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# Given mean insolation, can one place an upper bound on the daily peak insolation?

Photovoltaic panels are rated under standard conditions: eg a $100\mathrm{W}$ panel if irradiated with $1000\mathrm{W}/\mathrm{m}^2$ at $25^\circ\mathrm{C}$.

When the cost effectiveness of photovoltaics is discussed one is often presented with daily irradiation maps averaged yearly or possibly monthly. This is fine for cost effectiveness calculations.

But in designing the power conversion electronics or effects of grid injection one cannot work with average values - they must be able to sustain peak powers that are possibly larger than the nominal power of the panels, right?

If so, by what factor are these electronics oversized? I cannot find any similar maps of the average daily peak irradiance.

Given the daily solar energy incident ($\mathrm{kW hr /m^2\cdot day}$) averaged either annually or monthly, can one place an upper bound on the daily peak irradiance ($\mathrm{W / m^2}$)?

I am particularly interested in Mediterranean countries.

You're absolutely right that annual mean power is by no means a useful guide to the peak power a panel might generate. You can find places on Earth where annual mean insolation is well below $100~W/m^2$, and places where it's higher than $250~W/m^2$: but at both those places, there will be times when peak insolation is $1000~W/m^2$ or more. And so all PV is rated to that peak insolation, at the very least. So, mean annual insolation (whether in $W/m^2$ or $kWh/m^2\cdot day$) is no guide at all to peak insolation.

For more on the peak rated power of the panels, let's look more closely at specific technologies. PV comes in two distinct flavours, one is concentrating PV, the other is plain PV. They're very different beasts, so I'll treat them differently.

## Concentrating PV

This is designed to sit behind lenses which concentrate sunlight. The cell and the electronics are rated at much much higher insolation levels: the highest-efficiency cell according to the 2013 January review of solar cell efficiency records had an efficiency of $44.0\pm{3}\%$, and that was measured at 942 peak suns: i.e. $942~000~W/m^2$

## Plain PV

Panels are rated under very specific test conditions: 25° Celsius; light levels of $1000~W/m^2$; and a spectrum that represents the sun having passed through an air mass with an index of 1.5 (AM1.5).

Irradiation of the panel can be higher than $1000~W/m^2$: if the sun's rays are normal to the panel, and there's a lot of ambient light, those will add, and insolation will be above $1000~W/m^2$.

If the temperature is below 25° Celsius, then panel output will be higher: each panel has a coefficient that you can usually get from the manufacturer's data sheet, which represents the % change in output for each one degree change in temperature: the lower the temperature, the higher the output. Different panel technologies have different factors:: the First Solar Series-3 thin-film panel has a temperature coefficient of maximum peak power of $-0.25\% / ^oC$, so a reduction of 20°C below standard test conditions will give an extra 5% power.

At higher altitudes, the air mass is lower, and that will give higher insolation levels in direct beam. You're also above the snow line, which means it's colder, and there's a lot of reflected light from the snow, as well as a lot of light from the direct beam. And people do take PV to the top of very high mountains, and the electronics have to survive. So they're typically rated to a much higher power than the panel's nominal rated output.

The electronics are typically very cheap, compared to the rest of the panel's components, so typically they'll be very highly over-sized, to account for all sorts of extreme operating conditions. A full PV array on a mountain in Switzerland was recording power at 130% of rated output without any problems.


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# Differential Equations . When storage elements such as capacitors and inductors are in a circuit that is to be analyzed, the analysis of the circuit will yield differential equations. This section will deal with solving the types of first and second order differential equations which will be encountered in

Differential equations step by step. sinus sin(x), cosine cos(x), tangent tan(x), cotangent ctan(x) exponential functions and exponents exp(x)

2ttci. 3. 3 where /, g  cos ax dx = a sin ax − a sin ax p. Item should read. ∫ x sin ax dx = − x a cot ax + a CHAPTER – ORDINARY DIFFERENTIAL EQUATIONS p. Item should  Why is mg sin theta in the x-sum for an inclined plane Foto Ordinary Differential Equations (Updated 9/6/10) Foto. Gå till.

2. = +. + q Differential and integral calculus sin x cos x cos x sin. - x tan x x. 2. 2 cos.

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## The order and degree of the differential equation in sinx(dx+dy)=cosx(dx−dy) are: A. 1,1. B. 0,0. C. 1,2. D. 2,1. Easy. Answer. Correct option is. A. 1,1

What is their order? d7F dx7. = 3F(x), y.

### cosh(x) = ex + e − x 2 sinh(x) = ex − e − x 2 So, another way to write the solution to a second order differential equation whose characteristic polynomial has two real, distinct roots in the form r1 = α, r2 = − α

DIFFERENTIAL EQUATIONS 379 vHe who seeks for methods without having a definite problem in mind seeks for the most part in vain. – D. HILBERT v 9.1 Introduction In Class XI and in Chapter 5 of the present book, we discussed how to differentiate a given function f with respect to an independent variable, i.e., how to find f¢(x) for a given function f at each x in its domain of definition. Click here👆to get an answer to your question  Solve the following differential equations: x sin [ yx ] dydx = y sin [ yx ] - x Solutions: Applications of Second-Order Differential Equations 1. By Hooke’s Law k(0.6) = 20 so k = 100 APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS 3 is the spring constant and the differential equation is 3x00 + 100 3 x = 0. ¡ 10 The general solution is x(t) = c1 cos 3 t ¢ + c2 sin ¡ 10 3 t ¢ .

Info. Shopping. Tap to unmute. If playback doesn't begin shortly, try restarting your device. Up Next. 2018-05-29 · Transcript. Misc 8 Find the equation of the curve passing through the point 0 , 4 whose differential equation is sin cos + cos sin =0 sin x cos y dx + cos x sin y dy = 0 sin x cos y dx = cos x sin y dy sin cos = sin cos Integrating both sides sin cos = sin cos sin sin = sin sin = log = + log + log = c log .
Fredrik gummesson

Polar Curves and Differential Equations.pdf from MATH CALCULUS at University of St Andrews. 1. Problem 3 Given: = sin + cos To simplify the problem, let’s prove that this is the Solve a System of Differential Equations. Solve a system of several ordinary differential equations in several variables by using the dsolve function, with or without initial conditions. To solve a single differential equation, see Solve Differential Equation.

Second Order Linear Nonhomogeneous Differential Equations with Substituting this back into the differential equation produces: \[ {- 4A\cos 2x – 4B\sin 2x (iii) The highest order derivative present in the differential equation is y′′′, so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined. EXERCISE 9.1 Determine order and degree (if defined) of differential equations given in Exercises 1 to 10.
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### Since y1/y2 = cot ωx, ω ≠ 0, is not constant, y1 and y2 are linearly independent. We therefore have the following general solution: y = e–ax/2 (A cos ωx + B sin ωx ).

-- cos au + C. +C a3 a u. Ekvationen/ The equation x2 + px + q = 0 har rötterna/ has the roots x1 = − p. 2.

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### av A Wu · 2009 — Proof. For any given ϕ ∈ B, we consider the following almost periodic differential Since x(t) and x∗(t) are solutions of Eq.(1.1), combining with (3.5)-(3.6), (H2) and sin t cos t sin t+sin 2t. 2 cos t cos t cos t cos t+cos 3t. 2 sin t..

D. 2,1. Easy. Answer. Correct option is. A. 1,1  5 May 2017 In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees.

## Differential equations are very common in physics and mathematics. Without their calculation can not solve many problems (especially in mathematical physics). One of the stages of solutions of differential equations is integration of functions. There are standard methods for the solution of differential equations.

Since f is even we need to consider the cosine series f(x) = a0. 2 The general solution of the differential equation is X(t) = a sin 2. √ λt + b cos  An ordinary differential equation or ODE is a differential equation containing a function or functions of one independent variable and its  2 sin ωt − 2 π.

The order of a diﬀerential equation is the highest order derivative occurring. The cos β leg is itself the hypotenuse of a right triangle with angle α; that triangle's legs, therefore, have lengths given by sin α and cos α, multiplied by cos β. The sin β leg, as hypotenuse of another right triangle with angle α, likewise leads to segments of length cos α sin β and sin α sin β. dx* (x^2 - y^2) - 2*dy*x*y = 0.

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Definition 24.25.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A diffential graded $\mathcal{A}$-module $\mathcal{I}$ is said to be graded injective1 if $\mathcal{M}$ viewed as a graded $\mathcal{A}$-module is an injective object of the category $\textit{Mod}(\mathcal{A})$ of graded $\mathcal{A}$-modules.

[1] This may be nonstandard terminology.

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## Electronic Communications in Probability

### Fractional Poisson field and fractional Brownian field: why are they resembling but different?

#### Abstract

The fractional Poisson field (fPf) is constructed by considering the number of balls falling down on each point of $\mathbb R^D$, when the centers and the radii of the balls are thrown at random following a Poisson point process in $\mathbb R^D\times \mathbb R^+$ with an appropriate intensity measure. It provides a simple description for a non Gaussian random field that is centered, has stationary increments and has the same covariance function as the fractional Brownian field (fBf). The present paper is concerned with specific properties of the fPf, comparing them to their analogues for the fBf. On the one hand, we concentrate on the finite-dimensional distributions which reveal strong differences between the Gaussian world of the fBf and the Poissonnian world of the fPf. We provide two different representations for the marginal distributions of the fPf: as a Chentsov field, and on a regular grid in $\mathbb R^D$ with a numerical procedure for simulations. On the other hand, we prove that the Hurst index estimator based on quadratic variations which is commonly used for the fBf is still strongly consistent for the fPf. However the computations for the proof are very different from the usual ones.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 11, 13 pp.

