Question

Please answer the following:

1. United Dairies, Inc., supplies milk to several independent grocers throughout Dade County, Florida. Managers at United Dairies want to develop a forecast of the number of half gallons of milk sold per week. Sales data for the past 12 weeks are:

Week	Sales
1	2,750
2	3,100
3	3,250
4	2,800
5	2,900
6	3,050
7	3,300
8	3,100
9	2,950
10	3,000
11	3,200
12	3,150

1. Compute four-week and five-week moving averages for the time series.

1. 1. Compute the MSE for the four-week and five-week moving average forecasts.
   2. What appears to be the best number of weeks of past data (three, four, or five) to use in the moving average computation? The MSE for three weeks is 23527.78
2. Show the exponential smoothing forecasts using α = 0.1.
   1. Applying the MSE measure of forecast accuracy, would you prefer a smoothing constant of α=0.1 or α= 0.2 for the United Dairies sales time series?
   2. Are the results the same if you apply MAE as the measure of accuracy?
   3. What are the results if MAPE is used?

3.    Use exponential smoothing with a α = 0.4 to develop a forecast of demand for week 13. What is the resulting MSE?

           

Answer 1

Soln

1)

Four Month Moving Average

Week	Sales	Four-Month Moving Average	Deviation	Error	Error2
1	2,750
2	3,100
3	3,250
4	2,800
5	2,900	2,975	-75	75	5,625.00
6	3,050	3,013	38	37.5	1,406.25
7	3,300	3,000	300	300	90,000.00
8	3,100	3,013	88	87.5	7,656.25
9	2,950	3,088	-138	137.5	18,906.25
10	3,000	3,100	-100	100	10,000.00
11	3,200	3,088	113	112.5	12,656.25
12	3,150	3,063	88	87.5	7,656.25
			Mean	117.19	19,238.28

Four Month Moving Average = (Sum of Previous 4 Months Value)/4

Error = Sales - Four Month Moving Average

Five Month Moving Average

Week	Sales	Five-Month Moving Average	Deviation	Error	Error2
1	2,750
2	3,100
3	3,250
4	2,800
5	2,900
6	3,050	2,960	90	90	8,100.00
7	3,300	3,020	280	280	78,400.00
8	3,100	3,060	40	40	1,600.00
9	2,950	3,030	-80	80	6,400.00
10	3,000	3,060	-60	60	3,600.00
11	3,200	3,080	120	120	14,400.00
12	3,150	3,110	40	40	1,600.00
			Mean	101.43	16,300.00

a)

MSE for Four Month Moving Average = Average of Error² = 19238.28

MSE for Five Month Moving Average = Average of Error² = 16300

b)

The best number for past weeks’ computation seems to be Five Month Moving Average as MSE is the least for Five Month Moving Average.

2)

Exponential Smoothing = αA_t + (1-α) F_t

Week	Sales	Exponential Smoothing
		(alpha = 0.1)	Error	Error 2
1	2,750	2,750	0	0
2	3,100	2,750	350	1,22,500
3	3,250	2,785	465	2,16,225
4	2,800	2,832	32	992
5	2,900	2,828	72	5,134
6	3,050	2,836	214	46,004
7	3,300	2,857	443	1,96,281
8	3,100	2,901	199	39,495
9	2,950	2,921	29	833
10	3,000	2,924	76	5,772
11	3,200	2,932	268	72,026
12	3,150	2,958	192	36,687
		Mean		61,829

Alpha = 0.2

Week	Sales	Exponential Smoothing
		(alpha = 0.2)	Error	Error 2
1	2,750	2,750	0	0
2	3,100	2,750	350	1,22,500
3	3,250	2,820	430	1,84,900
4	2,800	2,906	106	11,236
5	2,900	2,885	15	231
6	3,050	2,888	162	26,296
7	3,300	2,920	380	1,44,193
8	3,100	2,996	104	10,771
9	2,950	3,017	67	4,486
10	3,000	3,004	4	13
11	3,200	3,003	197	38,863
12	3,150	3,042	108	11,601
		Mean		46,257

a)

For Exponential Smoothing with alpha = 0.1 , MSE = 61,829

For Exponential Smoothing with alpha = 0.2 , MSE = 46,257

Since, MSE for alpha = 0.2 is less, hence it is appropriate

b)

For Exponential Smoothing with alpha = 0.1 , MAE = 186.58

For Exponential Smoothing with alpha = 0.2 , MSE = 129.53

PS: we are only allowed to answer 4 parts per question