Time Truncated Sampling Plan under Hybrid Exponential Distribution

Transcription

1 Journal of Unceran Syses Vol.1, No.3, pp , 16 Onlne a: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon S. Sapah 1,, S. M. Lalha 1 Deparen of Sascs, Unversy of Madras, Chenna, Inda Deparen of Maheacs, Sr Sara Insue of Technology, Chenna, Inda Receved 9 July 15; Revsed 1 May 16 Absrac The desgn of accepance saplng plans for hybrd exponenal dsrbuon under a runcaed lfe es s consdered n hs paper. In hs work, experenal values are reaed as observed values of exponenal rando varables whose ean s assued o be a fuzzy varable n he sense of Lu [5]. A new chance dsrbuon called hybrd exponenal dsrbuon s consdered n hs paper and s properes are nvesgaed. Under he chance dsrbuon, he queson of developng e runcaed saplng plan s consdered. For varous accepance nubers, consuer s confdence levels and values of he rao of he fxed experenal e o he specfed edan lfe, he nu saple szes requred o ensure he specfed edan lfe are obaned. The operang characersc funcon values of he gven saplng plans and assocaed producer s rsk are presened n he fuzzy envronen. The resuls are llusraed wh exaples. 16 World Acadec Press, UK. All rghs reserved. Keywords: accepance saplng plan, hybrd exponenal dsrbuon, operang characersc funcon value, consuer s rsk and producer s rsk 1 Inroducon Desgnng of suable Accepance Saplng Plans for varous suaons s an poran exercse n he sudy of Sascal Qualy Conrol syses. Accepance saplng plans help us o exane wheher he anufacured producs ee he pre-specfed qualy levels. They are prarly used n sascal qualy conrol when s no possble o perfor coplee nspecon of he anufacured producs for varous reasons lke, he anufacured producs beng desrucve n naure or coplee nspecon ay be a e consung process. Bascally, accepance saplng plans help us o assess he qualy level of he produc based on sapled es. Accepance saplng plans can be broadly classfed as, Saplng plans for Arbues and Saplng plans for Varables. If he qualy level of he produc s easured n ers of arbues lke defecves or non-defecves, hen he saplng plans for arbues are used. On he oher hand, f he anufacured producs are nspeced by eans of easureens lke lengh, hegh, lfe e, ec., hen saplng plans for varables used. Characerscs of an accepance saplng plan are suded anly wh he help of probably dsrbuons whch nvolve ceran paraerc values. For exaple, n saplng plans for arbues, dsrbuons lke Bnoal, Posson, Hyper-Geoerc, ec. play poran roles. In he case of accepance saplng plans for varables, dsrbuons lke Noral, Exponenal, Gaa, Log-noral, ec. fnd wde applcaons. Several researchers have conrbued o he developen of saplng plans for varables under suaons nvolvng randoness. Soe of he are Zer and Burr [4], Owen [7], Guenher [14], Anzadeh [], Soundararajan and Chrsna [37], Erc e al. [35], Geeha and Vjyaraghavan [9], ec. I s o be noed ha varous ypes of accepance saplng plans for varables are avalable n he leraure lke, chan saplng, connuous saplng, skp-lo saplng, e runcaed saplng, ghened noral ghened saplng, relably saplng, ec. A class of accepance saplng plans known as e runcaed saplng plan ha has receved he aenon of any researchers s o be consdered n hs paper. Te runcaed saplng plans have been consdered by any auhors under varous probably dsrbuons. Such e runcaed saplng plans were developed by Epsen [8] n exponenal case, Sobel and Tschendrof [36] for Exponenal dsrbuon, Goode and Correspondng auhor. Eal: sapah1959@yahoo.co (S. Sapah).

