Mixed Acceptance Sampling Plans for Multiple Products Indexed by Cost of Inspection

Transcription

1 Mied ace Samplig Plas for Multiple Products Ideed by Cost of Ispectio Paitoo Howyig ), Prapaisri Sudasa Na - Ayudthya ) ) Dhurakijpudit Uiversity, Faculty of Egieerig (howyig@yahoo.com) ) Kasetsart Uiversity, Faculty of Egieerig (fegpsa@otri.ku.ac.th) Abstract ace samplig pla is a tool that used for cotrollig the quality of materials that put ito productio processes or deliver to the cosumer. The selectio of samplig pla depeds o a characteristic of the data. There are two types of lot for lot acceptace samplig plas: the former is the attribute samplig pla that is used for samplig the products with accept or reject characteristic of quality, the latter is variable samplig pla that is used for measurig variable data such as weight, legth or volume. The others acceptace samplig pla, i geerally, Dodge ad Romig s samplig pla which is used for protectig the cosumer s risk, the ANSI/ASQC.4 ad the ANSI/ASQC.9 which are used for protectig the producer s risk. They do ot cocer with samplig cost which occur i practice such as a uit cost, salary of workers, materials or tools for ispectio ad the budget for ispectio are limited. This article proposes the method to desig the acceptace samplig plas for multiple products that are restricted by the budget. They ca be desiged by applied the dyamic programmig with objective of the pla is to miimize the maimum cosumer s risk. Itroductio rmally, acceptace samplig pla ca be categorized ito two groups; attribute samplig pla ad variable samplig pla. For attribute samplig pla, it ca be divided ito sigle samplig pla which decisio ca be made by takig oe sample. Double samplig pla that the decisio ca be made by takig at most two samples. The aother attribute pla which complicate to use is multiple samplig pla that the decisio usually ca be made by takig at most seve samples. For variable samplig pla, it ca be divided ito two groups, the former is the pla that used for cotrollig the process mea which are sigle ad double specificatio limit, the latter is the pla that used for cotrollig the percet defective. These samplig plas ca be desiged by followig.. Desig sigle attribute samplig plas by usig the operatig characteristic curve (OC- Curve) passes through the two poits ( - α, AQL ) ad (, LTPD ) as show i figure. where as α The probability of rejectig a lot that meets the stated quality level (The producer s risk) The probability of acceptig a udesirable lot (The cosumer s risk) AQL The average quality level LTPD The lot tolerace percet defective X The umber of defective uits that are foud i ispectio which has a biomial distributio Ac The acceptace umber Re The rejectio umber The sample size.

2 Fig. The operatig characteristic curve passes through the two poits ( - α, AQL ) ad (, LTPD ). It ca be writte as equatio ad. Pr ( X Ac ) Pr ( X Ac p AQL) - α () Pr ( X Ac ) Pr ( X Ac p LTPD) () X < Ac? X > Re? Reject Fig. Geeral procedure for sigle samplig plas.. Desig double attribute samplig plas by usig the operatig characteristic curve (OC - Curve) passes through the two poits ( - α, AQL ) ad (, LTPD ) ca be calculated samplig plas by solvig equatio 3 ad 4. Ac ( X Ac ) + Pr( X i) *Pr( X Ac i) α Pr where p AQL. (3) i Ac Ac Pr( X Ac ) + Pr( X i) * Pr( X Ac i) where p LTPD. (4) i Ac

3 Where as X The umber of defective i first sample X The umber of defective i secod sample Ac The acceptace umber i first sample Re The rejectio umber i first sample Ac The acceptace umber i secod sample Re The rejectio umber i secod sample The sample size i first sample The sample size i secod sample ( ). X < Ac? X > Re? Reject X +X < Ac? X +X > Re? Reject Fig.3 Geeral procedure for Double samplig plas. 3. Desig variable acceptace samplig plas to cotrol a process mea. 3. Sigle (oe sided) specificatio limit.desig variable acceptace samplig plas whe sigle specificatio limit is used ad the operatig characteristic curve passes through the two poits ( α, X ), (, X ) where as α The probability of rejectig a lot that meets the specified quality level The probability of rejectig a lot that does ot meet the specified quality level X The average value of the quality characteristics for that the probability of acceptace is high X The average value of the quality characteristics for that the probability of acceptace is low X a The acceptace limit. Whe variace is kow ad process mea has ormal distributio. We ca be calculated samplig plas by solvig equatio 5 ad 6 simultaeously. Xa α (5)

