Precedence constrained TSP arising in printed circuit board assembly

Transcription

1 int. j. prod. res., 2004, vol. 42, no. 1, Precedence constrained TSP arising in printed circuit board assembly E. DUMANy* and I. ORz Component placement sequencing is a challenging problem that arises in automated assembly of printed circuit boards. While for some placement machines all placement sequences are acceptable, in other cases some sequences are not allowed because of the shape of the placement head. In such cases, while the head moves down to perform a placement, it might damage a previously placed component, and the problem of determining a minimum cost and at the same time acceptable sequence leads to a Precedence Constrained Travelling Salesman Problem formulation. In this study, a solution procedure to such a formulation is developed and its implementation in a real PCB assembly environment is discussed. 1. Introduction Computer or numerically controlled electronic component placement machines have brought speed, precision and reliability to printed circuit board (PCB) assembly operations. However, some serious operations research problems have also arisen through their use that need to be solved efficiently to achieve the maximum machine utilization. The major problems arising in such environments are placement sequencing, feeder configuration, machine workload balancing and board production scheduling, which are in many cases interdependent (McGinnis et al. 1992, Or and Duman 1996, Duman 1998, Ji and Wan 2001). The formulation, size and complexity of these problems depend mainly on the architectures of the machines used and the characteristics of the production environment. Component placement sequence refers to the order in which components are placed on the PCB; in other words, which paths are to be traversed by the placement head (or carrier board) to complete the placement of all components. For most machine architectures, the placement sequencing problem is formulated as the Travelling Salesman Problem (TSP) with the Chebyshev distance measure (McGinnis et al. 1992, Chan 1993, Or and Duman 1996, Duman 1998), while for some others it may also be formulated as a Rural Postman Problem (Ball and Magazine 1998). Additionally, in some machine architectures, some placement sequences may result in component damage (a placement head damaging a previously placed component during a placement operation) and these are not acceptable. Accordingly, the solution to be found should take account of some precedence constraints and thus leading to a Precedence Constrained Travelling Salesman Problem (PCTSP) (Chan 1993, Duman 1998). Revision received February ydepartment of Industrial Engineering, Dogus University, Istanbul, Turkey. zdepartment of Industrial Engineering, Bogazic i University, Istanbul, Turkey. *To whom correspondence should be addressed. eduman@dogus.edu.tr International Journal of Production Research ISSN print/issn X online # 2004 Taylor & Francis Ltd DOI: /

2 68 E. Duman and I. Or If the components are fed to the placement head from a feeder apparatus (a mechanism where component types are stored), then determining the configuration of the feeder is an important issue to be studied, since it directly influences the time between successive placements. Depending on the architecture of the machine used, the feeder configuration problem is mostly formulated as a linear or quadratic assignment problem (Ball and Magazine 1988, Or and Duman 1996, Duman 1998). If the manufacturer deploys more than one machine to populate a single PCB, then it is necessary to distribute the assembly workload to machines as equally as possible to avoid machine idle times. Together with machine workload balancing, it is also necessary to determine the production schedule of different board types to meet customer due dates or some other facility specific performance criteria. The formulation and solution of these problems are mainly dependent on the production characteristics (Carmon et al. 1989, Ben-Arieh and Dror 1990, McGinnis et al. 1992, Duman 1998, Hillier and Brandeu 2001). In the present study, the focus is on the PCTSP. In section 2, the operations of the placement machine leading to the PCTSP are described in detail and the placement-sequencing problem is defined. A solution procedure named Damage Reduction Procedure (DRP) is developed and described in section 3. In section 4, the implementation of DRP in a real PCB assembly environment and the results obtained are discussed. Finally, in section 5, the major conclusions are stated. 2. Description of the problem The component damage problem is usually faced in the use of machines placing radial leaded components or any other type of machine where the geometry of the placement head is in the form of an inclined plane. In the Turkish electronics industry, we have observed two such machine types that actually have the potential to cause component damage: Dynapert and Universal radial placement machines. This study is based on the observation and data collected about one such machine type: Universal brand, model 6346A radial placement machine. First the operations of this machine are described and then the placement-sequencing problem is defined Universal 6346A machine These machines have online sequencer mechanisms (also called the dispenser) located behind the machine, with room for up to 60 component types. The components are cut by the dispensing heads (DH) and dropped on a conveyor belt (for each feeder/sequencer position, there is a corresponding DH and both are numbered in the same manner), which carries the components to the placement head in the placement order (figure 1). The placement head is stationary in the x y plane and the alignment of the placement point under the head is performed by the independent and simultaneous motions of the carrier board (the platform that the PCB is placed on) in x and y dimensions. After the placement point is aligned beneath, the placement head picks up the next component from the conveyor belt, moves down and makes the placement. A more precise, step by step operation of the machine is as follows:. Carrier board makes the x y movement.. Loader moves forward and feeds the component to the placement head.

