Inventory position

If expedients with respect to raw materials were used to get to first manufacture (e.g., a single supplier of a critical material, a risky inventory position on another, etc.), those need to be addressed immediately, particularly if expanded output is desired. On the other hand, superfluous requests to qualify new suppliers (usually coming from Materials Management) should be rejected for the time being. 

In this context, in addition to the order size, we need a second parameter, the reorder point. Associated with the latter is the notion of inventory position, defined as the on-hand inventory minus any back orders plus any outstanding orders—orders that have been placed but have not yet arrived due to the lead time. Hence, whenever the inventory position falls below the reorder point, an order is placed. Note that since demand now fluctuates randomly and there is a lead time between placing and receiving an order, it is not desirable, in general, to order only when the inventory drops down to zero, as in the case of the EOQ model. 

Closely related, but more general, is the (5, s) model, which works as follows whenever the inventory position falls below the lower threshold, s, place an order to bring the inventory position to the upper threshold S. When demand comes in batches, typically there will be an undershoot (of random size) when the inventory position falls below s. Hence, in the (S, i) model, the order size is random, in contrast to the constant order size in the Q, R) model. 

A widely studied inventory control mecheuiism is base-stock control, which works as follows whenever the inventory position drops below R, a constant partuneter, place an order to bring the inventory position back to R. This way, every demand will trigger the placement of an order, such that the overall inventory position is tdways maintained at a constemt level, R, which is called the base-stock level. 

Consider an (S, s) inventory model with periodic review. A review takes place at the beginning of each period t, at which time the inventory position is updated and a replenishment decision is made... 

Observe that the inventory position, X, as defined above, always takes values between s and S. In particular, if an arriving demand brings the on-hand inventory level to below s, the inventory position is immediately brought up to 5 through placing a replenishment order. 

To characterize the on-hand inventory and the back orders, the key is to observe that by time t aU orders that are placed before or at t — L will have arrived. In other words, the on-order quantities included in the inventory position att — L will aU have turived by t. As before, let D(f - L, t) denote the total demand over the L -I- 1 periods t — L,t — L + 1,. . . , t. We have... 

Since in most applications the (S, s) values are large or moderately large, we can use the linear asymptote in (25) as an approximation for the renewal function. This way, the probability distribution of the inventory position X in Section 5.1 becomes ... 

That is, is the reorder point, since following the DRP logic, an order Q,-i is placed (at t — L) if and only if A, > 0. From the above expression. A, is the required quantity to bring the inventory position at t - L back to (This is consistent with the base-stock mechanism when there are no order-size restrictions.)... 

Now, suppose the replenishment policy is a dynamic (S s) rule. Specifically, when A, > 0, we want to bring the inventory position to (> Hence, an additional amount, S, i — is needed, and the order quantity is ... 

An (s,S) inventory system involves the periodic review of the level of inventory of some discrete unit. If the inventory position (units in inventory plus units on order minus units backordered) at a review is found to be below s units, then enough additional units are ordered to bring the inventory position up to S units. When the inventory position at a review is found to be above s units, no additional units are ordered. One possible goal is to select the values of s and S that minimize inventory cost. The third example is the (s, S) inventory model in Koenig and Law (1985) and Law and Kelton (2000). 

For convenience, the initial inventory position is taken to be / = S, that is, the inventory position is initially at its maximum. The demand X, t = 1, 2,. . . is modeled as a sequence of i.i.d. Poisson random variables with mean 25 units. 

Performing a steady-state simulation experiment implies that the analyst is interested in long-run performance of the model that is independent of the initial conditions of the simulation replication or run. In the inventory simulation the parameter of interest is the long-run expected cost per period of inventory policy which does not depend on the inventory position in period 1. 

The initial conditions for a simulation experiment are the starting values assigned to the variables in the model and the events scheduled at the beginning of each run or replication. Since the parameters of interest in a steady-state simulation do not depend on the initial conditions, initial conditions are often chosen for programming convenience. In the inventory simulation the initial inventory position is / = S, for example. Some care in setting the initial conditions can greatly reduce the bias they cause. 

Sales forecasting, which projects point-of-sale consumer demand, is one of the collaboration tasks associated with this activity. The retailer task here is Point of Sale (POS) Forecasting and the manufacturer task is Market Data Analysis. The other collaboration task is Order Planning/Forecasting which uses factors, such as transit lead times, sales forecast, and inventory positions to determine future product ordering and requirements for delivery. The associated retailer task is Replenishment Planning, and Demand Planning is the associated manufacturer task. 

Given the above background, we can look at inventory policies to answer the question of when to order. This inventory policy tells us when the level of inventory or the inventory position of an item should be reviewed to identify whether an item should be ordered. Until the computer took over most inventory functions, the inventory policy usually used was a periodic review policy. With this policy the inventory manager would count the amount of inventory available at prescribed times. Now that computers can continuously review the inventory status of thousands of item, most firms use a continuous review policy. This means that every time an item is used, the computer calculates the balance in stock (i.e., determines the item s inventory position) and evaluates whether this inventory position is at or below a reorder point. 

The initial inventory position of each DC at the beginning of the period is known. 

Table 2 shows the results using initial inventory positions (3, 3, 3). Suppose that current inventory positions are (3, 3, 0), and a customer from market 3 places an order. From Table 2, since + T(2,3,0) = 1 + 2.78 < c 3+ T(3,2,0) = 1 + 2.88, we need to make a transshipment from DC 1 to market 3. We will illustrate how to compute T S, S, Sj) after describing heuristic rule 3. 

Even though the results from Table 2 were derived from a simple setting, some observations become evident. The optimal system cost does not necessarily decrease with the reduction of inventory positions, which seems counter-intuitive. It is explainable because we assume that the local shipment cost is zero. Thus, the more balanced the inventory positions, the lower the probability of atransshipment—hence the smaller the optimal cost. 

Suppose that current inventory positions are (2, 0, 3), and a customer from market 2 places an order. According to the Nearest Rule, the retailer needs to make a transshipment from DC 3 since C >C... 

To illustrate, consider an infinite-horizon variant of the newsvendor game with lost sales in each period and inventory carry-over to the subsequent period, see Netessine et al. 2002 for complete analysis. The solution to this problem in a non-competitive setting is an order-up-to policy. In addition to unit-revenue r and unit-cost c we introduce inventory holding cost h incurred by a unit carried over to the next period and a discount factor 0. Also denote by x the inventory position at the beginning of the period and by yj the order-up-to quantity. Then the infinite-horizon profit of each player is... 

Periodic-review settings We discuss supply chains that work in a periodic-review fashion. In other words, inventories are monitored at the beginning of each fixed time interval (a period). This assumption has no significant limitation in practice, as the length of a period selected for the control policy can vary arbitrarily. Usually, the type of inventory that we will be interested in is the so-called inventory position - i.e., the inventory that was ordered but not yet received at the member s location ( outstanding orders ), plus the inventory on hand at the location, minus any accumulated backlogs. We assume fiill backlogging. 

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. 

The dynamic program in Proposition 8 represents the decision-maker s problem as an attempt to find the best inventory position level, given a multidimensional information state Xt- this type of problem, a state-dependent... 

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