Deterministic Constant Demand Models

Transcription

1 Determstc Costat Demad Models George Lberopoulos Ecoomc Order uatty (EO): basc model 3 4 vetory λ λ Parts to customers wth costat rate λ λ λ

2 EO: basc model Assumptos/otato Costat demad rate: λ (parts per ut tme) Shortages ot permtted Ifte producto/repleshmet rate (stataeous repleshmet) Zero lead tme Varable ut producto/order cost: c ( per part) Fxed setup producto/order cost: K ( per producto ru/order) Iterest rate: I ( per vested per ut tme) Computato Ivetory holdg cost rate: h = Ic ( per part per ut tme) Decso Reorder quatty: (parts per producto ru/order) Referece Harrs, F. W. 990 (reprt from 93). How may parts to make at oce. Operatos Research 38 (6) EO: basc model vetory λ tme Computato Reorder perod (cycle legth): = /λ (tme per cycle) Reorder frequecy: N = / = λ/ (orders/cycles per ut tme) 4

3 EO: basc model Ma ssue radeoff betwee fxed setup cost ad vetory holdg cost vetory tme 5 EO: basc model (Ucostraed) optmzato problem Mmze ( ) G K c h av. varable order cost total average cost G() av. fxed order cost av. vetory holdg cost K λ/ h / cλ 6 3

4 Soluto Solve optmalty codto EO: basc model dg( ) K h : 0 0 d K K h h G G K h c ( ) Isght: K, h 7 EO: basc model Sestvty G '( ) K h G'( ) Kh partal average cost Suppose a arbtrary order quatty s chose G'( ) G'( ) Example: G'( ).5 G'( ) I words: 00% error choosg 5% crease cost Cocluso: Performace s ot very sestve to errors the decso varable 8 4

5 EO: basc model Costraed optmzato problem Suppose that m max max m,, G() G() costr max m G() m max m max m max Alteratvely, suppose that m max m max costr max m, max, m 9 Homework problem: EO: basc model A bakery bakes pes a ove that holds 0 pes. It sells the pes at a almost costat rate of 50 per moth. he pes cost each to make. Before each bakg ru, the ove must be cleaed out at a cost of 0. Ivetory costs are based o 8 percet aual terest rate. he pes have a maxmum shelf lfe of 3 moths.. How may pes should the bakery bake at each producto ru? What s the aual setup ad holdg cost for the pes?. he ower of the bakery s thkg of buyg a ew ove that has half the cleag cost ad twce the capacty of the exstg ove. What s the optmal feasble umber of pes to be baked each tme the ew ove? 3. he buyg cost of the ew ove s 50. How may years would t take for the ew ove to pay for tself? 0 5

6 EO: basc model No-zero order lead tme τ Same as EO wth zero lead tme except that order s placed whe vetory reaches reorder pot R, where, f R ( mod ), f vetory vetory τ mod R λ τ R λ τ Order placed tme Order placed tme EO: model wth backorders Assumptos/otato Same as basc EO, except that: Shortages are permtted Backorder cost rate: b ( per part short per ut tme) Decso Reorder quatty: (parts per producto ru/order) Fll rate (percetage of demad covered from stock) F 6

7 EO: model wth backorders Ivetory / backorders F ( F) λ tme Ucostraed optmzato problem F ( F) Mmze GF (, ) K c h Fb ( F) F, :0F Oly costrat: 0 F K c h b F ( F) 3 Soluto EO: model wth backorders GF (, ) K hf b( F) 0 0, F : GF (, ) 0 hf b( F) 0 F b K h b K h b F h b h b h b max vetory level: F max backorder level: ( F ) b K h b h b G G(, F ) Kh c h b h K h b b 4 7

8 EO: model wth backorders Lmtg cases F b h b lm lm b b K h b K h b h lm lm EO formula! b b F b h b lm lm 0 h h K h b K b h h b b lm lm EO formula ( stead of )! b h 5 EO: model wth backorders ad perturbed demad Assumptos Same as EO wth backorders, except that: Backorder cost rate s ot cosdered drectly Demad rate depeds o fll rate: λ(f) Reward rate: r ( per part sold) Maxmum demad rate a cycle f there are o stockouts: λ 0 (parts per ut tme) Number of parts that a customer who faces a stockout wll ot buy each of a (possbly fte) umber of future perods: R (parts ot bought per part mssg) Referece Swartz, B.L A ew approach to stockout pealtes. Maagemet Scece () B538-B

