# The Evolution of Forecasts _for a sports drink_ by Rabia06

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```									The Evolution of Forecasts
(for a sports drink)

David Heath
Department of Mathematical Sciences
Carnegie Mellon University

Note: some details have been altered!
My charge (as I interpreted it):
 Discuss the use and usefulness of
mathematics to solve a real problem
 Present a specific, real example in which
relatively simple mathematics helped make
something better
 Convey a bit of the “flavor” of doing
mathematics motivated by applications
The story:
 Client contacted Peter Jackson and me
 Client made and distributed a “sports drink”
 Several years before they had run out,
resulting in:
 Lost sales

 Angry customers (potentially lost!)

 They were in a bind
An earlier solution
 They had studied the variability of
consumption (related to weather, etc.)
 They built a simple model and adopted a
very conservative (i.e., lots of inventory)
inventory policy.
 Since then, consumption had grown
 They couldn’t find enough warehouse space
to hold all of the inventory
Why the problem was difficult
   Most sales are in June, July and August
   Production facility had maximum production rate
which could produce (each month) perhaps 12
percent of annual sales
   The policy was to be sure that at the beginning of
each month there would be enough product in
inventory to cover the next 1.6 months’ forecast
demand. This meant that on July 1 they needed to
have more than 50% of a year’s sales in inventory
   This meant that on June 1 they needed lots of
inventory (because of their limited production
capabilities)
What they wanted from us
   Find a way to make things better: increase
“product availability” and/or decrease cost
What we needed to do
 Understand their current forecasting
methodology
 Determine whether current forecasts were
“best possible”, or, if could be improved,
improve them
 Build a model to allow computation of
system performance. This model should be
compatible with their current production
planning techniques
Complexity

Lots of warehouses, months, product sizes, product flavors
Their production planning
   Each month:

 Of each type

 In each region

 In each month

 Obtain forecasts for demands

   Use linear programming to solve for optimal
production levels for this month for each plant.
Solution is for this period and “all” future periods.
   Carry out the current month of the plan; move to
We first examined their forecasts
   How can you tell if forecasts are bad?
 Good forecasts should be a martingale

 What’s a martingale?

 Example: fortune of a gambler playing
a fair game.
 Example: Conditional expected values
of a future (unknown) quantity given
One simple property:
 Martingales have “uncorrelated increments”
 You can perform simple tests (regression,
for example) to determine whether observed
increments (i.e., changes in forecasts for a
given product, region, and date) are
uncorrelated.
 If they’re correlated, you can use the
correlations to improve the forecasts
We checked our data and found
 Forecast changes had fairly high negative
correlations.
 That meant that if the forecasts were raised,
it was likely that later they would be
reduced.
 This meant that the forecasts were far from
optimal.
The forecasts were constructed by:
 Initially, forecasts (for the sales in some future
month) were simply set equal to a multiple
(like 1.06) of last year’s demand.
 As a given month grew closer, the sales
estimates for that month were updated
 Here’s an example of how the forecasts varied
over time (and how forecasts eventually
became “past data”)
Actual          Forecast             Trivial forecast
Delivery      March   April    May       June   July     August   Sept
Month
Observation
date

March 1 20            25       40        60     100      90       40
April 1       22 A 26          42        63     100      90       40
May 1         22A     24 A     41        62     102      90       40
June 1        22 A 24 A        45 A      66     110      92       40
July 1        22 A 24 A        45 A      65 A   108      91       41
Aug 1         22 A 24 A        45 A      65 A   110 A    92       42
Sept 1        22 A 24 A        45 A      65 A   110 A    88 A     40
 We decided to find out how they were done
forecasts
 It turned out that forecasts came from the
sales department.
 The job of forecasting was given to the
newest employee in the sales department!
 (We asked why. They said that this gave
new employees a good introduction to
“operations”.)
We went to the client
 Our advice: You have to fix the forecasting
problem!
a better forecasting technique.
 Remember that there were several plants
(and warehouses). If one plant were short
inventory, product could be shipped to it by
truck.
   The firm had tried to understand how many trucks
they needed to handle these shipments.
   For this they needed forecasts of demand.
   The produced forecasts as follows:
 For each “customer” (a “customer” might be a
chain of supermarkets, for example)
 They knew when the customer had last

 Based on season, temperatures, previous

sales etc., they could estimate when this
customer would reorder (and reorder, …)
 provided forecasts of future demand.
 We checked to see whether the successive
forecasts looked like martingales
(regression analysis again) and found that
they did (i.e., we couldn’t improve the
forecasts).
 But we weren’t done!
What should we recommend to the client?

