Question

Both min and max values?

nge 7.2.13 Find the minimum and maximum values of z 3x+8y, if possible, for the following set of constraints. 4x + 3y 2 12 x+3y 26 x20, y20 rkec Sco ts: Select the correct choice below and, if necessary, fill n the answer box to complete your choice. O A. The minimum value is「, (Round to the nearest tenth as needed.) O B. There is no minimum value.

Answer 1

here constraints have $\geq$ sign so function has only minimum value for maximum value solution is unbound

find minimum value

subject to

Iteration-1		Cj	3	8	0	0	M	M
B	CB	XB	x1	x2	S1	S2	A1	A2	MinRatio XB/x2
A1	M	12	4	3	-1	0	1	0	12/3=4
A2	M	6	1	(3)	0	-1	0	1	6/3=2→
Z=18M		Zj	5M	6M	-M	-M	M	M
		Zj-Cj	5M-3	6M-8↑	-M	-M	0	0

Positive maximum Zj-Cj is 6M-8 and its column index is 2

Minimum ratio is 2 and its row index is 2.

The pivot element is 3.

Entering =x2, Departing =A2

$R_2 \leftarrow R_2 *\frac{1}{3}$

$R_1 \leftarrow R_1-3R_2$

Iteration-2		Cj	3	8	0	0	M
B	CB	XB	x1	x2	S1	S2	A1	MinRatio XB/x1
A1	M	6	(3)	0	-1	1	1	6/3=2→
x2	8	2	1/3	1	0	-1/3	0	2/(1/3)=6
Z=6M+16		Zj	3M+8/3	8	-M	M-8/3	M
		Zj-Cj	3M-1/3↑	0	-M	M-8/3	0

Positive maximum Zj-Cj is 3M-1/3 and its column index is 1

Minimum ratio is 2 and its row index is 1

The pivot element is 3.

Entering =x1, Departing =A1

$R_1 \leftarrow R_1 *\frac{1}{3}$

$R_2 \leftarrow R_2 -R_1 *\frac{1}{3}$

Iteration-3		Cj	3	8	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio
x1	3	2	1	0	-1/3	1/3
x2	8	4/3	0	1	1/9	-4/9
Z=50/3		Zj	3	8	-1/9	-23/9
		Zj-Cj	0	0	-1/9	-23/9

all $Zj-Cj\leq 0$

optimal solution is arrived

${\color{Red} x_1=2,\:\:\:x_2=\frac{4}{3} }$

${\color{Red}Min \:\:\:Z=\frac{50}{3} }$