The quality characteristics in the wafer fabrication process are diverse, variable, and fuzzy in nature. How to effectively deal with multiresponse quality problems in the wafer fabrication process is a challenging task. In this study, the fuzzy technique for order preference by similarity to an ideal solution (TOPSIS), one of the fuzzy multiattribute decision-analysis (MADA) methods, is proposed to investigate the fuzzy multiresponse quality problem in integrated-circuit (IC) wafer fabrication process. The fuzzy TOPSIS is one of the effective fuzzy MADA methods for dealing with decision-making problems under uncertain environments. First, a fuzzy TOPSIS methodology is developed by considering the ambiguity between quality characteristics. Then, a detailed procedure for the developed fuzzy TOPSIS approach is presented to show how the fuzzy wafer fabrication quality problems can be solved. Real-world data is collected from an IC semiconductor company and the developed fuzzy TOPSIS approach is applied to find an optimal combination of parameters. Results of this study show that the developed approach provides a satisfactory solution to the wafer fabrication multiresponse problem. This developed approach can be also applied to other industries for investigating multiple quality characteristics problems.
Wafer fabrication process plays an important role in the quality of integrated-circuit (IC) semiconductor components. The quality characteristics of IC semiconductor components are diverse, variable, and fuzzy in nature. The design and development of wafer fabrication process should take many aspects into consideration as it takes so much time and cost to solve the quality problem. How to effectively deal with the multiresponse quality problems in the wafer fabrication process is important and challenging.
Taguchi method, design of experiments, or response surface method have been helpful and prevailing in the industry for finding good combinations of parameters for manufacturing processes [
Several analytical methods have been proposed for dealing with the multiresponse quality problem. In the early years, the weighted method, regression analysis, desirability function, or loss function were used to simplify the multiresponse problem as a single-response problem with limited results [
Multiattribute decision analysis (MADA) methods deal with decision problems with known alternatives and have been applied in assessment and evaluation problems, such as assessment of design quality [
There are several methods that can be used for solving MADA problems, such as simple additive weight, technique for order preference by similarity to ideal solution (TOPSIS), elimination et choice translating reality (ELECTRE), analytic hierarchy process (AHP), weighted product, median ranking, linear assignment, and hierarchical additive weighting [
The objectives of this study are threefold. First, a fuzzy TOPSIS method with fuzzy entropy weight measure is developed for the fuzzy multiresponse quality problem. Second, Taguchi method is combined with the developed fuzzy TOPSIS method to select satisfactory combinations of parameters for wafer fabrication process. Finally, the developed approach is implemented to show the benefits of improving the quality problem in wafer fabrication process.
The method of technique for order preference by similarity to an ideal solution (TOPSIS) was proposed by [
The fuzzy TOPSIS method is one of the fuzzy MADA methods in which the fuzzy set theory is used to modify the TOPSIS method for decision problems under fuzzy environments. Fuzzy set theory is applied to provide fuzzy number for describing any fuzzy subset of responses and preferences with suitable membership functions. Then, the developed fuzzy TOPSIS method is applied to rank all possible alternatives by the criteria of relative closeness to positive and negative ideal solutions. The following is the detailed procedure for the developed fuzzy TOPSIS method.
Construction of fuzzy decision matrix is the first step for solving fuzzy multiattribute decision problems. Every alternative is evaluated under each criterion to obtain its performance measure. Suppose there are
In order to be comparable among each criterion, the collected data under different criteria should be normalized into the scale between 0 and 1. At least two ways of normalization can be used in this purpose, that is, vector normalization and linear normalization [
The entropy method provides a subjective way to estimate weight measure for each criterion [
The intuitive meaning of (
Since each criterion can have different preferences, the obtained fuzzy weight measures are used to reflect such facts by multiplying into the normalized decision matrix, as shown in (
One metric used in the fuzzy TOPSIS method to assess alternatives is based on the measure of distance to positive and negative ideal solutions, respectively. The determination of positive and negative ideal solutions may consider three situations regarding the characteristics of quality responses, that is, the larger the better, the smaller the better, and the nominal the better. To determine the fuzzy positive ideal solution (
where
Similarly, to determine the fuzzy negative ideal solution (
Let
Let
The obtained fuzzy closeness coefficients are averaged and used to rank all possible alternatives. The larger the mode of
In this study, one of the wafer fabrication processes, the deposition process, is used as a case study for showing the implementation of the developed fuzzy TOPSIS method. During the deposition process, three quality responses place significant effects on the quality of wafers, that is, defect number on wafer surface, density of wafer, and deposition rate. The defect number is the type of the-smaller-the-better response, the density of the-larger-the-better response, and the deposition rate of the-nominal-the-better response.
