Chapter 9, Problem 9PA

(a)

Summary Introduction

Interpretation: The forecast for next quarter, using a four period moving averages needs to be calculated.

Concept Introduction: The moving average method takes the average of the recent number of observations in any time series. The average is taken based on the k number of previous observations.

Explanation of Solution

B.

Following formula can be used to calculate the forecast values as per moving average method

  $\begin{array}{l}{F}_{t+1}=\frac{{\displaystyle \sum \left(A_t+A_{t-1}+A_{t-2}+A_{t-k+1}\right)}}{k}\\ where,\\ {\text{F}}_{\text{t+1}}\text{ = Forecasted Value for current period}\\ {\text{A}}_{\text{t}}\text{ = Actual Value for previous period}\end{array}$

The forecast for next quarter using a four-period moving average is as follows:

Year	Quarter	Sales	Forecast
1	1	4
1	2	2
1	3	1
1	4	5
2	1	6	3
2	2	4	3.5
2	3	4	4
2	4	14	4.75
3	1	10	7
3	2	3	8
3	3	5	7.75
3	4	16	8
4	1	12	8.5
4	2	9	9
4	3	7	10.5
4	4	22	11
5	1	18	12.5
5	2	10	14
5	3	13	14.25
5	4	35	15.75
6	1	Forecast	19

  ${\text{F}}_{November}=\frac{A_{(5,1)}+A_{(5,2)}+A_{(5,3)}+A_{(5,4)}}{4}=\frac{18+10+33+35}{4}=19$

Therefore, the forecast for next quarter, using a four-period moving average is 19.

(b)

Summary Introduction

Interpretation: The good value of a for a single exponential smoothing model needs to be determined .

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

			Forecast
Year	Quarter	Sales	Alpha = 0.1	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	3.8	2.8	7.84
1	4	5	3.52	1.48	2.1904
2	1	6	3.668	2.33	5.438224
2	2	4	3.9012	0.1	0.00976144
2	3	4	3.91108	0.09	0.007906766
2	4	14	3.919972	10.1	101.6069645
3	1	10	4.9279748	5.07	25.72543963
3	2	3	5.4351773	2.44	5.93008858
3	3	5	5.1916596	0.19	0.036733398
3	4	16	5.1724936	10.8	117.2348942
4	1	12	6.2552443	5.74	33.00221844
4	2	9	6.8297198	2.17	4.710115974
4	3	7	7.0467479	0.05	0.002185362
4	4	22	7.0420731	15	223.739578
5	1	18	8.5378658	9.46	89.53198432
5	2	10	9.4840792	0.52	0.266174285
5	3	13	9.5356713	3.46	12.00157356
5	4	35	9.8821041	25.1	630.9086924
6	1		12.393894	SUM	1264.182935
				MSE	66.53594394

At a = 0.2

			Forecast
Year	Quarter	Sales	Alpha = 0.2	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	3.6	2.6	6.76
1	4	5	3.08	1.92	3.6864
2	1	6	3.464	2.54	6.431296
2	2	4	3.9712	0.03	0.00082944
2	3	4	3.97696	0.02	0.000530842
2	4	14	3.981568	10	100.3689797
3	1	10	5.9852544	4.01	16.11818223
3	2	3	6.7882035	3.79	14.35048591
3	3	5	6.0305628	1.03	1.062059718
3	4	16	5.8244503	10.2	103.5418127
4	1	12	7.8595602	4.14	17.14324172
4	2	9	8.6876482	0.31	0.097563671
4	3	7	8.7501185	1.75	3.062914867
4	4	22	8.4000948	13.6	184.9574208
5	1	18	11.120076	6.88	47.33335619
5	2	10	12.496061	2.5	6.230318954
5	3	13	11.996849	1	1.006312832
5	4	35	12.197479	22.8	519.9549713
6	1		16.757983	SUM	1036.106677
				MSE	54.53193036

