Linear Programming
Class 12 Mathematics · Mathematics Part II · Chapter 6
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History
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Geography
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Summary
Linear programming finds the optimal value of a linear objective function subject to linear constraints. A typical problem maximises or minimises \(Z=ax+by\) subject to inequalities such as \(c_1x+c_2y\leq k\) together with the non-negativity constraints \(x\geq0,\,y\geq0\). The set of points satisfying all constraints is the feasible region, a convex polygon (possibly unbounded). The key result is that an optimal value of \(Z\), when it exists, occurs at a corner (vertex) of the feasible region. The Corner Point Method therefore evaluates \(Z\) at each vertex and selects the largest or smallest value. For a bounded feasible region both a maximum and a minimum exist; for an unbounded region an optimum may fail to exist, which must be checked. The chapter solves such problems graphically in two variables.
Formulation of LPPConstraints and feasible regionCorner Point MethodBounded and unbounded regionsGraphical solution
Key terms
Objective function
The linear function \(Z=ax+by\) to be optimised.
Constraints
The linear inequalities restricting the variables.
Feasible region
The common region satisfying all constraints, including \(x,y\geq0\).
Corner point
A vertex of the feasible region, the intersection of two boundary lines.
Optimal solution
A feasible point giving the maximum or minimum of \(Z\).
Bounded region
A feasible region that can be enclosed within a circle.
Extra questions & answers
Try one from this chapter
A linear programming problem optimises a: