DOI 10.1007/s10845-008-0100-xOptimal assembly plan generation: a simplifying approach Michel Martinez · Viet Hung Pham · Joël Favrel Received: 14 March 2008 / Accepted: 14 March 2008 / P
Trang 1DOI 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: 8 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
differ-ent) 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
Hanoi University of Science, 334 Nguy˜ên Trãi, Thanh Xuân,
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 hun-dreds or thousands of elementary components, planning of assembly, and also disassembly or maintenance, is still very complex and costly (Henrioud and Bourjault 1998) Computer-aided 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 engineer-ing (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 pos-sible 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) Selec-tion of the “best” sequence and (c) AllocaSelec-tion of assembly operations to assembly resources
In this approach, the main problem is the combinatorial explosion of possible solutions The number of different fea-sible 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 4: L n = n∗(n − 1)/2!
Trang 2In most methods of assembly planning, it is necessary to
take into account all these trees When the product is
com-posed 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
insignifi-cant 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
num-ber 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
How-ever, early elimination of certain entire classes of valid plans
can result in hiding potentially interesting solutions and
reduc-ing flexibility (Rajan and Nof 1996)
In any case, it is desirable to decrease the number of
gen-erated 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
con-straints and criteria in the generation process is underlined in
section “Assembly constraints and criteria” Assembly
oper-ation selection and plan generoper-ation steps are described in
sections “Determination of assembly operations” and
“Gen-eration of optimal and alternative assembly plans” In section
“Experimentation”, the generation process is applied to
dif-ferent 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
func-tional 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
of assembly states This method, called liaison-sequence, was simplified byDe 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 Bour-jault 1988) made another improvement with a dual approach based on product components.Homem de Mello and Sander-son(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 Geome-try-based representations allow to capture the surface mating
constraints (fit, coplanar, etc) to establish the relations of pre-cedence 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 byCampagne and Favrel(1984) who introduced the concepts of “parent sequence” and “parent bill
of materials” Other models of product family were proposed
byStadzisz and Henrioud(1995) and then byAdamou et al (1998) andDe Lit et al.(1999) Other solutions based on the
Bill of Material approach can be found inWortmannm et al (1997);Svensson(2001) andDu et al.(2005) An analysis
of global methods is presented in Table1
Local methods aim at determining “good” plans Their
interest lies in their low search time However, these methods
do 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 Table2for more details
Preliminary analysis
The presented method belongs to the group of global methods which generate all acceptable plans as regards to assem-bly constraints and then select the optimal sequence accord-ing to a simple or multiple criteria The difficulty for the methods of this group is the combinatorial explosion which
Trang 3Table 1 Methods of assembly plan determination: the global group
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
All Global Combinatorial explosion
to the expert Component 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 )
Determination of plans by decomposing intermediate components until obtaining the elementary components (approach said
“disassembly”)
Orientation : Operation
All Global Combinatorial explosion
Product families 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 )
Determination of the “parent”
plan from the components and associated operations, then generation of a
“specific” plan for a specific product
All Global Family modeling for
complex products
Orientation : Similarity of
products (CAGT)
Generation of the
“ specific ” plan from the
“ parent ” plan
Table 2 Methods of assembly plan determination: the local group
Generic Bonneville et al ( 1995 ),
Lebkowski ( 1997 )
Determination of “good”
plans from a limited set of plans, said Population, according to a generation rule
A limited set of plans Local Determination of the
population
generation rule Heuristic Mascle and Figour ( 1990 ),
Laperrière and ElMaraghy
( 1992 ), Shin and Cho ( 1994 ),
Pham et al ( 1998 )
Gradually selection of operations satisfying predefined constraints
A limited set of plans Local Constraint determination
Structure Chakrabarty and Wolter ( 1987 ) Incremental development of
a “parent” plan by merging “children” plans
A limited set of plans Local Modelling of product
structure Orientation : Product
structure
Merging algorithm and data base complexity Balancing Huang and Lee ( 1991 ),
Martinez and Campagne
( 1995 ), Martinez et al.
