Related terminology

Objective Function

The function which is to be optimized (maximized/minimized) is called an objective function.

Constraints

The system of linear inequations (or equations) under which the objective function is to be optimized is called constraints.

Non-negative Restrictions / Decision variables

The variables involved in the objective function are called decision variables.

All the variables considered for making decisions assume non-negative values.

Mathematical Description of a General Linear Programming Problem

A general LPP can be stated as (Max/Min) z = clxl + c2x2 + … + cnxn (Objective function)

Subject to constraints

and the non-negative restrictions

xl, x2,….., xn ≥ 0 where all al1, al2,…., amn; bl, b2,…., bm; cl, c2,…., cn are constants and xl, x2,…., xn are variables.

  1. Solution of a LPP : A set of values of the variables xl, x2,…., xn satisfying the constraints of a LPP is called a solution of the LPP.
  2. Feasible Solution of a LPP: A set of values of the variables xl, x2,…., xn satisfying the constraints and non-negative restrictions of a LPP is called a feasible solution of the LPP.
  3. Optimal Solution of a LPP: A feasible solution of a LPP is said to, be optimal (or optimum), if it also optimizes the objective function of the problem.
  4. Graphical Solution of a LPP: The solution of a LPP obtained by graphical method i.e., by drawing the graphs corresponding to the constraints and the non-negative restrictions is called the graphical solution of a LPP.
  5. Unbounded Solution: If the value of the objective function can be increased or decreased indefinitely, such solutions are called unbounded solutions.
  6. Fundamental Extreme Point Theorem: An optimum solution of a LPP, if it exists, occurs at one of the extreme points (i.e., corner points) of the convex
    Polygon of the set of all feasible solutions

Solution of Simultaneous Linear Inequations:

The graph or the solution set of a system of simultaneous linear inequations is the region containing the points (x, y) which satisfy all the inequations of the given system simultaneously.

To draw the graph of the simultaneous linear inequations, we find the region of the xy-plane, common to all the portions comparing the solution sets of the given inequations. If there is no region common to all the solutions of the given inequations, we say that the solution set of the system of inequations is empty.

Note The solution set of simultaneous linear inequations may be an empty set or it may be the region bounded by the straight lines corresponding to given linear inequations or it may be an unbounded region with straight line boundaries.