Dell’s pick-to-light: partially assembled PC rolls to operator. Behind are a series of drawers containing components. A light on a drawer indicates that a component from the drawer should be installed. Once removes and shuts drawer, light goes out, & if another component is to be installed next, a light will go on. Once no more lights, PC is ready for next station. Drawers are replenished from the back when inventory gets below a certain point. Result is fast assembly flow (e.g., assembly, test, and box time less than 2 hours).
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Managing the sell side of a business Supplier Supply-Demand Management Customer Relationship "Make, Move, Store" Relationship Management Management "Buy" Plant "Sell" Plant Warehouse CustomersSuppliers Plant 4-2
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Key questions1. What is the scope of demand management?2. What does order processing involve; why is it an important area for management attention?3. What is customer profit potential, & how is it relevant for influencing demand?4. What are 5 alternatives for improving forecast accuracy, what do they mean, & how can they be applied?5. How do the tactics of part standardization & postponement of form or place help improve forecast accuracy?6. What is the difference between long term & short term forecasting?7. What are 4 long term forecasting methods; what are the risks of salesperson/customer input?8. What are the components of demand, & which component is not forecasted?9. How do the moving average, Winters, & focus forecasting methods work?10. What is the role of the number of periods in the moving average method, & the smoothing parameters in the Winters method?11. What is the purpose of filtering, & why is it important for computer-based forecasting?12. What do the following principles of nature mean & how are they relevant for demand management? (1) law of large numbers, (2) trumpet of doom, (3) recency effect, (4) hockey stick effect, (5) Pareto phenomenon13. What are the managerial insights from the chapter? 4-3
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting 4-4
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Scope of demand management• So what is demand management? Concerned with processing, influencing, and anticipating demand• We’ll begin with processing demand or, in more common terms, order processing or order fulfillment 4-5
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Processing Demand Order processing• Order processing is usually viewed to span order booking to order shipment• Example steps? Customer validation, order entry, credit checking, pricing, design changes, availability checks, delivery time estimation, notification of shipment, notification of delays 4-6
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Processing Demand CUSTOMER ORDER ENTRY AND CHECKING ER Customer Validation P Credit Control Operations… ORDERRETURNS INTERRUPTION ORDER PICKING AND ASSEMBLY CUSTOMER SERVICE SHIPPING INVOICING 4-7
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Processing Demand Characteristics• Can be a complex & time consuming process dealing largely with information flow Susceptible to ad hoc modifications over time in response to problems (e.g., extra credit check added due to expensive nonpaying customer a few years ago)• A major customer contact point with organization → Can significantly impact customer perceptions• IT advances & high customer impact 4-8
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Processing Demand Example 1Benetton• Electronic loop linking sales agent, factory, & warehouse• If not available, measurements transferred to knitting machine for production• Benetton uses a single warehouse Staffed by 8 people & about 230,000 pieces shipped daily 4-9
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Processing Demand Example 2K-Mart and MasterLock• Policy for mistake in shipment or invoice Strike 1: $10,000, Strike 2: $50,000, Strike 3: lose business 4-10
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Processing Demand Example 3 – customer tools• Amazon online order tracking 4-11
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Processing Demand Example 4 – customer tools• UPS online order tracking 4-12
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting 4-14
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Influencing Demand Measure customer profit potential A simple idea• Some customers are more profitable than others• Advancing technologies → more practical to estimate profit potential of individual customers• Can guide efforts/investments for customer retention & acquisition . . . investments to influence demand• E.g., Electronics manufacturer: reviews historical customer profit before sending service contract renewal Wireless phone firm: churn scores & lifetime value estimates influence # of customer contacts & attractiveness of offerings 4-15
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting 4-16
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Forecasting Alternatives Motivating example 1SunbeamImproved forecasting led to 45% reduction ininventory Included estimates from top 200 customers 4-17
