Finco Investment Corporation must determine an investment strategy for the firm during the next

three years. Currently $($time $0),$ $$100,000$ is available for investment. Investments A$,$ B$,$ C$,$ D$,$ and

$E$ are available. The cash flow associated with investing $$1$ in each investment is given in Table.

For example, $$1$ invested in investment B requires a $$1$ cash outflow at time $1$ and returns $50$ at

time $2$ and $$1$ at time $3.$ To ensure that the company's portfolio is diversified, Finco requires that

at most $$75,000$ be placed in any single investment. In addition to investments $A-E,$ Finco can

earn interest at $8\%$ per year by keeping uninvested cash in money market funds. Returns from

investments may be immediately reinvested. For example, the positive cash flow received from

investment $C$ at time $1$ may immediately be reinvested in investment $B.$ Finco cannot borrow funds,

so the cash available for investment at any time is limited to cash on hand. Formulate an LP that

will maximize cash on hand at time $3.$

$*$Note: Time $0=$ present; time $1-=1$ year from now; time $2=2$ years from now; time $3=3$ years

from now.

Decision variables,

$x_1=$ dollars invested in investment $A$

$x_2=$ dollars invested in investment $B$

$x_3=$ dollars invested in investment C

$x_4=$ dollars invested in investment $D$

$x_5=$ dollars invested in investment $E$

$S_t=$ dollars invested in money market

funds at time $t(t=0,1,2)$

The mathematical model

maxZ$=x_2+1\mathrm{.}9x_4+1\mathrm{.}5x_5+1\mathrm{.}08S_2$

$s.t.$

$x_1+x_3+x_4+S_0=100,000$

$0\mathrm{.}5x_1+1\mathrm{.}2x_3+1\mathrm{.}08S_0=x_2+S_1$

$x_1+0\mathrm{.}5x_2+1\mathrm{.}08S_1=x_5+S_2$

$x_{1}75,000$

$x_{2}75,000$

$x_{3}75,000$

$x_{4}75,000$

$x_{5}75,000$

Write the GAMS code of the given mathematical model.

Answer the following questions.

$1.$ What are the optimum Z value and decision variable values?

$($For Z$,$ x$\_1,$x$\_2,$x$\_3,$x$\_4,$x$\_5,$s$\_0,$s$\_1,$s$\_2$ specify the values separately$)$

$2.$ Which one$($s$)$ of the constraint$($s$)$ is$($are$)$ binding? Which is$($are$)$ non$-$binding? Why?

$3.$ What are the shadow prices for all constraints? What do they mean for this model? Specify one by one.

$4.$ What are the reduced costs for all variables? What do they mean for the model? Specify one by one.

$5.$ If the investment budget was $$90,000($instead of $$100,000),$ would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values?

$6.$ If Finco can earn interest at $10\%($instead of $\%8)$ per year by keeping uninvested cash in money market funds? Why? If the current optimal basic solution changes, what are the new values?

$7.$ If $$1$ invested in investment C required again a $$1$ cash outflow at time $0$ but returned $$1\mathrm{.}5$ at time $1,$ would the current optimal basic solution change? Why? If the current optimal basic solution changes, what are the new values?

$8.$ If there is a $6$th investment opportunity and the cash flow is: $$1$ invested in investment F required a $$1$ cash outflow at time $1$ and returned $$2$ at time $2,$ should Finco invest money on investment F$?$