Production System Design and Control
Production System Design and Control MFE 654
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Inventory models with uncertain demand I Motivations I Causes of uncertain demand I Two approaches to model uncertainty I Use of probability distributions I Stochastic inventory models Motivations I Demand forecasts are always wrong only estimate I Uncertain random demand represents realistic situations I Effect of randomness propagates throughout the system In the case of dependent demand Example of dependent demand Sources of uncertainty I Consumer preferences I Trends in the market I The availability and cost of labor and resources I Vendor re supply times I Weather and its ramifications on operations logistics I Financial variables stock prices interest I Demand for products and services Causes of inventory a study at HP Two main approaches I Assume constant demand and then use some factors to account for randomness deterministic models I Explicitly represent randomness in model statistical models Deterministic model 39 D Dds Dran om deterministic component 0 random component I Can be used when variance of Dan is small relative to D The predictable variation is more important than the random variation eg aggregate planning problem structure is too complex to represent randomness eg MRP Deterministic model continued I Drawbacks For many situations random component is too signi cant to be ignored Factors to include randomness are dif cult to stimate I Stochastic inventory models are needed to capture necessary randomness Stochastic models I Demand is represented in terms of a probability distribution I These models are used to determine reorder point order quantity Stochastic inventory models I Newsboy model order quantity I ContinuousReview Q R model order quantity and reorder point Newsboy Model Assumptions I Random demand I a single replenishment I A single product is to be ordered at the beginning of a period and can only be used to satisfy demand during that period Determine the order quantity Newsboy Model I Base model I For discrete demand I Nonzero starting inventory I Interpretation ofthe overage and underage costs I Infinite horizon with full backordering I Infinite horizon with lost sales I with a fixed ordering cost An example Sales of the compmerdournal The critical ratio I The probability of satisfying all the demand during the period if Q units are purchased at the start ofthe period Histogram 7 B a 5 z a E 2 l El El 5 lEI l5 ZEI Ein Answer set I IQ CuCu Co single infinite horizon infinite horizon period w backorder wo backorder cU pS C p pS C cO hC h h Newsboy Model Insights I the appropriate productionorder quantity depends on the distribution of demand the relative costs of overproducing vs underproducing I mean and variance of total cost will increase with variability of demand QR Model Assumptions I Demands occur one at a time I Any demand not filled from stock is backordered I Replenishment lead times are fixed and known I Either a xed setup cost for a replenishment order a constraint on the number of orders per year I Determine the reorder point amp the order antity QR Model Insights I Cycle stock increases as replenishment frequency decreases I Safety stock provides a buffer against stockouts I Increasing the average demand durin a replenishment lead time tends to increase the optimal reorder point I Increasing the variability ofthe demand process tends to increase the optimal reorder point I Increasing the holding cost tends to decrease the optimal replenishment quantity and reorder point Inventory models studied I EOQ model I EPL model I Discount models I Newsboy model I QR model Basic Insights I There is a tradeoff between setups replenishment frequency and inventor r the more setups the less ihvehtory I There is a tradeoff between customer service and inventory 7 the higher levels of safety stock the higher custom er serVice levels I There is a tradeoff between variability and inventory e e higher the variability ofdernarid the more Ii iyel itoi yW must carry to meet the same repiehishrheht frequency and 3 o ABC Analysis I Also called Pareto analysis I Ranking by annual dollar volume of sales I A the top 20 ofthe items account for about 80 of the annual dollarvolume I B the next 30 ofthe items forthe next 15 of the sales I C the remaining 50 forthe last 5 of dollarvolume Inventory Policies Continuous Periodic Review Review A items s S R s S B items s Q R 5 Assignment 4 I Chapter 5 7 I Chapter 5 10 I Chapter 5 13 I Chapter 5 15 Inventory Control o the oldest result of the scientific management efforts 0 inventory plays a key role in the logistical behavior of virtually all manufacturing systems 0 the classical inventory results are central to more modern techniques such as MRP JIT TBC time based competition etc Inventory 125 trillion 1250000000000 in 1st quarter of 1995 USA 20 to 25 of total annual GNP Why inventory o economies of scale 0 uncertainties o transportation 0 technological restrictions 0 quantity discounts Inventories in a manufacturing plant raw materials components