Analysis for the Cost Difference

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Table 6- 12. The Values of LSL* for Each Parameter

 OT (a_I,b_I,₁,₂) (A, R_n, IC)

level 1 1.023 1.023 1.023 0.633

level 2 1.023 1.023 1.023 1.414

difference 0 0 0 0.781

Table 6- 13. The Values of yield for Each Parameter *

 OT (a_I,b_I,₁,₂) (A, R_n, IC) level 1 0.8774 0.8774 0.7557 0.9524 level 2 0.8774 0.8774 0.9991 0.8024

difference 0 0 0.2434 0.1500

Table 6- 14. The Values of ECIns* for Each Parameter

 OT (a_I,b_I,₁,₂) (A, R_n, IC) level 1 1245.7 1881.8 1794.0 1282.3 level 2 1645.3 1009.2 1097.0 1608.7

difference 400.4 872.6 697 326.4

6.3 Analysis for the Cost Difference

Table 6-15 represents the EC_Ins*EC_PC * for each level combination with respect to the common parameters for the model of process control and product inspection.

We consider the parameter as significant to EC_Ins*EC_PC* if the difference of the average of EC_Ins*EC_PC* between two levels is greater than 100. Fig. 6-3 and Table 6-7

shows that λ, OT, and (a_I,b_I,₁,₂) are the significant parameters for EC_Ins*EC_PC* and it comes the following findings:

(1) A larger λ indicates a larger EC_Ins*EC_PC*. (2) A larger OT indicates a larger EC_Ins*EC_PC*.

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Table 6- 15. The Values of EC_Ins*EC_PC* for Each Level Combination with Common Parameters Level

Combination λ OT (a_I,b_I,₁,₂) EC_Ins*-EC_PC*

1 0.1 25 (2,2,1.5,1.5) 967.1

2 0.1 25 (2,2,1.5,1.5) 1421.2 3 0.1 25 (2,2,1.5,1.5) 1017.1 4 0.1 25 (2,2,1.5,1.5) 1452.5

5 0.1 25 (5,5,2.5,2.5) 775.0

6 0.1 25 (5,5,2.5,2.5) 355.8

7 0.1 25 (5,5,2.5,2.5) 808.7

8 0.1 25 (5,5,2.5,2.5) 389.4

9 0.1 15 (2,2,1.5,1.5) 525.1

10 0.1 15 (2,2,1.5,1.5) 964.0 11 0.1 15 (2,2,1.5,1.5) 473.2 12 0.1 15 (2,2,1.5,1.5) 900.9

13 0.1 15 (5,5,2.5,2.5) 80.2

14 0.1 15 (5,5,2.5,2.5) -95.8

15 0.1 15 (5,5,2.5,2.5) 44.6

16 0.1 15 (5,5,2.5,2.5) -148.6 17 0.5 25 (2,2,1.5,1.5) 1627.6 18 0.5 25 (2,2,1.5,1.5) 1803.7 19 0.5 25 (2,2,1.5,1.5) 1708.3 20 0.5 25 (2,2,1.5,1.5) 1887.4 21 0.5 25 (5,5,2.5,2.5) 825.2 22 0.5 25 (5,5,2.5,2.5) 319.4 23 0.5 25 (5,5,2.5,2.5) 970.3 24 0.5 25 (5,5,2.5,2.5) 436.6 25 0.5 15 (2,2,1.5,1.5) 782.5 26 0.5 15 (2,2,1.5,1.5) 932.5 27 0.5 15 (2,2,1.5,1.5) 748.4 28 0.5 15 (2,2,1.5,1.5) 899.8 29 0.5 15 (5,5,2.5,2.5) 671.8 30 0.5 15 (5,5,2.5,2.5) 391.5 31 0.5 15 (5,5,2.5,2.5) 627.7 32 0.5 15 (5,5,2.5,2.5) 353.6

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Figure 6- 3. The Response Graph for EC_Ins*EC_PC* for Each Common Parameter.

Table 6- 16. The Values of EC_Ins*EC_PC * for Each Common Parameter.

 OT (a_I,b_I,₁,₂) level1 620.63 1047.82 1131.95

level2 936.65 509.46 425.33

difference 316.02 538.36 706.61

Fig.6-4 shows that, in a reasonable range of , 0.050.5, ECPC < ECIns. If let ECIns*-ECPC*= 0, we cannot locate the solution of λ. It means that under other parameters fixed, we should monitor the process by using the economic statistical X control chart to obtain the lower expected total cost rather than conduct inspecting products.

