Comparing the four cases expected profit of middleman for middleman instrument with

1. When δ > 0, middleman profit is larger for producer taking perfect repair action than producer taking sell low price action. So it is more favorable for middleman to buy the item is perfectly repaired.

2. When 0 < δ3 < 0.124, middleman profit is larger for producer taking sell low price action than producer taking perfect repair action. Therefore middleman should buy perfectly repaired items unless producer instrument measurement error is very large.

3.4.2 Comparing the four cases expected profit of middleman for middleman instrument with measurement error.

Since we want to find the significant parameters for the expected profit difference (

1

EPR

₋ ) for two actions under middleman instrument with measurement error. So we will calculate the difference in profits for the two actions first, that is,

1 R W .1 S W .1

EPR

₋ to find which parameters are significant.

‧

EPR

₋

values for different combinations of parameters

No. δ δ

3

δ

4

σ

x

IC ν

1

ν

2

γ

From Table 3.19, we found that,

1. when δ=2, δ3=0.9, and δ4=0.8 (See Nos.7, 10, 21), then the difference in middleman profits when producer adopts the action of perfect repair or sell low price is large.

2. when middleman instrument with measurement error, the middleman profit is larger for producer taking perfect repair action than producer taking sell low price action.

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Figure 3.12 Response figure of

^{E P R *R S .D− 1}

,

^{EPR *R S .D− 2}

,

^{E P R *R S .D− 3}

and

^{EPR *R S .D− 4}

Figure 3.12 (a) Response figure of E P R*R−S .D₁for each parameter Figure 3.12 (b) Response figure of EPR *R S .D− ₂ for each parameter

Figure 3.12 (c) Response figure of E P R *R S .D− ₃ for each parameter Figure 3.12 (d) Response figure of EPR *R S .D− ₄ for each parameter

Table 3.20 Response table of

^{E P R *R S .D− 1}

,

^{EPR *R S .D− 2}

,

^{E P R *R S .D− 3}

and

^{EPR *R S .D− 4}

(a) E P R*R−S .D₁ (b) E P R*R−S . D₂

δ δ3 δ4 σx IC ν1 γ δ δ3 δ4 σx IC ν1 ν2 γ

level1 0.066 0.056 0.401 0.952 0.904 0.984 1.012 0.067 0.056 0.413 0.974 0.929 1.021 0.978 1.045 level2 0.314 0.155 0.104 0.948 0.943 0.874 0.907 0.326 0.155 0.105 0.982 0.976 0.888 0.978 0.932 level3 2.404 2.573 2.28 0.884 0.937 0.926 0.865 2.465 2.647 2.341 0.902 0.954 0.949 0.901 0.882 diff 2.338 2.517 2.176 0.069 0.039 0.111 0.146 2.398 2.591 2.236 0.081 0.047 0.132 0.077 0.163

(c) E P R*R−S .D₃ (d) E P R*R−S . D₄

δ δ3 δ4 σx IC ν1 γ δ δ3 δ4 σx IC ν1 ν2 γ

level1 0.064 0.055 0.38 0.909 0.861 0.939 0.966 0.065 0.056 0.393 0.932 0.888 0.988 0.946 1.01 level2 0.293 0.155 0.101 0.903 0.898 0.832 0.864 0.307 0.155 0.103 0.948 0.941 0.847 0.936 0.891 level3 2.298 2.444 2.174 0.842 0.896 0.883 0.824 2.37 2.531 2.245 0.862 0.912 0.906 0.86 0.841 diff 2.234 2.389 2.072 0.067 0.037 0.107 0.142 2.304 2.475 2.143 0.085 0.053 0.141 0.086 0.169

According to the Table 3.20, if the difference of the maximal and minimal values of the three levels is larger than 1.5, then the parameter is determined to be significant, so δ, δ3 and δ4 are significant parameters of E P R *R S .D− ₁, EPR *R S .D− ₂, E P R *R S .D− ₃ and EPR *R S .D− ₄.

