Modeling and Analysis of a Variety of Exchangeable-Item Repair Systems with Spares

Transcription

1 בס"ד Modelg ad Aalyss of a Vaey of Echageale-em Repa Sysems wh Spaes Thess sumed paal fulfllme of he equemes fo he degee of "DOCTOR OF HLOSOHY" y Mchael Deyfuss Sumed o he Seae of Be-uo Uesy of he ege July 5 Bee Shea

2 בס"ד Modelg ad Aalyss of a Vaey of Echageale-em Repa Sysems wh Spaes Thess sumed paal fulfllme of he equemes fo he degee of "DOCTOR OF HLOSOHY" y Mchael Deyfuss Sumed o he Seae of Be-uo Uesy of he ege Appoed y he adsos Appoed y he Dea of he Kema School of Adaced aduae Sudes July 5 Bee Shea

3 Ths wo was caed ou ude he supeso of of. Moo ose of. E. Koach he Depame of dusal Egeeg ad Maageme Faculy of Egeeg

4 Asac The goal of hs eseach s o aalyze a aey of models echageale em epa sysems wh geeal epa dsuo ad ample sees. The goal each sysem s o fd he ume of spaes whch opmzes some ojece fuco. The eseach eploes he ehao of wo dffee sysems whe seeal classes of ems o compoes ae assumed. The fs sysem assumes ha a cusome gs a poduc cossg of seeal classes whch fals oly a oe of he ems ERSOF ad he secod sysem assumes ha he poduc ca fal moe ha oe em ERSMF. Ths hess eploes he ehao of seeal opmzao models fo he fs sysem. oe class of model we opmze.e. ma o m a sece ceo such as he fllae he aeage wag me he aeage ume of cusomes he sysem o he poaly ha a cusome was loge ha a specfed me ude oe o seeal lea cosas. a secod model class we opmze mmze he amou of moey esed fo spaes ude oe o seeal sece-cea cosas. a hd class of model we opmze seeal goals ude seeal lea cosas dued goal pogammg oday's leaue. We pode a full soluo of a eample ad demosae dffee algohms o sole he models. Sesy aalyss s added o asss eplog he ehao of he model. Fo he secod sysem a aalyc fomula s deeloped fo he wag me dsuo fo a sysem coag wo em classes. Ths assumpo ca e elaed easly fuhe eseach. Eesos o ul aals scappg mul-echelo sysems ae also aalyzed ad ca e oduced easly o he asc model. As pa of he aldao ad also as a ew geeal echque a ege pogammg appoach s used o show how o sole all of ou smalle polems. Fally a em-ased appoach s deeloped o fd he dsuo of he ume of cusomes he sysem whch was aleady peseed coecly y Hausma ad Cheug.

5 Acowledgmes Fs wsh o ha of. ose fo hs help fo hs adce ad especally fo hs fedless ad eadess o help me dug hs eseach. also wsh o ha of. Koach fo hs help ad gudace dug my sudes a Be uo Uesy. Thd waed o ha D. D Shaay ad he ohe efeees fo he helpful commes whch helped me o mpoe hs hess. Fally wa o ha my wfe who suppoed me wh loe ad udesadg dug hs eseach. Whou he could o hae fshed hs eseach.

6 Tale of Coes Asac... Acowledgmes... Tale of Coes... Ls of Fgues... Ls of Tales... a.... oduco..... oduco..... Leaue eew Ojeces of cue eseach The Basc Model The echageale-em epa sysem The asc aalycal model The wag me dsuo The sofwae pacage deeloped fo he asc model The Echageale-em Repa Sysem wh Seeal em Classes oduco The model fo a echageale - em epa sysem whe falues ae fom a sgle class ERSOF Sece measues Defg he opmzao model Opmzg he ERSOF model The fllae of he sysem The aeage wag me of a cusome he sysem Algohms: Valdao of he esuls of he algohms os-opmzao umecal eample... 35

7 3.8.. Sesy aalyss aou he udge allocao M Sesy aalyss o aal ae Model aldao Modelg a Echageale-em Repa Sysem wh moe ha oe Falue Class ERSMF oduco Sece measues The Wag Tme Dsuo The Aeage Wag Tme The Fllae The ume of cusomes sysem The aeage ume of cusomes he sysem...53 a Opmzg Addoal Sece Cea oduco Solg he fllae polem usg dyamc pogammg A appomae Dyamc ogammg algohm fo lage scale polems Deelopg aohe algohm fo he fllae The oal ume of cusomes he sysem The poaly of wag less ha Dffee ERSOF-Models oduco The Aeage Wag Tme Valdao of he algohm The fllae Coecg he cosa ad he ojece fuco The fed me pealy a ERSOF-model The fed me pealy a ERSOF-model fo a ge peod The fed me pealy a ERSOF-model fo a fe peod The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model 74

8 The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model fo a ge peod τ The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model fo a uouded eal The fed pealy fo delay loge ha me a ERSOF model The fed pealy fo delay loge ha me a ERSOF model fo a ge fed me peod The fed pealy fo delay loge ha me a ERSOF model fo oe a uouded eal Mulple oals oduco Comg wo ojeces Mssg he goals oal pogammg ege ogammg oduco The aeage wag me The fllae The poaly of wag me Sesy aalyss The poaly of wag moe ha fo wag cusomes Bul sysems oduco A ERSOF - sysem whee eey cusome gs eacly faled ems of he same class A ERSOF-sysem whee each cusome gs eacly faled ems of he same class whch faled A ERSOF-sysem whee each cusome gs a adom ume B of faled ems of he same class Deelopg ERSMF-models oduco... 4

9 .. A ERSMF-model whee all he cusomes ae of ype A ERSMF-model whee he sysems coas CT CT ad CT Whe he CT3 s ag Whe he CT s ag Whe CT s ag The Wag Tme Dsuo of a adom cusome Sece Measues Model Eesos Seeal model eesos Scappg he asc model Mul-echelo sysems Ample Sees Summay ad fuhe eseach oao summay... 5 Apped A: Reseach o sole he ERSOF model usg omal Appomao Apped B: Usg he sofwae deeloped fo he asc model Apped C: The eo of Hausma ad Cheug Apped D: Smulag he umecal esul of a ERSOF-model... 6 Apped E: oof of he wag me dsuo Apped F: oof of he Wag Tme Dsuo of a ERSMF-model Apped : Deelopg fomula Apped H: oof of he wag me dsuo of a ERSMF-model Apped : Chec mechasm fo he ERMF-fomula... 7 Refeeces... 73

10 Ls of Fgues Fgue.: Lfecycle of a em... Fgue.: Dso of he Leaue eew of Repa Facles... 5 Fgue.: Lfecycle of a em... 9 Fgue.: Load Daa Dalog o ge he daa fom he use Fgue.3: Resul Ma of Opmaly... 5 Fgue.4: eface of he sofwae... 5 Fgue 3.: Model of ERSOF... 8 Fgue 3.: Eample of he ehao of he fllae... 6 Fgue 3.3: A eample of he aeage Wag Tme of a asc sysem... 8 Fgue 3.4: Coeg gaph of Fgue 3.3 o pece-wse lea... 9 Fgue 3.5: pu ad Oupu daa of he sofwae... 3 Fgue 3.6: Resuls of he secod algohm... 3 Fgue 3.7: The Cumulae wag me fo he umecal eample Fgue 3.8: The desy fuco of he wag me he sysem Fgue 3.9: The dsuo of he ume of cusomes he sysem Fgue 3.: The aeage wag me fo he dffee udges Fgue 3.:The fllae of he opmzed sysem whe addg ceasg he udge Fgue 3.: A eample of he wag me dsuo whe he udge chages4 Fgue 3.3: All dffee wag me dsuo whe he udge chages... 4 Fgue 3.4: The effec of he udge o he wag me dsuo... 4

11 Fgue 3.5: The effec of he aal ae o he aeage wag me ad he fllae... 4 Fgue 3.6: The effec of he aal ae o he cumulae wag me dsuo... 4 Fgue 3.7: The oupu of he smulao fo he ume of cusomes he sysem Fgue 3.8: Compaso of he oupus of he smulao ad he aalyss Fgue 3.9: The wag me dsuo of he smulao Fgue 3.: Compaso of he wag me desy fuco of he smulao ad aalyss Fgue 4.: The epa facly wh seeal em classes ad seeal falues Fgue 5.: aph whch shows he esuls ge Tale Fgue 5.: The Opmal Fllae whe calculaed wh dffee seps Fgue 5.3: The Fllae of a su-sysem Fgue 5.4: Oupu fo he fllae whe dollas ae aalale... 6 Fgue 7. A eample of he aeage wag me esus udge allocao Fgue 7.: A eample of udge allocao esus he aeage wag me Fgue 7.3: Eample of he eface of he fs pogam wh dollas Fgue 7.4: Eample of he eface of he secod pogam wh he AWT ge.. 67 Fgue 7.5: Eample of he eface of he fs pogam wh 5 dollas Fgue 7.6: Eample of he eface of he secod pogam wh he AWT ge.. 68 Fgue 7.7: aph of he fllae esus dollas of ss... 7 Fgue 7.8: Moey esus he fllae of ss... 7 Fgue 7.9: Toal cos of ss Fgue 8.: Eample of he ehao of he fllae... 8

12 Fgue 8.: aph of he secod pa of he equao... 8 Fgue 9.: Opmal spaes eco fo he ege olem Fgue 9.: Opmal eco fo he Lea olem Fgue 9.3: Opmal spaes eco fo he fllae usg ege pogammg... 9 Fgue 9.4: Opmal spaes eco fo he fllae usg lea pogammg Fgue 9.5: aph of he oeall aeage wag me... 9 Fgue 9.6: aph of he oeall fllae... 9 Fgue 9.7: aph of he oeall fllae wh smalle eals Fgue 9.8: Opmal spaes eco fo he poaly of o wag Fgue 9.9: The poaly of wag less ha wh dffee udge allocao. 94 Fgue 9.: Wag Tme Dsuo ased o dffee udge allocao Fgue 9.: Oupu of he codoal poaly of wag moe ha Fgue 9.: The codoed poaly of wag moe ha Fgue 9.3: Sesy aalyss of dffee udges fo a ge Fgue.:Cusome ag ad leag afe +... Fgue.: Cusome ag + ad hag hs em epaed efoe Fgue.: aph of cusome ag du efoe ad em of class o epaed ad em of class o epaed efoe Fgue.: Cusome ag du ad hag he em of class epaed efoe + ad he em of class epaed afe Fgue.3: Cusome ag ad hag hs em of class epaed efoe + ad hs em of class epaed afe Fgue.4: Cusome ag ad hag hs em of class epaed efoe + ad hs em of class epaed efoe +...

13 Fgue.5: Cusome ag du afe ad hag hs em of class epaed efoe + ad hs em of class epaed efoe +... Fgue.6: Cusome ag du afe ad hs em of class epaed ad hs em of class o epaed efoe Fgue.7: aph of cusome ag duafe ad em of class epaed ad em of class o epaed efoe Fgue.8: Cusome ag ad hag hs em of class ad hs em of class epaed afe

14 Ls of Tales Tale 3.: Speadshee of a umecal eample Tale 3.: Fou dffee appoaches o sole he polem Tale 3.3: Daa of he sysem wh 3 classes of ems Tale 3.4: Sece-leel measues fo he umecal eample Tale 3.5: The esuls of he sofwae ad of he smulao Tale 4.: Eample of he cusome ypes wh wo em classes he sysem Tale 5.: Tme secods fo dffee amous of classes ad udge Tale 9.: Eample of he Speadshee Tale 9.: Tale of he aeage wag me fo each class ad fo each alue j Tale 9.3: Values of he fllaes fo each ss used he ege pogammg Tale.: Eample of he cusome ypes wh wo em classes he sysem... 4

15 a

16 . oduco.. oduco A echageale-em epa sysem s a sysem o whch cusomes g a faled poduc cossg of a ume of ems fo epa ad ecee seceale oes eu whch ae he esalled o he poduc. ems ae cosdeed echageale he sese ha cusomes ae eady o ae ay seceale em of he same d hey ough o he sysem. The sysem opeaes epa facles whch he faled ems ae epaed f o scapped ad ued o seceale oes. ew ems ca e added dug wo ad emoed ased o he eeds of he sysem. The ag cusomes jo a queue he cusome queue- o wa fo seceale ems f a he aal he sysem does o hae aalale ems o he sheles soc. The faled em s se o a epa facly whee wll e scapped o epaed. Afe epa he em wll e soced o he sheles fo fuhe use. Ths soc coas ew ems whch ae called spaes a he egg of ug he sysem ad added ems dug he pocessg of he sysem ad epaed ems dug wo. A cusome wll ge a em fom hs soc wheee hee ae ems he soc. Afe geg such a seceale em whch s he seed o he poduc he cusome leaes he sysem. Oously whe he sysem coas spaes he cusome s seed ee ad quce. Based o hs sysem he followg model was depced Fgue.:

17 Fgue.: Lfecycle of a em Ths asc model coas o scapped ems ad o ew spaes. Ths meas ha dug wo o ew spaes ca e added. A he egg hee ae eacly spaes o he shelf. Hece he ume of ems he sysem wll always e a leas ecause eey cusome gs eacly oe em ad ulmaely ecees oe u does o hae o e he oe he ough o he sysem. Thus f hee ae cusomes wag he queue hs meas ha hee ae o good ems o he shelf ad heefoe he ume of ems he epa facly equals he ume of cusomes wag plus he ume of spaes. f hee ae o cusomes wag he ems wll e ehe he epa facly o o he shelf. Hece Toal ume of ems he sysem = + ume of cusomes wag he queue.. The epa facly coas a ample ume of sees. Ths meas ha he epa me of a em s depede of he ume of ems he epa facly. The cusome wag me wll e deeloped lae as a fuco of he ume of spaes he sysem.

18 The epa me measued fom he sa of aal has a dsuo. wh mea. Codog o he ume of spaes ad assumg equlum codos peal we le W e he me a cusome was ul he ges a seceale em ad X he coespodg ume of cusomes he sysem. The W < s he poaly ha a cusome was less ha me > ad X = s he poaly ha hee ae cusomes he sysem seady sae =.... To complee he model we wll assume ha cusomes ae depedely wh he me ewee aals dsued epoeally wh mea. Thus cusome aals follow a homogeous osso pocess wh ae. A sysem such as ha desced mus offe ways o measue s effcecy a some sece leel. ca e measued y focusg o he cusome o o he sysem. mos cases hey ae equale. A eample of a "cusome sece-leel" s he poaly ha a cusome was less ha half a hou s moe ha 95%. Aohe eample fo he sece leel of he sysem s he aeage ume of cusomes wag he sysem... Leaue eew Whe epa facles ae eamed seeal aspecs ae cosdeed. Dffee sysems ca e dded o dffee classes whch shae commo popees. The fs dso o mae s he popey of spaes. We ca deal wh sysems wh spaes o whou spaes. ehe case echagealy of he faled ems s assumed. Ths meas ha a cusome who aes a a epa locao gg a faled poduc whch s composed of seeal ems does o md geg ohe ems as log as hey ae seceale ad ca e cluded hs poduc. Ths class of sysem s commo oday s epa sysems. Ou eseach goal s o aalyze sysems wh spaes echageale-em epa sysems whch ae moe comple ad whch deal wh a aey of dffee assumpos ad appoaches. Whe a echageale-em epa sysem s eamed ca e ha he ems ae o epaed a hs epa facly u a aohe locao so ha hese faled ems mus e se fuhe wo o mul-echelo sysems. Fo eample he amy cealzed 3

19 depos ae ofe used o eplesh local depos o mmze he ume of spaes fo he weapos sysems Sheooe [6][7][8] ad Algh [][3]. Mul-echelo sysems hae eceed cosdeale aeo he leaue ecause of he mpoace fo dusy o he amy. Seeal opcs such as achg Lee H ad K. Mozadeh [] dffee shpme mes aes [][][3] ad poy shpme Dada [9] coe may dffee possles of mul-echelo sysems wh some possle aaos. We wll fs cosde sysems whee ems ae epaed a oe locao o se fuhe u wh eglgle addoal delay sgle-echelo sysem. To ceae models of sgleechelo sysems seeal assumpos ae made such as: Assumg geeal epa me dsuo of a em o some specal oe epoeal: ose [6] Dadua [] ad ohes. Assumg ha aals of cusomes gg poducs follow a osso seam o o. Assumg ha he sysem coas a small ume of sees fe: Beg & ose[6] Dadua [] o a lage amou ample Beg & ose [5] Assumg ha he ems of he poduc ca e scapped o hey all mus e epaed. Assumg ha he cusomes wa fo sece ad do o leae o hey leae f hey acpae ha hey may hae o wa oo log. Assumg seady-sae ehaou o o. Assumg dyamc epa sysems whch clude scappg ad uyg ew spaes o o. Assumg fed shelf-les whe ems ae peshale ey ad ose [4] o o. Assumg ha he poduc whch he cusome gs cludes oe em o seeal. Assumg ha he poduc whch he cusome gs cludes seeal ems u he poduc faled ecause OE of he ems faled o seeal ems ca fal. 4

20 Repa Facles Echageale Sysemsspaes Oe-o oe epa whou spaes Repa a oe place Repa a seeal places Mul-echelo sysems Sheooe[-3]Algh[-3] Aal osso Aal o osso eeal epa Epoeal epa Ample sees Fe sees Fe sees ose[6] Dadua[9] Oe ype of em Seeal ypes Oe em ose[5] Seeal ems ose[5] Oe falue Seeal falues Hausma[5] Fgue.: Dso of he Leaue eew of Repa Facles he leaue may of hese ad ohe assumpos ae made ode o ceae models fo specal cases. A geeal model cludg ALL he dffee assumpos s sll o feasle due o s eeme compley as well as he compley of he mahemacs whch wll e equed o sole hese models. Specal cases whch clude seeal of hese assumpos ca e foud he leaue. Fgue. depcs a dagam of he dffee eseach felds of dffee models of Repa Facles. We coceaed ou eseach o he M class of models. These classes of models hae commo ha cusome aals follow a osso seam he epa mes of ems follow ay geeal dsuo ad he sysem cludes a fe ume of sees. Sheooe [-3] deeloped models opmzg he aalaly of he poducs ad he ume of poducs wag fo epa a mul-echelo sysem. The wag me of he poduc was o cluded o ee meoed. ose ad Beg [5] deeloped 5

21 he mahemacal fomulas fo he wag me whch wee appaely o used. They all deeloped a model fo poducs cossg a sgle em class whch may also clude aals ul he poduc coas moe ha oe of he same em class. Hausma ad Cheug [6] ed o sole a moe comple model. They deeloped a model fo a poduc wh mulple falues ypes; hs meas ha a cusome aes o a sysem wh a poduc whch ca fal seeal ways ad was ul hs poduc s epaed. We foud ha he fomulas wee o coec ad hs s show Apped C. Aohe eesg model peseed y Jg-Sheg Sog [3] s o a epa sysem u a odeg sysem wh spaes whee he cusome ca choose dffee ems. fac he models of epa ad odeg deal wh smla sece measues ad polems. Solg epa sysems ca affec odeg sysems ad ce-esa. Thus a deep ad oad owledge of dffee sysems s equed fo a eseache sag o deelop ew models. Rece eseaches hae focused he wo o mul-echelo sysems addg dffee assumpos egadg sece cea [33] [8] lmed capaces [7] ad moe [3]..3. Ojeces of cue eseach As eplaed he leaue eew a model fo a echageale em epa sysem wh spaes whch deals wh oly OE em class aleady ess. Bu wha happes f he poduc coas moe ha oe em whch s he case almos all poducs? Thee ae o aswes fo ha polem; Hausma ad Cheug dd appoach hs polem ad ed o sole ale usuccessfully. A poduc ca fal hough eacly oe em falue o hough mulple em falues. Theefoe hs wo we wll coceae o echageale em epa sysems wh spaes whee each poduc ough o he epa sysem cludes moe ha oe em. Spaes ae added o he sysem o mpoe he sece leel. Due o he fac ha he sysem has a echagealy popey whch meas ha cusome do o md geg aohe em as log as seceale addg spaes o he sysem wll auomacally mpoe he aalaly of ems o eplace he faled oes. The assumpos fo all he models hs eseach ae: 6

22 Assumpo : Assumpo : Assumpo 3: Assumpo 4: Assumpo 5: The epa mes fo all ems follow a geeal dsuo fuco. The aals of he cusomes gg poducs follow a osso seam. The sysem coas a ample ume of sees. The ems of he poduc ae epaed ad o scapped. The cusomes wa fo sece ad do o leae. Mos models assume ha he epa me s dsued epoeally o ease mahemacal aalyss. Hee he epa me ca follow ay ype of dsuo fuco. Aohe assumpo we mae s ha he cusome aals ae depede. Ths meas ha hey do o ae goups a ge mes u hey ae a adom. Ths s called leaue a osso ocess. Whe cusomes ae o he sysem hey ee a cusome queue ad wa ul hey ge sece. They wll wa ad mgh leae he sysem efoe sasfaco. Dealg wh polems whee cusomes leae he sysem efoe sasfaco s eyod he cue eseach scope. We ally assume ha he epa facly coas ample sees. All hese assumpos ceae a small feld of classes of models whch ca e eploed. hs educed feld we foud wo acles whch f o hese assumpos. Hausma ad Cheug [6] ad Beg ad ose [5]. Sep y sep we wll aalyze ad ela assumpos so ha he eade ca easly follow he aalyss of he dffee models. a we wll aalyze moe sadad polems wheeas a fom Chape 5 we wll deelop moe comple models. Thus Chape we wll pese he asc model deeloped y Beg ad ose [5] ad dffee aspecs whch ca help us he fuhe eseach. Chape 3 we wll pese ou fs sysem: The ERSOF-sysem. Ths s a sysem o whch cusomes g a poduc whch faled oly OE em. We wll defe wo models: Oe whee he ojece fuco s o mamze he fllae ad oe whee he ojece fuco s o mmze he aeage wag me ude a udgeay cosa o epedues fo spaes. Ufouaely usg sadad mahemacal echques he model wh he fllae ca o e soled. Chape 3 we wll loo a a specfc cea ype of polem we wll mamze he sece leel sujec o a udge lmao. Afe aalyzg oe 7

23 ERSOF-model we wll loo a he ERSMF-model. A ERSMF-model s a model of a sysem o whch cusomes gs poducs whch faled a moe ha oe em. Chape 4 we wll pese he sysem ad s sece cea. Due o he fac ha hs model cao e soled usg sadad echques we wll leae hs model o Chape o e aalyzed ad soled. Ths cocludes a whch peses asc deas ad smple models of he eseach. a of he eseach whch sas a Chape 5 we wll oduce moe complcaed models deelop ew aalyc fomulas ad pese ew echques fo solg geeal polems. sho he eseach ecomes moe comple u also moe eesg fo he eade. Chape 5 we wll opmze a model wh he fllae as he sece leels cea ad also he ume of cusomes sysem as aohe ceo. Chape 6 we wll aalyze a secod cea ype of polems we wa o mmze he oal esme sujec o a sece leel. sead of a o-lea ojece fuco wh lea cosas we loo a models whee he ojece fuco s lea ad he cosas ae o-lea. Chape 7 we wll focus o mulple goals. We wll defe a hd cea ype of polems we wa o mmze a aggegae ojece fuco wh o cosas; hs meas ha a maage ca defe mulple goals whch s a oely hs eseach. Aohe oely s he echque o sole polems y ege pogammg o e peseed Chape 8. Thoughou we should o foge ha oe of he mo goals of hs eseach s o coec he fomula of Hausma ad Cheug. Fo hs pupose we eed o deelop he wag me dsuo fo ul aals of a ERSOF sysem Chape 9 ad he wag me dsuo fo he ERSMF sysem Chape. We wll pese a echque o coec he fomula of Hausma ad Cheug. Fally we wll oduce ew feaues he model such as scappg ad aalyze he mpac of mul-echelo sysems ad he ample sees assumpo. Ths s wha s awag you dea eade he followg chapes. Ths eseach comes ew echques ce eamples mahemacal deelopmes ad ew fomulas wh suppo wheee possle usg sophscaed sofwae whch was deeloped o llusae umecal eamples ad aldae he algohms. 8

24 . The Basc Model.. The echageale-em epa sysem A echageale-em epa sysem s a sysem o whch cusomes g a faled poduc cossg of ems fo epa ad ecee seceale oes eu whch ae esalled o he poduc. ems ae cosdeed echageale he sese ha cusomes ae eady o ae ay seceale em of he same d hey ough o he sysem. The sysem opeaes epa facles whch faled ems ae epaed f o scapped ad u hem o seceale oes. ew ems may e added dug wo o emoed ased o he eeds of he sysem. A ag cusome jos a queue he cusome queue o wa fo a seceale em; f upo aal he sysem does o hae ay aalale ems o he shelf he wll hae o wa. Bu f hee ae good ems o he shelf he aes oe ad pepaes o leae. Fgue.: Lfecycle of a em The faled em s se o a epa facly whee wll e scapped o epaed. Afe 9

25 epa he em wll e placed oo he shelf fo fuhe use. Ths shelf coas oly ew ems whch ae called spaes a he egg of ug he sysem ad added ems dug he pocessg of he sysem ad epaed ems dug wo. A epaed em s cosdeed as good as ew. Oously whe he sysem coas spaes he cusome s seed quce. Based o hs sysem he followg model was ceaed: Ths model he asc model coas o scapped ems ad o ew spaes. Tha meas ha dug wo o ew spaes ca e added. A he egg hee ae eac spaes o he shelf. The ume of ems he sysem wll heefoe always e a leas ecause eey cusome gs eacly oe em ad eeually ges oe whch eed o o e he oe he ough o he sysem. Thus f hee ae cusomes wag he queue hs meas ha hee ae o ems o he shelf ad heefoe he ume of ems he epa facly s equal o he ume of cusomes wag plus he ume of spaes. f hee ae o cusomes wag he ems wll e ehe he epa facly o o he shelf. Toal ume of ems he sysem = + ume of cusomes wag he queue.. The epa facly coas a ample ume of sees. Ths meas ha he me a em eques o complee epa follows a geeal dsuo. ad s depede of he ume of ems he epa facly. The cusome wag me wll e deeloped lae as a fuco of he ume of spaes he sysem. Sece leel cea: Fllae FR The fllae s he poaly ha a cusome gg a poduc o a sysem soced wh spaes ges he faled em hs poduc eplaced sagh away. Ths meas ha he cusome s o equed o wa fo sasfaco. The moe spaes hee ae he hghe he poaly ha a cusome ges hs eplaceme em decly. Aeage Queue sze L

26 The aeage queue sze s he aeage ume of cusomes wag fo a seceale em a sysem wh spaes. Aeage Wag Tme - W The aeage wag me s he aeage me a cusome mus wa ul he ecees a seceale em a sysem wh spaes. The oaly of wag moe ha - W The aeage wag me may o e he ma cea. Aohe mpoa ceo s he poaly ha a cusome wll wa loge ha mues. The maage wll o e happy whe say % of he cusomes wa moe ha mues alhough he aeage wag me s secods. mos cases he aeage wag me s a good dcao how he sysems wos u o how he cusome s seed. Thus aohe measue fo he sysem s he poaly ha a adom cusome wll wa moe ha some specfed. The oaly of wag moe ha ude he codo ha he cusome was- W W Wha happes whe 9% of he cusomes do o wa ad oly % of he cusomes wa moe he mues? The maage of he epa facly wll h ha hs facly wos fe. Bu whe loog moe specfcally a all cusomes who do wa we may see ha moe ha 5% wa loge ha mues. Theefoe he poaly of wag moe ha may o e adequae cea cases. Thus we oduce he poaly of wag loge ha ge ha he cusome mus wa W W... The asc aalycal model The asc aalycal model was deeloped y Beg ad ose [5] ad s summazed elow.

27 ... The wag me dsuo a echageale-em FFO sysem wh spaes ha sasfes he assumpos of he asc model Assumpos -5 age 8 we defe W as he seady-sae wag me of a ag cusome wh desy f ad c.d.f F. Thus W F F Y Y Y Y <. whee Y ad Y ae depede geec osso adom aales ad s he cumulae epa me dsuo. Hee Y ad Y ae osso wh paamees λ ad λ especely whee u u du.3 ad u du.4 u F s he poaly ha he cusome wll ge a em y me ge spaes. Theefoe wh poaly -F he mus wa loge ha. Fo a eplaao please see Beg ad ose [5]... The aeage wag me ad he aeage ume aclogged The aeage wag me s he aeage me a cusome was ul he ges a seceale. Thus W f d

28 W F d.5 f d F F d F d.6 Le X e he seady sae ume of cusomes he sysem ad X = he poaly ha hee ae cusomes he sysem. The X s equal o he poaly ha hee + ems he sysem. sce eey cusome gs oe faled em ad hee ae spaes he sysem. Le Y e he ume of ems he sysem. Thus he poaly ha hee ae Y ems he sysem s osso wh paamee sce he model fo ems s M ad Y y! y y e fo y = 3. By. X Y. A ay me he ume of cusomes wag ad he ume of spaes wll e equal o he oal ume of ems he sysem soc ad epa. Thus X ad X! Y e = 3..7 X whee s he ume of spaes. The aeage ume of cusomes he sysem s y defo: 3

29 L * X.8 ad he aace of he ume of cusome he sysem s V L X * X..9 These wo alues ae mpoa fo he sece leel cea. Lle s fomula coecs W ad L y L *. W.3. The sofwae pacage deeloped fo he asc model A flele sofwae pacage was deeloped o suppo fuhe eseach o echageale-em epa sysems. Sce he fucos.-.5 ae geeally dffcul o calculae ecause of he lage ume of opeaos he sofwae pacage asssed ou udesadg of how hose fucos ehae. addo sesy aalyss was added o complee ou udesadg of he asc model. As a eample o calculae he wag me dsuo we eed daa aou he epa me ad he aal ae. The sofwae eales he use o choose dffee epa me dsuos such as omal amma ad Epoeal dsuo. Fg. Fgue.: Load Daa Dalog o ge he daa fom he use. 4

30 The sofwae he ges a use-fedly eface o hadle ge daa ad o suppo decsos such as he Resul Ma Fg.3 whch also podes dealed sesy aalyss of he mea epa me ad he aal ae of he cusomes. Fgue.3 shows Fgue.3: Resul Ma of Opmaly Fgue.4: eface of he sofwae he esul ma. O he lef sde hee ae dffee aal aes ad o he op dffee mea epa mes. he eample show he mea epa me was hou ad he falue 5

31 ae 5hou. hs eample a alue was chose fo he fllae. The esul ma shows he ume of spaes o uy o achee hs codo. You osee ha he opmal alue s. The fgue shows he sesy ma whch suppos he decso mae. Aohe scee of he sofwae shows addoal maeal such as gaphs of he wag me dsuo he epa me ad addoal sece leels Fgue.4. ew sofwae feaues wee also added such as uos o easly chage he oal ume of spaes+- ad sece leel cosas o mae he sofwae ee moe use fedly. Ths sofwae pacage was o modfed o clude all ew esuls of he eseach u seed as a asc ool udesadg he ge sucue of he asc model ad aalyzg moe comple models. 6

32 3. The Echageale-em Repa Sysem wh Seeal em Classes 3.. oduco A echageale-em epa sysem wh seeal classes of falue s a sysem o whch cusomes g oe poduc whch has faled. Howee he cause of he falue may e a ay oe of seeal ems wh he poduc. a modula sese he poduc may e composed of a ume of compoe modules ems ad a falue ay compoe esuls he falue of he poduc. Whe he cusome aes wh a faled poduc o he sysem he poduc s aalyzed ad he faled ems ae se fo epa. Fo ee sece he cusome would le o wa less he sysem eeps a ume of spaes fom each compoe class. Each class of em has a soc of ems ad a epa facly whee he ems ae epaed. fac hese socs ad epa facles eed o e physcally sepaaed. Afe epa he ems jo he soc. The cusome leaes he sysem whe hs poduc s epaed. hs followg chape we wll deal wh a aey of models. Oe of he sece measues he aeage wag me wll see as he ojece fuco o e opmzed ude a udgeay cosa. The we wll oduce he fllae as he ojece fuco ad show why we cao use KKT-mehods. The ERSOF-model wh he fllae as ojece fuco wll e aalyzed he secod pa. Fally we wll oduce he aeage wag me as a sece-leel ad sole he model cludg pos-opmaly hough a umecal eample. 3.. The model fo a echageale - em epa sysem whe falues ae fom a sgle class ERSOF hs model we mae he assumpo ha oly oe compoe module em ca fal a a me ad heefoe he followg model was deeloped. Cusomes ae o he sysem wh he poduc a he ma des whee he faled em class s defed ad he oued fo epa. We suppose hee ae classes of falue possle ad he oeall aal ae λ s paoed o aes λ λ λ 3 λ coespodg o he ddual falue 7

33 DESK aes. We wll defe a as he popoo of class- falues = so ha λ = λ ad a. Thus sce he poduc aal seam s osso wh ae λ he class- em falues fom a osso seam of ae λa. These ae se o a su- ample see epa facly dealg oly wh class- falues whch s soced wh spaes a leel. We call hs susysem susysem o ss. Aga ecause of he ample see assumpo all epas ae caed ou depedely ad we ca ew he oeall facly as paoed o susysems. Ths s depced Fgue 3.. Cusome ges a em fom he soc The epaed em o he soc Su-Sysem ss STOCK Repa Facly λ λ λ Su-Sysem ss STOCK Repa Facly Cusome wh a faled em The faled em o he epa facly Fgue 3.: The ERSOF-model 8

34 Hee we ca mage he cusome accompayg hs specfc class- falue o ss ad epeece hee pecsely wha he asc model dcaes. f we assume ha des me falue defcao ad esall mes ae eglgle he he asc model ges he equed susysem pefomace measues ems of he susysem paamees. Thus fo ss = we hae epa o epoduco me dsuo fuco. wh mea μ. We wll desgae he wag me of a cusome a ss y. T = ad he oeall me he sysem yw....we wll desgae he ume of cusomes a specfc ss y. X ad he oeall ume of cusomes he whole sysem y X. We ow we F F T ad hs s oaed y appopae use of he asc model. Thus he fs wo momes of he wag me he ume of cusomes ss ae ge y: T F d F d 3. T F d F d 3. L * X 3.3 fo =. 3.. Sece measues he sequel sece measues ae oduced o desce he pefomace of he ge sysem. Each su-sysem coa spaes ad we we The wag me dsuo The wag me dsuo of a adom cusome ag a he sysem s ge y: W F F 9

35 a T a F The wag me momes The aeage wag me chaacezed y spaes eco s W of a adom cusome comg o he sysem W a T d a a T T. d 3.5 whee T s he aeage wag me of a cusome gg em class o ss. E W a T 3.6 whee T s he secod mome of a cusome gg em class o ss ad V W E W W The Fllae Fo a sysem chaacezed y spaes eco we defe ha a adom cusome oas mmedae sasfaco: FR as he poaly FR whee a FR FR s he fllae coespodg o ss. Thus fom.

36 FR F Y Y sce. Thus Y Y Y else ad sce we hae FR Y fo = e! The wag me W C ude he codo ha he cusome was Ofe a maage of a sysem does wa o ae decsos ased o he wag me dsuo u ahe o he wag mes of cusome who wa. Le wag me a ss fo a cusome who was. Theefoe fo C T e he C T T fo = T Thus he oeall codoed wag me dsuo s ge y C T W. 3. a FR By usg he aoe equaos he momes ca also e easly calculaed. We leae hs o he eade.

37 3..5. The aeage ume of cusomes he sysem The aeage ume of cusomes he sysem s he sum of he aeage umes of cusomes wag fo epa of all compoe classes. L s he aeage ume of cusomes ss ad s ge y * X L 3. whee! e X fo =3. 3. X X. Thus L he oal epeced ume of cusomes he sysem s ge y L L The ume of cusomes he sysem Aohe mpoa sece leel coespods o he dsuo of he ume of cusomes he sysem. Le Q e he oal ume of cusomes he sysem. Whe hee ae wo su-sysems we ge ha X X Q Whe hee ae hee su-sysems X X X Q Wh moe ha hee su-sysems we may lely assume a omal appomao ad hus X V L Q ~. 3.4

38 3.3. Defg he opmzao model Afe oducg a aey of sece leel measues we ow wa o opmze he soluo o he model. A aual goal s o pode he mamum sece leel possle o a cusome who aes a he sysem. Fo hs pupose we wll assume ha a udgeay allocao of M dollas s a ou dsposal. We mae he assumpo ha M s used oly o uy spaes. ally hee ae o spaes he sysem ad we puchase spaes fo ss a u cos c whou dscous. he followg chapes we wll use ma a seceleel whch meas ha we wa o opmze a ge sece measue. cea cases such as he aeage wag me hs oo wll e eed ecause hee we would wa o mmze he aeage wag me. Thus we use ma as a geeal emology. The polem he ecomes Ma Sece leel 3.5 s..: c M. =3 fo =. whee he sece leel depeds o F =. ge y F Y Y Y Y 3.6 whee Y ad Y ae geec osso adom aales wh paamees λ ad λ ad u u du 3.7 u u du 3.8 3

39 3.4. Opmzg he ERSOF model As eplaed seco 3. hee ae seeal sece leel pefomace cea whch ca see as he ojece fuco fo he model. We wll apply wo of hem he followg seco. The ohes wll e eploed fuhe eseach as a ass fo ew deelopmes. The fs s he fllae of he sysem epessed as he aeage oeall su sysems ad he secod s he aeage wag me of a adom cusome also epessed as he aeage oe all susysems The fllae of he sysem We wsh o mamze he fllae FR. Hece ma FR a * FR 3.9 whee FR s he fllae of ss wh spaes. f FR s ehe a cocae o a coe fuco he he Kuh Tuce [9] codos ae oh ecessay ad suffce codos o achee a opmal soluo fo he couous eso of hs model. The sum of cocae fucos s also a cocae fuco. Thus f each FR s cocae fo eey = he FR wll also e cocae. Lemma 3.: leas oe FR s a o a cocae fuco. As meoed aoe each FR mus e cocae so ha FR s cocae. Thus f a FR s o cocae FR s o auomacally a cocae fuco. By defo he secod dffeece of ay fuco mus e less ha zeo fo he fuco o e cocae. Sce FR s o a couous fuco he fs dffeece s calculaed as follows: ' FR FR FR 3. whee y 3.8 FR T Y Y Y Y fo each =

40 5 Thus ' FR FR FR Y Y Y! e 3. whch s a osso dsuo wh paamee. The secod dffeece s calculaed as follows: ' ' '' FR FR FR!! e e.! e 3.3 The fuco s cocae as log as sasfes '' FR fo all. Thus! '' e FR. Sce! e we hae o. 3.4 Ths meas ha fo he fuco s cocae ad fo he fuco s coe. Cosequely eey FR has a fleco po ad heefoe has a cocae pa ad a coe pa. QED. Fgue 3. shows he fllae of a sysem whe chages as geeaed y he sofwae. deed ca e see ha he fs pa of he fuco s coe ad he secod pa cocae.

41 Fllae Fgue 3.: Eample of he ehao of he fllae Thus he Kuh-Tuce mehod cao e used. A aleae mehod used o sole hs polem s dyamc pogammg ad wll e addessed chape The aeage wag me of a cusome he sysem The secod sece ceo we eame s he aeage wag me of a cusome he sysem. m W a * T whee T s he aeage wag me of su-sysem wh spaes. Sce T L we hae W * T L. f W s a coe fuco he he Kuh-Tuce codos ae oh ecessay ad suffce codos o achee a opmal soluo fo he couous model. Thus f each T s coe fo eey = he W wll also e coe. 6

42 7 Lemma 3.: W s a coe fuco. f eey L fo eey = s coe W s also a coe fuco. To chec hs we wll calculae he secod dffeece. The fs dffeece of L s calculaed as follows: ' L L L! *! * e e!!!! *! *! *! * e e e! e 3.5 The secod dffeece of L s he calculaed as follows: ' ' '' L L L!!!!! e e e e e whch s clealy ue fo eey. Theefoe eey L s a coe fuco ad hus W s also a coe fuco. Thus solg he followg model fo hs couous eso ca e doe y applyg he KKT codos:

43 m W at 3.6 s..: c M. =3 fo =. Ths opmzao model s a ege Kapsac polem whch ca e soled usg dyamc pogammg. Bu whe coss ae low ad he udge M hgh whch meas ha he model wll hae a lage ume of saes dyamc pogammg ecomes mpaccal. Theefoe usg he coey popey of he fuco we decded o mae he ojece fuco couous ad pece-wse lea y mag all T fo = pece-wse lea. Ths appoach s meagful due he fac ha fom a aalycal po of ew geg a esul such as = 4.5 may mea ha oe yea we choose 4 ad he e yea 5. Fgue 3.3 ad Fgue 3.4 demosae how we mae he dscee fuco pece-wse lea. Aeage Wag Tme Fgue 3.3: A eample of he aeage Wag Tme of a asc sysem 8

44 Aeage Wag Tme Fgue 3.4: Coeg he gaph Fgue 3.3 o a pece-wse lea cue The mahemacal defo of he added les s as follows: = + Δ whee Δ whch meas ha s he ege compoe of. T T T T T whee Δ fo all = ad s he couous aale equale of he ege aale. By mag he ojece fuco o a couous fuco he cosa wll ecome dg ad hus c M. To sole hs model a Lagage mulple θ was added. W * m a * T M c W * a * ' T c. W * Thus M c. a * T ' = c 9

45 Apped A we ed o sole a wo-em class polem aalycally usg he omal appomao fo he osso em. Bu he esuls ae o paccal as ca e see ha apped. Theefoe we deeloped wo appomae algohms o sole he polem Algohms: Algohm : The fs appomae algohm sas a. Ths meas ha a he egg o ems ae chose ad o moey spe. The ased o he opmaly a T codo we hae fo = 3.8 c * ' The m s chose whch meas ha we ow hae he es sece leel pe dolla ad s he esulg em class seleced ad added. These seps ae mplemeed ul he whole udge s depleed. The algohm s as follows: a. 3. = whee s he ume of spaes of class ad m=m whee m s he moey lef fo spaes.. Calculae a fo =. c * T ' c. Choose sasfyg m whee s he esulg em class. d. f m c he = +mc. Fsh. e. Ohewse se m=m-c = + o o sep. The sofwae whch was deeloped fo hs algohm ceaes a queue of ems o choose fom odeed fom he smalles θ o he lages. Each me he fs em s ae ou of he queue has he smalles cue θ he queue s eodeed y θ. hs way he pogam aods calculag all θ a each eao. Ths pogam also foud comaos up o ems whe he udge s hgh ad he u coss ae small wh a secod whch s eglgle. 3

46 Fgue 3.5: pu ad Oupu daa of he sofwae Fgue 3.5 shows he pu ad he oupu of he sofwae fo a adom eample of 3 em classes. The pu daa s odeed o hee colums: mea epa me aal ae ad pce pe u. Afe he pu daa he oal udge allocao s lsed. The las le s he oupu: The h ume ges he opmal ume of spaes * fo =.3. Fally he las ume s he aeage wag me of a adom cusome ag a he sysem. ca e see ha all alues of he esuls ae ege ecep oe. Algohm : The secod heusc algohm s ased o he dea ha a ay sage we ca mpoe he ojece fuco whle sll sasfyg he cosa. We do hs y "mog" moey fom oe class o aohe. Fo eample f he u cos of oe em of he fs class s 6 dolla ad of he secod 8 he we choose o moe oe dolla fom he fs class o he secod ad hus = - 6 ad = + 8. Due o he fac ha we ae dealg wh pece-wse lea fucos we ca add ad emoe facos fom he ume of spaes he sysem. We poceed hs way ul we each a local opmum. he case of he aeage wag me we ow ha hee s oly oe local opmum ad hs wll cosequely also e he gloal oe. The sofwae deeloped fo hs pupose chooses oe ou of he hee dffee opos o sa. Thee could e may moe. To dde he udge M equally amog he ems. Ths meas ha =Mc =M*c fo all dffee class of ems

47 All ems hae he same ume of spaes a he egg. Ths meas ha M c The mos epese em class ges all he ems a he egg. M c ma c. 3.3 c The eac algohm loos le: a. 3. ae alzed as meoed y oe of he hee opos whee s he amou of spaes of class chose.. Calculae a fo =.. c * T ' c. Choose l ad h sasfyg m ad he ma whee l ad h ae he em class dces chose. d. f l h fsh. l e. f o calculae c l l l cl ad c h h h c h f. Choose so ha c =mc l c h g. l l c cl ad h h c ch. o o. h. Fgue 3.6: Resuls of he algohm 3

48 The eample shows ha oh algohms ge decal esuls fo he ge eample. The fs algohm s used whe o al soluo s aalale. ca also e show ha lage sysems hs algohm s effce. The secod algohm s used whe a al soluo s aalale ad eeds o e chaged; e.g.; whe he pces o aales of oe o wo su-sysems chage Valdao of he esuls of he algohms s dffcul o aldae he esuls of he algohm ecause he coec aswe s o always ow. The polem s a Kapsac polem. Sce he Kapsac polem s -Had s had o compae he heusc soluo o he opmal oe due o lage compuaoal equemes. fac hee could e =Mmc c c 3 c spaes fo some class esulg dffee soluos whee may e ey lage. spe of hs fac a speadshee was ceaed o aldae he esg pogam. Usg Sole of he Ecel pogam we soled he ge polem Tale 3. also ad foud a eac soluo. Ths s o always guaaeed. Bu Chape 8 we wll use ege ogammg o aldae all he esuls. Tale 3. depcs fou appoaches o solg hs pacula polem. The fs ow shows he esuls usg he Sole o sole as a ege polem. The secod ow shows he esuls usg he sofwae deeloped fo hs polem usg o-ege alues. The hd ow aes he esul of ow wo ad asfoms o ege alues as we wll show. The fouh ow shows he esuls usg a L sofwae. Howee lage sysems wh lage udges he L ad he Kapsac algohms use moe me esouces due o he compley. Fom Tale 3. we ca see ha he Sofwae yelded a coec soluo. g polems me s a cosa so ha good quc algohms ae equed. Theefoe we use he Sofwae deeloped as a good ad effce heusc algohm. Cosde a sysem wh 3 classes o ecessay dsc classes whee he epa mes ae dsued omally wh ge meas ad sadad deaos. Oously he poduc fals a a sgle class. addo he aal ae of em o ss ad he pce of em ae also ge Tale 3.. The mea epa me ad he sadad deao wee chose so ha he poaly of a egae epa me s eglgle. The las colum shows he ume of spaes o uy deemed y he sofwae usg a ge udge. 33

49 em class μ Λ ce λμ moey spe es of chose udge Aal ae a he sysem 69 Ojece Fuco M Tale 3.: Speadshee of a umecal eample Kapsac Sofwae Sofw. K L Tale 3.: Fou dffee appoaches o sole he polem 34

50 3.7. os-opmzao Afe opmzao s ow possle o aswe he followg quesos:. Wha s he fllae of a cusome ag o he sysem?. Wha s he poaly ha a cusome was less ha fo hs poduc 3. Wha s he aeage wag me of a cusome he sysem? 4. Wha s he aeage ume of cusome he sysem? 5. Wha s he poaly ha hee ae moe ha a ge ume of cusomes he sysem? 6. Wha s he poaly a cusome has o wa less ha f s ow ha he has o wa? These ad moe quesos ca e asweed whe s ge. Seeal of hese quesos cao e asweed y Sheooe ad Algh [-3] [-3] whch deal wh dffee mul-em sysems ecause hey dd o use owledge of F he wag me dsuo. Ths s he ma oely of ou wo; podes he capaly o aswe a lage aey of dffee sece quesos. Seco 3.8 podes aswes o all he aoe quesos hough a woed ou eample umecal eample Fo hs seco a compleely ew sofwae was ceaed whch was ased o he peous sofwae. To chec he ew sofwae we also deemed he fllae he aeage wag me ad he aeage queue sze aalycally. Thus y 3.8 FR Y e! Theefoe y.7 ad L e!.8 35

51 36!! e e!! e e!! e e. Hece y Lle's ule!! e e L W fo =.. We wll eame he eample sysem Tale 3.3 elow ad aswe he quesos meoed he peous seco. Eample: R. em Repa Dsuo Mea epa me Sde Aal Rae ceem ume_spaes ORMAL ORMAL ORMAL ORMAL ORMAL ORMAL ORMAL ORMAL ORMAL. 5 6 ORMAL.5. 5 ORMAL ORMAL ORMAL. 5 6 Toal udge dollas. Tale 3.3: Daa of he sysem wh 3 classes of ems. Usg he sofwae deeloped fo hs eseach we foud he sysem measues Tale 3.4. We dd' aalyze he mpac of he sadad deao o he esuls ad fac hs could e pa of a fuhe eseach. All he esuls ae pa of he sofwae whch poduces

52 he gaphs Fgue o llusae he sysem ehao. Tale 3.4 summazes all sece measuemes. Fgue 3.7 ad Fgue 3.8 show how he wag me s dsued. Fgue 3.8 shows ha he desy may hae mulple local mama ad cosequely s upedcale. The aeage fllae of a cusome The aeage ume of cusomes he sysem The aeage wag me of a cusome.377 The codoed aeage wag me.5974 The sadad deao of he wag me Coeffce of aao : Sadad Deaomea.697 Tale 3.4: Sece-leel measues fo he umecal eample W< Fgue 3.7: The Cumulae wag me dsuo fo he umecal eample 37

53 Desy Fuco Fgue 3.8: The desy fuco of he wag me he sysem Aohe mpoa aspec of he sysem s he ume of cusomes he sysem. Fom Chape we ow he dsuo of he ume of cusomes each susysem. The oal ume of cusomes he sysem s he sum of he umes of ems each su-sysem; s dsuo s ge as he sum of osso ems ad easoaly follows a omal dsuo wh mea 5.68 ad he aace Fgue 3.9 shows he gaph of he ume of cusomes he sysem Fgue 3.9: The dsuo of he ume of cusomes he sysem 38

54 3.8.. Sesy aalyss aou he udge allocao M Afe aalyzg he sece-leel measues of he sysem we wa o ow how sese he sysem s o he udge. Theefoe sesy aalyss was added o show he ehao of he dffee sece leels as a fuco of he udge. Fgue 3. shows he Aeage Wag Tme fo dffee udges M Aeage Wag Tme M$ Fgue 3.: The aeage wag me fo he dffee udges fllae M$ Fgue 3.:The fllae of he opmzed sysem whe addg ceasg he udge Aohe eesg sece leel measue s he fllae depced Fgue 3. fo ayg M. The gaphs 3. ad 3. show ha hee ae eamples whee a hghe udge deceases he aeage wag me u does OT cease he fllae 4-7$. Thus he fllae ad he aeage wag me ae o ecessaly coeced ad mus e deal 39

55 wh sepaaely. Ths meas ha o opmze he fllae we eque ohe mehods such as Dyamc ogammg. The fac ha a hghe udge wll o decly mpoe a ge sece leel ca also e see Fgue 3. whee he poaly of wag me of a adom cusome s less ha.7 ad smlaly Fgue 3.3 fo ohe poales W.7 M$ Fgue 3.: A eample of he wag me dsuo whe he udge chages M Fgue 3.3: Dffee wag me dsuos whe he udge chages W. W. W.3 W.4 W.5 W.6 W.7 W.8 W.9 W. W. W. W.3 W.4 W.5 W.6 W.7 W.8 W.9 W. W. W. W.3 4

56 The same daa ca e show aohe way ad we ca see he effec of he udge o he wag me dsuo. Fgue 3.4 shows he daa fom he pespece of he udge. W Fgue 3.4: The effec of he udge o he wag me dsuo The o-smooh chaace of he gaphed fucos s somewha supsg ad wll e susequely esgaed. 4

57 3.8.. Sesy aalyss o aal ae Afe checg how he soluo s sese o he udge faco we ow chec he sesy of he soluo o he aal ae faco. Thus we cosde 3% % % lowe ad % % ad 3% hghe o he aal aes. Fgue 3.5 shows he effec of he aal ae o he aeage wag me ad he fllae assumg he umes of spaes Tale 3.3. Fgue 3.5: The effec of he aal ae o he aeage wag me ad he fllae W %less %less %less ogal %moe %moe 3%moe Fgue 3.6: The effec of he aal ae o he cumulae wag me dsuo 4

58 3.9. Model aldao Apped D shows he smulao usg AREA of he ERSOF-model ad he esuls ae summazed Tale 3.5 elow: Smulao Sofwae Su-Sysem T L T L Smulao Sofwae W L Tale 3.5: The esuls of he sofwae ad of he smulao The ale compaes he aeage wag me T ad aeage ume of cusomes L of su-sysem ad of he whole sysem. The smulao also aldaes ou assumpo ha he ume of cusomes s dsued omally. Alhough he alues ae o eac we ca say ha he dffeeces come fom he sho-ug smulao. The eo of he smulao was o ge a dea f we ae "o he way" o o ad o hae a ool o ge measues wheee eeded. Fgue 3.7 ad Fgue 3.8 ge he oupu of he smulao ad shows ha he ume of cusomes s dsued omally. Fgue 3.7 wh smalle cell szes depcs he ume of cusome dsuo desy ad s cumulae. Tag he daa fom Fgue 3.7 ad compag o he soluo fom aalyss we ge Fgue

59 Fgue 3.7: The oupu of he smulao fo he ume of cusomes he sysem ume of Cusomes Oupu of he Aalyss Oupu of he Smulao Fgue 3.8: Compaso of he oupus of he smulao ad he aalyss Fgue 3.8 shows ha oupus of he aalyss ad of he smulao ae clealy close ad heefoe he aalyss ca e deemed o pode a good esul. Afe hag compaed he ume of cusomes he sysem we wa o compae he wag me dsuo of he aalyss ad of he smulao. 44

60 Fgue 3.9: The wag me dsuo of he smulao Fgue 3. shows how close he smulao ad he aalyss ae. Thus he model should e cosdeed o e aldaed Oupu of he Aalyss Oupu of he Smulao Fgue 3.: Compaso of he wag me desy fuco of he smulao ad aalyss 45

61 To summaze we ca say ha we ca easly appomae all he sece measues ge a spaes eco. addo we see ha opmzg he fllae does' auomacally ge a opmal soluo fo he aeage wag ad ce esa. Fally he wag me dsuo cao e ow adace; depeds o dffee facos such as he spaes eco he epa me mea ad dsuo ad aal aes of he cusomes o each su-sysem. A hs po of he eseach we leae he ERSOF-model ad aalyze aohe model whch cusomes ca g poducs whch faled moe ha oe class. The a we wll deepe ou udesadg of ERSOF-models y aalyzg a aey of dffee models. Fs we wll sole he fllae y dffee mehods ad he we wll add cosas ojece fucos ad mae he model "moe eesg". 46

62 DESK 4. Modelg a Echageale-em Repa Sysem wh moe ha oe Falue Class ERSMF 4.. oduco Afe hag aalyzed he ERSOF-model we ow moe o o aalyss of a ERSMF-model. Bu wha s a ERSMF-model? As he peous chape we also deal wh a echageale-em epa sysem; u wh seeal em classes. Ths s a sysem o whch cusomes g a poduc cossg of a aey of faled ems fo epa ad ge seceale oes eu fo all faled ems. We suppose hee ae classes of falue Cusome ges ems fom he soc The epaed em o he soc Su-Sysem ss STOCK λ λ Su-Sysem ss Repa Facly STOCK λ Repa Facly The faled em o epa facly The cusome gs faled ems ad leaes wh fed oes. Fgue 4.: The epa facly wh seeal em classes ad seeal falues 47

63 possle ad he oeall aal ae λ s paoed o aes λ λ λ 3 λ coespodg o he ddual compoe falue aes. Thus sce he em aal seam s osso wh ae λ he class- compoe falues fom a osso seam wh ae λ a. These ae se o a epaale ample see epa facly dealg oly wh class- falues whch s soced wh spaes. We call hs susysem o aga ss. A;sp ecause of he ample see assumpo all epas ae caed ou depedely ad we ca ew he oeall facly as paoed o susysems. Ths s depced Fgue 4.. We assume ha he sysem coas oly wo classes of ems. The cusomes whch ae a he des ca e dded o hee caegoes. Oe caegoy o cusome ype gs oly em class hs s aalogous o a poduc whch faled oly em class aohe caegoy cusome ype gs oe em of class ad he hd caegoy o cusome ype 3 gs oe of each Tale 4.. Caegoy em class em class Cusome ype Cusome ype Cusome ype 3 Tale 4.: Eample of he cusome ypes wh wo em classes he sysem We deoe he hee cusome ypes y j j j especely 3 ad le J e he se of all 3 comaos. Le Λ e a eco.. of all dffee cusomes ag o he sysem J j j j 3. Theefoe he poaly of specfc comao js specfc cusome ype o ae o he sysem s: a a j aa a a aa j ad j3. a a a a eealzg he cusome ca g eey aey of ems whch cludes all possle comaos of ems j s J. The poduc whch he cusome gs o he sysem may coa comaos of all dffee ems of he sysem. Thus he ume of comaos J wll e S. Usg hsocal daa we wll calculae he poaly of a cusome ag o he sysem gg a specfc poduc wh specfc 48

64 falues. Le Λ e a eco.. of all dffee cusomes ag o he sysem J j j... j s... j S whee a specfc cusome gs a poduc whch ca e desced y eco js js... js whee he poaly of a faled em o e j s s a a. Thus j s = a j j s s. Theefoe he poaly of specfc comao specfc cusome ype ag o he sysem s: j s j s a j a s a j a j s s a j j s s 4. whee s s he assged cusome ype ume ewee ad. Fo eample f s = 5 he he ay code s. Ths meas ha cusome ype 5 gs a poduc whch faled em ype ad em ype 3. Thus he poaly of a cusome ype 5 o ae a he sysem s a a a a a a... a Sece measues Befoe we coue o uld a opmzao model we fs aalyze how o oa measues fo such a model. We we F F =T oag hesey appopae use of he asc model. Thus he aeage wag me su-sysem s ge y: T F d 4. We assumed ha s clea o he eade wha meas "o wa". Bu he e seco we wll show ha wag me ca hae dffee meags. 49

65 4... The Wag Tme Dsuo 4... The Wag Tme of a specfc cusome ype The wag me of a specfc cusome ca e defed wo dffee ways: The fs sasfaco me W F j s Ths s he me ha a specfc cusome gg js o a sysem wh spaes eco mus wa ul he ges a leas oe hose ems. Ths s also called he fs espose me. How log mus he cusome wa ul he ges somehg fom he epa facly. Ths s doe fo a specfc cusome ype. The las sasfaco me W L j s Ths s he me ha a specfc cusome gg js o a sysem wh spaes eco mus wa ul he ges ALL ems epaed o eplaced. Ths s geeally wha we defe as he wag me. We do o cae f he cusome eceed some of he ems mmedaely u ahe we do cae whe he ca acually leae he sysem The wag me of a adom cusome The wag me of a adom cusome ca e defed wo dffee ways: The fs sasfaco me W F Ths wag me s he me ha a adom cusome ag o a sysem wh spaes eco mus wa ul he ges a leas oe em. Thus S F W j F j s s s W. 4.3 The las sasfaco me W L The wag me s he poaly ha a adom cusome ag o a sysem wh spaes eco mus wa ul he ges all ems ad ca he leae. Thus 5

66 S L W j L j s s s W. 4.4 fac we ae alg heoec ems. We sll do' ow he poales W ad W so ha we ca dee geeal fomulas fo a F j s L j s adom cusome. Ths pa of he puzzle wll e deeloped Chape u a he mome we assume ha hey ae ow deelopg ou sece measues The Aeage Wag Tme 4... The aeage wag me of a specfc cusome The aeage wag me of a specfc cusome ca e defed wo dffee ways: The epeced fs sasfaco me W F j Ths s he epeced me a specfc cusome gg eco mus wa ul he ges he fs em. Thus s js o a sysem wh spaes W W F j s F js d 4.5 The epeced las sasfaco me W L j Ths s he epeced me a specfc cusome gg eco mus wa ul he ges all ems. So s js o a sysem wh spaes W W L j s L js d The aeage wag me of a adom cusome The aeage wag me of a adom cusome ca e defed wo dffee ways: The epeced fs sasfaco me W F 5

67 Ths s he epeced me a adom cusome mus wa ul he ges hs fs em. Thus W W F j j. 4.7 s F s j J s The epeced las sasfaco me W L So Ths s he epeced me a adom cusome mus wa ul he ges he las em. S W L W L js js. 4.8 s The Fllae The Fllae of a specfc cusome The fllae of a specfc cusome ca e defed wo dffee ways: The Fllae ased o me o fs sasfaco FR F j Ths fllae s he poaly ha a specfc cusome gg js s o a sysem wh spaes eco ecees a leas oe em whou ay delay. The Fllae ased o me o ecee all ems FR L j Ths fllae s he poaly ha a specfc cusome gg js s o a sysem wh spaes eco wll o hae wa a all whe he aes a he sysem The Fllae of a adom cusome The Fllae ased o me o fs sasfaco FR F The fllae s he poaly ha a adom cusome who aes a he sysem ges a leas oe em sagh away: 5

68 FR W W j j. 4.9 F F F j s j J s s s The Fllae ased o me o ecee all ems FR L Ths fllae s he poaly ha a adom cusome wll o hae o wa a all whe he aes a he sysem. Thus FR W W j j. 4. L L L j s js J s s The ume of cusomes sysem The ume of cusomes he sysem cao e calculaed decly. Ths wll e doe a whee we aalyze he ERSMF-model geae deal. Thee we wll e ale o show a way o coec he eo of Hausma ad Cheug The aeage ume of cusomes he sysem The aeage ume of cusomes he sysem ca e calculaed wo ways. Fs hough he ume of cusomes dsuo o secod hough Lle's fomula whe he aeage wag me s ow. Ufouaely a hs sage of he eseach ehe ca e doe ad heefoe hs s pospoed ul Chape. hs seco we showed ha a ERSMF-model ca hae may dffee measues fo he sece leel. Ou goal s o defe a opmzao model u hee s o aalyc fomula fo ay of hese sece measues a hs sage. Theefoe we eed o wa fo a of hs eseach o deeme hese. 53

69 a 54

70 5. Opmzg Addoal Sece Cea 5.. oduco Chape 3 we aalyzed a model whee he aeage wag me seed as he ojece fuco ad he cosa was he aalale udge fo spaes. We oaed a appomae soluo ad a way o calculae seeal sece measues. Bu wha happes f he maage does o wa o mmze he aeage wag me? Wha happes f fo eample he was o mamze he fllae? Wha ca he do? s he fac ha he fllae coas a coe pa as well as a cocae pa a osacle fo opmzao? seco 5. we wll use a mehod of solg o lea o-couous ojece fucos - - dyamc pogammg o mamze he fllae. seco 5. we wll pese ways o mpoe he dyamc pogam o ge moe qucly o a appomao of he opmal soluo ad seco 5.3 we wll pese a ee algohm fo he fllae mamzao. The we wll focus he eseach o opmzg wo ohe secemeasues - - he oal ume of cusomes he sysem ad he poaly of wag moe ha me - - o e peseed secos 5.4 ad 5.5 especely. 5.. Solg he fllae polem usg dyamc pogammg seco 3.4 we assumed ha hee s o aalycal soluo o he polem of he fllae. To ecall we wee ge a ERSOF-sysem whee he ojece fuco s o he aeage wag me u he fllae. hs case we waed o mamze he peceage of cusomes who ge sece whou wag. Tha s we waed o sole he followg model: ma FR * FR 5. s..: c M 55

71 Hee we mae a assumpo ha wheee we eed o mae he 's couous o smplfy he peseao he we do so. Ou sadad eample peseed seco 3.8 coas 3 classes ad a udge of M = dollas. Theefoe due o he fac ha he pces of ems of all classes ae us of dollas we deeloped he followg dyamc pogammg model. FRT s he oal fllae fo dollas whe dsued opmally amog a specfc se of classes. Usg he dea of dyamc pogammg we use he ecuse fuco FR j he fllae of su-sysem wh -j dollas allocaed opmally amog s classes. Thus FRT ad FR j FRT ma FRT j fo j= ad = M =.. j We sa wh = ad do ecuso ul we sped M dollas. A each sage we opmze y addg aohe class. Fo eey em we eed M sucos ad heefoe he compley of he algohm s M. Thus he polem FRT M s soled usg a eac pseudo-polyomal algohm. The followg ale shows a eample of he ume of he sofwae o a sadad eum 4 compue. Tale 5. ad Fgue 5. llusae ha he moe classes hee ae he sysem ad he hghe he udge he me o calculae he opmal soluo ehs polyomal gowh as epeced. Dollas 5 5 Classes

72 3 4 8 Tale 5.: Tme secods fo dffee amous of classes ad udge sec $ 5 5 Classes Fgue 5.: aph whch shows he esuls ge Tale 5.. Fgue 5. shows ha as epeced he me gows wh ceasg udge o classes. classes wh dollas s a medum polem whch aes moe ha mues of ug me o ou C. Howee he acual algohm s clealy adequae o sole ey lage polems. 5.. A appomae dyamc pogammg algohm fo lage scale polems The algohm deeloped he peous polem ges a eac soluo o he polem. deed fo lage sze polems appomaos ae good eough as log as he polem s soled a easoale me. Theefoe he followg algohm was deeloped whch s appled o lage sze polems. sead of usg seps of dolla we wll ae seps of s dollas whee s aes fom o o depedg o he sze polem. A complee aalyss of he algohm was o doe u a eample o llusae he effcacy of he algohm s sll useful o show he powe of he ool. As eplaed aoe all he eamples wee ae fom he same sample so we do hae eough daa 57

73 o ge defe esuls; u ay case he followg eample demosaes he qualy of ou eample. The polem cludes dollas ad he seps wee y dolla Fllae Fgue 5.: The Opmal Fllae whe calculaed wh dffee seps Fgue 5. shows ha lage seps do o lead he algohm o mss he opmal soluo. Oly whe seps appoach whch s % of he oal amou s he opmal soluo mssed. ay case due o he fac ha a ee algohm wll e deeloped he couao of he aalyss of usg dyamc pogammg wll e aadoed Deelopg aohe algohm fo he fllae The dea of deelopg aohe algohm was coceed seeal sages. he fs sep we cocluded ha s mpossle ha hee e o algohm whch does o ae o accou he shape of he fuco. Ths pesumpo ough us o cosde a 58

74 ew algohm. As meoed aoe he fllae of he sysem s defed as he weghed sum of he fllae of each su-sysem. Each su-sysem ehaes le Fgue M Fgue 5.3: The Fllae of a su-sysem fac he opmal algohm es o use he seepes slope whee fllae esus dollas s he hghes. a fuco whch s oly coe le he aeage wag me hs ca e doe easly. Theefoe he al soluo sas a he fleco po whee he slope s he geaes ad es o fd ee soluos wo ways. The fs way s o fd a local opmum y echagg ems. Ths meas ha f he algohm sees ha addg a em o class j ad ag off oe em fom class mpoes he fllae we wll eep o echagg ul hee s o fuhe opo fo echagg ems. fac hs s doe usg dollas sead of ems due o he fac ha hee we made he ems couous. a secod sep we y o fd ohe local opma y ag off seeal ems of class j ad eplacg hem y ohe ems. Algohm 3: Ths algohm sas a he fleco po fo each class. The y echagg ems of dffee classes we mpoe he fllae ad ge he local opmum. Ths s a "aea". By fdg all ohe aeas ad fdg he local opma we ca fd he gloal opmum. 59

75 a. 3. ae alzed as meoed Calculae ' a * FR fo =.. c c. Se l m ad he h ma whee l ad h ae he dces of he em classes whch wee chose. d. f l h we hae a local opmum; go o h. e. f o calculae c l l l cl ad c h h h c h f. Choose he smalle alue of c l ad c h : c =mc l c h g. l l c cl ad h h c ch. o o.. h. a * s j ' FR c s j fo s =. ad =... Se l m ad h ma whee l ad h ae he dces of he em classes whch wee chose. j. f l h = j go o.. Compae all local opma o fd he gloal opmum. Fgue 5.4 shows he oupu of he sofwae whch mplemes he algohm. Fgue 5.4: Oupu fo he fllae whe dollas ae aalale 6

76 Fgue 5.4 shows he oupu of he sofwae deeloped fo ha pupose. shows also ha he soluo s sa. eesg o oe ha he esul ca e coss-checed wh he peous soluo usg dyamc pogammg The oal ume of cusomes he sysem Afe hag soled he model wh he fllae as he ojece fuco we ow sole a model wh he oal ume of cusomes he sysem as ojece fuco. Cosde a ERSOF-sysem whch eey cusome gs a poduc whch faled eacly oe compoe. Le... e he spaes eco of he sysem L he epeced ume of cusomes ss hag spaes ad sysem. Thus we hae L he oal ume he m L L 5. s..: c M By Lle s fomula we ow ha L T so ha m L T ad hus he model o opmze s ow m L T 5.3 s..: c M he same model as

77 5.5. The poaly of wag less ha. The sece leels dscussed ul ow dd o focus o he cusome wag me he sysem. hs ERSOF model we wa o mamze he poaly F ha a adom cusome was o loge ha me. Thus we hae ma z F F 5.4 s..: c M Ths polem wll e soled lae seco 8.3 usg ege pogammg. To summaze we ca say ha we deeloped a effce algohm fo he fllae. We showed how o sole he model wh he fllae as he ojece fuco. Aohe sece measue we wa o use ou opmzao model s he poaly of wag less ha u wh sadad aalycal mehods we cao sole. Bu we wll sole hem lae Chape 8 whee we deelop a mehod o sole ay polem usg ege pogammg. Bu fo he mome we leae he ojece fucos ad ela aohe cosa of ou opmzao model whch s he assumpo ha we hae eacly oe cosa. 6

78 6. Dffee ERSOF-Models 6.. oduco Chape 5 we cosdeed a ERSOF-model whch we waed o mamze a sece ceo wh a udge cosa. A ERSOF-model s a model of a sysem o whch cusomes g a poduc coag dffee compoes. Bu he poduc ca fal oly a oe of s compoes. hs chape we wll dscuss models whch we wsh o mmze he udge wh a cosa o he sece ceo such as he aeage wag me o he fllae. 6.. The Aeage Wag Tme We ow cosde a ERSOF-model whch we wa o mmze he udge equed ude he cosa of a desed aeage wag me. We hae heoecally a fe udge a ou dsposal u le ay dusy we do wa o sped moe ha ecessay ad heefoe we wa o es well ad uy ems fom classes whch hae a majo mpac o he aeage wag me. fac hs model s some sese he dual of he peous model dscussed seco 3.4. The model wh udge as cosa loos le: m z T 6. s..: c M. 6. Chape 3 we dscussed he model 6. whee we mmzed he aeage wag me ude he cosa of aalale moey 6.. Thee we desgaed hs as he fs model: s dual ude he aeage wag me cosa as he secod: 63

79 z m c 6.3 s..: T SL whee SL s he mposed lm equed fo he aeage wag me of he sysem. 6.4 The followg heoems wll poe ha he models ae que smla he sese ha he eco s he same whe pug a pa of alues. Fo eample f wh M dollas we ca achee a sece leel of SL he we also hae o pay M f a lm of SL s equed fo he aeage wag me. Ths mgh seem oous u eeds o e poe ha he opmal eco wll e he same oh cases. Tha s why we al aou he dual polem. We loo a he eac same polem fom a dffee pespece. Lae hs wll help us o aldae ou aalyses y solg oh. The dea ehd Theoem s ha a ge pa M SL s echageale wh he pa SL M he sese ha y defg he fs alue as a cosa M fo he fs model SL fo he secod model he secod alue wll e he esul of he ojece fuco SL fo he fs model M fo he secod. Ths dea leads o a gaph whee he -as s he udge ad he y-as he sece leel. see Fgue 6.. Moe ha ha he eco models. * wll e he same oh Theoem : * * The opmal soluo z M wh paamee M fo he fs model s also he * * opmal soluo * z SL wh paamee SL fo he secod model whee z SL * * * * * z M. Tha s z SL M = z M SL. oof: By coadco. * * * Suppose ha eco... s he opmal soluo o he fs model. Ths meas ha he eco sasfes due o he assumpo ha s couous he equao c M ad mmzes z T wh alue SL. Ths same eco sasfes 64

