J Intell Manuf (2009) 20:15–27 DOI 10.1007/s10845-008-0100-x Optimal assembly plan generation: a simplifying approach Michel Martinez · Viet Hung Pham · Joël Favrel Received: 14 March 2008 / Accepted: 14 March 2008 / Published online: April 2008 © Springer Science+Business Media, LLC 2008 Abstract The main difficulty in the overall process of optimal assembly plan generation is the great number of different ways to assemble a product (typically thousands of solutions) This problem confines the application of most existing automated planning methods to products composed of only a limited number of components The presented method of assembly plan generation belongs to the approach called “disassembly” and is founded on a new representation of the assembly process, with introduction of a new concept, the equivalence of binary trees This representation allows to generate the minimal list of all non-redundant (really different) assembly plans Plan generation is directed by assembly operation constraints and plan-level performance criteria The method was tested for various assembly applications and compared to other generation approaches Results show a great reduction in the combinatorial explosion of the number of plans Therefore, this simplifying approach of assembly sequence modeling allows to handle more complex products with a large number of parts Keywords Assembly · Assembly sequence · Assembly process representation · Binary tree · Binary tree equivalence M Martinez (B) Université de Lyon, Bât Nautibus 8, Boulevard Niels Bohr, Villeurbanne Cedex 69622, France e-mail: martinez@univ-lyon1.fr V H Pham ˜ Trãi, Thanh Xuân, Hanoi University of Science, 334 Nguyên Hanoi, Vietnam e-mail: hung-pv@hipt.com.vn J Favrel INSA de Lyon, Bâtiment Blaise Pascal 7, Avenue Jean Capelle, Villeurbanne Cedex 69621, France e-mail: joel.favrel@inso-lyon.fr Introduction In the manufacturing industry, process planning for assembly is a critical step in the overall product development process In the case of complex products comprising tens or even hundreds or thousands of elementary components, planning of assembly, and also disassembly or maintenance, is still very complex and costly (Henrioud and Bourjault 1998) Computeraided assembly planning is a promising solution to reduce the effort necessary to produce assembly plans while improving their quality and production cost More, with the arrival of new efficient development techniques such as concurrent engineering (Kusiak 1992), automated planning methods are needed to reduce time to market and to supply a feedback to product designers from the manufacturing point of view Since the eighties, a large amount of research has been devoted to the design of various methods of computer-aided assembly planning For this task, the two main difficulties are the complexity of products composed of a large number of elementary components and the multiplicity of the possible assembly sequences for a given product The principal generation methods are presented in section“State of the art”, followed by an analysis and classification In a general way, the design process of an optimal assembly plan includes three stages: (a) Determination of all possible sequences, (b) Selection of the “best” sequence and (c) Allocation of assembly operations to assembly resources In this approach, the main problem is the combinatorial explosion of possible solutions The number of different feasible plans, generally represented by binary trees, is high For an “ideal” product, that is a product without assembly constraints, see (Bourjault and Henrioud 1987), made up of n elementary components, De Fazio and Whitney (1987) estimated this number by the following formula, see also chapter : L n = n ∗ (n − 1)/2! 123 16 In most methods of assembly planning, it is necessary to take into account all these trees When the product is composed of a significant number of elementary components, the number of feasible plans becomes exceedingly high, usually a few tens or even hundreds of thousands of solutions This is mainly because the great majority of these solutions, said “redundant” (Baldwin et al 1991), differ only by insignificant differences, for example two assembly operations that can be performed in indifferent order In this case, assembly process planning becomes complex and in practice forces to limit the application of many assembly plan generation approaches only to products composed of a restricted number of components In other planning methods, a pragmatic solution to reduce the number of possible plans consists in reinforcing the optimality constraints, said “strategic” in (Jones and Wilson 1996), in addition to the inherent assembly mating constraints such as geometric feasibility, assembly stability, etc However, early elimination of certain entire classes of valid plans can result in hiding potentially interesting solutions and reducing flexibility (Rajan and Nof 1996) In any case, it is desirable to decrease the number of generated plans by avoiding the generation of “redundant” plans For this purpose, we propose in section “Representation of assembly plans” a new representation of assembly plans with introduction of the new concept of equivalence of binary trees The exact number of equivalence classes of binary trees for an “ideal” product without constraint is formally defined It represents