Dates
Accepted: 7 February 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315550

Digital Object Identifier
doi:10.1214/ECP.v18-1939

Mathematical Reviews number (MathSciNet)
MR3033594

Zentralblatt MATH identifier
1319.60105

Rights

#### Citation

Biermé, Hermine; Demichel, Yann; Estrade, Anne. Fractional Poisson field and fractional Brownian field: why are they resembling but different?. Electron. Commun. Probab. 18 (2013), paper no. 11, 13 pp. doi:10.1214/ECP.v18-1939. https://projecteuclid.org/euclid.ecp/1465315550

#### References

• Bassan, Bruno; Bona, Elisabetta. Moments of stochastic processes governed by Poisson random measures. Comment. Math. Univ. Carolin. 31 (1990), no. 2, 337–343.
• Biermé, Hermine; Estrade, Anne; Kaj, Ingemar. Self-similar random fields and rescaled random balls models. J. Theoret. Probab. 23 (2010), no. 4, 1110–1141.
• Breton, Jean-Christophe; Dombry, Clément. Functional macroscopic behavior of weighted random ball model. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 177–196.
• Beghin, L.; Orsingher, E. Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14 (2009), no. 61, 1790–1827.
• Chainais P., phInfinitely Divisible Cascades to Model the Statistics of Natural Images, IEEE Trans. Pattern analysis and Machine intelligence, 29, 12, 2105–2119, (2007).
• Cioczek-Georges, R.; Mandelbrot, B. B. A class of micropulses and antipersistent fractional Brownian motion. Stochastic Process. Appl. 60 (1995), no. 1, 1–18.
• Cohen, Serge; Taqqu, Murad S. Small and large scale behavior of the Poissonized Telecom process. Methodol. Comput. Appl. Probab. 6 (2004), no. 4, 363–379.
• Guyon, Xavier; LeÃ³n, José. Convergence en loi des $H$-variations d'un processus gaussien stationnaire sur ${\bf R}$. (French) [Convergence in law of the $H$-variations of a stationary Gaussian process in ${\bf R}$] Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 265–282.
• Heinrich, Lothar; Schmidt, Volker. Normal convergence of multidimensional shot noise and rates of this convergence. Adv. in Appl. Probab. 17 (1985), no. 4, 709–730.
• Istas, Jacques; Lang, Gabriel. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 407–436.
• Kaj, Ingemar; Leskelä, Lasse; Norros, Ilkka; Schmidt, Volker. Scaling limits for random fields with long-range dependence. Ann. Probab. 35 (2007), no. 2, 528–550.
• Kaj, Ingemar; Taqqu, Murad S. Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In and out of equilibrium. 2, 383–427, Progr. Probab., 60, Birkhäuser, Basel, 2008.
• Peccati, Giovanni; Taqqu, Murad S. Central limit theorems for double Poisson integrals. Bernoulli 14 (2008), no. 3, 791–821.
• Perrin E., Harba R., Jennane R. and Iribarren I. phFast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises, Signal Processing Letters IEEE, Vol.9 (1), 382–384, (2002).
• Sato, Yumiko. Distributions of stable random fields of Chentsov type. Nagoya Math. J. 123 (1991), 119–139.
• Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman &Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
• Stoyan, D.; Kendall, W. S.; Mecke, J. Stochastic geometry and its applications. With a foreword by D. G. Kendall. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. 345 pp. ISBN: 0-471-90519-4
• Takenaka, Shigeo. Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991), 1–12.


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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.1 pp.103-119
DOI : https://doi.org/10.7232/iems.2020.19.1.103

# The Development of Small and Medium-Sized Enterprises in the Modernization of Industrial Production in the Case of the Power Complex Enterprise

Galiya S. Ukubassova*, Kalyash K. Primzharova, Adaskhan K. Daribayeva, Alma H. Galiyeva, Almagul Sh. Nurgaliyeva
Department of Economics, Kazakh University of Economics, Finance and International Trade, Nur-Sultan, Republic of Kazakhstan
Department of Public Administration, Narxoz University, Almaty, Republic of Kazakhstan
Department of International Trade and Law, Kazakh University of Economics, Finance and International Trade, Nur-Sultan, Republic of Kazakhstan
Department of Management and Tourism, Kazakh University of Technology and Business, Nur-Sultan, Republic of Kazakhstan
*Corresponding Author, E-mail: gal-ukub@murdoch.in
November 13, 2019 January 11, 2020 January 21, 2020

## ABSTRACT

Small enterprises cannot organize significant money holdings and thus the modernization becomes a project of longterm financial and economic planning. The success of the project implementation can affect both individual segments of the market and the market in general. The purpose of the article is to illuminate the planning methodology, show its stages and form an analysis of possible deviations from the set goals, as well as determine how to correct the declared changes. As research methods, the authors applied simulation technology to solve the problems of resource allocation. In result of research analysis model was developed and a verification mechanism was provided. The practical significance of the research is determined by the possibility of forming a high-quality component that will make it possible to create an instrument for the development of small and medium-sized enterprises in the regions. It is consistent with the goals of the financial and economic development of the state and with programs aimed at stimulation of entrepreneurial activity.

## 1. INTRODUCTION

For any enterprise that relates to the field of small or medium-sized business, the smooth functioning is extremely important, which, in addition to the entrepreneurial initiative, is also involved in the formation of finished products (Kingsley and Klein, 1998). In this regard, the task of supporting the enterprise, its development on the basis of entrepreneurial initiative, which in turn forms the possibility of modernization and gradual development in a competitive environment, becomes urgent. It is nec-essary to analyze the processes supporting the activities of the enterprise in the context of its subsequent modernization (Petukhova, 2011). One of the driving elements of the organization of production in industrial enterprises is their logistic support. In the current economic environment of industrial enterprises, the efficiency and optimality of each process are extremely important (Toxanova et al., 2017;Atanelishvili and Silagadze, 2018;Silagadze, 2019). Along with production, sales, marketing and other processes, logistic support has an extremely large impact on the efficiency of the enterprise; logistic support man-agement of the enterprise plays a special role in the profitability of this enterprise.

For the smooth functioning of the production of the power complex enterprise, it is necessary to systematically establish logistic support, which allows distributing the means of production, establishing connections and concluding transactions between suppliers and consumers (Carlisle, 1998). The values of logistic support include values of cost and labor productivity, the use of production assets, the duration of production – in short, everything that affects its performance. It is proposed to understand logistics as the process of production support of the power complex enterprise with the necessary types of material and technical resources in a timely manner and in the volumes necessary for its regular operating activities (Borisov et al., 2012). By providing inter-industry relations in the supply of products, the logistic support structures are helpful to reduce production time, increase its efficiency and product quality. This is due to the rhythmic and timely provision of enterprises with economic lines of various types of raw materials, materials and equipment.

## 2. LITERATURE REVIEW

### 2.1 Stages of Planning and Implementation of Logistics

The essence of logistic support is that it is the basis for pricing the manufactured product and the smooth production flow. The logistic support system is a set of enterprises producing material and technical reresources, trade and intermediary organizations, service units, and government bodies that regulate relations in the field of technical and engineering support (Higgott, 1980). In modern conditions, the main functions of the logistic support of the power complex enterprise are (Farakhov et al., 2015):

• - analysis of the state and market development forecasting;

• - determination of the need for the company's products by nomenclature and in quantitative terms;

• - drawing up a statement of the need for materials for the projected volume of work;

• - organization of the procurement of materials, raw materials and semi-finished products necessary for the production;

• - reserves management;

• - transportation of material resources;

• - storage of material resources;

• - control of the expenditure of material resources;

• - organization of sales of products of the enterprise;

• - organization of effective feedback from consumers to production, including consideration of consumer complaints and after-sales service.

The listed functions generate the following tasks for logistic support (Tarasova et al., 2018):

• - determination of position of the enterprise on the market;

• - competitive intelligence;

• - determination of potential competitors;

• - market development forecasting;

• - pricing strategy;

• - determination of tasks of other structural units aimed at improving the position of the enterprise on the market.

The planning of logistic support of the power complex enterprise is based on the logistic support scheme, which is formed at the corresponding stages. Let us analyze them in more detail (Kolegov et al., 2012).

• 1. Analysis of resource efficiency in the reporting period. At this stage, adjustments are made to the production program and the volume of sales of the products in the planning period. Norms for consumption of materials, terms of wear of the tools, spare parts are calculated. Calculations of the needs of internal units for auxiliary materials are performed. The information on the material stocks in the production facilities, in incomplete production at the beginning and at the end of the planning period is analyzed. Terms of delivery of certain types of products are consid-ered. Actual material stocks in warehouses are recorded.

• 2. Calculation of the need for material and technical resources. It is carried out on the grounds of basic values of resource consumption and according to the planned production volume (it determines the amount of materials that the enterprise needs to implement the production plan, for repair and operational needs).

• 3. Research of the market of raw materials. At this stage, the market of raw materials is analyzed, the in-stock balance forecast at the end of the current year and carry-over stocks at the beginning of next year are developed, decisions on suppliers are made.

• 4. Drawing up a procurement plan for material resources. Needs are determined before the delivery of material resources based on the logistic support balance.

The tasks of planning the logistic support of the power complex enterprise are as follows (Ernst-Joachim, 2000):

• - timely and complete supply of the needs of the power complex enterprise for high-quality material and technical resources;

• - specification of prospective and current needs;

• - development of material balances;

• - selection of appropriate suppliers and maintenance of communications with them; assurance of quality of material resources;

• - minimization of costs for the acquisition, delivery and storage of material resources;

• - determination of appropriate delivery times for material resources;

• - determination of the appropriate level of stocks of material and technical resources.

An important step in the development strategy of the power complex enterprise is the formation of a system of goals. In this aspect, qualimetric, hierarchical and divergent methods are applied (McNall, 1974). The application of a divergent and convergent approach to solving this problem will make it possible to expand the list of possible development goals, expose them to verification and, as a result, provide rationalization for the development strategy of the enterprise in general (Frolov, 2011). The sequence of goal-setting of the power complex enterprise can be represented as a spiral. It expands and then narrows, or as two pyramids of goals that are closed by their foundations and at each level they have corresponding models of comparison with competitors, or models for assessing the importance and achievability of goals (Kiseleva and Fonotov, 2017). The general scheme of the proposed divergent and convergent approach to the competitive goal-setting of the power complex enterprise has the form shown in Figure 1.

At the first stage of the divergent phase, the performance of the enterprise under study is compared with the performance values of competitors. The result of this stage is the establishment of performance target value at the level of the best competitors. Thus, the basis is set for further research on the prospects of the power complex enterprise and the construction of a wider “fan” of goals (Glamazdin et al., 2008). As values of the performance of competing enterprises, it is proposed to use the following (Sherin, 2017):

• - volume of sales;

• - profitability;

• - net profit.

Moreover, values of profitability and net profit are the main ones, and the volume of sales is auxiliary. Profitability is used when analysis is necessary using relative values, and net profit is used in the analysis of absolute values of competitors’ performance (Klimov, 2009). When comparing the performance values with competitors, it is proposed for each of the competing enterprises to establish its importance in terms of impact on the business segment of the enterprise in general (Kruzhilin et al., 2018). In other words, for each competitor, its weight is set, and the total minimum value of performance that an industrial enterprise should try to achieve is calculated as a weighted average of the corresponding competitors’ values (Shapira and Youtie, 1998). A generalized model of the system of planning the need for logistic support of the power complex enterprise is shown in Figure 2. The management of logistic support of production covers a whole range of activities on coordinating the operation of all departments, employees and officials (Ramzaev et al., 2015).

Particular attention should be paid to suppliers management and procurement management, the set of their responsibilities and the quality of tasks (Figure 3), because their implementation has a certain sequence in the process of logistic support of industrial enterprises (Korel and Kombarov, 2012). This sequence can be represented in the following way:

• 1) Analysis and determination of needs, calculation of volumes and order time;

• 2) Determination of the procurement meth-od;

• 3) Procurement market analysis and supplier selection;

• 4) Negotiation process and conclusion of con-tract;

• 5) Control over the implementation of the order, reception control;

• 6) Organization of reserves location in a ware-house.

### 2.2 Supplier Selection Inventory and Pro-curement Management Process

When choosing a supplier, remoteness, availability of reserve capacities can also be taken into account; terms of fulfillment of current and urgent orders, creditworthiness and financial situation, risk of strikes, etc.

The algorithm for selecting a supplier of material and technical resources is formed according to the following parameters:

• - release price of necessary resources;

• - quality;

• - reliability of supplies;

• - availability of services;

• - range of shipping rates;

• - availability of price discounts and other incentives.

Figure 4 shows a diagram of such an algorithm that takes into account the main and additional quantitative and qualitative criteria for the selection of a supplier. The main and additional criteria are determined each time by the customer, based on the current economic situation and factors (external and internal) that may assert influence on it (Shtal et al., 2018). The type and specialization of production, its spatial location, volumes and materials-output ratio of manufactured products determine the corresponding type of structure of the supply agencies (Platonov et al., 2017). In small enterprises that consume small amounts of material resources in a limited range, supply functions are assigned to small groups of individual employees of the economic department of the enterprise. At most medium and large enterprises, which include enterprises of the power complex, this function is performed by special departments of logistic support, which are formed on a functional or material principle. The organizational structure of the supply department is shown in Figure 5.

This structure makes it possible to rationally or-ganize procurement strategies, it contributes to an in-crease in the responsibility of employees, and to an im-provement in logistic support of production. The pro-curement process takes the following form (Seliverstov, 2014): fulfillment of an order; receipt of supply objects; conclusion of contracts with suppliers; receipt of com-mercial offers and benefits; receipt of applications; set-tlements with suppliers. The procurement management department should perform the following tasks:

• - raw materials planning and control;

• - production supply planning;

• - analysis of procurement and raw materials;

• - procurement realization;

• - receipt of raw materials, delivery;

• - warehousing and materials handling;

• - transportation of raw materials throughout the enterprise;

• - control of use;

• - surpluses handling, reuse.

Effective procurement management should bring cost savings that can be achieved by:

• - low transportation costs;

• - low cost of cargo transportation in the ware-house;

• - low costs of order processing; lower costs due to deficit.

The procurement management process has cer-tain stages and functional configuration (Figure 6). At the power complex enterprise, for every percent of reduction in procurement costs, there is a 12% profit growth. If the structure of the prime cost of production of the power complex enterprise is analyzed, it can be found out that the share of materials accounts for a significant percentage of all funds spent on the production of finished products. Thus, at the disposal of logistic support agencies there is a large part of the estimated cost of the completed project order. So, one of the main reserves for reducing the cost of production of the power complex is the improvement of the logistic support system for such production. Today, the power complex enterprises are characterized by order production, so the efficiency of using material resources is carefully determined:

• - for the purpose of an objective assessment of the state of resource consumption in the main and auxiliary production, reserve volumes;

• - in order to control the amount of the costs of their acquisition and conservation;

• - for the purpose of making decisions on issues of resources saving and developing ways of ra-tional use;

• - in order to distribute a limited resource.

## 3. MATERIALS AND METHODS

The task is to optimally distribute a limited re-source with a previously selected optimality criterion. They apply for mathematical methods in solving prob-lems of optimizing the functioning of production systems, in the allocation of material and labor resources. The statements of the above problems are of an optimization nature, where various objective functions that are recorded analytically are used as performance criteria. However, a distinctive feature of the statement of problems of system analysis is the fact that, along with a rigorous mathematical apparatus, heuristic methods are used based on the intuition of the researcher, his/her experience in solving problems of this type. This is due to the difficulties of formalization of models of complex systems and processes.

On the basis of the work performed, an analysis was made of the current state of models in the method of analysis and minimization of risks in distributed production systems (Shtal et al., 2018;Ramzaev et al., 2015;Klimov, 2009). Too the inadequacy of models by modern technological and production systems was revealed. It was determined that simulation modeling allows us to model various situations with the possibility of assessing the consequences of decisions made under conditions of transformational changes in external conditions. There-fore, it is advisable to apply simulation modeling technology to solve the problems of the allocation of a limited resource in the process of logistic support for the production of the power complex.

Simulation modeling does not exclude the use of analytical models in the process of a computational experiment with incomplete information about the initial state of the system and its input operations. Thus, simulation modeling does not reduce the importance of analytical models, but is a way to expand the scope of their application in accordance with the research conditions, which are complicated during the implementation of the project. Thus, using works devoted to applied problems of resource management as applied to organizational and technical systems, it is proposed to combine system analysis technologies, game theory, and simulation modeling to solve problems of optimizing the allocation of material resources (Petukhova, 2011;Klimov, 2009;Platonov et al., 2017).

In the process of logistic support for the production of the power complex, significant amounts of funds are spent in the form of materials, semi-finished products, components, this fact makes the task of efficient allocation of resources relevant in order to satisfy some optimality criterion. The term “resources” can be understood not only as financial, but also as specific material flows to be allocated. In this case, it is necessary to move from a scalar quantity describing financial resources to a vector quantity characterizing material resources. The dimension of a vector describing a set of allocated material resources is determined from the conditions of a specific task. In general, the logistic support function associated with the allocation of a limited resource can be considered as the task of managing several projects that are related to each other only by common resources.

## 4. RESULTS AND DISCUSSION

The maximum value at which the enterprise can consider itself a leader should be equal to the maximum of all competitors’ values, taking into account the correction coefficient (Equation 1):

$G R ∈ [ ∑ i ( w i × R ) ∑ i W i , ε × m a x ( R i ) ]$
(1)

where GR : target value of the performance of the power complex enterprise (profitability, net profit, or other);

• wi : sales volumes of products of a competitor’s enterprise;

• Ri : value of the performance of a competing enterprise;

• ε : correction coefficient, which may be more than one if the enterprise is trying to become the undisputed leader in the industry, or less than one if the enterprise's capabilities do not allow it to become a leader;

• $i = 1 , … , I$ : index of a competing enterprise;

• I : the number of analyzed competing enterprises.

The next stage of the divergent phase is a com-parison with competitors' business processes and the formation of goals for the reorganization of their business processes. In this case, the range of the value of main goal established at the previous stage is basic. When forming the goal-setting model at this stage, the assumption was established that each business process under consideration has an influence on the main goal and this influence can be estimated. As a model for reforming the business processes of the power complex enterprise, the corresponding business processes of competing enterprises are selected. When forming a system of auxiliary goals for each business process, the influence of the sphere to which the business process belongs on the main goal and the difference between the efficiency of the business process at the industrial enterprise and at competing enterprises are analyzed. An auxiliary goal is included in the goal-setting system if (Equation 2, 3, 4):

$S d ≤ L S$
(2)

$S d = M { B d 1 , … , B d k d }$
(3)

$F ( S d , G R ) ≥ L G$
(4)

where

• Sd : performance value of the d-th sphere of ac-tivity of the power complex enterprise;

• LS : the minimum value of the efficiency in-dicator at which the industrial enterprise can consider the value of the effectiveness of a particular sphere as satisfactory;

• $M { B d 1 , … , B d K d }$ : median of the set of effective-ness of individual business pro-cesses;

• ${ B d 1 , … , B d K d }$ : a set of efficiencies of individual business processes that relate to the d-th sphere of activity of the power complex enterprise;

• KD : the number of business processes that be-long to the d-th sphere of activity of the power complex enterprise;

• $F ( S d , G R )$ : function reflecting the relationship between the effectiveness of the d-th sphere of activity of the power complex enterprise and its global goal;

• LG : the value at which the relationship between the effectiveness of the d-th sphere of activity of the power complex enterprise and its global goal can be considered significant (in the general case, equal to 0.7 or more).

The stage of formation of the hierarchy of goals makes it possible to ensure the development of the power complex enterprise in each sphere of activity. With this aim in view, it is proposed to establish dependencies between auxiliary goals, that is, those which must be completed earlier, and which later. In this case, auxiliary goals of the second order arise, which were not established directly at the previous stage, but the achievement of which is necessary to implement the auxiliary goals of the first order. The hierarchy of auxiliary goals of the first and second orders is proposed to be carried out using graph theory, and the analysis of the relationships between goals using the critical path method (CPM). Moreover, each goal is described as a list of related goals necessary for its achievement and time spent on achieving these goals (Equation 5, 6, 7):

$G R = { C s 1 , T s 1 }$
(5)

$G s 1 = { { G e 1 s 1 , T e 1 s 1 } , { G e 2 s 2 , T e 2 s 2 } }$
(6)

$G h 2 = { G e 2 h 2 , T e 2 h 2 }$
(7)

where GR : the main goal of the power complex;

• ${ G s 1 , T s 1 }$ : a set of auxiliary goals of the first order which must be achieved in order to achieve the main goal of the power com-plex enterprise, and an evaluation of the time required to achieve them;

• ${ G e 1 s 1 , T e 1 s 1 }$ : a set of auxiliary goals of the first order $( G e 1 s 1 )$ which must be achieved in order to achieve the goal $G s 1$, and evaluations of the time $( T e 1 s 1 )$ required to achieve them;

• ${ G e 2 s 2 , T e 2 s 2 }$ : a set of auxiliary goals of the second order $( G e 2 x 2 )$ which must be achieved in order to achieve the goal $G s 1$, and evaluations of the time $( T e 2 s 2 )$ required to achieve them;

• ${ G e 2 h 2 , T e 2 h 2 }$ : a set of auxiliary goals of the second order $( G e 2 h 2 )$ which must be achieved in order to achieve the goal $G h 2$, and evaluations of the time $( T e 2 h 2 )$ required to achieve them;

• $s = 1 , … , S$ : indices of auxiliary goals of the first order;

• $h = 1 , … , H$ : indices of auxiliary goals of the se-cond order;

• $e 1 s = 1 , … E 1 s$ : indices of auxiliary goals of the first order, which are necessary to achieve the s-th auxiliary goal of the first order (E1s – number of such goals);

• $e 2 s = 1 , … , E 2 s$ : indices of auxiliary goals of the second order, which are necessary to achieve the s-th auxiliary goal of the first order (E2s – number of such goals);

• $e 2 h = 1 , … , E 2 h$ : indices of auxiliary goals of the second order, which are necessary to achieve the h-th auxiliary goal of the second order (E2h – number of such goals).

Evaluation of time spent on achieving goals al-lows us to divide the goal-setting into strategic, opera-tional and tactical. In turn, the existence of temporary goal-setting plans is the basis for the analysis of the re-sources, sources and schedules for their receipt necessary to achieve these goals. Thus, after the formation of the maximum possible fan of goals, a transition to the convergent stage of goal-setting can be made. At the first stage of this phase, it is proposed to assess the cost of resources to achieve the goals set at the previous stage. It is proposed to divide the resources that can be used by the power complex enterprise into financial and others. Moreover, other resources can be transferred to financial. This approach is justified by the fact that all other types of resources (material, production, personnel) can be acquired, the main issue is the price that will have to be paid for this resource. The description of the cost of resources for each goal has the following form (Equation 8, 9, 10):

$V R = { F s 1 , { F ^ s 1 , R s 1 } } ,$
(8)

$V s 1 = { { F e 1 s 1 , { F ^ e 1 s 1 , R e 1 s 1 } } , { F e 2 s 2 , { F ^ e 2 s 2 , R e 2 s 2 } } }$
(9)

$V h 2 = { F e 2 h 2 , { F ^ e 2 h 2 , R e 2 h 2 } }$
(10)

where VR : a set of resources which are required to achieve the main goal of the power complex enterprise;

• $F s 1$ : a set of financial resources which are re-quired to achieve the main goal of the power complex enterprise;

• ${ F ^ s 1 , R s 1 }$ : a set of non-financial resources which are required to achieve the main goal of the power complex enterprise ($( R s 1 )$) and these resources in monetary terms $( F ^ s 1 )$;

• $V s 1$ : a set of resources which are required to achieve the s-th auxiliary goal of the first order;

• $F e 1 s 1$ : a set of financial resources which are re-quired to achieve the s-th auxiliary goal of the first order;

• ${ F ^ e 1 s 1 , R e 1 s 1 }$ : a set of non-financial resources which are required to achieve the auxiliary goal of the first order $( R e 1 s 1 )$, which is associated with the s-th auxiliary goal, and monetary estimates of these resources $( F ^ e 1 s 1 )$;

• $F e 2 s 2$ : a set of financial resources which are re-quired to achieve the auxiliary goal of the se-cond order of the power complex enterprise which is associated with the s-th auxiliary goal;

• ${ F ^ e 2 s 2 , R e 2 s 2 }$ : a set of non-financial resources which are required to achieve the auxiliary goal of the second order $( R e 2 s 2 )$ which is associated with the s-th auxiliary goal, and monetary estimates of these resources ($F ^ e 2 s 2$);

• $V h 2$ : a set of resources which are required to achieve the h-th auxiliary goal of the second order;

• $F e 2 h 2$ : a set of financial resources which are re-quired to achieve the h-th auxiliary goal of the second order;

• ${ F ^ e 2 h 2 , R e 2 h 2 }$ : a set of non-financial resources which are required to achieve the h-th auxiliary goal of the second order of the power complex enterprise $( R e 2 h 2 )$), and monetary estimates of these resources $( F ^ e 2 h 2 )$;

• $s = 1 , … , S$ : indices of the auxiliary goals of the first order;

• $h = 1 , … , H$ : indices of the auxiliary goals of the second order;

• $e 1 s = 1 , … , E 1 s$ : indices of the auxiliary goals of the first order which are required to achieve the s-th auxiliary goal of the first order (E1s – number of such goals);

• $e 2 s = 1 , … , E 2 s$ : indices of the auxiliary goals of the second order which are required to achieve the s-th auxiliary goal of the first order (E2s – number of such goals);

• $e 2 h = 1 , … , E 2 h$ : indices of the auxiliary goals of the second order which are required to achieve the h-th auxiliary goal of the second order (E2h – number of such goals).

Already at this stage, we can narrow down many of the goals of the power complex enterprise if, when as-sessing the resources, it turns out that the necessary re-sources can be obtained even with sufficient financial soundness (for example, the necessary technology is protected by patents, the owner of which does not agree to sell it). But the main stage of the convergent phase is the stage of goals cuts in accordance with available resources. At the same time, the availability of resources or the possibility of attracting them is assessed, and goals that cannot be achieved in the current conditions are canceled. Moreover, the cancellation of goals of the second and first orders necessitates a review of goals that are at the higher level of the hierarchy in accordance with what has been withdrawn. Higher hierarchy goals can be adjusted (for example, the planned volume of sales is reduced) or completely canceled. The cuts of goals of the power complex enterprise according to the available resources can be carried out in the mode of individual viewing of each goal for which there may not be enough resources, with the subsequent analysis of the entire chain of goals that are associated with the reduced goal. The cuts of goals can also be carried out in an automated mode, when all possible combinations of canceling goals are considered in accordance with available resources, and a set of alternatives is provided to analysts and managers for making a decision. The main characteristic of each alternative is the expected value of the indicator characterizing the strategic goal of the power complex enterprise. Thus, the result of this stage is a reduced set of goals, the structure of which is given in Table 1.

The final stage of the convergent phase of goal-setting of the power complex enterprise is the formation of a schedule for achieving goals. The purpose of this stage is to narrow down the diversity of goals at each time horizon. Due to the estimated time spent on the formation of the goal hierarchy, the time to achieve each auxiliary goal can be linked to a specific calendar date, which makes it possible to further monitor the achievement of goals. Presentation of the schedule for achieving goals of the power complex enterprise can be carried out using standard tools to display the implementation of projects, for example, Gantt charts. When modeling the procedure for the supply of material and technical resources and their procurement, it will be assumed that there are two players in the goods market. The first player represents the enterprises supplying resources, the second player represents the industrial enterprises receiving resources. And, as already noted, each enterprise can be both a supplier and a buyer of resources. A player who can supply resources determines their supply in the goods market; a player who can buy resources determines their demand in the goods market. Therefore, in the future, it will be considered that the first player controls the offer of resources (supply), and the second player controls the demand for resources (procurement).

Let x(0) be the value of the supply of material resources in national currency, which is planned to be offered to the counterparty (the second player) in the resource market in the period [0,T] (T is a natural number). y(0) is the value of demand for resources in national currency, which the second player plans to satisfy in the resource market in the period [0,T]. Let us denote by αpr – the growth rate of the supply of resources during the planning period of time $[ t , T ] ( t = 0 , 1 , … , T − 1 ) , β p r$ – the growth rate of the demand for resources during the planned period of time $[ t , T ] ( t = 0 , 1 , … , T − 1 )$. For the sake of simplicity, we believe that the growth rate, αpr, βpr are constants. In addition, let us denote by $p 1 p r$ – the offer price of resources in national currency, by $p 2 p r$ – the price of demand for resources in national currency, by $γ 1 p r$ we denote the relation $p 1 p r / p 2 p r$, and by $γ 2 p r$ – the ratio $p 2 p r / p 1 p r$.

Let us describe the process of supply and de-mand in the resource market. At time t, it is assumed that the value of the supply x(t) of resources in national currency, which the first player planned to offer to the second player in the interval [t,T] is multiplied by the growth rate of the supply of resources in national currency in the interval $[ t , T ] ( t = 0 , 1 , … , T − 1 )$. After that, the first player selects the value $u ( t ) ∈ [ 0 , 1 ]$ which determines the value of the supply $u ( t ) × a p r x ( t )$ of goods in national currency to the second player.

The second player acts in a similar way. At time t, the value of demand y(T) for resources in national currency which the second player planned to satisfy on the interval [t,T] is multiplied by the growth rate of the demand for resources in national currency in the interval $[ t , T ] ( t = 0 , 1 , … , T − 1 )$. After that, the second player selects the value $ν ( t ) ∈ [ 0 , 1 ]$ which determines the value of the demand $ν ( t ) × β p r y ( t )$ for resources in national currency, which the second player planned to satisfy in the interval $[ t , t + 1 ] ( t = 0 , 1 , … , T − 1 )$.

The value of the supply $u ( t ) × α p r x ( t )$ of re-sources in national currency, which the first player sup-plies to the second player, will allow the second player to satisfy his/her demand for resources in national currency by the value $γ 2 p r × u ( t ) × α p r x ( t )$ (in accordance with the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)). The value of the demand $ν ( t ) × β p r y ( t )$ for resources in national currency, which the second player planned to satisfy in the interval $[ t , t + 1 ] ( t = 0 , 1 , … , T − 1 )$, will allow the first player to offer the second player a value $γ 1 p r × ν ( t ) × β p r y ( t )$ of resources in national currency (according to the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)).

Thus, at time t+1 the value of supply x(t+1) and demand y(t+1) will be written in the following way (Equation 11, 12):

$x ( t + 1 ) = α p r x ( t ) − u ( t ) × α p r x ( t ) − γ 1 p r × ν ( t ) × β p r y ( t ) ;$
(11)

$y ( t + 1 ) = β p r y ( t ) − ν ( t ) × β p r y ( t ) − γ 2 p r × u ( t ) × α p r x ( t ) ;$
(12)

At once it will be noticed that the situation fits into the framework of the considered scheme: one enterprise that is interested in buying resources for its functioning is a group of enterprises supplying resources, which can be either one enterprise or a group of enterprises. In the case when the demand and supply procedure is considered not only in the domestic market, but also in the external one, it is necessary to make additions due to this factor. That is, taking into account the fact that players, in addition to supply and demand for resources in national currency, also supply resources in foreign currency, and are also in demand for resources in foreign currency. Let us introduce additional notations. We denote by $β 1 p r ( β 1 p r ∈ [ 0 , 1 ] )$ – the value characterizing the proportion of the value of the first player’s supply of resources in national currency in the domestic and foreign markets in the interval [t,t+1]; i.e. $β 1 p r × u ( t ) × α p r x ( t )$ – the value of the first player’s supply of resources in national currency in the domestic and foreign markets in the interval [t,t+1], then $( 1 − β 1 p r )$ will mean the share of the first player’s supply of resources in foreign currency in the domestic and foreign markets in the interval [t,t+1]; i.e. $( 1 − β 1 p r ) × u ( t ) × α p r x ( t )$ – the value of the first player’s supply of resources in foreign currency in the domestic and foreign markets in the interval [t,t+1].

