Mohammad Saber Fallah Nezhad * & Hasan Hosseini Nasab

Transcription

1 Intrnational Journal of Industrial Enginring & Prodution Rsarh Sptbr 0, Volu Nubr 3 pp ISSN: Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 Dsigning a Singl Stag Aptan Sapling Plan Basd on th Control Thrshold Poliy Mohaad Sabr Fallah Nhad * & Hasan Hossini Nasab Mohaad Sabr FallahNhad, Dpartnt of Industrial Enginring, Faulty of Enginring, Yad Univrsity, Yad, Iran Hasan Hossini Nasab, Dpartnt of industrial Enginring, Faulty of Enginring, Yad Univrsity, Yad, Iran, KEYWORDS Quality Control, Aptan Sapling Modl, Control Thrshold Poliy. Introdution Aptan sapling is a produr usd for sntning inoing baths. Sapling plan onsist of a sapl si and a dision aking rul. Th sapl si is th nubr of its to sapl or th nubr of asurnts to tak. Th dision aking rul involvs th aptan thrshold and a dsription of how to us th sapl rsult to apt or rjt th lot. Aptan sapling plans ar also pratial tools for quality ontrol appliations, whih involv quality ontrating on produt ordrs btwn th vndor and th buyr. Thos sapling plans provid th vndor and th buyr ruls for lot sntning whil ting thir prst rquirnts on produt quality. Various thods of insption to iprov th quality of its ar prsntd by any rsarhrs. Klassn [] proposd a rditbasd aptan sapling syst in whih th rdit of th produr was dfind as th total nubr of its aptd sin last rjtion. Tagaras [] studid th pross ontrol and ahin aintnan probl of a Markovian dtriorating ahin. Kuo [3] dvlopd an adaptiv ontrol poliy for ahin * Corrsponding author. Mohaad Sabr FallahNhad Eail: fallahnhad@yaduni.a.ir Papr first rivd Jun., 0, and in rvisd for April. 5, 0. ABSTRACT In this rsarh, a nw ontrol poliy for th aptan sapling probl is introdud. Dision is ad basd on th nubr of dftivs its in an insptd bath. Th objtiv of th odl is to find a onstant ontrol lvl that iniis th total osts, inluding th ost of rjting th bath, th ost of insption and th ost of dftiv its. Th optiiation is prford by approiating th ngativ binoial distribution with Poisson distribution and using th proprtis of binoial distribution. A solution thod along with nurial donstration on th appliation of th proposd thodology is prsntd. Furthror, th rsults of snsitivity analysis show that th proposd thod nds a larg sapl si. 0 IUST Publiation, IJIEPR, Vol., No. 3, All Rights Rsrvd. aintnan and produt quality ontrol. Frrll and Chhokr[] proposd an onoially aptan sapling plan basd on Taguhi loss funtion to quantify dviations btwn a quality haratristi and its targt lvl. Parn and Wu [5] introdud a variabl sapling plan for unilatral prosss basd on th on-sidd pross apability indis. Niaki and Fallahnhad [] usd th Baysian infrns onpt to dsign an optial sapling plan. Thy forulatd th probl into a stohasti dynai prograing odl to inii th ratio of total disountd syst ost to a disountd syst orrt hoi probability. Moskowit and Tang [7] proposd aptansapling plans basd on Taguhi loss funtion and Baysian approah. Ainadh [8] proposd aptan-sapling plans basd on th assuption that onsutiv obsrvations on a quality haratristi ar auto orrlatd. H obtaind th sapling plans basd on th ARMA odl and suggstd two typs of aptan sapling plans: () non-squntial aptan sapling and () squntial aptan sapling basd on th onpt of squntial probability ratio tst (SPRT). MWillias t al. [9] providd a thod of finding at dsigns for singl sapl aptan sapling plans. Willia t al. [0] dvlopd athatial odls that an b

