Question

Find the objective function and the constraints, and then solve the problem by using the simplex method. A confectioner has 600 pounds of chocolate, 100 pounds of nuts, and 50 pounds of fruits in inventory with which to make three types of candy: Sweet Tooth, Sugar Dandy, and Dandy Delite. A box of Sweet Tooth uses 3 pounds of chocolate, 1 pound of nuts, and 1 pound of fruit and sells for $8. A box of Sugar Dandy requires 4 pounds of chocolate and 1 2 pound of nuts and sells for $5. A box of Dandy Delite requires 5 pounds of chocolate, 3 4 pounds of nuts, and 1 pound of fruit and sells for $6. How many boxes of each type of candy should be made from the available inventory to maximize revenue?

Answer 1

Solution:

MAX Z = 8x1 + 5x2 + 6x3
subject to
3x1 + 4x2 + 5x3 <= 600 chocolate
x1 + 1/2x2 + 3/4x3 <= 100 nuts
x1 + x3 <= 50 fruit
and x1,x2,x3 >= 0

Given data x1 = number of boxes of sweet tooth
x2 = number of sugar dandy boxes

x3 = number of dandy delight boxes
Solution:
Problem is

Max Z = 8 x1 + 5 x2 + 6 x3

subject to

3 x1 + 4 x2 + 5 x3 ≤ 600 x1 + 0.5 x2 + 0.75 x3 ≤ 100 x1 + x3 ≤ 50

and x1,x2,x3≥0;

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

1. As the constraint-1 is of type '≤' we should add slack variable S1

2. As the constraint-2 is of type '≤' we should add slack variable S2

3. As the constraint-3 is of type '≤' we should add slack variable S3

After introducing slack variables

Max Z = 8 x1 + 5 x2 + 6 x3 + 0 S1 + 0 S2 + 0 S3

subject to

3 x1 + 4 x2 + 5 x3 + S1 = 600 x1 + 0.5 x2 + 0.75 x3 + S2 = 100 x1 + x3 + S3 = 50

and x1,x2,x3,S1,S2,S3≥0

Iteration-1		Cj	8	5	6	0	0	0
B	CB	XB	x1	x2	x3	S1	S2	S3	MinRatio XBx1
S1	0	600	3	4	5	1	0	0	6003=200
S2	0	100	1	0.5	0.75	0	1	0	1001=100
S3	0	50	(1)	0	1	0	0	1	501=50→
Z=0		Zj	0	0	0	0	0	0
		Zj-Cj	-8↑	-5	-6	0	0	0

Negative minimum Zj-Cj is -8 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 50 and its row index is 3. So, the leaving basis variable is S3.

∴ The pivot element is 1.

Entering =x1, Departing =S3, Key Element =1