Linear Programming (Shadow Pricing) Question Watch

jackie11
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Hi there

Please could someone confirm if Im correct or not.

So the question carries on from Question 1, so I will give you both qs;

Question 1
A manufacturer produces two types of radio, the profit contribution of Type 1 radio is £11 and £10 for Type 2 radios, if Type 1 = x, and Type 2 = y, the stock constraints are;

Component A: 2x <= 160
Component B: 3x + 2y <= 320
Component C: x + 3y <= 240
Component D: 2x + 4y <= 360

So to maximise profit - the optimum solution is to produce 70x + 55y which makes £1320 profit

Question 3 refers to Q1 and asks:

If the manufacturer could obtain additional suppliers of component B, determine the maximum amount of them he should consider buying.


So my answer:

So as component B is binding:
3x + 2y = 320
So we are looking to solve 3x + 2y = 321
and 2x + 4y = 360(the next line up on the graph)

so: 3x + 2y = 321
2x + 4y = 360

gives x = 70.5 and y = 54.75

pluggin these values into the original equation: 11x + 10y gives:
11*70.5 + 10*54.75 = £1323, so the increase in profit is £3, so under no circumstances should we pay more than £3 for an extra component B

The upper limit on the RHS of the component B constraint is achieved when the line moves up so that it undoes its own vertix and brings the another into the feasible region: and this new vertix is where lines Component A and component D intercept

line A: x = 80
line D: 2x + 4y = 360
so 160 + 4y = 360
4y = 200
y = 50

(80, 50) - coordinates of this new vertex

so plugging these values into the equation of component B;
3*80 + 2*50 = 340

which means an increase by 340-320 = 20

So the manufacturer should consider purchasing a maximum of 20 extra component Bs


Is this correct?
Many thanks
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