Cyclic Just-In-Time Sequences Are Optimal

Consider the Product Rate Variation problem. Given n products 1, ..., i, ..., n, and n positive integer demands d1, ..., di, ..., dn. Find a sequence α = α1, ..., αT , T = ∑n i=1 di, of the products, where product i occurs exactly di times that always keeps the actual production level, equal the number of product i occurrences in the prefix α1, ..., αt, t = 1, ..., T , and the desired production level, equal rit, where ri = di/T , of each product i as close to each other as possible. The problem is one of the most fundamental problems in sequencing flexible just-in-time production systems. We show that if β is an optimal sequence for d1, ..., di, ..., dn, then concatenation βm of m copies of β is an optimal sequence for md1, ...,mdi, ...,mdn.