Posted: November 19th, 2022

Linear programming

A linear programme is a problem consisting in maximizing or minimizing a linear function while
satisfying a finite set of linear constraints. When you are trying to maximize a function such as revenue
,you give priority to the variable with the largest coefficient .conversely when you have a minimization
function such as cost ,you then give a priority to the variable smallest co efficient . A linear program is a problem that involves maximizing or minimizing a linear function while adhering to a set of linear constraints. When attempting to maximize a function such as revenue, you prioritize the variable with the highest coefficient. When you have a minimization function, such as cost, you prioritize the variable with the lowest coefficient.An example of
maximized function may be shown as below

Maximize Σ n j
=1 c j x j
Subject to: Σ n j=1 a I j x j ≤ bi for all 1 ≤ i ≤ m
x j ≥ 0 for all 1 ≤ j ≤ n.

From the above equation , all constraints are inequalities (and not equations).
and all variables are non- negative .The variables x j are referred to as decision
variables . The function that has to be maximized is called the problem objective function.
To solve a standard maximization problem using the simple method, we take the following
Step 1 . Convert to a system of equations by introducing slack variables .
Step 2. Write down the initial tableau.
Step 3. Select the pivot column: Choose the negative number with the largest magnitude in the
bottom row excluding the rightmost entry.
Step 4. Select the pivot in the pivot column.
Step 5. Use the pivot to clear the column in the normal manner.
On the other hand a mi

However in minimization problem ,We solve it using simple method, convert it

into a maximization problem. If you need to minimize c, instead maximize p = -c. A

minimization problem is in standard form if objective function
w = c1x1 + c2x2..+ . . . + cnxn

Example
The minimization LP problem:
Minimize C = 3x + 4y – 8z subject to the constraints
3x – 4y ≤ 12,
x + 2y + z ≥ 4
4x – 2y + 5z ≤ 20
x ≥ 0, y ≥ 0, z ≥ 0
can be replaced by the following maximization problem:

Maximize P = -3x – 4y + 8z subject to the constraints
3x – 4y ≤ 12,
x + 2y + z ≥ 4
4x – 2 y + 5z ≤ 20
x ≥ 0, y ≥ 0, z ≥ 0.

References

W.J. Cook, W.R. Pulley blank ,and A . Schrijve(1998) . combinatorial optimization.Wiley-Interscience.

M. sakarovitch(1984). optimization combinatoire : Graphes et Programmation lineaire. H