subassemblles ormaterlals that are purchased from outslde work in process WIP all unrlnlshed parts orproducts that have been released to a productlon llne nished goods inventory FGI rllshed product that has not been sold a spare parts components used to malntaln or repalrproductlon equlpment Reasons for holding Raw Materials batching quantlty dlscounts llmlted capaclty ofthe plant s purchaslng functlorl economles ofscale ln delwenes randomness schedule problems quallty problems obsolescence changes ln demand changes ln deslgn Reasons for holding WIP queueing alob ls waltlng fora resource a processing a ob ls belng worked on by a resource a waiting for batch alob has to walt for otherlobs to arrwe ln order to form a batch moving alob ls belng transported between resources a waiting to match when components walt for thelrcounterparts to arrlve forarl assembly Reasons for holding FGI a customer responsiveness makertorstoc polle batch production outputwlll sometlme not match customer orders forecast errors production variability seasonality bulltrahead myehtory Reasons for holding Spare Parts o o 0 service purchasing lead times production lead times batch replenishment economies of scale a a a Functional classifications of inventories Cycle inventories to patch processlhg Congestion stocks Due to ltems competmg rorllmlted capaclty To allow roruhcertamty oroemaho and supply Anticipation inventory yahce orah expected peallt m sales Pipeline or WIP inventory Decoupling stocks To permlt the separatloh or declsloh malltmg at olrrereht echelohs locatlohs 0 ll lad The ABC classification SKU stockkeeping unit Ah ltem orstocllt that ls completely specmeo as to ruhctloh style slze color and usually locatloh ClassA The rst 540 percent or the SKUS accouht roraoout 50 percent ormore or the total annual dollarusage Class B More than 50 percent or total SKUS account for most orthe remammg 50 percent orthe total annual dollarusage Class C The relatlvely humerous remammg SKUS that make up only a mmorpart ortotal dollarusage The Economic Order Quantity Model o 0 Ford W Harris 1913 How Many Parts to Make at once example ofa metalworking shop that produced copper connectors setup cost as the sum total ofthe labor and material cost to ready the shop to produce a product Dilemma large lots reduce setup costs small lots reduce inventory The EOQ Model Assumptions 0 Production is instantaneous Delivery is immediate Demand is deterministic Demand is constant over time A production run incurs a constant setup cost Products can be analyzed singly The Notation D Demand rate in units per year v Unit variable cost not counting setup or inventory costs in dollars per unit A Constant setup ordering cost to produce purchase a lot in dollars r the carrying charge in llunit time h is the holding cost Q Lot size in units this is the decision variable TRCQ the total relevant costs per unit time EOQ 0 also referred to as the economic lot size The Key Insights 0 There is a tradeoff between lot size and inventory 0 Holding and setup costs are fairly insensitive to lot size Inventory Systems 0 Demand constant vs variable known v unknown 0 Lead time 0 Review time 0 Excess demand backordered lost Inventory Costs 0 Holding cost 0 Order cost 0 Penalty cost Holding cost 0 Cost of physical space 0 Taxes and insurance 0 Breakage spoilage deterioration obsolescence o Opportunity cost of alternative investment Order cost 0 cost of procuring x items 0 fixed proportional components Penalty cost 0 cost when demand exceeds supply per unit of excess demand Economic Production Lot EPL Model Assumptions 0 No Instantaneous production but a nite constant and deterministic production rate 0 Immediate delivery 0 Deterministic demand 0 Constant demand 0 Known constant setup costs 0 Single product or separable products Quantity Discount Models o All units discount model 0 Incremental discount model 0 Carload discount model All Units Discount Model A solution techni ue D rrhihe the largest realizable EOO value Curhpute th E qurthe iovvestpncerirst and cuhtihue vvith the hexthigherprice Step Wheri the first EDD value is realizable i e Withiri the currectiriterval ompare the vaiue orthe average annuai cost at the iargest realizable E00 and at all ofthe price breakpoints that are greater than the largest realizable EOO The optimai o is the point atWhich the average annuai cost is a minimum Incremental Discount Model A solution technique Determine an algebraic expression forCOcorrespohdihg to each price ihterVal Use that to determine an algebraic gtlt 13 3 a 2 Substitute the expressions derived rorco yo into the derining equation roroo compute the minimum vaiue or 0 corresponding to each prioe intervai separateiv Determine Which minima computed in the previous step are realizable l e raii into the correct interval compare the vaiues or the average annuai costs at the realizable EOO vaiues and pick the lowest Carload Discount Model A carload consists ofM units The supplier charges a constant c per unit up until you have paid for the cost the remaining units in that carload Once the rst carload is full