Optimization of the parameters of the stochastic model of inventory control

The following model of inventory control is considered: 1. The level of the stock of the product at the initial moment t = 0 is equal to J0. 2. Change in the level of stocks occurs instantly. 3. Customer requests form a simple events flow of intensity X. 4. The values of the applications are independent and have the same distribution with the probability density f( x) and mathematical expectation a. 5. Replenishment of stocks occurs at regular intervals of the length T up to the level J0. It is assumed that the cost of storing a unit of product per unit of time is equal to c1, the penalty for the deficit of a unit of product per unit of time is equal to c2, the cost of supplying of x units of the product is cjr + c4. The costs in this model are sufficient to investigate over a period of time (0, T). Let C(t) be the cost of storing the product and the penalties for its deficit at a time interval (0, t), 0 < t < T. Then the total cost of the time period (0, T), taking into account the cost of replenishment, is equal to C_m = C ( k) + Сз( J0 - J ( k)) + c,. The mathematical expectation of a quantity Cfull is considered as a cost function. It is required to find the value J0 at which the cost function takes the smallest value. It is not difficult to show that the mathematical expectation of the stock level on a time interval (0, t), 0 < t < T has the form M (J(t)) = J -Xat , so the cost function takes the form M(C_full) = M(C(T)) + cXaT + c, . This means that the cost function is minimal if the value M(C(T)) is minimal. It is shown that the probability distribution function of the stock level P(J(t) < x) satisfies the integro-differential equation Я +ОД -P(J(t) < x) = -XP(J(t) < x) + X j P(J(t) < v + x)f (v)dv ^St 0 in the band 0 < t < T, - ю < x < ю, and the initial condition 1, если x > J₀, P(J (0) < x) = < 0, если x < J₀. The mathematical expectation of storage costs and penalties on a time interval (0, T) is obtained M(C(k)) = jq(J₀ -Xat) - (q + c) j xm(x,t)dxjdt, where u(x, t) is the distribution density of the stock level. In the case of the exponential distribution of the quantity of requests, the density distribution of the inventory level is obtained, and hence the average cost, the minimum average cost and the optimal value of the initial inventory level J0.

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