80 T SL u we wll assume ha does mmze c. Because of he z assumpo ha 6.3 s o opmzed we ca do ee he sese ha we ca echage ** a leas oe em of class y aohe em of class j o ge sll sasfyg he cosa. Thus * c * ** ad z z ** c such ha ** * z z whee he cosa T SL s oh cases sasfed. Bu due o he fac ha z M we * ** hae z M < M. Howee he fs model s opmzed so yg o mmze he ojece fuco of he fs model wh oly z dollas we wll ge a sece leel ** * z * whch s smalle ha SL. Bu y defo z SL ; hus we hae a coadco. Hece fo eey specfc udge we wll ge a specfc sece leel ad ce esa. Q.E.D. Le us ow poceed o sole he secod model. Sadadze he polem y calculag he Lagaga as follows: T L c SL whece L c ad hece T ' =. c. 6.5 T ' The wo models ae fac he same u ae ewed hough a dffee pespece. Fgue 6. ad Fgue 6. show hese dffee pespeces. f we calculae he dffeeces slope of each gaph we wll ge 3.7 ad 6.5 whch epeses he dffee gaphs. Tha s why we al aou dffee pespeces of he same model. fac he gaph of he aeage wag me fo each su-sysem loos le Fgue 6. 65

81 wheeas he gaph of he udge allocao loos le Fgue 6.. me Aeage Wag Tme $ Fgue 6. A eample of he aeage wag me esus udge allocao $ Aeage Wag Tme me Fgue 6.: A eample of udge allocao esus he aeage wag me 66

82 Boh gaphs hae he same popees such as coey u wh dffee slopes. Thus we ca use he algohm deeloped seco 3.5 wh a slghly dffee θ o mmze he moey ojece fuco Valdao of he algohm hs seco we pese a eample of he algohm of 6.. Fo ha pupose we deeloped wo pogams. The fs pogam was deeloped o sole he fs model ad he secod fo he secod model. Oously we epec o ge he soluo of he dual model whch was peseed he peous chape. The pu o he fs pogam s he udge allocaed ad he oupu wll e he opmal spaes eco ad he aeage wag me. he secod pogam we pu he aeage wag me equed ad oupu he dollas ha he sysem would cos ad he coespodg opmal spaes eco fo he ems o uy. We pese oh eamples ad assess he esuls..eample: Fgue 6.3: Eample of he eface of he fs pogam wh dollas Fgue 6.4: Eample of he eface of he secod pogam wh he AWT ge he fs pogam he sofwae a wh a pu of dollas whch meas ha we hae a dsposal dollas o sped o spaes. The ume of spaes fo each susysem sag fom lef o gh s fo he fs su-sysem 6 fo he secod ad so o. 67

83 The Aeage Wag Tme of hs sysem coag hese umes of spaes s Fo he secod u wh he pu whch s he mamum allowale aeage wag me pemed we waed o ow wha udge s equed. The esul oaed was dolla as epeced. Alhough hese ae wo dffee pogams he esulg eco s he same as epeced..eample: Fgue 6.5: Eample of he eface of he fs pogam wh 5 dollas Fgue 6.6: Eample of he eface of he secod pogam wh he AWT ge Oeall hese pogams wee u o 5 dffee eamples ad he esuls wee ALL ually decal oh pogams. The eamples aoe show small elea dffeeces ecause of heay mahemacal calculaos wh esula oudg opeaos pefomed y he sofwae. 6.. The fllae We fs aalyzed he aeage wag me mmzed wh he cosa of he udge ad s ese whee we waed o mmze he moey spe wh he cosa of a ge aeage wag me. ow we wa o focus ou eseach o aohe seceleel whch was aalyzed seeal mes dug hs eseach amely he Fllae. Recall 68

84 ha he fllae s he poaly of a adom cusome oag sece mmedaely whou wag. seco 3.4. we dscussed he fs model elow whch mamzes he aeage fllae: Ma z FR 6.6 s..: c M. 6.7 f we loo a he ese of hs model he as he peous case wh he aeage wag me we ge he followg model whee ValueFR s he equed fllae of he sysem: M z c 6.8 s..: FR ValueFR. 6.9 hs model we wa o mmze he epedues o he spaes o achee a ge sysem fllae. z s ems of "poaly" wheeas z s ems of udge. he peous seco we showed ha he wo models fom a pa whee oe alue s epessed he fom of udge allocao ad he secod he fom of sece leel. hs seco we wll show ha he same dea ca also e appled o he aeage fllae of he sysem. he peous seco he gaphs of he aeage wag me esus he udge allocao ad he udge allocao esus he aeage wag me wee oh coe. To llusae he ehao of oh models he fllae of susysem s gaphed Fgue

85 Fllae of susysem $ Fgue 6.7: aph of he fllae esus dollas of ss Fgue 6.8: Moey esus he fllae of ss A smla algohm o he oe deeloped seco 6. ca e deeloped fdg he * opmal eco. Ou sag po s also he fleco po whee we ge he mos fllae fo he moey he sese ha a cemeal mpoeme of he fllae s he 7

86 cheapes. We dd o pusue hs le of eseach u we ae cofde ha he models ae also echageale as wh he aeage wag me he sese ha he soluo of oe model s also adequae as a dual fo he ohe Coecg he cosa ad he ojece fuco Ul ow we aalyzed ERSOF models wh a aey of cosas cludg aalale fuds. Bu oday s usess wold a plae of a epa facly eed o hae a specfc sum of moey a hs dsposal u ahe he ca assg a pealy fo a low leel of sece. hs class of models we mmze he oal cos of he sysem cludg he pce of he ems as well as a mpled Shoage-Cos cued fo low qualy sece The fed me pealy a ERSOF-model A cusome aes o he sysem ees he queue f he does o ge full sasfaco whee full sasfaco meas ha he ges all hs faled compoes whch faled eplaced ad leaes he sysem afe eceg full sasfaco. Bu wha s he moao of he maage of he sysem o uy spaes o mpoe he sysem? The moao s eal o eeal. eal meas ha he maage feels ha he cao le cusomes wa defely so he uys spaes. Eeal meas ha he compay has sged a coac olgag o pay a cea amou of moey fo pologed delay of sasfaco. ou case hs efes o he wag me. f he epa facly s opeaed y he compay he poduc whch faled s o usale ad cus ohe loss of pofs. Thus all hese cases ca e esmaed how much s woh o he sysem o pode quce sece. hs model we assume ha s woh dollas fo eey u of me a cusome was. Tha s why he maage mus decde how may spaes o uy. he followg secos we wll deal wh wo dffee models. Oe whee we assume ha he equed polcy fo uyg spaes s fo a ge peod such as oe yea o yeas o whee he spaes ae eeually scapped ad o woh ayhg a he ed of he peod. The ohe model wll deal wh a fe peod ag o accou he fac ha poduc falues may also occu oe a log me y usg ees ae ad pese alue cosdeaos. 7

87 The fed me pealy a ERSOF-model fo a ge peod Whe a cusome aes a he sysem he ees he queue ad wll susequely leae he sysem afe sasfaco. Hs wag me ss wll cu a pealy of pe u me fo he sysem. Thus fo a peod a aeage of cusomes wll ae a he sysem. Each cusome who aes a su-sysem ss wll hae o wa T o aeage ad cu a aeage cos of T dollas due o hs wag. Thus he oeall polem educes o c T T TC * m whee c s he amou spe fo he spaes. Teag he fo he pese as couous aales * * ' ' ' c TL c T L c T T TC. Fom 3.5 we hae! ' e L so ha T c e! o T c e!. Thus.! T c e

88 73 Bu we ow ha he omal dsuo s a good appomao fo he osso dsuo whe. hs case we ca use sead ~ X. Thus c X The fed me pealy a ERSOF-model fo a fe peod Whe a cusome aes a he sysem he ees he queue ad leaes he sysem afe sasfaco. Hs wag me wll cu a pealy fo he sysem: Fo each ss a cusome who was hee wll cos T dollas o aeage. O aeage he fs cusome aes a me he secod a ad he h -cusome a me. Ule he peous polem he polcy we ae loog fo s fo. f he ees ae s he pese alue of dollas a me s. Thus he oal aeage cos of uyg he spaes ad all peales fo all cusomes who wll ae o he sysem s: c T TC Hece c T TC 6. Fo he deelopme of 6. see Apped Thus mmzg TC ' ' c L c T TC So ha '! e c L. Bu we ow ha he omal dsuo s a good appomao fo he osso dsuo fo. Thus ~ X ad

89 c c X f ad oly f. Cosequely. c f c hs meas ha he pealy s so low ha s o woh uyg ay spaes The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model hs model eey cusome gg a poduc whch faled compoe ad who aes a he sysem ad does o ge mmedae sasfaco wll ge dollas The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model fo a ge peod τ hs model we wa o mmze he oal aeage cos. The fllae podes he poaly ha a cusome o ss s mmedaely sasfed. Thus he aeage ume of o mmedaely sasfed cusomes dug peod s FR. The oal cos fo he whole sysem oe wll e TC Thus FR c. TC FR ' c. Fom 3. FR e '! c. f X s osso wh paamee ad s deal as couous he X c. 6. Fom omal ales we ca fd he alues fo. ן 74

90 Bu whee s he opmal alue of he fllae? Fom 6. we coclude ha he opmal alue fo each ss ca e calculaed depedely ad so we wll aalyze each ss sepaaely. M c* -FR TC Fgue 6.9: Toal cos of ss Fgue 6.9 shows a ypcal eample of he gaph of ss. ou eample we see wo mma oe a ad oe he mddle. The gaph s he sum of a sagh le ad he fllae whch s dded o a coe ad cocae pa. Thus he fuco s also dded o a coe ad a cocae pa. Thus hee ae a mos local mma. Hee TC s he oal cos whe spaes ae ough fo ss. Thus TC. The aswe o ou polem wll e m TC TC wh calculaed y 6.. Hece TC m FR c m FR c The fed pealy fo ay o mmedaely sasfed cusome a ERSOF model fo a uouded eal Whe a cusome aes a he sysem he ees he queue of he class whch faled hs poduc ad leaes he sysem afe sasfaco. f he does o ge mmedae 75

91 sasfaco he sysem cus a pealy. f he ees ae s he he pese alue cos of ay cusome o ss s FR V. The poaly of o geg sa sasfaco s FR. Thus he oal aeage dscoued cos fo he sysem TADC s ge y TAC Hece FR c. TAC FR FR ' c ad c FR FR '. Hece FR Thus c. FR o c c FR whch leads o FR c fo. FR s a moooc ceasg fuco ewee ad. Hece FR s also a moooc ceasg fuco. Thus o sole hs d of equao we ca use ay oe dmesoal seach mehod. 76

92 We se FR ad h c ow we choose a al ad calculae h j j o ge. We coue ul j coeges The fed pealy fo delay loge ha me a ERSOF model As eplaed he oduco o hs chape he maage was o ow how may ems o uy fo he whole sysem. hs model he moao fo hm o uy ems s he fac ha eey cusome wh em who aes a he sysem ad has o wa loge ha me wll cos as compesao fo wag "oo log" The fed pealy fo delay loge ha me a ERSOF model fo a TAC ge fed me peod The oal aeage cos of he sysem fo me peod wll e F c. Hece o fd he opmal spaes eco we dee hs ojece fuco. F c F c TAC fo =.. To fd he opmal we wll chec whee F F c fo he fs me The fed pealy fo delay loge ha me a ERSOF model fo oe a uouded eal As wh he peous models we ow deal wh a fe me model whee he alues of fuue peales ae calculaed ad aggegaed he fom of he pese alue. Theefoe he oal aeage dscoued cos s: TADC fom whch F c 77

93 TC Thus F ad ' F F c ' F c. Hece F c. F c c F F. Sce c F. Theefoe we ca we F as F c We se ad F h c. ad use a umecal algohm o sole. F s a moooc ceasg fuco ewee ad. Thus F s also a moooc ceasg fuco. The o sole h we choose a al ad calculae h j j ul j coeges. 78

94 7. Mulple oals 7.. oduco Afe all hese deelopmes we wll ow poceed o mulple goals. Ul ow we had oe o moe cosas ad oly oe ojece fuco ehe he aeage wag me o he fllae o he poaly of wag a cea me. seco 7. we come wo ojece fucos o e oe ew ojece fuco. seco 7. we see wha happes f a goal s pesced ad he sysem s pealzed fo mssg he goal ad seco 7.3 we poceed o oal ogammg. 7.. Comg wo ojeces Wha happes f he maage of a compay was o mamze he fllae ad mmze he aeage wag me as well? Thus he would ge he followg model: Ma z FR 7. M z s..: c M T 7. fo all =. Oe possly fo dealg wh hs d of polem s o ceae a sgle ojece fuco copoag elae weghs of he ojeces assumg of couse ha hey ae ow o ha hey ca e esmaed. hs case we would hae Ma z a s..: c M FR a T

95 fo all =. whee a ad a a + a = ae he elae weghs of he fllae ad he aeage wag me especely To ge he opmal soluo we fom he Lagaga a L FR T M c 7.4 a ad dee: L M c 7.5 L a FR ' '. T 7.6 a Befoe coug he aalyss we loo a he popees of he wo pas of he fuco. The fs pa he fllae 7. has oe fleco po. The secod pa 7. he aeage wag me has oe. The fs pa wll ehae le Fgue 7. ad he secod pa le Fgue 7.. The fllae has a coe pa ad a cocae pa ad he aeage wag me has oly a cocae pa. Theefoe he sum of oh fucos wll hae a cocae pa ad possly a coe pa. Fllae of susysem $ Fgue 7.: Eample of he ehao of he fllae 8

96 M Fgue 7.: aph of he secod pa of he equao Thus fom 7.6 we hae '' '' T a a FR L whch educes o!! e a e a. Thus!!!! a a e a e a

97 8.!! a a a a a a a a f a = meag ha we ae dealg wh he ogal model whou Aeage Wag Tme we ge he fleco po o e a whch we hae aleady foud seco 3.3. f a s lage he he wegh fo he Aeage Wag Tme s so g ha he fllae s o mpoa a all whch case he fuco s scly cocae. Bu fo whch alues of a wll he fuco e cocae ad hae o fleco po?. a a a a a a f a a he hee wll e o fleco po ad we ca deal wh he fuco as f wee a cocae fuco. seco 3.4. we aleady deal wh hs polem. f a a he hee s eacly OE fleco po ad we ca deal wh he fuco as f wee smla o he fllae. The aeage wag me does o play a majo ole ecep mog he fleco po he deco of he og. The he algohm deeloped seco 5.3 ca e appled.

98 7.. Mssg he goals Aohe way o loo a mulple goals s o wegh he mssg of goals peseed y he decso mae. The decso mae peses a alue z as a goal fo he fllae ad z fo he aeage wag me. fac we wa o mmze he dffeeces of he sece leels fom he ge leels z ad z. a s he pealy fo mssg he desed fllae ad s he amou of dollas pe peceage of fllae we mssed ad a s he pealy of mssg he aeage wag me ad s he amou of dollas pe u me mssed. Thus he ojece fuco wll e epessed ems of dollas whch we wa o mmze. Mz a z s..: c M. FR a T z 7.7 The ojece fuco ca e lghly asfomed whou chagg he soluo o M z a FR a T f we asfom he polem o a mamum we wll ge Ma z a FR a T fac alhough we hough ha we hae hee a ew polem we eded up wh eacly he peous polem. hs sese we soled aohe polem decly oal pogammg Ths seco wll deal wh seeal goals whee he goals ae o weghed u odeed y he poes. We wa o mamze wo hee o fou sece cea. : ma sece ceo : ma sece ceo s..: udge allocao 7.8 Le s aalyze hs polem usg a eample. Le s say we wa fs o mamze he fllae ad he mmze he aeage wag me. Bu fom 3.4. we ow ha 83

99 mamzg he fllae wll ge us OE eco. Usg he echques of goal pogammg we mus opmze dffee sysems sequeally. Bu afe he fs sep we aleady hae a eco whch cao e chaged whou chagg he alues of he sece ceo. Thus due o he fac ha he sece cea ae led we ca oly sole he followg sysems: : sece ceo SL : sece ceo SL 3: sece ceo 3 SL s..: udge allocao whch meas ha he maage was o achee he sece leels of cea alues. Fo eample we asfom he fs equaly o a equaly y addg slac ad suplus aales s s especely. sece ceo s s SL. ou case ecause he lef-sde of he equaly s gge ha he gh sde s s desale. Because we wa he sece-ceo o e as g as possle u a leas SL we asfom he polem o a goal pogammg polem ad ge: m z o s..: s s ma s sece ceo s sece ceo s udge s allocao s s SL SL 7. f s he s possle o ge a model whch sasfes a leas hs ojece fuco u eeually also ohes. f s s mpossle o fd a soluo whch sasfes hs ojece fuco ad heefoe we mus chage he model y chagg he alues SL o y educg ojece fucos. ow we wa o opmze he model y usg sececeo. Thus we sole he followg sysem: m z s s..: udge allocao 84

100 sece ceo s sece ceo s s s s s s s s SL SL 7. Ad so o. f we ca sasfy all he ojeces we foud he soluo o he whole model. f o we eed o eh wha should e he alue fo SL. By choosg SL SL ellgely we ca achee cea goals o dffee sece-cea. 85

101 8. ege ogammg 8.. oduco he peous chape we ofe coeely assumed ha a ege aale ca e ewed as pece-wse lea ad ca heefoe e deal as f wee couous. Bu wha s he pce of hs assumpo? To wha ee ae we mssg he opmal soluo? Ths chape wll deal wh hese quesos. Bu how ca we sole such a ege pogam? 8.. The aeage wag me seco 3.4. we peseed he model whee we waed o mmze he aeage wag me ude a udge cosa. So fs le us defe he ege model: m z T 8. s..: c M. hs case s a ege aale of he model we wa o opmze. Bu how ca we sole? seco 3.5 we made hese aales pece-wse lea o ge a soluo. ow we wa o use aohe echque o ge a lea model. We defe ew aales as follows: We say j = f =j ad j ohewse. Clealy j so ha fo each j oe ad oly oe of he j s. We defe T j AW j whee j s he ume of spaes o e ough. Hece we ca ow we he ojece fuco as m z AW j j ad model 8. ca e ecofgued as follows: j 86