3 Precedence constrained TSP arising in PCB assembly 69 Component Tape Component Placement Head Conveyor Belt Figure 1. PCB Carrier Board Universal 6346A Radial Placement Machine.. Loader pulls back.. Component to be placed next is moved before the loader.. Placement head moves down (rotating if necessary).. Cut and clinch head moves up.. Pusher pushes the component down to the board.. Component is cut and clinched.. Placement head moves up.. Cut and clinch head moves down. These itemized tasks are performed sequentially and all but the first one are independent of the component type and placement sequence (having a constant duration of 0.34 s per placement cycle, based on technical documentation and actual observation and timing data). The placement head can make placements at one of the three rotation angles: 0, þ 90 or 90. As mentioned above, the head can perform the necessary rotation action while it is moving down and there is no time loss due to this action. The components to be placed on a board can have either horizontal or vertical orientation and they can be electrically polar or non-polar. The carrier board of the radial machine cannot rotate, so if there are any polar components to be placed at 180, that component should be allocated a second slot in the dispenser into which it should be stocked in reverse polarity (which then can be placed at 0 ), or if there is no available additional slot in the dispenser, it should be placed manually Problem definition With such machines, since the components are available for placement when needed, there is no feeder configuration problem (the position of the component types around the sequencer is not critical) and the problem reduces to finding the optimal component placement sequence.

4 70 E. Duman and I. Or Disregarding the component damage problem, the placement-sequencing problem can be formulated as a regular TSP with the Chebyshev distance measure (Duman 1998): min XN i¼1 X N j¼1 j6¼i t ij x ij ð1þ s:t: X N i¼1 i6¼j x ij ¼ 1 j"n ð2þ X N j¼1 j6¼i X x ij ¼ 1 i"n ð3þ X i2s j2s, j6¼i x ij jsj 1 for all S N x ij ¼ 0or1 i, j¼ 1,..., N i 6¼ j, where, N set of all placement points, S any non-empty proper subset of the set N, x ij ¼ {1, if placement at point i precedes placement at point j; 0, otherwise), t ij time between the completion of consecutive placements at points i and j. Here, t ij is the Chebyshev distance (time) between two placement points i and j. That is, if t 0 actions taking a fixed time, t 1 ij carrier board movement time in x direction between points i and j, t 2 ij carrier board movement time in y direction between points i and j, then t ij turns out to be the following: ð4þ t ij ¼ t 0 þ maxðt 1 ij, t 2 ijþ: ð5þ However, the placement-sequencing problem of the radial placement machine cannot be formulated as an ordinary TSP. This is because due to the physical shape of the placement head, in some placement sequences the head may crush and damage previously placed components, if they are sufficiently close to the back side of the head (figure 2). The exact definition of the component damage area is essential to develop a solution procedure to avoid damage. Obviously, the closeness of components to one another is a potential damage indicator; if a component is inside the projected area of the placement head from the top view, it may be damaged (figure 3). Together with the closeness, the heights of the nearby components are also important since they determine whether or not the 45 inclined plane of the head will touch them. Considering also the width of the head, the damage area can be taken as the rectangular area behind the current placement, which has a varying length (depending on component height). More precisely, having a fixed width, the length of the rectangular damage area (d) is determined by the multiplication of the component

