Single-machine group scheduling with resource allocation and learning effect q Zhanguo Zhu a,d,⇑ , Linyan Sun a , Feng Chu b , Ming Liu c a School of Management, State Key Laboratory for Mechanical Manufacturing Systems Engineering, The Key Lab of the Ministry of Education for Process Control and Efficiency Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi Province 710049, PR China b Laboratoire d’informatique, biologie intégrative et systèmes complexes (IBISC), FRE CNRS 3190, Université d’Evry Val d’Essonne, 40 Rue du Pelvoux, CE1455 Courcouronnes, 91020 Evry Cedex, France c School of Economics & Management, Tongji University, Shanghai 200092, PR China d Université de technologie de Troyes, Institut Charles Delaunay, FRE CNRS 2848, Laboratoire d’optimisation des systèmes industriels (LOSI), 12 rue Marie Curie – BP 2060, 10010 Troyes Cedex, France article info Article history: Received 25 August 2010 Received in revised form 20 October 2010 Accepted 21 October 2010 Available online 30 October 2010 Keywords: Scheduling Group technology Resource allocation Learning effect abstract This paper addresses single-machine scheduling problems under the consideration of learning effect and resource allocation in a group technology environment. In the proposed model of this paper the actual processing times of jobs depend on the job position, the group position, and the amount of resource allo- cated to them concurrently. Learning effect and two resource allocation functions are examined for min- imizing the weighted sum of makespan and total resource cost, and the weighted sum of total completion time and total resource cost. We show that the problems for minimizing the weighted sum of makespan and total resource cost remain polynomially solvable. We also prove that the problems for minimizing the weighted sum of total completion time and total resource cost have polynomial solutions under cer- tain conditions. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, by relaxing the classical assumption of constant job processing times, a lot of work has been done on models that reflect variable job processing times in scheduling problems. The assumption of fixed job processing times are indeed not appropri- ate in many situations of scheduling systems due to some realistic settings such as the skill levels of workers are influenced by learn- ing effects, resource allocation constraints, and group technology. The skill levels of workers can be improved due to the repetition of similar jobs/tasks, and the improvement usually brings a decrease of the actual job/task processing times. This process is known as the learning effect in the literature. Wright (1936) first discovered the learning effect in the aircraft industry and many researchers have confirmed the existence of learning effects in pro- duction environments, such as Cochran (1960), Bevis, Finniear, and Towill (1970), Venezia (1985). Moreover, Biskup (1999), Cheng and Wang (2000) were the pioneers that initiated the study of learning effects in the scheduling environment. Biskup (1999) discussed two objectives of minimizing the deviation from a common due date and minimizing the sum of flow times based on the well- known learning model in which the actual processing time of job j if it is scheduled in position r in the sequence is p jr = p j r a , where a is the negative learning index. Cheng and Wang (2000) dealt with the learning effect in scheduling problem with a model based on the volume-dependent processing time function. They proved that the scheduling problem for the objective to minimize the maxi- mum lateness is NP-hard in the strong sense and proposed two heuristics. Since then, much work has been done on scheduling problems of other environments, involving multi-machines, differ- ent learning effect models, and so on. Mosheiov and Sidney (2003) considered scheduling problems with job-dependent learning ef- fects. Biskup and Simons (2004) studied scheduling problems with autonomous and induced learning, and provided a polynomially bound solution procedure. Kuo and Yang (2006a) introduced an- other learning effect model, based on the total normal processing time of the previous scheduled jobs, to the single-machine sched- uling problem for the objective to minimize the total completion time. Wang and Cheng (2007a, 2007b) analyzed the simultaneous effects of deterioration and learning on single-machine scheduling based on different practical examples, respectively. Mosheiov (2001) studied flow-time minimization on parallel identical machines and provided polynomial solutions for this problem. Lee and Wu (2004), Eren and Güner (2008) considered the learning effect in a two-machine flow shop for the objectives of minimizing the total completion time and minimizing a weighted sum of total 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.10.012 q This manuscript was processed by Area Editor T.C. Edwin Cheng. ⇑ Corresponding author at: School of Management, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi Province 710049, PR China. E-mail addresses: zhanguo.zhu@utt.fr (Z. Zhu), lysun@mail.xjtu.edu.cn (L. Sun), feng.chu@ibisc.univ-evry.fr (F. Chu), minyivg@gmail.com (M. Liu). Computers & Industrial Engineering 60 (2011) 148–157 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie