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# Building a model for expected cost function to obtain double

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### Building a model for expected cost function to obtain double

1. 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), © IAEME282BUILDING A MODEL FOR EXPECTED COST FUNCTION TOOBTAIN DOUBLE BAYESIAN SAMPLING INSPECTION1Dr. Dhwyia Salman Hassan, 2Nashaat Jaisam Al-Anber, 3Dr.Suaad Khalaf Salman1Department of Statistics, College of Economic and Administration, University of Baghdad,Baghdad2Department of Information Technology, Technical College of Management-Baghdad,Foundation of Technical Education, Iraq3Department of Mathematics, College of Education for Pure Science, University of Karbala,IraqABSTRACTThis paper deals with building a model for total expected quality control cost fordouble 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 acceptancenumber, then accept and stop, if (‫ݔ‬ଵ ൒ ‫ݎ‬ଵ), were (‫ݎ‬ଵ) is rejection number, then take a decisionto reject and stop sampling, if (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), draw a second sample (݊ଶ) and check thenumber of defective (‫ݔ‬ଶ) in (݊ଶ), if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൑ ܽଶ, accept and stop, if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൒ ‫ݎ‬ଶ rejectand 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 isfound Beta prior with parameters (ܵ, ܶ).The model is applied on the set of real data obtainedfrom some industrial companies in Iraq, were the percentage of defective of (140) lots arestudied and tested, ݂ሺ‫݌‬ሻ is found Beta prior (ܵ, ܶ). These parameters are estimated bymoment estimator. After checking the distribution ݂ሺ‫݌‬ሻ and all the parameters of qualitycontrol 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 doubleBayesian sampling inspection plan from minimizing (ܴଶ), for various values of processaverage of quality (‫,)݌‬ with different lot size N.Keywords: Bayesian sampling Inspection, prior distribution, average of quality, standardregret function for single sampling plan. standard regret function for double Bayesiansampling plan.INTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING 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.aspJournal Impact Factor (2013): 5.8376 (Calculated by GISI)www.jifactor.comIJARET© I A E M E
2. 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), © IAEME2831- AIM OF RESEARCHThe aim of this research is to build a model for total cost of double Bayesian samplingplans, then determine the parameters of this double Bayesian sampling plans(݊ଵ, ܽଵ, ‫ݎ‬ଵ, ݊ଶ, ܽଶ, ‫ݎ‬ଶ) from minimizing the total expected cost function.2- NOTATIONSHere 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.ܲ௔ሺଵሻሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision fromfirst sample (݊ଵ).ܲ௥ሺଵሻሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejectingfrom first sample (݊ଵ).ܲ௔ሺଶሻሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision fromsecond sample (݊ଶ).ܲ௥ሺଶሻሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejectingfrom first sample (݊ଶ).ܲ௦ሺଶሻሺ‫݌‬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݌‬ሻܵଵ : 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. INTRODUCTIONIn rectifying inspection plans, each item produced by a process is inspected accordingto some plan and then according to the number of defectives units in the sample, the lot iseither 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 themodel of expected total quality control cost depends on cost of inspection and sampling and
