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Transcript of 1 1 ISyE 6203 Variability Basics John H. Vande Vate Fall 2011.
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ISyE 6203Variability Basics
John H. Vande Vate
Fall 2011
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Agenda
• Forecasting
• Assignable-Cause vs Common variation
• Review of Probability
• Review of Regression
• Forecasting
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Forecasting
Forecasting is the effort to determine what we can about the future from the past.
We will focus on Quantitative Methods– i.e., not opinions, judgments, “markets”, etc.
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Laws of Forecasting
• Law 1: Forecasts are wrong• Law 2: Forecast Demand not Sales• Law 3: It is generally easier to forecast
aggregate data than it is to forecast the details. (Big Idea)
• Law 4: It is generally easier to forecast a short time into the future that to forecast far into the future
• Law 5: Simpler forecasts are generally better forecasts
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General Framework
• Future = (Past) + Residual Error
• The Specifics– What aspects of the past are relevant
– What form of to use
• The Issues– Accuracy: Is the E[Residual Error] ~ 0?
– Precision: Is Residual Error small?
– Complexity and Cost!
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Variability
Residual Error is the “noise” or unpredictable variation
• Our focus is on managing this
• What tools are available for managing unpredictable variation?
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Types of Variability• Predictable or Assignable Cause
Variations– Weekends and holidays– Major scheduled events, promotions– Seasonality & Growth– Other Causal relationships
• Unpredictable or Common Cause Variations – Inherent randomness
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Types of Variability• Predictable or Assignable Cause
Variations
• Unpredictable or Common Cause Variations
• Avoid unnecessary variability
• Trade-off of capturing predictable variation while managing complexity
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Review of Probability
• Random Variable: A function that associates a value with each point in a sample space.
• Two Random Variables X and Y are independent if and only if for any two subsets A and B of their sample spaces,
P(XA and YB) = P(XA)P(YB)
• Mean:
E[X] =
• Variance: E[(X-E[X])(X-E[X])]
( ) , where p(x) is the density of Xxp x dx
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Review of Probability• Covariance: Cov(X,Y) = E[[X-E[X]][Y-E[Y]]• If X and Y are independent, Cov(X,Y) = 0• Correlation Coefficient
XY ranges between -1 and 1• If XY = 0, X and Y are uncorrelated• If X and Y are independent, they are uncorrelated.
Uncorrelated random variables need not be independent.
• More details at http://en.wikipedia.org/wiki/Correlation_and_dependence
YX
YXCovXY ),(
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Uncorrelated vs Independent• If X and Y are independent, they are
uncorrelated. Uncorrelated random variables need not be independent.
• Example: – X is Uniformly distributed on [-1,1]
– Y is X2
• E[X] = 0
• E[Y] = 1
2 13
1
/ 2X dx
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Uncorrelated vs Independent• If X and Y are independent, they are
uncorrelated. Uncorrelated random variables need not be independent.
• Example: – X is Uniformly distributed on [-1,1]
– Y is X2
• Cov(X,Y) = 2 31 1
3 3
3 13
13
1
[ ( )] [ ]
[ ] [ ]
/ 2 0
E X X E X X
E X E X
X dx
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Covariance and Variance• Constant α• E[αX] =• Var[α X] = 2
α X =• Stdev[α X] =
α X =• E[X + Y] = • Var[X + Y] = 2
X+Y = • Stdev[X + Y] = X+Y =
αE[X]α22
X
αX
E[X]+E[Y]
2X + 2
Y + 2*Cov(X,Y)
XYYXYXYX YXCov 2),(2 2222
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The Big Idea
• When combining random variables, some of the “noise” cancels. How much cancels depends on the correlation.
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Stdev[X + Y]
• Var[X+Y] =• If
• So
2 2 2 22 ( , ) 2X Y X Y X Y XYCov X Y
2 2 2
2 2
2 2 2
1, [ ] 2 ( )
0, [ ]
1, [ ] 2 ( )
XY X Y X Y X Y
XY X Y
XY X Y X Y X Y
Var X yY
Var X yY
Var X yY
2 2
2 2
2 2
1, [ ] 2
0, [ ]
1, [ ] 2 | |
XY X Y X Y X Y
XY X Y
XY X Y X Y X Y
Stdev X yY
Stdev X yY
Stdev X yY
No reduction in variation
Reductions in variation
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The Big Idea
• When combining random variables, some of the “noise” cancels. How much cancels depends on the correlation.
