A new approach of chain sampling inspection plan

A new approac h of c hain sampling insp ection plan Harsh T ripat hi a and Mahendra Saha a ∗ a Departmen t of Statisti cs, Central Univ ersit y of Ra jasthan, Ra jasthan, India Abstract T o dev elop decisio n rules rega rding accepta nce or rejec tion of pro du ction lots based on sample data is the pu rp ose of acceptance sampling in s p ection plan. Dep endent sampling procedu res cumulate results from sev eral preceding pro du ction lots when testing is exp en s iv e or destructiv e. This c haining of past lots redu ce the sizes of the required s amp les, essen tial for acceptance or rejection of p ro duction lots. In this article, a new approac h for c haining the past lot(s) results prop osed, named as mo d ified chai n group acceptance sampling insp ection plan is more effectiv e than wid ely used plan. S ev eral prop erties of op erating c haracteristic cur v es are deriv ed. A comparison stu dy has b een done b et w ee n the p rop osed and group acceptance samplin g insp ection plan as well as single acceptance sampling insp ection plan. A example h as b een giv en to illustrate the prop osed plan in a go o d mann er. Keyw ords: Consumer’s risk, life test, o p erating c haracteristic curv e, pro ducer’s risk. ∗ Corresp o nding author e-mail: mahendr asaha@c ura j.ac.in 1 Abbreviatio n s: ASIP : Acceptance sampling insp ection plan. SASIP : single acceptance sampling insp ection plan. D ASIP : Double acceptance sampling insp ection plan. GASIP : Group acceptance sampling insp ection plan. SEASIP : Sequen tial acceptance sampling insp ection plan. ChSP : Chain sampling plan. MChSP : Mo dified c hain sampling plan. MChGSP : Mo dified c hain g roup sampling plan. OC : Op erating c haracteristic. 1 In tro ducti o n ASIP is classifies in tw o broad areas: sampling insp ection plan by attribute a nd sam- pling insp ection plan b y v ariable. In literature man y attribute sampling insp ection plans are a v ailable viz., SASIP , DASIP , GASIP , SESAIP etc., while v ariable sampling plan uses the accurate measuremen ts of qualit y characteristics for decision-making rather t ha n classifying the pro ducts a s conforming or non-conforming. Both type (attribute and v ariable) sampling insp ection plan are used for sen tencing a lot based on sample of that lot. Plan para meters of b oth (attribute and v ariable) sampling insp ection plan are determined with the help of t w o p oin t approa ch. Man y research er hav e discussed the time truncated SASIP and some of them listed here, namely , G upta(1962), Gupta et al.(1961), Rosaiah et al. (2005), Ts ai et al. (2006), Baklizi et al. (2004), Balakrishnan et al.(2 0 07), Aslam et al. (2010) and Al-omari (2015). Man y authors ha ve discussed the time truncated D ASIP namely , GS Rao (2011), Ramasw amy et al. (2 0 12) and G ui (20 14). Also, man y r esearche r s ha ve studied GASIP with time truncated life test and r eaders may refer to Aslam et al. (2009) fo r gamma distribution, Aslam et al. (2009 , 2011) for W eibull and Birn baum-Saunders distributions, Ra o (2 011) fo r Marshall-Olkin extended expo nen- tial distribution. 2 ChSP first in tro duced by Do dge (1955), a lso kno wn as ChSP-1 plan. ChSP-1 is a plan with zero acceptance num b er sampling insp ection plan a nd dev elop ed fo r the inspec- tion b y attribute as we ll as by v ariable [see, Go vindara ju (2006) and Govindara j u, Balam ura li (19 98)]. ChSP-1 insp ection plan dep ends on c hain past lot results, i.e., qualit y o f pa st lot or insp ection of past lot play s an impor tan t role in t he deci- sion making pro cess of sen tencing a lot. Balamurali and Usha (2013), Go vindara j u and Subramani (1993 ) ha v e done the computation of tables and results consider- ing ChSP-1 plan. In ChSP-1 uses past results only when a non-confo rming unit is observ ed in curren t sample. Go vindra ju and Lai (19 98) hav e dev elop ed a mo dified v ersion of ChSP-1 plan, is know n as MChSP-1 plan. Ho w