Explain the meanings of the terms slack and surplus variable

Explain the meanings of the terms slack and surplus variables in the simplex algorithm. Use the simplex algorithm to find the maximum value of z = 5x + 4y given that 3x + 4y lessthanorequalto 40, 2x + y lessthanorequalto 20, x Greaterthanorequalto 0 and 0 lessthanorequalto y lessthanorequalto 6.

Solution

Here our objective is to maximise

z = 5x+4y

subject to

3x+4y+s = 20 and 2x+y+t = 20, where s and t are positive values.

x>=0, 0<=y<=6

To eliminate the inequality sign we introduced here s and t two slack variables

Now solving the equation

3x+4y+s = 20. Multiply second equation by 4.

8x+4y+4t = 80

Subtract to get

5x +4t-s = 60

x <= 12

Since 3x+4y <=40, y <=1

Thus y <=1 and x<=12 is the solution

max z = 60+4 = 64