentage of defectives in production. ‫݌‬ : percentage of defectives in sample. ܲ௔ ሺଵሻ ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from first sample (݊ଵ). ܲ௥ ሺଵሻ ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting from first sample (݊ଵ). ܲ௔ ሺଶሻ ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from second sample (݊ଶ). ܲ௥ ሺଶሻ ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting from first sample (݊ଶ). ܲ௦ ሺଶሻ ሺ‫݌‬ሻ : Probability of continue sampling after the first sample is chosen and the value of (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), so we cannot take a decision to accept or reject according to first sample. ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample, where; ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ ܵଵ : Cost of inspecting item from first sample. ܵଶ : Cost of repairing or replacement of item in sample (݊ଵ). ܴଵ : Cost of sampling and testing unit in rejected quantity (ܰ െ ݊). ܴଶ : Cost of repairing or replacement unit in rejected quantity (ܰ െ ݊). ‫ܣ‬ଵ : Cost of accepting good units which not represents penalty cost. ‫ܣ‬ଶ : Cost of accepting defective units. 3. INTRODUCTION In rectifying inspection plans, each item produced by a process is inspected according to some plan and then according to the number of defectives units in the sample, the lot is either accepted or the lot is rejected. Gunether [1977] explained how to obtain the parameters (݊, ܿ) for single sampling plan for ordinary sampling and for Bayesian sampling, using the model of expected total quality control cost depends on cost of inspection and sampling and
  • 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 284 cost due to wrong decision, which is accepting bad lot and rejecting good one. Another researchers gives model for obtaining parameters of single sampling inspection plan (݊, ܿ) under truncated time to failure, see Epstein, B. [1962]. Goode and Kao [1960]. Gupta, S. S. [1962] who designed sampling inspection plans for normal and lognormal distribution according to pre – determined level of percentage of defectives. Also Gupta and Kunda [1999] discuss the properties of generalized exponential distributions and how to use it is constructing Bayesian sampling. Baklizi, A. [2003] introduced how to obtain acceptance sampling plans based on truncated life tests in the Pareto distribution of second kind. The research in this subject were developed when Balakrishnan, N. [2007] designed a group of inspection sampling for generalized Birnbaum – Saundres distribution. In [2012], Dhwyia S. Hassun, et, built a model for total expected quality control cost, and compared it with the model of Schmidt – Taylor [1973]. Also Hald, A. [1968] gives a model for Bayesian single sampling attribute plans for continuous prior distribution. The parameters of single sampling inspection plans (݊, ܿ) are obtained from minimizing total expected quality control cost. Now, in this paper we introduce ( a mixed – Binomial) quality control expected cost function, to obtain the parameters of double Bayesian sampling inspection plans, from minimizing total expected regret function, according to conditions of producers and consumers risk. Building Model of Total Expected Cost Function The model comprises: a- ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample; ݇‫ݏ‬ሺ‫݌‬ሻ ൌ ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ …………………………………………….. (1) ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ : Average cost of inspecting units of second sample, under sampling distribution ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ for number of defectives units in first sample (݊ଵ). ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൌ ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ……………………………………………… (2) And; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ‫ݎ݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൌ ‫ݕ‬ሻ ‫׊‬௬ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ …………………………………………….. (3) or; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ‫׊‬௫మ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ …………………………………………………. (4) If we assume that [‫ݔ‬ଷ ൌ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ], then the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is; ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ ൌ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ|‫ݔ‬ሻ‫݌‬ሺ‫ݔ‬ሻ ………………………………………………. (5) ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ∑ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ‫׊‬௫య ………………………………………………...(6) And; ‫݌‬ሺ ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ݃௡మ ሺ ‫ݔ‬ଶ|݊ଵ, ‫ݔ‬ଵሻ ……………………………………………….. (7) The average inspection cost for second sample is equal to:
  • 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 285 ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ………………………………………………. (8) In case of accepting produced lot from first sample, the average cost is; ‫ܧ‬ሼሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵሻ‫ܣ‬ଶ|‫ݔ‬ଵሽ ൌ ሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ‫ܧ‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫ܣ‬ଶ ൌ ሺܰ െ ݊ଵሻሼ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሺ‫ݔ‬ଵሻሽ ൌ ሺܰ െ ݊ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ The average of acceptance cost for second sample is; ‫ܧ‬ሼሺܰ െ ݊ଵ െ ݊ଶሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶሻ‫ܣ‬ଶ|‫ݔ‬ଵ, ‫ݔ‬ଶሽ ሺܰ െ ݊ଵ െ ݊ଶሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሿ …………………………………. (9) After explaining all notations, we can show that the lot size N, in case of double sampling can divide into; ܰ ൌ ݊ଵ ൅ ሺܰ െ ݊ଵሻቀ‫݌‬௔ ሺଵሻ ൅ ‫݌‬௥ ሺଵሻ ቁ ൅ ݊ଶ‫݌‬ௌ ሺଵሻ ൅ሺܰ െ ݊ଵ െ ݊ଶሻቀ‫݌‬௔ ሺଶሻ ൅ ‫݌‬௥ ሺଶሻ ቁ … (10) Assume (݇݉), is the smallest average cost, happened according to results of decisions of accepting and rejecting, where acceptance happened at (‫݌‬ ൑ ‫,)ݎ݌‬ (i.e ‫,ݎ݌‬ break even quality level) and rejection when (‫݌‬ ൐ ‫,)ݎ݌‬ (‫)݌‬ is random variable have [݂ሺ‫݌‬ሻ] then (݇݉) is equal to; ݇݉ ൌ ‫׬‬ ሺ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ௣௥ ଴ ‫׬‬ ሺܴଵ ൅ ܴଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ………………..... (11) Now the construction of total expected quality control cost [݇ଶሺ‫݌‬ሻ] is given by; ݇ଶሺ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔ ሺଵሻ ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ ሺଵሻ ሺ‫݌‬ሻቅ ൅݊ଶ݇‫ݏ‬ሺ‫݌‬ሻ‫݌‬ௌ ሺଶሻ ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔ ሺଶሻ ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻቅ ……….…(12) When ሺ‫݌‬ሻ is random variable have [݂ሺ‫݌‬ሻ], then the average cost is; ݇ଶ ൌ ‫׬‬ ݇ଶሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ݇ଶ ൌ ݊ଵ݇ܵ ൅ ሺܰ െ ݊ଵሻ ቐ ෍ ݃ሺ‫ݔ‬ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ௔భ ௫భୀ଴ ቑ ൅ ෍ ݃ሺ‫ݔ‬ଵሻ݇‫ݎ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ௡భ ௫భୀ௥భ ൅ ∑ ݃ሺ‫ݔ‬ଵሻൣ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ൧ ௥భିଵ ௫భୀ௔భାଵ ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቐ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ݇ܽሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ ௔మି௫భ ௫మୀ଴ ቑ ൅ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻሾ݇‫ݎ‬ሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ ௡మ ௫మୀ௔మି௫భାଵ ………………………..