Because of strong competition and limitation of resources in our environment, scheduling is a very important decision-making process in production and service industries. In common flowshop scheduling, we have two main elements, namely a group of M machines and a set of N jobs to be processed on this group of machine . Assembly flowshop scheduling is a type of flowshop that at first each of n jobs has to be processed at the first stage consisting of m different parallel machines and then assembled at the second stage including only one assembly machine . Assembly-type production systems have evolved partially as an answer to the market pressure for larger product variety . Most of studies considered a two-stage assembly flowshop scheduling problem (AFSP) defined as follows. M machines are available in the first stage and only one machine is available in the assembly stage. There are n jobs, which should be scheduled and each of them includes m + 1 operations. The first m operations of a job are performed at the first stage in parallel by m machines and the final operation is conducted at the second stage. Each of m operations of a job at the first stage is performed by a different machine, and the assembly operation on the machine at the second stage starts when all m operations at the first stage are completed. Each machine works just on one job at a time. It should be noted that when there is only one machine at the first stage . In the two-stage AFSP, assumed collecting and transferring time of components from the first stage to assemble is negligible. This is unrealistic especially when a two-stage assembly problem is used to simulate production systems with a multi-facilities plant and a final assembly plant. But to have more realistic environments of a production system, it is required that the intermediate operation is devoted to collect and transport the manufactured parts from the various production areas to the assembly line. This stage is important especially when parts are manufactured in multiple production sites. The three-stage AFSP is the extended model of two-stage assembly flowshop that the collecting and transferring actions are regarded as the second stage, and assembly machine is in the third stage .
Suppose there are n jobs for scheduling, in which each job includes m components. At the first stage, there are m parallel and independent machines, in which each machine can process just one component. When all of m components of each job are processed on the first stage machines, they will be collected and transferred to the assembly machine (i.e., third stage) by passing the second stage (i.e., transportation stage). Then the machine at the third stage assembles m components of job that are transferred from the first stage together for completing a job. Koulamas and Kyparisis  proposed this type of an assembly line problem with the objective of minimizing the makespan. Hatami et al.,  developed this model with sequence-dependent setup time for first stage machines.
In this paper, we consider a three-stage AFSP with blocking times and sequence dependent setup times. To make this type of assembly flowshop more realistic our research added the blocking times limitation (buffer = 0) to the model presented in . Sequence-dependent setup time says that setup time of a job in position i on machine j depends on the current job and the previous job on this machine. Once its processing is completed on a processor in the first or second stage, a product is transferred directly to either an available processor in the next stage (or another downstream stage depending on the product processing route), or a buffer ahead of that stage when such an intermediate buffer is available. However, when an intermediate buffer is unavailable, the product remains blocking the processor until a downstream processor becomes available . In general, blocking scheduling problems arise in modern manufacturing environments with limited intermediate buffers between processors, such as just-in-time production systems or flexible assembly lines, and those without intermediate buffers, such as surface mount technology (SMT) lines in the electronics industry for assembling printed circuit boards, which includes three different stages in the following sequence: solder printing, component placement and solder reflow .
Yokoyama and Santos  presented a branch-and-bound method for three-stage flowshop scheduling with assembly operations to minimize the weighted sum of product completion times where there is only one machine in each stage. Koulamas and Kyparisis  analyzed a three-stage assembly scheduling problem by minimizing the makespan and analyzed the worst-case ratio bound for several heuristics for this problem. Hatami et al.,  extended the three-stage assembly flowshop model presented in  with sequence-dependent setup time by minimizing the mean flow time and maximum tardiness and they proposed two meta-heuristics, namely simulated annealing (SA) and tabu search (TS). Allahverdi and Al-Anzi  addressed a two-stage AFSP with setup time by minimizing the total completion time and they proposed a dominance relation and three heuristics, such as Ntabu, SDE and NSDE.
Lee et al.,  studied a two-stage AFSP with considering two machines at the first stage. Al-Anzi and Allahverdi  considered a two-stage AFSP with the objective of minimizing the weighted sum of makespan and maximum lateness and presented heuristics namely TS, PSO, and SDE. Cheng et al.,  studied two-stage differentiation flowshop consisting of a common critical machine in stage one and two independent dedicated machines in stage two by minimizing the weighted sum of machine completion times. Ng et al.,  proposed a branch-and-bound algorithm for solving a two-machine flow shop problem with deteriorating jobs. Ruiz and Allahverdi  minimized the bi-criteria of makespan and maximum tardiness with an upper bound on maximum tardiness of the flowshop scheduling problem. Sun et al.,  addressed powerful heuristics to minimize makespan in fixed, 3-machine, assembly-type flowshop scheduling.
In some environments, there are limited buffers or zero buffers between stages. Hall and Sriskandarajah  reviewed machine scheduling problems with blocking and no wait in process. Qian et al.,  presented an effective hybrid algorithm based on deferential evolution (DE) for multi-objective flow shop scheduling with limited buffers. Liu et al.,  solved flow shop scheduling with limited buffers with an effective hybrid PSO-based algorithm to minimize the maximum completion time. Wang et al.,  introduced a hybrid genetic algorithm (GA) for flowshop scheduling with limited buffers with the objective to minimize the total completion time. Grabowski and Pempera  developed a fast tabu search (TS) algorithm to minimize the makespan in a flow shop problem with blocking.
Ronconi  analyzed the minimization of the makespan criterion for the flowshop problem with blocking by proposing constructive heuristics, namely MM, MME and PFE. Tavakkoli-Moghaddam et al.,  presented an efficient memetic algorithm (MA) combined with (NVNS) to solve the flexible flow line with blocking (FFLB). Sawik  addressed a new mixed integer programming for the FFLB. Norman  explored a flowshop scheduling problem with finite buffer and sequence-dependent setup times and proposed a TS method. Ronconi and Henriques  introduced a GRASP-based heuristic method for a scheduling problem with blocking to minimizing the total tardiness. Tozkapan et al.,  developed a lower bounding procedure and a dominance criterion incorporated into a branch-and-bound procedure for the two-stage AFSP to minimize the total weighted flowtime. Yagmahan and Yenisey  offered a multi-objective ant colony algorithm for flowshop scheduling to minimizing the makespan and total flow time. Sung and Kim  developed a branch-and-bound algorithm for two stage multiple assembly flowshop to minimize the sum of completion times. Yokoyama  considered flowshop scheduling with setup and assembly operation and to solve used pseudo-dynamic programming and a branch-and-bound. Liu and Kozan  studied scheduling flowshop with combined buffer condition considering blocking, no-wait and limited-buffer. Lee et al.,  brought the concept of blocking into the deteriorating job scheduling problem on the two-machine flow- shop. They proposed A branch-and-bound algorithm incorporating with several dominance rules and a lower bound as well as several heuristic algorithms. Gong et al.,  studied two-stage flow shop scheduling problem on a batching machine and a discrete machine with blocking and shared setup times. Wang et al.,  proposed a HDDE algorithm for solving a flowshop scheduling with blocking to minimize the makespan.
Since blocking has been never considered in three-stage assembly flowshop so we add blocking as a constraint to the Hatami’s problem . Thus according to the new objective functions (i.e., weighted mean completion time and makespan) and blocking, we present a new mathematical model for this case.
The rest of this paper is come up as follows. In the next section, we explain the new mathematical model. In Section 3, we propose a meta-heuristic method based on SA to solve the given problem. Section 4 discusses the computational results and finally, the conclusion is presented in Section 5.