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## Lot Size

The lot size (LS) (also known as the replenishment quantity (RQ) or the cycle stock (CS)) is the number of units that arrive in a replenishment lot or are produced in a manufacturing lot (Points 1, 2, and 3 in the figure). The average replenishment quantity (ARQ) is the average size of lot size replenishments derived by dividing the total replenishment quantity over a particular period of time by the number of replenishments received during that time.

## Economic Order Quantity

The economic order quantity (EOQ) is the lot size that minimizes the sum of ordering cost and inventory carrying cost associated with the size of the order (see figure). The higher the order quantity, the greater the inventory level.  However, the higher the order quantity the fewer the number of orders and the lower the resulting ordering cost.

The economic run quantity (ERQ) is the production lot size (or run quantity) that minimizes the total of setup/changeover costs and the inventory carrying costs associated with the inventory produced by the run length. The tradeoffs between manufacturing setup cost and inventory carrying costs for determining optimal production run sizes for a large textiles client are illustrated in the figure below. Note in the example that the optimal run length is 3 or 4 rolls per setup for that particular SKU. As is often the case with EOQ modeling, the total cost curve is fairly flat near the optimal solution. The key, as is often the key, is to make decisions that are at least in the “ballpark of optimal”. Unfortunately we often find that lot sizing is off by 200% or 300%.

The formula to compute the EOQ for a purchased item is as follows:

EOQ = {(2 x FAD x POC) / (UIV x ICR)}1/2

For example, if an item has an annual demand of 3,000 units per year; a purchase order cost of \$300 per purchase order; a purchase price of \$2,100 per unit; and an inventory carrying rate of 30% per year then its EOQ is

EOQ = [(2 x 3,000 x \$300)/(\$2,100 x 30%)] ½ = [(1,800,000)/(630)] ½ = [2,857]1/2 = 53 units

The formula to compute the EOQ for a manufactured item, sometimes referred to as the economic run quantity (ERQ) is as follows:

ERQ = {(2 x FAD x SUC) / (UIV x ICR)}1/2

For example, if an item has an annual demand of 5,000 units per year; a setup cost of \$3,200 per setup; a standard cost of \$85.00 per unit; and an inventory carrying rate of 25% per year then its EOQ is

EOQ = [(2 x 5,000 x \$3,200)/(\$85 x 25%)] ½ = [(32,00,000)/(21.25)]½ = [1,505,882]1/2 = 1,227 units

EOQ is considered passé, outdated, and nearly pre-historic in many inventory circles. Yet, in our work with the most advanced supply chain organizations around the world we are finding great profit, service, and operational improvements with EOQ.

## Unit Fill Rate (UFR)

The unit fill rate (UFR) for an item is the portion of the total number of units requested with inventory available to fill the request. It is distinct from and higher than line fill rate (% of lines shipped complete) and order fill rate (% of orders shipped complete). The target unit fill rate is a decision, not an outcome. It is perhaps the most important inventory planning decision of all.

As discussed previously, the higher the unit fill rate, the lower the lost sales cost.  However, the higher the unit fill rate, the greater the inventory required to provide it, and the greater the resulting inventory carrying cost.  There are many ways to determine optimal target unit fill rates. One method is to choose the unit fill rate that minimizes expected inventory policy cost. Another method is to choose the unit fill rate that maximizes expected GMROI. Still another method is to choose the unit fill rate that maximizes IVA. What do we do? It depends on the financial, service, and operational goals. The ability to visualize and simulate those relationships as demonstrated in the figure from the RightStock™ Inventory Optimization System is the key and often missing piece in the inventory strategy puzzle.

As explained earlier, fill rate requirements go a long way toward determining overall inventory requirements.  Simply put, all things being equal, the higher the fill rate requirement, the higher the inventory level required to support it. The higher inventory levels are the result of additional safety stock inventory.

An example inventory and fill rate analysis from a recent engagement in the health and beauty industry is provided in Figure 2. Note that as fill rate increases (from 50% to 99.95%) the required inventory investment increases accordingly from \$4,646,094 to \$8,644,548. At the same time, lost sales cost declines from a high of \$17,953,234 at a 50% fill rate to a low of \$17,953 at a 99.95% fill rate.

The current inventory investment in the example was \$8,300,000 and the lost sales cost was \$3,949,712. The inventory investment that should have yielded a 99.9% fill rate only yielded an 87% fill rate. The discrepancy turned out to be a major mis-deployment of inventory.

## Safety Stock Inventory

The literal definition of safety stock inventory (SSI) is the inventory on-hand when a replenishment arrives (Point 7 in Figure 2.21). The average safety stock is the average on-hand inventory at the end of several replenishment cycles. Safety stock is required to support promised levels of inventory availability when the demand during a leadtime or the length of a leadtime is variable.  For example, if a replenishment is delayed or if the demand during a leadtime is much greater than normal, safety stock is in place to fulfill demand until the replenishment arrives or to satisfy some portion of the excess demand.  There would be no need for safety stock if we knew exactly what customers wanted, when they wanted it, and exactly when replenishments arrive.  To the extent there is uncertainty in any of those three variables, safety stock is required to provide anything better than 50% inventory availability.

Safety stock is required to support promised levels of inventory availability when the demand during a leadtime or the length of a leadtime are variable.  For example, if a replenishment is delayed or if the demand during a leadtime is much greater than normal, safety stock is in place to fulfill demand until the replenishment arrives or to satisfy some portion of the excess demand.  There would be no need for safety stock if we knew exactly what quantity the customers wanted, when they wanted it, and exactly when a replenishment would arrive.  To the extent there is uncertainty in any of those three variables, we will need safety stock to provide anything better than a 50% inventory availability.

## Reorder Point

The reorder point (ROP) is the inventory level at which a replenishment order is placed (Point 8 in the figure).  As a rule, the reorder point is set at the leadtime demand plus safety stock.

ROP = LTD + SSI

There are a variety of other inventory control policies and variables including the use of order-up-to-levels (OUL), the level of inventory a replenishment quantity should yield when it is placed; review time periods (RTP), the fixed time between inventory reviews; and a wide mix of programs for joint item replenishment.