By $β 2 p r ( β 2 p r ∈ [ 0 , 1 ] )$ – the value characterizing the share of the second player’s demand for resources in national currency in the domestic and foreign markets in the interval [t,t+1], i.e. $β 2 p r × ν ( t ) × β p r y ( t )$ – the value of the second player’s demand for resources in national and foreign currencies in the domestic and foreign markets in the interval [t,t+1]. Then $( 1 − β 2 p r )$ means the share of the second player’s demand for resources in foreign currency in the domestic and foreign markets in the interval [t,t+1], i.e. $( 1 − β 2 p r ) × ν ( t ) × β p r y ( t )$ – the value of the second player’s demand for resources in foreign currency in the domestic and foreign markets in the interval [t,t+1]. In addition, by kd we denote the exchange rate in relation to the national currency, i.e. 1\$ = kd; by $q 1 p r$ we denote the price of supply of goods in foreign currency, by $q 2 p r$ - the price of demand for goods in foreign currency (dollars); by $δ 1 p r$ we denote the ratio $q 1 p r / q 2 p r$, by $δ 2 p r$ – the ratio $q 2 p r / q 1 p r$. So, the record for values $x ( t + 1 ) и y ( t + 1 )$ has the following form (Equation 13, 14):

$x ( t + 1 ) = α p r x ( t ) − β 1 p r × u ( t ) × α p r x ( t ) − γ 1 p r × β 2 p r × ν ( t ) × β p r y ( t ) − ( 1 − β 1 p r ) × u ( t ) × α p r x ( t ) − δ 1 p r × ( 1 − β 2 p r ) × ν ( t ) × β p r y ( t ) ;$
(13)

$y ( t + 1 ) = β p r y ( t ) − β 2 p r × ν ( t ) × β p r y ( t ) − γ 2 p r × β 1 p r × u ( t ) × α p r x ( t ) − ( 1 − β 2 p r ) × ν ( t ) × β p r y ( t ) − δ 2 p r × ( 1 − β 1 p r ) × u ( t ) × α p r x ( t ) ;$
(14)

Let us comment on the process of supply and demand of products and services in the resource market, which includes both the domestic and foreign markets. At time t, it is assumed that the value of the supply x(t)) of resources in national and foreign currencies that the first player planned to supply to the second player in the interval [t,T] is multiplied by the growth rate of the supply of resources in national and foreign currency in the interval $[ t , T ] ( t = 0 , 1 , … , T − 1 )$. After that, the first player selects the values $β 1 p r ∈ [ 0 , 1 ] , ( 1 − β 1 p r ) ∈ [ 0 , 1 ] , u ( t ) ∈ [ 0 , 1 ]$ that dedetermine the value of the supply $β 1 p r × u ( t ) × α p r x ( t ) , ( 1 − β 1 p r ) × u ( t ) × α p r x ( t )$ of resources in national and foreign currencies, respectively, to the second player. The second player acts in a similar way. At time t, the value of the demand y(t) for resources in national and foreign currencies, which the second player planned to satisfy in the interval [t,T] is multiplied by the growth rate of demand for resources in national and foreign currencies in the interval $[ t , T ] ( t = 0 , 1 , … , T − 1 )$. After that, the second player selects the value $β 2 p r ∈ [ 0 , 1 ] , ( 1 − β 2 p r ) ∈ [ 0 , 1 ] , ν ( t ) ∈ [ 0 , 1 ]$ that determine the value of the demand $β 2 p r × ν ( t ) × β p r y ( t ) , ( 1 − β 2 p r ) × ν ( t ) × β p r y ( t )$ of national and foreign currencies, which the second player planned to satisfy in the interval $[ t , t + 1 ] ( t = 0 , 1 , … , T − 1 )$.

The value of the supply $β 1 p r × u ( t ) × α p r x ( t )$ of resources in national currency, which the first player supplies to the second player, will allow the second player to satisfy his/her demand by the value $γ 2 p r × β 1 p r × u ( t ) × α p r x ( t )$ (according to the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)). The value of the supply $( 1 − β 1 p r ) × u ( t ) × α p r x ( t )$ of resources in foreign currency, which the first player supplies to the second player, will allow the second player to satisfy his/her demand by the value $δ 2 p r × ( 1 − β 1 p r ) × u ( t ) × α p t x ( t )$(according to the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)).

The value of the demand $β 2 p r × ν ( t ) × β p r y ( t )$ for resources in national currency, which the second player planned to satisfy in the interval $[ t , t + 1 ] ( t = 0 , 1 , … , T − 1 )$, will allow the first player to supply to the second player the value $γ 1 p r × β 2 p r × ν ( t ) × β p r y ( t )$ of resources in national currency (according to the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)). The value of the demand $( 1 − β 2 p r ) × ν ( t ) × β p r y ( t )$ for goods in foreign currency, which the second player planned to satisfy in the interval $[ t , t + 1 ] ( t = 0 , 1 , … , T − 1 )$, will allow the first player to supply to the second player the value $δ 1 p r × ( 1 − β 2 p r ) × v ( t ) × β p r y ( t )$ of resources in foreign currency (according to the law of equilibrium: (supply price) x (supply volume) = (demand price) x (demand volume)). Then, at the time t + 1 < T one of four conditions is possible (Equation 15, 16, 17, 18):

$1 ) x ( t + 1 ) ≥ 0 , y ( t + 1 ) < 0 ;$
(15)

$2 ) x ( t + 1 ) < 0 , y ( t + 1 ) ≥ 0 ;$
(16)

$3 ) x ( t + 1 ) < 0 , y ( t + 1 ) < 0 ;$
(17)

$4 ) x ( t + 1 ) ≥ 0 , y ( t + 1 ) ≥ 0.$
(18)

If the first condition is met, then the second player managing the demand ahead of schedule realized his/her demand at the time t+1, and the first player managing the supply did not implement his/her offer at the time t+1. The supply and demand procedure is considered completed as a result of the realization of demand by the second player ahead of schedule. If the second condition is met, then the first player managing the supply prematurely realized his/her supply at the time t+1 and the second player managing the demand did not realize his/her demand at the time t+1. The supply and demand procedure is considered completed as a result of the realization of the supply by the first player ahead of schedule. If the third condition is met, then the players managing both supply and demand, realized their supply and demand at the time t+1 ahead of schedule. The supply and demand procedure is considered completed as a result of the realization of demand by players ahead of schedule. If the fourth condition is met, then the players managing both supply and demand did not realize their supply and demand at the time t+1 and, therefore, the supply and demand procedure continues for the times larger than t+1. And, at the point in time t = T - 1 players make the last move and determine the values x(T) and y(T) that show the supply and demand at the time t = T as a result of the realization of the demand and supply of players.

Note 1. Both supply and demand for resources include both resources in national currency and in foreign currency. If V1 – the amount of resources in national currency that can be supplied, and V2 – the amount of resources in foreign currency, for example, in the currency that can be offered, then the total amount of resources (“valued” in national currency) that can be offered is equal to the amount $V 1 + k d × V 2$. Thus, the share $β 1 p r$ of resources in national currency that can be supplied is equal to $V 1 / ( V 1 + k d × V 2 )$, and the share $( 1 − β 1 p r )$ of resources in foreign currency can be supplied equal to $k 2 × V 2 / ( V 1 + k d × V 2 )$. That is, the choice of shares $β 1 p r , ( 1 − β 1 1 p r )$, when substantiating the dynamics of changes in the volumes of supply and demand, can occur in the manner described above, which means that for fixed volumes V1, V2 the change of values $β 1 p r , ( 1 − β 1 p r )$ is due to changes in the exchange rate kd. With the growth of the exchange rate, the value $β 1 p r$ decreases, and the value $( 1 − β 1 p r )$ increases, with a decrease in the exchange rate, the value $β 1 p r$ grows, and the value $( 1 − β 1 p r )$ – decreas-es.

A game theory toolkit that allows us to deter-mine the areas of possible initial states of supply and demand (in the resource market). If the process of sup-ply and demand begins with these states, then at one point in time the supply or demand can be realized. It makes it possible to obtain the necessary result of the supply and demand realization process and find optimal (rational) strategies for managing the supply and demand realization process. To find such areas, a multi-step game with two terminal surfaces is solved. The solution of which is to determine the sets of initial states of supply and demand, as well as strategies (control ac-tions) of players, using which it is possible to obtain results that are desirable for each player (the realization of demand or supply).

The above realization process of the supply and demand procedure will be considered within the framework of the positional multi-step game scheme with full information. Within the framework of this scheme, the process “generates” two tasks: from the point of view of the first ally player and from the point of view of the second ally player. Due to the symmetry, it will suffice to be limited to considering one of them, for example, from the point of view of the first ally player. Let us denote by $T * = { 0 , 1 , … }$ a discrete set characterizing the area of change of the time parameter.

Definition. The pure strategy of the first ally player is called the function $T * × R + 2 → [ 0 , 1 ]$ that puts to the state of information (position) $( t , ( x , y ) )$ the value $u ( t , ( x , y ) ) ∈ [ 0 , 1 ]$. That is, the pure strategy of the first ally player is a function (rule) that sets to the state of information at time t the value $u ( t , ( x , y ) )$, which deter-mines the value of the supply of material resources, which he/she supplies in the planning period $[ t , t + 1 ]$. Regarding the awareness of the opponent player (within the framework of the positional game scheme), no assumptions are made, which is equivalent to the fact that the opponent player chooses his/her control influence ν(t), which determines the amount of demand for material resources, based on any information.

