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# SPTP COGNITIVE SYSTEMS ENGNG ISEN 689

Texas A&M

GPA 3.56

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This 63 page Class Notes was uploaded by Ms. Hipolito Willms on Wednesday October 21, 2015. The Class Notes belongs to ISEN 689 at Texas A&M University taught by Lewis Ntaimo in Fall. Since its upload, it has received 12 views. For similar materials see /class/226196/isen-689-texas-a-m-university in Industrial Engineering at Texas A&M University.

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Date Created: 10/21/15

ISEN 689 LargeScale Stochastic Optimization Modeling Examples of Stochastic Programs Dr Lewis Ntaimo September 73 2007 L N roimo TAMU 2007 Outline p Doko ro Furni rure o Delerminis rio Model o Expeo red Volue Solu rion 0 Scenario Anolysis o Fo r Solu rion TwoSloge SLP Recourse Model gt Some Clossiool SLP Exomples Closs Hondou r L NTaimO TAMU 2007 DakoTa Furniture The DakoTa FurniTure Company manufacTures desks Tables and chairs The manufacTure of each Type of furniTure requires lumber and Two Types of skilled labor finishing and carpenTry The cosT and amounT of each resource needed To make each Type of furniTure and The corresponding demand are given in i DeTermine how much of each iTem should be produced and The corresponding resource requiremenTs in order for DakoTa To maximize iT profiTs TC 1 Resource RequiremenTs for DakoTa FurniTure Resource Desk Table Chair CosT Lumber bd fl 8 6 i 2 Finishing hrs 4 2 15 4 CarpenTry hrs 2 i 5 05 52 Demand 1 50 i 25 300 Selling Price 8 60 40 10 This excellenT example problem is Taken from Higle 2005 and will be used as an illusTraTive example ThroughouT The course The original version of The problem is given in WinsTon and VenkaTaramanan 2003 L N raimo TAMU 2007 Dakota Furniture How to solve There are a number of ways to solve this problem Example Simple per item profit analysis TC 2 Production Solution for Dakota Item Prod Selling Protit Demand Prod Cost Price Item Desk 4240 6000 1760 150 150 Table 2780 4000 1220 125 125 Chair 1030 1000 060 300 0 TC 3 Resource Solution for Dakota Resource Amount Lumber bd ft 1950 Finishing hrs 850 Carpentry hrs 4875 L Ntdimo TAMU 2007 Dakota Furniture Deterministic Linear Programming Decision Variables ml Number of bd ft of lumber acquired wf Number of labor hours acquired for finishing a C Number of labors hours acquired for carpentry yd Number of desks produced yt Number of Tables produced yo Number of chairs produced LP Formulation MGX 2wl 4wf 52wc ST al wf wc ml 33f we 601d 8961 4yd 2961 yd yd 40 65 2yt 15yt yt yt 1090 90 1590 0590 yo 10 s 0 g 0 g 0 150 125 300 L Ntaimo TAMU 2007 Dakota Furniture Deterministic Linear Programming Use the CPLEX LP Solver to solve LP Solution is same as shown in Tables 2 3 Objective value 4165 ml 1950 a f 850 a C 425 yd 150 yt 125 1020 L N raimo TAMU 2007 DakoTa Furniture Introducing Uncertainty p Even Though The basic daTa can Change The sTruoTure of The problem remains The same gt Eg increase selling price of ohairfrom 10 To Si 1 WhaT happens p WhaT can you do if The daTa changes gt PosTopTimaliTy invesTigaTion L NTaimO TAMU 2007 DakoTa Furniture Introducing Uncertainty p EveryThing else remaining The same increasing The selling price of chairs To Si 1 makes 0 ProducTion of chairs profiTable p WhaT if The price remains aT 10 and demand for chairs changes from 300 To 200 p SensiTiviTy analysis is used To sTudy robusTness of The soluTion To an LP model gt How should you proceed if The soluTion varies when The daTa is changed L Ntaimo TAMU 2007 Dakota Furniture Introducing Uncertainty p Sensitivity analysis offers a false sense of security when there is uncertainty in the problem data gt Stochastic programming models are intended for problems in which data elements are difficult to predict or estimate p In the