During the experiment, six control factors are selected, that is, deposition temperature, deposition pressure, nitrogen flow, silane flow, setup time, and cleaning method. Each control factor has three levels. By the Taguchi experimental design method, the
Number | (A) Temperature | (B) Pressure | (C) Nitrogen flow | (D) Silane flow | (E) Setup time | (F) Cleaning method |
---|---|---|---|---|---|---|
1 |
|
|
|
|
|
None |
2 |
|
|
|
|
|
CM2 |
3 |
|
|
|
|
|
CM3 |
4 |
|
|
|
|
|
CM3 |
5 |
|
|
|
|
|
None |
6 |
|
|
|
|
|
CM2 |
7 |
|
|
|
|
|
CM3 |
8 |
|
|
|
|
|
None |
9 |
|
|
|
|
|
CM2 |
10 |
|
|
|
|
|
None |
11 |
|
|
|
|
|
CM2 |
12 |
|
|
|
|
|
CM3 |
13 |
|
|
|
|
|
CM2 |
14 |
|
|
|
|
|
CM3 |
15 |
|
|
|
|
|
None |
16 |
|
|
|
|
|
CM2 |
17 |
|
|
|
|
|
CM3 |
18 |
|
|
|
|
|
None |
For each combination of experiment, three wafers were randomly selected from one batch of 50 wafers and were inspected in terms of defect number, density, and deposition rate on three positions of each wafer, that is, top, central, and bottom. Experimental data were collected and summarized to provide a fuzzy decision matrix. The obtained fuzzy decision matrix is normalized. The fuzzy entropy method is then applied to estimate fuzzy weights for each quality response, provided in Table
Fuzzy weights for each quality response.
Response | Fuzzy weight |
---|---|
Defect number |
|
Density |
|
Deposition rate |
|
Weighted fuzzy decision matrix.
Number | Defect number ( |
Density ( |
Deposition rate ( |
---|---|---|---|
1 | (0.0000, 0.0000, 0.0002) | (0.0147, 0.0167, 0.0167) | (0.0156, 0.0179, 0.0180) |
2 | (0.0004, 0.0004, 0.0138) | (0.0395, 0.0449, 0.0446) | (0.0395, 0.0453, 0.0451) |
3 | (0.0057, 0.0188, 0.0304) | (0.0448, 0.0506, 0.0511) | (0.0446, 0.0509, 0.0515) |
4 | (0.0017, 0.0018, 0.0034) | (0.0160, 0.0182, 0.0180) | (0.0390, 0.0447, 0.0444) |
5 | (0.1111, 0.1409, 0.1898) | (0.0346, 0.0391, 0.0386) | (0.0793, 0.0903, 0.0895) |
6 | (0.0385, 0.0741, 0.2274) | (0.0217, 0.0247, 0.0247) | (0.0533, 0.0611, 0.0613) |
7 | (0.0718, 0.0737, 0.1697) | (0.0254, 0.0288, 0.0287) | (0.0827, 0.0947, 0.0945) |
8 | (0.1361, 0.2238, 0.4772) | (0.0329, 0.0386, 0.0398) | (0.1100, 0.1300, 0.1345) |
9 | (0.1841, 0.2147, 0.4938) | (0.0298, 0.0338, 0.0333) | (0.1249, 0.1425, 0.1408) |
10 | (0.0000, 0.0000, 0.0003) | (0.0258, 0.0292, 0.0289) | (0.0269, 0.0307, 0.0304) |
11 | (0.0000, 0.0001, 0.0003) | (0.0191, 0.0217, 0.0215) | (0.0217, 0.0247, 0.0246) |
12 | (0.0045, 0.0219, 0.0733) | (0.0435, 0.0496, 0.0488) | (0.0422, 0.0484, 0.0478) |
13 | (0.0020, 0.0121, 0.0465) | (0.0204, 0.0233, 0.0231) | (0.0573, 0.0658, 0.0654) |
14 | (0.0019, 0.0070, 0.0091) | (0.0216, 0.0244, 0.0241) | (0.0496, 0.0565, 0.0560) |
15 | (0.1593, 0.1747, 0.2186) | (0.0241, 0.0272, 0.0273) | (0.0593, 0.0675, 0.0678) |
16 | (0.0003, 0.0022, 0.0033) | (0.0235, 0.0266, 0.0262) | (0.0834, 0.0952, 0.0939) |
17 | (0.1615, 0.1836, 0.5477) | (0.0298, 0.0346, 0.0358) | (0.1105, 0.1294, 0.1343) |
18 | (0.2532, 0.2908, 0.8081) | (0.0272, 0.0308, 0.0303) | (0.0993, 0.1132, 0.1119) |
The fuzzy positive and negative ideal solutions for each quality response are computed and shown in Table
Positive and negative ideal solutions for each quality response.