4 ad Xa (6) Fig. 4 Relatioship of X ad X to X a i the samplig distributio of X. 3. Double (two-sided) specificatio limit. For desig of variable samplig plas whe double specificatio limits are obtaied ad OC Curve passes two poits ( α, X ), (, X L ) ad (, X U ). The sample size (), the upper acceptace limit ( X U ) ad the lower acceptace limit X ) solvig equatio (7), (8), (9) ad (0). Where as - X X X + X L U. Ua α / (7) - X La α / (8) XLa L (9) XUa U (0) ( L are calculated by Fig. 5 Relatioship betwee the various parameters for a pla usig double specificatio limits. 4. Desig variable acceptace samplig plas to cotrol the percet defective. 4. Variace kow. Desig variable samplig plas by usig the operatig characteristic curve (OC Curve) passes through the two poits ( - α, AQL ) ad (, LTPD ). Where as α The probability of rejectig a lot that meets the stated quality level. The probability of acceptig a udesirable lot. AQL The average quality level. LTPD The lot tolerace percet defective. k The critical value. L The sigle lower limit.

5 Whe variace is kow ad radom variable has ormal distributio ca calculate samplig pla from solvig equatio ad. If variace is ukow, we ca calculate samplig plas from equatio (3) ad (4). α + () α k, k where lot if L k, otherwise reject it. 4. Variace ukow. k + α + k α + L k, otherwise reject it. α + k k + k () (3) (4) lot if a. Distributio of quality characteristic, X b. Distributio of the sample average, X Fig. 6 Method Whe we use attribute or variable samplig plas for multiple products with the objective is miimize the maimum cosumer s risk ad subject to budget cost costrait. It ca be writte i form of the mathematical model as follow. Objective Fuctio : Miimize T Subject to : T i m ici B ad i ; i,,.., m. i where as i The cosumer s risk that occurs with product i. c i The cost of samplig for product i ( Baht / uit ). i The sample size for samplig product i ( uit ). B The budget which ca be available for samplig m products ( Baht ). m The umber of products which is ispected. j Whe used sigle samplig plas or variable samplig plas Whe used double attribute samplig plas ( ). The algorithm for desigs the multiple products samplig plas is show below.. Calculate sample sizes that ca possibly be i the pla regardig of the budget costraits as of equatio (5). i B j m c i + j ci i j ci i,,, m (5)

6 . Calculate the cosumer s risk for all possible sample sizes from equatio (5) for every product as follows.. Sigle ad double acceptace samplig plas for attribute. For each iterval of the product sample sizes, calculate the acceptace umber (Ac) for sigle acceptace samplig pla ad double acceptace samplig pla, regardig of equatio () ad (3), respectively. If the feasible sample size of product is 6 ad specific the AQL at %, LTPD at 0% ad the producer s risk is 5% Whe : varies the acceptace umber util equatio is satisfied. Ac with Ac 0 : 0.0 ( 0.0) > the, ad Ac 0. 0 Calculate the cosumer s risk for product whe use the sample size( ) ad the acceptace umber ( Ac ) 0. From equatio (), suppose that the LTPD 0%, we got the cosumer s risk as follow ( 0.) Whe : varies the acceptace umber util equatio is satisfied. Ac With Ac 0 : 0.0 ( 0.0) > so, ad Ac 0. 0 Calculate the cosumer s risk for product item whe the sample size is ad the acceptace umber is 0. Supposed that the lot tolerace percet defective is 0% the put ito equatio,the cosumer s risk will be 8.0% ( 0.) Calculate the cosumer s risk util 6. whe 6 : The ace umber is varied so the calculated probability are ivestigated as follows: Ac 6 6 Ac ( 0.0) < (ot valid) 6 6 Ac 0.0 ( 0.0) > (valid), therefore, 6 ad Ac. 0 Calculate the cosumer s risk for product item whe the sample size is 6 ad the acceptace umber is. Supposed that the lot tolerace percet defective is 0% ad put ito equatio,the cosumer s risk will be 88.5% ( 0.) Variable samplig plas for sigle specificatio limit to cotrol process mea. We ca calculate the acceptace limit ( X a ) from equatio (6) ad the cosumer s risk ( ) from equatio (7) ad (8). Xa X + α (6) Xa (7) Pr( < ) (8).3 Variable samplig plas for double specificatio limits to cotrol process mea. We ca calculate the upper acceptace limit ( X Ua ) ad the lower acceptace limit ( X La ) as XLa X α / (9) ad XUa X + α /. (0) So that, the cosumer s risk ca be obtaied from equatio () or (). XLa L ()

