Leasing Warehouse Space. Cox Cable Company needs to lease warehouse storage space for five months at the start of the year. Cox knows how much space will be required in each month, and the company can purchase a variety of lease contracts to meet these needs. For example, Cox can purchase one-month leases in each month from January to May. The company can also purchase two-month leases in January through April, three-month leases in January through March, four-month leases in January and February, or a five-month lease in January. In total, there are fifteen possible leases the company could use. Cox must decide which leases to purchase and how many square feet to purchase on each lease. 

Since the space requirements differ month to month, it may be economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the monthly cost for leasing space for additional months is much less than for the first month, so it may be desirable to lease the maximum amount needed for the entire five months. Another option is the intermediate approach of changing the total amount of space leased (by adding a new lease and/or having an old lease expire) at least once, but not every month. Two or more leases for different terms can begin at the same time.
The space requirements (in square feet) are shown in the table below:

Leasing costs (in dollars per thousand square feet) are given in the next table:

The task is to find a leasing schedule that provides the necessary amounts of space at the minimum cost.
a. Determine the optimal leasing schedule. What is the optimal total cost and the corresponding schedule?
b. Describe the qualitative pattern in the solution.
c. Use the pattern in (b) to trace the effects of increasing the space required for January. How will the leasing schedule change? How will the total cost change?
d. For how much of a change in January’s requirement will the pattern persist?

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Month Jan Feb Mar Apr Required space 15,000 10,000 20,000 5,000 May 25,000