Dakota problem what elements can be random o Selling prices o resource costs o product demands gt We can reasonably assume that there is no uncertainty in the resource requirements for each of the three products L Ntoimo TAMU 2007 Dakota Furniture Introducing Uncertainty p Assume that the selling prices ond resource costs ore known but demand is rondom gt Also for simplicity ossume that there ore 3 demond outcomes or scenarios depicted in the following scenorio tree L N roimo TAMU 2007 Dakota Furniture Introducing Uncertainty Low pl 2 03 Medium pm 2 04 High pk 03 L N raimo TAMU 2007 Dakota Furniture Introducing Uncertainty p The Dako ra demand scenarios are given in Table 4 TC 4 Resource Solu rion for Dako ra Low Medium High Desks 50 150 250 Tables 20 i 10 250 Chairs 200 225 500 p The collec rion of demand scenarios and The corresponding probabiliiies form a muliivaria re probabiliiy dis rribuiion gt So wha r should we do nexi L Ntoimo TAMU 2007 Dakota Furniture Introducing Uncertainty p Expected Volue Solution gt Scenario Anolysis gt Fot Solution p TwoStoge SLP Recourse Model L NTOimO TAMU 2007 deon Furniture Expected Value Solution p The expeCTed demand is dCTUdIIy The demand we iniTidlly used 0 desks 0350 04150 03250 150 0 Tables 0320 04110 03250 125 0 Chairs 03200 04225 03500 300 gt So The soluTion for The deTerminisTiC case is The some 08 The expeCTed soluTion gt Many deTerminisTic models may dcTudlly represenT The average case L N rdimo TAMU 2007 Dokoio Furniture Scenario Analysis Solution p In This cose we will cred re copies of resource dnd produc rion decisions for edch scendrio s E 1 2 3 where 12 0nd 3 correspond To The scendrios Low Medium dnd High respeciively Decision Variables C 8 8 8 lt 90 00 HO 0 C yii Number of bd ii of lumberocquired under scenario 5 Number of labor hours acquired for finishing underscenorio s Number of Idbors hours acquired for corpeniry under scenario 5 Number of desks produced under scenario 5 Number of Tables produced under scenario 5 Number of choirs produced under scenario 5 Demand Ddid df d3 dig for each scenario 5 is given in Tobie 4 L N raimo TAMU 2007 Dakoia Furniture Scenario Analysis Solution Scenario 5 LP Formulation Max 2xls an 52x 650yfi s r wf 8yfi w 4y w m2 9 33 33f we yd 40 i 6ygS i 2ygS 15yf 10y y 15y 05yg vi 9 L N rdimo TAMU 2007 Dakoia Furniture Scenario Analysis Solution gt Solving The scenario LPs we oblain The resul rs summarized in Table 5 TC 5 Produc rion Solu rion for Individual Demand Scenarios Resource Demand Quan ri ries Expecied Low Medium High Lumber bd fl 1950 520 1860 3500 Finishing Labor 850 240 820 1500 Carpen rry Labor 4875 130 465 875 Desks 150 50 150 250 Tables 125 20 l 10 250 Chairs 0 0 0 0 Pro i S 41 65 l l 24 3982 7450 L NTdimo TAMU 2007 DokoTo Furniture Scenario Analysis Solution p NoTe Tth The bdsic sTrucTure of The LP hos hoT chdhged Some observoTions con be mode 1 All scehdrios indicoTe Tth choirs should hoT be produced 2 The resource dhd producTion quthiTies under The ExpecTed column ore simply The expecTed volues of The resource dhd producTion quthiTies in edch of The individudl scehdrio columns a Recoil SensiTiviTy dhdlysis of The soluTion of The origihdl LP indicoTes Tth The some bdsis will be opTimdl for oil These demdhd scehdrios 0 Thus This siTuoTion should drise gt Remember The scehdrio soluTions ore bdsed on The ossumpTiOh of perfecT informoTion L NTdimo TAMU 2007 DokoTo Furniture Scenario Analysis Solution The Two observoTions offer some consoldTion o The bdsic sTrucTure of The soluTions is The some no choirs produced 0 There is d reldTionship beTween The scendrio dnd expecTed soluTions This mdy seem To suggesT Tth deon use The Low scendrio soluTion This soluTion would hove deon produce 150 desks 0nd 125 choirs dnd purchdse exochy enough resource To dccomplish This Tdsk However There is d 30 chdnce Tth deon will