hmwk9sol - ISyE 3232 Stochastic Manufacturing and Service...

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ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2011 H.Ayhan Solutions to Homework 9 1. Let { X n : n 0 } be the stock level at the evening of day n . The problem states an ( s,S ) inventory policy, i.e., if the stock level is s or less than s at the review time, we make orders to bring the stock level up to S ; we do not order otherwise. We can establish the relation between X n +1 and X n as follows: X n +1 = ± S - D if X n s X n - D if X n > s . (1) By Equation (1), { X n } is a discrete-time Markov chain. Note that we allow negative stock levels, which indicate unfilled orders at review times. ( a ) In this case, s = 2 and S = 6. We have the state space S = {- 1 , 0 , 1 , 2 , 3 , 4 , 5 , 6 } . The transition probability matrix can be written as, P = 0 0 0 . 01 . 04 . 3 . 15 . 5 0 0 0 . 01 . 04 . 3 . 15 . 5 0 0 0 . 01 . 04 . 3 . 15 . 5 0 0 0 . 01 . 04 . 3 . 15 . 5 . 01 . 04 . 3 . 15 . 5 0 0 0 0 . 01 . 04 . 3 . 15 . 5 0 0 0 0 . 01 . 04 . 3 . 15 . 5 0 0 0 0 . 01 . 04 . 3 . 15
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hmwk9sol - ISyE 3232 Stochastic Manufacturing and Service...

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