Mixed-model assembly line balancing problem: variants and solving techniques. Cintia Copaceanu. Abstract

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1 Stud.Cercet.Stiint., Ser.Mat., 16 (2006), Supplement Proceedings of ICMI 45, Bacau, Sept.18-20, 2006, pp Mixed-model assembly line balancing problem: variants and solving techniques Cintia Copaceanu Abstract The paper presents a survey of the literature on mixed-model assembly line balancing. Several existing approaches for modelling and solving different types of mixed-model assembly line balancing problems are reviewed considering the layout possibilities: serial lines, U-shaped lines and lines with parallel workstations. The main purpose of the study is to determine new opportunities of applications and a generalization of the solving instruments. Key words: assembly line balancing, mixed-model production, literature survey, solving methods. 1. INTRODUCTION Assembly lines are flow-line production systems that are encountered in the industrial assembly of volume standardized commodities ([57]). According to the classification proposed in [11], two types of flow lines are distinguished. The first type is dedicated to the production of one single product (single model lines). The second type is dedicated to the assembly of more than one model (mixed and multi flow lines). Serial assembly systems have traditionally been used for the production of a single product. The balancing problem, known as single model assembly line balancing (ALB) problem has been studied extensively. With an increasing requirement for flexibility of production, motivated by fast changes in technology and by customer demand for greater product variety, mixed-model assembly lines are replacing the traditional mass production assembly lines. Mixed-model production is important to respond to diversified expectations of today customer perspective. The mixed-model assembly line balancing problem is more difficult than the single model ALB problem and has many features that are significantly different. The mixed-model production is related to solving two related problems: the mixed-model line balancing and the mixed-model sequencing problems ([57]). The line balancing is the problem of assigning 337

2 different tasks to workstations on the line, and the model sequencing is the problem of determining the way to sequence models on the line. The both problems are known as NP-hard optimization problems ([62]). The paper presents a survey of the literature on mixed-model line balancing with a special emphasis on U-shaped assembly line balancing. It is focused on the identification of the types of ALB problems in mixed-model assembly lines and the methods used to solve these problems as well. The review takes into account the difficulties occurred in solving these types of ALB problems and the reported performances of the methods in terms of solution quality, the size of problem instances and computational times. The main purpose of the study is to determine new opportunities of applications and a generalization of the solving instruments. The extending of the proposed methods to solve certain versions of the mixed-model ALB problems in order to handle practice features is also investigated. The literature review is organized taking into account the classification of the mixed-model ALB problems by their main characteristics: the layout of the production system and the variability of operation times. Section 2 introduces some characteristics of the mixed-model assembly lines and presents the balancing and sequencing problems, which arise in the context of operating mixed-model assembly lines. Several existing approaches in modelling and solving different types of mixed-model ALB problems are reviewed considering the layout possibilities: serial lines, U-shaped lines, lines with parallel workstations, and multi-model lines. In section 3, some approaches dedicated to mixed-model assembly line sequencing are presented. The mixed-model ALB problem with stochastic processing times is considered in section 4. Last section summarizes the work. 2. DETERMINISTIC MIXED/MULTI MODEL LINE BALANCING The weaknesses of a single-model line are that it becomes inefficient when demand falls or rises, and that it is only efficient when producing the model for which is designed. If market demand changes so that other products are required, other products need to be produced. Installing separate lines for other products can do this but this is only economic when the additional lines are running efficiently in fulfilment of greater demand. For the problem of fluctuating demand two solutions are used: mixed-model assembly lines and multi-model assembly lines. On mixed-model lines, more than one product with similar production characteristics or different models of a product are assembled on the same line. The basic premise is that multiple products are handled by each 338

