Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems Of Stocks

International Journal of Business Marketing and Management (IJBMM) Volume 5 Issue 10 October 2020, P.P. 01-51 ISSN: 2456-4559 www.ijbmm.com

Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems Of Stocks Mihai Petrov Republic of Bulgaria, town Burgas, University "Asen Zlatarov", Department of Real Sciences, section of physics and mathematics.

Abstract: Econophysics as an integrated platform of physics together with other economic sciences has a broad perspective of phenomenological physics description of the processes of economic activities. This paper suggests methods of phenomenological physics of mechanical kinematics and the model of gravitational acceleration for the description of the activity of microeconomical systems of stocks. A criterion of continuous instant stability of microeconomic systems is established by the description of the phase trajectory which is a necessary condition that this shape of the trajectory to be unchangeable with time. The conception of the econophysical acceleration is described which is related to the sold inventory. Bigger is the sold inventory then the smaller is the acceleration. The following formulation of the interconnection between the acceleration and the sold inventory is suggested: The continuous decreasing of the acceleration with time is the indicator of the continuous increasing of the sold inventory. The validation of the acceleration concept is performed by the real example of the sold inventory. The result of the average acceleration coincides with value of the rating coefficients of the stocks and respectively with the values of thermodynamical temperatures. key words: econophysics, distribution of Pareto, phase trajectory, econophysical acceleration, sold inventory.

I.

Introduction to econophysics. Prerequisites of the continuous development of econophysics

Technical and scientific progress involves an integrational development of various scientific fields in order to solve new major goals and proxies in the field of medicine, economy, pharmaceutical industry, high modern technologies, social processes and the Human being in the new life conditions taking into account the evolution of climatic and ecological conditions. Also new philosophical conceptions about Life imply a widespread application of knowledge from different fields of science and eventual their application into a new integrative scientific fields such as: biophysics, bioinformatics, econophysics, bioeconophysics etc.

Econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics. Some of its application to the study of financial markets has also been termed statistical finance referring to its roots in statistical physics. [1] Econophysics was started on 1990s by several physicists working in the subfield of statistical mechanics.

Unsatisfied with the traditional explanations and approaches of economists – which usually prioritized simplified approaches for the theoretical models to matching financial data sets, and then to explain more general economic phenomena.

The worldwide scientist Harry Eugene Stanley has developed the contributions to statistical physics and is one of the pioneers of interdisciplinary science and is one of founding fathers of econophysics. Stanley has developed the term of econophysics for the description of the large number of papers written by physicists in the problems of markets and presented in a conference on statistical physics in Kolkata in 1995 and first appeared in its proceedings publication in Physica A 1996.[1][2] The inaugural meeting on econophysics was organized in 1998 in Budapest by János Kertész and Imre Kondor.

The multidisciplinary field of econophysics uses theory of probabilities and mathematical methods developed in statistical physics to study statistical properties of complex economic systems consisting of a large number of complex units or population (firms, families, households, etc.) made of simple units or humans. [3] Consequently, Rosario Mantegna and Eugene H. Stanley have proposed the first definition of econophysics as a multidisciplinary field, or “the activities of physicists who are working on economics problems to test a variety of new conceptual approaches deriving from the physical sciences”. “Economics is a pure subject in

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. statistical mechanics,” said Stanley in 2000: “It’s not the case that one needs to master the field of economics to study this.” Econophysics is a sociological definition, based on physicists who are working on economics problems. [4] Another, more relevant and synthetic definition considers that econophysics is an “interdisciplinary research field applying methods of statistical physics to problems in economics and finance”.

[5] A main peculiarity related to econophysics is its distinctiveness from the mainstream economics, although both sciences share the same subject of research. It seems quite strange, since physics has long been a source of inspiration for economists. Unquestionably, in the second half of the 19th century, physics significantly accelerated the development of economics by providing a necessary methodological framework. [6] A lot of scientists working on the subjects of econophysics define various points of view regarding the econophysics. For example the physicist A. Leonidov noted that "The study of economics as a quantitative science is one of the urgent, exciting and complex problems of cognition. The depth and diversity of the problems that arise makes the subject of study extraordinarily attractive for specialists in various fields of knowledge, from psychologists to mathematicians. Of course, representatives of one of the most developed and successful quantitative disciplines, physics, could not stand aside. [7] The term “econophysics” [8] was introduced also by analogy with similar terms, such as astrophysics, geophysics, and biophysics, which describe applications of physics to different fields. Particularly important is the parallel with biophysics, which studies living creatures, which still obey the laws of physics. It should be emphasized that econophysics does not literally apply the laws of physics, such as Newton’s laws or quantum mechanics, to humans, but rather uses mathematical methods developed in statistical physics to study statistical properties of complex economic systems consisting of a large number of humans. So, it may be considered as a branch of applied theory of probabilities. However, statistical physics is distinctly different from mathematical statistics in its focus, methods, and results. Originating from physics as a quantitative science, econophysics emphasizes quantitative analysis of large amounts of economic and financial data, which became increasingly available with the massive introduction of computers and the Internet. Econophysics distances itself from the verbose, narrative, and ideological style of political economy and is closer to econometrics in its focus. Studying mathematical models of a large number of interacting economic agents, econophysics has much common ground with the agent-based modeling and simulation. Correspondingly, it distances itself from the representative-agent approach of traditional economics, which, by definition, ignores statistical and heterogeneous aspects of the economy. Two major directions in econophysics are applications to finance and economics, statistical distributions of money, wealth, and turnover among interacting economic agents.

Econophysics that is a new branch of the study of economy includes not only proper sense of econophysics as usual but also physical economics [9] that explains the economical processes by the application of physical phenomena and has a large priority to choose the adequate physical model for the quantitative description of the processes of pharmaceutical marketing. [10] Physics (from Ancient Greek: υυσική (ἐπιστήμη), translit. physikḗ (epistḗmē), lit. 'knowledge of nature', from υύσις phýsis "nature") is the natural science that studies matter, its motion, and behavior through space and time, and that studies the related entities of energy and force. So, physics studies the general laws of nature and explains phenomena with appropriate patterns using mathematical methods. The traditional question of physics is: why does this phenomenon happen? And the answer is given according to the appropriate model. The question arises logically, but why physics? What physics, which is a very widespread science with modern new compartments, is not enough of its own domain? Surely, the development of physics has reached such limits that it is now becoming interdisciplinary. The human being always at different historical stages is accustomed to observing phenomena in nature and studying them in detail, to explain why these phenomena occur and the cause of their defense. So, namely physics is called science that deeper insight studies the essence of all things in the Nature. Logically, we can ask ourselves in the following way, since physics explains the essence of all things, then it really does explain everything like: historical evolution of society and eventually statistical repetition of some historical processes, economic phenomena, periodical physico-statistical variations of some social and economical processes, market processes described by analogical physical laws, etc.

Physics aims to observe the given phenomenon, and as a result of observation, the quantitative mathematical apparatus is performed, the final result of which is the quantitative law that contains the numerical parameters describing the given phenomenon.

It is worth noting that many now-famous economists were originally educated in physics and engineering. The well known Italian scientist Vilfredo Pareto that is considered as a parent of modern science of econophysics earned a degree in mathematical sciences and a doctorate in engineering at the ends of 19th century. Working as a civil engineer, he collected statistics demonstrating that distributions of turnover and wealth in a society follow a power law [11].

The word economy from Greek translation means order and discipline inside the house. Keeping this sense, then this order and discipline can be created somehow by the application of the principle of Pareto, especially if we

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. are referring to the stock markets. Nowadays, this principle is largely applied not only for economical systems, but for social, healthcare and organizational activity.

Nowadays the Pareto’s principle also has become a popular area of focus in the world of business and management and the statement of this principle is: 80 percent of effects always come from 20 percent of the causes. Pareto first observed this ratio when he realized that 80 percent of land and wealth in Italy was owned by 20 percent of the population. He then went on to observe the same phenomenon in his garden: 80 percent of peas came from 20 percent of pea pods. [11] Since he published these findings, the magical ratio of 80:20 (or the “80-20 rule”) has been found to be scattered throughout society and nature. The 80 percent of any company’s profits come from 20 percent of their best products, 80 percent of traffic comes from 20 percent of roads, 80 percent of food production comes from 20 percent of the best crops. The ratio is everywhere and frequently even tipped to a 90-10 or 95-5 division. However the 80-20 phenomenon is the distribution most often cited as a universal baseline and especially the application to the practice of hospital medicine [12]:  80 percent of the clinical and problematic issues on any given day will arise from 20 percent of the patients.  80 percent of telephone calls and pages will always come from 20 percent of nurses.  80 percent of valuable medical information that is received will come from only 20 percent of what are communicating.

Healthcare has its own Pareto principle: 80% of healthcare costs are attributed to 20% of the populace: the chronically ill.

The Pareto principle last time is applied largely and is combined with ABC analysis for supply management purpose. [13] Therefore the effective supply management ensures uninterrupted availability of quality approved, safe and effective products. The econophysical studies that include the principle of Pareto were reflected in [10] which shows that each stock article of pharmaceutical products is characterized by so-named econophysical temperature and this term of econophysical temperature is the capacity of the generating power of turnover (revenues) during one day of one stock article and respectively for each rating marketing groups A, B, C, X, Z of the stocks these values of temperatures are KA=21; KB=13; KC=8; KX=5; KZ=3 that coincide with the numbers of Fibonacci which stay on the basement of so-named “Golden ratio” of Nature’s structures and economical structures [10], [14]. The Fibonacci sequence are applicable for various kinds of the stocks.

The econophysical studies presented in [10] apply the physical model of the “ideal gas” of the pharmaceutical stocks and this model is related to the marketing state of hyper competition. The sold and reserve inventory of stocks is described by the equation of marketing state [10]:  P  p Ntot  Nart  K (1) here

 P  p is the average price of one pharmaceutical product; N tot - total amount of products of the

N art - total amount of the names of articles; K - the value of econophysical temperature and for the full ensemble of stocks this value is K  5,65 which is calculated on the base of KA=21; KB=13; KC=8; KX=5; KZ=3 by the consideration of the peculiarities of ABC analysis and this value K  5,65 is a worldwide inventory;

constant that is independent on national currencies [10].

Similar expression like (1) is described in the paper [15]. The difference is that the econophysical temperature is the volatility in [15]. The greater the volatility, the greater the opportunity to sell the stocks at high prices. [16] Otherwise, the higher the econophysical temperature K described by expression (1) , the greater the opportunity to sell the stocks at high prices.

It is clear that in order to be a good specialist in the field of econophysics, is necessary the fundamental initial studies in the fields of physics, statistics and economics. Only then can one understand the processes that are described this scientific integrative complex system.

Generalizing the introductory information, then the definition of econophysics could be given as follow: Econophysics is a multidisciplinary philosophical scientific integrative system that studies the general laws of the evolution of economical and social processes by the application of physics - mathematical and statistical methods of philosophical, social and economical sciences.

Econophysics like physics could also contain the similar chapters like mechanics, thermodynamics and statistics, electricity, optics, quantum mechanics, etc., exactly as phenomenological conception of econophysics that is described in [15] According to the point of view described in the paper [15] the equilibrium and crises in economies are explained well by phenomenological conception of econophysics.

Logically, the first chapter could be mechanics. Historically, classical mechanics emerged first and is originated with Isaac Newton's laws of motion in the paper [17] "Philosophie Naturalis Principia Mathematica".

Classical mechanics describes the general laws of the motion of macroscopic material bodies.

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems ..

Microeconomics is the study of individuals, households and firms, behavior in decision making and allocation of resources. It generally studies the markets of goods and services and deals with individual and economic issues [18]. It is focused on the study deals with what choices people make, what factors influence their choices and how their decisions affect the goods markets by affecting the price, the supply and demand [19].

The behavior of the activity of the markets of stocks is studied last time not only by classical theories of economics but also by modern integrative means of interdisciplinary branches of sciences like statistical mathematics and theory of probabilities, and modern statistical integrative science that is called econophysics.

[20], [21], [22].

II.

Application of the distribution of Pareto to the ABC analysis for the description of supply-demand marketing processes. The equation of the state of microeconomical systems of stocks

According to the conception that is developed in the paper [15] the motor force of the prosperity and good activity of the microeconomical systems is namely the Human being. He makes plans and orders of the activity and this order depends both on the customers and the dealers or the sellers. The main aim of the microeconomical research is to find such reasonable equilibrium between the supplied and demanded quantities.

In this topic the application of the principle of Pareto combined with ABC analysis will give the possibility to obtain the quantitative analytical expression that contains the information about the prices of one product, quantity of articles and quantity of packing products of each respective rating marketing groups A, B, C, X, Z. ABC analysis [23]-marketing tool that improves the efficiency of the activity of the markets. This analysis is performed in order to analyze the sales and priorities in the management of marketing activity. ABC analysis that is a part of marketing starts from policies of marketing mix [24], [25], [26],[27] which is a complex of controlled marketing varieties that the market uses in order to achieve the desire result and increasing of turnovers by attending to consumption necessity of customers (buyers).

The VI-th Congress of Pharmacy with International Participation [13] and III-rd International Conference of Econophysics [28] presented the information about the rating of the stocks by statistical distribution of Pareto with ABC analysis [29], [30]. The distribution of Pareto allows to describe quantitatively these rating groups A, B, C, X, Z of the stocks by special parameter K named rating coefficients of the stocks [13], or econophysical temperatures [10] and have the meaning of the power of the turnovers of one stock article during one day.

In order to present the generalized information about the amount quantities of stock articles in the form of relative position of the stock articles in the distribution of Pareto the modification was performed [13] like:

F ( x)  1  (1  x) K , 0  x  1

(2) where F is the cumulative turnovers , x is the relative position of the stock articles.