(13) Multiplying equation (10) by (݇݉) where ሾܰ݇݉ ൌ ݇݉ሿ and apply [ሺ݇2 െ ݇݉ሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ], we have obtained the equation of standard cost function (ܴଶ). ܴଶ ൌ ݊ଵ݀‫ݏ‬ ൅ ሺܰ െ ݊ଵሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ ቋ ൅ ቊන ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ቋ ൅݊ଶ ቄ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬ௌ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ‫׬‬ ߜሺ‫݌‬ሻ‫݌‬ௌ ሺଵሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ଴ ௣௥ ଴ ቅ ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥ ሺଶሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ න ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ ሺଶሻ ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ଵ ௣௥ ௣௥ ଴ ቋ … (14)
  • 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 286 ‫ݎ݌‬ ൌ break even quality level; ‫ݎ݌‬ ൌ ஺మିோమ ஺భିோభ Equation (14), now transformed to standard cost function in terms of conditional distribution of percentage of defectives, which is [݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ] especially these parts obtained from second sampling, where; ݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ ൌ ௣௥ሺ௔భழ௫భழ௥భ|௣ሻ௙ሺ௣ሻ ∑ ௕௪ሺ௫భ,௡భሻ ೝభషభ ೌభశభ ൌ ቄ௣ೌ ሺమሻ ሺ௣ሻା௣ೝ ሺమሻ ሺ௣ሻቅ௙ሺ௣ሻ ‫׬‬ ቄ௣ೌ ሺమሻ ሺ௣ሻା௣ೝ ሺమሻ ሺ௣ሻቅ௙ሺ௣ሻௗ௣ భ బ ……… (15) This because when [ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ] a second sample (݊ଶ) is drawn, and according to this sample either we accept or reject. The final formula for standard regret function (ܴଶ) used to obtain the parameters of second sampling plan is; ܴଶ ‫כ‬ ൌ ܴଵ ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ െ ሺܰ െ ݊ଵሻ ෍ ݃ሺ‫ݔ‬ଵሻ݀‫ݎ‬ሺ‫ݔ‬ଵሻ ௥భିଵ ௫భୀ௔భାଵ ൅ ∑ ݃ሺ‫ݔ‬ଵሻܴଵሺܽଶ െ ‫ݔ‬ଵ, ݊ଶ, ܰ െ ݊ଵ|‫ݔ‬ଵ ሻ௥భିଵ ௫భୀ௔భାଵ ……………………………….… (16) Where; ݀‫ݎ‬ሺ‫ݔ‬ଵሻ ൌ න ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ௣௥ ଴ ܴଵ ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ ൌ ௄ሺே,௡,௖ሻ ି ே௞௠ ௞௦ ି ௞௠ ………………………………… (17) Where; ‫ܭ‬ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ቂሺ‫ܣ‬ଶ െ ܴଶሻ ௌ ௌା் ‫ܨ‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻቃ ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ቂሺܴଵ ൅ ܴଶሻ ௌ ௌା் ቃ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ ቀ ௌ ௌା் ቁ , ‫݌‬ ൌ ் ௡ା் ݇ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ሺܰ െ ݊ሻሺ‫ܣ‬ଵ ൅ ܴଶ‫݌‬ሻ ൅ ሺܰ െ ݊ሻሺܴଵ െ ‫ܣ‬ଵሻ Which can be simplified to; ݇ሺܰ, ݊, ܿሻ ൌ ݊ሺ݇‫ݏ‬ െ ݇݉ሻ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ ൅ሾሺܴଵ െ ‫ܣ‬ଵሻሾ1 െ ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿሿ ൅ ܰ݇݉ ……………... (18) Therefore the equation of standard expected cost (Regret function ܴଵ) is; ܴଵሺܰ, ݊, ܿሻ ൌ ௞ሺே,௡,௖ሻ ି ே௞௠ ௞௦ ି ௞௠ Which is the first part of equation (ܴଶ ‫כ‬ ) defined in equation (16), so first of all and after determining the parameters of single Bayesian sampling plan [given in table (1)] we work on minimizing (16) to find the parameters of double Bayesian sampling inspection plan given in table (4). Equation (16) shows that the average of standard cost function for double sampling plan, is represented by standard cost function for single sampling plan and also standard conditional cost function weighted by the probability distribution of number of defective units [݃ሺ‫ݔ‬ଵሻ], from first sample, it is obtained from; ‫݌‬ሺ‫ݔ‬ଵሻ ൌ ݃௡భ ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ‫׊‬௫మ
