Forecasting Contd.

 

The graph below shows a series of points plotted on the graph of Demand D_i vs. the general parameter X_i .

 

 

 

 

 

          D_i

 

 

                                                X_i               

 

D_i = f (Xi)

where D_i is the dependent and Xi is the independent variable.

 

The demand being a function of  X_i which can be any variable, shows an approximate trend of some sort when looked at. So on the basis of our intuition we select a family of curves and find out the parameters from the data given minimizing the mean-square error.

         This general approach has yield many positive results and many standard procedures have been devised in order to study the relationships.

 

Below we try different functions that can be used as an estimate of f(X) under different conditions:

 

1) Let f(x) = a; a constant function that describes the trend of the observations,

then the deviation of the function from the actual graph is given by the following ‘Error’ expression:

                                                Error = å(D_ti – a)²

 

We choose the parameter ‘a’ such that it minimizes the error function. For finding out the minimum, differentiate the error function with respect to the parameter and equating it to

zero.

Þ (d E/ d a) = 0

Þ  a^ = å D_ti / n

 

2) Let f(X) = a + bX, a linear function with ‘a’ and ‘b’ as the parameters that gives the line of best fit. Going by the above approach of minimizing the error function, we get the error as:

Error =  å(D_ti – a - bX)²

 

Thus we minimize the error term by taking the partial differentials of f(X) w.r.t. ‘a’ and ‘b’ respectively and equating to zero.

(dE / da) = 0 = (dE / db)

performing the above function on the error function we get the following values of ‘a’ and ‘b’ :

a = D -  b t

b =  å(D_ti t_i – n D t)²

     å t_i² – n ( t )²

We say that the real function describing the distribution of data points is given by the following function:

 

D = a + bX + e

Where the  term ‘e’ indicates the random variation of data above and below the function. We find that the term ‘e’ is distributed normally with the distribution function N(0,s²).  The value of s for the function D is given by

 

s =  Ö 1/(n-2)å(D_ti  – D)²

 

This implies that as n Þ ¥, the function D is also normally distributed with N ( a + bX, s²).

 

In case Demand is a function of 2 variables then it can be expressed as,

D = a + bX + cY

where a, b and c are constants to be evaluated.

 

 

AGGREGATE PLANNING AND PRODUCTION

 

There is a variation in demand as D1, D2,…….,D12. There is a need for developing a strategy in order to counter these variations. In an industry decision making is required at all levels (the scope of decisions being different). Decisions taken in an organization are broadly divided into the following 3 categories:

 

1. Strategic Planning Decisions: These comprise of decisions that include the vision and plans of the company for the next few years. Generally involving the top management, they require a large amount of investment and change in the long term. Some of the typical decisions taken by the top management are the following:

 

·        Location and sizing of new plants

·        Acquisition of new equipment

·        Design of logistics system

 

2. Tactical Planning Decisions

 

3. Operational Planning Decisions

 

(Details about the points 2 and 3 are given in the next lecture)