Periodic order quantity

Periodic order quantity and target stock levels... 

An alternative way to deal with variable demand is to use the periodic order quantity. Here, the reorder quantities are revised more frequently. The method uses mean time between orders CTBO), which is calculated by dividing the EOQ by the average demand rate. In the above example, the EOQ is 1,000 and the average demand 410. The economic time interval is therefore approximately 2. An example shown in Table 6.2 illustrates the same situation as in Table 6.1 in terms of demand changes and safety stock level. However, the reorder quantity is based on total demand for the immediate two weeks of history. This reorder method is called periodic order quantity (POQ). 

This is a single-period problem. Demand, D, is random, with a distribution function F x) that is known at the beginning of the period. The actual realization of the demand will not be known until the end of the period. The problem is to decide the order quantity Q at the beginning of the period, under the following cost assumptions Each unit of demand supplied earns a profit (selling price minus cost) of p, each unit of unmet demand incurs a penalty of rr, and each surplus (i.e., unsold) unit at the end of the period carries a net loss of (i.e., cost minus any salvage value). The objective is to maximize the expected net profit ... 

The distinction between the requirements. A, and the constrained order quantity, 2, is important. Whereas A, represents the quantity that is needed at the beginning of period t, Q, reflects what is feasible, taking into account lead time constraints and order quantity restrictions. For example, if A, = 40 and the maximum order quantity is 30, then Q, would be set to 30. 

At the beginning of each period f, the following sequence of events takes place. First, replenishment (including any scheduled receipt) arrives, that is, Q, the constrained order quantity (which relates to A, and will be specified below). These units are used first to satisfy the back orders, if any, left from the previous period. Next, demand of the period, D, is realized and filled, which brings us to the end of the period, when 1, (on-hand inventory) and B, (back orders) are updated. 

Min-max With Q and being the (given) lower and upper limit on order quantities, Q, = min max ((2min> A,), 2max > becomes the order quantity planned for period t... 

EOQ (economic order quantity) Q, = S/2CE(D,)lh, the classical EOQ formula (refer to Section 1.1), where C is a fixed cost for placing a replenishment order, and h is the inventory holding cost per period. 

Finally, suppose the order lead time is L (periods). Then the recommended order quantity, Q,m period t, corresponds to the calculated (constrained) order quantity L + t periods later ... 

So the recommended order quantity at the start of period t is whatever the constrained order quantity is for period t -H L. 

To summarize, when the order rule is lot-for-lot, that is, Q, = A, for all t, the required quantity needed at the beginning of period t (A,), and hence the recommended order quantity at the beginning of period t — L should be... 

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... 

The inventory management follows local (sj,5j) policies at both production sites (i = 1,2 ) and the harbour (i = h). Six parameters have to be determined to fidly specify the SC s raw material inventory management. If the available stock at location i e 1,2, h in period t (say l ) falls below the corresponding order point Sj, an order is placed with the order quantity qu = Si-l u- The available stock comprises the local inventory level flu) and all non-delivered orders ir)- U orders are placed at the production sites, they are immediately fulfilled via pipeline as long as a sufficient quantity of Naphtha is available at the harbour tanks. 

For example, if the weekly demand for laptop toy is 500 units and set-up cost to initiate the order is 200, and a laptop has a variable cost of 5 per unit, assuming a holding cost of 10 % of variable cost per period, we can find the optimal order quantity ... 

Using an Excel worksheet, simulation of the orders received at the central warehouse generates an average demand across 100 periods oi = 101.07 and the standard deviation, crc = 141.57. With this mean and standard deviation, and using the same formulas as before, the corresponding values of reorder level and order quantity at the central warehouse are as follows ... 

The first term in this equation [i.e., (D/Q)S] represents the annual demand for an item (DJ divided by the order quantity (Q). The result of this division is the number of times an order is placed for an item each year. Since the cost of placing this order is S, the first term represents the cost of placing orders for an item over the period of a year. The second term in the equation (Q/2)H says that the average inventory (Q/2) multiplied by the cost of holding a unit in inventory (H) is the expected annual inventory carrying cost for a particular item. 

Fixed reorder cycle inventory model—A form of independent demand management model in which an order is placed every n time units. The order quantity is variable and essentially replaces the items consumed during the current time period. Let M be the maximum inventory desired at any time, and let x be the quantity on hand at the... 

Usually in the newsvendor setting, it is assumed that if any inventory remains at the end of the period, one discount is used to sell it or it is disposed of. Khouja [79] extends the newsvendor problem to the case in which multiple discounts with prices under the control of the newsvendor are used to sell excess inventory. They develop two algorithms, under two different assumptions about the behavior of the decision maker, for determining the optimal number of discounts under fixed discounting cost for a given order quantity and realization of demand. Then, they identify the optimal order quantity before any demand is realized for Normal and Uniform demand distributions. They also show how to get the optimal order quantity and price for the Uniform demand case when the initial price is also a decision variable. 

Many research problems that address pricing and production decisions with fixed production set-up cost fall within the area of the Economic Order Quantity (EOQ) model). The general EOQ model inventory model has been studied frequently in inventory literature (see [164] for a review). The problem consists of multiple periods in a fixed time horizon, with a stationary deterministic function in each period ordering or production costs have a fixed and variable component. Since demand is deterministic, the optimal policy will leave zero inventory at the end of each time cycle, so each period may be considered... 

The solution to this problem will be the order quantity for that period. In other words, in a myopic policy, we ignore the future consequences (beyond a lead time) of the actions we take today. Recall that in the elementary i.i.d. demand case this is indeed optimal to do. But, as expected, myopic policies may perform poorly if demand patterns fluctuate significantly from one period to another. To illustrate this, suppose that during a peak period the demand... 

In keeping with the framework we posed earlier, after describing demand we must describe the relevant costs. Typically, some of these costs will be well-specified (e.g., the cost of the item), and some will not (e.g., the cost of failing to fulfill all demand, and perhaps the post-selling-period value of goods left over if the order quantity exceeds the demand). We will address the nature of these costs in the models that follow. 

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