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Noise Canceling
• X1, X2 independent, identically distributed rvs
• Var[2X1] = 4 2X
• Stdev[2X1] = 2X
• Var[X1 + X2] =• Stdev[X1 + X2] =
• About 30% of the variability canceled
22X + 2*Cov(X,X)
XX XXCov 2),(22 2
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The Big Idea?
• That’s a “Big Idea”?• So what?!• Where’s the beef?
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Exam Review
• Rules of the game:– I probably made mistakes grading. – If you think I made mistakes grading your
exam, write a brief note indicating what you would like me to review, turn it in. I will review it.
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Results
• Average ~71• Std Dev ~18
0
1
2
3
4
5
6
7
8
31-40 41-50 51-60 61-70 71-80 81-90 91-100
ABC
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Question 1
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Question 1
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Question 2
• Where? – Indianapolis
• Why?– Co-locating the cross dock and the plant eliminates
$1.2 million in cycle inventory of monitors saving us $222 thousand annually
– Other reasons
• What I wanted you to see– We have a different model of cost if the cross dock is
at a plant– So check those locations separately
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Question 3
• Admittedly challenging.
• To simplify discussion assume the context in which the number and locations of cross docks and pools is fixed.
• We’ll address the case where we can close cross docks and pools subsequently
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What we have to do
• Manage flows of components to the cross docks – that’s no longer the same in every solution
• Balance the flows of components into and products out of the cross docks
• Manage the flows of finished goods from the cross docks to the pools
• Balance the flows of each product at each pool• Manage the assignments of stores to pools• Manage the single sourcing of products to the pools• Account for trucks between the cross docks and the
pools• Manage the frequency requirements (for each product)
to the pools
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Flows of Components
• We can identify the components and the plants
• CompFlow(i,j) to represent the units of component made a plant i shipped to cross dock j each week. Non-negative
• Multiply by $1*Distance between plant i and cross dock j*weight of product made at plant i and divide by 30,000 lbs to get trucking cost to the cross dock.
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Balance at Cross Docks
• We have to balance the flow of each component i to each cross dock j with the volumes of finished products assembled there.
• Let Assemble(p,j) be the units of finshed product p assembled at cross dock j each week. Non-negative
• Let Recipe(p,i) be the number of components from plant i in a unit of finished product p
• How to express the balance?
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Balance
• Assemble(p,j)*Recipe(p,i) = units of component made at plant i needed to produce finished product p at cross dock j.
• Sum over the finished products to get the total units of component made at plant i needed at cross dock j
Productsin
j),CompFlow(i i)Recipe(p,*j),Assemble(pp
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Flow of Finished Goods
• Let’s call the units of finished product p from cross dock j to pool k each week, ProdToPool(p, j, k)
• We want all the assembled product p at cross dock j to flow out to pools
j),Assemble(pk)j,(p,ProdToPool Poolsin
k
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Balance Flow at Pools
• Let’s use Assign(s,k) to indicate whether or not store s is assigned to pool k
• The demand for product p at each store each week is StoreDemand(p)
• The demand for product p at pool k implied by the assignments is
• The total units of product p available to meet this demand at pool k is
Docks Crossin
k)j,(p,ProdToPoolj
),(Assign*d(p)StoreDemanStoresin
kss
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Balance Flow at Pools
• These should balance for each product at each pool
docks Crossin Storesin
k) j, (p,ProdToPool),(Assign*d(p)StoreDemanjs
ks
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Assignments of Stores to Pools
• Assign each store s to one and only one pool
• So far everything is pretty standard.
1),(AssignPoolsin
ksk
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Manage Single Sourcing at Pools
• Each pool k should get all of each product p from one and only one cross dock
• Let Source(p, j, k) indicate whether or not pool k sources product p from cross dock j
• Single Sourcing
1),,(Sourcedocks Crossin
kjpj
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But it’s a bit more complicated
• We don’t know the demand at the pool except as a function of the Assign variables so we need to shut down ProdToPool variables based on the Source decisions. To do this we need an upper bound on ProdToPool.