3 workstation without stops to change over between them. This permits a random launch sequence so that the products can be made according to market demands. Since assembly process and processing times are not the same for different products, the problem of mixed-model assembly line balancing become more complex. One difficulty is that the work content at each workstation differs from model to model. Another one is that the unused time at each workstation varies from time to time depending on the sequence of models along the line. The performance of the line cannot be measured by workload balance alone. Some important performance measures depend on the quality of the model sequences (model changes require a significant amount of setup time and cost), and the sequences are influenced by the line balancing decisions. Therefore, in mixed-model assembly lines, the problems of line balancing and mixed-model sequencing are tightly interrelated with each other ([57]). A comprehensive literature review of different methods used for mixed-model ALB problems is presented in [57]. The mixed-model assembly line balancing is the problem of assigning tasks of different models to workstations so that some constraints are satisfied and some performance measures are optimized. Such performance measures are minimizing the number of workstations for a given cycle time, minimizing the cycle time for a given number of workstations, smoothing work overload and maximizing the efficiency of lines. The mixed-model sequencing is the problem of determining the production sequence of models produced on the line by optimizing some performance measure ([57]). The demands of all models are known by the production program of a planning period and combination of demands for all models is referred to as model mix. The performance measures usually refer to the variation of the station utilizations with respect to the models. Such measures are smoothing the usage rate of parts used in the line, minimizing the total cost of setups and minimizing the deviation of workloads across workstations. Almost all-existing approaches transform the mixed-model ALB problem into a single model ALB problem. In such approaches, the tasks that are common to several models need to be performed in the same workstations, the procedure balancing a single combined model ([5], [9], [37], [39], [57], [63]). A few approaches consider that tasks common to different models may be assigned to different workstations ([10], [56]), and other approaches take into account secondary objectives in order to reduce line inefficiencies such as work overload and unused time ([9], [63]). 339

4 An example of a mixed-model ALB problem is presented bellow. A set of three similar models is simultaneously assembled on the line, the demands for each model being 8, 10 and 5 units. The precedence diagrams for the three models with the task times presented near the nodes are given in figure 1, where each node represents a task and each arc represents a precedence relation between two tasks. The combined precedence diagram with the cumulated processing times is presented in figure 2. Assuming that the available time during the planning period is 112 time units (the cycle time is C=4), a possible assignment of tasks to three stations is S 1 ={1, 3}, S 2 ={2, 4}, and S 3 ={5, 6}. If the sequence of the models is model 1, model 2, model 3, the workflow along the line for individual models is shown in figure 3. The task 1 of model 1 is performed first in station 1 with the demand d 1 =8 units. Then, the task 1 of model 2 is completed in station 1 with d 2 =10 units. The task 3 of model 1 (d 3 =5 units), the task 3 of model 2 and the task 3 of model 3 are then performed in the same station. Figure 1. Precedence diagrams of the models Figure 2. The combined precedence diagram for the problem 340

5 Figure 3. The workflow of each model Traditionally, balancing and sequencing assembly line have been considered as two separate but related problems ([62]). In the first studies ([12], [37], [63]) the model sequence have been ignored. Other approaches such as those proposed in [3], [54], [64] and [71] solve different mixed-model sequencing problems assuming that the line balancing is predetermined. Even if the balancing and the sequencing can be solved optimally, the optimality of the overall solution may not be obtained by solving the two problems separately. Hierarchical approaches for solving the mixed-model ALB problems that integrates sequentially the balancing and sequencing problems are discussed in [20], [40], [57] and [62]. In [20] and [62] the balancing procedures are similar to single-model ALB problem procedures, but it assumes a model-mix for which the combined workload is balanced for the duration of the shift and not on the basis of station cycle times. The greedy procedure presented in [40] aims to smooth the station times of different models for each station in order to avoid operating inefficiencies like work overload or unused time. In [57] it is discussed a hierarchical planning approach in order to integrate sequencing aspects into balancing decisions. Different approaches to simultaneously deal with the integration of balancing and sequencing problems using genetic algorithms are proposed in [33], [34], [35] and [45]. On multi-model lines different models are produced in batches one after the other. Before producing a batch the line equipment (tools, human operators, and material supply) is set up to suit the model or variant of the model required. This process takes time. The batch of products is then produced according to a production program. The benefit of a multi-model line is that once set up for a particular model it is as efficient as a conventional line. The drawback is that setting-up takes time that means lost production and inefficiency. Two problems need to be solved. The first one is the way of balancing of the line for each product separately. This means that a function of technological feasibility is followed by application of a standard balancing method. For instance, the methods proposed in [24] or in [49]. The second problem is the way of sequencing the product batches to minimize product changeover losses. The batch production leads to scheduling and lot sizing problems. 341