The respective graphic is presented on Fig. 1. The ABC analysis combined with Pareto analysis can be represented into one diagram [13] as shown on the Fig. 2 Total shares of the stocks ABC gives approximately 80% of total turnovers and this total stock ABC includes 20% from the total stock articles of all products. The rating coefficients K of the stocks is calculated from expression (2) for the intervals of times from unspecified random first day till several months like 72 months for the pharmaceutical products. [13]

K

ln(1  F ( x)) ln(1  x)

(3)

Fig. 1. The modified theoretical distribution of Pareto: F ( x)  1  1  x for different numerical values of K

к

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems ..

Fig. 2. Rating shares of the stocks on the diagram of Pareto The found values of K of the respective rating stock groups A, B, C, X, Z depend on the interval of time as is represented on Fig. 3 There is an important peculiarity of the dependence K=f(t) that is important to be mentioned. The starting time can be chosen randomly and the same values of K are obtained during the same interval of time Δτ as shown schematically on the Fig. 4. This situation might corresponds to the one of criteria of instant progressive activity of the market.

These values of K are arranged on stationary numerical series of Fibonacci numbers (KAst=21, KBst=13, KCst=8, KXst=5, KZst=3) for relative big intervals of time as of order of 72 months. These stationary values represent the average turnovers of the selling per one stock article during a day and if these values of the average turnovers are divided by the price P0j of one packing product, then it means the result of sold packing products N0j of the respective stock article during a day [13].

The index j corresponds to the respective rating group A, B, C, X, Z , so (Z≤ j≤A). So, the quantity of sold products N0j of respective stock article during a day is calculated as follow:

N0 j 

Kj P0 j

; Z jA

(4)

Fig.3. The dependence of rating stock coefficients K on the interval of time

Fig.4. The independence of the starting time of K=f(t) The turnover from the selling of one stock article per day is:

K j  P0 j N0 j ;

Z jA

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(5)

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems ..

The sum of all turnovers of the selling per one stock article T0j during the interval of time Δt is the definite integral from zero till Δt: t

Toj   K j (t )dt  P0 j N 0 j t;

Z jA

(6)

0

N art j , then

Taking into consideration, that one rating group of the stocks contains quantity of stock articles

the turnover of entire rating group Tj during the interval of time Δt is the sum of all turnovers per stock articles: t

T j  Toj N art j  K j (t )dt  N art j Toj  N art j P0 j N 0 j t;

Z jA

0

(7) The average value <Kj> during the interval of time Δt is calculated as: t

 K (t )dt j

 K j 

0

t Nart j   K j  Nart j N0 j P0 j ;

Then:

The total quantity of packing products

;

Z jA

(8)

Z jA

(9)

N tot j for the full rating group is:

Ntot j  Nart j N0 j ;

Z jA

(10)

Then the expression 10 is written as:

Nart j  K j  Ntot j P0 j ;

Z jA

(11)

For the big systems of quantities of stocks is better to use the average price of one packing product the respective rating group j, and this average price

 Pj  p 

N

 Pj  p for

 Pj  p is calculated as:

P

0j 0j

(12)

Ntot j

The expression 5 can be generalized by the sum of the right and the left part of whole rating group j:

The sum

 K

 K

j

j

   P0 j N0 j ;

Z jA

(13)

 is repeated N art j times and, then:

 K

j

  Nart j  K j 

(14)

Taking into consideration the expression 12 and 14, then the expression 13 can be written as:

Nart j   K j  Pj  p Ntot j Taking into consideration that

Ntot j  N art j N0 j , then the expression 15 can be written as:  K j  Pj  p N0 j ;

Z jA

More important moment is the average quantity of the packing products this value can be calculated as:

(15)

(16)

 N0 j  per one stock article and

N art j N 0 j Ntot j N art j N 0 j N  N 0 j   0 j     N0 j N art j N art j N art j N artj (17)

So, the expression 16 can be written as:

 K j  Pj  p   N0 j ;

Then the average price of one stocking product is:

 Pj  p 

 Kj   N oj 

Z jA

; Z jA

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(18)

(19)

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This expression 19 describes quantitatively the process of supply and demand of the stocks, which states that if the prices of products are decreased the demanded stocks from customers are increased and vice versa [31], (  Pj  is inverse proportional to N oj and respectively to  N oj  ).

The process of interaction between the seller and the customer is a stochastic process and the final result is the event of the purchasing of the demanded products. Suppose a situation of such type of existence of substitutes products of the same type but with different price. The seller has the tendency to offer the expensive one in order to have more revenues. So, supply process is related to the seller, dealers and producers. The dealers and producers supply the products depending on the turnovers and salaries of the customers and as the salaries of customers are increased they supply more expensive substitutes. Always the tendency exist that customers are demanded more cheapest substitutes but dealers supply the more expensive substitutes. As the result of this complex stochastic situation there is a equilibrium point where the price P* and quantity Q* are stable. Such equilibrium point is obtained when the supply and demand shapes are joined in one diagram and the point of intersection of the shapes is equilibrium point E as shown in Fig. 5.

Generally speaking, an equilibrium is defined to be the price-quantity pair where the quantity demanded is equal to the quantity supplied. The analysis of equilibrium is a fundamental aspect of microeconomics: Market Equilibrium is a situation in a market when the price is such that the quantity demanded by consumers is correctly balanced by the quantity that firms wish to supply. In this situation, the market clears.

[32] The equation (15) is named the equation of the state of microeconomical system of stocks. It is a expression of interdependence of the prices of one product and the quantities of articles and the quantity of products of the respective articles at the fixed values of rating coefficient of the stocks K .

Regarding the expression (11), it can be observed that : P0 j Noj  K j  . Here, there is an inverse proportionality between

N oj and P0 j for the fixed stable value  K j  const at the respective moment of

time. The respective graphic of the dependence of

P0 j  f ( Noj ) is represented on the Fig. 6.

Fig. 5. Equilibrium of supply and demand

Fig. 6. The dependence of the price P0j of one packing product vs. the demanded quantity of products N0j on one stock article

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The process of the prices creation is influenced by a lot of very complicated factors such as the initial costs of the materials, the performed work, etc., and it is expected that the planned value of the prices can be validated in practice namely by the supply-demand process.

It is important to mention that the curve of demands have the form of hyperboles that are represented on the Fig. 6 showing the consequently decreasing of the prices of the products Poj with the increasing of the demanded amounts Noj, and experimentally it will be expected to have namely such hyperbolic forms and is described by such dependence

Poj 

 Kj  N oj

. Logically, the average price of one product

 Pj  p can be found for each

respective rating marketing group A, B, C, X, Z and is expected qualitatively that the price of one product is highest for the A group than of Z group. Respectively, the rating coefficients of the stocks  K j  are higher for A than of Z. It is known that if the demand is instant higher then the prices of products are fixing consequently to higher values or have the tendency of increasing in comparison with those which have small demand. If the demand is higher then the stock reserve of the respective items will have the planning of the increasing or they are in great quantity. So, it is expected that for the group A the stock reserve will be higher than of Z group.

The qualitative estimation of the amounts of products of each rating group allows to represent the curve of supply S on the Fig. 7

Fig. 7. The dependence of the price P0j of one packing product vs. the demanded quantity of products N0j on one stock article ; S - the curve of supply The intersection points of the curve of supply with those of demands allow to obtain the information of optimal stock reserve. The minimal limits of the stock reserves are the values N A , N B , NC , N X , N Z and the respective prices of one product are

PA , PB , PC , PX , PZ that are represented on Fig. 7. Real observation of

such position of points are expected to be almost real.

In such a way the Pareto distribution combined with ABC analysis gives two very important topics: 1) equation of the state of microeconomical systems of stocks; 2)The curve of demands-supply gives the real idea about the numerical values of the equilibrium prices and the respective quantities of the rating groups.

III.

Kinematics phenomenological econophysics of microeconomical systems of stocks

3.1. The definition of econophysical kinematics. The notion of the speed of movement, displacement and the vector. The instantaneous speed. The prerequisites of the possible development of the oscillator model of the inventory Kinematics is the chapter of mechanics dealing with the study of the coordinates of the moving bodies and how these coordinates are variable with the time. Mechanics is the science concerned with the motion of

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. bodies under the action of forces, including the special case in which a body remains at rest. Force is nothing but the ability of the customer to purchase the products by payments.

In order to give a mechanical model to the system of stocks, it is necessary to mention that the system is described by four main parameters that are resulting from the equation of the state of microeconomical system of stocks: 1) the total amount of the products Ntot; 2) average price of one product <P>p; 3) quantity of stock article (varieties of the names) Nart; 4) the rating coefficient of the stock Kj (or the econophysical temperature of the system of the stocks). The system could be into the "rest" state or into the state of the "movement". This system is considered as the "material" body that moves with time into the space of the coordinates Ntot. Namely, the change with time of Ntot means that the system "moves". Changing with the time of the quantities Ntot means the changeable stock reserve (changeable inventory).

The chapter which study the movement of the body without taking into consideration the reason of the emerging of the movement is named kinematics.

Kinematics is the part of mechanics that studies the motion of a particle (body), ignoring its causes.

A particle is a point-like mass having small size. The econophysical mass is nothing but the margin (or the profit), or the difference between the selling price and the price of dealers. For example, an inventory of 100000$ has a mass of about 20000$. This econophysical mass is comparative smaller in comparison with the value of the inventory.

The movement of the body could be of two types: 1) uniform motion; 2) non-uniform motion.

1)This type of the motion is defined as such motion of the body which coordinate Ntot is variable with the same constant value ΔNtot in equal intervals of time Δt .

Regarding this type of the motion it is necessary to define the speed of motion V. Namely, if the stock reserve that is determined by the value Ntot is changeable with the time, then is a criterion of the selling of stocks. The quantity of stocks that are sold during one unit of time is the speed of the motion V of the system.The speed of the motion V is the path traveled in the unit of time. The expression of the speed V is written as:

Ntot (t2 )  Ntot (t1 ) (20) t2  t1 where Ntot (t1 ) and Ntot (t2 ) are the amounts of products of the stock reserve (inventory) respectively at the V

initial

moment

Ntot 

of

time

t1 and the final moment of time t2. If the respective variation Ntot (t2 )  Ntot (t1 ) is the same for the same interval of time Δt, then this motion is uniform. The

measurement unit of the speed of motion V is: (products/s; products/min; products/h; products/day; products/month; etc.). So, the speed of motion V is constant all time. (V=const) The Fig. 8 shows two cases when the system moves with the constant speed. The case (a) is referring to the case when the inventory is increasing uniformly. This case (a) could be the case when the supplying with new stocks is greater than the quantity of sold products. The case (b) is referring to the case when the inventory is decreasing uniformly due to of stable uniform selling of products. In this situation the selling products are in great quantity than the supplied quantity from dealers.

Fig. 8 The uniform variation of inventory: a) case of uniform increasing of inventory; b) case of uniform selling of products 2) The non-uniform motion is such motion of the system which coordinate Ntot is variable randomly in equal intervals of time Δt . Such type of the movement could be like the trajectory that is represented on Fig. 9

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Fig. 9. Example of non-uniform movement of the system.

This type of movement could be for the cases when the products have the seasonal character, finite life-cycle of some products, substitution with other similar products, seasonal character of the entire system.

When the variation of the amounts Ntot takes place then it is important to emphasize that the variation is not continuous (is not a slow transition from one state with those four parameters of the system into another state), but is discrete. The discrete transition is represented by black points on the Fig. 8 and 9. Only one difference exist between uniform and non-uniform movement. A linear straight transition from one state to another state takes place for the case of uniform movement, but for the case of non-uniform movement the chaotic transition from one point to another point takes place.

The points represent an event of the selling or purchasing. Usually, if the event of the purchasing takes place then the value Ntot is decreased and vice versa if the supply from dealers takes place then Ntot is increased. If another event of purchasing from customers takes place, then another transition into another point takes place.

The segment between the two points is considered the "rest" state of the system. The lengths of the segments of the rest states could be various due to of the stochastical character of the processes.

The case of uniform movement is very rare. It can only occur in relatively short time intervals. More often, nonuniform movements could occur. In the classical mechanics of physics, the use of the notion of vector is applied.

The vector is the right oriented segment that unites the initial and the final point. The orientation of the vector by the arrow shows the direction of the movement.

Respectively the transition from one state to another (from one point to another) is nothing but the displacement. The displacement in this case coincides with the traveled road.

In classical mechanics the notion of the reference body is used. The reference body is the body with respect to which the movement of the system is studied. The reference body coincides with the origin of coordinates O. The reference body O in this mechanical description will be none other than himself own microeconomic system. This reference body will be considered strictly as something very initially zero with zero stock and a initial moment of time fixed at the zero value.

Referring to the recent econophysical description, then the three dimensional system of coordinates will be applied (Ntot, No, Nart). The values of Ntot are dependent on No and Nart as Ntot= No∙Nart.

Why is necessary three coordinates? It will give more information, because the total amount Ntot is changeable as the result of the changes of No and Nart. Sometimes, the same value of Ntot could be for the case when No is not changeable but Nart could be changeable due to of the apparition on the market of the new product (new name) or could be withdrawn, or could be a situation that No is changeable but Nart is fixed. The changeable value of No could be for the cases when the amount of products for one stock article is variable due to of seasonable character of the product. Therefore, the application of three dimensional system is more informative.

The Fig. 10 represents schematically the possible variation of the inventory on three dimensional system. The

Z 1 and Z 2 shows the consequent positions of the states 1 and 2 of the system at the respective moment of time t1 and t 2 . The vector of the displacement is Z  Z 2  Z1 . This vector of positional vectors

displacement Z shows the direction of the variation of Nart, No and Nart on the Fig. 10. This exact example on this Fig. 10 shows that all components Nart, No and Nart are increasing. In general, such situations could be when two of them are increasing but another is decreased. For example if No is increasing and Nart is decreasing then the result of Ntot is increased due to of the fact that the increasing of No is several more times bigger than of Nart.

