$(16$ points. This problem does not require the use of Excel.$)$

During the Gulf War in $1991,$ the Terraco Motor Company produced an all$-$terrain vehicle for the US military. The company is now planning to sell a modified version of the vehicle to the public. It has five potential plants that can manufacture the vehicle and four regional distribution centers. The company is unsure of public demand for the vehicle, so it needs to decide on which plant$($s$)$ to open. When a plant is opened, a fixed cost needs to be paid independently of how many vehicles are produced at the plant. There are also variable unit transportation costs between the plants and the distribution centers. All the demand at the distribution centers needs to be satisfied. Finally, the plants do not have any storage space, so all the vehicles produced need to be sent to the distribution centers. The table below summarizes the problem's data.

$\backslash$table$[[$Plant$/$Distribution Center,Unit Shipping Costs,Capacity,Fixed Cost$],[,1,2,3,4,],[1,5171,2024,3407,6704,12872,216361],[2,1839,4710,746,3601,18530,84664],[3,1233,7277,4186,5276,14040,188770],[4,3005,2403,5119,2887,10595,110929],[5,4562,5191,2608,3142,16045,91071],[$Demand$,6117,14748,8036,10250,,]]$

$($a$)(4$ points$)$ Express the problem's decision variables.

$($b$)(4$ points$)$ Express the problem's objective.

$($c$)(4$ points$)$ Express the constraint corresponding to the capacity of plant $1.$

$($d$)(4$ points$)$ Express the constraint corresponding to the demand of distribution center $2.$