( 1997 ), Sawik ( 1997 )
Selection of operations for
“optimal” use of the equipment
One optimal plan Local Modeling of the assembly
process
Orientation : Equipment
utilisation
Algorithm and data base complexity
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
Definitions
An assembly process produces a finished product from a set
of elementary components It generates intermediate compo-nents To simplify, the finished product is considered as an intermediate component
Definition 1 (operation) An assembly operation creates an
intermediate component from two product components In
Trang 4the operation noted Op: A + B → C, A and B are the input
components and C the output component
Definition 2 (equivalent operations) Two operations of
assembly are noted “equivalent” if they have the same input
components and the same output component
Convention 1 (representation of operations) The input
components of an assembly operation must be ordered (for
example according to their identification number).
Definition 3 (assembly plan) An assembly plan of a product
constituted of n elementary components is an ordered suite
of n−1 operations such as (Homem de Mello and Sanderson
1991a):
(a) At the beginning, all components are elementary,
(b) The output component of the i th operation, 1≤i<n−1, is
one of the input components of one of the next assembly
operations,
(c) The output component of the last operation is the finished
product
To represent an assembly plan, (a) suite of operations must
verify the condition (b) which actually represents the
anteri-ority constraints applied to assembly operations
Representation of the assembly process by a suite
of operations
An assembly plan is an ordered suite (sequence) of
opera-tions satisfying the condiopera-tions of definition 3 However, two
different operations suites can correspond to the same
tree-like process Indeed, let us consider the product composed
of four elementary components a, b, c, d (noted by the suite
abcd) and the three following assembly operations: Op1:
ab + cd → abcd, Op2 : a + b → ab, Op3 : c + d → cd.
The two assembly operations sequences: P1 = (Op3, Op2,
O p1 ) and P2 = (Op2, Op3, Op1) are in fact equivalent and
can be represented by a binary tree, or more exactly by a class
of equivalence of binary tree This concept will be defined in
the following paragraph The binary tree of Fig.1represents
a process in which some operations (O p2 and O p3) can be
achieved in indifferent order or even in parallel
Definition 4 (equal assembly plans) Two assembly plans are
said “equal” if their operation sets are equal
Fig 1 Assembly tree
a b c d
Op1
To obtain a monovalent correspondence between
assem-bly process and assemassem-bly operation suites, we specify for
each process a unique suite of operations From this unique
suite, it will be easy to determine all other possible suites of
operations by authorized permutations of operations.
Convention 2 (plan representation) Operations in an
assem-bly sequence must be ordered (for example, according to their
identification number in decreasing order) with respect of all anteriority constraints defined on these operations (condition (b) of definition 3)
Representation of assembly plans
A binary tree is a suitable representation for an assembly process (Wolter 1992)
Equivalence of binary trees
In the proposed method, a plan is represented by a binary tree where vertexes correspond to product subsets and where sheets are elementary components However, in this case the same plan can correspond to several trees For example, the three trees of Fig.2are representants of a unique plan P =
(Op3, Op2, Op1), where Op1: bcd + a → abcd, Op2:
cd + b → bcd, Op3: c + d → cd This multivalent
cor-respondence between plans and binary trees leads to a new concept: the equivalence of binary trees
Definition 5 (equivalence of binary trees) Two binary trees
are said equivalents if all operations corresponding to their vertexes are equal
In the above example, the three trees are equivalent,
because their sets of operations are identical, namely S =
{Op3, Op2, Op1} To define a monovalent representation of
assembly plans by binary trees, we select a unique represen-tant for each class of equivalence
Convention 3 (representant of binary tree) To each vertex
of a representant of an equivalence class of binary tree S, son vertexes must be ordered (for example, according to the identification number of the input components of the corre-sponding assembly operation)
a b c d
Op2 Op3 Op1
a c d b
Op2 Op3 Op1
c d b a
Op2 Op3 Op1
Tree 1
(Class Representant) Tree 3 Tree 2
Fig 2 Equivalent trees
Trang 5In the same example, suppose that components
(elemen-tary and intermediate) are enumerated as follows: 1.abcd,
2.bcd, 3.cd, 4.a, 5.b, 6.c, 7.d Then, the representant of the
three trees of Fig.2is Tree 3: Indeed, at the first vertex the
input components of Op1 are bcd and a They are ordered
according to their component number (here components 2
and 4) The second and third vertexes also verify this
con-vention
Set of binary trees representants
To estimate the maximal bound of the number of possible
plans, let us consider the ideal product defined in (Bourjault
and Henrioud 1987) i.e., the product that has one functional
link between every pair of components and that does not
undergo any assembly constraint Let us note Anthe number
of equivalence classes of binary trees for the ideal product
composed of n elementary components.
