A New Markov Chain Based Acceptance Sampling Policy via the Minimum Angle Method

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1 Irnn Journl of Opertons Reserch Vol. 3, No., 202, pp. 04- A New Mrkov Chn Bsed Acceptnce Smplng Polcy v the Mnmum Angle Method M. S. Fllh Nezhd * We develop n optmzton model bsed on Mrkovn pproch to determne the optmum vlue of thresholds n proposed cceptnce smplng desgn. Consder n cceptnce smplng pln where tems re nspected nd when the number of conformng tems between successve defectve tems flls below lower control threshold vlue, then the btch s rejected, nd f t flls bove control threshold vlue, then the btch s ccepted nd f t flls wthn the thresholds, the process of nspectng the tems contnues. A decson s mde to ccept or reject the btch. We begn wth developng Mrkov model for determnng performnce mesures of smplng desgns, resultng n n cceptnce smplng pln optmzed bsed on the mnmum ngle method. Then, the performnce mesures of the cceptnce smplng pln re determned nd the optmum vlues of thresholds re selected n order to optmze the objectve functons. In order to demonstrte the pplcton of the proposed methodology, numercl exmples re llustrted. Keywords: Qulty control, Mrkovn model, Sttstcl process control, Acceptnce smplng pln. Mnuscrpt receved on 6/0/202 nd ccepted for publcton on 2/04/202.. Introducton Acceptnce smplng plns re sttstcl tools for rectfyng the qulty ssurnce. The smplng plns provde the vendor nd buyer wth decson rules for product cceptnce to meet the present product qulty requrements. Severl types of decson rules hve been proposed for the cceptnce smplng problem but work on usng the number of successve conformng tems to control qulty of the receved lot bsed on the mnmum ngle method s scre. The de of usng the run-lengths of successve conformng tems s n ndctor of process performnce hs been round for long tme (Bourke [2]). Clvn [6] proposed control chrt bsed on the run-lengths of successve conformng tems. Goh [] proposed chrtng procedure to control the low-nonconformty producton. Bourke [4] proposed montorng sttstcs bsed on the sums nd CUSUMs of such conformng runlengths for the cse of 00% nspecton. Bourke [2] noted tht bsed on the conventonl mesures of performnce of smplng plns such s the verge outgong qulty, verge frcton nspected, nd the proporton pssed under smplng nspecton, the de of usng the run-lengths of successve conformng tems s n ndctor of process performnce turns to better performnce. Also, Bourke [3] proposed swtchng rules bsed on cumultve sum of the observed run-lengths of conformng tems between successve defectve tems. In recent decdes, pplyng cceptnce smplng methods hve rsed number of questons n qulty control. The mn trget beng producton specfcton nd reducton of mnufcturng tolernces, n mny cses becuse of humn nd mnufcturng system errors, cceptnce smplng s desred method (Arshd Khmseh et l. []). Vrdermn (20) nd Schllng (6) worked on how much ccurcy of the cceptnce smplng desgns s used n prctcl envronments. Hmlton nd Lespernce (2) developed method for sngle nd mult vrble cceptnce smplng ssumng tht the process qulty cn be determned from the number of defects n the lot whle the vrnce nd men re known. * Deprtment of Industrl Engneerng, Yzd Unversty, Yzd, Irn. Eml: Fllhnezhd@Yzdun.c.r

