Derivation of the profit model for middleman

IC Inspection cost for producer or middleman

ν1 The discount factor, when middleman return goods ν2 The discount factor, when customer return goods

Cmr Fee of middleman return goods, which calculated by Cmr = ν1×PPM

Ccr Fee of customer return goods, which calculated by Ccr = ν2×PMC

3.2 Maximize the Profit Model to Determine the Optimal Specification Limits for Middleman under Producer Adopts Perfect Repair Action

The products that middleman received includes the observed conforming item and

nonconforming item but with perfect repaired by producer, so the true quality distribution (XM) of the middleman received products is the mixture distribution of true distribution of the observed

conforming item and Xr. The mixture distribution (Xm) have discussed in section 2.3.1-2. So XM is equivalent to Xm, that is,

3.2.1 Derivation of the profit model for middleman.

I. Consider middleman instrument without measurement error

First we will discuss how producer, middleman and customer deal with the conforming item and nonconforming item respectively, and then give a flow chart.

(1) Let producer instrument with measurement error, so the conforming item and nonconforming item for producer depends on the observed quality of producer (YP). The observed conforming item (YP,G) will be sold to middleman with price PPM, and the observed nonconforming item (YP,D) will be perfect repaired and then sell to middleman with price PPM.

(2) After middleman received the items, he will do complete inspection again to ensure the items are conform the specification limits, so middleman has the inspection cost, IC (here let the inspection cost of producer and middleman are the same). Let middleman instrument without measurement

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error, so middleman can recognize the item’s true quality. For the true conforming item, the item will be sold to customer with price, PMC; but for the true nonconforming item, then the item will be returned back to producer and middleman can take back the fee, Cmr from the producer.

(3) Since middleman instrument without measurement error, so for middleman there is no cost of customer return goods.

Figure 3.1 Flow chart of products for producer, middleman and customer

Describe the middleman profit under different situation of item,

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

MC PM

P −P −IC,

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,

[ ]

MC PM P M P

M

PM mr M P P

P P IC LSL X USL

π P C IC X LSL ,USL

− − ≤ ≤

=  

− + − ∉



(3.2)

‧

Then the expected profit per item for middleman’s instrument without measurement error is the sum of the profit of an item’s status multiply to the corresponding probability, that is,

( ) ( )

II. Consider middleman instrument with measurement error

Assume middleman instrument with measurement error, which is

ε

_M

~ N 0,σ

( _me² ), we also let the middleman observed quality is YM=XM + εM, so the conditional distribution of the observed quality characteristic YM given true quality characteristic XM=x is,

) ( _me²

M

M |X x~ N x,σ

Y = ,

and its conditional pdf is,

 

−∞< <∞ < <∞

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Here, considering the producer takes the same two actions for the nonconforming items when middleman instrument without measurement error. Further, the middleman and customer deal with conforming items and nonconforming items.

(1) For middleman, after middleman inspection, the observed conforming item will be sold to customer with price, PMC; but the observed nonconforming item will be return to producer and middleman can take back refund, Cmr, from the producer.

(2) For customer, since middleman instrument with measurement error, so the true quality of products that customer received will contain conforming item and nonconforming item. Customer may return the observed conforming item but with true nonconforming item; therefore there will exist the cost of customer return goods (Ccr) for middleman.

Figure 3.2 Flow chart of products for producer, middleman and customer

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Describe the middleman profit under different situation of item,

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

MC PM

P −P −IC.

If the item is true nonconforming item but observed conforming item for middleman, we may discuss four different cases for producer and middleman to deal with this cost (Ccr).

Case1. Middleman gives full refund to customer, and receive refund from producer

In this case middleman gives full refund (PMC) to the customer when customer return nonconformities. Middleman then return these nonconformities to the producer, and then take back the fee (PPM) that he buying from producer (See following figure).

The profit of per item for middleman is,

( ) ( )

r = PM C −PM C − PP M −PP M −IC = −IC π

Case2. Middleman gives partial refund to customer and receive refund from producer

In this case middleman gives a partial refund (Ccr) to the customer when customer return nonconformities. Middleman then return these nonconformities to the producer, and receives a refund (Cmr) from the producer.

