A note on due-date assignment and single machine scheduling with a learning/aging effect

This paper considers the learning/aging effect in an n job single machine scheduling problem with common due date. The objective is to determine the optimal common due date and the optimal sequence of jobs that minimizes a cost function in the presence of learning/aging effect. The cost function depends on the individual job earliness and tardiness values: i.e., Sigma(n)(j=1) {E([j]) + T([j])}. This is a well-known problem when the learning/aging effect is not considered and it is shown in earlier studies that there are more than one optimal sequence and optimal common due dates. It is shown in earlier studies that there are 2(r-1) optimal sequences to this problem if n is odd, and 2(r) optimal sequences if n is even. The value of r is (n + 1)/2 if n is odd, and the value of r is n/2 if n is even. In this paper, we derive two bounds B(alpha) and B(alpha)* for the learning index alpha. We show that when B(alpha) < alpha < 0, then the optimal sequence is unique and provide an O(n log n) algorithm to obtain this unique optimal sequence and the optimal common due date. We also show that when alpha <B(alpha)*, the optimal sequence is obtained by arranging the longest job in first position and the rest of the jobs in SPT order. Similarly, we derive two bounds A alpha and A(alpha)* for the aging index alpha. We show that when 0<alpha <A(alpha), then the optimal sequence is unique and provide an O(n log n) algorithm to obtain this unique optimal sequence and the optimal common due date. We also show that when alpha > A(alpha)* the optimal sequence is obtained by arranging the jobs in LPT order. We also present a numerical example for ease of understanding. (C) 2008 Elsevier B.V. All rights reserved.