the maximal bound of the number of really different assembly plans for a product The proposed generation method exploits this concept to generate the set of non-redundant assembly plans for a product from the set of feasible assembly operations The role of assembly constraints and criteria in the generation process is underlined in section “Assembly constraints and criteria” Assembly operation selection and plan generation steps are described in sections “Determination of assembly operations” and “Generation of optimal and alternative assembly plans” In section “Experimentation”, the generation process is applied to different examples of product and a comparison is done with other methods of assembly planning State of the art The different approaches of assembly sequence generation are divided into two main groups in function of the character of their optimum, local or global Global methods are able to provide an optimal plan according to a criterion The modeling of a product by a graph of functional links (contacts/connections) was used by Bourjault (1984) and then improved by other researchers A series of questions are asked to the expert to establish a directed graph 123 J Intell Manuf (2009) 20:15–27 of assembly states This method, called liaison-sequence, was simplified by De Fazio and Whitney (1987), (Whitney 2004) to reduce the necessary number of questions and thus to allow the study of more complex products (Henrioud and Bourjault 1988) made another improvement with a dual approach based on product components Homem de Mello and Sanderson (1990) proposed a representation method of assembly sequences by AND/OR graph, which was then improved by Baldwin et al (1991) This technique allows the visibility of all product assembly operations, but this advantage quickly becomes illusory when the number of components increases In this same group, other approaches are directed towards the automated capture of the mating constraints between components Some methods are founded on kinematics-based representations of the product (Nof and Rajan 1993; Rajan et al 1997; Sudarsan et al 2006) to capture the type of joint and the degrees of freedom associated to the joint Geometry-based representations allow to capture the surface mating constraints (fit, coplanar, etc) to establish the relations of precedence and feasibility (Wilson 1998; Sudarsan et al 2006) Feature based methods (Mascle 2002; Venugopal et al 2002) are used to support different activities involved in assembly See also the approaches based on the STEP standard of NIST (Baysal et al 2005; Sudarsan et al 2006) or the Design for assembly concept (Nof and Chen 2003) For the case of product families, the planning problem was initially instigated by Campagne and Favrel (1984) who introduced the concepts of “parent sequence” and “parent bill of materials” Other models of product family were proposed by Stadzisz and Henrioud (1995) and then by Adamou et al (1998) and De Lit et al (1999) Other solutions based on the Bill of Material approach can be found in Wortmannm et al (1997); Svensson (2001) and Du et al (2005) An analysis of global methods is presented in Table Local methods aim at determining “good” plans Their interest lies in their low search time However, these methods not guaranty to obtain an “optimal” plan, what can lead to the elaboration of a non-effective assembly system Local methods can be classified according to four approaches: generic (Bonneville et al 1995; Lebkowski 1997), heuristics (Mascle and Figour 1990; Laperrière and ElMaraghy 1992; Shin and Cho 1994), structure (Chakrabarty and Wolter 1987) and balancing (Huang and Lee 1991; Martinez et al 1995; Sawik 1997), see Table for more details Preliminary analysis The presented method belongs to the group of global methods which generate all acceptable plans as regards to assembly constraints and then select the optimal sequence according to a simple or multiple criteria The difficulty for the methods of this group is the combinatorial explosion which J Intell Manuf (2009) 20:15–27 17 Table Methods of assembly plan determination: the global group Approach Research work Description Plans obtained Optimum Problems Liaison Bourjault (1984), Bourjault and Henrioud (1987), De Fazio and Whitney (1987), Baldwin et al (1991), Rajan and Nof (1996), Whitney (2004) Determination of plans from the graph of connections between elementary components and constraints of anteriority Orientation : Action All Global Combinatorial explosion Determination of plans by decomposing intermediate components until obtaining the elementary components (approach said “disassembly”) Orientation : Operation Determination of the “parent” plan from the components and associated operations, then generation of a “specific” plan for a specific product Orientation : Similarity of products (CAGT) All Global High number of questions to the expert Combinatorial explosion All Global Component Product families Henrioud and Bourjault (1988), Homem de Mello and Sanderson (1990), Homem de Mello and Sanderson (1991a,b), Xu et al (1991), Cittolin (1997), Jones et al (1998), Sudarsan et al (2006) Campagne and Favrel (1984), Adamou et al (1998), Stadzisz and Henrioud (1995), De Lit et al (1999), Martinez et al (2000), Du et al (2005) Family modeling for complex products Generation of the “ specific ” plan from the “ parent ” plan Table Methods of assembly plan determination: the local group Approach Research work Description Plans obtained Optimum Problems Generic Bonneville et al (1995), Lebkowski (1997) Determination of “good” plans from a limited set of plans, said Population, according to a generation rule Orientation : Plan A limited set of plans Local Determination of the population Gradually selection of operations satisfying predefined constraints Orientation : Operations Incremental development of a “parent” plan by merging “children” plans Orientation : Product structure Selection of operations for “optimal” use of the equipment Orientation : Equipment utilisation A limited set of plans Local A limited set of plans Local Heuristic Structure Mascle and Figour (1990), Laperrière and ElMaraghy (1992), Shin and Cho (1994), Pham et al (1998) Chakrabarty and Wolter (1987) Balancing Huang and Lee (1991), Martinez and Campagne (1995), Martinez et al (1997), Sawik (1997) arises as soon as the product attains nearly ten elementary components This trouble constitutes a real obstacle during the phases of plan generation and subsequent optimal plan selection Our approach for the improvement of automatic plan generation consists in limiting the number of solutions by generating only the plans which are “really different”, also known as “non-redundant” (Baldwin et al 1991) This approach is founded on a new simplifying representation of assembly process One optimal plan Local Determination of the generation rule Constraint determination Selection of constraints Modelling of product structure Merging algorithm and data base complexity Modeling of the assembly process Algorithm and data base complexity Definitions An assembly process produces a finished product from a set of elementary components It generates intermediate components To simplify, the finished product is considered as an intermediate component Definition (operation) An assembly operation creates an intermediate component from two product components In 123 18 J Intell Manuf (2009) 20:15–27 the operation noted Op : A + B → C, A and B are the input components and C the output component Definition (equivalent operations) Two operations of assembly are noted “equivalent” if they have the same input components and the same output component Convention (representation of operations) The input components of an assembly operation must be ordered (for example according to their identification number) Definition (assembly plan) An assembly plan of a product constituted of n elementary components is an ordered suite of n − operations such as (Homem de Mello and Sanderson 1991a): (a) At the beginning, all components are elementary, (b) The output component of the ith operation, 1≤i Let P the ideal product made up of n components and C the set of its elementary components Since P assembly is not subject to any constraint, any non-trivial subset of C is a component of P Let A a non-trivial subset of C and B its complement Since A and B are components of P and product P is ideal, there is an operation A + B → P Let us suppose that A comprises k, ≤ k ≤ n − 1, elementary components Then B is composed of n − k elements As k and n − k are lower than n according to the reduction rules, the number of equivalence classes of binary trees for A is Ak and the number for B is An−k From convention in section “Definitions” one can suppose that k ≤ n/2 In addition, since there are Cnk possibilities of picking a subset of k elements out of a set of n elements, we obtain An = (Cn1 A1 An−1 +Cn2 A2 An−2 +· · ·+Cnn−1 An−1 A1 )/2 An illustration for the case n = is presented in section “Generation of optimal and alternative assembly plans” 19 assembly plans of a product made up of n elementary components does not exceed An This bound can be compared to the maximum number of the different assembly plans (redundant or not) of a product determined by DeFazio and Whitney with their representation method Proposition (De Fazio and Whitney 1987) The number of possible assembly plans of a product composed of n elementary components does not exceed L n , where Ln = n(n − 1) ! This number increases very quickly with n (see Table 3) For example, for an ideal product of elementary components, the maximum number of possible plans generated by the method of DeFazio and Whitney is 1.31*1012 If our simplifying representation is applied to the same product, we obtain only 945 really different assembly plans The ratio An /L n of Table corresponds to the reduction ratio of the number of plans taken into account The proposed method is all the more successful that the product is more complex However, the ideal product is a theoretical vision which leads to considerable rise in the number of possible plans For a real product, the reduction of the number of plans is obviously lower but in practice remains very high, see further in section “Assembly process of a motor” Maximal number of operations The number of assembly operations is an important factor for planning complexity Let P an ideal product of n parts and Nn the number of its components (elementary and intermediate) Proposition The number of different components (elementary and intermediate) that it is possible to obtain during the assembly of an ideal product P composed of n elementary components is determined by the formula Nn = 2n − Proof Let C the set of elementary components of P As P does not suffer any constraint, every nonempty subset of C is Maximal bound of the number of assembly plans of a product Table Reduction of the number of assembly plans Number of components An Ln (DeFazio) Reduction ratio of the number of plans (An /L n ) Let P a product of n elementary components Let us name P the set of non-redundant plans of product P and A the set of equivalence classes of binary trees for P From definitions and 5, we obtain, 10 15 15 105 945 34 459 425 2.131014 720 628 800 1.311012 1.201056 1.0810168 0.5 0.02083333 0.00002893 0.00007213 10−5 0.00002871 10−44 0.00001972 10−149 Proposition The number of non-redundant assembly plans of a product P and the number of not - equivalent binary trees for P are equal, i.e |P| = | A | The number of non-redundant 123 20 J Intell Manuf (2009) 20:15–27 an intermediary component of P Since there is Cnk subsets composed of k components, ≤ k ≤ n, the number of different components, elementary or intermediate, of P is equal to Nn = Cn1 +Cn2 +· · ·+Cnn = Cn0 +Cn1 +Cn2 +· · ·+Cnn −1 = 2n − Let now On the number of assembly operations of an ideal product P composed of n elementary components plans 80 64 Proposition The number of assembly operations for an ideal product of n elementary components is determined by: n On = Cnk (2k−1 − 1) 48 k=2 Proof An assembly operation of P produces an intermediate component from two components (definition 1) Let C an intermediate component composed of k elementary components, k > The number of operations with C as output component is equal to the number of bipartitions of C, that is from convention (Ck1 +Ck2 +· · ·+Ckk−1 )/2 = (2k −2)/2 = 2k−1 − As there is exactly Cnk components, we obtain the above result Corollary The number of assembly operations of a product composed of n elementary components does not exceed On The values of On and An for n ≤ 15 are presented in Table The curve of plans–operations dependence for a real product (see Fig 3) shows that the number of assembly plans and consequently generation complexity are strongly dependent on the number of operations selected The need for a preliminary analysis to obtain a reasonable set of operations will be considered in 32 24 16 14 18 22 27 31 operations Fig Plan–operation dependence An operation is said to be feasible if it respects the assembly constraints coming from product, assembly process or assembly facility Examples of which are the constraint of collision-free insertion motions, or the constraint of maximizing the degree of parallelism in the plan, etc Assembly constraints take into account the assembly operation (for example geometrical feasibility or stability of subassemblies .) or the process of optimal plan selection (for example minimizing time or maximizing the number of subassemblies) The nature and weight affected to a specific criterion are sensitive as they can lead directly to eliminating entire groups of assembly solutions apply to each operation independently of the advance of the plan Following constraints are among the most common: Geometrical feasibility (or “collision avoiding” in automated assembly): the mating of the two input components must be possible This is the strongest constraint Access for assembly tool: requires that sufficient space be available for the tool used to assemble Linear assembly: components must be assembled one at a time Assembly operation awkwardness: imposes to minimize the difficulty of assembly Stability state: requires that an intermediate component (sub-assembly) be in a stable state In our method, operation-level constraints are used during the phase of selection of the feasible operations Feasibility, for example geometrical or according to the available assembly resources, is a crippling constraint for product industrialization and manufacturing Operation-level constraints Plan-level constraints These constraints are the most fundamental They are also called “tactical” (Jones et al 1998) or “local” because they This type of constraint, also named “strategic” (Jones et al 1998), applies to all or a fraction of the plan They are used to Assembly constraints and criteria 123 J Intell Manuf (2009) 20:15–27 exclude certain operation sequences or as a criterion during the step of selection of the best plan Examples: Minimize Time: minimizes the time to perform overall assembly Minimize Cost: minimizes the overall process cost Minimize Directions: minimizes the number of insertion directions Maximize Parallel: maximizes the number of operations performed in parallel Minimize Tool Change: minimizes the number of changes of assembly tools Plan-level constraints operate in certain planning methods as a filter during the phase of assembly planning Finding a compromise between required constraints and manufacturing objectives is sensitive A high level of constraints makes it possible to quickly select a good plan with respect to one or several criteria A lower level allows to maintain the flexibility of the assembly process by making it possible to generate alternative solutions An excessive level of constraints conducts to planning fail In the case of large products, a practical solution to prevent the combinatorial explosion consists in reinforcing or adding constraints as early as possible to eliminate some classes of solutions Then, one cannot guarantee to produce a globally optimal plan, nor even interesting alternative solutions in case of manufacturing risks At the contrary, the philosophy of our method consists in first decreasing the combinatorial explosion of the number of generated plans by our representant-based representation of plans, before determining the optimal solution among the different classes of really different plans Above planlevel constraints are then used as global criteria of assembly quality or cost, during the phase of optimal plan selection 21 the operational process performance and utilizes his knowhow to evaluate the constraints found in the assembly shop floor In the case of complex products, the expert uses virtual assembly environments to verify operation feasibility These systems exploit geometry modelers, such as the DELMIA DPM Assembly simulation module (for collision detection) of CATIA V5 Computer-aided assembly environments call more and more on Virtual Reality techniques (Zhao and Madhavan 2006; Ikonomov et al 2001; Pingjun et al 2006) and Augmented Reality techniques (Ping et al 2002; Boud 1999; Pang et al 2006; Zauner et al 2003) To identify all feasible operations, we use the common approach said disassembly where the expert begins with the finished product back to the elementary components by successive operations of disassembly The expert is asked to evaluate the manufacturing parameters of each feasible operation, such as time, cost, difficulty of assembly, stability of the intermediate output component, direction of insertion, necessary fixings and tools, etc Assembly operation characteristics will be used as decision criteria in the next phase of optimal plan selection, see bellow Automated methods Automated capture of the feasible operations, directly from the product geometrical model is an interesting approach for the cost of assembly process design This approach is also able to provide an important feedback to help the actors of concurrent engineering product development: designers, supply chain managers, maintenance agents to improve product and process design from a manufacturing standpoint Our generation method is associable with any method able to select assembly operations and more particularly with the methods based on a representation capable of capturing the mating constraints between components, such as liaison-based, kinematical-based, geometric-based, or assembly feature-based approaches, see section “State of the art” Determination of assembly operations The selection of feasible assembly operations is the first step of the assembly planning method Selection of feasible operations The method is sufficiently flexible to accept any method of determination of feasible operations, as the system validates the list of operations before generation, see below To be able to evaluate the intrinsic performance of our plan representation and generation method independently of the quality of the method of selection of assembly operations, we used the common pragmatic expert approach In the manufacturing sector, the assembly expert is generally responsible of Validation of assembly operation list If the assembly operation list is non-coherent, the plan generator cannot find a feasible plan Before generation, it is necessary to check the completeness and consistency of the operation list For this, we have first to verify that for any generated intermediate component x, there is at least one operation having x as one of its input components (condition (b) of definition 3) and then that every intermediate component is the output component of at least one operation The property of consistency ensures that all selected operations will be used in the generation phase of product assembly plans 123 22 The AGAS software, for Analytical Generation of Assembly Sequences, which supports the method, includes a module for completeness and consistency checking It automatically verifies the correctness of the selected operations and displays the list of operations that are at fault Then, the expert is requested to remove the useless redundant operations recognized by the system or to complete the operation list Number of selected operations Subsequent section “Assembly process of a motor” will present the example of assembly of a stepping motor For this product, assembly constraints allowed the selection of 27 operations, which in turn gave 32 (really different) assembly plans For the same product, if one imposes more severe stability or physical constraints (positioning constraints particularly) the number of selected operations lowers For example, for 20 selected operations, the number of plans falls to With more relaxed constraints, the number of operations reaches for example 33 and the number of non-redundant assembly plans 80 Figure gives an estimation of the dependence of the number of generated plans on the number of selected assembly operations, applied to the case of product assembly considered in “Assembly process of a motor” (a stepping motor) The number of operations is included in the interval [n − 1, On ], n being the number of elementary components and On the maximal number of operations defined in section “Maximal number of operations” The number of really different plans is included in the interval [1, An ] with An maximal number of equivalence classes of assembly plans For