The purpose of this article is to present in a systematic way the methods of forecasting sales volumes that are most often used in economic practice. The main attention in the work is paid to the applied value of the considered methods, to the economic interpretation and interpretation of the results obtained, and not to the explanation of the mathematical and statistical apparatus, which is covered in detail in the special literature.

The simplest way to predict the market situation is extrapolation, i.e. the propagation of past trends into the future. Established objective tendencies of change economic indicators to a certain extent predetermine their size in the future. In addition, many market processes have some inertia. This is especially evident in short-term forecasting. At the same time, the forecast for the long-term period should take into account as much as possible the likelihood of changes in the conditions in which the market will operate.

Sales forecasting methods can be divided into three main groups:

• methods of expert assessments;
• methods of analysis and forecasting of time series;
• casual (causal) methods.

Expert assessment methods are based on subjective assessment of the current moment and development prospects. It is advisable to use these methods for market assessments, especially in cases where it is impossible to obtain direct information about any phenomenon or process.

The second and third groups of methods are based on the analysis of quantitative indicators, but they differ significantly from each other.

Methods of analysis and forecasting of time series are associated with the study of indicators isolated from each other, each of which consists of two elements: the forecast of the deterministic component and the forecast of the random component. The development of the first forecast does not present great difficulties if the main development trend is determined and its further extrapolation is possible. The prediction of a random component is more difficult, since its appearance can be estimated only with a certain probability.

At the heart of casual methods is an attempt to find the factors that determine the behavior of the predicted indicator. The search for these factors leads actually to economic and mathematical modeling - the construction of a model of the behavior of an economic object, taking into account the development of interrelated phenomena and processes. It should be noted that the use of multivariate forecasting requires solving a complex problem of choosing factors, which cannot be solved by a purely statistical means, but is associated with the need for a deep study of the economic content of the phenomenon or process under consideration. And here it is important to emphasize the primacy economic analysis before clean statistical methods studying the process.

Each of the considered groups of methods has certain advantages and disadvantages. Their application is more effective in short-term forecasting, since they simplify to a certain extent real processes and do not go beyond the present day. The simultaneous use of quantitative and qualitative forecasting methods should be ensured.

Let us consider in more detail the essence of some methods of forecasting the volume of sales, the possibility of their use in marketing analysis, as well as the necessary initial data and time constraints.

Sales forecasts with the help of experts can be obtained in one of three forms:

1. point forecast;
2. interval forecast;
3. forecasting the probability distribution.

A point sales forecast is a forecast for a specific number. It is the simplest of all forecasts because it contains the least amount of information. As a rule, it is assumed in advance that a point forecast may be erroneous, but the methodology does not provide for the calculation of the forecast error or the probability of an accurate forecast. Therefore, in practice, two other forecasting methods are often used: interval and probabilistic.

The interval forecast of the sales volume provides for the establishment of boundaries within which the predicted value of the indicator with a given level of significance will be located. An example is a statement of the type: "In the coming year, the sales volume will be from 11 to 12.4 million rubles."

The forecast of the probability distribution is associated with determining the probability of the actual value of the indicator falling into one of several groups at set intervals. An example would be a forecast like:

While there is a certain probability when making a forecast that the actual sales volume will not fall within the specified interval, forecasters believe that it is so small that it can be ignored when planning.

Intervals that take into account low, medium and high sales are sometimes called pessimistic, most probable, and optimistic. Of course, the probability distribution can be represented by a large number of groups, but the three indicated groups of intervals are most often used.

To identify the general opinion of experts, it is necessary to obtain data on the predicted values ​​from each expert, and then make calculations using a system of weighing individual values ​​according to some criterion. There are four methods of weighing different opinions:

The choice of method remains with the researcher and depends on the specific situation. None of these can be recommended for use in every situation.

The Delphi method allows avoiding the problem of weighing individual forecasts of experts and the distorting influence of the noted undesirable factors (see, for example,). It is based on the work on convergence of the points of view of experts. All experts are introduced to the assessments and justifications of other experts and are given the opportunity to change their assessment.

The second group of forecasting methods is based on time series analysis.

Table 1 presents a time series for the consumption of the Tarhun soft drink in decalitres (dal) in one of the regions since 1993. Time series analysis can be carried out not only on annual or monthly data, but quarterly, weekly or daily data can also be used. about sales volumes. For calculations was used software Statistica 5.0 for Windows.

Table 1
Monthly consumption of Tarhun soft drink in 1993-1999 (thousand dal)

According to table 1, let us build a graph of consumption of the drink "Tarhun" in 1993-1999. (Fig. 1), where the abscissa represents the observation dates, and the ordinate represents the consumption of the drink.

Rice. 1. Monthly consumption of the Tarhun drink in 1993-1999. (thousand dal)

Forecasting based on time series analysis assumes that changes in sales volumes that have occurred can be used to determine this indicator in subsequent periods of time. Time series, such as those shown in Table 1, usually serve to calculate four different types of changes in indicators: trend, seasonal, cyclical and random.

Trend- this is a change that determines the general direction of development, the main trend of the time series. Identifying the main development trend (trend) is called time series alignment, and methods for identifying the main trend are called alignment methods.

One of the simplest methods of detecting the general trend in the development of a phenomenon is the enlargement of the time series interval. The meaning of this technique is that the initial series of dynamics is transformed and replaced by another, the levels of which relate to long periods of time. For example, the monthly data in Table 1 can be converted to a series of annual data. The graph of the annual consumption of the Tarhun drink, shown in Figure 2, shows that consumption increases from year to year during the study period. The consumption trend is a characteristic of a relatively stable rate of growth of an indicator over a period.

Revealing the main trend can also be carried out using the moving average method. To determine the moving average, enlarged intervals are formed, consisting of the same number of levels. Each subsequent interval is obtained by gradually moving from the initial level of the time series by one value. Based on the aggregated data generated, we calculate the moving averages that refer to the middle of the aggregated interval.

Rice. 2. Annual consumption of the "Tarhun" drink in 1993-1999. (thousand dal)

The procedure for calculating the moving averages for the consumption of the Tarhun drink in 1993 is shown in Table 2. A similar calculation can be carried out on the basis of all data for 1993-1999.

table 2
Calculation of moving averages based on data for 1993

In this case, the calculation of the moving average does not allow us to conclude that there is a stable trend in the consumption of the Tarhun drink, since it is influenced by the intra-annual seasonal fluctuation, which can be eliminated only by calculating the moving averages for the year.

Studying the main development trend using the moving average method is an empirical technique of preliminary analysis. In order to give a quantitative model of time series changes, the method of analytical alignment is used. In this case, the actual levels of the series are replaced by theoretical ones, calculated according to a certain curve, reflecting the general trend of changes in indicators over time. Thus, the levels of the time series are considered as a function of time:

Y t = f (t).

The most commonly used functions are:

1. with uniform development - a linear function: Y t = b 0 + b 1 t;
2. when growing with acceleration:
   1. parabola of the second order: Y t = b 0 + b 1 t + b 2 t 2;
   2. cubic parabola: Y t = b 0 + b 1 t + b 2 t 2 + b 3 t 3;
3. at constant growth rates - exponential function: Y t = b 0 b 1 t;
4. when descending with deceleration - hyperbolic function: Y t = b 0 + b 1 x1 / t.

However, analytical alignment contains a number of conventions: the development of phenomena is determined not only by how much time has passed since the starting point, but also by what forces influenced the development, in which direction and with what intensity. The development of phenomena in time acts as an external expression of these forces.

The estimates of the parameters b 0, b 1, ... b n are found by the least squares method, the essence of which is to find such parameters for which the sum of the squares of the deviations of the calculated values ​​of the levels calculated by the sought formula from their actual values ​​would be minimal.

To smooth economic time series, it is inappropriate to use functions containing a large number of parameters, since the trend equations obtained in this way (especially with a small number of observations) will reflect random fluctuations, and not the main trend of the development of the phenomenon.

The calculated values ​​of the parameters of the regression equation and graphs of the theoretical and actual annual volumes of consumption of the Tarhun drink are shown in Figure 3.

Rice. 3. Theoretical and actual values ​​of the consumption of the drink "Tarhun" in 1993-1999. (thousand dal)

The selection of the type of function describing the trend, the parameters of which are determined by the least squares method, is performed in most cases empirically, by constructing a number of functions and comparing them with each other by the value of the mean square error.

The difference between the actual values ​​of the dynamics series and its equalized values ​​() characterizes random fluctuations (sometimes they are called residual fluctuations or statistical noise). In some cases, the latter combine trend, cyclical fluctuations and seasonal fluctuations.