2 18 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon Kao [1] for Webull dsrbuon, Gupa and Groll [17] for Gaa dsrbuon, Kana and Rosaah [1] for Half logsc dsrbuon, Kana e al. [] for Log-logsc, Rosaah and Kana [9] for Raylegh, Baklz [5] for Pareo dsrbuon of second knd, Baklz and El Masr [6] for Brnbau Saunders odel, and Balakrshnan e al. [7] for generalzed Brnbau-Saunders dsrbuons. Recenly, Asla and Shahbaz [4], Tsa and Wu [39], Al-Nasser and Al-Oar [1], Sngh e al. [34], Gu and Zhang [15] developed accepance saplng plans for runcaed lfe es for generalzed exponenal dsrbuon, nverse Raylegh dsrbuon, generalzed Raylegh dsrbuon, Exponenaed Freche dsrbuon, Copound Raylegh dsrbuon and Goperz dsrbuon. In convenonal accepance saplng plans, a rando saple s seleced fro he lo and he consuer decdes o accep or rejec he lo based on he nforaon obaned fro he saple. In lfe es saplng plans or e runcaed accepance saplng plans, uns are subjeced o lfe es and he nuber of falures up o a pre-specfed e pon s observed. If he nuber of falures reaches he accepance nuber whn he specfed e hen he nspecon s sopped and he lo s rejeced. On he oher hand, f he nuber of falures s less han or equal o he accepance nuber hen he lo s acceped. The qualy characersc consdered under he lfe e experen s a connuous varable n naure; we assue a probably dsrbuon for he lfe e dsrbuon wh eher ean/edan as a paraeer. Snce he qualy characersc s a varable, here exss eher a lower confdence l or an upper confdence l or boh whch esablsh he accepable values of hs paraeer. The prary objecve of e runcaed accepance saplng plan s o fx he lower confdence l on he ean/edan lfe of he produc and ake sure ha he acual ean/edan lfe of he produc sasfes he consuer s confdence level wh a nuu probably Invarably, he paraeers nvolved n hese probably dsrbuons are assued o be eher known or esaed hrough soe sascal echnques. In real lfe suaons, fndng good esaed values for paraeers reans as a challengng proble. The nroducon of he Fuzzy heory paved a way for an alernave soluon o hs proble. Varous approaches for desgnng of saplng plans for arbues usng fuzzy se heory have been consdered by several researchers ncludng Oha and Ichhash [6], Kanagawa and Oha [], Arnold [3], Grzegorzwsk [11, 1, 13], Hrynewcz [18], Jakhaneh e al. [19], Tong and Wang [38] ec. I s o be enoned ha he ajory of hese works are relaed o saplng plans for arbues and hey assue he presence of fuzzness n he paraeers relaed o he underlyng dsrbuons. Whle soe of hese works consdered fuzzness n producer s rsk and consuer s rsk, ohers consdered fuzzness n he subed lo qualy level. To sudy envronens nvolvng precse suaons, Lu and Lu [4] and Lu [5] nroduced a heory called credbly heory parallel o probably heory. Sapah [3, 31] and Sapah and Deepa [3] have appled chance heory developed by Lu [5] whch s an negraon of precseness and randoness n he heory of accepance saplng for desgnng fuzzy accepance saplng plans for arbues. Recenly, Sapah, e al. [33] have consdered he applcaon of hybrd noral dsrbuon (he noral dsrbuon where he paraeers nvolved are reaed as fuzzy varables) n developng a sngle saplng plan for varables for suaons nvolvng boh randoness and precseness. A horough revew of he leraure on lfe es saplng plans ndcaes ha he exponenal dsrbuon plays a val role n desgnng lfe es saplng plans under rando envronen. In hs paper, he queson of developng runcaed lfe es saplng plan for varables usng hybrd exponenal dsrbuon (exponenal dsrbuon where he paraeer s reaed as a fuzzy varable) s consdered. Developng an accepance saplng plan for Hybrd exponenal dsrbuon o ensure he edan lfee of he producs under nspecon exceeds a pre-deerned qualy provded by he consuer wh a nu probably s he an a of hs paper. The res of he paper s organzed as follows. In Secon, n order o anan he readably of he paper we gve a bref nroducon o Chance heory. In Secon 3, hybrd exponenal dsrbuon s developed under chance envronen and s properes are dscussed. The desgn of e runcaed accepance saplng plan for hybrd exponenal dsrbuon s consdered n Secon 4. Soe poran characerscs of he plan under chance envronen are suded. In Secon 5, nuercal exaples are gven for llusrang he use of heorecal developens ade n hs paper. Concludng rearks are gven n he fnal secon of he paper. Hybrdzaon of Credbly and Probably Theores The nroducon of chance heory requres an undersandng of he credbly heory ha provdes he foundaon for he nroducon of fuzzy varables and Probably heory.

