# How to formulate this Linear Programming Problem?

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? • In Plan B, will I get out 3 dollars after 3 years (from one invested dollar) or will I have to wait for the next year to pass in order to get out the 4 dollars? – Friedrich Philipp Mar 6 '16 at 13:06 • You have to wait for the next year to pass to get 4 dollars. – hcoder Mar 6 '16 at 13:35 • Somewhat like this one:math.stackexchange.com/questions/62535/… – NoChance Mar 6 '16 at 13:48 • Thanks @NoChance that will help. – hcoder Mar 6 '16 at 14:00 ## 1 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.