Dynamic lead time quotation under responsive inven

Authors: Secil Savasaneril Ece Sayin

Publish Date: 2016/05/19

Volume: 39, Issue: 1, Pages: 95-135

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Abstract

We address the lead time quotation problem of a manufacturer serving multiple customer classes Customers are sensitive to the quoted lead times and the manufacturer has the flexibility to keep inventory to improve responsiveness We model the problem as a Markov decision process and characterize the optimal lead time quotation rationing and production policies We then define internal and external service level measures and analyze the impact of inventory keeping decision on these measures Before analyzing the impact we first derive the relation between inventory level and lead time quotes We then show that the effect of inventory level on the service levels may not follow the intuition We also study alternative lead time quotation and production schemes We contrast the performance of these alternative policies with that of the optimal policy Specifically through a numerical study we quantify the value of controlled arrivals and customer rejection the value of information on customer status and the value of a farsighted policy Through the numerical study we also identify the effect of revenue mix and demand mix on the inventory keeping decisions and the performance measures Finally we measure the impact of lead time quotation on resource pooling ie inventory and capacity pooling We show that the value of resource pooling is limited under the optimal policy since lead time quotation is already effective in balancing the demand with capacity and in allocating the resources among different customer classesWe first discuss the existence of an optimal policy under the average reward criteria Establishing the existence under average reward criteria is not straightforward when state space is countable and onestep rewards are not bounded Weber and Stidham 1987 provides the sufficient conditions for existence of an average reward optimal policy Our Markov decision process satisfies the conditions therefore an average reward optimal policy exists Also the Markov decision process under consideration is a communicating multichain which implies that the optimal policy has a constant gain Puterman 1994 This implies that under the optimal policy there exists a single recurrent class and possibly a set of transient statesIn our problem we can assume without loss of optimality that production is stopped under a sufficiently large stock level say i=L B0 and that all customers are rejected when the cost due to the expected lateness is sufficiently high say i=L U0 This assumption can be made since the holding cost is convex increasing in stock level and lateness cost is convex increasing in number of customers in the systemIn the following we show that for iin mathbb Z vivi+1 is increasing in i We analyze the cases i0 and ige 0 separately Let Delta i=vivi+1 barDelta i=vivi1 Lambda =sum jlambda j gamma =Lambda +mu The operator used to determine the optimal production decision for a given nonnegative state is max vi1vi see Eq 2 Rearranging the terms one obtains max 0vivi1+vi1=max 0barDelta i+vi1 We have shown that barDelta i is decreasing in i This implies that there exists a state s such that for is to produce is the optimal decision and for ile s not to produce is the optimal decision This implies that optimal production policy is of controllimit typeSince barDelta i is decreasing in i ie Delta i increasing in i from Eq 8 it is possible to infer that the term max R jDelta i is equal to Delta i for sufficiently large values of i and the term is equal to R j for other values of i Thus for i0 there possibly exists a threshold K j such that for iK j arriving customers are accepted and for K jle ile 0 arriving customers are rejected when there is stock If on the other hand it holds that R jDelta i for ile 0 then K j does not exist and all arriving customers of class j are accepted whenever there is stock In that case there exists a threshold B j at which customers of class j are rejected whenever the backlog in the system is B j or more It is possible to infer observing the Eqs 8 and 12 that either a K j or a B j exists but not both

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