you again pay c per unit until the second carload is full and so forth Gentle Electric Company GEO Situation i purchase at 500 per unit ii guantity discount iii transportation cost Inventory models with uncertain dem and o Motivations o Causes of uncertain demand a Two approaches to model uncertainty I Use of probability distributions o Stochastic inventory models Newsboy problem Motivations 0 Demand forecasts are always wrong only estimate 0 Uncertain random demand represents realistic situations 0 Effect of randomness propagates throughout the system c In the case of dependent demand Example of dependent demand Sources of uncertainty a Consumer preferences Trends in the market I o The availability and cost of labor and resources Vendor re supply times Weather and its rami cations on operations logistics Financial variables stock prices interest rates I Demand for products and services I o o Causes of inventory a study at HP Two main approaches 0 Assume constant demand and then use some factors to account for randomness deterministic models 0 Explicitly represent randomness in model statistical models Deterministic model D Ddet Dran Dd deterministic component 0 random component 0 Can be used when variance of Dan is small relative to D u The predictable variation is more im ortant than random variation eg aggregate planning problem structure is too complex to represent randomness eg RP Deterministic model continued Drawbacks For many Situations random component l5 too Significant to be ignored Factors to include randomness are difficult to estimate Stochastic inventory models are needed to capture necessary randomness Three key issues for stochastic control system 0 How often the inventory status should be determined 0 When a replenishment order should be placed 0 How large the replenishment order should be To establish inventory policies How important is the item Can or should the stock status be reviewed continuously or periodically What form should the inventory policy take What specific cost or service objectives should be se How important 0 A B C classification 0 Chapter 7 B items 0 Chapter 8 A items 0 Chapter 9 C items a AB C Analysis Also called Pareto analysis Ranking by annual dollar volume of sales A the top 20 of the items account for about 80 of the annual dollar volume B the next 30 ofthe items for the next 15 of the sales C the remaining 50 for the last 5 ofdollar volum e Continuous vs Periodic Review 0 Continuous review The stock status is always know Transactions reporting 0 Periodic review The stock status is determined only every R time uni s 0 Continuous review Difficult in coordinating replenishments Difficult in predicting the level of the workload on the staff involved More expensive in terms of reviewing costs and reviewing errors Forthe same level of customer service it requires less safety stock Four types of inventory control systems 0 Orderpoint orderquantity sQ System 0 Orderpoint orderuptolevel sS System 0 Periodicreview orderup to level RS System 0 RsS System 0 Methods of selecting the safety stocks A simpleminded approach Based on minimizing cost Based on customer service Based on aggregate considerations A simpleminded approach 0 Equal time supplies 0 Fixed safety factor o 0 Based on minimizing cost Cost per stockout occasion B1 Fractional charge per unit cost B2 Fractional charge per unit short per unit time Ba Cost per customer line item backordered B4 Based on customer service Cycle service level speci ed probability of no stockout per replenishment cycle P1 Fill rate Speci ed fraction ofdemand to be satis ed directly from shelf P2 Ready rate speci ed ready rate P3 Speci ed average time between stockout occations TBS Based on aggregate considerations Allocation of a given total safety stock among items to minimize the expected total stockout occasions per year ETSOPY Allocation of a given total safety stock among items to minimize the expected total value of shortages per year ETVSPY Stochastic inventory model 0 Newsboy model order quantity Newsboy Model Assumptions Random demand a single replenishment A single product is to be ordered at the beginning of a period and can only be used to satisfy demand during that period Determine the order quantity Newsboy Model 0 Base model 0 For discrete demand 0 Nonzero starting inventory 0 Interpretation of the overage and underage costs 0 Infinite horizon with full backordering 0 Infinite horizon with lost sales 0 with a fixed ordering cost An example Sales ofthe Computer Journal The critical ratio 0 The probability of satisfying all the demand during the period if Q units are purchased at the start of the period Histogram 7 E E 5 a E 2 1 El El 5 in 15 ZEI Eln Answer set 0 IQ CuCu Co infinite horizon infinite horizon Newsboy Model Insights o The appropriate productionorder quantity depends on u the distribution of demand the relative costs of overproducing vs under producing Mean and variance of total cost will increase with variability of demand single period w backorder wo backorder cU pS C p pS C cO hC h h Reading assignments 0 Ch 76 0 Ch 7 7 0 Ch 78 o Ch 79