5 Precedence constrained TSP arising in PCB assembly 71 component to be placed placed component 45 o head movement Figure 2. Shape of the radial placement machine and previously placed components facing with damage. d a h D Figure 3. Component damage area. height (h) and the cotangent of the angle the placement head makes with the x y plane (a). However, there is an upper limit (D) to the length of the rectangular damage area, which is determined by the end of the inclined plane of the placement head. To avoid component damage, the component placements causing damage on others should be placed at an 180 opposite angle (possible only for þ 90 or 90 placements) or their placement order should be changed. Therefore, the placement sequencing problem turns out to be a kind of precedence constrained TSP (PCTSP). A similar precedence constrained TSP in the context of the assembly of PCBs is considered by Chan (1993), where the details of the proposed solution procedure are not given. The mathematical formulation of the PCTSP faced here is not straightforward since the constraints are not related with the placement sequence only. The fact that some damage can be avoided by placement angle or placement sequence changes does not allow a simple closed form mathematical formulation like the standard TSP case and solution of this constrained sequencing problem becomes an even more difficult task. Accordingly, the following approach is chosen: formulation of the problem as a pure Chebyshev TSP, and then the application of a subprocedure to disregard placement sequences leading to component damage within the solution procedure. This methodology is described in section Solution methodology As indicated above, the solution methodology developed is based on initially ignoring the precedence constraints, solving the problem as a pure Chebyshev TSP, and then applying a procedure to eliminate the damage conditions in the

6 72 E. Duman and I. Or resulting TSP tour. For the solution of the pure TSP, we used the Convex Hull and Or Opt algorithms and to avoid the damage, we developed a procedure named the Damage Reduction Procedure (DRP). These algorithms are described below Convex Hull and Or Opt algorithms The Convex Hull algorithm was first proposed independently by Or (1976) and Stewart (1977). It is a heuristic procedure that starts with a subtour consisting of the convex hull of all points to be visited. Then, at each iteration, a candidate point not on but closest to the current subtour is determined and included into the subtour by eliminating the closest arc and connecting its endpoints to the candidate point. The details of the Convex Hull algorithm can be found in Or (1976) and Stewart (1977). Starting with the solution found by the Convex Hull algorithm, the Or Opt procedure aims at improving a current tour by considering all possible relocations of every point, every two consecutive points and every three consecutive points along that tour (Or 1976). The stepwise description of the Or Opt implementation used in our study is as follows. Step 1. Starting with some first point in the given tour, consider all three consecutive points; temporarily remove them from the tour and consider inserting them in their normal order or reverse order between any two other consecutive points in the tour (while considering the point of insertion, start with the two points coming right after the removed three points and proceed clockwise). Make the first insertion that yields an improvement in tour cost permanent. Continue testing other three consecutive point exchanges until the start point is reached. Step 2. Repeat Step 1 for all two consecutive point exchanges. Step 3. Repeat Step 1 for every single point exchanges. According to the original work of Or (1976), there can be alternative strategies in the above procedure: In Steps 1 3, the insertion to be made permanent could be either the first one leading to a cost improvement, or (after all possibilities being investigated) the one resulting in the highest cost improvement. Some experimentations have been accomplished to test the performance of both these alternative strategies and the choice of accepting the first insertion leading to a cost improvement was slightly superior and thus it is acted upon Damage reduction procedure The damage reduction procedure (DRP) is designed with an eye to cause minimal increase on the unconstrained TSP tour cost. It first tries to eliminate damage by considering the placements causing damage, to be made with a 180 orientation (this is possible only for placements at þ 90 or 90 orientation and it does not increase the TSP cost); if this is unsuccessful in eliminating a potential damage, various alternatives involving route changes are then considered. The details of the procedure are given below. Step 1. Finding tour direction: find the number of component damage in the TSP tour generated by the Convex Hull and Or Opt algorithms, both in clockwise and counter-clockwise directions of traversal. Pick the one, which results in fewer number of damage, and continue to the next steps with that TSP tour.