produce more Thdn They will be dble To sell L NTdimo TAMU 2007 DokoTo Furniture Scenario Analysis Solution gt Simildrly if 125 Tdbles ore produced There is d 70 chdhce deon will produce more Thdh They ore dble To sell gt If deon pldhs for The expecTed demdhd dhd The demdhd Turns ouT To be Low There will be 100 desks dhd 105 Tdbles produced Tth will hoT be sold gt FurThermore ihsTedd of The 84165 profiT Tth The model suggesTs They should expecT deon will redlize d heT loss of 6035 occurring wiTh Cl 30 chdhce gt So WHY does This hdppeh L NTaimo TAMU 2007 DakoTa Furniture Scenario Analysis Solution p This is The resuIT of The false sense of securiiy ThaT sensiTiviTy analysis offers p The scenario soluTions are on an individual basis and are besT for one parTicular demand scenario gt IT is necessary To obTain a soluTion ThaT balances The impacT of The various scenarios ThaT may occur 20 L N roimo TAMU 2007 Dakota Furniture Fo r Solution gt In This cose we will creo re copies of cons rroih rs for eoch scenario 5 E 1 2 3 where 12 end 3 correspond To The scehorios Low Medium and High respeciively p Demond Do ro di dg dg for eoch scenario 5 is given in Tobie 4 2i L N roimo TAMU 2007 Dakota Furniture Fa r LP Formulation MGX 2wl 4w 52336 60yd 40yt 1010 51 33 8961 6915 yc S 0 wf 4yd 2915 1590 S 0 wc 211d 15 05 S 0 yd 9t 90 yd 9t 90 yd 9t 90 S S 50 20 200 150 110 225 250 250 500 mlwfwc Z 0ydytyc Z 22 L NTdimo TAMU 2007 Dakota Furniture FoT Solution Solving The LP we odein The soluTion summdrized in Tobie 6 TC 6 Resource and ProducTion FdT SoluTion Resource QuonTiTy Product QuonTiTy Lumber bd Tl 520 Desks 50 Finishing Ldbor 240 Tables 20 CorpenTry Ldbor T30 Choirs 0 ProfiT 8 1124 p The soluTion is driven by The demond scenorio Low Recoil driven by eXTreme evenTs p The soluTion is very conservoTive end is bosed on The demond scenorio Low gt Regordless of whoT scenorio occurs DokoTo mokes mokes decisions os shown in Tobie o 23 d f d d for each scenario w e Q is given in Tobie 4 L N roimo TAMU 2007 Dakota Furniture TwoStage Recourse Solution TwoStage Recourse Formulation MGX 2wl 4w 52336 Efwl 23f we ST mlwfw620 where for each outcome w e 9 fm 33f 330 w Max 6021 40y 102 ST8y j 6yf y S ml 421 221 15 wf 221 16y 052 s we y j S di 92 S d3 y S d231 w w w ydyytyyc 20 25 L N roimo TAMU 2007 Dakota Furniture TwoStage Recourse Solution DEP LP Formulation Max 2wl 431 52xC pw60yi 401 log 5 wEQ st wl8y j6yy 0wea acf 4112 211 151 S 0w e 9 wc 221 16y 06y s 0 w e 9 yzf d fwea y sd wea y sd wea wlwa702079371lfyygj 26 L N roimo TAMU 2007 Solving the DEP we obtoin the solution summorized in Tobie 7 7 Recourse Solution Dakota Furniture TwoStage Recourse Solution Demand Resource Quantities Product Low Medium High Lumber bd ft 1300 Desks 50 80 80 Finishing hrs 540 Tobles 20 HO HO Corpentry hrs 325 Choirs 200 0 0 Expected Profit 8 1730 27 L N roimo TAMU 2007 Dakota Furniture Output Summary for all Models TC 8 Solutions for Dakota Models Resource Scens Quantities Expected Fat Low Med High Recourse Lumber 1950 520 520 1 860 3500 1300 Finishing 850 240 240 820 1500 540 Carpentry 487 5 130 130 465 875 325 Production Scens Quantities Low Med High Desks 150 50 50 150 250 50 80 80 Tables 125 20 20 1 10 250 20 1 10 1 10 Chairs 0 0 0 0 0 200 0 0 Profit 8 4165 1 124 1 124 3982 7450 1 730 28 L NTaimO TAMU 2007 DakoTa Furniture Recourse Solution gt One can make several observaTions from Table 8 0 IT is difficulTy To find a meaningful connecTion beTween The soluTions To The scenario problems and The soluTions To The recourse problem in Terms of resource quanTiTies O In conTrasT There is a connecTion beTween The scenario problem soluTions and The expecTed mean value problem gt Unlike The expecTed value and scenario problems The objecTive values of The recourse problems correspond To expecTed profiT gt The flexibiliTy ThaT The recourse model affords