Comparing the middleman ordered quantity and middleman profit for middleman instrument

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From Table 4.3 we found that,

1. The optimal producer specification, dp*, are all equal to the upper bound of producer specification, this result is same to when we do not consider the order quantity that discussed in Chapter3, so under producer taking perfect repair action, dp* do not affect by the middleman order quantity.

2. The optimal producer sell price (P*PM) is equal to the lower bound of producer sales price, and the optimal middleman sell price (P*MC) is equal to the upper bound of middleman sales price.

3. The middleman ordered quantity may be approximately equal to the mean of the customer ordered quantity (µc).

4.3.3 Comparing the middleman ordered quantity and middleman profit for middleman instrument without and with measurement error under producer taking perfect repair action.

Since we want to find the significant parameters for the difference order quantity and profit of middleman instrument without and with measurement error. So first we will calculate the difference in order quantity (

Q

^*_{MQR D−}

= Q

^*_{MQR WO−}

− Q

^*_{MQR W−} ) and in profits (

EPR

^*_{MQR D−}

= EPR

^*_{MQR WO−}

− EPR

^*_{MQR W−} ) for

middleman instrument without and with measurement error. Then do the response figures and tables for

Q

^*M_{QR D}

− and

EPR

^*M_{QR D}

− to find which significant parameters

‧

EPR

M ₋

values for different combinations of parameters under producer taking perfect

repair action

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120 Figure 4.2 Response figures of *

MQR D

Q

₋

and EPR

*MQR D₋

Figure 4.2 (a) response figure of

*

MQR D

Q

− for each parameter Figure 4.2 (b) response figure of

EPR

*M_{QR D−} for each parameter

Table 4.5 Response tables of *

MQR D

Q

₋

and EPR

*MQR D₋

(a)

Q ^*

M_{QR D−}

δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) level1

0.663 0.001 1.997 1.277 1.514 2.175 2.014 0.497

level2

1.69 0.387 0.703 0.616 0.802 0.962 0.359 1.162

level3

0.869 2.834 0.522 1.329 0.906 0.085 0.849 1.563

diff

1.028 2.833 1.475 0.713 0.712 2.089 1.654 1.065

(b)

EPR

*M_{QR D−}

δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) level1

102.314 0.175 286.034 162.074 169.193 190.456 185.198 136.4

level2

234.561 65.156 109.173 140.919 145.191 153.394 128.792 158.931

level3

126.278 397.822 67.946 160.16 148.768 119.302 149.163 167.822

diff

132.246 397.646 218.087 21.155 24.002 71.154 56.406 31.423

1. From Table 4.5, if the difference of the maximal and minimal values of the three levels is large than 2.5, then the parameter is determined to be significant, so δ3 are significantly influence

*

MQR D

Q

− . When δ3 increases,

*

MQR D

Q

− increases.

2. If the difference between the minimal and maximal values of the three levels is larger than 100, the parameter is determined to be significant, so δ, δ3 and δ4 are significantly influenceE P R*MQ R₋D

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(1) When δ3 increases,

EPR

*M_{QR D−} increases. Since when δ3 increases, the profit decreasing rate for instrument with measurement error is larger than without measurement error.

(2) When δ4 increases, the middleman measurement error decrease, so

EPR

*M_{QR D−} decreases.

(3) When δ increases,

EPR

*M_{QR D−} increases first and then decreases.

4.4 Maximize the Profit Model to Determine the Optimal Specification Limits, Order Quantity, Buying and Selling Price for Middleman under Producer Adopting Sell Low Price Action.

