Lecture outline. Basic inventory problems The economic order quantity An inventory game Multiperiod lot sizing

Transcription

1 Lecure ouline Basic invenory problems The economic order quaniy An invenory game Muliperiod lo sizing» Mah programming formulaion» Heurisics» Wagner-Whiin algorihm 2013 W.B. Powell 1

2 Basic invenory problems Examples:» Producs: Cusomers consume producs over ime. Sore replenishes periodically.» People wih specialized raining: People randomly leave he company over ime. Company periodically hires new graduaes.» Waer (managemen of dams). Rainfall randomly replenishes reservoirs. Release waer from dam o mainain level.» Oil being sored in sorage anks: Oil is seadily consumed. Periodically is replenished from ankers W.B. Powell 2

3 Basic invenory problems Examples:» Housing sock Houses are coninually being purchased. Developers produce new developmens or aparmen buildings.» Financial resources (sarup company) Cash is used o build up he company. Periodically fresh capial is raised from venure capialiss.» Feaures in a sofware program: Accumulaing feaures in a sofware program in response o user requess and he ideas of developers, or due o bug fixes. Periodically ship a new version of he program.» Purchasing sock: Funds become available for invesmen. Periodically purchase new shares of sock W.B. Powell 3

4 Basic invenory problems 2013 W.B. Powell 4

5 Basic invenory problems Muual fund cash balance Sock marke Cash Invesor How much cash do we keep on hand o srike a balance beween he deposis and wihdrawals of invesors, and he behavior of he marke? 2013 W.B. Powell 5

6 Basic invenory problems There are a number of ways o refer o soring resources for he fuure:» Physical resources Invenory Sockpile Sock» Financial resources Savings Nes egg Reserve 2013 W.B. Powell 6

7 Basic invenory problems Reasons for holding invenories:» Economies of scale Baches of goods Discouns (purchasing) Transporaion economies (e.g. shipping in bulk)» Uncerainies Demand Order lead imes Supply/price of raw maerials (OPEC) Supply/price of componens (srikes) Qualiy conrol» Speculaion commodiies prices currency flucuaions 2013 W.B. Powell 7

8 Basic invenory problems Reasons for holding invenories» Transporaion In-ransi or pipeline invenories» Smoohing producion Respond o seasonal paerns in demand Seasonal producion of some iems Cerain foods Syrup Snow Sudens» Conrol coss Lower invenories requires more sophisicaed conrol sysems 2013 W.B. Powell 8

9 Basic invenory problems The lo sizing problem» Ofen, here are economies of scale when ordering new resources: Raising operaing capial There is a fixed cos o going o he capial markes Jus as much work o raise $1m as $5m Shipping he laes version of a sofware program New feaures are added over ime There is a fixed cos of shipping a new version of he code How many new feaures do you add before you ship he code? Ordering new produc for a sore shelf Fixed cos for placing and shipping an order 2013 W.B. Powell 9

10 Basic invenory problems An aging and replenishmen process (negaive drif):» Sae is invenory drif is due o cusomer demand. Time 2013 W.B. Powell 10

11 Basic invenory problems Indexing ime:» Deerminisic indexing Index based on when somehing happens.» Sochasic indexing Index based on when somehing becomes known.» Deerminisic indexing: Coninuous ime Discree ime D0 D1 D2 D R0, x0 1, 1 R x R2, x2 R3, x3 R4, x W.B. Powell 11

12 Basic invenory problems Basic invenory equaion» Deerminisic indexing R 1 max 0, R x D R x D Demand (random a ime ) New orders Sae (e.g. invenory level)» In deerminisic indexing, everyhing is modeled a he beginning of a ime period W.B. Powell 12

13 Basic invenory problems Sochasic indexing» Informaion arrives coninuously over ime» A variable indexed by conains exogenous informaion up hrough ime. Coninuous ime Discree ime D1 D2 D3 D R0, x0 1, 1 R x R2, x2 R3, x3 R4, x4» In sochasic indexing, everyhing is indexed a he end of a period W.B. Powell 13

14 Basic invenory problems Basic invenory equaion» Sochasic indexing R max 0, R x D R x D Demand (random a ime ) New orders Sae (e.g. invenory level) 2013 W.B. Powell 14