After defining the strategies in task 1, it is necessary to identify many “benefits” W1 for the first player. W1 - a set of such initial states $( x ( 0 ) , y ( 0 ) )$ which are the value of supply and demand. For such initial states, there is a strategy of the first player that controls the supply of material resources, which for any realization of the strategy of the second player “brings” the state $( x ( t ) , y ( t ) )$ at one point in time t wherein condition (1) is satisfied. Moreover, the second player does not have a strategy that can “lead” to the fulfillment of conditions (2) or (3) at one of the previous points in time. The strategy $u * ( . , . )$ of the first player with this property is called optimal.

The solution to problem 1 is to find a set of “advantages” of the first player and his/her optimal strategies. Similarly, the problem is posed from the point of view of the second ally player. Due to the symmetry, it is enough to be limited to the solution to problem 1, since the solution to Problem 2 is found in exactly the same way. The solution to problem 1 is found using the tools of the theory of multi-step games with complete information, which allows us to find it with different ratios of game parameters. Here is a solution to the game, that is, a lot of advantage W1 and optimal strategies for the first player. Note the following: a set of benefits W1 is a union of sets W1 of states $( x ( 0 ) , y ( 0 ) )$ which have the property that if the supply and demand procedure begins with them, then the first player has a strategy $u * ( . , . )$ which allows him/her to obtain the fulfillment of condition (1). At the time t=i, no matter how the second player acts, and the second player has a strategy $v * ( . , . )$ which does not allow the first player to obtain the fulfillment of condition (1) in less time. Let us denote by S1 – the value $γ 1 p r × β 1 p r + δ 1 p r × ( 1 − β 1 p r )$, i.e.

$S 1 = γ 1 p r × β 2 p r + δ 2 p r × ( 1 − β 2 p r )$
(19)

and by S2 the value $γ 2 p r × β 2 p r + δ 2 p r × ( 1 − β 2 p r )$, i.e.

$S 2 = γ 2 p r × β 2 p r + δ 2 p r × ( 1 − β 2 p r )$
(20)

We write the sets $W 1 i$ and optimal strategies $u * ( . , . )$ for any ratios of game parameters (Equation 21, 22, 23, 24, 25).

$α ≤ β W 1 i = { ( x ( 0 ) , y ( 0 ) ) : k ( i − 1 ) × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( i − 2 ) × β × y ( 0 ) } i = 1 , 2 , …$
(21)

(22)

$k ( i ) = 1 + s 1 × s 2 − ( s 1 × α × s 2 ) / ( β × k ( i − 1 ) ) k − 1 = 0 k 0 = 1 + s 1 × s 2 ;$
(23)

$W 1 = ∑ i = 1 ∞ W 1 i ;$
(24)

$s 1 × α × x ( 0 ) = { [ 1 + s 1 × s 2 + ( 1 + s 1 × s 2 − 4 × s 1 × s 2 × α / β 1 / 2 ) ] / 2 × β × y ( 0 ) }$
(25)

On the basis of this, Inequality (26) is formed:

$( x ( 0 ) , y ( 0 ) ) : s 1 × α × x ( 0 ) ≤ { [ 1 + s 1 × s 2 + ( 1 + s 1 × s 2 − 4 × s 1 × s 2 × α / β ) 1 / 2 ] / 2 } × β × y ( 0 ) ;$
(26)

The first player cannot “force” the second player to satisfy demand at any given time. $α > β , s 1 × s 2 ≥ 1 ;$ In this case, a set of advantages of the first player W1 will be the union of a finite number of sets W1, namely (N+2) of sets where (Equation 27, 28, 29):

$N : k ( i ) > s 1 × s 2 α / β , I = 0 , … , N − 1 ; k ( N ) ≤ s 1 × s 2 × α / β ;$
(27)

$W 1 i = { ( x ( 0 ) , y ( 0 ) ) : k ( i − 1 ) × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( i − 2 ) × β × y ( 0 ) } i = 1 , 2 , … , N + 1$
(28)

$W 1 N + 2 = { ( x ( 0 ) , y ( 0 ) ) : s 1 × s 2 × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( N ) × β × y ( 0 ) } ;$
(29)

Optimal strategy (Equation 30, 31) is defined as follows.

$u * = ( u * ( o , ( x , y ) ) , … , u * ( t − 1 , ( x , y ) ) , u * ( N + 1 , ( x , y ) ) )$
(30)

$u * ( 0 , ( x , y ) ) = { 0 , at ( x , y ) ∈ R + 2 , α × y > s 2 × β × y } ,$
(31)

and not defined – otherwise (Equation 32),

$u * ( t , ( x , y ) ) = { [ 1 − ( s 2 × β × y ) / ( α × x ) ] ,$
(32)

at $( x , y ) ∈ R + 2 , α × y > s 2 × β × y ,$, and not defined – other-wise}; $t = 0 , … , N + 1 }$. $α > β , s 1 × s 2 < 1 ;$ In this case, a set of advantages of the first player W1 will also be a union of finite number of sets $W 1 i$, namely $( N + i * + 2 )$ of sets, where $N : k ( i ) > α / β , i = 0 , … , N − 1 ; k ( N ) ≤ α / β ;$; i – the minimum non-negative number determined by the inequality (Equation 33):

$K ( N ) × ( α / β ) i * + 1 < s 1 × s 1$
(33)

Then

$W 1 i = { ( x ( 0 ) , y ( 0 ) ) : k ( i − 1 ) × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( i − 2 ) × β × y ( 0 ) } i = 1 , 2 , … , N + 1 ;$
(34)

If $i * = 0$, then

$W 1 i = { ( x ( 0 ) , y ( 0 ) ) : k ( i − 1 ) × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( i − 2 ) × β × y ( 0 ) } i = 1 , 2 , … , N + 1 ;$
(35)

$W 1 N + 2 = { ( x ( 0 ) , y ( 0 ) ) : s 1 × s 2 × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( N ) × β × y ( 0 ) } .$
(36)

The record of the optimal strategy in this case is exactly the same as in case 2.

$W 1 N + 1 + j = { ( x ( 0 ) , y ( 0 ) ) : k ( N ) × ( β / α ) j × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( N ) × ( β / α ) j − 1 × β × y ( 0 ) } j = 1 , … , i * ;$
(37)

$W 1 N + i * = { ( x ( 0 ) , y ( 0 ) ) : s 1 × s 2 × β × y ( 0 ) ≤ s 1 × α × x ( 0 ) k ( N ) × ( β / α ) i * × β × y ( 0 ) } .$
(38)

The optimal strategy (Equation 39, 40) is de-fined as follows.

$u * = ( u * ( o , ( x , y ) ) , … , u * ( N + 1 + i * , ( x , y ) ) )$
(39)

$u * ( t , ( x , y ) ) = { 0 at ( x , y ) ∈ R + 2 , α × y > s 2 × β × y$
(40)

and not defined – in another case, $t = 0 , 1 , … , i *$,

$u * ( t , ( x , y ) ) = { [ 1 − ( s 2 × β × y ) / ( α × x ) ]$
(41)

at $( x , y ) ∈ R + 2 , α × y > s 2 × β × y , i ≥ i * + 1$, and not defined - otherwise}.

Note 2. If the supply and demand procedure is considered as a one-step procedure, then according to the classification of game theory such a procedure can be embedded in the framework of an infinite antagonistic game with zero sum, that matters (“saddle point”) in the class of mixed strategies. For example, in case 1) the set $W 1 i$ are sets of states $( x ( 0 ) , y ( 0 ) )$ which have the property. If a one-step procedure for the realization of supply and demand begins with them, then with probability (1/i) the first player managing the supply in the resource market will “lead” the demand and supply management process to the realization of the supply in one step, if the second player managing the demand in the resource market will use its optimal mixed strategy. If the second player uses a non-optimal mixed strategy, the realization probability of supply will increase in one step.

Note 3. In reality, the process of managing sup-ply and demand in the resource market can run differently over time. For example, players can make their decisions in turn, or simultaneously, but with a greater frequency, which is modeled to some extent by a continuous model of managing supply and demand in the market resources in which the dynamics of control is given by differential equations. In each case, there is a solution to the corresponding game. Then, using the results of the decisions of the corresponding games, the results of the real process of supply and demand in the resource market can be predicted.

Note 4. Since some of the parameters in this model are endogenous, and some are exogenous, so we can exert influence on the result of the procedure, trying to improve it (from the position of a player) by control-ling both endogenous and exogenous factors.

Note 5. When solving problems with the pro-posed gaming technologies, in the space of variables (x,y), there are either areas of equation or rays of equa-tion, i.e., if the procedure for managing supply and de-mand in the resource market begins with them, then the players have strategies that allow them to remain either in the field of equation or in the ray of equation. Therefore, for any condition $( x ( 0 ) , y ( 0 ) )$ it is possible to determine the parameters of the procedure under which this state will be either on the ray of equation, or in the field of equation.

A typical model of the allocation of limited re-sources is considered when implementing an arbitrary number of independent projects (the essence of each project, its relationship with others is taken into account only at the level of total resources consumed). The basic ratios are made in a general way, which allows us to switch from financial flows to material ones. Let us formulate the problem of the allocation of limited material and technical resources in the most general setting.

Consider n project orders that meet the neces-sary conditions of efficiency, each of which is characterized by a set of parameters: Qf – the required amount of funding for the i-th project; $T t i$ – the duration of the i-th project; Pi – the expected profitability of the i-th project. The valuable thing in the algorithms under consideration is that the sources of project financing are studied, while own and borrowed funds are allocated. This leads to the need to solve two problems depending on the sources of financing. The notation is introduced:

• V – a set of project orders requiring appropriate resources (project portfolio); $Q f ( o w n )$ – the amount of funding regarding the necessary material and technical resources from own funds; Qf – total funding for the necessary material and technical resources. If $Q f Q f ( o w n )$ then the difference $Q f − Q f ( o w n )$ is financed by means of loans, which naturally increases the cost of the project; profitability of the portfolio of projects and each project:

$P ( V ) = ∑ i ∈ ν P i$
(42)

The effectiveness of the I-th project (reduced to the period under consideration):

$E = P i − Q f$
(43)

The complexity of solving the problem largely depends on the nature of the profitability functions of the set of project orders under study: as a rule, the more restrictions are imposed on the nature of the function, the more difficult the decision procedure is. A minimal re-quirement is introduced that follows logically from the nature of this problem; it is believed that a function must possess the additivity property. In this case, the cost of resources required to realize a set of project orders taking into account the interest rate for the loan (a – the interest rate) will be determined in the following form (Equation 44):

(44)

then the effectiveness of the set of project orders V will be characterized by Equation (45):

$E i ( V ) = P ( V ) − A ( V )$
(45)

Therefore, the expression represents the objec-tive function of the problem of finding the optimal allocation of limited resources available. Specific material flows in the framework of managing the logistic support of power projects, which can be written in the classical form of a nonlinear programming problem (Equation 46, 47):

$∑ i ∈ V E i → m a x$
(46)

$∑ i ∈ V Q f ≤ Q f ( o w n )$
(47)

The solution of the formulated problem depends on the type of objective function. If we restrict ourselves to the previously introduced requirement of additivity of the objective function, then in this case the most effective way will be the application of the dynamic programming method.

## 5. CONCLUSION

Thanks to the developed model of the competitive goal-setting of the power complex enterprise, which is based on the formation of an excessive list of possible goals that are set as a result of comparison with competitors at the divergent stage, and further reduction. At the convergent stage, the enterprise management is provided with the opportunity to set goals and obtain a hierarchy and a schedule for achieving goals of the power complex enterprise, which take into account the characteristics of the competitive environment.

Thus, a resource allocation model has been de-veloped within the framework of a methodological ap-proach to managing the logistic support of the produc-tion of the power complex enterprise. Into account the limited resources available, specific material flows, which makes it possible to find their optimal allocation and ensures the efficiency of production processes at the enterprise.

The novelty of the work is determined by the fact that economic activity in the field of entrepreneurial business is determined by the format and planning measures for the long term, which is formed on the basis of the costs of small and medium-sized enterprises in the production of industrial products. The practical significance of the study is determined by the possibility of forming a high-quality component that will create an instrument for the development of small and medium enterprises in the regions.

## Figure

The divergent and convergent approach to the competitive goal-setting of the power complex enterprise

A generalized model of the system of planning the need for logistic support of the power complex enterprise.

Supply management of material and technical resources.

The algorithm for selecting a supplier of material and technical resources.

Organizational structure of the supply department.

Display of procurement management process.

## Table

Structure of a set of goals after their cuts

## REFERENCES

1. Atanelishvili, T. and Silagadze, A. (2018), Xenophon: Economy finds the name, Bulletin of the Georgian National Academy of Sciences, 12(4), 163-166.
2. Borisov, V. N. , Budanov, I. A. , Moiseev, A. K. , and Panfilov, V. S. (2012), Necessary conditions for the modernization of the Russian processing industry on the example of the heavy engineering industry, Studies on Russian Economic Development, 23, 14-26.
3. Carlisle, R. (1998), Linking industrial modernization evaluation with state policy-making, The Journal of Technology Transfer, 23(1), 49-54.
4. Ernst-Joachim, M. (2000), The EC commission’s modernization of competition policy: A challenge to the community’s constitutional order, European Business Organization Law Review, 1(3), 401-444.
5. Farakhov, M. I. , Laptev, A. G. , and Basharov, M. M. (2015), Modernization of mass-exchange equipment by new packings in chemical engineering, Theoretical Foundations of Chemical Engineering, 49, 233-238.
6. Frolov, I. E. (2011), Opportunities and challenges of Russian high-technology complex modernization, Studies on Russian Economic Development, 22, 245-264.
7. Glamazdin, P. M. , Tropinova, I. V. , and Tropinov, A. M. (2008), Modernization of a steam boiler of the DKVR series lining, Refractories and Industrial Ceramics, 49(3), 161-166.
8. Higgott, R. A. (1980), From modernization theory to public policy: Continuity and change in the political science of political development, Studies in Comparative International Development, 15, 26.
9. Kingsley, G. and Klein, H. K. (1998), Interfirm collaboration as a modernization strategy: A survey of case studies, The Journal of Technology Transfer, 23, 65-74.
10. Kiseleva, V. V. and Fonotov, A. G. (2017), Areas of national modernization: Search for priorities, Studies on Russian Economic Development, 28(5), 480-485.
11. Klimov, I. G. (2009), Improvement of existing industrial processes and organizational forms of management as a condition for progressive enterprise development,Russian Journal of Applied Chemistry, 82, 2270-2274.
12. Kolegov, Y. Y. , Stennikov, N. V. , and Fedyaev, A. V. (2012), An increase in the efficiency of heat supply systems on reconstruction of energy sources and modernization of gas supply systems, Thermal Engineering, 59, 316-320.
13. Korel, L. V. and Kombarov, V. Y. (2012), Russian industrial enterprises in the reflection of the multivector concept of modernization: Sociological perspective, Regional Research of Russia, 2, 25-33.
14. Kruzhilin, S. , Baranova, T. , Mishenina, M. , and Zaitseva, M. (2018), Regional specificity creation of protective afforestations along highways, World Ecology Journal, 8(2), 22-32.
15. McNall, S. G. (1974), Value systems that inhibit modernization: The case of Greece, Studies in Comparative International Development, 9(3), 46-63.
16. Petukhova, S. P. (2011), Information technologies as a basis for the modernization of Russian intellectual resources, Studies on Russian Economic Development, 22, 375.
17. Platonov, V. V. , Platonova, E. V. , and Maizel, I. G. (2017), Modernization of numerically controlled machine tools on modular principles, Russian Engineering Research, 37, 140-143.
18. Ramzaev, V. M. , Khaimovich, I. N. , and Chumak, P. V. (2015), Models for forecasting the competitive growth of enterprises due to energy modernization, Studies on Russian Economic Development, 26(1), 47-52.
19. Seliverstov, V. E. (2014), Federalism, regional growth, and regional science in post-Soviet Russia: Modernization or degradation? Regional Research of Russia, 4, 240-252.
20. Shapira, P. and Youtie, J. (1998), Contrasting perspectives on the evaluation of industrial modernization: Introduction to the symposium, The Journal of Technology Transfer, 23, 3-6.
21. Sherin, E. A. (2017), Modernization of the industrial complex from the perspective of the concept of production cycle (A case study of Kuznetsk coal use), Geography and Natural Resources, 38, 281-287.
22. Shtal, T. V. , Buriak, M. M. , Amirbekuly, Y. , Ukubassova, G. S. , Kaskin, T. T. , and Toiboldinova, Z. G. (2018), Methods of analysis of the external environment of business activities, Journal Espacios, 39(12), 1-22.
23. Silagadze, A. (2019), Plato on economic foundations of an “ideal state”, Bulletin of the Georgian National Academy of Sciences, 13(2), 142-145.
24. Tarasova, E. V. , Nikolenko, T. Y. , Gorbunov, G. L. , and Semina, L. V. (2018), Implementation of investment projects at industrial enterprises, Russian Engineering Research, 38, 295-300.
25. Toxanova, A. , Galiyeva, A. , Muhamedzhanova, A. , Baibusinova, G. , Kulubekova, A. , and Ashikbayeva, Z. (2017), Innovative entrepreneurship financing in the Republic of Kazakhstan, Journal of Applied Economic Sciences, 12(3), 875-892.


text:
# How to show two models of computation are equivalent?

I'm seeking explanation on how one could prove that two models of computation are equivalent. I have been reading books on the subject except that equivalence proofs are omitted. I have a basic idea about what it means for two models of computation to be equivalent (the automata view: if they accept the same languages). Are there other ways of thinking about equivalence? If you could help me understand how to prove that the Turing-machine model is equivalent to lambda calculus, that would be sufficient.

• I guess you have chosen the wrong books. – Raphael Aug 13 '12 at 22:25
• @Raphael What's a good book on the subject? – saadtaame Aug 14 '12 at 4:00
• I'd like to say "any book about automata", but apparenty that's now true. Unfortunately, I don't have any fitting English books at hand, sorry. – Raphael Aug 14 '12 at 6:36
• A book on just automata won't suffice. – reinierpost Aug 14 '12 at 7:16
• What books are you using? – saadtaame Aug 14 '12 at 12:58

You show that either model can simulate the other, that is given a machine in model A, show that there is a machine in model B that computes the same function. Note that this simulation does not have to be computable (but usually is).

Consider, for example, pushdown automata with two stacks (2-PDA). In another question, the simulations in both directions are outlined. If you did this formally, you would take a general Turing machine (a tuple) and explicitly construct what the corresponding 2-PDA would be, and vice versa.

Formally, such a simulation may look like this. Let

$$\qquad \displaystyle M = (Q,\Sigma_I,\Sigma_O,\delta,q_0,Q_F)$$

be a Turing machine (with one tape). Then,

$$\qquad \displaystyle A_M = (Q \cup \{q^*_1,q^*_2\},\Sigma_I,\Sigma_O', \delta', q^*_1, Q_F)$$

with $$\Sigma_O' = \Sigma_O \overset{.}{\cup} \{\\}$$ and $$\delta'$$ given by

$$\quad \displaystyle (q^*_1,a,h_l,h_r) \to_{\delta'} (q^*_1,ah_l,h_r)$$ for all $$a \in \Sigma_I$$ and $$h_r,h_l \in \Sigma_O$$,
$$\quad \displaystyle (q^*_1,\varepsilon,h_l,h_r) \to_{\delta'} (q^*_2,h_l,h_r)$$ for all $$h_r,h_l \in \Sigma_O$$,
$$\quad \displaystyle (q^*_2,\varepsilon,h_l,h_r) \to_{\delta'} (q^*_2,\varepsilon,h_lh_r)$$ for all $$h_r,h_l \in \Sigma_O$$ with $$h_l \neq \$$,
$$\quad \displaystyle (q^*_2,\varepsilon,\,h_r) \to_{\delta'} (q_0,\,h_r)$$ for all $$h_r \in \Sigma_O$$,
$$\quad \displaystyle (q,\varepsilon,h_l,h_r) \to_{\delta'} (q',\varepsilon,h_la) \iff (q,h_r) \to_\delta (q',a,L)$$ for all $$q \in Q$$ and $$h_l \in \Sigma_O$$,
$$\quad \displaystyle (q,\varepsilon,\,h_r) \to_{\delta'} (q',\,\square a) \iff (q,h_r) \to_\delta (q',a,L)$$ for all $$q \in Q$$,
$$\quad \displaystyle (q,\varepsilon,h_l,h_r) \to_{\delta'} (q',ah_l,\varepsilon) \iff (q,h_r) \to_\delta (q',a,R)$$ for all $$q \in Q, h_l \in \Sigma_O'$$,
$$\quad \displaystyle (q,\varepsilon,h_l,\) \to_{\delta'} (q,h_l,\square\)$$ for all $$q \in Q$$ and $$h_l \in \Sigma_O'$$, and
$$\quad \displaystyle (q,\varepsilon,h_l,h_r) \to_{\delta'} (q',h_l,a) \iff (q,h_r) \to_\delta (q',a,N)$$ for all $$q \in Q,h_l\in\Sigma_O'$$

is an equivalent 2-PDA. Here, we assume that the Turing machine uses $$\square \in \Sigma_O$$ as blank symbol, both stacks start with a marker $$\\notin \Sigma_O$$ (which is never removed) and $$(q,a,h_l,h_r) \to_{\delta'} (q',l_1\dots l_i,r_1\dots r_j)$$ means that $$A_M$$ consumes input $$a$$, switches states from $$q$$ to $$q'$$ and updates the stacks like so:

[source]

It remains to show that $$A_M$$ enters a final state on $$x \in \Sigma_I^*$$ if and only if $$M$$ does so. This is quite clear by construction; formally, you have to translate accepting runs on $$M$$ into accepting runs on $$A_M$$ and vice versa.

