# Top 6 Methods of Sales Forecasting

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**This article throws light upon the top six methods of sales forecasting used in an organisation. The methods are: 1. Collective Opinion Method 2****. ****Economic Indicators ****3. ****Method of Least Squares ****4. ****Time Series Analysis ****5. ****Moving Averages Method of Sales Forecasting ****6. ****Exponential Smoothing and Moving Average Method. **

**Method # 1. Collective Opinion Method: **

In this technique the forecasting depends upon the salesmen opinion regarding the product and estimations to its demand for the next year for their respective areas. In view of the fact that salesmen are closest to the consumers, they can estimate more properly about the customer’s reaction regarding the product.

These estimates are received by the branch sales managers and they would review these figures and make certain adjustments, to reflect their knowledge of the individual salesmen.

Some of the later may have demonstrated in the past that they are consistently optimistic and their estimates may be revised downwards, other might be known to be somewhat pessimistic and their estimates may require upwards revision, the remainder of the salesmen might have proved to be realistic and their estimates may remain unaltered.

These adjusted figures of estimates are then made available to a committee responsible for making the final forecast. The members of this committee might include the firm’s sales manager, chief engineer, production manager, marketing manager and economist. They would review the estimates in the light of certain factors with which the sales men and branch sales managers would not be acquainted.

These might include such things as expected changes in product design, a plan for increased advertising, a proposed increase or decrease in selling prices, new production techniques which will improve the product’s quality, changes in competition, changes in economic conditions such as purchasing power of consumer, income distribution, credits, population and employment conditions etc.

Thus the method of collective opinion takes advantages of collective wisdom of salesmen and senior executives of various fields connected with sales management.

**Advantages: **

1. The method is simple since it is based on the collective wisdom of salesmen and senior executives having expertise in various fields and requires no statistical technique.

2. The demand estimates are based on the knowledge of salesmen who are directly responsible for meeting the sales targets hence are correct.

3. For launching the new products the method is quite useful.

**Disadvantage:**

1. Since no past data and statistical technique is used the method is useful only for short term forecasting.

2. The salesmen may under estimate the future sales if sales quotas are fixed for them.

3. The estimates made by this method may not be realistic since the salesmen have no knowledge about the economic changes.

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**Method # 2. ****Economic Indicators: **

This method of sales forecasting is based on the use of indicators which serve to describe the economic conditions which prevail during a given time period.

**Some of these economic indicators are the followings: **

1. Construction contracts awarded for the demand of building materials.

**This article throws light upon the top six methods of sales forecasting used in an organisation. The methods are: 1. Collective Opinion Method 2 . Economic Indicators 3. Method of Least Squares 4. Time Series Analysis 5. Moving Averages Method of Sales Forecasting 6. Exponential Smoothing and Moving Average Method. **

**Method # 1. Collective Opinion Method: **

In this technique the forecasting depends upon the salesmen opinion regarding the product and estimations to its demand for the next year for their respective areas. In view of the fact that salesmen are closest to the consumers, they can estimate more properly about the customer’s reaction regarding the product.

These estimates are received by the branch sales managers and they would review these figures and make certain adjustments, to reflect their knowledge of the individual salesmen.

Some of the later may have demonstrated in the past that they are consistently optimistic and their estimates may be revised downwards, other might be known to be somewhat pessimistic and their estimates may require upwards revision, the remainder of the salesmen might have proved to be realistic and their estimates may remain unaltered.

These adjusted figures of estimates are then made available to a committee responsible for making the final forecast. The members of this committee might include the firm’s sales manager, chief engineer, production manager, marketing manager and economist. They would review the estimates in the light of certain factors with which the sales men and branch sales managers would not be acquainted.

These might include such things as expected changes in product design, a plan for increased advertising, a proposed increase or decrease in selling prices, new production techniques which will improve the product’s quality, changes in competition, changes in economic conditions such as purchasing power of consumer, income distribution, credits, population and employment conditions etc.

Thus the method of collective opinion takes advantages of collective wisdom of salesmen and senior executives of various fields connected with sales management.

**Advantages: **

1. The method is simple since it is based on the collective wisdom of salesmen and senior executives having expertise in various fields and requires no statistical technique.

2. The demand estimates are based on the knowledge of salesmen who are directly responsible for meeting the sales targets hence are correct.

3. For launching the new products the method is quite useful.

**Disadvantage:**

1. Since no past data and statistical technique is used the method is useful only for short term forecasting.

2. The salesmen may under estimate the future sales if sales quotas are fixed for them.

3. The estimates made by this method may not be realistic since the salesmen have no knowledge about the economic changes.

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**Method # 2. ****Economic Indicators: **

This method of sales forecasting is based on the use of indicators which serve to describe the economic conditions which prevail during a given time period.

**Some of these economic indicators are the followings: **

1. Construction contracts awarded for the demand of building materials.

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2. Farm income for the demand of agricultural implements and other inputs.

3 Personal income for the demand of consumer goods.

4. Automobiles production/Registration of automobiles for the demand of accessories and petroleum products.

5. Employment position.

6. Gross national income.

7. Consumer prices.

8. Wholesale commodity prices.

9. Bank deposits.

10. Industrial production.

11. Steel production.

12. Business inventories.

Data of this type are compiled and published by various governmental agencies such as central statistical organization and by private group such as trade associations and business research organizations.