15 Basic invenory problems Invenory wih posiive drif» Cash is deposied in an accoun. Periodically he cash is invesed in bach amouns o reduce ransacion coss. Time 2013 W.B. Powell 15

16 Basic invenory problems Basic invenory wih posiive drif» Blood donaions, waer reservoirs, R R x D 1 1 Exogenous drif Endogenously conrolled shif Sae (e.g. accumulaion) 2013 W.B. Powell 16

17 Ouline Basic invenory problems The economic order quaniy An invenory game Muliperiod lo sizing» Mah programming formulaion» Heurisics» Wagner-Whiin algorihm 2013 W.B. Powell 17

18 The economic order quaniy Order coss» Fixed cos of placing an order Paperwork, forms, elephone Sending a ruck ou Seing up a machine Price srucure imposed by supplier» Variable cos of ordering a cerain amoun of produc Variable cos may be fixed (a linear funcion). Or here may be economies of placing larger orders W.B. Powell 18

19 The economic order quaniy Holding cos» Sorage (hea, elecriciy, supervision, ec.)» Taxes and insurance» Breakage, spoilage, deerioraion and obsolescence Careful wih hese - hese coss conver o reducion in quaniy.» Opporuniy cos (ineres) Hurdle rae for a company is generally much higher han bank ineres raes. Le I = ineres rae c p = purchase cos of iem c h = Ic p = holding cos (be careful wih unis; if I is ineres rae per year, c h is holding cos per year) W.B. Powell 19

20 The economic order quaniy Sockou cos» Cos of los cusomers» Cos of pushing orders o fuure ime periods Noes:» Sockou coss are no relevan in our simples invenory sysem, because hey canno happen.» Sockous arise when: Demand is random Demand varies over ime wih producion capaciies Order coss may vary as a funcion of ime, possibly exceeding he benefi of covering demand W.B. Powell 20

21 The economic order quaniy Some assumpions:» Demand is deerminisic wih rae per uni ime.» Rae is consan saionary process.» Coss are saionary.» Orders arrive immediaely.» All orders mus be filled W.B. Powell 21

22 The economic order quaniy The basic invenory equaion: where: Noaion: R [ R x D ] R x 1 1 D Invenory a sar of ime Amoun ordered a ime Demand during period [ x] max x, W.B. Powell 22

23 The economic order quaniy The cos funcion: c ( x, R) Toal coss during period given order quaniy x and iniial invenory R where: c c c c o p h h o h c ( x ) c ( R, x ) ( x ) Order coss Uni purchasing coss ( R, x ) Holding coss p K c x x 0 0 x 0 h c R 0 x z dz Uni holding coss per ime K Order quaniy x 2013 W.B. Powell 23

24 The economic order quaniy The cos funcion: We can ransform he nonlinear cos funcion ino a linear one: o p c ( x ) Ky c x where: x x y 0 My (0,1) M "big M" If y 0 hen we force x 0. Now we have ransformed a nonlinear cos funcion ino a linear one, bu we have added an ineger variable W.B. Powell 24

25 The economic order quaniy Infinie horizon problem: We would like o solve: min c R, x, y x, y 0 T x, y This is a really big number! A more formal way o wrie i is as an average cos: T 1 lim min cr, x, y T W.B. Powell 25

26 The economic order quaniy Q Inuiion suggess ha we le invenories drop o zero, and hen order up o an amoun Q: Invenory, R = reorder inerval 2013 W.B. Powell 26

27 The economic order quaniy Reformulae he problem in erms of cos per order inerval : Q order quaniy Demand rae per uni ime (assumed consan and deerminisic) c o ( Q) Order cos per order inerval K (since we always make one order per inerval). Lengh of order inerval Q c p ( Q) Purchase coss per order inerval p c Q 2013 W.B. Powell 27

28 The economic order quaniy h c ( Q) Holding cos per order inerval (holding cos)*(average invenory)*(lengh of inerval) h Q c 2 2 hqq hq c c 2 2 c( Q) Toal cos per order inerval o c ( Q) c ( Q) c ( Q) 2 p h Q K c Qc 2 p h 2013 W.B. Powell 28

29 The economic order quaniy How do we find he opimum value of Q? We can ry differeniaing C(Q) wih respec o Q: dc( Q) p h Q 0 c 2c 0 dq 2 Solving for Q gives us: * c Q!!!!! h c Wha wen wrong? p 2013 W.B. Powell 29