Proposition 1 An is calculated according to the formula
An = (C1
n A1An−1+ C2
n A2An−2+ · · · + C n−1
n An−1A1)/2 with A1= 1 Let us recall that C k
n are the coefficients of the binomial of degree n.
Proof The case n = 2 is trivial Let us suppose that n > 2.
Let P the ideal product made up of n components andC
the set of its elementary components Since P assembly is
not subject to any constraint, any non-trivial subset ofCis
a component of P Let A a non-trivial subset ofC 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 , 1 ≤ 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
1 in section “Definitions” one can suppose that k ≤ n/2 In
addition, since there are C n k possibilities of picking a
sub-set of k elements out of a sub-set of n elements, we obtain An=
(C1A1An−1+C2A2An−2+· · ·+C n−1
n An−1A1)/2 An
illus-tration for the case n= 4 is presented in section “Generation
of optimal and alternative assembly plans”
Maximal bound of the number of assembly plans
of a product
Let P a product of n elementary components Let us nameP
the set of non-redundant plans of product P andAthe set of
equivalence classes of binary trees for P From definitions 4
and 5, we obtain,
Proposition 2 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
assembly plans of a product made up of n elementary com-ponents 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 representa-tion method
Proposition 3 (De Fazio and Whitney 1987) The number of possible assembly plans of a product composed of n elemen-tary components does not exceed L n, where
L n=n (n − 1)
This number increases very quickly with n (see Table3) For example, for an ideal product of 6 elementary compo-nents, 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 Table3corresponds 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 4 The number of different components
(elemen-tary 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− 1
Proof Let C the set of elementary components of P As P
does not suffer any constraint, every nonempty subset of C is
Table 3 Reduction of the number of assembly plans
Number of A n L n Reduction ratio of the components (DeFazio) number of plans(A n /L n )
.
10 34 459 425 1.201056 0.00002871 10 −44
.
15 2.1310 14 1.0810 168 0.00001972 10 −149
Trang 6an intermediary component of P Since there is C n k subsets
composed of k components, 1 ≤ k ≤ n, the number of
differ-ent compondiffer-ents, elemdiffer-entary or intermediate, of P is equal to
Nn = C1+C2+· · ·+C n = C0+C1+C2+· · ·+C n−1 =
2n− 1
Let now Onthe number of assembly operations of an ideal
product P composed of n elementary components.
Proposition 5 The number of assembly operations for an
ideal product of n elementary components is determined by:
On=
n
k=2
C n k (2 k−1− 1).
Proof An assembly operation of P produces an
intermedi-ate component from two components (definition 1) Let C
an intermediate component composed of k elementary
com-ponents, k > 1 The number of operations with C as output
component is equal to the number of bipartitions of C, that is
from convention 1(C1
k +C2
k +· · ·+C k−1
2k−1− 1 As there is exactly C k
ncomponents, we obtain the above result
Corollary 3 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
Table8 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
depen-dent on the number of operations selected The need for a
preliminary analysis to obtain a reasonable set of operations
will be considered in 6
Assembly constraints and criteria
An operation is said to be feasible if it respects the
assem-bly constraints coming from product, assemassem-bly process or
assembly facility Examples of which are the constraint of
collision-free insertion motions, or the constraint of
maxi-mizing the degree of parallelism in the plan, etc
Assembly constraints take into account the assembly
oper-ation (for example geometrical feasibility or stability of
sub-assemblies .) or the process of optimal plan selection (for
example minimizing time or maximizing the number of
sub-assemblies) The nature and weight affected to a specific
cri-terion are sensitive as they can lead directly to eliminating
entire groups of assembly solutions
Operation-level constraints
These constraints are the most fundamental They are also
called “tactical” (Jones et al 1998) or “local” because they
1
14 18 22 27 31 4
8 16 24
64
48
32
80 plans
operations
Fig 3 Plan–operation dependence
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 assem-bly resources, is a crippling constraint for product industri-alization and manufacturing
Plan-level constraints
This type of constraint, also named “strategic” (Jones et al
1998), applies to all or a fraction of the plan They are used to
