QUESTION $2$

a$)$ An individual wishes to invest $$5000$ over the next year in two types of investment: Investment A yields $5\%,$ and investment B yields $8\%.$ Market research recommends an allocation of at least $25\%$ in A and at most $50\%$ in B$.$ Moreover, investment in A should be at least half the investment in B$.$ Formulate this problem as a linear programming model that will maximize returns while achieving the specified conditions.

$[8$ marks$]$

b$)$ United International Bank is in the process of formulating a loan policy that involves a maximum of $$12$ million. The following table provides the pertinent data about available loans.

$\backslash$table$[[$Table l$,,],[$Type of Loan,Interest rate,Bad$-$debt ratio$],[$Personal$,0\mathrm{.}140,0\mathrm{.}10],[$Car$,0\mathrm{.}130,0\mathrm{.}07],[$Home$,0\mathrm{.}120,0\mathrm{.}03],[$Farm$,0\mathrm{.}125,0\mathrm{.}05],[$Commercial$,0\mathrm{.}100,0\mathrm{.}02]]$

Bad debts are unrecoverable and produce no interest revenue.

Competition with other financial institutions dictates the allocation of at least $40\%$ of the 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 limits the overall ratio of bad debts on all loans to at most $4\%.$

The objective of the United International Bank is to maximize net return, the difference between interest revenue and lost bad debts. Interest revenue is accrued on loans in good standing. For example, when $10\%$ of personal loans are lost to bad debt, the bank will receive interest on $90\%$ of the loan $-$ that is$,$ it will receive $14\%$ interest on $0\mathrm{.}9x_1$ of the original loan $x_1.$

Total interest $=0\mathrm{.}14(0\mathrm{.}9x_1)+$dots

Bad debt $=0\mathrm{.}10x_1+$dots

Formulate this problem as a linear programming model.

c$)$ Solve the linear programming problem by the simplex method.

Maximize $Z=2x_1+x_2$

Subject $to3x_1+5x_{2}15$

$6x_1+2x_{2}24$

$x_1,x_{2}0$

$[15$ marks$]$