4.4.1 Derivation of the profit model with order quantity for middleman.

(I) Consider middleman instrument without measurement error By Figure 3.6 the profit function contains,

If the item is true conforming item and observed conforming item for producer and Q_CM ≤Q_MP, then middleman profit is,

(

P_MCH×Q_CM −P_PMH×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item and observed conforming item for producer and Q_CM >Q_MP , then middleman profit is,

(

P_MCL×Q_CM −P_PML×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item but observed nonconforming item for producer and Q_CM ≤Q_MP, then middleman profit is,

(

P_MCL×Q_CM −P_PML×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item but observed nonconforming item for producer and

CM MP

Q >Q , then middleman profit is,

(

P_MCL−P_PML

)

×Q_MP−C_s

(

Q_CM −Q_MP

)

−IC Q× _MP,

If the item is true nonconforming item but observed conforming item for producer and Q_CM ≤Q_MP, then middleman profit is,

(

−PP M H+Cm rH −IC

)

×QM P,

‧

If the item is true nonconforming item and observed nonconforming item for producer and

CM MP

Q >Q , then middleman profit is,

(

−PPM L+Cm rL−IC

)

×QM P.

Therefore the profit function of middleman when instrument without measurement error is,

( ) ( ) ; Then the expected total profit every week for middleman with measurement error is, sum of the profit of an item’s status multiply to the corresponding probability, that is,

( )

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(II) Consider middleman instrument with measurement error By Figure 3.7 the profit function contains,

If the item is true conforming item and observed conforming item for producer and middleman, and Q_CM ≤Q_MP, then middleman profit is,

(

P_MCH×Q_CM −P_PMH×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item and observed conforming item for producer and middleman, and Q_CM >Q_MP, then middleman profit is,

(

P_MCH−P_PMH

)

×Q_MP−C_s

(

Q_CM −Q_MP

)

−IC Q× _MP,

If the item is true conforming item and observed conforming item for producer and true conforming item but observed nonconforming item for middleman, then middleman profit is,

PMH mrH

P C IC

− + − ,

If the item is true conforming item but observed nonconforming item for producer and true conforming item and observed conforming item for middleman, and Q_CM ≤Q_MP, then middleman profit is,

(

P_MCL×Q_CM −P_PML×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item but observed nonconforming item for producer and true conforming item and observed conforming item for middleman, and Q_CM >Q_MP , then middleman profit is,

(

P_MCL−P_PML

)

×Q_MP−C_s

(

Q_CM −Q_MP

)

−IC Q× _MP,

If the item is true conforming item but observed nonconforming item for producer and true conforming item but observed nonconforming item for middleman, then middleman profit is,

PML mrL

P C IC

− + −

If the item is true nonconforming item but observed conforming item for producer and true nonconforming item but observed conforming item for middleman, and Q_CM ≤Q_MP, we considering two situations,

‧

If the item is true nonconforming item but observed conforming item for producer and true nonconforming item but observed conforming item for middleman, and Q_CM >Q_MP , we considering two situations,

If the item is true nonconforming item but observed conforming item for producer and true nonconforming item and observed nonconforming item for middleman, then middleman profit is,

PMH mrH

P C IC

− + −

If the item is true nonconforming item but observed conforming item for producer and true nonconforming item but observed conforming item for middleman, and Q_CM ≤Q_MP, we considering two situations,

‧

If the item is true nonconforming item but observed conforming item for producer and true nonconforming item but observed conforming item for middleman, and Q_CM >Q_MP , we considering two situations,

If the item is true nonconforming item and observed nonconforming item for producer and middleman, then middleman profit is,

PML mrL

P C IC

− + − .

Therefore the profit function of middleman when instrument with measurement error is, when Ccr < PPM,

‧

Then the expected profit per week for middleman instrument with measurement error is the sum of the profit of an item’s status multiply to the corresponding probability, that is,

when Ccr < PPM,

‧

‧

‧ 4.4.2 Determine the optimal specification limits, middleman order quantity, buying and selling

price for middleman with the data analyses.

Table 4.6 Three Levels of Each Parameters

Level

Since 9 parameters each with 3 levels, the parameters could be assigned to each combination of orthogonal array table^L27

( )

^{313 .}

‧

Table 4.7 The 27 combinations of these parameters by using an orthogonal array table

L₂₇(3 )¹³

.

