Cycling of Expenses

One of the enjoyable features of mathematical research at the old Bell Laboratories was the great variety of problems that would "walk in the door."They came not just from the rest of the Labs, but from all over the Bell System. Not all were interesting, but the variety was fine for any mathematician who liked more than just one area of mathematics.Any narrow picture of "applicable" mathematics was quickly dispelled.

Sometime in 1958, my long-term colleague Ed Gilbert and I were visited by Mr. A. M. Marca of theWestern Electric Company, which was at that time the manufacturing arm of the Bell System. He wanted to know if the method they used for solving a particular accounting problem could be simplified, and if he could, in fact, be assured that it would always work.

Here was the situation. Amanufacturing organization has a large number of producing departments, and it is easy to assign the costs of running such departments to the products they manufacture. There are also service departments, such as stock rooms, food service, janitorial service, and secretarial service. The costs of running these have to be allocated somehow to the products of the manufacturing departments. At first glance, this does not appear to be much of a problem: Manufacturing department A has a certain number of employees, and this determines its share of the cost of running the restaurant.

But the problem is more difficult: The secretaries also eat, and the management of the restaurant gets its letters typed. In other words, the service departments service not only the producing departments, but they also service each other! What is a fair way of taking this into account?