Linear programming

Basic Concepts and principles


A Linear programming problem in standard formulation is a problem in the form

Minimize \( C^t x \)

Subject to

\( \begin{matrix} Ax=b \\ x\geqslant 0 \end{matrix} \)

Donde

\( A \in \mathbb M _{mxn}, \, rnk(A)=m ,\,\, b \in R^{m}, \, C \in R^{n}, \,\, n \geqslant m \)
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
$$S = \{ x\in\mathbb{R}^n : Ax=b, x\geq 0 \} $$
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
$$S = \{ x\in\mathbb{R}^n : Ax=b, x\geq 0 \} $$
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

\( A=\begin{bmatrix} B \\ N\end{bmatrix} \)

and we can write too

\( Bw_{B} + Nw_{N} = b \)

And solutiion can be written as

\( x=\begin{bmatrix} X_{B} \\ X_{N}\end{bmatrix} \)

Theorem 2 (Extreme points characterization)

Lets the set $$ S = \{ x\in\mathbb{R}^n : Ax=b, x\geq 0 \} \subset \mathbb{R}^n $$ Where

\( A\in\mathbb{M}_{mxn} : rnk(A)=m, b \in \mathbb{R}^m , C \in \mathbb{R}^n \)

then     x is an extreme point if and only if \( \exists B \) submatrix of A with rank(B) = m such as

$$ x=\begin{bmatrix} B^{-1}b \\ 0 \end{bmatrix} $$

The following corollary gives us an upper bound for the number of extreme points.
Corolary

The maximum number of extreme points of S is

$$ \binom{n}{m} = \frac{n!}{(n-m)!} $$

As we will 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

\( x_{1}, x_{2}, ..., x_{k} \) Extreme points of S and lets

\( d_{1}, d_{2}, ..., d_{r} \) Extreme directions of S.

\( C^t = d_{j} \geq 0 ,\:\:\: \forall j =1,2, ...,r\)

In adittion one of 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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