ECOLE CENTRALE PARIS COORDINATED CONTROL OF INVENTORIES, AND BACKORDERS IN STOCHASTIC MANUFACTURING SYSTEMS

Transcription

1 ECOLE CENTRALE PARIS COORDINATED CONTROL OF INVENTORIES AND BACKORDERS IN STOCHASTIC MANUFACTURING SYSTEMS Strato Ioannd Par 26 March 29

2 General proble Bac coponent of a producton yte: Producton unt Sale departent Raw ateral Producton unt Fnhed Ite SaleDepartent Order arrval Pendng order Producton control a: Proft axzaton (or cot nzaton. Related queton: What hould be the producton rate at any gven te ntant. When hould be accepted a new order and when t hould be rejected.

3 Inventory and order adon control n a ngle tage yte under CONWIP producton control polcy Producton Syte Raw Materal Producton Unt Fnhed Ite μ Fnhed Ite Raw Materal Sale λ Deand Pendng Order c Proft and cot paraeter of the yte p unt proft: ellng prce le cot of purchang raw ateral and proceng per te r unt raw ateral holdng cot rate: cot per unt te per tewatng to be or beng proceed n the yte h unt holdng cot rate: cot per unt te per fnhed te held n the buffer b unt backlog cot rate: cot per unt te of delay for a pendng order.

4 Inventory and order adon control n a ngle tage yte under CONWIP producton control polcy Producton Syte Raw Materal Producton Unt Fnhed Ite μ Fnhed Ite Raw Materal Sale λ Deand Pendng Order c Average proft rate functon The average proft rayepfthe yte gven by J ( c pτη rr hh bb ΤΗ ean producton rate R ean raw ateral nventory level H ean nuber of fnhed te B ean nuber of pendng order

5 Raw Materal Producton Syte Producton Unt μ λ Fnhed Ite Fnhed Ite Raw Materal Sale Equvalent queueng yte wth fnte capacty total capacty (queueerver c μ λ queue erver Deand Pendng Order c Raw Materal (Epty... (Full ax{}... ax{} Fnhed Ite (Full... (Epty... Pendng Order c(full Equvalent Syte (Epty (Full

6 Approxaton of equlbru probablte D A K δ α Κ α α A n nuber of cutoer n the queueng yte Bac aupton: A( και D ( ( n K n K n K n P n δ α δ δ D και Model havng tatonary probablte of the prevou for and atfyng the bac aupton The M/M// queue. An approxaton of G/G// queue propoed n the book of Buzacott and Shanthkuar Stochatc Model of Manufacturng Syte. The approxaton odel for G/G// queue propoed bykouvato. A odel of a two achne tranfer lne wth contant proceng te and falure propoed by the ae reearcher and another odel propoed n the book of Gerhwn Manufacturng Syte Engneerng.

7 Aeent and optzaton of average proft rate [ ] ( ( ( n K n P n H α Κ [ ] n K n P n B 2 ( 2 ( ( δ Κ ΤΗ [ P(] n n K K n P n np R 2 ( ax( 2 ( ( ax( ( δ Κ Κ Theore. If condton A( and D ( hold then we have that where the optal value of for a fxed value of the capacty. Theore 2. If condton A( and D( are atfed and pμ > r then there ext fnte nuber S and M uch that a the yte proftable that J( > f < S b the optal value for bounded fro above by M that < M where ( / ] ax[( ax n( δ D r h p b S r b p M h r p S

8 Nuercal reult Copared producton control polce CONWIP/Bae-backlog: BB (propoed polcy CONWIP/Coplete backorderng: CB CONWIP/Lot ale: LS CONWIP/Rando Adon Control: RAC Make-to-order/Bae-backlog: MTO/BB Make-to-order/RandoAdon Control: MTO/RAC Producton yte under conderaton M/M// queueng yte Two tage producton lne wth geoetrcally dtrbuted falureand repar te

9 Nuercal Reult for the M/M// yte Proft Rate BB 85 LS 75 CB MTO/BB 65 RAC MTO/RAC λ 75 BB LS BB LS 6 CB MTO/BB 65 CB MTO/BB RAC RAC Proft Rate 45 3 MTO/RAC Proft Rate 55 MTO/RAC b h

10 Nuercal reult for a two tage producton lne 5 4 BB LS CB MTO/BB Proft Rate Arrval rate λa /(a b BB LS CB MTO/BB 3 25 BB LS CB MTO/BB Proft Rate 2 5 Proft Rate b h

11 Jont nventory and order adon control n producton network Producton yte producton faclty M N M j raw part p 2 M M 2 p 3 M 3 M p j fnhed te n F ale deand λ whenever an order accepted a new raw part enter the nput buffer pendng order n B Equvalent cloed queueng network M j M N raw part M p 2 p 3 M 2 M 3 M p j vrtual achne M deand vrtual buffer B

