Lead-time demand

Hence, while the lead time demand has the same mean as E(iV), its variance is different, unless the lead time is deterministic and D follows a Poisson distribution. This distinction is important, since it is N, not demand over lead time, that determines the inventory performance, in terms of on-hand inventory (1) and back orders (S) via the relations in (1). 

Non-stationary demand patterns with independent realizations. When the statistical distribution of the demand in different periods varies, it is no longer true that a newsvendor solution of the type (10.9) is optimal, even if is replaced with Gf - the period-dependent distribution function of the lead time demand Di, The main reason for this is that in addition to the impact that the decision variable zt has on the expected cost at the end of period t, it may also have a negative consequence on the expected costs in later periods. Specifically, it may be the case that after placing the order Zt and after the demand Dt is realized, the inventory position at the beginning of period t + 1 will be higher than desired. It is easy to see that in the i.i.d. case, in which the target level /3 is fixed, this can never happen. 

Let be an estimate of the lead time demand D, and suppose, as common in the literature on inventory management, that c qty Xt) can be written as a convex function c qt — Let 7 be the value of e that minimizes... 

In the linear state-space demand model described in 2.3, the estimate of the lead-time demand is given by... 

The value of 7 is interpreted as a fixed safety-stock level set for the endpoint. Thus, although the long-run optimal base-stock level P Xt) can take any functional form, we restrict ourselves to a target level (i which is a very specific linear function of the vector-estimate Xt. The reason we use a fixed safety-stock in (10.14) is that the level of uncertainty surrounding the lead-time demands is constant over time. 

It is instructive to note that the supplier s observable vector includes, in addition to the market signals observed by the supplier (i.e., the component), the retailer s order At i. Using the above linear state space representation, the supplier can calculate the estimates of future lead-time demands using the exact same mechanism as used in the retailer s forecasting process. Here, the values of Zt t-i and the error covariance matrices generated... 

Proposition 14 The estimates of the lead time demands, t) and are given by... 

Thus far, we provided a characterization of the forecast evolution and the forecasting errors associated with the estimates of the lead-time demands. As we shall see in the next section, this lays the foundation for devising an inventory policy for the channel, and for assessing the overall supply chain s cost performance. Using this tool, one can easily compare between the three settings -LMI, SMI, and CFAR (see 6.4 below). 

The terms and Ef stand for the forecast errors of the retailer s and the supplier s estimates of their lead-time demand. We are particularly interested in the lagged forecast errors i.e., that made by the supplier during a particular period, and that made by the retailer r periods later. Proposition 15 tells us that the cost performance of the supply chain is uniquely determined by the safety stock parameters 7 and 7, and by the characteristics of the joint distribution of the lagged forecasts E] and which is described in the next theorem. 

Moreover, given a value of the reorder point R, the expected number of units short in an order cycle under normally distributed lead-time demand can be expressed as... 

As indicated by Silver et al. (1998), the assumption of the normal distribution for lead-time demand is common for a number of reasons, particularly that it is "convenient from an analytic standpoint" and "the impact of using other distributions is usually quite small" (p. 272). 

A less rigorous approach to finding a (Q, R) solution would be to solve for Q and R separately. Note that z = R - MoltV dlt gives a fractile of the distribution of demand over the lead time. Thus, we could set R to achieve a desired in-stock probability, along the lines of the newsvendor problem solution discussed earlier (i.e., to accumulate a given amount of probability under the DLT distribution). In this setting, the in-stock probability is typically referred to as the cycle service level (CSL), or the expected in-stock probability in each replenishment cycle. Specifically, for normally distributed lead-time demand, DLT, we set... 

Again, the backlogging assumption is important. If unmet demand is lost as opposed to backlogged, then the replenishment quantity Q cannot be used to fulfill demand backlogged from previous cycles, and this inflates the expected total demand that is fulfilled in an average cycle from Q to Q + S R), such that, with normally distributed lead-time demand, we obtain P(R,Q) = 1 -... 

Note that we can use standard normal loss function because demand over the lead time (DLT) is the convolution of normal random variables for demand and lead time, and is therefore normally distributed. See the Further Readings section of this chapter for references that address situations where lead-time demand is not normal. 

Tyworth, J. E. and L. O NeiU. 1997. Robustness of the normal approximation of lead-time demand in a distribution setting. Naval Research Logistics. 44 166-186. 

Assuming the same replenishment time L, from the plant to the central warehouse, the lead time demand at the warehouse is given by... 

Option 1 (Three Regional DCs) Since the lead time is 1 week, the weekly demand and the lead time demands are the same. For 90% service level, the safety factor Ksi = 1.29. Using Equation 5.47, the safety stocks to be carried at the three DCs are as follows ... 

The daily usage of this part is 20 units. The company follows an inventory policy of maintaining a safety stock of 50% of the lead-time demand and uses an inventory-carrying charge of 25% per year (i.e., 0.25/ dollar/year). Defective parts encountered in manufacturing cost the company 5/unit. The company wishes to choose one of the suppliers using the TCO approach. 

Annual procurement cost = (10) (20) (365) = 73,000 Lead-time demand = (10) (20) = 200 units Safety stock = (50%) (200) = 100 units Average cycle inventory =. 555 = 250 units... 

Given that HA/DR delivery supply chains usually operate in highly uncertain environments, they must be engineered and executed in shorter periods of time so as to provide relief to the affected population as soon as possible (Ratliff 2007). Further, inventory management in HA/DR delivery supply chains is affected by unreliable, incomplete, or nonexistent information about lead times, demand levels, and locations (Beamon 2004). In terms of distribution network configuration, the number and location of distribution centers is uncertain. This makes cost assessment difficult in terms of planning financial flows. 

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