ev er the MChSP-1 pla n can only b e used for insp ection b y attributes and their selection is only studied under the condition o f a P oisson mo del. No w Luca ( 2018) has dev elop ed an extens ion o f MChSP-1 plan, known a s mo dified sampling (MChSP) plan. MChSP is applicable in b oth attribute and v ariable insp ection. In this article we hav e dev elop ed a new sampling insp ection plan whic h is com bina- tion of MChSP and ordinary GASIP sampling insp ection plan, named as MChGSP . Prop osed plan is applicable in case of ASIP by a t tribute. Rest of the article is organized as follows : In section 2, design of the prop ose plan is discussed. Descrip- tion of tables and the comparison of the prop osed pla n with GASIP and SASIP are discusse d in section 3. In sec tion 4, w e ha v e giv en a example to understand the metho dology and the applicability of the prop osed plan. Finally , concluding words ab out the findings of the prop osed study is placed in section 5. 2 Design o f mo difie d c hain group sampling plan In this section, we ha v e prop osed a new sampling insp ection plan, named as MChGSP . Plan parameters of MChGSP a r e the n umber o f groups ( f ), acceptance num b er ( c ) and n um b er of chained sample results ( i ) resp ectiv ely . A MChGSP plan is deter- mined b y the triple of natural n umbers ( f , c , i ). No w, the pro cedure o f MChGSP is follows : 3 1. Select n items f rom a particular lot a nd allo cate e items t o f g roups, i.e, n = e × f . Start with normal insp ection for pre-fixed exp erimen t time t . 2. Inspect all the g roups sim ultaneously and record the n um b er of non-conforming units (d) upto pre-fixed exp erimen t time. 3. Go to MChSP insp ection plan, If d ≤ c the lot is accepted prov ided that t here is at-most 1 lot among the preceding i lots in whic h the n umber of defectiv e units d exceeds the criterion c, otherwise reject the lot. No w, the probability of acceptance of GASIP is obtained by the f ollo wing Equation: E = c X i =0  ef i  p i (1 − p ) ( ef − i ) (2.1) where, p is the probability that observ ed n um b er of failures o ccurs b efore t he exp er- imen tal time t Expression of OC function of the prop osed plan is give n b elo w: E a ( p ) = c X i =0  ef i  p i (1 − p ) ( ef − i )   c X i =0  ef i  p i (1 − p ) ( ef − i ) ! i + i c X i =0  ef i  p i (1 − p ) ( ef − i ) ! ( i − 1) 1 − c X i = 0  e f i  p i ( 1 − p ) ( e f − i ) ! )   Therefore, w e can use tw o-p oint approac h (at AQL and LQL ) to determine the plan parameters of the prop osed plan by using the following non-linear optimization prob- lem: Minimize, ASN: n = f × e (2.2) E a ( p 0 ) = E 0 ( E i 0 + iE i − 1 0 (1 − E 0 )) ≥ (1 − α ) (2.3) 4 E a ( p 1 ) = E 1 ( E i 1 + iE i − 1 1 (1 − E 1 )) ≤ β (2.4) In ab ov e optimizing problem, our aim is to minimize sample size n and n dep ends on n umber o f groups g for g iven group size r , i.e., w e hav e to minimize the n um b er of gro ups g in suc h a w ay that g satisfies ab ov e optimization problem for giv en r . 3 Descripti on of T ables In this section, w e hav e describ ed t he ta bles. T able 1 r epresen t s the plan para meters of the prop osed plan for the group size ( r = 5) when consumer’s risk ( α = 0 . 0 5) and pro ducer’s risk ( β = 0 . 10) are prefixed and for the giv en v a lue of A QL ( p 0 ) and LQL ( p 1 ). Num b er of groups decreases for the fixed A QL and v arying v alues of LQL. T able 2 sho ws the comparison study among the prop osed plan, ordinary GASIP and SASIP . This comparison show s that the prop osed plan p erform b etter than ordinary G ASIP a nd SASIP . In some cases, prop osed plan required same num b er of gro ups as in ordinary G ASIP and SASIP for the same set of v alues of A QL and LQL for sen t encing the lot. W e w ould prefer to use prop osed plan ov er the ordinary GASIP and SASIP for the reason that past infor ma t ion pla ys significan t role to tak e a decision ab out lot in prop osed MChGSP plan ra ther than to tak e decision on the basis of curren t sample. 