A business executive has the option to invest money in two plans:
Plan A guarantees that each dollar invested will earn $0.70$ dollar a year later, and plan B guarantees that each dollar invested will earn $2$ dollars after 2 years.
In plan A, investments can be made annually, and in plan B, investments are allowed for periods that are multiples of two years only. How should the executive invest $\$100,000$ to maximize the earnings at the end of $3$ years?
Best Answer
Let $x_t$ be the amount of money that is available at the end of year $t\in \{1,2,3\}$, $\omega_{it}$ the amount of money invested for project $i\in \{A,B\}$ at the beginning of period $t\in \{1,2,3\}$.
Let $c_{it}$ be the profit made by investment $i\in \{A,B\}$ at the beginning of year $t\in \{1,2,3\}$.
$$ \mbox{Maximize }\quad Z= \sum_{i\in \{A,B\} }\sum_{t\in \{2,3\}}c_{it}\sum_{z\in \{1,2\}}\omega_{iz} $$ subject to: $$ x_0=100\,000\\ x_1=x_{0}-\sum_{i\in \{A,B\}}\omega_{i1}\\ x_2=x_1-\sum_{i\in \{A,B\}}\omega_{i2}+\sum_{i\in \{A,B\}}c_{i2}\omega_{i1}\\ x_3=x_2-\sum_{i\in \{A,B\}}\omega_{i3}+\sum_{i\in \{A,B\}}\sum_{t\in \{1,2\}}c_{i(t+1)}\omega_{it}\\ \omega_{B1}=\omega_{B3}=0\\ x_t,\omega_{it}\ge 0\\ $$
Note I am not sure how to interpret the fact that investments for plan B are allowed for periods that are multiples of two years only. The way I see it is we cannot invest in plan B at the beginning of years 1 and 3.