Agenda. Managing Uncertainty in the Supply Chain. The Economic Order Quantity. Classic inventory theory


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1 Agenda Managing Uncertainty in the Supply Chain TIØ485 Produkjons og nettverksøkonomi Lecture 3 Classic Inventory models Economic Order Quantity (aka Economic Lot Size) The (s,s) Inventory Policy Managing uncertainty Risk Pooling Managing uncertainty with contracts Forecasting Methods Case The Newsboy Problem (The News Vendor Problem) The Economic Order Quantity Classic inventory theory Assumptions: Constant demand rate of D items per day Fixed order quantity Q Fixed cost K every time an order is placed Inventory carrying cost h (holding cost) per unit and day Lead time between placing an order an receipt of goods is zero Initial inventory is zero Planning horizon is infinite 3 4
2 Determining the EOQ The (s,s) Inventory Policy, Introduction Total inventory cost per cycle T Inventory Level htq K + Q Average cost per unit of time KD hq Q + Cost minimizing quantity * KD Q = T h Time This is known as the Economic Order Quantity (EOQ) Basic description of the (s,s) inventory policy: Whenever out inventory level drops below a certain level, say s, we order (or produce) in order to increase the inventory level to S. Such a policy is referred to as an (s,s) or min max policy. s is called the reorder point and S the orderuptolevel 5 6 The (s,s) Inventory Policy, Random Demand The (s,s) Policy with Random Demand Additional Assumptions: Daily demand is random and normally distributed Fixed cost K for placing an order Inventory holding cost h is charged per item per unit time If an order is placed, it arrives after the appropriate lead time All orders that can t be satisfied from stock are lost A service level is defined as the probability that the retailer is not stocking out during lead time Additional information needed to calculate the AVG = Average daily demand STD = Standard deviation of daily demand L x STD = cumulative variance over time L = Replenishment lead time α = Service level (implying that the probability of stocking out is  α) z= Safety factor associated with the service level (can be taken from tables, e. g. Table 3, p. 6 in SKS) 7 8
3 Reorder Point s and Orderuptolevel S Reorder Point s and Orderuptolevel S Inventory Level Average demand during lead time Inventory position S L AVG Safety stock Q z STD L s z is taken from statistical table to ensure that the probability of a stock out during lead time is α Reorder point s Lead time L Time L AVG + z STD L P{demand during lead time L AVG+z STD L} = α 9 Recall from EOQ * K AVG Q = h Orderuptolevel S S = Q+ s Expected level of inventory before receipt after receipt on average Q z STD L Q+ z STD L + z STD L Example 37 (SKS p. 6) I Example 37 (SKS p. 6) II Define the inventory policy for a TV distributor given the following assumptions: Whenever an order is placed the distributor incurs a fixed cost of $45 (independent of order size) The cost of the TV to the distributor is $5 and annual inventory holding cost is about 8% of the product cost Replenishment lead time is weeks Desired service level is 97% Demand data for the last months: Month Sales Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug Average monthly demand is 9.7, standard deviation of monthly demand is Lead time is two weeks, transform monthly demand data into weekly demand data: Average monthly demand Average weekly demand = 4.3 Monthly standard deviation Standard deviation of weekly demand = 4.3 Table 3 (SKS, p. 6) determines the safety factor z for a 97% service level as.88
4 Example 37 (SKS p. 6) III Parameter Value Calculate Average demand during lead time Safety stock Reorder point Average weekly demand L AVG = = 89.6 L AVG + z STD L = = z STD L = = 85.9 Standard deviation of weekly demand 3.8 Example 37 (SKS p. 6) IV Weekly inventory holding cost.8 5 =.87 5 Order quantity Q K AVG Q = = = h.87 Orderuptolevel S = Q+ s= = 854 Average inventory level Q z STD L= = Introducing Variable Lead Times With a normally distributed delivery lead time and AVGL = Average lead time STDL = Standard deviation of lead time Reorder point s s = AVG AVGL + z AVGL STD + AVG STDL Risk pooling From SKS, and additional example Orderuptolevel S S = Q + AVG AVGL + z AVGL STD + AVG STDL 5 6
5 Risk Pooling Risk Pooling Consider these two systems: Warehouse One Supplier Warehouse Two Market One Market Two For the same service level, which system will require more inventory? Why? For the same total inventory level, which system will have better service? Why? What are the factors that affect these answers? Supplier Warehouse Market One Market Two 7 8 Portfolio thinking Example: Demand variance of portfolio of n regions with stochastic demand D n P= Di i= n n Var P= VAR (D i) + COV (D i,d j) i= i= j i If perfect correlation between the n regions (P=nD): VAR( P) = n VAR (D) The variation of total demand is reduced if the regions are negatively correlated. The more positively correlated, the higher the variation. Risk Pooling Example Compare the two systems: two products maintain 97% service level $6 order cost $.7 weekly holding cost $.5 transportation cost per unit in decentralized system, $. in centralized system week lead time The effect of centralized inventories comes from the fact that peaks does not appear simultaneuously in regions which are not perfctly correlated. 9
6 Risk Pooling Example Risk Pooling Example Week Prod A, Market Prod A, Market Prod B, Market Product B, Market Warehouse Product AVG STD CV Market A Market A Market B Market B Risk Pooling Example Warehouse Product AVG STD CV s S Avg. Inven. Market A % Dec. Market A Market B Market B Cent. A % Cent B % Risk Pooling: Important Observations Centralizing inventory control reduces both safety stock and average inventory level for the same service level. This works best for High coefficient of variation, which reduces required safety stock. Negatively correlated demand. Why? What other kinds of risk pooling will we see? 3 4
7 Safety stock Service level Overhead costs Customer lead time Transportation costs Managing uncertainty and the effect of Supply contracts 5 6 Contract design Example Pricing and volume discounts Minimum and maximum purchase quantities Delivery lead times Product or material quality Product return policies Will show: The effect of uncertainty on supply chain cooperation and risk sharing How to use contracts to manage risk When demand is uncertain simple coordination schemes based on single prices are not enough to ensure supply chain optimality From SKS Ch
8 SnowTime Sporting Goods Fashion items have short life cycles, high variety of competitors SnowTime Sporting Goods New designs are completed One production opportunity Based on past sales, knowledge of the industry, and economic conditions, the marketing department has a probabilistic forecast The forecast averages about 3,, but there is a chance that demand will be greater or less than this. Supply Chain Time Lines Jan Jan Jan Design Production Retailing Feb Sep Feb Sep Production 9 3 SnowTime Demand Scenarios SnowTime Costs Demand Scenarios Production cost per unit (C): $8 Selling price per unit (S): $5 Salvage value per unit (V): $ Fixed production cost (F): $, Q is production quantity, D demand Probability 3% 5% % 5% % 5% % Profit = Revenue  Variable Cost  Fixed Cost + Salvage Sales 3 3
9 SnowTime Best Solution What to Make? Find order quantity that maximizes weighted average profit. Question: Will this quantity be less than, equal to, or greater than average demand? Question: Will this quantity be less than, equal to, or greater than average demand? Average demand is 3, Look at marginal cost Vs. marginal profit if extra jacket sold, profit is 58 = 45 if not sold, cost is 8 = 6 So we will make less than average SnowTime Scenarios SnowTime Expected Profit Scenario One: Suppose you make, jackets and demand ends up being 3, jackets. Expected Profit Profit = 5(,)  8(,) , = $44, Scenario Two: Suppose you make, jackets and demand ends up being, jackets. Profit = 5(,)  8(,) , + () = $ 335, Profit $4, $3, $, $, $ 8 6 Order Quantity 35 36
10 SnowTime Expected Profit SnowTime Expected Profit $4, $3, Expected Profit $4, $3, Expected Profit Profit $, Profit $, $, $, $ $ Order Quantity Order Quantity SnowTime: Important Observations Tradeoff between ordering enough to meet demand and ordering too much Several quantities have the same average profit Average profit does not tell the whole story SnowTime Demand Scenarios Demand Scenarios Question: 9 and 6 units lead to about the same average profit, so which do we prefer? Probability 3% 5% % 5% % 5% % Sales 39 4
11 Probability of Outcomes Supply Contracts When we add the manufacturer Manufacturer Fixed Production Cost =$, % Variable Production Cost=$35 Probability 8% 6% 4% % Q=9 Q=6 Wholesale Price =$8 Selling Price=$5 Salvage Value=$ % 33 Cost 5 Manufacturer Manufacturer DC Retail DC No fixed cost for distributor (otherwise equal to previous example) Stores 4 4 Demand Scenarios Distributor Expected Profit Demand Scenarios 5 Expected Profit Probability 3% 5% % 5% % 5% % Order Quantity Sales 43 44
