The restrictive dynamic model of the inertial market of one goods with the optimal delivery of goods to the market in conditions of delay

We solve the problem of the mathematical description and simulation of the inertial market ofone goods with optimal control of the goods supply to the market in conditions of delivery timelag.The mathematical description of the market is given by the restrictive (because of restrictionsof inequalities type) dynamic model with the time-lag of control. It was shown that the optimal (inthe sense of the maximum profit of the seller) strategy of the goods delivery to the market is determiningby the market conditions (commodity deficit, overstocking of the market, or the dynamicbalance state of the market). These conditions have been formulated mathematically rigorously.There are examples of the simulation of such system.Let P(t) > 0 be the price of goods at the moment of the discrete-time t, t N = {0,1,2,...}. LetQO(t) ≥ 0 be the remainder of the unsold goods at this instant, QZ(t - ƒ) ≥ 0 be the strategy of thegoods delivery to the market (the volume of goods ordered at the time t - ƒ for the delivery at thetime t). It is assumed that the ordered goods enter into the market with a delay of ƒ time units. Letthe volume QD(t) of the demand for goods at a price P(t) at the time t has the form of the simplelinear relationship: ( D ) ( )Q t =Qm −aP t , where Qm > 0 and a > 0 - given constants. Let Q(t) -the sales volume, assumed for the sale at the moment of the time t, and QO(t) - the volume of theresidue of goods at the previous discrete time interval (and passed into the market at the momentt). Then Q(t)=QO(t)+QZ(t − τ ) .The sales volume QS(t) at the discrete time interval t is subject to restrictive relation:QS(t)=min(QD(t),QO (t)+QZ (t− τ )).The residue of the goods satisfies the recurrent relation:QO(t+1)=QO(t)+QZ (t−τ )−QS (t) .The aim is to maximize the profit J(t) of the seller (the difference between the receipts fromthe sale of goods and the cost of its acquisition and storage) at each discrete time interval:( ) ( ) ( ) ( ) ( ) ( ( ) ( ))( ) ( )21 2,1 s2S Z OP t QZJ t =Q t P t Q t τ P Q t P R P t P tt− − − − − − − ƒwhere P1 - the price of the purchase of goods (from the wholesale market or from the manufacturer),P2 - the price of the storage of the unit of goods not sold at the previous discrete time interval.The last term (with the coefficient R > 0) expresses "penal sanctions" for the price changeof goods and determines the inertia of the market - a sharp increase in prices can be accompaniedby legislative sanctions, for the sharp decrease in prices - "sanctions" of the competition, reflectedthe damage to the seller in the equivalent of the penalty function. Herewith natural limitations onthe value of a possible price P(t) must be executed: P1 < Pmin < P(t) < Pmax = Qm / a.When solving the optimization problem, areas of the scarcity of goods to the market (area 1,in which Q(t) < QD(t)), of overstocking of the market (area 2, in which Q(t) > QD(t)), and of thebalance of the demand and proposals (area 3, the region of the dynamic balance state, in whichQ(t) = QD(t)) are allocated. For each of these areas the next conditionally optimal (for fixed Q(t))values of the price P(t) were found:P t = P t +Q t = P tR− , ( ) ( 1) (2) ( )2aP t =Qm+RP t =P t+R−, P(t)=Qm Q(t)=P(3) (t)a−.Correspondingly, for each area the boundary values of Q(t) were found:Q(t)< R(Qm aP(t 1))=Q(1) (t)a+ R− −, ( ) ( ( 1)) (2) ( )2aQ t > R Qm aP t + aQm=Q t+R− −,Q(1) (t)≤Q(t)≤Q(2) (t) .Then optimal values of all variables P(t), Q(t), QZ(t - ƒ), QS(t), QO(t) and J(t) are determinedrecursively for each current discrete moment of time t.As examples, the numerical simulation of the following situations has been performed. Initially(for at least ƒ steps of discrete time), the market is in the balance state. Then at some point intime t = 0 it is deduced from the balance state position (the price of goods changes dramatically,for example, increasing with respect to the balance state value P* = (Qm + aP1) / (2a), so thatP(0) = P0 > P*, or, conversely, decreasing, so that P0 < P*). Calculations of the evolution of themarket with the optimization of the delivery of goods on the market were made for R = 50,Qm = 4, a = 0.4, T = 300, P0 = 7 or P0 = 5.5, P1 = 3, P2 = 0.1, Pmin = P1 + P2, Pmax = Qm / a,P* = 6.5, Q0 = 0, ƒ = 10, 20, 30. The dynamics of the market parameters were graphically displayed.The optimal strategy for delivering the goods to the market requires the prediction ofprice and consumer demand for forward on time-lag ƒ, which, as was shown in this paper, can bedone using the proposed simulation model of the market.

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