• Let TotalDemand(p) be the total demand each week for product p across all stores
• For each product p, cross dock j and pool k:•
k)j,Source(p,*d(p)TotalDeman k)j,(p,ProdToPool
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Trucks from Cross Dock to Pool
• Since there is a frequency requirement for serving the pools, we must keep track of the trucks we send each week
• We know the weight moving from each cross dock j to each pool k each week:
• So,
k)j,(p,ProdToPool*(p)ProdWeightProductsin p
/30000k)j,(p,ProdToPool*(p)ProdWeight),(Productsin pkjTrucks
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Frequency requirement
• At least one truck to pool k from each cross dock j for each cross dock that pool k sources SOME product from
• For each pool k, product p and cross dock j
),,(),( kjpSourcekjTrucks
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If we can shut down cross docks?
• Let OpenCD(j) indicate whether or not we will open cross dock j
• Clearly we want for each product p, pool k and cross dock j
OpenCD(j) k)j,Source(p,
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Alternate & Better model
• Integrate the sourcing decisions into a binary decision variable
Path(prod, cd, pool, store) = 1
if store receives prod from pool and pool receives prod from cd
• Keep
Assign(store, pool) = 1
if store receives supplies from pool
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Alternate & Better model
• Single sourcing at Store s
• Keep Assign(store, pool) = 1
if store receives supplies from pool• Ensure Path choices are consistent with
assignments, i.e., we get to store via assigned pools for all products
Poolsin
1 p) Assign(s,p
pool) re,Assign(sto store) pool, cd, Path(prod,docks Crossin
cd
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Alternate & Better Model
• Ensure single sourcing at pool• Introduce Source(prod, cd, pool) = 1 if pool
receives prod from cd
• Ensure consistency with Path decisions
1 pool) cd, d,Source(proDocks Crossin
cd
pool) cd, d,Source(pro store) pool, cd, Path(prod,
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Alternate & Better Model
• Volumes at the pool as before – based on Assign
• Volumes at the Cross Dock based on Path
• Is the requirement for comp at cd• …
Storesin Poolsin Productsin prod
store)) pool, cd, Path(prod,*comp) od,(Recipe(prstore pool
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Reno
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Examples
• Adding across customers or geographies
• Demand for a single sku at a single DC
Carlstadt
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Daily Sales: Single SKU• Sum of Stdev’s of all DCs: 1,957
• Stdev of Demand over all DCs: 1,553 (21% less)
Total
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Correlated?
• If daily sales were uncorrelated across all DCs then variances would add
• Stdev across all DCs would equal
Sqrt(Var[DC-1]+Var[DC-2]+Var[…]) = 1,150
• Actual Stdev is 1,553
• Conclusion?
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Adding across Products
• Sum of Stdev in Weekly Sales across all SKUs for vendor at a DC: 1,043
• By far the largest SKU has Stdev 999.
• Stdev in Total Weekly Sales for DC of all SKUs from vendor: 996
• Explain
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Adding across time
• Stdev in daily sales of SKU: 999.49
• Stdev in weekly sales of SKU: 3882.63
• Much lower than 5*Daily Stdev
• Higher than ?*Daily Stdev
• Conclusion?
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How can we:
• Add across customers?
• Add across products?
• Add across time?
• When do these conflict?
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Questions?
