A MATHEMATICAL MODEL OF NON-DESTRUCTIVE DISASSEMBLY PROCESS

7/30/2019 A MATHEMATICAL MODEL OF NON-DESTRUCTIVE DISASSEMBLY PROCESS

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62 Ile Mircheski, Tatjana Kandikjan& Bojan Prangoski

RELATED WORK

Many authors have developed different methods for determining of the optimal disassembly sequence and for

planning of the disassembly process. A disassembly hypergraph, called AND/OR graph, which can represent compactly all

disassembly sequences, is proposed by Homem de Mello et al. They developed an algorithm for generation of the

AND/OR graph based on a relational assembly model. The input data include answers to queries about the movability of

the components in order to determine the feasible disassembly sequences [14]. In the paper [13], F. Cappelli et al.

developed the methodology which provides the theoretical basis for creation of computer-aided design tool for optimizing

of the disassembly sequences of mechanical systems for improving of maintenance and recycling activities.

In the first step in the paper is investigated the physical constraints that oppose the movement of mechanical

element, starting from its three dimensional computer-aided design representation and creation of AND/OR graph of

mechanical assembly. The second step in the paper is representation of binaries trees which allows automatic exploration

of the set of all possible disassembly sequences. In the paper [18], T. C. Woo investigated the generation of sequences for

movements and removal of components from three-dimensional assembly with translator movement of robot arm during

remove components. Moving over disassembly tree T. C. Woo would like to find the minimal sequence for assembly and

disassembly of the product. Disadvantage of this paper is that disassembly operations are in same order as the assembly

operations. N. Shyamsunder and R. Gadh in the paper [17] presented the importance of selective disassembly of virtual

prototypes.

The maintenance and reuse of the components from the product require disassembly of individual components of

the product. In the paper is presented a method for determination of selective disassembly sequences and identification of

component in virtual prototype. A.J.D. Lambert et al. in series of papers [1, 3, 5] represents a research of method for linear

programming and determination of disassembly sequences for end-of-life products. The method with linear programming

gives a contribution in optimization of disassembly process, which is presented in the papers [3, 5].

A MATHEMATICAL MODEL OF NON-DESTRUCTIVE DISASSEMBLY PROCESS FOR

OBTAINING OF DISASSEMBLY SEQUENCES

The product disassembly is required both during the product life cycle and after the end of the product life. The

disassembly process can be destructive and non-destructive. Destructive disassembly represents a process where the stream

of end-of-life products is shredded in small fragments which are later separated according to their material composition

using special separation techniques. Destructive disassembly process is applied mostly at the product end-of-life, for

products that don't have hazard materials inside. Non-destructive disassembly process is applied during the exploitation of

the product and at product end-of-life. During the product exploitation, maintenance, service or replacement of some

nonfunctional components is needed. While for end-of-life products, non-destructive disassembly process is needed for:recovery of some functional components [2]; removing of hazardous materials from the product, which can have negative

influence on the recycling process, and can pollute the environment; extracting of the precious materials from the product;

remanufacturing, etc.

The goal of this paper is definition of the mathematical model of non-destructive disassembly process for

determination of optimal disassembly sequence.


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A Mathematical Model of Non-Destructive Disassembly Process 63

Figure 1: A Methodology for Product Design for Disassembly for Determination of Optimal Disassembly Sequence

As shown in Figure 1, the design for disassembly methodology is carried out in three main phases. In the first

phase, an algorithm identifies all fasteners, components, contacts between fasteners (F) and components (C) and contacts

between components (C) and components (C) in the product assembly, materials and weight of fasteners and components.

The identification is carried out directly from an assembly CAD model [15] with special developed software in

VBA. In the second phase, the optimal disassembly sequence is determined in several steps, such as, determination of

contact matrix (FC) between fasteners and components, contact matrix (CC) between components and components, all

subassemblies (SA) in product assembly (A), lists for SA, F, C, the disassembly operations, all disassembly sequences,

disassembly interference matrix, all possible disassembly sequences and the optimal disassembly sequence. Determination

of optimal disassembly sequence for virtual assembly model depends on disassembly times, costs and revenues. The third

phase is creative phase, which includes product redesign by selection of new fasteners, improvement of product structure

and the choice of other types of materials, in order to increase the overall return of the product without change in the

product functionality. The changes made in the redesign process, by feedback go to the CAD model of the product

assembly, and again pass through the first and second phase. In the third phase, also, the optimal disassembly sequences for

the original and the redesigned product are compared and a choice of best optimal solution for product structure from the

aspect of disassembly is made. The overall design for disassembly methodology is an iterative process which gives optimal

solution for product structure.

The disassembly process is started with the CAD model of the product assembly. The product consists of a

number of discrete components, such as, parts, fasteners, etc. Components can be grouped in subassemblies. A

subassembly is a connected set of components and fasteners. If components are physically linked, such link is called a

connection. If the components are nearly in touch with each other, this can be considered a virtual connection in some

cases [4, 6]. Connections restrict the freedom of motion of the components involved. This can be established in different

ways, the most uncomplicated way is mating [3, 7, 8]. In many cases, specialized components, or parts of components,

called fasteners, are used for connections. Fasteners can be discrete components such as screws, or non-discrete material

objects such as snap fits, press fits, etc [3]. The set of components can be given by the following expression:


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},...,,{ 21 nCCCC= (1)

The set of fasteners can be given by the following expression:

},...,,{ 21 mFFFF= (2)

where n is number of components in the product assembly and m is number of fasteners in the product. The

assembly of product is composed from all components and fasteners and mathematically can be explained with the

following mathematical expression mn FFFCCCA ...... 2121= .

In order to demonstrate proposed mathematical model of non-destructive disassembly process, the assembly

product in the figure 2 is used as example. The main goal of the methodology is the determination of the mathematical

model for obtaining of optimal disassembly sequence. Constituent components of the assembly are: C1 = Component C,

C2 = Component D, C3 = Component A, C4 = Component B, C5 = Component E. Discrete fasteners in the product

assembly are: F1 = Fastener M6x30, F2 = Fastener M6x30, F3 = Fastener M6x30, F4 = Fastener M6x50. In the Figure 2 is

shown CAD model of product assembly and its constituent elements with exploded view. For simplicity of calculations the

abbreviation names for components and fasteners (C1, C2, and F1, F2, ) are applied. Non-discrete fasteners are

material objects and represent the whole with component. For this reason the component which have non-discrete fastener

in this paper will be defined as fastener.

The set of components in the product for example is },,,,{ 54321 CCCCCC= where number of the component is

n=5 and the set of fasteners is },,,{ 4321 FFFFF= where number of fasteners is m=4. All assembly is represented with the

following expression C1C2C3C4C5F1F2F3F4.

The relationship between components and fasteners in an assembly is required in order to determine all

subassemblies in assembly. For this goal will be defined contact matrix and contact diagram between components and

fasteners and components and components. The contact diagram represent visualization tool for analyzing of

subassemblies in the product assembly. If the component is in contact with some fastener in the assembly the element FCi

in contact matrix will be equal of 1, in otherwise 0. If the component is in contact with other component in the assembly

the element Cij in contact matrix will be equal of 1, in otherwise 0. The contact matrix between components and fasteners

can be representing with follow equation:

Figure 2: Exploded View and Assembly of Product


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[ ] mjniij

CFFC,...,2,1

,...,2,1

=

== (3)

The contact matrix between components and components can be representing with follow equation:

[ ] njni

ijCCC,...,2,1

,...,2,1

=

=

= (4)

In the Figure 3 is given contact matrix and diagram for example shown in the Figure 2.

Figure 3: Contact Matrix and Diagram Between Components and Fasteners, and Components and Components in

the Product AssemblySubassemblies from the first level are generated on the base of the matrix represented in the Figure 3.

Subassemblies from the first level are composed from a components and one fastener. In the case of the product assembly,

the set of subassemblies at the first level is: SA1={C2C3F1, C3C4F2, C3C4F3, C1C3C5F4}. From the matrix shown in

Figure 3, components and fasteners which cross-section of rows and column is 1 are combined. The set SA1 obtain from

contact matrix in general can be represented with the follow expression:

}1),(...),(),(...{ 2111 ===== jijijiFCCSA kjkii(5)

where ik(k=1,...,n) represents k-th row in the matrix FCandj =1,, m represents column in the matrix FC.

The subassemblies from higher the order in general can be represented with the following expression:

}},...,1{},...,1{

.....................{1111111111

fesi

nmjljteenljjhiimjjteehiin

CCthatsuchtfhs

SAFFCCSAFFCCFFCCCCSA

=

=++++

(6)

wheren

SA is the set from n-th level and 1+nSA is the set from n+1 level. Members from the set 1+nSA are

1111.........

+mth jjeeiiFFCCCC which are obtained with combination of the two members njjii SAFFCC lh ...... 11 and

njjee SAFFCC mlt ++ 111 ...... and both members are from the set nSA where is needed to exist one component which is

contained in both membersfs ei

CC = . Note: with agreements the same component which exists in both members from the

set nSA can be written once in the following way C1C1C2F1F2=C1C2F1F2. In the framework of the example the set

from the second level is obtained from the members of the set from the first level. The set from the second level is


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SA2={C2C3C4F1F2, C2C3C4F1F3, C1C2C3C5F1F4, C3C4F2F3, C1C3C4C5F2F4, C1C3C4C5F3F4}. The set SA of all

subassemblies which are obtained from the product can be obtained with equation:

nSASASASA = ...21 (7)

In the framework of the example the set SA={C2C3F1, C3C4F2, C1C3C5F4, C3C4F3, C2C3C4F1F2,

C2C3C4F1F3, C1C2C3C5F1F4, C3C4F2F3, C1C3C4C5F2F4, C1C3C4C5F3F4, C2C3C4F1F2F3,

C1C2C3C4C5F1F2F4, C1C2C3C4C5F1F3F4, C1C3C4C5F2F3F4}. The total number of subassemblies in the example is

14 subassemblies. All subassemblies are obtained with software which is made for analyzing of the disassembly sequences

and for obtaining of optimal disassembly sequence.

A component or fastener is material entity that can be separated from a product via nondestructive and destructive

disassembly operations [3]. In this paper non-destructive disassembly process will be considered. The input data in

definition of disassembly operations are all feasible subassemblies. The disassembly operations are needed for the

definition of disassembly sequences, among which the optimal one can be searched for certain parameters are set. The set

of disassembly operations can be defined with the following equation:

}}...{}...{......}...{}...{

......}...,...{,{

111111

1111

==

=

tjjhjjvggvggtjjhjj

tjjhjjtjjhjjnum

FFFFSAFZFFZFFYFFXF

SAFYFSAFXFFYFFXFDODO(8)

Note: That component (C)inwee

CCY ...1

= which is repeated inkii

CCX ...1

= , will be deleted in Y. The setZis

function from union of the components from the product assembly which is obtained from the sets Xand Y.DOnum = 1, ...

,Don, whereDon represents total number of disassembly operations.

The example is calculated in the special software made for analyzing of the disassembly where are obtained all

possible disassembly operations. The total number of disassembly operations is 15. Because there are a lots of numbers of

disassembly operations there will be shown only part of them in the Figure 4.

Figure 4: List of Disassembly Operations

The transition matrix TM and AND/OR graph have the same meaning, and can be obtained if all feasible

subassemblies and disassembly operations are known. In the previous sections, all feasible subassemblies and disassembly

operations are defined. can be formulated in the following way: the generic element ij is -1 if the j action

disassembles parent subassembly i, and is +1 if thej actions create the son subassembly, component or fastener i. All other

elements are 0 [11, 13].

Referring to the product assembly shown in Figure 2, the segment from TM is presented in Table 1. The first row

0 C1C2C3C4C5F1F2F3F4

1 C1C3C4C5F2F3F4 C2F1

2 C1C2C3C4C5F1F3F4 F2

3 C1C2C3C4C5F1F2F4 F3

4 C2C3C4F1F2F3 C1C5F4

5 C1C3C4C5F3F4 C2F1F2

6 C1C3C4C5F2F4 C2F1F3

7 C3C4F2F3 C1C2C5F1F4


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in TM represents the product assembly, the second group of rows represents all feasible subassemblies, and the third and

fourth groups of rows represent the fasteners and the components in the product, respectively. The columns represent the

feasible disassembly operations. In the matrix, a disassembly operation of a certain subassembly is represented by -1 in the

row corresponding to that subassembly. For some subassemblies, there are several feasible disassembly operations, which

are shown in the matrix with -1 in the corresponding colons. The columns that show a parent subassembly have -1, and the

two or more components and fasteners which are children of that subassembly, have 1 in the same column as the parent

subassembly.

Table 1: Transition Matrix for Example

For example, the product assembly C1C2C3C4C5F1F2F3F4 can be divided with disassembly operation 2 of

subassembly C1C2C3C4C5F1F3F4 and fastener F2. In the row where there is complete assembly will be put -1 and in the

rows where complete assembly is divided will be put 1 in the case of subassembly C1C2C3C4C5F1F3F4 will be put 1 and

for fastener F2 will be put 1. After the subassembly C1C2C3C4C5F1F3F4 can be divided with 12-th disassembly

operation of subassemblies C1C2C3C5F1F4 and fastener F3 and component C4. In 12-th column in the row of the

subassembly C1C2C3C4C5F1F3F4which is divided will be put -1, and in the row of subassemblies C1C2C3C5F1F4 and

fastener F3 and component C4 which are obtained will be put 1. Non-destructive disassembly process continues until all

components and fasteners in the product are disassembled.

The order of the disassembly operations in a specific disassembly process is called the disassembly sequence [3].

Definition: The order from disassembly operations knnno ,...,,, 21 , where knnno


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directed line shows which subassembly becomes a parent subassembly in the i th operation, and the black directed lines

point to the two or more child subassemblies, components or fasteners, resulting from this operation. In Table 1, with

directed line arrows, is shown an example of obtaining of one disassembly sequence: 0, 2, 12, 23, 27.

The all generated disassembly sequences for example is shown in Figure 5.

Figure 5: The List from all Disassembly Sequences

Many disassembly operations are not possible from the aspect of the priority for removing of the components and

fasteners, what limits the number of generated disassembly sequences. The priorities for removing of the components and

fasteners are determined from the CAD model of the product assembly by assembly analysis. The analysis gives the

possible disassembly directions, represented in a disassembly interference matrix, for removing of components and

fasteners.

DISASSEMBLY INTERFERENCE MATRIX

Disassembly interference matrix is obtained based on the directions of x, y and z axis, respectively [19]. The

matrix is with dimension n+m x n+m which depends on the number of components n and fasteners m in the product. The

elements in the matrix are binary pairs of numbersiii

zyx where i = 1, , n+m. If interference exists between components

or fasteners iC or iF and jC or jF , wherej = 1, , n+m and i


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The process continues until all the components and fasteners from the product assembly are removed. In that way,

a list which represents possible orders of removing of components and fasteners is obtained for all disassembly sequences

from lists shown in the figure 5. One order for disassembly sequence 0, 2, 12, 23, 27 for removing of the components

and fasteners from the example shown in the Figure 2 with defined order of removing directions is obtained.

