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Review on Assembly line optimization

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A weighted approach for assembly line design with station paralleling and equipment selection Joseph Bukchin and Jacob Rubinovitz Review by Uzair Khaleeq uz Zaman, Department of Mechanical Engineering, CEME, NUST , Pakistan Email: ukz_uzzi@yahoo.com I. INTRODUCTION The paper by Bukchin and Rubinovitz (2002) focuses on minimizing the number of stations and the cost of an assembly line through station paralleling and equipment selection. There are numerous factors that account towards assembly line balancing but the most common and worked upon idea is the cycle time . The cycle time (C) of the assembly line is determined by the workstation with the maximum work content time. It can also be explained as the time between the exits of two consecutive products from the line1. Hence, two problem formulations stem from the concept of C; one, to minimize the number of workstations for a required cycle time, and two, to minimize the cycle time for a given number of stations. The objective of the paper is therefore, to study an environment with several equipment types, human operators capable of performing various operations, the availability of the freedom to go for paralleling of stations, and to minimize the associated cost. As highly mechanized equipment is largely dependent on the number of parallel stations due to equipment duplication, the preliminary study is split in to two sections. The first section focuses on the parallel station problem and the second section adds weights (cost) to the objective function. As the weights change, line configurations change. The later part of the study then compares parallel station problem with the equipment selection via a Branch and Bound (BB) algorithm developed by Bukchin and Tzur (2000). Lastly, the parallel stations and the equipment duplication, combined, are solved by the same BB algorithm. II. LITERATURE REVIEW Extensive search has been done in the past with the evolution of heuristic methods and solutions focusing on design, balancing and scheduling of single and mixedmodal assembly lines. Sequential mode of assembly lines was focused upon a lot but it was soon realized that performing the tasks in parallel had far more advantages and important benefits like reduction of station idle time, shorter cycle times, and increased reliability. Buxey (1974) suggested an algorithm which incorporated weight method and used parallel station only for the longer elements. Pinto et. Al (1975) developed branch and bound (BB) algorithm to minimize total cost and then in (1981), proposed the incorporation of paralleling two stations into the algorithm. BB uses upper and lower bounds to search the entire solution space for the best optimized solution of NP-discrete optimization problems through iteration. The iteration has three main components: selection of the node to process, bound calculation, and branching2.The same BB algorithm has been used by the authors of the paper in review as well. III. PROBLEM DESCRIPTION A. Problem Formulation A model for the assembly line is developed. It starts with minimizing the number of stations (allowing parallel stations) and then adds weight (cost) factors for different paralleling situations, in itself. These weight factors have the tendency to formulate varying costs for varying assembly system environments. The mathematical formulation for the model (P1) is as follows: 1 Rekiek, Brahim and Delchambre, A., Assembly Line Design: The Balancing of Mixed-Model Hybrid Assembly Lines with Genetic Algorithms , Springer, 2006 2 Clausen Jens, Branch and Bound Algorithms- Principles and Examples , March 12, 1999 Subject to: (a) Wk, the weight factor, represents the cost of a stage with k parallel stations (including additional equipment costs). When it varies, different line design objectives can be addressed using parallel stations. Bukchin mentions that when the sole objective is the minimization of stations without preferring sequential stations or parallel stations, the weight values are as follows: (g) (b) (c) (d) (e) (f) The equation shows that the cost is proportional to the capacity of a station. For instance, if W1 = 1 (for single station), then W2 = 2 (two parallel stations), W3 = 3 (three parallel stations), and so on. This shows that the cost of n stations would be exactly n times higher than the cost of a single station. It also exhibits that the cost of installing a parallel station would be the same as the cost of installing a station in sequence. These results normally tally with manual assembly where the equipment cost is not high. Moreover, if the objective is modified to include the minimization of the number of parallel stations along with the minimization of the total number of stations, the following adjustment is made to equation (g): (h) where, C ti xij Pi Wk yjk Jmax Kmax n cycle time duration of task i 1, if task i is assigned at stage j 0, otherwise set of tasks that must precede