🎓 Lesson 20
D5
Multi-Objective Optimization: Cost, Emissions & Throughput Trade-offs
Multi-objective optimization is finding the best balance between competing goals—like spending less money, producing fewer emissions, and moving more material—when designing a mining haulage system.
🎯 Learning Objectives
- ✓ Calculate Pareto-optimal haul truck fleet configurations using weighted sum and epsilon-constraint methods
- ✓ Analyze trade-off surfaces between diesel consumption (kg CO₂/t), unit transport cost ($/t), and annual throughput (Mt/yr) using real mine data
- ✓ Design a constrained optimization model in Python or Excel that incorporates payload, cycle time, fuel efficiency, and emission factors
- ✓ Explain how changes in road gradient, tire rolling resistance, and payload utilization shift the Pareto frontier
- ✓ Apply ISO 14064-1 and GHG Protocol Tier 2 emission factors to quantify scope 1 transport emissions per tonne-kilometer
📖 Why This Matters
In modern open-pit mines, haulage accounts for ~35–50% of total energy use and up to 70% of mobile equipment emissions. A fleet sized purely for lowest $/t may over-invest in large trucks—increasing idle time, maintenance costs, and NOₓ emissions—while a minimal fleet may bottleneck production and raise unit costs via overtime or underutilization. Engineers must navigate these tensions deliberately: regulators demand emissions reporting (e.g., Canada’s GHG Reporting Program), investors require ESG-aligned KPIs, and operations need predictable throughput. This lesson equips you to quantify and visualize those trade-offs—not choose one goal over another, but find the smartest compromise.
📘 Core Principles
Multi-objective optimization begins with defining three interdependent objectives: (1) minimize total cost (capital + fuel + maintenance + labor), (2) minimize emissions (CO₂, NOₓ, PM₁₀ per tonne-km), and (3) maximize throughput (tonnes moved per shift). These conflict because larger trucks reduce trips (lowering labor/fuel per tonne) but increase capital cost, tire wear, and idling emissions during low-demand periods. Constraints include road geometry (max grade, turning radius), pit layout (haul distance variability), equipment availability, and emissions caps (e.g., Chile’s SQM compliance thresholds). Solution methods include scalarization (weighted sum), evolutionary algorithms (NSGA-II), and constraint-based approaches (epsilon-constraint). The output is a Pareto frontier—a curve showing all non-dominated solutions where improving one objective worsens at least one other.
📐 Weighted Sum Objective Function
The weighted sum method converts multiple objectives into a single scalar function by assigning normalized weights reflecting strategic priorities (e.g., 0.4 cost, 0.3 emissions, 0.3 throughput). It’s intuitive, computationally efficient, and widely used for initial fleet sizing—but cannot capture concave regions of the true Pareto frontier.
Weighted Normalized Objective
Z = w₁·(Cᵢ − Cₘᵢₙ)/(Cₘₐₓ − Cₘᵢₙ) + w₂·(Eₘₐₓ − Eᵢ)/(Eₘₐₓ − Eₘᵢₙ) + w₃·(Tᵢ − Tₘᵢₙ)/(Tₘₐₓ − Tₘᵢₙ)Scalarized objective combining normalized cost (C), emissions (E), and throughput (T) using strategic weights (w₁+w₂+w₃=1). Lower Z indicates better overall performance.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Z | Composite objective score | dimensionless | Aggregated performance metric; minimized |
| w₁, w₂, w₃ | Strategic weights | dimensionless | Normalized importance assigned to cost, emissions, and throughput (sum = 1) |
| Cᵢ | Annual operating cost for configuration i | $ | Includes fuel, maintenance, tires, labor, and depreciation |
| Eᵢ | Annual CO₂-equivalent emissions for configuration i | kg CO₂e | Calculated using fuel type, volume, and IPCC emission factors |
| Tᵢ | Annual throughput for configuration i | tonnes | Total material moved, adjusted for availability and utilization |
Typical Ranges:
Open-pit copper mine: 0.15 – 0.35
Coal mine with rail-haul interface: 0.08 – 0.22
💡 Worked Example
Problem: A copper mine evaluates two haul truck options: (A) 190 t payload CAT 789D (fuel: 32 L/km loaded, 22 L/km empty); (B) 290 t payload CAT 797F (fuel: 48 L/km loaded, 34 L/km empty). Average haul distance = 3.2 km (loaded), 3.8 km (empty). Diesel density = 0.83 kg/L; CO₂ factor = 2.68 kg CO₂/kg diesel. Annual target throughput = 45 Mt. Weights: cost=0.5, emissions=0.3, throughput reliability=0.2. Normalize each metric on [0,1] scale where lower cost/emissions = better, higher throughput = better.
1.
Step 1: Compute annual fuel use per truck: (32×3.2 + 22×3.8) × (45,000,000 / payload) = For CAT 789D: (102.4 + 83.6) × (45e6 / 190,000) = 186 × 236.84 ≈ 44,052 L → CO₂ = 44,052 × 0.83 × 2.68 ≈ 98,500 kg CO₂. For CAT 797F: (153.6 + 129.2) × (45e6 / 290,000) = 282.8 × 155.17 ≈ 43,890 L → CO₂ ≈ 98,100 kg.
2.
Step 2: Compute annual cost: Assume $120k/truck/year OPEX (maintenance, tires, labor) + $0.85/L fuel → CAT 789D: 44,052×0.85 + 120,000 ≈ $157,400; CAT 797F: 43,890×0.85 + 120,000 ≈ $157,300.
3.
Step 3: Normalize: Min cost = $157,300 → w_cost = (157,400−157,300)/157,400 ≈ 0.0006; min CO₂ = 98,100 → w_emission = (98,500−98,100)/98,500 ≈ 0.0041; max throughput reliability = 1.0 for both (same target met). Weighted score = 0.5×0.0006 + 0.3×0.0041 + 0.2×1.0 = 0.0003 + 0.0012 + 0.2 = 0.2015 (797F) vs. 0.2018 (789D).
Answer:
CAT 797F scores slightly better (0.2015 vs. 0.2018), confirming its marginal advantage despite higher fuel use—due to superior payload utilization reducing total truck count and associated labor/maintenance overhead. Both fall within acceptable ranges, but only 797F lies on the Pareto frontier when considering full lifecycle cost.
🏗️ Real-World Application
At Newmont’s Boddington Mine (Western Australia), engineers applied multi-objective optimization to replace aging 170-t trucks with 290-t units. Using historical GPS cycle-time data, fuel telemetry, and IPCC Tier 2 emission factors, they modeled 12 fleet configurations across 3 payload classes. The Pareto frontier revealed that increasing average payload from 170 t to 240 t reduced CO₂/t-km by 18% and $/t by 12%, but beyond 260 t, diminishing returns emerged due to increased tyre wear and road degradation. The selected 290-t configuration (18 trucks) achieved 94% of max throughput while cutting scope 1 emissions by 22% versus baseline—validated against WA EPA reporting requirements and aligned with Newmont’s 2030 net-zero roadmap.
🔧 Interactive Calculator
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