3. 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), © IAEME284cost due to wrong decision, which is accepting bad lot and rejecting good one. Anotherresearchers 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 distributionaccording 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 isconstructing Bayesian sampling. Baklizi, A. [2003] introduced how to obtain acceptancesampling plans based on truncated life tests in the Pareto distribution of second kind. Theresearch in this subject were developed when Balakrishnan, N. [2007] designed a group ofinspection 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 themodel of Schmidt – Taylor [1973]. Also Hald, A. [1968] gives a model for Bayesian singlesampling attribute plans for continuous prior distribution. The parameters of single samplinginspection plans (݊, ܿ) are obtained from minimizing total expected quality control cost.Now, in this paper we introduce ( a mixed – Binomial) quality control expected costfunction, to obtain the parameters of double Bayesian sampling inspection plans, fromminimizing total expected regret function, according to conditions of producers andconsumers risk.Building Model of Total Expected Cost FunctionThe model comprises:a- ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample;݇‫ݏ‬ሺ‫݌‬ሻ ൌ ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ …………………………………………….. (1)݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ : Average cost of inspecting units of second sample, under sampling distributionሾ‫݌‬ሺ‫ݔ‬ଵሻሿ for number of defectives units in first sample (݊ଵ).݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൌ ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ……………………………………………… (2)And;‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ‫ݎ݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൌ ‫ݕ‬ሻ‫׊‬௬ൌ ݃௡భሺ‫ݔ‬ଵሻ …………………………………………….. (3)or;‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ‫׊‬௫మൌ ݃௡భሺ‫ݔ‬ଵሻ …………………………………………………. (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 ൌ ݃௡మሺ ‫ݔ‬ଶ|݊ଵ, ‫ݔ‬ଵሻ ……………………………………………….. (7)The average inspection cost for second sample is equal to:
4. 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), © IAEME285‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ………………………………………………. (8)In case of accepting produced lot from first sample, the average cost is;‫ܧ‬ሼሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵሻ‫ܣ‬ଶ|‫ݔ‬ଵሽൌ ሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ‫ܧ‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫ܣ‬ଶൌ ሺܰ െ ݊ଵሻሼ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሺ‫ݔ‬ଵሻሽൌ ሺܰ െ ݊ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿThe average of acceptance cost for second sample is;‫ܧ‬ሼሺܰ െ ݊ଵ െ ݊ଶሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶሻ‫ܣ‬ଶ|‫ݔ‬ଵ, ‫ݔ‬ଶሽሺܰ െ ݊ଵ െ ݊ଶሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሿ …………………………………. (9)After explaining all notations, we can show that the lot size N, in case of double samplingcan divide into;ܰ ൌ ݊ଵ ൅ ሺܰ െ ݊ଵሻቀ‫݌‬௔ሺଵሻ൅ ‫݌‬௥ሺଵሻቁ ൅ ݊ଶ‫݌‬ௌሺଵሻ൅ሺܰ െ ݊ଵ െ ݊ଶሻቀ‫݌‬௔ሺଶሻ൅ ‫݌‬௥ሺଶሻቁ … (10)Assume (݇݉), is the smallest average cost, happened according to results of decisions ofaccepting and rejecting, where acceptance happened at (‫݌‬ ൑ ‫,)ݎ݌‬ (i.e ‫,ݎ݌‬ break even qualitylevel) 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݌‬௔ሺଵሻሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ሺଵሻሺ‫݌‬ሻቅ൅݊ଶ݇‫ݏ‬ሺ‫݌‬ሻ‫݌‬ௌሺଶሻሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔ሺଶሻሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥ሺଶሻሺ‫݌‬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௔భ௫భୀ଴ቑ ൅ ෍ ݃ሺ‫ݔ‬ଵሻ݇‫ݎ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ௡భ௫భୀ௥భ൅ ∑ ݃ሺ‫ݔ‬ଵሻൣ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ൧௥భିଵ௫భୀ௔భାଵ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቐ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ݇ܽሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ௔మି௫భ௫మୀ଴ቑ൅ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫݌‬௥ሺଶሻሺ‫݌‬ሻሾ݇‫ݎ‬ሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ௡మ௫మୀ௔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݌‬௥ሺଵሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬௣௥଴ቋ൅ ቊන ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ሺଵሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ଵ௣௥ቋ൅݊ଶ ቄ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬ௌሺଵሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ‫׬‬ ߜሺ‫݌‬ሻ‫݌‬ௌሺଵሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ଵ଴௣௥଴ቅ൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥ሺଶሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ න ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔ሺଶሻሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ଵ௣௥௣௥଴ቋ… (14)
5. 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), © IAEME286‫ݎ݌‬ ൌ break even quality level;‫ݎ݌‬ ൌ஺మିோమ஺భିோభEquation (14), now transformed to standard cost function in terms of conditional distributionof percentage of defectives, which is [݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ] especially these parts obtainedfrom second sampling, where;݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ ൌ௣௥ሺ௔భழ௫భழ௥భ|௣ሻ௙ሺ௣ሻ∑ ௕௪ሺ௫భ,௡భሻೝభషభೌభశభൌቄ௣ೌሺమሻሺ௣ሻା௣ೝሺమሻሺ௣ሻቅ௙ሺ௣ሻ‫׬‬ ቄ௣ೌሺమሻሺ௣ሻା௣ೝሺమሻሺ௣ሻቅ௙ሺ௣ሻௗ௣భబ……… (15)This because when [ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ] a second sample (݊ଶ) is drawn, and according to thissample either we accept or reject.The final formula for standard regret function (ܴଶ) used to obtain the parameters of secondsampling plan is;ܴଶ‫כ‬ൌ ܴଵ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ െ ሺܰ െ ݊ଵሻ ෍ ݃ሺ‫ݔ‬ଵሻ݀‫ݎ‬ሺ‫ݔ‬ଵሻ௥భିଵ௫భୀ௔భାଵ൅ ∑ ݃ሺ‫ݔ‬ଵሻܴଵሺܽଶ െ ‫ݔ‬ଵ, ݊ଶ, ܰ െ ݊ଵ|‫ݔ‬ଵ ሻ௥భିଵ௫భୀ௔భାଵ ……………………………….… (16)Where;݀‫ݎ‬ሺ‫ݔ‬ଵሻ ൌ න ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬௣௥଴ܴଵ‫כ‬ሺܽଵ, ݊ଵ, ܰሻ ൌ௄ሺே,௡,௖ሻ ି ே௞௠௞௦ ି ௞௠………………………………… (17)Where;‫ܭ‬ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ቂሺ‫ܣ‬ଶ െ ܴଶሻௌௌା்‫ܨ‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻቃ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ቂሺܴଵ ൅ ܴଶሻௌௌା்ቃ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ ቀௌௌା்ቁ , ‫݌‬ ൌ்௡ା்݇ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ሺܰ െ ݊ሻሺ‫ܣ‬ଵ ൅ ܴଶ‫݌‬ሻ ൅ ሺܰ െ ݊ሻሺܴଵ െ ‫ܣ‬ଵሻWhich 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 (ܴଶ‫כ‬) defined in equation (16), so first of all andafter determining the parameters of single Bayesian sampling plan [given in table (1)] wework on minimizing (16) to find the parameters of double Bayesian sampling inspection plangiven in table (4).Equation (16) shows that the average of standard cost function for double samplingplan, is represented by standard cost function for single sampling plan and also standardconditional cost function weighted by the probability distribution of number of defectiveunits [݃ሺ‫ݔ‬ଵሻ], from first sample, it is obtained from;‫݌‬ሺ‫ݔ‬ଵሻ ൌ ݃௡భሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ‫׊‬௫మ
