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How is linear algebra used in linear programming for resource allocation?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

Linear algebra provides the essential framework for solving resource allocation problems through linear programming. In this context, linear algebra allows complex, multi-variable problems to be expressed in a compact, algebraic format that computers can efficiently process.12

Mathematical Formulation

To allocate resources optimally, the problem is translated into matrix notation. The objective—such as maximizing profit or minimizing costs—is defined by a vector $x$ of variables and a coefficient vector $p$ , represented as the product $p x$ .1

Constraints, such as limited labor, raw materials, or machine time, are expressed as a system of linear inequalities:
$A x \leq b$
Here, $A$ is a matrix representing the constraints, and $b$ is a vector of the available capacities or bounds.1

Solving the System

Linear algebra techniques, such as the simplex method, are used to navigate the "feasible region" defined by these linear constraints. By treating the problem as a geometric space, the algorithm systematically moves between extreme points (vertices) of the solution set to find the optimal allocation that maximizes or minimizes the objective function.34