12 Propoed control polcy The nventory poton of the yte (the total nuber of part n the yte le the nuber of pendng order kept contant and equal to where a nonnegatve nteger whch wll be referred to a the bae tock.. When a cutoer arrve the correpondng order rejected when the current nuber of pendng order equal to the bae backlog c. When an order accepted a new raw part placed n the nput buffer. Thu the nventory poton of the yte unaltered. Proft and cot paraeter of the yte p unt proft: ellng prce le cot of purchang raw ateral and proceng per te h unt holdng cot rate: cot per unt te per te n the yte b unt backlog cot rate: cot per unt te of delay for a pendng order. Average proft rate functon The average proft rate of the yte gven by J ( c pτη hh bb ΤΗ ean producton rate H average total nventory level B ean nuber of pendng order

13 N n N U G n n P ( ( Κ n n N n N U G... ( ( ( ( ( TH G G U n P > H E [n H ] P(n c (P(n c P(n B E [n B ] P(n c 2P(n c2 cp(n J( pτη hh bb U [U U U N ] U UΠ Π [p j ] Equlbru probablte and atheatcal expreon for varou perforance eaure of the yte under the BB polcy

14 Theore 3. (a functon J( concave n for any fxed and aue t axu value at the pont whch atfy the followng condton (b furtherore t hold that ( ( ( ( G G b h b G G < Theore 4. The yte proftable(that J( > f ax ( ( ρ h p G G h p < < and t both proftable and provable (proft rate ncreae a ncreae f < < ax ( ( ρ ρ h b b h p b h b G G b h p the above condton are neceary for proftablty and provablty but not uffcent.

15 Theore 5. Let M be the allet nteger that atfe h p G ( M G ( M 2 G ( M G ( M f J( J( for oe M then the proft rate decreae for all ubequent value of the WIP capacty.

16 > N N q q A A q c n P c n qp ( ( ( TH ρ ρ ρ ρ ρ ρ λ λ λ λ ρ U / U [U U U N ] U UΠ U λq Π [p j ] Equlbru probablte and atheatcal expreon for varou perforance eaure of the yte under the RAC polcy N N q q A A b h h bb hh 2 ( ( ρ ρ ρ ρ ρ ρ J RAC ( q pτη hh bb j j N A ( ρ ρ ρ

17 Nuercal reult Copared producton control polce Bae-backlog: ΒΒ (Propoed polcy Coplete backorderng: CB Lot ale: LS Randozed acceptance: RAC (econd propoed polcy Producton yte under conderaton A producton lne wth x achne Syte paraeter value λ 4. μ 6. μ 2 7. μ 3 5. μ μ μ p. h 8. b 8..

18 295 Proft rate BB CB LS RAC Deand λ 3 λ 4.95 λ 6.95 Polce c q J c q J c q J BB RAC CB LS Deand rate Proft rate BB CB LS RAC Proft rate BB CB LS RAC Unt backlog cot Unt holdng cot

19 Optal control of nventore and order n ngle tage yte wth etup Producton Faclty Producton: Setup: η Fnhed product: n ax( n Sale Deand: λ Order arrval Pendng order: n ax( n Syte tate: (n a a: Producton faclty State (a f t workng a f t dle n: Inventory (n > or pendng order (n <

20 Event and related decon Producton copleton: Producton faclty contnue to work (π or turned off (π Setup copleton: acceptance (π η or rejecton (π η Order arrval: acceptance (π λ or rejecton (π λ Cot paraeter of the yte q unt rejecton cot: nclude the net revenue (ellng prce le cot of purchang raw part and proceng per te and a penalty per cutoer rejected h unt holdng cot rate: cot per unt te per fnhed te held n the buffer b unt backlog cot rate: cot per unt te of delay for a pendng order w unt etup cot.

21 Tranton Probablte π ( π π λλ P(( n ( n P(( n ( n P(( n ( n ν ν ν λ( π η π ηη π λλ P(( n ( n λ P(( n ( n P(( n ( n ν ν ν ( π η η ( π λ λ P(( n ( n ν where vλη Long-run average cot J * l nf T u E [ ( ( ( ( ] T T h n( t bax n( t w ( t dt q ( t dn( t T π η π λ where N(t the total nuber of ncong order up to te t and u the control polcy

22 Bellan equaton for the long-run average cot V k (n a: nu total expected cot over the ft k event when the ntal tate (n a υ(n a: relatve cot of tartng n tate (n a for any fxed reference tate (n a whch gven by ( V ( n a V ( n a υ( n a l k k k { hn bn ηυ( n n[ υ( n υ( n ] λ n[ υ( n q υ( n] } * J υ ( n v v { hn bn υ( n η n[ w υ( n υ( n ] λ n[ υ( n q υ( n ] } * J υ ( n v v