9 EO: model wth backorders ad perturbed demad Dervato of λ(f) otal demad rate a cycle should equal to 0 ( F) R0 ( F) R ( F) 0 0 ( F) R ( F) ( F) ( FR ) 7 EO: model wth backorders ad perturbed demad Optmzato problem ( F) F Maxmze P, :0 ( ) r( F) K c( F) h F F total average proft 0 0 F ( rc) K h ( FR ) ( FR ) Soluto r 0 R hk F 0 R r 0 R K0 hk F R h Referece Lberopoulos, G., I. sks, S. Delkouras. 00. Backorder pealty cost coeffcet "b": What could t be? Iteratoal Joural of Producto Ecoomcs 3 ()

10 EO: model wth lost sales Assumptos/otato Same as EO wth backorders, except that: Customer that ecouter shortages are lost Lost sales cost: l ( per part short) Decso Reorder quatty: (parts per producto ru/order) Fll rate (percetage of demad covered from stock): F 9 Ivetory EO: model wth lost sales ( F) λ F F=/λ = /λf N = / = λf/ tme Ucostraed optmzato problem F Mmze GF (, ) K cfh Fl( F) F, :0F Homework Problem: ry to solve for ad F 0 0

11 EO: model quatty dscouts Case: Same dscout for all uts otal cost for buyg uts, C() = c, where C() c0 for b0 b cc for b b where c c c c for b 0 c c c 0 b 0 b b EO: model quatty dscouts otal average cost fucto for dscout level j Gj( ) K cj Icj, j 0,, Costraed optmzato problem G0( ) for b0 b Mmze G ( ) G( ) for b b total average cost G( ) for b G() G 0 () hj G () G () b 0 b b

12 EO: model quatty dscouts Soluto K Ucostraed EO for dscout level j: j, j 0,, Ic j Costraed optmal order quatty: j,costr max m j, bj, b j, j 0,, Note: b 3 = j arg m G ( ) G G( ) G ( ) j j,costr j j,costr j j,costr G() G 0 () G () G () b 0 b b 3 EO: model quatty dscouts Case: Icremetal quatty dscouts otal cost for buyg uts, C(), where c 0 for b0 b C( ) c0bc( b) ( c0 c) bc Cc for bb cb c( b b) c( b) ( c c) b( c c) b cc c for b 0 0 j Cj c c b where, for smplfcato, we have used the otato: ( ). Note: C0 0 C() c C c C c 0 b 0 b b 4

13 EO: model quatty dscouts otal average cost per part whe orderg uts, C()/, where c0 for b0 b C C ( ) ( c0 c) b C c c for b b C ( c0 c) b( cc) b C c c for b otal average cost per ut tme C ( ) C ( ) G ( ) K I equvalet to c equvalet to c 5 EO: model quatty dscouts G() G 0 () G () G () b 0 b b Cj Cj Gj( ) K cji cj IC j ( K Cj) cj Icj ( K C ) j j Ic j 6 3

14 EO: model quatty dscouts Fal soluto j arg m G ( ) : b b j j j j j j G G( ) G ( ) j j j 7 EO: Resource-costraed multproduct systems Assumptos products λ, K, c, h : parameters for product Budget or space or other costrat Average cost per ut tme for product G ( ) K c h,,,, K h otal average cost per ut tme G (,,, ) G ( ) 8 4

15 EO: Resource-costraed multproduct systems Costraed mmzato problem Mmze G (,,, ) subject to c C,,, E.g. C s budget/space cap (upper lmt) Soluto c C,costr c C case ) If costrat s ot actve case ) If costrat s actve ot feasble I ths case, we kow that the costrat s bdg at the optmal soluto Problem to solve Mmze G (,,, ) subject to c C,,, 9 EO: Resource-costraed multple product systems Soluto for case Itroduce Lagrage multpler θ K h Mmze G (,,,, ) c C,,,, Necessary codtos for optmalty G K h K 0, c 0,,,, () h c,costr G 0 c,costr C () Solve umercally: ry dfferet values of θ utl optmalty codtos () ad () hold 30 5

