Model of inventory control of homogeneous products with relay control of production rate and MMP-flow of sales moments

In this paper, we consider a mathematical model of a system of inventory management, on the input of which some resources (goods) come with a speed C(S(t)), where S(t) is the volume of accumulated resources in the system at the time t. Consumption of the resource (sales) is carried out at random moments of time by batches of random volume, having an arbitrary density distribution ф(x) and momentsM{x} = a, M {x²} = a₂. The moments of resource consumption time form a MMP-flow with n states and matrix of infinitesimal characteristics [q. ] . For a stationary distribution P(s) = P{S(t) < s;X(t) = Xj of process S(t) and intensity X(t) equation n C(s)P, (s) = -Xfi i.vi • V_(/ /> _m . j> _{l v} . .vicpi.vK/.v ) + Уqj'Pj (s) + *,JP (s + x i=i о is obtained. The main attention is paid to the case when the function C(S(t)) is determined by the relation C(S) = C at S < S₀ and C(S) = 0 with S > S₀, the magnitude C = (1 + 6)X₀a and the parameter 6 << 1. It is proved that in this case A . я A - S0) + O(6), s < S₀ 1+6 a⁴! Л Pi (s) = я., s > S„ X Cl n-1 1 n n where я is the final probability of the state X , A₁ = X₀ a, A, = ^{0 2} - a² V-V (X₀-X₍ )R_it У P (X₀-X , X₀ is the average 2 Tt 7=1 ' intensity of the flow of purchases, the matrix [r ] is the matrix of the eigenvectors of the matrix [q ] , the matrix [P ] is the matrix inverse to the matrix [ R ] . Based on a comparison of the proposed asymptotic distribution with the exact one in the case of a two-state flow and the exponential distribution of the quantities of purchases, it was concluded that the proposed asymptotic distribution can be applied. The dependence of the average profit on the quantities C and S has been analyzed.

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