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Forecasting Alternatives Motivating example 2AppleA history of problems forecasting demandMany components sourced from 1 supplier -accurate forecasts are criticalOver $1 billion in unfilled orders during thecrucial holiday season. The CEO (Spindler)ousted a few months later 4-18
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Forecasting Alternatives Motivating example 3IBMBadly misjudged demand in PC business in 1996– went from being profitable in 1995 to a $200million loss through 1st half of 1996 4-19
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Forecasting Alternatives Motivating example 4Christmas 1999 & e-commerce takesoffLarge unanticipated increase in Internet orders –didn’t ship on time E.g., Many Toys ‘R Us Christmas orders not delivered until March – “I will never buy online again” 4-20
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Forecasting Alternatives Improvement alternatives• Change the forecasting method Collect more or different data Analyze the information differently • E.g., involve more people, new forecasting software, spend more time manually reviewing, focus groups etc.• Change operations or operating policies Introduce early warning mechanisms Take advantage of the law of large numbers Reduce information delays & leadtimes (trumpet of doom) 4-21
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Forecasting Alternatives Early warning• Change policies so that some (or more) customers provide earlier commitment of future demand, e.g., Early bird program for builder markets – discount for 60-day advance order Invite large buyers to Aspen in February to view next year’s skiwear line, & encourage orders• “Commitment” ≠ asking customers how much they are likely to buy next quarter 4-22
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Forecasting Alternatives Law of large numbers Principle of Nature• As volume increases, relative variability decreases Postponement in form or place, e.g., • Dell – configure your own PC • From full product line at 12 regional DCs to full product line at a single super DC, with 10% of product line stocked at 11 regional DCs (i.e., fast movers that account for 70% of sales) Part standardization, e.g., • Arby’s sandwich wrappers; plastic lids with push down drink indicator • Intel Pentium processors all the same size - 2.8 GHz tests out below 2.8 spec can be sold as a 2.66 GHz chip (“down- 4-23
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Forecasting Alternatives Trumpet of doom Principle of Nature F o re c a s t E rro r R a n g e o ve r T im e• As forecast horizon P e rc e n tag e increases, accuracy F o rec ast 0 E rro r decreases, e.g., 0 T im e U n til F o re c as t E ve n t Reduce production & delivery leadtimes • Dell pick-to-light system for assembly Reduce information delays • EDI transmission of daily consumer demand up through multiple levels in the supply chain 4-24
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Forecasting Alternatives Reduce demand volatility 2 Principles of Nature• Beware of product proliferation Pareto analysis – separating the important few from the trivial many Periodic length of line analysis to critically assess whether to continually offer “slow movers” Principle of Nature: Pareto phenomenon – the lion’s share of an aggregate measure is determined by relatively few factors • E.g., “the 80-20 rule” – 80% of demand is due to 20% of product line• Beware of perverse cycle of promotions – customers wait for sale before buying, thereby forcing a sale A step further – dynamic pricing to stabilize demand & align with supply• Reduce the hockey stick effect… 4-25
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Forecasting Alternatives Hockey stick effect Principle of Nature• Volume tends to pick up towards the end of a reporting period . . . why?• Look for ways to lessen the effect – contributes to demand volatility, inefficiency, poor service Jan Feb 4-26
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Forecasting Alternatives Channel stuffing One contributor to the hockey stick effectLots of sales booked near the end of a quarter,then sales drop off at the start of the nextquarterE.g., A large brewer offered a vacation to the salesperson in each region who sold the most beer to stores over a 3 month period One winner was able to convince a few stores to free up backroom space and fill it entirely with beer 4-27
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Forecasting Alternatives Improvement alternatives• We’re about to focus on methods for predicting short pork bellies demand• But, important to remember . . . many creative ways to improve forecast accuracy that have nothing to do with method – E.g., early warning incentives, law of large numbers, trumpet of doom, reduce demand volatility 4-28
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting 4-29
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Long Term Forecasting Characteristics of long term forecasts• Single or multi-year horizon• Monthly or annual time bucket• Aggregate units Input to “long term” decisions• Accuracy generally more important than short term forecasts . . . why?• Tend to use expensive & time consuming methods . . . due to the preceding point & due to a PON . . . which is? 4-30