7 XUa U ().4 Calculate the cosumer s risk from equatio (3). ( ) α (3).5 Variable samplig plas i case of variace ukow to cotrol percet defective. We apply equatio (4) for obtaiig the cosumer s risk. ( + ) + (4 α + ( α α )) + ( + ) ( ) 3. The miimum maimum cosumer s risk for product,,, m which was ordered i step. These are ideed stage for calculatig samplig plas. The product which has maimum cosumer s risk will be ordered followig the cosumer s risk util the product which yields the miimum cosumer s risk will be assiged to the last stage or stage m. The sample size which was calculated from step, get these values to calculate the path which give miimum maimum cosumer s risk by backward dyamic programmig ad the recursive fuctio from equatio (5) ad (6). fm(i) m ( i) (5) ft (i) mi { ma [ t ( i), ft + ( j) ] } (6) j t ( i ) the cosumer s risk for stage t whe use i sample size. 5. Backtrackig to fid the path which has the miimum maimum cosumer s risk. The the sample size will m be checked that ici B. If the total cost is greater tha limited budget, decrease the sample size i stage by i oe ad assig cosumer s risk equal to ifiity the repeat step 5 util the total cost is less tha the budget. 0 (4) A eample The Five Stars maufacturig has 6 products to ispect before sedig the products to the cosumers. The products will be ispected by the attribute or variable samplig plas depeded o its characteristics.. The products that will be ispected by sigle attribute samplig plas are show as follow.. Product A00: AQL 5%, LTPD 0%, α 5% ad the samplig cost is $0 per uit.. Product A00: AQL %, LTPD 0%, α 5% ad the samplig cost is $50 per uit.. The products that will be ispected by variable samplig plas to cotrol process mea are follow.. Product A003: X L 80,000 N / m, XU 00,000 N / m ad the stadard deviatio is 5,000 N/m, α 5%.The samplig cost is $500 per uit.. Product A004: X Kg., X 0.55 kg. ad the stadard deviatio is 0.05 kg, α 5% with the samplig cost is $50 per uit. 3. The products that will be ispected by variable samplig plas to cotrol percet defective are follow. 3. Product A005: AQL %, LTPD %, α 5% ad the samplig cost is $30 per uit. 3. Product A006: AQL 5%, LTPD 0%, α 5% ad the samplig cost is $0 per uit. The budget for samplig ispectio is ot greater tha $50,000.What is the acceptace samplig plas for these products with miimize maimum cosumer s risk? To solve this problem, we employed the applied dyamic programmig algorithm that was show previously.the results of the calculatio are show i table : Table: The results of the calculatio. Product umber sample size decisio criteria cosumer s risk samplig cost A00 39 c % $,390

8 A00 74 c % $8,700 A003 8 XUa 86, N/m X La 93,465.5 N/m 0.0 % $4,000 A004 8 X a 0.63 Kg % * $,400 A k % $,40 A k % $880 All of the cosumer s risk that occur, the miimum maimum cosumer s risk is 0.03 %. The total cost that will be used i ispectio activity is $7,50. Coclusio The attribute ad variable acceptace samplig plas for samplig multiple products that are ideed by cost of ispectio ad yieldly the miimum maimum cosumer s risk so that the producer ad the cosumer ca be both appreciated. The producer ca obtai the samplig plas uder the budget cost which are restricted ad the cosumer will have the least chace of accept the worse products. Further studies may be solved as same as the previous algorithm such as restrict i time of ispectio. It ca be calculated by chagig the ispectio cost per uit by the time of ispectio per uit. Refereces [] Bak, J. : Priciples of Quality Cotrol. Joh Wiley ad So, Ic., New York. 643 p., 989. [] Dodge, H.F. ad H.G. Romig. : Samplig Ispectio Tables. Joh Wiley ad So, Ic., New York. 4 p., 998. [3] Duca, A.J. : Quality Cotrol ad Idustrial Statistics. Richard D. Irwi, Ic., Homewood, Illiois. 3p [4] Grat,E.L. ad R.W. Leaveworth. : Statistical Quality Cotrol. McGraw Hill,Ic., New York. 70p [5] Mitra, A. : Fudametal of Qualitty Cotrol ad Improvemet. Mac milla, Ic., New York. 690p [6] Motgomery, D.C. : Itroductio to Statistical Quality Cotrol. Joh Wiley ad So, Ic., New York. 677 p., 996. [7] Paitoo Howyig ad Prapaisri Sudusa Na - Ayudthaya; The attribute samplig plas for multiple products, First Natioal Symposium o Graduate Research, pp66-76, 000. [8] Paitoo Howyig ad Prapaisri Sudusa Na - Ayudthaya; The variable samplig plas for multiple products, IE Network Natioal Coferece, pp655-66, 00. [9] Wisto, W.L. : Operatio Research. Iteratioal Thomso Publishig. 378p. 994.