.hmrnal . / ,~lanu/ucturin~ Systems Vol 18.'No. 3 19t~9 @ A Petri Net Model to Determine Optimal Assembly Sequences with Assembly Operation Constraints Shang-Tae Yee and Jose A. Ventura, Dept. of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, Pennsylvania Abstract The assembly process in an automated assembly system is the execution of successive assembly operations in which each operation joins one component with another compo- nent to form a larger component. The selection of the assem- bly sequence of a product has a great effect on the efficien- cy of the assembly process. A systematic procedure is need- ed not only to generate all feasible assembly sequences but also to choose an optimal sequence. This paper describes a method for finding tight bounds on optimal sequences in an assembly system. A Petri net obtained from the AND/OR graph of a product can be formulated as a 0-1 integer linear program that minimizes the total assembly time or cost while satisfying three assembly operation constraints, namely, ease of component handling, ease of component joining, and tool changes. A Lagrangian dual formulation is then developed to obtain a lower bound. A dynamic programming algorithm provides a dual solution, and a subgradient opti- mization algorithm is used to maximize the lower bound obtained from the dual problem. The solution procedure is validated by determining the optimal assembly sequences of three products. Keywords: Automated Assembly System, Assembly Sequencing, Petri Nets, DynamicProgramming Introduction Flexibility in modern manufacturing systems is essential to respond to frequent model changes and mixed production. ~ In particular, assembly systems require flexibility in the execution of operations due to a variety of product types. Flexible assembly sys- tems provide one way of exploiting cost-saving potential in the assembly process to a greater extent. 2 As the first step toward providing flexibility for an automated assembly system, assembly sequences should be generated by a compact representation scheme that forms the foundation of efficient system planning and control. Several representation schemes have been developed for generation of assembly sequences? -6 Among these schemes, the AND/OR graph is the most compact structure because it can represent all feasible assembly sequences and each component appears only once. Feasible sequences should be evaluated by certain criteria because the assembly process has some restrictions in the execution of the operations. Several methods have been developed to evaluate feasible sequences generated by certain representa- tion schemes and to select the best ones. Heemskerk 7 proposed a heuristic procedure that generates and reduces the number of sequences. This procedure is based on accessibility in order to eliminate the sequences that cause collision among components, and on stability for the purpose of removing the sequences that contain unstable states. Baldwin et al. 8 developed a system that can choose good sequences by editing the set of all feasible sequences. An evaluation is performed with respect to refixturing and reorientation criteria. Huang and Lee 9 evaluated the assembly plan in terms of a cost function. The cost of each operation relates not only to fixture and tool changes but also to the assembly operation. When two components are assembled, the operation cost is the summation of all costs of the involved suboperations. Homem de Mello ~° intro- duced two criteria for selecting good sequences: one is to maximize the flexibility of assembly opera- tions, and the other is to minimize the assembly time. In the first criterion, the number of different sequences to be considered in assembly planning is maximized, and in the second criterion the amount of parallelism or simultaneity encountered in assem- bly operations is maximized. Jeong ~ proposed a heuristic evaluation procedure that considers assem- bly operation constraints, where the best sequences are selected from the set of feasible sequences that satisfy the constraints. Cao and Sanderson 12 devel- oped a task (operation) decomposition and analysis method that evaluates the possible sequences by employing a search algorithm based on feasible sys- tem states. The Petri net mapped from the AND/OR graph is enlarged by decomposing each operation 203