Problem $1$: Mixed Binary Integer Programming

The Toys$-$R$-4-$U Company has developed two new toys for possible inclusion in its product line for the upcoming Christmas season. Setting up the production facilities to begin production would cost $$55,000$ for toy $1$ and $$45,000$ for toy $2.$ Once these costs are covered, the toys would generate a unit profit of $$15$ for toy $1$ and $$20$ for toy $2.$ The company has two factories that are capable of producing these toys. However, to avoid doubling the start$-$up costs, just one factory would be used, where the choice would be based on maximizing profit. For administrative reasons, the same factory would be used for both new toys if both are produced. Toy $1$ can be produced at a rate of $50$ per hour in factory $1$ and $40$ per hour in factory $2.$ Toy $2$ can be produced at the rate of $40$ per hour in factory $1$ and $25$ per hour in factory $2.$ Factories $1$ and $2,$ respectively, have $500$ hours and $700$ hours of production time available before Christmas that could be used to produce these toys. It is not known whether these two toys would be continued after Christmas. Therefore, the problem is to determine how many units $($if any$)$ of each new toy should be produced before Christmas to maximize the total net profit.

$($a$)$ Formulate algebraically a mixed binary integer programming $($BIP$)$ model for this problem that the number sold should not exceed the number that can be produced. Define the decision variables, objective function, and constraints.

$($b$)$ Formulate and solve this mixed BIP model on a spreadsheet and solve using Excel$$s Solver $($Provide the corresponding $$Excel Spreadsheet$$ and the $$Answer Report$).$ Include $$managerial statements$$ of the optimal decision $($i$.$e$.,$ describe verbally the results$).$