Figure 6- 4. Graph of ECPC and ECIns in View of 

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OT is the period we set for comparison and OT>ET. Table 6-3 shows that the minimum ET* is 3.30 and the maximum ET* is 21.83. Fig 6-5 shows the expected total cost for process control and product inspection and the reasonable ranges of OT. We can find that EC_Ins* = EC_PC* when OT = 9.89. Figures 6-5 indicates that if 3.3 < OT < 9.89, then EC_PC* is greater than ECIns*. In this case, product inspection is preferred. However, if

9.89 < OT < 21.83, then EC_PC* is smaller than EC_Ins*, and we choose to conduct process control, not product inspection. We conclude that when OT is larger, ECPC* and ECIns* both become larger. This result and reason is the same as we discussed in Section 6.2.1 and 6.2.2.

Figure 6- 5. Graph of EC_PC and EC_Ins in View of OT

Table 6-17 represents the EC_Ins*EC_PC * for each level combination with respect to the uncommon parameters for the model of process control and product inspection.

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Table 6- 17. The Values of EC_Ins*EC_PC* for Each Uncommon Parameter Level

Combination (C₀,C₁) e ₁ ₂ T₀ T₁ T₂ c d Y W (A, R , IC) EC_n Ins*-ECPC* 1 (30, 40) 0.1 1 1 0.5 0.5 2 0.5 0.1 100 50 (5, 20, 0.5) 967.1 2 (30, 40) 0.1 1 1 0.5 1 0.5 0.5 1 50 50 (1, 100, 0.1) 1421.2 3 (30, 40) 0.1 1 1 1 0.5 0.5 5 0.1 50 10 (5, 20, 0.5) 1017.1 4 (30, 40) 0.1 1 1 1 1 2 5 1 100 10 (1, 100, 0.1) 1452.5 5 (20, 28) 1 1 0 0.5 0.5 2 0.5 0.1 100 50 (1, 100, 0.1) 775.0 6 (20, 28) 1 1 0 0.5 1 0.5 0.5 1 50 50 (5, 20, 0.5) 355.8 7 (20, 28) 1 1 0 1 0.5 0.5 5 0.1 50 10 (1, 100, 0.1) 808.7 8 (20, 28) 1 1 0 1 1 2 5 1 100 10 (5, 20, 0.5) 389.4 9 (20, 28) 0.1 0 0 0.5 0.5 2 0.5 1 50 10 (5, 20, 0.5) 525.1 10 (20, 28) 0.1 0 0 0.5 1 0.5 0.5 0.1 100 10 (1, 100, 0.1) 964.0 11 (20, 28) 0.1 0 0 1 0.5 0.5 5 1 100 50 (5, 20, 0.5) 473.2 12 (20, 28) 0.1 0 0 1 1 2 5 0.1 50 50 (1, 100, 0.1) 900.9 13 (30, 40) 1 0 1 0.5 0.5 2 0.5 1 50 10 (1, 100, 0.1) 80.2 14 (30, 40) 1 0 1 0.5 1 0.5 0.5 0.1 100 10 (5, 20, 0.5) -95.8 15 (30, 40) 1 0 1 1 0.5 0.5 5 1 100 50 (1, 100, 0.1) 44.6 16 (30, 40) 1 0 1 1 1 2 5 0.1 50 50 (5, 20, 0.5) -148.6 17 (20, 28) 1 0 1 0.5 0.5 2 5 0.1 50 50 (5, 20, 0.5) 1627.6 18 (20, 28) 1 0 1 0.5 1 0.5 5 1 100 50 (1, 100, 0.1) 1803.7 19 (20, 28) 1 0 1 1 0.5 0.5 0.5 0.1 100 10 (5, 20, 0.5) 1708.3 20 (20, 28) 1 0 1 1 1 2 0.5 1 50 10 (1, 100, 0.1) 1887.4 21 (30, 40) 0.1 0 0 0.5 0.5 2 5 0.1 50 50 (1, 100, 0.1) 825.2 22 (30, 40) 0.1 0 0 0.5 1 0.5 5 1 100 50 (5, 20, 0.5) 319.4 23 (30, 40) 0.1 0 0 1 0.5 0.5 0.5 0.1 100 10 (1, 100, 0.1) 970.3 24 (30, 40) 0.1 0 0 1 1 2 0.5 1 50 10 (5, 20, 0.5) 436.6 25 (30, 40) 1 1 0 0.5 0.5 2 5 1 100 10 (5, 20, 0.5) 782.5 26 (30, 40) 1 1 0 0.5 1 0.5 5 0.1 50 10 (1, 100, 0.1) 932.5 27 (30, 40) 1 1 0 1 0.5 0.5 0.5 1 50 50 (5, 20, 0.5) 748.4 28 (30, 40) 1 1 0 1 1 2 0.5 0.1 100 50 (1, 100, 0.1) 899.8 29 (20, 28) 0.1 1 1 0.5 0.5 2 5 1 100 10 (1, 100, 0.1) 671.8 30 (20, 28) 0.1 1 1 0.5 1 0.5 5 0.1 50 10 (5, 20, 0.5) 391.5 31 (20, 28) 0.1 1 1 1 0.5 0.5 0.5 1 50 50 (1, 100, 0.1) 627.7 32 (20, 28) 0.1 1 1 1 1 2 0.5 0.1 100 50 (5, 20, 0.5) 353.6