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Fig.10 The three - dimensional schematically representation of the variation of the inventory with time The projection of the displacement vector

Z  AM  CL  XN . The respective module of the vector

Z is:

Z  Z  Z 2  Z1 

CL2  MA2   XN 2



No 2  Nart 2  Ntot 2

(21)

Z Z  t2  t1 t

The vector of the speed of movement:

VZ 

The respective decomposition of the velocity vector

VZ by the components of the axes is:

(22)

VZ  VAM  VCL  VXN

(23)

The respective speeds components by axes are written as:

VAM 

AM AM CL ; VCL  ;  t2  t1 t t

VXN 

XN t

(24)

The respective modules of the vectors of speeds of the expression (24) are written as:

VAM 

AM AM ;  t t

VCL 

CL CL ;  t t

VXN 

XN XN  t t

(25) The module of the vector

VZ is written as:

VZ 

VAM 2  VCL 2  VXN 2

(26)

So, the transition from one state into another state is like a way that is travelled during the interval of time Δt.

Then, the speed

VZ is considered like average speed:

 VZ 

Z Z  t t

(27)

For two respective neighbour segments with the length ΔZ1 and ΔZ2, then the average speed:

 VZ 

Z1  Z 2 t1  t2

(28)

The respective segments are shown on the Fig. 11 with the two segments.

Fig. 11 The way of transitions with two segments.

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The expression (29) could be written for the case of arbitrary quantity of segments, and then the average speed n

is written as:

 VZ 

 Z i 1 n

 t i 1

i

(29) i

If the intervals of the time are very short

ti  0 , then the speed is limited to a point of the way and the

respective speed is called instantaneous speed V (the speed at the given moment of time) and this instantaneous

 n    Z i   VZ (t )  lim  i n1 t i  0    ti   i 1 

speed is a function of time:

(30)

The case of the short time interval is highly idealized. This is the case when each buyer is served one after the other without any rest of the system. This is the case when the buyers wait in queue without any disobeying of this queue. The expressions of the instantaneous speed as a function of time could be various like:

V (t )  at 2  bt  c , or in the form of exponential functions: VZ (t )  a  ebt ; where a, b, c are the constant coefficients. The traveled way also is the function of time ΔZ(t) and the expression of the instantaneous speed could be written in this case as:

VZ (t ) 

dZ dt

(31)

The values of the instantaneous speed could be variable in time also by sign. Sometimes the could be negative, sometimes positive values. The negative value of the instantaneous speed means that at this moment the reserve quantity of inventory is decreasing and if the instantaneous speed is positive, then the reserve inventory is increased. The increasing takes place by supplying of new stocks from the dealers.

The curve of the way in the case of very short time of transitions is a continuous curve without any rest states and without any fast thresholds Fig. 12.

Fig. 12. The continuous curve of the way for the case of continues serve of the customers The infinitesimal small interval of time dt corresponds to a very small traveled way d(ΔZ) and the respective momentary speed is calculated by the expression (31) The full way ΔZ (the variation of the inventory during the interval of time Δt (one day, one months, etc.) is found by the integration of the expression (31):

Z (t )  VZ (t )dt  C

(32)

where C is the constant of integration, that is find by initial condition. One of initial conditions could be like as for the initial moment of time t0=0 the value ΔZ0=78 products, then C=78 products.

The numerical value of the traveled way ΔZ can be calculated by the definite integral if the limits of the t2

integration are known:

Z   VZ (t )dt

(33)

t1

If the way ΔZ that is traveled during the interval of time Δt=t2-t1 is known (Fig.13), then the average speed can be calculated as:

 VZ 

Z (t2 )  Z (t1 ) t2  t1

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Fig. 13 The mean of the calculation of the average speed <V ΔZ> by the traveled way ΔZ The traveled way of the system from the moment of time t1 till the moment of time t2 according to the Fig. 13 is: t2

Z t2   Z (t1 )   VZ (t )dt and substituting into the expression (34), then: t1

t2

 VZ

VZ (t )dt Z (t2 )  Z (t1 ) t1   t2  t1 t2  t1

(35)

One very important moment is necessary to mention. What value must be taken into consideration Ntot or Z ?

Taking into consideration the expression (21) then:

No 2  Nart 2  Ntot 2  No  Nart 2  Nart 2  No 2  No 2 Nart 2  1 Nart 2 2 2 Here is necessary to mention that for the big values of Nart, the numerical value N art   1  N art  , Z 

because a microeconomical systems of stocks could contain an amount of order 1000 names or bigger and

Nart 2  1  Nart 2 . Then: 2 2 2 Z  No  N art   1 N art 



Nart 2  No 2  1  Nart  No 2  1

(36) For the case when the amount of products that corresponds to one article No is relative big numerical value, then:

No 2  1  No 2 and finally Z  Nart  No 2  1  Nart  No  Ntot

The task is the study of the value of No that makes the coincidence of the values of

(37)

Z and Ntot , and another

task is the precision of the expression (36) The numerical simulations of the expressions (36) and (37) for the fixed values of ΔNo and various values of ΔNart with the consequent representation on the graphic of the Fig. 14 allows to observe any peculiarity.

Fig. 14 The numerical simulation of the values

Z and Ntot as the function of N art for the fixed

values of No

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Z and Ntot are almost the same and practically coincide for values of N art  10000 . The coincidence of Z and N tot takes place for the values of N 0  5 and N art  10000 .

The graphic of the Fig. 14 shows that the values

In order to calculate the precision of the formula of ΔZ , first the average value <ΔZ> between ΔZ and ΔNtot is calculated. Then, the deviation

% 

devZ  Z  Ntot is calculated. After that, the relative error is calculated

devZ  100% . The dependence of  %  f (No ) is presented on the Fig. 15.  Z 

Fig. 15 The relative error of ΔZ as the function of ΔNo The graphic

 %  f (No ) shows that for the quantities of N0  10

the error has tendency to reach the zero

value. In order one method to be validated it's necessary the error do not exceed the value 20% [33].

In such a way both methods could be applied either ΔNtot or ΔZ. The method of position vector Z is more informative and gives more general information about how all three values Ntot, No and Nart are changeable with time. So, the values ΔZ ≈ ΔNtot and for the further description the values ΔZ are considered simply as the amount of inventory.

Example 1. The instantaneous speed of the variation of the inventory is described by the following function

VZ (t )  50  e0.2t . Find the analytical expression of the inventory ΔZ as a function of time. Calculate the inventory at the third day, if the inventory of the first day is 100 products and the unit of time is considered one day. Represent the graphic of the function of the inventory with the path 1 day as the function of time. Calculate the average speed of the movement from: a) third day till seventh day; b) third day till tenth day.

dZ ; The respective analytical expression of the inventory as the dt function of time is calculated as the integration like: Z (t )   VZ (t )dt  C ; Solution: The momentary speed is:

VZ (t ) 

Z (t )   50  e 0.2t dt  C  50 

1  e 0.2t  C  250  e 0.2t  C ; (0.2)

The constant of integration C is found from the initial conditions: t =1 day; ΔZ=100;

100  250  e 0.2t  C; C  250  e 0.2  100  100 

250 250  100   100  204.68  304.68 0.2 e 1.2214

; 0.2t

 304.68 ; Then: Z (t )  250  e The respective inventory of the third day is: Z (3)  250  e0.23  304.68  304.60 

250 250  304.6   304.6  137.204  167( products ) 0.6 e 1.8221

.

The respective graphic of the function is represented on the Fig. 16:

Z (t )  250  e0.2t  304.68 , it could be observed that for the values of time t   ; Z ()  304.68  305 products  .

Referring to this expression of the given solved example

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Fig. 16. One example of the dependence of the inventory with time The average speed of the movement from third day till seventh day is:

Z (7)  Z (3)  250  e0.27  304.68  250  e0.23  304.68 250  0.6 1.4 ;    e e 73 4 4 250 0.6 1.4   e  e   62.50.5488  0.2466  18.88  19 products / day  ; 4

a)  VZ

 VZ







The method of integration:

Z (10)  Z (3)  250  e0.210  304.68  250  e 0.23  304.68 250 0.6 2    e e ; b)  VZ  10  3 7 7 250  0.6  2  VZ   e  e  35.71  0.5488  0.1353  14.76  15 products / day  ; 7









It can be observed that the average speed in this case is decreasing gradually with the increasing of the interval of time from the initial moment.

The ways of the movement of the system that is characterized by variation with time of the inventory ΔZ(t) could be very various. The next figure 17 shows a possible type of the movement of the system.

Fig. 17 The possible variety of the movement of the microeconomical system of stocks Specifically, for this Fig. 17 is that the system has seasonal character. More selling of the stocks is for the period at the start of the summer (minimal value of the inventory ΔZ). If the seasonality is repeating instant all time (a lot of years), then characteristically for this system is that this system is more active during the summer. The system has sufficient financial resource to increase its inventory that is represented by maxima on the Fig. 17.

The respective policies of the marketing mix of this system are processed and stated. The stocks are checked by seasonality and the supplying is performed according to the respective seasonal demand.

Another interesting situation could be for the case that is represented on the Fig. 18. Suddenly, the system is forced to be transferred from one "macro-" state with big values of inventory ΔZ into another "macro-" state that is characterized by smaller values of inventory ΔZ. It is like a "change of the phase" of the system.

Characteristically is that when the system is transferred into another "phase" and if it lasts for a long time to stay into this new macro-state, then it means that the system is already adapted for new conditions. Possible transition could takes place due to of a lot of factors like: social and economical crises, demographic problem of the given geographical place of the microeconomical system. In these new conditions the new policies of marketing mix are elaborated.

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Fig. 18. The transition of the system from one "macro-" state into another If the values of ΔZ are permanently decreasing then the system will reach the situation when will not be able to continue its activity and has the peculiarity of default trend (Fig. 19) until the new policies of the marketing mix are elaborated.

Fig. 16 The default trend of the microeconomical system The next examples allow to understand better the suggested method of inventory and how it varies with time.

Example 2. One shop decides suddenly to sell all remaining inventory of 10000 products of various articles with the average quantity of one article ΔNo=3. How long time is necessary to sell all products from the moment of decision if the law of the variation of the inventory is Z (t )  30000  200  t . What is the average speed of the selling?. Find the instantaneous speeds for the first and third day from the moment of decision. Find the initial quantity of the products and the initial quantity of stock articles. The unit of time is considered one day.

Solution: . First is necessary to find the moment of time when the inventory contains 10000 products: 10000  30000  200  t; 200t  20000; t  100  th day .

The

day

when

all

inventory

is

sold

is

find

as

the

consideration

that

Z (t )  0 :

0  30000  200  t; t  150  th day .

So, the interval of time during which the remaining of inventory will be sold is 150-100=50 .

2. The instantaneous speed of the selling is found by first derivative with time:

VZ (t ) 

 products  d (Z )  . The sign minus of the speed indicates that the inventory every time  200 dt  day 

is decreasing. It remains all time the same. So, the instantaneous speed for the first and third day from the moment of decision is the same

 products   ;  200  day 

3. The average speed of the selling is the traveled way during the interval of time 50 days.

 Vz 

 products  Z (150)  Z (100) 30000  200  150  30000  200  100 200  (100  150)     200 50 50 50  day 

The average speed can be found also by the integration:

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 VZ

150 150  products  1 1  200t |150 200(150  100) 100  ;    VZ (t )dt    (200)dt    200 50 100 50 100 50 50  day 

4. The initially quantity of products (initially inventory) is found for the start moment of the time t=0;

Z (0)  30000( products )

The initially quantity of stock articles is found by

N art 

Z (o) 30000   10000(articles ) .

No 3

Example 3. One shop decides suddenly to sell all remaining inventory of 10000 products of various articles with the average quantity of one article ΔNo=4. How long time is necessary to sell all products if the law of the 0.02t

variation of the inventory is Z (t )  30000  e . What is the average speed of the selling. Find the instantaneous speeds for the first, ninth and fortieth day from the moment of decision. Find the initial quantity of the products and the initial quantity of stock articles. The unit of time is considered one day.

Solution: 1. First is necessary to find the moment of time when the inventory contains 10000 products:

1 10000  30000  e 0.02t ; e 0.02t  ; e0.02t  3; 0.02t  1; t  50  th day .

3 The day when all inventory is sold is find as the consideration that Z (t )  1 (formally considering one because

practically

will

1  30000  e 0.02t ; e 0.02t 

not

be

sold

till

absolute

zero

inventory):

1 ln(30000) ; e0.02t  30000; 0.02t  ln(30000); t   516  th 30000 0.02

So, the interval of time during which the remaining of inventory will be sold is 516 -50=466 2. The instantaneous speed of the selling is found by first derivative with time:

d (Z )  30000  0.02  e 0.02t  600  e 0.02t ; dt d (Z ) VZ (51)   600  e 0.0251  600  e1.02  600  0.3605  216 products / day  ; dt d (Z ) VZ (59)   600  e0.0259  600  e 1.18  600  0.30727  184 products / day  ; dt d (Z ) VZ (90)   600  e 0.0290  600  e1.8  600  0.1652  99 products / day  . dt

VZ (t ) 

The instantaneous speed is decreased gradually with time by absolute value. The decreasing takes place by the fact that the remaining reserve inventory is decreasing gradually.

3. The average speed of the selling is the traveled way during the interval of time from 50-th day till the uncertainty day.