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Long Term Forecasting Recency effect Principle of NatureHumans tend to overreact to (or be overlyinfluenced by) recent eventsE.g., Hughes Electronics Corp. developed an artificial intelligence based financial trading system. The developers did this by encoding the wisdom of Christine Downton, a successful portfolio manager. One motivation for creating the system is that it is immune to the recency effect, i.e., humans tend to get overly fixated on the most recent information. 4-31
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Long Term Forecasting Some alternative methods• Judgment• Salesperson & customer input Great information source, but beware of bias potential & recency effect = humans tend to be overly influenced by recent events• Outside services• Causal methods . . . examples? 4-32
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting Characteristics Components of demand Moving average Winters method Focus forecasting Filtering 4-33
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Short Term Forecasting Long term/short term characteristicsLong term forecasts Short term forecasts Single or multi-year horizon Weekly or monthly horizon Monthly or annual time bucket Daily & weekly time bucket Aggregate units (e.g., product/ Detailed units (e.g., SKU) service categories) Input to “short term” decisions Input to “long term” decisions Inexpensive & quick methods Expensive & time consuming methods • Accuracy importance • Accuracy importance • Trumpet of doom • Trumpet of doomCould argue using 2 different principles of nature that it’s [easier?/harder?] to beaccurate with short term forecasting than with long term forecasting 4-34
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Definition of the Forecasting Process• The Art and Science of Predicting Future Events Forecasting vs. Predicting Based on Past Data Economic vs. Demand Forecasting 4-35
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Elements of Demand Forecasting• Dynamic in Nature• Consider Uncertainty (Stochastic)• Rely on Information contained in Past Data• Applied to various time horizons short term medium term forecasts long term forecasts 4-36
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Steps in the Forecasting Process• Determine the Use of the Forecast• Select the Items to be Forecasted• Determine a Suitable Time Horizon• Select an appropriate Set of Forecasting Models• Gather Relevant Data• Conduct the Analysis• Validate the Model - Assess its Accuracy• Make the Forecast• Implement the Results 4-37
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Independent Demand: What a firm can do to manage it?• Can take an active role to influence demand FORECASTING• Can take a passive role and simply respond to demand 4-38
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Types of Forecasts• Qualitative (Judgmental)• Quantitative Time Series Analysis Causal Relationships Simulation 4-39
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Delphi Method1. Choose the experts to participate representing a variety of knowledgeable people in different areas2. Through a questionnaire (or E-mail), obtain forecasts (and any premises or qualifications for the forecasts) from all participants3. Summarize the results and redistribute them to the participants along with appropriate new questions4. Summarize again, refining forecasts and conditions, and again develop new questions5. Repeat Step 4 as necessary and distribute the final results to all participants 4-41
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Quantitative Forecasting Models• Both Pattern Based and Correlational Models rest on the assumption that the relationships of the past will continue into the Future• Both can Mathematically Characterize the Probabilistic Nature of the Forecast• Both Use Information from Relevant Time Frames 4-42
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting Characteristics Components of demand Moving average Winters method Focus forecasting Filtering 4-43
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Components of Demand• Average demand for a period of time• Trend• Seasonal element• Cyclical elements• Random variation• Autocorrelation 4-44
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Pattern Based Analyses• Definition Identifying an underlying pattern in historical data, describe it in mathematical terms, and then extrapolate it into the future• Uses a “Time Series” of Past Data 4-45
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Time Series Variation• Time Series of Demand Data Typically Contain Four Components of Variation About the Mean or Average• Pattern Based Forecasting Needs to Mathematically Characterize Each of these 4-46
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Finding Components of Demand Seasonal variation Seasonal variation x x x Linear x x Linear x x x x Trend TrendSales x x x x x xx x xx x x x x Average x x x x x x x Average x x x xxx x x x x xxxx x x x 1 2 3 4 Year 4-47
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Time Series Analysis• Time series forecasting models try to predict the future based on past data• You can pick models based on: 1. Time horizon to forecast 2. Data availability 3. Accuracy required 4. Size of forecasting budget 5. Availability of qualified personnel 4-48