The root-mean-square error, calculated according to the annual data on the consumption of the drink "Tarhun" for the equation of a straight line (Fig. 1), was 1.028 thousand decaliters. Based on the root mean square error, the marginal forecast error can be calculated. In order to guarantee the result with a probability of 95%, a coefficient equal to 2 is used; and for a probability of 99% this coefficient will increase to 3. So, we can guarantee with a probability of 95% that the volume of consumption in 2000 will amount to 134.882 thousand decalitres. plus (minus) 2.056 thousand decaliters

Calculations for the selection of functions describing the volume of consumption of the Tarhun drink in certain months from 1993 to 1999 showed that none of the above equations is suitable for predicting this indicator. In all cases, the explained variation did not exceed 28.8%.

Seasonal fluctuations- recurring from year to year changes in the indicator at certain intervals. Observing them over several years for each month (or quarter), you can calculate the corresponding averages, or medians, which are taken as characteristics of seasonal fluctuations.

When checking the monthly data from Table 1, you can find that the peak consumption of the drink occurs during the summer months. The volume of sales of children's shoes falls on the period before the start school year, increased consumption fresh vegetables and fruit occurs in the fall, an increase in volumes construction works- in summer, an increase in purchasing and retail prices for agricultural products - in winter period etc. Periodic fluctuations in retail can be found both during the week (for example, sales of certain food products increase before the weekend), and during any week of the month. However, the most significant seasonal fluctuations occur in certain months of the year. When analyzing seasonal fluctuations, the seasonality index is usually calculated, which is used to predict the indicator under study.

In its simplest form, the seasonality index is calculated as the ratio of the average level for the corresponding month to the overall average value of the indicator for the year (in percent). All other known methods for calculating seasonality differ in how the flattened average is calculated. Most often, either a moving average or an analytical model for the manifestation of seasonal fluctuations is used.

Most of the methods involve the use of a computer. A relatively simple method for calculating the seasonality index is the centered moving average method. To illustrate it, suppose that at the beginning of 1999 we wanted to calculate the seasonality index for the consumption of the Tarhun drink in June 1999. Using the moving average method, we would have to carry out the following steps in sequence:

Comparison of standard deviations, calculated for different periods of time, shows shifts in seasonality (growth indicates an increase in the seasonality of consumption of the drink "Tarhun").

Another method for calculating seasonality indices, often used in various kinds of economic research, is the seasonal adjustment method, known in computer programs as the census method (Census Method II). It is a kind of modification of the moving average method. A special computer program eliminates trend and cyclical components using a whole range of moving averages. In addition, random fluctuations have been removed from the average seasonal indices, since the extreme values ​​of the features are under control.

Calculating seasonality indices is the first step in forecasting. Usually, this calculation is carried out together with an assessment of the trend and random fluctuations and allows you to adjust the forecast values ​​of indicators obtained along the trend. It should be borne in mind that seasonal components can be additive and multiplicative. For example, sales of soft drinks increase by 2000 decalitres each year during the summer months, so 2000 decaliters must be added to the existing projections in these months to take into account seasonal fluctuations. In this case, seasonality is additive. However, during the summer months, the sale of soft drinks can increase by 30%, that is, the coefficient is 1.3. In this case, the seasonality is multiplicative, or in other words, the multiplicative seasonal component is 1.3.

Table 3 shows the calculations of indices and factors of seasonality by census methods and centered moving average.

Table 3
Seasonality indices of sales of the Tarhun beverage, calculated based on data for 1993-1999.

The data in Table 3 characterize the nature of the seasonality of consumption of the Tarhun drink: in the summer months, the volume of consumption increases, and in the winter months, it falls. Moreover, the data of both methods - the census and the centered moving average - give almost the same results. The choice of the method is determined depending on the forecast error mentioned above. So, the indices, or factors, seasonality can be taken into account when forecasting sales volumes by adjusting the trend value of the predicted indicator. For example, suppose that the forecast for June 1999 was made using the moving average method and it was 10.480 thousand decaliters. The seasonality index in June (according to the census method) is 115.1. Thus, the final forecast for June 1999 will be: (10.480 x 115.1) / 100 = 12.062 thousand decaliters.

If on the studied time interval the coefficients of the regression equation that describes the trend remained unchanged, then the least squares method would be sufficient to construct a forecast. However, during the study period, the coefficients may change. Naturally, in such cases, later observations are of greater information value compared to earlier observations, and therefore, they need to be assigned the greatest weight. It is precisely these principles that the exponential smoothing method meets, which can be used for short-term forecasting of sales volume. The calculation is carried out using exponentially weighted moving averages:

where Z- smoothed (exponential) sales volume;
t- time period;
a- smoothing constant;
Y- the actual volume of sales.

Using this formula sequentially, the exponential sales volume Zt can be expressed in terms of the actual sales volume Y:

where SO is the initial value of the exponential average.

When constructing forecasts using the exponential smoothing method, one of the main problems is the choice of the optimal value of the smoothing parameter a. It is clear that for different values ​​of a, the forecast results will be different. If a is close to one, then this leads to taking into account in the forecast mainly the influence of only the latest observations; if a is close to zero, then the weights by which the sales volumes are weighed in the time series decrease slowly, i.e. the forecast takes into account all (or almost all) observations. If there is not sufficient confidence in the choice of the initial prediction conditions, then an iterative method of calculating a in the range from 0 to 1. There are special computer programs for determining this constant. The results of calculating the sales volume of the Tarhun drink by the exponential smoothing method are shown in Figure 4.

The graph shows that the aligned series reproduces the actual sales data fairly accurately. In this case, the forecast takes into account the data of all past observations, the weights by which the levels of the time series are weighed decrease slowly, a

Table 5
The results of forecasting the volume of sales of the drink "Tarhun" in 1999

The technique for detecting cyclicality is as follows. Market indicators showing the greatest fluctuations are selected, and their time series are built for the longest possible period. Each of them excludes the trend, as well as seasonal fluctuations. Residual series reflecting only conjuncture or purely random fluctuations are standardized, i.e. are reduced to the same denominator. Then the coefficients of correlation are calculated, which characterize the relationship between the indicators. Multidimensional links are divided into homogeneous cluster groups. Plotted cluster assessments should show the sequence of changes in the main market processes and their movement in the phases of conjuncture cycles.

Casual sales forecasting techniques involve developing and using predictive models in which changes in sales are the result of changes in one or more variables.

Casual forecasting methods require determining factor indicators, assessing their changes and establishing a relationship between them and the volume of sales. Of all the casual forecasting methods, we will consider only those that can be used with the greatest effect to predict the volume of sales. These methods include:

• correlation and regression analysis;
• method of leading indicators;
• the method of surveying the intentions of consumers, etc.

Correlation-regression analysis is one of the most widely used casual methods. The technique of this analysis is discussed in sufficient detail in all statistical reference books and textbooks. Let's consider only the possibilities of this method in relation to forecasting the volume of sales.

A regression model can be built in which variables such as the level of consumer income, prices for competitors' products, advertising costs, etc. can be selected as factor signs. The multiple regression equation has the form

Y (X 1; X 2; ...; X n) = b 0 + b 1 x X 1 + b 2 x X 2 + ... + b n x X n,

where Y is the predicted (effective) indicator; in this case, the volume of sales;
X 1; X 2; ...; X n - factors (independent variables); in this case, the level of consumer income, prices for competitors' products, etc .;
n is the number of independent variables;
b 0 - free term of the regression equation;
b 1; b 2; ...; b n - regression coefficients that measure the deviation of the resultant attribute from its average value when the factor attribute deviates per unit of its measurement.

The sequence of developing a regression model for forecasting sales includes the following steps:

1. preliminary selection of independent factors that, according to the opinion of the researcher, determine the volume of sales. These factors must either be known (for example, when predicting the sales volume of color TVs (effective indicator), the number of color TVs currently in use can act as a factor indicator); or they are easily determined (for example, the ratio of the price of the investigated product of the firm with the prices of competitors);
2. collection of data on independent variables. In this case, a time series is built for each factor, or data are collected for a certain population (for example, a set of enterprises). In other words, it is necessary that each independent variable be represented by 20 or more observations;
3. determining the relationship between each independent variable and an outcome characteristic. In principle, the relationship between the features should be linear, otherwise the equation is linearized by replacing or transforming the value of the factor feature;
4. conducting regression analysis, i.e. calculating the equation and regression coefficients, and checking their significance;
5. repeating steps 1-4 until a satisfactory model is obtained. The criterion for the satisfactory nature of the model can be its ability to reproduce actual data with a given degree of accuracy;
6. comparison of the role of various factors in the formation of the modeled indicator. For comparison, you can calculate the partial elasticity coefficients, which show how many percent on average the sales volume will change when the factor X j changes by one percent with a fixed position of other factors. The coefficient of elasticity is determined by the formula

where b j is the regression coefficient for the j-th factor.

