Parallel system scheduling with general worker learning and forgetting David A. Nembhard a,1 , Frank Bentefouet b,n a Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, 209 Leonhard building, University Park, PA 16802, USA b Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, 244 Leonhard building, University Park, PA 16802, USA article info Article history: Received 25 January 2012 Accepted 17 May 2012 Available online 16 June 2012 Keywords: Unrelated tasks Scheduling Individual differences Worker performance Learning Forgetting abstract Learning and forgetting models available in the literature are generally not intended for math programming formulations, thus their incorporation within worker-task assignment problem models leads to non-linearity in the objective function, producing a Mixed Integer Nonlinear Program (MINLP) that is difficult to solve. Even though Dynamic Programming and other alternate approaches have been implemented to solve assignment problems in production systems with independent stations and independent jobs while maximizing output, such solutions are generally not optimal. In this paper, we develop results that reveal the form of the optimal solution, which allows us to solve the problem as a Mixed Integer Linear Program (MILP), rather than the more complex MINLP. We examine the effectiveness of this information to reduce the time complexity of the problem. As the presented methodology is independent of the productivity model, it has general application, irrespective of the specific learning/forgetting models employed. The approach is relevant in the context of a set of tasks with some task similarity, learning and forgetting among the workforce, and equal or unequal numbers of workers and tasks. Published by Elsevier B.V. 1. Introduction Workforce scheduling has been studied from a variety of perspectives, in particular in relation to task management based on processing times, minimizing completion or latency times, or prioritization (e.g., Leung, 2004). Resolving such problems can be difficult, time intensive, or problematic. In recent years, the scheduling and assignment of human operators to tasks has emerged as an important field of study. However, it has intro- duced the additional complexities of modeling human behavior and performance. Some common examples are models aimed at scheduling workers in call centers (Ingolfsson et al., 2007; Cezik and L’Ecuyer, 2008), bucket brigade systems in order picking (Bratcu and Dolgui, 2005), and calibrating workforce cross-train- ing in flowshops (Qin and Nembhard, 2007). For instance, call centers have been studied in several domains including capacity planning, staffing, and personnel scheduling. The staffing problem seeks to find the minimum size workforce that covers the underlying hourly requirements often defined by queuing models and forecasting methods. The scheduling pro- blem creates a work schedule for each of the customer represen- tatives such that constraints due to staffing and shift scheduling are satisfied. For staffing problems, simulation models and analy- tic queuing models are two commonly employed alternatives (Mehrotra and Fama, 2003; Mandelbaum and Zeltyn, 2006). Similarly, Ingolfsson et al. (2007) studied the staffing and sche- duling problem in single-skill call centers when the planning intervals are not assumed to be independent. Other recent literature on staffing models has studied multi-skill settings where calls of different types are served using service represen- tatives with different skills. The traditional approach to the scheduling problem is to formulate and solve a math program that identifies a minimum cost schedule. However, some have considered alternate approaches involving simulation, queuing, and search, to address these problems (e.g., Cezik and L’Ecuyer, 2008; Bhulai et al., 2008; Avramidis et al., 2009). This paper focuses on scheduling workers to a set of parallel tasks, whereby both the stations and the jobs are independent of one another, taking into account the significant human capacity for learning and the tendency to forget (alternately referred to in the literature as improvement and retention, respectively). It is hoped that the results of this study may enable solving a wider range of scheduling problems and improve upon current methods and practices in this field. The remainder of the paper is organized as follows. Section 2 is devoted to the review of extant literature related to the worker-task assignment, with specific reference to learning. Section 3 presents a mathematical formulation of the scheduling problem in an environment with parallel tasks and learning, along with the resultant structure of the optimal Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.ijpe.2012.05.024 n Corresponding author. Tel.: þ1 814 863 2447. E-mail addresses: dnembhard@psu.edu (D.A. Nembhard), bentefouet@psu.edu (F. Bentefouet). 1 Tel.: þ1 814 863 2447; fax: þ1 814 863 4745. Int. J. Production Economics 139 (2012) 533–542