3 Journal of Unceran Syses, Vol.1, No.3, pp , Credbly Theory Le be a nonepy se and be he power se of. Each eleen of s called an even. For every even A, we assocae a nuber denoed by Cr{ A }, whch ndcaes he credbly ha A wll occur and ha sasfyng he followng four axos: Axo 1 (Noraly) Cr( ) 1; Axo (Monooncy) Cr( A) Cr( B) whenever A B; Axo 3 (Self dualy) Cr( A) Cr( A c ) 1for any even A ; Axo 4 (Maxaly) Cr( A ) SupCr( A ) for any evens { A } wh SupCr( A ).5. Credbly easure: The se funcon Cr s called a credbly easure f sasfes he noraly, onooncy, self-dualy and axaly axos. Credbly space: Le be a nonepy se, be he power se of and Cr a credbly easure. Then he rple (,, Cr) s called a credbly space. Fuzzy varable: A fuzzy varable s a easurable funcon fro a credbly space (,, Cr) o he se of real nubers. Mebershp funcon: Le be a fuzzy varable on he credbly space (,, Cr). Then s ebershp funcon s derved fro he credbly easure by ( x) Cr x 1, x. Credbly dsrbuon: The credbly dsrbuon : [,1] of a fuzzy varable s defned by. Probably Theory ( x) Cr ( ) x. Le be a nonepy se and be he power se of. Each eleen of s called an even. For every even A, we assocae a nuber denoed by Pr{ A }, whch ndcaes he probably ha A wll occur. The axos of probably heory are as follows. Axo 1 (Noraly) Pr( ) 1; Axo (Nonnegavy) Pr( A) for any even A ; Axo 3 (Counable addvy) Pr( A) Pr( A) for every counable sequence of dsjon evens{ A }. Probably easure: The se funcon Pr s called a probably easure f sasfes he noraly, non-negavy, and counable addve axos. Probably space: Le be a nonepy se, be he power se of and Pr a probably easure. Then he rple (,,Pr) s called a probably space. Rando varable: A rando varable s a easurable funcon fro a probably space (,, Pr) o he se of real nubers. Probably dsrbuon: The probably dsrbuon : [,1] of a rando varable s defned by.3 Chance Theory ( x) Pr ( ) x. Usng he above defnons relaed o credbly and probably spaces, L and Lu [3] developed deas relevan for handlng suaons where boh precseness and randoness play sulaneous roles n he gven syse. The hybrd developen based on credbly and probably space has been naed as Chance heory. The followng defnons are due o L and Lu [3]. Chance space: Suppose ha (,, Cr) s a credbly space and (,,Pr) s a probably space. The produc (,, Cr) (,,Pr) s called a chance space.

4 184 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon Le (,, Cr) (,,Pr) be a chance space. A subse s called an even f ( ) for each. Chance easure: Le (,, Cr) (,,Pr) be a chance space. Then a chance easure of an even s defned as Cr f Cr sup Pr ( ), sup Pr ( ).5 Ch( ) (1) 1 supcr Pr c ( ), f supcr Pr ( ).5. To descrbe a quany wh boh fuzzness and randoness, he concep of hybrd varable s used. I s forally defned as follows. Hybrd varable: A hybrd varable s a easurable funcon fro a chance space (,, Cr) (,,Pr) o he se of real nubers. Tha s, for any Borel se B of real nubers, B (, ) (, ) B s an even. L and Lu [3] have denfed fve dfferen approaches o defnng Hybrd varable. The Model IV of L and Lu [3] wll be used n our furher dscusson. Ths odel s suable for dealng wh suaons where he paraeers nvolved n a gven probably dsrbuon are fuzzy by naure. The odel proposed by Lu s explaned below. Le be a rando varable wh probably densy funcon ( x; 1,,..., n) where ( 1,,,..., n) s a se of fuzzy paraeer varables. If 1,,,..., n have ebershp funcon 1,,,..., n respecvely, hen for any Borel se B of real nubers, he chance Ch( B) due o Qn and Lu [8] s gven by ( ) sup n ( x, 1,,..., n ) dx, 1 1,,..., n B ( ) f sup n ( x, 1,,..., n ) dx.5 1 1,,..., n B Ch( B) ( ) 1 sup n ( x, 1,,..., n ) dx, 1,,..., 1 n c B ( ) f sup n ( x, 1,,..., n ) dx ,,..., n B Chance dsrbuon: The chance dsrbuon : [,1] of a hybrd varable s defned by ( x) Ch (, ) (, ) x. Chance densy funcon: The chance densy funcon : [, ) of a hybrd varable s a funcon such ha x ( x) ( y) dy, x and ( y) dy 1 where s he chance dsrbuon of. The defnons presened are relevan for furher dscusson ade n hs paper. For ore dealed and exhausve dscusson, one can refer o L and Lu [3]. Expeced value: The defnon of he expeced value operaor of a fuzzy varable was gven by Lu and Lu [4]. Ths defnon s applcable boh for connuous fuzzy varables and also dscree ones. Le be a fuzzy varable. Then he expeced value of s defned by provded ha a leas one of he wo negrals s fne. 3 Hybrd Exponenal Dsrbuon E ch( r) dr ch( r) dr (3) The probably densy funcon of he rando varable X havng an exponenal dsrbuon wh ean s gven x by ( x, ) e, x,. Here we assue s a fuzzy varable. Clearly he above dsrbuon s a hybrd dsrbuon (randoness creaed hrough he rando varable X and fuzzness enerng n he for precseness creaed by he paraeer ). We shall denoe by he hybrd varable. If s a ebershp funcon assocaed wh hen had been shown by Qn and Lu [8], for any Borel se B of real nubers, he chance Ch ( B ) s gven by ()