6 Serial mixed-model line balancing In [1] it is considered the mixed-model lines as a modification for the "Computer method of sequencing operations for assembly lines" procedure proposed for the single model ALB problem. The mixed-model ALB problem is transformed into a single model ALB problem using the average processing times of tasks required by the models, which are calculated taking into account the relative frequency of each model. The times necessary for changing tools and operator position are also considered. No experimental results are reported. To minimize the number of stations of a mixed-model ALB problem, in [62] it is used the heuristic proposed in [30] for the single model ALB problem. The heuristic is applied to the combined precedence diagram divided in columns. The sequence of the models is determined by computing penalty costs of line inefficiency such as idleness, work deficiency, utility work and work congestion. The method yields to near-optimal solutions for the tested problems. In [56] it is developed an integer-programming model with the objective of minimization of the total idle time. In this approach the tasks common to different models may be assigned to different workstations. The total unused time is minimized and both cycle time and the number of workstations are assumed to be the same for all models. It is considered that this formulation has only a theoretical interest due to the excessive number of constraints and variables. A greedy procedure for the mixed-model ALB problem aiming to minimize the number of workstations is proposed in [63]. A secondary objective to equalize the distribution of the total work content of single models among the workstations is introduced. Precedence diagrams are combined, the problem being reduced to a single model problem. The procedure is similar to the heuristic of [26]. It is examined a number of feasible combinations of tasks for every station until a combination that falls into a pre-specified interval is obtained. The feasible combination that minimizes the deviation of the station time from the average station time is selected. The method proposed in [37] for the mixed-model ALB problem is also based on the combined precedence diagram. The tasks are assigned to the workstations on basis of shift duration using the heuristic proposed in [24] for single model ALB problem. 342

7 Different procedures for solving mixed-model ALB problem with the objective of minimizing the number of stations that are based on the combined precedence diagram are compared on their performance in [72]. It is proposed a heuristic procedure derived from the branch and bound algorithm developed in [5]. All heuristics transform the mixed-model ALB problem to a single model ALB problem. Testing the performance of the heuristics it is reported that the position of the common tasks has a significant effect on the required computational time and on the unequal distribution of the total work content of single models among the workstations. The extension of the approaches to single-model for mixed-model has been applied in [22] and [23]. In [23] it is developed a binary integer programming for the mixed-model ALB problem based on a combined precedence diagram for the models. The model is better than the shortest route model proposed in [56] in terms of size of the formulated network as number of tasks increases because common tasks of different models are assigned to the same stations. Optimal solutions for the problems with up to 40 tasks in the combined precedence diagram are reported. The mixed-model ALB problem is optimally solved in [22] using a shortest-path based procedure. Several performance measures for throughput of a mixed model assembly line are analyzed by simulation in [9]. The following measures have been evaluated: smoothness of a station, idle time, station time variation, bottleneck and model variability. The results of the study show that among the considered measures, the bottleneck measure performs better than the other and the quality of these measures decreased if line length increased. A heuristic procedure for mixed-model assembly line is developed in [17], taking into account the product mix and the production sequence of the products. A method achieving the distribution of work element of each model is proposed. In order to reduce the work load variations on the performance of mixed-model lines, two balancing functions based on weighted difference between the maximal station time and the stations times of all other models are proposed in [40]. The proposed greedy heuristic, similar to that developed in [26] for single model ALB problem has as primary objective those balancing functions, and the minimization of the number of stations as secondary goal. The balancing and sequencing problems are sequentially solved. The mixed-model ALB problem with the aim of minimizing the cycle time given a fixed number of stations and a production sequence consisting in 343