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Simple Moving Average Formula• The simple moving average model assumes an average is a good estimator of future behavior• The formula for the simple moving average is: A t-1 + A t-2 + A t-3 +...+A t- n Ft = n Ft = Forecast for the coming period n = Number of periods to be averaged A t-1 = Actual occurrence in the past period for up to “n” periods 4-49
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Simple Moving Average Problem (1) A t-1 + A t-2 + A t-3 +...+A t- n Ft =Week Demand n 1 650 Question: What are the 3- Question: What are the 3- 2 678 week and 6-week moving week and 6-week moving 3 720 average forecasts for average forecasts for 4 785 demand? demand? 5 859 Assume you only have 3 Assume you only have 3 6 920 7 850 weeks and 6 weeks of weeks and 6 weeks of 8 758 actual demand data for the actual demand data for the 9 892 respective forecasts respective forecasts 10 920 11 789 12 844 4-50
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Plotting the moving averages and comparing Plotting the moving averages and comparing them shows how the lines smooth out to reveal them shows how the lines smooth out to reveal the overall upward trend in this example the overall upward trend in this example 1000 900 Demand 800Demand 3-Week 700 6-Week 600 500 Note how the Note how the 1 2 3 4 5 6 7 8 9 10 11 12 3-Week is 3-Week is Week smoother than smoother than the Demand, the Demand, and 6-Week is and 6-Week is even smoother even smoother 4-52
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Simple Moving Average Problem (2) Data Question: What is the 3 Question: What is the 3 week moving average week moving average forecast for this data? forecast for this data?Week Demand 1 820 Assume you only have Assume you only have 3 weeks and 5 weeks 3 weeks and 5 weeks 2 775 of actual demand of actual demand 3 680 data for the data for the 4 655 respective forecasts respective forecasts 5 620 6 600 7 575 4-53
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Weighted Moving Average FormulaWhile the moving average formula implies an equalWhile the moving average formula implies an equalweight being placed on each value that is beingweight being placed on each value that is beingaveraged, the weighted moving average permits anaveraged, the weighted moving average permits anunequal weighting on prior time periodsunequal weighting on prior time periodsThe formula for the moving average is:The formula for the moving average is:Ft = w1A t-1 + w 2 A t-2 + w 3A t-3 +...+w n A t-n nwt = weight given to time period “t”wt = weight given to time period “t”occurrence (weights must add to one) ∑w i =1occurrence (weights must add to one) i=1 4-55
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Weighted Moving Average Problem (1) DataQuestion: Given the weekly demand and weights, what is Question: Given the weekly demand and weights, what isthe forecast for the 4th period or Week 4? the forecast for the 4th period or Week 4? Week Demand Weights: 1 650 2 678 t-1 .5 3 720 t-2 .3 4 t-3 .2 Note that the weights place more emphasis on the Note that the weights place more emphasis on the most recent data, that is time period “t-1” most recent data, that is time period “t-1” 4-56
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Weighted Moving Average Problem (2) DataQuestion: Given the weekly demand information andQuestion: Given the weekly demand information andweights, what is the weighted moving average forecastweights, what is the weighted moving average forecastof the 5th period or week?of the 5th period or week? Week Demand Weights: 1 820 t-1 .7 2 775 t-2 .2 3 680 t-3 .1 4 655 4-58
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Short Term Forecasting – Moving Average and Weighted Moving Average Some pros/cons1. Simple (+)2. Designated weights of history (-)3. History cut-off beyond m periods (-) 4-60
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Exponential Smoothing Model Ftt = Ft-1 + α(At-1 - Ft-1) F = Ft-1 + α(At-1 - Ft-1)Where :Ft = Forcast value for the coming t time periodFt - 1 = Forecast value in 1 past time periodAt - 1 = Actual occurance in the past t time periodα = Alpha smoothing constant• Premise: The most recent observations might have the highest predictive value• Therefore, we should give more weight to the more recent time periods when forecasting 4-61
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Exponential Smoothing Problem (1) Data Question: Given the Question: Given theWeek Demand weekly demand data, weekly demand data, 1 820 what are the what are the exponential smoothing exponential smoothing 2 775 forecasts for periods 2- forecasts for periods 2- 3 680 10 using α=0.10 and 10 using α=0.10 and 4 655 α=0.60? α=0.60? Assume F1=D11 Assume F1=D 5 750 6 802 7 798 8 689 9 775 10 4-62
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Answer: The respective alphas columns denote the forecastvalues. Note that you can only forecast one time period intothe future. Week Demand 0.1 0.6 1 820 820.00 820.00 2 775 820.00 820.00 3 680 815.50 793.00 4 655 801.95 725.20 5 750 787.26 683.08 6 802 783.53 723.23 7 798 785.38 770.49 8 689 786.64 787.00 9 775 776.88 728.20 10 776.69 756.28 4-63
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Exponential Smoothing Problem (1) PlottingNote how that the smaller alpha results in a smoother line Note how that the smaller alpha results in a smoother linein this example in this example 900 800 Demand Demand 700 0.1 600 0.6 500 1 2 3 4 5 6 7 8 9 10 Week 4-64
