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Lot-for-Lot (L4L) is a lot-sizing technique that orders exactly the quantity needed for a specific production period. This method minimizes inventory costs by producing or ordering only the required amount for each period. Inventory holding costs are zero, but setup or order costs might be higher with this method due to smaller quantities produced or ordered more frequently.

Economic Order Quantity (EOQ) is a popular lot-sizing technique that orders a specific quantity that minimizes the total cost of inventory holding and ordering. The EOQ formula is given by: EOQ = \(\sqrt{\frac{2DS}{H}}\) where D is the annual demand, S is the setup or ordering cost per order, and H is the holding cost per unit per year. The EOQ technique balances inventory holding and ordering costs for an optimal ordering quantity which reduces overall costs.

Period Order Quantity (POQ) is similar to EOQ but takes into account the production rate and demand rate while ordering. The POQ technique keeps the holding cost at a minimum level while ensuring sufficient inventory to meet the demand. The POQ formula can be represented as: POQ = \(\sqrt{\frac{2DS}{H}}\times\frac{P}{P-D}\) where P is the production rate. The POQ method minimizes holding and ordering costs while maintaining a smooth production flow.

Fixed Order Quantity (FOQ) is a lot-sizing technique where the same quantity of an item is ordered or produced every time an order is placed. This quantity is determined based on a predetermined reorder point. FOQ often results in higher inventory holding costs, but setup or ordering costs can be lower as a result of more predictable order quantities.

To determine which lot-sizing rule results in the least inventory, we will compare their inventory holding costs: 1. Lot-for-Lot: Zero holding costs, as inventory is produced or ordered exactly when needed. 2. Economic Order Quantity: Moderate holding costs, minimizes combined holding and ordering costs. 3. Period Order Quantity: Low holding costs, smooth production flow with minimized combined holding and ordering costs. 4. Fixed Order Quantity: Higher holding costs due to predetermined order quantities, potentially lower setup or ordering costs. From the comparison, we can conclude that the Lot-for-Lot (L4L) rule results in the least inventory holding costs as it orders or produces exactly the amount needed for each production period without carrying any extra inventory.