Regression models can be used to predict the demand for consumer goods and means of production. As a result of the correlation-regression analysis of the sales volume of the drink "Tarhun", the model

Y t + 1 = 2.021 + 0.743A t + 0.856Y t,

where Y t + 1 is the projected sales volume in the month t + 1;
A t - advertising costs in the current month t;
Y t - sales volume in the current month t.

The following interpretation of the multivariate regression equation is possible: the volume of sales of the drink increased on average by 2,021 thousand decaliters, with an increase in advertising costs by 1 ruble. the average sales volume increased by 0.743 thousand decalitres, while the sales volume of the previous month increased by 1 thousand decaliters, the sales volume in the following month increased by 0.856 thousand decaliters.

Leading Indicators- these are indicators that change in the same direction as the studied indicator, but ahead of it in time. For example, a change in the standard of living of the population entails a change in the demand for individual goods, and therefore, by studying the dynamics of indicators of the standard of living, one can draw conclusions about a possible change in demand for these goods. It is known that in developed countries as incomes rise, so does the need for services, and in developing countries, for durable goods.

Leading indicator method is more often used to predict changes in the business as a whole than to predict the volume of sales of individual companies. Although it cannot be denied that the level of sales of most companies depends on the general market situation in the regions and the country as a whole. Therefore, before forecasting their own sales, firms often need to assess the overall level of economic activity in a region.

Data from surveys of consumer intentions can serve as a significant justification for forecasting the volume of sales of consumer goods. They know more about their own prospect purchases than anyone, which is why many companies conduct periodic surveys of consumers' opinions about their products and the likelihood of buying them in the future. Most often, these surveys concern goods and services, the purchase of which is planned. potential buyers in advance (as a rule, these are expensive purchases such as a car, apartment or travel).

Of course, the usefulness of such surveys cannot be underestimated, but it is also impossible not to take into account that consumers' intentions for a particular product may change, which will affect the deviation of the actual consumption data from the forecast.

So, when forecasting the volume of sales, all the methods discussed above can be used. Naturally, the question arises about the optimal forecasting method in a particular situation. The choice of method is associated with at least three limiting conditions:

1. forecast accuracy;
2. availability of the necessary initial data;
3. availability of time for forecasting.

If a forecast with an accuracy of 5% is required, then all forecasting methods that provide an accuracy of 10% may not be considered. If there is no data necessary for forecasting (for example, time series data when predicting the volume of sales of a new product), then the researcher is forced to resort to casual methods or expert estimates. A similar situation may arise due to an urgent need for forecast data. In this case, the researcher should be guided by the time at his disposal, realizing that the urgency of the calculations can affect their accuracy.

It should be noted that a measure of the quality of a forecast can be a coefficient characterizing the ratio of the number of confirmed forecasts to the total number of forecasts made. It is very important to calculate this coefficient not at the end of the forecast period, but when making the forecast itself. To do this, you can use the inverse verification method by means of retrospective forecasting. This means that the correctness of a predictive model is tested by its ability to reproduce actual data in the past. There are no other formal criteria, knowledge of which would make it possible to state a priori about the approximating ability of the predictive model.

Sales forecasting is an integral part of the decision-making process; it is a systematic review of the company's resources, allowing you to more fully exploit its advantages and identify potential threats in a timely manner. The company must constantly monitor the dynamics of sales and alternative opportunities for the development of the market situation in order to best allocate available resources and choose the most appropriate areas of its activities.

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The mistake of many businessmen is blind sales. They do not make any sales forecasts, evaluating only the results of the reporting period. This scheme resembles a roller coaster: now a peak, then a long lull.

Why shouldn't you do this?

• If you do not make a forecast of sales, the staff will drop. There is no benchmark for what to strive for.
• Any number is assessed on the basis of "at least something".
• There is no spirit of competition, no leaders to look up to.

To achieve goals, they must first of all be set. To increase revenue, you need to make a forecast. The main thing is that the desired growth is realistic. Practice shows that forecast figures are achieved when the planned indicators differ from the real capabilities of your sellers by no more than 30-35%.

Pay attention to the following methods of forecasting:

1. Plus 10% of what has been achieved

This method is familiar to those who have studied the Soviet economy and its forecasting methodology. The main point of this method is to predict indicators 10-15% higher than was achieved in the previous reporting period.

This method works well when your company has already built a sales system, and each manager has minimum acceptable performance indicators.

However, with this method, it is important to establish the real capabilities of your salespeople. So that the forecast has a challenge, and does not contain indicators of the lower level of the acceptable.

2. Matching the best

It is a popular motivator for achieving your goals. The main essence of the method is to show that if someone was able to meet the expectations of the sales forecast, then others can.

However, as a guide to the numbers in the forecast, this method is not always effective. At least, because in any sales department there are "locomotives" and "candidates for dismissal." Therefore, to make the forecast more realistic and justified, you need to focus on something in between the results of these two categories.

3. We look at competitors

It is logical to make a forecast based on your own achievements, but periodically you need to compare yourself with competitors in order to take a leading position.

This is a great way to forecast sales if you have access to competitor information. To their strategy, business processes, purchase prices, discounts, and much more that is not written in commercial offers and is not described on the site.

You can get this information different ways... Including, conducting partisan methods of work. For example, call a competitor under the guise of a buyer and see how he has built a chain of work with a client.

4. Encourage our desires

One of the methods of making a forecast of sales is that you start from your real desires. Even if this does not correspond to common sense. But you set yourself certain numbers for your goal and select methods for its implementation.

5. Focusing on your sales funnel

This method can be used for forecasting if you have measurements of the results of all stages of sales. Those. you know all the numbers that affect sales in your business.

To get everything required indicators- analyze the work of your department. To make a forecast, figures are needed for a period of 2-3 months.

What information should you analyze:

• how much time is spent on average on one cold call,
• how much time is spent on average collecting information about a potential client,
• how many calls need to be made in order to reach the person, the decision,
• how many meetings a manager can actually hold a day,
• what percentage of appointments ends with an order,
• the number of repeat sales,
• average check.

With these numbers in hand, you can make a realistic forecast.

How to decompose a plan

It is necessary to determine the goals that you set in the forecasts. Further, it is important to decompose them into tasks for each employee.

Therefore, when making a sales forecast, break down the overall vision into specific areas that you need to work with to achieve a result.

The following plans need to be drawn up:

• For new clients;
• For new products;
• Increasing the share in current customers;
• From various channels;
• Customer churn;
• On non-return of receivables (if there is such a problem).

Break each figure in the plan in the following directions:

• By region;
• By department;
• By employees;
• By months / days;
• By intermediate performance indicators, taking into account indicators for in the funnel (current and new customer base).

The more accurately and in detail you disaggregate the numbers in each plan, the more likely the forecast will come true.

Decomposition example

Let's give an example of decomposition of the sales forecast to the level of daily indicators for each employee. But before you do this, make sure that the commercial structure is working optimally. It is necessary to conduct a small audit in 4 areas.

Clients. It is necessary to segment the current customer base in order to identify the main target groups and focus on working with the most profitable ones.

Channels. Analyze the conversion of each of them based on the average cost per lead and stop investing in something that does not bring results.

Employees. Only the most people should stay in the department. best shots... Screening will happen automatically if you implement 2 principles:

• the principle of "compound salary", in which the bonus part for fulfilling the sales forecast is at least 50%;
• the principle of "large thresholds", which regulates the payment of bonuses: if he did not fulfill up to 80% of the plan - he did not receive a bonus, 80-100% - plus 1 salary, overfulfilled the plan - plus 2 salaries.

Products. Get rid of illiquid and low-margin products. This will prevent wasting resources.

With the optimally tuned system in place, proceed with the decomposition following the plan below.

1. Determine the projected profit figure. Look at the profits of the previous periods. Eliminate one-time deals. Consider marketing impact and seasonality.

2. Knowing your marginality, calculate the revenue by the profit share.

3. Divide the revenue by the average bill and get the approximate number of trades that need to be closed in order to achieve the set profit.

4. Using the conversion rate from the application to the buyer, calculate the number of leads.

5. Based on the intermediate conversion in the funnel, calculate the total number of actions that need to be performed within the business process. We are talking about calls, meetings, presentations, repeat calls, sent commercial offers, invoices.

6. Once you have quantitative indicators of each stage, divide them by the number of working days of the forecast period (most often it is customary to speak of a month).

This way, you figure out what and how much each salesperson should do so that the entire department will eventually close the plan by the end of the month. Monitor the achievement of these indicators on a daily basis.

Sales forecasting: accurate calculation or fortune telling on the coffee grounds? When we were building the system in the Urban Group developer company, Commercial Director, Dmitry Usmanov, asked a question - will we sign under a specific number. We have named the number, date and time.

Three weeks later, at 12.15 pm, we sat in a cafe and watched the schedule of receipts. At 12.00, the parcels for the last day are posted. The forecast accuracy was 99.7%.

The most frequent question that clients ask us is: "How can you calculate the future sales volume so accurately?"

It's all about coffee) No, not about how you can find out the fate of your business, but about which we drink while we solve the forecasting problem for each specific enterprise.

Do not confuse sales forecasts based on detailed calculations with unscientific divination. Let's look at how to make the most accurate sales forecast and what tasks it solves.

What is a sales forecast for?

1. Setting goals ... The figure obtained according to the annual forecast is what the company should come to for the next year, the plan that needs to be fulfilled. This is part of the business plan for the company and a real, well-defined goal for the sales department, from which you can build on when calculating premiums and bonuses. Very often, the goal is set from desires, and not from real possibilities.
Therefore, before setting a goal, you must first make a forecast, and then set a goal. If the goal is higher than the forecast, then you need to understand through what changes the goal will be achieved.

2. Formation of the necessary base of labor and production resources. Based on the projected number of customers and sales. Objective: Plan procurement and determine the future equipment and personnel needs of the company.

3. Warehouse stock management ... At each moment of time, the production will have warehouse balance sufficient to complete tasks at a certain stage. No shortage or excess of materials in stock - only rational use of funds!

4. Increasing business mobility ... On the forecast chart (or in the table), you can see in advance the moments of a possible drawdown in sales (for example, due to the seasonality of the product) and take measures to correct the situation even before the end of the period. In addition, the chances are increased to instantly track an unplanned decline in sales, to quickly identify the reasons for the decline in performance and to correct the situation in a timely manner.

5. Control and optimization of costs ... Forecasting will show what costs the company will incur in general for the production and sale of products. This means that you can develop a budget and determine in advance what costs are to be reduced in case of non-fulfillment of the forecast for an increase in sales.

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Forecasting methods and how they work

There are 3 main groups of methods:

1. The method of expert assessments ... The basis for them is a subjective assessment of a certain group of experts who have their own vision of the current situation and development prospects. Heads of companies and top managers act as internal experts. External experts can be hired consultants and financial analysts.

This technique is chosen in the absence of a large amount of statistical data, for example, when a company introduces a new product or service to the market. Experts assess the problem based on intuition and logic. The generalized opinion of experts is becoming a forecast. The method is highly dependent on the experience of an expert in the industry. Sometimes this is the best way to predict. And this has nothing to do with fortune-telling. Intuition is a calculation in our brain that humans cannot track. The main thing is to be able to clear the intuition of prejudices.

Example.

Brainstorming is a collective method peer review, which is attended by the heads of sales, marketing, production and logistics. Each in turn voices the factors that can positively or negatively affect future sales. The forecast is formed according to the consolidated list of the ideas put forward.

But it must be borne in mind that each of the participants will have their own interests. Salespeople need to underestimate the plan in order to then heroically execute it. Marketers overestimate to show market prospects. Production can reduce the assortment to 1 unit and form an even schedule; logistics do not need peaks and valleys.

2. Methods for analysis and forecasting of time series . The best option for a company that has accumulated a sales database for several years. For simplified forecasting, you can use the standard Excel program. It compiles a table with monthly sales for each year and builds a graph based on this table.

The graph shows the main trend (increase or decrease in sales volumes) as well as seasonal fluctuations. It remains to extrapolate the curve to a month, a year, or any other period of time. You can extend this method with the following item.

3. Casual (causal) methods. They take into account the dependence of the level of sales on one or more variables. To build an adequate model, it is necessary to know the independent factors that affect demand.
What are these factors? Incomes of the population, prices of competitors, advertising efficiency, production volumes of related areas - that is, everything that determines consumer behavior.

Example.

The company sells sanitary ware. The first factor is the volume of construction in the region. They were down 15% last year, and sanitary ware sales fell 10%. Next year, the crisis in the construction sector will continue, which means that sales of toilets, sinks and bathtubs will also fall. The second factor is advertising. As the experience of the plumbing company in the past has shown, a 10% increase in advertising costs increases sales by 20%. And so on for each factor of influence.

The final score is calculated using a multivariate equation in which each variable is tested and its level of significance is verified.

The choice of method depends on what kind of raw data is available. The most effective solution is a combination of several methods.

It should be borne in mind that forecasting the amount of sales works best in short term, and not because of some peculiarities of the calculation, but because at the business level it is almost impossible to predict the change in external political and economic conditions... Remember who was prepared for the 2008 crisis? And what about sanctions because of the situation in Ukraine?

How to Calculate Sales Forecast - Business Checklist

Check out which forecasting algorithm we use before guaranteeing our customers a 20-200% increase in sales:

• We analyze the results of the enterprise for the previous period... We take monthly or weekly data for the previous three years. For a new product that does not have a sales history, we use expert assessment methods - we rely on the experience of our specialists who have worked with a similar business, we interview external experts and study competitors.

At the same stage, based on the information provided, we determine the elasticity of demand in order to understand how strongly the sales volume depends on the increase / decrease in prices, if there were any during these periods. Each extremum on the chart was explained by analyzing the turnover structure. Which customers bought more or less, why, what influenced. In 99% of cases, the answers are found effortlessly.

• Determining the market trend... It is possible to predict an increase in product sales only if the general market trend is growing or at least stable. You can see the current trends in YandexWordstat - we type a request corresponding to the client's product and study the graph.

If the demand curve is steadily declining and there is no information about the imminent end of the crisis in this industry, you should not count on sales growth. however, you can try to stay at the current level., the crisis does not last forever. And if you maintain market share, you will have a better start than the competition when you rise.

• We take into account the seasonality of the offered product / service... If you have information on past sales - great! If not, there is an easy way to find out the presence or absence of seasonal fluctuations - use the same graph for the dynamics of requests.

See how the seasonal fluctuations are clearly visible for the query "roofing materials": summer peaks and winter dips. For goods and services, the demand for which is characterized by a pronounced seasonality, it is necessary to calculate the seasonality coefficient for each planning period.

Example.

The company sells soft roofing rolls. In April last year, 100 rolls were sold, and already in June - 176 rolls. In April this year, the company sold 124 rolls, how many rolls will it sell in June? A simple task for primary school solved in one action: 176/100 * 124 = 218 rolls (where 176/100 = 1.76 is the seasonality coefficient). Similarly, you can calculate the coefficient for the whole market.

• We evaluate the current USP. For example, when selling an apartment, we evaluate the company's USP by 32 parameters, assign a weight to each characteristic and clearly understand the strength of our proposal. The quality is unique trade offer seriously affects the conversion. After competitive analysis we can say what will be the conversion on the site for a particular business - 2% or all 10%. If we finalize a frankly weak USP and clearly prescribe it in advertisements, you can significantly increase the number of requests

• We test the effectiveness of advertising for each sales channel... For offline stores, you can run a test advertising campaign in newspapers, on television channels in the region. For online stores - we place targeted advertising in social networks or contextual ads in Yandex.Direct (GoogleAdwords). Each advertising channel is assigned its own phone number or any other marker that allows us to determine what exactly worked.

Example.

The company sells metal doors in two stores in its city and an online store with delivery in the region. Advertising in newspapers is a coupon with 5% discount, which must be presented when contacting. V contextual advertising we place the phone and track the number of calls received on it. One ad increased the number of customers by 10%, and the second did not work? We use this information for planning and forecasting.

• We analyze the client base by individuals and legal entities, the average bill, the regularity of purchases. We take statistics on already completed transactions, calculate the average check for each group of clients. We have already figured out how many new customers the advertisement will bring us. We multiply their number by the average bill and we get the projected sales volume.

The calculation of future sales volumes for the B2B segment has its own peculiarities. As a rule, these are not one-time clients, but regular business partners who will buy goods throughout the year. Accordingly, in addition to the average check, it is necessary to determine the frequency of deliveries. The potential can be assessed using the 2gis.ru databases.

• Checking how sales managers work... Listening to how managers work with requests. If, following the results of communication with a potential client, the manager could not bring him to the order, you need to draw up effective scripts telephone conversations and conduct staff training. As a result, out of 10 requests, not 1 client will reach the purchase, but 3.

When we make a forecast for sales growth, we use this particular checklist, supplementing or modifying it depending on the type of business. As you can see, it contains elements of all three methods. An estimate is given for each hypothesis, but their combination provides a high forecast accuracy.

We can guarantee the most accurate forecasting provided that the client first provides us with as much initial data as possible, and then all implementations are clearly implemented. We will conduct an audit of any business and accurately determine the volume that your business is capable of and do not be offended if it is several times your current

This article discusses one of the main forecasting methods - time series analysis. On the example of a retail store using this method, the sales volumes for the forecast period are determined.

