In this question, you have to construct an example of integer programming problems, abbreviated as IPs, with some required properties. Your IP is required to have the maximization objective, and have two decision variables. $($Thus$,$ each of the feasible regions can be illustrated by drawing it on paper.$)$

Important: All constraints of your IP model and the objective function of your IP model must be linear, with the exception that some of the variables can be declared to be binary or integer. Thus, relaxing the integrality restrictions on the variables should give an LP model. The LP relaxation of an IP model is defined to be the LP obtained from the IP model by relaxing $($i$.$e$.,$ dropping$)$ the integrality restrictions on all variables.

You must justify that your IP and LP relaxations has the required properties. By writing down an informal proof, supplemented by drawings of the feasible regions, and you may $``$certify$\text{'}\text{'}$ properties of your LP relaxations using LP duality theory,

Give an example of an integer programming problem $($IP$)$ with a maximization objective that has two variables, such that $($IP$)$ has a unique optimal solution x$^=($x$\_1^*,$ x$\_2^),$ the LP relaxation of $($IP$),$ call it $($P$),$ has a unique optimal solution y$*=($y$\_1^*,$ y$\_2^*),$ and $||$x$^$y$^||\_1=|$x$\_1^$y$\_1^*|+|$x$\_2^$y$\_2^|>=10.$

In other words, the one$-$norm of the vector x$^$ y$^$ is required to be at least $10,$ where x$^$ is the unique optimal solution of $($IP$)$ and y$^*$ is the unique optimal solution of $($P$).$

To show that x$^$ is the unique optimal solution of $($IP$),$ you could argue that all other feasible solutions to $($IP$)$ have smaller objective value. To show that y$^$ is the unique optimal solution of $($P$),$ you could argue that all other extreme points of the feasible region of $($P$)$ have smaller objective value.