is apparehT from The sTrucTure of The soluTion 0 IT is The only model ThaT suggesTs ThaT chairs should be made when demand is Low 0 The soluTion purchases sufficienT resources To saTisfy demand aT some of The high levels 0 When The demand is Low DakoTa is able To recover much of Their resource expense by producing chairs which is preferable To producing and NOT selling desks and Tables 0 NoTe ThaT when demand is Low some of The lumber and carpenTry labor will go unused buT DakoTa will make The maximum use of The resources ThaT They have purchased 0 In all oTher cases DakoTa will be able To sell all iTems produced gt The soluTions associaTed wiTh The scenario problems are overly opTimisTic They are based on perfecT informaTion 29 L NTaimo TAMU 2007 DakoTa Furniture Recourse Solution In Table 9 we compare The objecTive value associaTed wiTh The expecTed soluTion faT soluTion individual scenario problems To The corresponding expecTed profiTs in The recourse seTTing TCl ble 9 ExpecTed ProfiT AssociaTed with Edch Model Solution Model ProliT Suggested Expected by model ProliT Recourse 1 730 ExpecTed 41 65 1545 FaT 1 124 1 124 Scenario Low 1 124 1 124 Medium 3982 1 702 High 7450 2050 30 L N raimo TAMU 2007 Dakota Furniture References Winston Wayne L and M Venkataramanan 2003 Introduction to Mathematical Programming 4th Edition Duxbury Press Belmont CA Higle JL 2005 Stochastic programming Optimization when uncertainty matters to appear as an INFORMS Tutorial Paper Nov 2005 Also see Higle JL and S W Wallace 2003 Sensitivity analysis and uncertainty in linear programming Interfaces vol 33 no 45360 31 ISEN 689 LargeScale Stochastic Optimization Introduction Dr Lewis Ntaimo Sept 4 2007 L N roimo TAMU 2005 Outline Deoision Moking in on Unoer roin World Appliooiions Decision Sioges Decision Trees DeoisionIVIoking Models Preliminories Probobili ry Spooes Rondom Voriobles Lineor Progromming L N roimo TAMU 2005 DecisionMaking in an Uncertain World gt Since The real world is uncerioin ii becomes impero rive To consider uncer roin ry in decision making Exomple Sources of Uncerioiniy Morke r produc r s rocks relo red o Finonciol relo red Technology relo red O o Compeiiiion relo red Weo rher relo red eg oirline rescheduling Co ros rrophic even rs occiden rs wor 91 1 e rc gt And mony more L N raimo TAMU 2005 Stochastic Programming Applications Example Applica rions Manufac ruring supply chain planning Transpor ra rion eg airline indus rry Telecommunica rions eg ne rwork design Elec rrici ry power genera rion eg power adequacy planning Heal rh care eg pa rien rresource scheduling Agricul rure fores rry eg wildfire emergency response Finance eg por rfolio op rimiza rion gt And many more L N roimo TAMU 2005 Sequential Decision Models Decision Trees 0 Grophicol represen ro rion of The decision process El Decision even r Q Uncer roin ry even r o gt Time progressing from lef r ro righ r egg Rondom even r is o poin r or which informo rion is reveoled provided 0 L N roimo TAMU 2005 Sequential Decision Models Decision Trees o Decision Trees help To keep Track of in rerploy be rween decisions and rondom even rs Decisions rho r follow info con odop r ro ii Decisions iho r precede i r conno r Decision H Observe H Decision gt 39 39 39 p In generol ii is bes r ro deloy The decision as long as possible mos r flexible for odop ring ro info There is no odvon roge ro deloy if no info is on ricipo red L NTaimo TAMU 2005 DecisionMaking Models gt STaTisTical Decision Theory SDT Waldllt50 o DeTermine besT levels of variables ThaT affecT The ouTcome of an experimenT o a e Xw e Q associaTed disTribuTion Fw and reward my w The basic problem is H1683 EwrxwF max rxwdFw i o Problem 1 is The fundamenTal form of sTochasTic programming Underlying assumpTions lead To major differences beTween The fields L NToimo TAMU 2005 DecisionMaking Models Decision