Linear Programming (LP) Model is a mathematical representation of a real-world problem in which the objective is to maximize or minimize a particular quantity (like profit or cost) subject to a set of linear constraints. The model provides a systematic and quantitative way to make optimal decisions.
Decision Variables:
These are the unknown quantities that represent choices to be made.
Example:
Let x1 = number of product A units produced,
and x2 = number of product B units produced.
These variables define the outcomes of the problem and form the basis of the LP model.
Objective Function:
The objective function expresses the goal of the problem — maximization (profit, production) or minimization (cost, time). It is a linear combination of decision variables.
Example:
Maximize Z = c1*x1 + c2*x2
Where c1 and c2 are coefficients representing contribution per unit of each product.
Constraints:
Constraints are the limitations or restrictions on resources such as labor, material, machine hours, or budget. They are represented as linear equations or inequalities.
Example:
a11*x1 + a12*x2 ≤ b1
a21*x1 + a22*x2 ≤ b2
Here, aij represents resource usage, and bi represents available resources.
Non-Negativity Constraints:
Decision variables cannot take negative values because negative production or resources are not possible.
x1 ≥ 0, x2 ≥ 0
5. Feasible Region
The feasible region represents all possible combinations of decision variables that satisfy all constraints simultaneously. The optimal solution of an LPP lies within this region usually at a corner point (vertex).
General Structure of an LP Model:
Maximize or Minimize Z = c1*x1 + c2*x2+…+cn*xn
Subject to:
a11*x1+a12*x2+…+a1n*xn (≤,=,≥)b1
a21*x1+a22*x2+…+a2n*xn (≤, = ,≥)b2