  • 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 287 To satisfy the aim of research which is to find the parameters ሺܽଵ, ‫ݎ‬ଵ, ܽଶሻ from minimizing (ܴଶ), we apply difference equation since the distribution of number of defectives in production is of discrete type, and also we consider; ∆ ܴଶ ൌ ∆ ௞మ ஺మିோమ The differences equations according to ሺ‫ݎ‬ଵ, ܽଵ, ܽଶሻ, required to rewrite (݇ଶ) as in equation (19) follows; Since; ݇ଶ ൌ ݇ଵሺܽଵ, ݊ଵ, ܰሻሺ‫ܣ‬ଶܴଶሻ ∑ ݃ሺ‫ݔ‬ଵሻ൛݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ௥భିଵ ௫భୀ௔భାଵ ݊ଶሻ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ൟ …………………………………….. … (19) Then using; ∆ ܴଶ ൌ ∆ ௞మ ஺మିோమ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ … (20) We can find [ ∆௥భ ܴଶሺ‫ݎ‬ଵሻ] as; ∆ ௥భ ܴଶሺ‫ݎ‬ଵሻ ൌ ݃௡భ ሺ‫ݎ‬ଵሻቐ݊ଶߜቂ‫݌‬௡భ ሺ௥భሻ ቃ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃௡మ ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻቂ‫݌‬௡భା௡మ ሺ௥భ, ௫మሻ െ ‫ݎ݌‬ቃ ௔మି௥భ ௫మୀ଴ ቑ … (21) And also; ∆ ௔భ ܴଶሺܽଵሻ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଶ െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ ௔మି௫భ ௫మୀ଴ ቑ ௥భିଵ ௫భୀ௔భାଵ … (22) And also the difference of [ܴଶሺܽଵሻ] with respect to ሺܽଵሻ; ∆௔భ ܴଶሺܽଵሻ ൌ ݃௡భ ሺܽଵ ൅ 1ሻ ቄ݊ଶሺ‫ܣ‬ଶ െ ܵଵሻቂ‫݌‬௡భ ሺ௔భାଵሻ ቃ െ ሺܵଵ െ ‫ܣ‬ଵሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ ൅ ݃௡భ ሺܽଵ ൅ 1ሻሺܰ െ ݊ଵ െ ݊ଶሻ ∑ ݃௡మ ሺ‫ݔ‬ଶ|ܽଵ ൅ 1ሻቂ‫݌‬௡భା௡మ ሺ௔భାଵା ௫మሻ െ ‫ݎ݌‬ቃ ௡మ ௫మୀ௔మି௔భ ቅ …………. … (23) From ∆௔మ ܴଶሺܽଶሻ we can find the value of parameter (ܽଶ) for given (݊ଵ, ݊ଶ ) which satisfy the inequality (24) follows. ‫̂݌‬௡భା௡మ ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫̂݌‬௡భା௡మ ሺܽොଶ ൅ 1ሻ …………………………………….. (24) Then; ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 ‫݄݊݁ݓ‬ ݇‫ݏ‬ ൌ ݇‫ݎ‬ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 ݂‫ݎ݋‬ ݇‫ݏ‬ ൐ ݇‫ݎ‬
  • 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 288 And when [ܽଶ ൌ ܽଵ], the second term of ∆௔భ ܴଶሺܽଵሻ is positive, so we assume that [ܽଵ ൌ ܽଵ ‫כ‬ ] is the value at which the first term of ∆௔మ ܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ ‫כ‬ሻ ൐ 0 and when ݇‫ݏ‬ ൌ ݇‫ݎ‬ ฺ ሺ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1ሻ, and we see that the average cost of inspection equal to average cost of reject, therefore we need only to find the parameters (ܽଵ, ܽଶ ) and from ∆௔భ ܴଶሺܽଵሻ we obtain ‫݌‬௡భ ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to ‫ݎ‬ଵ ൑ ݊ଵ ൅ 1. After completing model for double Bayesian sampling inspection plan, this model is applied to distribution of Beta – Binomial. From ∆௔మ ܴଶሺܽଶሻ we can find (ܽଶ) that satisfy the inequality (24). ‫݌‬௡భା௡మ ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫݌‬௡భା௡మ ሺܽොଶ ൅ 1ሻ ………………………………………. (25) When (ܽଵ ൌ ܽଶ), the second term of ∆௔భ ܴଶሺܽଵሻ is positive, so we assume that (ܽଵ ൌ ܽଵ ‫כ‬ ) is the value of which the first term of ∆௔మ ܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ ‫כ‬ ሻ ൐ 0. When; ݇‫ݏ‬ ൌ ݇‫ݎ‬ Then; ∆ ܴଶሺ‫ݎ‬ଵሻ ൑ 0 ฺ ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 And for ݇‫ݏ‬ ൐ ݇‫ݎ‬ ฺ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 And when ݇‫ݏ‬ ൌ ݇‫,ݎ‬ we saw that average cost of inspection equal average cost of reject, therefore we need only to find the parameters (ܽଵ, ܽଶ), and from equation ∆௔భ ܴଶሺܽଵሻ, we obtain ‫݌‬௡భ ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to (‫ݎ‬ଵ ൑ ݊ଵ ൅ 1). 4 - APPLICATION OF PROPOSED MODEL After the total cost function are built, it is necessary to apply this model for sampling distribution to show how the parameters of double sampling plans are obtained, to take a decision for accept or reject, assume (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) are three independent random variables follow Binomial distribution as: ‫ݔ‬ଵ~ܾሺ݊ଵ, ‫݌‬ሻ ‫ݔ‬ଶ~ܾሺ݊ଶ, ‫݌‬ሻ ‫ݔ‬ଷ~ܾሺ݊ଷ, ‫݌‬ሻ And (‫)݌‬ is random variable varied from lot to lot, and it's prior distribution [݂ሺ‫݌‬ሻ] is assumed here ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ, (Beta