Cyclic Just-In-Time Sequences Are Optimal

Abstract

Consider the Product Rate Variation problem. Given n products 1,...,i,...,n, and n positive integer demands d ₁,..., d_i,...,d_n. Find a sequence α=α₁,...,α_T, T = ∑ ⁿ_i=1 d _i, of the products, where product i occurs exactly d _i times that always keeps the actual production level, equal the number of product i occurrences in the prefix α₁,..., α_t, t=1,...,T, and the desired production level, equal r _i t, where r _i=d_i/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 d ₁,...,d_i,...,d_n, then concatenation β^m of m copies of β is an optimal sequence for md ₁,..., md_i,...,md_n.