6. 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), © IAEME287To satisfy the aim of research which is to find the parameters ሺܽଵ, ‫ݎ‬ଵ, ܽଶሻ fromminimizing (ܴଶ), we apply difference equation since the distribution of number of defectivesin production is of discrete type, and also we consider;∆ ܴଶ ൌ∆ ௞మ஺మିோమThe 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 െ௥భିଵ௫భୀ௔భାଵ݊ଶሻ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ௔మି௫భ௫మୀ଴ ൟ …………………………………….. … (19)Then using;∆ ܴଶ ൌ∆ ௞మ஺మିோమൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ௔మି௫భ௫మୀ଴ቑ௥భିଵ௫భୀ௔భାଵെ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ௔మି௫భ௫మୀ଴ቑ௥భିଵ௫భୀ௔భାଵ… (20)We can find [ ∆௥భܴଶሺ‫ݎ‬ଵሻ] as;∆௥భܴଶሺ‫ݎ‬ଵሻ ൌ ݃௡భሺ‫ݎ‬ଵሻቐ݊ଶߜቂ‫݌‬௡భሺ௥భሻቃ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃௡మሺ‫ݔ‬ଶ|‫ݔ‬ଵሻቂ‫݌‬௡భା௡మሺ௥భ, ௫మሻെ ‫ݎ݌‬ቃ௔మି௥భ௫మୀ଴ቑ… (21)And also;∆௔భܴଶሺܽଵሻ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ௔మି௫భ௫మୀ଴ቑ௥భିଵ௫భୀ௔భାଶെ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ௔మି௫భ௫మୀ଴ቑ௥భିଵ௫భୀ௔భାଵ… (22)And also the difference of [ܴଶሺܽଵሻ] with respect to ሺܽଵሻ;∆௔భܴଶሺܽଵሻ ൌ ݃௡భሺܽଵ ൅ 1ሻ ቄ݊ଶሺ‫ܣ‬ଶ െ ܵଵሻቂ‫݌‬௡భሺ௔భାଵሻቃ െ ሺܵଵ െ ‫ܣ‬ଵሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ ൅ ݃௡భሺܽଵ ൅1ሻሺܰ െ ݊ଵ െ ݊ଶሻ ∑ ݃௡మሺ‫ݔ‬ଶ|ܽଵ ൅ 1ሻቂ‫݌‬௡భା௡మሺ௔భାଵା ௫మሻെ ‫ݎ݌‬ቃ௡మ௫మୀ௔మି௔భቅ …………. … (23)From ∆௔మܴଶሺܽଶሻ we can find the value of parameter (ܽଶ) for given (݊ଵ, ݊ଶ ) which satisfythe inequality (24) follows.‫̂݌‬௡భା௡మሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫̂݌‬௡భା௡మሺܽොଶ ൅ 1ሻ …………………………………….. (24)Then;‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 ‫݄݊݁ݓ‬ ݇‫ݏ‬ ൌ ݇‫ݎ‬‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 ݂‫ݎ݋‬ ݇‫ݏ‬ ൐ ݇‫ݎ‬
7. 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), © IAEME288And when [ܽଶ ൌ ܽଵ], the second term of ∆௔భܴଶሺܽଵሻ is positive, so we assume that[ܽଵ ൌ ܽଵ‫כ‬] is the value at which the first term of ∆௔మܴଶሺܽଶሻ is positive therefore∆ ܴଶሺܽଵ‫כ‬ሻ ൐ 0 and when ݇‫ݏ‬ ൌ ݇‫ݎ‬ ฺ ሺ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1ሻ, and we see that the average cost ofinspection equal to average cost of reject, therefore we need only to find the parameters(ܽଵ, ܽଶ ) and from ∆௔భܴଶሺܽଵሻ we obtain ‫݌‬௡భሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to ‫ݎ‬ଵ ൑ ݊ଵ ൅ 1.After completing model for double Bayesian sampling inspection plan, this model isapplied to distribution of Beta – Binomial.From ∆௔మܴଶሺܽଶሻ we can find (ܽଶ) that satisfy the inequality (24).‫݌‬௡భା௡మሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫݌‬௡భା௡మሺܽොଶ ൅ 1ሻ ………………………………………. (25)When (ܽଵ ൌ ܽଶ), the second term of ∆௔భܴଶሺܽଵሻ is positive, so we assume that (ܽଵ ൌ ܽଵ‫כ‬) isthe value of which the first term of ∆௔మܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ‫כ‬ሻ ൐ 0.When;݇‫ݏ‬ ൌ ݇‫ݎ‬Then;∆ ܴଶሺ‫ݎ‬ଵሻ ൑ 0 ฺ ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1And for ݇‫ݏ‬ ൐ ݇‫ݎ‬ ฺ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1And 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 ∆௔భܴଶሺܽଵሻ, weobtain ‫݌‬௡భሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to (‫ݎ‬ଵ ൑ ݊ଵ ൅ 1).4 - APPLICATION OF PROPOSED MODELAfter the total cost