7th August 2002 Wednesday

ME 403: Production Planning and Control

 

2^nd Lecture on Forecasting

 

Authors: Anish Gupta and Gunaranjan Chaudhry

 

 

 

Exponential Smoothing Methods

 

 

 

 

 

 

 

 

 

 

 

If the data to be forecasted follows the line shown above, then we postulate that

D_t = m_o + at + e_t

Where m_o = Constant,

            e_t  = Noise (Not in our control) – we cannot predict it.

            a = Slope of the line

S­_t = Best guess of the expected demand at the time instant t (i.e. for the time instant t)

T_t = Best guess of the expected increment at the time instant t.

 

The forecast for the time interval (t+1) made at time t is given as:

F_{t, t + 1} = S­_t + T_t

Similarly, the forecast (made at time t) for m time periods ahead is given as:

F_{t, t + m} = S­_t + mT_t       

 

The closer the forecast in terms of time, the more accurate it is. This is because the values of S_t and T_t  are updated according to the observed demand.

 

S_t and T_t are updated as follows:

 

S­_{t + 1} = Best guess at the time instant t + 1 (i.e. for the time instant t + 1)

S_{t + 1} = aD_{t + 1} + (1 - a)(S_t + T_t)                      D_{t  + 1} = Actual demand (observed) at t + 1

        = S_t + T_t + a  (D_{t + 1} – S_t – T_t)

 

The updated value of S_t+1 is obtained by giving some weight to the observed demand (D_t+1) and some weight to the previous value.

 

The value of T_t+1 is updated in a similar manner:

T_{t + 1} = b(S_{t + 1} – S_t) + (1 - b)T_t

 

The updated value of T_t+1 is obtained by giving some weight to the observed trend (i.e. S_t+1 – S_t) and some weight to the previous best guess of the trend (T_t).

 

Finally, the new forecast can be calculated as follows:

F_{t + 2} = S_{t + 1} + T_{t + 1}

 

This model is known as the Additive Trend Model. It involves estimating the values of a and b.

If the noise is large, a and b are kept small because we want to give less weight to the observed values (which contain a significant random component). If the noise is small, i.e. the random component is small, a and b are made large so that greater weight is given to the observed values.

 

 

·

·

·

If the demand is such that it involves a strong seasonal component (e.g. for variations as shown below), the model used is the Ratio Seasonality Method.

 

 

 

 

 

 

 

In this case, we assume that the demand follows some underlying mean and some noise. We postulate that

 D_t. = d_t m + e_t

Where d_t = Expected demand in month t

                   Overall expected demand

m = Constant,

            e_t  = Noise (Not in our control) – we cannot predict it.

Hence, in this case, we will have best guess of d_t and m and add noise to D_t.

 

In this model, we incorporate seasonal factors.

I₁, I₂, - - - - - - - - - - - - - - I­_L are tthe seasonal factors.

L is the periodicity of the demand. For example, if we study the monthly demand of a commodity, L will be equal to 12.

 

In this model, at any time t, we have a best guess for S_t and for each of the seasonal factors. The demand D_t for the current period is observed and an update is performed as follows:

S_t  = a (D_t / I_{t – L}) + (1 - a) S_{t – 1}

As before, the best estimate of demand S_t is calculated by giving some weight to the observed demand during this period (D_t) and the previous best estimate (S_t-1). However, D­_t contains a seasonal component and cannot be directly used for calculating S_t. It must first be “deseasonalised”, i.e., rid of its seasonal component. This must be done because the seasonal factor for the next period is not the same as the seasonal factor for the current period.

D_t is deseasonalised by diving it by I_t-L, which is the seasonal factor for the current period (note that the seasonal factor for the period t-L is the same as the seasonal factor for the period t).

 

The seasonal factor must itself be updated for use L periods from now.

I_t = g  (D_t / S_t) + (1 -g) I_{t – L}

This is updated by giving some weight to I_t-L and some weight to the observed seasonal factor for the current period (demand/base value).

 

The forecast for the next period can finally be calculated as follows:

F_t = S_{t – 1} ´ I_{t - L}

The base value is multiplied by the seasonal factor of the period for which the forecast is to be made.

The forecast for the demand m periods in advance is given by:

F_{t + m} = S_t  ´ I_{t – L + m}

(I_t+m–L   is the seasonal factor for the period t+m)

 

 

Additive Trend and Ratio Seasonality Model                           

 

This model is used to forecast for a variation which has a trend as well as a seasonal component.

 

 

 

 

 

                                                                 t

 

The equations used to update this model are basically a combination of the equations used in the earlier two models.

 

St is updated as follows:

S_{t  + 1} = a (D_{t + 1} / I_{t – L + 1}) + (1 - a) [S_t  + T_t]

Weight is given to the observed demand for the current period (D_t+1, which has been adjusted for seasonality by dividing it by I_t-L+1) and to the best estimate of the base value for the current month (S_t + T_t)

 

Tt is updated as follows:

T_{t  + 1} = b(S_{t + 1} – S_t) + (1 - b) T_t

This is the same equation as was used in the additive trend model.

The seasonal factor It+1 is updated as follows:

I_{t + 1} = g(D_{t + 1} / S_t ) + (1 - g)I_{t – L +1}.

This is the same equation as was used in the seasonality of the model

 

Finally the forecast for the next period is calculated as:

F_{t + 1} = [S_t  + T_t] ´ I_{t – L +1}

The forecast is determined by multiplying the sum of the best estimate for demand and the current best estimate for trend with the seasonal factor for the next period.

 

The forecast for m periods ahead can be made as:

F_{t + m} = [S_t  + mT_t] ´ I_{t – L + m}

 

The process is continued by observing the demand and updating the best estimate for demand (St), the best estimate for trend (Tt) and the appropriate seasonal factor.

Observe D_{t  + 1} and update S_t and T_t

 

We can carry this process forward if we have an initial starting value. We need starting values for 2+L parameters (S_t, T_t and L seasonal factors).