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CHAPTER 1. INTRODUCTION
1.1 Literature Review and Study Motivation
A complete inspection plan, ensures that every outgoing item is inspected for conforming to the specifications of an interested quality variable. Given the specifications, the following papers discuss the determination of the optimal process mean. Hunter and Kartha (1977) considered the larger the better quality variable, and investigated the optimization of a target or mean when a lower
specification is fixed. They developed a net income model per item and assumed that a conforming item will be sold at the regular price but the nonconforming item will be sold at a reduced price in a secondary market. Then maximize net income per item to determine the optimal process mean.
Bisgaard, Hunter, and Pallesen (1984) extend Hunter’s and Kartha’s work to let the reduced selling price depends on the difference of volume and target of a can. They called this optimal problem as
“Quality selection” or “Economic selection”. Golhar (1987) presented the profit model of the canning problem, that is, let the conforming cans be sold at a regular price in primary market, and let the nonconforming cans be refilled and sold to the primary market. Then maximize profit model to determine the optimal process mean under lower specification limit is known.
Kapur (1987) and Kapur and Wang (1987) presented an economic model for determining the process mean and tolerance under minimizing per unit cost by using Taguchi (1984) quadratic loss function. Tang (1988) determined the profitable specification under complete inspection and presented an economic model. Tang and Lo (1993) and Lee and Kim (1994) proposed a profit model to
determine that the optimum process mean and specification limits under correlated variable is considered. Maghsoodloo and Li (2000) devised an economic model with asymmetric specification limits for minimizing expected loss per unit. Fathi (1990) designed both producer tolerance and consumer tolerance to minimize expected loss per unit.
The production inspection results may relate to the precision of the instrument, if the instrument existed the measurement error, the measurement of outgoing may be influenced. Therefore, when performing quality control, measurement errors should be considered. Numerous studies ( Biegel (1974); Bennett, Case, and Schmidt (1974); Mei, Case, and Schmidt (1975); Dooris (1977); Case
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(1980); Menzefricke (1984)) have discussed measurement errors in inspection tasks. However, these studies did not determining the process mean or specification limit. Using a complete inspection plan for a single quality characteristic and the Taguchi quadric loss function, Tang (1987) and Tang and Schneider (1990) investigated the economic and statistical effects of inspection error. They assumed that inspection was unbiased and that measurement imprecision was independent of the true value of the quality. Chen. and Chung (1994) determined the optimal target value of a production process using different measurement precision levels. Chen. and Chung (1996) extended Chen and Chung (1994) by using repeated readings to minimizing the risk caused by measurement errors. Ferrell and Chhoker (2002) focused on designing economically optimal acceptance sampling plans in the presence of inspection errors. Duffuaa and Siddiqui (2003) used ‘‘cut-o
ff points’’ as the decision points for the inspection rather than specification limits. Feng and Kapur (2006) presented models to develop
optimal specifications that minimize expected total per unit shipping costs rather than per unit production cost. They also considered the model of bivariate quality characteristics with inspection error. Chen (2008) added the inspection errors to Chen (2006)’s paper and determined the process mean and specification limits. Previous studies have addressed sell different prices for the conforming items and the nonconforming items, however none of the above papers mentioned taking perfect repair for the nonconforming items. In this study, we considered the action of performing perfect repair or sell low price for nonconforming items.
Shewhart introduced the concept of the control chart in 1924. Later Duncan (1956) proposed a selection process for control chart design parameters, he presented a cost model to design the
X
chart parameters and consider the process do not shut down when the assignable cause is searched.Montgomery (1985) presented two different manufacture models to design the
X
chart, which are the continuous process model and discontinuous model. The difference of these two models is when searching the assignable cause the process will be shut down or not. Lorenzen and Vance (1986) presented a unified cost model by using a dummy variable to combine the continuous and discontinuous models and to determine the optimal design parameters ofX
control chart.Regarding small process parameter shifts, EWMA-charts are known to be more sensitive than
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Shewhart-charts. In this study we may discuss the process control usingE W M AX −E W M AlnS2charts. In the past studies, Crowder and Hamilton (1992) illustrated the EWMA scheme ofln S2, because using the natural logarithm of sample variance is more normal than using the sample variance. Gan (1995) considered the joint EWMA charts of
X
and ln S2, and provided a numeric tool to calculate the average run length (ARL) of the joint charts. Stemann and Weihs (2001) compared theX − S
chart and E W M AX−Schart, and investigated the effects of measurement error. The ARL is applied to measure the performance of theX S
E W M A − chart. The ARL calculation is referred to Lucas and Saccucci (1990) using the Markov chain method.
