In problem $1,$ the manager must decide whether to lease warehouse space for the coming three years and the quantity to lease. The manager anticipates uncertainty in demand and spot prices for warehouse space over the coming three years. The long$-$term lease is cheaper but the space could go unused if demand is lower than anticipated. The long$-$term lease may also end up being more expensive if future spot market prices come down. The manager is considering three options:

$1.$ Get all warehousing space from the spot market as needed.

$2.$ Sign a three$-$year lease for a fixed amount of warehouse space $(100,000$ sq$.$ ft$.)$ and get additional requirements from the spot market.

$3.$ Sign a flexible lease with a minimum charge $($$$10,000$ upfront$)$ that allows variable usage of warehouse space up to a limit $($flexible between $60,000$ to $100,000$ sq$.$ ft$.$ of warehouse space at $$1$ per sq$.$ ft$.),$ with additional requirements from the spot market. This means Trips Logistics must pay $$60,000$ per year for the first $60,000$ sq$.$ ft$.$ and can then use up to $40,000$ sq$.$ ft$.$ on demand at $$1$ per sq$.$ ft$.$

We now discuss how the manager can evaluate each decision, taking uncertainty into account.

One thousand square feet of warehouse space is required for every $1,000$ units of demand, and the current demand at Trips Logistics is for $100,000$ units per year. The manager forecasts that from one year to the next, demand may go up by $20$ percent, with a probability of $0\mathrm{.}5,$ or go down by $20$ percent, with a probability of $0\mathrm{.}5.$ The probabilities of the two outcomes are independent and unchanged from one year to the next.

The general manager can sign a three$-$year lease at a price of $$1$ per square foot per year. Warehouse space is currently available on the spot market for $$1\mathrm{.}20$ per square foot per year. From one year to the next, spot prices for warehouse space may go up by $10$ percent, with probability $0\mathrm{.}5,$ or go down by $10$ percent, with probability $0\mathrm{.}5.$ The probabilities of the two outcomes are independent and unchanged from one year to the next.

The general manager believes that prices of warehouse space and demand for the product fluctuate independently. Each unit Trips Logistics handles results in revenue of $$1\mathrm{.}22,$ and Trips Logistics is committed to handling all demand that arises. Trips Logistics uses a discount rate of k $=0\mathrm{.}1$ for each of the three years.