Mining Economics

Linear Programming

Optimising a linear objective over a polygon of constraints — the optimum always sits at a corner of the feasible region.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Many mine-planning problems — blending, allocation, transport — maximise (or minimise) a linear objective subject to linear constraints. The feasible region is a convex polygon, and a fundamental result guarantees the optimum lies at one of its corner (vertex) points. The graphical method therefore reduces to finding the corners and evaluating the objective at each.

The Core Formula Matrix

Standard form: maximise

Z = c_{1} x + c_{2} y

subject to

a_{i} x + b_{i} y \leq d_{i}

x, y \geq 0

.

Corner-point theorem: the optimum of

Z

over a bounded feasible region occurs at a vertex.

Method: find all corner points (constraint intersections), evaluate

Z

at each, pick the best.

The ‘IIT Traps’

⚠
Optima are at vertices, not interior points. Don't test the centroid; test the corners.
⚠
Include the axes' intercepts and constraint intersections — and discard any vertex that violates another constraint.
⚠
Check feasibility of each candidate corner. A constraint intersection outside the region is not a valid vertex.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

For the objective

Z = 3 x + 2 y

, the value at the point

(2, 2)

is:

Q2MEDIUM2 Marks · NAT

Maximise

Z = 3 x + 2 y

subject to

x + y \leq 4

x \leq 3

x, y \geq 0

. The maximum value of

Z

is ______. (Round off to two decimal places.)

Q3HARD2 Marks · NAT

Maximise

Z = 5 x + 4 y

subject to

6 x + 4 y \leq 24

x + 2 y \leq 6

x, y \geq 0

. The maximum value of

Z

is ______. (Round off to two decimal places.)