4 Example Supp ose that the pro ducer’s risk ( α ) and consumer’s risk ( β ) are 0 . 0 5 and 0 . 1 0 resp ectiv ely . V alues of A QL ( p 0 ) and LQL ( p 1 ) are 0 . 05 and 0 . 14 resp ectiv ely whic h are known to exp erimen ter to apply the t wo p oint approac h for the es timation of the plan parameters of prop osed plan. F ro m table 1, plan para meters are f = 13, c = 6 and i = 2 f or the prefixed group size e = 5. Based on theses obtained plan parameters, MChGSP is: • Select a sample of size 6 5 from a submitted lo t. Allo cat e 5 items to 13 g roups, i.e, n = r × f and start with normal inspection. 5 • Inspect all the g roups sim ultaneously and record the n um b er of non-conforming units (d). • Go to MChSP insp ection plan, If d ≤ 6 the lot is accepted pro vided that there is at-most 1 lot among the preceding 2 lots in whic h the n um b er of defectiv e units d exceeds the 6, o t herwise reject the lot. 5 Conclusi ons In this ar ticle, we ha ve in tro duced a new sampling inspection plan, name a s MChGSP . W e compared the prop o sed plan with exist ordinary GASIP and SASIP . MChGSP pro vides flexibilit y to reac h a decision regarding lot acceptance or rejection with the minim um n umber of groups by using past results. 6 Reference 1. Aslam, M., Kundu, D., Ahmed, M.(2010). Time truncated acce ptance sam- pling pla ns for generalized exp onen tial distribution, Journal of Applie d Statis- tics , 37(4) , 555-566 . 2. A.I, Al-Omari.(2015). Time truncated acceptance sampling plans for general- ized in v erted exp onen tial distribution, Ele ctr onic Journal of Applie d Statistic al A nalysis, , 8( 1) , 1-1 2 . 3. Aslam, M., Jun, C.H. and Ahmad, M. (2009). A G roup sampling pla n based on truncated life test for gamma distributed items. P akistan Journal of Sta t istics. 2009; 2 5(3): 333-340. 4. Aslam, M., and Jun, C.-H. ( 2 009). A g roup acceptance sampling plan for truncated life test hav ing W eibull distribution. Journal of Applie d Statistics , 36(9), 1 021-102 7. 6 5. Aslam, M., Jun, C.-H. and Ahmad, M. (2 0 11). 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Acc eptance sampling plan based on truncated life tests for generalized Ra yleigh distribution, Journal of Applie d Satistics, 33 , 595-600 . 8 T able 1: The plan para meters (g,c,i) of MChGSP for r=5 and α = 0 . 0 5 and β = 0 . 10. A QL ( p 0 ) LQL ( p 1 ) g c i P a ( p 0 ) P a ( p 1 ) 0 . 01 0 . 02 120 10 3 0 . 9533257 0 . 0949371 0 . 03 66 7 2 0 . 980437 7 0 . 0900610 0 . 04 44 6 2 0 . 992817 6 0 . 0861985 0 . 05 32 4 1 0 . 976980 2 0 . 0938539 0 . 06 22 3 1 0 . 974961 9 0 . 0980303 0 . 07 15 2 1 0 . 960330 9 0 . 0967880 0 . 05 0 . 10 24 10 3 0 . 9573947 0 . 088370 0 0 . 12 18 8 2 0 . 96257 8 0 . 0962938 0 . 14 13 6 2 0 . 954920 5 0 . 0574218 0 . 18 10 5 1 0 . 962223 8 0 . 0928591 0 . 20 8 4 1 0 . 9 519717 0 . 0759 145 0 . 10 0 . 20 12 10 3 0 . 9624775 0 . 079595 9 0 . 25 9 8 3 0 . 9 670151 0 . 0543 979 0 . 30 7 7 2 0 . 9 796181 0 . 0328 563 0 . 35 5 5 2 0 . 9 666001 0 . 0826 247 0 . 38 4 4 1 0 . 9 568255 0 . 0726 116 0 . 15 0 . 30 8 10 3 0 . 9675257 0 . 07011 7 39 0 . 40 5 8 3 0 . 991839 0 . 05 43979 0 . 50 3 5 2 0 . 9 829121 0 . 04209 421 0 . 55 2 3 1 0 . 9 500302 0 . 09084 266 9 T able 2: Comparison of prop o sed plan with existing GASIP and SASIP r=5 r=1 A QL ( p 0 ) LQL ( p 1 ) MCh-GSP GASIP SASIP n = g × r n = g × r n = g × r 0 . 01 0 . 02 600 = 120 × 5 −− — 0 . 03 330 = 66 × 5 395 = 79 × 5 399 0 . 04 220 = 44 × 5 325 = 65 × 5 262 0 . 05 160 = 32 × 5 195 = 39 × 5 160 0 . 06 110 = 22 × 5 135 = 27 × 5 110 0 . 07 75 = 15 × 5 80 = 16 × 5 75 0 . 05 0 . 10 120 = 24 × 5 −− — 0 . 12 90 = 18 × 5 −− — 0 . 14 65 = 13 × 5 −− — 0 . 18 50 = 10 × 5 50 = 10 × 5 50 0 . 20 40 = 8 × 5 40 = 8 × 5 40 0 . 10 0 . 20 60 = 12 × 5 −− — 0 . 25 45 = 9 × 5 −− — 0 . 30 35 = 7 × 5 35 = 7 × 5 41 0 . 35 25 = 5 × 5 25 = 5 × 5 25 0 . 38 20 = 4 × 5 20 = 4 × 5 20 0 . 15 0 . 30 40 = 8 × 5 −− — 0 . 40 25 = 5 × 5 30 = 65 × 5 30 0 . 50 15 = 3 × 5 −− 17 0 . 55 10 = 2 × 5 −− — 10