Proceedings of the 5" IEEE International Symposium on Assembly and Task Planning BesanGon, France July IO-11,2003

Optimum disassembly sequence with sequence-dependent disassembly costs

A.J.D. Lambert

Technische Universiteit Eindhoven Dept. of Technology Management

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

Abstract

This research deals with disassembly sequencing, in particular the search for the optimum disassembly sequence. The existing exact method, which is based on mathematical programming is extended to deal with situations, in which the costs of disassembly operations are sequence-dependent. This is frequently appficabfe in practice, due to features such as tool exchange and reorientatiodredirection. Existing metho& have been compared with a new approach rhat is discussed and successfilly tested. Iteration has been applied, as it is the appropriate tool for the elimination of cyclic quasi-solutions. It is observed that a limited amount of iterations is required for eliminating cyclic quasi-solutions. A rapid convergence can be observed. This is ilfustrated here with a case from the literature. The new method is suitabk both in awembly, repair, and end-opfe disassembly, notably in considering selective disassembly and design for life cycle. By this, optimum disassembly sequences can quickly be obtained because the use of integer variables is avoided here. The model can be treated as a binary linear programming problem.

1 Introduction

Disassembly has become a promising topic of interest in the course of the past decades, because a solid insight in the disassembly process is beneficial to a variety of crucial applications: - Disassembly models are applied to the design of

assembly processes, by considering assembly as reverse disassembly within definite assumptions.

- Combined disassembly and reassembly is required in maintenance and repair.

- In end-of-life processing, selective disassembly is becoming an essential process.

- In design, the feasibility of both the assembly and the disassembly process should be accounted for, aimed at optimizing the product's life cycle performance.

Selective disassembly is the non-destructive detachment of components and subassemblies up to a definite disassembly depth, aimed at the removal of hazardous components, the recovery of valuable

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components, and the reduction of impurities. In general, the disassembly process is followed by dismantling and bulk recycling, which are destructive processes. Much work bas been done on determining the complete set of disassembly sequences and selecting the most appropriate sequence from this set. A disassembly operation is defined here as the separation of a (parent) subassembly, which is a connected subset of components into two (child) subassemblies. The optimum sequence can be determined with heuristic methods, metaheuristic methods, and exact methods. This paper deals with exact methods that are based on mathematical programming . By these methods, a profit fhnction is maximized, in which the disassembly costs that are assigned to disassembly operations are subtracted from the revenues of the resulting subassemblies or components.

Crucial in disassembly sequencing is the representation of the possible disassembly sequences in one network. Usually, the state diagram [I], the AND/OR graph [2], or the disassembly precedence graph [3] are applied. Each presentation has its pros and cons. In this paper, the AND/OR graph is applied, which is an acyclic hypergraph in which nodes are assigned to subassemblies and hyperarcs are assigned to operations. Each hyperarc points h m a parent subassembly toward two child subassemblies. If one of the child subassemblies is a single component, the corresponding branch of the hyperarc is not depicted. This is called the simplified notation. It avoids the creation of intricate graphs without loss of information. Every possible disassembly sequence is represented with a connected subgraph in this AND/OR graph. This is a possible starting point for selecting the optimum disassembly sequence.

2 Optimizing Disassembly Sequences

The number of possible disassembly sequences exponentially increases with the number of components in a product, which means that an NP- complete problem is encountered. If a complete search is carried out, a considerable CPU time is required, which becomes prohibitive even for modeling products with a modest number of

