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176 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

S 1 . Using these results and the first-year obligation of 430, we obtain the constraint
for year 1:
F 1:15B 1 1B 2 1:35B 3 S 1 ¼ 430 Year 1

Investments in bonds can take place only in this first year, and the bonds will be held
We do not consider until maturity.
future investments in The funds available at the beginning of year 2 include the investment returns of
bonds because the future 8.875 per cent on the nominal value of bond 1, 5.5 per cent on the nominal value of
price of bonds depends
on interest rates and bond 2, 11.75 per cent on the nominal value of bond 3, and 4 per cent on savings.
cannot be known in The new amount to be invested in savings for year 2 is S 2 . With an obligation of 210,
advance. the constraint for year 2 is:
0:08875B 1 þ 0:055B 2 þ 0:1175B 3 þ 1:04S 1 S 2 ¼ 210 Year 2

Similarly, the constraints for years 3 to 8 are:

0:08875B 1 þ 0:055B 2 þ 0:1175B 3 þ 1:04S 2 S 3 ¼ 222 Year 3
0:08875B 1 þ 0:055B 2 þ 0:1175B 3 þ 1:04S 3 S 4 ¼ 231 Year 4
0:08875B 1 þ 0:055B 2 þ 0:1175B 3 þ 1:04S 4 S 5 ¼ 240 Year 5
1:08875B 1 þ 0:055B 2 þ 0:1175B 3 þ 1:04S 5 S 6 ¼ 195 Year 6
1:055B 2 þ 0:1175B 3 þ 1:04S 6 S 7 ¼ 225 Year 7
1:1175B 3 þ 1:04S 7 S 8 ¼ 255 Year 8

Note that the constraint for year 6 shows that funds available from bond 1 are
1.08875B 1 . The coefficient of 1.08875 reflects the fact that bond 1 matures at the
end of year 5. As a result, the nominal value plus the interest from bond 1
during year 5 is available at the beginning of year 6. Also, because bond 1
matures in year 5 and becomes available for use at the beginning of year 6,
the variable B 1 does not appear in the constraints for years 7 and 8. Note the
similar interpretation for bond 2, which matures at the end of year 6 and has the
nominal value plus interest available at the beginning of year 7. In addition,
bond 3 matures at the end of year 7 and has the nominal value plus interest
available at the beginning of year 8.
Finally, note that a variable S 8 appears in the constraint for year 8. The retire-
ment fund obligation will be completed at the beginning of year 8, so we anticipate
that S 8 will be zero and no funds will be put into savings. However, the formulation
includes S 8 in the event that the bond income plus interest from the savings in year
7 exceed the 255 cash requirement for year 8. Thus, S 8 is a surplus variable that
shows any funds remaining after the eight-year cash requirements have been
satisfied.
The optimal solution to this 12-variable, 8-constraint linear programme is shown
in Figure 4.9. With an objective function value of 1728.79385, the total investment
required to meet the retirement plan’s eight-year obligation is E1 728 794. Using the
current prices of E1150, E1000 and E1350 for each of the bonds respectively, we can
summarize the initial investments in the three bonds as follows:

Bond Units Purchased Investment Amount
1 B 1 ¼ 144.988 E1 150(144.988) ¼ E166 736
2 B 2 ¼ 187.856 E1 000(187.856) ¼ E187 856
3 B 3 ¼ 228.188 E1 350(228.188) ¼ E308 054

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