Design and Development of Three Stages Mixed Sampling Plans for Variable Attribute Variable Quality Characteristics

Transcription

1 International Journal of Statistis and Systems ISSN Volume 12, Number 4 (2017), pp Researh India Publiations Design and Development of Three Stages Mixed Sampling Plans for Variable Attribute Variable Quality Charateristis S. Devaarul 1 and D. Senthil Kumar 2 1 Dr. S. Deva Arul, Assistant Professor, Government Arts College, Coimbatore, TamilNadu, India. 2. Mr. D. Senthil Kumar, a part time researh sholar at Government Arts College, affiliated to Bharathiar University, Coimbatore, India. Abstrat The mixed sampling plans are two stages sampling plans in whih variable and attribute quality harateristis are used in deiding the aeptane or rejetion of the lot. In industries variables play a vital role in making the deision about the proess or bathes beause of its effiieny. Many quality ontrol pratitioners insist that in a mixed sampling environment if a bath is not aepted based on first and seond stage sample results, then the deision should be based on the results of variable riteria in the third stage. Hene three stages mixed sampling plans are being developed. This paper presents a new algorithm to make a unique deision on the lot for mixed environment with Vari-Attri-Vari ombinations for independent plans. The Operating harateristis funtion and other related measures of the mixed sampling plans are derived by the authors. A new designing methodology for determining the parameters of the sampling plans are given. Comparisons are made between two and three stages of mixed sampling plans. Tables are onstruted to failitate easy appliation of 3D sampling plans in the Quality Control Setion of prodution industries. Keywords: Mixed Sampling, Three stages, Variable riteria, Attribute riteria, Designing V-A-V.

2 764 S. Devaarul and D. Senthil Kumar 1. INTRODUCTION The mixed sampling plans are produt ontrol tehniques in whih variable and attribute quality harateristis are ombined in deiding the aeptane or rejetion of the lot. Due to modern quality ontrol systems, mixed sampling plans are widely applied in various stages of prodution. In industries, variable riteria play a vital role in inspetion area. If attribute riteria are used for making a deision, then it is not effiient to ontrol quality of the produts when the parameters are measurable. Hene to offset the disadvantage a new three stage mixed sampling plans are developed and termed as Vari-Attri-Vari (VAV) mixed sampling plans. This paper presents a new algorithm to make a unique deision on the lot for mixed environment with Vari- Attri-Vari ombinations. The Operating harateristis funtion and other related measures of the mixed sampling plans are derived. A new designing methodology for determining the parameters of the sampling plans are given. Comparisons are made between two and three stages of mixed sampling plans. Tables are onstruted to failitate easy appliation of three stages sampling plans in the Quality Control Setion of prodution industries. Dodge (1932) introdued the onept of mixed sampling plans with variable and attribute quality harateristis. Bowker and Goode (1952) have extended the work of mixed sampling plans. Gregory and Resnikoff (1955) have given a proedure for mixed plans when the standard deviation is known. Savage (1955) has studied the mixed plans, for non-normal distribution. Kao (1966) has used the onept of item variability instead of item entral tendeny to sentene a lot. In the year 1967, Shilling presented a method for determining the operating harateristis of mixed variables-attributes sampling plans. DevaArul (1996 & 2004) has developed mixed sampling plans by inulating the blend of proess & produt ontrol measures to suit speifi industrial needs. Suresh and Deva Arul (2003) have developed Multi- Dimensional Mixed Sampling Plans. DevaArul. S (2009) has ontributed towards mixed sampling system with tightened inspetion in the seond stage. Suresh and Devaarul (2002) have designed Mixed Sampling Plans with Chain Sampling as attribute plan. Suresh and DevaArul (2002) have developed mixed sampling plans by ombining proess and produt quality harateristis to redue the sampling ost. DevaArul and Jemmy Joye (2010) have developed Mixed Sampling Plans for Seond Quality Lots. Asokan and Balamurali (2000) have developed Multi Attribute Single Sampling Plans indexed through AQL and LTPD.

3 Design and Development of Three Stages Mixed Sampling Plans for Variable FORMULATION OF V-A-V MIXED SAMPLING PLANS Let N : Lot size n 1 : First Stage Sample size n 2 : Seond Stage Sample size n3 : Third Stage Sample size k = Variable fator suh that a lot is aepted if x U kσ = The attributes aeptane number whih leads to third stage β = Probability of aeptane β1 = Probability of aeptane for Pi β 1 = Probability of aeptane assigned to () stage for perent defetive Pi σ = Population standard deviation Un = Extreme deviate from the sample mean in studentized form F n (u) = Cumulative distribution of the extreme deviate from the sample mean in studentized form from a sample of size n p = Fration defetive P(i,x) = Pa(p) = Joint probability of i and X Probability of aeptane x = Sample mean ZA = z value for aeptane limit for x Z(i) = z value for the i th ordered observation μ = Population mean (proess) Z u = z value of upper speifiation limit Z (W) = Standard normal deviate suh that Z = Mean of z values zi = Average of z value from ith sample 2π t2 dt zw e =w