16 EO: Resource-costraed multple product systems Specal Case: c c c c h h h h I ths case: K K () h c h c/ h c/ h (),costr,costr m,,,, where m c/ h C c,costr C c mc m c 3 Ecoomc Producto Lot (EPL): EO wth fte producto rate P P P vetory λ λ λ Parts to customers wth costat rate λ λ λ 3 6

17 I EPL Assumptos/otato Same as basc EO, except that: Fte producto/repleshmet rate P (parts per ut tme) wth P > λ Setup tme to produce a ew producto lot s max vetory P λ λ /P =/λ tme Maxmum vetory: I max = (P λ)/p = ( λ/p) = ( ρ), where ρ = λ/p utlzato factor, ρ fracto of tme mache s ot producg s 33 EPL Ucostraed optmzato problem Lmtg case: Mmze G ( ) K c h dg( ) K h( ) : 0 0 d K K h h ( ) ( ) G G K h c K K lm lm EO! P P h h P 34 7

18 EPL What about the setup tme s? Cycle tme must be large eough to accommodate s s s s s P setup tme P P cycle tme producto tme costr max(, ) m Alteratvely s s s P P s s max(, ) m costr m m 35 EPL What f there s a maxmum storage capacty I max? m I max /( ρ) max max m,, costr max m Sestvty aalyss Same as EO model: Cost ot very sestve to errors G'( ) G'( ) G'( ) G'( ) 36 8

19 EPL Power-of- heurstc for choosg Suppose that perod (cycle) legth s restrcted to be a power-of- multple of the base tme ut,.e., H = k, for some k = 0,,, k k k Whch k to choose? Rule: k: H k k k k How bad s the cost crease? Worst case: k k k G'( ) H If, 06 k k G'( ) k k k G'( ) H If, 06 k k G'( ) Cocluso: Usg the best H wll result a crease G of at most 6% wth respect to usg! k 37 H Ecoomc Lot Schedulg Problem (ELSP) 3 P P λ λ 3 λ λ λ λ 3 λ λ 3 λ Parts to customers wth costat rate λ 38 9

20 Ecoomc Lot Schedulg Problem (ELSP) Assumptos Same as EPL, except that products λ, K, c, h, s : parameters for product Cyclc schedulg: All products must be produced by the same mache a cyclc fasho Smple cycle: Each product s produced oly oce each cycle Cycle patter: ( ) Computato Utlzato factor for product : ρ = λ /P 39 ELSP vetory tme Strog depedecy amog products: hey all have the same cycle tme Oce s determed the the producto lot szes ca be computed: 40 0

21 ELSP Average cost per ut tme for product otal average cost per ut tme G (,,, ) G( ) Problem,,, G( ) K c h Soluto Replace by λ, ad formulate a mmzato problem wth respect to Mmze G (,,, ) subject to,,,, 4 ELSP New otal average cost per ut tme Mmze G( ) G ( ) K c h Optmal soluto K h c A B C (same form as EO model) K,,,, h 4

22 ELSP What about setup tmes s? Commo cycle tme must be large eough to accommodate all s s s s s P P s m max(, ) costr m,costr costr 43 ELSP More complcated cycles Assumpto Each product s produced m tmes each cycle m m Same approach as wth smple cycle Replace by λ /m, ad formulate a mmzato problem wth respect to 44

23 Average cost per ut tme Optmal soluto ELSP m Mmze G ( ) G( ) K c h m m m K h c m K,,,, m h m 45 ELSP What about setup tmes s? Commo cycle tme must be large eough to accommodate all s s s s s m m m m P m P m m s m max(, ) costr m,costr costr 46 3

24 ELSP How to choose good values for m k Use powers-of- method,.e. set m for some k {0,,,3, } for =,,, Algorthm for computg k Step : Compute ucostraed optmal cycle tme of each product solato ad fd the mmum of these tmes K,,,, h m m Step : Compute relatve producto frequecy of each product solato N,,,, m 47 ELSP Step 3: Roud N to the earest power-of-" usg the rule k roud k k N N Example: 0 roud 0 k 0 : N.44 N 0 roud k :.43 N.88 N roud k :.88 N N 4 roud Step 4: Fd the largest rouded frequecy ad call t N max roud Nmax Step 5: Compute multple m : m N roud Step 6: Compute wth these multples. roud wll be N max m Step 7: Compute max(, ) costr m 48 4

