T1B-3 09.20 Proceedings of the 4th IEEE International Symposium on Assembly and Task Planning Soft Research Park, Fukuoka, Japan May 28-29,2001

AUTOMATIC DETERMINATION OF 'TRANSITION MATRICES IN OPTIMAL DISASSEMBLY SEQIJENCE GENERATION

A.J.D. Lambert Department of Technology Management

Technische Universiteit Eindhoven P.O.Box 513 5600 MB Eindhoven NEDERLND

a.j. d. lambert@tm. tue. nl

Abstract

This paper presents a method to automatically derive all the feasible subassemblies and transitions between them from assembly drawings. The method uses repre- sentation of the precedence relations via Boolean expres- sions. As a result, the transition matrix is automatically generated, representing the assembly's structure in the optimal disassembly sequence generation procedure. The method is applied to an assembly from industry.

oped to support this task via automatic processing of data that ;ire gathered by asking queries [ 1,2]. This paper pres- ents a method to rapidly formalise assembly structures. This method is aimed at generating data for the transition matrix, which contains the information about the assem- bly structure that is needed for determining the optimal disassembly sequence by the LP method. This paper starts with a recapitulation of this method, and proceeds with a description of the automatic transition matrix generation.

2. Automatic disassembly sequence genera- tion

1. Introduction

A combination of growing environmental concern and rapid technological growth has made the problem of discarded complex consumer goods (such as cars, white goods, and brown goods) a more urgent one. To deal with this, one of the basic tasks is to investigate selective dis- assembly, which is required for the recovery of parts and materials. Besides that, selective disassembly is of grow- ing importance to minimise the amount of shredder resi- due. Insight into the process of selective disassembly not only enables the design of effective reuse/recycling proc- esses, but also supports new concepts of product design that are aimed at optimising the product's complete mate- rial life-cycle. This includes design for recycling.

One of the challenges in this research is determining the optimal disassembly sequence, starting eom a given product structure, the disassembly costs, and materials revenue. Up to now, a lot of papers on disassembly se- quencing has already appeared. These have resulted in studies on the determination of assembly sequences, which were carried out to support the design of automatic assembly lines. Early studies on disassembly were fo- cused on repair, maintenance, and the extraction of spare parts. More recent work has been carried out that supports the design and the operation of disassembly lines for rec- ollected discarded complex consumer goods. The most promising method for calculation of optimal disassembly sequences is by linear programming (LP) and mixed inte- ger (MIP) programming methods. This requires formal- ised information on the product structure, which is con- densed in a transition matrix. The essence of this infor- mation embraces a listing of the feasible subassemblies and their feasible transitions. On top of that, costs of the transitions and revenues of the subassemblies should be listed. Canying out this work manually is an extensive and time-consuming activity. Methods have been devel-

Many papers have been published on optimal disas- sembly sequence generation. For an extensive review of many of those papers, see [3]. The methods involves those: based on mathematical programming, graph meth- ods, Petri nets, and on heuristic methods including fuzzy logic, genetic algorithms and neural networks. Which of those: methods is preferable depends highly on the level of detail that has to be considered. Mathematical program- ming, methods include LP- and MIP-based models, short- est distance approaches, and modified travelling salesman problems. The LP method was originally proposed in [4] and it has been M e r elaborated and extended in [5 ] . This method is appropriate if a level of aggregation is considered that abstracts fiom a detailed part level. In disassembly, this is often reasonable because less preci- sion is required compared with assembly operations. The algorithm offers the possibility for a quick calculation of the optimal disassembly sequence based on a representa- tion of the assembly structure that is reduced to a formal- ised expression. The data on the assembly structure are parameters that are arranged in the transition matrix T and in two vectors r and c, that represent the revenues of the different subassemblies and the costs of the different transformations between these respectively. This keeps the formulation of the model proper. In this paper, the algorithm is briefly summarized. It is formulated as fol- lows:

Let the disassembly problem have I possible subas- sembdies and J possible transitions between these subas- sembdies. The transition matrix T of size I X J is defined with elements Tg that are -1 if action j destroys subas- sembly i and that are +I if action j creates subassembly i . Other elements are zero. r, is a component of the revenue vector that represents the revenue of the subassembly i . c, is a component of the vector that represents the cost of action j .

