Question

Weekly production schedules are needed for the manufacture of two products X and Y. Each unit of X uses one component made in the factory, while each unit of Y uses two of the components, and the factory has a maximum output of 80 components a week. Each unit of X and Y requires 10 hours of subcontracted work and agreements have been signed with subcontractors for a minimum weekly usage of 200 hours and a maximum weekly usage of 600 hours. The marketing department says that all production of Y can be sold but there is a maximum demand of 50 units of X, despite a long-term contract to supply 10 units of X to one customer. The net profit on each unit of X and Y is $200 and $300 respectively.

Linnear Programming step by step, show work manually without using excel, Include a graph, and corner. I would like to try to understand how the question is done step by step. Thank you so much

Answer 1

Let the weekly production are needed for the manufacture of two products X and Y be x units and y units respectively.

Then by the given conditions we have,

Maximize Z = 200x+300y

Subject to x+2y 80

10x+10y 200

10x+10y 600

x 50

x,y 0

i.e., Maximize Z = 200x+300y

Subject to x+2y 80

x+y 20

x+y 60

x 50

x,y 0

Introducing slack variables a,c,d and surplus variable b the problem becomes,

Maximize Z = 200x+300y+0a+0b+0c+0d

Subject to x+2y+a = 80

x+y-b = 20

x+y+c = 60

x+d = 50

x,y,a,b,c,d 0

COMPUTATION TABLE :

		cj	200	300	0	0	0	0
cB	B	b	a1	a2	a3	a4	a5	a6
0	a3	80	1	2	1	0	0	0
0	a4	20	1	1	0	-1	0	0
0	a5	60	1	1	0	0	1	0
0	a6	50	1	0	0	0	0	1
		zj-cj	-200	-300	0	0	0	0
0	a3	40	-1	0	1	2	0	0
300	a2	20	1	1	0	-1	0	0
0	a5	40	0	0	0	1	1	0
0	a6	50	1	0	0	0	0	1
		zj-cj	100	0	150	-300	0	0
0	a3	20	-1/2	0	1/2	1	0	0
300	a2	40	1/2	1	1/2	0	0	0
0	a4	20	1/2	0	-1/2	0	1	0
0	a6	50	1	0	0	0	0	1
		zj-cj	-50	0	150	0	0	0
0	a3	40	0	0	0	1	1	0
300	a2	20	0	1	1	0	-1	0
200	a1	40	1	0	-1	0	2	0
0	a6	10	0	0	1	0	2	1
		zj-cj	0	0	100	0	100	0

In the last tableau all z_j-c_j 0.

The optimal solution is : x = 40, y = 20 and Z_max = (200*40)+(300*20) = 14000

Therefore, the weekly production are needed for the manufacture of two products X and Y are 40 units and 20 units respectively and total profit is $14000.