$3.$ Assigning Machines to Tasks with Cost Minimization

A manufacturing plant has four machine categories assigned to five tasks. Each machine category has a different assignment cost per task, and some categories have restrictions on which tasks they can be assigned.

$($a$)$ Formulate a transportation model to minimize assignment costs. Include assignment restrictions. Use the table below to record costs and specify restricted assignments.

$\backslash$table$[[$Task Type,Cat $1$ Cost,Cat $2$ Cost,Cat $3$ Cost,Cat $4$ Cost$],[$Task $1,10,5,15,20],[$Task $2,2,10,5,15],[$Task $3,3,15,14,13],[$Task $4,15,2,7,-],[$Task $5,9,4,15,8]]$