0-7803-7004-W01/$10.00 0 2001 IEEE 220

The problem is then stated by the following expres- sions: Maximise the net revenue:

CC(TJ rl -'/)'I (la) I J

subject to: X T J (Ib)

J

with the initial condition: x, = 1 (1 c) It should be stressed here, that costs and revenues are

considered as generalised quantities. The requirement is that they are both quantificiable and comparable. Apart from financial units, also environmental performance, time, or another relevant quantity can be introduced here.

The transition matrix can be explained by the fol- lowing simple example, see figure 1.

(a) (b) (c) Figure 1. Simple assembly (a), irs connection diagram (b) and its disassembly graph (c).

Here the disassembly graph, which is an AND/OR graph, is presented in a simplified but complete represen- tation. The arc 5 that connects subassembly ABD with BD for instance, should be accompanied by its counter- part that points towards A. However, without introducing ambiguity, this additional arc can be omitted. Similarly, the single parts A through D are not represented in the diagram. The arcs from AD etc. pointing to the right rep- resent the separation into two single parts. It is seen that, because of the existence of precedence relations, not every combinatorically possible combination of parts re- sults in a feasible subassembly. This will be elaborated in the sequel of the paper.

The rows of the transition matrix T represent subas- semblies, the columns represent actions. For this example, the matrix reads:

1 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 ABCD 11 -1 -1 -1 ABD ACD BCD AD BD CD A B C D

1 -1 -1 1 -1 -1

1 -1 -1 1 1 -1

1 1 -1 1 1 -1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

The determination of the transition matrix thus pre- supposes an inventory of all possible subassemblies.

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From this inventory, the possible actions can easily be derived. The paper proceeds with a description of the method that generates all feasible subassemblies.

3. Formal description of the assembly struc- ture

Early papers principally deal with automatic genera- tion of feasible assembly sequences without automatic selection of the optimal one. In these papers, the AND/OR graph [6,7] and the connection diagram (liaison diagram) [ 1 ] were introduced. The connection diagram can be de- rived straight from the assembly drawing. The formal representation of the assembly has to be completed by a set of precedence relations. Originally, these are derived from a set of user queries. In [ 11, these queries were con- nection oriented. For each connection 1 between two parts, two queries had to be answered: a) #at connections must be established prior to establishing I . b) What con- nections must be left disestablished prior to establishing I. Each query could be answered using a Boolean expres- sion. All feasible sequences of operations could be de- rived from these formal expressions. An operation is de- fined here as the establishment of one connection and therefore is assembly oriented. However, ambiguities were introduced, because frequently the application of a part to a subassembly implies the simultaneous establish- ment of multiple connections.

A modified method has been presented in [2]. The queries in this method are related to detachability of a child subassembly from a parent subassembly (cut-set method). With a simple yes or no the queries had to, be answered here. However, the number of queries increases sharply, which can become restrictive for the investiga- tion of complicated assemblies, consisting of many parts. The recent paper of GUngOr and Gupta [8] addresses this problem from a heuristic point of view, presenting search algorithms that keep the search tree modest in size in the case the number of parts in the algorithm is considerable. In this paper, the size problem is dealt with by selecting the feasible subassemblies at beforehand. Therefore, the establishment of the appropriate set of precedence rela- tions has been addressed. Subsequently, this method has been tested on a frequently discussed case assembly: the so-called Assembly for Industry (AFI). The use of prece- dence graphs, as proposed by some authors (see, e.g., [9]) is less appropriate because of the complexity of the un- derlying logical relations.

With respect to the design of disassembly sequences, it has to be stressed that disassembly differs from assem- bly by an additional degree of freedom, namely the disas- sembly depth. Disassembly is usually carried out to a re- stricted extent (incomplete disassembly), because the op- timisation criterion is not completeness but rather profit maximisation within environmental constraints.

With this in mind, a method is presented here that is a) more appropriate to disassembly and b) needs a re- stricted number of queries. It proceeds as follows:

1.

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3.

4.

5.

6.

7.

The connection diagram is derived from the assembly drawing. A formal description is given for the parts that di- rectly prevent the removal of a definite part or con- nection. These expressions are equivalent to the set of precedence relations. An inventory of the coherent subassemblies is made using the connection diagram. With the precedence relations an inventory of the detachable subassemblies is made. The set of feasible subassemblies is the intersection of the set of coherent subassemblies and the set of detachable subassemblies. The set of feasible actions is the set of transitions between a feasible parent subassembly and two feasi- ble child subassemblies that are complementary with respect to the parent subassembly. With the sets of feasible subassemblies and actions, the elements of the transition matrix are established.

The term subassembly refers here to every set of single parts that is connected, which at least means that they touch each other, in other words, they are coherent. A further restriction on feasibility of a subassembly is that it has to be produced, thus it should be detachable. The restriction of stability is not relevant to disassembly, as instability can simply be dealt with by introducing a fur- ther disassembly action with no disassembly costs. This is in contrast to assembly, where often subassemblies are produced that, subsequently, are joined together. In disas- sembly, the concept of subassembly thus slightly differs fkom which is usual in assembly. Functional subassem- blies, such as an engine in a car or a disk drive in a com- puter, are usually referred to as modules.

It turns out that by this method the set of feasible subassemblies is straightforwardly generated. Constraints that result fi-om the specific requirements of the assembly process, such as the stability of the subassemblies, the needed precision of some actions, and the necessity of performing the assembly process to the full extent, are relaxed here. A justification of the method is presented in the following section, by applicating it to a frequently investigated assembly.

4. Case study

4.1. AFI without bolt connections

In this section, the method is both clarified and justi- fied. To this purpose, its application to the AFI is pre- sented in more detail. This will be done in two steps. First, the bolt connections such as applied in the original case description, are neglected. This increases the number of feasible subassemblies. In the subsequent section, the bolt connections are re-introduced. There the treatment of fasteners is elaborated. The assembly drawing of the AFI is presented in figure 2. The bolt connections that are indicated by M through Q in the figure are neglected in this section.

The connection diagram is derived from figure 2 and it is presented in figure 3. From this, an inventory of the co- herent subassemblies can be made. It appears, for in- stance, that AB is coherent and that AE is not coherent.

F E D C B J A G H K L

Q O P N M Figure 2. The AFI example. Source: [l].

J

Figure 3. The connection diagram of the AFI.

Additionally, the precedence relations are easily and unambiguously derived &om the assembly drawing. This results in the following 11 expressions: (& and RK and RJ or (RB and & and RD) + RA (RA and RG and R H and RJ) or (Rc and Rd + RB (RA and Rd or (RD and Rd * & (RA and or RE + RD (Rs and R d or RF + RE RF (RA and Rf3 or RH + RG & 01’ (RJ and R.J + RH (RB and RH) or RL + RJ RA or RL + RK RL

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In these expressions, R, is a Boolean variable that represents the removal of part i. RA is true if A is re- moved, and false if A- is not removed, etc. The 8* line means: part G or (part J and part L) have to be removed prior to the removal of part H. In other words: the r.h.s. of the expression can made true if and only if the 1.h.s. is true. It is clear &om this expression that disassembly has to start either with the removal of L or with the removal of F.

The justification of the detachability of a definite subassembly, say ABCDGH, proceeds as follows. Try to create ABCDGH by disassembling the original assembly, part by part. Therefore E, F, J, K, and L have to be de- tachable when ABCDGH is not removed. This implies that RA through RD, RG, and RH are false. Consequently, the relations that have to be tested reduce to the follow- ing: RF ---t RE RF

RL + RJ RL + RK RL

ABCDEGHJKL

Table I . The feasible subassemblies of the AFI example.