There are two potential outcomes concerning how the middleman received the refund from the producer.

(2-1) If Ccr < PPM, then producer may pay Ccr to the middleman (See following figure)

The profit of per item for middleman is,

( ) ( )

r = PM C −Cc r −IC − PP M −Ccr = PM C −PP M −IC π

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(2-2) If PMC > Ccr≧PPM, then producer may pay PPM to the middleman (See following figure)

The profit of per item for middleman is,

( ) ( )

r = PM C −Ccr −IC + PP M −PP M = PM C −Ccr −IC π

Case3. Middleman gives full refund to customer (PMC), and also endure the product cost (PPM) In this case middleman gives full refund to the customer when customer return nonconformities, but middleman do not return these nonconforming items to the producer.

The profit of per item for middleman is,

( )

r = PM C −PM C −IC −PP M = −IC−PP M

π .

Case4. Middleman gives partial refund (Ccr) to customer, and the product cost (PPM)

In this case middleman gives partial refund to the customer when customer return nonconformities, but middleman do not return these nonconformities to the producer.

The profit of per item for middleman is,

( )

r = PM C −Cc r −IC −PP M

π .

And for the item is observed nonconforming item for middleman, the middleman profit is,

PM mr

P C

− + .

‧

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

^{[ ]}

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

( ) ( )

^{p p}

‧

3.2.2 Determine the Optimal Specification Limits, Middleman Buying and Selling Price with Data Analyses.

Table 3.1 Three Levels of each Parameter

Level

‧

Since 7 parameters each with 3 levels, the parameters could be assigned to each combination of orthogonal array table^L27

( )

³¹³ (see Table 3.2).

Table 3.2 The 27 combinations of these parameters by using an orthogonal array table

L₂₇(3 )¹³

.

No. δ δ3 δ4 σx IC ν1 ν2

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

R W O

E P RM

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

R W O

E P RM ₋ with the constraint

0 ≤ d

_p

≤ 3 σ + σ

²_x ²_pe ,

PMLB PMH MCH MCU

P ≤P <P ≤P , by using “optim” routine in R program.

‧

When middleman instrument with measurement error, there will have four cases middleman profit model, so we define the notation for these four cases first. Let

1

E P RM ₋ be the expected profit per item of Case1 to Case4 for middleman instrument with

measurement error under producer taking perfect repair action. To determine optimal specification limits, middleman buying and selling price, we maximize

1

Table3.3 shows the optimal solutions for the 27 combination of parameters when the producer adopt perfect repair action. From the optimal results we find that,

1. The optimal producer specification, dp*, are all equal to upper bound of producer specification (since all

σ + σ

=3); therefore, producer specification is the larger the better. If producer specifications become wider, the probability that items conform to the producer specification increase. This implies that, the yield rate increases, causing the middleman’s profit to increase.

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

Therefore a low producer sale price (PPM) is favorable, and a high middleman sale price (PMC) is favorable.

3. The profit of middleman instrument without measurement error is larger than that of middleman instrument with measurement error, and the relation of the four cases of middleman instrument with a measurement error is,

2 1 4 3

R WO R W . R W . R W . R W .

* * * * *

M M M M M

EPR

₋

> EPR

₋

> EPR

₋

> EPR

₋

> EPR

₋

.

‧

Table 3.3 The optimal solutions of 27 combinations of parameters when P

PMLB

=50, P

MCU

=90, under producer taking perfect repair action.

No.

Middleman without measurement error Middleman with measurement error

R W O

‧

Middleman with measurement error

Case3 Case4

Then in section 3.2.2.1 and 3.2.2.2, we will discuss the parameters effect for the middleman profit.

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3.2.2.1 Comparing the expected profit per item 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 profit of middleman instrument without and with measurement error. So first we will calculate the difference in profits for middleman instrument without and with measurement error.

