Optimal Process Mean Research Articles

Optimising the Gaussian-distributed filling process parameters

A filling production system is that characterised by two contradicting nonconforming cost components. The first component is that incurred when the product specification exceeds the upper tolerance limit, called an overfill cost, whereas the second cost component is contributed when the product specification falls below the lower tolerance limit, called an underfill cost. The ratio of the two cost components affects to a high extent the choice of the process parameters, especially the process location parameter that minimises the overall costs. The work in this research is devoted to finding the filling process's optimum parameters; namely the process's mean and variance, considering a Gaussian-distributed process. Firstly, the process's variance is considered constant; and searching for the optimal process mean parameter takes place. An analytical, closed form, solution model is developed that realises such target. The developed model is ended up with a graph that can simply be used by concerned engineers and practitioners. The assumption of a constant process variance is then relaxed; and the developed model is extended to involve the variance reduction cost. An algorithm is introduced to solve the extended model that optimises the filling process' mean and variance.

The modified Huang’s cost model for process optimization

In 2001, Huang presented the problem of determining the optimum process mean and standard deviation based on considering the trade-off relationship between the process adjustment cost and the quality loss of product. They considered the normal quality characteristic and adopted the quadratic quality loss function for measuring the product quality. In this paper, we further propose the problem of process optimization and reconsider the modified Huang’s model under the specified process capability index value for determining the optimum process parameters. The symmetric quadratic, asymmetric quadratic, and asymmetric linear quality loss functions will be adopted for evaluating the product quality. Finally, the numerical example and the sensitivity analysis of parameters will be provided for illustration.

Economic manufacturing quantity, optimum process mean, and economic specification limits setting under the rectifying inspection plan

Economic manufacturing quantity, process mean, and specification limits setting are three important methods for the inventory and quality control problems. In the imperfect production system, we usually consider the manufacturing quantity for reducing the inventory cost, determine the process level for reducing the production cost, and select the specification limits for screening the products. In this paper, we propose the above integrated model based on the application of rectifying inspection plan for obtaining maximum expected total profit of product. The asymmetric quadratic quality loss function is adopted for measuring the product quality. The sensitivity analyses of parameters are provided for illustration.

Joint optimization of process target mean and tolerance limits with measurement errors under multi-decision alternatives

Determining the optimal target mean for a process has been identified as an important research area and a number of models have been proposed in the literature. This paper differs from previous studies of this problem in two ways. First, most previous studies address the process target problem through models seeking the optimal process mean using fixed tolerance settings of a process. Second, in real-world industrial settings, there are several markets often available with different price/cost structures. In this paper, we develop a model for jointly determining both the optimal process target mean and the optimal tolerance limits under the situation where there are several markets available with different price/cost structures. We then investigate the effects of measurement errors on the optimum process target and tolerance limits with multi-decision alternatives. A numerical example is given, and sensitivity analyses are also performed to study the effects of measurement errors on this model.

Determining the optimum process mean based on quadratic quality loss function and rectifying inspection plan

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.

Determining the optimal process means under mixture normal distributions

In this paper, we consider that the working environment has certain states, and in every state, the parameters of quality characteristics are different. Thus, if we set the characteristics parameters in a specified state, these parameters will change to another state. To describe this situation, we use a mixture of normal distributions, which comprise a flexible and powerful statistical-based modeling tool in practice. Under the step loss function and the piecewise linear loss function, we select the optimal means for the proposed manufacturing process.

Process mean, process tolerance, and use time determination for product life application under deteriorating process

Robust design can significantly improve a producer’s competitive ability to deliver high-quality products with low development cycle time, quality loss, failure cost, and tolerance cost. However, during usage by a consumer, the functional performance of a product or some of its components may change as use time passes, leading to unexpected product failure that is usually costly in both time and money. The cost of these failures may influence the determination of optimal use time and optimal initial settings as compensation for the possible process mean changes during consumer usage. In addition to finding use time and initial settings for proper quality performance, the determination of process mean and tolerance also needs to be considered. As is known, changes in process means acquired quality loss and variability, while process tolerance has an effect on tolerance-related costs and quality loss. Because there exists a dependency among use time, initial setting of process mean, process mean, and process tolerance, these values must be determined simultaneously. Thus, in this paper, an optimization model with an acceptable reliability value is developed to minimize total cost, including quality loss, failure cost, and tolerance cost, by determining optimal use time, initial settings, process mean, and process tolerance, simultaneously. Applications of single and multiple components are presented to explain the proposed models. Finally, sensitivity analysis and model discussions on some decision variables are performed.

