Standard Form of LPP in Matrix Form

Standard Form of LPP:

The Simplex Method, which is a process to solving linear programming models (LPP), is convenient to explain linear programming problems that are in a fixed format which is called the standard form of linear programming problem or LPP. In other words, the standard form of a linear programming problem is to develop the procedure for solving the general linear programming problems. The optimal solution of the ‘standard form of an LP problem’ is always equal to the optimal solution of the ‘original LP problem’.

The following steps outline the characteristics of a standard form of LPP:

Step 1. All constraints should be changed to equations, except non-negativity restrictions that remain as inequalities.

Step 2. The right side element of each constraint should be made positive. If the right side element is negative then multiply both sides by −1.

Step 3. All variables must have non-negative values. If a variable is unrestricted in sign then replace it with the difference of two positive variables. For example, if x is unrestricted, then it can be replaced by x’− x”, where x’, x” ≥ 0.

Step 4. The objective function should be taken in the form of a maximization. The minimization of a function z is equivalent to the maximization of the negative expression of this function, z, that is,

Min z = − Max (−z).

For example, the objective function

Min z = c₁x₁ + c₂x₂ + … + c_nx_n

is equivalent to

Max (−z) = Max z’ = − c₁x₁ − c₂x₂ − … − c_nx_n.

Hence, a linear programming problem or LPP in standard form can be expressed in mathematical form as follows:

Standard Form of LPP in Matrix Form:

The linear programming problem in standard form can be expressed in matrix form as follows:

Objective function:

Maximize z = cX^T

Constraint Equations:

Subject to   AX = b,      b ≥ 0

Non-negative Restriction:

X ≥ 0

where,

X = [x₁, x₂, … , x_n, s₁, s₂, … , s_m]

c = [c₁, c₂, … , c_n, 0, 0, … , 0],          

b = [b₁, b₂, … , b_m]

and

(Source – Various books from the college library)

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Tags: standard form of linear programming, linear programming in matrix form, linear programming problem in matrix form, matrix form of lpp, matrix form of linear programming problem

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