7 Precedence constrained TSP arising in PCB assembly 73 Step 2. Angle change in non-polar components: if the þ 90 ( 90 ) placement of a non-polar component is causing damage, then consider it being placed at 90 ( þ 90 ) and check whether the damage condition has been eliminated and no new damage arises. If positive, change the placement angle, if not leave as is. Step 3. Angle and DH change in polar components: if the þ 90 ( 90 ) placement of a polar component is causing damage and if that component type is assigned (or may be assigned) to two dispensing heads in the dispenser with reverse polar orientations, then consider a pick up from the alternative feeder location (with the reverse polar orientation) and a placement at 90 ( þ 90 ); check whether the damage condition has been eliminated and no new damage arises. If positive, reverse the placement angle and change the DH number from which the component is obtained for that placement, to the alternative DH number in which that component type is placed; if not, leave as is. Step 4. Placement sequence changes: find the remaining component damage and identify series. Change the order of the involved components in the placement sequence according to the following method. Trace the damaged component list marking consecutive components as a series. Consider each such series (of damaged components); reverse their order and put them temporarily after the first non-damaged component in the sequence. Then, using the Or Opt procedure, try to find a better place for this group in the overall placement route by considering inserting them in their normal order at other points of the route. If a better place cannot be found, make their temporary place permanent. Step 5. Further angle changes: apply Steps 2 and 3 to eliminate, if possible, some of the new damage introduced with the sequence changes of Step 4. Step 6. Repetition: if there are still remaining damaged components, apply Steps 4 and 5 repeatedly until all damage is eliminated or until a prespecified number of iterations have been accomplished. As terminology in this study, application of Steps 1 5 is called one application of DRP and then each application of Step 6 counts for an additional application of DRP. In Step 4, the term series means a group of consecutive components damaged. A series of damage is a common structure especially in dense PCBs. For example, due to required circuit design, a number of capacitors may be put one next to the other to obtain a higher capacity. However, for convenience, we use the term series even the preceding and succeeding components of a damaged component are undamaged. Regarding Step 4 of the DRP, note that component damage is generally caused by the succeeding placement operations. This is because damage condition is related to the closeness of the components and the components whose placement locations are close are usually placed consecutively due to the nature of the Convex Hull algorithm. Because of this, in Step 4, damaged components are put after the first non-damaged component in the route expecting that the damage condition(s) will probably be eliminated. If this is not the case and if a later component in the sequence is the actual cause of the damage, damage condition may not be eliminated at first but in later applications of DRP damaged components will be detected again and will eventually be placed after the component actually causing the damage.

8 74 E. Duman and I. Or The above-described DRP has two particular features. First, in implementing the Or Opt, it seeks for a better place only in normal order of components and, second, for the insertion of these components, only the positions till the end of the sequence are considered (that is, the damaged components are not considered for insertion between any two placements coming before them with respect to the TSP tour currently at hand). The logic behind the first feature is that if Or Opt was allowed to re-reverse the order of components, the damage condition may reappear. Although this time the number of consecutive damage will most probably be one fewer (this is because in serial damage conditions generally a component placement causes damage on the component placed just before it and as they are now all after the component causing the last damage, the number of consecutive damage decreases by one) and as the damage prevention rules are applied in a clockwise orientation, potential damage will be detected again and tried to be cleared, reverse Or Opt is not preferred since it will increase the computation time. On the other hand, the second feature avoids the repetition of the identified potential component damage. Note that if a damaged component is relocated to a point before the component damaging it in the current directed TSP tour, it will still be open to damage by the same reason. Thus, each such component should be placed after the placement that has the potential to damage it. The above description of DRP should not imply that it will eventually eliminate all damage conditions. Recall that Steps 2 and 3 are only taken if no new damage condition arises. If a new damage arises, DRP will fail to eliminate that damage in the current application. However, in later applications of DRP, due to the order changes made in previous applications, no new damage may arise and the DRP could become able to eliminate the damage that it could not do before. On the other hand, there may be cases where a component is damaged by the placement of multiple components. In these cases, it may not be possible to eliminate all damage algorithmically and a change in the circuit design may be required. A pilot implementation of the above-described DRP is made in a television sets manufacturing facility. The results of this study are presented below. 4. Implementation results To finalize the placement sequencing in the radial machines, through the application of the DRP, one further adjustment was necessary. In Step 1 of the DRP, while calculating the numbers of component damage, the manufacturing facility could not provide the data of component heights. Thus, it was assumed that any component closer than 12 mm (which is approximately the height of the tallest component) to the back side of the head is subject to damage. The placement head had a width of 9.5 mm and considering also the radius of the components, the width of the damage area is taken as 10 mm. In other words, if there were any components inside the mm rectangle at the back side of the head, it was assumed to be damaged. For this reason, note that the components identified as damaged by our algorithm may not be actually damaged and thus it is better to call them potential damage. At the time of this study, the television set manufacturing facility was using an optimization package which was using the nearest neighbour (NN) algorithm, where the distance between two placement points was taken as the Euclidean distance