25 roud N ELSP Note: o compute step 3, thk as follows: roud k k N, where k s the smallest teger k such that N he above equalty ca be wrtte as: k k k N k N k ln l N N l N l l l l l k where x floor of xlargest teger x e.g., 4.9 4, 4. 4, roud N ELSP Note: o compute step 3, thk as follows: roud k k N, where k s the smallest teger k such that N he above equalty ca be wrtte as: k k k N k N k ln l N N l N l l l l l k where x floor of xlargest teger x e.g., 4.9 4, 4. 4,

26 Seral EO systems Assumpto: fte producto rate (stataeous repleshmet) Defto: Nested polcy: Producto (repleshmet) does ot occur at stage uless t also occurs at all successor stages,,,, Result: For a -stage seral system, t s optmal to follow a ested polcy Sketch of proof stage vetory stage tme vetory tme However, t s possble that (e.g., ),.e., t may be optmal to order at stage but ot order a prevous stage + Referece: Muckstadt, J. A., R.O. Roudy Aalyss of Multstage Producto Systems. S.C. et al., Eds. Hadbooks OR ad MS, Vol. 4: Logstcs of Producto ad Ivetory. Elsever, Amsterdam, he Netherlads λ 5 I I Seral EO systems: -stage I I I + Coveet to use echelo stock ad echelo vetory holdg cost I o had vetory stage I "echelo" stock for stage I I,,, k k I I I I I λ h covetoal vetory holdg cost stage (assumpto: h h ) (measures total value added up to ad cludg stage ) h "echelo" vetory holdg cost stage (measures cremetal value added oly at stage ) h hh,,, ; h h 5 6

27 Seral EO systems: -stage I I I λ A ote o echelo vetory : Suppose that = I I I Cost h 4 4 I I I I Cost h / / I I Cost h 4 I I / / I Cost h 4 O-had vetory vs. tme Echelo vetory vs. tme 53 Seral EO systems: -stage I I I λ Icremetal echelo vetory holdg costs, h ad h : h h; h hh o-had vetory cost echelo vetory cost stage h h stage h h 4 otal av. o-had v. cost ( hh ) h ( hh) h h h 4 otal av. echelo v. cost h h h h otal cost are the same! I 54 7

28 Seral EO systems: -stage A reorder terval NLIP problem I K Mmze k, k I I λ K h h subject to, k {0,,, } ( ) k k L k k, {0,,, } ( L ) k 0 Cosder ts relaxato, R-NLIP K K Mmze h h, subject to 0 I 55 Seral EO systems: -stage. Suppose Soluto: ( K K) (-) ( h h). Partto the system to two subproblems G ad G, ad solve them separately, gorg the costrat G Soluto: G K K (), () h h Optmal soluto of problem s: K K () () costrat ot actve (), () h h K K () () costrat actve (-) (bdg) h h G

29 Seral EO systems: -stage Optmal soluto of the orgal NLIP problem: Soluto: roud k k, where k s the smallest teger k such that k l l Note: () () roud roud 57 Seral EO systems: -stage λ A reorder terval NLIP problem Mmze k,, k Cosder ts relaxato, R-NLIP K h k k subject to ( ),,, 0,,, k {0,,, },,, L K Mmze h,, k k subject to 0,,, 58 9

30 Seral EO systems: -stage G 7-8 G 5-6 G he above partto of the stages to clusters ad the correspodg reorder commo tervals (-4), (5-6), (7-8) provde a optmal soluto to the correspodg R-NLIP problem f ad oly f: ( K7 K8) ( K5 K6) ( KK K3 K4) ( ) (7-8), (5-6), (-4) ( h h ) ( h h ) ( h h h h ) ( ) (7-8) (5-6) (-4) ( ) Each cluster caot be futher parttoed to smaller clusters e.g., For cluster -4, ths meas that: (4) (-3) ( ) ( ) (3-4) (-) ( ) ( ) ( ) ( ) K4 h4 KK K3 hh h3 K3 K4 h3 h4 KK hh (-4) () ( K K3 K4) ( h h3 h4) K h 59 Seral EO systems: -stage Algorthm for fdg optmal partto C {}, (),,,, S {,,, } j C C C j ( j) NO S S \{ ( j)} j ( j) ( C ) ( C ) ( j) ( ( j)) YES j j NO ( j) 0 YES j YES NO l Redex clusters { C : S} so that S {,,, N} ad f jc, kc, j k l 60 30

31 Seral EO systems: -stage Fd soluto to problem R-NLIP k For each cluster C, k S, set ( ) ( ) ( ) ( ) k k C kk kh C C k For each C, set ( k) Fd soluto to problem NLIP roud roud k k k For each C, set ( L ), k where k s the smallest teger k such that ( k) k k l ( ) l 6 Example: Seral EO systems: -stage Fd the optmal power-of-two reorder tervals for each stage of a 5- stage seral EO system wth demad rate /ad the followg cost parameters:

32 Seral EO systems: -stage Soluto:. Fd optmal partto,,3,4,5 ; ; Iterato????? NO, ; \,3,4,5 ; 0;? 0 NO 3;?5 YES GOO NEX IERAION; Seral EO systems: -stage Iterato?? 3?,?? YES 4;?5YES GOO NEX IERAION; Iterato 3?? 4? 3?? YES 5;?5YES GOO NEX IERAION; 64 3

33 Seral EO systems: -stage Iterato 4?? 5? 4?? NO 4,5 ; \ 5,3,5 ; ; 5?0 YES?? 4,5? 3?? NO 3,4,5 ; \ 5,5 ; ; 5?0 YES?? 3,4,5?,? 6;?5NO EXI; Redex Clusters:,,3;,; 3,4,5;? YES 65 Seral EO systems: -stage. Fd soluto to problem R-NLIP 40 4,47 8 4, , Fd soluto to problem NLIP 4 4,47, ,0945 5,5 0,693,498 0,693,66 ;,07 3; 66 33

34 Dstrbuto EO systems: cetral warehouse, retalers I 0 0 I 0 I I I I I I I0 I I I I I,,, h 0 0 h hh,,, h 0 g h Wthout loss of geeralty, assume: K K K g g g 67 Dstrbuto EO systems: cetral warehouse, retalers A reorder terval NLIP problem Mmze k0,, k 0 k k subject to ( ), 0,, Cosder ts relaxato, R-NLIP 0,,, 0 k {0,,, },,, K g L K Mmze g,, k k 0 subject to 0,,,

35 Dstrbuto EO systems: cetral warehouse, retalers Form of optmal soluto of R-NLIP Retalers are dvded to categores:. Retalers the st category share a commo reorder terval wth the cetral warehouse.. Retalers the d category follow ther atural ucostraed reorder tervals. 69 Dstrbuto EO systems: cetral warehouse, retalers Optmal soluto of R-NLIP If, the all retalers belog to the d category ad, 0,,,. Else (f ), let be the smallest dex of retaler that belogs to st category,.e., retalers,,, share commo reorder terval wth cetral warehouse. argm,,,,,,0,,, 70 35

36 Dstrbuto EO systems: cetral warehouse, retalers Optmal soluto of NLIP l l l,,,,,0 l,,,, 7 Seral EO systems: -stage Bullwhp effect: he varace of orders may be larger tha that of sales, ad the dstorto teds to crease as oe moves upstream a pheomeo termed bullwhp effect. hs dstorto may be due to: Demad forecastg Lead tme Batch Orderg (lot szg) Prce fluctuato Referece Lee, H.L., V. Padmaabha, S. Whag Iformato dstorto a supply cha: he bullwhp effect. Maagemet Scece 43 (4)

37 Seral EO systems: -stage uatfcato of the Bullwhp effect due to lot-szg: D demad at stage ED [ ], ED [ ] Var[ D ] E[ D ] E[ D ] 0 he varace of the customer demad see by stage s zero. ( ) ED [ ], ED [ ] Var[ D] E[ D] E[ D] he varace of the demad see by stage s proportoal to ad to. Var[ D ] ( ) ( ) D ( ) (because ) Var[ ] ( ) ( ) he varace of the demad creases as oe moves upstream the supply cha