At a = 0.3

			Forecast
Year	Quarter	Sales	Alpha = 0.3	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	3.4	2.4	5.76
1	4	5	2.68	2.32	5.3824
2	1	6	3.376	2.62	6.885376
2	2	4	4.1632	0.16	0.02663424
2	3	4	4.11424	0.11	0.013050778
2	4	14	4.079968	9.92	98.40703488
3	1	10	7.0559776	2.94	8.667267892
3	2	3	7.9391843	4.94	24.39554175
3	3	5	6.457429	1.46	2.12409936
3	4	16	6.0202003	9.98	99.59640172
4	1	12	9.0141402	2.99	8.915358615
4	2	9	9.9098982	0.91	0.827914653
4	3	7	9.6369287	2.64	6.953393015
4	4	22	8.8458501	13.2	173.0316597
5	1	18	12.792095	5.21	27.12227379
5	2	10	14.354467	4.35	18.96137891
5	3	13	13.048127	0.05	0.002316168
5	4	35	13.033689	22	482.5188362
6	1		19.623582	SUM	973.5909376
				MSE	51.2416283

At a = 0.4

			Forecast
Year	Quarter	Sales	Alpha = 0.4	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	3.2	2.2	4.84
1	4	5	2.32	2.68	7.1824
2	1	6	3.392	2.61	6.801664
2	2	4	4.4352	0.44	0.18939904
2	3	4	4.26112	0.26	0.068183654
2	4	14	4.156672	9.84	96.89110612
3	1	10	8.0940032	1.91	3.632823802
3	2	3	8.8564019	5.86	34.29744345
3	3	5	6.5138412	1.51	2.291715033
3	4	16	5.9083047	10.1	101.8423142
4	1	12	9.9449828	2.06	4.223095632
4	2	9	10.76699	1.77	3.12225256
4	3	7	10.060194	3.06	9.364786175
4	4	22	8.8361163	13.2	173.2878344
5	1	18	14.10167	3.9	15.19697856
5	2	10	15.661002	5.66	32.0469421
5	3	13	13.396601	0.4	0.157292447
5	4	35	13.237961	21.8	473.5863558
6	1		21.942776	SUM	973.0225869
				MSE	51.2117151

At a = 0.5

			Forecast
Year	Quarter	Sales	Alpha = 0.5	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	3	2	4
1	4	5	2	3	9
2	1	6	3.5	2.5	6.25
2	2	4	4.75	0.75	0.5625
2	3	4	4.375	0.38	0.140625
2	4	14	4.1875	9.81	96.28515625
3	1	10	9.09375	0.91	0.821289063
3	2	3	9.546875	6.55	42.86157227
3	3	5	6.2734375	1.27	1.621643066
3	4	16	5.6367188	10.4	107.3975983
4	1	12	10.818359	1.18	1.396274567
4	2	9	11.40918	2.41	5.804146767
4	3	7	10.20459	3.2	10.26939607
4	4	22	8.6022949	13.4	179.4985014
5	1	18	15.301147	2.7	7.283805028
5	2	10	16.650574	6.65	44.23013094
5	3	13	13.325287	0.33	0.105811545
5	4	35	13.162643	21.8	476.8701419
6	1		24.081322	SUM	998.398592
				MSE	52.54729432

At a = 0.6

			Forecast
Year	Quarter	Sales	Alpha = 0.6	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	2.8	1.8	3.24
1	4	5	1.72	3.28	10.7584
2	1	6	3.688	2.31	5.345344
2	2	4	5.0752	1.08	1.15605504
2	3	4	4.43008	0.43	0.184968806
2	4	14	4.172032	9.83	96.58895501
3	1	10	10.068813	0.07	0.004735201
3	2	3	10.027525	7.03	49.38610931
3	3	5	5.81101	0.81	0.657737298
3	4	16	5.324404	10.7	113.9683495
4	1	12	11.729762	0.27	0.073028789
4	2	9	11.891905	2.89	8.363112465
4	3	7	10.156762	3.16	9.965145423
4	4	22	8.2627047	13.7	188.713281
5	1	18	16.505082	1.49	2.234780134
5	2	10	17.402033	7.4	54.79008896
5	3	13	12.960813	0.04	0.001535613
5	4	35	12.984325	22	484.6899351
6	1		26.19373	SUM	1034.121562
				MSE	54.42745061

