IMMD VII, University of Erlangen Nürnberg, Martensstr. 3, Erlangen, Germany.

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1 A DESIGN METHODOLOGY FOR KANBAN-CONTROLLED PRODUCTION LINES USING QUEUEING NETWORKS AND GENETIC ALGORITHMS Markus Ettl and Markus Schwehm IMMD VII, University of Erlangen Nürnberg, Martensstr. 3, Erlangen, Germany. One way to reduce costs in high volume production lines is to smooth and balance the material flow by means of controlled inventories. Kanban systems are now being implemented worldwide due to their ability of reducing inventories and production lead times. This paper addresses two fundamental design issues in kanban systems and presents an efficient heuristic method for designing such systems. An analytical technique for modelling kanban systems and a general-purpose genetic algorithm are integrated in a heuristic design methodology which evaluates the performance of kanban systems using alternative network partitions and allocations of kanbans. As we demonstrate, the heuristic method provides a useful procedure to evaluate the impact of design alternatives and can thus serve as a rough-cut decision support tool which assists managers in the planning of large-scale manufacturing systems. 1. INTRODUCTION The manufacturing industries have seen a dramatic change away from high product throughput and high capacity loads towards lower production lead times and work-in-process inventory. Pull-type production control aims at reducing costs through keeping the work-in-process inventory at a minimum level and thus improving the company s ability to adapt to changes, e.g. demand and production fluctuations. Kanban systems are often used to implement the pull-type control in production systems. The success of Toyota s kanban system has motivated considerable research of practitioners and analysts in order to gain a better understanding of the operation of pull systems. In this paper, a pull-type production line is considered which is composed of a sequence of production stages performing various process steps on parts. Each production stage consists of several work stations in tandem. The flow of parts through the overall facility is controlled through a combined push/pull control policy which is implemented by means of kanbans. A push-type policy is used to produce parts within each production stage. Between different production stages, however, the parts are pulled according to the rate at which parts are consumed by the downstream stages. Two well-known kanban systems appear as special cases of the above kanban control policy, i.e. the minimum blocking policy described by Mitra and Mitrani [1] and the Constant Work-in-Process (CONWIP) system proposed by Spearman et al. [2]. In order to effectively operate such systems, a design methodology is needed to assist managers in solving the following production plant configuration problems [3]: Partitioning Problem: Given a sequence of work stations, how many production stages should the production line be divided up into? In addition, how many work stations should be assigned to each individual production stage? Kanban Allocation Problem: Given a partition of the production line, how many kanbans should be assigned to each production stage such that a desired product throughput is achieved while the inventory carrying costs are minimized. The purpose of this paper is to develop a design methodology which produces reasonably accurate solutions to the above configuration problems. Such a methodology is likely to facilitate the design of kanban systems and should also enhance our understanding of their behavior. In our methodology, the optimal partition and kanban allocation is determined from a global optimization model in such a way that the average work-in-process is minimized while a predetermined throughput rate must be guaranteed. In favor of a computationally efficient design process, we use a closed queueing network representation of kanban systems which is evaluated using an approximate analytical method [4]. A general-purpose genetic algorithm package is then used in order to simultaneously determine the optimal network partition and kanban allocation. Applying the genetic algorithm is straightforward, since the output of the modelling process can directly be used as the objective function.

