Graph-Theoretic Generation of Assembly Plans Part II: Problem Decomposition and Optimization Algorithms

Efficient generation of assembly plans has significant implications on manufacturing costs, however, the underlying mathematical problem is both theoretically and computationally intractable. In the previous paper (Part I), we developed a graph-theoretic framework for the general problem, and we described a constraint satisfaction construct that generates correct assembly precedence graphs given establishment conditions. In this paper, we develop additional components for the framework: a problem decomposition scheme which decomposes a general set of establishment conditions into generalized 2-SAT problems. After mapping these satisfiability problems into specialized decision graphs, we develop a branch-and-bound algorithm which generates optimal precedence graphs and optimal assembly plans from the decision graph. Our method generates assembly precedence graph(s) which satisfy two basic criteria: (i) All sequences defined by the precedence graph are feasible according to design specifications, (ii) Given any graph-computable performance measure (sparsity, number of node-disjoint paths etc), the precedence graph is optimal. The latter links the graph-theoretic framework to practical applications where performance criteria motivated by line balancing and scheduling can be directly associated to assembly plan generation. Computational results show that the proposed algorithm can solve moderate-size problems within seconds. Finally, we investigate alternative decomposition schemes and explore their theoretical and computational implications, including redundancy, dominance, and global optimality. 1. Motivations Assembly Planning problems have received a lot of attention over the past fifteen years. The problems encompass modeling and representation of assembly constraints, (e.g. Bourjalt, 1984), generation of feasible assembly plans (Homem de Mello and Sanderson, 1990) and selection of assembly plans for final assembly (Bonneville et. al., 1995).. Any mechanical assembly process can be decomposed into a set of tasks, where each task involves joining two or more components or subassemblies together. Given a set of establishment conditions ( precedence constraints), the sequence generation problem involves generating one or more sequences in which all the tasks in this set can be performed in order to feasibly assemble the product. Similarly, the precedence graph generation problem involves generating precedence graphs, such that all assembly sequences defined by them are feasible. The sequence of tasks in which assembly is performed can have a significant impact on cost and efficiency through both quantitative and qualitative measures. Quantitative measures may be derived from resource allocation, line balancing, scheduling, or more direct measures such as number of re-orientations and number of tool changes required. Qualitative measures, or perhaps measures difficult to quantify may include ease and stability of assembly, fixturing requirements and complexity of operations. In this paper, we focus our attention on the optimal generation of precedence graphs. We generate and select optimal precedence graphs from a correct and complete set. The word complete implies that all possible assembly sequences