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We consider the parameter as significant to EC_Ins*EC_PC* if the difference of the average of EC_Ins*EC_PC* between two levels is greater than 100. Fig. 6-6 and Table 6-18

shows that (C0, C1) and δ2 are the significant parameters for EC_Ins*EC_PC * and it comes the following findings:

(1) A larger C0 or C1 indicates a smaller EC_Ins*EC_PC *. Because is C0 or C1 increases,

PC*

EC increases but EC_Ins* do not change.

(2) EC_Ins*EC_PC* will be larger If the process continues while engineer fix the out-of-control process.

Figure 6- 6. The Response Graph for EC_Ins*EC_PC* for Each Parameters in Process Control.

Table 6- 18. The Values of EC_Ins*EC_PC* for Each Parameter in Process Control.

(C0,C1) e δ1 δ2 T0 T1 T2 c d Y W

level1 665.80 769.82 787.16 863.11 771.67 790.79 776.63 789.04 806.07 779.96 749.65 level2 891.48 787.47 770.12 694.17 785.61 766.49 780.65 768.24 751.21 777.32 807.63 diff 225.68 17.65 17.04 168.94 13.94 24.30 4.02 20.79 54.85 2.64 57.97

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Figure 6-8 indicates that the relationship between EC PC and C₁C₀ is linear. Because no (C₀, C₁) exists in the model of EC_Ins, EC_Ins is irrelevant to (C₀, C₁). Furthermore, we can predict the ECIns* -EC PC * based on C₁C₀ by using a linear regression model approach;

that is, ˆ ˆ ( )

0 1 , 1 ,

0 C C

EC

EC_Ins _PC  _C  _C  . Hence, we obtain ˆ_0,C 2097.2 and ˆ_1,C  -113.18. We draw the regression line as shown in Fig. 6-8.

Figure 6- 7. Graph of EC_PC and EC_Ins in View of (C₀, C₁)

Figure 6-8 shows that we can obtain the solution of (C₁C₀)= 18.53 by letting EC_Ins*-EC_PC* = 0. We can draw the conclusion as follows:

If (C₁C₀)18.53, then EC_PC* = EC_Ins*. In this case, we can select to conduct either

the process control or product inspection. If (C₁C₀)18.53, then EC_PC* is larger than ECIns*. In this case, we choose to conduct a product inspection instead of process control;

however, if (C₁C₀)18.53, then ECPC* is smaller than ECIns*. We then choose to

perform process control, not product inspection. We conclude that when (C₁C₀) is larger, EC_Ins*-EC_PC* becomes smaller.

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Figure 6- 8. Regression Line of EC_Ins-EC_PC and C₁-C₀

Figure 6-9 shows that whether the process has stopped or continued during repairs, the expected total cost for product inspection is greater than that of process control (i.e., ECIns* >

ECPC*). When the process halts during repairs (₂ 0), we obtain ECPC* = 912.14.

However, if the process continues during repair (₂ 1), we obtain ECPC*= 925.49, indicating that if we wish to reduce the expected total cost of process control, we must stop production while the engineers repair the process.

Figure 6- 9. Graph of EC_PC and EC_Ins in View of δ₂

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We consider the parameter as significant to EC_Ins*EC_PC* if the difference of the average of EC_Ins*EC_PC* between two levels is greater than 100. Fig. 6-10 and Table 6-9 show that (A, Rn, IC) is the significant parameters for EC_Ins*EC_PC*.

Figure 6- 10. The Response Graph of EC_Ins*EC_PC* for (A, Rn, IC) of Production Inspection.

Table 6- 19. The Values of EC_Ins*EC_PC* for (A, R_n, IC) of Production Inspection.

(A, Rn, IC) level1 615.69 level2 941.59 difference 325.90

Figure 6-11 indicates that EC_Ins* and A has a linear relationship. In addition, when A = 1.77, then ECPC* = ECIns*. In this case, we can choose to either perform process control or product inspection. When A > 1.77, then EC_PC* is smaller than EC_Ins* and we would select to perform process control, not product inspection. However, when A < 1.77, then EC_PC* is greater than EC_Ins*. In this case, we choose to conduct a product inspection instead of process control. We conclude that when A is smaller, ECIns* is also smaller.