 Vz 

 products  Z (516)  Z (50) 0  10000    22 466 466  day 

The average speed by the integration: 516

 VZ 

  600  e

 0.02t

dt

50

466

516  products  300 300  500.02     e 0.02t dt   e 0.02t |516  22 50  64.37  e 233 50 0.02  233  day 

4. The initial quantities of products is found by: articles is found by:

N art

Z (0)  30000  e0  30000 . The initial quantity of stock

30000  7500(articles ) 4

The Fig. 17 that is like a oscillation movement represents an special interest. The values of ΔZ are changeable similar to Sinus or Cosinus laws with the amplitudes during summer-autumn each year. The system oscillates

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Namely, the practical major cases show that a lot of activities of microeconomical systems of stocks have seasonal periodical character if are not observable the criteria of the default trends as that represented on the Fig. 16 . This fact lead us to one of prerequisites of the emerging of the idea of the oscillator model of the system of stocks.

Considering that the values of ΔZ oscillates by Cosinus law, then: (38) Z (t )  Zech  A cos(t   ) where A - amplitude of oscillation; ΔZech - the equilibrium value of the inventory; ω-cyclical frequency; tinterval of time; υ-initial phase; The respective oscillations of the values ΔZ with time is represented on Fig. 17.

Fig. 17. The oscillation character of the stock inventory The equilibrium value of the inventory is such a value around which the values ΔZ are changeable within the interval [ΔZmin res; ΔZech+A]. The value ΔZmin res is the minimal value of the stock inventory. The minimal stock inventory ΔZmin res is such minimal reserve, when the system cannot fully satisfy buyers' needs and demands, therefore the system is supplied by the new stocks from the dealers.

VZ (t ) 

The instantaneous speed is calculated as:

The cyclical frequency ω expressed by period of oscillation:

T

dZ   A sin(t   ) dt 2



(39) (40)

The period of oscillation T for a lot of cases is one year as for the Fig. 17.

The average speed of movement during one period is:

1 A A A   VZ (t )dt     sin(t   )dt     sin(t   )d (t   )   cos(t   ) |T0  T 0 T 0 T 0 T T

 VZ (T ) 



T

T

A A A A  2  2   2   cos(T   )   cos   cos(2   )  cos    2   sin   sin   0; T T T T 2    2 

The fact that the average speed within one period of time is zero means that the system returns back to its initial state with the initially value of the inventory.

Another method of the calculation of the average speed by the method of displacement is:

 VZ (T ) 



Z (T )  Z (0) Z ech  A cosT     Z ech  A cos  A cos(2   )  cos      T 0 T 2

A  2  2   2  2  sin  sin 2 2    2

   0; 

The calculation of the average speed can be checked also on the intervals of time [t;t+T]:

 VZ (t  T ) 

1  T

t T

 VZ (t )dt   t

t T

A A   sin(t   )dt   cos(t   ) |tt T  T t T

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

A A A  cos( (t  T )   )   cos(t   )  cos(t  2   )  cost     T T T A  2  2t  2   2   2   sin   sin   0; T 2    2 

The calculation of the average speed by the method of displacement:

Z (t  T )  Z (t ) Z ech  A cos (t  T )     Z ech  A cos(t   )   T t t T A A  2t  2  2   2   cos (t  T )     cost     2   sin  sin   0; T T 2    2 

 VZ (t  T ) 

The average speed of the movement during one half of the period is:

 VZ (T / 2) 

1  T /2

T /2

 0

VZ (t )dt  

A  T /2

T /2

 sin(t   )dt  0

2A  cos(t   ) |T0 / 2  T

2A T 2A 2A cos(   )  cos    4  A  sin 2     sin     cos(    )   cos   T 2 T T T  2  2 4A   4A 4 A 2 A  sin     cos    cos    cos   0; ; T 2 T 2   2 A 2 A In the case when φ=0; then  VZ (T / 2)   cos 0    0; 





The sign "-" means that the inventory is decreasing during this interval of time [0;T/2].

The respective method of displacement:

 Z ech  A cos (T / 2)     Z ech  A cos  2 A   T   cos     cos     T /2 T   2   2A cos     cos     4 A  sin    sin    4 A cos    2 A cos   0 ;  T T 2 2 T    VZ (T / 2) 

The next example allow to understand all practical peculiarities about the suggested method of the oscillator model of the inventory and how it behaviors with time.

Example 4. One shop has the equilibrium permanent stock of 6500 articles with the average amount of products per articles No=4 products. It has seasonal character with the period of one year. The peak of inventory rises 36000 products. The minimum reserve within "inactive" period reaches 16000 products. The variation of the inventory takes place by Cosinus law. Calculate: a) equilibrium inventory expressed in products; b) the amplitude of oscillations of the inventory; c) calculate the initial phase υ if the starting moment has the inventory of 30000 products; d) calculate the cyclical frequency if the unit of time is one month; e) the variation of the stock inventory for the moments of time 6 months, 9 months from the start moment of time and 12 month; f) the average speed for the interval of time 6 month and 12 month; g) the instantaneous speed at sixth month and tenth month; e) represent the graphics of the dependence of inventory and instantaneous speed as the function of time on the same frame.

Solution: a) The equilibrium inventory expressed in products is ΔZech≈ΔNtot=No∙Nart=4∙6500=26000 products; b) The picture will give the idea how the inventory is changed:

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The amplitude A is the maximum deviation from equilibrium position A=ΔZmax-ΔZech=36000-26000=10000 (products) c) The starting moment of time has the inventory of Z (0)  30000( products ) ; The initial phase  is found from the relation:

Z (0)  Zech  A cos(0   ); A cos   Z (0)  Zech  30000  26000  4000; 4000 2 cos     0.4 ; 10000 5 The value 0.4 has the meaning as that for the starting moment of time the inventory "is somehow planned" to have such an inventory that consists 40% of the "future possible maximum of the inventory".   arccos(0.4)  1.1592 (rad)=66.450; d) the cyclical frequency



2 6.28   0.52(month 1 ) ; T 12

The meaning of the cyclical frequency is the quantity of radians that corresponds to one month. e) The variation of the stock inventory for 6 months from the start moment of time:

Z var (6)  Z (6)  Z (0)  Zech  A cos(6   )  Zech  A cos   Acos6     cos   

 6  2   6   6  0.52  2 1.16   6  0.52   3.12  2.32   2 A  sin  sin   20000  sin   sin   20000  sin  2 2  2   2     2     3.12   sin   20000  sin 2.72  sin(1.56)  20000  0.4092  0.999  8176( products ) ;  2  It means that the inventory is decreased during 6 moths with 8176 products.

The variation of the stock inventory for 9 months from the start moment of time:

Z var (9)  Z (9)  Z (0)  Zech  A cos(9   )  Zech  A cos   Acos9     cos   

 9  2   9   9  0.52  2  1.16   9  0.52   4.68  2.32   2 A  sin  sin   20000  sin   sin   20000  sin  2 2  2   2     2     4.68   sin   20000  sin 3.5  sin(2.34)  20000  (0.35038)  0.7184  5034( products )  2  It means that the inventory for the moment of time 9-th month is bigger with 5034 products higher than of the starting inventory.

The variation of the stock inventory for 12 months from the start moment of time:

Z var (12)  Z (12)  Z (0)  Zech  A cos(12   )  Zech  A cos   Acos12     cos   

 12  2   12   12  0.52  2  1.16   12  0.52   6.24  2.32   2 A  sin  sin   20000  sin   sin   20000  sin  2 2 2    2     2   

 6.24   sin   0( products ) ;  2 

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It means that the inventory of 12-th month coincides with the staring inventory. The starting inventory:

Z (0)  Zech  A cos   26000  10000  cos(1.1592)  26000  4000  30000( products )

f) The average speed for the interval of time 6 month:

Z (6)  Z (0) Acos6     cos   2A A  6  2     sin  sin 3    sin(3   ) sin(3 )  6 6 6 2  3  10000 10000 10000  sin 3  0.52  1.16sin(3  0.52)   sin(1.56  1.16) sin(1.56)   sin(2.72) sin(1.56)  3 3 3

 VZ (6) 



10000  products   0.4092  0.9999  1364  3  month 

The comment of this result is that during six months from the start moment the inventory is decreasing with 1364 products every month.

The average speed for the interval of time 12 month:

Z (12)  Z (0) Acos12     cos   2A A  12  2      sin  sin 6    sin(6   ) sin(6 )  12 12 12 2 6   10000 10000 10000  sin 6  0.52  1.16sin(6  0.52)   sin(3.12  1.16) sin(3.12)   sin(4.28) sin(3.12)  6 6 6

 VZ (12) 



10000  products   (0.9079)  0.02159  33  6  month 

The comment of this result is that during 12 months from the start moment the inventory is increasing with 33 products every month. g) The instantaneous speed at sixth month, tenth month and twelve month .

The expression of instantaneous speed: VZ (t )   A sin(t   ) ; The instantaneous speed at the moment sixth month is:

 products  VZ (6)  0.52 10000  sin(0.52  6  1.16)  5200 sin(3.12  1.16)  5200  (0.907)  4716 ;  month  Exactly, at this moment of the time, this result means that the inventory is increasing its quantity by 4716 (products/month).

The instantaneous speed at the moment tenth month is:

 products  VZ (10)  0.52 10000  sin(0.52 10  1.16)  5200 sin(5.2  1.16)  5200  0.076  395 ;  month  Exactly, at this moment of the time, this result means that the inventory is decreasing its quantity by 395 (products/month).

The instantaneous speed at the moment twelve month is:

 products  VZ (12)  0.52 10000  sin(0.52 12  1.16)  5200 sin(6.24  1.16)  5200  0.8987  4673 ;  month  Exactly, at this moment of the time, this result means that the inventory is decreasing its quantity by 4673 (products/month). e) The graphics of the dependence of the inventory and the instantaneous speed as the function of time is represented on Fig. 18.

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Fig. 18. The numerical simulation of the dependence of the inventory ΔZ(t) and the instantaneous speed VΔZ(t) vs. time So, the graphic of the speed is displaced with respect to the inventory with the phase difference 900. The point A that is the minimal value of the inventory corresponds to the zero value of the instantaneous speed (point K).

Major practical cases namely such situation takes place when the system of stocks sells firstly products without any payments. The decreasing of the inventory takes place (the segment MA) as the result of the selling. The financial resources are earned and they are spent for the new stocks (the speed is increasing on the segment NK).

The inventory is still increasing on the segment AB and the respective speed is continuing its increasing on the segment KE. The supplying with the new stocks gradually is decreasing (the segment ED) and the inventory slowly reaches its maximal value (the point C). The processes are repeating periodically. This is an ideal model of the oscillations and it takes place really for the every day big turnovers and continuous supplying with the new stocks exactly with the same amounts that were sold every day. In this case the expenses for passive assets are comparative small with respect to active assets and the movement of the system reaches the ideal case of harmonic oscillator. The phase trajectory in this case is an ellipse in the two - dimensional coordinate system (VΔZ; ΔZ) (Fig. 19).

Fig. 19 Phase trajectory of the microeconomical system of stocks The system starts the movement from тхе point 1 and consequently the speed is passing through the minimal value -Aω, then the value 0 and finally the maximal value Aω. The values of ΔZ are oscillating within the interval [ΔZmin; ΔZmax].

The trajectory of the three-dimensional spatial phase has a spiral shape located on the lateral surface of the cylinder with the height equal to the time interval and with the bases coinciding with the ellipses in the twodimensional space (VΔZ; ΔZ). (Fig. 20)

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Fig. 20 Phase trajectory of the perfect microeconomical system in three dimensional phase space If the oscillations are continuously with the same amplitudes without any attenuations then this cylinder is infinite and perfectly with the same bases as the form of ellipses. If attenuations emerge as the result of the decreasing of turnovers and the increasing of the passive assets then the final basement of the cylinder will have smaller area as the initial one (S1>S2) and if the final basement is continuously decreasing all time then the peculiarities of default of the system are observed. (Fig.21).

Fig. 21 The attenuated elliptical cylinder in the conditions of the default of the system The qualitative description of the activity of microeconomical systems of stocks by three dimensional phase space allow to conclude about the behavior of the system with time. Qualitatively, it could be stated that smaller instantaneous surfaces S(t) of the ellipses suggest about smaller turnovers in comparison with bigger instantaneous surfaces of ellipses of bigger turnovers (Fig. 22).

Fig. 22 The possible real seasonal character of microeconomical system of stocks

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3.2. The notion of the acceleration of the movement of systems. The acceleration as the degree of the turnovers and stability of the systems The previous topic describes the notion of the speed and the trajectory. And the speed is related to the change of the quantity of inventory expressed as the measurement unit (product/unit time). In the recent topic the situations when the speed is not constant (is changeable with time) are studied.

In physics, acceleration is the rate of the change of the velocity with respect to time. The IS unit in physics for the acceleration is meter per second squared (m⋅s−2). It is expected that the econophysical measurement unit is (product/unit time-2).

Let's examine the trajectory that the regarded system moves with changeable speed. Let

V1Z and V2 Z are the

movement speeds of the system at the moment of time t1 and t2 (Fig.22) and the respective small interval of time is t  t2  t1 .

Fig. 22 The trajectory of the movement of the system Imaginary the velocity vector is paralleled transferred from the point 2 into the point 1 and then according to the triangle rule we can see what is the velocity variation during this interval of time is:

VZ . The variation of the speed by the triangle rule

VZ  V2 Z  V1Z

(41)

aZ 

The vector size:

VZ t

(42)

is named the acceleration of the body at the moment of time t2. According to the definition, the acceleration is a vector. The system moves with acceleration every time when the vector of the speed VZ changes its direction, its value or both the value and its direction. These changes every time of the speed value and the direction of the speed leads to this fact that the acceleration is instantaneous for the fixed moment of time and respectively the acceleration and the module of acceleration is a function of time: aZ (t ) ; aZ (t ) .

The trajectory of the system represented on three dimensional system (Ntot, Nart, No) (Fig. 23) shows the vectors of acceleration

aZ (t ) for several moments of time t1, t2, t3 and t4.

The vector of the acceleration aZ (t ) that coincides with the direction of the variation of the vector 22) in general is not tangent to the trajectory but forms an angle as shown in the Fig. 23.

VZ (Fig.

