Procedia - Social and Behavioral Sciences 230 ( 2016 ) Joint Probability Distribution and the Minimum of a Set of Normalized Random Variables

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1 Available olie a wwwsciecedireccom ScieceDirec Procedia - Social ad Behavioral Scieces 30 ( 016 ) rd Ieraioal Coferece o New Challeges i Maageme ad Orgaizaio: Orgaizaio ad Leadership, May 016, Dubai, UAE Joi Probabiliy Disribuio ad he Miimum of a Se of Normalized Radom Variables Noura Yassie * Beiru Arab Uiversiy PO Box Riad El Solh, Beiru , Lebao Absrac Suppose ha ypes of compoes M 1, M M are combied o form ad iegraed obec I ad suppose ha y uis of he iegraed obec are required o be formed Assumig ha o all compoes ca be used i formig he iegraed obecs, le q be he perceage of usable compoes of he h ype, a radom variable havig a probabiliy desiy fucio f (q ) Le w be he ormalized radom variable obaied from q by w = q /μ, where μ is he expeced value of q Cosider he radom variable W=Mi{w, 1 } This paper describes he oi probabiliy disribuio of he se of he ormalized radom variables ad deermies he probabiliy disribuio of he miimum W of his se The expeced value of W is key o deermiig he umber of compoes eeded o form he y iegraed obecs A special case is preseed where he perceages of usable compoes are uiformly disribued The problem is applied o a producio model 016 The The Auhors Published Published by Elsevier by Elsevier Ld Ld This is a ope access aricle uder he CC BY-NC-ND licese (hp://creaivecommosorg/liceses/by-c-d/40/) Peer-review uder resposibiliy of he Ardabil Idusrial Maageme Isiue Peer-review uder resposibiliy of he Ardabil Idusrial Maageme Isiue Keywords: Radom Variables; Joi Probabiliy Disribuio; Usable Compoes; Uiform Disribuio 1 Iroducio Suppose ha ypes of compoes M 1, M,, M are combied o form oe iegraed obec I ad suppose ha y iegraed obecs are o be formed I is assumed ha each iegraed obec requires oe compoe of each ype ad o all compoes ca be used i formig he iegraed obecs Tha is, each se of compoes coais a perceage of usable iems This perceage is a radom variable havig a kow probabiliy desiy fucio The umber of compoes of each ype eeded o form he required y iegraed obecs is obaied by deermiig he probabiliy disribuio of he miimum of a se of ormalized radom variables obaied from he perceages of * Correspodig auhor Tel: address: ourayassei@bauedulb The Auhors Published by Elsevier Ld This is a ope access aricle uder he CC BY-NC-ND licese (hp://creaivecommosorg/liceses/by-c-d/40/) Peer-review uder resposibiliy of he Ardabil Idusrial Maageme Isiue doi:101016/sbspro

2 36 Noura Yassie / Procedia - Social ad Behavioral Scieces 30 ( 016 ) usable compoes This provides a exac soluio o a model iroduced by Kha ad Jaber (011) who proposed a approximae soluio based o a very resricive ad urealisic assumpio The aim of his paper is o solve he problem of deermiig he umber of compoes of each ype eeded o form he y iegraed obecs A special case is preseed where he umber of compoes is wo ad heir correspodig perceages of usable compoes are uiformly disribued A umerical example is provided o illusrae he special case The geeral soluio is applied o producio model ha ca be used o provide a exac soluio o he model of Kha ad Jaber (011) The Saisical Problem Cosider a se of idepede radom variables {q 1, q,, q }, ad le f (q ) be he probabiliy desiy fucio of q, 1 For each q, defie he ormalized radom