prior), so the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is; ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ሻ ൌ න ‫ܥ‬௫భ ௡భ ଵ ଴ ‫݌‬௫భ‫ݍ‬௡భ ି ௫భ‫ܥ‬௫మ ௡మ ‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫ܥ‬௫య ௡య ‫݌‬௫య‫ݍ‬௡య ି ௫య 1 ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ି ଵ ݀‫݌‬ ൌ ஼ೣభ ೙భ஼ೣమ ೙మ஼ೣయ ೙య ஻௘௧௔ ሺ௦,௧ሻ ‫׬‬ ‫݌‬௫భା௫మା௫యାௌିଵଵ ଴ ‫ݍ‬௡భ ା௡మ ା௡య ି ௫భି௫మି௫యା௧ିଵ ݀‫݌‬ ൌ ஼ೣభ ೙భ஼ೣమ ೙మ஼ೣయ ೙య ஻௘௧௔ ሺ௦,௧ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ሻ …. (26) Also we can find the posterior distribution ݂ሺ‫ݔ|݌‬ଵሻ and ݂ሺ‫ݔ|݌‬ଵ, ‫ݔ‬ଶሻ ൌ ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ, and; ‫݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ ݀‫݌‬ ଵ ଴ Since; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ݆‫ݐ݊݅݋‬ ሺ‫,݌‬ ‫ݔ‬ଵሻ ݃ሺ‫ݔ‬ଵሻ ݃ሺ‫ݔ‬ଵሻ ൌ න ‫ܥ‬௫భ ௡భ ଵ ଴ ‫݌‬௫భ‫ݍ‬௡భ ି ௫భ 1 ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ି ଵ ݀‫݌‬
  • 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 289 ݃ሺ‫ݔ‬ଵሻ ൌ ‫ܥ‬௫భ ௡భ ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ܵ, ݊ଵ െ ‫ݔ‬ଵ ൅ ‫ݐ‬ሻ And from equation below which is Beta distribution; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ௚ሺ௣,௫భ ሻ ௚ሺ௫భሻ The conditional distribution of (‫)݌‬ given (‫ݔ‬ଵ) is; ݂ሺ‫ݔ|݌‬ଵሻ ൌ ଵ ஻௘௧௔ሺ௫భାௌ,௡భ ି ௫భା௧ሻ ‫݌‬௫భାௌିଵ ‫ݍ‬௡భ ି ௫భା௧ିଵ …………………….. (27) And; ‫ܧ‬ሺ‫ݔ|݌‬ଵሻ ൌ ‫ݔ‬ଵ ൅ ܵ ݊ଵ ൅ ܵ ൅ ‫ݐ‬ Also we can find ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ; ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ ሻ ଵ ଴ ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ………………………………….. (28) ൌ ଵ ௞ ‫׬‬ ‫ܥ‬௫మ ௡మଵ ଴ ‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫݌‬௫భାௌିଵ ‫ݍ‬௡భ ି ௫భା௧ିଵ ݀‫݌‬ ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ଵ ௞ ‫ܥ‬௫మ ௡మ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ൅ ‫ݐ‬ሻ ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ௌା௫భ ௌା௫భା௫మ ஼ೣమ ೙మ஼ೄశೣభ ೙భశ೟శೄషభ ஼ೄశೣభశೣమ ೙భశ೙మశ೟శೄషభ ………………………………………. (29) Which is [݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ] Assume ‫ݔ‬ଷ ൌ ܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ‫ݔ‬ଷ~ ‫݈ܽ݅݉݋݊݅ܤ‬ ሺ݊ଷ, ‫݌‬ሻ Then the conditional distribution of (‫ݔ‬ଷ) given (‫ݔ‬ଵ, ‫ݔ‬ଶ) is; ‫ݎ݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ ଵ ଴ ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ݀‫݌‬ ……………………….. (30) To solve equation (30), we need to find ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ; Let ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ‫׶‬ ‫ݔ‬ଵ~ ܾሺ݊ଵ, ‫݌‬ሻ ‫ݔ‬ଶ~ ܾሺ݊ଶ, ‫݌‬ሻ And are independent, then; ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ~ ‫ܤ‬ሺ݊ଵ ൅ ݊ଶ, ‫݌‬ሻ ݃ሺ‫,݌‬ ‫ݕ‬ሻ ൌ ‫ܥ‬௬ ௡భା௡మ ‫݌‬௬ ‫ݍ‬௡భା௡మି௬ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ ‫݌‬ௌିଵ ‫ݍ‬௧ିଵ ‫׵‬ ݃ሺ‫݌|ݕ‬ሻ ൌ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ න ‫ܥ‬௬ ௡భା௡మ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మି௬ା௧ିଵ ݀‫݌‬ ଵ ଴ ݃ሺ‫݌|ݕ‬ሻ ൌ 1 ‫ܤ‬ሺܵ, ‫ݐ‬ሻ ‫ܥ‬௬ ௡భା௡మ ‫ܽݐ݁ܤ‬ ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ ‫׵‬ ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൌ ݆‫ݐ݊݅݋‬ ݉ܽ‫݈ܽ݊݅݃ݎ‬ ൌ భ ಳሺೄ,೟ሻ ஼೤ ೙భశ೙మ௣೤శೄషభ௤೙భశ೙మష೤శ೟షభ భ ಳሺೄ,೟ሻ ஼೤ ೙భశ೙మ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ ൌ ଵ ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మି௬ା௧ିଵ 0 ൑ ‫݌‬ ൑ 1 …………. (31) Then ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ is distributed as Beta distribution with parameters (‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫.)ݐ‬ This distribution is necessary to find the distribution of number of defectives ሺ‫ݔ‬ଷ) after drawing the first and second sample, i.e to find the distribution of ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ.
