Question

You have three lines of training modules: Company Training (CT), On-line Training (OT), and Academic Training (AT). For each sold CT, you will receive $1,000 in revenue, while for each sold OT, you will receive $800 and for each AT, you will receive $700. Each module lasts for one month. To deliver the module CT, UQ-HDTC requires 100 hours of data scientist and computer programmer time. The module OT requires 300 hours of data scientist and 500 hours of computer programmer time, while AT requires 200 hours of data scientist and 100 hours of computer programmer time. Suppose you has purchased 1,000 hours of data scientists time and 800 hours worth of computer programmer time for each month. How many CT, OT, and AT modules you should sell per month, so as to maximize your revenue, given the constraints on data scientist and computer programmer time? Please form the problem as an LP problem and solve it using Tableu form of Simplex method.

Answer 1

Let x CT modules, y OT modules and z AT modules be sold per month and w is the total revenue for each month.

Then, by the given conditions, the LP problem becomes,

Maximize w = 1000x+800y+700z

Subject to 100x+300y+200z 1000

100x+500y+100z 800

x,y,z 0

After introducing slack variables u and v, we rewrite the problem in the standard form as

Maximize w = 1000x+800y+700z+0u+0v

Subject to 100x+300y+200z+u = 1000

100x+500y+100z+v = 800

x,y,z,u,v 0

We then construct the simplex tableau.

		cj	1000	800	700	0	0
cB	B	b	a1	a2	a3	a4	a5
0	a4	1000	100	300	200	1	0
0	a5	800	100	500	100	0	1
			-1000	-800	-700	0	0
0	a4	200	0	-200	100	1	-1
1000	a1	8	1	5	1	0	1/100
			0	4200	300	0	10

The solution is optimal as z_j-c_j 0 for all j. The optimal solution is also unique as for all non-basic vectors z_j-c_j 0 for all the basis vectors. Hence, the required solution is :

x = 8, y = 0, z = 0 and w_max = 1000*8+800*0+700*0 ,i.e., w_max = 8000

So, 8 CTmodules must be sold in each month to maximize the total revenue per month and the maximum revenue is $8000.