2 Previous research in this area often uses simulation to study various performance and implementation aspects of pull systems [2, 5, 6, 7]. Analytical models of kanban systems have been developed only recently, see for example [1, 8, 9, 10]. The issue of determining the number of kanbans has been addressed by Davis and Stubitz [11] and Philipoom et al. [7] by means of simulation. Jothishankar and Wang [12] developed a stochastic Petri net model to find the optimal kanban allocation in a twostation kanban system. Siha [13] derived a continuous time Markov model to investigate various kanban allocation patterns for up to five-station kanban systems. Little attention has yet been devoted to the partitioning problem. Tayur [3] studied the partitioning problem in serial kanban systems by means of sample path methods. Johri [14] discussed the problem of shop floor control (without kanbans) in semiconductor wafer fabrication and proposed to partition the set of process steps to be performed on the wafer into individual stages called zones. The optimal partition is determined using a dynamic programming formulation of the problem. For practical purposes, however, it is evident that powerful heuristics are necessary in order to solve problems of realistic size. The present paper is a step in this direction. The paper proceeds as follows. In Section 2, we describe the operation of kanban-controlled production lines and discuss the design problem under consideration. The approximate algorithm used in the heuristic methodology to evaluate the performance of kanban systems is also briefly reviewed. in Section 3, a genetic algorithm is presented which is applied to solve the design problem for kanban systems. Section 4 provides numerical experiments and Section 5 concludes the paper. 2. KANBAN-CONTROLLED PRODUCTION LINES The production system studied here is a tandem network of N work stations which are distributed among S production stages (Figure 1). Each production stage is assumed to consist of one or several work stations in tandem, each equipped with an unlimited local buffer to store unfinished parts. In production stage i, there are K i kanbans and N i work stations. A part must acquire a free kanban in order to enter production stage i, which is attached to the part as long as it is in that stage. When the processing of a part has been completed in production stage i, the finished part is moved to an output buffer where it waits for admission to the downstream stage (stage i + 1). The kanban associated with a finished part is detached as soon as the part is withdrawn by the downstream stage. The unattached kanban is then returned without delay to the input buffer where it serves as a pull signal for the upstream stage (stage i? 1). Production Stage 1 Input Buffer Production Stage 2 Output Buffer Production Stage 3 Raw Parts Finished Parts K1 Kanbans K2 Kanbans K3 Kanbans Figure 1: Illustration of a Three-Stage Kanban System. The kanban system produces only one type of parts. It is further assumed that an unlimited supply of raw parts is available at the first stage, and an inexhaustible demand for finished parts is present at the output of the final stage. As a consequence, no input buffer is required at the first production stage, and no output buffer is required at the final stage. All processing time distributions are represented in terms of the first two moments denoted by b j (i) and b (2) j (i) for i = 1;... ; S and j = 1;... ; N i. Statement of the Problem The problem of determining the appropriate network partition and kanban allocation can be stated as an optimization problem where the objective is to minimize the inventory carrying costs and the constraint is the desired throughput rate. Since the inventory carrying costs are directly associated with the workin-process level, it is reasonable to choose the average work-in-process as the objective function. The overall design problem can be stated as a nonlinear integer optimization problem as follows:

3 min w(n 1 ;... ; N S ; K 1 ;... ; K S ) (1) s.t. SX i=1 SX i=1 N i = N; N i 1 and integer (2) K i K; K i 1 and integer (3) t(n 1 ;... ; N S ; K 1 ;... ; K S ) t 0 (4) where w() denotes the average work-in-process inventory and t() denotes the throughput rate of a kanban system with the network partition N 1 ;... ; N S and the kanban allocation K 1 ;... ; K S. The desired throughput rate t 0 is a strategic decision variable that should be selected by management so as to meet the long-term requirements of the company. The above formulation of the optimization problem has an additional constraint on the maximum number of kanbans allowed, K, which may be due to costs related to providing buffers or floor space restrictions. Considering the issue of partitioning the production plant, it is assumed that the number of stages, S, is predetermined by the nature of the production process or by other management considerations. Various further constraints on the kanban number and/or partitioning can be included in a straightforward manner. The main difficulty in the above optimization problem is that the performance metrics of kanban systems do not have closed form solutions. As numerical techniques or simulation often become computationally intractable, in particular for problems of realistic size, we have decided to use an approximate procedure to obtain estimates of the performance metrics. We use the approximation algorithm proposed in [4] to solve the kanban systems under consideration. The algorithm has been shown to provide fairly accurate results in many cases and requires low computation times (less than one second). In the next subsection, we briefly describe the procedure. Approximate Analysis of Kanban Systems The approximation algorithm decomposes the original kanban system into a set of subnetworks representing the individual production stages and solves each subnetwork in isolation. The subnetworks are extended by additional queues to account for delays due to blocking and starvation. Since the service time distributions at the additional queues are not known in advance, an iterative algorithm must be used to establish the performance metrics of the entire kanban system. Let us denote the subnetworks by T(i) for i = 1;... ; S. Each subnetwork T(i), i = 2;... ; S? 1, consists of N i work stations, an additional "upstream" queue S u (i) representing the part of the system upstream of stage i, and an additional "downstream" queue S d (i) representing the part of the system downstream of stage i. The sojourn time at queue S u (i) models the amount of time that an unattached kanban remains in the input buffer of stage i until a finished part is supplied by stage i? 1. In a similar way, the sojourn time at queue S d (i) models the amount of time that a finished part remains in the output buffer of stage i until it is withdrawn by stage i + 1. As an unlimited supply of raw parts and an unlimited finished-parts demand is assumed, no upstream queue is present in subnetwork T(1) and no downstream queue is present in subnetwork T(S). The decomposition is illustrated in Figure 2 for a three-stage kanban system. Notice that the number of customers in subnetwork T(i) is given by the number of kanbans K i, respectively. Downstream Queue S (1) d Upstream Queue S (2) u Downstream Queue S (2) d Upstream Queue S (3) u Subnetwork T(1) K1 customers Subnetwork T(2) K2 customers Subnetwork T(3) K3 customers Figure 2: Decomposition of the Three-Stage Kanban System.

