Question

Solve the following optimization problem (Be sure to include the statement of the optimization problem and a graph of the feasible in your solution):

Jamie has joined a building contest. A dog shape requires 3 small blocks and one large block to build. A robot shape requires 5 small bricks and 5 large bricks to build. Jamie has a supply of 240 small bricks and 100 large bricks.

If a dog is worth 2 points and a robot is worth 7 points, how many shapes of each type should Jamie build to maximize the points?

Answer 1

from the given data make table

	small blocks	small blocks	points
x1 = number of dog	3	1	2
x2 = number of robot	5	5	7
available	240	100

.

system is

subject to

After introducing slack variables

subject to

Iteration-1		Cj	2	7	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio XB/x2
S1	0	240	3	5	1	0	240/5=48
S2	0	100	1	(5)	0	1	100/5=20→
Z=0		Zj	0	0	0	0
		Zj-Cj	-2	-7↑	0	0

Negative minimum Zj-Cj is -7 and its column index is 2.

Minimum ratio is 20 and its row index is 2.

The pivot element is 5.

Entering =x₂, Departing =S₂

Iteration-2		Cj	2	7	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio XB/x1
S1	0	140	(2)	0	1	-1	140/2=70→
x2	7	20	0.2	1	0	0.2	20/0.2=100
Z=140		Zj	1.4	7	0	1.4
		Zj-Cj	-0.6↑	0	0	1.4

Negative minimum Zj-Cj is -0.6 and its column index is 1.

Minimum ratio is 70 and its row index is 1

The pivot element is 2.

Entering =x₁, Departing =S₁

Iteration-3		Cj	2	7	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio
x1	2	70	1	0	0.5	-0.5
x2	7	6	0	1	-0.1	0.3
Z=182		Zj	2	7	0.3	1.1
		Zj-Cj	0	0	0.3	1.1

Since all

Hence, the optimal solution has arrived

70 dog shape

6 robot  shape

maximum point is 182