When δ (or δ3) increase, E P R *R S .D− ₁, EPR *R S .D− ₂ , E P R *R S .D− ₃ and EPR *R S .D− ₄ increase.

When δ4 increases, E P R *R S .D− ₁, EPR *R S .D− ₂, E P R *R S .D− ₃ and EPR *R S .D− ₄decrease first and then increase.

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For significant parameters of E P R *R S .D− ₁, EPR *R S .D− ₂, E P R *R S .D− ₃ and EPR *R S .D− ₄ , where contains, δ, δ3 and δ4, we plot the levels value corresponding to E P R *R S .D− ₁, EPR *R S .D− ₂ , E P R *R S .D− ₃ and

R S .D4

EPR * − , then calculate the intersection points on x-axis to determine under which range the producer middleman is more larger for producer taking perfect repair action or taking sell low price action.

Figure 3.13 (a)

^{E P R *R S .D− 1}

,

^{EPR *R S .D− 2}

,

^{E P R *R S .D− 3}

and

^{EPR *R S .D− 4}

under different δ value

Figure 3.13 (b)

^{E P R *R S .D− 1}

,

^{EPR *R S .D− 2}

,

^{E P R *R S .D− 3}

and

^{EPR *R S .D− 4}

under different δ

3

value

Figure 3.13 (c)

^{E P R *R S .D− 1}

,

^{EPR *R S .D− 2}

,

^{E P R *R S .D− 3}

and

^{EPR *R S .D− 4}

under different δ

4

value

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From Figure 3.13 we found that, when the middleman instrument with measurement error, middleman should buy a product from the producer adopting perfect repair action, rather than sell low price action.

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CHAPTER 4. DERIVE THE MIDDLEMAN PROFIT MODEL WITH CONSIDERING ORDER QUANTITY UNDER PRODUCER TAKING TWO ACTIONS

In Chapter 3, we found that no matter producer take perfect repair or sell low price action for middleman instrument without or with measurement error, the optimal specification are all equal to the constraint upper bound. In this chapter we want to know whether the optimal specification will be affect by the order quantity or not. And in chapter 3 when middleman instrument with measurement error, the middleman profit in the second case is larger than other cases, so in this chapter we only consider the second case. Here we also assume the unit of order quantity is one week.

4.1 Notations

QMP The per week product quantity that middleman order from producer.

QCM Per week customer demand quantity.

µ

c Consumer expected demand quantity

c 2

σ

Variability of customer demand quantity.

Ch Per item holding cost of middleman, when QMP > QCM

Cs Goodwill loss for per unit of stock-out of middleman, when QCM > QMP

4.2 Describe the Sell Price and Order Quantity Relationship among Producer, Middleman and Customer

For the producer and middleman sell price depends on the action that producer take for the nonconforming items, this have discussed in Chapter 3. And in this Chapter we want to find the optimal middleman ordered quantity, optimal producer and middleman sell price, and the optimal specification limits under customer demand (QCM) following uniform distribution, Q_CM ~ U a, b

( )

.

Figure 4.1 The relationship of producer, middleman and customer when considering quantity.

‧

4.3 Maximize the Profit Model to Determine the Optimal Specification Limits, Order Quantity, Buying and Selling Price for Middleman under Producer Adopting Perfect Repair Action.

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

We will derive the profit models by considering middleman instrument without and with measurement error.

