Taking sell low price action for observed nonconforming items

國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

23

From the Table 2.8, we found that,

1. the optimal specification limits width became wider in Nos. 10, 18, 24, and for others the optimal specification limits are the same.

2. IS1

*

EPR

P is larger for high η value.

2.3.2 Taking sell low price action for observed nonconforming items.

If the quality of the product satisfies the producer’s specifications, then it will be sold at a high price, (PPMH). However, if the quality does not satisfy the producer’s specifications, then the cost of the nonconforming item (Csc) is involved, and the product will be sold at low price, (PPML > Csc).

Figure 2.9 Producer specification based on Y

P

, and sold at low price for nonconforming item

Figure 2.10 Quadratic loss function based on X

P

‧

2.3.2-1 Derivation of the profit model per unit time

For in-control process, the expected profit per item contains,

(1) The expected profit for per conforming item is the producer high selling price minus the expected loss for per conforming item, that is,

[ ] ( )

(2) The expected profit for per nonconforming item is, the probability of per nonconforming item multiply to the per item profit of nonconforming item, (PPML-Csc), that is,

1

^p

( ) [ ]

(3) Inspection cost of per item for producer is, IC.

Hence, the expected in-control profit per item for producer is E(PI), that is, producer selling price minus the cost of producer,

[ ] [ ]

14444444 4244444444 3 1444444 424444444 3

( ^{p x} ) ( ^{p x} ) 1 ( 2 1) ^{2 p x p x} 1 The expected out-of-control profit per item contains,

(1) The expected profit of per conforming item is, producer high selling price minus the expected loss for per conforming item, that is,

[ ] ( )

‧

(2) The expected profit of per nonconforming item is, the product of the probability of nonconforming item and the profit of per nonconforming item, (PPML-Csc), that is,

1

^p

( ) [ ]

(3) Inspection cost of per item for producer is, IC.

Then the expected in-control profit per item for producer is the sum of profit of per conforming item and the nonconforming item and minus the inspection cost per item, that is,

[ ] [ ]

14444444 4244444444 3 1444444 424444444 3

1 1

The expected profit of per unit time is the sum of the in-control profit and out-of-control profit divided by m unit time, that is,

2

‧

Since we want to compare the yields of XP andYP under nonconforming items are sell with low price, so we need to derive the yield formula first.

1. Yield of XP is,

2.3.2-2 Determine the optimal inspection specification limits and data analyses

Table 2.9 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

( ) ³

¹³ (see Table 2.10).

Let the expected profit of per unit time under producer selling low price for nonconforming item be

EPR

PIS2. To determine optimal producer specification limits, we maximize

IS2

EPR

P with the constraint 0≤d_p ≤

3

σ²_x +σ²_pe by using “optim” routine in R program.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

27

Table 2.10 The 27 combinations of these parameters by using an orthogonal array table

L₂₇(3 )¹³

.

