Chapter 5 of 8

Transportation and Network Optimization

3 sections · ~9 min read · 3 practice questions · part of Supply Chain Analytics

Learning objectives

  • Assign demand regions to distribution centers on a least-cost basis and total the resulting outbound cost
  • Trade fixed facility cost and inventory pooling against outbound transportation savings
  • Compute a truckload versus less-than-truckload breakeven weight and interpret it as a shipment policy

The three cost families in a network decision

Choosing how many distribution centers to operate and where to place them is a trade-off among three cost families that respond differently to the number of facilities. Outbound transportation cost falls as facilities multiply, because each customer region is served from a closer node and the average delivery distance shrinks; the marginal saving is largest going from one facility to two and diminishes rapidly thereafter. Fixed facility cost rises roughly linearly with the count, covering lease or depreciation, management, systems, and the baseline labour needed to keep a building open regardless of throughput. Inventory cost rises sub-linearly, and the reason is the square root law of inventory pooling: when demand is split across independent locations, total system safety stock scales approximately with the square root of the number of stocking locations, so doubling from one facility to two multiplies safety stock by about 1.414 rather than by 2. The result of these three shapes is a shallow U-shaped total cost curve. There is usually a broad region of near-equivalent designs rather than a sharp optimum, which means the decision often turns on factors the cost model omits, such as delivery lead time, labour availability, or the flexibility to exit a lease. A responsible analysis states the size of the gap between the top options and compares it to the model's own uncertainty.

Assignment, and the assumptions hiding in a rate table

Given candidate facility locations, the assignment step is straightforward when capacity is not binding: send each demand region to whichever open facility serves it most cheaply per unit, then multiply by regional volume and sum. Two assumptions do heavy lifting here. First, that per-unit rates are constant, meaning no volume breakpoints and no minimum charges; real carrier tariffs have both, and a region assigned a small volume may not achieve the rate the model assumed. Second, that capacity is unconstrained, so no region gets pushed to a more expensive facility because the cheap one filled up. Adding capacity limits turns the problem from arithmetic into an optimization that needs a solver, and it can change assignments materially. Units also require attention: outbound rates are quoted per unit, per hundredweight, per pallet, or per shipment, and a model that mixes bases produces confident nonsense. Convert everything to cost per unit of the same item before combining.

Mode choice and the breakeven weight

Beneath the network structure sits a recurring shipment-level decision: send a load as less-than-truckload, priced per hundredweight and shared with other shippers, or as a dedicated truckload at a flat rate for the lane. LTL costs scale with weight, so it is cheaper for small loads; truckload is a fixed charge, so its effective cost per hundredweight falls continuously as the trailer fills. Setting the two equal gives a breakeven weight above which truckload is cheaper. Below it, LTL wins. This single number is more actionable than any rate negotiation, because it converts to a simple operating rule: consolidate orders on a lane until the accumulated weight crosses the breakeven, then dispatch a truckload. It also exposes the worst outcome in freight, the load just above the breakeven weight tendered as LTL, which pays truckload-level money for LTL-level service. Note that the breakeven ignores transit time and handling; LTL involves terminal transfers and more handling touches, so it carries a higher damage rate and a longer, more variable transit, and those belong in the decision even though they do not appear in the rate comparison.

Worked example

Problem

Marlow Instruments ships 600,000 units per year to three regions: North 180,000 units, Central 260,000 units, South 160,000 units. Option 1 operates a single central distribution center with outbound rates of 2.85 dollars per unit to North, 0.95 to Central, and 2.60 to South. Option 2 operates two distribution centers, one north and one south, with rates from the north facility of 0.80 to North, 1.60 to Central, and 2.40 to South, and from the south facility 2.55 to North, 1.45 to Central, and 0.85 to South. Fixed cost is 420,000 dollars per facility per year in either option. System safety stock is 31,000 units with one facility, and the square root law applies. Unit cost is 24 dollars and the annual holding rate is 25 percent. Compare total annual cost.