ABCDEFGHK ABCDEGHJK ABCDGHJKL

11 parts (1) ABCDEFGHJKL

5 parts (10) ABCDE ABCDG ABCDK ABCGH ABCGK ABGHJ ABGHK AGHJK BCDEF GHJKL

It is evident that all parts can be removed sequentially, e.g. according to the sequence: L, K, J, F, E. Thus ABCDGH is a detachable subassembly and should, con- sequently, appear in the disassembly graph.

Following the same procedure, ABDFGL appears to be undetachable. For instance, detachment of C is not possible because the Boolean expression in the 1.h.s. of the related precedence relation is false - A, B, and D are not removed. This can be confirmed by investigating the original assembly drawing. The calculation results in a list of 83 coherent and detach- able subassemblies. This is 4 % of the maximum of 2,047 subassemblies, which can be calculated by inserting N = 11 in the expression 2N - I for the maximum number of subassemblies [6] . The feasible subassemblies are listed in table 1.

As the amount of feasible subassemblies thus con- sists of only a minor part of the amount of coherent ones, the disassembly graph is reduced to a reasonable size. From this, the T-matrix is derived and, subsequently, the LP solver generates the optimum disassembly sequence within calculation time that can be considered negligible.

10 parts(2) I gparts(4) I 8 parts (6) ABCDEFGHJK I ABCDEFGHJ I ABCDEFGH

ABCDEFGK ABCDEGHJ ABCDEGHK ABCDGHJK ABCGHJKL

ABCG ABCK ABGH ABGK AGHJ AGHK BCDE CDEF GHJK HJKL

GHK

AG AK BC CD EF GH HJ HK JL

7 parts (8) ABCDEFG ABCDEFK ABCDEGH ABCDEGK ABCDGHJ ABCDGHK ABCGHJK ABGHJKL

1 part (11) A B C D E F G H J K L

6 parts (9) ABCDEF ABCDEG ABCDEK ABCDGH ABCDGK ABCGHJ ABCGHK ABGHJK AGHJKL

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The AFI example was thoroughly studied as a typical example of an important class of assemblies. This re- vealed some information about general properties of as- sembly structure. One of these properties involves the share of the feasible subassemblies with respect to the theoretical maximum. Generally, this share decreases sharply with an increasing number of particles. This is presented in figure 4. A nearly symmetrical curve ap- pears. This reflects a fundamental property of complex assemblies.

share of feasible subassemblies

1 111 11111 9

Ii -

0.5-

- - -

81330 101462 - 0

1 1 1 1 1 1 1 1 1 1

1 1 1 0 9 8 7 6 5 4 3 2 I

__.+ number of parts

Figure 4. The share of feasible subassemblies as a func- tion of the number ofparts, for the AFI case.

4.2. Introduction of fasteners

The presence of fasteners, e.g. bolt connections, is discussed in [lo] for the AFI. The authors added the original five bolt connections to the disassembly graph, which resulted in 5 additional parts and 10 additional connections (see figure 5).

Although the number of potential subassemblies thus increases fiom 2,047 to 65,535, the number of feasible subassemblies effectively decreases. This contradiction is met by considering the connective parts (e.g., bolts) as quasi-parts. Note that in a number of papers, such as in [ 1 11, the reverse takes place: a weld connection is consid- ered to be a part, which increases the number of potential subassemblies.

By considering connections as quasi-parts, the precedence relations are extended in two ways:

For each connection, a precedence relation is added. 0 The 1.h.s. of the precedence relations of the real parts

is extended by the restrictions due to the fixed con- nections.

J

Figure 5. Connection diagram for the AFI with fasteners included.

