MATHEMATICAL MODEL OF THE MARKET

Transcription

1 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN MATHEMATICAL MODEL OF THE MARKET Yu. K. Mashunin, I. A. Mashunin Vladivostok. Far Eastern Federal University, Russia. Abstract. The paper presents a methodology of creating a mathematical model of the market as a vector (multiobjective problem of mathematical programming or a vector optimization problem (VOP. The model is structured in such a manner that it takes into account, first, the balance of supply and demand, secondly, purposefulness of each market participant (a manufacturer and a consumer. The methods based on normalization of criteria and principles of guaranteed result are applied to solve the VOP. The methodology of modeling is shown through solving practical tasks of vector optimization - models of the one- product market with two manufacturers and consumers. Key words: Marketing Model, Modeling, Market, Supply and demand, Management, Decisions making, Vector optimization, Introduction Marketing model construction and simulation of its behavior is one of the main problems of econometrics. This question has been of much attention in the world s advanced countries since the middle of the nineteenth century. The majority of market models were made on the basis of competitive equilibrium establishment, about which Walras wrote in his work (874. In the 950s Arrow, Debreu (954, McKenzie (954, Gale (967, Nikaido (959 gave the mathematical justification of a Walras s hypothesis in their works. Further there were research works on improvement of models and their generalization. Their descriptions are given fully enough in the monographs of Morishima (964, Nikaido (968, and of contemporary authors, such as Scherer and Ross (990, Tirole (99. The majority of these works examined the balance of the cumulative supply and demand (that is market equilibrium, Nikaido (968. The models found the balance between supply and demand, but they could not be considered marketing models, as, first, they did not take into account competition both between manufacturers, and between consumers; secondly, they did not include purposefulness of the market participant actions (manufacturers and consumers, which is the basis of competition. Game models were also of no effect. Marketing model should include not only contradiction between manufacturers and consumers, but also contradiction (competition between each manufacturer; the same can be said about consumers. As a rule, what is good for one manufacturer (consumer is bad for another and vice verse. Thus, the marketing model should reflect not only balance between supply and demand but also purposefulness of each market participant and their interrelation. So the vector (multiobjective problem of mathematical programming is that mathematical model which can reflect balance along with the purposefulness of each market participant. Some methods of solution are developed. They are based on normalization of criteria and a principle of guaranteed result, Mashunin, (986, 000. Thus, it is important to solve the following problems: to develop a marketing mathematical model (simulator. At first there must be a one - product model, then a multiproduct one; to use marketing models in economic development of region, nation. The purpose of the present work is to create a model of the market as a vector problem of mathematical programming, which includes the balance between supply and demand and the purposefulness of each market participant. The work is designed for solution of practical tasks of market modeling. For this purpose the work gives the methods of solution of vector optimization problems, which underlie model of the one-product market with 45

2 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN two manufacturers and two consumers (Model and their testing. Model construction of the market The model examines the market with one product. Q manufacturers and L consumers of the product are participate: q =, Q, - is an index (q and set (Q of the manufacturers (firms. l =, L - is an index and set of the consumers. Fig.. Initial data market on time tt Person reshegny decision maker tt Manufacturer Supply Model of the market Vector problem: Opt F(X(t+, G(X(tB, X(t+0. Goods Goods ( service Manufacturer q Goods Price Price Manufacturer Q Fig.. Market scheme and the relationship with the model Results of the solution - the market at the moment time (t+t Market - The object of study Demand Consumer Consumer l Consumer L To construct a mathematical model of the market let s create for each manufacturer and consumer: a vector variable, restrictions imposed on production operation and their purposes (criteria of functioning. The manufacturers (firms X = {x ql, q = } is a vector variable, determining volume of production made by qq firm and, Q, l =, L sold to ll consumer for some final period of time tt, (the time is not specified; c q is a cost per unit established by qq firm in the market (for all consumers is the same; c q x ql is an amount of money received by qq firm in the market from ll consumer; L f q (X= c q x ql is the value, which characterizes the amount of money received by qq firm in the l market from all consumers. Resource (costs characteristics: i = - is an index and set of all resources used by qq firm for production;, M q i = - is an index and set of material resources used for production, M mat M q ;, M mat a iq, i =, M mat - costs of im mat resource per unit of output produced by qq firm; We assume that there is a linear, functional dependence of costs increase on output: L g i (X= a iq x ql, i =, M mat, l where g i (X - costs of im mat resource for the whole volume of production. q=,q equivalent q=,,, Q. 46

3 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN The following calculations are carried out similarly: labour expenses (cost of labour per unit of production. i=, M - an index and set of labour resources used for production, M lab M q ; direct costs connected with production capacities per unit of any product, M cap M q. With the above-mentioned calculations, the cost price per product unit (variable costs is computed as follows: a qv = M mat i c a i iq M lab i c a i iq M cap i c a i qj, q =, Q, where c i is cost of material, labour and capacity resources respectively. The same scheme is used to calculate overheads (constant per product unit - a qo. In the whole cost price of unit production is estimated as: a q = a qv + a qo, q =. (. L l, Q We assume that there is also a linear, functional dependence of costs increase in (. on output: a q x ql, q =, Q. p q = c q - a q is profit received by firm from the product unit, q = b iq, i = a planned period. b q = M, M, q =, Q i q lab, Q. - is value of im q resource available at qq firm and used to produce a product for c i b iq - financial opportunities of the firm for production, q = imposed on the firm during production are estimated as follows: L l a q x ql b q, qq. (., Q. From here the restrictions The purpose of any manufacturer is to sell the maximal volume of goods at the highest price possible in order to get the highest profits possible. It can be described as a problem of mathematical (linear programming: L max f q (X= p q x ql, (.3 l with restrictions (., qq. The consumers X = {x ql, q = } is a vector variable, determining volumes of the product bought by ll, Q, l =, L consumer from qq manufacturer (firm, which is the same with a vector variable, determining actions of the manufacturer. The actions of the of the consumer are restricted by minimal b min l, l =, L and maximal b max l, l =, L volume of financial assets which he can allocate to buy a product from different firms: Q b min l c q x ql b max l, l=, L, (.4 q where c q is cost per unit established by qq firm in the market. The main "to buy - to not buy "actions of the consumer when he purchases a product are: the price c q established qq by the manufacturer. The consumer tries to choose the product with c q, qq as small as possible; Currently, there are several alternative models of firm behavior: a model of profit maximization, maximize sales, maximize growth, a model of managerial behavior; maximization model value added (Japanese model. Seo (

4 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN characteristics of the product: quality, location of outlets, time of access, advertising and so on. Let's notice, that the quality rating of the product q, which is established by all the consumers for the q-th manufacturer, can be fixed in 00-point scale, i.e. q 00. At the same time the consumer wants to choose the goods of the highest quality. The goal of any consumer is to buy the necessary volume of goods of the highest quality at the lowest price possible. It can be expressed as a problem of mathematical (linear programming (PMLP: Q min f l (X = c q x ql, (.5 q with restrictions (.4, ll. The problem of mathematical programming (.5 is the model of behavior of any ll consumer. In the model (.5 consumer takes into account quality and other characteristics of the product implicitly in its price. Model of one-product market As a whole, let s present a mathematical model of the one-product market as a vector problem of mathematical (in this case linear programming considering both the purposes of all manufacturers (.3 and consumers of a product (.5, and the restrictions imposed on their opportunities (. and (.4: opt F(X = {max F (X={max f q (X = b min Q l L l x ql 0, q= q a q x ql b q, q = min F (X={min f l (X = L l Q q p q x ql, q=, (.6, Q c q x ql, l= }}, (.7, L c q x ql b max l, l=, L, (.8, Q, l=, L, Q, (.9, (.0 where F(X is a vector criterion function. Thus, (.6-(.0 is problem of vector optimization, where K=QL is the set of criteria - manufacturers and consumers, respectively; in (.6 F (X - vector criterion "Q" producers, each component of which is the manufacturer, maximizing their profits f q (X, q=, Q ; in (.7 F (X - vector criterion "L" consumers who minimize their costs, due to the cost for purchased products; (.8 is budgetary constraints (financial opportunities "L" consumers; (.9 is restriction concerning production capacities of the "Q" manufacturers; (.0 - limitations associated with no negative volumes sold. In the model (.6-(.0, quality and other characteristics of the product, which the consumer takes into account, are determined by the priority of this or that criterion. The concept of criterion priority is given Mashunin, (986, 000. The following chapter contains in brief the methods of solving a vector problem of linear programming (.6- (.0 based on normalization of criteria and a principle of guaranteed result. For more detail see Mashunin, (986, Methods of the solution of the problem of vector optimization In general view a vector problem of mathematical programming or vector optimisation problem (VOP is considered as follows: Opt F(X = {max F (X = {f k (X, k =, K }, (. 48

5 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN min F (X = {f k (X, k =, K }}, (. G(X B, X0, (.3 where X = {x j, j = } is the vector of unknown quantities;, N F ( X { f ( X, k } is the vector criterion function, or criterion vector, its every component is k to be maximised; F ( X { f ( X, k } is the same but every component of the function is to be minimised; k K K K is the set of indices; G( X B, or{ gi ( X bi, i, M} the vector of constraints. Let s assume, that the problem (.-(.3 belongs to convex problems, and the set of allowable points S in restrictions (.3 is not empty and presents a compact. N S { X R X 0, G( X B} (.4 Then there exists a solution vector, X, if the problem (.-(.3 is solved against each criterion individually. To solve the problem (.-(.3 the axioms of equality, equivalence and criterion priority have been worked out, which are based on normalisation of criteria and the principle of a guaranteed result. This allows to divide a Pareto optimal set S o into a point subset where all the criteria are equivalent (the subset contains one point X o S o and subsets S o q, q=, where a criterion qk has priority over the rest: K q o o S q X =S o, S o q So S, q=. Therefore, the solution of the problems is reduced to selecting a point from the corresponding subset. Thus, the decision problem (.-(.3 is brought to the solution of the problem: o = k (X, k= max X G(XB, X 0, where min p q k k(x, (.5 kk k (X=(f k (X-f o k /(f k -f o k, k=, XS, (.6 is the relative estimate of a point XS against the k-th criterion; f k, k= the k-th criterion if VOP (.-(.3 is solved against each criterion individually; f o k k-th criterion: f o k = min f k (X, kk, f o k = X S kk is the relative estimation k (X, k = max f k (X, kk ; XS of the q-th criterion over the k-th, p q k = q(x/ k (X, qk, k = lays within the limits of 0 k (X, k =, with p q k is the optimal value of, k= - the worst value of the ; p q k - is the priority =, qk, k = if VOP is solved with the equivalent criteria. The vector of priorities P q ={p q k, k= }, qk is calculated in an interactive procedure of taking solutions from a ratio: q (X o / k (X o p q k q (X q / k(x q, qk, k=. (.7 Entering of an additional variable = min p q k k(x, (or p q k k(x, qk, k= will transform a maximin problem (.4 into one criterion -problem: o = max, (.8 kk 49