Step by step

  1. Annual holding cost per unit = 24 x 0.25 = 6.00 dollars per unit per year.
  2. Option 1 outbound: North 180,000 x 2.85 = 513,000; Central 260,000 x 0.95 = 247,000; South 160,000 x 2.60 = 416,000. Total outbound = 513,000 + 247,000 + 416,000 = 1,176,000 dollars.
  3. Option 1 fixed = 420,000 dollars. Option 1 safety stock cost = 31,000 x 6.00 = 186,000 dollars.
  4. Option 1 total = 1,176,000 + 420,000 + 186,000 = 1,782,000 dollars per year.
  5. Option 2 assignment, taking the cheaper facility per region: North goes to the north facility at 0.80 (versus 2.55); Central goes to the south facility at 1.45 (versus 1.60); South goes to the south facility at 0.85 (versus 2.40).
  6. Option 2 outbound: 180,000 x 0.80 = 144,000; 260,000 x 1.45 = 377,000; 160,000 x 0.85 = 136,000. Total outbound = 144,000 + 377,000 + 136,000 = 657,000 dollars. Outbound saving versus Option 1 = 1,176,000 - 657,000 = 519,000 dollars.
  7. Option 2 fixed = 2 x 420,000 = 840,000 dollars, an increase of 420,000 dollars.
  8. Option 2 safety stock = 31,000 x square root of 2 = 31,000 x 1.4142 = 43,840.6 units, an increase of 12,840.6 units. Safety stock cost = 43,840.6 x 6.00 = 263,043.72 dollars, an increase of 77,043.72 dollars.
  9. Option 2 total = 657,000 + 840,000 + 263,043.72 = 1,760,043.72 dollars per year.
  10. Difference = 1,760,043.72 - 1,782,000 = negative 21,956.28 dollars, so Option 2 is cheaper by 21,956.28 dollars per year.
  11. Per unit: Option 1 = 1,782,000 / 600,000 = 2.9700 dollars per unit; Option 2 = 1,760,043.72 / 600,000 = 2.9334 dollars per unit; the gap is 3.66 cents per unit, or 1.23 percent.
  12. Breakeven fixed cost per facility: set 519,000 = fixed cost + 77,043.72, giving fixed = 441,956.28 dollars. Check at that value: Option 1 = 1,176,000 + 441,956.28 + 186,000 = 1,803,956.28; Option 2 = 657,000 + 883,912.56 + 263,043.72 = 1,803,956.28. The two are equal.

Answer. Option 2, the two-facility network, costs 1,760,043.72 dollars per year against 1,782,000 dollars for the single central facility, an advantage of 21,956.28 dollars or 3.66 cents per unit. The mechanics are that 519,000 dollars of outbound savings are almost entirely consumed by 420,000 dollars of additional fixed cost and 77,043.72 dollars of additional pooled safety stock, leaving a thin margin. This is a genuinely close call and should not be decided on the cost model alone: a fixed cost above 441,956 dollars per facility, a fifteen percent error in one of the larger outbound rates such as the 2.85 to North or the 0.95 to Central under Option 1, or a slightly different safety stock base would flip the ranking. Not every rate is that powerful: fifteen percent on Option 2's 0.85 South rate moves only 20,400 dollars and would leave the ranking intact, which is itself worth knowing, since it says the decision hangs on the rates carrying the most volume. The tiebreakers that belong in the decision are the shorter delivery transit times Option 2 gives to the North and South regions and the reduced exposure to a single facility outage, weighed against the operating complexity of running two buildings.

Practice

Work each question before opening the solution.

  1. A consultant proposes four distribution centers, arguing that pooling losses are modest. Compute the safety stock cost at four facilities and state what outbound saving would be needed to justify it against Option 1.

    Show solution for question 1

    Safety stock = 31,000 x square root of 4 = 31,000 x 2 = 62,000 units, costing 62,000 x 6.00 = 372,000 dollars, an increase of 186,000 dollars over Option 1. Fixed cost rises by 3 x 420,000 = 1,260,000 dollars. Total additional cost before transport = 1,446,000 dollars, which exceeds the entire Option 1 outbound spend of 1,176,000 dollars. Even free outbound delivery could not justify four facilities at these fixed costs, so the proposal fails without any further analysis.

  2. On one lane, truckload costs a flat 1,850 dollars with a 22,000-pound trailer limit, while LTL is quoted at 9.40 dollars per hundredweight. Compute the breakeven weight and the cost of a 14,000-pound shipment under each option.

    Show solution for question 2

    Breakeven weight: set 9.40 x (W / 100) = 1,850, so W = 1,850 / 9.40 x 100 = 19,680.9 pounds. Above roughly 19,681 pounds, truckload is cheaper. A 14,000-pound shipment costs 9.40 x 140 = 1,316 dollars as LTL versus 1,850 dollars as truckload, so LTL saves 534 dollars. Note that a full 22,000-pound truckload has an effective rate of 1,850 / 220 = 8.41 dollars per hundredweight, so the truckload option is only attractive when the trailer is genuinely well filled.

  3. In Option 2, the Central region is served from the south facility at 1.45 dollars per unit rather than the north facility at 1.60. Explain why a strict least-cost assignment might still be the wrong operating choice.

    Show solution for question 3

    The 15-cent gap on 260,000 units is worth 39,000 dollars per year, a thin margin that a fuel surcharge change or a rate renegotiation could erase. More importantly, sending the largest region entirely to one facility concentrates 420,000 of 600,000 units at the south building, creating a labour and capacity peak there and leaving the network without a practical fallback if that facility is disrupted. Splitting Central between the two facilities costs a modest premium and buys balanced utilization and a working contingency, which is often a better purchase than the last 39,000 dollars of freight savings.