  • 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 290 ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ න ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶ ൌ ‫ݕ‬ሻ݀‫݌‬ ଵ ଴ ൌ න ‫ܥ‬௫య ௡య ଵ ଴ ‫݌‬௫య‫ݍ‬௡యି ௫య 1 ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ ‫݌‬௬ାௌିଵ ‫ݍ‬௡భା௡మା௧ି௬ିଵ ݀‫݌‬ ൌ ‫ܥ‬௫య ௡య ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ න ‫݌‬௫యା௬ାௌିଵ ଵ ଴ ‫ݍ‬௡భା௡మା௡యି௫యା௧ି௬ିଵ ݀‫݌‬ ൌ ‫ܥ‬௫య ௡య ‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ ൌ ‫ܥ‬௫య ௡య Γሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ Γ ሺ‫ݕ‬ ൅ ܵሻΓ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ ൅ ܵሻ Γ ሺܵ ൅ ‫ݐ‬ ൅ ݊ଵ ൅ ݊ଶ ൅ ݊ଷሻ ൌ ஼ೣయ ೙యሺ௡భା௡మା௧ାୗିଵሻ! ሺ௬ାௌିଵሻ!ሺ௡భା௡మା௧ି୷ିଵሻ! ൈ ሺ௫యା௬ାௌିଵሻ!ሺ௡భା௡మା௡యି௫యି௬ା௧ିଵሻ! ሺௌା௧ା௡భା௡మା௡యିଵሻ! ൌ ஼ೣయ ೙య஼೤శೄ ೙భశ೙మశ೟శೄషభ ሺ௫యାௌା௬ሻ ஼ೣయశೄశ೤ ೙భశ೙మశ೙యశ೟శೄషభ ሺ௬ାௌሻ Which is mixed hypergeometric distribution.We can obtain the conditional mean of ሺ‫ݔ‬ଷሻ given ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ; ‫ܧ‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ݊ଷ‫݌‬௡భା௡మ ሺ௫భ,௫మሻ Where, ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫݌‬௡భା௡మ ሺ௫భା௫మሻ 5- CASE STUDY The following data represents the percentage of defectives of (140) lots from some industrial companies in Iraq, these percentages are grouped in frequency table as follows; Table (1): The frequency table for percentage of defectives of 140 lots Percentage of defectives Observed frequency ሺ݂௜ሻ Class mark ‫݌‬௜ ‫݌‬௜݂௜ ‫݌‬௜ ଶ ݂௜ 0.0 - 6 0.01 0.06 0.0116 0.02 - 13 0.03 0.39 0.0117 0.04 - 17 0.05 0.85 0.0425 0.06 - 24 0.07 1.68 0.1176 0.08 - 18 0.09 1.62 0.1456 0.10 - 16 0.11 1.76 0.1936 0.12 - 15 0.13 1.95 0.2535 0.14 - 10 0.15 1.50 0.2250 0.16 - 9 0.17 1.53 0.2601 0.18 - 7 0.19 0.33 0.2527 0.20 - 3 0.21 0.63 0.1323 0.22 – 0.24 2 0.23 0.46 0.1058 Total 140 13.699 1.7413
  • 10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 291 Then we draw the histogram and frequency curve, the curve indicate it is Beta distribution with parameters (S, T), which are estimated by moments method using the equations; ܵመ ൌ ‫ݔ‬௣ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ ଶ ሿหܵ௣ ଶ ܶ෠ ൌ ‫ݔ‬௤ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ ଶ ሿหܵ௣ ଶ Where; ‫ݔ‬௣ ൌ ∑ ௣೔௙೔ ∑ ௙೔ ൌ 0.09785 ܵ௣ ଶ ൌ ∑ ௣೔ మ ௙೔ ∑ ௙೔ െ ‫݌‬ ଶ ൌ 0.00286 Then; ܵመ௠௢௠ ൌ 2.9 ؆ 3 ܶ෠௠௢௠ ൌ 26.9 ؆ 27 Then the distribution of (‫݌‬௜ሻ is tested using ( ߯ଶ ) test, where the cumulative probabilities are computed from; ‫ܨ‬ሺ‫݌‬ሻ ൌ ‫׬‬ ଵ ஻ሺௌ,ఛሻ ௣ ଴ ‫ݑ‬ௌିଵ ሺ1 െ ‫ݑ‬ሻ்ିଵ ݀‫ݑ‬ ൌ ‫ܧ‬ሺܵ, ܵ ൅ ܶ െ 1, ‫݌‬ሻ ൌ ‫ݎ݌‬ሺ‫ݔ‬ ൐ ܵሻ ൌ ∑ ‫ܥ‬௫ ௌା்ିଵௌା்ିଵ ௫ୀௌ ‫݌‬௫ ‫ݍ‬ௌା்ି௫ିଵ Where; ܵመ ൌ 3 , ܶ෠ ൌ 27 The