ISEN 315
Spring 2011
Dr. Gary Gaukler
A First Operations Model: Capacity
Strategy
Fundamental issues:
– Amount. When adding capacity, what is the optimal
amount to add?
• Too little
• Too much
– Timing. What is the optimal time between adding
new capacity?
– Type. Level of flexibility, automation, layout,
process, level of customization, outsourcing, etc.
Capacity Expansion Cost
Dynamic Capacity Expansion
Suppose demand exhibits a linear trend:
y: current demand (= current capacity)
D: rate of increase per unit time
Dynamic Capacity Expansion
Capacity leads demand
Optimal Expansion Size
• Need to satisfy all demands
• x is the time interval between expansions
• Hence, at the time of expansion, the expansion size
should be:
• Cash flows:
Sum of Discounted Costs
• Cost = C(x) = f(xD) + f(xD)e-rx + f(xD)e-2rx + ...
• After some algebra:
– Cost = C(x) = f(xD)/(1-e-rx)
• Want to find: min C(x) s.t. x>=0
• Result: rx / (erx-1) – a = 0
• Numerical solution only!
Graphical Solution
The solution is given by x that satisfies the
equation:
rx
a
rx
e 1
This is a transcendental equation, and has no
algebraic solution. However, using the graph
on the next slide, one can find the optimal
value of x for any value of a (0 < a < 1)
The function f(u) = u / (eu-1)
To Use: Locate the value of a
on the y axis and the corresponding value
of x on the x axis.
Recall: Model Assumptions
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•
•
•
•
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Infinite planning horizon
Demand grows linearly
Capacity expansion allowed at any time point
Any size capacity expansion allowed
No shortages allowed
Continuous discounting at rate r
Capacity expansion is instantaneous
Expansion cost for expanding by size x is f(x)=kxa
(0<a<1)
Introduction to Forecasting
• What is forecasting?
– Primary Function is to Predict the Future
• Why are we interested?
– Affects the decisions we make today
• Examples: who uses forecasting in their jobs?
– forecast demand for products and services
– forecast availability of manpower
– forecast inventory and materiel needs daily
What Makes a Good Forecast
•
•
•
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It should be timely
It should be as accurate as possible
It should be reliable
It should be in meaningful units
Forecasting Time Horizons
 Short-range forecast
 Up to 1 year, generally less than 3 months
 Purchasing, job scheduling, workforce levels,
job assignments, production levels
 Medium-range forecast
 3 months to 3 years
 Sales and production planning, budgeting
 Long-range forecast
 3+ years
 New product planning, facility location, research
and development
Characteristics of Forecasts
• They are usually wrong!
• Aggregate forecasts are usually
accurate
• Accuracy
as we go further into the
future
Aggregated Forecasts
Forecasting Approaches
Qualitative Methods
 Used when situation is vague
and little data exist
 New products
 New technology
 Involves intuition, experience
 e.g., forecasting sales on Internet
Jury of Executive Opinion
 Involves small group of high-level
managers
 Group estimates demand by working
together
 Relatively quick
 Disadvantage:
Sales Force Composite
 Each salesperson projects his or
her sales
 Combined at district and national
levels
 Sales reps know customers’ wants
 Disadvantage:
Delphi Method
 Iterative group
process, continues
until consensus is
reached
Staff
 3 types of
participants
(Administering
 Decision makers
 Staff
 Respondents
Decision Makers
(Evaluate
responses and
make decisions)
survey)
Respondents
(People who can
make valuable
judgments)
Consumer Market Survey
 Ask customers about purchasing
plans
 Sometimes difficult to answer
 Disadvantage:
Forecasting Approaches
Quantitative Methods
 Used when situation is ‘stable’
and historical data exist
 Existing products
 Current technology
 Involves mathematical
techniques
 e.g., forecasting sales of LCD
televisions
Quantitative Methods
• Stationary demand:
– moving average
– exponential smoothing
• Trend:
– Regression
– Double exponential smoothing
• Seasonality:
– Winter’s method
Notation Conventions
Let D1, D2, . . . Dn, . . . be the past values of the
series to be predicted (demand). If we are
making a forecast in period t, assume we have
observed Dt,, Dt-1 etc.
Let Ft, t + t  forecast made in period t for the
demand in period t + t where t = 1, 2, 3, …
Then Ft -1, t is the forecast made in t-1 for t and
Ft, t+1 is the forecast made in t for t+1. (one step
ahead) Use shorthand notation Ft = Ft - 1, t .
Evaluation of Forecasts
The forecast error in period t, et, is the
difference between the forecast for demand
in period t and the actual value of demand in
t.
For a multiple step ahead forecast: et = Ft - t, t Dt.
For one step ahead forecast: et = Ft - Dt.
MAD = (1/n) S | e i |
MSE = (1/n) S ei 2
Biases in Forecasts
• A bias occurs when the average value
of a forecast error tends to be positive
or negative.
• Mathematically an unbiased forecast is
one in which E (e i ) = 0.
Forecast Errors Over Time
Figure 2.3
Forecasting for Stationary Series
A stationary time series has the form:
Dt = m + e t where m is a constant and e t
is a random variable with mean 0 and
var s2 .
Two common methods for forecasting
stationary series are moving averages
and exponential smoothing.
Moving Averages
In words: the arithmetic average of the n
most recent observations. For a onestep-ahead forecast:
Ft = (1/n) (Dt - 1 + Dt - 2 + . . . + Dt - n )
(Go to Example.)
Moving Average Example
Actual
3-Month
Month
Shed Sales
Moving Average
January
10
February
12
March
13
April
16
May
19
June
23
July
26
Shed Sales
Graph of Moving Average
Moving
Average
Forecast
Actual
Sales
Moving Average Lags a Trend
Figure 2.4
In-class exercise
 In the example, we created the onestep-ahead forecast, e.g., forecast
August sales, given July and older data
 What if we are in July and want to
forecast September sales?
Potential Problems With Moving Average
 Increasing n smooths the forecast but
makes it less sensitive to changes
 Do not forecast trends well
 Require extensive historical data
Summary of Moving Averages
• Advantages of Moving Average Method
– Easily understood
– Easily computed
– Provides stable forecasts
• Disadvantages of Moving Average Method
– Requires saving all past N data points
– Lags behind a trend
– Ignores complex relationships in data
Exponential Smoothing Method
A type of weighted moving average that applies
declining weights to past data.
1. New Forecast = a (most recent observation)
+ (1 - a) (last forecast)
or
2. New Forecast = last forecast
a (last forecast error)
where 0 < a < 1 and generally is small for
stability of forecasts ( around .1 to .2)
Exponential Smoothing (cont.)
In symbols:
Ft+1 = a Dt + (1 - a ) Ft
= a Dt + (1 - a ) (a Dt-1 + (1 - a ) Ft-1)
= a Dt + (1 - a )(a )Dt-1 + (1 - a)2 (a )Dt - 2 + . . .
Hence the method applies a set of exponentially
declining weights to past data. It is easy to show
that the sum of the weights is exactly one.
(Or
Ft + 1 = Ft
- a (Ft - Dt)
)
Weights in Exponential Smoothing
Exponential Smoothing Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant a = .20
Forecast for next period:
Multiple-step-ahead forecasts:
Comparison of ES and MA
• Similarities
– Both methods are appropriate for stationary series
– Both methods depend on a single parameter
– Both methods lag behind a trend
• Differences
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