An Indefinite - Quadratic - Programming Model Fora Continuous - Production Problem

A model is presented for a problem of scheduling the lengths of N production periods on one machine, which will manufacture q products. The problem is to choose production periods so as to minimize the sum of inventory costs for the q products in the presence of given demands. Mathematically, the problem is one of minimizing an indefinite quadratic function subject to linear constraints. A numerical example concludes the paper. 1. Definitions Suppose there are q products, and these have sales rates SI and production rates PI> i= 1, ... , q. The products are to be produced on one machine according to a sequence denoted by a, a 2N-tuple of integers aI' a2, ... , a2N, such that at the jth production period a product al' ° ~ al ::::;; q, is being produced. If al = 0 then the machine is idle. Given the production sequence a, the activity of the machine can be represented by an activity matrix A = [al}] such that al} is the rate of increase of product i in period j. The elements of A are clearly given by where Ol,k denotes the Kronecker delta. Finally, a convention is adopted that the machine is idle after each production period, i.e. a2j = 0; j = 1, 2, ... , N, (1.2) a2}+ I =j::. 0; j = O, 1, 2, ... , N 1. (1.3) This allows the possibility of enforcing a changeover-time constraint. With two products and six production periods a machine schedule is shown in fig. 1. In this case the sequence a is given by (1, 0, 2, 0, 1, 0). There are three production periods and three idle periods. The activity matrix takes on the form ( PI-SI -sI, -SI -SI PI-SI -SI ). -S2 -S2 P2-S2 -S2 -S2 -S2 PROGRAMMING FOR CONTINUOUS-PRODUCTION PROBLEM 245