Mineral Economics, Mine Planning and Systems Engineering

Critical path method network analysis

The critical path method (CPM) models a project as a network of activities linked by precedence relations, with each activity given one fixed duration. The forward pass gives the earliest start of an activity as the maximum of the earliest finish times of its immediate predecessors, so every merge node takes a maximum. The project duration is the length of the longest path, and the activities on that path have zero float. CPM assumes single-valued durations and is therefore deterministic. PERT, by contrast, uses optimistic (t_o), most likely (t_m) and pessimistic (t_p) estimates, so it is probabilistic. The expected duration is t_e = (t_o + 4t_m + t_p)/6 and the variance is \sigma^2 = ((t_p - t_o)/6)^2. The expected duration of the project is the length of the longest path when each activity uses its t_e. The variance of the project completion time equals the sum of the variances of the activities on the critical path, and the standard deviation is its square root. If the required project completion time is fixed, the unknown duration of one activity can be found by forcing the forward-pass result on the critical path to equal that target. Crashing shortens an activity at extra cost. Only activities on the critical path can reduce the project duration, and the cheapest critical activity should be crashed first. Crashing a non-critical activity does not shorten the project.

• Float (slack) — the delay an activity can absorb without delaying the project. Critical activities have zero float.
• Crash cost per week — the extra cost of reducing an activity by one time unit, which is used to choose the cheapest critical activity.

Depreciation and amortization allowances

Depreciation is the allowance for the wear and loss of value of tangible fixed assets such as machinery, and it is spread over the declared useful life of the asset. Amortization is the systematic write-off of intangible assets such as patents and goodwill, and it is spread over a specified period. Under the sum-of-the-years-digits method, the charge in year t is \frac{n-t+1}{S}(C-S_v), where S = \frac{n(n+1)}{2}, C is the cost and S_v is the salvage value. This relation can be inverted to find the salvage value when the charge in a given year is known. Under straight-line depreciation the annual charge is (C - S_v)/n. When the charge is deducted for tax, the after-tax cash flow is (\text{Revenue} - \text{Operating cost} - D)(1-T) + D, where D is the depreciation and T is the tax rate, so the depreciation acts as a tax shield. Under the declining balance method, the rate is d = 1 - (S_v/P)^{1/n}, and the book value after k years is P(1-d)^k.

• Declining balance — a method in which the charge is a fixed percentage of the book value at the start of each year, so the charge falls each year.
• Tax shield — the reduction in tax obtained from a depreciation charge that is deductible from taxable income.

Mine production scheduling and NPV

The net present value (NPV) of a mining project is the sum of the discounted annual net cash flows less the initial capital. Each year's net cash flow equals the revenue from ore sold minus the ore mining cost and the waste mining cost for that year. When there is a gestation period g before production starts, the cash flow of production year k is discounted by (1+i)^{g+k}, so a longer gestation reduces NPV. The NPV of a block model is found by discounting the economic value of each block by the year in which the block is extracted, so the extraction sequence determines the result. A mechanics item sometimes appears among mining questions. Newton's second law states that acceleration equals the net force divided by the mass, a = F_{net}/m, a basic relation used in estimating haulage acceleration.

• Gestation period — the time between the start of investment and the start of production.
• Net cash flow — revenue minus all operating costs in a given year, before tax and depreciation are considered.

Transportation problem least cost method

A transportation problem is balanced when total supply equals total demand, and a dummy source or destination is added when it is not. The least-cost (matrix minima) rule finds the cell with the lowest unit cost, allocates the largest possible quantity to it, and crosses out the exhausted row or column. The process repeats until all supplies and demands are met. Vogel's approximation method uses penalties, which are the differences between the two smallest unit costs in each row and column, and allocates to the cell in the row or column with the largest penalty. The north-west corner rule starts from the top-left cell and ignores transportation costs, so it usually gives a higher initial cost than the least-cost or Vogel methods.

• Basic feasible solution — an allocation with m+n-1 positive cells, where m is the number of sources and n is the number of destinations.
• Penalty — the cost difference that indicates the cost of missing the cheapest allocation in a row or column.

Linear programming optimization

A linear programming problem maximizes or minimizes a linear objective subject to linear constraints. Under the graphical method, each constraint is a half-plane, the feasible region is their intersection, and the optimum lies at a corner point of the region. A blending requirement such as a minimum average grade is written as a weighted-average inequality. Cross-multiplying by the total tonnage converts it into a linear constraint, and the optimal blend lies at the vertex where that constraint holds with equality. If the feasible region extends indefinitely in a direction in which the objective improves, the problem has an unbounded solution. An unbounded region does not by itself make the problem infeasible.

• Feasible region — the set of points that satisfy all constraints, including non-negativity.
• Corner point — a vertex of the feasible region, where two constraint boundaries intersect.

Machine owning and operating costs

The owning cost of a machine consists of fixed charges that accrue whether or not the machine is used, including depreciation, interest on the invested capital, insurance, taxes and storage. The operating cost depends on use and includes fuel, lubricants, tyres, repairs and maintenance, and operator labour. Operating cost is therefore not part of owning cost. The fuel consumption rate of a truck usually depends on speed. If the hourly cost is A + Bx^2 at speed x, the total cost of a trip of distance d is (A + Bx^2)\frac{d}{x}, so C(x) = \frac{Ad}{x} + Bdx. Setting dC/dx = 0 gives the speed that minimizes total cost, x^* = \sqrt{A/B}.

• Fixed (owning) cost — a charge per hour or per year that does not change with the number of hours worked.
• Variable (operating) cost — a charge that increases with the hours of use or the distance travelled.

