J Intell Manuf (2007) 18:331–342 DOI 10.1007/s10845-007-0026-8 Mathematical modeling and heuristic approaches to flexible job shop scheduling problems Parviz Fattahi · Mohammad Saidi Mehrabad · Fariborz Jolai Received: March 2006/ Accepted: October 2006 / Published online: July 2007 © Springer Science+Business Media, LLC 2007 Abstract Scheduling for the flexible job shop is very important in both fields of production management and com- binatorial optimization. However, it is quite difficult to achieve an optimal solution to this problem in medium and actual size problem with traditional optimization approaches owing to the high computational complexity. For solving the realistic case with more than two jobs, two types of approaches have been used: hierarchical approaches and inte- grated approaches. In hierarchical approaches assignment of operations to machines and the sequencing of operations on the resources or machines are treated separately, i.e., assign- ment and sequencing are considered independently, where in integrated approaches, assignment and sequencing are not differentiated. In this paper, a mathematical model and heuristic approaches for flexible job shop scheduling prob- lems (FJSP) are considered. Mathematical model is used to achieve optimal solution for small size problems. Since FJSP is NP-hard problem, two heuristics approaches involve of integrated and hierarchical approaches are developed to solve the real size problems. Six different hybrid searching structures depending on used searching approach and heu- ristics are presented in this paper. Numerical experiments are used to evaluate the performance of the developed algo- rithms. It is concluded that, the hierarchical algorithms have better performance than integrated algorithms and the algo- P. Fattahi (B ) Department of Industrial Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran e-mail: fattahi@basu.ac.ir M. Saidi Mehrabad Department of Industrial Engineering, Iran University of Science & Technology, Tehran, Iran F. Jolai Industrial Engineering Department, Faculty of Engineering, University of Tehran, Tehran, Iran rithm which use tabu search and simulated annealing heuris- tics for assignment and sequencing problems consecutively is more suitable than the other algorithms. Also the numerical experiments validate the quality of the proposed algorithms. Keywords Flexible job shop · Scheduling · Tabu search · Simulated annealing · Hierarchical approach Introduction Scheduling problems occur in all the economic domains, from computer engineering to manufacturing techniques. Most scheduling problems are complex combinatorial opti- mization problems and very difficult to solve. The job shop scheduling is a branch of production scheduling, which is among the hardest combinatorial optimization problems. The job shop scheduling problem is to determine a schedule of jobs that have pre-specified operation sequences in a multi- machine environment. In the classical job shop scheduling problem (JSP), n jobs are processed to completion on m unre- lated machines. For each job, technology constraints spec- ify a complete, distinct routing which is fixed and known in advance. Each machine is continuously available from time zero, and operations are processed without preemption. The general JSP is strongly NP-hard (Garey, Johnson, & Sethi, 1976). In order to match nowadays market requirements, manufacturing systems have to become more flexible and efficient. To achieve these objectives, the systems need not only the automated and flexible machines, but also the flex- ible scheduling systems. The flexible job shop scheduling problem (FJSP) extends JSP by assuming that, for each given operation, there is at least one instance of the machine type necessary to perform it. The scheduling problem of a FJSP 123