$6\mathrm{.}3$ Selecting an Investment Portfolio: An investment manager wants to determine an optimal portfolio for a wealthy client. The fund has $$2\mathrm{.}5$ million to invest, and its objective is to maximize total dollar return from both growth and dividends over the course of the coming year. The client has researched eight high$-$tech companies and wants the portfolio to consist of shares in these firms only. Three of the firms $($S$1$S$3)$ are primarily software companies, three $($H$1$H$3)$ are primarily hardware companies, and two $($C$1$C$2)$ are Internet consulting companies. The client has stipulated that no more than $40\%$ of the investment be allocated to any one of these three sectors$.$ To ensure diversification, at least $$100,000$ must be invested in each of the eight stocks. Moreover, the number of shares invested in any stock must be a multiple of $1000.$

The table below gives estimates from the investment company$$s database relating to these stocks. These estimates include the price per share, the projected annual growth rate in the share price, and the anticipated annual dividend payment per share.

Stock

S$1$ S$2$ S$3$ H$1$

Price per share $$40$ $$50$ $$80$ $$60$ Growthrate $0\mathrm{.}050\mathrm{.}100\mathrm{.}030\mathrm{.}04$ Dividend $$2\mathrm{.}00$ $$1\mathrm{.}50$ $$3\mathrm{.}50$ $$3\mathrm{.}00$

H$2$ H$3$ C$1$ C$2$

$$45$ $$60$ $$30$ $$250\mathrm{.}070\mathrm{.}150\mathrm{.}220\mathrm{.}25$ $$2\mathrm{.}00$ $$1\mathrm{.}00$ $$1\mathrm{.}80$ $$0\mathrm{.}00$

a$.$ Determinethemaximumreturnontheportfolio.Whatistheoptimalnumberofsharesto buy for each of the stocks? What is the corresponding dollar amount invested in each stock?

b$.$ Comparethesolutioninwhichthereisnointegerrestrictiononthenumberofshares invested. By how much $($in percentage terms$)$ do the integer restrictions alter the value of the optimal objective function? By how much $($in percentage terms$)$ do they alter the optimal investment quantities?