Question 4 of 7: Parametric Shaft and Block-Caving Cost Estimation with Escalation to 2012

Question 4: Parametric Shaft and Block-Caving Cost Estimation with Escalation to 2012 (15 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.

Given.

Find. The component and total 1989 shaft capital/operating costs (4.1), the component and total 1989 block-caving mining capital/operating costs excluding the shaft (4.2), and the total shaft-plus-mining capital and operating costs escalated to 2012 (4.3), with commentary on model adequacy throughout.

Approach. Substitute X = 20,000 and D = 2,000 into every component formula of Table 4.1 (shaft) and Table 4.2 (block-caving mine) to get the 1989 component and "Total" line costs; then multiply the 1989 shaft and mining totals by the ratio of the 2012 to 1989 capital and operating indices from Table 4.3.

1. Part 4.1 — shaft capital cost components, 1989. Each Table 4.1 capital term is evaluated at X=20,000, D=2,000 (Lumber, Fuel and Tires are marked "NAp" for a shaft and contribute nothing):

   Table 4.1 shaft capital cost, 1989 (USD)

   Category	Formula	Cost (USD)
   Labor	75D·X0.399	7,801,900
   Equipment	350X + 65D·X0.386	12,944,900
   Steel	25D·X0.373	2,010,300
   Lube	6D·X0.342	354,900
   Explosives	5D·X0.389	471,100
   Construction material	9D·X0.522	3,165,300
   Electricity	4D·X0.230	78,000
   Sum of components	—	26,826,400
   Total (Table 4.1 formula)	371X + 180D·X0.404	27,095,200

   Equipment (Equipment is the largest single item at roughly 48% of the sum of components) and Labor dominate, consistent with a mechanised shaft-sinking crew and permanent hoisting/guide/service installation; the printed "Total" formula (independently regressed against the historical cost database, not a re-sum of the component lines) comes out about 1% above the arithmetic sum of components — the two should always be checked against each other as a sanity cross-check, and here they agree closely.
2. Part 4.1 — shaft operating cost components, 1989.

   Table 4.1 shaft operating cost, 1989 (USD/short ton)

   Category	Formula	Cost (USD/st)
   Labor	2010/X	0.1005
   Equipment	0.325D/X	0.0325
   Steel	0.00014D	0.2800
   Lube	0.090D/X	0.0090
   Construction material	200/X	0.0100
   Electricity	0.0014D	2.8000
   Sum of components	—	3.2320
   Total (Table 4.1 formula)	2343/X + 0.44D/X + 0.00163D	3.4211

   At 20,000 st/d × 365 d/yr this is an annual shaft operating cost of about USD 24.97 million. The Electricity term (2.80/st, over 80% of the operating total) is disproportionately large compared with Labor and Equipment for what is normally a modest hoisting/pumping/ventilation power draw at a single shaft — a value this dominant is worth flagging as a possible scaling artefact in the underlying 1989 regression (an operating-cost model fitted mainly to shallower, lower-capacity shafts in the sample can extrapolate poorly to a 2,000 ft, 20,000 st/d case), and should be sanity-checked against an independent kWh-based estimate before being relied on for a real feasibility budget.
3. Part 4.2 — block-caving mining capital cost, 1989 (excluding shaft). All ten Table 4.2 categories are pure functions of X alone (no depth term):

   Table 4.2 block-caving mining capital cost, 1989 (USD)

   Category	Formula	Cost (USD)
   Labor	27,900X0.646	16,752,100
   Equipment	25,600X0.812	79,556,600
   Steel	4,410X0.685	3,896,200
   Lumber	149X0.902	1,129,100
   Fuel	10.6X0.897	76,400
   Lube	4.54X0.897	32,700
   Explosives	1,040X0.737	1,537,800
   Tires	1.87X0.946	21,900
   Construction material	31,100X0.591	10,831,000
   Electricity	50.4X0.748	83,100
   Sum of components	—	113,917,000
   Total (Table 4.2 formula)	64,800X0.759	119,139,800

   Equipment again dominates (about 70% of the component sum), consistent with the mechanised LHD/crusher/conveying fleet a block-caving operation of this scale requires; the Total-formula result runs about 4.6% above the component sum, a wider gap than the shaft model's, which is expected since the mining Total was regressed across a capacity range (4,000–40,000 st/d) rather than summed directly — both figures should be carried forward and the difference treated as part of the estimate's inherent uncertainty band, not resolved arbitrarily in favour of one.
4. Part 4.2 — block-caving mining operating cost, 1989 (excluding shaft).

   Table 4.2 block-caving mining operating cost, 1989 (USD/short ton)

   Category	Formula	Cost (USD/st)
   Labor	60.0X-0.305	2.9264
   Equipment	4.40X-0.230	0.4510
   Steel	0.217X0.0	0.2170
   Lumber	0.310X0.0	0.3100
   Fuel	0.894X-0.239	0.0838
   Lube	0.545X-0.253	0.0445
   Explosives	0.183X-0.0	0.1830
   Tires	0.412X-0.151	0.0924
   Construction material	2.83X-0.182	0.4667
   Electricity	1.36X-0.060	0.7507
   Sum of components	—	5.5255
   Total (Table 4.2 formula)	48.4X-0.217	5.6432

   Labor is the single largest term (about 53% of the component sum), the pattern expected of an underground block-caving labour force, and the negative exponents on every category reflect the intended economy of scale — unit cost falling as capacity X rises. That every exponent is negative (or zero) is itself a useful adequacy check: a positive exponent here would produce a rising unit cost with capacity, which is not physically sensible for this class of model, and none of the ten terms shows that failure.
5. Part 4.3 — escalation factors, 1989→2012 (Table 4.3). $$\text{capital factor} = \dfrac{256.8}{95.5} = 2.689, \qquad \text{operating factor} = \dfrac{210.8}{91.1} = 2.314$$
6. Part 4.3 — total shaft and mining costs escalated to 2012. Applying the 2012 factors from the previous step to the Table-formula 1989 totals of Steps 1–4:

   2012 escalated totals (USD, using the Table 4.1/4.2 "Total" formulas)

   Item	1989	× factor	2012
   Shaft capital	27,095,200	2.689	72,859,100
   Shaft operating (/st)	3.4211	2.314	7.9163
   Mining capital	119,139,800	2.689	320,367,500
   Mining operating (/st)	5.6432	2.314	13.0579

   $$\boxed{C_{cap,2012} = 72.86\text{M} + 320.37\text{M} = 393.23\text{M USD}}$$ $$\boxed{c_{op,2012} = 7.9163 + 13.0579 = 20.97\ \text{USD/st}\ \ (\times 20{,}000 \times 365 \approx 153.1\text{M USD/yr})}$$
7. Part 4.3 — adequacy of the 2008, 2009 and 2012 estimates. Table 4.3's capital index climbs from 95.5 (1989) to a local peak of 245.9 in 2008, falls back to 228.0 in 2009, then recovers to 256.8 by 2012 — a swing of roughly 8% down and 13% back up in just four years, driven by the 2008 construction/commodity cost boom and the subsequent 2009 financial-crisis pullback. A single scalar escalation factor, by construction, cannot represent that a Labor-heavy line item and an Equipment- or Steel-heavy line item did not move together through that cycle (structural steel and commodity-linked inputs spiked and crashed harder than site labour rates), so the escalated 2008/2009 figures understate the true cost dispersion across categories even though the blended total tracks roughly right. The estimate is further weakened by extrapolating a 1989-fitted regression 19–23 years forward and, for the shaft model, outside any depth range the source explicitly bounds — both are order-of-magnitude/pre-feasibility tools (per the question's own framing) and are adequate for screening-level comparison between design options, but should never substitute for vendor quotes once a project reaches a bankable feasibility study.