By Suresh P. Sethi
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Additional resources for Average-Cost Control of Stochastic Manufacturing Systems
On the one hand, the monotonicity property can be used to solve some optimal control problems in a closed form, which are otherwise diﬃcult to handle. On the other, it can greatly reduce computations needed for numerical approaches for solving the problem. The plan of the chapter is as follows. 2 we precisely state the production–inventory model under consideration. 3 we establish a systematic approach of constructing the ergodic (stable) control policies. In addition, we develop some estimates for the value function of the corresponding discounted cost problems.
Un (t)) ∈ n+ , x(t) = (x1 (t), . . , xn (t)) ∈ n , z = (z1 , . . , zn ) ∈ n+ denote the rates of production of n diﬀerent products, the vector of their surpluses, and the rates of their demand, respectively. 6) where ri > 0 (i = 1, 2, . . , n) is the amount of capacity required to process one unit of the ith product at rate 1. 7) denoting the vector of initial surplus levels. 1. In the model formulated here, simultaneous continuous production of diﬀerent products is allowed. 2. 2. Without any loss of generality, we may set r1 = 1.
The optimality of the control u∗ (·) in the (natural) class of all admissible controls. Let u(·) ∈ A(k) be any control and let x(·) be the corresponding surplus process. Suppose that J(x, k, u(·)) < λ. 44) Set f (t) = E[h(x(t)) + c(u(t))]. Without loss of generality we may assume that t 0 f (s) ds < ∞, for each t > 0, or else, we would have J(x, k, u(·)) = ∞. Note that J(x, k, u(·)) = lim sup T →∞ while 1 T ∞ ρJ ρ (x, k, u(·)) = ρ T f (s) ds, 0 e−ρs f (s) ds. 44) to obtain lim sup ρJ ρ (x, k, u(·)) < λ.
Average-Cost Control of Stochastic Manufacturing Systems by Suresh P. Sethi