8 several copies of each model is considered in [29]. It is assumed that the models are currently performed in all stations. The problem is solved by a modified priority rule based procedure. The mixed-model ALB approach developed in [10] assumes that common tasks to different model types are assigned to different workstations and the goal is to minimize both the station cost and task cost. The task cost is related to the task assignment assumption and reflects the costs of tool duplication, the cost of worker training and the cost of inventory management. The problem is solved with a branch and bound-based algorithm that yield to optimal or near optimal solutions. Experimental tests show that the procedure performs efficiently even for large instance problems. In [33] it is proposed a combined mixed-model balancing and sequencing problem aiming to minimize the amount of work that is not completed within the given length of workstation. The genetic approach simultaneously deals with the integration of balancing and sequencing problems, proposes methods to select symbiotic partners and evaluating fitness. The experimental results demonstrate that the method can explore the solution space effectively providing better solutions than a hierarchical approach. The mixed-model ALB problem aiming to minimize the number of stations taking into account several additional objectives ([8]) is solved in [9] using a three stage approach. The tasks are divided into two groups, one for the tasks that are assigned to the same workstations for all models and the other for the tasks that are assigned to different workstations for different models. In the first stage of the procedure, the problem is reduced to a single model ALB problem (combined precedence diagram), and is solved considering the set of tasks that are common to all models and are performed at the same station. In the second stage, each model type is separately balanced assigning the tasks of the second group taking into account the assignments made in the first stage. In order to compare the quality of solutions, a performance measure of the line cycle time is considered. In third stage, the obtained solution is improved using a neighbourhood search procedure, which changes the assignments of the first group of tasks. The mixed-model ALB problem with the objective of minimizing the cycle time of the line given a fixed number of stations is presented in [55]. The increasing the uniformity of tasks at the stations is considered as a secondary objective. To solve the problem, a priority rule based procedure and a tabu search method is used. 344

9 The number of studies conducted on the mixed-model version of the ALB problem is considerably less. There are potential applications to extend the proposed models by incorporating other practical features related to restrictions on the locations of tasks or assigning combinations of tasks, equipment selection in the balancing process and refining the traditional objective function of minimizing the deviation of workloads across workstations. Most proposed approaches are dedicated to the effects of line balancing decisions on workload assuming that customer orders are sequenced randomly. There are also opportunities to generalize the proposed approaches to consider other aspects related to processing times of customer orders in order to improve the performance of the line. The combinatorial nature of the problem makes it difficult to solve when the problem size increases and forces up to develop efficient approximate algorithms. It seems that some improvements can be achieved by improving the solution approaches for the proposed models using metaheuristics such as genetic algorithms, tabu search and simulated annealing. U-shaped mixed-model line balancing U-lines have become very popular in manufacturing environment as a consequence of improvement and cost reduction efforts of just-in-time production. Many benefits of U-lines utilized in a just-in-time environment are reported in the literature ([15], [44], [47], [48]), including increasing productivity, reduced work-in-process inventory, shorter throughput and improved quality. A successful implementation of a mixed-model U-line requires solutions for the two interrelated with each other problems: mixedmodel U-shaped line balancing and the mixed-model U-shaped line sequencing ([34], [60]). The first study dedicated to balancing mixed-model U-shaped lines is proposed in [60]. It is developed a model of the mixed model U-shaped ALB problem and a heuristic procedure to solve this problem based on the branch and bound algorithm developed in [46] for the single model U-shaped balancing. Travel times between tasks are also considered. The proposed approximate algorithm combines each model precedence diagram into a single precedence diagram and solves problems with up to 25 tasks. In order to reduce the model imbalance, a smoothing algorithm is proposed, by interchanging tasks between adjacent stations. The objective of the smoothing algorithm is to minimize the absolute deviation of workloads across stations by solving the U-shaped balancing problem and U-shaped sequencing problem sequentially. The model sequencing is used to reduce the effect of the 345

