ORIGINAL ARTICLE Minimizing makespan in a two-machine no-wait flow shop with batch processing machines Shanthi Muthuswamy & Mario C. Vélez-Gallego & Jairo Maya & Miguel Rojas-Santiago Received: 15 August 2011 / Accepted: 5 January 2012 # Springer-Verlag London Limited 2012 Abstract Given a set of jobs and two batch processing machines (BPMs) arranged in a flow shop environment, the objective is to batch the jobs and sequence the batches such that the makespan is minimized. The job sizes, ready times, and processing times on the two BPMs are known. The batch processing machines can process a batch of jobs as long as the total size of all the jobs assigned to a batch does not exceed its capacity. Once the jobs are batched, the processing time of the batch on the first machine is equal to the longest processing job in the batch; processing time of the batch on the second machine is equal to the sum of processing times of all the jobs in the batch. The batches cannot wait between two machines (i.e., no-wait). The problem under study is NP-hard. We propose a mathematical formulation and present a particle swarm optimization (PSO) algorithm. The solution quality and run time of PSO is compared with a commercial solver used to solve the mathematical formulation. Experimental study clearly highlights the advantages, in terms of solution quality and run time, of using PSO to solve large-scale problems. Keywords No-wait flow shop . Batch processing machines . Particle swarm optimization . Makespan 1 Introduction Batch processing machines (BPMs) can process several jobs simultaneously. BPMs are commonly found in electronics manufacturing, metal working, food processing, and pharmaceutical industries among many other applications. This research was motivated by a cleaning process observed at a sensor manufacturing facility. The substrates used for automobile sensors are made of ceramic and the electronic circuitry is typically established using a wire bonding machine. After the bonding operation, the substrate needs to be cleaned to remove any residuals (residuals lead to defects in subsequent operations). Cleaning takes place in two sets of baths. Multiple parts can be cleaned at the same time in the first bath—the parts are attached to a fixture and the fixture is dipped in the bath for a predetermined amount of time. The time taken here depends on the composition of the batch. Typically, the cleaning time is equal to the part which requires the longest cleaning. This time can be estimated a prior for each job and it depends on several factors such as the number of electrical connections made, type of bonding machine used, type of wire used to bond, among others. The S. Muthuswamy (*) Department of Technology, Northern Illinois University, Dekalb, IL 60115, USA e-mail: smuthuswamy@niu.edu M. C. Vélez-Gallego : J. Maya Departamento de Ingeniería de Producción, Universidad EAFIT, Carrera 49 #7 sur 50, Medellín, Colombia M. C. Vélez-Gallego e-mail: marvelez@eafit.edu.co J. Maya e-mail: jmaya@eafit.edu.co M. Rojas-Santiago Departamento de Ingeniería Industrial, Universidad del Norte, Km.5 Vía Puerto Colombia, Barranquilla, Colombia e-mail: miguelrojas@uninorte.edu.co Int J Adv Manuf Technol DOI 10.1007/s00170-012-3906-9