(I) Consider middleman instrument without measurement error By Figure 3.1 the profit function contains,

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

(

P_MC×Q_CM−P_PM×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

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

(

P_MC−P_PM

)

×Q_MP−C_s

(

Q_CM−Q_MP

)

−IC Q× _MP,

If the item is true nonconforming item for middleman, then middleman profit is,

PM mr

P C IC

− + −

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

( )

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

p

( )

‧

(II) Consider middleman instrument with measurement error

By Figure 3.2 the profit function contains,

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

(

P_MC×Q_CM−P_PM×Q_MP

)

−C_h

(

Q_MP−Q_CM

)

−IC Q× _MP,

If the item is true conforming item and observed conforming item for middleman and

CM MP

Q >Q , then middleman profit is,

(

P_MC−P_PM

)

×Q_MP−C_s

(

Q_CM−Q_MP

)

−IC Q× _MP,

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

CM MP

Q ≤Q , we considering two situations, 1. when Ccr < PPM, then middleman profit is,

‧

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

CM MP

Q >Q , we considering two situations, 1. when Ccr < PPM, then middleman profit is,

(

PM C×QM P −Ccr×QC M −IC×QM P

) (

− PP M ×QM P −Ccr×QC M

)

−Cs⁽QC M −QM P⁾

2. when PMC > Ccr≧PPM, then middleman profit is,

(

PM C×QM P −Ccr×QC M −IC×QM P

) (

− PP M ×QM P−PP M ×QC M

)

−Cs⁽QC M−QM P⁾

If the item is true nonconforming item for middleman, then middleman profit is,

(

−PP M +Cm r

)

×QM P

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

‧

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

when Ccr < PPM,

LSL USL USL USL USL

X ,Y X ,Y X ,Y PM mr MP

LSL USL LSL LSL LSL

h x,y dydx h x, y dydx h x, y dydx P C Q

LSL USL USL USL USL

X ,Y X ,Y X ,Y PM mr MP

LSL USL LSL LSL LSL

h x,y dydx h x, y dydx h x, y dydx P C Q

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4.3.2 Determine the optimal specification limits, middleman order quantity, buying and selling price for middleman with the data analyses.

Table 4.1 Three Levels of Parameters Levels

Level

parameter 1 2 3

δ 0 1 2

δ3 0.3 0.5 0.9 δ4 0.4 0.5 0.8 σx 0.2 0.5 0.7

IC 0.05 0.1 1

ν1 0.75 0.9 1

ν2 0.4 0.7 0.8 (Ch, Cs) (0.2, 1) (0.5, 2) (1, 5)

Since 8 parameters each with 3 levels, the parameters could be assigned to each combination of orthogonal array table^L27

( )

^{313 .}

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

EPR

_{MQR WO}

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

EPR

_{MQR 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

M_{QR W−} . To determine optimal specification limits, middleman order quantity, buying and selling price, we maximize

EPR

M_{QR 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.

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Table 4.2 The 27 combinations of these parameters by using an orthogonal array table

L₂₇(3¹³)

.

No. δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) 1 1 0.9 0.4 0.7 0.05 0.75 0.4 (1, 5) 2 2 0.5 0.4 0.5 0.05 0.9 0.4 (0.5, 2) 3 0 0.5 0.8 0.2 0.1 0.9 0.4 (1, 5) 4 2 0.3 0.5 0.5 1 1 0.4 (1, 5) 5 0 0.3 0.4 0.7 1 0.9 0.7 (0.5, 2) 6 0 0.5 0.8 0.5 1 1 0.8 (0.5, 2) 7 2 0.9 0.8 0.2 0.05 1 0.7 (0.5, 2) 8 2 0.3 0.5 0.7 0.05 0.75 0.8 (0.5, 2) 9 0 0.9 0.5 0.2 1 0.75 0.4 (0.5, 2) 10 2 0.9 0.8 0.7 1 0.9 0.8 (1, 5) 11 0 0.5 0.8 0.7 0.05 0.75 0.7 (0.2, 1) 12 1 0.5 0.5 0.7 1 0.9 0.4 (0.2, 1) 13 2 0.3 0.5 0.2 0.1 0.9 0.7 (0.2, 1) 14 0 0.9 0.5 0.5 0.05 0.9 0.8 (0.2, 1) 15 1 0.3 0.8 0.5 0.05 0.9 0.7 (1, 5) 16 1 0.5 0.5 0.2 0.05 1 0.8 (1, 5) 17 2 0.5 0.4 0.7 0.1 1 0.8 (0.2, 1) 18 2 0.5 0.4 0.2 1 0.75 0.7 (1, 5) 19 1 0.3 0.8 0.7 0.1 1 0.4 (0.5, 2) 20 0 0.3 0.4 0.2 0.05 1 0.4 (0.2, 1) 21 2 0.9 0.8 0.5 0.1 0.75 0.4 (0.2, 1) 22 0 0.3 0.4 0.5 0.1 0.75 0.8 (1, 5) 23 1 0.9 0.4 0.2 0.1 0.9 0.8 (0.5, 2) 24 1 0.3 0.8 0.2 1 0.75 0.8 (0.2, 1) 25 0 0.9 0.5 0.7 0.1 1 0.7 (1, 5) 26 1 0.9 0.4 0.5 1 1 0.7 (0.2, 1) 27 1 0.5 0.5 0.5 0.1 0.75 0.7 (0.5, 2)

