Minimizing maximum cost on a single machine with two competing agents and job rejection Baruch Mor 1 and Gur Mosheiov 2 * 1 Department of Economics and Business Administration, Ariel University, 40700 Ariel, Israel; and 2 School of Business Administration, The Hebrew University, 91905 Jerusalem, Israel The classical Lawler’s Algorithm provides an optimal solution to the single-machine scheduling problem, where the objective is minimizing maximum cost, given general non-decreasing, job-dependent cost functions, and general precedence constraints. First, we extend this algorithm to allow job rejection, where the scheduler may decide to process only a subset of the jobs. Then, we further extend the model to a setting of two competing agents, sharing the same processor. Both extensions are shown to be solved in polynomial time. Journal of the Operational Research Society (2016). doi:10.1057/s41274-016-0003-8 Keywords: scheduling; single machine; minmax; precedence constraints; job rejection; two agents 1. Introduction In his seminal paper, Lawler (1973) solves the following problem: ‘‘Suppose n jobs are each to be processed by a single machine, subject to an arbitrary given precedence constraints. Associated with each job j is a known processing time p j , and a monotone non-decreasing cost function f j (t), giving the cost that is incurred by the completion time of that job at time t. The problem is to find a sequence which will minimize the maximum of the incurred costs.’’ Lawler introduces an efficient solution algorithm that finds an optimal sequence in O(n 2 ). In this note, we first extend Lawler’s Algorithm to allow the option of job rejection. This option has become a popular topic in the last decade among scheduling researchers, as reflected in the many references provided in the recent survey of Shabtay et al (2013). The survey states that ‘‘in many practical cases, mostly in highly loaded make-to-order production systems, accepting all jobs may cause a delay in the completion of orders which in turn may lead to high inventory and tardiness cost. Thus, in such systems, the firm may wish to reject the processing of some jobs by either outsourcing them or rejecting them all together.’’ Each unprocessed (‘‘rejected’’) job incurs a (job-dependent) penalty, and the cost of the rejected jobs becomes a factor in the objective function. We refer the reader to the early papers of Sengupta (2003) who studied job rejection with lateness and tardiness criteria, and Hoogeveen et al (2003) who focused on job rejection with preemption on related and unrelated machines. Some of the more recent papers (published after Shabtay et al 2013) are as follows: Zhao et al (2014) who studied job rejection and due-date assignment with position-dependent job processing times, Shabtay (2014) who addressed the problem of job rejection with batching to minimize total completion time, and Ou et al (2015) who focused on minimizing makespan on parallel identical machines with the option of job rejection. Following the original minmax model considered by Lawler, we focus here on a minmax objective containing rejection cost. In our knowledge, in all published papers dealing with scheduling with job rejection, rejection costs were always considered additive, ie, total rejection cost (incurred by all the rejected jobs) was assumed. We assume here a minmax cost with respect to the rejected jobs as well, ie, the rejection cost is that of the largest rejected job. This is the case, eg, when the scheduler outsources a set of jobs, which are processed (by the external firm) on a parallel batching machine. Recall that in this setting (see eg, Tian et al, 2011; Zhang et al, 2012; Li and Yuan, 2012; Fan et al, 2013; Fu et al, 2014; Geng and Yuan, 2015), the machine can process a number of jobs simultaneously, and the processing time of the batch is the longest processing time of any job in the batch. Assuming that the rejection cost is proportional to the job size, a minmax rejection cost objective appears to be appropriate. Thus, in this model, we look for the sequence that has minimum cost among all the jobs (processed or rejected). An efficient polynomial time O(n 2 log n) algo- rithm is introduced, where n is the number of jobs. We further extend Lawler’s problem to a setting of two competing agents. This setting, which has attracted many researchers in recent years, consists of two agents (producers) that share the same processor (machine). Each of the agents has a set of jobs, and each has his own objective function, which depends on the completion times of his jobs. We refer the reader to the recently published book of Agnetis et al (2014), which summa- *Correspondence: Gur Mosheiov, School of Business Administration, The Hebrew University, 91905 Jerusalem, Israel. E-mail: msomer@huji.ac.il Journal of the Operational Research Society (2016) ª 2016 The Operational Research Society. All rights reserved. 0160-5682/16 www.palgrave.com/journals