One of the main responsibilities of any manager is to competently plan the work of his company. The world and business are now changing very rapidly, and it is not easy to keep up with all the changes. Many events that cannot be foreseen in advance change the plans of the company (for example, the release of a new product or group of goods, the appearance of a strong company on the market, the unification of competitors). But you need to understand that often plans are needed only in order to make adjustments to them, and there is nothing wrong with that.

Any forecasting process, as a rule, is built in the following sequence:

1. Formulation of the problem.

2. Gathering information and choosing a forecasting method.

3. Application of the method and assessment of the resulting forecast.

4. Using the forecast for decision making.

5. Forecast-fact analysis.

It all starts with the correct formulation of the problem. Depending on it, the forecasting problem can be reduced, for example, to an optimization problem. For short-term production planning, it is not so important what the sales volume will be in the coming days. It is more important to distribute production volumes as efficiently as possible across the available capacities.

The cornerstone limitation when choosing a forecasting method will be the initial information: its type, availability, processing capability, homogeneity, volume.

The choice of a specific forecasting method depends on many factors. Is there enough objective information about the predicted phenomenon (does this product or analogues exist for a long time)? Are qualitative changes in the studied phenomenon expected? Are there dependencies between the studied phenomena and / or within data sets (sales volumes, as a rule, depend on the volume of investments in advertising)? Is the data a time series (information about the ownership of borrowers is not a time series)? Are there recurring events (seasonal fluctuations)?

Regardless of what industry and area of ​​economic activity the company operates in, its management constantly has to make decisions, the consequences of which will manifest themselves in the future. Any decision is based on one way or another. One of these methods is forecasting.

Forecasting- this is a scientific definition of the likely paths and results of the upcoming development of the economic system and the assessment of indicators characterizing this development in a more or less distant future.

Consider forecasting sales volume using a time series analysis method.

Forecasting based on time series analysis assumes that changes in sales volumes that have occurred can be used to determine this indicator in subsequent periods of time.

Time series - it is a series of observations carried out regularly at regular intervals: a year, a week, a day, or even minutes, depending on the nature of the variable under consideration.

Usually a time series consists of several components:

1) trend - the general long-term trend of changes in the time series, which underlies its dynamics;

2) seasonal variation - short-term regularly recurring fluctuations in the values ​​of the time series around the trend;

3) cyclical fluctuations that characterize the so-called business cycle, or the economic cycle, consisting of economic recovery, recession, depression and recovery. This cycle repeats regularly.

To combine individual elements of a time series, you can use multiplicative model:

Sales Volume = Trend × Seasonal Variation × Residual Variation. (one)

In the course of drawing up a sales forecast, the indicators of the company over the past few years, the forecast of market growth, and the dynamics of competitors' development are taken into account. Optimal sales forecasting and forecast adjustments provide a complete sales report of the company.

Let us apply this method to determine the sales volume of the "Hours" salon for 2009. 1 shows the sales volumes of the "Watch" salon, which specializes in the retail sale of watches.

Table 1. Dynamics of the sales volume of the "Watch" salon, thousand rubles.

For the data shown in table. 1, we note two main points:

existing trend: the volume of sales in the corresponding quarters of each year is growing steadily from year to year;

• seasonal variation: in the first three quarters of each year, sales grow slowly but remain relatively low; the highest annual sales values ​​are always in the fourth quarter. This dynamics is repeated from year to year. This type of deviation is always called seasonal, even when it comes to, for example, a time series of weekly sales volumes. This term simply reflects the regularity and short-term nature of the deviations from the trend compared to the length of the time series.

The first step in time series analysis is plotting the data.

In order to make a forecast, you must first calculate the trend and then the seasonal components.

Trend calculation

A trend is a general long-term trend in the time series underlying its dynamics.

If you look at fig. 2, then through the points of the histogram, you can draw an upward trend line by hand. However, there are mathematical methods for this that allow you to assess the trend more objectively and accurately.

If the time series has seasonal variation, the moving average method is usually used. The traditional method of predicting the future value of the indicator is averaging n its past values.

Moving averages (which serve as an estimate of the future value of demand) are expressed as:

Moving Average = Sum of demand for previous n-periods / n. (2)

Average sales for the first four quarters = (937.6 + 657.6 + 1001.8 + 1239.2) / 4 = 959.075 thousand rubles.

When a quarter ends, the sales data for the last quarter is added to the sum of the data for the previous three quarters, and the data for the earlier quarter is discarded. This results in smoothing out short-term irregularities in the data set.

Average sales for the next four quarters = (657.6 + 1001.8 + 1239.2 + 1112.5) / 4 = 1002.775 thousand rubles.

The first calculated average shows the average sales volume for the first year and is in the middle between the sales data for the II and III quarters of 2007. The average for the next four quarters will be placed between the sales volume for the III and IV quarters. So the data in column 3 is the trend of the moving averages.

But to continue analyzing the time series and calculating seasonal variation, it is necessary to know the trend value at exactly the same time as the original data, therefore, it is necessary to center the obtained moving averages by adding adjacent values ​​and dividing them in half. The centered mean is the value of the calculated trend (calculations are presented in columns 4 and 5 of Table 2).

Table 2. Time series analysis

	Sales volume, thousand rubles	Four Quarter Moving Average	The sum of two adjacent values	Trend, thousand rubles	Sales volume / trend × 100
I quarter. 2007 year
II quarter. 2007 year
III quarter. 2007 year
IV quarter. 2007 year
I quarter. 2008 r.
II quarter. 2008 r.
III quarter. 2008 r.
IV quarter. 2008 r.

To make a forecast of sales for each quarter of 2009, it is necessary to continue the trend of moving averages on the chart. Since the smoothing process eliminated all fluctuations around the trend, it will not be difficult to do this. The spread of the trend is shown by the line in Fig. 4. According to the schedule, you can determine the forecast for each quarter (Table 3).

Table 3. Forecast of the trend for 2009

2009 r.	Sales volume, thous.rub.

Calculating seasonal variation

In order to make a realistic forecast of sales for each quarter of 2009, it is necessary to consider the quarterly dynamics of the sales volume and calculate the seasonal variation. If you look at the sales data for the previous period and neglect the trend, you can see the seasonal variation more clearly. Since for the analysis of the time series will be used multiplicative model, you must divide each sales measure by the trend value, as shown in the following formula:

Multiplicative Model = Trend × Seasonal Variation × Residual Variation × Sales Volume / Trend = Seasonal Variation × Residual Variation. (3)

The calculation results are presented in column 6 of the table. 2. In order to express the values ​​of indicators in percent and round them to the first decimal place, we multiply them by 100.

Now we will take the data for each quarter in turn and set how much, on average, they are more or less than the trend values. The calculations are shown in table. 4.

Table 4. Calculation of the average quarterly variation, thousand rubles.

	I quarter	II quarter	III quarter	IV quarter
Unadjusted mean

Uncorrected data in table. 4 contain both seasonal and residual variation. To remove the element of residual variation, it is necessary to correct the means. In the long term, the amount of sales overtreated in good quarters should be equalized with the amount of sales undertrend in unlucky quarters so that the seasonal components add up to about 400%. In this case, the sum of the unadjusted means is 398.6. Thus, each average must be multiplied by a correction factor to add up the averages to 400.

Correction factor is calculated as follows: Correction factor = 400 / 398.6 = 1.0036.

The calculation of seasonal variation is presented in table. 5.

Table 5. Calculation of seasonal variation

Based on the data in table. 5, one can predict, for example, that in the first quarter the volume of sales on average will be 96.3% of the trend value, in the fourth quarter - 118.1% of the trend value.

Sales forecast

When making a sales forecast, we proceed from the following assumptions:

the dynamics of the trend will remain unchanged compared to previous periods;

the seasonal variation will retain its behavior.

Naturally, this assumption may turn out to be incorrect; adjustments will have to be made, taking into account the expected expert change in the situation. For example, another large watch dealer may enter the market and bring down the prices of the "Hours" salon, the economic situation in the country may change, etc.

Nevertheless, based on the above assumptions, it is possible to forecast sales by quarters for 2009. To do this, the obtained values ​​of the quarterly trend must be multiplied by the value of the corresponding seasonal variation for each quarter. The calculation of the data is given in table. 6.

Table 6. Making a forecast of sales by quarters of the "Watch" salon for 2009

From the obtained forecast, it is clear that the turnover of the "Watch" salon in 2009 may amount to 5814 thousand rubles, but for this the enterprise needs to carry out various measures.

Read the full text of the article in the "Economist's Handbook" magazine # 11 (2009).

The cornerstone of inventory management and a huge managerial headache. How to do it in practice?

