The Big Max grocery store sells three brands of milk in half$-$litre cartons$$its own brand, a local dairy brand, and a national brand. The profit from its own brand is $$0\mathrm{.}97$ ?a carton, the profit from the local dairy brand is $$0\mathrm{.}83$ ?per carton, and the profit from the national brand is $$0\mathrm{.}69$ ?per carton. The total refrigerated shelf space allotted to half$-$litre cartons of milk is $3\mathrm{.}5$ ?square metres per week. A half$-$litre carton takes up about $100$ ?square centimetres of shelf space. The store manager knows that they always sell more of the national brand than the local dairy brand and their own brand combined each week, and they always sell at least three times as much of the national brand as their own brand each week. In addition, the local dairy can supply only $10$ ?dozen cartons per

week. The store manager wants to know how many half$-$litre cartons of each brand to stock each week to maximize profit. Formulate and solve a linear programming model for this problem.

If Big Max could increase its shelf space for half$-$litre cartons of milk, how much would profit increase per carton?

If Big Max could get the local dairy to increase the amount of milk it could supply each week, would it increase profit?

Big Max is considering discounting its own brand to increase sales. If it does so$,$ ?it would decrease the profit margin for its own brand to $$0\mathrm{.}86$ ?per carton but it would cut the demand for the national brand relative to its own brand in half. Discuss whether or not the store should implement the price discount.