ELSEVIER European Journal of Operational Research 105 (1998) 509-524 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Impact of learning and fatigue factors on single machine scheduling with penalties for tardy jobs V. Reddy Dondeti, Bidhu B. Mohanty * Department of Decision Sciences, Norfolk State Unirersit); 2401 Corprew At'enue, Norfolk, VA 23504, USA Received I June 1996; accepted I December 1996 Abstract We consider a single machine scheduling problem in which the machine experiences the effects of leaming or fatigue as it continues to work and the jobs have due dates and are subject to penalties if they are not completed on time. Because of the effects of learning or fatigue, the performance rate of the machine varies over time. As a result, the processing time of a job depends on its work content as well as the total work content of the jobs completed prior to its loading. In this paper, we prove that even when the machine works at a variable rate, the pair-wise interchange of jobs minimizes the maximum tardiness and a simple modification to the well-known Moore-Hodgson's algorithm yields the minimum number of tardy jobs. Further, we formulate the problem of minimizing the total penalty for tardy jobs as a 0-1 knapsack problem with nested constraints, and solve it by using dynamic programming recursion as well as the maximum-weighted network path algorithm. Then we combine these two techniques and solve the 0-1 knapsack problem, by inducing a nested constraint structure and constructing a network with fewer nodes. 9 1998 Elsevier Science B.V. Keywords: Variable performance rate; 0-1 nested knapsack problem; Job penalties; Machine scheduling 1. Introduction Single machine scheduling problems have been studied extensively in the literature (Baker, 1974; French, 1982; Morton and Pentico, 1993; and Pinedo, 1995). They are important for several reasons. They form the foundation for the development of the theory of scheduling and help us understand the complexities involved in scheduling flow shops and job shops. They also find applications in managing facilities having a single highly valuable resource. Examples include single-doctor medical offices, single-mechanic automobile repair shops, small hospitals with just one operation room, etc. A common objective in many scheduling problems is to minimize the average waiting time or penalties for the tardy jobs. In many such problems, it is also commonly assumed that the processing time of a job is constant and known in advance. In other words, there are no " Corresponding author. Tel.: + 1-757-683-8005; fax: + 1-757-683-2506; e-mail: b-mohanty@vger.nsu.edu. 0377-2217/98/SI9.00 9 1998 Elsevier Science B.V. All rights reserved. PII S0377-2217(97)00070-2