AbstractAn investigation was carried out to establish a method of determining the allocation between certain factories of a known production requirement so as to minimize the total expense of production. Each factory was made up of a number of departments or services whose total expense varied non-linearly with production level.The method was to divide the production range of each department in each factory into regions of approximate linear expense, and consider the associated total company expense connected with the optimum allocation for each possible group of regions (one region in each department). In theory the simplex technique could be applied to each group to find its optimum allocation, but as the number of groups is prodigious for quite simple problems this is not practicable. (In the actual problem for which the method was produced there was of the order of 100,000 groups.)A procedure was devised using the transportation technique together with various lemmas proved in the text and feasibility considerations, by which it was possible to reduce rapidly the number of groups needed to be considered in detail to just a few.Other problems should be amenable to this method provided they are similar to the one above in two respects.(1) The departmental variation of expense with production from one region to the next is in general continuous with decreasing rate of change of expense, i.e. the variation is concave.(2) It should be possible to put the requirements into one of the units of capacity usage in which there are important physical or maximum capacity restrictions.In addition, the method is particularly suitable when the number of factories is not large.As the method required repeated applications of the transportation model, a technique was devised for obtaining an optimum solution to this model more quickly than by the usual techniques. This technique will be described in another paper.An example of a whole problem is given after the Appendix. A full solution of this problem is shown in which occur most of the points dealt with in the text.The problem for which the method was devised, was one of minimizing the total expense of production of twenty common products between three factories. Each factory had seven departments or services, in most of which two regions were required to describe satisfactorily a department's variation of expense with production, in linear terms. A complete solution was obtained in a few days using the method described without recourse to an electronic computer (not counting the initial computations of costs, etc. required).