6 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN , k=, - p q k (f k(x-f o k /(f k -f o k G(XB, X0. The -problem (.8 is solved by standard methods. Solution of the -problem (.8 will result in getting a point of an optimum X o estimation o, such as o k (X o, k=, X o S, (.9 in VOP with equivalent criteria, where p q k =, qk, k= ; o p q k k(x o, qk, k=, X o S, (.0 and maximal relative in VOP with the given priority of the q-th criterion over the rest. Thus, o is the maximal lower level for all the relative estimations k (X o, k=, (guaranteed result in relative units, and point X o S is optimum against Pareto according to the theorem.3 (Mashunin, (986, p.3. Changing p q k, qk, k=, in (.7 makes it possible to calculate any point from Pareto set with the given accuracy, Mashunin, (03. Technique of solution of VOP (.-(.3 with equivalent criteria and given priority of the q-th criterion over others according to Mashunin, (986, 03 for model of the one-product market is shown in Chapter Solution of the vector optimization problem, underlying the model of the one-product market Let's examine the solution of VOP (.6-(.0, representing the mathematical model of the one-product market. Solving the problem (.6-(.0 with equivalent criteria on the basis of normalization of criteria and principle of guaranteed result, we shall calculate: a point of an optimum X o ={x ql, q=, Q, l=, L }, which determines output of a product sold by each manufacturer to each consumer; values of criterion functions f k (X o, k= =QL, including, f q (X o, q=,q, which determine the income (benefits of each manufacturer; f l (X o, l=, L, which determine expenditures of each buyer; The maximal relative estimation o, where o k (X o, k=, X o S, where k (X o =(f k (X o -f o k /(f k -f o k, k=, Xo S is normalized criterion (the relative estimate, in which f k - is the best solution against criterion kk, f o k - is the worst one accordingly=ql is the set of criteria. The maximal relative estimation o can be interpreted as a maximum level of the mutual interests of all manufacturers and consumers in relative units. Any increase of interests (criterion of any manufacturer or consumer changes for the worse for all the rest participants of the market (manufacturers and consumers. The received point is an optimum against Pareto. A vector function F (X={f q (X o, q=, Q } is a function of supply (offer, and a vector function F (X={f l (X o, l= L, } is a function of demand. The theorem concerning the analysis of results of the VOP (.-(.3 with equivalent criteria (and vector function F (X and F (X analysis in particular is given below. Theorem. (Theorem of the most Contradictory criteria in VOP with equivalent criteria. In a convex vector problem of mathematical programming (.-(.3 with equivalent criteria solved on the basis of normalization of criteria and a principle of guaranteed result, in an optimum point X o there will always be 50

7 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN two criteria with indexes of qk, pk (let s call them the most contradictory criteria of all criteria k=, for which the following equality is true: o = q (X o = p (X o, q, p K, XS, (3. for the rest of criteria, the following inequalities are true: o k (X o, kk, q p k. The proof of the theorem, Mashunin, (986, 03. Let s look into the methodology of the solution of the vector problem (.6-(.0, simulating the one-product market in the form of test examples. In the first test example the market is p[resented by two manufacturers and two consumers (we shall call the model of such market - model with same parameters (with the same prices for the goods and costs. Model was solved, first, with equivalent criteria, secondly, with the given priority of criterion, where the priority of criterion simulates affinity (distance one of the manufacturers to the consumers or quality of the goods. 4. Model of the one-product market with two manufacturers and two consumers (Model 4.. Construction of the model of the one-product market For the analysis of market structures consider the model (.6-(.0 for the most about my situation, when the market with the same item participate, two manufacturers, and two of the consumer of the product. The closest to this situation duopoly model Cournot, Stackelberg, Bertrand and Adiwarta, Seo (000, which consider the relationship only two manufacturers (firms. Let us enter designations: x, x are volumes of a product sold by the first manufacturer, x 3, x 4 the same for the second manufacturer to the first and the second consumer accordingly; c = c = c 3 = c 4 = 0 is the price for a product; a = a = a 3 = a 4 = 8 are the costs of production for both manufacturers; p = p = p 3 = p 4 = (c -a = - profit from production and sales of a product for both manufacturers; b min l = 700, b max l = 000, l =, are minimal and maximal volumes of financial assets, which the first and second consumer can allocate to buy the product from different firms. b q = 000 are financial opportunities of a firm for manufacturing the product, q =,. The entered designations indicate, that each firm can make no more than x max q = b q /a q = 5, q =, units of the product, hence: considering the cost c q x max q = 50 it is more, than minimal sum of financial assets, which one consumer can allocate to buy the product from different firms b min = b min = 700, but it is less, than the first and second consumer allocate together b min + b min = 400; considering the sum as x max + x max = 50 and its costs as c q (x max + x max = 500 it is more, than maximal sum of financial assets, which the first and second consumer can allocate to buy the product from different firms b max + b max = 000. Let us present the mathematical model of the market of two manufacturers and two consumers (model as a vector problem of linear programming: opt F(X= {max F (X ={max f (X = p x + p 3 x 3, max f (X = p x + p 4 x 4 }, (4. min F (X = {min f 3 (X = c x + c x, min f 4 (X = c 3 x 3 + c 4 x 4 }}, (4. with restrictions 700 c x + c x 000, 700 c 3 x 3 + c 4 x 4 000, (4.3 a x + a 3 x 3 000, a x + a 4 x 4 000, (4.4 x, x, x 3, x 4 0. (4.5 Let us solve the vector problem (4.-(4.5 on the basis of normalization of criteria and principle of guaranteed result with equivalent criteria and given priority of criterion. 5

8 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN In the market model (4.-(4.5 depending on the changes of parameters of model - commodity prices and resource costs are considered: model with the same parameters that corresponds to the model of perfect competition; model of oligopoly in which the parameters can be changed, i.e. in some companies there is the possibility of domination over other firms at prices of resources and other parameters. model monopoly, which has one criterion (4. manufacturers and multiple criteria (4. users (in our example, two; model of monophony in which there are many criteria (4. manufacture lei and one criterion (4. of the consumer. Below is a model of perfect competition and model of oligopoly in which the decision is submitted at the same criteria and specified the priority criteria. 4.. Solution of a vector problem simulating the market with equivalent criteria The economic essence of the solution of a vector problem (4. - (4.5 is that all the purposes (criterions of both manufacturers and consumers are equalized as relative estimations in joint optimization. Let us present the solution of the vector task (4.-(4.5 with equivalent criteria on the basis of normalization of criteria and principle of guaranteed result as a sequence of steps according to Mashunin (986, 03. Step,. The problem (4.-(4.5 is solved against each criterion separately. Let us look for the best solution (X k and the worst solution (X o k, i.e. for kk =Q is determined the maximum and minimum (f k = max f k (X, XS f o k = min f k (X, kk, and for kk =L the minimum and maximum (f k = min f k (X, f o k = max f k (X, kk ; X S are searched accordingly. The solution of the problem (4.-(4.5 results in: Criterion. max: X ={x =6.5, x =6.5, x 3 =37.5, x 4 =37.5}, f (X =50.0, f (X =50.0, f 3(X =000.0, f 4(X = 000.0, min: X o ={x = 7.5, x = 7.5, x 3 =6.5, x 4 =6.5}, f (X o = 30.0, f (X o = 50.0, f 3(X o = 700.0, f 4(X o = Criterion. max: X ={x =37.5, x = 37.5, x 3 =6.5, x 4 = 6.5}, f (X = 50.0, f (X = 50.0, f 3(X = 000.0, f 4(X = min: X o ={x = 6.5, x = 6.5, x 3 =7.5, x 4 = 7.5}, f (X o =50.0, f (X o = 30.0, f 3(X o = 700.0, f 4(X o = Criterion 3. min: X 3 ={x =35.0, x =50.0, x 3 =35.0, x 4 = 50.0}, f (X 3 = 70.0, f (X 3 = 70.0, f 3(X 3 =700.0, f 4(X 3 = 000.0, max: X o 3 ={ x =50.0, x =50.0, x 3 =50.0, x 4 = 50.0}. f (X o 3 = 00.0, f (X o 3 = 00.0, f 3(X o 3 = 000.0, f 4(X o 3 = Criterion 4. min: X 4 ={x =50.0, x =35.0, x 3 = 50.0, x 4 = 35.0}. f (X 4 = 70.0, f (X 4 = 70.0, f 3(X 4 =000.0, f 4(X 4 = max: X o 4 ={x = 50.0, x =50.0, x 3 = 50.0, x 4 = 50.0}. f (X o 4 = 00.0, f (X o 4 = 00.0, f 3(X o 4 = 000.0, f 4(X o 4 = The economic sense of the solution of the vector problem (4.-(4.5 at the first step is that every market participant are given the most favorable conditions, i.e. calculations take into account only their restrictions and 5 X S XS

9 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN determine their optimum opportunities. In the result we receive the optimum solutions f k =f k (X k, k= - which are purposes every market participant wants to achieve. Step 3. The standard normalization of criteria is carried out: k (X = (f k (X - f o k /(f k - f o k, k =, X S. In the problem (4.-(4.5 kк the relative estimation k (X, k= lies within the limits of 0 k (X, k К. At this step the analysis of optimum points is carried out. (Note, the points lye on the frontier of Pareto set. Let us carry out the analysis of the results of the solution against each criterion on the basis of normalization: k (X q =(f k(x q -f o k /(f k -f o k, (q, k =,,3, The economic essence of the solution of the vector problem (4.-(4.5 at the third step is that the purposes of each market participant are leveled in numerical value. In the result we receive the normalized purposes k = k(x k = (00%, kк, which each market participant wants to achieve. Step 4. Construction of a -problem. Let us XS determine a level = min k (X or k (X, k = and maximize it against XS. The result is a maximin problem with normalized criteria, o = max min k (X. XS kk This problem is transformed to a -problem: o = max, (4.6 - (f k (X - f o k /(f k - f o k 0, k =, (4.7 G(X B, X 0. (4.8 In the appendix to the problem (4.-(4.5 - problem (4.6-(4.8 will take the following form: o = max, (4.9 with restrictions - (p x + p 3 x 3 - f (X o /(f (X -f (X o 0, (4.0 - (p x + p 4 x 4 - f (X o /(f (X -f (X o 0, - (c x + c x - f 3 (X o 3 /(f 3(X 3 -f 3(X o 3 0, - (c 3 x 3 + c 4 x 4 - f 4 (X o 4 /(f 4(X 4 -f 4(X o 4 0, 700 c x + c x 000, 700 c 3 x 3 + c 4 x 4 000, a x + a 3 x 3 000, a x + a 4 x 4 000, x, x, x 3, x 4 0. The economic essence of the solution of the -problem (4.9-(4.0 is that the -problem tends the criteria (relative estimations of each market participant to the optimum (X k =, k=. Step 5. The solution of the -problem. The -problem is a standard problem of convex programming and the standard algorithms of optimization are used to solve it. The solution of the -problem results in receiving an optimum point X o, the values of objective function f k (X o, k= and maximal relative estimation o, which meets the conditions (.9, i.e. o is the maximal lower level for all relative estimations k (X o - (guaranteed result in relative units. 53 kk