If the enterprise or organisation finds that there is a relationship between one or some combination of such economic indicator and sales of some of its products, this approach to sale forecasting can be utilized.

Further the selected or relevant economic indicator may prove to be a leading, lagging or coincident.

A leading indicator is one whose value for a given period will affect sales in a subsequent period. For example, a manufacturer of school bags may find that his sales during a given year are affected by the number of children born three or four year earlier. This is most desirable type of economic indicator because its value will be known at the time the forecast of future sales is being made.

A lagging indicator is one whose value for a given period will reflect sales in some preceding period. For example the manufacturer of desert cooler bodies may find that data on inventories of the bodies for a given period are related to his sales during some earlier period.

A coincident indicator is one whose value for a given period will affect sales in that period. For example the manufacturer of baby milk feeders may find that the volume of production of feeders during a given period is affected by population growth rate during the same period. This is less desirable type of indicator because its value must be estimated for the future period for which sales forecast is being done.

The sales forecasting is done with the help of least squares equation.

**Limitation: **

1. Need for finding an appropriate indicator. In some cases a given indicator may be correct, but in other cases no one indicator will be obviously applicable and trial error approach may be required which is tedious and time consuming.

2. Appropriate indicator may vary with product or product group under consideration.

3. After investigating all the possible alternatives the company may find that no single or composite indicator is suitable. In such cases this method of forecasting cannot be used. However the firm may find that although its sales are not correlated to any economic indicator the organisations sales exist.

This would be the case when the company’s share of the market is fluctuating widely. Collective opinion approach may be used.

4. Another difficulty stems from the fact that the relevant indicator may be say an annual index, whereas the company may like to forecast sales on a month by month basis.

5. A further limitation of this method is that it does not lend itself to a forecast of sales for a new product because no past data exist on which correlation analysis can be based.

#### Relationship between two Variables:

**Suppose a industry’s record of past three years production and manufacturing cost is: **

If we plot the output which is independent variable v/s manufacturing cost which is a dependent variable for each of the three years.

The graph reveals that all the three points fall on a line of best fit. Since the line is a straight line thus there is a strong linear correlation between production output and manufacturing cost.

**Suppose now the manufacturing cost varies as indicated in the following table: **

The graph of this relationship appears as indicated by dotted line. The nature of these points is such that they do not fall on the line of best fit. However they are close to it and we can say that a nearly linear correlation exist between the two variables not as strong as displayed by first set of data.

Therefore we can less precisely forecast the manufacturing cost for some given level of production or output. Similarly in actual practice there may be a curvilinear correlation between the two variables.

A relatively simple method of fitting the line is method of least squares.

This method of least squares yields an equation which describes and locates the line of best fit.

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**Method # 3. ****Method of Least Squares: **

The method of least squares provides an equation which gives two characteristics of the line of best fit. A straight line can be described in terms of two things i.e., its slope and Y intercept. Y intercept is the point on Y axis of graph between two variables where the line intersects the Y-axis.

**If we know the Y-intercept and slope of the line, the equation of the line can be determined from the general expression for the equation of any line which is as follows: **

Y’ = m x + a

where Y’ is the calculated value of the dependent variable which is to be forecasted.

a = Y intercept of the line of best fit.

m = slope of the line of best fit.

x = given value of independent variable in terms of which value of dependent variable is to be forecasted.

In this way all this serves to describe only what the equation of line is and equation can be determined if we have already located the line.

But usually the points do not fall on a straight line hence we must decide where the line should be located. This requires first determining the equation of the line of best fit and then locate the position of the line by using this equation.

The method of least squares can help us to find out the equation of the line by working directly with the original data of dependent and independent variables by making appropriate substitutions in the following expressions.

∑Y = na + m∑x

∑xY = a∑x + m∑x^{2 }

where x = given values of the independent variable, which may be the economic indicator.

Y – Given value of the dependent variable which may be sales of the product in this case.

n = number of given paired observations.

**Again: using the previous set of data for three year we have: **

Substituting values of ∑x, ∑Y, ∑xY

∑x and n = 3 in equations (1) & (2)

We have 18 = 3a + 12m

80 = 12a + 56m

Solving these two equations for a &m

We get a = 2m = 1

The equation of the line because

Y = 1 * × + 2

With the second set of data where the correlation is not so liner we have

Substituting the value in the equations (1) & (2) we have

20 = 3a + 12m

92 = 12a + 56m

Solving theses two equations for a and m we get

a = 2/3 m = 2/3

The equation of the line becomes

Y^{i}_{c} = [3/2x+2/3]

The equation of this line can be drawn in correct position by finding minimum two points and connecting them.

**Properties of Least Squares Line: **

If all our points do not fall on the line and a curvilinear correlation is indicated as we have seen with second set of data, substitution of our given values of x in the equation of the line we obtain by method of least squares will not yield calculated values of Yc equal to our actual values.

**If we substitute our given values of output as 2, 6, 4 in equation (4) we shall not obtain the corresponding actual values of manufacturing cost of 4, 10, 6 as follows: **

For x = 2 Y_{c}= 3/2 x 2 + 2/3 = 3⅔

For x = 6 Y_{c }= 3/2 x 6 + 2/3 = 9⅔

For x = 4 Y_{c} = 3/2 x 4 + 2/3 = 6⅔

**As shown from this table following are the properties of the least squares Line: **

1. Sum of the deviations i.e., difference between actual and calculated values of dependent variable will always be zero.