At a = 0.7

			Forecast
Year	Quarter	Sales	Alpha = 0.7	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	2.6	1.6	2.56
1	4	5	1.48	3.52	12.3904
2	1	6	3.944	2.06	4.227136
2	2	4	5.3832	1.38	1.91324224
2	3	4	4.41496	0.41	0.172191802
2	4	14	4.124488	9.88	97.52573726
3	1	10	11.037346	1.04	1.076087554
3	2	3	10.311204	7.31	53.45370276
3	3	5	5.1933612	0.19	0.037388544
3	4	16	5.0580084	10.9	119.7271812
4	1	12	12.717403	0.72	0.514666355
4	2	9	12.215221	3.22	10.33764448
4	3	7	9.9645662	2.96	8.788652906
4	4	22	7.8893699	14.1	199.1098827
5	1	18	17.766811	0.23	0.054377128
5	2	10	17.930043	7.93	62.88558655
5	3	13	12.379013	0.62	0.385624871
5	4	35	12.813704	22.2	492.2317348
6	1		28.344111	SUM	1071.391237
				MSE	56.38901248

At a = 0.8

			Forecast
Year	Quarter	Sales	Alpha = 0.8	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	2.4	1.4	1.96
1	4	5	1.28	3.72	13.8384
2	1	6	4.256	1.74	3.041536
2	2	4	5.6512	1.65	2.72646144
2	3	4	4.33024	0.33	0.109058458
2	4	14	4.066048	9.93	98.68340234
3	1	10	12.01321	2.01	4.053012894
3	2	3	10.402642	7.4	54.7991074
3	3	5	4.4805284	0.52	0.26985076
3	4	16	4.8961057	11.1	123.2964691
4	1	12	13.779221	1.78	3.165627849
4	2	9	12.355844	3.36	11.26169048
4	3	7	9.6711688	2.67	7.135143001
4	4	22	7.5342338	14.5	209.2583926
5	1	18	19.106847	1.11	1.225109736
5	2	10	18.221369	8.22	67.590914
5	3	13	11.644274	1.36	1.837993339
5	4	35	12.728855	22.3	496.0039097
6	1		30.545771	SUM	1104.256079
				MSE	58.11874101

At a = 0.9

			Forecast
Year	Quarter	Sales	Alpha = 0.9	Error	Squared Error
1	1	4
1	2	2	4	2	4
1	3	1	2.2	1.2	1.44
1	4	5	1.12	3.88	15.0544
2	1	6	4.612	1.39	1.926544
2	2	4	5.8612	1.86	3.46406544
2	3	4	4.18612	0.19	0.034640654
2	4	14	4.018612	9.98	99.62810641
3	1	10	13.001861	3	9.011170664
3	2	3	10.300186	7.3	53.29271739
3	3	5	3.7300186	1.27	1.612852726
3	4	16	4.8730019	11.1	123.8100876
4	1	12	14.8873	2.89	8.336502365
4	2	9	12.28873	3.29	10.81574514
4	3	7	9.328873	2.33	5.423649459
4	4	22	7.2328873	14.8	218.0676175
5	1	18	20.523289	2.52	6.366986015
5	2	10	18.252329	8.25	68.10093183
5	3	13	10.825233	2.17	4.729611994
5	4	35	12.782523	22.2	493.6162714
6	1		32.778252	SUM	1128.731901
				MSE	59.40694213

After evaluating the MSE of all forecasting models above, The good value of a for a single exponential smoothing model is found at 0.4 as the MSE is the least for this model.