## Abstract

The reliability of electronic assemblies is a vital criterion used to assure product quality over its lifetime. Weibull distribution is the most common distribution utilized to describe the reliability data. Most of the studies use the Weibull scale parameter, or characteristic life, to compare alternatives and make a selective decision. This may not lead to achieving the optimal parameters which can be problematic because this method doesn’t consider the variability behavior of the fatigue life. In this study, a new approach for process parameters selection is proposed to find the optimal parameter values that improve the micro-optimality selection process based on reliability data. In this study, a new approach is proposed based on examining the solder joint reliability by using a multi criteria analysis. The fuzzy logic is utilized as a tool to solve the multi criteria problem that is presented from the proposed approach. The reliability of microelectronic connections in thermal cycling operating conditions is used as a validation case study. In the validation case study, the optimal process parameters are found for ball grid array electronic components. Two levels of the solder sphere materials, three levels of the surface finish, and 10 levels of solder paste alloys are studied as process parameters. Using the proposed approach, four quality responses are employed to assess the reliability data, including the scale parameter, the B10 (life at 10% of the population failure), mean-standard deviation response, and the signal to noise ratio (SNR). The fuzzy logic is applied to solve the multiresponse problem. An optimal process parameter setting that considers different quality characteristics was found for the validation case study. ENIG surface finish, SAC305 solder sphere, and material six were the optimal factor levels that are obtained for the aged CABGA208 component using the proposed approach.

## Introduction

Assessing the reliability of a product is one of the methods used to maintain the product quality within an acceptable region along its lifetime. Different distributions are used to describe the reliability performance of products such as Weibull, exponential, and log-normal. Two-parameter Weibull distribution, with scale and shape parameters, is the most common statistical distribution used to fit reliability data [1, 2]. Using the scale parameter, called characteristic life, only to compare alternatives may cause improper selection of the optimal choice. The scale parameter-based selection neither considers the data variability nor the chance of earlier failure. For example, Fig. 1 represents the Weibull reliability distributions for three products (A, B, and C). If the selection is based on the characteristic life, product C is more reliable than products A and B. On the other hand, product A has a lower probability of earlier failure compared with products A and C. Therefore, using different responses that consider these criteria is an efficient way to find the global optimal operation factor levels. To solve this problem, a suitable multiresponse technique should be utilized. Different techniques were implemented to optimize the process performance for multiresponse problems, for example, Fuzzy logic, artificial neural networks (ANN), grey fuzzy, fuzzy regression utility, data envelopment analysis, fuzzy regression, principal component analysis, and goal programming [3–11].

The fuzzy logic is one of the artificial intelligence tools that are implemented to analyze stochastic data. Different studies utilized the fuzzy logic in analyzing the fatigue behavior for different products. Those studies assume that the reliability of different components depends on random (stochastic) events [12]. For example, Zafiropoulos and Dialynas developed a new methodology of predicting the reliability and the failure model effects and criticality analysis by employing the fuzzy logic. The switched-mode power supply was used as a validation study in their approach [13]. Jarrah et al. explored the fatigue behavior of unidirectional glass fiber/epoxy composites by applying neuro-fuzzy modeling. Using this method, the relationship between input and output variables of the fatigue behavior was constructed. The performance response was fatigue life, and the process parameters were fiber orientation, maximum stress, and stress ratio [14]. Chen et al. explored the fatigue behavior of SAC 305 and 63Sn-37Pb solder joint under a rapid thermal cycling test. In their study, the fuzzy logic was used as a controller of the cooling and heating process where fatigue life was determined under controlled conditions [15]. Chien et al. utilized the Taguchi method and fuzzy logic to optimize the moisture gain weight and the adhesion strength in IC packages. The process parameters were the operating temperature, relative humidity, soaking time, and solder mask thickness [16]. In this study, the Mamdani fuzzy system is utilized to solve the multiresponse problem. The resultant is a single response problem represented by the comprehensive output measure (COM) response.

The case study used to validate this approach illustrates the reliability of the aged solder joints at thermal cycling operating conditions. Using the solder joints as an interconnection material leads to an increase in the number of interconnection points between the electronic parts, which in turn increase the electronic assembly’s performance compared with the traditional (leads) connections [17–20]. Several studies investigated the reliability of different solder joint materials at different aging conditions. Ma et al. explored the effect of isothermal aging on lead-free solder alloys’ mechanical behavior. A better creep response was observed for the leaded solder alloys compared with SAC alloys at elevated temperatures [21]. Lau investigated the thermal, vibrational, and mechanical behavior of the plastic ball grid array and flip chip. The thermal response was evaluated using the fracture mechanics methods, nonlinear finite element, and the Coffin-Manson model [22]. Darveaux and Banerji developed a method for predicting the fatigue life of the flip-chip bonds under different thermal cycling conditions by utilizing the finite element simulation and Coffin-Manson model [23].