the particular case of n − operations, there is only one equivalence class of plans We note on Fig that the curve converges quickly towards the infinite with the increase in the number of operations because of the disproportion between the increase speed of the number of plans and the increase speed of the number of operations (see Table 8) In practice, the choice of a number of operations in the order of ∗ n seems to provide a satisfying number of nonredundant plans (optimal and alternative plans) Generation of optimal and alternative assembly plans The production of an optimal plan is achieved in two steps: (a) Generation of all possible (non-redundant) plans according to the list of feasible operations (b) Classification of plans in function of a performance criterion, simple or multiple 123 J Intell Manuf (2009) 20:15–27 Plan generation is constrained by one of the two strategic criteria: parallelism-oriented generation or structure-oriented generation According to the mode of traversal in the (implicit) operation tree, that is breadth-first search or depth-first search, it is possible to respectively generate parallelism-oriented assembly plans (emphasizing parallelizable or sequenceable operations) or structure-oriented plans (emphasizing product subassemblies), see the industrial interest of structured assembly in (Nof et al 1997) and also (Rea et al 1998) After structuring of the process according to subassemblies (certain sub-assemblies highlighted by the system are not immediately obvious) the method can be re-applied for refining the assembly of sub-assemblies The method generates all possible really different plans from the list of selected operations As mentioned above in section “Maximal number of operations” only the plans representant of a whole class of possible plans are generated (the differences between plans belonging to the same class are minor) Thus, the waste of expert time for the examination of very numerous plans can be prevented (all are valid but the immense majority of them are “redundant”) To refine a solution, for example to solve an assembly line balancing problem (Sawik 1997), it is then possible to generate all processes belonging to the same class by authorized permutations of the operations of the class representant To determine the optimal plan, generated plans are then ordered according to a global performance criterion (simple or multiple) evaluated from operation parameters, such as manufacturing lead time, cost, ease of assembly, adaptability to automated assembly, adaptability to the available resources, resource occupation, or other parameter In practice, the examination of the best plans by the expert is useful to understand the complexity of large product assemblies The overall plan generation process is in fact carried out in an interactive way The expert in charge of the production must evaluate the adequacy of solutions to assembly shop realities According to results, he can be led to adjust his optimization criteria, or even to reconsider upstream constraints of selection of the feasible operations by enforcing, or at the contrary, relaxing certain constraints To give flexibility to assembly process, alternate assembly plans are necessary Two cases have to be considered Selecting an alternative plan in the list of the “best” assembly plans provides sufficient robustness in the case of expected change of the working environment, such as activity reorganization, equipment overload, or adaptation to resource availability of partner firms Some cases require more flexibility, particularly the policies of dynamic assembly (reactive piloting) or in the occurrence of an unexpected event such as disfunctioning or accident during the process, for instance an equipment breakdown, delay, etc Competition between really different assembly processes of the same product is not allowed J Intell Manuf (2009) 20:15–27 23 because it may result in a dead-lock (Fanti et al 2002) due to concurrent allocation of components or tools To guarantee system safety, we propose the following solution: (a) Determination of the optimal plan, which is in reality the representant of a whole class of optimal plans (b) Generation of a limited number of alternative plans (typically less of ten or so) by authorized permutations, that is, respecting the precedence constraints of assembly operations This restricted set of compatible plans provides alternatives in the case of process collapse (Sun et al 2002) It also allows to design dynamic assembly systems, optimizing resource allocation or lead-time or other criterion (Sedqui et al 1999) In the occurrence of unavailable critical resource, component shortage or failure of unique equipment, it is only possible to perform partial plans Disassembly process (Xu et al 1991) is a special case with an elevated probability of disassembly failures Paradoxically, this type of process is not subject to deadlock attributable to product component allocation In this case, really-different competing assembly processes can be selected to appreciably increase the rate of successful, or partially successful, product disassembly (Martinez et al 1997) This is also the case for servicing processes Experimentation The method was tested for different products and then compared with other methods Assembly process of a motor Let us consider the example