$($Portfolio optimization$)$ Many types of financial investment problems are modeled and solved using

linear optimization. Such portfolio investment $($allocation$)$ models have the basic characteristics of

blending models.

Model and solve the following example:

In$-$X Investments is a small, family$-$owned business that manages personal financial portfolios. The

company manages six mutual funds. The characteristics of the funds are given below:

The company has a client who has acquired $$500,000$ from an inheritance and has asked In$-$X to help

them invest this inheritance in the best possible way.

In$-$X Investments uses a proprietary algorithm to establish a measure of risk for its funds based on the

historical volatility of the investments. The higher the volatility, the greater the risk. After the initial

meeting with the client, the company recommends that no more than $$200,000$ be invested in any

individual fund, that at least $$50,000$ be invested in each of the multinational and balanced funds, and that

the total amount invested in income equity and balanced funds be at least $40\%$ of the total investment, or

$$200,000.$ The client would like to have an average return of at least $5\%$ but would like to minimize risk.

What portfolio would achieve this?

a$)$ Develop a spreadsheet model and solve.

b$)$ Express the model in algebraic form.