2 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 usd to dsign both 00% insption and singl sapling plans. In thir rsarh, insption rror is pliitly inludd in th odl as is th ability to itigat th onsquns by th pnding rsours. In this rsarh, a nw poliy for th aptan sapling probl is introdud. Th objtiv of th odl is to find a onstant ontrol lvl that iniis total osts, inluding th ost of rjting th bath, th ost of insption and th ost of dftivs its. With attribut sapling plans, ths apt/rjt dision prfors basd on ounting of th nubr of dftiv its in a bath. Th assuptions and drivations of th proposd thod ar prsntd in stion, th solution algorith along with nurial donstration on th appliation is prsntd in stion 3, and disussion and onlusion of th rsults ar prsntd in stions.. Proposd Modl W suppos a bath of si n is rivd whih its proportion of th dftivs its is qual to p. For a bath of si n, rando variabl Y is dfind as th nubr of insptd its and is dfind as th nubr of its lassifid as 'dftiv' aftr insption. Th nubr of insptd its has an uppr thrshold qual to. For Y,,..., insptd its ( n ) th bath will b rjtd if whr is th uppr ontrol lvl for bath aptan. In th othr words, whn th nubr of dftiv its in th insptd its gts or than th ontrol thrshold thn dision aking pross stops and th bath is rjtd. Th probability distribution funtion of Y is dtrind by th following quations, dftiv it has bn Y th insptd it thus in this as, th probability distribution funtion of Y follows a ngativ binoial distribution. Th ptd an of th nubr of insptd its is dtrind as follows: EY [ ] ( ) p p 0 p ( p) Y Y Y p ( p) Y p ( p) 0 Y Y Y p ( p) Y () Y Y Sin Pr{ Y } p ( p) Y < is a ngativ binoial distribution thus using th approiation thod of stiating ngativ binoial probabilitis with Poisson distribution [], following is onludd, Pr Y (3) Γ { Y } Poisson ( ) ( Y ) Pr { Y } Pr{ } 0 Pr{ } Pr{ } Y p ( p) 0 p ( p) Y Y p ( p) Y < () whr p is th paratr of Poisson p distribution. In ordr to iprov th auray of this approiation, and should b suffiintly larg nubrs. Using th abov approiation thod, following is onludd, EY [ ] Y Γ Y p ( p) Y 0 Y () In quations, Y indiats that all its ar insptd thrfor, th nubr of dftiv its has bn lss than or th dftiv it has bn th insptd it. For th as Y <, th Now, lt P dnots th probability of rjting th bath. Th bath is rjtd if th nubr of dftiv its is or than or qual to thus th valu of P is dtrind by th following quation, Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

3 5 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 P p p ( ) (5) In ordr to alulat th total ost, inluding th ost of rjting th bath, th ost of insption and th ost of dftiv its, assu R is th ost of rjting th bath, is th insption ost of on it and ' is th ost of on dftiv it, so th total ost, C, is dtrind by onditioning C on two vnts of rjting or apting th bath, thus th objtiv funtion is writtn as follows: C E ( C Rjt th bath) P ( Rjt th bath) E ( C Apt th bath) P ( Apt th bath) P ( R E [ Y ] ) ( np ' E [ Y ] )( P ) P R np ' ( P ) E [ Y ] Thus w hav, 0 C P R np' P p p Y Y Y ( ) Γ Y 0 Y R p ( p) p p np Y ( ) ( ' ) Y Γ ( Y ) () In quation (), E [ Y ] is th total ost of insption and np ' is th total ost of dftiv its. Th optial valu of is dtrind by iniiing th valu of objtiv funtion C. Using th optiiation thods, it is onludd that, Δ 0 C C C R p p ( np ') p ( p) Y Y Y Γ ( Y ) ( ) R p ( p) ( np ') p ( p) 0 Y ( ) Y Y Γ( Y ( ) ) (7) To valuat abov quation, following quality is onsidrd, Y Y ( ) Y Y Y Y Y Γ Y Γ Y ( ) ( ) Y Γ Y Γ ( ) (8) Sin is a suffiintly larg nubr thus th valu ( ) of is approiatly qual to Γ( ( ) ) ro thrfor it is onludd that, ( ) Δ C R p ( p) ( ) ( np ') p ( p ) Y Y Γ( Y ) ( np ' R ) p ( p) Y Γ Y ( Y ) ( ) (9) To nsur that iniis th objtiv funtion (), it is nssary to find th valu of that satisfis following inqualitis: Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