10 remaining imbalance. Because such a heuristic may become trapped at a local minimum, the use of a local search technique such as simulated annealing is recommended to reduce the model imbalance. The study also includes the extension of the proposed algorithm for solving the mixed-model multiple U- shaped line problem. This problem of determining the number of multi-lines stations whose tasks from many U-shaped lines are performed given the identical cycle time for all U-lines is first formulated in [59]. A multi-line station is defined as the station including tasks from two adjacent U-lines. In [42] it is proposed a dynamic programming approach for a U-line facility to balance many U-lines connected by multi-line stations. The cycle time for each U-shaped line is given. The approach leads to optimal solutions for small problem instances (up to 22 tasks on each individual U-shaped line. The multiple U-line problem is also investigated in [59]. The U-shaped lines are operated at the same cycle time and include multi-line stations. Several heuristic procedures are proposed to determine the multi-line stations. Problems with up to nine individual U-shaped lines that do not have more than 18 tasks each are solved. The effect of the U-shaped layout on the effectiveness of the U-line when breakdowns occur is investigated in [43]. The U-shaped line is preferred when buffer inventories are considered at all contact points between stations. In [34] it is developed a genetic approach to solve the mixed-model U- shaped ALB problem dealing simultaneously with the balancing and sequencing problems. The aim is to minimize the absolute deviation of workloads for a given number of workstations. The solution obtained from balancing problem is input to sequencing problem. The algorithm constructs two populations that represent sub-problems: a balancing population and a sequencing population. It is reported that the approach based on the integration of balancing and sequencing problems provides better solutions than a hierarchical approach. The U-shaped, mixed-model, asynchronous line in a just-in-time environment is analyzed in [45]. A genetic algorithm is designed to search good solution to the joint problem of the balancing and model sequencing problems, which is described as a mixed, zero-one integer, non-linear problem. The model aims to minimize the absolute deviation of workloads across workstations and the deviation of part production quantities in a just-intime system to facilitate level production. The computational study proves a good quality of the solutions found by the genetic algorithm. For instances 346

11 of practical size, the computational requirements are large but not unreasonable. An evolutionary approach called endosymbiotic evolutionary algorithm that efficiently solves the balancing and sequencing problems in a mixed-model U-shaped line is developed in [35]. The algorithm uses three types of populations. A balancing population and a sequencing population composed of individuals representing the task assignment and the model sequences. These individuals are called symbionts, each of them being a partial solution to the mixed-model U-shaped ALB problem. The last population consists of individuals obtained by combining the above two types of symbionts. These individuals are called endosymbionts that represent an entire solution to the problem. Efficient genetic representations and genetic operators are used. The procedure provides better quality solutions than other symbiotic evolutionary algorithms, and traditional hierarchical approaches as well. Significantly, less research has been conducted to the U-shaped mixedmodel ALB procedures. The combinatorial nature of the problem makes it difficult to be solved when the problem size increases and forces up to develop efficient approximate algorithms. Considerable improvements may be achieved by using improved solution approaches such as genetic algorithms and simulated annealing. Mixed-model line balancing with parallel stations For the situation of high production rates, where some task processing times exceed the specified cycle time, the use of parallel workstations, where multiple workers perform an identical set of tasks, is the common remedy. Even if paralleling results in additional equipment costs it allows more flexibility in assigning tasks and permits to deal with shorter cycle times. An approach for solving mixed-model ALB problem with stochastic times and paralleling of tasks is developed in [39]. The method incorporates different task selection rules for assigning tasks to workstations using the output performance measures and requirements of workers and equipments. The obtained layouts are simulated and the performance results are analyzed. In [2] it is proposed a nonlinear integer-programming model for the line balancing problem. The serial production system is addressed to mixedmodel production and is composed of serial stages with parallel workstations at each stage. A greedy heuristic method that takes into account taskdependent equipment cost and station paralleling simultaneously in task 347

