Production, Manufacturing and LogisticsA note on minimum makespan assembly plans
Introduction
In robotic assembly systems, one or more types of items are produced by assembling together several parts given in input according to a properly defined assembly plan, where an assembly plan is a sequence of feasible assembly operations which allow one to derive the desired product from the given parts. Systems of this type are required to have a high degree of flexibility in order to allow for quick switches from one type of product to another or to recover easily from errors or unexpected events which may require changes in the assembly plan. Crucial to this point is the capability of generating and updating assembly plans in an efficient way.
In the literature, alternative ways to represent relations among the feasible assembly operations have been proposed (De Fazio and Whitney, 1987; Abell et al., 1991). In particular, the use of hypergraphs has been suggested to model assembly plans (Homem De Mello and Sanderson, 1990; Gallo and Pallottino, 1992). However, no algorithm has been proposed so far to select an optimal assembly plan, under certain optimality criteria, and to schedule its operations on the machines, in the case in which k parallel identical machines are available for the execution of the operations.
In this paper, we investigate the problem of finding an assembly plan, and scheduling its operations on k parallel identical machines, where 1<k<∞, in such a way to minimize the makespan. We propose a general heuristic based on the classical list scheduling (LS) rule, and show that the heuristic produces solutions with a bounded approximation in a particular but still relevant case. Then, some computational results on randomly generated instances are discussed.
Section snippets
Assembly plans
Let Q be a set of distinct parts. An assembly is an item obtained by properly connecting some of the parts in Q. Its cardinality is the number of parts it contains (a part is considered as an assembly of cardinality 1). For the sake of the notational simplicity we will consider only binary operations, i.e., the assumption will be made that an assembly having a cardinality greater than one is obtained by connecting two subassemblies. The representation of connections involving three or more
LS heuristic for MMAP
The algorithm we propose consists of two phases. In the first phase, the bound on the number of machines is relaxed, and the minimum makespan assembly plan is found for the relaxed problem (i.e., k=∞). This is done in linear time by computing a minimum distance hyperpath. Let be the computed minimum distance hyperpath (or, equivalently, the computed minimum makespan assembly plan with unbounded parallelism).
In the second phase, the feasible assembly operations in are scheduled
Implementation and computational results
In order to perform a computational experimentation, we have defined a special type of assembly plan problem which, although simplified with respect to the specification of MMAP, is quite realistic, and suitable for a random generation. This type of problem, which we call Graph structured assembly plan problem, has to be interpreted as an auxiliary structure, used to generate assembly hypergraphs in our experimentation. The structure of the problem is completely represented by means of an
Conclusions
Based on hypergraph models, we have presented and analyzed some special classes of the optimal assembly plan problem. We have shown that, depending on the objective function chosen, an optimal assembly plan problem can be easy (in fact solvable by means of rather simple optimal hyperpath computations) or NP-hard. A computationally fast heuristic algorithm for finding a minimum makespan assembly plan has been proved to give solutions with a bounded approximation when the operations' durations
Acknowledgements
We are grateful to Maria Paola Scaparra for her programming and testing work.
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