Problem $2(45$ pts$)$

You've looking to create a business manufacturing tablets for children. You want to manufacture low$-$end and high$-$end tablets, to ensure you reach a maximum of customers. You expect the demand for types of tablets to be about the same: $900$ each month. You have found two potential locations that could manufacture both types of tablets for you: Syracuse, NY and Springfield, MA$.$ Each month, the plant in Syracuse has the capacity to produce at most $800$ tablets, while the plant in Springfield could produce up to $1,000$ tablets. At the Syracuse plant, one low$-$end tablet would require $2$ hours of skilled labor; a high$-$end tablet would also require $2$ hours of skilled labor. At the Springfield plant, one low$-$end tablet would require $3$ hours of skilled labor, while a highend tablet would require $4$ hours of skilled labor. Each plant has a capacity of $4,000$ hours of skilled labor. You estimate you could make a profit of $$600$ for each low$-$end tablet made in Syracuse, $$800$ for each high$-$end tablet made in Syracuse, $$1,000$ for each low$-$end tablet made in Springfield, and $$1,300$ for each high$-$end tablet made in Springfield.

a$)$ Formulate a linear program to determine how many of each tablet type should be produced monthly at each plant to maximize profit. Describe in words what each constraint written in the LP represents. $(10$ pts$)$

b$)$ Apply the Simplex method to the LP found in part a through $3$ iterations $($i$.$e$.,4$ tableaus$).$ Only for the Simplex method $($not for Excel Solver$)$: make your demand constraints $"$at most" constraints. $(10$ pts$)$

c$)$ Solve the LP from part a using Excel Solver. Provide a screenshot of the solved LP$,$ as well as of the Excel Solver window $($pre$-$solved$).(5$ pts$)$

d$)$ You think you may have overestimated the demand for the low$-$end tablets. Without re$-$solving the LP$,$ if the demand for the low$-$end tablets was $850,$ by how much will that impact the optimal objective value found in part $c?$ Explain and show work $($including screenshots of any reports used$).(10$ pts$)$

e$)$ If Syracuse was able to produce $10$ tablets more at a higher cost, how much would you be willing to pay for those extra tablets $($without re$-$solving the LP$)?$ Explain and show work $($including screenshots of any reports used$).(10$ pts$)$