The purpose of these notes is not to present the theory of forecasting - there are many books. The goal is to concisely and, if possible, without deep and rigorous mathematics, to give an overview different methods and application practices specifically in the field of inventory management. I tried not to "get into the jungle", to consider only the most common situations. The notes are written by a practitioner and for practitioners, so you shouldn't look for some sophisticated techniques here, only the most general ones are described. That is to say, mainstream in its purest form.

However, as elsewhere on this site, participation is encouraged in every possible way - add, correct, criticize ...

Forecasting. Formulation of the problem

Any forecast is always wrong. The whole question is how wrong he is.

So we have sales data at our disposal. Let it look like this:

In the language of mathematics, this is called a time series:

A time series has two critical properties

the values ​​are necessarily ordered. Rearrange any two values ​​in places, and get another row

it is assumed that the values ​​in the series are the result of measurements at regular fixed intervals; predicting the behavior of a series means getting a "continuation" of a series at the same intervals for a given forecast horizon

Hence the requirement for the accuracy of the initial data - if we want to get a weekly forecast, the initial accuracy should be no worse than weekly shipments.

It also follows that if we "get" sales data from the accounting system on a monthly basis, they cannot be used directly, since the amount of time during which shipments were made is different in each month and this introduces an additional error, since the sales volume is approximately proportional to this time ...

However, this is not such a difficult problem - let's just bring this data to average daily.

In order to make any assumptions about the further course of the process, we must, as already mentioned, reduce the degree of our ignorance. We assume that our process has some kind of internal flow patterns that are completely objective in the current environment. In general terms, this can be represented as

Y (t) is the value of our series (for example, sales volume) at time t

f (t) is a function that describes the internal logic of the process. In what follows, we will call it the predictive model

e (t) - noise, error associated with the randomness of the process. Or, which is the same, related to our ignorance, inability to take into account other factors in the f (t) model.

Now our task is to find such a model so that the magnitude of the error is noticeably less than the observed value. If we find such a model, we can assume that the process in the future will go approximately in accordance with this model. Moreover, the more accurately the model describes the process in the past, the more confidence we have that it will work in the future.

Therefore, the process is usually iterative. Based on a simple look at the graph, the forecaster chooses a simple model and selects its parameters in such a way that the value

was, in a sense, the minimum possible. This quantity is usually called "residuals", because it is what is left after subtracting the model from the actual data, what the model could not describe. To assess how well the model describes the process, it is necessary to calculate a certain integral characteristic of the magnitude of the error. Most often, to calculate this integral value of the error, the mean absolute or root-mean-square value of the residuals over all t is used. If the error is large enough, an attempt is made to “improve” the model; choose a more complex type of model, take into account a larger number of factors. As practitioners, we should strictly adhere to at least two rules in this process:

Naive forecasting methods

Naive methods

Simple mean

In the simple case, when the measured values ​​fluctuate around a certain level, it is obvious that the average value is estimated and that real sales will continue to fluctuate around this value.

Moving average

In reality, as a rule, the picture is at least a little, but "floats". The company is growing, the turnover is increasing. One of the modifications of the mean model that takes this phenomenon into account is to discard the oldest data and use only a few of the last k points to calculate the mean. The method is called "moving average".

Weighted Moving Average

The next step in modifying the model is the assumption that the later values ​​of the series more adequately reflect the situation. Then each value is assigned a weight, the more the more recent the value is added.

For convenience, you can immediately select the coefficients so that their sum is one, then you do not have to divide. We will say that such coefficients are normalized to unity.

The results of forecasting for 5 periods ahead using these three algorithms are shown in the table

Simple exponential smoothing

In English-language literature, the abbreviation SES is often found - Simple Exponential Smoothing

One of the varieties of the averaging method is exponential smoothing method... It differs in that a number of coefficients are chosen here in a completely definite way - their value decreases exponentially. Let's dwell here in a little more detail, since the method has become ubiquitous due to its simplicity and ease of calculations.

Let us make a forecast at the moment of time t + 1 (for the next period). Let's denote it as

Here we take the forecast of the last period as the basis for the forecast, and add an amendment associated with the error of this forecast. The weight of this adjustment will determine how “harshly” our model will react to changes. It's obvious that

It is believed that for a slowly changing series it is better to take a value of 0.1, and for a rapidly changing one - to select in the region of 0.3-0.5.

If we rewrite this formula in a different form, we get

We got the so-called recurrent relation - when the next term is expressed through the previous one. Now we express the forecast of the past period in the same way through the value of the series before last, and so on. As a result, it is possible to obtain the forecast formula

As an illustration, we will demonstrate smoothing at different values ​​of the smoothing constant.

Obviously, if the turnover is growing more or less monotonously, with this approach we will systematically receive underestimated forecasts. And vice versa.

And finally, the anti-aliasing technique using spreadsheets. For the first value of the forecast, we will take the actual, and then using the recursion formula:

Predictive model components

Obviously, if the turnover is growing more or less monotonously, with this “averaging” approach, we will systematically receive underestimated forecasts. And vice versa.

In order to more adequately model the trend, the concept of "trend" is introduced into the model, ie. some smooth curve, which more or less adequately reflects the "systematic" behavior of the series.

Trend

In fig. shows the same series assuming approximately linear growth

This trend is called linear - by the shape of the curve. This is the most commonly used form; polynomial, exponential, logarithmic trends are less common. Having chosen the type of curve, specific parameters are usually selected using the least squares method.

Strictly speaking, this component of the time series is called trend-cyclical, that is, it includes fluctuations with a relatively long period, for our tasks - about ten years. This cyclical component is characteristic of the world economy or the intensity of solar activity. Since we decide not like that here global problems, we have smaller horizons, then we will leave the cyclical component outside the brackets and henceforth we will talk about the trend everywhere.

Seasonality

In practice, however, it is not enough for us to model behavior in such a way that we imply the monotonic nature of the series. The fact is that consideration of specific data on sales quite often leads us to the conclusion that there is one more pattern - periodic repetition of behavior, a certain pattern. For example, looking at ice cream sales, it is obvious that they tend to be below average in winter. This behavior is completely understandable from the point of view of common sense, so the question arises, is it possible to use this information to reduce our ignorance, to reduce uncertainty?

This is how the concept of "seasonality" arises in forecasting - any change in value that is repeated at strictly defined intervals. For example, a surge in sales Christmas tree decorations in the last 2 weeks of the year can be considered as seasonality. Typically, the rise in supermarket sales on Friday and Saturday compared to the rest of the days can be seen as seasonal with weekly frequency. Although this component of the model is called "seasonality", it is not necessarily associated with the season in the everyday sense (spring, summer). Any frequency can be called seasonality. From the point of view of the series, seasonality is characterized primarily by the period or lag of seasonality - the number through which the repetition occurs. For example, if we have a series of monthly sales, we can assume that the period is 12.

There are models with additive and multiplicative seasonality... In the first case, the seasonal adjustment is added to the original model (in February we sell 350 units less than the average)

in the second - there is a multiplication by the seasonality coefficient (in February we sell 15% less than the average)

Note that, as mentioned at the beginning, the very existence of seasonality should be explainable from the point of view of common sense. Seasonality is a consequence and manifestation product properties(characteristics of its consumption in a given point of the world). If we can accurately identify and measure this property of this particular product, we can be confident that such fluctuations will continue in the future. Moreover, the same product may well have different characteristics (profiles) of seasonality, depending on the place where it is consumed. If we cannot explain such behavior in terms of common sense, we have no reason to presumably repeat such a pattern in the future. In this case, we must look for other factors external to the product and consider their presence in the future.

It is important that when choosing a trend, we must choose a simple analytical function (that is, one that can be expressed by a simple formula), while seasonality is usually expressed as a tabular function. The most common case is the annual seasonality with 12 periods by the number of months - this is a table of 11 multipliers representing the adjustment relative to one reference month. Or 12 coefficients relative to the monthly average, only it is very important that the same 11 remain independent, since the 12th is uniquely determined from the requirement

The situation when the model contains M statistically independent (!) parameters, in forecasting is called a model with M degrees of freedom... So if you come across special software, in which, as a rule, you need to set the number of degrees of freedom as input parameters, this is where it comes from. For example, a model with a linear trend and a period of 12 months will have 13 degrees of freedom - 11 from seasonality and 2 from trend.

How to live with these components of the series, we will consider in the following parts.

Classic seasonal decomposition

Decomposition of a number of sales.

So, we can quite often observe the behavior of a series of sales, in which there are components of trend and seasonality. We intend to improve the quality of the forecast given this knowledge. But in order to use this information, we need quantitative characteristics... Then we can exclude the trend and seasonality from the actual data and thereby significantly reduce the amount of noise, and hence the uncertainty of the future.

The procedure for extracting non-random model components from the actual data is called decomposition.

The first thing we will do on our data is seasonal decomposition, i.e. determination of the numerical values ​​of seasonal coefficients. For definiteness, let us take the most common case: sales data are grouped monthly (since a forecast with an accuracy of up to a month is required), a linear trend and multiplicative seasonality with a lag of 12 are assumed.

Smoothing a row

Smoothing is a process in which the original series is replaced by another, smoother, but based on the original. The purpose of this process is to assess general trends, trend in a broad sense. There are many methods (as well as targets) of anti-aliasing, the most common

enlargement of time intervals... Obviously, a monthly aggregated sales row behaves smoother than a daily sales row.

moving average... We have already considered this method when we talked about naive forecasting methods.

analytical alignment... In this case, the original series is replaced by some smooth analytic function. The type and parameters are selected expertly to minimize errors. Again, we already discussed this when we talked about trends.

Next, we will use moving average smoothing. The idea is that we replace a set of several points with one according to the principle of "center of mass" - the value is equal to the average of these points, and the center of mass is, as you might guess, in the center of the segment formed by the extreme points. So we establish a certain "average" level for these points.

As an illustration, our original series smoothed over 5 and 12 points:

As you might guess, if there is averaging over an even number of points, the center of mass falls in the interval between the points:

What am I leading all this to?

In order to spend seasonal decomposition, classic approach suggests first smoothing a series with a window that exactly coincides with the seasonality lag. In our case, lag = 12, so if we smooth over 12 points, most likely, the disturbances associated with seasonality are leveled and we get an overall average level. Then we will begin to compare actual sales with smoothed values ​​- for the additive model we will subtract the smoothed series from the fact, and for the multiplicative model we will divide. As a result, we get a set of coefficients, for each month, several pieces (depending on the length of the row). If the smoothing is successful, these coefficients will not spread too much, so averaging for each month would not be such a stupid idea.

There are two important points to note.

• Averaging the coefficients can be done both by calculating the standard average and the median. The latter option is highly recommended by many authors, since the median does not react so strongly to random outliers. But in our tutorial we will use a simple average.
• We will have a seasonality lag of 12, even. Therefore, we will have to do one more smoothing - replace two adjacent points of the series smoothed for the first time with an average, then we will get to a specific month

The picture shows the result of re-smoothing:

Now we divide the fact into a smooth series:

Unfortunately, I only had data for 36 months, and when smoothing by 12 points, one year, respectively, is lost. Therefore, at this stage, I received only 2 seasonality coefficients for each month. But there is nothing to do, it is better than nothing. We will average these pairs of coefficients:

Now we recall that the sum of the multiplicative seasonality coefficients should be = 12, since the meaning of the coefficient is the ratio of monthly sales to the monthly average. This is exactly what the last column does:

Now we have completed classic seasonal decomposition, that is, we got the values ​​of 12 multiplicative coefficients. Now it's time to tackle our linear trend. To assess the trend, we will remove seasonal fluctuations from the actual sales by dividing the fact by the value obtained for a given month.

Now we will plot the data with the adjusted seasonality on the graph, draw a linear trend and draw up a forecast for 12 periods ahead for interest as the product of the trend value at a point by the corresponding seasonality coefficient

As you can see from the picture, the data cleared of seasonality does not fit very well into a linear relationship - there are too large deviations. Perhaps, if you read off the baseline data from outliers, things will get much better.

For a more accurate determination of seasonality using classical decomposition, it is highly desirable to have at least 4-5 complete data cycles, since one cycle is not involved in calculating the coefficients.

What if, for technical reasons, there is no such data? It is necessary to find a method that will not discard any information, will use all available to estimate seasonality and trend. Let's try to consider this method in the next part.

Exponential smoothing based on trend and seasonality. Holt-Winters method

Coming back to exponential smoothing ...

In one of the previous parts, we have already considered a simple exponential smoothing... Let us briefly recall the main idea. We assumed that the forecast for point t is determined by some average level of previous values. Moreover, the way in which the predicted value is calculated is determined by the recurrence relation

In this form, the method gives digestible results, if the series of sales is sufficiently stationary - there is no pronounced trend or seasonal fluctuations... But in practice, such a case is happiness. Therefore, we will consider a modification of this method that allows working with trend and seasonal patterns.

The method was named Holt-Winters by the names of the developers: Holt proposed a method of accounting trend Winters added seasonality.

In order not only to understand the arithmetic, but also to "feel" how it works, let's turn our head a little and think about what changes if we introduce a trend. If for simple exponential smoothing the forecast estimate for p-th period was done as

where Lt is the "general level" averaged according to the well-known rule, then in the presence of a trend, an amendment appears

,

that is, a trend estimate is added to the overall level. Moreover, we will average both the general level and the trend independently using the exponential smoothing method. What is meant by trend averaging? We assume that there is a local trend in our process that determines a systematic increment at one step - between points t and t-1, for example. And if, for linear regression, a trend line is drawn across the entire set of points, we believe that later points should contribute more, since the market environment is constantly changing and more recent data is more valuable for forecasting. As a result, Holt suggested using two recurrence relations - one smoothes the general level of the series, the other smooths out trend component.

The smoothing technique is such that at first the initial values ​​of the level and trend are selected, and then a pass is made along the entire row, at each step calculating new values ​​according to the formulas. From general considerations, it is clear that the initial values ​​should somehow be determined based on the values ​​of the series at the very beginning, but there are no clear criteria here, there is an element of voluntarism. Two approaches are most commonly used in the selection of "reference points":

The initial level is equal to the first value of the series, the initial trend is zero.

We take the first few points (5 pieces), draw a regression line (ax + b). The initial level is set as b, the initial trend as a.

By and large, this issue is not fundamental. As we remember, the contribution of early points is scanty, since the coefficients decrease very quickly (exponentially), so that with a sufficient length of the initial data series, we are likely to get almost identical forecasts. The difference, however, can show up when estimating the model error.

This figure shows the smoothing results for two seed selections. It is clearly seen here that the large error of the second option is due to the fact that the initial trend value (taken at 5 points) turned out to be clearly overestimated, since we did not take into account the growth associated with seasonality.

Therefore (following Mr. Winters) let us complicate the model and make a forecast taking into account seasonality:

In this case, we, as before, assume multiplicative seasonality. Then our system of smoothing equations gets one more component:

where s is the seasonality lag.

And again, note that the choice of the initial values, as well as the values ​​of the smoothing constants, is a matter of the will and opinion of an expert.

For really important predictions, however, it is possible to propose to compile a matrix of all combinations of constants and by exhaustive search to choose those that give a smaller error. We will talk about methods for assessing the error of models a little later. In the meantime, let's start smoothing our series by Holt-Winters method... In this case, the initial values ​​will be determined according to the following algorithm:

The initial values ​​are now defined.

The results of all this mess:

Conclusion

Surprisingly, such a simple method gives in practice very good results, quite comparable to much more "mathematical" - for example, linear regression. And at the same time, the implementation of exponential smoothing in information system an order of magnitude easier.

Forecasting rare sales. Croston's method

Forecasting rare sales.

The essence of the problem.

All the well-known predictive mathematics, which the authors of textbooks are happy to describe, is based on the assumption that sales are in some sense "even". It is with such a picture, in principle, that concepts such as trend or seasonality arise.

But what if the sales look like this?

Each bar here - sales for the period, there are no sales between them, although the product is present.
What "trends" can we talk about here, when about half of the periods have zero sales? And this is not the most clinical case yet!

Already from the graphs themselves, it is clear that it is necessary to come up with some other prediction algorithms. I would also like to note that this task is not sucked from the finger and is not some kind of rare. Almost all aftermarket niches deal with just this case - auto parts, pharmacies, providing service centers, ...

Formulation of the problem.

We will solve a purely applied problem. I have sales data point of sale accurate to days. Let the supply chain reaction time be exactly one week. The minimum task is to predict the speed of sales. The maximum task is to determine the size of the safety stock based on the service level of 95%.

Croston's method.

Analyzing the physical nature of the process, Croston (J.D.) suggested that

• all sales are statistically independent
• whether a sale happened or not, obeys the Bernoulli distribution
  (with probability p the event occurs, with probability 1-p not)
• in case a sell event has occurred, the buy size is distributed normally

This means that the resulting distribution looks like this:

As you can see, this picture is very different from the "bell" of Gauss. Moreover, the top of the hill depicted corresponds to buying 25 units, whereas if we calculate the average over a series of sales head-on, we get 18 units, and the calculation of RMS gives 16. The corresponding “normal” curve is drawn here in green.

Croston proposed to make an assessment of two independent values ​​- the period between purchases and the actual size of the purchase. Let's look at the test data, I just accidentally have real sales data at hand:

Now we will divide the original row into two rows according to the following principles.

original	period	the size
0
0
0
0
0
0
0
0
0
0
4	11	4
0
0
4	3	4
5	1	5
...	...	...