4 The approximation is based upon estimating the first two moments of the service time that a job experiences at the additional queues. For that purpose, we condition on the position of customers in subnetwork T(i). To obtain a consistent numbering scheme, we shall use the subscript 0 to denote quantities related to the upstream queues, and the subscript N i + 1 to denote quantities related to the downstream queues as necessary. Let L j (i) be the number of customers present at work station j in subnetwork T(i) and define! mx n;m (i) = P L m (i) > 0; L p (i) = K i : (5) p=n Let b u (i + 1) and b (2) u (i + 1) denote the first and second moment of the service time at queue S u (i + 1). Using the probabilities defined above, we can express the first two moments of the service time at queue S u (i + 1) entirely in terms of quantities related to subnetwork T(i). We get b u (i + 1) = j=0 X b (2) u (i + 1) = N i j=0 2 0;j (i) 4^b j (i) + 0;j (i) 2 p=j+1 4^b (2) j (i) + 2^b j (i) b p (i) 3 q=j+1 5 (6) b q (i) + p=j+1 b (2) p (i) + 2b p(i) q=p+1 b q (i) 13 A 5 (7) where ^b (2) j (i) and ^b j (i) denote the first and second moment of the residual service time at queue j in subnetwork T(i). Consider now queue S d (i? 1) and let b d(i? 1) and b (2) d (i? 1) denote the first and second moment of the corresponding service time. Proceeding in a similar fashion as above, we get b d (i? 1) = b (2) j=1 X d (i? 1) = 1;j (i) j=1 Ni+1 2 1;j (i) 4^b j (i) + 2 p=j+1 4^b (2) j (i) + 2^b j (i) b p (i) 3 q=j+1 5 (8) b q (i) + p=j+1 b (2) p (i) + 2b p(i) q=p+1 b q (i) 13 A 5 (9) Assuming for a moment that the service time parameters of the additional queues are known, we can solve any subnetwork T(i) using standard approximation techniques for general closed queueing networks. In particular, we utilize the product-form approximation technique proposed by Marie [15] which to date appears to be the most accurate approximation technique for general closed queueing networks. To compute the performance of the entire kanban system, a two-pass algorithm is developed. The algorithm is initialized by setting the downstream parameters of the subnetworks T(i) to some starting values. The main loop consists of a forward pass moving from stage one to stage S? 1, and a backward pass moving in turn from stage S to stage two. On the forward pass, it is assumed that the downstream parameters of subnetwork T(i) are known from the previous iteration of the backward pass. The moments of the service time at the upstream queue of subnetwork T(i + 1) are revised according to (6) and (7). On the backward pass, it is assumed that the upstream parameters of subnetwork T(i) are known from the previous iteration of the forward pass. The moments of the service time at the downstream queue of subnetwork T(i? 1) are then revised according to (8) and (9). Furthermore, it is demonstrated in [4] that the probabilities n;m (i) required in (6) to (9) can be calculated directly from the analysis of subnetwork T(i) once we assume that the stages operate independent of each other. The algorithm iterates between the two passes until an appropriate convergence criterion is met, e.g. if the difference of the throughput rates obtained in the forward and backward pass is less than a given tolerance level (< 10?5 ). The performance metrics of interest of the kanban system can then be derived from the performance metrics of the subnetworks. The complete algorithm is summarized in the Appendix.

