What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

What is a linear programming problem?

A problem of finding the optimal (maximum or minimum) value of a linear objective function subject to linear constraints with non-negative variables.

What is the feasible region?

The set of all points satisfying every constraint, including the non-negativity restrictions \(x\geq0,\,y\geq0\).

Where does the optimal value of \(Z\) occur?

At a corner point (vertex) of the feasible region, when such an optimal value exists.

Describe the Corner Point Method.

Find all vertices of the feasible region, evaluate \(Z\) at each, and pick the vertex giving the largest (or smallest) value of \(Z\).

What is the objective function?

The linear function \(Z=ax+by\) (\(a,b\) constants) that is to be maximised or minimised.

What are decision variables?

The variables (such as \(x\) and \(y\)) whose values are chosen to optimise the objective function, subject to constraints.

Does a bounded feasible region always have an optimum?

Yes; if the feasible region is bounded, \(Z\) attains both a maximum and a minimum, each at a corner point.

Can an unbounded feasible region lack an optimum?

Yes; for an unbounded region a maximum or minimum may not exist and must be verified using the open half-plane test.

What does it mean if two corner points give the same optimal value?

Then every point on the segment joining them is also optimal, giving infinitely many optimal solutions.

What shape is a feasible region?

It is always a convex region (a convex polygon when bounded).

Linear Programming

Class 12 Mathematics · Mathematics Part II · Chapter 6

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CBSE Class 12 · Mathematics

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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:

Flashcards & interactive practice