Now we apply a simple exponential smoothing to each of the resulting series and get the expected values ​​of the interval between purchases and the amount of purchases. And dividing the second by the first, we get the expected intensity of demand per unit of time.
So, I have daily sales test data. Selecting rows and smoothing with a low constant gave me

• expected period between purchases 5.5 days
• expected purchase size 3.7 units

hence the weekly sales forecast will be 3.7 / 5.5 * 7 = 4.7 units.

Actually, this is all that Croston's method gives us - a point estimate of the forecast. Unfortunately, this is not enough to calculate the required safety stock.

Croston's method. Refinement of the algorithm.

Disadvantage of Croston's method.

The problem with all classical methods is that they model behavior using a normal distribution. And there is a systematic error here, since the normal distribution assumes that the random variable can vary from minus infinity to plus infinity. But this is a small problem for a fairly regular demand, when the coefficient of variation is small, which means that the likelihood of negative values ​​appearing is so insignificant that we may well turn a blind eye to it.

Predicting rare events, when the expectation of the size of a purchase is of little value, is another matter, and the standard deviation in this case may well turn out to be at least of the same order:

To get away from such an obvious error, it was proposed to use the lognormal distribution, as more "logically" describing the picture of the world:

If someone is confused by all sorts of scary words, don't worry, the principle is very simple. The original series is taken, the natural logarithm of each value is taken, and it is assumed that the resulting series already behaves normally distributed with all the standard math described above.

Croston's method and safety stock. Demand distribution function.

I sat down here and thought ... Well, I got the characteristics of the demand flow:
expected period between purchases 5.5 days
expected purchase size 3.7 units
expected demand rate 3.7 / 5.5 units per day ...
even though I got the RMSE of daily demand for non-zero sales - 2.7. What about safety stock?

As you know, the safety stock must ensure the availability of goods when sales deviate from the average with a certain probability. We've already discussed service level metrics, let's talk about the first kind first. The strict formulation of the problem is as follows:

Our supply chain has a response time. The total demand for a product during this time is a random quantity that has its own distribution function. The condition "probability of non-zeroing of the stock" can be written as

In the case of infrequent sales, the distribution function can be written as follows:

q - probability of zero outcome
p = 1-q is the probability of a nonzero outcome
f (x) - density of the purchase size distribution

Note that in my research the previous time I measured all these parameters for a daily series of sales. Therefore, if my reaction time is also equal to one day, then this formula can be successfully applied right away. For instance:

suppose f (x) is normal.
suppose that in the domain x<=0 вероятности, описываемые функцией очень низкие, т.е.

then the integral in our formula is sought from the Laplace table.

in our example p = 1 / 5.5, so

the search algorithm becomes obvious - by specifying SL, we increase k until F exceeds the given level.

By the way, what is in the last column? That's right, the level of service of the second kind, corresponding to a given stock. And here, as I have already said, there is a certain methodological incident. Let's imagine that sales happen approximately once every ... well, let it be 50 days. And let's also imagine that we keep a zero stock. What level of service will it be? Kind of like zero - no stock, no maintenance. The stock control system will give us the same figure, since there is a constant out of stock. But after all, from the point of view of banal erudition, in 49 cases out of 50, the sale exactly matches the demand. That is does not lead to loss of profit and customer loyalty, but for nothing else service level and is not intended. This somewhat degenerate case (I feel the dispute will begin) is simply an illustration of why even a very small stock with a rare demand gives high levels of service.

But these are all flowers. But what if my supplier changed, and now the reaction time was equal to a week, for example? Well, here everything becomes quite funny, for those who do not like "multiformulas", I recommend not reading further, but waiting for an article about Willemain's method.

Our task now is to analyze the amount of sales for the period of the system reaction, understand its distribution, and from there pull out dependence of the service level on the amount of stock.

So, the distribution function of demand for one day and all its parameters are known to us:

The result of one day is still statistically independent of any other.
Let a random event consist of something that happened in n days smooth m facts of non-zero sales. According to Bernoulli's law (oh well, I'm sitting here and copying from a textbook!) The probability of such an event

where is the number of combinations from n to m, and p and q are again the same probabilities.
Then the probability that the amount sold in n days as a result of exactly m sales facts does not exceed the value z, is

where is the distribution of the amount sold, that is, the convolution of m identical distributions.
Well, since the desired result (total sales does not exceed z) can be obtained for any m, it remains to sum up the corresponding probabilities:

(the first term corresponds to the probability of a zero outcome for all n trials).

Something further I'm too lazy to tinker with all this, those who wish can independently build a table similar to the one above applied to the normal probability density. To do this, we just need to remember that the convolution of m normal distributions with parameters (a, s 2) gives a normal distribution with parameters (ma, ms 2).

Forecasting rare sales. Willemain's method.

What's wrong with Croston's method?

The fact is that, firstly, it implies the normal distribution of the purchase size. Second, for adequate results, this distribution should have low variance. Third, although it is not so fatal, the use of exponential smoothing to find the characteristics of the distribution implicitly implies the nonstationarity of the process.

Well, God bless him. For us, the most important thing is that real sales don't even seem close to normal. It was this thought that prompted Thomas R. Willemain and the company to create a more versatile way. And the need for such a method was dictated by what? Correctly, the need to predict the need for spare parts, especially car parts.

Willemain's method.

The essence of the approach is to use the bootstrapping procedure. This word was born from the old saying “pull oneself over a fence by one“ s bootstraps ”, which almost literally corresponds to our“ pull yourself out by your own hair. ”The computer term boot, by the way, is also from here. the essence contains the necessary resources to transfer itself to another state, and, if necessary, it is possible to launch such a procedure.This is the process that happens to the computer when we press a certain button.

When applied to our narrow problem, the bootstrapping procedure means calculating the internal patterns present in the data, and is performed as follows.

According to the conditions of our problem, the response time of the system is 7 days. We DO NOT know and DO NOT TRY to guess the type and parameters of the distribution curve.
Instead, we randomly "pull out" days from the whole series 7 times, summarize the sales of these days and write down the result.
We repeat these steps, each time writing down the amount of sales for 7 days.
It is desirable to carry out the experiment many times to obtain the most adequate picture. 10 - 100 thousand times will be very good. It is very important here that the days are chosen randomly EQUALLY over the entire analyzed range.
As a result, we should get "as it were" all possible sales outcomes for exactly seven days, and taking into account the frequency of occurrence of the same results.

Next, we divide the entire range of the resulting values ​​of the sums into segments in accordance with the accuracy that we need to determine the stock. And we are building a frequency histogram that will show the real probability distribution of purchases. In my case, I got the following:

Since I have sales of piece goods, i.e. the size of the purchase is always an integer, then I did not break it into segments, I left it as it is. The height of the bar corresponds to the share of total sales.
As you can see, the right, "non-zero" part of the distribution does not resemble the normal distribution (compare with the green dotted line).
Now, based on this distribution, it is easy to calculate the service levels corresponding to different stock sizes (SL1, SL2). So, by setting the target level of service, we immediately get the required stock.

But that's not all. If we enter into consideration the financial indicators - the cost price, the forecast price, the cost of maintaining the stock, it is easy to calculate the profitability corresponding to each size of the stock and each level of service. I have it shown in the last column, and the corresponding graphs are:

That is, here we find out the most efficient stock and level of service in terms of making a profit.

Finally (once again) I would like to ask: "why do we base the level of service on ABC analysis? "It would seem that in our case optimal level of service the first kind is 91%, regardless of which group the product is in. This secret is great ...

Let me remind you that one of the assumptions on which we were based - sales independence one day from another. This is a very good retail assumption. For example, the expected sales of bread today do not depend in any way on its yesterday's sales. Such a picture is generally typical where there is a large enough client base. Therefore, randomly selected three days can give such a result.

such

and even this

It is quite another matter when we have relatively few customers, especially if they buy infrequently and in large quantities. in this case, the probability of an event similar to the third option is practically zero. In simple terms, if I had large shipments yesterday, it is likely that there will be a lull today. And the option when the demand is high for several days in a row looks absolutely fantastic.

This means that the independence of the sales of neighboring days in this case may turn out to be bullshit, and it is much more logical to assume the opposite - they are closely related. Well, that won't scare us. We just won't pull out the days by chance we'll take the days coming contract:

Everything is even more interesting. Since our series is relatively short, we don't even need to bother with a random sample - it is enough to run a sliding window in the size of the reaction time along the series, and we have a ready-made histogram in our pocket.

But there is also a drawback. The point is that we get much fewer observations. For a window of 7 days, 365-7 observations can be obtained per year, while with a random sample of 7 out of 365, this is the number of combinations 365! / 7! / (365-7)! It’s too lazy to count, but it’s much more.

And a small number of observations means unreliable estimates, so save up the data - they are never superfluous!