J Intell Manuf (2010) 21:843–851 DOI 10.1007/s10845-009-0260-3 Parallel machine scheduling problem to minimize the earliness/tardiness costs with learning effect and deteriorating jobs M. Duran Toksarı · Ertan Güner Received: 5 May 2008 / Accepted: 2 March 2009 / Published online: 19 March 2009 © Springer Science+Business Media, LLC 2009 Abstract The focus of this work is to analyze parallel machine earliness/tardiness (ET) scheduling problem with simultaneous effects of learning and linear deterioration, sequence-dependent setups, and a common due-date for all jobs. By the effects of learning and linear deterioration, we mean that the processing time of a job is defined by an increas- ing function of its starting time and a decreasing function of the position in the sequence. We develop a mixed integer programming formulation for the problem and show that the optimal sequence is V-shaped: all jobs scheduled before the shortest jobs and all jobs scheduled after the shortest job are in a non-increasing and non-decreasing order of processing times, respectively. The developed model allows sequence- dependent setups and sequence-dependent early/tardy pen- alties. The illustrative example with 11 jobs for 2 machines and 3 machines shows that the model can easily provide the optimal solution, which is V-shaped, for problem. Keywords Earliness/tardiness · Learning effect · Deterioration jobs · V-shaped property · Parallel machine · Sequence-dependent setup time · Common due date Introduction In this paper, we have considered parallel machine sched- uling problem with learning and deterioration effects, and a M. D. Toksarı (B ) Industrial Engineering Department, Erciyes University, Kayseri, Turkey e-mail: dtoksari@erciyes.edu.tr E. Güner Industrial Engineering Department, Gazi University, Ankara, Turkey common due-date for all jobs where the objective is minimization of the weighted sum of earliness and tardiness penalties of jobs. The weighted earliness tardiness problem arises in a simplified JIT production environment (Mondal and Sen 2001). In a JIT scheduling environment, a job that completes early must be held in finished goods inventory until its due date, while a job that completes after its due date may cause a customer to shut down operations. Therefore, an ideal schedule is one in which all jobs finish exactly on their assigned due dates (Baker 1994). In the ET literature, the types of penalty cost functions used in the objective function include linear, nonlinear, uni- form penalties, and penalty differences among jobs. Penalty differences between earliness and tardiness are important assumption. They can be classified on four basic forms as job dependent (Baker 1990; Bank and Werner 2001; Mosheiov and Shadmon 2001; Zhu and Heady 2000; Hendel and Sourd 2006), unequal (Zeng et al. 1993; Ventura et al. 2005), equal and penalty cost with job dependent proportional weight (Sun and Wang 2003; Arkin and Roundy 1991; Szwarc 1993; Bauman and Jozefowska 2006). In determining penalty func- tions, different penalty functions are more appropriate in that earliness and tardiness may be undesirable at the same pro- portion. Almost all ET articles assume that setup time is either immaterial or sequence-independent. However, sequence- dependent setups are very important to some industries. For instance, in the chemical industry, machine setup times vary considerably as a function of the sequence of processing jobs (Zhu and Heady 2000). A review of the literature indicates that only a few ET studies (Zhu and Heady 2000; Colemon 1992; Sourd 2005) address the sequence-dependent setups. The article published by Zhu and Heady (2000) was focused on ET problem with multi machines (parallel machines) and sequence-dependent setups. The main difference between the 123