components. Many authors are relaxing this problem via the application of heuristic or metaheuristic methods. Heuristic methods offer the possibility of finding near optimum solutions at the cost of moderate CPU time. Metaheuristic methods also return near optimum solutions. Further drawbacks are that these methods require adapted software, and that the needed CPU time is considerable. Fortunately, the disassembly sequence optimization problem can be solved with exact methods by formulating it as a binary l ines programming problem, which even relaxes into a linear programming problem [4]. By this, the optimization procedure requires negligible CPU time, because linear programming bypasses the problem of NP-completeness. Although on a rather low level of detail at a first glance, this method offers a good insight in the disassembly process; A concise description of &is method is as follows: First of all, the product is defined as' a set of connected components, which are indicated with a letter. The connections between the components are represented with a connection diagram. Subassembties are connected subsets of the components that are present in the product. Feasible subassemblies are those that can be obtained via disassembly only. These are the nodes of the AND/OR graph. The structure of this graph is represented with a transition matrix T with elements Z, [4,5]. hthis'notation, i is an index for subassemblies, j is an index for operations. The element Tu equals 1 if subassembly i is created via operationj; it equals -1 if subassembly i is destroyed via operation j; it is zero in other wes.

Flow variables .yare assigned to every hyperarc representing a disassembly operation. A flow variable equals 1 if an operation is performed, and 0. in other cases. Although thus being a (0,l)-variable, the flow variables can be relaxed into a continuous variable because in a solution they automatically adapt an extreme value, i.e. 0 or 1, if no capacity constraints are added. For every node i a node constraint has to be formulated, which reads:

' .

-

- I .Thenode constraints guarantee that (1) a subassembly cannot be destroyed if it is not previously created, and (2) a subassembly will be destroyed by no more than one action. This is important because a typical subassembly can be destroyed in multiple ways. If complete disksembly is aimed at,.the 2 sign has to be replaced with the = sign.

- . Further model parameters are the cost and revenue vectors with components cj and r,, respectively. The 'vector component cj represents the cost of performing the disassembly operation j ; the vector component r, represents the revenue of selling

'

the subassembly i to a customer for reuse or recycling purposes. Its value is negative if a fee has to be paid. Although in many cases financial costs and revenues are considered, generalized costs and revenues can also be dealt with. These might account for environmental performance, complexity, and so on, or a combination of these features. The profit L' is the objective that has to be maximized. It is defined via:

An initial operation with the index 0 is added, which is represented by an extra flow.variable xo that is put equal to 1. As can be observed, the optimization problem thus can be expressed with a linear programming formulation. Solving the LP problem returns the values of xj that correspond to a Connected hypergraph, h which is a subgraph of the disassembly AND/OR graph. The matrix T, which reflects the structure of the AND/OR graph, guarantees that the returned hypergraph is connected indeed.

3 Optimizing with Sequence-dependent cos t

A principal drawback of the optimization method is that it does not deal with the sequence dependency of the disassembly costs. In practice, disassembly operations are composed of disassembly task. Some of these, such as tool exchange and product reorientation, are typically dependent on the preceding action, because this determines which tool and orientation are available at the beginning of the new operation. As a matter of fact, costs increase if additional tasks have to be performed. Thus only those sequences that require as few as passible reorientations or tool exchanges have to be accounted for.

So far, several authors have~dealt with this type of optimization problem. A heuristic is applied in [a], in which disassembly cost . i s proportional to the disassembly time. The cost of a particular disassembly operation, such as the detachment of a specific component, can be calculated via the analysis of all the tasks that are required for this operation. The heuristic that is applied by these authors uses penalty functions for quantifying the ease of disassembly. Two extra penalties, which are ,called the direction factor and the joint type change factor, are added to these costs. These account for product reorientation and tool exchange, respectively. These authors apply the heuristic to a product (personal computer) with 17 components called A throughQ.

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The detachment of these components is submined to the constraints that are depicted in the disassembly precedence graph of figure 1. The cost of detaching a component is assumed here independent of the subassembly from which it is taken. . .%.

Figure 1. Disassembly precedence graph of the PC, see [6].

An exact method for dealing with the sequence- dependent cost problem is in 171. In this paper, the method is applied it to a printer that is structured according to a hierarchical tree approach. This study is an extension of the method that has already been described in section 1. Use has been made of partial flow variables wJb which are analogous to the flow variables xi, but that equal 1 if and only if the disassembly operation k is performed immediately after the execution of the disassembly operation j. Costs c,~ are assigned to each partial flow variable, which is the cost of operation k i f it is performed next to operation j. This definition results in a cost matrix C rather than a cost vector. The authors use task precedence graphs instead of AND/OR graphs, which are directed graphs with operations as nodes and partial flow variables as arcs. For instance, wJk is pointing from node j toward node k. A source node with index 0, corresponding with the initial action, and a sink node s, which is reached when the disassembly process is stopped, are added. Partial flow variables are always pointing from the source node and, alternatively, toward the sink node. Intermediate nodes, which correspond with a real disassembly operation, can have both incoming and outgoing partial flows. The already defined flow variables xk follow fiom:

xk = x w j k V k # 0 i

(3)

A feasible sequence always starts with the source node, next it might visit one or more intermediate nodes but not more than once, and it ends with arriving at the sink node. Not every w,t is possible due to precedence constraints. However, in c o n t r a with the conventional AND/OR graph that is acyclic, cyclic quasi-solutions are not excluded in this case. This is relevant in parallel disassembly operations, when two subassemblies are created that each can be further separated. No precedence relations exist between the separation of each of these two subassemblies. If, e.g. a parent subassembly ABCD (letters correspond with components) is separated in the child subassemblies AB and CD, there is no precedence relation between the separation of AB in A and B, and that of the separation of CD in C and D. If the indices j and k are assigned to these operations, the optimal quasi-solution of the problem might include, apart from a linear sequence from the source to the sink, a cyclic sequence that goes back and forth between the nodes j and k. In this case, wik = w, = 1 for this particular j and k. Because two nodes are involved in this cycle, it is called a 2-fold cycle. Multiple cycles and m-fold cycles are also possible if various parallel disassembly operations exist.

Several methods can be applied to deal with the problem of cycles, each with is . specific advantages and drawbacks.

1. The use of a mixed integer programming trick, by which counters are added to the- nodes, for guaranteeing that every node is visited not more than once. This needs two additional integer variables per node, which are acting as counters. This, however, results in a CPU time that strongly increases with the size of the problem. This method. which has been discussed in [7l, is mathematically sound and can be performed with standard mathematical programming software.

2. The use of a state diagram instead of an AND/OR graph. In such a diagram, the nodes correspond with states rather than subassemblies. A state is a partition of the set of components that represents the original product. If the previously discussed subassembly ABCD is considered, its possible states are ABCD; AB,CD; A,B,CD, AB,C,D, and A,B,C,D. No cycles can occur in a state diagram, as it more sharply discerns between the order in which parallel operations are performed. The drawback of the state diagram is the larger number of nodes, because this number typically increases much stronger with the number of components in the original product, compared with the corresponding AND/OR graph. For example, if there are 10

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components, the theoretical maximum number of nodes in the state diagram is already more than the 10-fold of that of the corresponding AND/OR graph. Thus, although the required CPU time is strongly reduced because the absence of integer variables, the size of the model increases, which might be a serious drawback.

3. The use of additional constraints in the model that are aimed at suppressing the cycles explicitly. The I-fold cycles w,, are represented by the diagonal elements of the flow matrix and have been initially put equal to zero; 2-fold cycles are impeded by adding the constraint:

wlk + wb I1 Vjb’k ( 4 4 Three-fold cycles are impeded with:

w , ~ + wH + wb 1 2 VjVkVI (4b)

etc. Although this method is mathematically sound and the modeling transparent, the calculation requires, unfortunately, a CPU time that strongly increases with the size of the graph because of the exploding number of constraints.