9 Precedence constrained TSP arising in PCB assembly 75 between the points. The optimized solution obtained by the NN algorithm was taken as the input data to our algorithm to be improved further. The software developed for this purpose was named VOPT. Twelve television main board types were considered and best placement sequences were found by VOPT. The simulated solutions of NN and VOPT were compared and the percentage improvements determined. Several earlier tests made indicated that the comparisons made based on simulated solutions almost exactly reflected the real life situation and, thus, the simulation results could be deployed effectively. The full simulated results of a one application of the DRP on these 12 types are shown in table 1. In table 1, VOPT before DRP stands for the TSP solution using Convex Hull and Or Opt algorithms, whereas VOPT after DRP shows the assembly time obtained when DRP is applied once to the result obtained by Convex Hull and Or Opt algorithms. Note that VOPT after DRP assembly times are slightly larger since prevention of damage may increase the unconstrained TSP tour cost. In all 12 boards, the component placement area is a cm rectangle and the number of components (n) is from seven to 280. In comparing the NN and VOPT results, two different indicators are used. The first is the improvement in the total assembly time, which we call the real improvement ; the other one is the comparison after the constant 0.34 s per placement is disregarded from the cost of the solutions. Since nothing can be done about the constant times, this second performance measure offers a more unbiased comparison of the said methods and we call it the TSP improvement. There are various criteria to test the performance of the VOPT solution procedure. The real and TSP improvements for the tour costs, the performance of damage reduction rules, the number of times the damage reduction procedure should be applied are some of these performance measures. As can be seen in table 1, when VOPT is used and the damage reduction procedure applied only once, the real improvements range from 0 to 8.53% (with a simple average of 2.13%), whereas the range of TSP improvements is from 0 to 29.23% (7.02%). The weighted averages of these improvements (based on the number of components placed) are 1.96 and 6.47% for real and TSP improvements, respectively. Figure 4 shows the scatter diagram of the TSP improvements for the radial machine. The improvements tend to stabilize around 6% as the number of components increases. Regarding the 12 radial placement test problems, the average number of potential component damage is 36.8 after the application of the NN algorithm. After the damage reduction procedure is applied once (in VOPT), this number is reduced to 3.6, which means a 90% improvement in component damage (see table 1 for details). Note that since in an actual implementation the NN placement sequence will almost certainly be manually altered by the machine technician to avoid the potential 36.8 damage, the production time will most likely be increased and, thus, the above-mentioned improvements will be larger (both the real and the TSP improvements). Furthermore, the manual intervention of the operator in this case is likely to introduce human error and major increases in set-up time. When the damage reduction procedure is applied more than once, the amount of potential component damage is reduced further and, on average, after four

10 No. of damages Assembly time (s) VOPT Board name n VOPT NN NN Before DRP After DRP Constant time (s) Imp. damages (%) Real Imp. (%) TSP Imp. (%) ak17n ak ak ak ak ak ak ak ak sb pip tk average E. Duman and I. Or Table 1. Implementation run results.