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Figure 6- 11. Graph of EC_PC and EC_Ins for A

Figure 6-12 indicates that the relationship between EC_Ins* and R_n is linear. When Rn = 6.93, then EC*= ECIns*. In this case, we can choose to perform either process control or product inspection. If R_n > 6.93, then EC* is smaller than EC_Ins* and we choose to perform process control, but not product inspection. However, if 0 < Rn < 6.93, then EC* is greater than EC_Ins*. In this case, we opt to conduct a product inspection rather than the process control. When Rn decreases, ECIns*-EC* also decreases.

Figure 6- 12. Graph of EC_PC and EC_Ins for R_n

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Figure 6-13 indicates that EC_Ins* and IC has a linear relationship. When IC = 0.57, then ECPC* = ECIns*. In this case, we can opt to either perform process control or product

inspection. If IC > 0.57, then EC_PC* is larger than EC_Ins*, and we choose to conduct a

product inspection, and not process control. However, if 0 < IC < 0.57, then ECPC* is smaller than EC_Ins* and we choose to perform a process control instead of a product inspection.

Figure 6- 13. Graph of ECPC and ECIns in for IC

We summarize the above discussion of the relationship of the significant parameter between the expected total cost in process control and product inspection. Here, we use the same notation introduced in Table 2-5. In addition, if the parameter is significant to optimal design parameter and they have a negative linear relationship, we use the notation “○^—”.

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Table 6- 20. The Relationship Between the Significant Parameter and ECPC* Common

parameters ECPC* Parameters

only in SPC ECPC* Parameters only

in Inspection ECPC*

OT ○⁺ C₁ - C₀ ○⁺ A ×

 ○⁺ δ2 ○⁺ Rn ×

IC ×

Table 6- 21. The Relationship Between the Significant Parameter and ECIns* Common

parameters EC_Ins* Parameters

only in SPC EC_Ins* Parameters only

in Inspection EC_Ins*

OT ○⁺ C1 - C0 × A ○⁺

 ○⁺ δ2 × Rn ○⁺

IC ○^—

Table 6- 22. The Relationship Between the Significant Parameter and EC_Ins*EC_PC * Common

parameters ECIns*-ECPC* Parameters

only in SPC ECIns*-ECPC* Parameters only

in Inspection ECIns*-ECPC*

OT ○⁺ C1 - C0 ○^— A ○⁺

 ○⁺ δ2 ○^— Rn ○⁺

IC ○^—

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CHAPTER 7.

CONCLUSION AND RECOMMENDATIONS FOR FUTURE STUDY

We have studied the economic and economic statistical design of X control charts for monitoring the mean and variance of a Weibull distributed process. The Taguchi’s quadratic loss function approach has been used to represent the cost of producing products for

in-control and out-of-control period. So far, no researcher considers the determination of economic (or economic statistical) X control chart and the specification limit

simulatneously. However, we combine these two topics together from an economic

viewpoint. Hence, this study can provide a suggestion for the producers to conduct process control and/or products inspection. An example was given to illustrate the solution procedure and application of the economic or economic statistical design of X control chart and complete inspection plan. In our data analysis we have observed the following results:

1. In the economic statistical design, higher shift of variance increases the probability of type II error.

2. Higher in-control and out-of-control quality cost results in higher expected cycle cost, expected cost or expected total cost.

3. Higher numbers of production in unit time (or lower inspection cost per unit item) increases the expected cycle cost, expected cost or expected total cost.

4. Higher time of discovering the assignable cause and repairing the process increases the expected cycle cost, expected cost or expected total cost.

5. A frequent occurrence of assignable cause leads to a higher expected cycle cost, expected cost or expected total cost.

6. The optimal lower specification limit is only relevant to the shape and scale parameters of in-control Weibull distribution.

7. Including constraints of the probability of type I and II error leads to increase in the optimal sampling interval and the width of X control charts.

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8. We have found that the inspection plan is always preferred because the reduction of expected cycle cost. This is because the coefficient of Taguchi’s quadratic loss function (k) we set is very small. However, if k becomes larger, we may prefer to only monitor the process by using economic statistical X control chart.

9. We have also explored the economic statistical designs of X control chart are always preferred to economic designs by the producer who desire the lower expected cycle cost or expected cost.

In the future, this study can be extended to design the economic or economic statistical

EWMAX chart, which is sensitive to small shifts in the mean and variance. Furthermore, the study can be extended to determine the optimal Weibull distribution for ideal process

improvement.

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