Questions $1$ to $10$ are based on the situation given below.

National Insurance Associates carries an investment portfolio of stocks. Currently, $$500,000$ is available

and at least $$200,000$ must be considered for new investment opportunities in the following stocks.

Stocks A B C D

Price per share $$100$ $$50$ $$80$ $$40$

Annual rate of return $0\mathrm{.}120\mathrm{.}080\mathrm{.}060\mathrm{.}10$

Risk measure per dollar invested $0\mathrm{.}100\mathrm{.}070\mathrm{.}050\mathrm{.}08$

Risk measure per share $100*0\mathrm{.}1=1050*0\mathrm{.}07=3\mathrm{.}580*0\mathrm{.}05=440*0\mathrm{.}08=3\mathrm{.}2$

The risk measure indicates the relative uncertainty associated with the stock in terms of its realizing the

projected annual return $($higher values indicate greater risk$).$ National$$s top management has stipulated

the following guidelines. The annual rate of return on the portfolio must be at least $$18,000.$ The total

amount invested in stocks B and C must be at least $$100,000.$ The amount invested in stocks A or D

cannot exceed $$50,000.$ The return on investment from C should not exceed $50\%$ of the total return on

investment. Develop an investment portfolio that minimizes risk.

Problem Formulation

Decision Variables

A: Number of shares of stock A in portfolio

B: Number of shares of stock B in portfolio

C: Number of shares of stock C in portfolio

D: Number of shares of stock D in portfolio

Objective Function: Minimize z $=10$A $+3\mathrm{.}5$B $+4$C $+3\mathrm{.}2$D

Constraints

Amount Available: $100$A $+50$B $+80$C $+40$D $<500,000$

Minimum investment: $100$A $+50$B $+80$C $+40$D $>200,000$

Minimum Return: $12$A $+4$B $+4\mathrm{.}8$C $+4$D $>18,000$

Minimum B and C: $50$B $+80$C $>100,000$

Maximum A and D: $100$A $+40$D $<50,000$

C Return on investment: $-6$A $2$B $+2\mathrm{.}4$C $2$D $<0$

A$,$B$,$C$,$D $>0$

The sensitivity report obtained on solving the problem is given below. Answer the questions based on

the sensitivity report. Each question is independent of the other questions. While answering

questions, numbers should be rounded to $4$ decimal places. If the risk measure per share of stock A increases to $11,$ which of the following statements is true?

a$.$ This is a change in the objective function coefficient of a basic variable within the allowable

limits$,$ hence the optimal solution is unchanged but OFV will change.

b$.$ This is a change in the objective coefficient of a basic variable within the allowable limits$,$

hence optimal solution and OFV are unchanged.

c$.$ This is a change in the objective coefficient of a non$-$basic variable within allowable limits$,$

hence both optimal solution and objective function value are unchanged.

d$.$ This is a change in the objective coefficient of a non$-$basic variable within the allowable

limits$,$ hence optimal solution is unchanged but the OFV will change and can be calculated.

e$.$ This change is not among the permitted changes and hence we must re$-$solve to obtain the

new optimal solution and OFV.