By Yao D.D., Zheng S.

ISBN-10: 0387954910

ISBN-13: 9780387954912

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**Additional info for Dynamic Control of Quality in Production-Inventory Systems**

**Example text**

14). Denote this limit as Vα (1, y). 19), we have lim Vαm (2, y) m→∞ = min{CI + r(y, 1) + αE[Vα (1, f1 (y, D)], r(y, 0) + αVα (1, f0 (y))}. 17). 16). 6 (i), we have Vα (1, y) = Vα (y). 17). 17). 20) and if ai = 0 is chosen, then choose bi = 1 if and only if CI + r(y, 1) + αE[Vα (1, f1 (y, D))] ≤ r(y, 0) + αVα (1, f0 (y)). 14). 6, any policy that chooses control actions in this manner is optimal. Therefore, we have the following. 8 The stationary policy πα as speciﬁed previously is optimal; in particular, Vα (πα , y) = Vα (y), for any state y ∈ S.

9. (ii) Suppose ai−1 = 0. 1 we can write Yi = f1 (Yi−1 , Di (Yi−1 )). Note the following: P[Di (y) = k] = yp1 (k) + (1 − y)e−λ p0 (k) + (1 − y)(1 − e−λ )p2 (k) = y{p1 (k) − p2 (k) + e−λ [p2 (k) − p0 (k)]} +e−λ p0 (k) + (1 − e−λ )p2 (k). , N . 9, where the stronger likelihood ratio ordering was established. Because f1 (·, ·) is increasing in both components, it follows that Yi is stochastically increasing in Yi−1 . When ai−1 = 1, we have Yi = f (0, Di (Yi−1 )), and the same argument applies. (iii) It is easy to see that E[Yi | Yi−1 , ai−1 = 0, bi−1 = 0] = f0 (Yi−1 ) = ≥ Yi−1 + (1 − Yi−1 )(1 − e−λ ) ✷ Yi−1 .

We will reveal more structural properties of the optimal policy and eventually establish its threshold nature. Observing that P[Dn+1 = d + 1|Dn = d] = E[Θn (d)] = E[Θd+1 (1 − Θ)n−d ] , E[Θd (1 − Θ)n−d ] we can also express Ψn (d) as Ψn (d) = ci + [cr + Vn+1 (d + 1)]E[Θn (d)] + Vn+1 (d)(1 − E[Θn (d)]). Yet another expression for Ψn (d), which will be used later, is: Ψn (d) = ci + cr E[Θn (d)] + E[Vn+1 (Dn+1 )|Dn = d]. 11) For each 0 ≤ n ≤ N − 1, deﬁne Sn := {d : 0 ≤ d ≤ n, Φn (d) ≤ Ψn (d)}, Sn := {d : 0 ≤ d ≤ n, Φn (d) ≥ Ψn (d)}.

### Dynamic Control of Quality in Production-Inventory Systems by Yao D.D., Zheng S.

by Ronald

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