Debtors' Payments: Fuzzy Approach to Planning

The financial aspect is essential to any kind of business. How a company receives funding and incomes determines its overall welfare. For any B2B company, one of the major concerns is the control over the payments of the non-bank debtors, i.e. the payments resulting from sales of goods and/or services. Indeed, this inflow enables the company to assess its efficiency, playing the role of the factor underlying the company's profits. Having produced some goods or services, the company sells the ware, receiving money for the ware -- which becomes income. The company crucially needs this income in order to be able to buy some raw materials and equipment needed to produce new portions of goods. Thus, it is essential that the company receives income regularly. What is regularity, in this case? It's in fact receiving the money on a predetermined schedule. The one that has been formed with a necessity in mind to meet the company's needs in financing its expenditures. However, we are living in a REAL world, which means that, inevitably, there are delays in debtors' payments. This, in turn, can lead to a complete breakdown of the financial plan. The latter may cause a non-reversible failure of the company. Effective planning of these delays is the key to successful financial management.

Given the stated facts, we arrive at the importance of a system that would be able to forecast potential delays in debtors' payments. Errors (deviations of the actual payment dates from forecasted dates) should be minimal in order for such a system to be considered effective. Now this is a tough point. Existing works show that ordinary statistical models cannot bear really effective results that would be stable in time. From our viewpoint, the best way to solve this issue is to use the so-called "fuzzy approach", which is based on the fuzzy set theory, originally suggested by L. Zadeh.

The basics of the fuzzy sets are explained in a huge amount of articles and books -- use web search engines to find out what fuzzy logic is and how it all works, if there's such a need. Here, we only suggest a ready-to-use principle of forecasting debtors' payments, basing on the fuzzy approach. The principle suggested in this article has been realized in the form of a computer program. The program has been tested on real data of a real company. The mean-square deviation thus calculated estimated 3, which suggests the idea that the principle presented herein is rather effective, but can be subject to further improvement.

Given a relational database (which may be in fact realized in any way, including but not limited to, MS Access, MS Excel DB-like data sets etc.) containing info on invoices, their birth dates, adjournment periods for each of the debtors, actual dates of debtors' payments that have occurred in the past, we can view the statistics "Past payment delays". The density function of this statistics can be viewed as a subnormal fuzzy set. This set, labeled "A", will be the first of the three fuzzy sets to be components of the resulting fuzzy set "payment date forecast". The density function can give us a general idea about the "payment discipline" of a specific debtor in the past. The density function, in a general case, will be containing several "waves" because it's usually not a trend-containing characteristic as to how many days a debtor will be evading from paying the debt.

Firstly, in most cases the amount of days of a payment delay is a random variable. It can be fluctuating within a couple days' limits. Secondly, statistical forecasts of delays may be differing significantly for different periods of time. This is because B2B relationships are not static, they are developing all the time. Sometimes, the selling company comes to "shaking hands" with the buying company for the latter to pay a couple days earlier, whereas sometimes the buying company may be facing temporary financial problems (e.g., resulting from a huge credit to be returned to a bank by the buying company), so that the buying company warns the selling company that there may be slight delays of payments. This is reflected in another component of the resulting fuzzy forecast, -- fuzzy set "C". It is in fact a linguistic variable "Payment delay most likely" fuzzy set. The linguistic variable may take one of the following values: "Neutral" (which means that there are no specific anticipations of the payments delay value for the specific debtor), "A slight delay is possible", "A slight delay is most likely", "A large delay is most likely", "An on-time payment is most likely", "Payment in advance is most likely". Each of these term-values has its own membership function. A corresponding membership function is used each time when building a forecast for a specific debtor. The membership functions for the term-values of the linguistic variable "Payment delay most likely" are given below:

"An on-time payment is most likely": y=SQRT(1-ABS(x)/2), x belongs to [-2;2]

"A slight delay is most likely": y=SQRT(1-ABS(x-4)/3), x belongs to [1;7]

"A slight delay is possible": y=(1-ABS(x-4)/3)**2, x belongs to [1;7]

"A large delay is most likely": y=SQRT(0.25-(12-x)/24)+0.5, x belongs [6;12]

y=(0.71-(6-x)/4.23)**2, x belongs to [3;6)

y=0, x<3

y=1, x>12

"Neutral": y=0.5

"A payment in advance is most likely": y=1/SQRT(ABS(x)), x<0

y=0, in any other cases

In the listed formulae SQRT(x) means "square root of x", ABS(x) means "the absolute value of x", x**y means "y-th power of x".

Finally, the fuzzy set B which is also used in calculating the resulting forecast, is in fact a triangular fuzzy number with the peak equal to the expectation function based on a "4 last payments" sample (the number of payments may be altered, depending on which value leads to a better efficiency of the forecast). The left and the right borders of the triangular fuzzy number "B" are respectively the minimum and the maximum values of the sample.

Now, as we have the three components of the forecast (fuzzy set A and fuzzy numbers B and C), we are able to build a resulting forecast. To do so, we transform the fuzzy sets A, B and C, so that their domains are identical, and calculate a weighted sum of these fuzzy sets (since in terms of our task these fuzzy sets are discrete, we are viewing their sum as the sum of vectors a, b and c, components for these vectors being the discrete values of the fuzzy sets A, B and C). Weighting coefficients are set for each of the fuzzy sets A, B and C; these weighting coefficients are defined as the coefficients that are providing best results; their values should be recalculated regularly (e.g., once a month).

In our tests, we used actual data of a Russian industrial company. In these tests, best results were achieved with coefficients 0.1, 0.6 and 0.3 for the fuzzy sets A, B and C, respectively.