Unit Summary

This section gathers together everything the chapter has covered on the linear programming problem (LPP) — one of the core topics of the CBSE Class 12 Applied Mathematics syllabus.

1. A linear programming problem deals with optimising (minimising or maximising) a linear function of several variables, subject to a number of conditions on those variables expressed as linear inequalities or equations.
2. An LPP has three essential components: (i) decision variables, (ii) the objective function, and (iii) the linear constraints.
3. The decision variables represent the competing activities (or limited resources) that share the available resources.
4. The objective function is a real-valued linear function written as $Z = ax + by$, where $a$ and $b$ are constants and $Z$ is to be maximised or minimised.
5. The conditions $x \ge 0,\ y \ge 0$ are the non-negativity restrictions on the decision variables.
6. The important types of LP problems studied are the Manufacturing problem, the Diet problem, the Transportation problem, and the Assignment problem.
7. A solution is any set of values of the decision variables that satisfies the constraints of the LPP.
8. A feasible solution satisfies all the constraints and the non-negativity conditions; a set of values that does not is an infeasible solution.
9. The feasible region is the common region determined by all the constraints (including non-negativity); every point in it is a feasible solution.
10. An optimal solution is a feasible solution that optimises (maximises or minimises) the objective function.
11. An LPP can have no solution, a unique optimal solution, or more than one optimal solution.
12. Theorem 1 (Corner-Point Theorem). Let $R$ be the feasible region for an LPP and $Z = ax + by$ the objective function. When $Z$ has an optimal value (maximum or minimum), that value must occur at a corner point (a vertex where two boundary lines meet) of $R$. …