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Agenda
• Assignable-Cause vs Common variation
• Review of Probability
• Review of Regression: A tool for capturing predictable variation
• Forecasting
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Correlation Example• Truck load shipments from Green Bay and
Denver to Indianapolis• Assemble Products in Indianapolis and
distribute by full truckload from there to stores• What will happen to costs compared to direct
full truck load shipments?– Transportation– Pipeline– At plants– At Indianapolis Warehouse/Cross Dock– At Stores
Tree Height
Trunk Diameter Correlation Coefficient: 0.886
y x35 849 927 733 660 1321 745 1151 12
Tree Height vs Trunk Diameter
0
10
20
30
40
50
60
70
4 5 6 7 8 9 10 11 12 13 14
Trunk Diameter (inches)
Tre
e H
eig
ht
(fee
t)
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Regression
• Explain or model the relationship between the dependent variable (e.g., tree height) and the independent variables (e.g., trunk diameter)
• Linear Regression Model
y = β0 + β1x +
Assignable cause variation
Common cause variation
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RegressionTree Height vs Trunk Diameter
y = 4.5413x - 1.3147R2 = 0.7854
0
10
20
30
40
50
60
70
4 5 6 7 8 9 10 11 12 13 14
Trunk Diameter (inches)
Tre
e H
eig
ht
(fee
t)
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Regression• The model
y = 4.5413 x - 1.3147
is the “best fit” linear model for the relationship
• It is not based on physical laws or causality (e.g., thin trees don’t have negative height)
• It does “explain” about 78% of the variability in the data R2 = 0.7854
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Explained vs Unexplained Variation
For Linear Regression
Coefficient of Determination
)yy( )y ( )y (
Squares of Sum Explained Squares of Sum Residual Squares of Sum Total
estimateith y
nsobservatio theof averagey
nobservatioith
2i
2ii
2
i
i
i
yy
y
Squares of Sum TotalSquares of Sum Residual
Squares of Sum TotalSquares of Sum Explained2 1R
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Coefficient of Determination• 0 ≤ R2 ≤ 1• Note in our example the Coefficient of
Determination R2 is equal to the square of the Correlation Coefficient r2
R2 = 0.7854 = r2 = 0.8862
• This is generally the case for simple linear regression (1 independent variable)
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Standard Error • A sort of standard deviation about the
regression line. • How widely dispersed are observations
about the regression line
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P-values
• Indicates how likely it is to see this value if the true value of the coefficient is 0 and there’s as much noise as we see in the data.
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.886R Square 0.785Adjusted R Square 0.750Standard Error 6.635Observations 8.000
ANOVAdf SS MS F Significance F
Regression 1.000 966.736 966.736 21.960 0.003Residual 6.000 264.139 44.023Total 7.000 1230.875
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept -1.315 9.149 -0.144 0.890 -23.701 21.072 -23.701 21.072X Variable 1 4.541 0.969 4.686 0.003 2.170 6.913 2.170 6.913
Strong evidence there’s a relationship
Weak evidence the intercept isn’t 0
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Multiple Linear Regression• With more than one independent variable, e.g.,
Sales = β0 + β1 GDP + β2 Unemployment Rate + …
• Need to watch out for– Non-linearity: The relationship might not be linear,
e.g., weight of the tree vs trunk diameter. – Multi-colinearity: one independent variable is a
linear function of another (eliminate one)– Over specified model: Adding more independent
variables increases R2, but reduces the degrees of freedom in the fit. Adjusted R2 attempts to account for this.
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Static Regression
Salaries
Independent Data
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Excel’s Linest
• Array Function Linest(Y-array, Array of X’s, [const],[stat]) outputs the β’s
• One column for each β• Remember Array Functions are entered
with Ctrl-Shift-Enter• Allows you to perform running
regressions• Coefficients come out in reverse order
(Go figure)
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LinEst
• Regression.xls
Faculty Salary Gender Rank Dept Years Merit SUMMARY OUTPUT1 38 0 3 1 0 1.47
2 58 1 2 2 8 4.38 Regression Statistics3 80 1 3 3 9 3.65 Multiple R 0.948 4 30 1 1 1 0 1.64 R Square 0.899 5 50 1 1 3 0 2.54 Adjusted R Square 0.878 6 49 1 1 3 1 2.06 Standard Error 3.765 7 45 0 3 1 4 4.76 Observations 30.000 8 42 1 1 2 0 3.059 59 0 3 3 3 2.73 ANOVA
10 47 1 2 1 0 3.14 df SS MS F Significance F11 34 0 1 1 3 4.42 Regression 5.000 3,037.212 607.442 42.859 0.000 12 53 0 2 3 0 2.36 Residual 24.000 340.155 14.173 13 35 1 1 1 1 4.29 Total 29.000 3,377.367 14 42 0 1 2 2 3.81
15 42 0 1 2 2 3.84 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%16 51 0 3 2 7 3.15 Intercept 0.323 4.810 0.067 0.947 (9.604) 10.250 (9.604) 10.250 17 51 1 2 1 8 5.07 Gender 8.050 1.495 5.386 0.000 4.965 11.134 4.965 11.134 18 40 0 1 2 3 2.73 Rank 8.430 0.849 9.930 0.000 6.678 10.183 6.678 10.183 19 48 1 2 1 1 3.56 Dept 9.220 0.885 10.413 0.000 7.392 11.047 7.392 11.047 20 34 1 1 1 7 3.54 Years 0.059 0.311 0.191 0.850 (0.583) 0.701 (0.583) 0.701 21 46 1 2 1 2 2.71 Merit 3.107 1.032 3.010 0.006 0.977 5.237 0.977 5.237 22 45 0 1 2 6 5.1823 50 1 1 3 2 2.66
24 61 0 3 3 3 3.7 Merit Years Dept Rank Gender Intercept25 62 1 3 1 2 3.75 3.106969168 0.059 9.220 8.430 8.050 0.323
26 51 0 1 3 8 3.96 {=Linest(B2:B31, C2:G31)}27 59 0 3 3 0 2.8828 65 1 2 3 5 3.3729 49 0 1 3 0 2.8430 37 1 1 1 9 5.12
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Questions?
