Chapter 2 of 4

Network Design and Warehouse Siting

3 sections · ~7 min read · 3 practice questions · part of Logistics & Transportation

Learning objectives

  • Estimate the optimal number of warehouses for a given customer base using a square-root law
  • Compute transportation cost savings from adding a warehouse
  • Recognize the trade-offs between inventory pooling and customer proximity

Why network design matters

Once shipments are flowing, the largest source of structural logistics cost is the number and location of warehouses. Each warehouse adds fixed cost (rent, labor, IT, equipment) and adds inventory, but reduces outbound transportation cost by moving stock closer to customers. Two different square-root relationships get muddled in practice, and it is worth separating them. The first is the inventory square-root law, which is exact under standard assumptions: total system safety stock across n facilities is roughly sqrt(n) times the safety stock a single pooled facility would need. This one is used later in the chapter. The second is a planning heuristic, not a law: because outbound transport savings taper off while fixed and inventory costs keep rising, the cost-minimizing facility count tends to grow with roughly the square root of demand or customer count rather than in proportion to it. Under that heuristic, doubling the customer base raises the indicated warehouse count by about 41 percent (sqrt(2) - 1), not by 100 percent. Treat the heuristic as a sanity check on a network model's output, not as a substitute for one; it ignores geography, and geography is usually what decides the answer. It does explain a common pattern, though: direct-to-consumer brands typically run one warehouse far longer than intuition suggests, and add a second only when density on the far side of the country justifies duplicating the fixed cost.

Inventory pooling and the value of a central DC

Pooling inventory at one central location rather than splitting it across many regional sites reduces total safety stock because the variance of the sum is less than the sum of variances. If two warehouses each face demand with standard deviation sigma, holding them separately requires safety stock of 2 x k x sigma; pooling them at one site requires k x sigma x sqrt(2) = 1.414 k sigma, a saving of about 29 percent. The trade-off is that pooling raises outbound transportation cost and lengthens customer lead time. The right network design balances pooled inventory savings against split-network transportation and service benefits. For low-variance, high-service products the answer usually tilts toward pooling; for high-variance, urgent products the answer tilts toward regional stock.

A simple siting calculation

A useful first-pass siting method is the center-of-gravity calculation. Place every supplier and customer zone on a coordinate grid, weight each point by its annual volume (or by volume times freight rate per mile, if inbound and outbound rates differ materially), and compute the volume-weighted average of the x coordinates and of the y coordinates. The weighted centroid minimizes a weighted sum of squared distances; it does not generally minimize freight cost, which depends on actual routes and rates. Treat it as a starting point. It may land somewhere unsuitable, so list feasible sites nearby and compare their inbound and outbound costs across all zones. The center of gravity narrows the search; the candidate evaluation picks the winner. The method ignores real-world constraints (labor markets, real estate, taxes, natural hazard risk) but is good enough to narrow the field from a country to a region. A more refined analysis uses a discrete-event simulation or a mixed-integer optimization, both of which require good cost data and a clear objective function. Evaluate candidate sites under future demand scenarios as well as current demand, especially when leases and capacity commitments extend several years.

Worked example

Problem

A firm has 144 customer zones. Demand in each zone over the replenishment period is normally distributed with a mean of 1,000 units and a standard deviation of 300 units, and zones are independent of one another. Compare (i) 12 regional warehouses, each serving 12 zones and holding its own safety stock, with (ii) one central DC serving all 144 zones from pooled stock. Use k = 1.65 and a holding cost of $4 per unit of safety stock per year. Compute annual safety stock cost under each design and the ratio between them.

Step by step

  1. Because zones are independent, variances add. A facility serving n zones faces sigma = sqrt(n) x 300 units.
  2. Split design: each of the 12 warehouses serves 12 zones, so sigma per warehouse = sqrt(12) x 300 = 3.4641 x 300 = 1,039.2 units.
  3. Safety stock per warehouse = 1.65 x 1,039.2 = 1,714.7 units. Across all 12 warehouses = 12 x 1,714.7 = 20,576 units.
  4. Pooled design: one DC serves all 144 zones, so sigma = sqrt(144) x 300 = 12 x 300 = 3,600 units. Safety stock = 1.65 x 3,600 = 5,940 units.
  5. Annual cost at $4 per unit: split = 20,576 x $4 = $82,306. Pooled = 5,940 x $4 = $23,760.
  6. Saving from pooling = $82,306 - $23,760 = $58,546 per year, a reduction of 71.1 percent.
  7. Check the result against the square-root law: consolidating 12 facilities into 1 should cut safety stock by a factor of sqrt(12) = 3.464, and 20,576 / 5,940 = 3.464. The two agree, which is the point of the law: the saving depends only on the number of facilities being merged, not on the demand figures.
  8. Caveat: this is safety stock only. The pooled design adds outbound transportation cost and lengthens customer lead time, and those two effects are what usually decide a real network case.

Answer. Split design safety stock cost = $82,306 per year; pooled design = $23,760 per year. Pooling saves $58,546 per year, or 71 percent, and the ratio is exactly sqrt(12) as the square-root law predicts. The number to carry forward is the ratio rather than the dollars: merging n facilities into one cuts safety stock by a factor of sqrt(n), so the first consolidation buys far more than the last. Going from 12 sites to 6 recovers about 29 percent of safety stock; going from 2 to 1 recovers another 29 percent of a much smaller base.

Practice

Work each question before opening the solution.

  1. Using the square-root law, if a firm currently runs 9 warehouses optimally for its current customer base, by what factor would the customer base need to grow to justify 16 warehouses?

    Show solution for question 1

    Ratio of warehouses = sqrt(customer ratio). 16/9 = 1.778. Customer ratio = 1.778^2 = 3.16. Customer base would need to grow by about 3.16x.

  2. Two regional warehouses each hold 60 days of safety stock for a slow-moving industrial part. Quantify what pooling them at one site would save, in days and as a percentage.

    Show solution for question 2

    Let S be each site's safety stock, so the network holds 2S, equivalent to 120 site-days of cover. Pooling gives sqrt(2) x S = 1.414S, equivalent to 84.9 days. The reduction is 2 - 1.414 = 0.586 of one site's stock, which is 0.586 / 2 = 29.3 percent of the network total, or about 35 days of cover removed. Two things temper the answer for an industrial part. First, the pooling result assumes the two regions' demand is independent; if both regions serve the same seasonal end market, correlation is positive and the saving shrinks. Second, a slow-moving industrial part is often bought on urgency, and the extra day of transit from a central site can cost more in expedite and downtime than 35 days of carried stock is worth.

  3. Why does pooling safety stock work even when average demand at the central site is exactly the sum of the regional averages?

    Show solution for question 3

    Safety stock is driven by the standard deviation of demand, not by its mean, and those two aggregate differently. Means add: two sites at 1,000 units become one site at 2,000. Variances also add, but standard deviation is the square root of variance, so two sites at sigma = 300 become one site at sqrt(2 x 300^2) = 424 rather than 600. The mean doubled while the standard deviation rose by only 41 percent, so the buffer needed per unit of demand falls. High and low deviations in the two regions cancel each other out inside a shared pool; held separately, each site has to be protected against its own worst case.