102 m z AW j * 8. s..: j j j fo all j j c j * j M whee j =. j=. Tale 8. shows a eample of he aales j. Eey colum desces he em classes fom 3 ad eey ow he ume of ems ough fo ha class. Fo eample 4 = whch meas ha fo class 4 we wll uy ems. Tale 8.: Eample of he Speadshee We he calculae AW j fo = j=. Tale 8. shows AW j whch wll e used o calculae he ojece fuco. 87

103 Tale 8.: Tale of he aeage wag me fo each class ad fo each alue j. The soluo yelds he followg eco whch s hghlghed. Fgue 8.: Opmal spaes eco fo he ege olem To compae we pese he esul of he lea polem fom seco 3.8. Fgue 8.: Opmal eco fo he Lea olem Fom Fgue 8. we oa he opmal eco fo he lea polem ad he compae wh he esuls Fgue 8.. The speadshee appoach o oly aldaes he lea soluo u ges us also he possly o sole polems whch could o e aalyzed ul oday such as he poaly of wag moe ha a specfed me. 88

104 oh cases he pu was dollas. We ca see ha hee ae o esseal dffeeces ewee he esuls. Due o escos of ou sofwae s mpaccal o sole lage sze polems. Thus wheee smalle polem hae o e soled we ca do so a speadshees u fo lage polems we mus deelop ohe echques. cocluso we see ha we ca choose lea pogammg sead of ege pogammg. Alhough hs s o a full poof cealy seems plausle. 8.. The fllae The same dea of ege pogammg was appled o he fllae. seco 3.4. we calculaed he fllae whch leads o he followg ale Tale 8.3: Values of he fllaes fo each ss used he ege pogammg. ma z FR j * 8.3 s..: j j j fo all j j c j j M whee j =. The opmal spaes eco s foud afe solg he polem y seg j * j j. was eesg o see ha alhough he fllae has oh coe ad cocae pas he ege pogam fds he opmal soluo decly. 89

105 Fgue 8.3: Opmal spaes eco fo he fllae usg ege pogammg Fgue 8.3 depcs a eample of he ge sysem. As meoed alhough hee ae oh coe ad cocae pas whch meas ha hee ae mulple local mama he Speadshee Sole foud he opmal soluo. hs case he lea pogam aleady peseed chape 3 foud he same soluo u wh small dffeeces. Fgue 8.4: Opmal spaes eco fo he fllae usg lea pogammg Aeage Wag Tme M$ Fgue 8.5: aph of he oeall aeage wag me 9

106 Ul ow we had almos o dea how he oeall aeage wag me o he oeall fllae ehaes whe ceasg o deceasg he udge. We decded o aalyze hs fuhe. Thus we soled he aoe models fo seeal udge allocaos yeldg he followg gaphs Fgue 8.5 ad Fgue 8.6. The gaph of Fgue 8.5 s wha we epeced. The aeage wag me of each ss ehaes somewha epoeally. Ths eplas why we heefoe epeced he oeall aeage wag me also o loo hypeepoeal Fgue Fllae M$ Fgue 8.6: aph of he oeall fllae Fgue 8.6 shows he cue of he oeall fllae of he sysem. A he egg we epeced fleco pos u afe aalyzg he ehao of he sysem we came o he cocluso ha he aey of dffee ss affecs he oeall fllae so ha does o coa ay fleco po. Fom hs ew loos a smooh cue u whe choosg smalle eals see Fgue 8.7 he fac ha he aales ae ege causes he gaph o o e smooh. Also he fac ha hee ae coe ad cocae pas also affecs he gaph hs way. 9

107 .5 Fllae M$ Fgue 8.7: aph of he oeall fllae wh smalle eals The poaly of wag me Chape 4 we dscussed he model whee he sece-leel ojece fuco s o mmze he poaly of wag moe ha. Hee m z W * 8.4 s..: j j j j fo all j j c j * j M whee j =. Ul ow we had o echque o sole hs sysem. Bu usg we ca sole he model. all he models we used Vsual Basc o calculae he fomulas fo hs model usg he speadshee sole. Fs we calculaed. Of couse hs s he fllae ad oously we go he same esul as show Fgue 8.8. The Speadshee W 9

108 peseed ca e chaged fo ay o fd he opmal eco of wag. We chose = o aldae he soluo. Whe = fac we deal wh he fllae whch we soled seco 8.. Fgue 8.8: Opmal spaes eco fo he poaly of o wag he e seco we use he Speadshee o u dffee eamples o ge some dea of he shapes of he fucos. The shapes ae ae fom a adom eample Sesy aalyss hs seco we wa o aalyze he sesy of he model: Fo hs pupose we chose seeal udge amous ad a he model wh dffee. We pese he oupus he followg fgues. Fgue 8.9 show he fluece of udge allocao o he wag me dsuo. O he oe had whe o udge s aalale he wag me dsuo depeds sogly o he epa me dsuo. O he ohe had he gge he udge he hghe he poaly of o wag. 93

109 .9.8 W< Fgue 8.9: The poaly of wag less ha wh dffee udge allocao..9 W< =. =.5 =.. $ Fgue 8.: Wag Tme Dsuo ased o dffee udge allocao. 94

110 As a secod sep we waed o eploe he fluece of udge allocao o he wag me dsuo. Oously he moe moey we es he less we wll wa. The gge he legh of me we agee o wa.8 he hghe s he poaly o e sasfed wh ha eal The poaly of wag moe ha fo wag cusomes Oe of he mos mpoa sece cea s he poaly of wag moe ha ude he codo ha he cusome was. a sece facly ca happe ha oly % mus wa a all u hey ealy hae o wa ey log. Ths s o desale fo a maage who sees o opmze hs sysem pefomace. He was he cusomes who mus wa o wa as sho as possle. Thus m z W W * 8.5 s..: j j j fo all j j c j * j M whee j = ad whee F W W. F Ths s o a lea fuco a all ad cealy a mos complcaed fuco o opmze. Bu wh ou mehod of ege pogammg ecomes ey smple. The followg gaphs show a eample wh sesy aalyss. 95

111 Fgue 8.: Oupu of he poaly of wag moe ha fo wag cusomes. Fgue 8. shows he oupu of he poaly ha he cusome was moe ha.5 ude he codo ha he mus wa whe dollas ae aalale fo spaes. Fgue 8. depcs he wag me dsuo fo wag cusome wheeas Fgue 8.3 he poaly of wag moe ha.3 fo dffee udge amous W> W> Fgue 8.: The codoed poaly of wag moe ha. 96

112 Fgue 8.3: Sesy aalyss of dffee udges fo a ge. W>.3 W> ay case he oduco of ege pogammg opeed up a ew mehod of solg polems. We eploed a ew sece leel he poaly of wag moe ha a cea me. fac all pulshed acles aou spaes deal eclusely wh he aeage wag me o wh he fllae u oe wh he wag me dsuo. Ths pa epeses a esseal couo o he eseach hs feld. A addoal model o e soled y ege pogammg cludes dscoug. Whe uyg mulple spaes hee ae ofe dscous whch fluece he decsos of a maage. may models hs assumpo cao e oduced u he appoach of ege pogammg eales us o ee sole models cludg dscoug. $ 97

113 9. Bul sysems 9.. oduco he pecedg chapes we deal wh sysems whee a cusome gs a poduc whch faled a sgle class. Due o he fac ha eey em class coas eacly oe em we could easly deelop fomulas ad mehods o sole he model fo hs sysem. Bu wha happes f he poduc coas say wo of he same class ad oh fal. How does ha fluece modelg of he sysem? Hausma ad Cheug [6] peseed a cusome-ased appoach u faled o deelop coec fomulas fo he ume sysem. hs chape we wll deelop a em-ased appoach ad fd coec soluos fo he wag me dsuo ad fo he ume of cusomes sysem. Beg ad ose [5] efly deal wh he dea of ul aal sysems whee a cusome may g moe ha oe em. They poded a soluo fo a ase model whee he ume of ems ough s also a adom aale. seco 9. we fs dscuss a ERSOF-model whee eey cusome gs a poduc whch faled eacly wo ems of he same class. Recall ha a ERSOF-model s a model olg oly class falue a he same me. 9. we geealze he model o whee eey cusome gs a poduc whch faled eacly ems of he same class whee s a cosa. Fally 9.3 we geealze he model o deal moe fully wh adom ul aals. 9.. A ERSOF - sysem whee eey cusome gs eacly faled ems of he same class A cusome aes a me wh ems of class equg epa ad was ul he ges full sasfaco ad leaes he sysem. W s he wag me of he cusome who aes a me. W s he ee ha a agged cusome who aes a me was less ha. Ths cusome who aes a me wll wa less ha f y me + all he cusomes who had ough ems of class efoe hm hae eceed sasfaco ad a leas ems ae lef. Thus 98

114 y me + ou agged cusome eaches he fo of he queue ad W hee ae a leas ems o he shelf fom class. To see all cusomes fo of ou agged cusome as well as he agged cusome hmself we eed he oal ume of ems ag a he shelf fom spaes o fom epa dug + o eceed he ume of ems equed o sasfy all he cusomes who aed me cludg ou agged cusome. Thus o. ems aed o shelf + + o. ems fom cusomes who W aed. We wll defe he followg adom aales: Z s he ume of ems epaed y me + fom ou agged cusome who aed a. Clealy Z Z Z s p Z.. Thus The ume of ems aed o he shelf s equal o he ume of spaes plus he ume of ems epaed y me +. Le S = he ume of ems epaed + fom all cusomes apa fom ou agged cusome. The ume ems ag o he shelf + + ume ems ough W y ohe cusomes <=> +S++Z + 9. ode o deeme he dsuo of S+ we fs defe he followg adom aales: s he ume of ems whch aed efoe me ad complee epa afe me +. s he ume of ems whch aed afe me ad complee epa efoe me +. S : The ume of ems epaed me + s he sum of he ume of ems epaed fom cusomes who came efoe me plus hose fom he cusomes who aed afe me. Theefoe 99

115 S. 9. Le s loo a. du + Fgue 9.:Cusome ag ad leag afe + The poaly of a em ag du o e epaed y me + s +-u. Thus he poaly of a em ag du compleg epa afe + s u. The poaly of aal du s du. Thus he poaly of oe em of a adom cusome ag efoe ad hag hs em epaed afe + s u du whch ca e we as p d 9.3 wh q p. X s he ume of ems of cusome uodeed who aed whch ae o epaed y me +. X { }. Thus he ume of ems epaed afe + whch aed efoe s he sum of decal adom aales X whee eey X ~omalp ad ~ ossoλ. Thus =X +..+X = X. We wll use geeag fucos o deeme he dsuo of. The geeag fuco of a omal p aale s z e. Thus he geeag fuco of s z p q ad of he osso s

116 E... X X X z EE z E Ez Bu wha happes whe goes o fy? We hae p d. Clealy lm p. Bu p lm p lm d d. 9.4 Thus [ ] [[ zq ] ] lm e p [ ] lm d z p z p z e e e dz. 9.5 Bu hs s he geeag fuco of a osso aale wh paamee p. Thus whe appoaches s osso wh paamee p. Tha s p p e =.! Thus E Ez X E [ p z q ] [[ pz q ] ] e 9.6

117 ow cosde : du + Fgue 9.: Cusome ag + ad hag hs em epaed efoe +. LeY e he ume of ems of cusome uodeed ha we ough + ad whch ae he epaed y me +. Y { }. The poaly of a em ha aed du s epaed y me + s +-u. Thus he poaly ha ay oe em fom adom cusome ag afe ad he hag hs em epaed + s u du whch ca e we as p p d 9.7 wh q p. Thus s he sum of adom aale Y whee eey Y p ad ~ osso so ha Y Y... Y Y ~ Due o he fac ha he dsuo of he ume of ag cusomes + s he same as fo a homogeous osso ocess we ca edefe as equale o ad

118 3 e q p! =.. ow eug o ou ogal polem we hae he equalece Z Z S W Fo smplcy we oduce ew adom aales D D oposo: a echageale-em FFO sysem he osaoay delay dsuo of a cusome ha aes a me wh eacly wo ems of class s ge y D D D W F 9.8 Thus he delay dsuo of a cusome ha aes a me wh eacly wo ems s ge y D D D W W F 9.9!!!! e q p e e q p e q p

119 The saoay delay dsuo of a cusome s F W D D D 9. whee D. Fo he poof see Apped E. Oously we ae o eesed such a esced ul sysem whch cusomes gs pecsely wo ems of he same class u we wa a moe geeal model whee a cusome gs ems of he same class. seco 9. we wll aalyze a sysem whee cusome gs eacly ems ad seco 9.3 we geealze hs o a sysem whee he cusome gs a adom ume of ems. 9.. A ERSOF-sysem whee each cusome gs eacly faled ems of he same class whch faled Afe hag aalyzed he case we ow geealze o a ul sze of. A cusome aes a me ad was ul he ges sasfaco ad leaes he sysem wh all faled ems eplaced. W s he wag me of he cusome who aes a me. W s he ee ha a agged cusome who aes a me was less ha. We wll assume ha he cusome ough ems of class so ha we ca deal wh hs polem sepaaely. A he ed we wll geealze he model so ha he cusome ca g of ay class. A cusome who aes a me wll wa less ha f y me + all cusomes who aed efoe hm hae goe sasfaco ad a leas ems ae lef. y me + ou agged cusome eaches he fo of he queue ad W hee a leas ems o he shelf of class. To see all he cusomes fo of ou agged cusome as well ou agged cusome we eed he oal ume of ems ag a he shelf fom spaes o fom epa dug 4

120 + o eceed he ume of ems equed o sasfy all he cusomes who aed me as well as ou agged cusome. Thus ume ems aed o shelf + + ume ems fo W cusomes who aed. We wll defe he followg adom aales: Z s he ume of ems epaed y me + fom ou agged cusome who aed a. Thus Z =.. The ume of ems ha aed o he shelf s equal o he ume of spaes plus he ume of ems epaed y me +. S = he ume of epaed ems y me + fom all ohe cusomes. ems aed o shelf + + ems fo cusome me = +S++Z +. Hece W S Z. ode o deeme S+ we fs defe he followg adom aales: s he ume of ems whch aed efoe me ad complee epa afe me + ad s he ume of ems whch aed afe me ad ae complee epa efoe me +. The ume of ems epaed y me + s he sum of he ume of ems epaed fom cusomes who came efoe me plus hose fom cusomes who aed afe me : S Le s fs loo a. Fom he peous seco we aleady ow ha =X +..+X ad hus X ~ omalp ~ ossoλ 5

121 6 p was aleady calculaed 9.4. We wll use geeag fuco o calculae. The geeag fuco of a omalp aale s q z p ad of he osso z e. Thus he geeag fuco of s... ] [ X X X q z p E z E E z E E z E q z p e Bu wha happes whe goes o fy? Fom he peous seco we ow ha lm p ad d p lm. ow lm q z p. lm lm lm lm z p z p z p z p Bu lm d p sce lm p. Hece we ge lm z d z p z E Thus s dsued osso wh paamee d so ha

122 7 p e p! =. sce p d ow le us loo a. s he ume of ems ag afe ad epaed efoe +. V e he ume of ems epaed efoe + fom some cusome uodeed. he eal + hee ae cusomes ag. Thus =V +.V We wll use geeag fucos o deeme he dsuo of.... ] [ V V V q z p E z E E z E E z E q z p e Bu wha happes whe goes o fe? We ow ha lm p ad d p lm. ow lm q z p z p z p z p z p lm lm lm lm Bu lm d p sce lm p. Hece we ge

123 8 lm z d z p z E Thus s dsued osso wh paamee d so ha p e p! =.. ow le us deelop he osaoay ad he he saoay dsuos. Z Z Z S W F W Z Ths s he wag me dsuo fo ay sysem.

124 9 oposo: a echageale-em FFO sysem wh classes he osaoay delay dsuo of a cusome ha aes a me wh ems of he same class s ge y D W F 9. whee D The saoay wag me dsuo of a adom cusome s D W F 9. hs seco we aalyzed a sysem whee eey cusome gs eacly ems. he e seco we wll geealze hs model so ha a cusome ca g a adom ume of ems fom he same class A ERSOF-sysem whee each cusome gs a adom ume B of faled ems of he same class. We ow geealze o a ul sze of B whee B s a adom aale. Cusome uodeed aes a me wh B ems equg epa ad was ul he ges sasfaco ad leaes he sysem. W s he wag me of he cusome who aes a me. W s he ee ha hs agged cusome who aes a me was less ha. We wll assume ha cusome ough B ems of class so ha we ca deal wh hs polem sepaaely. A he ed we wll geealze o cusomes gg B of ay class. Ths cusome who aes a me wll wa less ha f y me + all he cusomes fo of hm hae goe sasfaco ad a leas B ems ae lef o sece hs demad fo B ems. The poduc coas Q ems of he class we ae dealg wh ad heefoe B...Q.

125 W y me + ou agged cusome eaches he fo of he queue ad hee a leas B ems o shelf of class. To see all he cusomes fo of ou agged cusome as well ou cusome we eed he oal ume of ems ag a he shelf fom spaes o fom epa dug + o eceed he ume of ems equed o sasfy all he cusomes who aed me ad as well as ou agged cusome. Due o he fac ha ou agged cusome s he + h cusome ems aed o shelf + B + +ume of ems fo cusomes W who aed me. We wll defe he followg adom aales: Z s he ume of ems epaed y me + fom ou agged cusome. Clealy he dsuo of Z B ~ = p=. Z B =.. The ume of ems aed o he shelf + s equal o he ume of spaes plus he ume of ems epaed y me +. Le S = he ume of ems epaed +. ume of ems aed o shelf + B + ume of ems ough y cusomes = +S++Z B + +B + +B. Thus we ge W S Z B ode o deeme S+ we fs defe he followg adom aales: s he ume of ems whch aed efoe me ad complee epa afe me +. s he ume of ems whch aed afe me ad complee epa efoe me +.