2 A New Mrkov Chn Bsed Acceptnce Smplng 05 Tgres (8) proposed n economcl model for the sngle vrble cceptnce smplng pln bsed on the Tguch loss functons n the bsence of nspecton errors. Klssen (3) proposed credtbsed cceptnce smplng system. The credt of the producer ws defned s the totl number of tems ccepted snce the lst rejecton. In ths system, the smple sze from lot s gven by smple functon on the lot sze, the credt nd the chosen gurnteed upper lmt on the outgong qulty. Nk nd Fllhnezhd (5) ppled Byesn nferences concept to desgn n cceptnce smplng desgn. They used stochstc dynmc progrmmng model to mnmze the rto of the system cost to the system correct choce probblty. Fllhnezhd nd Hossennsb (7) proposed sngle stge cceptnce smplng pln bsed on the control threshold polcy. The objectve of ther model s to desgn n economc cceptnce smplng model. Fllhnezhd nd Nk (7) proposed Mrkov model for sngle smplng pln bsed on the control threshold polcy consderng the run-lengths of successve conformng tems s n ndctor of the process performnce. Fllhnezhd et l. (9) extended ths pproch to the sum of run-lengths of successve conformng tems. Fllhnezhd et l. (0) proposed decson tree pproch for desgnng the economc models of smplng plns. Some uthors ppled the Mrkovn models to mchne mntennce polcy to derve the optml process control. Tgrs (9) studed the jont process control nd mchne mntennce problem of Mrkovn deterortng mchne. Kuo (4) developed n optml dptve control polcy for the jont mchne mntennce nd product qulty control. Bowlng et l. (5) proposed Mrkovn pproch to determne the optml process trget levels for mult-stge serl producton system. Here, generl model for cceptnce smplng plns s developed ncorportng the number of conformng tems between successve defectve tems n ts desgn. It s ssumed tht when the number of conformng tems between successve defectve tems s more thn control threshold vlue, then the btch s ccepted nd when t s less thn control threshold vlue, then the btch s rejected. Ths pper provdes n optmzed cceptnce smplng pln for gven vlues of the cceptble qulty level nd lmtng qulty level usng the mnmum ngle technque. The rest of the pper s orgnzed s follows. We present the prelmnres n Secton 2 nd gve the model development n Secton 3. The proposed methodology s descrbed n Secton 4. Secton 5 provdes summry of results for the proposed method. 2. Prelmnres Our nottons re summrzed below: p: the proporton of the defectve tems n the btch AQL: the mxmum cceptble level of the btch qulty (Accepted Qulty Level) LQL: the mnmum rejectble level of the btch qulty (Lmtng Qulty Level) P: trnston probblty mtrx Q: squre mtrx contnng trnston probbltes of gong from ny non-bsorbng stte to ny other non-bsorbng stte R: mtrx contnng ll probbltes of gong from ny non-bsorbng stte to n bsorbng stte (.e., ccepted or rejected btch) A: n dentty mtrx representng the probblty of styng n stte O: mtrx representng the probbltes of escpng n bsorbng stte (lwys zero)

3 06 Fllh Nezhd M: fundmentl mtrx contnng the expected number of trnstons from ny non-bsorbng stte to ny other non-bsorbng stte before n bsorpton F: the bsorpton probblty mtrx contnng the long run probbltes of the trnston from ny non-bsorbng stte to ny bsorbng stte p : probblty of gong from stte to stte j n sngle step j mj p : expected number of trnstons from ny non-bsorbng stte to ny other non-bsorbng stte (j) before bsorpton occurs when proporton of the defectve tems s p f j p : long run probblty of gong from ny non-bsorbng stte to ny bsorbng stte j when proporton of the defectve tems s p. 3. Model Development The model s to develop Mrkovn pproch for determnng n optml vlue of threshold for the cceptng or rejectng the lot. Assume tht n n cceptnce smplng pln, Y s defned s the number of conformng tems between the successve (-)th nd th defectve tems. The decson rule s defned s follows: If Y U, then the lot s n good stte nd ccepted. If Y L, then the lot s n bd stte nd rejected. If U Y L, then nspecton of the tems contnues; where, U s n upper control threshold nd L s lower control threshold vlue. The sttes of the problem s defned s follows: Stte ( U Y L ) : the vlue of Y s between the control thresholds L nd U nd thus nspecton of the tems contnues. Stte 2 ( Y U ) : the btch s n good stte nd ccepted. Stte 3 ( Y L ) : the btch s n bd stte nd rejected. Thus, t s concluded tht: probblty of nspectng more tems = p P U Y L, probblty of cceptng the btch = p P Y U, () 2 probblty of rejectng the btch= p3 P Y L r where P Y r p p,, wth p denotng the proporton of the defectve tems n the lot. The trnston probblty mtrx mong the sttes of the lot s determned s follows: P p p p 0 0. (2) The mtrx P s n bsorbng Mrkov chn wth sttes 2 nd 3 beng bsorbng nd stte beng trnsent. To nlyze ths mtrx, frst the trnston probblty mtrx s rerrnged nto the followng form: A O R Q. (3) Rerrngng the P Mtrx results n the followng mtrx:

4 A New Mrkov Chn Bsed Acceptnce Smplng (4) p2 p3 p The fundmentl mtrx M cn be determned s follows (Bowlng et l., 2004): M m p I Q, (5) p P U Y L where I s the dentty mtrx. The vlue m p denotes the expected number of tmes n the long run tht the trnsent stte s occuped before bsorpton occurs (.e., the btch ccepted or rejected), gven tht the ntl stte s. The long-run bsorpton probblty mtrx F s clculted s follows (Bowlng et l., 2004): p2 p 3 F M R f 2 p f 3 p (6) p p f p, f p denote the probbltes of the btch beng ccepted nd The elements of mtrx F, 2 3 rejected, respectvely. The prctcl performnce of ny smplng pln s determned through ts opertng chrcterstc curve. When producer nd consumer re negottng for desgnng smplng plns, t s mportnt speclly to mnmze the consumer s rsk. In order to mnmze the consumer s rsk, the del OC curve could be mde to pss s closely through [AQL, α] nd [AQL, β]. One pproch to mnmze the consumer s rsks for del condton s mnmzton of ngle between the lnes jonng the ponts [AQL, α], [AQL, β] nd [AQL, α], [LQL, β]. In ths cse, the vlue of the performnce crteron n the mnmum ngle method s: Cot where, P LQL P AQL, LQL - AQL (7) P LQL P AQL re the probbltes of cceptng the btch when the proporton of defectve tems n the btch re respectvely LQL, AQL. Assume A s the pont [AQL, α], B s the pont [AQL, β] nd C s the pont [LQL, β]. Thus the smller the vlue of Cot, the closer the ngle pprochng zero, the del condton for the chord AC pprochng AB. For exmple, n desgn, we hve P LQL P AQL shown n Fgure , The smple plot of the OC curve s It s seen tht s pproches zero, the chord AC pproches the, rechng the del condton. The vlues of P LQL, P AQL re determned s follows: P U Y p AQL P AQL f 2 AQL P U Y L (8) P U Y p LQL P LQL f 2 LQL P U Y L Snce the vlues of LQL, AQL re constnt nd LQL AQL, therefore the objectve functon s determned s to be,

5 08 Fllh Nezhd V Mn P LQL P AQL. (9) L, U P AB AC p Fgure. The smple plot of the OC curve Another performnce mesure for the cceptnce smplng pln s the expected number of nspected tems. Snce smplng nd nspectng usully ncur cost, therefore desgns mnmzng ths mesure whle stsfyng the frst nd the second error nequltes re consdered to be optml smplng plns. Snce the proporton of defectve tems s not known n the strt of the process, n order to consder ths property n desgnng the cceptnce smplng plns, we try to mnmze the expected number of nspected tems for cceptble nd rejectble lots smultneously. Therefore, the optml cceptnce smplng pln should hve certn propertes: t should hve mnmzed vlue for the objectve functon of the mnmum ngle method tht s resulted from the del OC curve. It should lso mnmze the expected number of nspected tems ether n the decson of rejectng or cceptng the lot. Therefore, second objectve functon s defned s the expected number of tems m p, where nspected. The vlue of ths objectve functon s determned bsed on the vlue of m p s the expected number of tmes n the long run tht the trnsent stte s occuped before bsorpton occurs. Snce for vsts to trnsent stte, the verge number of nspectons s p, the expected number of tems nspected s gven by m p. Now, the objectve functons, W nd Z p re defned s the expected number of tems nspected respectvely n the cceptble condton p AQL p LQL : nd rejectble condton W Mn m AQL, L, U AQL (0) Z Mn m LQL. L, U LQL One pproch to optmze the objectve functons smultneously s to defne control lmts for the objectve functons Z, W nd then try to mnmze the vlue of the objectve functon V. For

