Minimizing the Weighted Sum of Quadratic Completion Times on a Single Machine Federico Della Croce, Wlodzimierz Szwarc? Roberto Tadei, Paolo Baracco, and RdTaele Di Tullio zyxwv Dipartimento di Automatica ed Informatica, Politecnico di Torino, Torino, Italy, and %chool zyxwvuts o f Business Administration, University o f Wisconsin,Milwaukee, Wisconsin, zyx USA zy This article discusses the scheduling problem of minimizing the weighted sum of quadratic completion times on a single machine. It establishes links between orderings of adjacent and nonadjacent jobs that lead to a powerful branch and bound method. Computational results show that this method clearly outperforms the state ofthe art algorithm. zyx 0 1995 John Wiley &Sons. Inc. I, INTRODUCTION This article considers the problem of scheduling n jobs on a single machine to minimize the weighted sum of quadratic completion times of the jobs. Then the objective function is defined byf( S) zyxwvuts = zyxwvu I:=, ckkc: where zyxwv Ck is the completion time ofjob zyx k in schedule S a n d ck and pI are the weight and processing time of job k respectively. Schedule S is a processing arrangement of jobs 1, 2, . . . , n. If S = 1, 2, . . . , n then C, = C h=, pk. Towsend first formulated this problem and presented a branch and bound method to solve it. Several improvements of Towsend’s method are discussed in [ 1-31 (the last reference deals with a more general function C ( ckCi + ciCk)), but are able to handle only small size problems where n zyxwvuts I 20. Szwarc et al. [4] develop a powerful decomposition procedure based on a special type of precedence relations of adjacent jobs, called adjacent orderings, and solve 19 1 out of 200 problems of sizes 15 I n I 100 on a PC bypassing enumeration. Szwarc [ 51 generalizes the findings of [ 41 to a wider class of quadratic models for which he designs a general branch and bound algorithm. He reports computational experience for Towsend’s model on 60 less decomposable problems for 20 I n I 50 where his lower bound greatly outperforms that of Towsend. References [2] and [ 31 examine nonadjacent orderings (when the pairs of jobs may not be adjacent) and identify conditions where job i appears earlier in the schedule than job j. These conditions, however, occur for a limited number of pairs i, j. By comparison, adjacent orderings occur for every pair ofjobs. In our work we establish links between adjacent and nonadjacent orderings. As a result, a criterion is provided when some adjacent orderings turn nonadjacent. This important finding Ieads to a number of properties that drastically improve Szwarc [ 51 branch and bound procedure for less decomposable problems. The advantage of our method over that of [5] is demonstrated by solving 160 problems divided into three categories, where the maximum sizes n are 400, 150 and 100 respectively. This article is organized as follows. Section 2 discusses some known properties of the model. In Section 3 we present our results and their implications. Section 4 outlines and Naval Research Logistics, Vol. 42, 1263-1270 (1995) Copyright 0 1995 by John Wiley & Sons, Inc. CCC 0894-069)</95/08 1263-08