Uncertain Demand

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Topic 10 Uncertain Demand

• Classification of forecasting models
• Models for short-term demand forecasts
• Time series models
• Newsvendor inventory modeling
• Performance measures

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Forecasting For Operational Decisions
Long time frame Aggregated data Lots of
uncertainty

• Assist in strategic (long term) decision-making
• When will demand require that we expand our
  facility?
• Assist in tactical (medium-term) decision-making
• How many employees will we need to hire next
  quarter?
• Assist in shop control (short-term)
  decision-making
• How much raw material should we order next
  week?

Short time frame Disaggregated data Less
uncertainty
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Forecasting Methods
Time frame of decision
short
long
much
Quantitative tools that can incorporate current
information. - Averaging - Exponential smoothing
Quantitative tools that can incorporate a great
deal of historical information. - Causal
methods - Time series
Experience with decision
Qualitative tools that use very rich sources of
data. - Judgement methods - Market research
none
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Patterns in Time-Series Data
TREND
Sale of fuel oil in gallons
CYCLE
SEASON
RANDOM
Time
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Forecasting Methods
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• Goals
• Predict future from past
• Smooth out noise
• Standardize forecasting procedure

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Time Series Forecasting
Forecast
Historical Data
Time series model
A(i), i 1, ,t
f(tt), i 1, 2,
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Moving Average

• Average most current values to predict future
  outcomes. The trend-cycle can be estimated by
  smoothing the series to reduce random variation.
• Assumptions
• No trend
• Equal weight to last m observations
• Model

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Example calculations
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Two-point WMA
Three-point WMA
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Calculating Forecast Error
Cumulative Forecast Error (CFE)
Mean Square Error (MSE)
Standard Deviation (SD)
Mean Absolute Deviation (MAD)
Mean Percentage Error (MPE)
Mean Absolute Percent Error (MAPE)
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Calculating Forecast Error
Cumulative Forecast Error (CFE)
Mean Square Error (MSE)
Standard Deviation (SD)
Mean Absolute Deviation (MAD)
Mean Percentage Error (MPE)
Mean Absolute Percent Error (MAPE)
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The tracking signal measures whether the
forecasting method is biased over time.
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Exponential Smoothing

• Estimate next outcome with a weighted combination
  of the forecast for previous period and the most
  recent outcome
• Assumptions
• No trend
• Exponentially declining weight given to past
  observations
• Model

F(t) Forecast in period t x(t) Actual demand
in period t smoothing constant is alpha
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Exponential Smoothing with a Trend

• Assumptions
• Linear trend
• Exponentially declining weights to past
  observations/trends
• Model

F(t) Forecast in period t x(t) Actual demand
in period t ? ? smoothing constants (values
between 0 and 1) T(t) Trend component in period
t
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Double Exponential Smoothing

• Special case of the Holts linear method - where
  the two parameters (? ?) assumed to be equal.
• Model

F(t) Forecast in period t x(t) Actual demand
in period t ? smoothing constant
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Regression of original and deseasonalized data
253.05 2.5615 291.4 (291.4).93 271.0
Reseasonalized (intercepttrend)season
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Regression Equation for Deseasonalized data
253.05 2.56P
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Forecasting example You must prepare a forecast
of product demand in order to plan for
appropriate production quantities. You receive
the following historical information from
marketing Month Sales
Advertising in thousands
in thousand s 1 264 2.5 2 116 1.3 3 165
1.4 4 101 1.0 5 209 2.0
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Year
Month
Sales
Deseasonalized Sales
12-point Moving Average
Centered Moving Average
Sales /CMA
Seasonal Index
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Conclusions

• Sensitivity Lower values of m or higher values
  of a will make moving average and exponential
  smoothing models (without trend) more sensitive
  to data changes (and hence less stable).
• Trends Models without a trend will underestimate
  observations in time series with an increasing
  trend and overestimate observations in time
  series with a decreasing trend.
• Smoothing Constants Choosing smoothing constants
  is an art the best we can do is choose constants
  that fit past data reasonably well.
• Seasonality Methods exist for fitting time
  series with seasonal behavior (e.g., Winters
  method), but require more past data to fit than
  the simple models given here.
• Judgement No time series model can anticipate
  structural changes not signaled by past
  observations these require judicious overriding
  of the model by the user.

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Process Flow at Wolf Neck Farms
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Wolf Neck Farms Timeline and Economics

• Economics
• On average, WNF sells organic beef for p 6.00
  per lb
• WNF pays c 4.00 per lb
• Discounted (C-code) beef sells for v 2.50 per
  lb

Generate Demand Forecast
Beef Harvested
Buy Beef Contract
Beef Discounted (C-code)

• The too much/too little problem
• Contract too much and left-over beef is sold as
  commercial grade
• Contract too little and sales are lost.
• Assume for May, forecast for sales is 90,000
  pounds.

