Scheduling problems with two competing agents to minimized weighted earliness–tardiness Enrique Gerstl, Gur Mosheiov n School of Business Administration, The Hebrew University, Jerusalem 91905, Israel article info Available online 23 June 2012 Keywords: Scheduling Parallel machines Two-agents Earliness–tardiness Common due-date abstract We study scheduling problems with two competing agents, sharing the same machines. All the jobs of both agents have identical processing times and a common due date. Each agent needs to process a set of jobs, and has his own objective function. The objective of the first agent is total weighted earliness– tardiness, whereas the objective of the second agent is maximum weighted deviation from the common due date. Our goal is to minimize the objective of the first agent, subject to an upper bound on the objective value of the second agent. We consider a single machine, and parallel (both identical and uniform) machine settings. An optimal solution in all cases is shown to be obtained in polynomial time by solving a number of linear assignment problems. We show that the running times of the single and the parallel identical machine algorithms are O(n mþ3 ), where n is the number of jobs and m is the number of machines. The algorithm for solving the problem on parallel uniform machine requires O(n mþ3 m 3 ) time, and under very reasonable assumptions on the machine speeds, is reduced to O(n mþ3 ). Since the number of machines is given, these running times are polynomial in the number of jobs. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction In classical Just-in-Time (JIT) scheduling problems, the objec- tive is minimum earliness–tardiness cost of the job completion times from their due dates. Most of the early studies (see e.g. Baker and Scudder [1]) focused on scheduling problems where all the jobs share a common due date. Among these studies, some considered minsum objectives (where the scheduler goal is to minimize the total cost incurred by all the jobs). Others research- ers focused on minmax objectives (where the goal is to minimize the cost of the worst scheduled job). In their seminal paper, Hall and Posner [2] proved that the single machine problem to minimize the weighted deviation of the jobs completion times from a common due date is NP-hard. This minsum version is known as the Weighted Earliness–Tardiness (WET) problem. Cheng and Li [3] proved that the problem of minimizing the maximum weighted deviations among the job completion times from a common due date is NP-hard. This minmax version is known as the Minimum Weighted Absolute Lateness (MWAL) problem. In recent years, scheduling researchers have focused on a setting of two competing agents. In this setting, two agents who need to process their own sets of jobs, compete on the use of a common resource. Each one of the two agents has his own objective function, and the goal is to find the joint schedule that minimizes the value of the objective function of one agent, subject to an upper bound on the value of the objective function of the second agent. Baker and Cole Smith [4] introduced the first scheduling paper dealing with two agents sharing a single processor. They focused on minimizing makespan, maximum lateness and total weighted completion time. Agnetis et al. [5] extended significantly the list of scheduling measures and the machine settings, and were followed by: Ng et al. [6], Cheng et al. [7], Lee et al. [8], Agnetis et al. [9], Gawiejnowicz et al. [10], Leung et al. [11], Mor and Mosheiov [12], Lee et al. [13], Li and Hsu [14], Li and Yuan [15], and Mor and Mosheiov [16], among others. In this paper we study a two-agent scheduling JIT problem, where the objective is to minimize the minsum measure of the first agent (WET), subject to an upper bound on the minmax measure of the second agent (MWAL). This setting of objective functions is a generalization of some of the settings introduced in [5], since the weighted earliness–tardiness objective (of the first agent) is clearly an extension of weighted tardiness or of total completion times studied in Agnetis et al., and maximum weighted deviation (of the second agent) is an extension of maximum weighted completion time assumed in Agnetis et al. The general problem (i.e. assuming general job-dependent processing times) is NP-hard, since, as mentioned above, even Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2012.05.019 n Corresponding author. Tel.: þ972 2 588 3108; fax: þ972 2 588 1341. E-mail address: msomer@mscc.huji.ac.il (G. Mosheiov). Computers & Operations Research 40 (2013) 109–116