2. The second characteristic of the least squares line is that sum of the deviations squared is minimum.

This indicates that if the has been drawn in any other position the total of the squares of the resultant deviations would have been greater than the sum obtained with the least squares line.

In case equations (1) and (2) are solved for a and m the following expression are obtained.

**Coefficient of Correlation:**

It is the quantitative measure of the strength of relationship described by the least squares line. The magnitude of the coefficient of correlation will vary with degree of correlation which exists between the variables under consideration.

**The expression from which this coefficient of correlation is determined is as follows: **

where Y_{a} = is used for actual values of the dependent variable like manufacturing cost.

Yc = the corresponding calculated values of the dependent variable found from the least squares line.

Y̅ = average of the actual values of the dependent variables.

In equation (5) the value of the numerator (Y_{a} – Y_{c})^{2} can never be less than zero since it is a squared sum. This can maximum becomes zero when the actual values of the dependent variable are equal to calculated values. In that case all the points lie on the line of best fit. In such cases the numerator is zero and the coefficient of correlation reaches its maximum value i.e., 1. This will occur only when the two variables are perfectly correlated.

The minimum value of coefficient of correlation can be zero which indicates the absence of correlation between the two variables.

The division between a high and low degree of correlation is difficult to make but the following tables gives the generally accepted values of coefficient of correlation r.

In practice a different form of equation (6) is used for finding the value of r.

This equation permits us to depict the value of coefficient of correlation by working directly with the original data hence it is a simple method of calculating the value of r.

**Application of the Least Squares Method: **

**Example 1: **

**A firm finds that relationship exists between the rupees sales of one of its product group and a given economic indicator. Specifically a comparison between the past sales and the corresponding values of the economic indicator reveals the following: **

(a) Determine the strength of the relationship by computing the value of coefficient of correlation for the two variables.

(b)Determine the equation of the line of best fit by means of the method of least squares.

(c) If the value of the economic index for a future period is expected to be 112 what sales can be expected during that period. (Industrial Engineering, R.U., 1980)

**Solution:**

Assuming a linear forecaster of the form Y’ – mx + a where m & a are constants for this to be line of best fit

∑Y_{a }= na + m∑x …(i)

∑x Y_{a} = a∑x + m∑x^{2} …(ii)

Putting the values from table in equations (i) and (ii)

19.6 = 10a + 1025m

2067.1 = 1023a + 105673m

Solving these equations we get

a = -7.78

m = 0.0951

Hence equation of the line of best fit is

Y = -7.78 + 0.0951x Ans.

For future period when economic index is 112 substituting x = 112

Y = -7.78 + .0951 x 112

= 2.8712 Expected sales = Rs. 28712 | Ans.

Using the relation.

**Multiple Regressions:**

In the previous example it has been assumed that the sales of a product group is dependent on the value of one economic indicator only, hi many cases however sales of a product or product groups may be a function of a combination of indicators.

**If the relationship between the sales and these economic indicators or some other indicators is linear it can be described by means of an equation of the following general form: **

Y’ – a + m_{1} + m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}

in which a, m_{1}, m_{2} are constants and x_{1}, x_{2}, x_{3}., are variables/indicators on which sales forecast is to be based. The unknown constants can be determined by solving simultaneous equations. The procedure involved is multiple regression analysis.

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**Method # 4. ****Time Series Analysis:**

This method of sales forecasting is considered similar to economic indicator method since it also requires regression analysis. A time series is a chronological data which has some quantity such as sales volume or sales in rupees as the dependent variable and time as independent variable.

These time series available with established organization are analysed before making the forecast. There is a common technique which is generally employed is called as “project the trend.” In this method the trend line is projected by least squares method.

**The variations of the dependent variable may be segregated as: **

(a) Long period changes.

(b) Short period changes.

The long period tendency of data to change i.e., increase or decrease is called basic trend which may be linear or non-linear.

**The short period changes may be of two types: **

(i) Regular

(ii) Irregular

**Regular fluctuations are those which occur at regular intervals of time. These may be: **

(a) Seasonal variations.

(b) Cyclic variations.

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**Seasonal Variations**:

The most common periodic variation is the seasonal variation which occurs with some regularity in a span of time weather conditions, social customs and festivals etc. These effect sales of various product (normally of consumer utilization).

**Cyclic Variations: **

These changes show periodicity and occur over a shorter period of time. Like seasonal variations cyclic variations also regular. But whereas seasonal variations occur within a period of one year or less cyclic variations repeat at intervals of 5 to 10 years.

**Irregular Variations: **

Theses variation occurs without any particular rhythm. They can be caused by causes operating in a casual and irregular fashion. Causes may be like droughts, floods, wars, strikes and earth quakes etc.

In time series analysis technique of sales forecasting an organization analyses its past sales to find out if there is some trend. This trend is then projected into the future and the resultant indicated sales are used as a basis for a sales forecast. This method will be clear with the help of following illustrations.