4 766 S. Devaarul and D. Senthil Kumar 3. Algorithm for three stages (v-a-v) mixed sampling plans Step 1: Draw a random sample of size n1. Step 2: Determine x 1. Step 3: If x 1 U kσ, aept the lot. Step 4: Ifx 1 U kσ, go to next step. Step 5: Draw another sample of size n 2. Step 6: Count the number of defetives d. Step 7: If d, now go to next step else if d >, rejet the lot Step 8: Draw another sample size n 3 and determine x 2. Step 9: Ifx 2 U kσ, aept the lot otherwise rejet the lot. 4. Operating Charateristis and Assoiated Measures of Mixed Plans The four prinipal urves, whih desribe the performane of an aeptane sampling plan for various frations defetive, are the Operating Charateristi urves, the ASN urves, the AOQ and the ATI urves. The operation of the mixed plans annot be properly assessed until formulae for the ordinates of eah of these measures are defined for the known values of perent defetive. The Operating Charateristis funtion of VAV mixed sampling plan is defined as Proof: P a (p) = P n1 [x 1 A] + P n1 [x 1 > A] [ e n 2p (n2 p)x ] P x! n3 [x 2 A] The lot will be aepted in the following ases Case(i) x=0 From the sample of size n 1 if x 1 A, aept the lot Case (ii) From the sample of size n 2 if d and also from the sample of size n 3 if 2 x A, the lot will be aepted Case (i) & (ii) are mutually exlusive ase. By the law of addition on probability we get

5 Design and Development of Three Stages Mixed Sampling Plans for Variable 767 P a (p) = p(i) + p(ii) = P[x 1 A] + P n1 [x 1 > A]P[d ]P[x 2 A] In ase of type II probability, P a (p) = P n1 [x 1 U k 1 σ] + P n1 [x 1 > A] e n 2p (n2 p)x x! In ase of Binomial distribution, x=0 P n3 [x 2 u k 2 σ] P a (p) = P n1 [x 1 U k 1 σ] + P n1 [x 1 > A] ( n x ) px q n x P n3 [x 2 u k 2 σ] In ase of lot size is known then P a (p) = P n1 [x 1 U k 1 σ] + P n1 [x 1 > A] [ x=0 (Np np ) x=0 ( N Np n np )] ( N n ) {P n3 [x 2 u k 2 σ]} 5. Designing Through AQL Step1: Let the three stages be independent. Let the first stage probability aeptane be β 1. Let the seond stage probability aeptane be β 1. Let the third stage probability aeptane be β 1 Step2: Determine the sample size n1 for the known β 1 Step3: Calulate the aeptane limit k1 for the existing proess average AQL= p1 k1 = z(p1) + z(β 1 ) n 1 where z(w) is the standard normal variate orresponding to w suh that 1 2π z(w) e 1 2 z2 dz Step4: Now determine β 1 the seond stage probability of aeptane suh that β 1 = β 1 β 1 1 β 1

6 768 S. Devaarul and D. Senthil Kumar Step5: Determine the seond sample size n2 and aeptane number suh that e n 2p (n2 p)x x! x=0 β 1 Step 6: Now determine the third stage probability of aeptane β 1 suh that β 1 = β 1 β 1 ( β 1 β 1 ) 1 β 1 ( β 1 β 1 ) Step 7: Determine the third stage parameters n3 and k2 suh that k2 = z(p1) + z(β 1 ) n 3 For pratial reasons we set n1 = n2 = n3 the values are known. Hene k2 an be easily determined. Table 1: Values of the parameters k1, k2 and for the known n1, n2 and n3for the known AQL and by assuming first stage probability of aeptane is 65% having total probability of aeptane with 95% AQL n1=n2=n3= 50 n1=n2=n3= 100 n1=n2=n3= p1 k1 k2 k1 k2 C k1 k

7 Design and Development of Three Stages Mixed Sampling Plans for Variable 769 Illustration 1: A prodution proess turns out 0.1% proess fration defetive. Determine three stages mixed sampling plan with 95% probability of aeptane. Solution: It is given that AQL = 0.1%. From table 1, one an find the required parameters for 95% probability aeptane. For pratial reasons if n1=n2=n3=100 then from table 1 we get, k1= , k2= , C = 0. Operating proedure: Step 1: Draw a random sample of size n1= 100. Step 2: Determine x 1. Step 3: If x 1 U σ, aept the lot. Step 4: Ifx 1 > U σ, go to next step. Step 5: Draw another sample of size n 2.= 100. Step 6: Count the number of defetives d. Suppose if d > 0, rejet the lot. Step 7: If d = 0, now go to third stage (i.e) next step. Step 8: Draw another sample size n 3 =100 and determinex 2. Step 9: If x 2 U σ, aept the lot otherwise rejet the lot. 7. Designing through LQL Step 1: Let the first stage probability aeptane be β 2 Let the seond stage probability aeptane be β 2 Let the third stage probability aeptane be β 2 Step 2: Determine the sample size n1 for the known β 2 Step 3: Calulate the aeptane limit k1 for the existing proess average LQL= p2 k1 = z(p2) + z(β 2 ) n 1 where z(w) is the standard normal variate orresponding to w suh that 1 2π z(w) e 1 2 z2 dz Step 4: Now determine β 1 the seond stage probability of aeptane suh that β 1 = β 1 β 1 1 β 1

