# HSC Higher Mathematics 2nd Paper Note 2nd Chapter Linear Programming

HSC Higher Mathematics 2nd Paper Note 2nd Chapter Linear Programming. Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.

## HSC Higher Mathematics 2nd Paper Note 2nd Chapter Linear Programming

Linear programming is the process of taking various linear inequalities relating to some situation and finding the “best” value obtainable under those conditions. A typical example would be taking the limitations of materials and labor and then determining the “best” production levels for maximal profits under those conditions. In “real life”, linear programming is part of a very important area of mathematics called “optimization techniques”. This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These “real life” systems can have dozens or hundreds of variables or more. In algebra, though, you’ll only work with the simple (and graphable) two-variable linear case.

The general process for solving linear-programming exercises is to graph the inequalities (called the “constraints”) to form a walled-off area on the x,y-plane (called the “feasibility region”). Then you figure out the coordinates of the corners of this feasibility region (that is, you find the intersection points of the various pairs of lines), and test these corner points in the formula (called the “optimization equation”) for which you’re trying to find the highest or lowest value. Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. So, to find the solution to this exercise, I only need to plug these three points into “z = 3x + 4y”.

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