Trang 7exclude 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
manufactur-ing 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
flexibil-ity of the assembly process by making it possible to generate
alternative solutions An excessive level of constraints
con-ducts to planning fail
In the case of large products, a practical solution to prevent
the combinatorial explosion consists in reinforcing or
add-ing constraints as early as possible to eliminate some classes
of solutions Then, one cannot guarantee to produce a
glob-ally 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
num-ber of generated plans by our representant-based
representa-tion of plans, before determining the optimal solurepresenta-tion among
the different classes of really different plans Above
plan-level constraints are then used as global criteria of assembly
quality or cost, during the phase of optimal plan selection
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
repre-sentation and generation method independently of the quality
of the method of selection of assembly operations, we used
the common pragmatic expert approach In the
manufactur-ing sector, the assembly expert is generally responsible of
the operational process performance and utilizes his know-how 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 Mad-havan 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 suc-cessive operations of disassembly The expert is asked to evaluate the manufacturing parameters of each feasible oper-ation, such as time, cost, difficulty of assembly, stability of the intermediate output component, direction of insertion, necessary fixings and tools, etc Assembly operation charac-teristics 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 captur-ing the matcaptur-ing constraints between components, such as liai-son-based, kinematical-based, geometric-based, or assembly feature-based approaches, see section “State of the art”
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 compo-nent is the output compocompo-nent 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
Trang 8The AGAS software, for Analytical Generation of
Assembly Sequences, which supports the method, includes a
module for completeness and consistency checking It
auto-matically 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
opera-tions recognized by the system or to complete the operation
list
Number of selected operations
Subsequent section “Assembly process of a motor” will
pres-ent 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
sta-bility or physical constraints (positioning constraints
particu-larly) the number of selected operations lowers For example,
for 20 selected operations, the number of plans falls to 8 With
more relaxed constraints, the number of operations reaches
for example 33 and the number of non-redundant assembly
plans 80
Figure3gives an estimation of the dependence of the
num-ber of generated plans on the numnum-ber of selected assembly
operations, applied to the case of product assembly
consid-ered 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
“Max-imal 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− 1 operations, there is only one
equiv-alence class of plans
We note on Fig.3that 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 Table8)
In practice, the choice of a number of operations in the
order of 2∗ n seems to provide a satisfying number of
non-redundant 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
accord-ing to the list of feasible operations
(b) Classification of plans in function of a performance
cri-terion, simple or multiple
Plan generation is constrained by one of the two strategic
cri-teria: 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 prod-uct subassemblies), see the industrial interest of strprod-uctured assembly in (Nof et al 1997) and also (Rea et al 1998) After structuring of the process according to subassemblies (cer-tain sub-assemblies highlighted by the system are not imme-diately 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 rep-resentant 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 balanc-ing problem (Sawik 1997), it is then possible to generate all processes belonging to the same class by authorized permu-tations of the operations of the class representant
To determine the optimal plan, generated plans are then
ordered according to a global performance criterion (sim-ple or multi(sim-ple) evaluated from operation parameters, such