12 assignment is developed. Computational results show that the method provides good feasible solutions. The mixed model ALB problem aiming to minimize the number of stations for a fixed cycle time taking into account parallel stations and assignment restrictions is considered in [67]. The procedure has as additional objective the balancing of workloads between and within workstations. It is shown that simulated annealing technique can be adapted relatively easily to the ALB problem and it achieves good quality solutions in acceptable computational times. Because the computation time to reach an optimal solution can be very large for practical instances, a lower bound for the number of workstations is introduced. Experimental results proved that the procedure produces good quality solutions in reasonable times even for large problems. In [51] it is developed an ant technique to solve the mixed-model ALB problem considering parallel stations and stochastic processing times. The comparison with the results obtained from other heuristic methods in terms of output performance measures show that the ant methodologies perform well and even outperform other heuristic for small problem sizes. The procedure exploits the properties of this ALB problem and provides good solutions in reasonable times based on features of ant techniques. The review on ALB with parallel workstations indicates that a great potential of meta-heuristics approaches (simulated annealing and ant techniques) exists in order to effectively find good solutions. Multi-model line balancing The problem of assigning multi models to a given number of lines is considered in [36]. The proposed heuristic for this problem aims to minimize the total assembly cost given a maximum capacity for each assembly line. To balance a multi-model line with small lot sizes in [69] it is proposed the use of a method for mixed-model lines considering that every repetition of a task is performed in the same workstation. In the case of a multi-model line with large lot sizes, a procedure for a single model line balancing is successively applied, taking into account the model with the largest frequency. To solve the multi model ALB problem, in [12] it is developed a balancing procedure that considers the labour, machine setup and inventory costs. The models are produced in batches. The precedence diagrams of models are combined to reduce the problem to a single model problem. As in other approaches ([37], [63]), the model sequence is ignored. Placing buffers 348

13 between two adjacent workstations, it is allowed the batch sizes to vary between stations. In [5], it is developed a branch and bound algorithm with a truncated search for the multi-product ALB problem. It is considered that the multi products have only one order for execution of the tasks. The common tasks of the models are performed first. The approach exploits a lower bound procedure as well as a partitioning scheme. Several node generation schemes are developed and tested. The method leads to good results for the multiproduct ALB problem. In order to reduce the production inefficiency due to the fact that models are significantly different, in [61], the use of a bypass sub-line adjacent to the main line is proposed. The tasks of models with relatively longer times are performed on this sub-line. A dynamic programming technique is used to assign the tasks to workstations. The solution is improved by a tabu search technique. A parallel computing approach for solving a combined balancing and sequencing problem is developed in [6]. The approach incorporates different extensions of the ALB problem and the consequences of disturbances during the execution of the production process (material bottleneck, machine breakdown). The proposed real time oriented control approach of assembly lines has as consequence a large decrease of additional costs in production process. Mixed-model sequencing problem The products assembled in mixed-model lines involve different assembly tasks and processing task times, require different facilities, parts and raw materials. The model changes often require a significant amount of setup time and cost that depend on model sequence. The mixed model assembly line sequencing is investigated in [31] for the first time. Various objectives are reported in the literature in determining the optimal sequence for a mixed model assembly line. The common objectives are: minimizing the overall line length ([3], [4]), [18], [19], [32]), minimizing the risk of stopping a conveyor ([54]), minimizing the total utility work ([4], [71]), keeping a constant rate of part usage ([21], [38], [41], [48], [50]). Surveys on several procedures for different versions of the mixed model-sequencing problem are provided in [3], [57], [70]. The problems of balancing and sequencing are tightly interrelated with each other. The optimality of model sequencing depends on the results of line balancing, which is affected by the model sequence. Some mixed-model 349