variables o be w = q /μ, where μ is he expeced value of q Le g (w ) deoe he probabiliy desiy fucio of he radom variable w ad le G ( be is a cumulaive disribuio; ie, G ( = Prob(w The, w 1, w,, w are also idepede ad heir oi probabiliy disribuio w 1, w,, w ) is he produc g 1 (w 1 )g (w ) g (w ) Also, he expeced value of w is equal o oe ad he is sadard deviaio is μ / Defie he radom variable W o be he miimum of w 1, w,, w Tha is, W = Mi{w, 1 } = Mi{q /μ, 1 } The cumulaive disribuio G( of W is give by G ( Prob( W w1, w,, w ) dw1 dw dw, B where he regio B is he iersecio bewee he domai of w 1, w,, w ) ad he regio represeig W Usig he fac ha w 1, w,, w are idepede ad he fac ha B is a cube-like regio, we have G( Prob( W B w, w,, w ) dw dw dw 1 w1 ) dw1 1 w ) dw w ) dw G1 ( G( G ( (1) Oce he cumulaive disribuio G( is deermied, he probabiliy desiy fucio is obaied by differeiaig G( ad he expeced value μ of W ca be calculaed by E [ W ] d () To illusrae, we cosider he case where each q is uiformly disribued over [a, b ], so ha μ = E[q ] = (a +b )/ The, w = q /μ is uiformly disribued over [a /μ, b /μ ] ad E[W ] = 1 Sice E[W ] = 1 is he midpoi, he ierval [a /μ, b /μ ], we ca wrie [a /μ, b /μ ] = [1 m, 1+m ], where m = (b a )/(a +b ) Hece, he probabiliy desiy fucio of w is g (w ) = 1/(m ) ad is cumulaive probabiliy desiy fucio is give by 0 if m m 1 G ( if 1 m 1 m (3) m 1 if 1 m From (1), G( is obaied from G 1 (G ( G ( ad he probabiliy desiy fucio of W is = dg(/d Equaio () ca he be used o calculaed he expeced value of W

3 Noura Yassie / Procedia - Social ad Behavioral Scieces 30 ( 016 ) For he case whe =, he oi disribuio w 1, w ) = g 1 (w 1 )g (w ) = 1/(4m 1 m ), defied over he recagular regio R = {(w 1,w ): 1 m 1 w m 1, 1 m w 1 + m } Assumig m 1 m ad usig equaios (1) ad (3), he cumulaive disribuio G( of W is give by 0 if 1 m ( m 1)/(m) if 1 m 1 G ( G1( G( (1 (1 m (4) 1 if 1 1 4m 1 if 1 The probabiliy desiy fucio for W is obaied by differeiaig G( i (4) From (), we have ha he expeced value is give by 3m E[ W ] 1 (5) 1m 3 Deermiig he Number Compoes Model Cosider he problem of deermiig he umber of compoes of each ype M 1, M,, M eeded if y iegraed obecs are o be formed, where each iegraed obec I requires oe compoe of each ype I is assumed ha o all compoes ca be used i formig he iegraed obecs so ha each se of compoes coais a perceage of usable compoes Le q be he perceage of usable compoes of ype, a radom variable havig a kow probabiliy desiy fucio f (q ) The umber of compoes of each ype eeded o form he required y iegraed obecs, u, is obaied by deermiig he probabiliy disribuio of he miimum of a se of ormalized radom variables obaied from he perceages q of usable compoes To see his, we firs se he umber of available compoes of ype, 1, o be u = y/μ Hece, each compoe of ype coais Z = q u = q y/μ compoes ha ca be used o form used he y iegraed obecs Hece, he expeced value of Z is E[Z ] = E[q u ] = E[q y/μ ] = (y/μ )E[q ] = (y/μ )μ = y Le Z be he acual umber of iegraed obecs ha ca be formed from he available usable compoes The, Z Mi{ Z,1 } Mi{ yq /,1 } ymi{ q /,1 } (6) The, Z is a radom variable whose probabiliy desiy fucio is deermied by he oi disribuio of he ormalized radom variables w = q /μ, 1 Le μ deoe he expeced value of W = Mi{w = q /μ, 1 } From (6), we have ha he expeced umber of iegraed obecs I ha ca be formed from he available usable compoes is E [ Z] y (7) The expeced value μ is obaied from equaios (1) ad () I he case of = ad each q is uiformly disribued, equaios (5) ad (7) give 3m E [ Z] y1 1 (8) m

