Answer to Question #175707 in Operations Research for Nomi

Question

The Crumb and Custard Bakery makes both coffee cakes and Danish in large pans. The main ingredients are flour and sugar. There are 25 pounds of flour and 16 pounds of sugar available and the demand for coffee cakes is 8. Five pounds of flour and 2 pounds of sugar are required to make one pan of coffee cake, and 5 pounds of flour and 4 pounds of sugar are required to make one pan of Danish. One pan of coffee cake has a profit of $1, and one pan of Danish has a profit of $5. Determine the number of pans of cake and Danish that the bakery must produce each day so that profit will be maximized. Formulate a linear programming model for this problem and solve using the simplex method.

Answer 1

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 :

$x_1=0,x_2=4$

$Max \ Z=20$