11 Precedence constrained TSP arising in PCB assembly 77 percent improvement number of components Figure 4. TSP improvements for the radial machine. BOARD n init. time 1. time last time Dam(NN) ak17n ak ak ak ak ak ak pip Table 2. Convergence of potential component damages by VOPT. iterations, it converges to either zero or an acceptable small number (table 2). The meanings of column titles in table 2 are as follows: BOARD name of the board, n number of components to be placed, Init. time cost of the TSP tour before the application of DRP, 1. time cost of the TSP tour after one application of DRP-A, Last time cost of the TSP tour after 20 applications of DRP-A, Dam(NN) number of potential damage according to the NN sequence, k¼ 1,..., 20 number of potential damage after k applications of DRP-A. 5. Summary and conclusions In this study, the placement sequencing optimization problem of Universal brand radial placement machines was undertaken. The placement sequencing problem of such machines turns out to be a kind of precedence constrained TSP, for which it is not possible to provide a closed form mathematical formulation due to the special structure of the precedence constraints. Accordingly, it is formulated and solved as a pure Chebyshev TSP and then the Damage Reduction Procedure (DRP) is applied to eliminate the component damage in the resulting TSP tour. The developed DRP is quite successful in eliminating the component damage (the number of damages were decreased by 97.5% against the output of the NN

12 78 E. Duman and I. Or algorithm) and the cost of eliminating these damages was only a 0.9% increase in the unconstrained pure Chebyshev TSP solution. The pilot implementation carried out in the television sets production facility not only resulted in about a 2% direct increase in throughput (against the NN algorithm), but also helped to provide radial placement sequences which have only a very minor number of potential damages (which may not actually be damaged as stated above). Considering the fact that the machine technicians were spending almost one shift manually altering the given NN placement sequence, in order to avoid potential damage before starting the serial production, the effective increase in throughput is actually quite larger. On the other hand, as compared with manual correction of damage and thus being open to errors due to human judgement and being dependent to the existence of experienced technicians, the suggested approach removes such uncertainties and facilitates full automation of operations. References BALL, M. O. and MAGAZINE, M. J., 1988, Sequencing of insertions in printed circuit board assembly. Operations Research, 36, BEN-ARIEH, D. and DROR, M., 1990, Part assignment to electronic insertion machines: two machine case. International Journal of Production Research, 28, CARMON, T. F., MAIMON, O. Z. and DAR-EL, E. Z., 1989, Group set-up for printed circuit board assembly. International Journal of Production Research, 27, CHAN, D., 1993, Precedence constrained TSP applied to circuit board assembly and no wait flow-shop. International Journal of Production Research, 31, DUMAN, E., 1998, Optimization issues in automated assembly of printed circuit boards. PhD thesis, Bogazici University. HILLIER, M. S. and BRANDEU, M. L., 2001, Cost minimization and workload balancing in printed circuit board assembly. IIE Transactions, 33, JI, P. and WAN, Y. F., 2001, Planning for printed circuit board assembly: the state-of-the-art review. International Journal of Computer Applications in Technology, 14, McGINNIS, L. F., AMMONS, J. C., CARLYLE, M., CRANMER, L., DEPUY, G. W., ELLIS, K. P., TOVEY, C. A. and XU, H., 1992, Automated process planning for printed circuit card assembly. IIE Transactions, 24, OR, I., 1976, Traveling salesman type combinatorial problems and their relation to the logistics of blood banking. PhD thesis, Northwestern University. OR, I. and DUMAN, E., 1996, Optimization issues in automated production of printed circuit boards: operations sequencing, feeder configuration and load balancing problems. Proceedings of the Workshop on Production Planning and Control, Mons, Belgium, pp STEWART, W. R., JR, 1977, A computationally efficient heuristic for the traveling salesman problem. Proceedings of the 13th Annual Meeting of S. E. TIMS, pp