Question

A pharmacist has been monitoring sales of a certain over-the-counter pain reliever. Monthly sales during the last 15 months were:

Month	Number sold		Month	Number sold		Month	Number sold
1	36		6	49		11	52
2	38		7	50		12	55
3	42		8	49		13	54
4	44		9	52		14	56
5	48		10	48		15	57

1. Using 3-Month Weighted Moving Average using 0.2, 0.3, and 0.5 with the heaviest weights applied to the most recent months, develop forecasts for months from 11 to 16.
2. Using Exponential Smoothing using an α = 0.4 and the 10^th Month forecast of 50, develop forecasts for months from 11 to 16.  
3. Using Trend Projection, develop forecasts for months from 11 to 16.
4. Justify which method you would suggest using to predict future sales.

Answer 1

a. 3-Month Weighted Moving Average using 0.2, 0.3, and 0.5 with the heaviest weights applied to the most recent months, the forecasts for months 11 to 16 are

Month Number	Sold	3 month weighted moving average
1	36
2	38
3	42
4	44
5	48
6	49
7	50
8	49
9	52
10	48
11	52	49.4
12	55	50.8
13	54	52.7
14	56	53.9
15	57	55.2
16		56.1

Forecast for month 11 = 0.2*49+0.3*52+0.5*48 = 49.4

Similarly the forecast for months 12 to 16 are calculated

Calculating Error (Mean Absolute Deviation)

Month Number	Sold	3 month weighted moving average	|D-F| for moving average
1	36
2	38
3	42
4	44
5	48
6	49
7	50
8	49
9	52
10	48
11	52	49.4	2.6
12	55	50.8	4.2
13	54	52.7	1.3
14	56	53.9	2.1
15	57	55.2	1.8
16		56.1
Total			12

The absolute value of Demand - Forecast is calculated. The sum is 12 as shown in the above table.

MAD = ∑|D-F| / n = 12/5 = 2.4

The demand for 16th month is not available, hence n = 5.

Excel Formulae

b. Exponential Smoothing using an α = 0.4 and the 10th Month forecast of 50, the forecasts for months 11 to 16 are

Month Number	Sold	Exponential Smoothing Using α =0.4
1	36
2	38
3	42
4	44
5	48
6	49
7	50
8	49
9	52
10	48	50
11	52	49
12	55	50
13	54	52
14	56	53
15	57	54
16		55

Forecast for 11th Month = Demand for Month 10 * α + Forecast for Month 10 * (1- α)

= 48*0.4+50*(1-0.4) = 49

Similarly the values for other months can be calculated.

Calculating MAD:

Month Number	Sold	Exponential Smoothing Using α =0.4	|D-F| for exponential smoothing
1	36
2	38
3	42
4	44
5	48
6	49
7	50
8	49
9	52
10	48	50.0
11	52	49.2	2.8
12	55	50.3	4.7
13	54	52.2	1.8
14	56	52.9	3.1
15	57	54.1	2.9
16		55.3
		Total	15.2

MAD = ∑|D-F| / n = 15.2/5 = 3.04

Excel Formulae

c. Trend Projection:

Month Number (x)	Sold (y)	xy	x^2
1	36	36	1296
2	38	76	1444
3	42	126	1764
4	44	176	1936
5	48	240	2304
6	49	294	2401
7	50	350	2500
8	49	392	2401
9	52	468	2704
10	48	480	2304
11	52	572	2704
12	55	660	3025
13	54	702	2916
14	56	784	3136
15	57	855	3249
120	730	6211	36084

The xy and x^2 are calculated to help in finding the trend equation

Trend Equation:

Y = a + bX

b = (∑xy – nx̄ȳ) / (∑(x^2)-n (x̄^2))

a = ȳ - b x̄

x̄ = ∑x /n = 120/15 = 8

ȳ = ∑y/n = 730/15 = 48.66

b = (6211-15*8*48.66)/(36084-15*8^2) = 0.0106

a = 48.66-0.0106*8 = 48.57

Equation is

y = 48.57+0.0106x

replacing values of x in the equation

Month Number (x)	Sold (y)	xy	x^2	Forecast	|D-F|
1	36	36	1296
2	38	76	1444
3	42	126	1764
4	44	176	1936
5	48	240	2304
6	49	294	2401
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