J Intell Manuf (2011) 22:179–190 DOI 10.1007/s10845-009-0270-1 Minimizing total weighted flowtime subject to minimum makespan on two identical parallel machines Johnny C. Ho · Francisco J. López · Alex J. Ruiz-Torres · Tzu-Liang (Bill) Tseng Received: 9 July 2008 / Accepted: 13 May 2009 / Published online: 6 June 2009 © Springer Science+Business Media, LLC 2009 Abstract We study the problem of scheduling n jobs on two identical parallel processors or machines where an optimal schedule is defined as one with the shortest total weighted flowtime (i.e., the sum of the weighted completion time of all jobs), among the set of schedules with minimum make- span (i.e., the completion time of the last job finished). We present a two phase non-linear Integer Programming formu- lation for its solution, admittedly not to be practical or useful in most cases, but theoretically interesting since it models the problem. Thus, as an alternative, we propose an optimization algorithm, for small problems, and a heuristic, for large prob- lems, to find optimal or near optimal solutions. Furthermore, we perform a computational study to evaluate and compare the effectiveness of the two proposed methods. Keywords Parallel machines scheduling · Hierarchical criteria · Weighted flowtime · Makespan J. C. Ho (B ) Turner College of Business, Columbus State University, Columbus, GA 31907, USA e-mail: ho_johnny@colstate.edu F. J. López School of Business, Macon State College, Macon, GA 31206, USA A. J. Ruiz-Torres Departamento de Gerencia, Facultad de Administración de Empresas, Universidad de Puerto Rico-Rio Piedras, San Juan, PR 00931, USA T.-L. (Bill) Tseng Department of Mechanical and Industrial Engineering, University of Texas at El Paso, El Paso, TX 79968, USA Introduction This paper considers the following scheduling problem: a set N ={1, 2,..., n} of n jobs available at time zero is to be processed on two identical parallel machines. It is desired to minimize the makespan of the schedule (maximum com- pletion time) as the primary objective and to minimize total weighted flowtime (sum of the weighted completion times of all jobs) as the secondary objective. Specifically, the sched- uling objective is to minimize the weighted flowtime subject to the constraint that makespan does not increase. Both makespan and weighted flowtime performance mea- sures have significant impact on a schedule’s cost. The former generally represents the amount of resources attached to a set of jobs while the latter is a practical indicator of the amount of work-in-process. The concept of parallel machines is a generalization of the single-machine model. It is important because the parallel distribution or assignment of resources or tasks is widely used in the real-world. Applications include the scheduling of a set of tasks to be performed on a num- ber of simultaneously available processors, like in the case of scheduling a set of jobs on a number of milling machines (Sule 1997). Additional examples where identical parallel resources are used to process tasks include assembly opera- tions with multiple cells of similar capabilities, each assigned different customer orders; parallel repair stations, each assigned items with individual repair requirements; and par- allel manufacturing facilities in a supply chain (Ruiz-Torres et al. 2006) There are two general approaches for multiple criteria scheduling, namely simultaneous and hierarchical. For simultaneous optimization, there exist two methods. The first method generates all efficient schedules, where an efficient schedule is one in which any performance improvement with respect to one of the criteria causes a worsening of one of 123