ORIGINAL ARTICLE Scheduling problems with the nonlinear effects of learning and deterioration M. Duran Toksarı & Ertan Güner Received: 19 October 2008 / Accepted: 10 March 2009 / Published online: 31 March 2009 # Springer-Verlag London Limited 2009 Abstract In this paper, we present both nonlinear job deterioration and nonlinear learning which exist simultaneous- ly. Job deterioration and learning co-exist in many realistic scheduling situations. By the effects of learning and deterio- ration, we mean that the processing time of a job is defined by the increasing function of its execution start time and position in the sequence. The following objectives are considered: single-machine makespan and sum of completion times (square) and the maximum lateness. For the single-machine case, we derive polynomial time optimal solutions. For the case of an m-machine permutation flowshop, we present polyno- mial time optimal solutions for some special cases of the problems to minimize makespan and total completion time. Keywords Scheduling . Single machine . Flowshop . Position-based learning effect . Deterioration jobs 1 Introduction Although scheduling with deteriorating jobs and learning effect has been widely investigated, scheduling research has seldom considered the two phenomena simultaneously. This paper addresses some scheduling problems, the nonlinear effects of learning and deterioration. A scheduling problem is very important for a manufacturing system. So numerous schedul- ing problems have been studied by researchers. In the classical scheduling theory, job processing times are considered to be constant. In practice, however, we often encounter a setting in which processing times increase or decrease over time. Most researchers assumed that the time of task is independent from learning of worker(s) for repetition jobs. However, in many realistic settings, workstations improve continuously as a result of repeating the same or similar activities. Thus, the processing time of a job is shorter if it is scheduled later, rather than in the sequence. Mosheiov [1] determined that this phenomenon is known in the literature as a “learning effect.” The learning effect has been examined on scheduling problems by many researchers in recent years [1–8]. Biskup [2] and Cheng and Wang [3] were among the pioneers that introduced the concept of learning to schedul- ing problems. Biskup [2] assumes that production time of a single item under learning effect decreases depending on the number of repetitions of its production. There is a growing interest in the literature to study scheduling problems of deterioration jobs, i.e., jobs whose processing times are increasing functions of their starting times. Mosheiov [9] pointed out another case searching for an object under worsening weather or growing darkness. Although Gupta and Gupta [10] provided an example of steel rolling miles where the temperate of an ingot, while waiting to enter the rolling machine, drops below a certain level, required the ingot to be reheated before rolling. Kunnathur and Gupta [11] gave a fire fighting example where the time and effort required to control a fire increases if there is a delay in the start of the fire fighting effort. Scheduling problems with deterioration jobs have re- ceived increasing attention in recent years. Scheduling problems with time-dependent processing times have been initiated independently by Gupta and Gupta [10] and Browne and Yechiali [12]. They proposed models which depend on the processing time function. Alidae and Womer Int J Adv Manuf Technol (2009) 45:801–807 DOI 10.1007/s00170-009-2016-9 M. Duran Toksarı (*) Industrial Engineering Department, Erciyes University, Kayseri, Turkey e-mail: dtoksari@erciyes.edu.tr E. Güner Industrial Engineering Department, Gazi University, Ankara, Turkey