The acceleration aZ (t ) can be represented for each point of the trajectory as the sum of two components:

aZ (t )  aZ n (t )  aZ (t )

(43)

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Fig. 23 The trajectory of the movement of the system of stocks on the three dimensional space (Ntot, Nart, No) The component

aZ (t ) lies on the tangent to the trajectory and is called the tangential acceleration. The

meaning of the tangential acceleration is the change in the magnitude of the velocity per unit time. If the velocity is increasing, the direction of the tangential acceleration coincides with the direction of the velocity or with the direction of the travel (Fig. 23, p. 1, p. 4). If the speed decreases, the direction of this acceleration is opposite to the direction of the speed. (Fig. 23, p. 2) The component

aZn (t ) is called normal acceleration. This acceleration indicates only the change of the

direction of the speed per unit time. It is always directed to the center of the curvature of the trajectory. (Fig. 23) Only for the case of linear motion, the normal (centripetal) acceleration is zero because in this case the velocity direction does not change (Fig. 23, p. 3). Characteristically for the p. 3 of the Fig. 23 is that the radius of the curvature of the trajectory r is very big (r→∞) and the normal acceleration

aZn (t ) tends to zero. It is important

to mention that if the inventory has the continuous tendency of the increasing as for the point 4 of the Fig. 23, then the resultant

aZ (t ) is oriented up (in the direction of the increasing of Ntot). For example, the point 2 of

the Fig. 23 has the tendency of the decreasing of the value Ntot and therefore the resultant aZ (t ) is oriented down.

In order to see better how each component of N0, Nart and Ntot varies separately as the dependence of the orientation of the acceleration resulting vector

aZ (t ) , it is necessary to project this acceleration vector

aZ (t ) on the plane (N0; Nart) and on the axis Ntot. (Fig. 24) The vector

AB that corresponds to the resulting vector aZ (t ) for the moment of time 1 has the

projection A1B1 on the plane (N0; Nart). The orientation of the vector quantity N0 (vector (vector

A1B1 indicates on the increasing of the

A2 B2 ) and the decreasing of Nart (vector A3 B3 ), so that the result of Ntot is the decreasing

A4 B4 ). (Fig. 24). The another moment of time 2 is characterized by the vector CD . Its projection on

the plane (N0; Nart) indicates on the vector

C1D1 and its projections on the axes N0 and Nart has the vectors

C2 D2 and C3 D3 . The increasing of N0 is more bigger than of the decreasing of Nart, so that the final result gives the increasing of Ntot (vector C4 D4 ) than in comparison with the previous case of the moment of time 1.

In this way it is solved the problem how each of the components of the inventories Ntot, Nart, No varies according to the projections of the acceleration resulting vector The absolute value of the acceleration is :

aZ (t ) on the coordinate axes.

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a Z (t ) 

a

Z

 

(t )  aZ n (t ) 2



2

(44)

dVZ (t ) is the first derivative with time. Only for the case when the dt speed is constant (the case of uniform movement), then the acceleration aZ is zero. The normal acceleration The tangential acceleration

aZ (t ) 

depends on the radius of the curvature of the trajectory r and is determined by

aZ n (t ) 

VZ (t ) 2 . r

Fig. 24 A modality of the explanation of the components quantities variations of N0; Nart ; N tot according to the projections of the resulting acceleration vector

aZ (t )

Taking into consideration the expression (37): Z  N art  N o  Ntot , then the momentary speed is

dZ (t ) dNtot (t ) (45)  dt dt So, the changeable in time of the inventory Ntot (t ) depends both on the amounts of products N o (t ) of one article and the quantity of articles N art (t ) . The quantity of articles N art (t ) also in general is dependent on written as:

VZ (t ) 

the time because the articles could have in general the seasonal character (during summer more various articles for example, but during winter more limited to a limited quantity). In general the supply-demand processes has the seasonal character.

In order to describe quantitatively the supply-demand processes the following system formed of two subsystems can be examined: 1) the subsystem of supplier (dealers); 2) subsystem of demander (shops, pharmacies, etc).

(Fig. 25) The processes inside of this complex supplier-demander system are stochastical. The stochastical processes are such random processes which evaluate in time and are variable with time. [34].

Let, the quantity of products is N0st of one article of the first subsystem of dealers at the initial moment of time t =0 . This value of products of one article N0st is well planned statistically due to of the long period of activity of the system and due to of statistical processes and analyses of the data. This is like a stationary value of the products of one article.

The processes of receiving of the stocks by the shops evaluate with time and during the time the quantity of products is increased. Which type of functional law of the amounts of products as a function of time takes place? The result of the amount of transferred products for the respective interval of time is influenced by a lot of factors: the price of product, socio-economical status of the customers (patients), geographical place, stock reserves that are supplied at this respective moment of time, weather conditions, so all the factors that are described by marketing mix policies. For example, if ten thousand products are sold during ten months, then it means one hundred products averagely within three days and the linear functional law takes place:

N 0 

100  t ; t - days ( 1-st day, 2 -nd day,.....).

3

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Fig.25. Schematical representation of the interaction between demander and supplier with the result of transfer of stocks from first subsystem to the second subsystem Is this linear law is valid for all rating marketing groups? Or, has this linear law some limits of application ?

The research papers [35], [36], [37], [38], [39], [40], [41], [42] explain these processes and after the processing of big amounts of statistical data and analyses the following exponential expression with asymptotical increasing of inventory level of one article N o (t ) is suggested:

No (t )  N0 st  (1  ebt )

(46)

N 0 st is saturation value of inventory level of one article; t - interval of time; b - exponential rate constant that depends on a lot of factors regarding marketing mix policies. The meaning of this exponential rate constant b is the inverse interval of time during which the quantity of products of the first subsystem is decreased e times. (e ≈2.71). The measurement unit of b is [b]=day-1, month-1, year-1, etc.

The respective graphic (Fig. 26) of the expression (46) with asymptotical increasing is:

Fig. 26 The graphic of

No (t ) with asymptotical increasing of the inventory model

It can be observed from Fig. 26 that for the small values of time the shape has the linear segment and for the bigger values of time then it is increased till the saturation value of N 0 st . The exponential function has linear approximation for the small values of b∙t : [43]

e bt  1  bt

 No (t )  Nost  (1  ebt )  N ost b  t

(47) The question about which segments of time is valid for such linear approximation can be answered when the comparison of linear and exponential graphs are plotted on the same plane Fig. 27.

It can be seen from the Fig. 27 that if the value of b is increased then the segment of linear approximation is decreased and also if the value of N 0 st is increased, the segment of linear approximation is decreased too.

Referring to Fig. 25, we have that the supplier subsystem at initial moment of time (t=0) contains the amount of products N 0 st and this subsystem after the interaction with the subsystem of demanders evaluate with time like as the dependence that is represented on the Fig. 28:

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Fig. 27. The numerical simulation of the expression with asymptotical increasing in comparison with the linear approximation as the dependence of time for two values of exponential rate constants b

Fig. 28. The evaluation with time of the amount

N 0 of products of supplier

The process of the purchasing from the supplier to the demander of some amount of products ΔN during the interval of time t is represented schematically on the Fig 29 :

Fig. 29 The evaluation with time of the amount of products of the supplier and the demander So, according to the Fig. 29 the amount of the products of the supplier at the moment of time t is N ost  N and the respective amount of the products that are transferred from the supplier to the demander is

N . The total sum of the amounts of the first subsystem and the second subsystem is constant with time: N st  N  N  const  Nost (48)  o   I  subsystem

II  subsystem

The amount of products of the first subsystem at the moment of time t is :

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No (t )  Nost  No (t ) ; but the value N o (t ) is: No (t )  Nost ebt

No (t )  No (t )  Nost ;

Referring to Exp. 48, then : (50) and the final expression of



Nost  e

bt

(49)

 No (t )  Nost

N (t ) for the second subsystem at this moment of time t is:

N (t )  Nost (1  ebt )

(51)

Referring to the marketing rating groups A, B, C, X, Z with the stocks articles and the amount of stock articles is Noj of one article, then the Exp. 51 can be written analogically as:

Noj (t )  Nost j (1  e

b j t

); Z  j  A

(52)

The values of ΔNoj are increased like as the shapes of the dependences that are represented on the Fig. 27 and asymptotically reaches the stationary value N o st for the interval of time t   . Such form of asymptotically j

reaching of the dependence is explained by the fact that some products have seasonal characters and finite product life cycles of some products of the system of stocks [44], [45]. If the demand is continuously and permanently and the product exist on the market permanently for a very long time then the increasing of the amounts takes place by linear function. [44], [45].

The numerical simulation of the Exp. 52 for various values of the stationary amounts for one stock article N o st and various values of exponential decay constants b is represented on the Fig. 30. j

Fig. 30. The numerical simulation of the behavior with time of sold amounts ΔNoj at various values of exponential decay constants b The results of numerical simulation that is represented on the Fig. 30 can take place in general for all rating marketing groups. It is observed from the graphic that if the value b is increased the saturation till the value N0st is reached more quickly with the interval of time shorter than for the small values of b. For the value b=0.1 we have that the stationary value N0st = 2000 is reached during greater interval of time in comparison with the value N0st = 500.

Rating coefficients of the stock of the rating groups that show the capacities of turnovers from one stock article also reach stationary states coinciding with the series of Fibonacci numbers.

The expression (52) as:

Noj (t )  Nost j (1  e

N oj (t )  N ost j (1  e

b j t

) 

b j t

); Z  j  A allows to represent the dependence of Kj(t)

K j (t )  P p



K j st  P p



 1 e

 b j t





 K j (t )  K j  1  e st

 b j t



(53) Schematically, this dependence on time for the various values of exponential decay constants b is represented in the (Fig.31).

Also, it is observed that if the value of the exponential decay constant b is increased the reaching of the saturation takes place at more shorter interval of time (Fig. 31).

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Fig. 31. The simulation of the rating coefficients of the stocks as the function of time for different values of exponential decays constants b Regarding the items (stock articles) the similar dependence in the form of asymptotical increasing takes place that is represented on the Fig.32.

Fig. 32. The behavior with time of the items of the rating groups with asymptotical decreasing The expression about the amount of stock articles as the function of time can be written analogically as the

N art j (t )  N art st  (1  e

expression (52) like:

j

B jt

); Z  j  A

(54)

the rate exponential constant of the stock articles in general can be different of exponential decay constant b and is signed as B. The measurement unit of B is the same as for b.

It is necessary to mention that the scheme of the transferring of products from the dealers (supplier) to the demander is valid also for the case of the interaction between the shops and the customers. In this situation the shop plays the role of supplier but the customers play the role of demanders.

The rating marketing group A has bigger exponential rate constant B and the level of saturation is situated higher than of B, C, X, Z (Fig.32). The respective interval of time is shorter for the bigger value of B in comparison with the smaller one.

In order to understand the processes that takes place as the result of the selling, then the Fig. 30 that shows numerical simulation can be applied for some examples of products. For example, one OTC pharmaceutical product is researched with continuous permanently demand with big fluctuations within the values from 15 till 55 products each month. (Fig. 33, a) These sold products of the respective month are signed by the value ΔNom (meaning momentary amounts of sold products of the respective month). In order to use these model of asymptotical increasing it is necessary to sum previous values of ΔNom till the respective last moment of time and then the amount of sold products for the respective stock article is

No (t )   Nom .

The graphic of the function

N0  f (t ) in general could be linear or with asymptotical saturation. For this

OTC product the following linear graphic is obtained that is represented on Fig. 33, b. Nevertheless that the

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. demand values ΔNom are uncertainty with big instant fluctuations all time, the linear dependence of cumulative amounts N0  f (t ) is obtained and the coefficient of correlation R=0.998 indicates the strong belonging to the linear function. The proportional coefficient of this linear function shows the average amount sold during one month (≈32 products per month), (Fig. 33, b). The graphic of the function

 N st  ln    f (t ) of  N st  N o 

course for this case is nonlinear (fig. 33, c) due to of the fact that the dependence ΔNo(t) is not with the asymptotical saturation.

Fig. 33. The application of the model with asymptotical increasing to the real example of OTC product.

 N st  ln    f (t ) gives the answer about the processes  N st  N o   N st  that are developed with time. If such dependence ln    f (t ) is not linear, then we can conclude  N st  N o  In general we can remark that the dependence

that the activity is expected to be instant and stable with the stable demand just if the big fluctuations exist and the respective interval of time of these fluctuations is stable in time (Fig. 33,a). Only the case of linear form of the dependence

 N st  ln    f (t ) suggests the seasonal character of the process or in some case could  N st  N o 

be just finite life cycle of the products.

Another example is about well known product Panthenol spray. This product has seasonal character and the values Nom have the peaks that are represented on the Fig. 34, a. The peaks represent the great demand at the respective moment of time (7-th - 8-th months of the year). If we take only the interval of time one year then an asymptotical saturation is observed on the Fig. 34, b. The graphic

 N st  ln    f (t ) that is represented  N st  N o 

on Fig. 34, c contains two linear segments: first till 7-th month and another till 12-th month. As two linear segments exist then the conclusion is that this interval of time of one year contains one peak at seventh month that means the great demand at this moment of time (Fig. 34,c). The Fig. 34, d contains six peaks corresponding to the peaks of demand. The graphic of Fig. 34, e gives more detailed information. Beside that, it gives the information about six peaks during the entire period of time and also the answer about of the linearity of the

 N st  ln    f (t ) is given. One important moment we can remark, that if N   N o  st  N st  ln    f (t ) is linear then the criterion of the saturation takes place (seasonal character or finite  N st  N o 

graphic

life cycle).

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Fig. 34. The application of the model of asymptotical increasing for the real example of Panthenol spray products Regarding the product with the finite life cycle, we have that the demand of this product is decreased gradually with time with some exceptions that sometimes the fluctuations from minimal till maximal values take place but finally the demand reaches zero (Fig. 35, a).