Suppose a manufacturer of painting equipment (may be paint roller frames) decides to forecast next year sales of its product. He starts by collecting the data for the last four/five years.

The manufacturer knows from the past experience that sales of his product fluctuates due to seasonal variations. In fact he has found from the past data that the market demand for the product is at its minimum during the first quarter of the year results in an increase in sales due to improved weather conditions.

Similarly a greater increase in sales take place during third quarter of the year due to further improvement in weather conditions and season of festivals. However with the on set of less favourable weather conditions delete the demand for the product goes down in the fourth quarter.

As a result of these quarterly variations the company decides to develop a forecast by quarter for production planning purposes.

**The application of sales forecasting by time series analysis technique will be clear with the help of following illustration: **

In spite of limited amount of data available. Determine the equation of the trend line. With the equation, calculate the trend values of quarterly sales for the 4^{th} year. Then adjust these values to provide for expected seasonal variations. (K.U.K. (Non-Departmental). May, 1995, B.Tech, May, 1998)

**Solution:**

In other words the actual sales in the first quarter were 77% of calculated sales.

**Similarly calculated sales for other quarters are as follows:**

For adjusting the trend values to provide for expected seasonal variations we determine the magnitude of this off season adjustment factor by finding the average of the past variation during first quarter of each year i.e., in 1^{st}, 5^{th} and 9^{th} quarters.

Actual sales as percentage of calculated sales values for four quarters.

The last column of this table gives the values of seasonal adjustment factor for four quarter of the year as 0.7097, 0.8663, 1.120, 1.30 i.e., for 1^{st}, 2^{nd}, 3^{rd} and 4^{th} and multiplication of the calculated sales for the four quarters of the 4^{th} year with the adjustment factors shall yield adjusted sales forecasts for the year under consideration.

**Advantages of Time Series Analysis: **

1. This technique is less subjective than collective opinion method and method of economic indicators since its application is not dependent on the organization’s ability to find appropriate indicator.

2. In comparison with method of collective opinion and method of economic indicators which may yield only an annual forecast that must thereby be broken down into shorter periods, the organization can forecast sales by year by analysing past annual sales, by month by analysing past monthly sales or even by week by analysing past weekly sales.

**Limitation of Time Series Analysis: **

1. This technique cannot be used for forecasting the sales of a new or relatively new product since no past data or in sufficient past data are available.

2. If the significant fluctuation in level of demand occur month to month in a year due to seasonal variations or, 12 adjustment factors may be required for adjusting the forecasting in a year.

3. The impact of changes in selling prices, product quality, economic conditions, marketing methods and sales promotional efforts made by the organizations cannot be incorporated into the method in a satisfactory way.

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**Method # 5. ****Moving Averages Method of Sales Forecasting: **

In this method the sales forecasting is obtained by taking average of past sales over a desired number of past periods (may be years, months or weeks). Extending the moving average to include more periods may increase the smoothening effect but decreases the sensitivity of forecast.

Long periods provide too many opportunities for significant changes to occur in demand pattern. To reduce this risk, the organizations can base its forecast on the average demand during short periods say three months. The application of this technique will be clear with the following illustration.

**A forecast based on un-weighted moving averages for number of customers:**

This forecast is based on the post two weeks average number of customers.

Therefore the unadjusted forecast for the 9th week is 512. At the end of week 9 the forecast for the l0th week would be based on the average number of customer actually visiting during 7 weeks, 8 and 9 and so on. The result is a series of moving averages enlisted in the table above.

**Weighted Moving Averages: **

The moving averages as calculated in the preceding part are known as un-weighted because the same weight is assigned to each of the numbers whose average is being ascertained. Some enterprises base their forecast on a weighted moving average.

Let us assume that the number of customers who visit during two weeks interval provides a sound basis for third week forecast and let us further assume that first week is less important than second and consequently we assign weights of 0.4 to first week and 0.6 to second week. The weighted average for 9th week would be

0.4 X 549 + 0.6(474) = 220 + 284 = 504

**Similarly the weighted moving averages for other weeks are enlisted in the following table:**

A forecast based on weighted moving averages for number of customers.

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**Advantages of the Moving Average Method: **

1. This technique is simpler than the method of least squares.

2. This method is not affected by personal prejudice of the people using it.

3. It the period of moving average is equivalent to the period of the cycle. The cyclic variations are eliminated.

4. If the trend in the data if any is linear the moving average gives a good picture of long term movement in data.

5. The moving average technique has the merit of flexibility i.e., if a few years are added the entire calculations are not changed due to adoption of new conditions.

**Limitations of the Moving Average Method: **

**Following are the drawbacks of this method of forecasting:**

1. It does not result in mathematical relations which may be used for sales forecasting.

2. There is a tendency to cut corners which results in the loss of data at the ends

3. A great deal of care is needed for the selection of the period of moving average since the wrong periods selected would not give the correct picture of the trend.

4. In case of the sharp turns in the original graph, the moving average would reduce the curvature.

5. It is very sensitive even to small movement in the data.

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**Method # 6. ****Exponential Smoothing and Moving Average Method: **

This method of sales forecasting is a modification of the moving average method or in better words it IS an improvement over the moving average method of forecasting. This method tries to eliminate the limitations of moving averages and removes the necessity of keeping extensive past data it also tries to remove the irregularities in demand pattern.

This method represents a weightage average of the past observations. In this case most recent observations is assigned the highest weightage which decreases in geometric progression as we move towards the older observations.

Since the most recent observations which are likely to reflect more up- to-date information or average of the series are given more weightage so it becomes one of the most accurate statistical method of sales forecasting. This method keeps a running average of demand and adjusts it for each period in proportion to the difference between the latest actual demand figure and the latest value of the average.

When there is no trend in the demand for a product or service, sales are forecasted for the next period, by means of the exponential smoothing method by using the expression

Forecast for the next period = a (latest actual demand) + (1 – α) old estimate of latest actual demand where a represents the value of a weighting factor which is referred to as a smoothing factor.

**This method follows the equation **

F_{n}= F_{n -1 }+ α (D _{n-1 }– F _{n-1})

where F_{n}= forecast for the next period

F_{n-1} = forecast for previous period

D _{n-1} = demand in previous period.

If a is equal to 1. then the latest forecast would be equal to previous period actual demand In practice, the value of a is generally chosen between 0.1 and 0.3. The application of technique is demonstrated by using data of moving averages method of sales forecasting on page 78. In the application of the method, we would use the value of a as 0.10.

Using equation (7), if the actual demand for 3rd week is 487, the forecast for the 4th week will be

0.10(487) + (1.00 – 0.10)550 = 544

Similarly, if the actual demand for 4th week is 528 customers, the forecast for the 5th week will be

0.10 (528) + (1.00 – 0.10) (544) = 542

If this procedure had been applied during the entire 8 week period the results are shown in the following table. The unadjusted forecast error is also indicated under column D = B – C. If the value of a is not given; it can be determined by an approximate relation of a.

α = 2/ Number of periods in moving average + 1

In equation (7) in so far as the weight factors a is concerned, it can assume a minimum value 0 and a maximum value of 1. The greater the value of a, the greater is the weight placed on recent data. When the value of a is 1, the forecast will be equal to the demand experienced during the last period.

Although the value of a varies from product to product but most organization have found that a value between 0 06 to 0.20 usually proves to be satisfactory.

When attempting to find out what value of a should be used for a product or service the organization/enterprise can select various values, examine the past forecasts with the use of these values and adopt for future use the one which would have minimized forecast errors in the past.

In this way we go close of the description of exponential smoothing as it is applied when a trend in sales/service is available. In case trend exist, a trend adjustment can be made with this technique but its application becomes bit difficult.

2. Farm income for the demand of agricultural implements and other inputs.

3 Personal income for the demand of consumer goods.

4. Automobiles production/Registration of automobiles for the demand of accessories and petroleum products.

5. Employment position.

6. Gross national income.

7. Consumer prices.

8. Wholesale commodity prices.

9. Bank deposits.

10. Industrial production.

11. Steel production.

12. Business inventories.

Data of this type are compiled and published by various governmental agencies such as central statistical organization and by private group such as trade associations and business research organizations.

If the enterprise or organisation finds that there is a relationship between one or some combination of such economic indicator and sales of some of its products, this approach to sale forecasting can be utilized.

Further the selected or relevant economic indicator may prove to be a leading, lagging or coincident.

A leading indicator is one whose value for a given period will affect sales in a subsequent period. For example, a manufacturer of school bags may find that his sales during a given year are affected by the number of children born three or four year earlier. This is most desirable type of economic indicator because its value will be known at the time the forecast of future sales is being made.

A lagging indicator is one whose value for a given period will reflect sales in some preceding period. For example the manufacturer of desert cooler bodies may find that data on inventories of the bodies for a given period are related to his sales during some earlier period.

A coincident indicator is one whose value for a given period will affect sales in that period. For example the manufacturer of baby milk feeders may find that the volume of production of feeders during a given period is affected by population growth rate during the same period. This is less desirable type of indicator because its value must be estimated for the future period for which sales forecast is being done.

The sales forecasting is done with the help of least squares equation.

**Limitation: **

1. Need for finding an appropriate indicator. In some cases a given indicator may be correct, but in other cases no one indicator will be obviously applicable and trial error approach may be required which is tedious and time consuming.

2. Appropriate indicator may vary with product or product group under consideration.

3. After investigating all the possible alternatives the company may find that no single or composite indicator is suitable. In such cases this method of forecasting cannot be used. However the firm may find that although its sales are not correlated to any economic indicator the organisations sales exist.

This would be the case when the company’s share of the market is fluctuating widely. Collective opinion approach may be used.

4. Another difficulty stems from the fact that the relevant indicator may be say an annual index, whereas the company may like to forecast sales on a month by month basis.

5. A further limitation of this method is that it does not lend itself to a forecast of sales for a new product because no past data exist on which correlation analysis can be based.

#### Relationship between two Variables:

**Suppose a industry’s record of past three years production and manufacturing cost is: **

If we plot the output which is independent variable v/s manufacturing cost which is a dependent variable for each of the three years.

The graph reveals that all the three points fall on a line of best fit. Since the line is a straight line thus there is a strong linear correlation between production output and manufacturing cost.

**Suppose now the manufacturing cost varies as indicated in the following table: **

The graph of this relationship appears as indicated by dotted line. The nature of these points is such that they do not fall on the line of best fit. However they are close to it and we can say that a nearly linear correlation exist between the two variables not as strong as displayed by first set of data.