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Table 4.3 The optimal solutions of 27 combinations of parameters when P

PMLB

=50, P

MCU

=90, a=0, and b=300 under

producer taking perfect repair action.

Middleman without measurement error Middleman with measurement error No. ^{d*PQ R−W O Q*M PQ R W O− PP M* Q R W O− PM C* QR W O− E P R*MQ R W O− d*PQ R W− Q*M PQ R W− PP M* Q R W− PM C* Q R W− E P R*MQ R W−}

1

2.333 140.36 50 90 2395.455 2.333 130.024 50 90 1542.549

2

3 136.054 50 90 2553.648 3 135.138 50 90 2380.982

3

1.2 140.312 50 90 2400.011 1.2 140.312 50 90 2399.999

4

5 137.5 50 90 2274.966 5 137.5 50 90 2274.821

5

7 132.973 50 90 2425.931 7 132.971 50 90 2425.656

6

3 132.973 50 90 2425.942 3 132.973 50 90 2425.931

7

0.667 136.051 50 90 2499.341 0.667 135.399 50 90 2327.798

8

7 136.053 50 90 2553.675 7 136.051 50 90 2553.378

9

0.667 132.961 50 90 2424.788 0.667 128.736 50 90 2041.603

10

2.333 137.093 50 90 2209.174 2.333 135.585 50 90 2007.043

11

4.2 134.704 50 90 2608.057 4.2 134.704 50 90 2608.039

12

4.2 131.579 50 90 2481.567 4.2 131.474 50 90 2461.7

13

2 134.539 50 90 2601.287 2 134.538 50 90 2601.059

14

1.667 134.699 50 90 2607.161 1.667 132.975 50 90 2297.335

15

5 140.469 50 90 2407.015 5 140.469 50 90 2407.003

16

1.2 140.469 50 90 2407.027 1.2 140.469 50 90 2391.785

17

4.2 134.539 50 90 2601.263 4.2 134.539 50 90 2462.221

18

1.2 137.499 50 90 2274.891 1.2 135.293 50 90 2062.21

19

7 135.892 50 90 2546.918 7 135.892 50 90 2546.913

20

2 134.704 50 90 2608.052 2 134.704 50 90 2607.824

21

1.667 133.64 50 90 2510.183 1.667 131.1 50 90 2272.414

22

5 140.312 50 90 2399.997 5 140.309 50 90 2399.626

23

0.667 135.846 50 90 2537.936 0.667 131.44 50 90 1862.408

24

2 131.579 50 90 2481.551 2 131.578 50 90 2481.536

25

2.333 140.312 50 90 2399.353 2.333 140.302 50 90 2172.452

26

1.667 131.57 50 90 2474.321 1.667 131.465 50 90 1953.716

27

3 135.892 50 90 2546.92 3 135.633 50 90 2520.055

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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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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

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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.

