Fleet Sizing & Vehicle Composition
Mixed-Integer Programming · Vehicle Mix
A logistics operator must decide how many vehicles of each type (small van 500 kg, medium truck 2 t, large articulated 10 t) to acquire. Fleet over-sizing wastes €50K–€200K/year in idle capital; under-sizing forces expensive spot-market hiring at 3× normal cost.
The Problem
Mixed-integer fleet composition under weekly demand uncertainty
A logistics company operates 3 vehicle types: small vans (500 kg capacity, €25K acquisition, €0.45/km operating), medium trucks (2 t capacity, €65K acquisition, €0.72/km operating), and large articulated vehicles (10 t capacity, €140K acquisition, €1.10/km operating). Each vehicle can serve a fixed number of delivery routes per week based on its type.
The objective is to find the optimal integer fleet composition that minimizes total annual cost (acquisition amortization + operating costs) while meeting a weekly demand profile that varies across a planning horizon of 12 weeks. Any shortfall must be covered by spot-market vehicles at 3× the normal operating cost.
Because the number of vehicles must be a non-negative integer, the problem is a Mixed-Integer Program (MIP). The integrality constraint prevents simple LP relaxation from yielding the optimal solution directly — branch-and-bound or cutting-plane methods are required.
subject to
Σj qj · ujw + sw ≥ Dw // meet demand each week w
ujw ≤ nj // can't use more than owned
nj ∈ ℤ≥0 // integer vehicle counts
ujw, sw ≥ 0 // non-negative usage & spot
Where nj is the number of type-j vehicles to acquire, ujw is utilisation of type-j in week w, sw is spot-market tonnage in week w, qj is the capacity of type j, and Dw is weekly demand in tonnes.
See Integer & Combinatorial Optimization theoryTry It Yourself
Optimize fleet composition across vehicle types and demand scenarios
Fleet Composition Optimizer
3 Types · 12 Weeks| Type | Capacity (t) | Acquisition (€K) | Op. Cost (€/km) | Routes/Week |
|---|
Ready. Select a scenario, then click “Optimize Fleet.”
| Algorithm | Acquisition (€K) | Annual Op (€K) | Utilisation % | Time |
|---|---|---|---|---|
| Click Optimize Fleet | ||||
The Algorithms
Three approaches to fleet composition
Greedy (Cheapest per Capacity)
O(T · W) | Greedy fill by cost-efficiencyRanks vehicle types by acquisition cost per tonne of capacity and greedily assigns the most cost-efficient type first. For each week, it fills demand starting with the cheapest-per-tonne vehicle until capacity is met, purchasing additional units only if existing fleet cannot cover peak demand. Fast but ignores operating cost trade-offs and may over-invest in large vehicles that sit idle during low-demand weeks.
Min Acquisition
O(T · W) | Minimize capital outlayMinimizes upfront acquisition cost by purchasing the fewest vehicles that can cover average-week demand, then relying on spot-market hiring for peak weeks. This strategy keeps capital expenditure low but can result in high operating costs when spot premiums compound over many weeks. Best suited for operators with tight capital budgets but flexible operating margins.
Min Total Cost (MIP)
O(2n) worst case | Branch-and-bound — globally optimalFormulates the full mixed-integer program with integer vehicle counts, weekly utilisation variables, and spot-market slack. Solves via enumeration over integer combinations with LP relaxation bounds to prune the search tree. Guarantees the globally optimal fleet composition that minimizes total annualized cost (acquisition amortization + operating + spot penalties). For 3 vehicle types and reasonable fleet sizes, the search space is tractable.
Real-World Complexity
Why fleet sizing goes beyond the basic MIP
Beyond Basic Fleet Sizing
- Stochastic demand — Real demand is uncertain; robust or stochastic MIP formulations hedge against worst-case scenarios with chance constraints
- Vehicle age & replacement — Fleet renewal decisions must account for maintenance cost curves that increase with vehicle age and mileage
- Multi-depot operations — Vehicles are stationed at multiple depots; fleet sizing becomes coupled with depot location and assignment problems
- Emission regulations — Euro-7 standards and urban low-emission zones constrain which vehicle types can serve which routes
- Driver availability — Fleet size is bounded by licensed driver headcount and working-time regulations (EU Regulation 561/2006)
- Seasonal peaks — Holiday demand spikes (Black Friday, Christmas) create extreme peaks that may justify leasing vs ownership trade-offs
- Electric transition — BEV range limitations and charging infrastructure add range constraints and depot charging capacity constraints
Key References
Foundational works in fleet sizing and vehicle routing
- (2010). “Industrial aspects and literature survey: Fleet composition and routing.” Computers & Operations Research, 37(12), 2041–2061.
- (1993). “Incorporating vehicle routing into the vehicle fleet composition problem.” European Journal of Operational Research, 66(3), 313–330.
- (1977). “On a fleet sizing and allocation problem.” Management Science, 23(9), 972–977.
- (1984). “The fleet size and mix vehicle routing problem.” Computers & Operations Research, 11(1), 49–66.
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