12 Distributor Expected Profit Supply Contracts (cont.) Expected Profit Distributor optimal order quantity is, units Distributor expected profit is $47, Manufacturer profit is $44, Supply Chain Profit is $9, Order Quantity IS there anything that the distributor and manufacturer can do to increase the profit of both? Supply Contracts Retailer Profit (Buy Back=$55) Fixed Production Cost =$, 6, Variable Production Cost=$35 5, Wholesale Price =$8 Manufacturer Manufacturer DC Retail DC Selling Price=$5 Salvage Value=$ Stores Retailer Profit 4, 3,,, Order Quantity
13 Retailer Profit (Buy Back=$55) Manufacturer Profit (Buy Back=$55) Retailer Profit 6, 5, 4, 3,,, 6 7 $53,8 8 9 Order Quantity Manufacturer Profit 6, 5, 4, 3,,, Production Quantity Manufacturer Profit (Buy Back=$55) Supply Contracts Manufacturer Profit 6, 5, 4, 3,,, 6 7 $47,9 8 9 Production Quantity Fixed Production Cost =$, Variable Production Cost=$35 Wholesale Price =$8 Manufacturer Manufacturer DC Retail DC Selling Price=$5 Salvage Value=$ Stores 5 5
14 Retailer Profit (Wholesale Price $7, RS 5%) 6, 5, 4, 3,,, Retailer Profit Order Quantity 53 Retailer Profit (Wholesale Price $7, RS 5%) 6, 5, 4, 3,,, $54,35 Retailer Profit Manufacturer Profit (Wholesale Price $7, RS 5%) 7, 6, 5, 4, 3,,, 55 Order Quantity 54 Manufacturer Profit Production Quantity Manufacturer Profit (Wholesale Price $7, RS 5%) 7, 6, 5, 4, 3,,, $48,375 Manufacturer Profit Production Quantity 56
15 Supply Contracts Supply Contracts Strategy Retailer Manufacturer Total Sequential Optimization 47,7 44, 9,7 Buyback 53,8 47,9 985,7 Revenue Sharing 54,35 48, ,7 Fixed Production Cost =$, Variable Production Cost=$35 Wholesale Price =$8 Selling Price=$5 Salvage Value=$ Manufacturer Manufacturer DC Retail DC Stores Supply Chain Profit Supply Chain Profit Supply Chain Profit,,,, 8, 6, 4,, Production Quantity Supply Chain Profit,,,, 8, 6, 4,, 6 7 $,4, Production Quantity
16 Supply Contracts Supply Contracts: Key Insights Strategy Retailer Manufacturer Total Sequential Optimization 47,7 44, 9,7 Buyback 53,8 47,9 985,7 Revenue Sharing 54,35 48, ,7 Global Optimization,4,5 Effective supply contracts allow supply chain partners to replace sequential optimization by global optimization Buy Back and Revenue Sharing contracts achieve this objective through risk sharing 6 6 Other Contracts Quantity Flexibility Contracts Supplier provides full refund for returned items as long as the number of returns is no larger than a certain quantity Sales Rebate Contracts Supplier provides direct incentive for the retailer to increase sales by means of a rebate paid by the supplier for any item sold above a certain quantity Forecasting (Hillier & Lieberman Ch, SKS Ch 3) 63 64
17 Forecasting Forecasting Methods Basic Principles of Forecasting: The forecast is always wrong The longer the forecast horizon, the worse the forecast Aggregate forecasts are more accurate Forecasting methods can be split into four general categories: Judgment methods, Market research methods, Causal methods, and Timeseries methods Judgment Methods Market Research Methods Panels of experts A group of experts is assembled in order to agree upon a forecast. It is assumed that sharing information and communicating allows for reaching a consensus. Delphi method A group of experts makes independent forecasts. The results of all forecasts is presented to the experts and they are asked to refine their forecasts based on the now available knowledge. The process is repeated a specified number of times or until the expert s forecast are within a certain margin. Market testing Groups of potential customers are assembled and tested for their response to products. Their response is extrapolated to the whole market and used for estimating demand. When new models are launched in the automotive industry, potential customers are assembled to check their response to the planned design long before production starts. Market survey Customer response is estimated by gathering data from various potential customer groups. Data is usually collected through interviews, telephonebased surveys, written surveys, etc
18 Causal Methods TimeSeries Methods Causal methods try to generate forecast based on data other than the data being predicted. Using a causal forecasting method, you may try to forecast sales for the next three months based on inflation GNP advertising/marketing activities weather etc. Timeseries based methods use a variety of past data to estimate future data. Examples of timeseries methods: LastValue forecasting Averaging forecast Moving average Exponential smoothing Methods for data with trends Methods for seasonal data etc Eample of time series LastValue forecasting Volume Demand F = t+ xt Variance is large because of small sample size ( data point) Worth considering if Constantlevel is not likely World is changing so fast that most of the previous observations are irrelevant Often called the naive method (by statisticians) May be the only relevant value (under rapid changes) Month 7 7
