Question 8 of 11: Pareto cost analysis and capital-cost power laws (20 marks)

Question 3: Pareto cost analysis and capital-cost power laws (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

3.1 Pareto's Law applied to mine operating costs

Pareto's Law (the 80/20 rule) holds that roughly 80% of a cost centre's total spend is concentrated in about 20% of its individual line items; applying it means pulling one year's full accounting detail for drilling, trucking and shovel loading, ranking EACH cost centre's line items by dollar value (not by count), and attacking the top few items first, since they dominate the achievable savings and any effort spent chasing the long tail of small line items yields little.

3.1.1 Drilling (blast-holes): typically dominated by (1) drill-bit/rod consumables (~30–35%, directly tied to rock abrasiveness), (2) fuel/power (~20–25%), and (3) labour/maintenance labour (~20%) — consistent with the exam's own illustrative split.

3.1.2 Trucking (haulage): (1) fuel (~30–35%, the single largest and most volatile line), (2) tires (~15–20%, second only to fuel on large rigid-frame trucks), and (3) maintenance labour and parts (~20–25%).

3.1.3 Shovel (loading): (1) wages and benefits for the operating crew (~25–30%, shovels run continuously across shifts), (2) scheduled maintenance/wear parts (dipper teeth, cables, ~20–25%), and (3) power/fuel (~15–20%).

3.1.4 Role of inflation indexes: once the dominant 2–3 items per cost centre are identified, a SEPARATE index tracking each item's own market (a fuel-price index for haulage diesel, a labour-wage index for crew costs, a steel/consumables index for bits and tires) gives a far more accurate forward cost estimate than one blended mining-cost index, precisely because Pareto's ranking has already shown that these specific items — not the average of everything — drive the total.

3.2 1997 capital costs from power-law relations

Given. $P=aX^b$ for each machine class, with $a,b$ as tabulated and $X$ in the STATED (non-SI) unit for each formula.

Find. 1997 capital cost of each machine. Approach. Convert each given size to the formula's OWN unit system first (this is a deliberate mixed-unit trap in the source data), then evaluate the power law.

1. Drill (3.2.1). $X = 55{,}000\ \text{kg} \times 2.20462\ \text{lb/kg} = 121{,}254\ \text{lb}$. $$P_{drill} = 200\,(121{,}254)^{0.67} = \boxed{\$509{,}500}$$
2. Shovel (3.2.2). $X = 53\ \text{m}^3 \times 1.30795\ \text{yd}^3/\text{m}^3 = 69.32\ \text{yd}^3$. $$P_{shovel} = 520{,}000\,(69.32)^{0.75} = \boxed{\$12{,}492{,}600}$$
3. Truck (3.2.3). $X = 300\ \text{t} \times 1.10231\ \text{st/t} = 330.7\ \text{short tons}$. $$P_{truck} = 20{,}000\,(330.7)^{0.90} = \boxed{\$3{,}702{,}700}$$

3.3 Today's cost and realism

Given. Cost index: a 1997 dollar is worth 1.33 today's dollars. Find. Today's cost of each machine.

1. Escalate. $$P_{today}=P_{1997}\times1.33$$ giving drill $=\boxed{\$677{,}600}$, shovel $=\boxed{\$16{,}615{,}200}$, truck $=\boxed{\$4{,}924{,}500}$.

Are the values realistic? Only partly. The escalated drill cost ($0.68M) and truck cost ($4.9M) sit in a believable range for a large blast-hole rig and a 300 t-class rigid-frame haul truck respectively, but the shovel figure ($16.6M for a 53 m³/≈69 yd³ bucket) is on the high side for a machine of that bucket class even allowing for two decades of escalation — a strong hint that the 1997 power-law CONSTANTS given for the shovel (a=520,000, b=0.75) describe an unusually large-frame class of shovel, or that the "cost index of 1.33" understates true equipment-cost inflation over the intervening period (per the discussion in 1.3.c, a single blended index cannot be trusted to scale a capital-heavy item like a mining shovel).