Fig. 35. The application of the model of asymptotical increasing to the real example of the product with the final life cycle The cumulative value ΔNo(t) reaches asymptotically the saturation (Fig. 35, b). The total answer about the degree of asymptotical saturation gives the graphic of the dependence

The approximated linear dependence of

 N st  ln    f (t ) , (Fig. 35, c).  N st  N o 

 N st  ln    f (t ) that is represented on the Fig. 35, c shows  N st  N o 

the character of the finite life cycle of this product. The exponential rate constant b can be found from this linear dependence by the slope to the axis x.

This value b is b=0.099(months-1), (Fig. 35, c). This constant b can also be found from Fig. 35, b taking into consideration only the linear segment corresponding to the small values of the interval of time 27 months.

No (t )  Nost  (1  ebt )  Nost  (1  1  bt )  N0 st  b  t  No (t )  Nost  b  t (56)

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The

values

No  770 ;

Nost  1040 ;

t  27 (months)

from

the

Fig.

35,

b,

and:

770  0.027(months 1 ) . The value b from Fig. 35, c that corresponds strictly to the linear 27.1040 0.8 dependence gives: b   0.03(months 1 ) 26 b

The correlation calculations for the entire linear segment of the Fig. 35, c gives b=0.099(months-1).

The full segment till 70 months almost is good described by linear dependence. The value of correlation coefficient R2 states about the strong connection to the linear dependence. The question is which values of b are valid? Both values are valid. Just if we have planned stock reserve, the interval of time for total selling of this stock reserve is calculated as:

t

No 1 1    33 (months) Nostb b 0.03

or

t

No 1 1    10 (months) Nostb b 0.099

(57) It means that the full stock reserve with the quantity Nost=1040 can be sold minimum during ten months till maximum thirty three months.

The respective amounts during 10 months is: No  1040  0.03  10  312 ( products ) So, the question is again, why all this information is necessary? Each activity is based on experience and practice. In order to have more performed the activity it is necessary to have a large information about previous activity till the recent moments. We can forecast the activity for the future if we have the value of the rate exponential constant b. Just if the forecasted amounts of products deviate from real ones then the remained reserve will be used forward with the condition if the expiration date is far. And therefore it is necessary to consider as long as is the interval of time the probability is bigger to have small deviations from real amounts.

So, the values of exponential rate constant b serve as the criterion of levelling of the forecasted amounts and if the interval of time is bigger then the more real results are obtained.

The next example is about the subsystem of two products: the Panthenol spray and one of OTC product. Both product with their momentary amounts Nom are represented on Fig. 36, a. The panthenol spray has seasonal character but OTC one has all time the demand with big fluctuations. The cumulative value in this case is calculated as the average of two products:

N o (t ) 



N om (t )   N om (t )

panthenol

OTC

2

(58)

The respective graphic ΔNo(t) is represented on Fig.36, b. In general this graphic is linear and flexible points are observably corresponding to their six peaks of the product panthenol that are similar to the Fig. 34, d. The peaks are attenuated by the fluctuations of OTC product but the peaks are bigger than the fluctuations and therefore the thresholds are visible on Fig. 36, b that corresponds to six seasons of the entire interval of time.

The points on Fig. 36, b are almost arranged on the straight line and the coefficient a of the linear function shows the average amount of both product per month.

Fig. 36. The application of the model of asymptotical increasing to the subsystem of two products: Panthenol and one of OTC product The following question is about if there is no a big error for the forecasting if the value a is considered for all items. The found value a is more real for Panthenol than for the second product of OTC due to of the fact that the second product has the big fluctuations. Just if this value a is applied for the forecasting of the amounts of

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. the OTC product then maybe the eventually surplus of stock is used forward. In order do not have the big errors during the process of the forecasting it is necessary to have small interval of time for the forward like several days till several weeks. Another important moment it is necessary to mention that promotional packages also influences on the planning of necessary amounts. So, in the conditions of high hyper competition the forecasting politics based on the found value of exponential decay plays an important role. In general it would be avoidable to use the value of exponential rate constant for the full rating group and this value will serve for the forecasting of each item for small interval of time taking into consideration promotional packages.

Generally speaking, each item has a lot of peculiarities in the real conditions of competition and a series of factors act on the demand of products like seasonal character, finite life cycle of product due to of the fact that a lot of generics substitute the previous ones. The prices of generics are usually significantly lower than the corresponding originals. The appearance of generic products on the market often leads to a decrease in the price of the original as well as of the individual competing products. Generally, a generic product is offered at a price that is many times lower than the original price of the original. There are about ≈25% in Bulgaria of generic products, and ≈75% original pharmaceutical products [46].

It was mentioned before that the exponential rate constant b has the property of the leveling of the amounts for each item that serves as the criteria of the forecasting. For example, the subsystem of eight items that is represented on the Fig. 37, a shows the items which have different behaviors, some of them are seasonal, some with finite life cycle, and some with fluctuations of the demand all time.

Fig. 37. The application of the model of asymptotical increasing to the subsystem of eight products Nart

The average cumulative amount ΔNo(t) of the item is calculated as:

N o (t ) 

 N i 1

omi

N art

(t ) (59)

All these peculiarities in time of the items lead to such asymptotical increasing represented on Fig. 27,b. The respective dependence

 N st  ln    f (t ) is almost linear for this subgroup of products meaning that the  N st  N o 

model of asymptotical increasing is valid and applicable.

Regarding the stock articles the situation has the specific peculiarities. The amount of the stock articles can be various as the dependence of the specific interval of time and could have the seasonal character. Sometimes the set of articles is fixed and non changeable but sometimes it could be in the state of the decreasing of the set or in the state if increasing. The inventory in this case could contains initially the quantity of articles N artst and this j

inventory is evaluating in time depending on the value

N art j (t ) .

Referring to the Exp. 54, it is possibly to calculate the value of exponential rate constant B if the quantity of the stock articles at the stationary state N artst and the quantity of the items N art j that are sold during the interval j

of time t (expressed in quantities of months) for the respective rating group j is known. So, the expression of the rate constant B of the articles is written as:

  N artst j  ln   N artst j  N art j (t )  B t

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

The graphic of the function ln 

 N artst j

   f (t ) gives the possibility to find the value B by the slope  N art j (t )  

N artst

j

of this graphic of the Fig. 38. Such graphic allows to conclude about the seasonality of the respective item for the respective interval of time.

Fig. 38. The dependence of

  N artstj ln    f (t ) for different situations:  N artstj  N art j (t ) 

a - The new articles are appeared on the market with continuously permanent demand and the instant keeping of the demand of the old ones; b - Seasonal character of the demand of the articles; c - limited life cycle of the article (sometimes the articles could be returned back to the dealers-dotted curve); d Periodical seasonality of the articles.

Referring to the Fig. 38, a it could be concluded that the amount of the articles have a high level of the saturation. The permanent new other articles could appear on the market with continuously permanent demand and the instant keeping at the same time of the demand of the old ones. In this case, the evaluation with time of the sold articles has a linear function of the logarithm due to of the fact that N art j (t ) is permanently increasing with time till the limit planned value





N artst N art j (t )  N artstj . Only for very big interval of

time a situation could be when the amount of the items is established and then the inventory could reach some saturation due to of the fact that with the increasing of the time the amount of the articles N art j remains almost constant, Fig. 38, a. This situation could be valid for the big systems of stocks like hypermarkets. The case of the Fig. 38, b has another peculiarity. The level of the saturation is reached more quickly. After the reaching of the saturation the system operates only with the limited quantity of the articles and the value N art j remains unchangeable with time. This situation has the seasonal character. The situation differs drastically for the case Fig. 38, c. This case could be when the activity of the system is very weak. The articles have limited life cycle.

In general the full system has limited life cycle. It reaches quickly some level of the saturation and then all time remains at the same level (no activity). Another situation could be that all the stock is returned back to the dealers (the dotted curve) Fig. 38, c.

More interesting case is for such situation when the increasing of N art j takes place seasonality Fig. 38, d.

Three seasons are observed on this figure. After the finishing of the first season the amount of articles stops to be changed. The system operates some interval of time with the well determined amount of the articles. The amount of articles after the first season starts its increasing and the second threshold is observed, and so on.

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The task of the kinematics is the study of the movement of the material body as a function of time. It is necessary to obtain the general law of the movement of the system of stocks as a function of time. The laws of the variation of the quantities of one stock article as well as the quantity of the stock articles were discussed by the expression (52) and (54). Taking into consideration these expression, then the law of variation of the inventory can be written as:





Z (t )  Nart t  No t   Ntot t   Nost (1  ebt )  Nartst 1  e Bt  Nost  Nartst  (1  ebt )(1  e Bt )

(61) For the special case when the values

b  t and B  t are small numbers, then: Z (t )  Nost  N artst  b  B  t 2

The expression (62) has an analogy to the kinematical equation of the way

(62)

S t  

a  t2 with the constant 2

acceleration a, and then this analogy can be written as:

Z (t )  N ost  N artst  b  B  t 2 

a t2 ;  aZ  2  N ost  N artst  b  B 2

(63) So, the acceleration of the movement of the system of stocks depends on the quantity of the stocks articles of the inventory N artst and on the average quantity of products N ost referring to one article, and finally depends on very special parameters as the exponential rate constants b and B. The difference between the kinematical mechanics is that the acceleration which is determined by the expression (63) could not be in general a constant value, because of that the values of b and B are dependent in their turn on time. In order the acceleration aZ to be an unchangeable value is necessary that the rate exponential constants b and B to be constant with time. The exponential rate constants b and B are dependent on time. An approximation of constant acceleration can be considered only for the case when the values of b and B are considered as the constant values during very small interval of time [t; t+dt]. This value of acceleration is instantaneous acceleration and is constant only for this small interval of time [t; t+dt]. The respective instantaneous acceleration is found by the derivative of the instantaneous speed VZ (t ) :

aZ (t ) 

dVZ (t ) d  dZ  d 2 Z (t )    2  Nost  N artst  b  B  dt dt  dt  dt 2

(64) The respective expression of the instantaneous speed is:

VZ (t ) 

dZ  2  N ost  N artst  b  B  t dt

(65)

Considering an idealized case when the acceleration is constant during the interval of time t, then the trajectory with such constant acceleration is shown on the Fig. 39. For the initial moment of time t=0 the initial speed is zero. During the interval of time t the speed is increased till the value VΔZ.

Fig. 39. The case of uniform accelerated movement (aΔZ = const) The starting speed of the initial moment of time is zero (VΔZ(0)=0). It is considered that for the initial moment of time (starting moment) the torrent of customers is null and therefore the initial speed of the system is null (VΔZ(0)=0). The "movement" of the system starts when the torrent of customers acts on the system. The speed of the system is increased equally in equal time intervals ( aZ =const).(Fig.39).

The acceleration of uniform accelerated movement for this case is found by the expression:

aZ 

VZ (t )  VZ (0) VZ (t )  t t

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The average speed of the movement <VΔZ> during the interval of time t is determined by the expression:

 VZ 

Z aZ  t 2 aZ  t VZ (t )    t 2t 2 2

(67)

The case of uniform accelerated movement is an ideal case and this case can be observed sometimes for some intervals of time. Taking into consideration the expressions (63) and (67), then:

Z  b  B  Nost  Nartst  t 2  VZ  t;   Vz  b  B  Nost  Nartst  t (68) According to the expression (68) it can be seen that the average speed is a function of time. The exponential rate constants are also function of time due to of seasonal character of the stocks. Then, the expression of the average speed can be written as: (69)  Vz (t )  b(t )  B(t )  Nost  N artst  t The value of average speed is changeable complexly with time. Sometimes, it could be decreasing or increasing as the dependence of the values of the exponential rate constants b and B. In general the "trajectory" ΔZ(t) could be with the increasing shape with the fluctuations that are related to the seasonal character (Fig.40)

Fig. 40 The example of the "trajectory" as the function of time In this case the values of average speeds of each interval of time are different. Each interval of time is considered from the origin of coordinates.

 VZ (t1 ) 

Z1 Z 2 Z 6 ;  VZ (t2 )  ;.......................;  VZ (t6 )  t1 t2 t6 (70)

 VZ (t1 )  VZ (t2 )  ....  VZ (t6 ) 

(71)

The calculation of the average speed for the entire interval of time till the value t 6 (Fig. 40) is performed as follow:

Z  Z 5 Z1 Z 2  Z1  t1   (t 2  t1 )  ... 6  t6  t5   V1  t1   V2  (t 2  t1 )  ......  V6  t6  t5  t1 t 2  t1 t 6  t5  VZ (t6 )    t6 t6 

Z 6 t6

here <V1>, <V2>,<V3>,...., <V6> are the respective average speeds for the intervals of time [0; t 1], [t1; t2], [t2; t3],...., [t5; t6].

So, the average speed for the entire interval of time is determined only by the last value of ΔZ6 and t6.

Taking into consideration the expressions (64) and (69), then:

aZ (t )  2  Nost  N artst  b(t )  B(t ) 

2  VZ (t )  t

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The expression (72) gives very important statement, that the acceleration never equals to zero. The value of

aZ  0 only for the case when the average speed  Vz (t )  remains almost constantly all time (  Vz (t )  const ) and the interval of time t   . The pure physical mechanics states that if the acceleration

speed of the movement is constant all time then the acceleration is zero. Here, for the case of mechanics econophysical conception, the acceleration has the tendency of gradually decreasing if the average speed is almost constant all time. This is the principally difference between the development of mechanics econophysical conception and pure physical mechanics.

There are following situations that can take place (Fig.41). The first situation is referring to the stable and fast increasing of ΔZ with time. This situation can takes place as the result of suddenly increasing of demand. The average speed of the entire interval of time is high. The respective acceleration of this interval of time is high.