4. Straightforward optimization with the a posteriori rejection of cycles. If the problem is considered a generalization of the optimization problem with sequence independent cost, it can be modeled as a binary linear programming problem. This can be solved, but the ‘optimum’ solution that is revealed is typically a quasi-solution, as it might include one or more cycles, thus having no physical significance. For avoiding this, the model is modified by adding an extra constraint that inhibits one of these cycles. Subsequently, the calculation is redone. Via iteration, one will finally arrive at a realistic optimum solution that does not contain cycles. Although one iteration can be performed in negligible CPU time, even if rather large models are considered, it can be feared that many iterations are required. Fortunately, only a. few iterations are required in practice. This seems’ contradictory with the large amount of cycles that are basically possible. However, if one considers an ANLYOR-graph with J operations, the maximum number of acyclic solutions is &en by:

, . - , . -

The maximum number of cyclic quasi-solutions is given by the expression:

Although both figures grow exponentially with J, the ratio AdCc tends to J with increasing J. Apart from this, it can be concluded from considering the terms of (Sa) separately, that the majority of the acyclic solutions incorporates a large number of operations. 4 CaseStudy

The above-mentioned method will be illustrated with a case that is adapted from Bourjault’s ballpoint [SI. The corresponding assembly drawing and connection diagram are depicted in figure 2. This figure also gives the symbols for the different components and connections.

F 0 E

E

F/

Figure 2. Assembly drawing of Bourjault’s ballpoint and its connection diagram.

ABCDEF

ABCDE ABCDF

Figure 3. The disassembly AND/OR graph of Bourjault’s ballpoint, with enumeration of operations and subassemblies.

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The corresponding AND/OR graph shows 15 subassemblies and 13 real actions, which are enumerated according to figure 3.

An additional flow variable xo is assigned to the virtual action that corresponds to the source node. The model reads as follows:

Maximize the profit:

-70 61 53 6 29 14 63 52 21 81 23 63 55 70 81 70 I I 81 56 94 51 65 91 48 36 52 45 55 79 9 26 45 69 47 79 3 16 57 33 43 46 25 14 14 5 51 53 7 72 48 25 75 84 31 54 26 56 97 43 1 17 84 73 88 53 86 36 20 21 53 7 86 17 16 29 21 34 69

4 21 5 24 16 8 73 80 22 9 5 96 4 96 23 55 12 19 5 81 49 9 58 30 61 3

I 88 70 98 16 99 32 3 54 96 54 3 41 72 83 19 70 84 17 84 18 72 53 70 17 54 60 97 73 91 54 28 10 48 62 67 10 3 82 38 19 32 93 68 17 60 50 80 98 98 41 73 40 16 66 83 73 38 82 87 ,88 16 84 31 62 57 79 4 46 68 5 98 41 48 69 20 21 5

subject to: Partial node constraint:

Z ( W b - w,) 2 0 Vj # 0 k

Aggregate node constraint, see (I): CT,~.X, 2 0 V i I

Ordering constraint

Slk - wIk 2 0

xk = wjk t lk

Vj Vk Flow aggregation, see (3):

I Initialization:

xo = I

The model parameters are: r,, c , ~ Tv and s,X; binary variables are: wlk; variables are x,. The index i refers to subassemblies and the indices j and k refer to operations.

The sequence matrix S refers to the operations that can be performed next to a definite operation. The row elements S,, are given by (0 0 0 0 0 0 0 0 1 1 1 0 0 0), which means that the operation 7 can only be followed by the operations 8 , 9, or 10, which follows straight from figure 3.

The transition matrix T has already been described in section 2. For example, the row etements T3> are p e n by (0 0 1 0 -1 0 0 0 0 0 0 -1 0 0), which means that subassembly 3 or ABCDF is created via the disassembly operation 2 and destroyed via the disassembly operations 4 or 11.

