Introduction to Modeling and Optimization

HOMEWORK ASSIGNMENT$-1$

Due July $15,2024$

$($will be submitted to MS Teams as $.$PDF file with the following format:

Name$\_$Surname$\_$Mod$\_$and$\_$Opt$\_$HW$\_1.$PDF$.$ Typing $/$ hand writing are both allowed$)$

Q$-1)(40$ points$)$

Consider the following linear program:

maximize $Z=2x_1+x_2$

subject to:

$-x_1+x_{2}5$

$x_1-x_{2}0$

$x_{1}5$

$x_{1}0,x_{2}0.$

a$)$ Graph the feasible region and identify all corner point feasible solutions. $(20$ points$)$

b$)$ Identify the optimal solution and the optimal objective function value. $(20$ points$)$

Q$-2)(60$ points$)$

Consider a firm that produces a type of highly expensive shoes at two plants located in a particular country. Customers $($e$.$g$.,$ retailers$)$ from three neighbor countries have placed orders from the firm. The table below shows unit price charged for shoes shipped to each of those customers $($the assumption is: unit prices vary due to differences in shipping costs, customs quota, etc.$)$

The number of units that will be produced at each plant and the number of units ordered by each customer are also given in the table.

$\backslash$table$[[$To$,$Unit prices,$],[$Customer$,\backslash$table$[[$Customer$],[1$