4 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 Δ C C C > 0 Δ C C C < 0 Hn, Δ C Y ( ) > 0 Γ ( np ' R ) p ( p) Y ( Y ( ) ) Δ C Y < 0 Γ ( np ' R ) p ( p) Y ( Y ) ( ) (0) () Sin is a suffiintly larg nubr thus following is onludd, Y Y ( ) Y Y Γ Y Γ Y () Now If np ' < R, thn, ( ) Y Y Y p ( p) > R np ' > p ( p ) Γ( ) ( ) (3) Sin with inrasing th valu of th valu of binoial distribution with paratrs and p drass thus aording to th proprtis of binoial > p thrfor, distribution, it is onludd that th optial valu of is dtrind using th following forula, Also Th objtiv funtion,, should b iniid rgarding two onstraints on Typ-I and Typ-II rrors assoiatd with th aptan sapling plans. Typ-I rror is th probability of rjting th bath whn th nononfority proportion of th bath is aptabl. Typ-II rror is th probability of apting th bath whn th nononforing proportion of th bath is not aptabl. Thn, in on hand, if p δ, th probability of rjting th bath should b lss than ε. On th othr hand, in as whr p δ, th probability of apting th bath should b lss thanε whr δ is th aiu aptabl lvl of th bath quality (Aptd Quality Lvl (AQL)) and δ is th iniu allowabl lvl of th bath quality (Lot Tolran Proportion Dftiv (LTPD)) andε is th probability of Typ-I rror and ε is th probability of Typ-II rror in aking a dision, thrfor, th optial valu of is dtrind using th following forula, C ; > p; Y Y Γ( Y ) > ( R ( np' )) Min p ( p) δ ( δ) ε δ ( δ) ε 0 (5) Whn np ' > R, It is onludd that quation () is positiv for all valus of so 0. In this as, if on dftiv it is found in an insptd sapl thn th bath would b rjtd. In this as, th rjtion ost R is lss than th total ost of inspting its and th ost of dftiv its, hn rjting th bath would b th optial dision. Howvr, in prati th rjtion ost R is usually big nough so that, w ovrlookd that as. ; > p; Y Y ( Y Γ ) Min > ( R ( np ')) p ( p ) () ( ) Δ C ( np' R) p ( p) () 3. Nurial Eapl In this stion, w giv a nurial apl to illustrat th forulas for alulating a onstant ontrol lvl of that iniis th total ost, inluding th ost of rjting th bath, th ost of inspting and th ost of dftiv its. In this apl, w tak th Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

5 7 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan bath si, bath rjtion ost, insption ost for on it and th ost of on dftiv it n 00, R 000, 7, ' 30 rsptivly whr δ 0.05, δ 0., ε 0.05, and ε 0..Th optial valus of for diffrnt valus of,p ar shown in Tabl. Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 (, p ) Tab.. Th optial valu of for diffrnt valus of (, p) optial valu of Optial Dsign Typ-I rror Probability Typ-II rror Probability Lowr Bound for th Eptd Nubr of Insptd Its (50,0.) 7 (50,7) (55,0.) 7 (55,7) (5,0.) No fasibl solution (0,0.) No fasibl solution (80,0.05) 8 (80,8) (0,0.05) 7 (0,7) (70,0.05) 8 (70,8) (90,0.05) 9 (90,9) Basd on th rsult shown in Tabl, as th valu of inrass, th optial valu of ontrol thrshold,, inrass. Th optial aptan thrsholds ar prsntd in olun 3. For instant, th dsign in row 7 of olun 3 indiats that, whr 0, p 0.05, th bath si of 0 will b rjtd if th nubr of dftiv its is 7or or. For 70, p 0.05 (row 8), if 8 its out of 70 insptd its wr dftiv thn th bath will b rjtd. It ans that whn at last prnt of th bath was dftiv thn th bath should b rjtd. For 90, p 0.05 (row 9), if 9 its out of 90 insptd its wr dftiv, thn th bath will b rjtd, it ans that whn at last 0 prnt of th bath was dftiv thn th bath should b rjtd. It is onludd that slting th valu of sapl si for insption strongly affts on th final rsult of th proposd odl. Figur shows as a funtion of whr p It is infrrd fro Figur that by inrasing th valu of sapl si, th valu of onvrgs to a onstant nubr also sin th drivation of th odl was basd on assuing a suffiintly larg nubr for th sapl si thus inrasing th valu of sapl si ould inras th auray of th odl / Fig.. Efft of hanging sapl si on whr p 0.05 Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

6 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan 8 Tab.. Th optial valu of for diffrnt valus of and ' whr 80, p 0.05 (, ') Optial valu of Typ-I rror Probability Typ-II rror Probability Lowr Bound for th Eptd Nubr of Insptd Its Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 (,) (,30) (5,5) (5,30) (5,0) (,30) (7,30) Snsitivity Analysis A snsitivity analysis on th diffrnt valus of, ' for 80, p 0.05 is arrid out in this stion. Th rsults ar shown in Tabl. 0 8 As it is lar in Tabl, by inrasing th ost of a dftiv it and th ost of insption, th optial valu of th ontrol lvl, drass, as it was ptd. Figur shows th optial valu of ontrol thrshold as a funtion of and ' ' Fig.. Efft of hanging and ' on whr 80, p Coparativ Study In this stion, th prforan of th proposd thodology is opard to th on using an aptan-sapling plan, in whih n its of a bath ar insptd. In this plan, if th nubr of nononforing its is lss than a lowr ontrol thrshold, th bath is aptd. If this nubr is or than a ontrol thrshold, th bath is rjtd, and if th nubr of nononforing its lis within th thrsholds and, th pross of inspting th bath ontinus ( Fallahnhad and Niaki []). Proposd sapling plan is opard with abov sapling plan with sapl si n 50 and p 0., basd on th asurs suh as probabilitis of Typ-I and Typ-II rror and th ptd nubr of insptd its. Y Sin Y thus th lowr and Y Γ( Y ) uppr bound for th ptd nubr of insptd its in proposd sapling thod is dtrind by th following quations, Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

