Theory and Methodology Complexity of parallel machine scheduling with processing-plus- wait due dates to minimize maximum absolute lateness T.C. Edwin Cheng a, * , Mikhail Y. Kovalyov b a Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, People's Republic of China b Institute of Engineering Cybernetics, National Academy of Sciences of Belarus, Surganova 6, 220012 Minsk, Belarus Received 1 March 1997; accepted 1 March 1998 Abstract We study the problem of scheduling n jobs on several parallel identical machines. An optimal combination of a job schedule and processing-plus-wait (PPW) due dates is to be determined so as to minimize the maximum absolute lateness. The problem is shown to be strongly NP-hard if the number of machines is variable and ordinary NP-hard if it is a constant greater than one. For the single machine case, the problem is shown to be solvable by a graphical approach in On log n time. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Parallel machine scheduling; Due date assignment 1. Introduction In this paper, we study the parallel machine scheduling problem with processing-plus-wait (PPW) due dates to minimize the maximum ab- solute lateness, which may be stated as follows. There are n independent non-preemptive jobs to be scheduled for processing on m identical parallel machines. It is assumed that each machine handles jobs from time zero onwards without idle time and at most one job at a time. Each job j has an integer processing requirement p j > 0 and it is assigned a PPW due date d j  kp j  d , where k P 0 and d are decision variables which have to be de- termined. Due to the no idle time assumption, a schedule is completely characterized by the se- quences of jobs on the machines. Given a schedule, the completion time C j of each job j is easily de- termined. Given a schedule and values of k and d , the lateness of job j is de®ned as L j  C j  d j . The objective is to ®nd an optimal schedule and opti- mal values of k and d so as to minimize the max- imum absolute lateness maxfjL j jg. Here and below we assume that each maximum and minimum is taken over all jobs j if not de®ned dierently. Motivation of this problem stems from the construction industry. A contractor plans to bid for n construction projects (i.e. jobs) from a de- veloper. The projects will be assigned to m teams of construction crew (i.e. machines) to handle. In European Journal of Operational Research 114 (1999) 403±410 * Corresponding author. Tel.: 852 2766 5216; fax: 852 2356 2682; e-mail: mscheng@polyu.edu.hk 0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 1 1 - 8