10 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN The solution of the -problem (4.9-(4.0. (Solution with equivalent criteria. o = X o ={ x o = , x o = , x o 3 = , x o 4 = }. Objective functions (criteria in the optimum point f k (X o, k = : f (X o = 63.57, f (X o =63.57, f 3 (X o = , f 4 (X o = , Resource Costs A(X o : Financial costs, consumers: c x o + c 3x o 3 =87.858, c x o + c 4x o 4 =87.858; Financial costs, Manufacturer: a x o + a x o =654.8, a 3x o 3 + a 4x o 4 = Relative valuations at optimum k (X o, k = : (X o = , (X o =0.6074, 3 (X o =0.6074, (X o =0.6074, that meet the conditions of (.9 o q (X o, q =,,3,4, but all the criteria are contradictory according to (.0. Any improvement (increase of one of them will results in deterioration of the others Model of the market, with the given priority of criterion The result received in the previous Chapter, is determined on the basis of criterion equivalence in the problem (4.-(4.5, i.e. criteria of both manufacturers and consumers do not a priority (preference over each other. In real conditions there can be the priority of some criterion, for example, let us assume for the problem (4.-(4.5 that the second manufacturer (q= is closer to the consumers than the first one, i.e. he has a priority in sales over him, qk (K=4 in the VOP (4.-(4.5. In this case the solution is taken out of a subset of points with priority on the q-th criterion S o q So, lying between points X o and X q, qk in Pareto set So. The axiom of the choice of the criterion qk priority and of the method of solution of a vector problem with given priority of criterion based on this axiom are stated by Mashunin, (986, 03. Let us present this method as a sequence of steps on the example of the problem (4.-(4.5. Step. The problem (4.-(4.5 with equivalent criteria is solved. The algorithm of the solution is given in the previous Chapter. The solution result in receiving: points of an optimum X k, k= and value of criterion functions in these points f k =f k (X k, k =,K, which are the frontier of Pareto set; the worst unchangeable part of criterion f o k, k= ; The solution of the maximin problem and -problem, made on its basis, results in receiving X o which is the point of an optimum of the solution of VOP with equivalent criteria and maximal estimation o, which is the guaranteed level for all the criteria in relative units. Let us calculate values of all the criteria q= :, K in each point X k, k= {f q (X k, q= }, k=, (4. which show the value of each qk criteria on moving from one optimum point to another. In the point X o we shall calculate value of criteria f k (X o and all relative k (X o, k=,k, which meet the inequality o k (X o, k=,k, X o S, i.e. the point X o S o S is the "center" of Pareto set in relative units, where all the criteria are above or equal to o. In other points XS the smallest in relative units criterion = min k(x is kk always less than o, i.e. o is a maximum point of a dome made of the minimal relative estimations in set of points optimum on Pareto. We shall calculate a matrix of relative estimations in points of an optimum out of (4. { q (X k = (f q(x k - fo /(f q - fo, q = }, k =. (4. 54

11 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN The matrix (4. shows values of all the criteria in relative units on the frontier of Pareto set. This information is that very basis for further study of structure of Pareto set. The priority criterion qk is chosen on the basis of this information. The data of the -problem (4.6-(4.8 are stored. Step. As a priority, we choose the second criterion, ie, the second producer has priority over other market participants. This was reported on the display. Computer processes the data that are related to the second criterion, and outputs to the display value criteria and relative valuations in the points X and X o : f k (X f k (X f k (X o, k (X k (X k (X o ; ( f (X 63.57, (X ; 50.0 f (X 63.57,.0 (X ; f 3 (X , (X ; f 4 (X , (X ; The vector of priorities of the second criterion over the others of P k (X = (X/ k (X, k=,,3,4 in the points of extremum X, Xo P k (X P k P k (Xo, k =, 4. is calculated, i.e. the limits of change of a vector of priorities are determined: In the given problem the vector of priorities lies within the limits: P k ={.8333 p.0,.0 = p =.0, p 3.0, p 4.0}. These data are the basic information for decision-making. Step 3. After analyzing the priority criterion, the person, who makes the decision, specifies the numerical value f q desirable to be received from the matrix (4.3: 50.0 f (X Let us assume that the desirable value of the second criterion is equal to f =00. Step 4. The relative estimation is calculated as follows: = (f - f o /(f -f o = Step 5. Let us calculate the proportional factors between q (X o, q, q (X q assuming the linear character of change of criterion f q (X in (4. and to a relative estimation q (X in (4. accordingly: = ( q - q (X o / ( q (X q - q(x o, qk. In our example it is: = ( - (X o /( - (X o = Step 6. Assuming linear character of change of a vector of priorities P q k (X, k=, we calculate it for using received factor of proportionality : p q k = p q k (Xo + (p q k (X q - p q k (Xo, k =,K, q K. In our example: p k = {p =.35, p =.0, p 3 =.0, p 4 =.0}. Step 7. Construction of a -problem. The vector of priorities p k, k=, 4 is entered into the -problem (4.6- (4.8, which results in receiving a -problem with a priority of the q-th criterion o = max, (4.4 - p q k k (X 0, k=, (4.5 k=,k ; G(X B, X 0 (4.6 Step 8. Solution of the -problem. The solution results in: Point of an optimum X o = {x o j, j=, N } and maximal relative estimation o where o p q k (X o k (X o, 55

12 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN values of criterion functions f k (X o, k= and relative estimations k (X o, k= ; The value f q (X o, qk is not usually equal to given f q. The error of choice f q is defined by the error of linear approximation of the real change (X, k= K, qk, carried out at steps 5 and 6. of relative estimations k (X o, k=,k and the vector of priorities p q k If the error f q = f q (X o - f q is less than f, then one should move to step 0, if f q f, then the next step is carried out. (To solve our example one step is made, the results are at the end of the algorithm. Step 9. The vector of priorities of criterion qk is determined in point X o p q k (Xo = q (X o / k (X o, k=, qk. The new vector of priorities is under construction p q kw = (p q k + p q k (Xo /, k=, qk. Move to step 7. Here the -problem is solved with a new vector of priorities p q kw, k=, qk instead of p q k, k=, qk. Two manufacturers and two consumers of the product participate in the one-product market for a period tt. Let us consider the second case with the different in value parameters for two manufacturers (the resource expenses and cost - model of oligopoly and without priority of criterion determining quality of a product. The designations and values of parameters are similar to 5.. Only the differences are shown: 56, Step 0. Will the VOP be solved with priority criterion of another value? If the answer is "yes", then move to step 3, if - "no ", the following step is carried out Step. Will the VOP be solved with another priority criterion? If the answer is "yes", then move to a step, if no follow the next step. Step. The end. At step 8 the solution of the -problem results in the following: o = , X o = {x o = 4.88, x o = 34.6, x o 3 = 45.59, x o 4 = 53.86}, Objective functions (criteria in the optimum point f k (X o, k = f (X o =55.0, f (X o =98.9, f 3 (X o = , f 4 (X o = , Resource Costs A(X o : : Financial costs, consumers: c x o + c 3x o 3 =884.83, c x o + c 4x o 4 =884.83; Financial costs, Manufacturer: a x o + a x o =60.05, a 3x o 3 + a 4x o 4 = Relative valuations at optimum k (X o, k = : (X o =0.5687, (X o = , 3 (X o = , 4 (X o = Check: o p (X o, (X o, p 3 3 (X o, p 4 4 (X o, where {p (X o = (X o = p 3 3 (X o = p 4 4 (X o =0.7678}. The error of linear approximation f q = f q (X o - f q = =.09 (0,545 %. Carrying out the subsequent steps of the algorithm can reduce it. (In the example the steps 0, are not carried out The given example has shown a practical opportunity of choice of any point from Pareto set with the given accuracy. 5. Model of the one-product market with different parameters for the manufacturers 5.. Construction of marketing model with different parameters (model of oligopoly and its solution with equivalent criteria

13 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN c = c 3 = 0, c = c 4 = are prices for the product for the first and the second manufacturer accordingly; a = a 3 = 8, a = a 4 = 0 are expense for manufacture of a product for both manufacturers. These data are inserted in the VOP (4.-(4.5. The result shall be marketing model with different parameters. Let us present algorithm for its solution with equivalent criteria. Step,. The problem (4.-(4.5 is solved against each criterion separately. As a result of the solution we shall receive: Criterion. max: X ={x =6.5, x =6.5, x 3 =3.0, x 4 =3.0}, f (X =50.0, f (X =4.0, f 3 (X =997.0, f 4 (X = 997.0, min: X o ={x = 0.0, x =0.0, x 3 = 50.0, x 4 =50.0}, f (X o = 40.0, f (X o = 00.0, f 3 (X o = 700.0, f 4 (X o = f (X o = 40.0, f (X o = 00.0, f 3(X o = 700.0, f 4(X o = Criterion. max: X ={x =3., x =3., x 3 = 50.0, x 4 = 50.0}, f (X = 8.4, f (X = 00.0, f 3 (X = 9., f 4 (X = 9., min: X o ={x =6.5, x = 6.5, x 3 = 6.5, x 4 = 6.5}, f (X o =50.0, f (X o = 5.0, f 3 (X o = 700.0, f 4 (X o = Criterion 3. min: X 3 ={x =33.55, x =45.6, x 3 =30.37, x 4 = 37.3}, f (X 3 = 58.34, f (X 3 = 35.37, f 3 (X 3 =700.0, f 4 (X 3 = , max: X 3 o ={ x =48., x =48.3, x 3 =43.7, x 4 = 35.39}. f (X 3 o = 93.06, f (X 3 o = 57., f 3 (X 3 o = 000.0, f 4 (X 3 o = Criterion 4. min: X 4 ={x =45.6, x = 33.55, x 3 =37.3, x 4 = 43.7}. f (X 4 = 58.34, f (X 4 = 35.37, f 3 (X 4 =904.0, f 4 (X 4 = max: X 4 o ={x = 48.33, x =48.9, x 3 = 35.39, x 4 = 43.7}. f (X 4 o = 93.06, f (X 4 o = 57., f 3 (X 4 o =908.0, f 4 (X 4 o = Step 3. The standard normalization of criteria and the analysis of the results of the solution against each criterion are carried out: Step 4. Construction of a -problem in form of (4.9-(4.0. Step 5. The solution of the -problem (4.9-(4.0 with equivalent criteria. The solution of the -problem. (Solution with equivalent criteria. o = 0. 6, X o = {x o = , x o = , x o 3 = 3.986, x o 4 = 3.986}, Objective functions (criteria in the optimum point f k (X o, k = : f (X o = 68.33, f (X o = 3.94, f 3 (X o = 86.66, f 4 (X o = 86.66, Resource Costs A(X o : Financial costs, consumers: c x o + c 3x o 3 =86.66, c x o + c 4x o 4 =86.66; Financial costs, Manufacturer: a x o + a x o =673.33, a 3x o 3 + a 4x o 4 = Relative valuations at optimum k (X o, k = : (X o = 0.6, (X o =0.6, 3 (X o =0.6, (X o =0.6, that is the conditions (.9 o q (X o, q =,,3,4 are met, and according to (.0 all the criteria are contradictory. Any improvement of one of them results in deterioration of the other. Thus, the increase of the product price of the second manufacturer has resulted in decreasing the quantity of the sold products from 8.78=x +x 4 (see Unit 4. to 65.97=x +x 4. The difference of sales volume was compensated by the first manufacture whose price for the goods is less. 57

14 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN Marketing Model with a priority of criterion, determining the quality of the goods The models of the one-product market presented in the previous Chapters do not take into account the goods quality and solution is carried out on the assumption of equivalence of criteria in VOP (4.-(4.5. In real life the consumers pay no less attention to the goods quality than to its price. Therefore, in the basic model of the oneproduct market (.6-(.0 the description of the goods quality appreciated by the consumers can be presented as a priority of criterion of the manufacturer with the higher goods quality. Let us examine how the goods quality influences results of modeling of the one-product market with two manufacturers and two consumers (4.-(4.5. Unit 5.3 presents the solution with a priority of the second criterion in model (4.-(4.5. This approach will be used for estimation of the goods quality. Let us carry out the analysis of the problem solution with equivalent criteria given in the previous chapter and compare it with the results of the solution of model with identical parameters. The results show, that sales volumes of the second manufacturer (q= has decreased from x +x 4 =8.786 (the value of Unit 5. to x +x 4 =65.97 (see previous unit. As a consequence, the profits of the second manufacturer have fallen. There are the questions: "Is it possible to increase sales volume at the expense of quality?" and "How much should the quality of goods be improved to make level the profits of the first and second manufacturers? " Let us present the solution of VOP (4.-(4.5 with a priority of the second criterion as a sequence of steps to give the answers to these questions. Step. VOP (4.-(4.5 with equivalent criteria is solved. The algorithm is given in the previous unit. The data of -problem (4.6-(4.8 are stored. Step. The second criterion is chosen as the priority one. The data connected to the second criterion is being processed: f k (X f k (X f k (X o, k (X k (X k (X o : ( f (X 68.34, 0.57 (X 0.6; 00.0 f (X 3.94,.0 (X 0.6; 9. f 3 (X 86.66, (X 0.6; 9. f 4 (X 86.66, (X 0.6; The vector of priorities of the second criterion over others p k (X= (X/ k (X, k=, 4 in extremum points X, Xo, is determined: p k (X = [ p k (X o = [ o o =.7673 = o o =.0 = 3 3 o o =.5855 = 4 4 o o =.5855] i.e. the limits of change of a vector of priorities P k (X P k P k (Xo, k=, 4. In the present problem the vector of priorities lies within the limits:.7673 p.0,.0 = p =.0,.5855 p 3.0,.5855 p 4.0. These data are the basic information for taking the decision. Step 3. After analyzing priority criterion, we determine numerical value f q. Let us assume, that the desirable value of the second criterion is f = Step 4. The relative estimation is calculated as follows: = (f - f o /(f -f o = Step 5. Let us calculate the factor of proportionality between q (X o, q, q (X q assuming linear character of the change of criterion f q (X (4. and according to a relative estimation q (X (4.: = ( - (X o /( - (X o = = ] 58

15 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN Step 6. Assuming linear character of change of a vector of priorities p q k (X, k=, we shall calculate it for k, k=, using factor of proportionality : P q k =P q k (Xo +(P q k (X -P q k (Xo, k=, qk. In the given problem the vector of priorities determined as follows: P k = {p =. 7673, p =.0, p 3 =. 5855, p 4 =. 5855}. Step 7. Construction of a -problem. Let us enter the given vector of priorities p k, k=, 4 into the data of - problem (4.6-(4.8., The result is a -problem with a priority of q-th criterion: Step 8. The solution of a -problem. The results of a -problem are the following: o = , X o = {x o = 37.5, x o = 3., x o 3 = 3., x o 4 = 44.45}, Objective functions (criteria in the optimum point f k (X o, k = : f (X o = 36.93, f (X o = 67.76, f 3 (X o = , f 4 (X o = , Resource Costs A(X o : Financial costs, consumers: c x o + c 3x o 3 =845.64, c x o + c 4x o 4 =845.64; Financial costs, Manufacturer: a x o + a x o =547.74, a 3x o 3 + a 4x o 4 = Relative valuations at optimum k (X o, k = : (X o = 466, (X o = 0.858, 3 (X o = , 4 (X o = Check: o p (X o, (X o, p 3 3 (X o, p 4 4 (X o, where { p (X o = 0.858, p (X o = 0.858, p 3 3 (X o = 0.858, p 4 4 (X o = 0.858}. Conclusions. The volumes of production of both manufacturers have matched and are equal to sales volume of purchases. The demand and supply are equal. 6. Analysis of the model of the one-product market with the aggregated criteria Estimating model (.6-(.0 arises a desire to join efforts of both the manufacturers in (.6 and consumers (.7, as such method is standard. Let us check the criteria on models of the previous units before summing up them and solving a problem. In Unit 4. the model of the one-product market with two manufacturers and two consumers is constructed as a vector problem. Its solution is given in Unit 4.. Let us enter criteria with the sums on both manufacturers and consumers into VOP (4.-(4.5: max f 5 (X = p x + p x + p 3 x 3 + p 4 x 4, (6. min f 6 (X = c x + c x + c 3 x 3 + c 4 x 4, (6. Let us solve the problem (4.-(4.5 with criteria (6. and (6.. Criterion (6.. max: X 5 ={x =50.0, x =50.0, x 3 =50.0, x 4 =50.0}, f (X 5 =00.0, f (X 5 =00.0, f 3(X 5 =000.0, f 4(X 5 = 000.0, f 5 (X 5 = 400.0, f 6(X 5 = min: X o 5 ={x = 35.0, x = 35.0, x 3 = 35.0, x 4 =35.0}, f (X o 5 = 40.0, f (X o 5 = 40.0, f 3(X o 5 = 700.0, f 4(X o 5 = f 5 (X o 5 = 80.0, f 6(X o 5 =