of the assembly of a stepping motor (type SS MØ61 — FDØ8E, Fig 4) The motor is composed of the 15 following elementary components: Front bearing plates Wedge Stopping ring Shaft Stator Rubber joint Label a: Shaft b: Rotor c: Front stopping ring d: Rear stopping ring e: Front ball bearings f : Rear ball bearings g: Casing h: Stator i: Wedge j: Front bearing plates k: Rear bearing plates l: Seal m: Screws n: Label o: Rubber joint The first stage of the method consists in determining the table of assembly operations of the product As mentioned above, the selection of the feasible operations is achieved by the expert who takes into account the operation-level constraints, also said of tactical scope (Bourjault 1984; Jones et al 1998) Then, the completeness and consistency of the operation list are verified as described in section “Determination of assembly operations” The 27 feasible operations selected for motor assembly are given Table From the operation table, the software tool generates all non-redundant plans by seeking operations satisfying the conditions stated in definition 3, section “Definitions” The assembly software carries out the search of candidate operations As can be seen in Table 4, in the “disassembly” approach the finished product is progressively decomposed in intermediate components until obtaining the elementary components (See Fig 5) In the considered case, the software tool generates the output file of 32 assembly plans of Table These 32 plans represent the minimal list of the really different plans By simple permutations of operations in each plan with respect of the anteriority constraints (implicitly contained in the list of 27 feasible operations, see Table 4) it is then possible to generate 361,800 plans exactly The almost totality of these plans are only variations regarding the 32 non-redundant generated plans For this real product, the reduction ratio of the number of different plans that have to be taken into account is considerable, better than 1/10,000 Comparison with other methods Screws Ball bearings Rotor Fig Motor components Casing Seal Rear bearing plates As another illustration of the efficiency of our method of non-redundant representation, we compare its results with those of the liaison-sequence method and then we consider the extreme case of the ideal product Case of a ball pen assembly Let us consider the example of the assembly of a ballpoint pen proposed in (Bourjault 123 24 J Intell Manuf (2009) 20:15–27 Table Motor assembly operations : abcde f ghi jklno + m → abcde f ghi jklmn : abcde f ghiklno + j → abcde f ghi jklno : abcde f ghi jo + kln → abcde f ghi jklno : abcde f i j + ghklno → abcde f ghi jklno : abcde f ghklno + i → abcde f ghiklno : abcde f kln + gho → abcde f ghklno : ghklno + abcde f → abcde f ghklno : abcde f i j + gho → abcde f ghi jo : abcde f + kln → abcde f kln 10 : abcde f + i j → abcde f i j 11 : gho + kln → ghklno 12 : abcde + f → abcde f 13 : abcd f + e → abcde f 14 : abcd + e → abcde 15 : abcd + f → abcd f 16 : abc + d → abcd 17 : abd + c → abcd 18 : gh + o → gho 19 : ho + g → gho 20 : kn + l → kln 21 : ab + c → abc 22 : ab + d → abd 23 : i + j → i j 24 : g + h → gh 25 : h + o → ho 26 : k + n → kn 27 : a + b → ab Table Motor M∅61- Generated assembly plans : (O p27, O p21, O p16, O p26, O p14, O p24, O p20, O p12, O p18, O p9, O p6, O p5, O p2, O p1) : (O p27, O p22, O p17, O p26, O p14, O p24, O p20, O p12, O p18, O p9, O p6, O p5, O p2, O p1) : (O p27, O p21, O p16, O p26, O p15, O p24, O p20, O p13, O p18, O p9, O p6, O p5, O p2, O p1) : (O p27, O p22, O p17, O p26, O p15, O p24, O p20, O p13, O p18, O p9, O p6, O p5, O p2, O p1) : (O p27, O p21, O p16, O p26, O p14, O p25, O p20, O p12, O p19, O p9, O p6, O p5, O p2, O p1) : (O p27, O p22, O p17, O p26, O p14, O p25, O p20, O p12, O p19, O p9, O p6, O p5, O p2, O p1) : (O p27, O p21, O p16, O p26, O p15, O p25, O p20, O p13, O p19, O p9, O p6, O p5, O p2, O p1) : (O p27, O p22, O p17, O p26, O p15, O p25, O p20, O p13, O p19, O p9, O p6, O p5, O p2, O p1) : (O p27, O p21, O p16, O p26, O p24, O p14, O p20, O p18, O p12, O p11, O p7, O p5, O p2, O p1) 10 : (O p27, O p22, O p17, O p26, O p24, O p14, O p20, O p18, O p12, O p11, O p7, O p5, O p2, O p1) 17 : (O p27, O p21, O p16, O p14, O p24, O p23, O p12, O p26, O p18, O p10, O p20, O p8, O p3, O p1) 18 : (O p27, O p22, O p17, O p14, O p24, O p23, O p12, O p26, O p18, O p10, O p20, O p8, O p3, O p1) 19 : (O p27, O p21, O p16, O p15, O p24, O p23, O p13, O p26, O p18, O p10, O p20, O p8, O p3, O p1) 20 : (O p27, O p22, O p17, O p15, O p24, O p23, O p13, O p26, O p18, O p10, O p20, O p8, O p3, O p1) 21 : (O p27, O p21, O p16, O p14, O p25, O p23, O p12, O p26, O p19, O p10, O p20, O p8, O p3, O p1) 29 : (O p27, O p21, O p16, O p26, O p24, O p15, O p20, O p18, O p23, O p13, O p11, O p10, O p4, O p1) 30 : (O p27, O p22, O p17, O p26, O p24, O p15, O p20, O p18, O p23, O p13, O p11, O p10, O p4, O p1) 31 : (O p27, O p21, O p16, O p26, O p25, O p15, O p20, O p19, O p23, O p13, O p11, O p10, O p4, O p1) 32 : (O p27, O p22, O p17, O p26, O p25, O p15, O p20, O p19, O p23, O p13, O p11, O p10, O p4, O p1) Ink Head Cap Tap Barrel Cartridge Fig Ball pen (Bourjault 1984), Fig Motor—Assembly plan generation 1984) and composed of six elementary components: a: Cap, b: Head, c: Cartridge, d: Ink, e: Barrel, f: Tap, see Fig For this