The values of ΔZ are almost arranged on the straight linear function with the exception of small fluctuations of ΔZ for the second case (Fig. 41). The value of average speed is almost constant for the entire interval of time.

The acceleration has the tendency to reach zero value for the case of very big interval of time with the condition that the average speed is constant with time. This qualitative description of the behavior of the numerical values of acceleration allows to differentiate substantially its meaning from pure classical mechanic and econophysical one. It suggests that the proper change of the value of acceleration denotes a change of the value of the average speed. The change of the value of the average speed denotes in its turn the change of the amount of products that are sold during some fixed interval of time. Only the case of the stable amount of sold products with the exception of small fluctuations leads to the limit value of zero of the acceleration when the interval of time is very big. Strictly, the zero value of acceleration could be only for the case when the selling activity is not started yet.

Fig. 41. The possible real cases of sold amounts ΔZ with time Regarding the fourth case (Fig.41), the saturation is reached more quickly at more low level than of the previous case and the respective average value of the speed is lower than the previous case. The smaller value of the acceleration till zero is reaching at the more shorter interval of time in comparison with the previous one.

The fact that the speed shows the amount of sold products per one unit of the interval of time and the respective acceleration is the indication of the change of the speed for the different moments of time, then the criteria of the stability of the stocks systems can be discussed on the base of the results that are represented on the Fig. 42.

Regarding the case 1, Fig. 42, it can be seen that the value of ΔZ of the sold products is established at the some level of the "saturation". The respective values of the speeds of the selling and of acceleration are gradually decreasing. Such situation could be referred to the case of the instability of the systems. It could be the case when the expenses for the liabilities are bigger than of the assets. This is the situation when the financial resources are not enough to restore the initial levels of the inventory.

Referring to the case 2, then the values of ΔZ are arranged almost on the straight line. The proportional coefficient of the correlational linear dependence is the average speed. It is almost constant all time with the exception of small fluctuations. The respective acceleration is decreasing gradually with time. If there are no any saturation all time, then it means that the system is stable, nevertheless that the value of acceleration is decreased. It is more important to mention that the financial resources must be enough for all expenses.

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More interesting situation is for the case 3, Fig. 42. Here the increasing of Î&#x201D;Z has some thresholds that are related to the seasonal character. The speed at first is increasing, then it is decreasing for the case when it reaches some level of "saturation" for the interval of time of about 30-th till 50-th months. After that, the speed is increasing but the acceleration is decreasing. It just can be stated that the value of the speed is more important parameter than of the acceleration. It has more prerequisite for the description of the stability than of the acceleration. The criteria of stability could be formulated as follow. The system is stable when the speed is at least constant with time or is increasing with time. So, the behavior of acceleration with time is not an enough criterion of the formulation of the stability. It only shows the criterion of the behavior of the speed with time.

The case 4 is similar to the first case. It is necessary to mention that the interval of time of the keeping of the level of "saturation" could reach such critical value of the time when the financial resources will not be enough to recover all expenses. Therefore the special dynamics econophysical description could give the answer which interval of time in the conditions of the decreasing of the speed and of acceleration gives the criterion of the default of the system.

The case 5, Fig. 42 shows the increasing both of the speed and the acceleration. This situation could be for the case of gradually increasing of the demand. The financial resources are increasing and various types of assets and liabilities can be largely utilized.

The difference between third case and sixth case shows that in the sixth case the acceleration is increasing for the last interval of time than in comparison with the third case.

Fig. 42 Examples of the sold products Î&#x201D;Z as the dependence on time

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The results of the calculations of the speeds and accelerations of each case are presented on the Table 1.

Resulting all that is described, then the acceleration is really an important characteristics that describes the processes of turnovers. The application of the expression (15) would give the possibility to estimate the time t of the selling of some quantity of the articles.

 P  p Z  K  N art ;   P  p 

t

Then:

Cases

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

aZ (t )  t 2 2  K  N art 2  K  N art  t  K  N art ; t 2   2  P  p aZ (t )  P  p 2  VZ  K  N art ;  P  p   VZ 

Table 1. The results of the speeds and accelerations Average speed Interval of time; Interval of ΔZ;  Vz (t )  ; months products (products/month) 2 1300 650 9 2900 322.22 23 3330 144.78 72 3400 47.22 2 4000 1689.4 9 16000 1689.4 23 38000 1689.4 72 120000 1689.4 21 15800 752.38 31.8 58000 1823.9 48 68000 1416.67 72 120000 1666.66 18.5 9300 502.7 21 15000 714.28 25.8 20000 775.19 48 26000 541.67 72 27000 375 18 20000 1111.11 48 300000 6250 60 715000 11916.67 72 1500000 20833.33 7 40000 5714.28 15.5 410000 26451.62 54 660000 12222.22 72 1515000 21041.67

(73)

Acceleration aZ (t ) ; (products/months2) 650 71.6 12.58 1.31 1689.4 375.42 146.9 46.92 71.65 114.71 59.02 46.29 54.34 68.02 60.09 22.57 10.42 123.45 260.04 397.22 578.7 1632.65 3413.11 452.67 584.5

So, the result of the time t that is necessary for the selling of some inventory depends on the amount of the articles Nart , average price of one product <Pp>, average speed <VΔZ>:

t

K  N art ;  P  p   VZ 

(74)

Bigger is the average speed <VΔZ>, then the shorter is the time t for the selling of the given amount of stocks articles Nart. Smaller is the quantity of the sold articles Nart, then the shorter is the time t for the selling at the fixed value of the average speed <VΔZ>, and all of these statements have logical interconnection. Here, according to the expression (74) there is an interesting paradox that if the higher average price <Pp>, then the shorter is the time of the selling. The explanation of this moment which seems that is paradox is of such order: The demanded products that have really more higher demand then the time is shorter for this high demand. Higher is the demand, then the higher is the price of the product.

Considering that

aZ (t ) 

2  VZ  2  VZ  ; t , then the expression (74) is transforming as: t aZ (t )

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2  P  p  VZ   2  VZ  K  N art  ;  aZ (t )  aZ (t )  P  p   VZ  K  N art

2

(75) The quantity of the sold inventory with time is determined by the factor

K (t )  N art (t ) . Bigger is the sold

inventory K (t )  N art (t ) , then the smaller is the acceleration aΔZ. Then the following formulation of the interconnection between the acceleration and the sold inventory is: The continuous decreasing of the acceleration with time is the indicator of the continuous increasing of the sold inventory.

The conception of the equation of marketing state related to the interval of time t by the expression (74) can be more better understood by the following example.

Example 5. Let an amount of articles Nart=2000 (2000 various names of articles) are sold during some interval of time. The average price of one product is 4$. Considering that the speed of the selling is <VΔZ>=200 (products/day) and the rating coefficient of the stocks is K=5.65 ($ /article), calculate how long estimative time is necessary to sell all articles.

Solution: The expression of the time:

t

K  N art 5.65  2000 56.5    14(days )  P  p   VZ  4  200 4

The respective amount of articles sold during one day is:

2000  143(articles / day ) . This result satisfies 14

the possible real average daily turnover of microeconomical system of stocks.

Another important that is related to the notion of acceleration is the method of natural logarithm of acceleration, that will give the possibility to find by this method the average speed. If the value of the acceleration depends with time by the expression aZ (t )  2  VZ (t )  , then the natural logarithm of acceleration as a function of t time must have a form of the linear dependence as follow (Fig. 44).

Fig. 44 The theoretical dependence of

ln aZ  ln 2  VZ   ln t

The crossing of the straight line with the ordinate axis will give the value of ln 2  VZ  .

The average value of the acceleration for the interval of time [a, b] could be found by the expression: b

1  az   aZ (t )dt b  a a The respective graphic of the expression aZ (t ) 

2  VZ  is represented on Fig. 45. t

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Fig. 45 Calculation of average value of acceleration on the interval of time [a, b] b 1 1 2  VZ  dt 2  VZ  2  VZ   b   az    aZ (t )dt    ln t |   ln    a b  a  ba a ba a t ba a b

b

Here, in this expression of the calculation of the average acceleration, there is a peculiarity of such order that for the initial moment of time a=0 there is a result of ln(a)→-∞ and the respective average value of acceleration is ∞. Therefore, it would be better to make such approximation of the order that the initial time could be considered as a=1 (day). Then, the expression of the average value of acceleration could be written as:

 az 

2  VZ   b  2  VZ  ln b  ln 1 2  VZ   ln b  ln     b 1 b 1 b 1 1 2  VZ   ln t  az (t )  ; t 1 t 1

(76)

It is important to mention regarding the application of expression (76), that if the value of t is measured in days, then the value of t must be t >1 day, but if the months are applied then t >1month.

For the limit case when t=0, then:  az (0)  lim t 0

2  VZ   ln t () , and this is an  2  VZ   t 1 1

indetermination. This indetermination could be solved by the application of L'Hospital's rule: the derivatives of numerator and denominator.

2  VZ  ln t  1  2  VZ   lim   / t 0 t 0 t t  1 The graphic of the dependence of  az (t )  f (t ) is represented on the Fig. 46. /

 az (0)  lim

(77)

Fig. 46 The dependence of average acceleration  az (t )  on time Considering that

aZ (t ) 

2  VZ  , then the expression (76) is transforming as: t

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 az (t ) 

2  VZ   ln t aZ (t )  t  ln t  aZ  (t  1)  ;  aZ (t )  t 1 t 1 t  ln t (78)

 (t  1)   . It gives the  t  ln t 

The expression (78) allows to represent the graphic of the dependence of aZ (t )  f 

possibility to observe the respective segments of the time where the average acceleration is "constant". The Fig.

 (t  1)   . It contains two segments. The initially values  t  ln t 

47 shows schematically the dependence of aZ (t )  f 

 (t  1)   , but high values of time are corresponding to small values of  t  ln t 

of time correspond to high values of 

 (t  1)    . The respective slope for the high values of time has smaller average acceleration, and bigger average  t  ln t  acceleration corresponds to the smaller values of time (Fig. 47).

The average acceleration for the initial times is found by formula:

acceleration of the slow segment of time is:

 a 2 

 a 1 

a1 and the respective average x1

a2 . For the convenience it would be better to use x x2

 (t  1)  .  t  ln t 

instead of 

 (t  1)    t  ln t 

Fig. 47 The dependence of aZ (t )  f 

The respective similar dependences could be represented for all marketing groups.

Experimental confirmation of the kinematics econophysical model 1. The determination of the acceleration by the expression aZ (t )  2  VZ (t )  . The validation of t kinematics econophysical model In order to calculate the accelerations of each marketing groups A, B, C, X, Z it is necessary to have the information about the sold products ΔZ with time. The graphic of the dependence of ΔZ as a function of time is represented on the Fig. 47 for each respective marketing group A, B, C, X, Z till 72 months. The points that are corresponding to the values of sold products ΔZ are almost arranged on the straight line. The correlational calculations show a strong linear dependence. The proportional coefficients of the linear dependence are the

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Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems .. values of the average speeds. The respective average speeds are calculated for each marketing rating group. The Table 2 contains the calculated values of accelerations with time.

Table 2. The values of accelerations for each marketing group Marketing group

A

B

C

X

Z

Entire system

Average speed <VÎ&#x201D;Z>; (products/month)

t; interval of time; month

786 786 786 786 786 786 786 522 522 522 522 522 522 522 357 357 357 357 357 357 357 479 479 479 479 479 479 479 92 92 92 92 92 92 92 2235 2235 2235 2235 2235 2235 2235

10 20 30 40 50 60 72 10 20 30 40 50 60 72 10 20 30 40 50 60 72 10 20 30 40 50 60 72 10 20 30 40 50 60 72 10 20 30 40 50 60 72

aď &#x201E;Z (t ) ; (products/month2) 157.2 78.6 52.4 39.3 31.44 26.2 21.83 104.4 52.2 34.8 26.1 20.88 17.4 14.5 71.4 35.7 23.8 17.85 14.28 11.9 9.92 95.8 47.9 31.93 23.95 19.16 15.97 13.3 18.4 9.2 6.13 4.6 3.68 3.06 2.55 447 223.5 149 111.75 89.4 74.5 62.08

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When the acceleration is calculated for the entire system of the stocks than the respective acceleration at the given moment of time is the sum of the accelerations of each separated marketing group (Table 3). This is a Z

property of additively:

aZ (t )   aZ j (t ) jA

Table 3 The property of additivelly of acceleration t; interval of time; months

Values of accelerations aΔZ ( t ); (products/month2) A

B

C

X

Z

157.2 78.6 52.4 39.3 31.44 26.2 21.83

104.4 52.2 34.8 26.1 20.88 17.4 14.5

71.4 35.7 23.8 17.85 14.28 11.9 9.92

95.8 47.9 31.93 23.95 19.16 15.97 13.3

18.4 9.2 6.13 4.6 3.68 3.06 2.55

The values of accelerations are calculated by the expression

aZ (t ) 

10 20 30 40 50 60 72

the full system 447 223.5 149 111.75 89.4 74.5 62.08

Sum of each group A+B+...+Z 447.2 223.6 149.06 111.8 89.44 74.53 62.1

2  VZ (t )  and the respective values t

az (t )  t 2 . The real values and the calculated values of ΔZ 2 az (t )  t 2 are represented on the same plot of the Fig. 48. The calculated values of Z (t )  are arranged on 2 of ΔZ are re-calculated by the expression

Z (t ) 

the straight line that coincides almost with the real values that are situated on the straight line. The Fig. 48 contains the graphics only for the group A and the full group.

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Fig. 47 The dependence of sold products ΔZ with time for each marketing groups The similar dependence is for other groups B, C, X, Z. In such way the idea of kinematics mechanical econophysical model is validated and confirmed.

Fig. 48 The confirmation of the kinematics mechanical econophysical model of the stocks Another validation of the kinematics mechanical econophysical model of the stocks is the representation of the graphic Z calc  f (Z ) .(Fig.49) It must be a straight line with the proportional theoretical coefficient equals to one.

