Question 8 of 11: Pareto's Law, Cost Indexing, and Equipment Capital-Cost Estimation

Question 3: Pareto's Law, Cost Indexing, and Equipment Capital-Cost Estimation (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. Pareto's Law (the “80/20 rule”) observes that in most cost or activity distributions, a small fraction of the categories (typically around 20 %) accounts for the large majority (typically around 80 %) of the total. Applied to mine operating costs, it means the engineer does not need to analyse every line item in the accounting system with equal effort: sorting each cost centre's line items by dollar value, largest first, and computing the cumulative percentage of total cost quickly shows that the top handful of items — consumables, labour, and major replacement parts, typically — already account for 70–85 % of the centre's cost. Management attention and cost-reduction effort should concentrate on that short list first, since chasing the long tail of minor items yields little for the analysis time spent.

Approximate major cost items by cost centre (illustrative splits, consistent with the pattern given for blasting in the question):

3.2 — Inflation indexes and future cost estimates. A single-year cost figure decays in usefulness the moment it is used to plan a future project, because labour, steel, fuel and equipment prices do not move at the same rate as general inflation. Applying a cost-centre-specific inflation index (rather than a blanket CPI figure) to each of the Pareto-identified major items lets an estimator escalate a known historical actual to a defensible future estimate item by item — the same logic already used for the capital-cost indexing in 3.3–3.4 — and, because 3.1 has already identified which few items dominate the total, only those need index-quality tracking to keep the whole estimate current, rather than indexing every minor line item.

3.3 — capital cost estimation.

Given. Power-law cost curves $P=aX^b$ for trucks, shovels and drills (constants above). Each machine's capacity is stated in the question in a different unit system than the formula requires: truck capacity in metric tonnes vs. the formula's short tons; shovel bucket in cubic metres vs. cubic yards; drill pull-down in kilograms-force vs. pounds-force. Check: this three-for-three unit mismatch reads as deliberate — each sub-part is converted to the formula's own native unit below before substitution, using 1 short ton = 0.90718 t, 1 yd³ = 0.76455 m³, 1 kgf = 2.20462 lbf.

Find. 1997 capital cost of each machine (3.3.1–3.3.3), and today's cost after a 2.50× escalation factor (3.4).

Approach. Convert each given capacity to the formula's stated unit, then evaluate $P=aX^b$ directly.

1. 3.3.1 — 300-tonne truck. Convert to short tons: $X = 300/0.90718 = 330.69$ st. $$P = 20{,}000\,(330.69)^{0.90} = \boxed{\text{CAD }3{,}702{,}653}$$
2. 3.3.2 — 53 m³ shovel. Convert to cubic yards: $X = 53/0.76455 = 69.32$ yd³. $$P = 540{,}000\,(69.32)^{0.75} = \boxed{\text{CAD }12{,}973{,}112}$$
3. 3.3.3 — 55,000 kg pull-down drill. Convert to pounds-force: $X = 55{,}000\times2.20462 = 121{,}254$ lbf. $$P = 400\,(121{,}254)^{0.67} = \boxed{\text{CAD }1{,}018{,}913}$$
4. 3.4 — escalate to today's dollars (1997 dollar × 2.50): $$P_{\text{truck}}=3{,}702{,}653\times2.50=\boxed{\text{CAD }9{,}256{,}631}, \quad P_{\text{shovel}}=12{,}973{,}112\times2.50=\boxed{\text{CAD }32{,}432{,}780}, \quad P_{\text{drill}}=1{,}018{,}913\times2.50=\boxed{\text{CAD }2{,}547{,}282}$$

Are the values realistic? As order-of-magnitude figures for large mining-class equipment they are broadly plausible — ultra-class haul trucks and large electric mining shovels in this size class do run into the multi-million and low-tens-of-millions range respectively in current dollars — but a single blanket escalation factor applied uniformly to all three machine types is a genuine oversimplification: it assumes truck, shovel and drill prices have all inflated at exactly the same rate since 1997, ignores technology-driven cost changes (AC drive systems, autonomous-haulage retrofits, tier-4 emissions equipment) that have added real cost independent of general inflation, and ignores that a single power-law curve fit to 1997-era machines of this size may not extrapolate reliably to today's largest-class equipment, which sits well outside the range the original regression was fitted to. The figures are therefore a reasonable planning-level starting point, not a substitute for a current vendor quotation.