product, the liaison-sequence method of (Bourjault 1984) for the determination of operation precedence constraints led to the selection of the seventeen feasible operations of Table From these data, the proposed method allows to generate the 10 (non-redundant) assembly plans of Table 123 By analyzing the result of the generation obtained in (Bourjault 1984) with 12 different plans, one can note that three plans (according to our classification), P4 = (O p16, O p15, O p11, O p4, O p1), P11 = (O p15, O p16, O p11, O p4, O p1 and P12 = (O p16, O p11, O p15, O p4, O p1) correspond to the same tree (more exactly to the same class of equivalence) represented Fig They are considered as “equal” according to definition of “Representation of the assembly process by a suite of operations” Our method of tree representation automatically identifies P11 and P12 plans as simple alternatives of plan P4 This simple example highlights again the interest of the representation method J Intell Manuf (2009) 20:15–27 25 Table Ball pen assembly operations : bcde f + a → abcde f : abcde + f → abcde f : bcde + f → bcde f : bcd + e f → bcde f : bcd + e → bcde 10 : bce + a → abce 13 : be + c → bce 16 : b + c → bc : bcde + a → abcde : bce + d → bcde 11 : bc + d → bcd 14 : be + a → abe 17 : b + e → be : abce + d → abcde : abe + c → abce 12 : bc + e → bce 15 : e + f → e f Table Ball pen – Generated assembly plans : (O p16, O p11, O p7, O p3, O p1) : (O p16, O p12, O p8, O p5, O p2) : (O p16, O p12, O p8, O p3, O p1) : (O p17, O p13, O p8, O p3, O p1) : (O p16, O p15, O p11, O p4, O p1) : (O p16, O p11, O p7, O p5, O p2) : (O p17, O p13, O p8, O p5, O p2) : (O p17, O p14, O p9, O p6, O p2) : (O p16, O p12, O p10, O p6, O p2) 10 : (O p17, O p13, O p10, O p6, O p2) Table Ideal product – Estimate of the numbers of operations and plans N On An Estimation (by n) (Components) (Operations) (Plans) On 10 11 12 13 14 15 25 90 301 966 3025 9330 28501 86526 261625 788990 2375101 7141685 15 105 945 10395 135135 027 025 34 459 425 6.55108 1.381010 3.611011 7.901012 2.131014 in the general case of assembly processes including operations which are feasible in an indifferent order, or in parallel This is the case here of Op11 and Op15 In fact, these operations correspond to the assembly of independent subassemblies (here Head–Cartridge–Ink subassembly and Barrel–Tap subassembly) By using our method of non-redundant representation and plan generation, Bourjault would have obtained the same results as ours The most of industrially assembled products are structured in subassemblies, and more especially the complex products made up of a high number of parts We recall that in the presence of parallel assemblies, our method proposes a unique representant for a whole class of plans from which it is easy to obtain all other plans of the class, see sections “Representation of the assembly process by a suite of operations” or “Generation of optimal and alternative assembly plans” Case of the ideal product The “ideal” product is the extreme case of comparison For the ideal product P composed of four elementary components a, b, c, d, the exact number of really different plans and the exact number of assembly ∼ 0.5 × 3n Comparison An (An /On ) ∼0.1 × n0.87n 0.5 0.6 1.1666 3.1395 10.7608 44.6727 217.2588 1209.060 7572.2543 5.27104 4.57105 3.33106 2.98107 Fig Corresponding tree Op1 Op4 Op11 Op16 b Op15 c d e f a operations of P according to propositions 1, and on On and An , are respectively 25 and 15 These theoretical results were experimentally verified by the list of assembly plans automatically generated by the proposed method Table gives the values of the number of operations and number of plans in function of the number of elementary components As noted in the case of the stepper motor of Fig 6, we notice that the rates of increase at the infinity for On and An are very different In function of the number of product elementary components, noted n, the number of operations can be estimated at approximately 0, × 3n whereas the number of plans is about 0.1 × n 0.87 n (see Table 8) 123 26 Conclusion In this study, we defined and formalized a new model of assembly process coupled to an assembly plan generation method which answer to the problem of combinatorial explosion in automated planning methods Our main contribution is the simplifying representation of assembly process based on the original concept of “equivalence” of binary trees, which allows to generate only the minimal list of “nonredundant” (really different) assembly plans of manufactured products This concept allows to dramatically decrease the number of generated binary trees (division by a factor higher than 10,000 for example) and consequently to take into account more complex products The possibility to prohibit the generation of the very numerous redundant plans (presenting only non-significant differences) of a product represents a considerable saving of time in the preparation of the operational process The method is easy to implement and requires only a user interface and computationally inexpensive algorithms of generation The user 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Determination of “good” plans from a limited set of plans, said Population, according to a generation rule Orientation : Plan A limited set of plans Local Determination of the population Gradually