Operations Research

Que. 2. Express the following L.P. model in standard form with slack and surplus variables. 

Maximize :- Z= 2x1 + 3x2 + 5x3 
S.t. x1 + x2 - x3 = -5
-6x1 + 7x2 - 9x3 ³ 4
x1 + x2 + 4x3 = 10
x1, x2 ³ 0
x3 Unrestricted

Ans. 

Now, by taking the brief knowledge of slack and surplus variables: 

· Slack Variables : It is used to (convert) correct less than or equal to type (£) constraints in to an equation.

OR

constraint of the form g(x) <= b, the slack is b-g(x), which is designated by the slack variable, s. Then, the original constraint is equivalent to the defining equation, g(x) + s b, plus s >= 0.

· Surplus Variables : It is used to (convert) correct greater than or equal to type (³) constraints in to an equation. 

In our equation the slack variable s1 is introduced to convert (£) type constraints in to an equation (2).

Now our L.P. Model can be expressed as : 

Maximize Z = 2x1 + 3x2 + 5x3 
s.t. x1 + x2 - x3 = -5
-6x1 + 7x2 -9x3 + s1 = 4
x1 + x2 + 4x3 = 10
x1, x2, s1 ³ 0

x3 Unrestricted

As per the standard form of L.P. Model all variables should be ³ (greater than or equal to) 0. In above L. P. model x3 is unrestricted, therefore it is necessary to convert it in positive, for that assume

X3 = x4 - x5 where x4, x5 ³ 0.

By placing all assumption value of x3 in above L. P. model we get : - 

Maximize Z = 2x1 + 3x2 +5 (x4 - x5) 
= 2x1 + 3x2 + 5x4 - 5x5
s.t. x1 + x2 - (x4 - x5) = -5
\ = x1 + x2 - x4 + x5 = -5
\ = -6x1 + 7x2 -9 (x4 - x5) + s1 = 4
\ = -6x1 + 7x2 - 9x4 + 9x5 + s1 =4
\ = x1 + x2 +4 (x4 - x5) =10
\ = x1 + x2 + 4x4 -4x5 = 10
x1, x2, x4, x5, s1 ³ 0

\ standard form of above L. P. model is as follows:-

Maximize Z = 2x1 + 3x2 + 5x4 - 5x5 
s.t. x1 + x2 - x4 + x5 = -5
-6x1 + 7x2 - 9x4 + 9x5 + s1 = 4
x1 + x2 + 4x4 - 4x5 = 10
x1, x2, x4, x5, s1 ³ 0

This is the standard form above L. P. Model described in que 2.