Chapter 2.7, Problem 24P

a.

Summary Introduction

To calculate:The observed weekly sales of ball peen hammers at one of the town hardware shop over an eight week period have been 14, 9, 30, 22, 34, 12, 19, and 23. To get the solution of the one step ahead forecast for weeks through 8.

Introduction:The current forecast is the weighted average of the last forecast and the current value of demand

  $\begin{array}{l}{F}_t=\alpha D_{t-1}+\left(1-\alpha \right)F_{t-1}\\ where0\le \alpha \end{array}$

Answer to Problem 24P

The one step ahead forecast from week 4 to week 8 are F₄=17.67, F₅= 20.33, F₆= 28.67, F₇= 22.67, F₈= 21.67

Explanation of Solution

Four month average of one step ahead forecast from week 4 to week 8 is calculated below:

Week	Demand	MAD
1	14
2	9
3	30
4	22	17.67
5	34	20.33
6	12	28.67
7	19	22.67
8	23	21.67

Therefore, one step ahead forecast from week 4 to week 8 are F₄=17.67, F₅= 20.33, F₆= 28.67, F₇= 22.67, F₈= 21.67.

b.

Summary Introduction

To calculate:The exponential smoothing forecasts for weeks 4 through 8.

Introduction:Forecasting is the main function of predicting the future using the information available for decision making. It is a mechanism for planning decisions based on the predicted information.

Answer to Problem 24P

Week	Demand	MAD	Exponential Smoothing	$\left|\text{Error}\right|$	MAD   $\left|\text{Error}\right|$
1	14
2	9
3	30
4	22	17.67	17.67	4.33	4.33
5	34	20.33	18.32	15.68	13.67
6	12	28.67	20.67	8.67	16.67
7	19	22.67	19.37	0.37	3.67
8	23	21.67	19.31	3.69	1.33
Total				32.74	39.67

Explanation of Solution

The exponential smoothing forecasts for weeks 4 through 8 are calculated below:

  $\begin{array}{c}{F}_t=\alpha D_t+\left(1-\alpha \right)F_{t-1}\\ F_4=17.67\\ F_5=\left(0.15\right)\left(22\right)+\left(0.85\right)\left(17.67\right)\\ =\boxed{18.32}\\ F_6=\left(0.15\right)\left(34\right)+\left(0.85\right)\left(18.32\right)\\ =\boxed{20.67}\\ F_7=\left(0.15\right)\left(12\right)+\left(0.85\right)\left(20.67\right)\\ =\boxed{19.37}\end{array}$

  $\begin{array}{c}{F}_8=\left(0.15\right)\left(19\right)+\left(0.85\right)\left(19.37\right)\\ =\boxed{19.32}\end{array}$

c.

Summary Introduction

To calculate: Based on the MAD, to find out the method which did better.

Introduction: Forecasting error can be evaluated using the following formula:

  $MAD=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|e_i\right|}$

Answer to Problem 24P

  $\begin{array}{c}MAD=\frac{1}{5}39.67\\ =\boxed{7.93}\end{array}$

Explanation of Solution

MAD for Smoothing constant, $\alpha$ = 0.15

  $\begin{array}{c}MAD=\frac{1}{5}39.67\\ =\boxed{7.93}\end{array}$

So, the exponential smoothing method is better than moving average method for the five weeks.

d.

Summary Introduction

To calculate: The exponential smoothing forecast made at the end of week 6 for the sales in week 12.

Introduction: Forecasting error can be evaluated using the following formula:

  $MAD=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|e_i\right|}$

Answer to Problem 24P

The forecast made through exponential smoothing is in week 6 for sales in week 12 is 20.67.

Explanation of Solution

The exponential smoothing constant forecast made in the week 6 is the same for the demand in week 7 for sales of in week 12. Therefore, the forecast made through exponential smoothing in week 6 for sales in week 12 is 20.67.