國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
1
CHAPTER 1. INTRODUCTION
1.1 Research Motivation
Control charts are widely used in statistical process control, and their design parameters must be pre-determined for their use, such as sample size, sampling time, and control limits. Economic design of control charts have been widely used to determine these parameters from economic viewpoint. However, to reduce product quality loss, the 100% inspection with specification limits is always performed. To determine the specification limits, a widely use method is to minimize products cost.
In the technology industry, like the product’s insulation property, its loss of heat is the smaller the better. In the service industry, the service time is also the smaller the better.
How to determine their specification limits and control chart to monitor the quality variable with maximum profit is an important issue. In this project, we simultaneously determine the specification limits and the control chart parameters with maximize expected profit per unit time. Most articles on economic design of specification and control charts consider normal distribution of quality variable, but in this article, we consider gamma distribution for the smaller the better quality variable; hence, we determine only an upper specification limit in the gamma distribution. For wide use of different shift scales and because of the asymmetric gamma distribution, we design the asymmetric economic EWMAX-bar control chart and the asymmetric economic VSI EWMAX-bar control chart. Hence, we combine product cost and control chart cost in a profit model and then we maximize this profit per unit time to determine the optimum upper specification limit and the design parameters of the EWMAX-bar
control chart and the VSI EWMAX-bar control chart.
1.2 Literature Review
Economic design of control charts have been widely used to determine control chart parameters from economic viewpoint. Duncan (1956) first proposed the concept of an economic design for the X-bar control chart. He considered a process that does not shut down when the assignable cause is searched, and developed a process cost model that includes the cost of sampling and finding the assignable cause when it exists or when none exists. He also demonstrated how to determine control chart parameters. Montgomery (1980) presented a review and literature survey in the economic design of control charts. Panagos, Heikes, and Montgomery (1985) described two continuous and discontinuous manufacturing process models, where the
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
2
continuous process model is consistent with that developed by Duncan. They showed that the wrong choice of a process model would have a potentially serious economic result.
Control charts typically take a fixed number of samples with a fixed sampling time and plot them on the control chart with a fixed control limit. To improve control chart performance, the adaptive control chart has been developed such as variable sampling interval control chart. Reynolds et al. (1988) proposed the VSI X-bar control chart. Bai and Lee (1998) considered the economic design of the VSI X-bar control chart. They preferred to use only two sampling interval lengths with two sub-regions between the two control limits.
The EWMA chart is widely used for detecting small process shifts. Roberts (1959) first introduced the exponentially weighted moving average chart. Crowder (1987) and Saccucci and Lucas (1990) discussed the ARL calculation of the EWMA control chart. Montgomery et al. (1995) presented a statistically constrained economic design of the EWMA control chart. They minimized the cost model, subject to statistical constraints on average run length or average time to signal, to determine the design parameters of the EWMA control chart. Chou et al. (2006) proposed an economic design of VSI EWMA charts. They considered two sampling interval lengths and derived the cost model to determine the parameters of VSI EWMA control charts using the genetic algorithm.
To reduce product quality loss, the 100% inspection with specification limits is necessary. Kapur (1988) considered three types of quality characteristics; the smaller the better, the larger the better, and the nominal the best and used three types of loss function to evaluate the loss of three types of quality characteristics with normal distribution. Phillips and Cho (1998a) used the truncated quadratic loss function on the smaller the better quality characteristic, which follows gamma distribution.
Phillips and Cho (1998b) used linear empirical loss function and quadratic empirical loss function for the quality variable, which follows normal distribution. Feng and Kapur (2006) considered asymmetric quadratic loss function and asymmetric piecewise linear loss function for the quality variable with normal distribution. These four articles minimize the expected cost per product to determine the specification limits. Hong et al. (2006) considered the larger the better quality characteristics of normal distribution. They used two types of profit models, unconformable items that are reprocessed and unconformable items that are sold at a discount price, to maximize the profit model to determine the optimum process mean and specification
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
3
limit. Hong and Cho (2007) considered several available markets with different price structures. They derived the model of expected profit per item and maximized it to determine the process mean and tolerance. They also investigated the effects of measurement errors on the process mean and tolerance.
1.3 Research Method
This study simultaneously determines the upper specification limit and the design parameters of EWMAX-bar control chart with maximal profit. In Chapter 2, we consider the smaller the better quality variable with in-control and out-of-control gamma distributions. To measure the performance of the proposed EWMAX-bar chart, we let in-control average run length (ARL0) equal to 370 by using the Markov chain approach and then find the best reference value of λ, which is a weight of EWMAX-bar
statistic, and factors of control limits of EWMAX-bar chart to minimizing out-of-control average run length (ARL1). In Chapter 3, we derive the profit model per unit time with only EWMAX-bar chart but without producer tolerance. We then give an example to determine the optimum design parameters of the EWMAX-bar chart by maximizing the expected profit per unit time and present a sensitivity analysis to find the significant parameters. We also compare the performance of EWMAX-bar chart with λ=1 which is equivalent to the X-bar probability chart. In Chapter 4, we derive the profit model per unit time with EWMAX-bar chart and producer tolerance. We give an example to determine the optimum upper specification limit and the design parameters of the EWMAX-bar chart by maximizing the expected profit per unit time and compare the performance to EWMAX-bar chart without producer tolerance. Finally, we present a sensitivity analysis and compare the performance to EWMAX-bar chart with λ=1. In Chapter 5, we determine the best λin the EWMAX-bar chart under six different shift scales by maximizing the expected profit per unit time and conclude the better λ for different shift scales in the mean and variance. In Chapter 6, we consider the VSI EWMAX-bar chart and calculate average time to signal (ATS) to measure the performance of this chart. We also derive the profit model and then give two examples to determine the upper specification limit and the design parameters of the VSI EWMAX-bar chart by maximizing the expected profit per unit time and conducting sensitivity analysis. We also compare the performance with the FSI EWMAX-bar chart.
Finally, we summarize the results in Chapter 7. In this study, we use the R program to perform all calculations, including using the “uniroot” command to solve the one-dimensional root and using the “DEoptim” command to find the global optimum value of the expected profit per unit time using the differential evolution algorithm.
We also use the R program to plot all of the figures.
‧
2.1 In-control Sampling Distribution of X-bar under Gamma Distribution
Construct the EWMAX-bar control chart based on the sample mean, in this section, we need to derive the X-bar distribution.
We take the in-control gamma distribution as follows:
, , ,
22.2 Out-of-control Sampling Distribution of X-bar under Gamma Distribution
To choose out-of-control distribution, we first compare different gamma distributions.Figure 2-1. The p.d.f Comparison of Different Gamma Distributions
According to Figure 2-1, if a or b increases, the p.d.f. shifts right and both mean and variance of the distribution increase. Because the considered quality variable is the smaller the better, we assume that both a and b of the out-of-control distribution are bigger than those of the in-control distribution.