Analysis DA Roiffo 1968 o PorTiculor porT of SDT Emphasis is on Acquiring informoTion obouT possible ouTcomes EvoluoTing The uTiIiTy ossooioTed wiTh possible ouTcomes Defining o limiTed seT of possible ouTComes usually in The form of o decision Tree L Nioimo TAMU 2005 DecisionMaking Models Dynomic Programming DP ond Morkov Decision Processes iDP DA eg Bellmon 1957 Ross 1983 o Seoroh for opiimol ooiions io ioke oi discreie poinis in iime o Aciions influenced by rondom ouicomes ond oorry one from some sioie oi some sioge t io onoiher sioie oi sioge t 1 Emphosizes ideniifying finiie or oi leosi low dimensionol sioie ond ociions spoces in ossuming some Morkovion siruoiure L N roimo TAMU 2005 DecisionMaking Models p Op rimol Siochds ric Con rrol 0 Models of ren similor To s rochos ric programming models 0 Problem dimensions ore lower 0 Emphasizes oon rrol rules 0 More res rrioiive cons rroini ossump rions L N raimo TAMU 2005 DecisionMaking Models Siochas ric Programming SP o Basically generaliza rions of deierminis ric ma rhema rical programs in which uncon rrollable da ra are no r known wi rh cer rain ry o No re The cer rain ry assump rion in linear programming LP is viola red o Siochas ric linear programming SLP deals wi rh linear programs wi rh random da ra course focus o Siochas ric mixed inieger programming SMIP deals wi rh mixed in reger programs wi rh random da ra L NTaimo TAMU 2005 DecisionMaking Models SToohasTic Programming SP 0 Key feaTures Many decision variables wiTh many poTenTiaI values DiscreTe Time periods for decisions 9 Use of expeoTaTion and oTher risk measures funoTionaIs for objeCTives Known or parTiaIIy known probabiliTy disTribuTions o The reIaTive imporTanoe of These feaTures oonTrasTs wiTh The oTher models L N raimo TAMU 2005 Preliminaries Some Basic Probabili ry Firs r w is an ou rcome of a random experimen r we will use a To deno re a mul rivaria re random variable vec ror No re The iex rbook by Birge amp Louveaux uses 5 and g 0 Q is The se r of all possible ou rcomes sample space 0 A is collec rion of random ou rcomes even rs of Q L NToimo TAMU 2005 Preliminaries ProbebiliTy Spaces For each A e A There is o probobiliTy measure or disTribuTion P Thc1T Tells The probobiliTy wiTh which A e A occurs 9 0 S PA S 1 9 PG 2 1PD O PA1 U A2 2 HA PA2 if A1 A2 Q o The TripIeT Q A P is called 0 probobiliTy space L N raimo TAMU 2005 Preliminaries Random Variables o A random variable rv a on a probabili ry space 9 A P is a real valued funciion w w e 9 such Thai w Dw g 13 is an even r for all fini re 13 o For The random variable a we define iis cumularive disrribufion by F5 2 P D g as o Discrefe random variables Take on a finiie number of values wk k e K wiih associa red probabiliiies ock 2 135 2 wk wi rh ZkEK ask 1 L N raimo TAMU 2005 Preliminaries Random Variables o Con rinuous rv s are described by a densify func rion fw Probabili ry of w being in an in rerval a b is Pm g a g b f bdd 01pm Fb Fa o Con rrary To The discre re case P as 0 L N raimo TAMU 2005 Preliminaries More 0 Expec red value of a is Discre re case M 2 E5 ZkeK wk wk Con rinuous case M E 10 mmch 10 am 0 Variance of a is Vard EJJ m2 L NTaimo TAMU 2005 Preliminaries p Consider The following linear program LP Min cTa sT A33 2 b Ta 2 7 a Z 0 where o 1 6 ER is The decision variable veoTor o c 6 ER is The oosT veCTor o A e Ele L1 is The consTrainT maTrix o b e Ele is The RHS veoTor o T e 3W2 1 is The Technology maTrix o r e 3W2 is The RHS veoTor 2a 2b 20 2d L NTaimo TAMU 2005 Preliminaries p In LP we deal wiTh The following oonoest FeasibiliTy OpTimaliTy BoTh These concest are Clear In faoT sensiTiviTy analysis in LP deals wiTh These Two oonoest BuT suppose T r oonTains random variables T f Whai should we do L Nioimo TAMU 2005 Preliminaries gt Assump rions o Recrl vcrlue of T r is no r known 9 Uncer rcrin ry if expressed by probobili