Quantitative Methods Answers

Suppose you are offered the opportunity of sharing a small stand at a trade fair. The service provided by the stand is to assist job seekers in securing employment. The cost of the stand will be R20000, the cost of promotional material will be R2000, and the partners include a cost of R1000 for their time. The organisers of the fair claim that the likely number of enquiries will average out at 3000. Past exxperience has given a conversion rate of 5% of enquries into actual jobs, and the partners work on an average profit of R200 per job. Assume that the enquries are normally distributed with a standard deviation of 300. Would you take up the offer? Show details of your calculations.

The optimal relaxed solution for an ILP has X1 = 3.6 and X2 = 2.9. If we branch on X1, what constraints must be added to the two resulting LP problems?

Given an initial guess x(0) consider the general fixed point iteration x(n+1)=g(x(n))

Show that the fixed point iteration with g defined by g(x)= (5-e^x-2x)/4 cannot be guaranteed to converge to x(*), when x(0) exists in [0,1]

With f(x) defined as f(x) = e^x+6x-5 show that the fixed point iteration
x(n+1) = (5-e^(x(n)))/6, n=0,1,... must converge to x(*) with x(0) exist in [0,1]

I think that probably the solution is based on the contraction theorem!