A project manager is considering a portfolio of $5$ ?project investments. The estimated profit

for investment opportunity $j=1,2,$dots,$5$ ?is $$10,$$$8,$$$12,$$$7,$ ?and $$9,$ ?respectively.

Moreover, the estimated capital required for the projects is $$15,$$$10,$$$18,$$$12,$ ?and $$14,$

respectively.

The total amount of capital available for these investments is $$50.$

Investment opportunities $3$ ?and $4$ ?are mutually exclusive $($if project $3$ ?is funded,

then project $4$ ?must not be funded$).$

Project $5$ ?cannot be undertaken unless either $3$ ?or $4$ ?is undertaken.

At least two and at most four investment opportunities have to be undertaken.

If project $2$ ?is funded, then project $5$ ?must be funded.

If project $3$ ?is funded, then projects $1$ ?and $2$ ?must not be funded.

Formulate this problem using an integer programming model.