The main aim of the study is providing a systematic methodology for optimizing the process parameters of the microelectronic interconnection materials under high stochastic fatigue life behavior. In this study, the solder joint reliability was examined by using the proposed approach. Instead of using one or two parameters to demonstrate the solder joint reliability, the proposed approach generated four different criteria for reliability assessment. The fuzzy logic was used as an effective algorithm to analyze the outcomes from the proposed approach. The suggested approach is valid to examine the reliability of any application that its life data follows a Weibull distribution. A validation case study of CABGA208 component aged for 12 mo was utilized to examine the proposed approach. The life data of the solder joints for different solder paste alloys, solder sphere alloys, and surface finishes at a thermal cycling condition are analyzed using four different quality responses. The fuzzy logic is then utilized to convert all responses into a single response (COM) value. The optimal factor levels are found based on the highest COM values.

Fig. 2 shows the flow chart for the proposed fuzzy methodology to assess the reliability of the solder joints by using the multi criteria-based decision. The new methodology provides a higher fatigue life, acceptable variability, and a low earlier failure opportunity compared with the use of the characteristic life alone for comparison.

## Experimental Setup

The test vehicle for the validation study is shown in Fig. 3. FR-4 glass epoxy was used to fabricate the printed circuit board (PCB) with a nonsolder mask defined copper pads.

The test vehicle contains different electronic components (QFNs, SMRs, and CABGAs). The reliability CABGA208 at 12 mo of aging time and 125°C aging temperature was investigated by implementing the thermal cycling test. The main purpose of the used aged component (CABGA208) is to examine solder joint life behavior at harsh environmental condition. The thermal cycling test was performed using a highly controlled thermal chamber (Fig. 4). The thermal cycling temperature was between 125 and −40°C. The detailed thermal profile is shown in Fig. 5.The dwell time was 15 min, the ramp time was 50 min, and the total cycle time was 115 min. The test vehicles were cycled for 5,000 cycles and any component that didn’t fail was considered as a censored data point. Ten solder paste materials were studied with three different surface finishes for SAC105 and SAC305 solder spheres. All testing boards were aged using the same aging parameters. Five samples were utilized at each experimental combination. Table I shows the solder pastes, solder spheres, and the surface finishes that were used in this test. MINTAB 16 software was used to construct the test matrix (orthogonal array) for the current experimental conditions. The L_{60} experimental orthogonal array that was used in this test is shown in Table II. The failure in the component was identified based on the change in the resistance where a notable change in the resistance value was obtained from the electrical open circuit due to failure in the interconnection parts (solder joint).

## Results and Analysis

Two-parameter Weibull analysis was performed to illustrate the fatigue behavior of the solder joints under thermal cycling conditions for each experimental combination. The proposed approach worked under the assumption that the solder joint life follows a Weibull distribution at all studied conditions. Fig. 6 represents the Weibull probability plots for SAC105 and SAC305 solder spheres at different surface finishes for material 1 solder paste. The maximum likelihood method was used to estimate the Weibull distribution parameters. Eq. (1) represents the Weibull distribution function, where θ is the scale parameter, β is the shape parameter, and *t* is the number of cycles. The Weibull distribution was applied at each experimental combination, and their scale and shape parameters were determined. B10 was computed as well at each combination by using eq. (2), which describes the earlier failure opportunity at a 10% probability of surviving. The characteristic life and B10 were considered as two different responses [24, 25].

Another way to analyze the fatigue data are by calculating the mathematical average of the fatigue life and its standard deviation at each experimental combination. To combine the mean and standard deviation in one response, that is called the mean-SD response. First, the averages and the standard deviations were normalized to keep the contribution of each one equal by using eqs. (3) and (4), where *M _{i}* and

*SD*are the fatigue life average and standard deviation at experiment

_{i}*i, M*max and

*SD*max are the maximum average and standard deviation among all combinations, and

*M min*and

*SD*min are the minimum average and standard deviation at all experimental conditions.

The Mean-SD response was determined by subtracting the normal value of the average from the normal value of standard deviation as shown in eq. (5). This is because the increase in the average means better performance and the decrease in the standard divination leads to increase the consistency in the fatigue life date [26].

The last type of analysis discussed in this study is the Signal to Noise Ratio (SNR). SNR is commonly used in the Taguchi method for optimizing process performance when the process has one quality response, which is used for optimizing the mean and variance at the same time. To compute the SNR, the class of the quality response should be identified. There are three mean classes for the quality response, which are larger-the-better (LTB), nominal-the-best (NTB), and smaller-the-better (STB). The fatigue life always follows the LTB class. The SNR is determined at each experiment by using eq. (6), where *K* is the number of replicates at each experiment, *SNR (i)* is the SNR at experiment *i*, *y** _{ik}* is the fatigue life of the solder joints at replicate

*k*and experiment

*i*[27]. The optimal operating process parameters were found by computing the average of SNR at each factor level, as shown in Table III. Table III represents the averages of the COM values for the validation study. The optimal factor levels based on the SNR were identified by selecting the factor levels with the largest value of the SNR average from Table III (bold values). The optimal operating factor levels were level 6 for factor 1, level 2 for factor 2, and level 2 for factor 3.