Cut-off grade determination

The cut-off grade is the minimum grade at which material is classed as ore. Material at or above the cut-off is sent to the mill, and material below it is treated as waste. The tonnage of ore is the cumulative tonnage above the cut-off, and the mine life is set by whichever of the mine or the mill has the lower annual capacity, because the slower element limits the run. For a grade density function f(x) on [0, x_{max}], the proportion of ore above a cut-off x_c is \int_{x_c}^{x_{max}} f(x)\,dx. This equals the total proportion minus the proportion below the cut-off, \int_0^{x_{max}} f(x)\,dx - \int_0^{x_c} f(x)\,dx.

• Grade-tonnage curve — a plot of the tonnage of material above each grade, used to find the cut-off that gives the desired ore tonnage.
• Mill cut-off — the cut-off grade set by the processing cost and the recovery at the mill.

Break-even stripping ratio

The break-even stripping ratio is the maximum volume of waste per tonne of ore at which mining remains economical. It equals the net value of ore per tonne divided by the cost of removing one cubic metre of waste. Because it is a static relation between price, cost and stripping, it determines the economic pit boundary and the volume of overburden, subject to the safe pit slope. It does not depend on time, so it does not reflect the life of the mine. If the net value of ore is a linear function of grade, the break-even stripping ratio is also linear in grade. Dividing the net value line by the waste cost per cubic metre gives the stripping ratio as a function of grade, with the intercept and slope scaled by the waste cost.

• Net value — the selling value of ore per tonne minus the ore mining and processing costs.
• Economic pit limit — the boundary of the pit within which the stripping ratio is below the break-even value.

Ore sale value calculation

The sale value of ore is calculated from the contained metal, the recovery in processing and the selling price. The contained metal in one tonne equals the grade multiplied by the tonnage, and the recovered metal equals the contained metal multiplied by the recovery fraction. The sale value is the recovered metal multiplied by the price per unit. The break-even cut-off grade is the grade at which the revenue from one tonne of ore equals the cost of mining and processing that tonne. For a grade of g % (where 1 % of one tonne is 10 kg of metal), the revenue per tonne is 10g \times \text{recovery} \times \text{price}, so g = \text{cost per tonne}/(10 \times \text{recovery} \times \text{price}).

• Overall recovery — the fraction of the contained metal in the ore that is finally recovered as saleable product.
• Break-even cut-off — the grade at which revenue just covers the cost of mining and processing, with all other costs ignored.

Assignment problem Hungarian method

An assignment problem allocates one job to one machine or worker, with each job assigned to exactly one resource. The Hungarian method subtracts the row minima and then the column minima from the cost matrix. It then covers the zeros with the fewest lines. If the number of lines equals the matrix size, an optimal assignment exists among the zeros. Otherwise, the smallest uncovered entry is subtracted from all uncovered rows and added to all doubly covered columns, and the process is repeated. A forbidden assignment is treated by giving that cell a very large cost, so that it cannot appear in the optimal solution. Enumerating the feasible assignments then gives the minimum total cost.

• Dummy cell — a cell with a very large cost that represents an assignment that is not allowed.
• Optimal assignment — the one-to-one allocation of jobs to resources with the lowest total cost.

MMDR Act 1957 minor mineral leasing

Under the Mines and Minerals (Development and Regulation) Act, 1957, minor minerals are regulated by the State Government. The State Government is authorised to grant permits, prospecting licences and mining leases for minor minerals, and the Central Government does not give these permits. The Central Government's role in the Act covers major minerals and the framing of rules, and no central consent is needed for a State grant of a minor mineral lease.

• Minor mineral — a mineral that the Act classes as minor, such as building stone, sand and clay, and whose leasing is controlled by the State.
• Major mineral — a mineral that the Act classes as major, whose leasing is governed by the Central Government's rules.

Cost-revenue vs production curves

In a plot of cost or revenue against production level, revenue is proportional to output, so its line passes through the origin with a slope equal to the selling price per unit. Total cost is the sum of fixed and variable costs, so its line has a positive intercept and a slope equal to the variable cost per unit. Indirect operating cost does not vary with output and is therefore a horizontal line with a positive intercept.

• Break-even point — the production level at which the revenue line crosses the total cost line.
• Fixed cost — a cost that remains constant over the range of production, so it appears as a horizontal line.

System reliability of parallel components

In parallel airways or branches, the pressure drop across each branch is the same, and the air quantities add. For a branch with resistance R carrying quantity Q, the pressure drop is \Delta P = RQ^2, so the quantity in each branch is Q_i = \sqrt{\Delta P / R_i}. The equivalent resistance satisfies 1/\sqrt{R_{eq}} = \sum_i 1/\sqrt{R_i}. To supply a required quantity to one branch, a regulator is added to the other branch so that the pressure drops become equal.

• Regulator — an artificial resistance placed in a branch to control the share of air that flows through it.
• Minimum permissible quantity — the least air flow that a face or panel must receive under the applicable regulations.

Mineral royalty basis

Under the Indian mining legislation, royalty on a mineral is levied on the quantity removed, dispatched or sold, not on the quantity merely raised or held in stock. Ore that is produced but not sold or dispatched is not charged royalty, so the royalty base is the quantity sold.

• Royalty — a payment to the State that is calculated per unit of mineral removed, dispatched or sold.
• Dispatch — the movement of mineral from the mine for sale or use, which is the event on which royalty is assessed.

NPV crossover discount rate

The NPV profiles of two projects are the NPV of each project plotted against the discount rate. The crossover rate is the rate at which the two NPVs are equal, and it is found by equating the two NPV expressions and solving the resulting equation in the discount rate. The positive root is the valid crossover rate, because a negative discount rate has no economic meaning.