Question $21(40$ points$)$

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Textbook Problem $7$:

Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U$.$S$.$ Oil and H shares of Huber Steel. The annual return for U$.$S$.$ Oil is $$3$ per share and the annual return for Huber Steel is $$5$ per share. U$.$S$.$ Oil sells for $$25$ per share and Huber Steel sells for $$50$ per share. The portfolio has $$80,000$ to be invested. The portfolio risk index $(0\mathrm{.}50$ per share of U$.$S$.$ Oil and $0\mathrm{.}25$ per share a maximum of $700.$ In addition, the portfolio is limited to a maximum of $1000$ shares of U$.$S$.$ Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:

Max $3$U $+5$H Maximize total annual return

s$.$t$.25$U $+50$H $<=80,000$ Funds available

$0\mathrm{.}50$U $+0\mathrm{.}25$H $<=700$ Risk maximum

$1$U $<=1000$ U$.$S$.$ Oil maximum

U$,$ H $>=0$

The computer solution of this problem is shown in Figure $3\mathrm{.}14.$

What is the optimal solution, and what is the value of the total annual return?

Find the objective function ranges $($range of optimality$).$

c$.$ Which constraints are binding?

d$.$ What are the dual values for each of the constraints? What are the implications of the numbers?

e$.$ Find the range of feasibility for each constraint.

f$.$ Would it be beneficial to increase the maximum amount invested in U$.$S$.$ Oil? Why or why not?

FIGURE $3\mathrm{.}14$ THE SOLUTION FOR THE INVESTMENT ADVISORS PROBLEM

Optimal Objective Value $=8400\mathrm{.}00000$

Variable Value Reduced Cost

$--------------$

$---------------$

$-----------------$

U

$800\mathrm{.}00000$

$0\mathrm{.}00000$

H

$1200\mathrm{.}00000$

$0\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$--------------$

$---------------$

$-----------------$

$1$

$0\mathrm{.}00000$

$0\mathrm{.}09333$

$2$

$0\mathrm{.}00000$

$1\mathrm{.}33333$

$3$

$200\mathrm{.}00000$

$0\mathrm{.}00000$

Variable

Objective

Coefficient

Allowable

Increase

Allowable

Decrease

$----------$ U

$------------$

$3\mathrm{.}00000$

$----------$

$7\mathrm{.}00000$

$----------$

$0\mathrm{.}50000$

H

$5\mathrm{.}00000$

$1\mathrm{.}00000$

$3\mathrm{.}50000$

Constraint

RHS Value

Allowable

Increase

Allowable

Decrease

$----------$

$1$

$------------$

$80000\mathrm{.}00000$

$----------$

$60000\mathrm{.}00000$

$----------$

$15000\mathrm{.}00000$

$2$

$700\mathrm{.}00000$

$75\mathrm{.}00000$

$300\mathrm{.}00000$

$3$

$1000\mathrm{.}00000$

Infinite

$200\mathrm{.}00000$