The role of the middleman is considered in this study. Few previous relevant studies considered to determine both the producer and middleman specifications, and producer and consumer specifications.
Fathi (1990) proposed a concept of different tolerance for producer and consumer. Given the consumer tolerance, Fathi used a simple graphical procedure to determine the optimal producer tolerance.
Similar to Fathi, Ferrell and Chhoker (2002) determined the optimal producer tolerance according to a fixed consumer tolerance, the difference are Ferrell and Chhoker consider the effect of inspection error, and the purpose of his paper is designing economically optimal acceptance sampling plans. Sheng Sheng (2012) studied the economic design of the
X − S
chart and determined the optimal producer and consumer tolerances with minimal expected costs. She used Taguchi quadratic loss function toexamine the different relationships among producer and consumer losses and tolerances, such as equivalent loss with large producer tolerance or large consumer tolerance, a large coefficient for
producer loss with large producer tolerance or large consumer tolerance, large coefficient for consumer loss with large producer tolerance or large consumer tolerance.
In this study we assume that the specifications of the producer and the middleman are the same.
Therefore, when middlemen determine their specifications, the producers’ specifications are also determined. Regarding the maximization of the middleman’s profit, nonconforming items that bought from a producer pose two potential situations (Perfectly repaired by producer or sold at a low price), which is discussed in the maximizing producer profit section. In the situation of buying perfectly repaired items, we should find the mixture distribution of true quality that middleman received, this is
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because the middleman received items include true conforming ( the original nonconforming items are repaired perfectly) and others are observed conforming items. After maximizing middleman profit and determining the decision variables, we compare these two actions to determine which one has higher profit.
We are also interesting in whether quantity ordered may affect producer specifications, buying price and selling price of the middleman, we refer to Chen and Liu (2007) and use their traditional inventory management system as the profit model then maximize middleman profit to determine the ordered quantity, buying and selling prices. The purpose of this study is to determine the producer specifications that maximized producer and middleman profit under producer instrument with measurement error. We also compare the profits of producer takes complete inspection and quality control in the process.
1.2 Research Method
Chapter 2 derives the profit model to determine the optimal specification limits for a producer instrument with measurement errors, we compare the expected profit per unit time of producer who only conducts a complete inspection with a producer who only conducts PC. In Chapter 3 the producer and middleman are classified into a vertical integration scheme, the condition of maximized producer profit and the presence of producer instrument with measurement errors. We then compare the
expected profit per item of middleman instrument with and without a measurement error. We also compare the expected profit per item of two different middleman purchasing situations. Chapter 4 we elaborate on Chapter 3 by using a traditional vendor model to evaluate the order quantity of the middleman and customer. Chapter 5 offers a summary and suggestions for future study. In this study, we use R programs to perform all calculations. For optimal problems, we call the routine “optim” to find the global optimum value and solution.
For easily read and understand the study, the framework of this article is illustrated in the Table 1.1
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Table 1.1 The framework of this article
Producer Instrument with Measurement Error
Complete
profit per unit time to Determine the specification limitsMaximize expected profit per unit time to Determine the design parameters
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Chapter 3 - -
Maximize expected profit per item to determine the specification limits, middleman buying and selling prices
Maximize expected profit per item to determine the specification limits, middleman buying and selling prices
Chapter 4 - -
Maximize expected profit per week to determine the specification limits, middleman ordered quantity and buying and selling prices
Maximize expected profit week determine the specification limits, middleman ordered quantity and buying and selling prices