Mixed Integer Binary LP$.$ The Gadget Phone Company makes a type of cellular phone and just received an order for $1,500$ phones. The phone can be made on either of two machines, or both of them, that each have limited capacities to make the phone. Machine $1$ can make up to $1,000$ units, and Machine $2$ can make up to $800$ units of the product. Machine $1$ has a variable cost of $$100$ per phone and a fixed setup cost of $$800.$ Machine $2$ has a variable cost of $$200$ per phone and a fixed setup cost of $$1,200.$ How many phones should be made on each machine to minimize the total cost of producing the $1,500$ phones?

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a$.$ Formulate this problem as a mixed integer LP problem.

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Minimize:$$

X$1+$ X$2+$ Y$1+$ Y$2$

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Subject to:

X$1+$ X$2=$ X$1-$ Y$1$ X$2-$ Y$2$ X$1$ X$2$

Y$1$binary$,$ Y$2$binary

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b$.$ How many phones should be produced on each machine to minimize cost? What is the total cost of production?

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Machine $1$: Produce phones.

Machine $2$: Produce phones.

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Minimum cost $=$ $

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c$.$ What are the binding constraints in the optimal solution?

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Requirement to make $1,500$ units $($Click to select$)$ Yes No Capacity of Machine $1($Click to select$)$ Yes No Capacity of Machine $2($Click to select$)$ Yes No

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