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Fig. 49 The confirmation of the validation of kinematics econophysical model of the stocks

ln aZ  ln 2  VZ   ln t that is represented on the Fig. 50. The correlational calculations shows that ln 2  VZ   7.3598 for the marketing group A, and ln 2  VZ   8.405 for the full system. (Table 4) The average speed <VΔZ> also could be found by the graphic of

Table 4 The calculated average speeds by natural logarithm of acceleration Marketing group ln 2  V   V ; ( products / month) Z

A Full system

Z

7.3598 8.405

785.76 2234.68

The results of the speeds that are presented on the Table 4 coincide with the results of the speeds from the Table 2.

Fig. 50 The dependences of accelerations and their natural logarithms as a function of time 2. The determination of the average acceleration The

application

of

the

expression

(76)

of

the

determination

of

average

acceleration

2  VZ   ln t  az (t )  ; t  1 , gives the possibility to represent the graphic of the dependence of t 1  az (t )  f (t ) . (Fig. 51)

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It could be seen from the Fig 51 that the average acceleration is highest for the marketing group A. The values of average accelerations are decreasing gradually with time. First they are decreased fast for the first months and starting from fifth month they are almost at the same numerical level.

Fig. 51 The dependence of average acceleration on time for different marketing groups In order to observe the segments of the time where the average accelerations could be constant it would be better

 (t  1)   (Fig. 52)  t  ln t 

to represent the graphic of the dependence of aZ (t )  f 

 (t  1)   for all marketing groups  t  ln t 

Fig. 52 The graphic of the dependence of aZ (t )  f 

The Table 5 contains the information about the calculated values of average accelerations. The exact signs like as for the Fig. 47 are applied.

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 (t  1)    t  ln t 

Table 5 The calculations of average acceleration by the aZ (t )  f  Marketing group

Δx1

Δx2

Δa1

Δa2

A B C X Z Full system

0.256 0.25 0.255 0.259 0.281

0.145 0.152 0.115 0.17 0.15

550 350 241 321 66

0.258

0.145

1480

220 75 48 81 13.9

<a>1; (products/mont h2) 2148.43 1400 945.09 1239.38 234.87

<a>1; (products/day2 ) 2.38 1.55 1.05 1.37 0.26

<a>2; (products/month 2 ) 1517.24 493.42 417.39 467.47 92.66

<a>2; (products/day 2 ) 1.60 0.548 0.46 0.529 0.103

315

6561.02

7.29

2172.41

2.41

The last results of about average accelerations for the full system of stocks present a great interest. It is a possibility to calculate the full average of the acceleration <a>f for the total interval of time by the expression:

 a f 

x1  a 1 x2   a  2 ; x1  x2

 products  0.258  7.29  0.145  2.41 1.881  0.349    5.53 2 0.258  0.145 0.403  day   products   is very This result of average acceleration for the entire system of stocks  a  f  5.53 2  day   a f 

important and very interesting. It coincides somehow with the results of the econophysical temperature K=5.65 presented in the paper [10] of the full system of the model of "ideal gas", with the rating coefficient of the stocks KABCXZ =5.65 presented in [13] and with the value of econophysical gravitational acceleration 5.65 for the suggested econophysical gravitational model that is presented in [47].

Such important results lead to the statement that the big ensemble of stocks articles will give one and the same result within the limits of errors of exactitudes without any dependence of national currencies of the countries.

So, supposing that for some moment of time the acceleration coincides with the value of the rating coefficient of the stock, then the equation of the state of microeconomical system of stock is written as:

 P p 

aZ  t 2 t2  K  N art ; aZ  aZ  K  5.65;   P  p   N art ; 2 2

Suppose, for an example Nart=4000; <P>p=10 $; then

t

 t

2  N art  P p

2  4000  800  28.28(days ) is estimate 10

necessary time to sell these articles.

On the other hand, the equation of the state of microeconomical system, could be written also as:

 P p 

aZ  t 2 2  VZ  t 2  K  N art ;   P  p  K  N art ;   P  p   VZ  t  K  N art 2 2t

; and for the case when K=5.65, then: So,

both

results

of

the

t

interval

K  N art 5.65  4000   1.02(month)  30.6(days ) ;  P  p   VZ  10  2235 of

time

are

almost

coincident.

The

average

time

is

28.28  30.6  t   29.44(days ) ; The error of the time is: t  29.44  30.6  1.16(days ) ; 2 t 116 100%   3.94% . This obtained result of relative error means that The relative error  %  t  29.44 the model conception of the econophysical gravitational acceleration is valid and can be applied.

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IV.

Conclusions

Complex economic studies evolve dynamically in time. Permanently the statistical methods of the data processing are optimized together with the integrative applications of the scientific platform of econophysics.

The present methodical work can be optimized and performed in time and with the widening of the spectrum of concrete examples. The most important moment is that the value of the econophysical acceleration coincides with the values of the rating coefficients of the stocks (econophysical temperatures) and is an universal one world widely of the microeconomical systems.

References [1].

[2].

[3].

[4].

[5].

[6].

[7].

[8].

[9].

[10].

[11].

[12].

[13].

[14].

[15].

[16].

[17].

[18].

[19].

[20].

[21].

[22].

[23].

[24].

[25].

[26].

[27].

Interview of H. E. Stanley on Econophysics (Published in "IIM Kozhikode Society & Management Review", Sage publication (USA), Vol. 2 Issue 2 (July), pp. 73-78 (2013)) 2. Econophysics Research in India in the last two Decades (1993-2013), (Published in "IIM Kozhikode Society & Management Review", Sage publication (USA), Vol. 2 Issue 2 (July), pp. 135-146 (2013)) Gheorghe Săvoiu, Ion Iorga Simăn “History and Role of Econophysics in Scientific Research”, Econophysics, 2013, R. Mantegna, H. E. Stanley, “An introduction to econophysics, correlations and complexity in finance”, Cambridge University Press, (2000) A. A. Dragulescu and V. M. Yakovenko "Statistical mechanics of money", The European Physical Journal B, v. 17, pp. 723-729 (2000), A. Jakimowicz, “Econophysics as a New School of Economic Thought: Twenty Years of Research”, Proceedings of the 8th Polish Symposium of Physics in Economy and Social Sciences FENS, Rzeszów, November 4–6, 2015 Водолазский А., А., "Начала эконофизики и количественная определенность первых экономических законов", Новочеркасск: “НОК”, 2013. - 227 с.. 2013 Victor M. Yakovenko, “Econophysics, Statistical Mechanics Approach”, Department of Physics, University of Maryland.

P. Richmond, J. Mimkes, and S. Hutzler, “Econophysics and Physical Economics” (2013) Print ISBN-13: 9780199674701, Published to Oxford Scholarship Online: December 2013 M. Petrov “The Thermodynamical Processes of the Model of Ideal Gas for the Description of the Activity of the Microeconomical Systems”, European Journal of Business and Management, Vol 10, No 26 (2018) Pareto V (1897), Cours d’Economie Politique. ´ L’Universit´e de Lausanne.

“The 80-20 rule: Apply Pareto’s principle to hospital medicine”, www.kevinmd.com, december 9, 2014 M. Petrov, V. Petrova, VI-th Congress of Pharmacy with International Participation, Bulgaria, Sandanski, October 13-16, The study of the principle of Pareto in the pharmaceutical activity (2016).

Jason Marshall, “What is the Golden Ratio and how is it Related to the Fibonacci sequence?” May 5, 2010, https://www.quickanddirtytips.com.

Ion Spanulescu, Anca Gheorghiu, "Trends in econophysics", www.enec.ro, proceedings - archive.

"Volatility", Masterinvest.com "The Mathematical Principles of Natural Philosophy", Encyclopædia Britannica, London A. Mas-Colell, "Microeconomic Theory", Oxford University Press, USA (1995) W. Nicholson, "Microeconomic Theory: Basic Principles and Extensions", South Western Educational Publishing, (2004) U. Garibaldi, E. Scalas, "Finitary Probabilistic Methods in Econophysics", Cambridge University Press, (2010) G. Savoiu, "Econophysics: Background and Applications in Economics, Finance, and Sociophysics",by Academic Press (2012) B. G. Sharma, Sadhana Agrawal, Malti Sharma, D. P. Bisen , Ravi Sharma, "Econophysics: A Brief Review of Historical Development, Present Status and Future Trends", Cambridge University Press, (2010) Подиновский В.В., Ногин В.Д. Парето-оптимальные решения многокритериальных задач, Физматлит, p. 14-17 (2007).

A. Chernev, Strategic Marketing Management, 8-th edition, Cerebellum Press, USA, p. 5-9, (2014) D. Bowman, H. Gatignon, Marketing Response and Marketing Mix Models, Foundations and Trends(r) in Marketing, p.11-18 (2010) Невяна Кръстева, "Фармацевтичен маркетинг", Авангард Прима, p. 34-39 (2014)

International Journal of Business Marketing and Management (IJBMM)

Page 50

Mechanics Phenomenological Econophysics For The Description Of Microeconomical Systems ..

[28].

[29].

[30].

[31].

[32].

[33].

[34].

[35].

[36].

[37].

[38].

[39].

[40].

[41].

[42].

[43].

[44].

[45].

[46].

[47].

[48].

[49].

[50].

[51].

[52].

[53].

[54].

[55].

Златка Димитрова, Илко Гетов, "Основи на аптечна практика и бизнес", Ун. Изд. "Свети Климент Охридски", p. 23-25 (2009) M. Petrov, III-rd International Conference of Econophysics, Greece, Volos, September, 26, probabilistic and thermodynamical approximation of the model of ideal gas for the microeconomical description of ABC marketing analysis, (2017) Chris Caplice ESD.260/15.770/1.260, Logistics Systems Sept 2006, Supply Chain Fundamentals and Segmentation Analysis, p. 1-27 (2006) François Geerolf, A Theory of Pareto Distributions, UCLA October 2016, p.1- 48 (2016) Пиндайк Р., Рабинфельд Д. Микроэкономика. / Пер. с англ. СПб.: Питер, 608 с: ил. (Серия «Учебники для вузов») (2002) Mankiw, N.G.; Taylor, M.P. Economics (2nd ed., revised ed.). Andover: Cengage Learning. p. 24-28 (2011) "Measurements and Error Analysis", College Physics Lab, The university of North Carolina, https://www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html I. I. Gikhman, A. V. Skorokhod, "Introduction to the Theory of Random Processes", Courier Corporation, (1969). p. 1. ISBN 978-0-486-69387-3.

Mosanao Aoki, Hiroshi Yoshukawa, "Demand saturation-creation and economic growth", Journal of Economic Behaviour and organization, Vol. 48, (2002).

M. Hook, Li Junche, Noriaki Oba, S. Snowden, "Descriptive and predictive growth curves in energy system analysis", Natural Resources Research,Vol. 20, issue 2, (2001), pp. 103-116 N. Fokas, "Growth functions, social diffusion and social change", Review of Sociology, Vol. 13, (2007), 1, pp.5-30 R. K. Hobbie, B. J. Roth, "Exponential growth and decay", Intermediate Physics for Medicine and Biology, (2015), XX, pp. 623-636 J. Grashman, G. Van Straten, "Predictability and Nonlinear Modelling in Natural Sciences and Economics", 1994 Gary Lilien, Arvind Rangaswammy, Arnaud De Bruyn, "Principles of Marketing Engineering", (2007) , pp. 36-45 R. Begum, S. K. Sahu, R. R. Sahoo, "An inventory Model with Exponential Demand Rate, Finite Production Rate and shortages", Journal of Scientific Research, 1, (3), pp. 473-483, (2009) M. Ekramol Islam, Shirajul Isla Ukil, Md. Sharif Uddin, "A time dependent Inventory Model for Exponential Demand Rate with Constant Production", Open Journal of Applied Sciences, (2016), 6, pp.

36-48 M. Abramowitz, Irene A. Stegun, "Handbook of Mathematical Functions with Formulas, graphs and mathematical tables", USA, National Bureau of Standards, Applied Mathematical Series, 55, (1972) D.Simchi-Levi, S. David Wu, Z. Jun Shen, "Handbook of quantitative supply chain analysis, Modelling in the E-Bussines Era", London (2004) Kee Kuo Che, Ching-Ter Chang, "A seasonal demand inventory model with variable lead time and resource constraints", Applied Mathematical Modelling, Vol. 31, issue 11,(2007), pp.2433-2445. www.export.gov/article/Bulgaria-Pharmaceuticals 8/23/2017 Mihai Petrov, "Econophysical gravitational mechanical model of microeconomical systems of stocks", 12-th Edition of International Conference, ENEC - 2019, Romania, 1 June, Econophysics, New economy and Complexity.

Isard, Walter, "Location Theory and Trade Theory: Short-Run Analysis". Quarterly Journal of Economics. 68 (2): 305–320. doi:10.2307/1884452. JSTOR 1884452. (May 1954) Nicholas Alexander, Mark Rhodes, Hayley Myers, (2011) "A gravitational model of international retail market selection", International Marketing Review, Vol. 28 Issue: 2, pp.183200, https://doi.org/10.1108/02651331111122669 Christine Evans-Pughe, "We should treat money like energy...", Engineering and Technology Magazine, volume 6, issue 6, 13 June 2011, pages 41-43, publication of the UK Institution of Engineering and Technology.

Rachael A. Lancor, "The Many Metaphors of Energy: Using Analogies as a Formative Assessment Tool", Journal of College Science Teaching, Vol. 42, No. 3 (January/February 2013), pp. 38-45 52. John Roe, Russ deForest, Sara Jamshidi, "Mathematics for Sustainability, Text for quantitative critical thinking", (2018), Springer. https://www.coursehero.com/file/19942659/Slides-Day-9/ , Slides_Day_9 - "An analogy between energy and money Work..."

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