Notations:

(1)

EPR

^*_{R D− 1} is the difference in profits of middleman instrument without measurement error and the Case 1 under middleman instrument with measurement error (

EPR

^*_{R D− 1}

= EPR

^*_{MR WO−}

− EPR

^*_{MR W .− 1}), (2)

EPR

^*_{R D− 2} is the difference in profits of middleman instrument without measurement error and the

Case 2 under middleman instrument with measurement error (

EPR

^*_{R D− 2}

= EPR

^*_{MR WO−}

− EPR

^*_{MR W .− 2}), (3)

EPR

^*_{R D− 3} is the difference in profits of middleman instrument without measurement error and the

Case 3 under middleman instrument with measurement error (

EPR

^*_{R D− 3}

= EPR

^*_{MR WO−}

− EPR

^*_{MR W .− 3}), (4)

EPR

^*_{R D− 4} be the difference in profits of middleman instrument without measurement error and the

Case 4 under middleman instrument with measurement error (

EPR

^*_{R D− 4}

= EPR

^*_{MR WO−}

− EPR

^*_{MR W .− 4}), The response figure and table are applied to find the significant parameters of

1

*

EPR

R D₋ ,

2

*

EPR

R D₋ ,

3

*

EPR

R D₋ and

4

*

EPR

R D₋ .

From Table 3.7 we find that, when (δ, δ3, δ4) equal to (1, 0.9, 0.4) this combination (see No. 1, 23, and 26), middleman’s profit difference is large.

We want to find the significant parameters of the response variables (EPR *R D− ₁, E P R *R−D₂ ,

R D3

EPR * − , or E P R *R−D₄ ) and also want to know the trend of response variable when the significant parameters change, so we using the response figure and table to find the significant parameters.

‧

EPR

R D₋

values with different parameters combination under producer taking perfect repair action

No. δ δ3 δ4 σx IC ν1 ν2

EPR

^*R D₋₁

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R D1

EPR * −

,

R D2

E P R * −

,

R D3

EPR * −

, and

R D4

E P R * −

Figure 3.3 (a) Response figure of E P R *R−D₁ for each parameter Figure 3.3 (b) Response figure of E P R *R−D₂ for each parameter

Figure 3.3 (c) Response figure of E P R *R−D₃ for each parameter Figure 3.3 (d) Response figure of E P R *R−D₄ for each parameter

Table 3.5 Response table of

^{EPR *R D− 1}

,

^{E P R *R−D2}

,

^{EPR *R D− 3}

and

^{E P R *R−D4}

R D1

E P R * − E P R *R−D₂

δ δ3 δ4 σx IC ν1 δ δ3 δ4 σx IC ν1 ν2

level1

1.433 0.002 3.982 2.205 2.272 2.465 1.431 0.002 3.97 2.176 2.241 2.422 2.377

level2

3.267 0.906 1.528 2.046 2.1 2.123 3.255 0.906 1.527 2.005 2.06 2.103 1.926

level3

1.766 5.557 0.955 2.214 2.094 1.877 1.687 5.466 0.877 2.192 2.072 1.848 2.071

diff 1.834 5.555 3.027

0.168 0.178 0.588

1.824 5.463 3.093

0.186 0.18 0.574 0.451

R D3

E P R * − E P R *R−D4

δ δ3 δ4 σx IC ν1 δ δ3 δ4 σx IC ν1 ν2

level1

1.435 0.003 4.004 2.259 2.326 2.52 1.433 0.002 3.99 2.23 2.293 2.461 2.416

level2

3.289 0.906 1.53 2.1 2.154 2.177 3.275 0.906 1.529 2.046 2.102 2.157 1.98

level3

1.904 5.719 1.093 2.269 2.148 1.931 1.813 5.612 1.002 2.244 2.126 1.902 2.125

diff 1.854 5.716 2.91

0.168 0.178 0.589

1.841 5.61 2.988

0.197 0.191 0.559 0.436

Based on Table 3.5, if the difference between the maximal and minimal values of the three levels is larger than 1.5, the parameters are determined to be significant, so δ, δ3, δ4 are significantly

influence EPR *R D− ₁, E P R *R−D₂ , EPR *R D− ₃and E P R *R−D₄ .

(1) When δ increases, EPR *R D− ₁, E P R *R−D₂ , EPR *R D− ₃and E P R *R−D₄ increases first and then decreases.

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(2) When δ3 increases, EPR *R D− ₁, E P R *R−D₂ , EPR *R D− ₃and E P R *R−D₄ increases. Why when δ3

increases, the difference in middleman profits increases, the reason is describe as follows.

From the Figure 3.4 (a), we found that δ3 is a significant parameter of producer

specification limits, and when δ3 increase then producer specification will become narrow, and the distribution of observed quality may similar to the distribution of true quality. However because producer specification become narrow, the yield of true quality decreases. Since if the yield of true quality decreases, then it will increase the cost of middleman return goods.

Therefore, if δ3 increases, then the average middleman profit may decrease. And by Figure 3.4 (c) when δ3 increases, the profit decreasing rate are larger for middleman instrument with

measurement error than without measurement error. Therefore profit differences may increase when δ3 increases.

Figure 3.4 (a) Response figure of δ

3

to

d^*_p

for each parameter

Figure 3.4 (b) Producer specifications and distribution of true quality of producer under different δ

3

value

Figure 3.4 (c) Response figure of δ

3

to the

R W O

E P RM ₋

and

R W

E P RM ₋

‧

EPR

R D₋ decrease. This suggests that when the middleman instrument measurement error decreases, middleman instrument with

measurement error profits are similar to the profits of middleman instrument without measurement error.

↑︰represent if parameter is increasing then response is significant and with increase trend.

↓︰represent if parameter is increasing then response is significant and with decrease trend.

↑↓︰represent if parameter is increasing then response is significant and with increase first and then decrease trend.

N︰represent parameter is not significant for response.

-

︰Not available

3.2.2.2 Comparing the four kinds of middleman profits under middleman instrument with measurement error and the producer taking perfect repair action.

Since we want to find the significant parameters for the different combination of the difference profit of the four cases that middleman instrument with measurement error. So we will calculate the difference in profits for these four cases middleman instrument with measurement error first.

Notations:

‧

The response figure and table are applied to find the significant parameters of

1

From Table 3.7 and Table 3.8 we found that, the four profit outcomes exhibit few differences.

Therefore, we may choose one case that is easiest for the middleman to manage or execute.

(1) Regarding large-quantity production, Case 2 is more favorable for the middleman. Although per item profit does not exhibit a large difference, when production is considerably large, there may be a substantial difference in total profit.

(2) For small-quantity production with high per unit price, the middleman may adopt Case 2 or Case 4.

If the middleman wants to maximize profit, Case 2 is the favorable choice. However, if the

middleman is more concerned with the ease of managing business operations, Case 4 might be the favorable choice.

‧

different combinations of parameters for producer taking perfect repair action

No. δ δ3 δ4 σx IC ν1 ν2 1

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R .W D1

EPR * −

,

R .W D2

E P R * −

,

R .W D3

EPR * −

,

R .W D4

E P R * −

,

R .W D5

EPR * −

and

R .W D6

EPR * −

Figure 3.5 (a) Response figure of E P R *R .W−D₁ for each parameters Figure 3.5 (b) Response figure of E P R *R .W−D₂ for each parameters

Figure 3.5 (c) Response figure of E P R *R .W−D₃for each parameters Figure 3.5 (d) Response figure of E P R *R .W−D₄ for each parameters

Figure 3.5 (e) Response figure of E P R *R .W−D₅for each parameters Figure 3.5 (f) Response figure of E P R *R .W−D₆for each parameters