16 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN Criterion (7.. min: X 6 ={x =35.0, x = 35.0, x 3 =35.0, x 4 = 35.0}. f (X 6 = 40.0, f (X 6 = 40.0, f 3(X 6 =700.0, f 4(X 6 =700.0, f 5 (X 6 = 80.0, f 6(X 6 = max: X o 6 ={x = 50.0, x =50.0, x 3 = 50.0, x 4 = 50.0}. f (X o 6 = 00.0, f (X o 6 = 00.0, f 3(X o 6 = 000.0, f 4(X o 6 = f 5 (X o 6 = 400.0, f 6(X o 6 = We estimate the results on the relative estimated at optimum points X 5, X o 5, X 6, X o 6 respectively The results in relative estimations show, that the limits of changes of production of the first and second manufacturers lie within the limits: 00f (X 40, (X 0.5; 00f (X 40, (X0.5. Thus, the aggregated criterion (6., (6. does not reflect in full measure the purposes of the manufacturers, which lie within the limits:.0 (X 0. These drawbacks of the aggregated criterion were repeatedly mentioned in literature. Let us analyze reasons for it. When solving VOP (4.-(4.5 in a point of an optimum against the first criterion the second criterion meets only 0.54 of its optimum value. In the worst point against the first criterion the second criterion reaches the optimum value. Thus, the purposes of the manufacturers are directly opposite, what is good for one is bad for another, and on the contrary, what is bad for one is good for another. This is competition. The sum of criteria cannot reflect these inconsistent purposes. Let us estimate the solution of the problem (4.-(4.5 with additional criteria (6., (6. and restrictions: 400 c x + c x + c 3 x 3 + c 4 x 4 000, (6.3 a x + a x + a 3 x 3 + a 4 x ( As a result of the decision of a -problem is received: o = 0.5, X o = {x =4.5, x =4.5, x 3 =4.5, x 4 =4.5}, f (X o =70, f (X o = 70, f 3 (X o =850, f 4 (X o = 850, f 5 (X o =340, f 6 (X o = 700. (X o =0.6364, (X o =0.6364, 3 (X o =0.5, 4 (X o =0.5, 5 (X o =0.5, 6 (X o =0.5. The received result also shows that the aggregated criterion inexactly reflects the purposes both manufacturers and consumers, whose general level = min k (X, XS is raised up to kk o = min k (X o =0.6074, X o S in the results of the kk solution of the VOP (4.-(4.6, reflecting individual purposes. Conclusions. The work presents the construction of model of the one-product market as a vector problem of mathematical programming. The model takes into account not only balance between supply and demand, but also purposefulness of each market participant. The methodology of modeling of the market is shown through the solution of a number of practical problems presented by vector problems of linear programming. 60

17 9 th May 04. Vol.5 No JITBM & ARF. All rights reserved ISSN References. K. J. Arrow and G. Debreu, "Existence of an equilibrium for a competitive economy", Econometrica,, N3, D. Gale,: "On optimal development in multisector economy", Rev. Econ. Studies, 34, N, Yury K. Mashunin, Techniques and Models of Vector Optimization, Moscow, Nauka, 986, 4 p. 4. Yury K. Mashunin. Control Theory. The mathematical apparatus of management of the economy. - M: Logos, p. 5. Yury K. Mashunin, "Engineering system modeling on the base of vector problem of nonlinear optimization", Control Applications of Optimization. Preprints of the eleventh IFAC International workshop. CAO 000. July 3-6, 000. Saint - Petersburg, 000, L. W. McKenzie, "On equilibrium in Graham's model of world trade and other competitive market", Econometrica,, N, Morishima, Michio, Equilibrium Stability, and Growth. A Multi - Sector Analysis, Oxford, Clarendon press, J. von Neumann, A Model of General Economic Equilibrium Behavior, Review of Economic Studies, X, -9, H. Nikaido, "Stability of equilibrium by the Brown - von Neumann differential equation", Econometrica, 7, N4, H. Nikaido, Convex structures and economic theory, Academic press, New York and London, F. M. Scherer, and D. Ross, Industrial Market Structure and Economics Performance, Houghton Mifflin Company, Boston, 698 p, K. K.Seo, Managerial Economiks. Text, Problems, and short cases. IRWIN, Homewood, IL Boston, VF , 67 p. 3. Tirole, Jean, The theory of Industrial Organization, The MIT Press. Cambridge, Massachusetts, London, England, L. Walras, "Elements d'economie Politique Pure", Lausanne, Elements of Pure Economies, London, Mashunin Yury Konstantinovich (Vladivostok, Russia doctor of economic Sciences, associate Professor - Far Eastern Federal University, Professor of the Department «State and municipal management», (690068, Vladivostok, street of Kirov, 34-37, ( , Mashunin@mail.ru 7. Mashunin Ivan Aleksandrovich (Vladivostok, Russia - Far Eastern Federal University, chair of «Economics and management at enterprise», (690068, Vladivostok, street of Kirov, 34-37, ( , Mashunin_ia@primorsky.ru. 6