function are built, it is necessary to apply this model for samplingdistribution to show how the parameters of double sampling plans are obtained, to take adecision for accept or reject, assume (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) are three independent random variablesfollow Binomial distribution as:‫ݔ‬ଵ~ܾሺ݊ଵ, ‫݌‬ሻ‫ݔ‬ଶ~ܾሺ݊ଶ, ‫݌‬ሻ‫ݔ‬ଷ~ܾሺ݊ଷ, ‫݌‬ሻAnd (‫)݌‬ is random variable varied from lot to lot, and its prior distribution [݂ሺ‫݌‬ሻ] isassumed here ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ, (Beta prior), so the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is;‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ሻ ൌ න ‫ܥ‬௫భ௡భଵ଴‫݌‬௫భ‫ݍ‬௡భ ି ௫భ‫ܥ‬௫మ௡మ‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫ܥ‬௫య௡య‫݌‬௫య‫ݍ‬௡య ି ௫య1‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ‫݌‬ௌିଵ‫ݍ‬௧ି ଵ݀‫݌‬ൌ஼ೣభ೙భ஼ೣమ೙మ஼ೣయ೙య஻௘௧௔ ሺ௦,௧ሻ‫׬‬ ‫݌‬௫భା௫మା௫యାௌିଵଵ଴‫ݍ‬௡భ ା௡మ ା௡య ି ௫భି௫మି௫యା௧ିଵ݀‫݌‬ൌ஼ೣభ೙భ஼ೣమ೙మ஼ೣయ೙య஻௘௧௔ ሺ௦,௧ሻ‫ܽݐ݁ܤ‬ 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,݌‬ ‫ݔ‬ଵሻ݃ሺ‫ݔ‬ଵሻ݃ሺ‫ݔ‬ଵሻ ൌ න ‫ܥ‬௫భ௡భଵ଴‫݌‬௫భ‫ݍ‬௡భ ି ௫భ1‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ‫݌‬ௌିଵ‫ݍ‬௧ି ଵ݀‫݌‬
8. 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), © IAEME289݃ሺ‫ݔ‬ଵሻ ൌ‫ܥ‬௫భ௡భ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ܵ, ݊ଵ െ ‫ݔ‬ଵ ൅ ‫ݐ‬ሻAnd from equation below which is Beta distribution;݂ሺ‫ݔ|݌‬ଵሻ ൌ௚ሺ௣,௫భ ሻ௚ሺ௫భሻThe conditional distribution of (‫)݌‬ given (‫ݔ‬ଵ) is;݂ሺ‫ݔ|݌‬ଵሻ ൌଵ஻௘௧௔ሺ௫భାௌ,௡భ ି ௫భା௧ሻ‫݌‬௫భାௌିଵ‫ݍ‬௡భ ି ௫భା௧ିଵ…………………….. (27)And;‫ܧ‬ሺ‫ݔ|݌‬ଵሻ ൌ‫ݔ‬ଵ ൅ ܵ݊ଵ ൅ ܵ ൅ ‫ݐ‬Also we can find ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ;‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ ሻଵ଴݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ………………………………….. (28)ൌଵ௞‫׬‬ ‫ܥ‬௫మ௡మଵ଴‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫݌‬௫భାௌିଵ‫ݍ‬௡భ ି ௫భା௧ିଵ݀‫݌‬‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌଵ௞‫ܥ‬௫మ௡మ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ൅ ‫ݐ‬ሻ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌௌା௫భௌା௫భା௫మ஼ೣమ೙మ஼ೄశೣభ೙భశ೟శೄషభ஼ೄశೣభశೣమ೙భశ೙మశ೟శೄషభ ………………………………………. (29)Which is [݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ]Assume ‫ݔ‬ଷ ൌ ܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ‫ݔ‬ଷ~ ‫݈ܽ݅݉݋݊݅ܤ‬ ሺ݊ଷ, ‫݌‬ሻThen the conditional distribution of (‫ݔ‬ଷ) given (‫ݔ‬ଵ, ‫ݔ‬ଶ) is;‫ݎ݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻଵ଴݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ݀‫݌‬ ……………………….. (30)To solve equation (30), we need to find ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ;Let ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ‫׶‬ ‫ݔ‬ଵ~ ܾሺ݊ଵ, ‫݌‬ሻ‫ݔ‬ଶ~ ܾሺ݊ଶ, ‫݌‬ሻAnd are independent, then;‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ~ ‫ܤ‬ሺ݊ଵ ൅ ݊ଶ, ‫݌‬ሻ݃ሺ‫,݌‬ ‫ݕ‬ሻ ൌ ‫ܥ‬௬௡భା௡మ‫݌‬௬‫ݍ‬௡భା௡మି௬1‫ܤ‬ሺܵ, ‫ݐ‬ሻ‫݌‬ௌିଵ‫ݍ‬௧ିଵ‫׵‬ ݃ሺ‫݌|ݕ‬ሻ ൌ1‫ܤ‬ሺܵ, ‫ݐ‬ሻන ‫ܥ‬௬௡భା௡మ‫݌‬௬ାௌିଵ‫ݍ‬௡భା௡మି௬ା௧ିଵ݀‫݌‬ଵ଴݃ሺ‫݌|ݕ‬ሻ ൌ1‫ܤ‬ሺܵ, ‫ݐ‬ሻ‫ܥ‬௬௡భା௡మ‫ܽݐ݁ܤ‬ ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ‫׵‬ ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൌ݆‫ݐ݊݅݋‬݉ܽ‫݈ܽ݊݅݃ݎ‬ൌభಳሺೄ,೟ሻ஼೤೙భశ೙మ௣೤శೄషభ௤೙భశ೙మష೤శ೟షభభಳሺೄ,೟ሻ஼೤೙భశ೙మ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻൌଵ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ‫݌‬௬ାௌିଵ‫ݍ‬௡భା௡మି௬ା௧ିଵ0 ൑ ‫݌‬ ൑ 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 defectivesሺ‫ݔ‬ଷ) after drawing the first and second sample, i.e to find the distribution of ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ.