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LTL Rates
• Complicated to work with rate engines• Alternative: model the rates• Challenge:
– Model for rates = Max{Min Charge, Min{Intercept1 + Rate1*Weight*Distance, Intercept2 + Rate2*Weight*Distance}}
• Come up with estimates for – Min Charge, Intercepts, Rates
• Winner: Minimum sum of square errors
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LTL RatesObservation Distance Weight Price
1 165 87 112.52$ 2 165 125 161.66$ 3 333 107 103.11$ 4 165 299 386.70$ 5 165 322 416.44$ 6 561 389 409.77$ 7 165 1,754 1,344.97$ 8 165 3,345 2,303.70$ 9 333 5,997 2,443.78$
10 561 5,000 2,345.00$ 11 165 20,000 2,680.00$ 12 333 10,000 3,270.01$ 13 333 20,000 3,434.00$ 14 561 20,000 2,932.00$
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Agenda
• Assignable-Cause vs Common variation
• Review of Probability
• Review of Regression
• Forecasting
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Forecasting
Forecasting is the effort to determine what we can about the future from the past.
We will focus on Quantitative Methods– i.e., not opinions, judgments, “markets”, etc.
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General Framework
• Future = (Past) + Residual Error
• The Specifics– What aspects of the past are relevant
– What form of to use
• The Issues– Accuracy: Is the E[Residual Error] ~ 0?
– Precision: Is Residual Error small?
– Complexity and Cost!
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Examples• Autoregressive or time-series
– Past = Historical values of the process we are forecasting, e.g., past demand forecasts future demand
• Causal– Past = Historical values are “leading
indicators” like GDP, employment, housing starts, etc.
• Regression, Maximum Likelihood– Past may include both historical values and
leading indicators
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Past & Future• We lump data into time periods
– Average, total or sample in some period– Reduces data requirements– Averaging and totaling smooth the data (remember
the Big Idea)– Actionable
• What we’re forecasting– yt = value at time t
• Past– Autoregressive: yt-1, yt-2, yt-3,…– Leading indicators: xi,t-1, xi,t-2, xi,t-3,…
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Specifics• Autoregressive or Time-Series
– Moving average• Past = past n observations yt-1, yt-2, yt-3,… yt-n
(yt-1, yt-2, yt-3,… yt-n ) =
• More a tool for understanding the past than for forecasting the future
– Exponential Smoothing• Past = “all” past observations yt-1, yt-2, yt-3,…
(yt-1, yt-2, yt-3,… ) =
1
1
n
t ini
y
11 2 3
1
(1 ) (1 ) ( , ,...)it i t t t
i
y y f y y
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Specifics• Exponential Smoothing with Trend
– Past = “all” past observations yt-1, yt-2, yt-3,…
– Forecast uses exponential smoothing to estimate• The “Level”: weighted average of observation and past
estimate
• The “Trend”: weighted average of observation and past estimate
• Forecast m periods in the future = Level + m*Trend
– Details at Engineering Statistics Handbook
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Specifics• Exponential Smoothing with Trend & Seasonality
– Past = “all” past observations yt-1, yt-2, yt-3,…
– Forecast uses exponential smoothing to estimate• The “(De-seasonalized) Level”: weighted average of the de-
seasonalized observation and past de-seasonalized estimate
• The “Trend”: weighted average of observation and past estimate
• The “Seasonal factors”: weighted average of observation and past estimate
• Forecast m periods in the future
= (Level+m*Trend)*Seasonal Factor
– Details at Engineering Statistics Handbook
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Specifics• Exponential Smoothing with …
– You get the idea.