The modified precedence relations read as follows:

RM and RN and Rp and [(RG and RK and RJ or (RB and Rc and RD)] -+ RA Ro and [(RA and RG and RH and RJ) or (Rc and Rd] -+ RB RN and [(RA and Rd or (RD and R d ] + Rc Rp and [(RA and Rd or R E ] -+ RD Ro and RQ and [(RB and R d or R F ] --* RE

(RA and RS) or RH -+ & RG or (RJ and RJ ---t RH (RB and RH) or RL --* RJ

RQ -+ RF

RA or RL -+ RK

Rh4

RF --* Ro

Re

RM -+ RL

RD -+ RN

RE -+ Rp

This implies that disassembly should start with the de- tachment of the connections M or Q. Subsequently, part removal becomes possible. Additional restrictions result in a decrease of the number of detachable subassemblies. Consequently, subassemblies such as EF no longer are detachable, because RA = true presupposes Rp = true, which in turn presupposes RE = true. However, RE = false, because EF is considered.

In this case, addition of the bolt connections leads to a reduction of the list of feasible subassemblies by 7, namely: BCDEF, BCDE, CDEF, BCD, CDE, CD, and EF. By considering the connections as quasi particles, an unnecessary extension of the problem’s size is avoided.

4.3. Inventory of the actions

Once the possible subassemblies have been listed, the transitions (actions) between these subassemblies can be dletermined. We restrict ourselves to the separation of a pare:nt subassembly into no more than two subassemblies.

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Transitions can take place by detaching of a single part, a subassembly consisting of 2, 3 parts, etc. However, the child subassemblies siiould be feasible. This means that only those feasible subassemblies can be considered that are complementary with respect to the parent subassem- bly. This heavily restricts the number of actions, because only 4 YO of the theoretical maximum number of subas- semblies is feasible. For instance, ABCDEFGHJKL can be transformed into ABCDEF and GHJKL, which are both feasible subassemblies and complementary with re- spect to the parent subassembly.

For the studied example, with fasteners included, 132 different actions by which a single part, is removed, appear possible. Besides that, there are 52 actions through which a two-part subassembly is removed, 27 actions through which a three-part subassembly is removed, and 8 through which a four-part subassembly is removed. Fi- nally, there are 2 transitions through which a five-part subassembly is detached. The full problem thus consists of 76 subassemblies and 221 actions. This is exactly in accordance with the results that were presented in [2]. In our study, the original assembly as an additional node. Consequently, the T-matrix has 663 elements that are dif- ferent fkom 0 and the optimal disassembly sequence can be determined, if costs and revenues are assigned to the actions and the subassemblies, respectively.

5. Conclusions

The determination of optimal disassembly sequences can be carried out via a linear programming method. This requires the determination of the transition matrix, which presupposes the determination of all the feasible subas- semblies and the transitions between them. This task can be performed automatically by using the connection dia- gram and the formal representation of the assembly’s ge- ometry, which is obtained through a restricted set of precedence relations. Removal of a part rather than dis- establishment of a connection is considered a basic op- eration here. This reduces the complexity of the prece- dence relations and guarantees their unambiguity. Fasten- ers can be treated as quasi-parts, each of which adds a precedence relation and is included in the precedence relations of the real parts.

Justification by applying the method to a standard assembly frequently used as a reference in the literature revealed the complete disassembly graph. It was found that the precedence relations could be established without any complication. The connection graph and the prece- dence relations revealed the complete information for the generation of all coherent and detachable subassemblies. Based on this set, the inventory of the feasible actions can be established. This provides us with the complete infor- mation for assigning values to the elements of the trans- formation matrix in the linear programming model, which represents the structure of the assembly.

Acknowledgement

The research presented in this paper makes up part of the research on re-use in the context of the TMR project REVersed LOGistics sponsored by the European Union (ERB 4061 PL 97-650) in which take part the Aristoteles University of Thessaloniki (GR), Erasmus University Rotterdam (NL), MSEAD (F), the Otto-von-Guericke- Universitat Magdeburg (D), Technische Universiteit Eindhoven (NL) and the University of Piraeus (GR).

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