5 Journal of Unceran Syses, Vol.1, No.3, pp , ( ) ( ) sup (, ), sup (, ).5 x dx f x dx B B Ch( B) ( ) ( ) 1 sup ( x, ) dx, f sup ( x, ) dx.5. c B B ( x, ) s an exponenal probably densy funcon. Therefore, x x ( ) 1 ( ) 1 sup dx, f sup dx.5 e e B B Ch( B) x x ( ) 1 ( ) 1 1 sup 1 dx, f sup dx.5. e c e B B In hs paper we shall assue s a rangular ebershp funcon over ( abc,, ). Tha s, a, f a b b a b ( ), f b c b c, oherwse. For hybrd exponenal dsrbuon, he dsrbuon funcon s gven below x x ( ) 1 ( ) 1 sup dx, f sup.5 e dx e, x x ( ) 1 ( ) 1 1 sup 1 dx, f sup dx.5. e e By akng no accoun of hs, we ge he dsrbuon funcon as ( ) ( ) sup 1 e, f sup 1 e.5, ( ) ( ) 1 sup e, f sup 1 e.5. The followng heore gves expressons for he chance dsrbuon consdered above. (4) (5) (6) (7) (8) Theore 1: The dsrbuon funcon of a hybrd varable whch follows he hybrd exponenal dsrbuon s gven below, f 1 1 e, f bln() Ch( ) (, ) 1, f b ln() 1 e, f bln() where 1 and are he soluons of ( ) =1 e, ( ) = e, Proof: The dsrbuon funcon of hybrd exponenal saed n equaon (8) s respecvely.

6 186 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon ( ) ( ) sup 1 e, f sup 1 e.5, ( ) ( ) 1 sup e, f sup 1 e.5. In order o fnd,, over dfferen values of, we need o exane he behavor of ( ) and 1 e over he perssble values of and. Noe ha, R and ac, he axu value aaned by ( ) (whch s ndependen of ) s 1, and 1 e s decreasng n for a gven. The curve 1 e wll eher nersec wh ( ) dependng on he choce of. I ay be noed ha he curve 1 e s non-decreasng n for a gven as shown n Fgure 1 for Fgure 1: Ipac of he value of on he nersecon of ( ) and 1 Therefore, he curve ay le enrely above ( ) or wll nersec a wo dfferen pons. The hrd possbly s ha he curve ay ouch he rangle a only one pon. Ths case wll arse when 1 e = ( ). Noe ha ( ) =1 f = b. In hs case, we have 1 ha 1 e b =1, solvng for, we ge bln e wll be greaer han 1 for all as long as bln we conclude ha Ths ples ha When bln, ( ) ( ) ( ) 1 e e. Hence, we conclude and s less han.5 f, for all bln. ( ) 1 Sup 1 e and 1 e nersec a wo dfferen pons, say, for all bln. 1 and bln. Therefore,. Snce ( ) s a rangle and 1 e s a onoone curve (n ers of ) we conclude ha nersecons wll ake place on dfferen sdes of he rangle as shown n Fgure. Evdenly, where 1 s he soluon of ( ) For all bln, ( ) sup 1 e 1 e 1 =1 e. Therefore, we have (, ) 1 1 (, ) sup e d = sup 1 e for all bln e.. (9)

