 # Linear programming

### Basic Concepts and principles

A Linear programming problem in standard formulation is a problem in the form
Min Ctx
Subject to
Ax = b
x ≥ 0
Where A∈ Mmxn, and rank(A)=m, b∈Rm, C∈Rn
From now on we will call P to this problem.
C will be the costs vector, A the constraints matrix and b the constraints vector.
The feasible set
S = {x∈Rn : Ax = b, x≥ 0}
is a polyhedral set (semispaces intersection), in particular is a convex set. The convex theory join linear programming theorems says that if P has finite optimal solution it will be into the extreme points set of S.

### Extended Theory

The feasible Set
S = {x∈Rn : Ax = b, x≥ 0}
Can be characterized in terms of its extreme points and its extreme directions
Theorem 1 (extreme points existence)
The feasible set, se has at least one extreme point.
If B is submatrix of A with dimension mxm then we can write
A = [B, N] and the linear programming equations system can be write as follows
BXB + NXN = b
and we can write too
x = [XB, XN]
There exists a characterization of the extreme points of feasible Region S, There is a characterization of extreme points of feasible set S, the intuitive idea is that any extreme point is a solution of a linear system of equations constructed from a submatrix of A of range m.
Teorema 2 (Extreme points characterization)
Lets S = {x∈Rn : Ax = b, x≥ 0} ⊂ Rn.
Where
A∈ Mmxn, con range(A)=m, b∈Rm, C∈Rn
then     x is an extreme point if and only if ∃ B submatrix of A with r(B) = m such as
x = [B-1b, 0]t

The following corollary gives us an upper bound for the number of extreme points.
Corolary
The maximum number of extreme points of S is (n m) = n!/m!(n-m)!
(We apologize for the notation, we have no way to write combinatorial numbers)

As we shall see, not every linear programming problem has finite optimal solution, but if one then there is a characterization for it in the next theorema:

Theorem 3 (Finite optimal solutions characterization for problem P)
Lets
x1, x2, ... xk los Extreme points of S and lets
d1, d2, ... dr las Extreme directions of S.
P has finite optimal solution if and only if Ctdj ≥ 0, j =1,2, ..., r
And one of the extreme points is the solution for the problem P.

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### Mary Navarro:

2019-02-11 00:41:26
Hello, can anyone please help me to solve this problem using the linear programming method. I need the whole computation of the problem. Your response is highly appreciated, thank you. 1. ABC farms plans to introduce two (2) new gift packages of fruit for the holiday season. Box A will contain 20 oranges and 15 mangoes, Box B will contain 40 oranges and 20 mangoes. The farm has 18,000 oranges and 12,000 mangoes available for packaging. They believe that all fruit boxes can be sold. Profits are estimated at P56 for Box A and 70 for box B. Determine the number of boxes A and B that should be prepared to maximize the profit. 2. A manufacturer makes two (2) types of product: X and Y. Three (3) machines A, B, and C are required for the manufacture of each product. One unit of X requires 2 hours on A, 1 hour on B and 6 hours on C. While one unit of Y requires respectively 2 hours, 5 hours and 2 hours on A, B, and C. In a given period, there are 24 hours available on A. 44 hours on B and 60 hours on C. The profit per unit on X is Php60 and Php 90 on Y. Given that the machines are available when required, how many units of each product should be made in order to maximize the profit?

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