14 sequencing approach aiming to find a sequence of units that satisfies the demands of all models, for a given line balance, are reviewed in the following. The mixed-model ALB problem is applicable in just-in-time manufacturing environments, because attempts have been made to find sequences resulting in minimal work in process levels. The goal chasing method introduced in [48] is concerned with the sequencing of multi-product assembly lines taking into account the stability of parts usage rates. The processing time at workstation are smoothed by sequencing models so that a model with short processing time follows soon after a model with long processing time at the same workstation. In [3], the sequencing problem is formulated as an integer programming involving two objectives: minimizing the line length and keeping a constant rate of part usage. Taking into account various multiproduct assembly line configurations, the proposed analysis indicates that solutions minimizing the line length are not significantly different from the solutions that minimize the throughput of the line. A mathematical model formulation for mixed-model sequencing problem with the goal of minimizing the total work overload is proposed in [57]. To solve this problem a branch and bound procedure is developed. A branch and bound procedure for sequencing mixed model assembly lines aiming to minimize the line length and level the part usage rate is proposed in [4]. The solution is improved using a tabu search approach. The multiple objectives sequencing problem is solved in [25] using a genetic algorithm approach. The considered objectives are minimizing the total utility work, level the part usage and minimizing the total setup cost. The Pareto optimally technique is used to obtain near-optimal solutions to multiple objective sequencing problems. The tabu search procedure developed in [58] uses a pattern-based vocabulary building strategy for solving the problem of sequencing mixedmodel assembly lines. Vocabulary units represent partial sequences of model units. Applying the tabu search procedure, a solution of the sequencing problem is obtained sequencing the vocabulary units. It is experimentally demonstrated the importance of the use of the vocabulary building as a fundamental component of a more effective method for solving sequences problems with large demand values. To solve the mixed-model sequencing problem considering both objectives of setups and material usage rate minimization, a simulated annealing technique is presented in [50]. Comparing the results with those 350

15 obtained by a tabu search approach, superior results of the simulated annealing approach are reported. An ant colony optimization approach to solve the same problem is developed in [38]. The performance of this approach is demonstrated by comparing the obtained results with those of other meta-heuristics, such as simulated annealing, tabu search and genetic algorithms. Determining the optimal sequence is an important problem for efficient use of mixed model assembly lines. The review of mixed-model sequencing procedures reveals that there is a large potential of optimization with respect to the developing of new efficient solution procedures to industrial problems. The combinatorial nature of the problem makes it difficult to solve when the problem size increases and the use of search metaheuristics, such as simulated annealing, tabu search and ant colony can be exploited to solve these types of real problems. Stochastic mixed/multi model line balancing The processing time variability is especially the case of automated flow line where varying production rates may result from machine breakdowns. Significant variation may also result from human operators ([57]) with respect to work rate, skill and motivation. These variations may considerably influence the performance of the system. In [68] it is modified the heuristic proposed for single model stochastic ALB problem to allow parallel workstations and mixed-model lines. The data obtained by the single model ALB heuristic are utilized to determine the proportional occurrences of the model variations. The results obtained for different cycle times values are compared with those obtained in [49], requiring a lower number of operators. The procedure leads to smaller smoothness indices and lower total processing costs. In [28], several modifications that can be implemented in the proposed branch and bound procedure dedicated to single model ALB problem are mentioned. Such modifications reflect mixed model lines and the variability of the task times. For solving the mixed-model stochastic ALB problem, in [13] two procedures are developed. The first procedure is based on the heuristic proposed in [24] and the second procedure uses a consecutive sequencing of the tasks. The balancing approach takes into account the process inventory between workstations and has as objective minimizing the total operating cost of the line. The latter procedure that is a shortest path heuristic yields designs with lower total costs. 351