7 Journal of Unceran Syses, Vol.1, No.3, pp , e Fgure : Inersecon of ( ) and 1 Snce s srcly ncreasng n for a gven, by followng he lnes of earler arguens, we undersand he scenaro prevalng n hs case, wll be as shown n Fgure 3. e Evdenly, Fgure 3: Inersecon of ( ) and ( ) sup e e where s he soluon of ( ) = e. Therefore, we have (, ) Thus we have proved Theore 1. Expeced Value of Hybrd Exponenal Dsrbuon 1 e e for all bln. Snce he chance varable correspondng o hybrd exponenal dsrbuon assues only non-negave values, he expeced value of he exponenal hybrd varable s calculaed usng he forula, E = 1, (1) d (11) The value of he negral gven n (11) canno be heorecally copued. Hence s decded o nvesgae he value of he above negral on akng use of rapezodal rule for nuercal negraon. The nerval of negraon s paroned no 1 sub nervals where he lower l of he nerval consdered for negraon s aken as zero and he upper l s deerned by a very large value whose value s closer o zero. I ay be noed ha he negrand s a decreasng funcon. Table 1 furnshed below gves he expeced value of hybrd exponenal dsrbuon for dfferen choces of a, b, c were a and c are deerned by fxng he value of b and defnng ab and c b. Four dfferen choces were used for, say,.5,.1,.15 and.. The qualy of produc whose lfe e has skewed dsrbuon can be ore eanngfully assessed usng he edan of he dsrbuon raher han s ean [16]. Hence, s worhwhle o deerne he value of edan n

8 188 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon h hybrd exponenal dsrbuon. The edan of he hybrd exponenal dsrbuon denoed by s obaned by solvng, 1. Table 1: Expeced values of hybrd exponenal dsrbuon b ε= Table : Medan and ean of hybrd exponenal dsrbuon B Medan Mean Rao

9 Journal of Unceran Syses, Vol.1, No.3, pp , Fro Theore 1, can be seen ha, he edan of he hybrd exponenal dsrbuon s bln. I ay be recalled ha n he case of crsp exponenal dsrbuon, he edan s a consan ulpled by he ean of he dsrbuon. Tha s, we have ln. Now, we shall exane wheher here exss any such relaonshp beween he expeced value and edan n he case of hybrd exponenal dsrbuon. Towards hs, for dfferen values of b, he rao of he edan value o he expecaon of hybrd dsrbuon were copued and was found h ha n he case of hybrd exponenal dsrbuon (.67) E[ ]. Table gves he values of edan, expecaon and rao of edan o he expecaon for hybrd dsrbuon for dfferen values of b. 4 Desgn of Te Truncaed Accepance Saplng Plan for Hybrd Exponenal Dsrbuon In e runcaed accepance saplng plans, n es fro he lo are nspeced over a gven perod of e, say,. The lo s acceped f he nuber of observed falures ll e pon does no exceed a pre-specfed accepance nuber c, and he es s ernaed wh rejecon of he lo f he nuber of falures observed before he e perod exceeds he accepance nuber c. The nspecon e s a pre-specfed quany. The saplng plan should use a carefully chosen value for. I s usually aken as a ulple of a argeed edan lfe e of he produc, say,. Tha s, we ake a, where a s also a pre-deerned quany whch ndcaes he nuber of cycles needed o guaranee specfed edan lfe e of he produc. Ths s based on he reasonng ha nspecon over varous cycles where he nuber of cycles s ade dependen on he gven edan lfe e wll ensure a nu qualy level expressed ners of a desred edan lfe e. I ay be noed ha arrvng a a decson based on a e runcaed accepance saplng plan s equvalen o akng a decson whle esng he null hypohess H agans he alernave hypohess H a : 1 : level of sgnfcance1 P, whch s nohng bu he consuer s rsk. Here, denoes he acual edan lfe e whch s n general unknown. The consan rejecng a bad lo. Here, we have and / 1 P P rejecng a lo P referred o as consuer s confdence level s he lower bound for (level of sgnfcance- consuer s rsk) (1) / P P rejecng a lo (consuer s confdence level) (13) The paraeers of he e runcaed accepance saplng plan are he nuber of es n o be drawn fro he lo, an accepance nuber c, he e rao, where he specfed edan lfe e and s he pre-assgned esng e. Sybollcally, he saplng plan s denoed by he rple n, c,. Any se of values for he paraeers of a e runcaed accepance saplng plan s expeced o sasfy he condons saed n (1) and (13). Here, we resrc ourselves o hose saplng plans sasfyng nequaly relaed o consuer s rsk. I ay be noed ha several se of plan paraeer values sasfyng hs requreen. Hence, we look for a saplng plan by fxng he nspecon e, edan lfe e and accepance nuber c for a gven P. When hese values are fxed, one can fnd several n for whch he consuer s rsk nequaly s sasfed. Hence, we look for a sall posve neger n such ha c n n p 1 p 1 P (14) where p s he probably ha an e fals before he e. In our sudy, where we negrae randoness and precseness, nsead of probably p, we use he chance of an e fals before e. Ths chance value s copued usng he chance dsrbuon obaned earler, naely,