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Exponential Smoothing Problem (2) Data Question: What are Question: What are Week Demand the exponential the exponential 1 820 smoothing forecasts smoothing forecasts 2 775 for periods 2-5 using for periods 2-5 using 3 680 a =0.5? a =0.5? 4 655 5 Assume F11=D11 Assume F =D 4-65
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Seasonal Adjustments• Applied to Moving Averages and Time Series Regression• First, Calculate a Seasonal Index (SI) Factor for Each Relevant Time Period (day, week, month, quarter)• Each Seasonal Period’s SI is Calculated by Averaging the Ratio of its Actual Demand to the Forecast Demand for all Corresponding Periods 4-67
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Seasonal Adjustments• Forecast for Future Periods is Calculated by Multiplying the Unadjusted Moving Average or Time Series Forecast for a given Period by the Corresponding Seasonal Index for that Period• i.e. if the SMA forecast for the month of March is 27 and the SI for March is 1.125, then • Emar = 27*1.125 = 30.375 4-68
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Seasonal Adjustment Example Seasonal Adjustments Sales Demand Monthly Overall SI AdjustedMonth 1993 1994 Seasonal Index Average Average Forecast Jan 80 100 90.00 94.00 0.96 86.17 Feb 75 85 80.00 94.00 0.85 68.09 Mar 80 90 85.00 94.00 0.90 76.86 Apr 90 110 100.00 94.00 1.06 106.38 May 115 131 123.00 94.00 1.31 160.95 Jun 110 120 115.00 94.00 1.22 140.69 Jul 100 110 105.00 94.00 1.12 117.29 Aug 90 110 100.00 94.00 1.06 106.38 Sep 85 95 90.00 94.00 0.96 86.17 Oct 75 85 80.00 94.00 0.85 68.09 Nov 75 85 80.00 94.00 0.85 68.09 Dec 80 80 80.00 94.00 0.85 68.09Average 87.92 100.08 Expected Demand for 1995 = 1153.23 4-69
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Seasonal Adjustments Example Graph Seasonal Adjusted Forecasting 1993 1994170 SI Adjusted Forecast150 Overall Average130110907050 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 4-70
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Evaluating Forecast Accuracy• Use of Residuals Analyses Residuals are the Difference Between the Forecast and the Actual Demand for a Given Period• Assessed by Several Measures Mean Absolute Deviation - MAD Mean Squared Error - MSE Tracking Signal 4-71
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The MAD Statistic to Determine Forecasting Error n 1 MAD ≈ 0.8 standard deviation ∑A t=1 t - Ft 1 standard deviation ≈ 1.25 MADMAD = n • The ideal MAD is zero which would mean there is no forecasting error • The larger the MAD, the less the accurate the resulting model 4-72
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MAD Problem DataQuestion: What is the MAD value given Question: What is the MAD value giventhe forecast values in the table below? the forecast values in the table below? Month Sales Forecast 1 220 n/a 2 250 255 3 210 205 4 300 320 5 325 315 4-73
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MAD Problem Solution Month Sales Forecast Abs Error 1 220 n/a 2 250 255 5 3 210 205 5 4 300 320 20 5 325 315 10 40 n Note that by itself, the MAD ∑A t=1 t - Ft 40 Note that by itself, the MAD only lets us know the mean only lets us know the meanMAD = = = 10 error in a set of forecasts error in a set of forecasts n 4 4-74
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Tracking Signal Formula• The Tracking Signal or TS is a measure that indicates whether the forecast average is keeping pace with any genuine upward or downward changes in demand.• Depending on the number of MAD’s selected, the TS can be used like a quality control chart indicating when the model is generating too much error in its forecasts.• The TS formula is: RSFE Running sum of forecast errorsTS = = MAD Mean absolute deviation 4-77
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting Characteristics Components of demand Moving average Winters method Focus forecasting Filtering 4-79
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Short Term Forecasting – Winters Old man wintersWinters method used to forecast one period into the futureSee how method detects patterns & adapts to market changes overtime Old Man Winters in Action 600.00 500.00 400.00 Volum e Actual 300.00 Forecast 200.00 100.00 0.00 0 20 40 60 80 100 Tim e 4-80
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Short Term Forecasting – Winters Key to Winters method• Winters is an exponential smoothing method• Smoothing is based on a key idea For each component (which are?), a portion of difference between estimate & actual is due to randomness & certain portion due to real change 4-81
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Short Term Forecasting – Winters Smoothing in action...• New estimate = old estimate + (some percentage)(error)• Smoothes out peaks & valleys (i.e., randomness) of actual 4-82
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting Characteristics Components of demand Moving average Winters method Focus forecasting Filtering 4-83
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Short Term Forecasting – Focus Bernie’s insight… …or what is focus forecasting?• An intuitive & successful idea• Regularly use a # of different methods to generate forecasts• Maintain historical accuracy information on each method• Use the most accurate method to generate “official” forecasts 4-84