9. 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), © IAEME290‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ න ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶ ൌ ‫ݕ‬ሻ݀‫݌‬ଵ଴ൌ න ‫ܥ‬௫య௡యଵ଴‫݌‬௫య‫ݍ‬௡యି ௫య1‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ‫݌‬௬ାௌିଵ‫ݍ‬௡భା௡మା௧ି௬ିଵ݀‫݌‬ൌ‫ܥ‬௫య௡య‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻන ‫݌‬௫యା௬ାௌିଵଵ଴‫ݍ‬௡భା௡మା௡యି௫యା௧ି௬ିଵ݀‫݌‬ൌ‫ܥ‬௫య௡య‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ െ ‫ݕ‬ሻൌ‫ܥ‬௫య௡యΓሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ െ ‫ݕ‬ ൅ ‫ݐ‬ሻΓ ሺ‫ݕ‬ ൅ ܵሻΓ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻΓ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ ൅ ܵሻΓ ሺܵ ൅ ‫ݐ‬ ൅ ݊ଵ ൅ ݊ଶ ൅ ݊ଷሻൌ஼ೣయ೙యሺ௡భା௡మା௧ାୗିଵሻ!ሺ௬ାௌିଵሻ!ሺ௡భା௡మା௧ି୷ିଵሻ!ൈሺ௫యା௬ାௌିଵሻ!ሺ௡భା௡మା௡యି௫యି௬ା௧ିଵሻ!ሺௌା௧ା௡భା௡మା௡యିଵሻ!ൌ஼ೣయ೙య஼೤శೄ೙భశ೙మశ೟శೄషభሺ௫యାௌା௬ሻ஼ೣయశೄశ೤೙భశ೙మశ೙యశ೟శೄషభሺ௬ାௌሻWhich is mixed hypergeometric distribution.We can obtain the conditional mean of ሺ‫ݔ‬ଷሻgiven ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ;‫ܧ‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ݊ଷ‫݌‬௡భା௡మሺ௫భ,௫మሻWhere,‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫݌‬௡భା௡మሺ௫భା௫మሻ5- CASE STUDYThe following data represents the percentage of defectives of (140) lots from someindustrial companies in Iraq, these percentages are grouped in frequency table as follows;Table (1): The frequency table for percentage of defectives of 140 lotsPercentage ofdefectivesObservedfrequency ሺ݂௜ሻClass mark‫݌‬௜ ‫݌‬௜݂௜ ‫݌‬௜ଶ݂௜0.0 - 6 0.01 0.06 0.01160.02 - 13 0.03 0.39 0.01170.04 - 17 0.05 0.85 0.04250.06 - 24 0.07 1.68 0.11760.08 - 18 0.09 1.62 0.14560.10 - 16 0.11 1.76 0.19360.12 - 15 0.13 1.95 0.25350.14 - 10 0.15 1.50 0.22500.16 - 9 0.17 1.53 0.26010.18 - 7 0.19 0.33 0.25270.20 - 3 0.21 0.63 0.13230.22 – 0.24 2 0.23 0.46 0.1058Total 140 13.699 1.7413
10. 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), © IAEME291Then we draw the histogram and frequency curve, the curve indicate it is Beta distributionwith parameters (S, T), which are estimated by moments method using the equations;ܵመ ൌ ‫ݔ‬௣ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ଶሿหܵ௣ଶܶ෠ ൌ ‫ݔ‬௤ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣ଶሿหܵ௣ଶWhere;‫ݔ‬௣ ൌ∑ ௣೔௙೔∑ ௙೔ൌ 0.09785ܵ௣ଶൌ∑ ௣೔మ௙೔∑ ௙೔െ ‫݌‬ଶൌ 0.00286Then;ܵመ௠௢௠ ൌ 2.9 ؆ 3 ܶ෠௠௢௠ ൌ 26.9 ؆ 27Then the distribution of (‫݌‬௜ሻ is tested using ( ߯ଶ) test, where the cumulative probabilitiesare computed from;‫ܨ‬ሺ‫݌‬ሻ ൌ ‫׬‬ଵ஻ሺௌ,ఛሻ௣଴‫ݑ‬ௌିଵሺ1 െ ‫ݑ‬ሻ்ିଵ݀‫ݑ‬ൌ ‫ܧ‬ሺܵ, ܵ ൅ ܶ െ 1, ‫݌‬ሻ ൌ ‫ݎ݌‬ሺ‫ݔ‬ ൐ ܵሻ ൌ ∑ ‫ܥ‬௫ௌା்ିଵௌା்ିଵ௫ୀௌ ‫݌‬௫‫ݍ‬ௌା்ି௫ିଵWhere;ܵመ ൌ 3 , ܶ෠ ൌ 27The hypothesis tested is;‫ܪ‬௢: ‫݌‬ ~ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ‫ܪ‬ଵ: ‫݌‬ ‫؂‬ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻUsing Chi – square test, which depend on the following formula;߯ଶൌ ෍ሺܱ௜ െ ‫ܧ‬௜ሻଶ‫ܧ‬௜௡௜ୀଵThe following table represents the necessary computations for goodness of fit.Table (2): The necessary computations for goodness of fitProportion ofdefectivesObservedfrequency (ܱ௜ሻ‫ݎ݌‬ሺ‫݌‬ଵ ൏ ‫݌‬ ൏ ‫݌‬ଶሻ‫ܧ‬௜ሺܱ௜ െ ‫ܧ‬௜ሻଶ‫ܧ‬௜0.0 - 6 0.0231 3.234 2.36570.02 - 13 0.0974 13.636 0.02960.04 - 17 0.1662 23.268 1.63840.06 - 24 0.1612 22.568 0.09080.08 - 18 0.1500 21.00 0.42850.10 - 16 0.1252 17.528 0.13320.12 - 15 0.0827 11.578 1.01140.14 - 10 0.0715 10.01 0.000010.16 - 9 0.0435 6.09 1.29040.18 - 7 0.0373 5.222 0.06050.20 - 3 0.0252 3.528 0.07900.22 – 0.24 2 0.0167 2.338 0.0488Total 140 1 140 7.1931
11. 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), © IAEME292Comparing [ ߯௖௔௟ଶൌ 7.1931ሿ with ሾ߯௧௔௕ሺ଴.଴ହ,଼ሻଶൌ 15.507ሿ, we find that, ߯௖௔௟ଶ൏ ߯௧௔௕ଶ, weaccept the hypothesis that, ‫݌‬௜ ~ ‫ܽݐ݁ܤ‬ distribution with parameters ሺS,Tሻ. Then the priordistribution 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 qualitycontrol cost for each produced units, which are;‫ܣ‬ଵ ൌ 0, ܴଵ ൌ 10.000 ‫ܦܫ‬ܵଵ ൌ 10,000 ‫ܦܫ‬ܴଶ ൌ 28,000 ‫ܦܫ‬ܵଶ ൌ 28,000 ‫ܦܫ‬‫ܣ‬ଶ ൌ 153.000 ‫ܦܫ‬‫ݎ݌‬ ൌோభି஺భ஺మିோమൌ 0.08, which is the break even quality control point.According to the above parameters we find the parameters of single Bayesian samplinginspection plan from;݊ଵ‫כ‬ൌ ߣଵ√ܰ ൅ ߣଶ‫ܥ‬ଵ‫כ‬ൌ ݊ଵ‫כ‬‫ݎ݌‬ ൅ ‫ܤ‬ଵ‫ܤ‬ଵ ൌ ܶ෠‫ݎ݌‬ െ ܵመ‫ݎݍ‬ െଵଶ‫݌‬ ൌ௣௣௥ൌ 1.2‫ݎ݌‬ ൌ 0.08And;ߣଵଶൌ௣ೝೄ ௤ೝ೅ሺ஺మିோమሻଶ஻ሺௌ,்ሻሺ௞௦ି௞௠ሻ݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬௣௥଴ߣଶ is a constant obtained according to the prior distribution of percentage of defectiveas;ߣଶ ൌ ሾ3ሺܵ ൅ ܶሻଶെ 11ሺܵ ൅ ܶሻ െ 2 െ ሺ3ܵ െ 1ሻܵ|‫ݎ݌‬ െ ሺ3ܶ െ 1ሻܶ|‫ݎݍ‬ െ 1| ‫ݎݍݎ݌‬ሿ 12⁄And݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬௣௥଴Since ܵመ ൌ 3 , ܶ෠ ൌ 27ߣଵ ൌ 4.663, ߣଶ ൌ െ27, ܰ ൌ 200The parameters of single Bayesian sampling inspection plan are;݊‫כ‬ൌ ߣଵ√ܰ ൅ ߣଶ ൌ 39‫ܥ‬‫כ‬ൌ ݊‫כ‬‫ݎ݌‬ ൅ ‫ܤ‬ଵ ൌ 2‫׵‬ parameters of single Bayesian sampling plan is; ሺ݊‫כ‬ൌ 39, ‫ܥ‬‫כ‬ൌ 2 ሻ, for differentvalues of ሾ ‫݌‬ ൌ 0.07ሺ0.05ሻ0.09ሿ for ሺܰ ൌ 200, 225, 250,275,300ሻ, the following tablesummarize single sampling plans.
