Abstract
The paper studies the production inventory problem of minimizing the expected discounted present value of production cost control in a manufacturing system with degenerate stochastic demand. We establish the existence of a unique solution of the Hamilton-Jacobi-Bellman (HJB) equations associated with this problem. The optimal control is given by a solution to the corresponding HJB equation.
Highlights
Many manufacturing enterprisers use a production inventory system to manage fluctuations in consumer demand for the product
J p E e−ρt h xt pt[2] dt Journal of Probability and Statistics subject to the dynamics of the state equation which says that the inventory at time t is increased by the production rate and decreased by the demand rate can be written according to dxt pt − yt dt, x0 x, x > 0, 0 ≤ pmin ≤ pmax, 1.2 and the demand equation with the production rate is described by the Brownian motion dyt Ayt dt σyt dwt, y0 y, y > 0, 1.3 in the class P of admissible controls of production processes pt with nonnegative constant pt ≥ 0
1.4 defined on a complete probability space Ω, F, P endowed with the natural filtration Ft generated by σ ws, s ≤ t carrying a one-dimensional standard Brownian motion wt, xt is the inventory level for production rate at time t state variable, yt is the demand rate at time t, pt is the production rate at time t control variable, ρ > 0 is the constant nonnegative discount rate, A is the nonzero constant, σ is nonzero constant diffusion coefficient, x0 is the initial value of inventory level, and y0 is the initial value of demand rate
Summary
Many manufacturing enterprisers use a production inventory system to manage fluctuations in consumer demand for the product. 1.4 defined on a complete probability space Ω, F, P endowed with the natural filtration Ft generated by σ ws, s ≤ t carrying a one-dimensional standard Brownian motion wt, xt is the inventory level for production rate at time t state variable , yt is the demand rate at time t, pt is the production rate at time t control variable , ρ > 0 is the constant nonnegative discount rate, A is the nonzero constant, σ is nonzero constant diffusion coefficient, x0 is the initial value of inventory level, and y0 is the initial value of demand rate This optimization control problem of production planning in manufacturing systems has been studied by many authors like Fleming et al 1 , Sethi and Zhang 2 , Sprzeuzkouski 3 , Hwang et al 4 , Hartl and Sethi 5 , and Feichtinger and Hartl 6.
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