24-MMP-A5 Surface Mining Methods and Design · May 2016
Question 2 of 11: Special Record (IJK) Addressing of a 3-D Block Model
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, May 2016 — 09-MMP-A5, Surface Mining Methods and Design. Three hours, closed book; one hand-written, double-sided 8.5×11″ reference sheet and an approved Sharp or Casio calculator are permitted. Question 1 is compulsory (six parts, 40 marks); candidates then choose three of the five optional questions (2–6, 20 marks each) for a 100-mark paper — only the first three optional answers appearing in the answer book are graded. All six parts of Question 1 and all five optional questions are answered here, because this set is a study resource rather than an exam script.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
W. Hustrulid, M. Kuchta & R. Martin, Open Pit Mine Planning and Design, 3rd ed. — block modelling and record addressing (Ch. 4–5), pit-limit optimization: floating cone and Lerchs–Grossmann (Ch. 12–13), truck–shovel systems, match factor and dispatch (Ch. 15–16), slope design (Ch. 14).
SME Mining Engineering Handbook, 3rd ed. — open-pit design and materials-handling chapters; mine closure and reclamation planning.
H. Lerchs & I. F. Grossmann, “Optimum Design of Open-Pit Mines,” CIM Bulletin (1965).
Y. Lizotte, “The Economics of Computerized Open Pit Design,” International Journal of Surface Mining, Reclamation and Environment, 2:59–78 (1988) — the floating-cone procedure cited directly on this paper's Question 1.3.
BC Health, Safety and Reclamation Code for Mines — mine closure planning and reclamation security (Canadian regulatory context for Question 5).
Question 1.2: Special Record (IJK) Addressing of a 3-D Block Model (8 marks)
Given. The addressing scheme is explicit in the worked page-1 example: block indices $X_b, Y_b, Z_b$ (all starting at 0) locate a block along each axis, and the special record number is $IJK = X_b\, n_Y n_Z + Y_b\, n_Z + Z_b$ — i.e. a mixed-radix (base-$n_Y$/base-$n_Z$) integer, exactly analogous to how a day-of-year number encodes month and day. Storing records in ascending IJK order means the file is simultaneously sorted by X, then Y, then Z, so a direct-access read of record IJK retrieves the block at that (X,Y,Z) with no index lookup — the origin of the “minimize file size, maximize access speed” design goal stated in the question.
Quantity
Value
Model size $(n_X, n_Y, n_Z)$
46, 56, 51
Origin block centre floor $(x_0, y_0, z_0)$
1510, 1210, 1000 m
Block size $(\Delta x, \Delta y, \Delta z)$
20, 20, 8 m
Find. 1.2.1 — the IJK number for the block centred at $(1970, 1950, 1208)$. 1.2.2 — the block centre-floor coordinates for $IJK = 6028$.
Approach. Convert real-world coordinates to block indices via $X_b = \mathrm{round}[(x-x_0)/\Delta x]$ (and similarly for $Y_b, Z_b$), then apply the mixed-radix formula in each direction, inverting it with integer division/modulo for 1.2.2.
Assemble the special record number. With $n_Y n_Z = 56\times51 = 2856$ and $n_Z = 51$:
$$IJK = X_b\,n_Yn_Z + Y_b\,n_Z + Z_b = 23(2856) + 37(51) + 26 = 65688+1887+26 = \boxed{67601}$$
1.2.2 — invert the formula for $IJK=6028$. Integer-divide by $n_Yn_Z=2856$ to recover $X_b$, then repeat on the remainder for $Y_b, Z_b$:
$$X_b = \left\lfloor\frac{6028}{2856}\right\rfloor = 2, \qquad N = 6028 - 2(2856) = 316$$
$$Y_b = \left\lfloor\frac{316}{51}\right\rfloor = 6, \qquad Z_b = 316-6(51) = 316-306 = 10$$
Convert indices back to real-world centre-floor coordinates.
$$x = 1510+2(20)=1550, \qquad y = 1210+6(20)=1330, \qquad z = 1000+10(8)=1080$$
$$\boxed{(x,y,z) = (1550,\ 1330,\ 1080)\ \text{m}}$$