8 770 S. Devaarul and D. Senthil Kumar Step 5: Determine the seond sample size n2 and aeptane number suh that e n 2p (n2 p)x x! x=0 β 1 Step 6: Now determine the third stage probability of aeptane β 1 suh that β 1 = β 1 β 1 ( β 1 β 1 ) 1 β 1 ( β 1 β 1 ) Step 7: Determine the third stage parameters n3 and k2 suh that k2 = z(p1) + z(β 1 ) n 3 Sine n1 = n2 = n3 the values are known. k2 an be easily determined. Table 2: Values of the parameters k1, k2 and for the known n1, n2 and n3 for the known AQL and by assuming first stage probability of aeptane is 65% having total probability of aeptane with 95% LQL n1=n2=n3= 50 n1=n2=n3= 100 n1=n2=n3= 150 p2 k1 k2 k1 k2 k1 k Illustration 2 A prodution proess turns out 3% proess average fration defetive. Determine three stages mixed sampling plans for the known LQL at 10% Probability of aeptane. Solution: It is given that LQL = 3%.

9 Design and Development of Three Stages Mixed Sampling Plans for Variable 771 Let the probability of aeptane at LQL is 3%. From table 2, one an find the required parameters for 10% probability aeptane. For pratial reasons if n1=n2=n3=200, then from table 2, k1= , k2= , =2. 8. Average Sample Number (ASN) The expeted number of sample size required to make a deision on the lot is defined as E(n) = n1 + n2 [ Pn1( x > A)] + (n1+n2) Pn1 ( x > A) P(d ) 9. Average Outgoing Quality (AOQ) The expeted outgoing quality after the inspetion is defined as AOQ = p Pa(p) CONCLUSION: In this researh artile, a new three stages mixed sampling plans are developed with final deision based on variable riteria whih will lead to exat and effiient deision. The operating proedure is given and the related measures suh as OC, ASN, AOQ are derived for the first time. The new algorithm is easy to operate in quality ontrol setion espeially when the deision should be taken based on the basis of variable quality harateristis. Utilizing the new designing proedure tables are onstruted to failitate quality ontrol engineers for easy seletion of sampling plans. It is found that the 3-stage mixed sampling plan is more sensitive towards deterioration of the quality. Whenever lot quality is not maintained then there is a derease in probability of the aeptane of the lot. Hene this sampling plan oeres the produer to maintain the standard quality so that the ustomer is satisfied. The omparison between 2stages and 3stages mixed sampling plan shows that the average sample number is more eonomial with respet to sample size in ase of newly developed mixed sampling plans. REFERENCES: [1] ASOKAN M.V. and BALAMURALI.S(2000), Multi Attribute Single Sampling Plans, Eonomi Quality Control, Vol.15, No:3/4, pp , Germany. [2] DEVAARUL, S (2004), Certain Studies relating to Mixed Sampling plans and Reliability based Sampling Plans, Ph.D., Thesis, Department of

10 772 S. Devaarul and D. Senthil Kumar Statistis, Bharathiar University, Coimbatore, Tamilnadu, India. [3] DEVAARUL, S (1996), A Study on the Speial Purpose Mixed Aeptane Sampling Plans, M.Phil. Dissertation, Department of Statistis, Bharathiar University, Coimbatore, Tamilnadu, India. [4] DEVAARUL, S (2009), Mixed sampling system with tightened inspetion in the seond stage, International Journal of Artifiial Intelligene, Vol.2, No. S09, pp [5] DEVAARUL, S. AND JEMMY JOYCE, V. (2010), Seletion of Mixed Sampling Plans for Seond Quality Lots, Eonomi Quality Control, Germany. [6] SAVAGE I.R. (1955), Mixed Variables and Attributes Plans, The Exponential Case, Tehnial Report No.23, Applied Mathematis and Statistis Laboratory, Stanford University, Stanford, California. [7] SCHILLING, E.G.(1967), A general method for determining the OC of Mixed variable-attributes Sampling plans, Single sided speifiations, SD known, Ph.D. Thesis, Rutgers The state university, New Brunswik, New Jersey. [8] SCHILLING E.G. and DODGE H.F (1969), Proedure and Tables for Evaluating Dependent Mixed Aeptane Sampling Plans, Tehnometris, pp [9] SURESH K.K. and DEVAARUL S (2002), Designing and Seletion of Mixed Sampling Plans with Chain Sampling as attribute plan - Vol.15, No:1, pp , Quality Engineering Journal, U.S.A. [10] SURESH K.K. and DEVAARUL S.(2002) Combining Proess and Produt Control for Reduing Sampling Costs Vol. 17, No.2, , Eonomi Quality Control, Journal and Newsletter for Quality and Reliability Germany. [11] SURESH K.K. AND DEVA ARUL. S (2003), Multi Dimensional Mixed Sampling Plans, Vol:16. No.2, pp , Journal of Quality Engineering, U.S.A.