بررسی میانگین فرایند بهینه بر اساس تابع درجه دو از دست دادن کیفیت و اصلاح طرح بازرسی

In 1996, Pulak and Al-Sultan presented a rectifying inspection plan for determining the optimum process mean. However, they did not consider the quality cost for the product within the specification limits and did not point out whether the non-conforming items in the sample of accepted lot is replaced or eliminated from the lot. In this paper, we propose a modified Pulak and Al-Sultan’s model with quadratic quality loss function of product within the specification limits. Assume that the non-conforming items in the sample of accepted lot are replaced by conforming ones. Finally, the numerical results and sensitivity analysis of parameters of modified model and those of Pulak and Al-Sultan are provided for illustration.

Traditionally, the product characteristic within specifications is the conforming item. Taguchi [33] redefined that the product quality is the total loss to the society. According to Taguchi’s opinion, a producer needs to manufacture a product based on its target value in order to reduce the society’s loss. Taguchi [33] proposed the quadratic quality loss function for evaluating product quality. If the process mean approaches the target value and the process standard deviation approaches zero, then the process is under optimum control. Taguchi’s quadratic quality loss function has been successfully applied in on-line and off-line quality control. The economic selection of optimum process parameters is a major problem for the filling/canning industry. Recently, there are considerable attentions to the study of economic selection of process mean, e.g., [21], [20], [14], [5], [32], [7], [8], [9], [25], [27], [35], [18] and [17]. Lee and Elsayed [21] presented a two-stage screening procedure for obtaining the optimum process mean and screening limits of the surrogate variable. They assumed that the performance and surrogate variables are jointed normally distributed. Lee et al. [25] extended Lee and Elsayed’s model and emphasized two surrogate variables are used simultaneously in single screening procedure for determining the process mean and screening limits. Kim and Cho [20] adopted the truncated Weibull quality characteristic and quadratic quality loss function for determining the optimum process mean. Duffuaa and Siddiqui [14] considered the three-class screening of product and the presented the effect of measurement error on the optimum process mean, the expected profit per item, and the inspection policy. Bowling et al. [5] first presented the problem of setting the optimum process mean for a multi-stage serial production system. They adopted the Markovian approach for formulating the model. Assume that the quality characteristic is normally distributed and consider the product with both-sided specification limits. When the work-in-process product performance falls below a lower specification limit or above an upper specification limit, it is necessary to be reworked or scrapped, respectively. If its performance falls within the specification limits, the work-in-process product goes on to the next stage until the product finished. Chen and Chung [11] presented the quality selection problem to imperfect production system for obtaining the optimum production run length and target level. Rahim and Tuffaha [32] further proposed the modified Chen and Chung’s model with quality loss and sampling inspection. Chan and Ibrahim [7], [8] and [9] addressed the multivariate quadratic quality loss functions applied in the determination of optimum process mean for nominal-the-best, smaller-the-better, and larger-the-better quality characteristics, respectively. Li [27] proposed the optimum process mean setting by considering the asymmetric linear quality loss function in the application of industry. Teeravaprug [35] considered the two grades of product. The product may be sold in the primary market or secondary market if the product quality is accepted. If the quality cannot be accepted by the secondary market, then the product is scrapped. He adopted the quadratic quality loss function for evaluating the quality cost of product and obtained the optimum process mean based on maximizing the expected profit per item. Hariga and Al-Fawzan [18] proposed an integrated inventory and target level problem for determining the optimum process mean and production cycle time. They considered the quality characteristic of product may be sold to different quality requirements of customers from the same market. Feng and Kapur [17] adopted the asymmetric quadratic quality loss function and asymmetric piecewise linear loss function for jointly determining the optimum process mean and economic specification limits for 100% inspection policy of product. Some works [26], [22], [23], [21] and [25] have addressed the 100% screening in the filling process. However, the 100% inspection policy cannot be executed in some situations. Hence, one needs to consider the use of sampling inspection plan for deciding the quality of a lot. In rectifying inspection plan, one needs to do a 100% inspection for the product of the rejected lot and replace the non-conforming items with conforming ones. The non-conforming items in the sample of accepted lot are usually replaced by conforming ones. It can provide assurance of the product level shipped to the customer. Carlsson [6] first proposed the application of the variable sampling plan for determining the optimum process mean. Subsequently, Boucher and Jafari [4] considered the attribute single sampling plan applied in the selection of process target. They assumed that items which belong to accepted lots can be sold at a certain price while those items that belong to rejected lots can be sold at a reduced price in a secondary market. Tang and Tang [34] discussed a general screening procedure for product quality. Screening is becoming an attractive practice for removing non-conforming items. Al-Sultan [1] extended Boucher and Jafari’s model to series production. In 1996, Pulak and Al-Sultan extended Boucher and Jafari’s model and presented rectifying inspection plan for determining the optimum process mean. However, they did not consider the quality cost for the product within the specification limits and did not point out whether the non-conforming items in the sample of accepted lot is replaced or eliminated from the lot. In Pulak and Al-Sultan’s model, they considered rectifying inspection based on the fact that rejected lots can have considerably different numbers of defective units, and it is not realistic that they can be sold for the same reduced price as in the Boucher and Jafari’s model. Al-Sultan and Pulak [2] further considered the effect of process variance on Pulak and Al-Sultan’s rectifying inspection plan. Juran [19, pp. 23–26], Duncan [15, pp. 371–372], Feigenbaums [16, pp. 751–757], and Montgomery [28, pp. 566–567] pointed out that rectifying inspection plan is commonly used in many industries. Pulak and Al-Sultan [29] did not consider the quality cost of a product within specification limits in their model. In 2006, Chen propose a modified Pulak and Al-Sultan’s model for determining the optimum process parameters under the rectifying inspection plan with average outgoing quality limit protection. The quality cost within specification limits has been addressed in Chen’s model. Some works (e.g., [12], [3], [31], [13], [30], [20], [24], [7], [8], [9], [32], [35], [10] and [17] and others) have addressed the quadratic quality loss function applied in the determination of optimum process target. In this paper, we shall present a modified Pulak and Al-Sultan’s model for determining the optimum process mean under a rectifying inspection plan. Assume that the non-conforming items in the sample of accepted lot are replaced by conforming ones. The 100% inspection is executed in the rejected lot and the perfect replacement of a product is considered. Taguchi’s quadratic quality loss function is applied in the evaluation of the product quality. The motivation behind this work stems from the fact that the neglect of the quality loss within the specification limits should have the overestimated expected profit per item for the production system. The next section gives a literature review of the Pulak and Al-Sultan’s model and discusses the modified Pulak and Al-Sultan’s model and the solution procedure. In Section 3, a numerical example and the sensitivity analysis of parameters are provided. The numerical results of modified model and those of Pulak and Al-Sultan [29] are compared. Some discussion about modified model is illustrated. Finally, the conclusions and areas for further research are given in Section 4.

In this paper, we have presented the optimum process mean setting based on rectifying inspection plan and quadratic quality loss function of product. The non-conforming items in the sample of accepted lot are replaced by conforming ones in the modified model. We also performed the sensitivity analysis of parameters and compared the solutions between the modified model and Pulak and Al-Sultan’s one. The modified model for product having quality cost within specification limits always has smaller expected profit per item than that of Pulak and Al-Sultan’s one. The extension to the modified Pulak and Al-Sultan’s model by considering different cases of the non-conforming cost of accepted lot may be left for further study.