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Newsvendor model implementation steps

• Gather economic inputs
• Selling price, production/procurement cost,
  salvage value of inventory
• Generate a demand model
• Use empirical demand distribution or choose a
  standard distribution function to represent
  demand, e.g. the normal distribution, the Poisson
  distribution.
• Choose an objective
• e.g. maximize expected profit or satisfy a fill
  rate constraint.
• Choose a quantity to order.

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Using historical A/F ratios to choose a Normal
distribution for the demand forecast

• Start with an initial forecast
• Wolf Neck Farms forecast is 90,000 lbs.
• Evaluate the A/F ratios of the historical data
• Wolf Neck Farms historical data
• Average A/F ratio 1.085
• Standard deviation of A/F ratio 0.256
• Set the mean of the normal distribution to
• 1.085(90,000)97,650
• Set the standard deviation of the normal
  distribution to
• 0.256(90,000)23,040

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Too much and too little costs

• Co overage cost
• The cost of ordering one more unit than what you
  would have ordered had you known demand.
• In other words, suppose you had left over
  inventory (i.e., you over ordered). Co is the
  increase in profit you would have enjoyed had you
  ordered one fewer unit.
• For WNF Co Cost Salvage value c v 4.00
  2.50 1.50 per lb
• Cu underage cost
• The cost of ordering one fewer unit than what you
  would have ordered had you known demand.
• In other words, suppose you had lost sales (i.e.,
  you under ordered). Cu is the increase in profit
  you would have enjoyed had you ordered one more
  unit.
• For WNF Cu Price Cost p c 6.00 4.00
  2.00 per lb

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Incremental Analysis of Newsvendor Costs at Wolf
Neck

• Ordering one more unit increases the chance of
  overage
• Expected loss on the Qth unit Co x F(Q)
• F(Q) Distribution function of demand
  ProbDemand lt Q)
• but the benefit/gain of ordering one more unit
  is the reduction in the chance of underage
• Expected gain on the Qth unit Cu x (1-F(Q))

As more units are ordered, the expected benefit
from ordering one unit decreases while the
expected loss of ordering one more unit
increases.
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Newsvendor expected profit maximizing order
quantity

• To maximize expected profit order Q units so that
  the expected loss on the Qth unit equals the
  expected gain on the Qth unit
• Rearrange terms in the above equation -gt
• The ratio Cu / (Co Cu) is called the critical
  ratio.
• Hence, to maximize profit, choose Q such that we
  dont have lost sales (i.e., demand is Q or
  lower) with a probability that equals the
  critical ratio

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Wolf Necks expected profit maximizing order
quantity using the Normal distribution

• Inputs p 6.00 c 4.00 v 2.50 Cu 2.00
  Co 1.50
• critical ratio 0.5714
• mean m 97,650 standard deviation s
  23,040
• Look up critical ratio in the Standard Normal
  Distribution Function Table
• If the critical ratio falls between two values in
  the table, choose the greater z-statistic
• Choose z 0.18
• Convert the z-statistic into an order quantity

To optimize long term profits, set the order
quantity high enough to satisfy .18 standard
deviations more than expected demand
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Newsvendor model performance measures

• For any order quantity we would like to evaluate
  the following performance measures
• Expected lost sales
• The average number of units demand exceeds the
  order quantity
• Expected sales
• The average number of units sold.
• Expected left over inventory
• The average number of units left over at the end
  of the season.
• Expected profit
• Expected fill rate
• The fraction of demand that is satisfied
  immediately
• In-stock probability
• Probability all demand is satisfied
• Stockout probability
• Probability some demand is lost

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Expected Lost Sales at Wolf Neck

• Definition
• e.g., if demand is 103,000 and Q 102,000, then
  lost sales is 1000 units.
• e.g., if demand is 101,000 and Q 103,000, then
  lost sales is 0 units.
• Expected lost sales is the average over all
  possible demand outcomes.
• If demand is normally distributed
• Step 1 normalize the order quantity to find its
  z-statistic.
• Step 2 Look up in the Standard Normal Loss
  Function Table the expected lost sales for a
  standard normal distribution with that
  z-statistic L(0.18)0.3154
• Step 3 Evaluate lost sales for the actual normal
  distribution

The expected amount (in standard deviations) that
demand exceeds Q
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Measures that follow expected lost sales

• Expected sales m - Expected lost sales
• 97,650.00 7,266.82 90,383.18
• Expected Left Over Inventory Q - Expected Sales
  
• 101,797.20 90,383.18 11,414.02
• Expected Profit (Price-Cost) x Expected
  sales-
• (Cost-Salvage value) x Expected left over
  inventory
• 2.00(90,383.18) - 1.50(11,414.02)
  163,645.33
• Expected Fill Rate Expected sales / Expected
  demand
• 1 - (Expected lost sales / Expected demand)
• 1 - (7,266.82 / 97,650.00) 92.56