Let the nonzero elements of the cost matrix be put equal to: co,l=6; c0,2=2; c1,3=9, CI.II= 9; ~ 2 . ~ 3 ;

C ~ J F I ; ~ 3 . 5 4 ; C#,I~=I; ~ ~ , 2 = 2 ; ~,,13=6; ~ 5 . ~ 1 ; c,&; C,l0=7; C7,8=5; C7,9=7; C, lF2; CS,IO*; C9,F8; C9,10=3; clo,s+; C10,9=1; C11.6=~; Cll,lo=7; Ciz,F3; Cl2,9=4; c13,s=5; c13,10=2. The components of the revenue vector have been chosen as follows: r1=-40; r2=-35; r3=-30; r ~ - 2 5 ; r5=-20; r ~ - 1 5 ; r,=-10; rp-5; r9=O; rlo=5; rll=lO; r12=15; rls=20; rl,=25; r l ~ 3 0 . In this case, an optimum profit of 96 is obtained with the

non-zero partial flow variables: ~ 0 . 2 , w2.1, ~ 1 . 5 , w5.1. w , , ~ ~ . This results in the sequence of operations: 0-2- 4-5-7-10. No cycles are encountered in this particular case.

.For validating the approach, an unconstrained model i s also considered, which is a modification of (6af). Many cycles are potentially possible here. The model reads as follows:

Maximize the Drofit:

subject to:

C W b > C W l k Vj#O (7b)

xk = c w j k Vk i

xo = I (70

The elements of the cost matrix cIk are generated with Maple's random generator. The following matrix has been used, with j running from 0 through 13, and k running from 1 through 13:

Initially, an objective equal to 1136 is returned, which is a quasi-solution that consists of the linear sequence 0-10-2-3-13-7-1-5, the 4-cycle 44-12-84, and the 2-fold cycle 9-1 1-9. After forbidding the 2-cycle, an objective equal to 1129 is found, with a quasi-solution consisting of the linear sequence 0-10-2-9-5, the 4-fold cycle 1-3-13- 7-1, the 3-fold cycle 6-12-8-6, and the 2-fold cycle 4- 11-4.

With the additional 2-fold cycle forbidden, the objective equals 1128 is found, resulting from a quasi-solution that consists of the linear sequence:

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0-8-4-6-12-1 1-9-5, and the 6-fold cycle 2-3-13-7-1- 10-2. With the 6-fold cycle forbidden, an objective equal to 1126 is found, which is the result of a feasible linear solution that is visiting all the operations, viz.: 0-8-4-

Consequently, a solution is found in 4 iterations, each being carried out within a negligible (less than 1 second on a 100 MHz machine) CPU time. It can be observed that rather large sequences of actions are persistent during the iteration, for instance 10-2-3-13- 7-1-5, 4-6-12, and 9-11. This might point to a link with genetic algorithms 191. An unconstrained model with 20 operations has also been also solved within 4 iterations, although the number of potential cycles exceeds IOL9. By this it becomes apparent that the presence of cycles does not become an obstacle in the search for the optimum solution of the optimum disassembly sequence with sequence-dependent costs.

6-12-9-1 1-10-2-3-13-7-1-5.

5 Conclusions and Further Research

Different .methods in optimum disassembly sequencing with sequence-dependent cost have been discussed in this paper. An extension of the already available exact optimization methods has been demonstrated, which could be solved via a rapidly converging iteration process, thus dealing with the possibility of cyclic quasi-solutions. Each iteration includes a binay linear programming optimization process, which can. be quickly executed. Iteration is a promising altemative compared with earlier described methods, which are based on heuristics, or exact methods that makeuse of integer programming. This method can be applied. in a variety of disassembly sequencing problems, including those that deal with models of an increasing size, as no NP-completeness is encountered. By this, the modeling of disassembly processes becomes closer to practice. Further research will deal with the class of cases that are represented with a disassembly precedence graph, such as discussed in 161, and with products that are arranged according to a hierarchical tree structure. It is recommended that the application of “heurist ic methods, such as genetic algorithms, to this class of problems should also be considered.

References

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