you are responsible for determining the optimal mix of assets for one of the firm's flagship mutual funds. The assets are cash, bonds, real estate, and shares. To determine the mix of investments, you must pick x$\_$a more than or equal to $0,$ the fraction to invest into asset a$,$ where the sum of x$\_$a $=1,$ a in A$.$ However, you need to decide how to choose the mix to ensure high future returns.

Since future returns of each asset are uncertain, your team of analysts have produced six possible scenarios. More specifically, let r$\_$ia be the return of asset a in scenario i$,$ for a in A and i in $\{1,...,6\}.$ Denote r$\_$i to be the vector $($r$\_$ia$)\_$a for each scenario i$.$ Then the return of a portfolio mix x in scenario i is r$\_$i transpose x $=$ the sum of r$\_$ia $*$ x$\_$a$.$

You must determine x before knowing which scenario will occur; using only the estimates you have for each of the scenarios.

You devise the following strategy to pick the optimal portfolio mix:

Let mu transpose x $=1/6*$ the sum of r$\_$i transpose x be the mean return across all scenarios, and let $1/6*$ the sum of $|$r$\_$i transpose x $-$ mu transpose x$|$ be the mean absolute deviation of the returns across scenarios. The optimal mix should ensure that the mean absolute deviation is at most $50\%$ of the mean return.

In at least four out of six scenarios, the return r$\_$i transpose x in that scenario must exceed $70\%$ of the mean return mu transpose x$.$

Given the two constraints above, your objective is to maximize the mean return mu transpose x$.$

Formulate a mixed integer program to find the optimal portfolio mix. For the second constraint, you want the model to pick the four scenarios automatically. You will need to specify big$-$M comstraints and describe what appropriate M values should be$.$