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.