16 In [39], a previous approach addressed to deterministic processing times is modified to solve a mixed-model ALB problem with stochastic processing times. The mixed-model ALB problem with fuzzy processing time is considered in [7] and [27]. To solve this problem, a heuristic that aggregates fuzzy time and transform the problem into a fuzzy single model ALB problem is proposed. The procedures consider both cycle time and precedence constraints and its performance is proved by experimental tests with good results. In [52] and [53] important design factors and their effects on the performance of U-shaped flow line is analyzed. The stochastic U-shaped ALB problem aiming to minimize the cycle time given a fixed number of stations is investigated in [53]. Several bounds and approximations for the cycle time are proposed assuming that task times are random variables. An analysis related to optimally allocate workers to machine on a U-shaped line with multiple workers in order to find the minimum cycle time is presented in [52]. It is showed that reducing the task variability increase the throughput of several workers with multi-functions, each of them being responsible for several machines. The approach presented in [14] takes into account the performance of straight line and U-line configurations under stochastic task times. The obtained results lead to the conclusion that the optimal U-shaped layout is less productive under the stochastic times consideration. The U-shaped ALB problem with stochastic times is formulated as a stochastic program in [66] combining stochastic straight-line models and deterministic U-shaped line models. The tasks are assigned to stations so that the probability of completing the tasks within a fixed cycle time is at least a fixed value. To formulate the stochastic U-shaped ALB problem, the deterministic model proposed in [65] is adapted and the objective of minimizing the number of stations exceeding the lower bound on the number of stations is considered. The methodology presents different approaches to identify the lower bound. The procedure finds optimal solutions to problems up to 24 of tasks with commercially software and can be modified to treat realistic situations. A hybrid heuristic for U-shaped ALB problem with stochastic times is proposed in [16]. The problem is solved using priority rule based procedures and an improvement procedure to improve the current solution. Although the stochastic mixed-model ALB problem reflects more realistically modern 352

17 manufacturing environments, the amount of studies dedicated to this type of problem is much smaller. Concluding remarks The paper presented a review on the mixed-model assembly line balancing problem. This study identified the types of ALB problems in mixed-model environment and investigated the proposed solution procedures. In order to distinguish the problem types, the mixed model ALB problems are reviewed considering the layout of the production system and the variability of operation times. Due to the importance of the mixed-model assembly lines in modern industry, an increasing interest in the research of mixed-model assembly lines is noticed. The mixed-model ALB problem is much more complex by additional considerations related to high variability of demands and relatively small volume for each model. The overall interest in the research is to design more flexible assembly systems, which include additional characteristics in order to reflect as much as possible the industrial reality. The study reveals on one hand, the modelling of different types of ALB problems in the mixed-model environment, and on the other hand, it is needed more research in developing efficient solution procedures for these complex ALB problems. However, the number of studies conducted on the mixed-model version of the ALB problem is small. The study leads to the conclusion that the meta-heuristic procedures can be successfully applied in order to solve large problem instances. Taking into account their performance, the genetic algorithms, simulated annealing and ant colony techniques worth further investigations. The genetic algorithm approaches as well as the genetic approaches combined with simple local optimization procedures show promising results to a variety of ALB problems, including multiple objectives and the U-shaped assembly line. There is also the opportunity to generalize the approaches addressed to mixed-model ALB problems. The research trend to the improvement of the solution approaches for the proposed models using meta-heuristic procedures in order to solve practical problems. The extension of the proposed models through the incorporation of different factors encountered in practice, such as equipment selection in the balancing process, additional restrictions (related to the locations on tasks or to assigning of combinations of tasks, station length constraints), many U-lines or other types of lines, and stochastic processing task times is still needed for further research. Although the variability in the processing time of tasks can significantly affect the performance of the 353

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