126 The ume of ems epaed y me + s he sum of he ume of ems epaed fom cusomes who came efoe me plus hose fom cusomes who aed afe me ecludg ou agged cusome whch we wll deal wh sepaaely. B S Le s loo a. Y s he ume of ems of ay cusome who aed whch ae o epaed y me +. B Y.... Thus he ume of ems epaed afe + whch aed efoe s he sum of adom aale Y whee eey Y ~ omalb p ad ~ossoλ. We ca he we =Y +..+Y. We wll use geeag fucos o deeme he dsuo of. ep ep ep ] [ ep ] [... z p B z p B q z p B q z p E e z E E z E E z E B B z E Y Y Y Y whee B z s he geeag fuco of he adom aale B. Bu wha happes whe goes o fy? Fom he peous seco we ow ha lm p u d p lm. Thus

127 lm lm lm B z p B z p B z p Bu lm d p Hece we ge lm B z p B z p Thus s dsued osso wh paamee d B d B V. Thus V e V!. ow le us cosde a. Fom he peous seco we ow ha s a compoud osso ocess wh cusomes whee each cusome gs B ems. Thus e q p B B!. ow le us defe he osaoay ad he saoay dsuos B B Z B Z B B B Z B B B W F

128 3 To smplfy we defe a ew adom aale: D oposo: a echageale-em FFO sysem he osaoay delay dsuo of a cusome ha aes a me wh B ems s ge y B D W F 9.3 The saoay wag me dsuo of a adom cusome s B D W F 9.4 Fally we oa o he wag me dsuo fo each su-sysem ou ERSOFmodel whe ul aals ae allowed. These calculaos ca e cluded o he eseach aoe o mae he model moe ealsc ad moe flele. Chape 9 we aalyzed ul models whch meas ha eey cusome ca g a ume of ems of he same class. Bu wha happes f he cusome gs seeal ems fom dffee classes? Ths wll e doe Chape.

129 . Deelopg ERSMF-models.. oduco he peous chape we aalyzed a ERSOF-model wh ul aals. Ths meas ha eey cusome gs seeal ems fom he same class whch ae he epaed ad eplaced. hs chape we wa o aalyze models whee cusomes ca g ems of dffee classes o e epaed ad eplaced. Hee we wa o deelop he laguage ad asc fomulas fo a ERSMF-model. chape 4 we defed he asc laguage of he model. Thus he aalyss of hese opcs s pogessely deeloped seps oe seeal secos. hs chape we wll oly deal wh a model wh em classes. Clealy hese deas ca e geealzed ad hs wll e pusued fuhe eseach. The cusomes whch ae o he sysem ca e dded o hee caegoes. Oe caegoy o cusome ype gs oly em class hs s aalogous o a poduc whch faled oly em class aohe caegoy cusome ype gs oe em of class ad he hd caegoy o cusome ype 3CT3 gs oe of each Tale.. em class em class Aal ae Cusome ype CT λ Cusome ype CT λ Cusome ype 3CT3 λ Tale.: Eample of he cusome ypes wh wo em classes he sysem seco. we wll loo a a sysem whch coas oly cusome ype 3 ad seco. we cosde a geeal sysem wh cusome ype ad 3... A ERSMF-model whee all he cusomes ae of ype 3 A ERSMF-model s a model whee he cusome ag o he epa facly gs a poduc whch faled moe ha oe em class. Fo smplcy we wll assume ha eey class coas eacly oe em. Fuhe o ease he aalyss of he model we wll assume ha he poduc coas oly wo classes. A adom cusome aes a 4

130 me o hs sysem wh a poduc whch faled hough class ad hough class. He was ul he ges full sasfaco whch meas ha oh ems ae eplaced ad leaes he sysem. W s he wag me of he cusome who aes a me. fac he ees a ual queue fo each em. Wheee he ecomes compleely sasfed y oh susysems he leaes he sysem. y me + ou agged cusome eaches he fo of oh queues ad W hee s a leas oe em o each shelf. To see all he cusomes fo of ou agged cusome as well as ou agged cusome we eed he oal ume of ems ag a each shelf fom spaes o fom epa dug + o eceed he ume of ems equed o sasfy all he cusomes who aed me as well as ou agged cusome. Thus ume of ems aed o shelf + +ume of ems fom W cusomes who aed ad ume of ems aed o shelf + +ume of ems fom cusomes who aed. We wll defe he followg adom aales: Z s a eco of ems epaed y me + fom ou agged cusome who aed a. Thus w. p * w. p * Z. w. p w. p * whee s he cumulae epa me dsuo of ss ad s he cumulae epa me dsuo of ss. The ume of ems aed o he shelf s equal o he ume of spaes plus he ume of ems epaed y me +. Le S = he ume of epaed ems of ss y me +. S = he ume of epaed ems of ss y me +. Fom. we hae +S ++Z + +S ++Z +. 5

131 We defe he followg adom aales: s he ume of ems whch aed ha complee epa oh ss ad ss y me +. s he ume of ems whch aed wh epaed ss ad he ohe o ss y me +. s he ume of ems whch aed wh epaed ss ad he ohe o compleed epa ss y me +. s he ume of ems whch aed ad oe of he ems s epaed y me +. Thus =. s he ume of ems whch aed + whee he fs s epaed ss ad he secod ss y me +. s he ume of ems whch aed + whee he fs s epaed ss ad he secod o ss y me +. s he ume of ems whch aed + whee he fs s o epaed ss ad he secod epaed ss y me +. s he ume of ems whch aed + whee ehe he fs o he secod epaed ss y me +. Thus =. The ume of ems epaed y me + s equal o he ume of ems whch aed efoe ad epaed efoe + plus he ems whch aed afe ad ae epaed efoe +. We wll oa he fomulas fo each class. Fo class : S Fo class : S 6

132 7 Thus Z Z Z Z Z S Z S W Z Z Z Z We cosde each adom aale u. : Fgue.: aph of CT3 ag du efoe ad oh ems epaed afe + A cusome who aed o du+ has a poaly of u u of o hag each of he ems epaed ul me +. Theefoe he poaly of a adom cusome ag ad o hag oh hs ems epaed y me + s du u u o d du +

133 Cusomes ae as a osso pocess wh ae λ. Thus s dsued osso d wh paamee. Whe we ge paamee d. : du + Fgue.: CT3 ag du ad hag he em of class epaed efoe + ad he em of class epaed afe + A cusome who aed o du+ has a poaly of of o hag u hs em of class epaed ad of hag epaed he em ul me +. u Theefoe he poaly of a adom cusome ag of hag hs em of class epaed u o he em of class y me + s du u u ha s d. Sce cusomes ae as a osso ocess wh ae λ heefoe s dsued osso wh paamee d. Whe we ge paamee d. 8

134 : Fgue.3: CT3 ag ad hag hs em of class epaed efoe + ad hs em of class epaed afe + A cusome who aed o du+ has a poaly of of hag hs u em of class epaed ad of o hag hs em of class epaed ul u me +. Theefoe he poaly of a adom cusome ag hag hs em of class epaed ad o hag hs em of class epaed y me + s du + u u du o d. Sce cusomes ae as a osso ocess wh ae λ. Thus s dsued osso wh paamee d. Whe we ge paamee d. 9

135 : du + Fgue.4: CT3 ag ad hag hs em of class epaed efoe + ad hs em of class epaed efoe + A cusome who aed o du+ has a poaly of of hag u hs em of class epaed ad of hag hs em of class epaed efoe u me +. Theefoe he poaly of a adom cusome ag ad hag hs em of class epaed ad hs em of class epaed y me + s du u u o d. Cusomes ae as a osso ocess wh λ. Thus d s dsued osso wh paamee. Whe we ge d. ow le us aalyze he secod goup of adom aales.

136 : du + Fgue.5: CT3 ag du afe ad hag hs em of class epaed efoe + ad hs em of class epaed efoe + A cusome who aed o du+ has a poaly of of hag u hs em of class epaed ad of hag hs em of class epaed y u me +. Theefoe he poaly of a adom cusome ag ad hag oh hs ems of class epaed ad hs em of class epaed y me + s u u du o d. The cusomes ae as a osso pocess wh ae λ. Thus s dsued osso wh paamee d. : du + Fgue.6: CT3 ag du afe ad hs em of class epaed ad hs em of class o epaed efoe +.

137 A cusome who aed o du+ has a poaly of of hag u hs em of class epaed ad of o hag hs em of class epaed y u me +. Theefoe he poaly of a adom cusome ag ad hag hs em of class epaed ad o hag hs em of class epaed y me + s du u u o d. Cusomes ae as a osso ocess wh ae λ. Thus s dsued osso wh paamee d. : du + Fgue.7: aph of CT3 ag duafe ad em of class epaed ad em of class o epaed efoe +. A cusome who aed du+ has a poaly of of o u hag hs em of class epaed ad of hag hs em of class epaed u y me +. Theefoe he poaly of a adom cusome ag ad o hag oh hs em of class ad hs em of class y me + s du u u o d. Cusomes ae as a osso pocess wh paamee λ. Thus s dsued osso wh paamee d. :

138 du + Fgue.8: CT3 ag ad hag hs em of class ad hs em of class epaed afe + A cusome who aed du+ has a poaly of of hag u hs em of class epaed ad of hag hs em of class epaed y u me +. Theefoe he poaly of a adom cusome ag ad o hag oh hs ems epaed y me + s u u du o d. The cusome ae as a osso ocess. Thus s dsued osso wh paamee d. 3

139 4 Summay Tale: aamee -> d d d d d d d d d d d d d d d d

140 5 Fom. we ge fo W.3 whee j j Fo he poof see Apped E. To smplfy he aoe we defe he followg adom aales: 4 3 Sce all j ae osso heefoe j j = 4 ae all osso. The paamees ae ge elow he summay ale. s he sum of wo depede osso adom aales. Thus s also dsued osso wh paamee d + d = d

141 s he sum of wo depede osso adom aales. Thus s also dsued osso wh paamee d + d = d s he sum of wo osso adom aales. Thus s also dsued osso 3 wh paamee d + d = d s he sum of wo osso adom aales. Thus s also dsued osso 4 wh paamee d + d = d. Summay Tale: 3 4 aamee d d d d 6

142 7 Thus fom.3 we ge fo W.4 We defe he followg adom aales: D D.6 Thus fom.4 we ge fo D D D D D D D W.7 D has he mea d - d. d d d d f >> we ca use he omal appomao.

143 D has mea d - d. d d d d d f >> we ca use he omal appomao... A ERSMF-model whee he sysems coas CT CT ad CT3... Whe he CT3 s ag Whe we ae dealg wh a sysem whee he cusome ca g up o wo ems hee ae 3 dffee cusome ypes. The fs gs oly oe of class he secod gs oly oe of class ad he hd gs fom each class. Hausma ad Cheug [6] ed o sole hs model u wee o ale o do successfully. Apped C we eplaed he fudameal eos he deelopme. hs seco we deelop he wag me fo CT3 he e seco he fomula fo CT. To eep all he adom aales ode we wll use supescps. They wll defe o whch goup of adom aale hey elog. As a eample s he aal ae of cusome ype 3 he aal ae fo CT ad he aal ae of CT. Thus y me + ou agged eaches he fo of oh ual queues ad hee W s a leas oe em o each shelf. W ume of ems aed oo shelf + + ems fo cusome me ad ems aed oo shelf + + ems fo cusome me 8

144 We defe he followg adom aales: Z Z s a eco of ems epaed y me + fom ou agged cusome. w. p w. p w. p w. p * * * whee s he cumulae epa me dsuo of ss ad s he cumulae epa me dsuo of ss. The ume of ems aed o he shelf + s equal o he ume of spaes plus he ume of ems epaed y me +. S = he ume of ems epaed y me + ss. S = he ume of ems epaed y me + ss. s he ume of cusome ype who aed o he sysem y me +. s he ume of cusome ype who aed o he sysem y me +. s he ume of cusome ype 3 who aed o he sysem y me +. s he ume of ems whch aed fom CT3 ad ae epaed ss ad ss y me +. s he ume of ems whch aed fom CT3 wh epaed ss ad he ohe o ss y me +. s he ume of ems whch aed fom CT3 wh epaed ss ad he ohe o ss y me +. s he ume of ems whch aed fom CT3 wh ehe epaed y me +. Thus = s he ume of ems whch aed fom CT3 + ad ae epaed ehe ss ad ss y me +. 9

145 s he ume of ems whch aed fom CT3 + ad ae epaed ss u o ss y me +. s he ume of ems whch aed fom CT3 + ad ae epaed ss u o ss y me +. s he ume of ems whch aed fom CT3 + u ae epaed ehe ss o ss y me +. Thus = s he ume of ems whch aed fom CT ad ae o epaed me +. s he ume of ems whch aed fom CT ad ae o epaed y me +. s he ume of ems whch aed + fom CT ad ae epaed y me +. s he ume of ems whch aed + fom CT ad ae epaed y me +. W ume of ems aed oo shelf + + ems eeded fo cusomes me ad ume of ems aed o shelf + + ems eeded fo cusome me = +S ++Z + + +S ++Z + +. The ume of epaed ems y me + s he oal ume of ems whch aed efoe ad ae epaed efoe + plus he ume of ems whch aed afe ad ae epaed y me +. Thus Fo class : 3

146 3 S Fo class : S Thus fo... W.8 Fo he oof see Apped H. Bu sll he fomulas ae awwad ad heefoe o smplfy we oduce he followg adom aales:. 4 3

147 3 Each s he sum of a ume of osso adom aales ad hus s self osso. The meas ae summazed elow: s osso wh paamee d + d + d = d. s osso wh paamee d + d + d = d 3 s osso wh paamee d + d + d + d = d + d + d 4 s osso wh paamee d + d + d + d = d + d + d. Thus W.9

148 33 Bu sll he fomula ca e made loo smple y defg he followg adom aales: D } { D. 4 3 D } { D. Thus D D D D D D D W. D s he dffeece of wo depede osso adom aales each of whch s he sum of depede osso adom aales. Thus has he mea d - d = d + d = D s he dffeece of wo depede osso adom aales each of whch s he sum of depede osso adom aales. Thus s also dsued wh mea d + d + d - d - d - d = d - d + d + d - d

149 = d - + = - +. Apped we demosae a chec mechasm fo hs fomula.... Whe he CT s ag he peous seco we deal wh a sysem whch coas wo class of ems ad whee he cusome s of CT3. Bu he same sysem hee may e cusomes who g oly oe em of class o oe of class. We wll aalyze a cusome who gs oly oe of class. As he peous seco we wll sa wh he ee of a adom cusome who aed a me ad wll wa less ha. y me + ou agged cusome eaches he fo of he queue ad hee W s a leas oe em o he shelf. W ume of ems aed o shelf + +ume of ems fo all cusome me We wll defe he followg adom aales: Z s he ume of ems epaed y me + fom ou agged cusome. Z The ume of ems aed oo he shelf s equal o he ume of spaes plus he ume of ems epaed y me +. S = he ume of epaed ems y me +. W ume of ems aed oo shelf + + ems fo cusome me = +S ++Z

150 35 Fo class : S Z Z Z S W Thus W We wll defe he followg adom aales: M M Thus whe we ge M M M M W.3 Oously hs s he same fomula as Beg ad ose [5] whee he adom osso aales ae adjused o he model. M s he sum of hee osso adom aales. Thus s also dsued osso wh paamee d + d + d = d. Smlay M s he sum of hee osso adom aales. ad s also dsued

151 36 osso wh paamee d + d + d = d. Due o he fac ha M ae sums of osso adom aales we ca deal wh hem as f hey wee dsued omal as follows: d M M ~..3. Whe CT s ag. Aalogously he wag me of a cusome gg oly he secod em wll e M M M M W.4 whee 3 M 4 M ae geec osso aales wh paamees d ad d especely...4. The wag me dsuo of a adom cusome hs seco we aalyze he wag me dsuo of a adom cusome. Fo hs pupose we defe he followg adom aales: M M D M M D D D

152 D has mea d - d = d d d d. D has mea d + d + d - d + d + d = d d d d d. d. D 3 has mea D 4 has mea d - d =. d - d = d d d d. Thus he wag me dsuo of a adom cusome wll e ge y: 37

153 D D D D D D D D D D D W Sece Measues As all he models we wa o calculae he dffee sece measues fo he ge sysem. The Wag Tme Dsuo he peous seco we deeloped he fomula fo he Wag Tme Dsuo. Thus f we wa o ow wha s he poaly ha a cusome was moe ha me F W W. The Fllae As he ERSOF-models oe of he mos useful sece-measues s he fllae. hs case s he poaly of las sasfaco whch meas ha s he poaly ha a cusome ges mmedae sasfaco upo aal;.e. fo = so ha.theefoe 3 D D D W 4 D.6 We wll ow aalyze wha happes o he adom aales D.

