How Linear Programming Helps Companies Like FedEx Optimize Shipping Routes
Companies like FedEx handle millions of packages every day, so it is important for them to find the best way to deliver them as quickly and cheaply as possible.
Getting a package from Point A to Point B sounds simple. In real life, it’s a giant puzzle: millions of parcels, flight schedules, truck capacity, traffic, weather, cut-off times, and delivery promises.
These companies deal with complex logistical challenges, such as minimizing shipping costs, reducing delivery times, maximizing resource utilization, and ensuring customer satisfaction.
Operations Research (OR) techniques, especially Linear Programming (LP), play a critical role in optimizing shipping routes for logistics companies like FedEx. Linear Programming (LP), is a method that helps them make better decisions about their shipping routes.
This article explains, how OR—especially linear programming (LP)—helps plan and improve shipping routes. Here is how companies like FedEx benefits from LP, that is, how OR techniques like LP optimizes their shipping routes.
Table of content
What is Operations Research?
Operations Research is the science of making better decisions using math and data. It turns real-world decisions into models you can solve on a computer. In shipping, that means finding routes, assigning vehicles, and scheduling flights so that cost is low and deliveries are on time.
Why Routing Is Hard (and Why We Need OR)
- So many choices: thousands of trucks and planes, many hubs and cities.
- Many limits: vehicle capacity, driver hours, airport slots, time windows.
- Changing conditions: storms, road closures, last-minute orders.
Trying all possibilities by hand is impossible. OR models explore the options and find the best one according to a goal—usually minimum cost or fastest delivery with given cost.
Linear Programming in One Minute
Linear programming is a way to choose the best numbers for decisions when your goal (cost, time, distance) and your rules (capacity, demand, time limits) are all linear.
- Decision variables – the “how much” choices (e.g., how many packages to send on each route).
- Objective function – the thing to minimize or maximize (e.g., total shipping cost).
- Constraints – the rules that must be true (e.g., every city’s demand is met; capacity isn’t exceeded).
When the decision is “how much,” plain LP works. When the decision is “yes/no” (use this route or not), we use integer or mixed-integer LP (MILP). Shipping problems often mix both.
The Classic “Transportation Problem” (Made Simple)
A perfect starter model for shipping is the transportation problem:
- Supply points (hubs, warehouses, airports)
- Demand points (delivery areas, cities)
- Cost per unit between each pair
- Goal: ship at minimum total cost without breaking capacity or leaving any demand unmet
Decision variable: xij = number of packages sent from hub i to city j.
Objective: Minimize Total Cost = ╬г (cost per package × packages shipped).
Constraints:
- For each hub: shipments out ≤ hub capacity.
- For each city: shipments in = city demand.
- All flows ≥ 0.
Solve this, and you get the cheapest flow of packages through the network.
How a Company Like FedEx Applies LP in Real Life
FedEx runs a hub-and-spoke network: local pickups move to regional hubs, then to major hubs, then out to destination hubs and finally to customers. OR and LP support decisions at several levels:
1) Network Flow & Linehaul Planning
What it does: decides how much volume should flow along each lane (e.g., from City A to Hub X).
LP role: a transportation or minimum-cost flow model balances cost, capacity, and due times.
2) Aircraft Routing & Loading
What it does: picks which planes fly which legs and how to fill them.
MILP role: adds yes/no choices with constraints for range, payload, and maintenance schedules.
3) Truck Dispatch & Linehaul Schedules
What it does: plans highway moves between hubs and stations.
MILP role: assigns tractors/trailers, respects driver hours, and meets cut-off times.
4) Last-Mile Delivery (Vehicle Routing Problem)
What it does: builds delivery tours for vans with time windows.
MILP + heuristics: selects visit order, bundles stops, and fits shift length and capacity.
5) Real-Time Re-Routing
What it does: updates plans when flights delay or roads close.
LP/MILP + fast heuristics: re-optimizes flows and swaps loads across alternate routes.
6) Forecast-Driven Planning
What it does: uses demand forecasts to plan ahead.
LP role: decides how much capacity to schedule, where to position trailers, and when to add overflow flights.
A Tiny Example (No Heavy Math)
Imagine two hubs (H1, H2) and two cities (C1, C2). Each hub has limited capacity; each city needs a certain number of packages. Costs per package differ on each lane.
- The LP “flows” packages on the cheapest lanes first.
- If a cheap lane hits capacity, it diverts to the next best lane.
- All city demands are met; hub capacities are not exceeded.
- Result: the cheapest plan that obeys all rules.
Typical Constraints in Shipping LP Models
- Capacity: plane payload, truck cubes/weight, number of containers.
- Time windows: pickup/delivery deadlines; hub cut-off times.
- Balance: what goes in must come out; every package is assigned.
- Regulations: driver hours, crew duty limits, aircraft maintenance.
- Service levels: overnight vs. two-day must meet promised times.
- Robustness: keep buffers so small delays don’t break the plan.
Tools Beyond Basic LP:
- Mixed-Integer Programming (MIP): adds yes/no route and assignment choices.
- Network Flow & Shortest Path: fast methods for large route graphs.
- Vehicle Routing Problem (VRP): last-mile routing with time windows.
- Heuristics & Metaheuristics: smart shortcuts (savings, tabu, genetic).
- Simulation: tests “what if” (e.g., a storm) without risking real operations.
- Stochastic/Robust Optimization: plans for uncertainty in demand and travel times.
These methods often work together: forecasts → LP for flows → MIP for assignments → heuristics for fine-tuning → simulation for stress tests.
What Companies Gain from LP-Based Routing
- Lower cost: chooses cheap lanes first, avoids waste, fills vehicles better.
- Faster, more reliable delivery: meets deadlines with smart buffers.
- Higher asset use: better aircraft/truck utilization and fewer empty miles.
- Greener operations: fewer miles and better loads reduce fuel and emissions.
- Consistency at scale: same logic works for 100 stops or 100,000.
Step-By-Step Modeling Template
Use DOCS to remember the flow:
- D — Decisions: What are you choosing? (flows on lanes, route selections)
- O — Objective: What are you optimizing? (minimize cost or time)
- C — Constraints: What rules must hold? (capacity, demand, time windows)
- S — Sensitivity: What happens if demand or costs change? (robustness)
If you can write DOCS for a shipping question, you’re most of the way to a full model.
Common Exam Questions (and Short Answers)
Q1: Why is linear programming suitable for shipping?
Because costs and limits can be written as linear sums of flows, so LP finds the cheapest feasible plan fast.
Q2: When do we need integer variables?
When choices are yes/no or counts (use this route, assign this truck, open this hub). That’s MILP.
Q3: How does hub-and-spoke reduce cost?
Consolidating volume at hubs fills vehicles better and cuts the number of long, thin direct routes.
Q4: What is the transportation problem?
An LP that ships goods from multiple sources to multiple destinations at minimum cost while meeting demand and respecting capacity.
Q5: How do we handle time windows?
Add constraints linking departure and arrival times, and ensure each delivery happens inside its window.
Mini Case: Weather Disruption
- Detect: a hub/airport becomes unavailable.
- Re-optimize: the LP/MILP model blocks those lanes and reroutes flows through alternate hubs.
- Respect limits: checks capacity, crew duty time, and arrival deadlines.
- Deliver: high-priority shipments get first access to the tightest paths.
The result: delays are reduced, promises are kept when possible, and the plan remains feasible.
Hence, OR turns the messy world of shipping into clean math problems that computers can solve. Linear programming is the backbone: it helps companies like FedEx choose the best routes, fill planes and trucks wisely, and deliver on time—even when plans need to change mid-day.
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