Problem $1.$ Starting from the capactiated facility location model covered in class, model the following two problems:

$($a$)$ Modify the definition of the variable $x_{ij}$ to represent the fraction of customer jinJ $\text{'}$s demand that is served by facility$/$supplier iinI. How will the model change? Show a complete formulation.

$($b$)$ Using the model developed in part $($a$),$ show how the redundant, but strengthening, constraints covered in class will change.

$($c$)$ Modify the model developed in part $($a$)$ to develop another model that can help an organization decide on where each customer will get their demand served from assuming the entirety of a customer's demand has to be served by a single facility.