Table 3.8 Response table of

^{EPR *R .W−D1}

,

^{E P R *R .W−D2}

,

^{EPR *R .W−D3}

,

^{E P R *R .W−D4}

,

^{EPR *R .W−D5}

and

^{EPR *R .W−D6}

(a) E P R *R .W−D₁ (b) E P R *R .W−D₂

δ δ3 δ4 σx IC ν1 ν2 δ δ3 δ4 σx IC ν1

level1 -0.001 0 -0.012 -0.028 -0.031 -0.043 -0.043 0.002 0 0.022 0.054 0.054 0.054

level2 -0.012 0 -0.001 -0.041 -0.04 -0.019 -0.029 0.022 0 0.002 0.054 0.054 0.054

level3 -0.078 -0.092 -0.078 -0.023 -0.021 -0.029 -0.02 0.138 0.162 0.138 0.054 0.054 0.054

diff 0.077 0.092 0.077 0.018 0.019 0.024 0.024 0.136 0.161 0.136 0 0 0

(c) E P R *R .W−D₃ (d) E P R *R .W−D₄

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

level1 0.001 0 0.008 0.026 0.021 -0.004 -0.004 0.003 0 0.034 0.082 0.085 0.097 0.097

level2 0.007 0 0.001 0 0.001 0.035 0.025 0.034 0 0.004 0.095 0.094 0.074 0.083

level3 0.047 0.055 0.047 0.029 0.033 0.025 0.035 0.217 0.253 0.217 0.077 0.075 0.083 0.074 diff 0.046 0.055 0.046 0.029 0.032 0.039 0.039 0.213 0.253 0.213 0.018 0.019 0.024 0.024

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(e) E P R *R .W−D₅ (f) E P R *R .W−D₆

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

level1 0.002 0 0.02 0.054 0.052 0.039 0.039 -0.001 0 -0.014 -0.028 -0.033 -0.058 -0.058 level2 0.02 0 0.002 0.041 0.041 0.054 0.054 -0.014 0 -0.001 -0.054 -0.053 -0.02 -0.029 level3 0.125 0.147 0.125 0.052 0.054 0.054 0.054 -0.091 -0.107 -0.091 -0.025 -0.021 -0.029 -0.019 diff 0.123 0.147 0.123 0.013 0.013 0.015 0.015 0.09 0.107 0.09 0.029 0.032 0.039 0.039

3.3 Maximize the Profit Model to Determine the Optimal Specification Limits for Middleman under Producer Adopts Sell Low Price Action.

3.3.1 Derivation of the profit model for middleman

(I) Middleman instrument without measurement error

First we will discuss when middleman instrument without measurement error, how the producer, middleman and customer deal with the conforming items and nonconforming items.

For producer, the observed conforming item will be sold to middleman at high price (PPMH) and the observed nonconforming item will be sold at low price (PPML).

For middleman, since producer do not adjust the quality of products, so the true quality of middleman (XM) is equivalent to the true quality of producer (XP), that is,

X

_M

≡ X

_P

~ N ( µ , σ

_{x x}²

)

. Since

middleman instrument without measurement error, so after inspection, middleman could know the true quality of products. For true conforming item with high buying price (PPMH) will be sold to customer at a high price (PMCH), and the true conforming item with low buying price (PPML) will be sold to customer at a low price (PMCL), but for true nonconforming item middleman will return to the producer and take back part refund. The refund depends on the middleman’s buying price, if the middleman’s buying price is high, then middleman receives high refund (CmrH); otherwise the middleman receives low reimbursement

(CmrL).

‧

Figure 3.6 Flow chart of products for producer, middleman and customer

Describe the middleman profit under different situation of item,

If the item is true conforming item and observed conforming item, then middleman profit is,

MCH PMH

P −P −IC.

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

MCL PML

P −P −IC .

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

PMH mrH

P C IC

− + − .

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

PML mrL

P C IC

− + − .

Therefore the profit function of middleman without measurement error is

;

‧

The expected profit per item for middleman without measurement error is the sum of the profit of an item’s status multiply to the corresponding probability, that is,

( )

(II) Middleman instrument with measurement error

In this situation the action for producer deal with the conforming item and nonconforming item is same to when middleman instrument without measurement error, the differences are middleman and customer how to deal with the conforming item and nonconforming item.

