EBK OM
6th Edition
ISBN: 9781305888210
Author: Collier
Publisher: YUZU
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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.
Expert Solution & Answer
Explanation of Solution
Following formula can be used to calculate the forecasted values:
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
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.
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