a$)$ Describe any $3$ assumptions of a Linear Programming model.

$($b$)$ A manufacturer produces two types of fruit juice for filling and distributing, the X$-$S fruit juice and the Super fruit juice. A batch of X$-$S costs $$375$ to manufacture and a container of the Super costs $$420.$ The manufacturer wishes to establish the weekly production plan that minimizes cost. Production of these products is limited to processing, testing and material.

Each batch of X$-$S requires $4$ hours processing whereas each batch of Super requires $2$ hours of processing. Further each batch of X$-$S requires $4$ hours of testing compared to $6$ hours of testing for a batch of Super. A batch of X$-$S and a batch of Super each require $1$ litre of material.

Processing and testing have a maximum of $100$ and $180$ hours available respectively and the total material available is $40$ litres. Because of a distribution agreement, the sales of X$-$S are limited to a weekly maximum of $20$ batches and to honour an agreement with a loyal distributor, at least $10$ batches of Super must be sold each week.

construct and graphically solve an appropriate linear programming model for the question above