Therefore we can less precisely forecast the manufacturing cost for some given level of production or output. Similarly in actual practice there may be a curvilinear correlation between the two variables.

A relatively simple method of fitting the line is method of least squares.

This method of least squares yields an equation which describes and locates the line of best fit.

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**Method # 3. ****Method of Least Squares: **

The method of least squares provides an equation which gives two characteristics of the line of best fit. A straight line can be described in terms of two things i.e., its slope and Y intercept. Y intercept is the point on Y axis of graph between two variables where the line intersects the Y-axis.

**If we know the Y-intercept and slope of the line, the equation of the line can be determined from the general expression for the equation of any line which is as follows: **

Y’ = m x + a

where Y’ is the calculated value of the dependent variable which is to be forecasted.

a = Y intercept of the line of best fit.

m = slope of the line of best fit.

x = given value of independent variable in terms of which value of dependent variable is to be forecasted.

In this way all this serves to describe only what the equation of line is and equation can be determined if we have already located the line.

But usually the points do not fall on a straight line hence we must decide where the line should be located. This requires first determining the equation of the line of best fit and then locate the position of the line by using this equation.

The method of least squares can help us to find out the equation of the line by working directly with the original data of dependent and independent variables by making appropriate substitutions in the following expressions.

∑Y = na + m∑x

∑xY = a∑x + m∑x^{2 }

where x = given values of the independent variable, which may be the economic indicator.

Y – Given value of the dependent variable which may be sales of the product in this case.

n = number of given paired observations.

**Again: using the previous set of data for three year we have: **

Substituting values of ∑x, ∑Y, ∑xY

∑x and n = 3 in equations (1) & (2)

We have 18 = 3a + 12m

80 = 12a + 56m

Solving these two equations for a &m

We get a = 2m = 1

The equation of the line because

Y = 1 * × + 2

With the second set of data where the correlation is not so liner we have

Substituting the value in the equations (1) & (2) we have

20 = 3a + 12m

92 = 12a + 56m

Solving theses two equations for a and m we get

a = 2/3 m = 2/3

The equation of the line becomes

Y^{i}_{c} = [3/2x+2/3]

The equation of this line can be drawn in correct position by finding minimum two points and connecting them.

**Properties of Least Squares Line: **

If all our points do not fall on the line and a curvilinear correlation is indicated as we have seen with second set of data, substitution of our given values of x in the equation of the line we obtain by method of least squares will not yield calculated values of Yc equal to our actual values.

**If we substitute our given values of output as 2, 6, 4 in equation (4) we shall not obtain the corresponding actual values of manufacturing cost of 4, 10, 6 as follows: **

For x = 2 Y_{c}= 3/2 x 2 + 2/3 = 3⅔

For x = 6 Y_{c }= 3/2 x 6 + 2/3 = 9⅔

For x = 4 Y_{c} = 3/2 x 4 + 2/3 = 6⅔

**As shown from this table following are the properties of the least squares Line: **

1. Sum of the deviations i.e., difference between actual and calculated values of dependent variable will always be zero.

2. The second characteristic of the least squares line is that sum of the deviations squared is minimum.

This indicates that if the has been drawn in any other position the total of the squares of the resultant deviations would have been greater than the sum obtained with the least squares line.

In case equations (1) and (2) are solved for a and m the following expression are obtained.

**Coefficient of Correlation:**

It is the quantitative measure of the strength of relationship described by the least squares line. The magnitude of the coefficient of correlation will vary with degree of correlation which exists between the variables under consideration.

**The expression from which this coefficient of correlation is determined is as follows: **

where Y_{a} = is used for actual values of the dependent variable like manufacturing cost.

Yc = the corresponding calculated values of the dependent variable found from the least squares line.

Y̅ = average of the actual values of the dependent variables.

In equation (5) the value of the numerator (Y_{a} – Y_{c})^{2} can never be less than zero since it is a squared sum. This can maximum becomes zero when the actual values of the dependent variable are equal to calculated values. In that case all the points lie on the line of best fit. In such cases the numerator is zero and the coefficient of correlation reaches its maximum value i.e., 1. This will occur only when the two variables are perfectly correlated.

The minimum value of coefficient of correlation can be zero which indicates the absence of correlation between the two variables.

The division between a high and low degree of correlation is difficult to make but the following tables gives the generally accepted values of coefficient of correlation r.

In practice a different form of equation (6) is used for finding the value of r.

This equation permits us to depict the value of coefficient of correlation by working directly with the original data hence it is a simple method of calculating the value of r.

**Application of the Least Squares Method: **

**Example 1: **

**A firm finds that relationship exists between the rupees sales of one of its product group and a given economic indicator. Specifically a comparison between the past sales and the corresponding values of the economic indicator reveals the following: **

(a) Determine the strength of the relationship by computing the value of coefficient of correlation for the two variables.

(b)Determine the equation of the line of best fit by means of the method of least squares.