– Issues• Initialization data
• Choosing the weights
• Growing complexity
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Specifics
• Regression, Maximum Likelihood– Past = past observations yt-1, yt-2, yt-3,…
and leading indicators xi,t-1, xi,t-2…
(yt-1, yt-2, yt-3,… xi,t-1, xi,t-2…) is some function of these past values
• Examples: – Linear– Non-linear models
» Diffusion » Logit» Probit
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Bass Diffusion• Postulates a form for sales over the life of a
product• Three parameters
– m: The total potential market
– p, q: Shape parameters
( )
( )1
1( )
p q t
q p q tp
ee
N t m
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Questions?
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Top Down vs Bottom Up
• Often faced with forecasting 100’s of families and 1,000’s of SKUs
• Different Options:– Top Down:
• Develop an aggregate forecast and allocate it to more detailed level
– Bottom Up:• Develop individual detailed forecasts and
aggregate them up
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Laws of Forecasting
• Law 1: Forecasts are wrong• Law 2: Forecast Demand not Sales• Law 3: It is generally easier to forecast
aggregate data than it is to forecast the details. (Big Idea)
• Law 4: It is generally easier to forecast a short time into the future that to forecast far into the future
• Law 5: Simpler forecasts are generally better forecasts
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Examples from ABL
• Sales for SKU 8295
• Raw dataRaw Sales
0
200
400
600
800
1000
1200
1/4/
2010
1/18
/201
0
2/1/
2010
2/15
/201
0
3/1/
2010
3/15
/201
0
3/29
/201
0
4/12
/201
0
4/26
/201
0
5/10
/201
0
5/24
/201
0
6/7/
2010
6/21
/201
0
7/5/
2010
7/19
/201
0
8/2/
2010
8/16
/201
0
8/30
/201
0
9/13
/201
0
9/27
/201
0
10/1
1/20
10
10/2
5/20
10
11/8
/201
0
11/2
2/20
10
12/6
/201
0
12/2
0/20
10
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Variability?
• How do we forecast this?
• How do we assign a variability to this?
Not Actionable!
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Sales by Day
We can make ordering decisions on a daily basis:
Daily Sales
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1/4/
2010
1/18
/201
0
2/1/
2010
2/15
/201
0
3/1/
2010
3/15
/201
0
3/29
/201
0
4/12
/201
0
4/26
/201
0
5/10
/201
0
5/24
/201
0
6/7/
2010
6/21
/201
0
7/5/
2010
7/19
/201
0
8/2/
2010
8/16
/201
0
8/30
/201
0
9/13
/201
0
9/27
/201
0
10/1
1/20
10
10/2
5/20
10
11/8
/201
0
11/2
2/20
10
12/6
/201
0
12/2
0/20
10
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Weekly Sales
• Or on a weekly basis
Weekly Sales
0
5000
10000
15000
20000
25000
30000
35000
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
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Or a Monthly Basis
Units Sold by Month
0
10000
20000
30000
40000
50000
60000
70000
1 2 3 4 5 6 7 8 9 10 11 12
• Or on a monthly basis
• Which is appropriate?
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Compare the Variability
• .
•
Weekly Sales
0
5000
10000
15000
20000
25000
30000
35000
Daily Sales
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1/4/
2010
1/18
/201
0
2/1/
2010
2/15
/201
0
3/1/
2010
3/15
/201
0
3/29
/201
0
4/12
/201
0
4/26
/201
0
5/10
/201
0
5/24
/201
0
6/7/
2010
6/21
/201
0
7/5/
2010
7/19
/201
0
8/2/
2010
8/16
/201
0
8/30
/201
0
9/13
/201
0
9/27
/201
0
10/1
1/20
10
10/2
5/20
10
11/8
/201
0
11/2
2/20
10
12/6
/201
0
12/2
0/20
10
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The Big Idea
• Average Daily Sales: 1280.196• Std Dev. In Daily Sales: 1546.472
• Average Weekly Sales: 6400.981• Std Dev. In Weekly Sales: 5971.578
• Avg Weekly Sales = 5*Average Daily Sales
• What about the relationship between the variabilities?• 5*Std Dev. In Daily Sales = 7732.361
What does the Big Idea say we should expect?
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The Big Idea
• If sales from week to week are independent, we should see
• 5*Variance in Daily Sales = Variance in Weekly Sales
• Sqrt(5)*Std Dev in Daily Sales = Std Dev in Weekly Sales
• Sqrt(5)*Std Dev in Daily Sales = 3458.017 < 5971.578
• So, Sales from Day to Day are (auto) correlated
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Forecasting
• Try Simple Techniques:– Moving Average– Exponential Smoothing– Exponential Smoothing with Trend– Exponential Smoothing with Trend &
Seasonality– (Auto) Regression– Bass Diffusion
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Moving Average
• Average Error: 666.29 (Over Estimates)• Std Dev of Error: 4,094 (< Std Dev. In Sales)
Weekly Sales vs Moving Average
0
5000
10000
15000
20000
25000
30000
35000
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
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Exponential Smoothing
• Average Error: 786.13 (less accurate)• Std.Dev in Errors: 4,308 (less precise)
Weekly Sales vs Exponential Smoothing
0
5000
10000
15000
20000
25000
30000
35000
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52
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Exponential Smoothing w/ Trend
• Average Error: -167.58 (more accurate)• Std.Dev in Errors: 5,065 (less precise)
Weekly Sales vs Exponential Smoothing
0
5000
10000
15000
20000
25000
30000
35000
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Weekly Sales vs Exponential Smoothing w/ Trend
0
5000
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Exp. Smoothing w/ Trend & Seasonality
Seasonality by:Day of WeekWeek of MonthWeek of Year
Sales by Day of Week
Mon, 15%
Tue, 21%
Wed, 20%
Thu, 24%
Fri, 20%
Sales By Week of Month
1, 31%
2, 16%
3, 23%
4, 21%
5, 25%
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9393
Regression
Using
Previous 2 week
Previous 3 weeks
…
Might Use
Week of Month too
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Previous 2 Weeks
Average Error: 2,096Std Dev in Error: 4,929
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5,000
10,000
15,000
20,000
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30,000
35,000
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9595
Previous 3 Weeks
Average Error: 1,837
Std Dev in Error: 5,010
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Series1
Series2
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9696
Previous 4 Weeks
Average Error: 1664Std Dev in Error: 5367.29
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9797
Bass Diffusion
Average Error: -1536.17Std Dev in Error: 5324.07
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5,000
10,000
15,000
20,000
25,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 410
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Conclusion
• Simple moving average provides best forecast on a weekly basis
• Exponential smoothing better on a monthly basis
• Now let’s build a demand distribution given a forecast
• Idea: Compare Actual Sales to Forecasted Sales through the ratio
• Accuracy means Average should be 1• Precision means Std Dev should be small
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Demand Distribution
• We know the forecast is WRONG
• But it does give us some information
• What Actual Sales will be is uncertain, but we can develop a distribution for it
• What are the chances Actual Sales are larger than X? Smaller than Y? …
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Actual to Forecast Ratios
the Avg is 1.1 (What does that mean?)
• σ the Std Dev is 0.87
Frequency
0
2
4
6
8
10
12Ratio < 1 Over
forecast
Ratio > 1 Under forecast
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Translate Forecast into Demand Distribution
• Assuming things continue as they have…
• If the forecast is 100, what do we expect actual sales to be?
• So, if is the Average Actual/Forecast ratio & F is the forecast, Expected demand is F
• What is the spread of actual demand about this mean value?
110
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Translate Forecast into Demand Distribution
• If things continue as they have, it is natural to assume we will draw from the historical distribution of Actual/Forecast ratios.
• So, if = 1, If the forecast is 100, what should the distribution of Actual Sales be?– The mean should be?
– The std dev of actual demand about this mean value should be?
• Why?
100
100σ
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Demand Distribution
• The Actual/Forecast ratio has a distribution with mean and std dev σ
• If = 1, the distribution for Actual Sales is just F, the forecast, times the Actual/Forecast ratio so, it has mean F and std dev σF
• If ≠ 1, the distribution for Actual Sales is just F, the forecast, times the Actual/Forecast ratio so, it has mean F and std dev σF
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Common Cause Variability
• So, the common cause variability in demand for SKU 8295 that we will need to protect against with safety stock is about 87% of the Forecasted demand!
• Just working with Raw Sales– Std Dev/Average = 93%
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Common Cause Variability
• On the other hand, with monthly sales
• Working with Raw Sales– Std Dev/Average = 74%
• Working with Exponential Smoothing forecast– Std Dev/Average = 36%