ISEN 315 Spring 2011 Dr. Gary Gaukler

ISEN 315Spring 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 ExpansionSuppose demand exhibits a linear trend:

y: current demand (= current capacity)D: rate of increase per unit time

Dynamic Capacity ExpansionCapacity 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:

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)

1rx

rx ae

To Use: Locate the value of on the axis and the corresponding valueof on the axis.

ay

x x

The function f(u) = u / (eu-1)

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

(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

• 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

Used when situation is vague and little data exist New products New technology

Involves intuition, experience e.g., forecasting sales on Internet

Qualitative Methods

Involves small group of high-level managers

Group estimates demand by working together

Relatively quick Disadvantage:

Jury of Executive Opinion

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

3 types of participants Decision makers Staff Respondents

Staff(Administering

survey)

Decision Makers(Evaluate

responses and make decisions)

Respondents(People who can make valuable

judgments)

Consumer Market Survey

Ask customers about purchasing plans

Sometimes difficult to answer Disadvantage:

Forecasting Approaches

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

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 ForecastsThe 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 one-step-ahead forecast:

Ft = (1/n) (Dt - 1 + Dt - 2 + . . . + Dt - n )

(Go to Example.)

January 10February 12March 13April 16May 19June 23July 26

Actual 3-MonthMonth Shed Sales Moving Average

Moving Average Example

Graph of Moving AverageSh

ed S

ales Actual

Sales

Moving Average Forecast

Moving Average Lags a Trend Figure 2.4

In the example, we created the one-step-ahead forecast, e.g., forecast August sales, given July and older data

What if we are in July and want to forecast September sales?

In-class exercise

Increasing n smooths the forecast but makes it less sensitive to changes

Do not forecast trends well Require extensive historical data

Potential Problems With Moving Average

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)

or2. 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 MustangsActual demand = 153Smoothing 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– –