No. δ δ1 δ2 δ3 σx (θ, m) (R, Csc, IC, PPM) kp γ 1

1 2.5 1 0.9 0.2 (0.05, 150) (30, 20, 1, 90) 8 0.75

2

2 1.5 1 0.5 0.2 (0.1, 120) (30, 20, 1, 90) 4 0.9

3

0 1.5 2 0.3 0.5 (0.1, 120) (30, 20, 1, 90) 8 0.75

4

2 1 1.5 0.5 0.7 (0.01, 200) (30, 20, 1, 90) 8 0.75

5

0 1 1 0.9 0.7 (0.1, 120) (100, 10, 0.1, 65) 4 0.5

6

0 1.5 2 0.5 0.7 (0.01, 200) (500, 5, 0.05, 50) 4 0.75

7

2 2.5 2 0.3 0.2 (0.01, 200) (100, 10, 0.1, 65) 4 0.5

8

2 1 1.5 0.9 0.2 (0.05, 150) (500, 5, 0.05, 50) 4 0.75

9

0 2.5 1.5 0.3 0.7 (0.05, 150) (30, 20, 1, 90) 4 0.9

10

2 2.5 2 0.9 0.7 (0.1, 120) (500, 5, 0.05, 50) 8 0.5

11

0 1.5 2 0.9 0.2 (0.05, 150) (100, 10, 0.1, 65) 2 0.75

12

1 1.5 1.5 0.9 0.7 (0.1, 120) (30, 20, 1, 90) 2 0.5

13

2 1 1.5 0.3 0.5 (0.1, 120) (100, 10, 0.1, 65) 2 0.75

14

0 2.5 1.5 0.5 0.2 (0.1, 120) (500, 5, 0.05, 50) 2 0.9

15

1 1 2 0.5 0.2 (0.1, 120) (100, 10, 0.1, 65) 8 0.9

16

1 1.5 1.5 0.3 0.2 (0.01, 200) (500, 5, 0.05, 50) 8 0.5

17

2 1.5 1 0.9 0.5 (0.01, 200) (500, 5, 0.05, 50) 2 0.9

18

2 1.5 1 0.3 0.7 (0.05, 150) (100, 10, 0.1, 65) 8 0.9

19

1 1 2 0.9 0.5 (0.01, 200) (30, 20, 1, 90) 4 0.9

20

0 1 1 0.3 0.2 (0.01, 200) (30, 20, 1, 90) 2 0.5

21

2 2.5 2 0.5 0.5 (0.05, 150) (30, 20, 1, 90) 2 0.5

22

0 1 1 0.5 0.5 (0.05, 150) (500, 5, 0.05, 50) 8 0.5

23

1 2.5 1 0.3 0.5 (0.1, 120) (500, 5, 0.05, 50) 4 0.75

24

1 1 2 0.3 0.7 (0.05, 150) (500, 5, 0.05, 50) 2 0.9

25

0 2.5 1.5 0.9 0.5 (0.01, 200) (100, 10, 0.1, 65) 8 0.9

26

1 2.5 1 0.5 0.7 (0.01, 200) (100, 10, 0.1, 65) 2 0.75

27

1 1.5 1.5 0.5 0.5 (0.05, 150) (100, 10, 0.1, 65) 4 0.5

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

28

Table 2.11 Optimal solutions of the 27 combinations parameters.

No. d^*P

IS2

*

EPR

P Yield for Xp Yield for Yp

1 0.667 2017.991

99.01773 98.20875

2 1.2 2533.115

99.99683 97.72182

3 5 2321.076

100 99.73002

4 2.91 1835.217

99.99683 97.72182

5 2.333 6078.165

99.91419 99.73002

6 3.021 21965.6

99.99999 99.30991

7 2 6112.801

100 99.16434

8 0.667 21434.2

90.87887 88.49295

9 6.252 2232.667

100 99.73002

10 1.863 11763.75

89.99255 87.55468

11 0.667 6025.241

99.91419 99.73002

12 2.333 2045.02

99.01773 98.20875

13 5 5919.678

100 99.16434

14 1.2 24417.14

100 99.70322

15 1.2 6200.403

94.75043 75.63351

16 2 24006.72

100 98.44392

17 1.667 22421.45

90.87887 88.49295

18 0 5138.493

6.66E-14 1.11E-13

19 1.667 2482.318

99.01773 98.20875

20 2 2659.944

100 99.73002

21 3 2012.783

99.99683 97.72182

22 3 23057.32

100 99.73002

23 5 18828.91

99.9991 87.11712

24 3.336 21900.97

100 98.583

25 1.457 5844.869

99.45046 98.75392

26 4.2 5700.365

99.99997 99.35577

27 3 5605.566

99.99997 99.35577

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

29

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

d

^*_P and

EPR*

PIS₂.

(I) Response figure and table of

d

^*_P

Figure 2.11 Response figure of d

^*_P

for each parameter

Table 2.12 Response table of d

^*_P

δ δ1 δ2 δ3 σx (θ, m) (R, Csc, IC, PPMH) kp γ

level1 2.77 2.457 2.23 3.399 1.289 2.325 2.417 2.6 2.392

level2 2.6 2.099 2.758 2.526 3.199 2.288 2.206 2.793 3.015

level3 2.034 2.849 2.417 1.48 2.916 2.792 2.781 2.011 1.998

diff 0.736 0.75 0.528

1.919 1.91

0.504 0.575 0.783 1.017

Based on the Table 2.12, if the difference between maximal and minimal values of the three levels is larger than 1.5, the parameters, δ3 and σx are determined to be significant in relation to

d

^*_P .

The optimal producer specification is determined by the constraint of the variance of observed quality characteristics,

^σ

²_x

+ ^σ

²_pe. Therefore, δ3 and σx are significant parameters of

d

^*_P , and the δ3 and σx trends are,

(1) when δ3 increases then

d

^*_P decreases.

(2) when σx increases then

d

^*_P increases first and then decreases.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

30

(II) Response figure and table of

EPR

*P_IS2

Figure 2.12 Response figure of

*

PIS2

EPR

for each parameter

Table 2.13 Response table of

*P_IS2

EPR

δ δ1 δ2 δ3 σx (θ, m) (R, Csc, IC, PPMH) kp γ

level1 10511.34 10174.25 9826.195 9902.362 10600.84 10336.59 21088.45 10344.73 9260.23 level2 9865.362 10229.14 10371.23 10369.72 9832.664 9936.137 5847.287 9697.038 9560.919 level3 8796.832 8770.141 8976.104 8901.445 8740.027 8900.806 2237.792 9131.76 10352.38 diff 1714.504 1459.002 1395.127 1468.279 1860.813 1435.782 18850.66 1212.973 1092.151

Based on the Table 2.13, if the difference between the maximal and minimal values of the three levels is larger than 3000, the parameter (R, Cpr, IC, PPM) is determined to be significantly related to

*P_IS2

EPR

.