(c) If the value of the economic index for a future period is expected to be 112 what sales can be expected during that period. (Industrial Engineering, R.U., 1980)

**Solution:**

Assuming a linear forecaster of the form Y’ – mx + a where m & a are constants for this to be line of best fit

∑Y_{a }= na + m∑x …(i)

∑x Y_{a} = a∑x + m∑x^{2} …(ii)

Putting the values from table in equations (i) and (ii)

19.6 = 10a + 1025m

2067.1 = 1023a + 105673m

Solving these equations we get

a = -7.78

m = 0.0951

Hence equation of the line of best fit is

Y = -7.78 + 0.0951x Ans.

For future period when economic index is 112 substituting x = 112

Y = -7.78 + .0951 x 112

= 2.8712 Expected sales = Rs. 28712 | Ans.

Using the relation.

**Multiple Regressions:**

In the previous example it has been assumed that the sales of a product group is dependent on the value of one economic indicator only, hi many cases however sales of a product or product groups may be a function of a combination of indicators.

**If the relationship between the sales and these economic indicators or some other indicators is linear it can be described by means of an equation of the following general form: **

Y’ – a + m_{1} + m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}

in which a, m_{1}, m_{2} are constants and x_{1}, x_{2}, x_{3}., are variables/indicators on which sales forecast is to be based. The unknown constants can be determined by solving simultaneous equations. The procedure involved is multiple regression analysis.

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**Method # 4. ****Time Series Analysis:**

This method of sales forecasting is considered similar to economic indicator method since it also requires regression analysis. A time series is a chronological data which has some quantity such as sales volume or sales in rupees as the dependent variable and time as independent variable.

These time series available with established organization are analysed before making the forecast. There is a common technique which is generally employed is called as “project the trend.” In this method the trend line is projected by least squares method.

**The variations of the dependent variable may be segregated as: **

(a) Long period changes.

(b) Short period changes.

The long period tendency of data to change i.e., increase or decrease is called basic trend which may be linear or non-linear.

**The short period changes may be of two types: **

(i) Regular

(ii) Irregular

**Regular fluctuations are those which occur at regular intervals of time. These may be: **

(a) Seasonal variations.

(b) Cyclic variations.

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**Seasonal Variations**:

The most common periodic variation is the seasonal variation which occurs with some regularity in a span of time weather conditions, social customs and festivals etc. These effect sales of various product (normally of consumer utilization).

**Cyclic Variations: **

These changes show periodicity and occur over a shorter period of time. Like seasonal variations cyclic variations also regular. But whereas seasonal variations occur within a period of one year or less cyclic variations repeat at intervals of 5 to 10 years.

**Irregular Variations: **

Theses variation occurs without any particular rhythm. They can be caused by causes operating in a casual and irregular fashion. Causes may be like droughts, floods, wars, strikes and earth quakes etc.

In time series analysis technique of sales forecasting an organization analyses its past sales to find out if there is some trend. This trend is then projected into the future and the resultant indicated sales are used as a basis for a sales forecast. This method will be clear with the help of following illustrations.

Suppose a manufacturer of painting equipment (may be paint roller frames) decides to forecast next year sales of its product. He starts by collecting the data for the last four/five years.

The manufacturer knows from the past experience that sales of his product fluctuates due to seasonal variations. In fact he has found from the past data that the market demand for the product is at its minimum during the first quarter of the year results in an increase in sales due to improved weather conditions.

Similarly a greater increase in sales take place during third quarter of the year due to further improvement in weather conditions and season of festivals. However with the on set of less favourable weather conditions delete the demand for the product goes down in the fourth quarter.

As a result of these quarterly variations the company decides to develop a forecast by quarter for production planning purposes.

**The application of sales forecasting by time series analysis technique will be clear with the help of following illustration: **

In spite of limited amount of data available. Determine the equation of the trend line. With the equation, calculate the trend values of quarterly sales for the 4^{th} year. Then adjust these values to provide for expected seasonal variations. (K.U.K. (Non-Departmental). May, 1995, B.Tech, May, 1998)

**Solution:**

In other words the actual sales in the first quarter were 77% of calculated sales.

**Similarly calculated sales for other quarters are as follows:**

For adjusting the trend values to provide for expected seasonal variations we determine the magnitude of this off season adjustment factor by finding the average of the past variation during first quarter of each year i.e., in 1^{st}, 5^{th} and 9^{th} quarters.

Actual sales as percentage of calculated sales values for four quarters.

The last column of this table gives the values of seasonal adjustment factor for four quarter of the year as 0.7097, 0.8663, 1.120, 1.30 i.e., for 1^{st}, 2^{nd}, 3^{rd} and 4^{th} and multiplication of the calculated sales for the four quarters of the 4^{th} year with the adjustment factors shall yield adjusted sales forecasts for the year under consideration.

**Advantages of Time Series Analysis: **

1. This technique is less subjective than collective opinion method and method of economic indicators since its application is not dependent on the organization’s ability to find appropriate indicator.

2. In comparison with method of collective opinion and method of economic indicators which may yield only an annual forecast that must thereby be broken down into shorter periods, the organization can forecast sales by year by analysing past annual sales, by month by analysing past monthly sales or even by week by analysing past weekly sales.

**Limitation of Time Series Analysis: **

1. This technique cannot be used for forecasting the sales of a new or relatively new product since no past data or in sufficient past data are available.

2. If the significant fluctuation in level of demand occur month to month in a year due to seasonal variations or, 12 adjustment factors may be required for adjusting the forecasting in a year.