hypothesis tested is; ‫ܪ‬௢: ‫݌‬ ~ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ ‫ܪ‬ଵ: ‫݌‬ ‫؂‬ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ Using Chi – square test, which depend on the following formula; ߯ଶ ൌ ෍ ሺܱ௜ െ ‫ܧ‬௜ሻଶ ‫ܧ‬௜ ௡ ௜ୀଵ The following table represents the necessary computations for goodness of fit. Table (2): The necessary computations for goodness of fit Proportion of defectives Observed frequency (ܱ௜ሻ ‫ݎ݌‬ሺ‫݌‬ଵ ൏ ‫݌‬ ൏ ‫݌‬ଶሻ ‫ܧ‬௜ ሺܱ௜ െ ‫ܧ‬௜ሻଶ ‫ܧ‬௜ 0.0 - 6 0.0231 3.234 2.3657 0.02 - 13 0.0974 13.636 0.0296 0.04 - 17 0.1662 23.268 1.6384 0.06 - 24 0.1612 22.568 0.0908 0.08 - 18 0.1500 21.00 0.4285 0.10 - 16 0.1252 17.528 0.1332 0.12 - 15 0.0827 11.578 1.0114 0.14 - 10 0.0715 10.01 0.00001 0.16 - 9 0.0435 6.09 1.2904 0.18 - 7 0.0373 5.222 0.0605 0.20 - 3 0.0252 3.528 0.0790 0.22 – 0.24 2 0.0167 2.338 0.0488 Total 140 1 140 7.1931
  • 11. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 292 Comparing [ ߯௖௔௟ ଶ ൌ 7.1931ሿ with ሾ߯௧௔௕ሺ଴.଴ହ,଼ሻ ଶ ൌ 15.507ሿ, we find that, ߯௖௔௟ ଶ ൏ ߯௧௔௕ ଶ , we accept the hypothesis that, ‫݌‬௜ ~ ‫ܽݐ݁ܤ‬ distribution with parameters ሺS,Tሻ. Then the prior distribution of percentage of defective is Beta with ܵመ ൌ 3 , ܶ෠ ൌ 27. After we know that the prior distribution of percentage of defective ሾ݂ሺ‫݌‬ሻሿ is Beta – prior with the estimated parameters ሺܵመ, ܶ෠ ሻ we also obtain the parameters of quality control cost for each produced units, which are; ‫ܣ‬ଵ ൌ 0, ܴଵ ൌ 10.000 ‫ܦܫ‬ ܵଵ ൌ 10,000 ‫ܦܫ‬ ܴଶ ൌ 28,000 ‫ܦܫ‬ ܵଶ ൌ 28,000 ‫ܦܫ‬ ‫ܣ‬ଶ ൌ 153.000 ‫ܦܫ‬ ‫ݎ݌‬ ൌ ோభି஺భ ஺మିோమ ൌ 0.08, which is the break even quality control point. According to the above parameters we find the parameters of single Bayesian sampling inspection plan from; ݊ଵ ‫כ‬ ൌ ߣଵ√ܰ ൅ ߣଶ ‫ܥ‬ଵ ‫כ‬ ൌ ݊ଵ ‫כ‬ ‫ݎ݌‬ ൅ ‫ܤ‬ଵ ‫ܤ‬ଵ ൌ ܶ෠‫ݎ݌‬ െ ܵመ‫ݎݍ‬ െ ଵ ଶ ‫݌‬ ൌ ௣ ௣௥ ൌ 1.2 ‫ݎ݌‬ ൌ 0.08 And; ߣଵ ଶ ൌ ௣ೝ ೄ ௤ೝ ೅ሺ஺మିோమሻ ଶ஻ሺௌ,்ሻሺ௞௦ି௞௠ሻ ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬ ݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ ߣଶ is a constant obtained according to the prior distribution of percentage of defective as; ߣଶ ൌ ሾ3ሺܵ ൅ ܶሻଶ െ 11ሺܵ ൅ ܶሻ െ 2 െ ሺ3ܵ െ 1ሻܵ|‫ݎ݌‬ െ ሺ3ܶ െ 1ሻܶ|‫ݎݍ‬ െ 1| ‫ݎݍݎ݌‬ሿ 12⁄ And ݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ௣௥ ଴ Since ܵመ ൌ 3 , ܶ෠ ൌ 27 ߣଵ ൌ 4.663, ߣଶ ൌ െ27, ܰ ൌ 200 The parameters of single Bayesian sampling inspection plan are; ݊‫כ‬ ൌ ߣଵ√ܰ ൅ ߣଶ ൌ 39 ‫ܥ‬‫כ‬ ൌ ݊‫כ‬ ‫ݎ݌‬ ൅ ‫ܤ‬ଵ ൌ 2 ‫׵‬ parameters of single Bayesian sampling plan is; ሺ݊‫כ‬ ൌ 39, ‫ܥ‬‫כ‬ ൌ 2 ሻ, for different values of ሾ ‫݌‬ ൌ 0.07ሺ0.05ሻ0.09ሿ for ሺܰ ൌ 200, 225, 250,275,300ሻ, the following table summarize single sampling plans.