task i due to technological constraints weight (cost factor) for each paralleling situation of k identical parallel stations binary variable, which equals one when there are exactly k parallel station in stage j the maximal number of stages The maximum number of stations in parallel The number of tasks The objective function (P1) minimizes the weighted product, Wkyjk, of k identical parallel stations at stage j. Constraint (a) handles the precedence relationship between tasks. Constraint (b) ensures that each task is performed exactly once. The capacity constraint (c) ensures that the work content assigned to each stage does not exceed the associated capacity. Constraint (d) ensures that if a station is opened in stage j, the number of parallel stations at this stage is unique. where is a small constant to favor smaller number of parallel stations. It can also be considered as a small penalty cost for using identical parallel stations. The penalty can be understood by the idea that when there is a need to install an identical station in parallel, it would require additional cost for the duplication of tools and equipment. This cost would not be required if a new station is put in sequence. If the cost of equipment duplication is very large, then only parallel stations which are essential are opened. This normally happens when the time taken to perform certain tasks exceeds the defined cycle time. In this case, (g) is modified as: (i) where A is a large constant. B. P1 model formulation analogy to multi-equipment selection V. SCENARIOS A. The constraint (c) in P1 was related to the capacity content of a particular station. If a new binary variable, xijk, is introduced which equals one if task i is assigned at stage j with a configuration of k parallel stations, then the constraint (c) can be re-written as: Balancing improvement and design trade-offs For high production rates, the line must be balanced for short cycle times. To explain this concept, Bukchin and Rubinovitz used an example to minimize the number of stations for a given cycle time of 60 time units by the use of weights in the objective function. or alternatively as: (c) and constraints (a) and (b) can be written as: (a) (b) The (c) shows that each equipment alternative, k, performs task, i, in duration of ti/k. This means that adding stations in parallel, makes the system more efficient and task execution becomes faster. IV. BRANCH AND BOUND (BB) ALGORITHM The authors have used the BB algorithm developed for a general problem by Bukchin and Tzur (2000). Throughout the process, following steps happen: 1. Workstations are opened sequentially 2. Equipment types are selected and placed in any newly opened workstation 3. Tasks to be performed by the selected equipment are assigned to the last workstation opened in a given partial solution 4. Algorithm ends when all tasks are assigned to workstations and obtained solution value is not larger than the lower bound of all partial solutions. Figure 1: Input data of paralleling example along with the precedence diagram The example is a problem with twenty (20) tasks. At max, three (3) parallel stations can be accommodated. The important thing is that two stations in parallel are considered as a single station in this example. Therefore, the time executed becomes half for each task assigned to the stage. The subsequent columns two and three show the time being halved and further reduced to one third, respectively. Care must be taken for the tasks requiring expensive equipment. They should be treated in a single sequential station so that equipment duplication should be avoided. Table 1: Optimal solutions of example problem (C1 -C3) The problem in Figure 1 is solved using three sets of weights and the solution is displayed in Table 1 above. For C1, the weights are taken to be 100, 2000, and 30,000 for W1, W2, and W3 respectively. The resulting configuration includes 14 stations, all in sequence. The C2 solution weights each stage proportionally to its capacity using weight values of W1 = 100, W2 = 200, and W3 = 300. The objective in this second case is to reduce the total number of stations not caring much whether parallel or sequential stations are used. The solution resulted in 13 stations, with six in parallel. This is an improvement in C1 but maybe all the parallel stations obtained are not required. Therefore, C3 solution is generated which includes the penalty cost for the paralleling as well. With = 1, the weight values obtained are W1 = 100, W2 = 201, and W3 = 302.5. As seen in the Table 1, the stations are still 13, but the number of parallel stations is reduced by five, making C3 the best solution among all. B. Long task constraints If the tasks take longer duration to execute than the cycle time, parallel stations are required. The authors give example of a Television (TV) assembly line in which the processes like testing can take a couple of hours on a single unit. The Figure 2 shows a situation with stage j being the testing stage and the line cycle time is C time units. Figure 2: Parallel