6 A New Mrkov Chn Bsed Acceptnce Smplng 09 exmple, f prmeters Z, W re defned s the upper control lmts for Z, W, respectvely, then the optmzton problem cn be defned s follows: MxV L, U s. t. () Z Z, W W. Optml vlues of L, U cn be determned by solvng the bove nonlner optmzton problem usng serch procedures or other optmzton tools. Therefore, there re three objectve functons nd the optml desgn s bsed on the vlue of these objectve functons. The frst objectve functon s defned s, V Mn P LQL P AQL L, U where, P AQL P U Y. P U Y L re determned usng the fct tht Y follows geometrc dstrbuton wth the success probblty beng equl to p AQL. Also, P LQL s determned by smlr resonng. The second objectve functon s defned s W Mn m AQL, where m AQL. Agn, the probblty L, U AQL P U Y L The probbltes P U Y nd P U Y L P U Y L s determned usng the fct tht success probblty beng equl to p AQL rgument. 4. A Numercl Exmple Y follows geometrc dstrbuton wth the. The thrd objectve functon s defned by smlr To demonstrte the pplcton of the proposed methodology n n cceptnce smplng desgn, numercl exmple s solved. Consder smplng problem, where AQL 0.05, LQL 0.2. The vlues of the objectve functons for dfferent lterntve vlues of L nd U mong the exstng lterntves re obtned usng (9) nd (0). Tble shows 6 dfferent lterntve combntons of L nd U together wth ther objectve functons vlues. The optmzton model s used to determne the optml soluton. Assume tht the vlues of control lmts re Z 6, W 2. The fesble vlues of L nd U mong the exstng lterntves re obtned usng the constrnts of the optmzton model (). These fesble vlues re shown n Tble 2. Bsed on the results, the best combnton vlue s L 3 nd U 5 wth the mxmum vlue of V= As mentoned, the optml cceptnce smplng desgn s determned bsed on the vlues of the three objectve functons. In other words, the desgn wth the optmzed vlue for ech objectve functon s selected s optml. Snce the optmum of the objectve functons s not expected to occur t sngle pont, we need some compromsed methods n optmzng the objectve functons.

7 0 Fllh Nezhd Z Tble. Objectve functons vlues W V L U Tble 2. Fesble vlues of L nd U mong the exstng lterntves Z W V L U Methods used to optmze the objectve functons cn be nterctve to consder prortes of the objectves. In generl, snce the OC curve s the most mportnt performnce mesure of smplng plns, the method whch mxmzes the objectve functon V consderng the two nequltes of expected number of tems nspected s suggested to be ppled n prctce. 5. Concluson Here, new pproch for cceptnce smplng pln ws ntroduced. The number of conformng tems between successve defectve tems ws selected s the decson mkng crteron. Three objectve functons were developed to optmze the performnce mesures of the cceptnce smplng pln. Then, procedure ws proposed to optmze the objectve functons smultneously. In the cse tht group nspecton ws mpossble nd the tems were to be nspected consecutvely, the proposed pproch s sutble lterntve. Snce the proposed cceptnce smplng pln consders dfferent performnce mesures of cceptnce smplng pln, t s benefcl for the prcttoners to use t n the qulty control envronments. For further reserch, t s recommended to use the cumultve sum of the observed run-lengths of conformng tems between successve defectve tems s the ndctor of the process performnce. Also, economc desgn of cceptnce smplng pln usng the proposed pproch s nother subject for future reserch.

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