For middleman, since the middleman instrument with measurement error, so we let the distribution of the measurement error is,

(0 ² )

M m e

ε ~ N , σ ,

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and the distribution of observed product quality is,

(

^{2 2}

)

M M M p M x x me

Y = X + ε = X + ε ~ N µ , σ + σ

.

After middleman inspection, the observed conforming item with high buying price will be sold to customer at a high price, and the observed conforming item with low buying price will be sold to customer at a high price, but for observed nonconforming item will return to the producer and take back part refund.

For customer received products if the item is observed conforming item but true nonconforming item with middleman high selling price, then customer will return goods and middleman may have high return cost (CcrH). If the item is observed conforming item but true nonconforming item with middleman low selling price, then middleman may have low return cost (CcrL).

The flow chart of producer, middleman and customer deal with products will show in Figure3.9

Figure 3.7 Flow chart of products for producer, middleman and customer

‧

Describe the middleman profit under different situation of item,

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

MCH PMH

P −P −IC,

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

MCL PML

P −P −IC ,

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

PMH mrH

P C IC

− + − ,

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

PML mrL

P C IC

− + − ,

If the item is true nonconforming item but observed conforming item for middleman, we may discuss four different cases for producer and middleman to deal with the cost (Ccr) of customer return goods.

Case1. Middleman gives full refund to customer, and middleman receive refund from producer If the item is true nonconforming item but observed conforming item for producer, with the high buying price for middleman, then middleman profit of per unit is,

( ) ( )

1

r = PM C H −PM C H − PP M H −PP M H −IC = −IC π

If the item is true nonconforming item and observed conforming item for producer, with the low buying price for middleman, then middleman profit of per unit is,

‧

Case2. Middleman gives partial refund to customer and middleman receive refund from producer (2-1) If Ccr < PPM, then producer may pay Ccr to the middleman

If the item is true nonconforming item but observed conforming item for producer, with the high buying price for middleman, then middleman profit of per unit is,

( ) ( )

1

r = PM C H −CcrH −IC − PP M H −CcrH = PM C H −PP M H −IC π

If the item is true nonconforming item and observed conforming item for producer, with the low buying price for middleman, then middleman profit of per unit is,

( ) ( )

2

r = PM C L−CcrL−IC − PP M L−CcrL =PM C L −PP M L−IC π

(2-2) If PMC > Ccr≧PPM, then producer may pay PPM to the middleman

If the item is true nonconforming item but observed conforming item for producer, with the high buying price for middleman, then middleman profit of per unit is,

( ) ( )

1

r = PM C H −CcrH −IC + PP M H −PP M H = PM C H −Cc rH −IC π

If the item is true nonconforming item and observed conforming item for producer, with the low buying price for middleman, then middleman profit of per unit is,

( ) ( )

2

r = PM C L−CcrL−IC + PP M L−PP M L =PM C L −Cc rL−IC π

Case3. Middleman gives full refund to customer (PMC), and also endure the product cost (PPM) If the item is true nonconforming item but observed conforming item for producer, with the high buying price for middleman, then middleman profit of per unit is,

( )

1

r = PM C H −PM C H −IC −PPM H = −IC−PPM H

π

If the item is true nonconforming item and observed conforming item for producer, with the low buying price for middleman, then middleman profit of per unit is,

( )

2

r = PM C L −PM C L −IC −PP M L = −IC −PP M L

π

Case4. Middleman gives partial refund (Ccr) to customer, and the product cost (PPM)

If the item is true nonconforming item but observed conforming item for producer, with the high buying price for middleman, then middleman profit of per unit is,

( )

1

r = PM C H −CcrH −IC −PP M H

π

‧

If the item is true nonconforming item and observed conforming item for producer, with the low buying price for middleman, then middleman profit of per unit is,

( )

2

r = PM C L −CcrL−IC −PP M L

π

Therefore the profit function of middleman with measurement error is

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

( )

‧

‧

3.3.2 Determine the optimal specification limits, middleman buying and selling price with Data

在文檔中 生產者與中間商利潤模式有量測誤差下規格之最適設計 - 政大學術集成 (頁 74-102)