23 Value teraton algorth V V Let V (n a for any (n a and ( n { hn bn ηvk ( n n[ Vk ( n Vk ( n ] n[ Vk ( n q Vk ( n]} v k λ ( n { hn bn Vk ( n η n[ w Vk ( n Vk ( n] n[ Vk ( n q Vk ( n]} v k λ for k > then J * [ V ( n a V ( n a ] l k k k for every (n a

24 Optal control polcy Fro nuercal experent the optal polcy ee to be a threhold-type control Producton faclty wtched off when the nventory level reache. Setup hould tart when the nventory level drop to. When the faclty workng the ncong order are accepted fthe nuber of pendng order le than l. When the producton faclty dle ncong order are accepted f the nuber of pendng order le than c. where > c and c l. The paraeter c and l are called bae backlog and o the above polcy wll be referred to a double bae backlog polcy (DBB.

25 Statonary dtrbuton under DBB λ λ λ λ c λ λ l λ λ λ λ λ η η c Chapan Kologorov equaton: P( λ P( P(n λ P(n λ n P(n (λ η P(n λ n c P(c η P(c λ P( (λ P(2 P(n (λ P(n λ P(n n2 P(n (λ P(n λ P(n η P(n n c P(n (λ P(n λ P(n n c l P(l P(l λ.

26 Statonary dtrbuton under DBB P(n P( n P(n P(r n n c rλ/(λη P(c P(r c λ/η n ρ P( n P( ρ n Κ ρ P(n C ρ n Dr n n c DP( (λ η/(λη. C P( ρ ( λ λ λ λ λη λ P(n C 2 ρ n nc l C 2 C r c ρ (c [P( D] P( etated wth the ue of the noralzaton equaton l P( n a n a

27 Mnzaton of the average cot rate The average cot rate of the yte J( c l qr Lot wr Setup hh bb were H the average nventory B the average backlog R Lot average order rejecton rate and R Setup average etup rate. The average cot rate ay be explctly expreed a follow l [ P( c P( l ] w P( - h np( a b J ( c l qλ n a The optal producton cot rate for th cla of polce J DBB n > c l J ( c l n a the The optal average cot rate can be calculated by exhautve earch provded that we lt the earch pace n the pace of all adble threhold paraeter. np( n a

28 Nuercal reult Copared producton control polce Propoed polcy: Double bae backlog (DBB Bae backlog (BB Coplete backorderng (CB Lot Sale (LS Standard paraeter value λ 4. μ 5. η 2.5 w 2. h b 5. and q. Optal value of control paraeter and average cot of exaned control polce for varou value of λ and η Polcy Optal DBB BB CB LS λ η c l J c l J l J J J

29 Average Cot Rate Average Cot Rate DBB CB LS BB Deand Rate DBB CB LS Unt Order Rejecton Cot BB

30 Inventore and order adon control n ngle product yte wth two cutoer clae Producton: Fnhed te n f Sale to cutoer of cla Deand rate of the t cla: Deand rate of the 2 nd cla: n Pendng order of t cla cutoer n 2 Pendng order of 2 nd cla cutoer Sale to cutoer of cla 2

31 Proft and cot paraeter of the yte p proft fro ellng a fnhed te to a cutoer of cla h unt holdng cot rate b unt backlog cot rate for cutoer of cla. Average proft rate functon The average proft rate of the yte gven by J p ΤΗ p 2 ΤΗ 2 hh b B b 2 B 2 ΤΗ average rate of ale to cutoer of cla H average total nventory level B ean nuber of pendng order for cutoer of cla

32 Propoed polce Dependent double bae-backlog polcy (DDBB: Produce: untl nventory n p becoe equal to Satfy order: of the t cla f n f > of the 2 nd cla f n f > Accept new order: of t cla f n < c and of the 2 nd cla f n n 2 n f < c 2 Double bae-backlog polcy (DBB: Produce: untl nventory n p becoe equal to Satfy order: of the t cla f n f > of the 2 nd cla f n f > Accept new order: of t cla f n < c and of the 2 nd cla f n 2 < c 2 Bae-backlog polcy (ΒΒ: Produce and atfy order a n the prevou polce Accept new order: f n n 2 < c

33 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Syte tate: ( k : the nventory level equal to n f c (when >c and the t cla cutoer pendng order are n c (when < c k: nuber of 2 nd cla cutoer pendng order

34 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( ( P(( P( c c

35 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( ( P(( P( c c General oluton: P( K x K 2 x 2 c c where x ρ /( and x 2 are the root of the charactertc equaton.

