By David D. Yao
Appears at caliber administration in a brand new manner; bargains a truly diverse mathematical device set than normally hired for quality controls difficulties; Yao is a number one researcher within the quarter of stochastic modeling and the writer of a prior Springer publication
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Additional resources for Dynamic Control of Quality in Production-Inventory Systems: Coordination and Optimization (Springer Series in Operations Research and Financial Engineering)
Vn+1 (d) = Φn+1 (d) for any d ≤ n + 1. We then have Ψn (d) = ci + cr E[Θn (d)] + E[Vn+1 (Dn+1 )|Dn = d] = ci + cr E[Θn (d)] + E[φ(n + 1, Θn+1 (Dn+1 ))|Dn = d] = ci + cr E[Θn (d)] + E[φ(n + 1, Θn (d))] ≥ Φn (d), ¯ ¯ + 1, Θn (d)) ≥ Π(n, where the inequality follows from Π(n Θn (d)), for n ≥ n∗1 , which in turn follows from the fact that Π(n, θ) is increasing in n for n ≥ n∗1 (because Π(n, θ) is convex in n and reaches its minimum at n∗ (θ) ≤ n∗1 ). This implies d ∈ Sn . ✷ The next theorem establishes the following monotone property of the optimal policy: at each stage n, if it is optimal to continue inspection in state d, then it is also optimal to continue in state d + 1.
That is, the lower the quality of the batch (in terms of a larger θ), the more we need to inspect. It turns out that the key to this is the notion of K-submodularity deﬁned soon. 3): suppose x∗ (y) is the optimal solution to the minimization problem, minx g(x, y), for a given y; then x∗ (y) is increasing in y. However, here we are interested in a slightly diﬀerent problem: minx [Kxy + g(x, y)], where K > 0 is a constant [cf. 4)]. Because Kxy is a supermodular function, the submodularity of g will not guarantee the increasingness of the optimal solution x∗ (y) in y.
Our objective is to ﬁnd an optimal policy in revising the machine and inspecting the products so that the total discounted or average-cost is minimized. , the one that is produced in the ith period). Let (ai , bi ) denote the pair of control actions in period i, where ai = 1, 0 denotes the actions of revising and not revising the machine, and bi = 1, 0 denotes the actions of inspecting and not inspecting the batch produced. 1 depicts the sequence of decisions carried out over time, along with the production process.
Dynamic Control of Quality in Production-Inventory Systems: Coordination and Optimization (Springer Series in Operations Research and Financial Engineering) by David D. Yao