ry dis rribu rion eg scenorios PTf T rw pww E S This can also be expressed crs PTf T8r8 1985 2 1 where S Q a Probabili ry dis rribu rion known do rcr exper rs e rC L Nioimo TAMU 2005 Preliminaries p DecisionMoking Under Unoer roin ry Using Sioohos rio Lineor Programming or SLP c Find 13 hereondnow wi rhou r knowing The reol volue of T r bu r knowing i rs probobili ry dis rribu rion o Ta 2 r con be in rerpre red os o gool cons rroin r To be specified more precisely 20 L N rdimo TAMU 2005 Preliminaries Approaches p Expeo red Volue Solu rion o Repldoe Ta 2 f wi rh ETx 2 EV Ta 2 where T ZspsTs and F Zspsrs o Advon rdge Simple model de rerminis rio LP Disddvon rdge Risk is no r Token core of since T351 2 r8 for some scenarios only o thi else con you do here Apply sensi rivi ry dndlysis S rill poor model of deoisionmoking under unoer rdin ry See Higle dnd Wolldoe 2003 2i L N rdimo TAMU 2005 Preliminaries Approaches gt Fd r Soluiion 0 Replace Tx 2 fwiih T3513 2 r8s 1 S o AdVdn rdge De rerminis ric LF 0 Disddvon rdge This is Cl conservo rive expensive resiriciive model of ren no fedsible solu rion exis rs Yields overly conservoiive soluiions driven by exireme even rs no md rier how rdre 22 L N raimo TAMU 2005 Preliminaries Approaches gt Scenario Analysis o Solve Min chs Axs Z bTSxS 2 7 x8 for every scenario T8r8 s 1 S o Ge r solu rions 5138 s 1 8 Find an overall solu rion based on The scenario solu rions o Advan rage Each scenario solu rion is a de rerminis rio LP improvemen r over The expeo red value approach very popular approach Disadvan rage How do you find an overall solu rion 23 L N rdimo TAMU 2005 Preliminaries Approaches p Chdnce Probdbilis ric Cons rrdin rs o Repldce Ta 2 f wi rh PTx 2 f 2 a forsome specified relidbili ry level or E 051 o Advon rdge Risk is rdken core of explici rly 1 a is mdximdl dccep rdble risk o Disddvon rdge Difficul r To compu re discre re dis rribu rions mdy ledd To MIP model in generdl possibly nonconvex model 24 L N rdimo TAMU 2005 Preliminaries Approaches p TwoSidge Recourse Model 0 ln rroduce explici rly correc rive oc rions Reploce Tx 2 f wi rh Ta l Wy Z 77 where y 6 ER is The decision vec ror of o secondsioge LP problem 0 The volue of y depends on The reolizo rion of T f o Penolize correc rive oc rions colled recourse oc rions in SLF Minimize io rol expec red cos rs Decision Ob serve Decision Uncertainty y 39 Stage 1 Stage 2 25 L Nioimo TAMU 2005 Preliminaries p TwoSioge SLP wi rh Recourse Model discr disin S Min cTa Zpquys 31 s r A33 2 b TSx i Wys er 20318 ZO51S wi rh q uni r recourse cos rs Objec rive ch expec red recourse cos rs 30 3b 30 3d 26 L N rdimo TAMU 2005 Preliminaries TwoSioge SLP wi rh Recourse Model o Advon roge Risk is Token core of explioi rly expeo red recourse oos rs ldrgesoole LP model a Disodvon roge model moy be Too lorge To solve eg lO independen r rondom voriobles o reolizo rions eooh gt 8 610 m 60000000 SLP dimensions moirix A is m1 gtlt n1 W is 7712 x 712 SLP hos n1 712539 decision voriobles m1 mgs oons rroin rs Lorge soole model gt DECOMPOSITION opproooh To solve 27 L Nioimo TAMU 2005 Preliminaries p A generol rwos roge SLP wi rh recourse model con be wri r ren os follows Min cTa l E5 f13d s r A33 2 b a Z 0 where for ony reolizo rion w of a we hove Min qwTy s r Wwy 2 New Twx y 2 0 40 4b 40 50 5b 5C 28 L Nioimo TAMU 2005 Preliminaries gt Some siondord SP Terminology o Func rion Ef13d Expecied recourse funciion o Moirix Ww Recourse moirix Moirix Ww W fixed Fixed recourse course focus Moirix Ww rondom Rondom recourse Moirix W 1 1 Simple recourse o If fxa lt oo wp1 Va 6 ER Compleie recourse o If fxcD lt oo wp1 Va 6 XX 13 E ERnlAx 2 b Relo rively compleie recourse gt We will iolk more oboui These loier 29 L Nigimo TAMU 2005 Preliminaries gt NGXT o Siochgs ric programming modeling exgmples 30

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