stations for a long task situation The capacity of stage j is m times C where m is the number of stations in parallel. If the daily output of a line is 200 TV sets in an 8 hours shift, then the cycle time, C = (8*60)/200 = 2.4 minutes. If the testing time required for each TV is 2 hours, then 50 stations (120/2.4) of type j would be required to prevent blockage in stage j and meet the 200 TV target. By considering the problem of Figure 1 again, and making slight changes to the input data, the authors create a new example which contain tasks having duration longer than the required cycle time. They assume task 10 to have duration of 72 time units (> 60 time units), and task 17 to have duration of 90 time units (> 60 time units). Hence, a new Table 2 is generated. Table 2: Optimal solutions of example problem (C4-C6) By applying three different weight factors in this scenario, three different objective functions are obtained which result similarly, in three different solutions as shown above. The solution C4 requires only essential paralleling. The result is 15 stations with two parallel stations, each in stage 9 and stage 11. Solution C5 is similar to C2 where the objective is to minimize capacity subject to a given C. The result is 14 stations with seven parallel stations. Here, up to three parallel stations are allowed. Finally, solution C6 includes the penalty cost and it has the same number of stations as C5 but the number of parallel stations reduces to four. Hence, C6 turns out to be the best solution in the new modified example. C. Effect of problem parameters on balancing improvement All experiments performed by the authors are for problems with 20 tasks. The factors affecting balancing improvement are as under: 1. F-ratio It is a measure of the flexibility in creating different assembly sequences for a K elements assembly task. Relatively low F-ratios are usually kept in assembly tasks. Precedence diagrams with F-ratios 0.1, 0.3 and 0.5 are generated by the authors. 2. Average number of tasks per station (ATS) It is the ratio between number of tasks in the assembly and the minimal number of stations required to meet the production rate. The authors set ATS to 2 to 4 tasks per station. 3. Variability of task duration (VTD) Two tasks are examined. One was with small variance and the other was with large variance. 4. Maximal number of stations in parallel (MSP) Each problem is solved for MSP of two, three and four. The balancing improvement is the ratio between the number of stations obtained with paralleling and the number of stations obtained with no paralleling. This value will always be less than or equal to 1. Moreover, the authors use Analysis of Variance (ANOVA) to see the effect of F-ratio, ATS, and other parameters. ANOVA provides a statistical test of whether or not the means of several groups are equal. Also, they examine the effect of the limitation on the number of parallel stations on the balancing improvement. Seventy two (72) problems are solved for each limitation giving a total of two hundred eighty eight (288) problems. Figure 3: Balancing improvement as a function of F-ratio and ATS It is evident from Figure 3 that for a higher ATS, as Fratio increases from 0.1 to 0.3, balance improvement decreases and then slowly improves as the value is further increased to 0.5. For a lower ATS, the balance improvement increases continuously as F-ratios increase from 0.3 to 0.5. Figure 4: balancing improvement as a function of paralleling limitation The above figure shows that major improvement is obtained when the first parallel station is added (improvement of around 8.4%). There is very little improvement when two or more parallel stations are added. Hence, it can be concluded that a problem should be started with a small value of maximal number of parallel stations and then increase this value until no significant improvement is seen. VI. CONCLUSION The different scenarios taken into account by Bukchin and Rubinovitz, clearly solve many issues related to assembly line balancing as shown in the review. The factors for balancing improvement, and the objective functions incorporated in the study, hence forth, divide the research into distinct and work-oriented case studies. VII. REFERENCES [1] Rekiek, Brahim and Delchambre, A., Assembly Line Design: The Balancing of Mixed-Model Hybrid Assembly Lines with Genetic Algorithms , Springer, 2006 [2] Clausen Jens, Branch and Bound AlgorithmsPrinciples and Examples , March 12, 1999 [3] Bukchin, Joseph and Rubinovitz, Jacob, A weighted approach for assembly line design with station paralleling and equipment selection , May, 2002. [4] Bukchin, Joseph and Tzur, Michal, Design of flexible assembly line to minimize equipment costs , July, 1999 [5] Buxey, G.M., Assembly line balancing with multiple stations , 1974 [6] Pinto, P., Dannenbring, D.G. and Khumawala, B.M., A branch and bound algorithm for assembly line balancing with paralleling , International Journal of Production Research, 13(2), 183 196, 1975.

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