30 The economic order quaniy Q Cos per cycle =K plus quaniy proporional o green area 2013 W.B. Powell 30

31 The economic order quaniy Q 2013 W.B. Powell 31

32 The economic order quaniy Q 2013 W.B. Powell 32

33 The economic order quaniy The smaller Q ges, he lower our coss per cycle. Q 2013 W.B. Powell 33

34 The economic order quaniy Need o minimize cos per uni ime, no cos per order inerval. So, we wan o solve: CQ ( ) CQ ( ) CQ ( ) min C ( Q) Q/ Q Q K Q p h K c Q c p c c h Q 2 2 Q 2 Differeniaing wih respec o Q and seing o 0: h d C ( Q) K c 0 2 dq Q W.B. Powell 34

35 The economic order quaniy Finally, solving for Q gives us: 2K Q The Economic Order Quaniy (EOQ) h c Also called he Economic Lo Size. Properies of he opimal soluion: 1. Purchase coss do no ener he equaion (why?) 2. Order coss per uni ime holding coss per uni ime: Order coss per uni ime Holding coss per uni ime K K c K Q 2K 2 h c h h Q c 2K c K c h 2 2 c 2 h h 2013 W.B. Powell 35

36 The economic order quaniy The average cos funcion: Coss Order coss Toal cos c h K 2 Holding cos Q 2K h c 2013 W.B. Powell 36

37 The economic order quaniy Sensiiviy analysis Acual order quaniy Relaive increase in cos Acual order quaniy 2013 W.B. Powell 37

38 Ouline Basic invenory problems The economic order quaniy An invenory game Muliperiod lo sizing» Mah programming formulaion» Heurisics» Wagner-Whiin algorihm 2013 W.B. Powell 38

39 An invenory game The basics:» Random demand (uniform beween 0 and 10)» Demands are no revealed unil an order is enered.» Cos parameers: Order cos 25 Purchase 10 Holding 1 Sockou W.B. Powell 39

40 An invenory game Player: Joe Time Inveno Order DemanEnding Sold Los Order Purcha HoldingSocko Toal Cum

41 An invenory game Player Jimmie: Time Inveno Order DemanEnding Sold Los Order Purcha HoldingSocko Toal Cum

42 An invenory game Player: Time Inveno Order DemanEnding Sold Los Order Purcha HoldingSocko Toal Cum W.B. 0Powell

43 An invenory game Player: Time Inveno Order DemanEnding Sold Los Order Purcha HoldingSocko Toal Cum W.B. 0Powell

44 Ouline Basic invenory problems The economic order quaniy An invenory game Muliperiod lo sizing» Mah programming formulaion» Heurisics» Wagner-Whiin algorihm 2013 W.B. Powell 44

45 Mah programming formulaion Wha do we do when he demands are nonsaionary?»d = he forecased demand for ime period, 0 < < T. We are going o use a poin esimae of he demand, which produces a deerminisic model.»t = he planning horizon. Objecive funcion: T 1 0 min c R, x, y x, y 2013 W.B. Powell 45

46 T 1 Mah programming formulaion The opimizaion problem can be visualized as a nework: Producion arcs 0 D T 1 Consumpion Invenory 0 D 2013 W.B. Powell 46

47 Mah programming formulaion Ineger programming formulaion: Noaion: Aciviy variables: R invenory a beginning of period D demand during period saring a Parameers: c h Uni holding cos per ime period p c Uni purchase cos K Fixed order cos Decision variables: x amoun ordered in period y 1 x 0 0 x W.B. Powell 47 R, D x

48 Mah programming formulaion Objecive funcion: min xy, subjec o: T 1 p h Ky 0 c x c R x D R R x D x 1 My x D R x y 0 (0,1) (M big number) This is an ineger programming problem, which can be solved using commercial solvers such as Cplex and Gurobi W.B. Powell 48

49 Mah programming formulaion We implemen our mah programming formulaion as a rolling horizon procedure» Opimize over 0-4, implemen ime » Roll o ime 1, see new informaion, solve updaed problem for ime periods 1-5: » Roll o ime 2, see new informaion, solve updaed problem for ime periods 2-6: W.B. Powell 49

