Computational experience with an algorithm for the lock box problem

The lock box problem involves the location of post office boxes within a company's distribution area. Customer remittances are mailed to these boxes and the checks are processed by a local bank. The problem is to locate the boxes in a way that will minimize processing cost and the opportunity costs associated with the remittances while in transit (float costs).

For m potential lock box locations and n customer groups, the problem can be formulated as a zero-one integer programming problem with mn + n variables and m + n constraints. The problem, however, can be partitioned in a way that results in a zero-one integer programming problem with only m variables. Once values have been established for these m variables, values for the remaining mn zero-one variables can be determined by a trivial process. Thus the problem reduces to determining values for the m zero-one variables. This is accomplished by an implicit enumeration procedure. Computational results are reported for problems involving up to 5050 variables.