7 9 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 ( ) 0 EY [ ] p ( p) 0 p p (7) Tabl 3 shows diffrnt altrnativ obination valus of and togthr with thir probability of rjting or apting th bath, of whih th ons in bold ar fasibl. Basd on th rsults, th bst obination valu is and with th iniu valu of th ptd nubr of insptd its of 75.. Basd on th rsults of Tabl, th lowr bound and uppr bound for th ptd nubr of insptd its in proposd sapling thod is 8.8 and 83.8 whn 55, p 0. that is a satisfatory prforan with rgards to th rsults of th sond sapling plan. Tab. 3. Th probabilitis of rjting and apting th bath in th sond sapling plan Probability of Probability of Eptd Nubr of rjting th bath apting th bath Insptd Its whn δ whn 0. δ Th rsults in Tabl 3 show that th sond sapling plan is apabl to provid sapling dsigns with suffiintly sall probabilitis of Typ-I and Typ-II rror with a rasonabl sapl si but sin th prforan of proposd thod is satisfatory and onsidring this fat that proposd sapling plan is a singl stag sapling plan thus th rsults of th oparison studis ar in favor of th proposd thodology.. Conlusion A nw odl for th sltion of ost iniiing singl stag aptan sapling plans has bn prsntd. Th rlationship btwn th ost odl and a dision thory odl with binoial utilitis has bn invstigatd. Howvr, th aptan sapling plan, whih is drivd fro th optiiation of this odl, ay diffr substantially fro th plans that othr onoi approahs suggst but optiiation of th odl is sipl and ffiint, with ngligibl oputational rquirnts. Snsitivity analysis on th diffrnt valus of sapl si and rjtion osts has shown that th optial valu of th ontrol lvl drass along with inrasing th ost of dftiv it and th ost of insption. Also th rsults of th oparison studis ar in favor of th proposd thodology. Rfrns [] Klassn, C.A.J., Crdit in Aptan Sapling on Attributs, Thnotris, 3, 00, pp.. [] Tagaras, G., Intgratd Cost Modl for th Joint Optiiation of Pross Control and Maintnan, Journal of Oprational Rsarh Soity, 39, 988, pp [3] Kuo, Y., Optial Adaptiv ontrol Poliy for Joint Mahin Maintnan and Produt Quality Control, Europan Journal of Oprational Rsarh, 7, 00, pp [] Frrll, W.G., Chhokr, J.A., Dsign of Eonoially Optial Aptan Sapling Plans with Insption Error, Coputrs & Oprations Rsarh, 9, 00, pp [5] Parn, W.L., Wu, C.W., Critial Aptan Valus and Sapl Sis of a Variabls Sapling Plan for Vry Low Fration of Dftiv, Oga, 3, 00, pp Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3

8 M.S. Fallah Nhad & H. Hossini Nasab Dsigning a Singl Stag Aptan Sapling Plan 50 [] Niaki, S.T.A., Fallahnhad, M.S., Dsigning an Optiu Aptan Plan Using Baysian Infrn and Stohasti Dynai Prograing, Intrnational Journal of Sin and Thnology (Sintia Irania),,, 009, pp Downloadd fro ijipr.iust.a.ir at 7:0 IRST on Monday Novbr th 08 [7] Moskowit H., Tang, K., Baysian Variabls Aptan- Sapling Plans: Quadrati Loss Funtion and Stp Loss Funtion. Thnotris, 3, 3, 99, pp [8] Ainadh, M.S., Squntial and Non-Squntial Aptan Sapling Plans for Autoorrlatd Prosss using ARMA (p,q) Modls. Coputational Statistis,, 009, pp. 95. [9] MWillias, T.P., Saniga, E.M., Davis, D.J., On th Dsign of Singl Sapl Aptan Sapling Plans. Eonoi Quality Control,, 00, pp [0] Willia, G., Frrll Jr., W.G., Chhokr, A., Dsign of Eonoially Optial Aptan Sapling Plans with Insption Error. Coputrs & Oprations Rsarh, 9, 00, pp [] Hilb, J.M., Ngativ Binoial Rgrssion, Cabridg, UK: Cabridg Univrsity Prss, 007. [] Fallahnhad, M.S., Niaki, S.T.A., A Nw Aptan Sapling Poliy Basd on Nubr of Sussiv Conforing Its, To Appar in Couniations in Statistis-Thory and Mthods, 007. Intrnational Journal of Industrial Enginring & Prodution Rsarh, Sptbr 0, Vol., No. 3