4 38 Noura Yassie / Procedia - Social ad Behavioral Scieces 30 ( 016 ) The Producio Model The classical Ecoomic Producio Quaiy (EPQ) iveory model is based o several simplifyig assumpios ad do o cosider may facors ha may be ecouered i real life Recely, he classical EPQ models have bee exeded i may direcios o accou for imperfec qualiy iems Oe modellig approach was riggered by he work of Salameh ad Jaber (000) i which a model was developed o deermie he opimal lo size where each lo delivered by he supplier coais imperfec iems wih a kow probabiliy desiy fucio El-Kassar (009) preseed a iveory model wih imperfec qualiy fiished produc El-Kassar e al (010) developed a EPQ model ha accous for he producio coss ha occur a he various sages of producio The model did o accou for he cos or qualiy of he raw maerial/compoes eeded a he various sages of producio Arayssi, M, & Yassie, N (014) preseed a iveory model wih qualiy ad shor-erm fiacig A review of hose works ha exeded, modified or criiqued he work of Salameh ad Jaber (000) ca be foud i Kha e al (011) Salameh & El-Kassar (007) preseed a EPQ model ha uses oe ype of raw maerial Kha ad Jaber (011) preseed a wo-sage supply chai icorporaig imperfec iems of several ypes of raw maerial obaied from suppliers Their proposed opimal soluio was based o a very resricive assumpio I he followig, we propose a producio model based o he resuls of secios ad 3 ha gives a exac soluio ha elimiaes hese resricive assumpios Cosider he case of a producio process ha requires ypes of compoes/raw maerials o produce a sigle iem of he fiished produc Suppose ha a sigle iem of he fiished produc requires oe compoe of each ype Le D ad P be he fiished produc demad rae ad producio rae, where P>D, ad le y be he order size of fiished produc per producio cycle A he begiig of he iveory cycle, a order of size u = y/μ compoes of each ype, 1, is received We assume ha each order coais a perceage q of o-defecive compoes, a radom variable wih a kow probabiliy desiy fucio f (q ) Hece, each order coais Z = q u = q y/μ compoes of ype ha ca be used i producio of he fiished produc The acual umber of fiished produc produced usig he o-defecive compoes obaied from orders received a he begiig of he cycle is he give by (6) ad is expeced value is obaied from (7) For each, defie e = Z Z, he umber of o-defecive compoes of ype o used i he producio of he y uis of he fiished produc Sice E[Z ] = y ad E[Z] = yμ, we have ha E[ e ] E[ Z ] E[ Z ] y 1, (9) which is he same for compoes of all ypes Thus, y(1μ) is he expeced umber of o had iveory of odefecive compoes of ype o used i producio This reame gives a exac soluio o he problem of Kha e al (011) I he case whe each = ad each q, = 1,, is uiformly disribued, equaios (3) ad (9) give ha 3m [ 1] [ ] 1 E e E e y (10) m 5 Numerical Example Cosider he case where he y = 400 uis of a iegraed obec I are o be formed ad each iegraed produc requires oe ui of each of wo ypes of compoes, M 1 ad M Give ha ay se of compoes of ype 1 coais a perceage q 1 of o-usable iems, where q 1 is uiformly disribued over [70%, 90%] Similarly, compoes of ype are assumed o coai a perceage q of o-usable iems, a radom variable uiformly disribued over he ierval [60%, 90%] The parameers of his problem are: y =400; a 1 = 070, b 1 = 090, a = 060, ad b = 090 To obai he umber of compoes of each ype required for he 400 uis of iegraed obecs, we firs calculae μ 1 = (a 1 + b 1 )/ = 80%

