Have i correctly formulated the LP from a worded problem? Jack is an aspiring freshmen at a University. He realizes that “all work and no play make Jack a dull boy.” As a result, Jack wants to apportion his available time of about $10$ hours a day between work and play. He estimates that play is twice as much as fun as work. He also wants to study at least as much as he plays. However, Jack realizes that if he is going to get all his homework assignment done, he cannot play more than $4$ hours a day. How would Jack allocate his time to maximize his pleasure from both work and play? Formulate this problem as a linear programming problem. > This is my attempt: > > Let $x_1$ be the amount of time in hours of work, and $x_2$ be the amount of time in hours of play. > > for $(x_1 , x_2) \in \mathbb{R^2}$ > > Max $x_1 + 2x_2$ > > subject to > > $x_1 + x_2\leq10$ > > $x_1\geq x_2$ > > $x_2\leq4$ > > $x_1, x_2\geq0$ If anything in incorrect please correct me

The amount of fun from $x_1$ hours of work and $x_2$ hours of play is $x_1+2x_2$ "fun units." Or if you prefer, if an hour of work yields $k$ pleasure units, then an hour of play yields $2k$ pleasure units, and therefore $x_1$ hours of work and $x_2$ of play yield $kx_1+2kx_2$ pleasure units.

We want to maximize the amount of pleasure, so we want to maximize $kx_1+2kx_2$, or equivalently $x_1+2x_2$.

Your inequalities are fine, except for $x_2\ge 2x_1$, which was an incorrect attempt to express the fact that play is twice as much fun as work. That inequality should be removed from the list.