154 D whch ae dsued ~ osso ~ osso Thus D ~ osso. D Thus fo whch ~ osso ~ osso d osso d d D has mea d d D M 3 M d d. M fo whch M ~ osso ~ osso Thus D ~ osso. D M 4 M fo whch M ~ osso M 3 4 ~ osso Thus D ~ osso. 4 d d. 39

155 Thus fom.6 we ge D4. D D D D 3 The fllae ca ow e educed o W 3 D D D D The Aeage Wag Tme Aohe mpoa measue s he aeage wag me whch ca e calculaed wo ways. The fs way s y Lle s fomula whee W L whch L s he Aeage ume he sysem. Bu ou case hs s o ow ad so we wll calculae he aeage wag me decly y W also hae he aeage ume of cusomes he sysem. F d. Oously fom hs we wll he The Dsuo of he ume of cusomes he sysem hs seco we wll oule a mehod o deeme he dsuo of he ume of cusomes he sysem. Hee we wll show he famewo of how o poceed owad he coec fomulas u he wo wll e compleed a fuhe eseach. As eplaed seeal mes dug hs eseach hs was oe of he moaos of hs eseach ecause Hausma ad Cheug [5] faled o oa coec fomulas. Le us defe some adom aales whch we wll use he aalyss. C C The oal ume of cusomes wag he sysem. The oal ume of cusomes of ype wag he sysem. 4

156 C C j The oal ume of CT wag he sysem. The oal ume of CT3 wag he sysem. The ume of ems epa facly ss fom CT. The ume of ems epa facly ss fom CT. The ume of ems epa facly ss fom CT3. The ume of ems epa facly ss fom CT3. Whe cusomes ee he sysem ad sed he faled ems o epa facles ss ado ss hey m ay ecee spaes decly f aalale ad leae o may e equed o wa fo epa efoe leag. Hee hee ae ems he wo epa facles ss ad ss : epa facly hee ae + ems of class wheeas epa facly hee ae j+ ems of class. As used ofe hs hess we wll use fo he ume of spaes ss ad fo he ume of spaes ss. he followg we wll focus ou aalyss o ss u oously he same aalyss s ald fo ss. ss hee ae a he mome + ems of class. Someoe ough hem. Some of he cusomes ough ems ad lef ad some of hem ae sll wag. ss hee ae spaes of class ; f + > he hee ae cuely o spaes o he shelf. Bu who oo hem? We wll defe m as he ume of ems of class whch wee ae y CT3 fom ss. Thus assumg all ems ae equally elgle o hae eceed a spae mmedaely upo aal a ss he ume of aalale spaes m whch wee ae y cusomes of ype 3 fom he aalale has a hypegeomec dsuo. Thus we we hs as m m m fo m =. m. Aalogously f m s he ume of spaes of class whch wee ae y cusomes of ype 3 ss we hae j m m m j fo m =. m. j 4

157 4 f CT3 oo m spaes fom he shelf ss he CT oo he emag -m spaes ad lef he sysem. Also f CT3 oo m spaes fom he shelf ss he cusomes of ype oo -m spaes ad lef. Thus he ohe cusomes ae sll wag. Hece m m m C m ad j m m j m j j C m Bu wha s he poaly of 3 CT3 wag? Thee ae m equess fom ype 3 ems ss ad m fom ype 3 ems ss. Because of he FFO-polcy of sasfyg cusomes ma m m C m m C. Thus 3 3 j m m m m j C j C m m whee ohewse m m m m j C ma 3 3 ad j m m j m m whch ae deemed aoe. Bu wha s he oal ume of cusomes wag he sysem? C C C C. Hece j m m m m j m m j C C * whee ohewse C C C m m j C ad j m m j j m m m m j

158 whee s osso wh paamee ad j osso wh paamee. s sll o do ad wll e doe a eeual eseach. Ths appoach helps aod he depedeces ewee he susysems. The oly depedecy appeas whch s he mos dffcul pa he aalyss. Bu oce calculaed we ed o complee he aalyss of how o fd he dsuo of he oal ume of cusomes wag he sysem. The Aeage ume of Cusomes he Sysem Thee s o dec way o calculae he ume of cusomes he sysem. geeal we ca say ha he aeage ume of cusomes he sysem C s equal o he sum of ume of cusomes of each ype. Tha s C C C ways o calculae he aeage ume of cusomes of each ype. C. Thee ae wo Though he Aeage wag Tme. By Lle s fomula C W e calculaed usg he Wag Tme Dsuo. Ths s he umecal way. W ca Though he ume of cusomes he sysem. The polem wh hs way s ha he ume of cusome dsuo s sll o ow. hs chape we calculaed he wag me dsuo of a cusome a ERSMFmodel. We also showed a mehod o calculae he dsuo of he ume of cusomes he sysem. Ths s a mpoa oely of hs eseach. As eplaed seeal mes hs hess Hausma ad Cheug [5] ed o deelop a fomula fo he ume of cusomes sysem u faled o fd coec fomulas. Oously we oly showed a mehod o ge hee ad we wll eeually fsh hs wo. 43

159 . Model Eesos.. Seeal model eesos Oously he model eed o sad aloe. may e pa of a gge sysem. seco. we aalyze wha happes f hee s o oe epa facly u seeal whch we call a mul-echelo sysem. Sheooe [-3] Algh [-3] ad almos all eseaches he aea of spaes hae focused he aalyss o hs opc ad heefoe we wa o add hs eesos o ou model. seco. we wll deeme ude wha codos he fe sees assumpo hold... Scappg he asc model To geealze he asc model we ed o ela he assumpo of scappg. The fs pa of he eseach deals wh he opo o scap a em. Each me a cusome gs a faled em o he sysem hee s a poaly p ha he em ca e epaed ad hus s scapped. Ths meas ha sce he cusome mus e sasfed he sysem loses a em. To maa sysem pefomace ha s a cea leel of spaes o he shelf scapped ems mus somehow e eplaced. Ths eplaceme ca e accomplshed wo ways. eeal eplaceme polces wll e eamed ad dded o wo pas: Replace a em whe s scapped. Ths polcy us ou o f o he esg model y chagg he emology of epa me o epoduco me. Ths meas ha epoducg a em ca e doe ehe y epag o y odeg a ew oe. hs way he ewly defed epoduco me dsuo. wll clude he me of odeg a ew oe. Theefoe o ew aalyss eed e doe o clude hs opo. Ode ew ems a a cosa ae. The ume of scapped ems mus equal he ume odeed o aeage. f he sysem odes moe ew ems ha he ume scapped he ume of spaes o he shelf of he sysem wll gow cosaly ad eeually ew cusomes wll o hae o wa a all. f he sysem odes fewe ems ha he ume scapped he sysem wll ecome ceasgly sho ad wll eeually o ale o pode ay epaed oes. The ume of cusomes wag fo ems wll cease 44

160 eyod he oal ume of ems he sysem ad he queue wll eplode. Thus o maa alace we eed o ode ew ems a oe same ae ha ae scapped; he aeage ume scapped accumulaes o pλ y me whee p s he poaly of scappg a em. Fo eample cosde a model whee cusomes ae a ae λ= whle he scappg ae p=.. Ths meas ha em wll e scapped pe hou o aeage. By odeg ew em each hou o eey wo hous he scapped ems wll e eplaced. By me hee wll R scapped ems whee R s he amou of ems scapped. ow R s osso wh paamee pλ. f s lage he R s dsued omally wh mea pλ ad aace pλ. O he ohe had y me hee ae pecsely pλ eplacemes. Theefoe we wa o ow lm R p ha s he e chage he ume of ems he sysem he lm. Sce R s dsued omally R p Z s p so ha R p Z * p ad lm R p lm Z * p whch s + whe Z s pose ad - whe Z s egae. Ths popey was fs foud y smulao whee he aeage wag me aed fom eao o eao gg dcao ha he sysem s o sale. cocluso polcy cao e appled ad oly polcy s used. Thus he asc model eed o e chaged o clude scappg... Mul-echelo sysems Mos of he acles he aea of epa sysem deal wh mul-echelo sysems due o he fac ha ealy epa facles ae dsued amog dffee locaos. 45

161 3 4 The eplaceme me wll e he me measued fom whe a em leaes sysem ul s suseque eu o sysem. Fo eample cusomes ae o facly whee ems ae o e epaed. f a em cao e epaed hee s se o o facles 3 4 o 5 whee s epaed. p3wll e he poaly of a em o e se fom facly o facly 3 phe poaly of eg epaed a facly. Thus p p p3 p4. Whe a em s se o facly 3 o 4 he wag me a he specfc facly wll fac e he eplaceme me of he em. Thus he epa me dsuo a facly s: H p p F p 3 3 F p 4 F 4 whee s he epa me dsuo a facly ad F s he wag me dsuo a facly. Usg hs mehod we ca aalyze ad sole lage mul-echelo sysems sepaaely..3. Ample sees Oe of he mos mpoa assumpos of ou model s ha of ample sees. Ths meas ha hee s always a see aalale whe a cusome aes. Bu ealy hee ae o ample see sysems. Each see coss moey. So wha s he mmum 46

162 ume of sees so ha we ca defe he sysem as a ample-see sysem? The ume of sees ss s osso dsued wh paamee ad hus as a paccal ule fo each ss we may wa o ow how may sees ae eeded o hadle say 95% of he demad o ha see mmedaely. So wha s he ume of sees each ss so ha we ca defe ou sysem a ample see assumpo? The poaly ha hee ae sees occuped ss s! e fo =.. Thus he poaly ha sees wll e suffce s! e fo =.. Thus f each ss sasfes he codo ha e. 95 we ca ca defe! ou sysem as sasfyg a ample see assumpo. Fo he case ha we cao use he ample see assumpo we wll eed dffee fomulas see Beg ad ose [6]. 47

163 . Summay ad fuhe eseach hs hess we waed o aalyze a aey of echageale epa sysems wh spaes. Oously we dd o deal wh ALL sysems u wh some specfc sysems: The ERSOF-sysem ad he ERSMF-sysem. Fo he ERSOF-sysem hee ae a aey of models whch wee aalyzed ad peseed hs eseach. The asc opmzao model: The asc opmzao model opmzes a sece ceo ude a udge cosa. As eey laoaoy we wa o ge he mamum ou of ou esme. We soled hs model fo he aeage wag me as he ojece fuco cludg pos-opmaly aalyss ad afe solg we showed a a eample how o ge all he elea oupu daa fom he model. fac we showed ha whe he opmal spaes eco s deemed eeyhg ca e ow fom ou model. Seeal mehods such as ew algohms as well as dyamc pogammg ae used whe he fllae ad he oal ume of cusomes sysem ae desced as he ojece fuco. Mulple cosas: Ou asc opmzao model may o always sasfy he equess of he maage. may cases a maage may wa o sole a model whch cludes mulple cosas. Mahemacal poofs peseed hs hess lead o a algohm whch educes whou loss of geealy a model wh mulple cosas o he fom of ou asc opmzao model. The Dual model: The asc opmzao model has a sece ceo as ojece fuco ad aalale udge as cosa. Bu wha happes f he maage was o opmze hs udge allocao whe a ge sece leel s pesced? A dual model of he asc opmzao model s peseed cludg he aeage wag me ad he fllae as cosas whe he udge equed sees as he ojece fuco. To show how we ae ale o mapulae all dffee opmzao models mulple sece cea ased cosas ae also added. Opmzao model wh oly oe ojece fuco: cea cases he asc opmzao model may e adequae. The maage may wa o opmze a cea sece ceo u does o wa o oesped. Eey cusome o 48

164 sasfed wll cu a pealy ad he goal s he o mmze he oal cos of he sysem. Usg hs echque we deal wh oe ojece fuco whou cosas. Mulple oals: Oe of he ojeces of hs eseach was o aalyze models coag mulple goals. he leaue mulple goals ae meoed u ee elao o spaes posog. Usg seeal echques o come he ojeces cludg oal ogammg we show ha aous models wh mulple ojece fucos ca e soled. ege ogammg: ally used o aldae he esuls deeloped hs hess ege ogammg qucly demosaes ha may see as a mehod o sole polems decly. Oously he polem s -Had ad hus s a ool fo small sze polems. Comg ools such as Vsual Basc ad he Ldo Sole fo Ecel we soled dffcul polems such as he codoal wag me fo a ojece fuco. We demosaed a mehod o how o asfom a model wh o-lea ojece fuco o a lea ay ege polem whch ca e soled y a sadad Sole such as Ldo. Eesos such as dscoug wee also oduced o hs model. Bul: Oe mpoa eeso made he asc model s o accommodae ul aals. A cusome aes gg a goup of ems. Ul ow he wag me dsuo fo he cusome was o ow. hs hess we deeloped he wag me dsuo fo cusomes gg a goup of ems. Oously hs eeso ca e easly oduced o he ERSOF-opmzao model. The secod class of sysem aalyzed hs eseach was called ERSMF-sysem. A cusome gs a poduc whch ca fal a moe ha oe em ype. We deeloped fomulas fo he wag me dsuo ad also a mehod o ge he oal ume of cusomes wag he queue. Fally we elaed ohe mpoa assumpos. We showed how scappg ad mul-echelo sysems ca e oduced o he asc model. 49

165 Oously hs eseach s o fshed. Thee ae mulple decos whch ca e eploed fuhe. We ed o pese he acgoud of a feld whch seemed o e soled y majo eseaches such as Hausma ad Cheug u whch ued ou o e coec. Ths shows how comple ad how easy s o mae eos eplog hs pa of spaes eseach. As eplaed he oduco spaes posog s a mpoa aspec oday s epa polcy deelopme. Fuhe eseach s plaed he followg aeas: The Hausma ad Cheug polem. As meoed seeal mes hs hess Hausma ad Cheug faled o ceae coec fomulas fo a ERSMF-model. We ouled a mehod o sole u dd' pusue o compleo. Thus a fuhe eseach we wll pode coec fomulas. We poded fomulas fo he ERSMF-model u dd o acually aalyze ay specfc opmzao models. Ths wll e doe a fuhe eseach ecause ERSMF-models ae moe ealsc ha ERSOF-models. All he ERSMF-models hs wo coa eacly wo classes. Bu wha happes whe moe classes ae aalale. Wha ae he fomulas o wha ae he mehods o calculae he sece measues? Ths s sll ope. hs hess we peseed a ew mehod of copoag ege pogammg o sole dffee olea models. The polem s ow o e -Had. Thus wha ae s lmaos? Ae hee lmaos wh oday's compues? hs hess he ee aalale udge we o spaes. Bu a sysem hee may e ohe chaels whch o es: Fo eample people epa ools ec.. he leaue hese ohe chaels wee' cosdeed. Bu ealy maages do o oly es spaes u also he facoy mode epa ools ad ag people. How does ha affec he opmzao model? s hee a opmzao model whch cludes all hese dffee aspecs? To summaze we ca say ha mpoa deelopmes eseach of spaes posog wee peseed hs eseach. showed dffee mehods of how o deal wh complcaed models ehe aalycally o a appled usg appomao ad heusc algohms ege pogammg o dyamc pogammg. he ed he comao of mahemacal ools ad he compue made hs eseach a success. 5

166 oao summay c = Cos of oe em of class. F = Wag me dsuo of a cusome ude spaes cofguao. F Wag me dsuo a ss ge spaes. FR = The fllae of he sysem: The poaly ha a adom cusome ges hs poduc epaed sagh away. FR = The fllae a ss : The poaly ha hee s o wag me a ss. FR F j The fs fllae of a specfc cusome: The poaly ha a specfc FR s L j s cusome ges a leas oe em epaed whou wag. = The las fllae of a specfc cusome: The poaly ha a specfc cusome ges hs poduc epaed whou wag. FR F = The fs fllae of a adom cusome: The poaly ha a adom cusome ges a leas oe em epaed whou wag. FR L = The las fllae of a specfc cusome: The poaly ha a specfc cusome ges hs poduc epaed whou wag.. = The cumulae epa me dsuo.. = The cumulae epa me dsuo of ss. = ume of dffee em classes he sysem. j s = A specfc poduc coag dffee faled ems. J = The se of all possle poducs. Thee ae - dffee possle poducs. L L = The aeage ume of cusomes a seady-sae sysem wh spaes. L = The aeage ume of cusomes a ss coag spaes. M = The udge a ou dsposal o sped o spaes. = The ume of spaes he sysem.... = ume of spaes ss = 3 5

167 T = Wag Tme of a cusome a ss ge spaes. T = The aeage wag me of ss ge spaes. W W W = The seady-sae wag me of a cusome a sysem wh spaes. W = The aeage wag me of a cusome a seady-sae sysem wh W F j s W F spaes. = Fs sasfaco me of a specfc cusome. The cusome was ul he ges a leas oe em. = The fs sasfaco me of a adom cusome. W L j = Las sasfaco me of a specfc cusome. The cusome was ul he s W L W F W L ges all he ems. = The las sasfaco me of a adom cusome. = The epeced fs sasfaco me. = The epeced las sasfaco me. X = The seady-sae ume of cusomes a sysem wh spaes. X The seady-sae ume of cusomes ss. Y = The ume of ems he epa facly a sysem wh spaes. = The aal ae of cusomes o he sysem. λ = The ae of aal of faled ems a ss = μ = The mea epa me of a see. μ = The mea epa me of ss. 5

168 53 Apped A: Reseach o sole he ERSOF model usg omal Appomao As a eample le us cosde a sysem wh oly wo su-sysems wh spaes leels ad. Thus. ' ' c T a c T a M c c Sce ' '! * e L T a. Thus!! e e Ths specfc wo polem ca e soled y eumeao u we used he appomao of he omal dsuo. The omal dsuo wh mea ad sadad deao s a good appomao fo he osso dsuo wh ae λ μ whe λ μ >4. Thus. * * c c M X c c X omal omal whee X s a adom aale dsued omally. Thus

169 54 * * c c M Z c c Z omal omal A appomao fo he cumulae omal dsuo was deeloped y Ham Shoe [9] : ]} [ Dz Cz Ep B Ep L Ep wh D C B. Thus ]} * ] * [ [ { * ]} ] [ [ { c c M D c c M C Ep B Ep L Ep c c D C Ep B Ep L Ep should e soled fo. Solg we oa: l ] * ] * [ [ ] ] [ [ c c c c M D c c M C Ep B Ep L D C Ep B Ep L l l ] * ] * [ [ ] ] [ [ c c c c M D c c M C Ep B Ep D C Ep B Ep Oously hs equao cao yeld a closed fom soluo fo ad heefoe heuscs ae equed.

170 Apped B: Usg he sofwae deeloped fo he asc model The followg apped helps he eade o udesad how o use he sofwae eface. Whe sag he sofwae a wdow s opeed whch defes he daa of he ge model. Fgue B. shows 3 colums: Fgue B.: Loadg he daa fo he sofwae. The ame ad he D of he em. The daa fo he epa facly: The dsuo he mea ad he sadad deao. 3. The falue ae of he poducs. Afe defg he daa he sofwae sas ad he daa s peseed o he scee. Fgue B. shows he scee. O he lef dffee cues ae peseed whch ca e chaged usg F f uos. 55

171 Fgue B.: The eface of he sofwae O he gh he daa of em s peseed. cludes all daa of he pu of he em he fllae he aeage wag me ad he aeage ume aclogged. O he meu choose he opmzao o choose he opmzed sece leel. 5 dffee opos ae peseed o he use. Choose oe. We wll choose he fllae as a eample 56

172 Fgue B.3: Dalog o choose he fllae as he opmzao mode Fgue B.3 shows he wdow of he fllae. Choose he alues o opmze e.g.:.95 The esuls he use ca see he esul-ma. Fgue B.4: The esul-ma 57