When (R, Cpr, IC, PPM), increases

EPR

*P_IS2 decreases. Although PPM increases between the three levels, but the difference in PPM is not large. Moreover, R decreases between the levels, but the difference in R have large difference; therefore

EPR

*P_IS2 will be significantly affected by amount of production (R). This means when per unit profit is small, the producer should adopt mass production to increase profit.

‧

Table 2.14 Summary table for sensitivity analyses of perfect repair and sell different price

Inspection

↑︰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.

For comparing expected profit per hour of producer taking perfect repair with selling low price for nonconforming items, we need calculate the difference value of

d

^*_p for two actions and

difference value of

EPR for two actions first. Here let

^*_P parameters are significant to

dPIS

D

and

EPRPIS

D

.

From Table 2.15 and Table 2.16 we find that,

(1) when δ3=0.3 ,σx=0.7 (see No. 18, 24), there is large difference in the optimal producer specification limits under two actions of nonconforming items. Since δ3=0.3 represents producer instrument with largest measurement error between three levels, and σx=0.7 is the largest values between three levels, so the producer specification have significant difference. For other parameters combination, producer specifications of these two actions are almost the same.

(2) many results show that producers’ profit are higher when producer repairs nonconforming item Although the profit from the action of perfect repair is less than that of sell low price, the difference is not substantial (except Nos. 18, 24).

‧

(3) the observed yield for the action of perfect repair is larger than that of sell low price, because the observed yield of the perfect repair will be equal to 100%. For No. 1, 4, 8, 10, 17, 18, 19, and 20 in Table 2.14 the yield of true quality of perfect repair action is greater than sell low price action, but for other Nos. in the Table 2.14 the yield of perfect repair action is slight less than sell low price action.

Table 2.15 The 27 parameters combinations with

dPIS

‧

Table 2.16 Yield of true quality and observed quality for two actions that producer took

No.

Perfect Repair Sell Different Price

Yield for Xr and XP

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

34

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

D

d_PIS and

D

EPR_PIS . (I) Response figure and table of

D

d_PIS

Figure 2.13 Response figure of D

d_PIS

for each parameter

Table 2.17 Response table of D

d_PIS

δ δ1 δ2 δ3 σx (θ, m) (R, Cpr ,Csc, IC, PPMH) kp γ

level1

0.05 0.387 0.447 0.944 0 -0.028 0.288 0.403 -0.108

level2

0.403 0.441 0.073 -0.023 -0.005 0.944 0.442 0.055 -0.023

level3

0.355 -0.02 0.288 -0.114 0.813 -0.108 0.078 0.35 0.939

diff

0.353 0.461 0.375 1.058 0.818 1.052 0.364 0.348 1.047

Based on Table 2.17, if the difference between the maximal and minimal values of the three levels is larger than 1, the parameter is determined to be significant, so δ3, (θ, m) and γ are significantly influence

D

^d_PIS.

(1) When γ increases,

D

^d_PISincreases. Moreover, when γ is Level 3, the difference value is positive, this suggests that when the nonconforming item discount is small, the producer’s specification may be wider than perfect repair.

(2) When δ3 increases,

D

^d_PISdecreases. δ3 increase represent that producer instrument measurement error may decrease, and the producer specification will become narrow. We found that, when δ3 is Level 1, the difference value is positive. This means when measurement error is large, the producer specification of perfect repair may wider than sell different price.

(3) When (θ, m) increases,

D

d_PIS increases first and then decreases.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

35

(II) Response figure and table of

D

EPR_PIS

Figure 2.14 Response figure of D

EPR_PIS

for each parameter

Table 2.18 Response table of D

EPR_PIS

δ δ1 δ2 δ3 σx (θ, m) (R, Cpr ,Csc, IC, PPM) kp γ level1 ^{40.677 120.531 -335.591 -394.046 217.928 36.024 1017.005 37.869 997.603} level2 ^{31.061 -280.738 215.226 61.819 45.154 -168.141 -302.424 191.251 210.046} level3 ^{721.359 953.303 913.462 1125.324 530.015 925.213 78.516 563.976 -414.552} diff ^{690.298 1234.041 1249.053 1519.37 484.861 1093.354 1319.429 526.107 1412.154}

Based on Table 2.18, if the difference between the maximal and minimal values of the three levels is larger than 1000, the parameter is determine to be significant, so δ1, δ2, δ3 (θ, m), (R, Cpr ,Csc, IC, PPMH) and γ are significantly influence DE P R_PIS .

