LPP using simplex method Maximize Z=300x+200y Subject to:- 5x+2y ≤ 180 3x+2y ≤, 135 x,y ≥0

Question

ANSWER FOR BBA

Answer 1

GRAPHICAL METHOD

Graph 5x+2y=180 and 3x+2y=135.

Lines intersect at 2x=45, x=22.5, 2y=135-3x=135-67.5=67.5, y=33.75.

Since x,y≥0 we have a triangle ABC with vertices A(22.5,33.75), B(36,0), C(45,0).

Z=300x+200y for each point:

Z_A=13500; Z_B=10800; Z_c=13500. So max Z is achieved for A or C using this method.

SIMPLEX METHOD

TABLEAUX

ROW	x	y	s1	s2	Z	Result	Ratio	Op/Comment
R1	5✔️	2	1	0	0	180	36	Pivot row
R2	3	2	0	1	0	135	45
R3	-300	-200	0	0	1			Objective
R1	1	0.4	0.2	0	0	36		R1/5 (adjust pivot row)
R2	3	2	0	1	0	135
R3	-300	-200	0	0	1	0
R1	1	0.4	0.2	0	0	36	90
R2	0	0.8✔️	-0.6	1	0	27	33.75	R2-3R1 (next pivot row)
R3	0	-80	60	0	1	10800		R3+300R1
R1	1	0.4	0.2	0	0	36
R2	0	1	-0.75	1.25	0	33.75		R2/0.8 (adjust pivot row)
R3	0	-80	60	0	1	10800
R1	1	0	0.5	-0.5	0	22.5		R1-0.4R2
R2	0	1	-0.75	1.25	0	33.75
R3	0	0	0	100	1	13500		R3+80R2 (finish)

See following comment.

Comments

The tableaux are shown in the table, where each tableau is in a different colour. The row operations are described under Op/Comment. Checked values indicate the pivot element PE). The pivot column (PC) has the largest negative value in the bottom row; the pivot row (PR) is determined by the lowest value in the ratio column for all the rows excluding the bottom row, calculated by dividing the value in the Result column by the corresponding value in PC. PE is the value where the PC and PR intersect. PR, PC and PE change as the process progresses. s1 and s2 are the slack variables used to change the inequalities into equalities (equations).

AIM: Matrix operations: Force PE=1; force other values in PC to zero. When there are no negative values in the bottom row the process is complete. 1 in the variable columns (x, y and Z) indicates that the result (solution) are in the Result column.

Bold type has been used to indicate PR and PC.

Therefore:

x=22.5, y=33.75, Z=13500 which concurs with one of the results using the graphical method.