50 Mah programming formulaion Rolling horizon procedures» These are deerminisic approximaions of he problem over a planning horizon H Objecive funcion: min xy H p h Ky ' ' c ' x ' c R ' x ' D ' where x = x, y = y subjec o: ' ',..., H ' ',..., H,'1 ' ' ' ' R R x D x ' ' ' ' ' My x D R x y 0 ' (0,1) (M big number) 2013 W.B. Powell 50

51 Heurisics Silver-Meal heurisic (Leas average cos) Le C(s) average cos per uni ime if s 1 1 h K c D s 0 Calculae C(1), C(2),..., C(s). Se T s. s ime periods we order over he nex Sop when C(s 1) C(s). Order enough for he nex T ime periods.» One of he bes known and mos widely used heurisics in supply chain managemen W.B. Powell 51

52 Heurisics Leas uni cos Le C(s) average cos per uni produced if Se T s. s ime periods s1 h K c D 0 s1 D 0 Calculae C(1), C(2),..., C(s). we order over he nex Sop when C(s 1) C(s). Leas uni cos reflecs he way managers are acually measured. No - one is measured in erms of $/day (why no???) W.B. Powell 52

53 Wagner-Whiin algorihm Example: Seasonal TV demand» Parameers: Seup cos 8 Order cos 1 Holding cos 1.3» Demands (in 1000 s): Time Demand W.B. Powell 53

54 Wagner-Whiin algorihm A nework represenaion:» The se of decisions represens a shores pah problem over a specialized nework: » The cos on each arc is he cos of he decision, including order coss and all holding coss.» Wha does he opimal soluionn look like? 2013 W.B. Powell 54

55 Wagner-Whiin algorihm Properies of opimal soluion» We only need o make decisions when he invenory is zero.» This means our decision variable is no he quaniy, bu he number of ime periods ino he fuure ha we need o cover. Order enough jus for ime periods 0 and 1 Order enough jus for ime periods 2 and Order enough jus for ime period 0 The sae ha we are a ime period 2 wih zero invenory W.B. Powell 55

56 Wagner-Whiin algorihm Sar wih he final node: $43.7 $17.6 $43.7 $10 $29.7 $17.6 $10 $0 $ W.B. Powell 56

57 Wagner-Whiin algorihm Links ino ime period 3: $43.7 $33.9 $17.6 $43.7 $29.7 $17.6 $13 $10 $10 $0 $22.5 $ W.B. Powell 57

58 Wagner-Whiin algorihm Links ino ime period 2: $43.7 $15.9 $33.9 $17.6 $33.5 $43.7 $28.6 $29.7 $11 $17.6 $13 $10 $10 $0 $22.5 $ W.B. Powell 58

59 Wagner-Whiin algorihm Links ino ime period 1: $43.7 $15.9 $33.9 $17.6 $9 $33.5 $43.7 $28.6 $29.7 $11 $17.6 $13 $10 $10 $0 $22.5 $ W.B. Powell 59

60 Wagner-Whiin algorihm Finally, we walk forward in ime: We use he values compued in he backward pass o walk forward and compue decisions W.B. Powell 60

61 Wagner-Whiin algorihm Srenghs:»Very fas» Handles very general cos funcions You can use virually any shape order cos funcion.» Handles ime-dependen daa (e.g. seasonal daa, day of week effecs or hour of day paerns).» Handles forecass of he fuure (which is a form of ime-dependency) W.B. Powell 61

62 Wagner-Whiin algorihm Limiaions of his model:» Assumes demands are deerminisic!!! Opimal soluion is no really opimal. Drives invenories o zero, which will creae sockous. Have o reopimize as forecased demands change.» Limiaions: Compuaionally demanding when you have o solve 100,000 problems (Wal-Mar!). Soluions are no obviously beer han good heurisics under realisic condiions. Ges complicaed if you have muliple iems and join capaciy consrains (need o use ineger programming formulaion)» Bu: Serves as a useful subproblem in he conex of larger applicaions. Highlighs behavior of he problem W.B. Powell 62

63 Wagner-Whiin algorihm Imporan generalizaions:» Upper bounds on order quaniies Wha if we canno order more han u in ime period?» Upper bounds on producion and muliple iems: This is he problem ha acually arises in pracice. Called he capaciaed muli-iem lo sizing problem.» Limi on oal producion ime, in he presence of seup imes. The lieraure on seup imes is very sparse W.B. Powell 63