Inventory-Routing Problem
IRP · BELL ET AL. 1983
Logistics · Transportation · TacticalThe Inventory-Routing Problem jointly decides when each customer is replenished, how much is delivered, and which routes the vehicles take — a coordinated answer to two questions that inventory and routing would otherwise answer separately. Born from the Exxon gas-distribution work of Bell, Dalberto, Fisher, Greenfield et al. (1983), IRP is the canonical Vendor-Managed-Inventory problem: the supplier replaces the customer's reorder decision with a system-wide cost-minimising schedule. The best survey is Coelho, Cordeau & Laporte (2014), "Thirty years of inventory routing".
Why IRP matters
Integration pays
IRP is the canonical case for integrated decision-making in logistics. Solving inventory and routing separately (e.g., EOQ at every customer, then VRP over resulting daily demands) is globally suboptimal: the supplier's truck is a shared resource that couples the stocking decisions. Coelho-Cordeau-Laporte (2014) report 5–30% cost reductions against sequential heuristics on standard benchmarks.
Vendor-Managed Inventory (VMI) is the operational paradigm that makes IRP tractable: the supplier — not the customer — owns the replenishment decisions. This gives the supplier visibility of customer stock levels and freedom to combine deliveries across customers. IRP quantifies the cost savings of that authority.
Mathematical formulation
Multi-period MILP · after Archetti et al. (2007)
Notation
| Symbol | Meaning |
|---|---|
| $T$ | planning-horizon length (periods) |
| $N$ | set of customers, $N' = N \cup \{0\}$ with depot $0$ |
| $d_i^t$ | demand of customer $i$ in period $t$ |
| $I_i^t$ | inventory level of customer $i$ at end of period $t$ |
| $U_i, L_i$ | upper / lower stocking bounds at customer $i$ |
| $C$ | vehicle capacity per period |
| $h_i$ | holding cost per unit per period at customer $i$ |
| $c_{ij}$ | travel cost on edge $(i,j)$ |
| $q_i^t \ge 0$ | quantity delivered to $i$ in period $t$ |
| $y_i^t \in \{0,1\}$ | 1 if $i$ is visited in period $t$ |
| $x_{ij}^t \in \{0,1\}$ | 1 if edge $(i,j)$ is used in period $t$ |
Objective (total cost = holding + routing)
Constraints
ML / OR variants. Maximum-Level (ML) policy allows order-up-to-U decisions; Order-Up (OU) restricts $q_i^t = U_i - I_i^{t-1}$ when visited. The Single-Vehicle IRP uses one vehicle per period; multi-vehicle IRP requires an extra vehicle index on $x$ and $q$.
Solution methods. Branch-cut-price (Archetti-Bertazzi-Hertz-Speranza 2007); ALNS metaheuristics (Coelho-Laporte 2013); matheuristics combining MIP on $y$ with VRP solvers on the period sub-problems.
Key references
DOIs & permanent URLs