Lockbox Problem

A major credit card company $($call it $$MasterDebt$)$ receives checks from all different regions in the country on a daily basis. Once these checks are mailed, the time a check spends in the mail $($called $$float$)$ creates loss for MasterDebt, for as soon as they receive the checks they can cash them and collect interest on the funds. MasterDebt can make $15\%$ annual interest on their cash holdings $($since that$$s what they charge their customers in credit card interest$).$

In order to reduce the amount of float loss for these checks, MasterDebt is considering opening $$Lockbox$$ locations across the country where the checks can be received and processed. The locations and the projected annual cost of operations $($labor and overhead$)$ at each location are as follows:

ANNUAL COSTS $($X$$1000)$

Sacramento Denver Chicago Dallas New York Atlanta

$256035353035$

The average number of days that a check would float between each region and each lockbox location is shown in the following chart.

AVERAGE FLOAT DAYS

REGION Sacramento Denver Chicago Dallas New York Atlanta

Central $422233$

Mid$-$Atlantic $643422$

Midwest $323254$

Northeast $642523$

Northwest $235467$

Southeast $743242$

Southwest $236276$

The average daily payments received from each region are shown in the following chart $($in thousands of dollars$).$

REGION Payments

Central $45$

Mid$-$Atlantic $65$

Midwest $50$

Northeast $90$

Northwest $70$

Southeast $80$

Southwest $60$

The annual interest lost can be computed for each region$-$lockbox location by taking the average daily payments times the float time and multiplying by fifteen percent. For example, if payments from the Central region are sent to New York, then on any given day there is an average of $$135,000$ of undeposited checks, which costs MasterDebt $$20,250$ annually in interest.

Where should MasterDebt open lockbox locations in order to save the most money each year?

Which regions should be assigned to those lockbox locations? $($This will be implemented by providing different return addresses on the payment envelopes that are sent to the different regions.$)$