36 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( ( P(( P( c c General oluton: P( K x K 2 x 2 c c where x ρ /( and x 2 are the root of the charactertc equaton. Ue of boundary equaton: P( c ( P(c

37 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( ( P(( P( c c General oluton: P( K x K 2 x 2 c c where x ρ /( and x 2 are the root of the charactertc equaton. Ue of boundary equaton: P( c ( P(c gve P( K ρ c c.

38 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k k c 2 c c 2 k

39 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k k c 2 c c 2 k General Soluton: P( k D k z D k2 z 2 k c 2 c k where z / ρ and z 2.

40 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k k c 2 c c 2 k General Soluton: P( k D k z D k2 z 2 k c 2 c k where z / ρ and z 2. Ue of boundary equaton: P( k P( k k c 2

41 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k k c 2 c c 2 k General Soluton: P( k D k z D k2 z 2 k c 2 c k where z / ρ and z 2. Ue of boundary equaton: P( k P( k k c 2 yeld P( k D k ρ k c 2 c c 2 k.

42 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P(( P( P( c c 2 c

43 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P(( P( P( c c 2 c General Soluton: P( k A y A 2 y 2 c c 2 c where ( λ λ ± ( λ λ 2 2 4λ y 2. 2λ 2

44 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k P( k k c 2 2 kc c 2 c

45 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton: P( k( P( k P( k P( k k c 2 2 kc c 2 c General oluton: k k P ( k 2 j A k j j y k... c 2 k c c 2... c 2

46 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton (Boundary equaton: P( c ( P(c ( P(c P( c

47 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton (Boundary equaton: P( c ( P(c ( P(c P( c P( c k( P(c kp(c k P(c k k c 2

48 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton (Boundary equaton: P( c ( P(c ( P(c P( c P( c k( P(c kp(c k P(c k k c 2 P(c c 2 ( P(c c 2 P(c c 2

49 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton (Boundary equaton: P( c ( P(c ( P(c P( c P( c k( P(c kp(c k P(c k k c 2 P(c c 2 ( P(c c 2 P(c c 2 P(c k k( P(c c 2 k k P(c k k P(c k k k c 2

50 Statonary dtrbuton under DDBB c c 2 c 2 c 2 c k k c kk k c c c Chapan Kologorov equaton (Boundary equaton: P( c ( P(c ( P(c P( c P( c k( P(c kp(c k P(c k k c 2 P(c c 2 ( P(c c 2 P(c c 2 P(c k k( P(c c 2 k k P(c k k P(c k k k c 2 P(c ( P(c c 2 P(c.

51 Statonary dtrbuton under DBBB P( Kρ c c P( k D k ρ k c 2 k c P( A y A 2 y c c 2 c the reanng contant A k D k and K are etated wth the ue of the boundary condton and the noralzaton equaton where ρ /( ρ / ( k j j k c... c c k c... k y j k A P ( ( y λ λ λ λ λ λ ± y A A k k k k λ λ λ λ...k j y A y A A j k j k k k λ λ λ λ λ ( ( 2 c c c c k P k P

52 Aeent and axzaton of the average proft rate The average cot rate of the yte J( c c 2 p ΤΗ p 2 ΤΗ 2 hh b B b 2 B 2 The average cot rate ay be explctly expreed a follow J c ( cc2 pλ P( k p2λ2 P( k b c c 2 ( c P( k b2 k k The optal producton cot rate for th cla of polce J DDBB ax J ( c c2 c The optal average cot rate can be calculated by exhautve earch provded that we lt the earch pace n the pace of all adble threhold paraeter. Theore 6. c The functon J( c c 2 2 c k c c c c 2 k k kp( k h c c 2 k a qua-concave n for fxed value of c c 2 b qua-concave n c when c c 2 for fxed value of and c 2 c concave n and for fxed value of c c and c 2. ( c P( k

53 Nuercal reult Optal value of control paraeter and average proft of exaned control polce for varou value of λ Polcy Optal DDBB DBB SBB CB LS J c c J c c J c c J c c 2 3 J c 2 2 c 2 J c c 2

54 Average proft of exaned control polce for varou value of h and b Average Proft Rate Optal DDBB DBB SBB CB LS h Percentage Proft dfference DDBB DBB SBB CB LS h

55 Average proft of exaned control polce for varou value of h and b Average Proft Rate Optal DDBB DBB SBB CB LS b Percentage Proft dfference DDBB DBB SBB CB LS b

56 Extenon and future reearch Mult-part type producton yte. Invetgaton of the optal nventory and order adon control polcy tructure n two-tage producton lne. Study of order adon control n producton yte where other producton control polce are ued (Bae-tock Kanban etc. Optal producton and order adon control n yte wth perhable te.