Chapter 9, Problem 7PA

Summary Introduction

Interpretation: The best smoothing constant needs to be determined by evaluating the MSE .

Concept Introduction: Single Exponential Smoothing is a method which computes the weighted average of previous sales data to forecast the future sales value.

Explanation of Solution

Following formula can be used to calculate the forecasted values:

  $\begin{array}{l}{\text{F}}_{\text{t+1}}{\text{ = αA}}_{\text{t}}{\text{+(1-α)F}}_{\text{t}}\\ \text{where,}\\ \text{α = smoothing constant}\\ {\text{A}}_{\text{t}}\text{= Actual Sales of previous period}\\ {\text{F}}_{\text{t}}\text{= Forecasted Sales of previous period}\end{array}$

The Mean Square Error for various values of a is as follows:

At a = 0.1

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1614	159	25281
4	1386	1598.1	212.1	44986.41
5	1209	1576.89	367.89	135343.0521
6	1348	1540.101	192.101	36902.7942
7	1581	1520.8909	60.1091	3613.103903
8	1332	1526.90181	194.9018	37986.71554
9	1245	1507.411629	262.4116	68859.86303
10	1521	1481.170466	39.8295	1586.38907
11	1421	1485.153419	64.15342	4115.661232
12	1502	1478.738078	23.2619	541.1159916
13	1656	1481.06427	174.396	30413.96482
14	1614	1498.557843	115.442	13326.85536
15	1332	1510.102059	178.1021	31720.34325
16		1492.291853	SUM	442777.2685
			MSE	34059.79

At a = 0.2

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1605	150	22500
4	1386	1575	212.1	44986.41
5	1209	1537.2	367.89	135343.0521
6	1348	1471.56	123.56	15267.0736
7	1581	1446.848	134.152	17996.7591
8	1332	1473.6784	141.6784	20072.76903
9	1245	1445.34272	200.3427	40137.20546
10	1521	1405.274176	39.8295	1586.38907
11	1421	1428.419341	7.419341	55.04661791
12	1502	1426.935473	23.2619	541.1159916
13	1656	1441.948378	174.396	30413.96482
14	1614	1484.758702	115.442	13326.85536
15	1332	1510.606962	178.607	31900.44687
16		1474.88557	SUM	382227.088
			MSE	29402.08

At a = 0.3

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1596	141	19881
4	1386	1553.7	212.1	44986.41
5	1209	1503.39	367.89	135343.0521
6	1348	1415.073	67.073	4498.787329
7	1581	1394.9511	186.0489	34614.19319
8	1332	1450.76577	118.7658	14105.30812
9	1245	1415.136039	170.136	28946.27177
10	1521	1364.095227	39.8295	1586.38907
11	1421	1411.166659	9.83334	96.69457556
12	1502	1414.116661	23.2619	541.1159916
13	1656	1440.481663	174.396	30413.96482
14	1614	1505.137164	115.442	13326.85536
15	1332	1537.796015	205.796	42351.99973
16		1476.05721	SUM	378792.0421
			MSE	29137.84

At a = 0.4

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1587	132	17424
4	1386	1534.2	212.1	44986.41
5	1209	1474.92	367.89	135343.0521
6	1348	1368.552	20.552	422.384704
7	1581	1360.3312	220.6688	48694.71929
8	1332	1448.59872	116.5987	13595.26151
9	1245	1401.959232	156.9592	24636.20051
10	1521	1339.175539	39.8295	1586.38907
11	1421	1411.905324	9.83334	96.69457556
12	1502	1415.543194	23.2619	541.1159916
13	1656	1450.125916	174.396	30413.96482
14	1614	1532.47555	115.442	13326.85536
15	1332	1565.08533	233.0853	54328.77103
16		1471.851198	SUM	393495.819
			MSE	30268.909

At a = 0.5

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1578	123	15129
4	1386	1516.5	212.1	44986.41
5	1209	1451.25	367.89	135343.0521
6	1348	1330.125	17.875	319.515625
7	1581	1339.0625	241.9375	58533.75391
8	1332	1460.03125	128.0313	16392.00098
9	1245	1396.015625	151.0156	22805.71899
10	1521	1320.507813	39.8295	1586.38907
11	1421	1420.753906	9.83334	96.69457556
12	1502	1420.876953	23.2619	541.1159916
13	1656	1461.438477	174.396	30413.96482
14	1614	1558.719238	115.442	13326.85536
15	1332	1586.359619	254.3596	64698.81585
16		1459.17981	SUM	412273.2873
			MSE	31713.329

At a = 0.6

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1569	114	12996
4	1386	1500.6	212.1	44986.41
5	1209	1431.84	367.89	135343.0521
6	1348	1298.136	17.875	319.515625
7	1581	1328.0544	252.9456	63981.47656
8	1332	1479.82176	147.8218	21851.27273
9	1245	1391.128704	146.1287	21353.59813
10	1521	1303.451482	39.8295	1586.38907
11	1421	1433.980593	9.83334	96.69457556
12	1502	1426.192237	23.2619	541.1159916
13	1656	1471.676895	174.396	30413.96482
14	1614	1582.270758	115.442	13326.85536
15	1332	1601.308303	269.3083	72526.96216
16		1439.723321	SUM	427423.3071
			MSE	32878.71

At a = 0.7

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1560	105	11025
4	1386	1486.5	212.1	44986.41
5	1209	1416.15	367.89	135343.0521
6	1348	1271.145	17.875	319.515625
7	1581	1324.9435	256.0565	65564.93119
8	1332	1504.18305	172.1831	29647.00271
9	1245	1383.654915	138.6549	19225.18545
10	1521	1286.596475	39.8295	1586.38907
11	1421	1450.678942	9.83334	96.69457556
12	1502	1429.903683	23.2619	541.1159916
13	1656	1480.371105	174.396	30413.96482
14	1614	1603.311331	115.442	13326.85536
15	1332	1610.793399	278.7934	77725.75957
16		1415.63802	SUM	437901.8765
			MSE	33684.75

At a = 0.8

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1551	96	9216
4	1386	1474.2	212.1	44986.41
5	1209	1403.64	367.89	135343.0521
6	1348	1247.928	17.875	319.515625
7	1581	1327.9856	253.0144	64016.28661
8	1332	1530.39712	198.3971	39361.41722
9	1245	1371.679424	126.6794	16047.67646
10	1521	1270.335885	39.8295	1586.38907
11	1421	1470.867177	9.83334	96.69457556
12	1502	1430.973435	23.2619	541.1159916
13	1656	1487.794687	174.396	30413.96482
14	1614	1622.358937	115.442	13326.85536
15	1332	1615.671787	283.6718	80469.68301
16		1388.734357	SUM	443825.0609
			MSE	34140.38

At a = 0.9

Period	Observation	Forecast	Error	Squared Error
1	1623	1623
2	1533	1623	90	8100
3	1455	1542	87	7569
4	1386	1463.7	212.1	44986.41
5	1209	1393.77	367.89	135343.0521
6	1348	1227.477	17.875	319.515625
7	1581	1335.9477	245.0523	60050.62974
8	1332	1556.49477	224.4948	50397.90176
9	1245	1354.449477	109.4495	11979.18802
10	1521	1255.944948	39.8295	1586.38907
11	1421	1494.494495	9.83334	96.69457556
12	1502	1428.349449	23.2619	541.1159916
13	1656	1494.634945	174.396	30413.96482
14	1614	1639.863494	115.442	13326.85536
15	1332	1616.586349	284.5863	80989.39029
16		1360.458635	SUM	445700.1073
			MSE	34284.623

After evaluating the MSE of all forecasting models above, the best single exponential smoothing model is found at a = 0.3 as the MSE is the least for this model.

After comparing the MSEs of Exponential smoothing method performed in this question and moving average method in previous problem, The moving average method seems better option based on the given data.