Note the above equations hold for any demand
distribution
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Service measures of performance

• In-stock probability F(Q) F(z)
• Evaluate the z-statistic
• for the order quantity
• Look up F(z) in the Std. Normal
• Distribution Function Table F(0.18)
  57.14
• Stockout probability 1 F(Q)
• 1 In-stock probability 1 0.5714 42.86
• Note the in-stock probability is not the same as
  the fill rate
• Fill rate is the fraction of demand that is
  satisfied immediately
• In-stock probability is the probability that all
  demand is satisfied

Look familiar?
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Choose Q subject to a minimum in-stock
probability

• Suppose we wish to find the order quantity that
  minimizes left over inventory while generating at
  least a 99 in-stock probability.
• Step 1
• Find the z-statistic that yields the target
  in-stock probability.
• In the Standard Normal Distribution Function
  Table we find F(2.32) 0.9898 and F(2.33)
  0.9901.
• Choose z 2.33 to satisfy our in-stock
  probability constraint.
• Step 2
• Convert the z-statistic into an order quantity
  for the actual demand distribution.
• Q m z x s 97,650 2.33 x 23,040
  151,333.20

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Choose Q subject to a minimum fill rate constraint

• Suppose we wish to find the order quantity that
  minimizes left over inventory while generating at
  least a 99 fill rate.
• Step 1
• Find the lost sales with a standard normal
  distribution that yields the target fill rate.
• Step 2
• Find the z-statistic that yields the lost sales
  found in step 1.
• From the Standard Normal Loss Function Table,
  L(1.33)0.0427 and L(1.34) 0.0418
• Choose the higher z-statistic, z 1.34
• Step 3
• Convert the z-statistic into an order quantity
  for the actual demand distribution.
• Q m z x s 97,650 1.34 x 23,040
  128,523.60

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Newsvendor model summary

• The model can be applied to settings in which
• There is a single order/production/replenishment
  opportunity.
• Demand is uncertain.
• There is a too much-too little challenge
• If demand exceeds the order quantity, sales are
  lost.
• If demand is less than the order quantity, there
  is left over inventory.
• Firm must have a demand model that includes an
  expected demand and uncertainty in that demand.
• With the normal distribution, uncertainty in
  demand is captured with the standard deviation
  parameter.
• At the order quantity that maximizes expected
  profit the probability that demand is less than
  the order quantity equals the critical ratio
• The expected profit maximizing order quantity
  balances the too much-too little costs.

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The demand-supply mismatch cost

• Definition the demand supply mismatch cost
  includes the cost of left over inventory (the
  too much cost) plus the opportunity cost of
  lost sales (the too little cost)
• For WNF
• The maximum profit is the profit without any
  mismatch costs, i.e., every unit is sold and
  there are no lost sales
• The mismatch cost can also be evaluated with
• For WNF

Mismatch cost Maximum profit Expected profit
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Unlimited, but expensive reactive capacity

• WNF can pay a premium to buy organic beef for
  immediate slaughter (cost is 5.25 vs. 4.00 per
  pound).
• There are no restrictions imposed on the 2nd
  order quantity.
• How does this change the original order quantity

Beef Harvested
Beef Discounted (C-code)
Buy Short Contract for Immediate Slaughter
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Apply Newsvendor logic even with a 2nd order
option

• The too much cost remains the same Co c v
  4.00 2.50 1.50
• The too little cost changes
• If the 1st order is too low, we cover the
  difference with the 2nd order.
• Hence, the 2nd order option prevents lost sales.
• So the cost of ordering too little per unit is no
  longer the gross margin, it is the premium we pay
  for units in the 2nd order.
• Cu 5.25 4.00 1.25
• Critical ratio
• Corresponding z-statistic F(-0.11)0.4562,
  F(-0.12)0.4522, so z -0.11.

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• Expected lost sales if demand is normally
  distributed
• Step 1 normalize the order quantity to find its
  z-statistic.
• Step 2 Look up in the Standard Normal Loss
  Function Table the expected lost sales for a
  standard normal distribution with that
  z-statistic
• L(-.11)0.4564
• Step 3 Evaluate lost sales for the actual normal
  distribution

• Expected sales m - Expected lost sales
• 97,650.00 10,515.5 87,134.50
• Expected Left Over Inventory Q - Expected Sales
  
• 95,115.60 87,134.50 7,981.1

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Profit improvement due to the 2nd order option

• With a single ordering opportunity
• Optimal order quantity 101,797.20 lbs
• Expected profit 163,645.33
• Mismatch cost 31,654.67
• The maximum profit is unchanged 195,300.00
• With a second order option
• Optimal order quantity 95,115.60
• mismatch cost 195,300 170,183.97 25,116.03