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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.
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: 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)
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
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 3 types of participants Decision makers Staff Respondents Decision Makers (Evaluate responses and make decisions) Staff (Administering 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 | ei| MSE = (1/n) S ei2
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 (ei ) = 0.
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.)
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
Graph of Moving Average Moving Average Forecast Actual Sales Shed Sales
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) 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 = aDt + (1 - a ) Ft = aDt + (1 - a ) (a Dt-1 + (1 - a ) Ft-1) = aDt + (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) )
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