10 19 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon where 1 and are he soluons of ( ) =1, f 1 1 e, f bln() Ch( ) (, ) 1, f b ln() 1 e, f bln() = e. e and ( ) To be precse, p used n (14) s copued usng he relaon p, where s he desred edan lfe e and a where a s a pre-specfed consan. I ay be noed ha he cuulave dsrbuon funcon of crsp exponenal dsrbuon and hence he chance dsrbuon funcon s onooncally decreasng n edan. Hence fro he nequaly (14), we observe, f he nuber of falures less han or equal o c hen he chance of he o even, ), wll be P. Ths ensures ha, where s he rue or acual edan lfe e. The desred qualy level expressed n ers of he edan lfe e can be unquely deerned by he expeced value of he hybrd exponenal dsrbuon hrough he relaon ln, where s he expeced value of he hybrd exponenal dsrbuon. Hence, akng a s equvalen o a ln(). I s clear ha chance value p depends on he nspecon duraon. As enoned above, n our sudy, we ake a where s he edan of he chance dsrbuon, and he value a s a pre-specfed consan. We assgn dfferen values for a n hs work pursung he lnes of earler slar nvesgaons done under crsp suaon. Table 3 gves he nu values of n sasfyng he nequaly (14), for P =.75,.9,.95,.99 and =.68,.94, 1.57, 1.571,.356, 3.141, 3.97, These choces were ovaed by he works of Gupa and Groll [17], Kana [], Tsa and Wu [39], Balakrshnan e al. [7], and Asla and Shabaz [4]. I s well known ha, f he saple sze n s large and p s very sall hen bnoal s approxaed by Posson dsrbuon wh ean np. Table 4 gves he nu values of n under Posson approxaon for he sae se of values used n Table 3. Operang Characersc (OC) Funcons of he Te Truncaed Accepance Saplng Plan The operang characersc (OC) funcon descrbes he effcency of accepance saplng plans. I calculaes he effcency of a sascal hypohess es whch s desgned o accep or rejec a lo / produc. The OC funcon for he above e runcaed saplng plan n, c, s defned as n OC( p )= p 1 p p,. I s reaed as a funcon of he lo qualy paraeer. OC( p ) s a decreasng funcon of p where c n (15) whch decreases when decreases. For he gven e runcaed accepance saplng plan, he OC funcon values have been copued for dfferen cobnaons of Producer s Rsk P and and hey are lsed n Table 5. In he usual frae work, he producer s rsk s he probably of rejecon of a lo when. For a gven value of he producer s rsk, say,, n he gven saplng plan, one ay be neresed n knowng he nu value of he edan rao ha wll ensure he producer s rsk o be a os. The value of s he salles posve nuber for whch p sasfes he followng nequaly n n 1 p (16) c1 n p

11 Journal of Unceran Syses, Vol.1, No.3, pp , Table 3: Mnu saple sze necessary o asser he edan lfe o exceed a gven value, wh probably P and correspondng accepance nuber, c, usng bnoal probables P c

12 19 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon Table 4: Mnu saple sze necessary o asser he edan lfe o exceed a gven value, wh probably P and correspondng accepance nuber, c, usng Posson probables P c

13 Journal of Unceran Syses, Vol.1, No.3, pp , Table 5: Operang characersc values of gven saplng plan for hybrd exponenal dsrbuon c n = P