After computing the four quality responses (characteristic life, B10, mean-SD, and SNR), as shown in Table IV, it will become a multiresponse problem, and finding the optimal experimental combination will be more complicated. The optimal factor levels of the characteristic life, B10, and the mean-SD responses were determined by choosing the factor levels with the highest values of those responses. The shape parameter is the key value to identify the applied phase in the bathtub curve. If the shape parameter is less than one, the solder life behavior is in the infant mortality period. In addition, the solder life follows a constant failure rate phase in the bathtub curve when the shape parameter is equal to one. Finally, the wear out phase represents the solder life behavior when the shape parameter value is more than one. In order to reflect the results from the reliability analysis using a two-parameter Weibull distribution to bathtub curve, the obtained shape parameter at each experimental condition was analyzed. The shape parameters values at all conditions were more than one, which means the solder life behavior is represented by a wear out phase in the bathtub curve at any studied conditions. Thus, the solder joint reliability behavior at the optimal factor levels follows the wear out phase in the bathtub curve. Table V shows the optimal factor levels for all quality responses. Four different process parameters settings were obtained from four different responses. Therefore, it is vital to find a robust method to determine the global optimal factor levels.

The Mamdani fuzzy inference method was employed to find the COM to convert the multiresponse problem into a single response problem. The fuzzy system includes different functions, which are fuzzification (defining the inputs membership functions [MFs]), rule evaluation, output membership functions, and defuzzification method. The detailed fuzzy structure is presented in Fig. 7 [28–30].

The four responses obtained were used as inputs for the fuzzy system, as shown in Fig. 8. The first step in the fuzzy logic is the fuzzification of the input by identifying two MFs for each response (Low, High). Fig. 9 represents the MFs for the scale parameter. The fuzzy system rules were then established based on the number of fuzzy inputs, as shown in Table VI. The output MFs were defined based on the rule aggregation, as shown in Fig. 10. Five MFs (Lowest, Low, Mid, High, and Highest) were utilized in the fuzzy system. The last step is the defuzzification method, which is used to convert the fuzzy values into COM values. The center of gravity (COG) was used in this study as a defuzzification method. Fig. 11 displays an example of how the COM value was calculated from three output MFs. The COM values were determined by calculating the average of the values that were achieved from the COG method for the 16 rules. This was performed using a MATLAB (version R2018) code. Table VII shows the COM value at each experimental combination. Then the average of the COM values at each factor level was determined, as shown in Table VIII. The optimal factor levels were chosen by picking the level that has the highest average COM value for each factor (bold values), which are level 6 for factor 1 (Material 6), level 2 for factor 2 (ENIG surface finish), and level 2 for factor 3 (SAC305 solder sphere). In addition, the differences between the COM value between material 6 and 7 of solder paste material are not significant. Therefore, the solder paste material 7 can be considered as an alternative optimal factor level of the solder paste material.

## Conclusions

In this study, a new methodology to analyze the reliability data is proposed. This methodology is designed to find the best operating factor levels to obtain the global optimal experimental operating settings. Dealing with a multiresponse problem by extracting multiple outputs from the reliability data are the basis for this approach. To implement this methodology, a sequence of steps should be followed. The first step is fitting the reliability data to a Weibull distribution. From Weibull distribution, the characteristic life and B10 can be obtained, which are the first two responses. The third response is the results of subtracting the normal value of the average and the normal value of the standard deviation. The SNR for fatigue life is the last response, which is determined at each experimental combination. Different optimal factor levels may be obtained for each response. The Mamdani fuzzy inference method is implemented to transform those responses into a single response to define the optimal global solution in the multiresponse problem, which is called COM output. Calculating the average COM values at each factor level is the last step in this approach. The optimal factor levels are identified by determining the level with the largest COM value at each factor. The reliability of the aged solder joints under thermal cycling conditions is considered as a validation case study. Three experimental factors are studied: solder paste materials with 10 levels, surface finish with three levels, and solder sphere alloys with two levels. The reliability data are obtained, and the values of the four responses are calculated. The fuzzy logic is applied to achieve the COM value at each factor level. Finally, the optimal factor levels are found for this study are material six solder paste, ENIG surface finish, and SAC305 solder sphere.

## References

*Journal of the Brazilian Society of Mechanical Sciences and Engineering*

*Energy*

*Measurement*

*Silicon*

*Applied Mathematical Modelling*

*International Journal of Management Science and Engineering Management*

*Journal of Ambient Intelligence and Humanized Computing*

*Mechanical Systems and Signal Processing*

*International Journal of Quality & Reliability Management*

*Journal of Composite Materials*

*Soldering & Surface Mount Technology*

*Microelectronics and Reliability*

*Materials for Advanced Packaging*(Vol. 181)

*IEEE Transactions on Components, Packaging, and Manufacturing Technology*

*Journal of Microelectronics and Electronic Packaging*

*Microelectronics and Reliability*

*Mean-Variance-VaR based Portfolio Optimization*

*Proceedings of the Institution of Mechanical Engineers. Part E, Journal of Process Mechanical Engineering*

*Optics and Lasers in Engineering*

*Fuzzy Control and Identification*

*Fuzzy System Design Principles*