Al has $60,000 that he wants to invest now in order to use the accumulation for purchasing a retirement annuity in 5 years. After consulting with his financial adviser, he has been offered four types of fixed-income investments, which we will label as investments A,B,C, D.
Investments A and B are available at the beginning of each of the next 5 years (call them years 1 to 5). Each dollar invested in A at the beginning of a year returns $1.40 (a profit of $.40) 2 years later (in time for immediate reinvestment). Each dollar invested in B at the beginning of a year returns $1.70 three years later.
Investments C and D will each be available at one time in the future. Each dollar invested in C at the beginning of year 2 returns $1.90 at the end of year 5. Each dollar invested in D at the beginning of year 5 returns $1.30 at the the end of year 5.
Al wishes to know which investment plan maximizes the amount of money that can be accumulated by the beginning of year 6.
(a) All the functional constraints for this problem can be expressed as equality constraints. To do this let A t, B t, C t, and D t be the beginning of year t for each t where the investment is available and will mature by the end of year 5. Also let R t be the number of available dollars not invested at the beginning of year t (and so available for investment in a later year). Thus, the amount invested at the beginning of year t plus R t must equal the number of dollars available for investment at that time. Write such an equation in terms of the relevant variables above for the beginning of each of the years to obtain the five functional constraints for this problem.
(b)Formulate a complete linear programming model for this problem.
As investment 'A' matures every 2 yrs, therefore investments in `A' will be: A1, A2, A3, A4 at the beginning of years 1,2,3 and 4 respectively and the returns will be 1.4A1, 1.4A2, 1.4A3 and 1.4A4 at thends of year 2, 3, 4 and 5 respectively.
Invsetment B: matures after 3 yrs.
Therefore investments will be: B1, B2, B3 at the beginning of years 1, 2 and 3 respectively and returns will be 1.7B1, 1.7B2 and 1.7B3 at the ends of years ...
An investment problem is solved using linear programming.