a. The Naive Approach The naive approach assumes that the sales for September will be the same as the sales for August.

$Forecast for September=Sales in August=20\backslash text\{Forecast\; for\; September\}\; =\; \backslash text\{Sales\; in\; August\}\; =\; 20$

b. A Five-Month Moving Average To calculate a five-month moving average, we take the average of the sales of the last five months (April through August).

$Five Month Moving Average=\frac{Sales in Apr+Sales in May+Sales in June+Sales in July+Sales in Aug}{5}\backslash text\{Five\; Month\; Moving\; Average\}\; =\; \backslash frac\{\backslash text\{Sales\; in\; Apr\}\; +\; \backslash text\{Sales\; in\; May\}\; +\; \backslash text\{Sales\; in\; June\}\; +\; \backslash text\{Sales\; in\; July\}\; +\; \backslash text\{Sales\; in\; Aug\}\}\{5\}$​

$Five Month Moving Average=\frac{15+20+18+22+20}{5}=\frac{95}{5}=19\backslash text\{Five\; Month\; Moving\; Average\}\; =\; \backslash frac\{15\; +\; 20\; +\; 18\; +\; 22\; +\; 20\}\{5\}\; =\; \backslash frac\{95\}\{5\}\; =\; 19$

c. A Weighted Average For the weighted average, we use the given weights: 0.60 for August, 0.30 for July, and 0.10 for June.

$Weighted Average=(0.60\times Sales in Aug)+(0.30\times Sales in July)+(0.10\times Sales in June)\backslash text\{Weighted\; Average\}\; =\; (0.60\; \backslash times\; \backslash text\{Sales\; in\; Aug\})\; +\; (0.30\; \backslash times\; \backslash text\{Sales\; in\; July\})\; +\; (0.10\; \backslash times\; \backslash text\{Sales\; in\; June\})$

$Weighted Average=(0.60\times 20)+(0.30\times 22)+(0.10\times 18)\backslash text\{Weighted\; Average\}\; =\; (0.60\; \backslash times\; 20)\; +\; (0.30\; \backslash times\; 22)\; +\; (0.10\; \backslash times\; 18)$

$Weighted Average=12+6.6+1.8=20.4\backslash text\{Weighted\; Average\}\; =\; 12\; +\; 6.6\; +\; 1.8\; =\; 20.4$

d. Exponential Smoothing For exponential smoothing with a smoothing constant of 0.20 and a previous forecast of 19(000):

$Forecast for September=\alpha \times \left(Sales in Aug\right)+(1-\alpha )\times \left(Previous Forecast\right)\backslash text\{Forecast\; for\; September\}\; =\; \backslash alpha\; \backslash times\; (\backslash text\{Sales\; in\; Aug\})\; +\; (1\; –\; \backslash alpha)\; \backslash times\; (\backslash text\{Previous\; Forecast\})$Forecast for September=α×(Sales in Aug)+(1−α)×(Previous Forecast)

$Forecast for September=0.20\times 20+0.80\times 19\backslash text\{Forecast\; for\; September\}\; =\; 0.20\; \backslash times\; 20\; +\; 0.80\; \backslash times\; 19$

$Forecast for September=4+15.2=19.2\backslash text\{Forecast\; for\; September\}\; =\; 4\; +\; 15.2\; =\; 19.2$

Summary of Forecasts:

• Naive Approach: 20
• Five-Month Moving Average: 19
• Weighted Average: 20.4
• Exponential Smoothing: 19.2