General Linear Programming Problem (GLPP)

Maximize / Minimize Z = c₁x₁ + c₂x₂ + c₃x₃ +.................+ c_nx_n

Subject to constraints

a₁₁x₁ + a₁₂x₂ + .............+a_1nx_n (≤ or ≥) b₁

a₂₁x₁ + a₂₂x₂ + ...........+a_2nx_n (≤ or ≥) b₂

.

.

.

a_m1x₁ + a_m2x₂ + ..........+a_mnx_n (≤ or ≥) b_m

&

x₁ ≥ 0, x₂ ≥ 0,..., x_n ≥ 0

Where constraints may be in the terms of any inequality (≤ or ≥) or even in terms of an equation (=) and at the end fulfill the non-negativity restrictions.

Steps to change GLPP to SLPP (Standard LPP)

Step 1 - First of all, write the objective function in the maximization form. If the known objective function is in minimization form then multiply all through by -1 and write Max z^? = Min (-z)

Step 2 - Then, convert all the inequalities as equations.

o   If an equality of '≤' appears then add a variable called Slack variable. We can change it to an equation. For instance x₁ +2x₂ ≤ 12, we can write as

x₁ +2x₂ + s₁ = 12.

o   If the constraint is of '≥' type, we deduct a variable known as Surplus variable and change it to an equation. For instance

2x₁ +x₂ ≥ 15

2x₁ +x₂ - s₂ = 15

Step 3 - After that, the right side element of every constraint should be made non-negative

2x₁ +x₂ - s₂ = -15

-2x₁ - x₂ + s₂ = 15 (Multiplying throughout by -1)

Step 4 - All variables should have non-negative values.

  For instance: x₁ +x₂ ≤ 3

          x₁  > 0, x₂ is unrestricted in sign

  Then x₂ can be written as x₂ = x₂? - x₂?? where x₂?, x₂??   ≥ 0

  Thus the inequality takes the shape of equation as x₁ + (x₂? - x₂??) + s₁ = 3

Using all the above steps, we can change or write the GLPP in the form of SLPP.

Write the Standard LPP (SLPP) of the following 

Example 1

Maximize Z = 3x₁ + x₂

    Subject to

            2 x₁ + x₂ ≤ 2

            3 x₁ + 4 x₂ ≥ 12

    &     x₁ ≥ 0, x₂  ≥ 0

SLPP

    Maximize Z = 3x₁ + x₂

    Subject to

            2 x₁ + x₂ + s₁ = 2

            3 x₁ + 4 x₂ - s₂ = 12

            x₁ ≥ 0, x₂  ≥ 0, s₁ ≥ 0, s₂  ≥ 0

Example 2

Minimize Z = 4x₁ + 2 x₂

    Subject to

            3x₁ + x₂ ≥ 2

            x₁ + x₂ ≥ 21

x₁ + 2x₂ ≥ 30

    &     x₁ ≥ 0, x₂  ≥ 0

SLPP

Maximize Z^? = - 4x₁ - 2 x₂ 

Subject to

            3x₁ + x₂ - s₁ = 2

            x₁ + x₂ - s₂ = 21

x₁ + 2x₂ - s₃ = 30

            x₁ ≥ 0, x₂  ≥ 0, s₁ ≥ 0, s₂  ≥ 0, s₃  ≥ 0

Example 3

Minimize Z = x₁ + 2 x₂ + 3x₃

     Subject to

            2x₁ + 3x₂ + 3x₃ ≥ - 4 

            3x₁ + 5x₂ + 2x₃ ≤ 7

     &    x₁ ≥ 0, x₂  ≥ 0, x₃ is unrestricted in sign

SLPP

Maximize Z^? = - x₁ - 2 x₂ - 3(x₃^? - x₃^??)

Subject to

            -2x₁ - 3x₂ - 3(x₃^? - x₃^??) + s₁= 4 

            3x₁ + 5x₂ + 2(x₃^? - x₃^??) + s₂ = 7

            x₁ ≥ 0, x₂  ≥ 0, x₃^? ≥ 0,  x₃^?? ≥ 0, s₁ ≥ 0, s₂  ≥ 0

Some fundamental Definitions

Solution of LPP

Solution of LPP is any set of variable (x₁, x₂... x_n) which satisfies or fulfills the given constraint.

Basic solution

It is a solution achieved through setting any 'n' variable equivalent to zero and solving remaining 'm' variables. These 'm' variables are known as Basic variables and 'n' variables are known as Non-basic variables.

Basic feasible solution

A basic solution that is feasible or possible (all basic variables are non negative) is called basic feasible solution. There are two kinds of basic feasible solution.

1.      Degenerate basic feasible solution

If any of the basic variable of a basic feasible solution is zero (0) than it is referred to as degenerate basic feasible solution.

2.      Non-degenerate basic feasible solution

It is a basic feasible solution which has precisely 'm' positive x_i, where i=1, 2, ... m. In other terms all 'm' basic variables are positive and left over 'n' variables are zero.

Optimum basic feasible solution

A basic feasible solution is referred to be optimum when it optimizes (max / min) the objective function.

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