Answer to Question #291991 in Operations Research for jone

Question

The crumb and Custard bakery makes cakes and pies. The main ingredients are flour and sugar. The following linear programming model has been developed for determining the number of cakes and pies (x1 and x2 to produce each day to maximize profit.

Max Z=x1 + 5x2

Subject to:

8x1+10x2< 25(flour, lb)

2x1+4x2< 16(sugar, lb)

x1< 5(demand for cakes)

x1,x2 >0

Solve this model using simplex method.

Answer 1

Solution:

According to question:

$maximize\ \ Z=1x_1+5x_2\\subject\ to\\5x_1+5x_2\leq25\\2x_1+4x_2\leq 16\\x_1\leq8\\and\ x_1,x_2\geq0$

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '≤' we should add slack variable $S_1$

2. As the constraint-2 is of type '≤' we should add slack variable $S_2$

3. As the constraint-3 is of type '≤' we should add slack variable $S_3$

After introducing slack variables

$Max Z=x_1+5x_2+0S_1+0S_2+0S_3\\subject\ to\ 5x_1+5x_2+S_1=25\\2x_1+4x_2+S_2=16\\x_1+S_3=8\ and \\x_1,x_2,S_1,S_2,S_3≥0$

Negative minimum $Z_j-C_j$  is -5 and its column index is 2. So, the entering variable is $x_2.$

Minimum ratio is 4 and its row index is 2. So, the leaving basis variable is $S_2.$

∴ The pivot element is 4.

Entering =$x_2$ , Departing =$S_2$ , Key Element =4

Since all $Z_j-C_j≥0$

Hence, optimal solution is arrived with value of variables as :

$x_1=0,x_2=4$

$Max \ Z=20$