No. δ δ3 δ4 σx IC ν1 ν2 γ (Ch, Cs)

Let the expected profit of per item for middleman instrument without measurement error under producer taking perfect repair action be

EPR

_{MQS WO−} . To determine optimal specification limits,

middleman order quantity, buying and selling price, we maximize

EPR

_{MQS WO−} with the constraint

2 2

0 ≤ d

_p

≤ 3 σ + σ

_{x pe} , Q_MPL ≤Q_MP≤Q_MPU and P_PMLB ≤P_PMH <P_MCH ≤P_MCU , by using “optim” routine in R program.

‧

Let the expected profit of per item for middleman instrument with measurement error under producer taking perfect repair action be

EPR

_{MQS W}

− . To determine optimal specification limits, middleman order quantity, buying and selling price, we maximize

EPR

_{MQS W}

− with the constraint

2 2

0 ≤ d

_p

≤ 3 σ + σ

_{x pe} , Q_MPL ≤Q_MP≤Q_MPU, P_PMLB ≤P_PMH <P_MCH ≤P_MCU , by using “optim” routine in R program

Table 4.8 The optimal solutions of 27 combinations of parameters.

Middleman without measurement error Middleman with measurement error No. d^*P_{QS WO}

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From Table 4.8 we found that, the optimal solutions for the 27 combinations of parameters under producer adopt sell low price action. From the optimal results we found that,

The optimal specification limits, and the optimal middleman selling price are all equal the upper bound. The optimal middleman buying price are all equal the lower bound. And the optimal

middleman ordered quantity is approximately equal to µc. These result are similar to when producer take perfect repair action, which we found in section 4.3.2.

4.4.3 Comparing the middleman ordered quantity and middleman profit for middleman instrument with and without measurement error under producer taking sell low price action.

Since we want to find the significant parameters for the difference order quantity and profit of middleman instrument without and with measurement error. So first we will calculate the difference in order quantity (

Q

^*M_{QS D}

Q

^*M_{QS WO}

Q

^*M_{QS W}

−

=

−

−

− ) and in profits (

EPR

^*M_{QS D}

EPR

^*M_{QS WO}

EPR

^*M_{QS W}

−

=

−

−

− ) for middleman

instrument without and with measurement error. Then do the response figures and tables for

Q

^*M_{QS D}

−

and

EPR

^*M_{QS D}

− to find which significant parameters

‧

EPR

M ₋

values with different combinations of parameters under producer taking sell low

price action

‧ 國

立 政 治 大 學

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134 Figure 4.3 Response figure of *

MPQS D

Q

−

and

_{E P R *MQ S−D}

Figure 4.3 (a) response figure of

*

MPQS D

Q − for each parameter Figure 4.3 (b) response figure of _{E P R *MQ S−D} for each parameter

Table 4.10 Response table of *

MPQS D

Q

−

and

_{E P R *MQ S−D}

*

MPQS D

Q

₋

δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) γ level1

-0.082 -0.076 0.155 0.236 -0.102 0.104 -0.113 0.29 -0.063

level2

-0.163 0.184 0.138 -0.264 0.21 0.037 0.103 0.104 0.157

level3

0.502 0.149 -0.036 0.286 0.149 0.116 0.267 -0.136 0.163

diff

0.665 0.26 0.191 0.55 0.312 0.079 0.38 0.426 0.226

MQS D

EPR*

₋

δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) γ level1

425.227 867.242 956.204 724.15 766.212 708.994 711.124 748.142 799.089

level2

598.383 712.09 766.153 718.918 691.708 736.447 725.002 726.668 707.636

level3

1139.115 583.392 440.367 719.657 704.804 717.284 726.598 687.915 655.999

diff

713.888 283.851 515.837 5.232 74.504 27.453 15.475 60.226 143.091

1. From Table 4.10, there do not have significant parameters of

*

MPQS D

Q

− .

2. If the difference of the maximal and minimal values of the three levels is large than 400, then the parameter is determined to be significant, so δ and δ4 are significantly influence

EPR*

MQS D₋ .

(1) When δ increases,

EPR*

MQS D₋ increases.

(2) When δ4 increases, middleman instrument measurement error decreases, so

MQS D

EPR*

₋ decreases.

‧

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