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##### ISEN 315 Spring 2011 Dr. Gary Gaukler

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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