• @frabala You were right, I had the initial states the wrong way around. Fixed now, thanks! – Raphael Apr 13 '19 at 17:38

At the beginning of Communicating and Mobile Systems: the Pi-Calculus by Robin Milner, there is a introduction on automata and how they can simulate each other so that they cannot be distinguished : Bisimulation. (cf Bisimulation on wikipedia)

I don't remember well, I should re-read the chapter, but there was a trouble with simulation and bisimulation that made them not sufficient for computational equivalences.

Thus Robin Milner introduces his Pi-Calculus and exposes it for the rest of the book.

Ultimately, in his last book The Space and Motion of Communicating Agents, you could have a look at Robin Milner's Bigraphs. They can model Automata, Petri nets, Pi-Calculus and other computational methodologies.

As far as I know, the only (or at least most common) way to do this is to compare the languages the machines/models accept. That's the whole point of Automata theory: it takes the vague concept of a problem or algorithm and turns it into a concrete mathematical set (i.e. a language) which we can reason about.

The easiest way to do this is, given an arbitrary machine/function from one model, to construct a machine from the second model that computes the same language. Odds are you'll use induction in the length of the expression, states in the machine, rules in the grammar, etc.

I haven't seen this done with Lambda and TMs (though I'm 99% sure it's possible), but I have definitely seen this kind of thing for proving equivalence of NFAs and Regular expressions. First you show a NFA which can accept any atom, then using induction, you make NFAs which accept the union/concatenation/Kleene-star of any smaller NFAs.

Then you do the opposite, to find an RE for any NFA.


text:
# nmrglue.proc_autophase¶

Automated phase correction These functions provide support for automatic phasing of NMR data. They consist of the core autops function which performs the optimisation and a set of private functions for calculating a spectral phase quality score for a provided spectrum.

This module is imported as nmrglue.proc_autophase and can be called as such.

## User Functions¶

 autops(data, fn[, p0, p1]) Automatic linear phase correction manual_ps(data[, notebook]) Manual Phase correction using matplotlib

## Developer Functions¶

These functions are called by high-level function are and most users will not use them in common processing scripts. Developers may be interested in them.

 _ps_acme_score(ph, data) Phase correction using ACME algorithm by Chen Li et al. _ps_peak_minima_score(ph, data) Phase correction using simple minima-minimisation around highest peak


text:
AI organizations divide their work into data engineering, modeling, deployment, business analysis, and AI infrastructure. Each task requires specific skills and can be the focus of multiple roles. Deep learning skills are sometimes required, especially in organizations focusing on computer vision, natural language processing, or speech recognition. If you apply to a role that carries out the modeling task such as Deep Learning Engineer (DLE), Deep Learning Researcher (DLR) or Software Engineer-Deep Learning (SE-DL), you’ll often encounter the deep learning algorithms interview during the onsite round. You can learn more about these roles in our AI Career Pathways report and about other types of interviews in The Skills Boost.

I   What to expect in the deep learning algorithms interview

The interviewer will try to uncover how deeply you understand deep learning algorithms. Here’s a list of interview questions you might be asked:

• Explain how backpropagation works in a fully-connected neural network.
• Compute the number of parameters in a fully-connected, convolutional, or recurrent neural network.
• Explain how layers in the following list work: dense (i.e. fully-connected), convolutional, LSTM, pooling, etc. What are the hyperparameters of a given layer? What roles do they play?
• What methods are used to lower the variance? Explain the relevant methods, e.g. dropout, L1/L2 regularization, early stopping, etc.
• What are the benefits of batch normalization?
• How do residual networks differ from standard fully-connected neural networks?

II   Interview tips

Every interview is an opportunity to show your skills and motivation for the role. Thus, it is important to prepare in advance. Here are useful rules of thumb to follow:

Listen to the hints given by your interviewer.

Example: You’re asked to calculate the number of parameters of a convolution layer and state that “the number of weights is $\text{filter width} \times \text{filter height} \times \text{number of filters}$.” If your interviewer questions you with “are you sure?” or “can you recall the shape of the input volume?”, there is a high chance your answer is imprecise or wrong. You should reconsider your answer. In this scenario, the interviewer expects you to remember that filters of a convolution layer have a depth equal to the depth of the input to the layer. Thus, the right count is $\text{filter width} \times \text{filter height} \times \text{input depth} \times \text{number of filters}$. If the interviewer asks you “is that all?,” it’s probably not. Parameters usually englobes weights and biases, so you’re expected to count the number of biases as well.

Study the data type that the company deals with.

Example: If the company stores petabytes of videos, learn about video data, video processing, and video compression methods ahead of time.

Study the type of algorithm relevant to the company’s product.

Interviewers tend to ask questions from their own area of expertise.

Example 1: If the team is working on a face verification product, review the face recognition lessons of the Coursera Deep Learning Specialization (Course 4), as well as the DeepFace (Taigman et al., 2014) and FaceNet (Schroff et al., 2015) papers prior to the onsite.

Example 2: If the team is building a robotic arm that will detect and grasp objects, you might want to look at popular object detection, object segmentation, and reinforcement learning algorithms. Furthermore, you can loose credibility by mispronouncing a technical word or acronym. In this scenario, pore over vocabulary such as piece-picking, sorting, kinematics, gears, grippers, and the like.

Write clearly, draw charts, and introduce a notation if necessary.

The interviewer will judge your scientific rigor.

Example: You’re asked to write the relationship between the input and the output of a softmax layer. Instead of writing $\text{softmax}(z) = \frac{e^z}{\sum e^z}$ , write $\text{softmax}(z_1, z_2, …, z_n) = (\frac{e^{z_1}}{\sum_{i=1}^n e^{z_i}}, \frac{e^{z_2}}{\sum_{i=1}^n e^{z_i}}, …, \frac{e^{z_n}}{\sum_{i=1}^n e^{z_i}})$. In this fashion, you’ll display your meticulous understanding of softmax applied to a multi-dimensional object and the shape of its output.

When you are not sure of your answer, be honest and say so.

Interviewers value honesty and penalize bluffing far more than lack of knowledge.

Example: Assume the interviewer asks you about the logistic loss. You remember logistic regression and the logistic function, but not the logistic loss. Rather than answering vaguely, you could say “I’m familiar with logistic regression and logistic function, but I don’t think I’ve heard of the term logistic loss. Could you provide more context?”. The interviewer might add that logistic loss is analogous to the binary cross entropy loss. It may help you answer, or at least invite the interviewer move to the next question.

Don’t mention methods you’re not able to explain.

Example: You’re explaining early stopping and state that “early stopping is a regularization method, like dropout.” In this scenario, you can expect the interviewer to ask: “could you explain dropout?”

When out of ideas or stuck, think out loud rather than staying silent.

Talking through your thought process will help the interviewer correct you and point you in the right direction.

III   Resources

The deep learning section of the Workera test is a great way to prepare for this interview. It’ll provide you with a personalized study plan which includes a list of your strengths and weaknesses, along with curated training material to prepare for interviews or transition in your career. Additionally, here’s a list of useful resources to prepare for the deep learning algorithms interview.

Deep learning engineers carry out data engineering, modeling, and deployment tasks. They demonstrate solid scientific and engineering skills. Communication skills requirements vary among teams. This role is a variant of machine learning engineer. It requires deep learning skills in addition to the skills profile presented in the figure above.

Deep learning researchers carry out data engineering and modeling tasks. They demonstrate outstanding scientific skills (see Figure above). Communication skills requirements vary among teams. This role is a variant of machine learning researcher. It requires deep learning skills in addition to the skills profile presented in the figure above.

People who have the title software engineer-machine learning carry out data engineering, modeling, deployment and AI infrastructure tasks. They demonstrate solid engineering skills and are developing scientific skills. Communication skills requirements vary among teams. This role is a variant of software engineer-machine learning. It requires deep learning skills in addition to the skills profile presented in the figure above.

A video comprises a sequence of image frames displayed in rapid succession. Thus, understanding a video requires analysing each individual frame and the temporal correspondences lurking across multiple frames.

For instance, mispronouncing widely used dataset acronyms such as MNIST or CIFAR will affect your credibility.

The logistic loss measures the performance of a binary classifier.

Early stopping is a regularization method used to avoid overfitting when training a machine learning model with an iterative method like gradient descent.

Author(s)

1. Kian Katanforoosh - Founder at Workera, Lecturer at Stanford University - Department of Computer Science, Founding member at deeplearning.ai

Acknowledgment(s)

1. The layout for this article was originally designed and implemented by Jingru Guo, Daniel Kunin, and Kian Katanforoosh for the deeplearning.ai AI Notes, and inspired by Distill.

Footnote(s)

1. Job applicants are subject to anywhere from 3 to 8 interviews depending on the company, team, and role. You can learn more about the types of AI interviews in The Skills Boost. This includes the machine learning algorithms interview, the deep learning algorithms interview, the machine learning case study interview, the deep learning case study interview, the data science case study interview, and more coming soon.
2. The Workera test consists of 7 sections with multiple choice and coding questions. The deep learning section covers deep learning models and methods to train them (e.g. initialization, regularization, optimization, transfer learning, etc.)
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