By Yao D.D., Zheng S.

ISBN-10: 0387954910

ISBN-13: 9780387954912

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

**Example text**

Thompson and Koronacki [97], chapter 4): inspect the units one at a time; at each step update the cumulative sum of the log-likelihood estimate of the defective rate; continue if the sum falls within a prespeciﬁed interval, say [α, β]; stop inspection and ship the whole batch if the sum falls below α; and inspect all the remaining units if the sum exceeds β. Common to all these approaches is that the policy is prespeciﬁed; the issue then becomes essentially a parametric design problem of ﬁnding one or two threshold values: the breakeven point in the back-of-envelope analysis, the upper limit on the acceptable number of defective units along with the sample size in the acceptance-rejection approach, and the α and β values in the CUSUM technique.

Assume each unit in the batch of N is either defective or nondefective. A nondefective unit has a lifetime of X, and a defective unit has a lifetime of Y . Both X and Y are random variables. , P[X ≥ a] ≥ P[Y ≥ a] for all a ≥ 0. ) Assume an inspection procedure can identify whether a unit is defective at a cost of ci per unit. Each defective unit identiﬁed by the inspection is repaired, at a cost of cr per unit, and becomes a nondefective unit. The quality of the batch, before any inspection and repair, is represented by Θ, the proportion of defective units in the batch.

6 (i) The optimal discounted-cost function Vα (y) is the unique solution to the following equation: Vα (y) = min { CR + CI + r(0, 1) + αE[Vα (f1 (0, D))], CR + r(0, 0) + αVα (f0 (0)), CI + r(y, 1) + αE[Vα (f1 (y, D))], r(y, 0) + αVα (f0 (y)) }. 14) 46 4. 14) is a stationary optimal policy. (iii) Deﬁne Vα0 (y) = 0, and Vαm (y) = min {CR + CI + r(0, 1) + αE[Vαm−1 (f1 (0, D))], CR + r(0, 0) + αVαm−1 (f0 (0)), CI + r(y, 1) + αE[Vαm−1 (f1 (y, D))], r(y, 0) + αVαm−1 (f0 (y))}. 15) Then, limm→∞ Vαm (y) = Vα (y) for any initial state y .

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

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