Lecture 2

ISEN 315
Spring 2011
Dr. Gary Gaukler
A First Operations Model: Capacity
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
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
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
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
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
• Accuracy
as we go further into the
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
 Group estimates demand by working
 Relatively quick
 Disadvantage:
Sales Force Composite
 Each salesperson projects his or
her sales
 Combined at district and national
 Sales reps know customers’ wants
 Disadvantage:
Delphi Method
 Iterative group
process, continues
until consensus is
 3 types of
 Decision makers
 Staff
 Respondents
Decision Makers
responses and
make decisions)
(People who can
make valuable
Consumer Market Survey
 Ask customers about purchasing
 Sometimes difficult to answer
 Disadvantage:
Forecasting Approaches
Quantitative Methods
 Used when situation is ‘stable’
and historical data exist
 Existing products
 Current technology
 Involves mathematical
 e.g., forecasting sales of LCD
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
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
Shed Sales
Moving Average
Shed Sales
Graph of Moving Average
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)
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.
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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