The order is:

F2(+z), F3(+z), C4(+xyz), F1(+x), C2(+z), F4(+z),C3(-z), C5(+z), C1(xyz).

After determining the priority of detaching for components and fasteners in the product and after definition of lists

for all disassembly direction, all disassembly sequences are checked for possibility. Impossible disassembly sequences are

deleted and remain only possible disassembly sequences from which after that is obtained the optimal disassembly

sequence by some criteria.

The all possible generated disassembly sequences are shown in Figure 6.

Figure 6: The List from all Possible Disassembly Sequences

0 2 12 23 27

0 3 15 23 27

1C 2C nC...

1F mF...

1C

2C

nC

mF

1F

.........

++++++++++++

+++

+++

mnmnmnmnmnmnmnmnmnmnmnmn

mnmnmn

mnmnmn

zyxzyxzyx

zyxzyxzyx

zyxzyxzyx

,,,2,2,2,1,1,1,

,2,2,22,22,22,21,21,21,2

1,,1,12,12,12,11,11,11,1

......

...............

...............

...............

......

......

=xyzI

.........

(9)

(10)

(11)


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OPTIMAL DISASSEMBLY SEQUENCE

The optimal disassembly sequence is obtained based on disassembly times, revenues and costs in the disassembly

process. The disassembly time ofi th component or fastener in the product is calculated as:

EFWDISi TTTT ++= (12)

where: WT represents the time spent for tool change, tool placement, product placement for disassembling, tool

return and removal of a detached component or fastener; FT represents the time spent for separation of a specific fastener ;

and ET represents the time spent depending on the difficulty for detaching of the component or fastener (for example

corroded threaded connection). For different fastener types, time FT is calculated by different function. If a connection is

easily for remove, then time ET is zero.

The total disassembly time for the product is:

+

=

mn

i

DISiDIS TT (13)

Costs for i th operation itsC cos are:

LDISiits PTC =cos (14)

where LP represents the cost of manual labor in euro/hour.

The cost of complete disassembly is given with the following equation:

+

=

mn

i

itsts CC coscos (15)

The total revenue obtained from disassembly of components and subassemblies is:

+=k

i

Ri

g

i

iRRR (16)

where Ri represents the reuse value of i th subassembly or component in the product, RRi represents the

revenue from material recycling of the i th component, g is the number of reusable components and subassemblies, k

is the number of recyclable components and subassemblies.

The profit obtained from disassembly is:

tsCRP cos= (17)

For the disassembly sequences given in section 4, the profit function is calculated. The optimal disassembly

sequence is usually a partial disassembly sequence, because not all disassembly operations return profit.

The optimal disassembly sequence gives insight into the disassembly cost, the percent of recovered material and

other characteristics of the product. The lower the disassembly cost, the higher is the economic effect of the product


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recovery. The higher is the weight and volume of the recovered materials, the higher is the environmental benefit. These

criteria can give important information to the designer in order to compare the design variants and select those that return

higher value at product end-of-life and have lower negative effect to the environment.

CONCLUSIONS

In the paper, a mathematical model of non-destructive disassembly process for determination of the optimal

disassembly sequence is presented. The proposed methodology is applied in solving of a realistic problem in the product

design phase. With the developed software, all possible subassemblies and disassembly operations for the product can be

determined, based on the priority for detachment in different disassembly directions. Also, the optimal disassembly

sequence can be estimated based on the disassembly times, revenue and costs of disassembly. The input of geometric data

in the system is performed automatically by analysis of the CAD model of the product. The goal of the paper is to provide

a tool for DfD analysis of the product in the early phase of the product development, through generation of the optimal

disassembly sequence, fastener analysis and product structure examination. Such tool should help the designers during of

virtual design phase in order to satisfy the European waste directives, and to improve further the suitability of the new

products from the aspect of disassembly and recycling.

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A MATHEMATICAL MODEL OF NON-DESTRUCTIVE DISASSEMBLY PROCESS

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