مدل ریاضی برای بهینه سازی هزینه کیفیت Mathematical model for quality cost optimization

abstract Quality engineering uses robust design in order to improve quality by reducing the effects of variability. Variability of the product can be reduced by two stages. One is parameter design which is adjustable to the nominal value so that output is less sensitive to the cause of variability. Other one is tolerance design which is to reduce the tolerance in order to control variability. All costs incurred in a product life cycle can be divided into two categories—manufacturing cost before the sale to the customer and quality loss after the shipment of the product to the customer. It is very important to find the optimum tolerances for each of the characteristics. A balance between manufacturing cost and quality loss should be arrived at in the tolerance design for quality improvement and cost reduction. For the case of Nominal-The-Best, a mathematical model is developed in order to determine the optimum product tolerance and minimize the total cost which includes the manufacturing cost and the quality loss. Since the process capability index (Cpm) shows the balance of quality responsibility between the design and the manufacturing engineers, this is taken as the basis in developing the functional relationship between the variability of the product and the tolerance. Based on these relationships, the total cost of model can be expressed as a function of product tolerance from which the optimal tolerance limits can be found out. Finally, using this model a tolerance design approach that increases the quality and reduces the cost can be achieved in the early stages of the product process design stage itself. & 2008 Elsevier Ltd. All rights reserved. 1. Introduction Quality engineering uses robust design in order to improve quality by reducing the effects of variability. Variability of the product can be reduced by two stages [10]. One is parameter design, which is adjustable to the nominal value so that output is less sensitive to the cause of variability. Other one is tolerance design, which is to reduce the tolerance in order to control variability. All costs incurred in a product life cycle can be divided into two categories—manufacturing cost before the sale to the customer and quality loss after the shipment of the product to the customer [2]. Using parameter-design technique the optimum level of each control factor for the case of Nominal-The-Best quality characteristic is determined. There is no manufacturing cost associated with parameter design i.e., changing of the nominal value of the product parameters. During the tolerance design, the design engineer will systematically specify the performance levels of certain factors needed to meet the requirement of the quality characteristics. Designers can get the tolerance limit for each factor in order to achieve this design objective. The loss function is an expression of estimating the cost of quality with respect to the target value and the variability of the product characteristics in terms of monetary loss due to product failure in the hands of the customer [1]. The loss function is a way to show the economic value of reducing the variability and staying very close to the target value. Whereas in the case of manufacturing cost for a product, cost usually increases as the tolerance of the quality characteristic are close to the ideal value [4]. That is why there is a need for more refined and precise operations as the ranges of output are reduced. Therefore, a balance between manufacturing cost and quality loss should be arrived at in the tolerance design for product quality improvement and cost reduction. Since the process capability index (Cpm) [3] shows the balance of quality responsibility between the design and the manufacturing engineers, this is considered as a tool for the estimation of the product variability in terms of product tolerance. If the tolerances are very tight the manufacturing cost will be high and loose tolerances result in low manufacturing cost. The cost equation suggested by Mr. Spotts is A+B/t 2 [5], where t is the tolerance. It can be seen that tight tolerance specifications results in more manufacturing cost since additional operations cost, high precision equipment and machines and slower manufacturing ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/rcim Robotics and Computer-Integrated Manufacturing 0736-5845/$ – see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2008.03.018 Tel.: +9180 3370912(Res.); +9180 360 0822(Off.). E-mail address: nvrnaidu@gmail.com Robotics and Computer-Integrated Manufacturing 24 (2008) 811– ۸۱۵ rates. This tolerance cost equation is considered for the mathematical modeling. In addition to the manufacturing cost incurred, Dr. Taguchi’s [6] quality loss function L(y) ¼ K (yT) 2 which is associated with deviation from the target value T, is also considered. In general even though less manufacturing cost, loose tolerance indicates that the variability of the product characteristic will be high resulting in poor-quality and high-quality loss. On the other hand, a tight tolerance indicates that the variability of the product characteristic will be less, resulting in very good quality reducing quality loss but increasing manufacturing cost. In addition to these two costs, associated scrap/reworks costs are also considered when the quality characteristic falls outside the tolerance limits [9]. Hence the total cost that consists of quality loss and manufacturing cost is applied to find the most economical and efficient way of determining the tolerance limits.