INDR 372 Production Planning and Control _INDR 372 Review Exercises for the Midterm Exam.

INDR 372 Review Exercises for the Midterm Exam, Spring 2017 1. Let us assume that Dt = µ+ϵt (for t=1,2,...) where ϵt are independent and identically distributed normal random variables with mean 0 and variance σ2. Consider the following three forecasts: i) Ft = Dt-1 ii) Gt = 0.8Dt-1 + 0.2Dt-2, iii) Ht = Dt-1 + (Dt-2 - Dt-3). For all three forecasts, (a) Determine whether the forecast is unbiased. (b) Find the variance of the error et = Ft - Dt. 2. Let us assume that Dt = a+bt+ϵt (for t=1,2,...) where ϵt are independent and identically distributed normal random variables with mean 0 and variance σ2. (a) Assume that b is unknown (but constant). Show that 0.5(Dt-1 - Dt-2) + 0.5(Dt-2 - Dt-3) is an unbiased estimator for b. What is the variance of this estimator? (b) Assume a is known, consider the following forecast: Ft = a + (Dt-1 - Dt-2)t. Is this unbiased? 3. The Australian Beer Production Data (see the blackboard page) for 1991 starts with the following monthly production quantities: y1 = 164, y2 = 148, y3 = 152, y4 = 144,y5 = 155, y6 = 125. (a) Compute F4 using a moving average with a 3 period window. (b) Compute F4 using exponential smoothing with α = 0.8. (c) Compute F4 using double exponential smoothing with α = 0.8 and β = 0.5. You can use an initial estimator of slope and level aˆ = 150 and ˆb = 10. 4. John Kittle, an independent insurance agent, uses a five-year moving average to forecast the number of claims made in a single year for one of the large insurance companies he sells for. He has just discovered that a clerk in his employ incorrectly entered the number of claims made four years ago as 1,400 when it should have been 1,200. (a) What adjustment should Mr. Kittle make in next years forecast to take into account the corrected value of the number of claims four years ago? (b) Suppose that Mr. Kittle used simple exponential smoothing with α = 0.2 instead of moving averages to determine his forecast. What adjustment is now required in next years forecast? (Note that you do not need to know the value of the forecast for next year in order to solve this problem.) 5. The owner of a small brewery in Milwaukee, Wisconsin, is using Winterss method to forecast his quarterly beer sales. He has been using smoothing constants of α = 0.2, β = 0.2, and γ = 0.2. He has currently obtained the following values of the various slope, intercept, and seasonal factors: S10 = 120, G10 = 14, c10 = 1.2, c9 = 1.1, c8 = 0.8, c7 = 0.9. (a) Determine the forecast for beer sales in quarter 11. (b) Suppose that the actual sales turn out to be 128 in quarter 11. Find S11 and G11, and find the updated values of the seasonal factors. Also determine the forecast made at the end of quarter 11 for quarter 13. 6. Consider a monthly sales data that exhibits increasing trend with a 12 month seasonality. (a) What would be an appropriate ARIMA model based on this information only? (i.e. Dt = α1Dt-1 + α2et-1 + ... ?) (b) How would you analyze the data for a complete ARIMA study? 7. Mr. Meadows Cookie Company makes a variety of chocolate chip cookies in the plant in Albion, Michigan. Based on orders received and forecasts of buying habits, it is estimated that the demand for the next four months is 850, 1,260, 510, and 980, expressed in thousands of cookies. During a 46-day period when there were 120 workers, the company produced 1.7 million cookies. Assume that the number of workdays over the four months are respectively 26, 24, 20, and 16. There are currently 100 workers employed, and there is no starting inventory of cookies. 2