19 Averaging Forecasting t+ t xi = t i= Excellent if the process is stable Normally limited to young processes F Moving Average t+ Very simple forecasting method F t xi = n i= t n+ Calculating the average of previous demand over a given number of time periods Every past demand point is weighted equally Determining the number of time periods n to calculate the average demand is very important Only recent history and multiple observations Exponential Smoothing F = α x + α F ( ) ( ) t+ t t F = αx + α α x t+ t t i α smoothing constant (between and i ) Forecast is a weighted average of the previous forecast and the last demand More recent value receive more weight than past values Similar to the Moving Average Forecast, but tracks changes in the process faster Lags behind a continuous trend Choice of the smoothing constant α (weight of the last demand) is important i 75 76
20 Methods for Data with Trends Example Regression Analysis, tries to fit a straight line into data points. Thus, it s identifying linear trends. Holt s Method, combines the concepts of exponential smoothing with the ability to capture linear trends BoxJenkins BoxJenkins method BoxJenkins is a methodology for identifying, estimating, and forecasting ARIMA models Requires a large amount of data (historical observations of a variable) Alternative name: ARIMA method Iterative process 79 8
21 ARIMA models ARIMA components Describes a stochastic process or a model of one Explains the variable Y based only on the history of Y No other explanatory variables Not derived from economic theory Stands for "autoregressive integrated moving average" Made up of sums of autoregressive and movingaverage components An ARIMA process is stationary mean, variance and autocorrelation structure do not change over time 8 AR (Autoregressive) Process that can be described by a weighted sum of its previous values and a white noise error AR(): AR(p): t p t p t p= MA (Moving Average) Process that can be described by a constant plus a moving average of the current and past error terms MA(): MA(q): Y Y = αy + θ t t t n = α Y + θ Y = µ + βθ t Y µ βθ t q t q q= n = + t 8 ARIMA components () White noise error ARMA (p, q) May be nonstationary (integrated) Non constant mean, variance or covariance The ddifferences of a time series that are integrated of order d is stationary  differences of Y: Y t Y t ARIMA (p, d, q) p AR terms, integrated of order d and q MA terms Uncorrelated random error term with zero mean and constant variance Often assumed to be normally distributed 83 84
22 The BoxJenkins Methodology. Parameter estimation. Identification of the model (Choosing p, d and q). Parameter estimation of the chosen model Estimate the value of the p AR and the q MA parameters Simple problems: use least squares method Otherwise: nonlinear (in parameter) estimation methods Done by several statistical packages (SPSS, MiniTab, S Plus, EasyReg,..) 3. Diagnostic checking (Are the residuals white noise?) If no go to step If yes go to step 4 4. Forecasting Diagnostic checking 4. Forecasting Test if the estimated model fits the historical data In particular: test if the residuals are white noise If not the model is not appropriate and a new one must be estimated If they are the model may be good and used for forecasting Use the model to forecast new values Can be used as a test of the goodness of the model Outofsample tests Cross validation 87 88
23 Conclusions Requires no economic theory Can not make use of any economic theory + Can not misuse economic theory/ make false assumptions Usually provides accurate forecasts Requires a large amount of data Only needs data for one variable Regretion analysys Assume that a company s demand fucntion ca be written on the form Y = A+ Bx + Bx + B3I + B4Pr Y er dependent variable x, x, I, Pr Are examples on independent variables. We wish to estimate Based on historical data AB,, B, B, B Population regression curve Sales Advertising Line shows the value of E(Y X) Y = A+ B x The line is fitted so that squared deviation is minimized 9 9