12. 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), © IAEME293Table (3): Single Bayesian sampling inspection plansP 0.07 0.075 0.08 0.085 0.09N ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ150 10 0 71.6 15 0 80.16 23 1 102.4 17 0 70.9 20 1 72.1175 13 0 81.8 19 1 94.37 27 1 116.0 22 1 87.9 23 1 88.9200 17 0 95.4 23 1 107.9 30 2 126.2 27 1 104.9 27 1 105.7225 20 0 105.65 26 1 118.2 35 2 143.2 32 2 120.3 35 2 122.3250 23 1 115.82 29 1 128.4 38 2 153.6 36 2 135.6 38 3 137.1275 26 1 126.02 32 2 138.6 41 3 163. 6 40 2 149. 2 41 3 150.2300 29 1 135.22 35 2 148.8 45 3 177.2 43 2 159.4 45 3 161.4After we find the single Bayesian sampling inspection plans, from minimizing (ܴଵ), thenminimizing (ܴଶሻ gives the double Bayesian sampling inspection plans, which works onminimizing total expected quality control cost, derived in equation (10) and then transformedto standard cost function (ܴ2) in equation (14).Table (4): Double Bayesian sampling inspection plansP 0.07 0.075 0.08 0.085 0.09N ݊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 0.94175 13 2 0.9 21 2 0.986 25 2 1.06 21 2 0.986 22 2 0.96200 16 1 0.88 23 2 0.97 28 2 0.99 26 2 0.977 25 3 0.97225 19 2 0.85 25 2 0.85 32 2 0.97 30 3 0.967 30 3 0.99250 20 2 0.80 29 2 0.83 36 2 0.88 32 3 0.933 39 3 1.04275 23 1 0.75 31 3 0.79 40 3 0.86 36 3 0.942 40 3 1.65300 30 1 0.74 36 3 0.75 42 3 0.85 38 4 0.931 45 4 1.85From 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 ሺ݊ଵ‫כ‬ൌ 39, ‫ܥ‬ଵ‫כ‬ൌ 2ሻ,which defined as drawing at random from lot (ܰ ൌ 200ሻ, a sample of size ሺ݊ଵ‫כ‬ൌ 39ሻ, andtesting its units, if the number of defective is found (‫ܥ‬ଵ‫כ‬ൌ 2ሻ, then accept from first sample(‫ݔ‬ଵ ൑ 2ሻ but if ሺ‫ݔ‬ଵ ൒ 2) then reject from first sample and also stop, but according to themodel, we find the single Bayesian sampling inspection plans (݊ଵ, ‫ܥ‬ଵ, ܴଵሻ as tabulated intable (5), then the second Bayesian sampling inspection plans are found and put in table (4).CONCLUSION1- Finding plan which minimize the linear combination of two expected cost functionfor first sampling and second sampling required a procedure carried out by means ofdifference equation, since the cost function is of discrete type, and the minimization is donein two stages to find (ܴଵሻ and ሺܴଶ).2- We try also to find a family of optimum plans through maximizing the OC curve, but itdepend on probability of sampling under double sampling plans, and do not consider theparameters of cost function.3- The model was built under Beta – Binomial distribution since the distribution of process isBinomial (݊, ‫݌‬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 estimatedalso by any other statistical method.
13. 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), © IAEME2945- The decision to accept or reject obtained from second sample is more effective than thedecision 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 thisnumber, because of bad station of production in Iraq, but in spite of this the company work tosatisfy the required quality control level.7- The percentage of defectives in 140 lot are varied from to lot, so the distribution of thesepercentage 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 Paretodistribution of second kind", Advances and Applications in Statistics, Vol. 3, 33- 48.[2] Balakrishnan, N., Leiva, V. and Lopez, J. (2007), " acceptance sampling plans fromtruncated life test based on generalized Birnbaum – Saunders distribution", Communicationsin Statistics – Simulation and Computation, Vol. 36, 643 – 656.[3] Epstein, B. (1954), "Truncated life test in the exponential case", Annals of mathematicalStatistics, 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 Grifinand company LTD.[6] Gupta, S. S. and Groll, P. A. (1961), "Gamma distribution in acceptance sampling basedon 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 samplinginspection plans", Technometrics, Vol. 4, 497 – 513.[9]Hald, A. (1968), "Bayesian single sampling attribute plans for continuous priordistributions", Technometrics, Vol. 10, 667 – 683.[10] Nilesh Parihar and Dr. V. S. Chouhan, “Extraction of Qrs Complexes using AutomatedBayesian Regularization Neural Network”, International Journal of Advanced Research inEngineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 37 - 42, ISSN Print:0976-6480, ISSN Online: 0976-6499.