MODFLING PROBLEMS $($each worth $20$ points$)$

Note that in this part you are only asked to model the problem. Do NOT try to solve and find the optimal solution.

M$1)$ Metro Food Services Company delivers fresh sandwiches each morning to vending machines throughout the city. The company makes three kinds of sandwiches$-$ham and cheese, bologna, and chicken salad. A ham and cheese sandwich requires a worker $0\mathrm{.}45$ minutes to assemble, a bologna sandwich requires $0\mathrm{.}41$ minutes, and a chicken salad sandwich requires $0\mathrm{.}50$ minutes to make. The company has $960$ available minutes each night for sandwich assembly. Vending machine capacity is available for $2,000$ sandwiches each day. The profit for a ham and cheese sandwich is $$0\mathrm{.}35,$ the profit for a bologna sandwich is $$0\mathrm{.}42.$ and the profit for a chicken salad sandwich is $$0\mathrm{.}37.$ The company knows from past sales records that its customers buy as many ham and cheese sandwiches as the other two sandwiches combined, if not more so$,$ but customers need a variety of sandwiches available, so Metro stocks at least $200$ of each. Metro management wants to know how many of each sandwich it should stock to maximize profit. Please model this problem as a linear programming. Note that we assume the demand is large enough and all sandwiches in the machine will be sold daily.