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Square root law: The math behind why inventory centralization reduces safety stock

Published Aug 2024

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The Square Root Law is a mathematical principle used in supply chain management to explain how inventory centralization reduces safety stock. Here’s a breakdown of the concept and the math behind it:

Understanding Safety Stock

  • Safety Stock: This is the extra inventory kept to prevent stockouts due to demand and lead time variability or supply chain disruptions.
  • When inventory is decentralized across multiple locations, each location must carry safety stock to cover its own demand and lead time variability. This leads to higher overall safety stock across the system.

Centralization vs. Decentralization

  • Decentralized Inventory: Inventory is kept at multiple locations close to the customer.
  • Centralized Inventory: Inventory is kept at a single or fewer locations, serving multiple demand points.

The Square Root Law Formula

The Square Root Law provides a formula to estimate the reduction in safety stock when moving from a decentralized system to a centralized one.

Where:

  • sscentralized the safety stock in the centralized system.
  • ssdecentralized is the safety stock in the decentralized system.
  • ncentralized is the number of central locations.
  • ndecentralized is the number of decentralized locations.

Explanation of the Math

  • In a decentralized system, each location holds safety stock proportional to its demand and lead time variability.
  • By centralizing inventory, you pool the demand and lead time variability across all locations. Because demand and lead time variability across locations tends to average out, the combined variability (and hence safety stock) is lower.
  • The reduction is not linear but follows a square root function. This is because the standard deviation of the aggregated demand and lead time decreases at the rate of the square root of the number of locations.

Example Calculation

Suppose you have 4 decentralized warehouses, each holding safety stock of 100 units. If you centralize this inventory into one location, the new safety stock would be:

This shows a 50% reduction in safety stock due to centralization.

Key Impacts of centralization

Reduction in Safety Stock:Centralizing inventory leads to a significant reduction in safety stock. This decreases the overall inventory holding costs, freeing up capital and reducing the risk of obsolescence.

Impact on warehousing costs: Centralization reduces the number of warehouses, leading to savings on fixed costs like rent and labor. However the remaining warehouses may need to be larger or more advanced, which could increase costs at these locations. The net impact on warehousing costs depends on the balance between reduced locations and potentially higher costs at central sites.

Increased Lead Times and Service Level Impacts: While centralizing inventory can reduce safety stock, it may also increase lead times to customers. With inventory located further from end customers, the time to fulfill orders might increase, potentially impacting service levels. Companies need to balance the benefits of lower safety stock against the potential cost of slower response times.

Increased Transportation Costs: When inventory is centralized, products often need to be shipped over longer distances to reach customers. This can lead to higher transportation costs, as goods must be moved from a single, central location to various points of demand.

The increased transportation costs might offset some of the savings achieved through reduced safety stock. Additionally, transportation costs can fluctuate with fuel prices and other factors, introducing variability in total logistics costs. Therefore, it’s essential to analyze whether the savings in inventory costs outweigh the potential rise in transportation expenses.

Balancing these factors requires a comprehensive analysis to determine the optimal number and location of warehouses in a supply chain network. Companies might find that a hybrid approach—centralizing certain types of inventory while keeping others decentralized—offers the best trade-off between cost and service.

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