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Math Help - Linear Programming problem

  1. #1
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    Linear Programming problem

    hey guys could you please help me with this problem ? its my h.w and i dont know how to slove it :$
    thaanks




    (Bank Loan Policy) A financial institution, the Thriftem Bank, is in the process of formulating a loan policy involving a total of $12 million. Being a full-service faculty, the bank is obligated to grant loans to different clientele. The following table provides the types of loans, the interest rate charged by the bank, and the probability of bad debt at estimated from past experience:


    Probability of
    Type of Loan | Interest Rate | Bad Debt


    Personal | .140 | .10
    Car | .130 | .07
    Home | .120 | .03
    Farm | .125 | .05
    Commercial | .100 | .02


    Bad debts are assumed unrecoverable and hence produce no interest revenue.

    Competition with other financial institutions in area requires that the bank allocate at least 40% of the total funds to farm and commercial loans. To assist the housing industry in the region, home loans must equal at least 50% of the personal, car, and home loans. The bank also has a stated policy specifying that the overall ratio for bad debts on all loans may not exceed .04.
    The objective of the Thriftem Bank is to maximize its net return comprised of the difference between the revenue from interest and lost funds due to bad debts.
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  2. #2
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    Start by introducing variables for the amounts of dollars that the bank lends to the various categories, say v, w, x, y, z for Personal, Car, Home, Farm, Commercial respectively. The total amount lent is 12 million, so v+w+x+y+z = 12,000,000.

    The objective function (to be maximised) is the net return, which you calculate like this. For Personal loans, out of the total amount v dollars that the bank lends, there will be an income of 0.14v in interest (in a one-year period *). There will be a loss of 0.1v in bad debts (presumably that's also in a one-year period, though the question doesn't say so *). So the net income to the bank will be 0.04v. For the other four forms of loan, you can work out the net income in the same way, then add all five income streams together to get the objective function.

    There are some constraints on the different forms of loans. You need to translate these into inequalities among the variables v, w, x, y, z. Then use the simplex method to maximise the objective function subject to those constraints.

    * Note. As so often seems to happen, this is a badly worded problem. The bank's income from the five forms of loan is given in terms of interest rates, in other words the income per year. It doesn't tell you anywhere in the question that you should maximise the income over a one-year period, but that is what you have to assume in order to be able to do the problem.
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