Chapter 3, Problem 9P

a. Obtain the linear trend equation for the following data on new checking accounts at Fair Savings Bank and use it to predict expected new checking account for periods 16 through 19.

b. Use trend-adjusted smoothing with α = .3 and β = .2 to smooth the new account data in part a. What it the forecast for period 16?

Summary Introduction

To determine: Linear trend equation for the given data.

Introduction: Forecasting is the planning process that helps to predict the future aspects of the business or operation using present or past data. It uses certain assumptions based the knowledge and experience of the management. Linear trend forecasting is a simple forecasting technique that helps to predict the demand.

Explanation of Solution

Given information:

Obtain the linear trend forecast model for the following data on new checking accounts at a bank and use the model for predicting the new checking accounts for periods 16 through 19 as shown below.

Obtain the linear trend equation:

Plot the data on a graph and use Microsoft Excel to fit a linear trend on the graph. Obtain the equation to predict the new checking accounts as shown below.

The equation $y=7\times x+195.47$ can be used to forecast new accounts for future periods.

• For the period16, substitute the value of $x=16$ in the above equation to arrive at the expected checking accounts

  $y=7\times 16+195.47=307.47$

  Rounding off, the number of checking accounts for the period 16 is forecasted at 308.

• For the period17, substitute the value of $x=17$ in the above equation to arrive at the expected checking accounts

  $y=7\times 17+195.47=314.47$

  Rounding off, the number of checking accounts for the period 17 is forecasted at 315.

• For the period18, substitute the value of $x=18$ in the above equation to arrive at the expected checking accounts

  $y=7\times 18+195.47=321.47$

  Rounding off, the number of checking accounts for the period 18 is forecasted at 322.

• For the period19, substitute the value of $x=19$ in the above equation to arrive at the expected checking accounts

  $y=7\times 19+195.47=328.47$

  Rounding off, the number of checking accounts for the period 19 is forecasted at 329.

Summary Introduction

To determine: Forecast for period 16.

Introduction: Forecasting is the planning process that helps to predict the future aspects of the business or operation using present or past data. It uses certain assumptions based the knowledge and experience of the management. Linear trend forecasting is a simple forecasting technique that helps to predict the demand.

Explanation of Solution

The trend adjusted smoothing method of forecasting carries out two adjustments for smoothing and can be represented by the formulas shown below. The two adjustments are for both the average and for the trend. The formula carries two constants α and β

Denote the following:

S_t is the smoothed forecast for time period t

α is the smoothing constant

β is the trend constant

A_t is the actual demand for the time period t

T_t is the trend estimate for the time period t

TAF_t is the trend adjusted forecast for time period t

The trend adjusted forecast TAF_t+1 has two components, a smooth forecast S_t and a trend component T_t, as shown in the formula below.

$TA{F}_{\text{t}}=S_t+T_t$

The smoothed forecast S_t for period t is calculated by the formula shown below.

$S_t=TA{F}_{t-1}+\alpha \times \left(A_{t-1}-TA{F}_{t-1}\right)$ (1)

The trend T_t for period is given by the formula shown below.

$T_t=T_{t-1}+\beta \times \left(S_t-TA{F}_{t-1}\right)$ (2)

• Since initially for the period 1, there is no smoothed forecast S₁ and no trend T₁ given, assume that the smoothed forecast is same as the actual and the trend is zero.

  Therefore, $S_1=200$ and $T_1=0$. Calculate the TAF₁ as shown below.

  $TA{F}_1=200+0=200$

• Compute for the period 2, the value of S₂ using equation (1) above, by substituting the relevant values shown below.

  $S_2=TA{F}_1+\alpha \times \left(A_1-TA{F}_1\right)$

  $S_2=200\times 0.3\times \left(200-200\right)$

  Therefore, S₂ works out to 200.

  Compute the trend T₂ using the formula is equation (2) as shown below

  $T_2=T_1+\beta \times \left(S_2-TA{F}_1\right)$

  $T_2=0+0.2\times \left(200-200\right)=0$

  Compute TAF₂

  $\begin{array}{c}TA{F}_2=S_2+T_2\\ =200+0\\ =200\end{array}$

• Compute for the period 3, the value of S₃ using equation (1) above, by substituting the relevant values shown below.

  $S_3=TA{F}_2+\alpha \times \left(A_2-TA{F}_2\right)$

  $S_3=200\times 0.3\times \left(214-200\right)$

  Therefore, S₃ works out to 204.2.

  Compute the trend T₃ using the formula is equation (2) as shown below

  $T_3=T_2+0.2\times \left(S_3-TA{F}_2\right)$

  $T_3=0+0.2\times \left(204.2-200\right)=0.84$

  Compute TAF₂

  $\begin{array}{c}TA{F}_3=S_3+T_3\\ =204.2+0.84\\ =205.04\end{array}$

• Further calculations have been carried out using Microsoft Excel for subsequent periods and shown below in a tabular format

Observe that the trend adjusted forecast for the period 16 works out to 307.68 accounts.