5 3. OPTIMIZING THE NETWORK PARTITION AND KANBAN ALLOCATION The next step in the design process of kanban-controlled production lines is the search for an appropriate network partition and a corresponding optimal kanban allocation scheme. This problem is a combination of a combinatorial problem and an integer optimization? problem. Given N work stations, S production N P stages, and K kanbans, the search space has Ki=S? i S?1 S states. Enumeration of all possible states is feasible only for small production lines and a small number of kanbans, so a heuristic should be considered. Furthermore, the heuristic should be able to cope with additional constraints which may arise in real-world applications. Genetic Algorithms The genetic algorithm is a random search heuristic based on an abstract model of biological evolution [16], which has been successfully applied to combinatorial and discrete parameter optimization problems, see for example [17]. The basic idea is to maintain a population of candidate solutions which are modified at random, while selective pressure forces the population to increase in quality. Similar to other random search heuristics, the genetic algorithm provides mutation as random modification operator, but here the mutation operates on the binary representation of a given solution by inverting randomly chosen bits. A second random modification operator is especially useful in the case of combinatorial optimization: the crossover. Two candidate solutions are chosen from the population to recombine their properties. Again, this operator operates directly on the binary representation of the candidate solutions (parents) by simply concatenating alternate substrings of the parents binary representations. Since the genetic algorithm operates blindly on the binary representations, a general-purpose genetic algorithm can be applied to the given problem. We used a genetic algorithm package similar to the one described in [19]. The code is written in C++ and runs on Sun workstations and PCs. A parallel implementation of the genetic algorithm to be run on a cluster of workstations is in preparation [20]. The approximate analytical method described in Section 2 is used to compute the objective function (1) as well as the throughput rate required in constraint (4). To apply the genetic algorithm, it remains to provide a suitable binary representation for candidate solutions. Choosing a Binary Representation The genetic algorithm will work with any representation, provided the whole search space can be represented and the modification operators introduce enough variance into the population [18]. Therefore, a working prototype of the design methodology can be implemented very rapidly. For an efficient implementation, however, the binary representation must be chosen carefully, since this mapping significantly influences the speed of convergence of the genetic search. For example, if an integer parameter (e.g. kanbans per stage) is represented by standard binary encoding, there is a hamming cliff between the representation of the number 7 (00111) 2 and its successor 8 (01000) 2. It is unlikely for the genetic algorithm to proceed from 7 to 8 by random mutation of single bits. In this case, gray coding of the integer values is a better choice, so that two successive numbers always have a hamming distance of one. Thus the binary representation determines which modifications (possibly improvements) to the candidate solutions are likely to be applied. We have implemented several binary representations. Best results were achieved with the following: The bit string consists of two segments, corresponding to the two components of problem (1) to (4). The first segment describes the network partition by an integer vector (N 1 ;... ; N S ) which must fulfill the constraints (2). After substitution of N i by Ni the partitioning is represented as (gray-coded) un- P signed integer vector (N1 0 ;... ; N S 0 ) under the constraint S i=1 N i 0 = N? S. It is not efficient to include this remaining constraint into the objective function by punishing invalid solutions with a penalty, since computing time would be wasted by handling invalid solutions. Instead we used a voting-like scheme, which assigns the fixed number of work stations to production stages according to the number of votes N 0 i each stage has received. This segment of the binary representation uses S dld N? S + 1e bits. The second segment of the bit string addresses the kanban allocation problem by an integer vector (K 1 ;... ; K S ) under the constraints (3). Again, one could use a voting-like scheme as described in the previous paragraph. Although this scheme would result in a very short binary representation (S dld K? Se bits), it does not support the genetic algorithm in its search. Consider a mutation which moves one work station from one stage to the next. For the modified network partition, a different kanban allocation vector is likely to be optimal. Since a single mutation can either affect partitioning vector or kanban allocation vector, in most cases the fitness decreases and such a modification would

6 be rejected by the algorithm. Instead, we used the following representation: Each stage receives one kanban, while each of the remaining K? S kanbans is assigned directly to a work station s i by a kanban index vector (s 1 ;... ; s K?S ). Kanban i is assigned to work station s i, if s i 2 f1;... ; N g; kanban i is not used otherwise. The kanban allocation vector can be computed from the kanban index vector after each work station has been assigned to a stage by the network partition vector. A good kanban index vector can produce good results with many network partition vectors, thus decoupling the two optimization tasks. This segment of the binary representation uses (K? S) dld N + 1e bits. 4. NUMERICAL EXPERIMENTS We now proceed with numerical experiments in order to investigate the accuracy of the heuristic design method. Two sets of experiments are presented with a 12 work station production line. In the first set of experiments (Problems 1a-e), a