$2.$ Modifications of CFLP $(40$ points$)$ Write a modified formulation of CFLP with the following

changes. If you introduce any new notation, define it clearly. Explain your modification in words.

$($a$).(10$ points$)$ Shipping each item from a supplier to warehouse j incurs a unit cost $!.($The cost may be

different for each j$.)$

$($b$).(10$ points$)$ Processing each item at warehouse j incurs a unit cost $!.($The cost may be different for

each j$.)$

$($c$).(10$ points$)$ There is a fixed cost $"!$to ship items from warehouse j to customer i$.($The cost is

independent of the quantity shipped but may be different for each i and j$.)$

$($d$).(10$ points$)$ A fixed capacity$-$expansion cost $!$ is incurred if the demand served by facility j exceeds a

threshold $!.($Hint: you may have to introduce new decision variables to represent whether facility j needs

an expansion.$)$