Quality response | Positive ideal solution ( |
Negative ideal solution ( |
---|---|---|
Deposition rate |
|
|
Defect number |
|
|
Density |
|
|
|
Closeness coefficients for each combination of control factors.
Number | A | B | C | D | E | F |
|
m( |
---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 1 | (0.6972, 0.6987, 0.8673) | 0.6987 |
2 | 1 | 2 | 2 | 2 | 2 | 2 | (0.7466, 0.7481, 0.8905) | 0.7481 |
3 | 1 | 3 | 3 | 3 | 3 | 3 | (0.7517, 0.7413, 0.8896) | 0.7413 |
4 | 2 | 1 | 1 | 2 | 2 | 3 | (0.7446, 0.7465, 0.8923) | 0.7465 |
5 | 2 | 2 | 2 | 3 | 3 | 1 | (0.5647, 0.5258, 0.7600) | 0.5258 |
6 | 2 | 3 | 3 | 1 | 1 | 2 | (0.7280, 0.6673, 0.7075) | 0.6673 |
7 | 3 | 1 | 2 | 1 | 3 | 3 | (0.6993, 0.7242, 0.7854) | 0.7242 |
8 | 3 | 2 | 3 | 2 | 1 | 1 | (0.5241, 0.3691, 0.4239) | 0.3691 |
9 | 3 | 3 | 1 | 3 | 2 | 2 | (0.4140, 0.4063, 0.4063) | 0.4063 |
10 | 1 | 1 | 3 | 3 | 2 | 1 | (0.7213, 0.7227, 0.8798) | 0.7227 |
11 | 1 | 2 | 1 | 1 | 3 | 2 | (0.7099, 0.7112, 0.8739) | 0.7112 |
12 | 1 | 3 | 2 | 2 | 1 | 3 | (0.7478, 0.7334, 0.8600) | 0.7334 |
13 | 2 | 1 | 2 | 3 | 1 | 2 | (0.7893, 0.7839, 0.8958) | 0.7839 |
14 | 2 | 2 | 3 | 1 | 2 | 3 | (0.7707, 0.7682, 0.9035) | 0.7682 |
15 | 2 | 3 | 1 | 2 | 3 | 1 | (0.3764, 0.3999, 0.7198) | 0.3999 |
16 | 3 | 1 | 3 | 2 | 3 | 2 | (0.8629, 0.8629, 0.9449) | 0.8629 |
17 | 3 | 2 | 1 | 3 | 1 | 3 | (0.4501, 0.4581, 0.3428) | 0.4581 |
18 | 3 | 3 | 2 | 1 | 2 | 1 | (0.2495, 0.2479, 0.1063) | 0.2479 |
Averaged closeness value over each level of control factors.
Level | A | B | C | D | E | F |
---|---|---|---|---|---|---|
1 |
|
|
0.5701 | 0.6363 | 0.6184 | 0.4940 |
2 | 0.6486 | 0.5967 | 0.6272 |
|
0.6066 |
|
3 | 0.5114 | 0.5327 |
|
0.6063 |
|
0.6953 |
| ||||||
Best combination |
|
The quality problems arising in the IC semiconductor components are diverse, variable, and fuzzy. The design of wafer fabrication process should consider many aspects of the quality problem because it takes so much time and cost to solve the quality problem. A fuzzy TOPSIS method with fuzzy entropy weight measure is developed for the fuzzy multiresponse quality problem. Since during the deposition process in wafer fabrication, some undesired defect number on the wafer surface might occur quite often and the variability of the density within wafers and in-between wafers might be large, it is essential to investigate those causes and effects to improve the quality problem. In this study, the deposition process in the wafer fabrication is used as a case study to demonstrate the merit of the developed fuzzy TOPSIS method. Results of this study indicate that the developed approach provides a satisfactory combination of control factors for the wafer fabrication process.
There are several opportunities that can apply the developed fuzzy TOPSIS approach for future study. One can find that the demand for electronic product with higher quality is increasing and the environment under the various quality measurements is becoming uncertain and fuzzy. The developed fuzzy TOPSIS method can be applied for those manufacturing processes to solving multiresponse quality problems under uncertain environments. Also, one can investigate the possible benefits for applying different fuzzy MADA methods for solving the multiresponse quality problems.