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Short Term Forecasting – FocusAdvertisement appearing in APICS The Performance Advantage 4-85
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting Characteristics Components of demand Moving average Winters method Focus forecasting Filtering 4-86
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Short Term Forecasting – Filtering Two types of filters• An important feature of computer-based forecasting systems Large amounts of data – impractical to manually review all1. For data input errors (e.g., typos, scanner errors) If |“actual” - forecast| > limit, then report2. For unacceptable forecast errors (e.g., warranting management attention) If average absolute error > limit, then report 4-87
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Road map• Processing Demand• Influencing Demand• How to Improve Forecast Accuracy• Long Term Forecasting• Short Term Forecasting• Dependent Demand• Correlational Forecasting• Summary 4-88
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Demand Management Bill of Materials (BOM) Independent Demand: Finished Goods A Dependent Demand: Raw Materials, Component parts, Sub-assemblies, etc. B(4) C(2)D(2) E(1) D(3) F(2) 4-89
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Web-Based Forecasting: CPFR• Collaborative Planning, Forecasting, and Replenishment (CPFR) a Web-based tool used to coordinate demand forecasting, production and purchase planning, and inventory replenishment between supply chain trading partners.• Used to integrate the multi-tier or n-Tier supply chain, including manufacturers, distributors and retailers.• CPFR’s objective is to exchange selected internal information to provide for a reliable, longer term future views of demand in the supply chain.• CPFR uses a cyclic and iterative approach to derive consensus forecasts. 4-90
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Web-Based Forecasting: Steps in CPFR1. Creation of a front-end partnership agreement2. Joint business planning3. Development of demand forecasts4. Sharing forecasts5. Inventory replenishment 4-91
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Correlational Forecasting• Assumes an Outcome is Dependent an Existing Relationship Between the Demand Variable and Some other Independent Variable(s) Demand Variable is Dependent Variable Other Related Variables are Independent Variables Generally Expressed as a Multiple Linear Regression Model• Y = β0 + β1 X1+ β2 X2+ β2 X2+ . . . βnXn+ εi 4-92
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Simple Linear Regression ModelThe simple linear regression The simple linear regression Ymodel seeks to fit a line model seeks to fit a linethrough various data over through various data overtime a time 0 1 2 3 4 5 x (Time) Yt = a + bx Is the linear regression model Is the linear regression model - Yt is the regressed forecast value or dependent variable in the model -a is the intercept value of the the regression line, and - b is similar to the slope of the regression line. - However, since it is calculated with the variability of the data in mind, its formulation is not as straight forward as our usual notion of slope. 4-93
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Simple Linear Regression Formulas for Calculating “a” and “b” a = y - bx ∑ xy - n(y)(x) b= 2 2 ∑ x - n(x ) 4-94
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Simple Linear Regression Problem DataQuestion: Given the data below, what is the simple linear Question: Given the data below, what is the simple linearregression model that can be used to predict sales in future regression model that can be used to predict sales in futureweeks? weeks? Week Sales 1 150 2 157 3 162 4 166 5 177 4-95
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Answer: First, using the linear regression formulas, weAnswer: First, using the linear regression formulas, wecan compute “a” and “b”can compute “a” and “b” Week Week*Week Sales Week*Sales 1 1 150 150 2 4 157 314 3 9 162 486 4 16 166 664 5 25 177 885 3 55 162.4 2499 Average Sum Average Sum b= ∑xy - n( y)(x) = 2499 - 5(162.4)(3) = 63 = 6.3 ∑x - n(x ) 2 2 55 − 5(9 ) 10 a = y - bx = 162.4 - (6.3)(3) = 143.5 4-96
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97The resulting regression modelis: Yt = 143.5 + 6.3xNow if we plot the regression generated forecasts against theactual sales we obtain the following chart: 180 175 170 165 160 Sales Sales 155 Forecast 150 145 140 135 1 2 3 4 5 Period 4-97
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Statistical Assumptions of Multiple Linear Regression • The Error Term (the residual εi) is Normally Distributed • There is no Serial Correlation Among Error Terms • Magnitude of the Error Term is Independent of the Size of Any of the Independent Variables - Xi • Assumptions Can be Tested Through Analyses of the Residuals - εi 4-98
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Major Statistical Problems of Multiple Linear Regression• Multicolinarity• Use of Time-Lagged Independent Variables• Both of These Problems Result in Models with Potentially Valid Predictions, but the Reliability of the β Coefficients is Questionable 4-99