24 Explained variance and the constant of determination Variance in dependent variable : n ( Y Y) i= and i Y Y = ( Y Yˆ) + ( Yˆ Y) i i i i Explained variance : n i= ( Yˆ Y) i i= n Explained variance and the constant of determination Variation not explained by regression R = Total variation R = n i= ( Y Yˆ ) i i ( Y Y) i i Unexplained variance: n i= ( Y Yˆ ) i i Methods for Seasonal Data Seasonal data has to be accounted for.. Calculate seasonal factor average for the period Seasonal factor = overall average. Calculate seasonally adjusted values actual value Seasonal adjusted value = seasonal factor 3. Select timeseries forecasting method 4. Apply this method to the seasonally adjusted data 5. Multiply this forecast with the seasonal factor in order to get the next actual value 95 Example on Forecasting Seasonal Demand (HL p. 9) I CCW's Average Daily Call Volume Year Quarter Call Volume
25 Example on Forecasting Seasonal Demand (HL p. 9) II Example on Forecasting Seasonal Demand (HL p. 9) III Calculating seasonal factors Quarter 3 4 ThreeYear Average Total=37 37 Average= = Seasonal Factor = = = = Year Quarter Seasonal Factor Actual Call Volume Seasonally Adjusted Call Volume The Newsboy Problem Short problem description: Stochastic inventory models case: The newsboy problem You are selling newspapers. Each morning you go and buy the amount of newspapers you think you are going to sell in the course of the day. Unfortunately, you don t know exactly how many newspapers you are going to sell, your demand is uncertain. In addition, you have no chance of buying additional newspapers if you realize you face a higher demand than what you anticipated. So all demand you cannot satisfy from stock is lost. In the other case you have bought too many newspapers you will not be able to sell them later, because no one will be interested in yesterday s news. (But you can sell the papers with a loss as recycling material.) How many newspapers do you buy every morning? 99
26 The Newsboy Problem, Formulation by Rudi and Pyke We change notation here: W  cost of buying a newspaper R  selling price of the newspaper (R > W) S  return price (salvage value) of a newspaper (S < W) Q  number of newspapers bought in the morning D  stochastic demand In addition, we will need the following definition: y if y y + = otherwise The Newsboy Problem, Reformulation by Rudi and Pyke Define expected profit as function of Q + Π r ( Q) = E Rmin( D, Q) + S( Q D) WQ With min( D, Q) = D ( D Q ) + and Q= D ( D Q) + ( Q D) + + we reformulate Π r (Q) + + Π r ( Q) = ( R W) ED E ( R W)( D Q) + ( W S)( Q D) and call G r (Q) the cost of uncertainty Gr ( Q) The Newsboy Problem, Reformulation by Rudi and Pyke The first term ( R W)( D Q ) + represents expected number of units short times the opportunity cost per unit short. Thus, define the unit underage cost C u as: C = R W The second term ( W S)( Q D ) + represents expected number of left over units times the opportunity cost per left over unit. Thus, define the unit overage cost C o as: C = W S Rewrite G r (Q) as u o + + Gr( Q) = E Cu( D Q) + Co( Q D) 3 The Newsboy Problem, Reformulation by Rudi and Pyke In order to determine the order quantity that results in the highest expected profit, derive G r (Q) and find Q such that the derivative equals zero. + + Gr( Q) = E Cu( D Q) + Co( Q D) Derivative of the first term of G r (Q): C Pr( D> Q) u Derivative of the second term of G r (Q): C Pr( Q> D) o The derivative of G r (Q): G ( Q) = C Pr( D> Q) + C Pr( Q> D) r u o 4
27 The Newsboy Problem, Reformulation by Rudi and Pyke With Pr(D>Q)= Pr(D<Q), we can rewrite G r (Q) as [ ] G ( Q) = C Pr( D< Q) + C Pr( D< Q) r u o Equating this derivative to zero, we get the following Newsboy optimality condition Pr( D Cu < Q ) = C + C u This allows to find the optimal order quantity Q* o Introducing the Concept of Real Options into the Newsboy Problem The Newsboy problem is now extended by the concept of Real Options. It becomes more clear if you do not think of newspapers now, but of this year s favorite Christmas present, this summer s most popular fashion item, etc. Ahead of the selling season, the retailer buys q call options at unit cost c. Each of these options gives the retailer the right to buy one unit of the finished product at exercise price x after observing customer demand. The idea is for the manufacturer to induce the retailer to buy more finished products thus being able to capture more sales by sharing some of the risk of demand uncertainty with the retailer. 5 6 Introducing the Concept of Real Options into the Newsboy Problem Expected profit π = * r ( q ) ( R x)min( D, q) cq Reformulate π r (q) according to the regular Newsboy problem + + π r ( q) = ( R x c) ED E ( R x c)( D q) + c( q D) gr ( q) Express expected opportunity cost g r (q) with overage and underage cost + + gr( q) = E k u( D q) + k o( q D) What can you say about the manufacturers profit Without real options? With real options? Optimality Condition ku Pr( D< q) = k + k u o 7 8
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