Long Shot I Know Maths Related Question

[Simpson]

Got food posioning feel like total shit haven't slept for 3 days straight so if someone could point me in the right direction id be really pleased, I think ive got 75/85 of the marks done anyway but I'd prefer to nail the CW than rely on being able to do it in a Test.

Basically a simple linear programme with my found constraints,

a = argentina, profit 7, labour 11, capacity 15, material 6

b= Bath, profit 9, labour 7, Capacity 10, Material 6

Labour cost < 100

material <= 120

z = 7a +9b (optimal)

Solve using

11a + 7b <100 (Labour)

a + b <= 20 (Materials)

a <= 15 (capacity)

b<= 10 (capacity)

Solved graphically and using the simplex method to get z = 1200/11 a=30/11 b=10

Now for the question

Calc. the RANGE of profit per item made in Bath for which the optimal basis found in the simplex method remains optimal.

Is it to do with slack variables or..... I dunno! Something else? Im totally lost.

No useless replys, spanks

Edited December 5, 2010 by Simpson

[Simpson]

]

[JT!]

I would do this by comparing the gradient of z {(7, P), where P is the Bath Profit} with the tangent vectors to the constraints pointing at the optimum point: (1,0) from b≤10 and (-7,11) from 11a+7b≤100. So long as the gradient forms less than a right angle with these tangent vectors, it is pointing into the same corner as before:

(1,0) • (7,P) > 0 and (-7,11) • (7,P) > 0.

I'm too lazy to look up "slack variables" (the above in disguise?), but the above method works so long as as everything is linear. However, for more than 2 dimensions, those tangent vectors are the vectors along the 1-dimensional edges between vertices: intersections of several constraints.