Supply chains are optimization problems made physical. Raw materials move to factories; factories ship to distribution centers; distribution centers deliver to retailers or customers. Every flow, every inventory decision, every routing choice involves tradeoffs that operations research has been formalizing since the 1950s. The aggregate value of supply chain OR is estimated in the hundreds of billions of dollars annually.
Inventory exists to buffer against uncertainty and lead times. Hold too much and you pay carrying costs and risk obsolescence. Hold too little and you face stockouts, lost sales, and expediting costs. The core question: how much to order, and when?
The EOQ model (Harris, 1913) is one of the oldest and most enduring results in OR. It assumes:
The total annual cost as a function of order quantity Q:
TC(Q) = K*(D/Q) + h*(Q/2)
= ordering costs + holding costs
Ordering cost decreases with larger Q (fewer orders); holding cost increases (larger average inventory Q/2). Minimizing by taking the derivative and setting to zero:
Q* = sqrt(2KD/h) (the EOQ formula)
The optimal order quantity balances ordering and holding costs. At Q*, both costs are equal — a property called the square root law.
Example: Demand D = 10,000 units/year, K = $100/order, h = $2/unit/year. Q* = sqrt(2 × 100 × 10,000 / 2) = sqrt(1,000,000) = 1,000 units per order. Order frequency: 10 orders per year, every 5.2 weeks.
When demand is stochastic, the EOQ quantity is still used, but the reorder point includes a safety buffer:
Reorder point R = μ_L + z × σ_L
Where:
Safety stock = z × σ_L. It is inventory held purely to protect against demand variability and lead time uncertainty. Reducing demand variability (through better forecasting or shorter lead times) can dramatically cut safety stock requirements.
The newsvendor (or critical fractile) model addresses single-period inventory decisions with uncertain demand:
Optimal order quantity Q* satisfies:
F(Q*) = (p - c) / (p - s)
The right-hand side is the critical ratio — order up to the point where the probability of demand ≤ Q* equals the ratio of underage cost to total misalignment cost. This elegant result underlies fashion retail buying, option pricing, and any single-period capacity commitment.
The classic transportation problem is to ship goods from m supply origins to n demand destinations at minimum total cost, where each origin has a fixed supply capacity and each destination has a fixed demand requirement.
Minimize: sum_i sum_j c_ij * x_ij
Subject to: sum_j x_ij <= s_i for all i (supply constraints)
sum_i x_ij = d_j for all j (demand constraints)
x_ij >= 0
This is a network flow LP with totally unimodular structure — the LP always has an integer optimal solution. The transportation simplex or MODI method solves it efficiently.
Applications: Shipping costs between factories and warehouses; assigning workers to jobs (the assignment problem, a special case); balancing inventories across a network.
Real supply chains are networks with multiple echelons: supplier → regional distribution center → local distribution center → retailer. Inventory decisions at each level affect the others. Optimizing each echelon independently leads to the bullwhip effect — demand variability amplifies as it moves upstream.
Clark-Scarf model (1960): The foundational multi-echelon inventory model. For serial supply chains (each stage feeds exactly one downstream stage), an optimal policy exists with a decomposition property: each stage can be optimized semi-independently using an echelon inventory concept.
Key insight: Coordinate inventory decisions across the chain by sharing demand information, not just order information. Point-of-sale data flowing upstream reduces the bullwhip effect dramatically.
The vehicle routing problem (VRP) asks: given a depot and a set of customers with known demands and locations, find the minimum-cost set of routes for a fleet of vehicles such that each customer is visited exactly once and vehicle capacities are not exceeded.
VRP is NP-hard and generalizes the TSP. It is the core problem behind package delivery, grocery delivery, school bus routing, and service technician dispatch.
| Variant | Additional constraint |
|---|---|
| CVRP | Vehicle capacity limits |
| VRPTW | Time windows for each customer |
| VRPP | Pickup and delivery |
| DVRP | Dynamic arrivals in real time |
| MDVRP | Multiple depots |
Exact methods: Branch-and-cut for small instances (up to ~150 customers optimally). The cutting planes include capacity cuts and path elimination.
Heuristics: Clarke-Wright savings algorithm, sweep algorithm — fast but suboptimal.
Metaheuristics: Large neighborhood search (LNS), tabu search, adaptive LNS — the standard approach for large real instances. Modern LNS solvers handle thousands of customers in seconds.
FedEx, UPS, and Amazon route millions of deliveries daily using VRP variants with real-time updates. UPS's ORION (On-Road Integrated Optimization and Navigation) system saves an estimated 100 million miles and 10 million gallons of fuel annually by optimizing delivery sequences.
At the strategic level, supply chain decisions involve which facilities to open, where to locate them, and how to configure the network. Facility location problems are integer programs that balance fixed opening costs against service costs.
Uncapacitated Facility Location (UFL): For each potential facility site, decide whether to open it (binary variable) and which customers to serve from it (continuous fraction), minimizing total fixed and service costs subject to each customer being fully served from open facilities.
Greedy approximation gives a solution within (1 + ln n) of optimal. LP rounding gives a constant-factor guarantee. Commercial MIP solvers handle instances with thousands of facilities and customers.
| Company | Application | Impact |
|---|---|---|
| Dell | Build-to-order, component inventory management | Reduced inventory from 30 days to 7 days (1990s) |
| Walmart | Vendor-managed inventory, cross-docking | Structural cost advantage over competitors |
| Amazon | Multi-echelon fulfillment network, same-day inventory | Enabled Prime delivery promises |
| Zara | Rapid replenishment, deliberate limited stock | High freshness, low markdown losses |
| P&G | Collaborative planning with Walmart (CPFR) | Reduced bullwhip, improved service levels |
OR methods are increasingly applied to humanitarian supply chains — disaster relief, vaccine distribution, refugee resettlement. The optimization objectives differ from commercial logistics:
The UN Humanitarian Response Depot network — pre-positioned supply caches around the world — was designed using facility location optimization.