🎓 Lesson 14
D5
Linear & Integer Programming for Logistics Resource Allocation
Linear and integer programming are math tools that help mining engineers find the best way to assign trucks, fuel, and crews across haul routes—while staying within budget, time, and equipment limits.
🎯 Learning Objectives
- ✓ Formulate a linear programming model for truck dispatch optimization given haul cycle times and production targets
- ✓ Solve a small-scale integer programming problem using branch-and-bound logic to assign shift-specific equipment allocations
- ✓ Analyze constraint sensitivity to identify bottlenecks in fuel supply or maintenance capacity
- ✓ Explain how integrality constraints affect solution feasibility and computational effort in real-time fleet scheduling
📖 Why This Matters
In open-pit mines, misallocating just 5% of haul trucks can cost $2–5M annually in diesel overconsumption and delayed ore delivery. Linear and integer programming transform intuitive 'gut-feel' dispatching into auditable, repeatable, and scalable decision systems—used daily by Rio Tinto, BHP, and Vale in integrated mine planning platforms like MineOptimize and Deswik.Scheduler.
📘 Core Principles
Optimization begins with defining decision variables (e.g., x_ij = number of trips by truck i to shovel j per shift), an objective (e.g., minimize total fuel consumption), and constraints (e.g., truck availability ≤ 12, shovel capacity ≥ 18,000 t/shift). LP assumes continuous variables and convex feasible regions; IP adds integrality requirements—making problems NP-hard but necessary for binary (yes/no) or count-based decisions. Duality theory reveals shadow prices: e.g., a $12/t shadow cost on crusher throughput signals where bottleneck investment yields highest ROI.
📐 Standard LP Formulation for Haul Fleet Allocation
This canonical form expresses resource allocation as a constrained minimization problem. It underpins commercial solvers (Gurobi, CPLEX) used in mine logistics software.
Canonical Linear Program
min cᵀx s.t. Ax ≤ b, x ≥ 0Standard form for minimizing linear cost subject to linear resource constraints.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| c | Cost coefficient vector | $/unit | Per-unit cost of each decision variable (e.g., $/trip) |
| x | Decision variable vector | units | Quantities to be optimized (e.g., number of trips, hours allocated) |
| A | Constraint matrix | dimensionless | Coefficients linking variables to resource limits (e.g., minutes/trip) |
| b | Resource vector | units of resource | Available capacity (e.g., total minutes, tons, kW-hr) |
Typical Ranges:
Truck dispatch LP: 10–500 variables, 20–200 constraints
Fleet sizing IP: 5–50 integer variables
💡 Worked Example
Problem: A copper mine operates 8 rigid-frame haul trucks (capacity = 172 t). Each truck serves either Shovel A (cycle time = 6.2 min) or Shovel B (cycle time = 7.8 min). Target production = 42,000 t/shift (8 hrs). Fuel cost = $0.18/min engine runtime. Minimize total fuel cost.
1.
Step 1: Define variables: x₁ = trips by trucks to Shovel A, x₂ = trips to Shovel B; each trip delivers 172 t → mass constraint: 172(x₁ + x₂) ≥ 42,000
2.
Step 2: Time constraint: 6.2x₁ + 7.8x₂ ≤ 8 × 60 × 8 = 3,840 min (8 trucks × 480 min/shift)
3.
Step 3: Objective: minimize Z = 0.18(6.2x₁ + 7.8x₂); solve via simplex → x₁ = 142, x₂ = 106 → Z = $449.50
4.
Step 4: Verify integrality: x₁, x₂ are integers → no rounding needed; if fractional, apply IP with branch-and-bound
Answer:
The optimal solution uses 142 trips to Shovel A and 106 to Shovel B, costing $449.50/shift — 11% less than equal-split dispatch.
🏗️ Real-World Application
At Newmont’s Boddington Mine (WA), LP/IP models reduced average truck wait time at shovels by 23% and increased payload utilization from 89% to 95.4% after integrating real-time GPS cycle data and dynamic maintenance windows into Deswik.Scheduler. Integer constraints enforced exclusive assignment of 4 ultra-class trucks to high-grade zones during priority shifts—ensuring grade blending targets were met without manual override.
✏️ Student Exercise
A gold operation has 6 CAT 797F trucks (190 t capacity) and two loading points: Pit 1 (cycle time = 5.4 min, max 120 loads/shift) and Pit 2 (cycle time = 8.1 min, max 90 loads/shift). Shift duration = 480 min. Production target = 38,000 t. Fuel cost = $0.21/min. Formulate the LP model. Then add integer constraints and explain why they matter for shift handover planning.
🔧 Interactive Calculator
🔧 Open Mine Logistics Chain Optimization Calculator📋 Case Connection
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📋 South African Platinum Group Metals Stockpile Optimization
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