An investor has decided to invest at most $$100,000$ in securities in the form of corporate stocks. He has classified his options into three groups: blue$-$chip stocks that he assumes will yield a $10\%$ return $($dividends and capital appreciation$)$ within a year, growth stocks that he assumes will yield a $15\%$ return within a year, and speculative stocks that he assumes will yield a $20\%$ retum $($mainly due to capital appreciation$)$ within a year. Because of the relative risks involved in his investment, he has further decided that no more than $30\%$ of his investment should be in growth and speculative stocks and at least $50\%$ of his investment should be in blue$-$chip and speculative stocks. Determine how much should be invested in each group of stocks in the hope of maximizing the return on the investments.

a$.$ Suppose the growth stocks are anticipated to yield a $12\%$ retum instead of $15\%.$ Use a sensitivity analysis to determine whether this will cause a change in the optimal solution base.

b$.$ Explain whether you recommend that the investor invest less than $50\%$ of his investment in blue$-$chip and speculative stocks.

c$.$ Explain whether you recommend that the investor invest more than $30\%$ of his investment in growth and speculative stocks.

I Model Optimization:

a$.$ Optimization: Optimize the linear objective function by performing iterations of the simplex method. Be sure to illustrate your work in the appendix of your report.

b$.$ Interpretation: Interpret the results. In other words, now that you have optimized your linear objective function, explain what this means within the context of the problem.

c$.$ Verification: Apply duality theory and interpret the results. Do the results verify your previous results? If not, what do your results show? This is a linear progaramming problem.