14 194 S. Sapah and S.M. Lalha: Te Truncaed Saplng Plan under Hybrd Exponenal Dsrbuon equvalenly, c n p Hence, for a gven saplng plan n, c, wh specfed confdence level sasfyng he nequaly (16) are worked ou, and hey are presened n Table 6. n 1 p 1. (17) P, he nu value of Table 6: Mnu rao of rue edan lfe o specfed edan lfe for he accepance of a lo wh producer s rsk of.5 P c

15 Journal of Unceran Syses, Vol.1, No.3, pp , Abou he Tables Tables 3 hrough 6, presen he resuls of e runcaed accepance saplng plan when s assued ha he lfe e of es es follows he hybrd exponenal dsrbuon where he scale paraeer s reaed as rangular fuzzy varable. Now, we shall dscuss he uly of hese ables. Table 3 provdes nu saple sze as well as he accepance nuber c requred o ensure ha he edan lfe e exceeds a gven pre-specfed edan value wh consuer s confdence level P. The calculaons were perfored on usng bnoal approxaon by assung ha he lo s large enough and p s no very sall. Table 4 provdes slar resuls under he posson approxaon o bnoal. Operang characersc funcon values are shown n Table 5 for dfferen cobnaons of he edan rao, probably P, and he experenal e rao. Table 6 shows he nu raos of he acual edan lfe o he specfed edan lfe for he accepance of he lo wh producer s rsk of.5. Assung ha he lfe e dsrbuon follows hybrd exponenal dsrbuon and s decded o esablsh a nu edan lfe e of =1 hours wh probably P =.95 gven ha he lfe es ges ernaed a =68 hours. For hs suaon, fro Table 3, we ge he nu saple sze 16 and accepance nuber. Ths eans ha, ou of 16 es, f no ore han es fal durng 68 hours, hen he experener assures ha he acual edan lfe e of he es s a leas 1 hours wh confdence level of.95. Table 4 can be used n he sae anner when posson approxaon o bnoal probables s jusfed. Table 5 gves he values of he operang characersc funcon for he accepance saplng plan adoped fro Table 3, for dfferen values of and P. For exaple, when P =.95, =.68, c =, =4, he probably of accepng he lo s I ples ha, he lo s acceped wh probably.5897 when e runcaed saplng plan wh saples sze 16 and accepance nuber s used wh 4.68 =4 hours. For he accepance of a lo, Table 6 provdes he nu rao of he rue edan lfe o he specfed edan lfe when he producer s rsk s.5. For exaple, f P =.95, =.68, and c =, hen he able value of s Ths ples when , he lo wll be rejeced wh probably less han or equal o.5 5 Concluson Thus n hs paper, a new hybrd dsrbuon called hybrd exponenal dsrbuon s consdered based on he Lu s chance heory [5]. Theorecal and nuercal sudes have been carred ou o sudy s properes. The dsrbuon s developed usng he exponenal dsrbuon where he ean s reaed as a rangular fuzzy varable. In hs work, he nu saple sze requred o decde o accep/rejec a lo based on s specfed edan lfe e of he experenal uns have been abulaed assung he lfe e dsrbuon follows hybrd exponenal dsrbuon under e runcaed accepance saplng plan. The opal saple sze provdes he desred level of proecon for boh he cusoers as well as anufacurers. Apar fro hs, Operang Characersc funcon has been evaluaed for dfferen choces of he paraeers nvolved n he hybrd exponenal dsrbuon. Fnally, values of he nu rao of he rue edan lfe o he specfed edan lfe are also abulaed when he producer s rsk s.5. The auhors are nvesgang desgnng such plans under oher dsrbuons as well. References [1] Al-Nasser, A.D., and A.I. Al-Oar, Accepance saplng plan based on runcaed lfe ess for exponenaed fréche dsrbuon, Journal of Sascs and Manageen Syses, vol.16, pp.13 4, 13. [] Anzadeh, M.S., Inverse Gaussan accepance saplng plans by varables, Councaons n Sascs-Theory and Mehods, vol.5, pp , [3] Arnold, B.F., An approach o hypohess esng fuzzy, Merka, vol.44, pp , [4] Asla, M., and M.Q. Shabaz, Econoc relably es plans usng he generalzed exponenal dsrbuon, Journal of Sascs, vol.14, pp.5 59, 7. [5] Baklz, A., Accepance saplng plans based on runcaed lfe ess n he Pareo dsrbuon of second knd, Advances and Applcaons n Sascs, vol.3, pp.33 48, 3.

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