(1) When δ2 (or δ3) increases, DE P R_PIS increases. Mover when δ2 (or δ3) is Level 1, the difference value is negative. This means if the out-of-control process variance shift is small, or if the producer instrument measurement error is large, the producer’s expected profit for sell different price action may be larger than adopting the perfect repair.

(2) When γ increases, DE P R_PIS decreases. Moreover, when γ is Level 3, the difference value is negative. This means if the producer sells nonconforming items at a low price, and the discount of low selling price is small, then the expected profit may be larger than perfectly repaired acion.

(3) If δ1 (or (θ, m) or (R, Cpr ,Csc, IC, PPMH) ) increases,

D

E P R_PIS decreases first and then increases.

Therefore,

(i) when the out-of-control process mean shift is small or large, then the producer’s profit of

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

36

conducting perfect repair is larger than that of selling products at low price.

(ii) when (θ, m) equals (0.05, 150), the producer’s profit of taking sell low price is larger than taking perfect repair.

(iii) when (R, Cpr ,Csc, IC, PPMH) equals (100, 15, 10, 0.1, 50), the producer’s profit of taking sell low price is larger than that of taking perfect repair.

For significant parameters of

D

EPRP_IS , δ1, δ2, δ3, θ, m, R, Cpr ,Csc, IC and PPMH , we plot the levels

value corresponding to

D

EPRP_IS , and calculate the intersection points on x-axis to determine under which range the producer profit of taking perfect repair is larger.

Figure 2.15 (a) D

EPR_PIS

under different δ

1

Figure 2.15 (b) D

EPR_PIS

under different δ

2

(1) From Figure 2.15 (a) we found that, when 1.15 < δ1 < 1.727, the producer obtains larger profit by selling nonconforming items at a low price than selling perfectly repaired products. This implies that if producer takes sell different price tactics, then producer should control δ1 between a 1.15 and 1.727 standard deviation.

(2) From Figure 2.15 (b) we found that, when 0 < δ2 < 1.305, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if producer takes sell different price tactics, then producer should control δ2 less than 1.305.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

37

Figure 2.15 (c) D

EPR_PIS

under different δ

3

Figure 2.15 (d) D

EPR_PIS

under different θ

(3) From Figure 2.15 (c) we found that, when 0 < δ3 < 0.473, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if producer instrument measurement error is large, then producer should sell low price for

nonconforming item.

(4) From Figure 2.15 (d) we found that, when 0.017 < θ < 0.058, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if the expected time of occur assignable cause between 17.241 and 58.823 hour (since

1 1 1

17 241 58 823

0 058 0 017

. .

. θ .

≈ < < ≈

), then producer should sell low price for nonconforming item.

Figure 2.15 (e) D

EPR_PIS

under different m Figure 2.15 (f) D

EPR_PIS

under different R

(5) From Figure 2.15 (e) we found that, when 145.386 < m < 191.178, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if the comparing time is between 145.386 and 191.178 hour, then producer should sell low price for nonconforming item.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

38

(6) From Figure 2.15 (f) we found that, when 44.428 < R < 191.683, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if the per hour production is between 44.428 and 191.683, then producer should sell low price for nonconforming item.

Figure 2.15 (g) D

EPR_PIS

under different C

pr

Figure 2.15 (h) D

EPR_PIS

under different C

sc

(7) From Figure 2.15 (g) we found that, when 13.854 < Cpr < 26.908, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if now producer is taking sell different price tactics, then producer should control the perfect repair cost of per item between 13.854 and 26.908.

(8) From Figure 2.15 (h) we found that, when 8.854 < Csc < 17.939, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if now producer is taking sell different price tactics, then producer should control the cost of per nonconforming item between 8.854 and 17.939.

Figure 2.15 (i) D

EPR_PIS

under different IC Figure 2.15 (j) D

EPR_PIS

under different P

PM

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

39

(9) From Figure 2.15 (i) we found that, when 0.089 < IC < 0.814, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if now producer is taking sell different price tactics, then producer should control the inspection cost between 0.089 and 0.814.

(10) From Figure 2.15 (j) we found that, when 61.562 < PPM (PPMH) < 84.847, producers profit are larger for selling nonconforming items at a low price than selling perfectly repaired products. This implies that if now producer is taking sell different price tactics, then producer should control the conforming item selling price between 61.562 and 84.847.

Figure 2.15 (k) D

EPR_PIS

under different γ

(11) From Figure 2.12 (k) we found that, when 0.8 < γ < 1, producers profit are larger selling nonconforming items at a low price than selling perfectly repaired products. This means if the discount is large than 0.8, then producer should sell low price for the nonconforming item.

From section 2.1 to section 2.3 we have discussed producer do complete inspection plan, now we will consider producer do the process control but no do inspection. Then we will compare these two profits to find approximate ranges of process parameters and design parameters for producer taking complete inspection or conducting process control.

‧

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 34-51)