‧

4.5 Compare Expected Profit per Week for Middleman under Producer Take Different Actions.

In Section 4.3 we discussed maximizing middleman profit when the producer adopt perfect repair action (

Q R

*

E P RM ) and in section 4.4 we discussed maximizing middleman profit when producer adopt sell low price action (

Q S

*

E P RM ). Now in this section we will compare the middleman profits in the context of these two actions.

4.5.1 Comparing ordered quantity and expected profit of the middleman for middleman instrument without measurement error.

Since we want to find the significant parameters for ordered quantity difference (

^Q

^*_{QR QS.D− WO}) and the expected profit difference (

^EPR

^*_{QR QS.D− WO}) for two actions under middleman instrument without measurement error. So we will calculate the difference in order quantities and in profits for the two actions first, that is,

Q

QR QS.D^* _WO

Q

^*M_{QR WO}

Q

^*M_{QS WO}

1. the ordered quantity of these two actions are almost the same.

2. when δ=2 and δ3=0.9 (See No.7, 10, 21), then there have large difference in middleman profits for producer take two actions, and for others combinations of parameters the difference in middleman profits are small.

3. since the difference profit of these two actions (

WO

* QR QS.D

EPR

₋ ) are all positive, so we can find that the middleman profit is larger under producer taking perfect for nonconforming item than selling low price.

‧

EPR

₋

values for different combinations of parameters

No. δ δ3 σx IC ν1 (Ch, Cs) γ

Q

^*QR QS.D_{− WO}

Since the ordered quantity of these two actions are almost the same, so the following we only do the sensitivity analysis for difference of middleman profit (

WO

* QR QS.D

EPR

₋ ).

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137 Figure 4.4 response figure of

Q R Q S .DW O

E P R * −

for each parameter

Table 4.12 Response table of

Q R Q S .DW O

E P R * −

Q R Q S .DW O

E P R * −

δ δ3 σx IC ν1 (Ch, Cs) γ level1

2.292 3.749 31.421 28.791 30.835 29.177 29.177

level2

10.376 8.022 28.349 29.654 23.68 29.83 29.83

level3

71.122 72.02 24.021 25.346 29.275 24.784 24.784

diff

68.83 68.271 7.4 4.308 7.155 5.046 5.046

From Table 4.12, if the difference of the maximal and minimal values of the three levels is large than 50, then the parameter is determined to be significant, so δ and δ3 are significant parameters ofE P R *Q R Q S .D− _{W O} When δ (or δ3) increases, E P R *Q R Q S .D− _{W O} increases.

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For significant parameters of E P R *Q R Q S .D− _{W O} , where contains, δ and δ3, we plot the levels value corresponding to E P R *Q R Q S .D− _{W O} , then calculate the intersection points on x-axis to determine under which range the producer middleman is more larger for producer taking perfect repair action or taking sell low price action.

Figure 4.5 (a)

Q R Q S .DW O

E P R * −

under different δ Figure 4.5 (b)

Q R Q S .DW O

E P R * −

under different δ

3

From figure 4.5 (a) and (b), we find that,

1. when δ > 0, the middleman’s profit is larger for producer adopting perfect repair action than producer adopting sell low price action, so it is favorable for the middleman to buy an item that is perfect repaired.

2. when 0 < δ3 < 0.125, middleman profit is larger for producer adopting sell low price action than producer adopting perfect repair action. Therefore, middleman should purchase a perfect repaired item unless the producer instrument error is very large.

These result are similar to when we do not consider the order quantity, which discussed in Section 3.4.1.

4.5.2 Comparing ordered quantity and expected profit of the middleman for middleman instrument with measurement error.

Since we want to find the significant parameters for ordered quantity difference (

W

* QR QS.D

Q

₋ ) and the expected profit difference (

W

* QR QS.D

EPR

₋ ) for two actions under middleman instrument with measurement error. So we will calculate the difference in order quantities and in profits for the two

‧

EPR

₋ to find which parameters are significant.

Table 4.13

EPR

₋

values for different combinations of parameters

No. δ δ3 δ4 σx IC ν1 ν2 (Ch, Cs) γ

Q

QR QS.D^* _{− W}

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From Table 4.13, we found that,

1. when δ3 is equal to 0.9, and ν1 is between 0.75 and 0.9 (see Nos.1, 9, 21, 23), then the difference in ordered quantities for the perfect repair action and sell low price is large. For others combinations of parameters, the order quantity is very close.

2. when δ is between 0 and 1, δ3=0.9, δ4 is between 0.4 and 0.5, ν1=0.75, ν2=0.4, and γ is between 0.75 and 0.9 (See No.1 and 9), the middleman’s profit is larger for producer taking sell low price action than taking perfect repair action.

Figure 4.6 Response figure of

QR QS.DW

Q*

₋

and

E P R *_{Q R}−_{Q S .DW}

Figure 4.6 (a) response figure of

QR QS.DW

Q* ₋ for each parameter Figure 4.6 (b) response figure of E P R *Q R−Q S .D_W for each parameter

Table 4.14 Response table of

QR QS.DW

Q*

₋

and

E P R *_{Q R}−_{Q S .DW}

QR QS.DW

Q*

₋

δ δ

3

δ

4

σ

x

IC ν

1

ν

2

(C

h

,C

s

) γ

level1

-0.748 -0.081 -1.871 -1.059 -1.639 -2.196 -2.247 -0.313 -1.211

level2

-1.876 -0.211 -0.57 -0.989 -0.705 -0.982 -0.269 -1.073 -1.306

level3

-0.531 -2.864 -0.715 -1.107 -0.811 0.023 -0.639 -1.77 -0.639

diff

1.346 2.783 1.301 0.119 0.934 2.22 1.978 1.457 0.667

Q R Q S .DW

E P R * −

δ δ

3

δ

4

σ

x

IC ν

1

ν

2

(C

h

,C

s

) γ

level1

325.205 870.816 684.344 593.497 625.81 549.372 554.197 640.919 680.287

level2

374.199 654.956 661.626 606.348 576.171 606.733 627.023 597.567 570.001

level3

1083.959 257.59 437.392 583.517 581.382 627.257 602.143 544.877 533.074

diff

758.754 613.226 246.952 22.83 49.639 77.884 72.826 96.042 147.212

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From Table 4.14,

1. we found that δ3 significant to

QR QS .DW

Q*

₋ , and when δ3 increases,

QR QS .DW

Q*

₋ decreases.

2. If the difference between the maximal and minimal values of the three levels is larger than 200, the parameter is determined to be significant; therefore, δ, δ3, and δ4 are significant parameters

ofE P R *Q R−Q S .D_W

,

(1) When δ increase, E P R *Q R−Q S .D_W increases.

(2) When δ3 increases, E P R *Q R−Q S .D_W decreases, since when δ3 increases, the difference in order quantities decrease, so the difference of profits may decrease.

(3) When δ4 increases, the middleman instrument error will decrease, so E P R *Q R−Q S .D_W decreases.

For significant parameters of E P R *Q R−Q S .D_W , where contains, δ ,δ3 and δ4, we plot the levels value corresponding to E P R *Q R−Q S .D_W , then calculate the intersection points on x-axis to determine under which range the producer middleman is more larger for producer taking perfect repair action or taking sell low price action.

Figure 4.7 (a)

E P R *Q R−Q S .D_W

under different δ

Figure 4.7 (b)

Q R Q S .DW

E P R * −

under different δ

3

Figure 4.7 (c)

Q R Q S .DW

E P R * −

under different δ

4

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From Figure 4.7 (a) to (c) we found that, when considering ordered quantity and middleman instrument with a measurement error the middleman should buy products from a producer taking the action of perfectly repaired rather than taking the action of sell at a low price for nonconforming items.

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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

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

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