Determination of the optimum target value for a production process with multiple products

This paper considers the problem of determining the optimum target value of the process meant for a production process where multiple products are processed. Every outgoing item is inspected, and each item failing to meet the specification limits is scrapped. The quality characteristics of the products are assumed to be normally distributed with known variances and a common process mean. A profit model is constructed which involves selling prices, costs of production and inspection, and losses due to the scraps. The common process mean is determined so that the expected profit is maximized. A method of finding the optimum common process mean is presented and an example from electronic device (4L PBGA) production process is given.

The Modified Pulak and Al-Sultan's Model for Determining the Optimum Process Parameters

Economic selection of process parameters has been an important topic in modern statistical process control. The optimum process parameters setting have a major effect on the expected profit/cost per item. There are some concerns on the problem of setting process parameters. Boucher and Jafari (1991) first considered the attribute single sampling plan applied in the selection of process target. Pulak and Al-Sultan (1996) extended Boucher and Jafari's model and presented the rectifying inspection plan for determining the optimum process mean. In this article, we further propose a modified Pulak and Al-Sultan model for determining the optimum process mean and standard deviation under the rectifying inspection plan with the average outgoing quality limit (AOQL) protection. Taguchi's (1986) symmetric quadratic quality loss function is adopted for evaluating the product quality. By solving the modified model, we can obtain the optimum process parameters with the maximum expected profit per item and the specified quality level can be reached.

Joint optimization in process target and tolerance limit for L -type quality characteristics

Most models reported in the current process target literature assume a nominal-the-best type quality characteristic in determining the optimal process mean value. In real-world industrial settings, however, a number of quality characteristics of interest are of a larger-the-better type (L-type), and the problem of jointly determining the optimum process mean and tolerance limit for the L-type quality characteristic may be of significant practical importance, yet this has largely been overlooked in the literature. In this paper, cost-effective optimization models are developed, and the methods of finding optimum solutions are presented. Numerical examples and sensitivity analysis are given.

Optimal determination of production run and initial settings of process parameters for a deteriorating process

This paper presents a generalized model for the optimal determination of a production run and the initial settings of the process mean and process variance for a deteriorating production process. It is assumed that the process deteriorates due to tool wear-out. The probability that the process deterioration starts at a random point in time follows an exponential distribution. Quality loss from the target values is measured using Taguchi’s quadratic loss function. The time dependent maintenance cost and the salvage value of the equipment are included. The expressions for determining the optimal process mean and process variance are developed. Numerical examples are provided to demonstrate the application of the proposed model.

Designing the optimal process means and the optimal production run for a deteriorating process

The objective of this paper is to develop an integrated model for simultaneously determining the optimal initial process means and the optimal production run length in cases with multiple quality characteristics, in which the process means are assumed to shift from an in-control state to an out-of-control state by the occurrence of an assignable cause. It is assumed that the elapsed time until the occurrence of the assignable cause is exponentially distributed. A mathematical model for the expected cost per unit time is derived based on the framework of a multivariate quality loss function. A numerical example is provided to illustrate the application and usefulness of the proposed model. A sensitivity analysis is provided to study the impact of the input parameters on the model. Furthermore, some future research directions are outlined.

Determining the optimum process mean for a mixed quality loss function

In 1999, Wen and Mergen adopted the step loss function of a product to balance the costs of out-of-specification occurrences and to determine the optimum process mean. They considered a normal quality characteristic and assumed a known standard deviation for the process in their model. This paper proposes a modified Wen and Mergen [1] model with a mixed quality loss function to determine the optimum process mean. The quality loss of the product per item includes a quadratic loss within specifications and an out-of-specification piecewise linear loss.

Determining the Optimum Process Mean under a Log-normal Distribution

Wen and Mergen [Quality Engineering (1999) vol. 11, pp. 505–509] used the unbalanced step loss function for measuring the cost of the nonconforming item and adopted a trade-off model for determining the optimum process mean. They assumed that the quality characteristic is normally distributed, the process variance is constant, and the process mean is unknown. In this paper, we further present the modified Wen and Mergen’s (1999) model with log-normal distribution. The step loss function and the piecewise linear loss function of product are considered in the modified model.

