Design by Algorithm: A Mathematical Method of Designing Standard Assemblies for Minimum Manufacturing Cost

Summary: It is pointed out that in order to reduce manufacturing costs, it is often advantageous to make use of standard assemblies despite the waste incurred due to the non-use of parts of these standard assemblies in meeting specific requirements. A special prototype problem is considered based on that premise; it arose in practice, in the design for production of logic circuits. A problem is outlined which requires that 55 basic elements, with one to seven diode outputs each, be assembled into a number of standard assemblies, and with a minimum of waste of diode outputs. The problem considered is: Given the number required for each size for basic elements of sizes n, n-1,...,1. Further, for k = l, 2,..., n, a basic element of size k may be used instead of a basic element of size k or k-1..., or 1 but consequently incurring a waste proportional to 0, or 1,..., or k-1, respectively. Then the problem is to define a standard assembly in terms of the number and sizes of basic elements and the number of such standard assemblies necessary to fulfill the requirements, under the condition that the waste be a minimum. A simple algorithm is given for finding the make-up of the standard assembly. The Appendix gives a more rigorous mathematical generalization for the solution of uniqueness and minimality, and its proof; linear programming is used in this treatment.