SUMMARY AND FUTURE STUDY

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CHAPTER 5. SUMMARY AND FUTURE STUDY

From the recent literatures review, we found that a producer takes the action of perfect repair for nonconforming items for instrument with measurement error, under the producer may or may not provide a refund to middleman and the middleman may provide total or partial refund to customer then the optimal specification limits, buying price and selling price are determined to maximizing a profit model of the middleman, and considering both the order quantity and specification limits in the profit model of middleman have not been discussed.

In this study, we first determined the optimal specification limits to maximize profit model for producer taking the action of perfect repair or action of sell low price for observed nonconforming item under instrument with measurement error. Secondly, we considered the producer constructed

economic ²

P YP

Y lnS

EWMA − EWMA

chart to control process quality with maximal profit per unit time but did not take complete inspection. From the results of comparing the profits of complete inspection and control process quality in a fixed observed time, we found that under the combinations of small R, high Cpr, high Csc, and high PPM, the profit of conducting process control is higher than conducting a complete inspection. Hence, process control should be favorable by producer; otherwise the producer should take complete inspection.

The third, under the first situation we considered that a middleman bought products from the producer and sold products to a customer. Assume the specification limits of the producer and the middleman are the same, the middleman’s instrument may or may not exist measurement error, for returning nonconforming products the producer may or may not provide a refund to middleman and the middleman may provide total or partial refund to customer, and the buying price and selling price of products are decided by middleman. By maximizing a derived expected profit per item of the middleman, the optimal specification limits, buying price and selling price are determined. From the results of numerical analyses, we found that the larger specification limits and selling price the larger profit, but the lower buying price the larger profit; the expect profits per item are almost same for middleman provided total or partial refund to customer and the producer may or may not provide a refund to middleman; to have larger profit, middleman should buy products from producer took the

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action of perfect repair when middleman’s instrument existed measurement error or middleman’s instrument existed no measurement error but producer’s instrument existed small measurement error.

The fourth, under the middleman provides a partial refund for customer and get the refund from producer we investigated whether the order quantity may affect the optimal specification limits, since the optimal specification limits are all the same ( or equal to the upper bound of specification

constraint) in the third section. However, from the numerical analyses we found that the order quantity do not affect the optimal specification limits either.

For the future study, we may extend the provided model to consider the producer and middleman with different tolerances, for a complete inspection plan may let the ratio of the expected in-control time to m unit time be a decision variable in the profit model, adaptive economic process control scheme to improve out-of-control detection speed, and also consider the widely popular consignment policy to determine the ordered quantity for middleman.

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Appendix B. Calculate the Average Run Length

We calculate the ARL’s forEWM A_Y and ²

lnSYP

EWMA

individual by using the Markov chain approach (describe following), and by

YP lnS2 ,we can get the ARL value for combing the

YP

EWMA

and ²

lnSYP

EWMA

scheme The following will describe the ARL calculation procedure for the

YP

For in-controlEWMA_Y control charts is

YP

UCL

) and lower control limits (

LCL

EWMAYP) into g=2m+1 , the number of states , sub-intervals of width 2δ, where

YP YP

EWMA EWMA

UCL LCL

δ 2 g

= −

‧

interval. (Like the following figure)

Step3 The statistic

^{( )}

Step4 The transition probability matrix for the transient state ofY_P

[ ( )]

Expressed by Normal CDF For in-control

, then the transition probability matrix of

Y

which calculated by in-control normal distribution is

‧

, then the transition probability

matrix ofZ_Ywhich calculated by out-of-control normal distribution is

[ ( )] Step5 Assume that process begins from state 0 (zero state) ; thus

let

_{{ {}

m terms m terms

( 0,...,0,1, 0,...,0)

zs

=

p

,wherep_zsrepresent the zero state probability vector.

Step6

Where I is the g×g dimension identity matrix , and 1 is the g×1 dimension vector with all components

EWMA

control charts is

[ ( )] [ ( )]

‧

UCLEWMA ) and lower control limits

( lnS2

YPI

LCLEWMA ) into g=2m+1 , the number of states , sub-intervals of width 2δ, where ^{lnSY2PI lnSY2PI}

Step4 The transition probability matrix for the transient state of ²

YP

σ

is the standard deviation of ²

YP

ln S

, and

Φ(

⋅

)

is the cdf of standard normal distribution.

Expressed by Normal CDF

For in-control

( [ ( )] [ ( )])

ln S

which calculated by in-control normal distribution is

[ ( )]

2

YPI t 1,t

I,lnS

= p

₋

jk j,k = − m,..., 1,0,1,...,m −

R

‧

lnS ~ N E ln S ,Var ln S

, then the transition probability matrix of ²

YO

S

ln

which calculated by out-of-control normal distribution is

[ ( )]

Step5 Assume that process begins from state 0 (zero state) ; thus

let

_{{ {}

m terms m terms

( 0,...,0 ,1, 0,...,0 )

zs

=

p

,wherep_zsrepresent the zero state probability vector.

Step6

Where I is the g×g dimension identity matrix , and 1 is the g×1 dimension vector with all components.

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