as manufacturing lead time, cost, ease of assembly, adapt-ability to automated assembly, adaptadapt-ability 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 reali-ties According to results, he can be led to adjust his opti-mization 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 Select-ing 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
differ-ent assembly processes of the same product is not allowed
Trang 9because 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
(typi-cally less of ten or so) by authorized permutations, that is,
respecting the precedence constraints of assembly
oper-ations
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
dead-lock attributable to product component allocation In this
case, really-different competing assembly processes can be
selected to appreciably increase the rate of successful, or
par-tially 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
com-pared 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
com-ponents:
Front bearing plates Wedge Stopping ring Shaft Stator Rubber joint Label
Ball bearings
Fig 4 Motor components
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 con-straints, 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 “Determi-nation of assembly operations” The 27 feasible operations selected for motor assembly are given Table4
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 Table4, 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 Table5 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 Table4) it is then possi-ble to generate 361,800 plans exactly The almost totality of these plans are only variations regarding the 32 non-redun-dant generated plans
For this real product, the reduction ratio of the number of different plans that have to be taken into account is consid-erable, better than 1/10,000
Comparison with other methods
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
Trang 10Table 4 Motor assembly operations
1: abcdef ghi jklno + m → abcdef ghi jklmn 10: abcdef + i j → abcdef i j 19: ho + g → gho
2: abcdef ghiklno + j → abcdef ghi jklno 11: gho + kln → ghklno 20: kn + l → kln
3: abcdef ghi jo + kln → abcdef ghi jklno 12: abcde + f → abcdef 21: ab + c → abc
4: abcdef i j + ghklno → abcdef ghi jklno 13: abcd f + e → abcdef 22: ab + d → abd
5: abcdef ghklno + i → abcdef ghiklno 14: abcd + e → abcde 23: i + j → i j
Table 5 Motor M∅61- Generated assembly plans
1: (Op27, Op21, Op16, Op26, Op14, Op24, Op20, Op12, Op18, Op9, Op6, Op5, Op2, Op1)
2: (Op27, Op22, Op17, Op26, Op14, Op24, Op20, Op12, Op18, Op9, Op6, Op5, Op2, Op1)
3: (Op27, Op21, Op16, Op26, Op15, Op24, Op20, Op13, Op18, Op9, Op6, Op5, Op2, Op1)
4: (Op27, Op22, Op17, Op26, Op15, Op24, Op20, Op13, Op18, Op9, Op6, Op5, Op2, Op1)
5: (Op27, Op21, Op16, Op26, Op14, Op25, Op20, Op12, Op19, Op9, Op6, Op5, Op2, Op1)
6: (Op27, Op22, Op17, Op26, Op14, Op25, Op20, Op12, Op19, Op9, Op6, Op5, Op2, Op1)
7: (Op27, Op21, Op16, Op26, Op15, Op25, Op20, Op13, Op19, Op9, Op6, Op5, Op2, Op1)
8: (Op27, Op22, Op17, Op26, Op15, Op25, Op20, Op13, Op19, Op9, Op6, Op5, Op2, Op1)
9: (Op27, Op21, Op16, Op26, Op24, Op14, Op20, Op18, Op12, Op11, Op7, Op5, Op2, Op1)
10: (Op27, Op22, Op17, Op26, Op24, Op14, Op20, Op18, Op12, Op11, Op7, Op5, Op2, Op1)
.
17: (Op27, Op21, Op16, Op14, Op24, Op23, Op12, Op26, Op18, Op10, Op20, Op8, Op3, Op1)
18: (Op27, Op22, Op17, Op14, Op24, Op23, Op12, Op26, Op18, Op10, Op20, Op8, Op3, Op1)
19: (Op27, Op21, Op16, Op15, Op24, Op23, Op13, Op26, Op18, Op10, Op20, Op8, Op3, Op1)
20: (Op27, Op22, Op17, Op15, Op24, Op23, Op13, Op26, Op18, Op10, Op20, Op8, Op3, Op1)
21: (Op27, Op21, Op16, Op14, Op25, Op23, Op12, Op26, Op19, Op10, Op20, Op8, Op3, Op1)
.
29: (Op27, Op21, Op16, Op26, Op24, Op15, Op20, Op18, Op23, Op13, Op11, Op10, Op4, Op1)
30: (Op27, Op22, Op17, Op26, Op24, Op15, Op20, Op18, Op23, Op13, Op11, Op10, Op4, Op1)
31: (Op27, Op21, Op16, Op26, Op25, Op15, Op20, Op19, Op23, Op13, Op11, Op10, Op4, Op1)
32: (Op27, Op22, Op17, Op26, Op25, Op15, Op20, Op19, Op23, Op13, Op11, Op10, Op4, Op1)
Fig 5 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.6
For this product, the liaison-sequence method of (
Bour-jault 1984) for the determination of operation precedence
constraints led to the selection of the seventeen feasible
oper-ations of Table6
From these data, the proposed method allows to generate the
10 (non-redundant) assembly plans of Table7
Cap
Tap Ink
Cartridge Barrel
Head
Fig 6 Ball pen (Bourjault 1984),
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 = (Op16,
O p15 , Op11, Op4, Op1), P11 = (Op15, Op16, Op11,
correspond to the same tree (more exactly to the same class
of equivalence) represented Fig.7 They are considered as
“equal” according to definition 4 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 exam-ple highlights again the interest of the representation method