   Client’s questions:
 How much money were we talking about
(saving)?
 If the benefits were small, then the risks
(of having a new model) and the costs of
implementation might outweigh the
benefits.
 How should inventory levels be set?
 We needed to have a simulation. (The
production process was far too complex for
us to build analytic models.)
 To run the simulation we needed to be able
to generate forecasts which behaved “like”
the observed data (from the “store by store
customer” analysis).
 We wanted to be sure that stock-outs would
be infrequent, and costs would be low.
A simulation process
We need to produce “simulated
forecasts” to drive the simulation
 These simulated forecasts should be the
results of a good estimation procedure
 The forecasts should resemble, as closely as
possible, the observed forecast behavior.
 So we analyzed the “good” forecasts to
build a model for generating random
variables to use in simulations
Recall the forecast and actual values
Delivery      March   April   May    June   July      August   Sept
Month
Observation
date

March 1 20            25      40     60     100       90       40
April 1       22 A 26         42     63     100       90       40
May 1         22A     24 A    41     62     102       90       40
June 1        22 A 24 A       45 A   66     110       92       40
July 1        22 A 24 A       45 A   65 A   108       91       41
Aug 1         22 A 24 A       45 A   65 A   110 A     92       42
Sept 1        22 A 24 A       45 A   65 A   110 A     88 A     40
How can we construct these?
 How do the forecasts evolve? To
understand this, take differences
 Want to compare two forecasts for several
months of demand at two successive
forecast dates.
 Example:
 Months out:      1 2 3           4   5
 March        20 25 40 60 100
 April        22 26 42 63 100
 Difference: 2      1     2      3  0
For each month
   We look at changes (from each month to the next)
of the forecasts for each future month.
   This gives us a sequence of vectors. The number
of components that change from month to month
is, in our example, 4.
   The first such vector is (2,1,2,3), giving the
changes, from March 1 to April 1, of the forecast
values (and one actual value) for each month.
 This gets us a set of vectors (or curves).
 On average (over all curves) the changes
should be 0. (This is the martingale
property.)
 We’d like to generate new curves which
have essentially the same variance-
covariance structure as the data. So we
estimate this VCV structure and generate
samples for use in the simulation.
The first 4 differences:
   Mar – April       2, 1, 2, 3
   Apr – May         -2, -1, -1, 2
   May – June         4, 4, 8, 2
   June – July       -1, -2, -1, -1
   I’ll call these vectors “wiggles” (They’re the
“wiggles” we’ve observed in past forecast
changes.)
   In practice we’d get many such vectors, compute
their covariance, and then choose our random
variables for a simulation to have a Normal
distribution with mean 0 and VCV equal to that of
the vectors above.
We use “Principal Components”
 Principal components analysis finds
“fundamental wiggles”, based on the data.
 These “fundamental wiggles”, expressed as
vectors, are orthogonal.
 The fundamental wiggles vary in size; we
arrange them in decreasing order of size (so
“biggest first”)
 We sometimes, to simplify the model, decide
how many of these wiggles to keep (say M)
and keep only wiggles 1,…M
How to get the simulated demands
and demand forecasts
   We assume we have a current forecast (vector) for the
demand in each future period.
   To get the next forecast vector we
    Report the first entry in the forecast vector as the
current realized demand. We then “slide the remaining
forecasts to the left “one place”.
   For each “fundamental wiggle” we add (the wiggle),
multiplied by an independent standard normal random
variable, to our forecast vector
   The result is our new forecast vector.
   We feed these forecasts to the simulation
routine and get
The final presentation
   Peter Jackson and I presented our work
 Audience:

 client’s project team members

 The head of that group

 The head of that division

 I presented the theory and model, and
Peter presented the simulation results and
our recommendations
The room was quiet
 Everybody seemed quite attentive
 They were participants in the project

 The “higher-ups” were present

 At the end of Peter’s presentation he
asserted, “Based on this analysis, we
believe you can achieve better product
availability and, at the same time, save
There was a long moment of silence
 Nobody spoke
 Finally, the most senior manager present
said, “That’s almost a million a month. We
have to do it!”
 Then there was applause …
 (Reports of how things went in later years
were positive. But, as usual, we didn’t get
details.)
Leftovers:
 You can find the slides of this talk, as well
as the published article, at
www.math.cmu.edu/~heath
 By the way, we didn’t actually use the data
in the say I described. We found that
modelling changes in the logarithm of the
data worked better. But it required some