5 Noura Yassie / Procedia - Social ad Behavioral Scieces 30 ( 016 ) ad μ = (a + b )/ = 75% Nex, we se he umber of compoes of ype 1 o be u 1 = y/µ 1 = 3000 uis Similarly, for he ype compoes, u = y/µ = 300 uis To deermie he expeced umber of iegraed obecs o be formed, we use equaio (7) so ha E[Z] = μy Now W 1 = q 1 /μ 1 is uiformly disribued over [0875, 115] so ha m 1 = 015 ad g 1 (w 1 ) = 1/(m 1 ) = 4 Similarly, W = q /μ is uiformly disribued over [080, 10], where m = 00, ad g (w ) = 1/(m ) = 4 From (5), he radom variable W = Mi{W 1, W } has a expeced value of E[W] = μ = Therefore, (7) ad (10) give ha E[Z] = μy = 64 ad E[e 1 ] = E[e ] = 136 Noe ha he umber of o-defecive of compoes ou of he 3000 uis ype 1, Z 1, is uiformly disribued over [100, 700], ad he disribuio for Z is uiform over [190, 880] Also oe ha oly 64 iegraed uis of he required y = 400 are acually formed This problem ca be easily remedied by adusig u 1 ad u by a facor of 1/μ Hece, he umber of compoes of each ype eeded are u 1 = 3180 uis ad u = 339 uis 6 Coclusio This paper described he oi probabiliy disribuio of a se of ormalized radom variables ad deermied he probabiliy disribuio of he miimum of his se The expeced value of his miimum is key o deermiig he umber of compoes eeded o form a give umber of iegraed obecs obaied from several ypes of compoes A special case was preseed where he perceages of usable compoes are uiformly disribued The problem is applied o a producio model wih defecive compoes The saisical problem was illusraed usig uiform disribuio ad oly wo ypes of compoes For fuure research, we sugges ha he saisical problem be applied o he case where more ha wo ypes of compoes are required ad o he case whe he radom variables have o-uiform disribuios Refereces Arayssi, M, & Yassie, N (014) Shor-Term Fiacig of Ecoomic Order Quaiy (EOQ) Iveory Model Wih Probabilisic Qualiy Joural of Moder Accouig ad Audiig, 10(7), Chiu, P Y (003) Deermiig he opimal lo size for he fiie producio model wih radom defecive rae, he rework process, ad backloggig Egieerig Opimizaio, 35 (4), El-Kassar, ANM, (009) Opimal Order Quaiy for Imperfec Qualiy Iems Proceedigs of he Academy of Iformaio ad Maageme Scieces, 13(1), 4-30 El-Kassar, A N, Yassi, N, & Makieh, K (010) Lo Sizig a Muli-sage Producio Process The Busiess Review, Cambridge, 14(), El-Kassar, AN, Salameh, M & Biar, M, (01) EPQ model wih imperfec qualiy raw maerial Mahemaica Balkaica, 6, Hayek, P A, & Salameh, M K (001) Producio lo sizig wih he reworkig of imperfec qualiy iems produced Producio Plaig ad Corol, 1(6), Jaber, MY, & Kha, M, (010) A model for maagig yield i a serial producio lie wih learig ad lo spliig Ieraioal Joural of Producio Ecoomics, 14(1), 3-39 Kha, M, & Jaber, MY (011) Opimal iveory cycle i a wo-sage supply chai icorporaig imperfec iems from suppliers, Ieraioal Joural of Operaioal Research, 10(4), Kha, M, Jaber, MY, Guiffrida, AL, & Zolfaghari, S, (011) A review of he exesios of a modified EOQ model for imperfec qualiy iems, Ieraioal Joural of Producio Ecoomics, 13(1),1-1 Salameh, M K, & Jaber, M Y (000) Ecoomic producio quaiy model for iems wih imperfec qualiy Ieraioal Joural of Producio Ecoomics, 64, Salameh, MK & El-Kassar, AN, (007) Accouig for he Holdig Cos of Raw Maerial i he Producio Model I Proceedig of BIMA Iaugural Coferece, Sharah, 7-81