Keywords: assembly technology, process and product modelling, assembly sequence, Yager method, CAD, virtual environment.

This paper presents the concept of the determination of the assembly sequence of machine parts and sets, taking advantage of graph theory and heuristic methods of multicriterion optimisation. Algorithms for establishing possible assembly sequences of machine parts and sets on the basis of a mathematical product model created with graph theory and the selection of the most advantageous assembly sequence based on the Yager optimisation model [2] are discussed. Also presented is the simulation and visualization connection process algorithm using CAD systems and the virtual environment (VE).

The assembly process is one of the most responsible tasks in team designing and implementing technological processes. This concerns both assembly processes carried out by hand and with special equipment and industrial robots.

The design of an assembly process is not easy and it calls for the designers' thorough knowledge and rich experience. One of priorities is establishing such an assembly sequence as to minimise the time and cost of the production process. However, in order to obtain this, mathematical modelling which can be carried out by means of classical graph theory becomes necessary. While designing the technological process of the assembly sequence one should consider possible variants of the process. The selection of the best assembly sequence is difficult from a combinatorial perspective due to geometric, topological and technological restrictions citeciszak:1,ciszak:3. Most often a mathematical product model is required in the case of designing a technological assembly process with a resulting multi-variant operation sequence, no matter what methods have been used to generate it. The main concept behind the mathematical product model is a description of relations between its parts, contact surfaces and the limitations imposed by the designer and the technology. From the formal perspective the relationships of such a succession and antecedence are considered, which means that the sequence of all operations, events and activities must comply with strict rules. Out of several possibilities the Berge graph was selected which is simultaneously a digraph and a unigraph. On the basis of a design drawing one can establish relations between parts of the product, sub-assemblies and assemblies and, consequently, define sets of parts: anteceding and succeding elements. The structure of a product can be visualised in the form of a graph or in a form of a matrix which is more suitable for the purposes of mathematical analysis. Assembly sequences can be established within a graph by means of determining a path going through all its vertices. In the graph theory it is called a Hamilton circuit (sometimes also a Hamilton path). Determination of a possible sequence of assembling of specific parts is equivalent to finding a Hamilton path in the graph being considered. If there is no Hamilton path, the assembly is impossible. The algorithms for determining Hamilton paths as well as the components of strong connectivity - Leifman's algorithm have been presented.

Another stage of designing an assembly sequence is the assessment and selection of the most advantageous variant out of those obtained at the previous stage. The Yager method [2] has been used to achieve this, where the input data are the process assessment criteria m (the criteria may include for example: the producibility of a construction, basket, etc.), the number of variants of the technological assembly process n (received in the previous stage), the importance matrix of the selected criteria (determined by experts) as well as the score assessment matrix , which are determined for the i-th process variant according to the j-th criterion. Assessment of each variant of the technological process depending on the acceptance criterion, is carried out according to the scores expressed in points given by experts. The algorithm of Yager's method and its practical utilisation for evaluating and selecting assembly sequence variants are presented in this paper.

Due to the theoretical and practical research a computer application "Montaz" for the determination (graph theory) and selection (Yager's method) was developed.

The third step for determining the optimum assembly sequence is the simulation of assembly process using the 3D CAD systems in a virtual environment. The main steps in the process of modelling of the assemblies (products) and testing the sequences using the VR environment are presented in the paper.

The suggested methodology (presented in this paper in three main stages) of determining the assembly sequence of the technological process should secure the selection of the most advantageous solution from the technological point of view as well as the most profitable one from the economic perspective.

1

J. Zurek, O. Ciszak, "Modelowanie oraz symulacja kolejnosci montazu czesci i zespoów maszyn za pomoca teorii grafów". Wydawnictwo Politechniki Poznanskiej, Poznan, 1999 r.

2

St. Ponka, "Model optymalizacji wielokryterialnej struktury procesów technologicznych". Postepy Technologii Maszyn i Urzadzen, vol. 18, 1994r., s. 25-60.

3

O. Ciszak, J. Zurek, "Modelling and simulation of assembly sequence of machine parts and sets", Advances in Manufacturing Science and Technology, Polish Academy of Sciences, vol. 26, No 3, 2002, page 55-62

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