Determining the optimum process mean under a Gamma distribution

Wen and Mergen (1999) presented a trade-off problem, taking both the costs of falling below the lower specification limit and exceeding the upper specification limit into account, to determine the optimum process mean. In Wen and Mergen’s (1999) model, the quality characteristic, X, is assumed to be normally distributed and X is nominal-is-best with a target value. In this paper, we extend Wen and Mergen’s (1999) model to a gamma quality characteristic case by respectively using step loss function and piecewise linear loss function in the cost function. The modified Wen and Mergen’s (1999) model with gamma distribution is developed. A numerical example is provided for illustration.

Integrated model for determining the optimal initial settings of the process mean and the optimal production run assuming quadratic loss functions

Chen and Chung (1996) addressed the problem of the joint determination of the optimal process mean and production run for an industrial process. Their study considered a product with an upper and a lower specification limit. The optimal process mean and optimal production run were obtained by balancing the profit of meeting and not meeting the specification limits. However, Chen and Chung did not consider the quality cost for the product within the specification limits. The present paper revisits the problem and incorporates the quality cost by introducing a Taguchi loss function for determining simultaneously the optimal process mean and production run. As per Chen and Chung, the present paper assumes a 100% inspection scheme. It also investigates the differences between Chen and Chung's approach and the Reward Theorem approach. A sample inspection scheme is also proposed. Numerical examples are provided to demonstrate the application of the model. A sensitivity analysis of the model is provided. Some new directions for further research are also outlined.

Joint determination of target value and production run for a process with multiple markets

Setting the target value (process mean) for an industrial process when its outputs are sold to quality-dependent customers is an important and challenging decision for quality and manufacturing managers. The determination of an appropriate process mean does not only affect the output rate of conforming units but also affects other manufacturing decisions such as finished product and raw material lot sizing policies. In this paper, we address the integrated targeting-inventory problem to determine simultaneously the optimal production cycle time and the target value that maximizes the total expected profit per unit of time. Depending on its quality characteristic, the output of the manufacturing process may be sold to different customers from the same market but with different quality requirements. The output rejected by original customers may be stored for some time and then sold to other customers in the same market at a reduced price. We built the developed mathematical model in Microsoft Excel and used SOLVER to search for the optimal production run and process mean. We also carried out a sensitivity analysis to study the effects of some problem parameters on the quality and inventory decisions.

Determining the Optimum Process Mean of a One-sided Specification Limit with the Linear Quality Loss Function of Product

Wen & Mergen (1999) proposed a method for setting the optimal process mean when a process was not capable of meeting specifications in the short term. However, they neglected to consider the quality loss for a product within specifications in the model. Chen & Chou (2002) presented a modified Wen & Mergen's (1999) model, including the quadratic quality loss function for a one-sided specification limit. In this paper, we propose the modified Wen & Mergen (1999) cost model including the linear quality loss function of a product for determining the optimal process mean of a one-sided specification limit.

Simultaneous determination of the optimal process mean and cutting speed for a manufacturing process

The relationship between machining speed and the process mean for a manufacturing process has received little attention. This paper presents an analytical model and its solution procedure to simultaneously determine the optimal machining speed and process mean when the objective is to minimize the total cost per part. The total cost of the model includes replacement (resetting), production and quality costs. A global optimization algorithm is used to solve the proposed unconstrained model. Numerical examples to demonstrate the effectiveness of the proposed model are given and an extensive sensitivity analysis of the proposed model parameters is also provided. It has been concluded that machining speed and process mean are highly related. Therefore, the machining conditions should be taken into account when solving process mean targeting problems.

ECONOMIC SELECTION OF MEAN VALUE FOR A FILLING PROCESS UNDER QUADRATIC QUALITY LOSS

Consider a filling process where containers are filled with an important ingredient in a character. All containers are inspected, and the containers satisfying to meet the predetermined specification limits are sold in a regular market for a fixed price, and failing to meet them are emptied and refilled by the same filling process after some reprocessing. We assume that reprocessing cost is proportional to the quantity of the ingredients in a container that is not changed after reprocessing. An economic model is constructed on the basis of the selling price and the costs of production, inspection, reprocessing, and quality. We assume that the quality cost function is a quadratic function of the deviation from target and the quantity of the ingredients in a container is normally distributed with a known variance. Method for finding the optimum process mean is presented and a numerical example is given.