Download Average-Cost Control of Stochastic Manufacturing Systems by Suresh P. Sethi PDF

By Suresh P. Sethi

ISBN-10: 0387219471

ISBN-13: 9780387219479

Such a lot production platforms are huge, advanced, and function in an atmosphere of uncertainty. it's normal perform to control such structures in a hierarchical model. This publication articulates a brand new conception that exhibits that hierarchical determination making can in reality result in a close to optimization of method pursuits. the cloth within the booklet cuts throughout disciplines. it's going to entice graduate scholars and researchers in utilized arithmetic, operations administration, operations learn, and method and regulate thought.

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Bielecki and Kumar [20] dealt with a single machine (with two states: up and down), single product problem. They obtained an explicit solution for the threshold inventory level, in terms of which the optimal policy is as follows: Whenever the machine is up, produce at the maximum possible rate if the inventory level is less than the threshold, produce on demand if the inventory level is exactly equal to the threshold, and do not produce at all if the inventory level exceeds the threshold. Basak, Bisi, and Ghosh [12], and Ghosh, Aropostathis, and Marcus [63] incorporated both diffusion and jump Markov processes in their production planning model, and thus generalized Kimemia and Gershwin [79] as well as Sethi and Thompson [114] and Bensoussan, Sethi, Vickson, and Derzko [17].

38). 1 that equation d x(t) = u∗ (x(t), k(t)) − z, x(0) = x, dt has a unique solution x∗ (t), t ≥ 0, for each sample path k(t). Next we devote ourselves to proving that u∗ (·, ·) is a stable control. For this, we first derive some intermediate results. 1. Let Assumptions (A1)–(A5) hold. For each k ∈ M, we have inf V (x, k) > −∞. 5 Existence and Characterization of Optimal Policy 45 Proof. Let (xρ , kρ ) be the minimum point of the value function V ρ (·, ·). Then we can write V ρ (x, k) = V ρ (x, k) − V ρ (xρ , kρ ) + V ρ (xρ , kρ ) − V ρ (0, 0) ≥ V ρ (xρ , kρ ) − V ρ (0, 0).

Let u(t) = k(t), so that for t ≤ τ1 , t x(t) = 0 t [u(s) − z] ds = 0 [k(s) − z] ds, t ≤ τ1 , and u(t) = 0, x(t) = x + K − (t − τ1 )z, for τ1 < t ≤ τ1 + K/z. For convenience in notation, we write σ1 = τ1 + K/z. Proceeding in this manner, we can define the required control u(·) ∈ A(0) inductively: ⎧ ⎨ k(t), if σ n−1 < t ≤ τn , u(t) = ⎩ 0, if τ < t ≤ σ , n n where t x(t) = 0 [u(s) − z] ds, 0 ≤ t ≤ τn , τn = inf{t > σn−1 : x(t) = x + K} and σn = τn + K/z, n > 1. We set σ0 = 0 and τ0 = 0. The control u(·) can be characterized as follows: Use the maximum available production rate u(t) = k(t) to move the surplus process from 0 or x to x+K, and then use the zero production rate until the surplus process drops to the level x.

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