3. The impact of changes in selling prices, product quality, economic conditions, marketing methods and sales promotional efforts made by the organizations cannot be incorporated into the method in a satisfactory way.

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**Method # 5. ****Moving Averages Method of Sales Forecasting: **

In this method the sales forecasting is obtained by taking average of past sales over a desired number of past periods (may be years, months or weeks). Extending the moving average to include more periods may increase the smoothening effect but decreases the sensitivity of forecast.

Long periods provide too many opportunities for significant changes to occur in demand pattern. To reduce this risk, the organizations can base its forecast on the average demand during short periods say three months. The application of this technique will be clear with the following illustration.

**A forecast based on un-weighted moving averages for number of customers:**

This forecast is based on the post two weeks average number of customers.

Therefore the unadjusted forecast for the 9th week is 512. At the end of week 9 the forecast for the l0th week would be based on the average number of customer actually visiting during 7 weeks, 8 and 9 and so on. The result is a series of moving averages enlisted in the table above.

**Weighted Moving Averages: **

The moving averages as calculated in the preceding part are known as un-weighted because the same weight is assigned to each of the numbers whose average is being ascertained. Some enterprises base their forecast on a weighted moving average.

Let us assume that the number of customers who visit during two weeks interval provides a sound basis for third week forecast and let us further assume that first week is less important than second and consequently we assign weights of 0.4 to first week and 0.6 to second week. The weighted average for 9th week would be

0.4 X 549 + 0.6(474) = 220 + 284 = 504

**Similarly the weighted moving averages for other weeks are enlisted in the following table:**

A forecast based on weighted moving averages for number of customers.

**Advantages of the Moving Average Method: **

1. This technique is simpler than the method of least squares.

2. This method is not affected by personal prejudice of the people using it.

3. It the period of moving average is equivalent to the period of the cycle. The cyclic variations are eliminated.

4. If the trend in the data if any is linear the moving average gives a good picture of long term movement in data.

5. The moving average technique has the merit of flexibility i.e., if a few years are added the entire calculations are not changed due to adoption of new conditions.

**Limitations of the Moving Average Method: **

**Following are the drawbacks of this method of forecasting:**

1. It does not result in mathematical relations which may be used for sales forecasting.

2. There is a tendency to cut corners which results in the loss of data at the ends

3. A great deal of care is needed for the selection of the period of moving average since the wrong periods selected would not give the correct picture of the trend.

4. In case of the sharp turns in the original graph, the moving average would reduce the curvature.

5. It is very sensitive even to small movement in the data.

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**Method # 6. ****Exponential Smoothing and Moving Average Method: **

This method of sales forecasting is a modification of the moving average method or in better words it IS an improvement over the moving average method of forecasting. This method tries to eliminate the limitations of moving averages and removes the necessity of keeping extensive past data it also tries to remove the irregularities in demand pattern.

This method represents a weightage average of the past observations. In this case most recent observations is assigned the highest weightage which decreases in geometric progression as we move towards the older observations.

Since the most recent observations which are likely to reflect more up- to-date information or average of the series are given more weightage so it becomes one of the most accurate statistical method of sales forecasting. This method keeps a running average of demand and adjusts it for each period in proportion to the difference between the latest actual demand figure and the latest value of the average.

When there is no trend in the demand for a product or service, sales are forecasted for the next period, by means of the exponential smoothing method by using the expression

Forecast for the next period = a (latest actual demand) + (1 – α) old estimate of latest actual demand where a represents the value of a weighting factor which is referred to as a smoothing factor.

**This method follows the equation **

F_{n}= F_{n -1 }+ α (D _{n-1 }– F _{n-1})

where F_{n}= forecast for the next period

F_{n-1} = forecast for previous period

D _{n-1} = demand in previous period.

If a is equal to 1. then the latest forecast would be equal to previous period actual demand In practice, the value of a is generally chosen between 0.1 and 0.3. The application of technique is demonstrated by using data of moving averages method of sales forecasting on page 78. In the application of the method, we would use the value of a as 0.10.

Using equation (7), if the actual demand for 3rd week is 487, the forecast for the 4th week will be

0.10(487) + (1.00 – 0.10)550 = 544

Similarly, if the actual demand for 4th week is 528 customers, the forecast for the 5th week will be

0.10 (528) + (1.00 – 0.10) (544) = 542

If this procedure had been applied during the entire 8 week period the results are shown in the following table. The unadjusted forecast error is also indicated under column D = B – C. If the value of a is not given; it can be determined by an approximate relation of a.

α = 2/ Number of periods in moving average + 1

In equation (7) in so far as the weight factors a is concerned, it can assume a minimum value 0 and a maximum value of 1. The greater the value of a, the greater is the weight placed on recent data. When the value of a is 1, the forecast will be equal to the demand experienced during the last period.

Although the value of a varies from product to product but most organization have found that a value between 0 06 to 0.20 usually proves to be satisfactory.

When attempting to find out what value of a should be used for a product or service the organization/enterprise can select various values, examine the past forecasts with the use of these values and adopt for future use the one which would have minimized forecast errors in the past.

In this way we go close of the description of exponential smoothing as it is applied when a trend in sales/service is available. In case trend exist, a trend adjustment can be made with this technique but its application becomes bit difficult.