(Solved): This homework will have 4 parts. a) Is the following optimization problem linear? If not, why? ...

This homework will have 4 parts. a) Is the following optimization problem linear? If not, why? Variables are $x,y,z,t$ $\begin{array}{c}\min x+y+z+\max (x,t)\\ x+y\ge t\\ z-y\le 2\\ x-z\ge 1\\ t+y\ge 1.5\\ x,y,z,t\in R^{+}\end{array}$ b) Is the following optimization problem linear? If not, why? Variables are $x,y,z$. $\begin{array}{c}\min x+y+z+tx\\ x+z-y\ge t^2\\ 3x+5y-2z\le c\\ x+y-cz\le 1\\ x,y,z\in R^{+}\end{array}$ c) Is the following optimization problem linear? If not, why? $\begin{array}{c}\text{ Variables are }x,y,z\\ \min x+y+z+yz\\ x+y+z\le 12\\ 2xy\le 9\\ 3x+5y\ge 12\\ x,y,z\in R\end{array}$ d) Formulate the following problem as a linear optimization problem: You are in charge of a Window Company with 3 types of employees Wood specialists, Aluminium specialists and Generalists. Having received your education in a small town called Ithaca, you have a firm belief in the value of hand-crafted windows. You make two types, a wooden frame window and an aluminium framed window. The wood framed windows yield you $\$300$ in profit while the aluminium framed windows yield $\$150$. Wood specialists work on wooden frame windows only and can build 120 per month, Aluminium specialists work on aluminium framed windows only and can build 80 per month. The wooden framed windows require 12 square feet of glass each while the aluminium framed windows require 4 square feet of glass. Generalists can work on either type of window, they can build wooden windows at a rate of 20 per month and aluminium windows at a rate of 60 per month. You receive 96,000 square feet of glass per month. You have 100 employees of each type. You want to identify how many wooden and aluminium windows you should make. Formulate this problem as a linear optimization problem.