  • 12. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 293 Table (3): Single Bayesian sampling inspection plans P 0.07 0.075 0.08 0.085 0.09 N ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ 150 10 0 71.6 15 0 80.16 23 1 102.4 17 0 70.9 20 1 72.1 175 13 0 81.8 19 1 94.37 27 1 116.0 22 1 87.9 23 1 88.9 200 17 0 95.4 23 1 107.9 30 2 126.2 27 1 104.9 27 1 105.7 225 20 0 105.65 26 1 118.2 35 2 143.2 32 2 120.3 35 2 122.3 250 23 1 115.82 29 1 128.4 38 2 153.6 36 2 135.6 38 3 137.1 275 26 1 126.02 32 2 138.6 41 3 163. 6 40 2 149. 2 41 3 150.2 300 29 1 135.22 35 2 148.8 45 3 177.2 43 2 159.4 45 3 161.4 After we find the single Bayesian sampling inspection plans, from minimizing (ܴଵ), then minimizing (ܴଶሻ gives the double Bayesian sampling inspection plans, which works on minimizing total expected quality control cost, derived in equation (10) and then transformed to standard cost function (ܴ2) in equation (14). Table (4): Double Bayesian sampling inspection plans P 0.07 0.075 0.08 0.085 0.09 N ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ 150 7 1 1.00 17 1 0.996 20 2 1.008 18 2 0.996 20 2 0.94 175 13 2 0.9 21 2 0.986 25 2 1.06 21 2 0.986 22 2 0.96 200 16 1 0.88 23 2 0.97 28 2 0.99 26 2 0.977 25 3 0.97 225 19 2 0.85 25 2 0.85 32 2 0.97 30 3 0.967 30 3 0.99 250 20 2 0.80 29 2 0.83 36 2 0.88 32 3 0.933 39 3 1.04 275 23 1 0.75 31 3 0.79 40 3 0.86 36 3 0.942 40 3 1.65 300 30 1 0.74 36 3 0.75 42 3 0.85 38 4 0.931 45 4 1.85 From the data we find the mean of percentage of defective is ‫݌‬ ൌ 0.09785, ‫ݎ݌‬ ൌ 0.08, ܰ ൌ 200. The single Bayesian sampling inspection plan is found ሺ݊ଵ ‫כ‬ ൌ 39, ‫ܥ‬ଵ ‫כ‬ ൌ 2ሻ, which defined as drawing at random from lot (ܰ ൌ 200ሻ, a sample of size ሺ݊ଵ ‫כ‬ ൌ 39ሻ, and testing it's units, if the number of defective is found (‫ܥ‬ଵ ‫כ‬ ൌ 2ሻ, then accept from first sample (‫ݔ‬ଵ ൑ 2ሻ but if ሺ‫ݔ‬ଵ ൒ 2) then reject from first sample and also stop, but according to the model, we find the single Bayesian sampling inspection plans (݊ଵ, ‫ܥ‬ଵ, ܴଵሻ as tabulated in table (5), then the second Bayesian sampling inspection plans are found and put in table (4). CONCLUSION 1- Finding plan which minimize the linear combination of two expected cost function for first sampling and second sampling required a procedure carried out by means of difference equation, since the cost function is of discrete type, and the minimization is done in two stages to find (ܴଵሻ and ሺܴଶ). 2- We try also to find a family of optimum plans through maximizing the OC curve, but it depend on probability of sampling under double sampling plans, and do not consider the parameters of cost function. 3- The model was built under Beta – Binomial distribution since the distribution of process is Binomial (݊, ‫݌‬ሻ, and ሺ‫)݌‬ is random variable have [݂ሺ‫݌‬ሻሿ, and it found to be Beta (ܵ, ܶ). 4- The parameters of [݂ሺ‫݌‬ሻ], (ܵ, ܶሻ are estimated by moments method and it can be estimated also by any other statistical method.
  • 13. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME 294 5- The decision to accept or reject obtained from second sample is more effective than the decision just obtained from first sampling especially for expensive units. 6- The producer risk for the studied company was (‫ܤ‬ ൌ 0.10), but it is found greater than this number, because of bad station of production in Iraq, but in spite of this the company work to satisfy the required quality control level. 7- The percentage of defectives in 140 lot are varied from to lot, so the distribution of these percentage is considered, and it is found to be Beta prior with estimated parameters (ܵመ, ܶ෠), therefore the model is build for Bayesian sampling plan. REFERENCES [1] Baklizi, A. (2003), "Acceptance sampling plans based on truncated life tests in the Pareto distribution of second kind", Advances and Applications in Statistics, Vol. 3, 33- 48. [2] Balakrishnan, N., Leiva, V. and Lopez, J. (2007), " acceptance sampling plans from truncated life test based on generalized Birnbaum – Saunders distribution", Communications in Statistics – Simulation and Computation, Vol. 36, 643 – 656. [3] Epstein, B. (1954), "Truncated life test in the exponential case", Annals of mathematical Statistics, Vol. 25, 555 – 564. [4] Goode, H. P. and Kao, J.H.K. (1961), "Sampling plans based on the Weibull distribution", Proceeding of the seventh national symposium on reliability and quality control, Philadelphia, 24 – 40. [5] Guenther, W.C. (1977), "Sampling inspection in statistical quality control", Charles Grifin and company LTD. [6] Gupta, S. S. and Groll, P. A. (1961), "Gamma distribution in acceptance sampling based on life tests", Journal of the American statistical Association, Vol. 56, 942 – 970. [7] Gupta, S. S. (1962), "life test plans for normal and log – normal distributions", Technometrics, Vol. 4, 151 – 160. [8] Hald, A. (1962), "some limit theorems for the Dodge – Romige LTPD single sampling inspection plans", Technometrics, Vol. 4, 497 – 513. [9]Hald, A. (1968), "Bayesian single sampling attribute plans for continuous prior distributions", Technometrics, Vol. 10, 667 – 683. [10] Nilesh Parihar and Dr. V. S. Chouhan, “Extraction of Qrs Complexes using Automated Bayesian Regularization Neural Network”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 37 - 42, ISSN Print: 0976-6480, ISSN Online: 0976-6499.