$.$ Selecting an Investment Portfolio: An investment manager wants to determine an optimal portfolio for a wealthy client. The fund has $$2\mathrm{.}5$ million to invest, and its objective is to maximize total dollar return from both growth and dividends over the course of the coming year. The client has researched eight high$-$tech companies and wants the portfolio to consist of shares in these firms only. Three of the firms $($S$1-$S$3)$ are primarily software companies, three $($H$1-$H$3)$ are primarily hardware companies, and two $($C$1-$C$2)$ are internet consulting companies. The client has stipulated that no more than $40$ percent of the investment be allocated to any one of these three sectors$.$ To ensure diversification, at least $$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$$s database relating to these stocks. These estimates include price per share, the projected annual growth rate in the share price, and the anticipated annual dividend payment per share. S$1$ S$2$ S$3$ H$1$ H$2$ H$3$ C$1$ C$2$ Price$/$Share $$40$ $$50$ $$80$ $$60$ $$45$ $$60$ Growth Rate $0\mathrm{.}050\mathrm{.}1$ $$30$ Dividend $0\mathrm{.}03$ $$2\mathrm{.}00$ $$1\mathrm{.}50$ $$3\mathrm{.}500\mathrm{.}04$ $$3\mathrm{.}000\mathrm{.}07$ $$2\mathrm{.}000\mathrm{.}15$ $$1\mathrm{.}00$ $$250\mathrm{.}22$ $$1\mathrm{.}800\mathrm{.}25$ $$0\mathrm{.}00$ a$.$ Determine the maximum return on the portfolio. What is the optimal number of shares to buy for each of the stocks? What is the corresponding amount invested in each stock? b$.$ If there were no integer restriction on the number of the shares invested, how would your solution $($in part a$)$ change?

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