New Logistics, a provider of warehousing, is facing a decision regarding the amount of warehousing capacity to lease from a space provider $($such as the owner of an industrial building$)$ for the upcoming $2$ years. The management forecasts that the company will handle an annual demand of $1,000$ units in the first year. In the second year, the annual demand will increase to $1,200$ units with a probability of $50\%$ and will decrease to $800$ units with a probability of $50\%.$ Each unit of demand will require $1$ square feet of warehouse space.

New Logistics receives a revenue of $$1\mathrm{.}5$ for each unit of demand. The management is considering an option of signing a $2-$year lease with the space provider, by which New Logistics pays $x annually for each square feet of space contracted. Assume that the excessive demand beyond the leased capacity, if there is any, will have to be turned down.

New Logistics is evaluating two alternatives: $(1)$ Signing a lease with a constant amount of $1,000$ square feet over the $2$ years; $(2)$ Signing a lease with a constant amount of $1,100$ square feet over the $2$ years.

i$)$ Draw a graph of decision tree for New Logistics.

ii$)$ Assume x $=1\mathrm{.}2.$ Then, determine the optimal decision $($on choosing between the two alternatives$)$ for New Logistics and the resulting profit over the two years.

iii$)$ Now treat x as a variable. Then, determine the range for the value of x within which choosing the first alternative $($signing a lease with a constant amount of $1,000$ square feet over the $2$ years$)$ is optimal for New Logistics.