balanced production line is examined where it is assumed that the processing times at all stations are exponentially distributed with mean equal to one. The second set of experiments (Problems 2a-e) studies the effect of imbalance in the line. The processing times are represented in terms of their mean b i and squared coefficient of variation cv 2 i as shown in Table 1. In all experiments, it is assumed that the number of production stages, S, is equal to four and the number of kanbans allowed, K, is twice the number of work stations, i.e. K = 24. Work Station i b i cv 2 i Table 1: Processing Time Parameters Used in Problems 2a-e. For each problem, an exhaustive search procedure was run to find the optimal network partition and kanban allocation, given the constraints (2) to (4). To evaluate the performance of the genetic search, five instances were generated for each problem (a total of 50) and solved using the genetic algorithm package. The binary representation consists of 96 bits. A population of 16 candidate solutions was used with tournament selection (Size 3) and elitist replacement. A reproduction step consisted of either a mutation with mutation rate 0:01 (with probability 0.9) or a two point crossover (with probability 0.1). A computation time of 900 seconds was selected for a single run. All computations were carried out on a Sun SPARC 10. Table 2 compares the solutions found by the genetic algorithm to those obtained from exhaustive search for increasing values of the target throughput rate. Notice that the columns associated with the genetic algorithm represent the average of five problem instances. The quality of the suboptimal solutions are indicated by the ratio of the work-in-process produced by the genetic algorithm to the optimal one produced by exhaustive search. A total of 1,753,290 evaluations were required for each run during the exhaustive search. For comparison, the number of fitness evaluations of the genetic algorithm is also provided in the table. Exhaustive Search Genetic Algorithm Problem Target Throughput Rate WIP Fitness Throughput Rate WIP Ratio to No. Throughput Rate Achieved Achieved Eval. Achieved Achieved Optimal 1a b c d e ? 2a b c d e Target throughput rate not achieved Table 2: Comparison Between Exhaustive Search and Genetic Algorithm.

7 We can see from Table 2 that the genetic algorithm performs reasonably well in comparison to exhaustive search. All suboptimal solutions are within 7 percent of optimal for the entire set of problems considered and within 4.3 percent on average. Furthermore, if we take into account that on average less than 1500 evaluations were required for the genetic algorithm, the computational complexity of the heuristic approach is quite small (although it clearly depends on the efficiency of the approximation algorithm). We point out that the computation times of 900 seconds was chosen arbitrarily to demonstrate that the genetic algorithm converges quickly. There is a potential to obtain better solutions once the computation times are increased. For the entire set of problems considered above, additional runs were carried out with a time limit of 60 minutes which resulted in suboptimal solutions within 1.4 percent of optimal on average. To investigate the accuracy of the work-in-process and throughput estimates provided by the approximate analytical method, it is necessary to compute the exact values. Due to lack of any closedform solution for the performance metrics of kanban systems, a simulation model was implemented. Table 3 reports the differences between the analytical and simulated performance metrics of the optimal configurations that were established by exhaustive search. In each case, the simulation model was run with 200,000 jobs, using the independent replication method to obtain simulation point estimates. It can be seen from Table 3 that the simulated values and those produced by the analytical method match very well in both sets of experiments. Throughput Rate Work-in-Process (WIP) Problem No. (N 1,N 2,N 3,N 4) (K 1,K 2,K 3,K 4) Analyt. Simul. Rel. Err. Analyt. Simul. Rel. Err. 1a (5,5,2,1) (4,6,4,5) % % 1b (6,2,1,3) (6,4,6,6) % % 1c (5,2,2,3) (6,5,7,6) % % 1d (3,3,4,2) (4,6,8,5) % % 1e (3,3,2,4) (5,6,6,7) % % 2a (6,4,1,1) (5,5,5,5) % % 2b (6,3,2,1) (6,5,6,6) % % 2c (6,2,1,3) (7,5,5,7) % % 2d (2,4,1,5) (3,6,5,7) % % 2e (2,4,1,5) (3,7,5,9) % % Table 3: Analytical and Simulation Results for the Configurations Obtained in Exhaustive Search. 5. SUMMARY AND CONCLUSIONS This paper presented a heuristic design methodology to determine the optimal network partition and kanban allocation in complex production systems. The approach is based on a general-purpose genetic algorithm and an approximate procedure to evaluate the performance of kanban systems. As the numerical experiments demonstrated, the solutions produced by the heuristic method are reasonably accurate and they can be obtained with an acceptable amount of computational effort. The heuristic is well-suited in practice to quickly explore alternative configurations and kanban settings during the conceptual design phase of the production planning process. One limitation of the proposed methodology is that only suboptimal solutions can be guaranteed. Given the NP-hardness of the underlying nonlinear integer optimization problem, however, a heuristic approach is justified from a practical perspective, in particular if it is intended to solve problems of realistic size. In this context, the forthcoming parallel implementation of our genetic algorithm package promises to be useful. The results of this paper are preliminary and must be confirmed by further study. Future research is directed towards the development of tractable (heuristic) solution methods for different formulations of the kanban system design problem. For instance, it may be reasonable in certain production environments to consider the number of production stages as an additional decision variable. This can easily be accomplished in our approach by appropriately adjusting the binary representation used in the genetic algorithm.

8 ACKNOWLEDGEMENTS The authors would like to thank Dr. H. Fromm for valuable suggestions and comments on an early draft of this paper and Thilo Opaterny for implementing the genetic algorithm package. REFERENCES [1] Mitra D. and Mitrani I., Analysis of a Kanban Discipline for Cell Coordination in Production Lines I. Man. Sci., 36, , [2] Spearman M.L., Woodruff D.L. and Hopp W.J., CONWIP: A Pull Alternative to Kanban, Int. J. Prod. Res., 28, , [3] Tayur S.R., Properties of Serial Kanban Systems, Queueing Systems, 12, , [4] Ettl M., An Approximate Method for Evaluating the Performance of Serial Lines with Kanban Control, Technical Report 20/93, University of Erlangen-Nürnberg, 1993 (submitted for publication) [5] Huang P.Y, Rees L.P., Taylor III B.W., A Simulation Analysis of the Japanese Just-in-Time Technique (With Kanbans) for a Multiline, Multistage Production System, Dec. Sci., 14, , [6] Kimura O. and Terada H., Design and Analysis of Pull System: A Method of Multi-Stage Production Control, Int. J. Prod. Res., 19, , [7] Philipoom P.R, Rees L.P., Taylor III B.W. and Huang P.Y., Dynamically Adjusting the Number of Kanbans in a Just-in-Time Production System Using Estimated Values of Lead Time, IEE Trans., , [8] Deleersnyder J.L., Hodgson T.J., Muller H. and O Grady P.J., Kanban Controlled Pull Systems: An Analytical Approach, Man. Sci., 35, , [9] Di Mascolo M., Frein Y. and Dallery Y., Queueing Network Modeling and Analysis of Kanban Systems. 3rd Int. Conf. on Computer Integrated Manufacturing, IEEE Society Press, [10] So K.C., and Pinault S.C., Allocating Buffer Storages in a Pull System, Int. J. Prod. Res., 26, , [11] Davis W.J. and Stubitz S.J., Configuring a Kanban System Using a Discrete Optimization of Multiple Stochastic Responses, Int. J. Prod. Res., 25, , [12] Jothishankar M.C. and Wang H.P., Determination of Optimal Number of Kanbans Using Stochastic Petri Nets, J. Manufact. Syst., 11, , [13] Siha S., The Pull Production System: Modelling and Characteristics, Int. J. Prod. Res., 32, , [14] Johri, P.K. Optimal Partitions for Shop Floor Control in Semiconductor Wafer Fabrication. Europ. J. Oper. Res., 59, , [15] Marie R., An Approximate Analytical Method for General Queueing Networks, IEEE Trans. Soft. Eng., 5, , [16] Goldberg D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, [17] Bäck T., Hoffmeister F. and Schwefel H.-P., Applications of Evolutionary Algorithms, Technical Report SYS-2/92, University of Dortmund, [18] Atmar, W., Notes on the Simulation of Evolution, IEEE Trans. Neural Networks, 5, , [19] Schwehm M., A Massively Parallel Genetic Algorithm on the MasPar MP-1, Proc. Int. Conf. ANNGA 93, Springer Verlag, Wien, , [20] Schwehm M., Opaterny, T., A Distributed Parallel Genetic Algorithm Package, Working Paper, University of Erlangen-Nürnberg, APPENDIX: COMPUTATIONAL ALGORITHM initialization: for(i=1; i<s; i++)f set bd(i) and b (2) d (i) to some starting value. g main loop: dof forward pass: for(i=1; i<s; i++)f solve subnetwork T(i) using Marie s method. compute bu(i+1) and b (2) u (i+1) from equations (6) and (7). g backward pass: for(i=s; i>1; i--)f solve subnetwork T(i) using Marie s method. compute bd(i-1) and b (2) d (i-1) from equations (8) and (9). g g while(throughput rates obtained in forward and backward pass not approximately equal)