A Fuzzy Mathematical Programming Model for Cell Formation and Layout Design

A fuzzy linear programming (FLP) approach is taken to solve an extended mathematical linear programming model to handle two important problems in cellular manufacturing systems (CMS) simultaneously: Cell formation and layout design. We seek to minimize the total cost of inter-cell and intra-cell (forward and backward) movements and the cost of machines. This model also considers the minimum utilization level of each cell to achieve an improved performance of cell utilization. Over the past three decades, group technology (GT) has emerged as a useful scientific principle in improving the productivity of batch-type manufacturing systems in which many different types of products with relatively low volumes are produced in small lot sizes. Cellular manufacturing (CM) is a successful application of GT concepts. The design of a CMS usually begins with two fundamental grouping tasks: Part-family formation and machine-cell formation. Many authors (Liao (1994), Heragu and Kakuturi (1997), Nair and Narendran (1998), Xambre and Vilarinho (2003), Chiang and Lee (2004) and Keeling et al. (2007)) adopt either a sequential or a simultaneous procedure to group the parts and machines. The sequential procedure used in some of these studies determines the part families first, followed by machine assignments. On the other hand, the simultaneous procedure determines the part families and machine groups concurrently. Some newly developed models are more realistic and appealing to real-world applications (Heragu and Gupta (1994), Atmani et al.) 1995(, Kim and Suh (1998), Wu et al. (2007), Xiaodan et al. (2007) and Saidi-Mehrabad and Safaei (2007)) because they take more factors such as demands, processing times, space availabilities, material handling costs, and machine capacities into consideration.Most CMS models assume that the input parameters are deterministic and certain. In practical situations, however, many parameters such as processing time, part demand, and available machine capacity are uncertain and imprecise. Since sufficient data are not always available for predicting uncertain parameters, fuzzy logic is introduced as a powerful tool for expressing this uncertainty through the expert’s knowledge. CMS design problem, as a real life problem, can be investigated in a fuzzy environment due to the fuzzy design parameters (Shanker and Vrat (1999), Arikan and Gungor (2005) and Safaei et al. (2007)).We propose a mathematical programming model for an extended cell formation problem and layout design simultaneously with uncertain conditions. The fuzzy demand and fuzzy machine capacity are also considered in the proposed model. The proposed model determines the optimal cell configuration by minimizing inter-cell movement, intra-cell movement (forward and backward) and machine costs. The main advantage of the proposed model is in its consideration of uncertain conditions, batch material handling movements, sequence operation and minimum utilization level of each cell. The proposed model considers the fuzziness in part demands and machine capacities. The model is developed under the following assumptions:1- The number of cells is known.2- The upper bound and lower bound of the cell size are known. 3- Each part type has a number of operations to be processed according to a known sequence. Operations related to each part type must be processed in the order specified by their numbers.4- The processing times for all operations of part types on different machine types are known and deterministic.5- Parts are moved between and within cells in batches. Inter and intra-cell (forward and backward) batches have different sizes. Inter and intra-cell movement (forward and backward) costs are constant for all moves, but in each cell the distance travel from machines to is considered.6- The demand for each part type is given as a piecewise fuzzy number.7- The capability of each machine type is known. Also, the capacity of each machine is given as a piece-wise fuzzy number. The fuzzy capacity of machine is determined by the decision maker in terms of nominal capacity” and actual capacity”. The actual capacity is more realistic and applicable than the nominal capacity. In other words, in movng from the actual capacity towards the nominal capacity values, the risk related to the decision making process increases.8- The costs of each machine type such as constant cost and variable cost are known.