Using improved simulated anneal arithmetic to improve semiconductor packaging ball placement process capability

Abstract

This study used improved simulated annealing arithmetic to improve the semiconductor packaging ball placement process capability. The simulated annealing computing process was combined with an artificial neural network, Metropolis algorithm, and sequential Gaussian simulation. Grey relation analysis was used as the target value, which is intended to obtain the optimum parameter design of process capability. The results showed that the substrate design-Pad Open is 0.30 mm ball size, 0.275 mm Pad open, Profile type B, maximum temperature 236 (°C), solder melting time 53 (s), preheating temperature 176 (°C), and a solder ball component, as SAC405 can result in a parameter design for optimal solder ball shear strength of C_pk = 2.38 and $G_{0 i} = 0.87$ . This optimum parameter design process is better than other methods, and it reduces the cost waste of extra defectives.

Keywords

Semiconductor packaging ball placement improved simulated annealing grey relation analysis optimum parameter design process

Introduction

The purpose of packaging is to provide the finished product (package) with an interface, in order to connect the internal electrical signal via the packaging material to the system and to prevent the silicon chip from being damaged and corroded by external force, water, moisture, and chemicals. The main process of semiconductor packaging is as shown in Figure 1. This research scope is Ball Placement, and the purpose of Ball Placement is to place the Solder Ball on the substrate. The Solder Ball is the circuit connection and heat sinking conductor between substrate and mainboard. The ball placement process consists of a Ball Mount, IR-Reflow, and Cleaner. In terms of ball placement, the pad of the substrate is coated with flux, and the Solder Ball is placed on the Pad of the substrate at the high temperature (150°C–245°C) of IR-Reflow furnace, the Solder Ball reacts with flux and half of it melts on the substrate pad. The Cleaner removes the excess flux and the oxide after reaction from the substrate to avoid contaminating the substrate. In terms of the research setting range of ball placement, this study uses the reflow parameter as the research subject and uses solder ball strength process stability as the target response value. The thin and fine ball grid array (TFBGA) (Figure 2) product is validated, the solder ball shear strength (Figure 3), the fatigue life of the TFBGA can be obtained (larger-the-better, LTB), warpage (smaller-the-better), residual stresses (smaller-the-better) are the quality characteristics, and a higher solder ball shear is better, thus, it is a LTB characteristic. However, higher strength may be accompanied by IMC (intermetallic compound) layer embrittlement, which may result in another reliability problem; thus, it is not discussed in this study for the moment.^1,5–8

Figure 1.

Assembly process and research process.

Figure 2.

Solder bump structure.

Figure 3.

Solder ball shear test.

The purpose of ball placement is to lay the solder ball on the substrate. The solder ball is the conductor of electrical connection and heat dissipation between the substrate and the mainboard. The process of ball placement can be divided into three procedures: Ball Mount, IR-Reflow, and Cleaner. As for Ball Mount, it coats a layer of flux on the pad base of the substrate first, and then it mounts the solder ball on the substrate. Through the high temperature of the reflow oven (about 150°C–245°C), half of the solder ball will be fused to the pad of the substrate after it reacts with the flux. Cleaner is for removing the excess flux and the oxide after the reaction on the substrate to avoid contamination of the substrate. This study used the reflow parameter as the study subject and used the processing stability of the solder ball strength as the target response value. Taking the TFBGA product as the subject for verification, the thrust strength of the solder ball (Figure 3) was used as the quality characteristics, in which case the higher the thrust is of the solder ball, the better it is, and so it exhibits the LTB characteristic. However, when the strength rises, it may be associated with the embrittlement of the IMC layer, and it may cause another reliability problem. Thus, it is not included in the research scope of this study.^1–4

This study used the Taguchi method and multiple quality characteristics to calculate the signal-to-noise (SN) ratio and factor level combination from the original data measured after the soldering experiment and took multiple quality characteristics to work out analysis of variance (ANOVA) and contribution rate data, which are integrated into a cross-over analysis table in order to obtain four visualized optimal solutions. The SN ratio was used as an input value, the priority value of various schemes was found by the conversion mode of grey relation analysis, and a rough optimal solution was obtained. The optimum architecture of the relation of the standardized data was realized using the back-propagation neural network method. The approximately global optimum was achieved by employing a genetic algorithm and the improved simulated anneal arithmetic. Finally, the optimal solutions obtained by the aforesaid methods were validated by a confirmation experiment to find out the optimum parameter combination.^9–24

The higher the thrust of the solder ball, the better it is, and therefore, it exhibits an LTB characteristic²⁵

S N_{L T B} = - 10 \log_{10} (M S D) = - 10 \log_{10} [\frac{1}{n} \sum_{i = 1}^{n} \frac{1}{y_{i}^{2}}]

SN_LTB is SN (signal-to-noise) ratio of the LTB characteristic, MSD is mean square error of deviation from the target value, and y_i is observation data.

IMC

IMC is a chemical composition, and so the energy must be given to the formation of IMC, which is the reason why the solder paste needs to be heated during the welding process. In addition, in the composition of the solder paste, only the pure tin (Sn) has a diffusion reaction with the copper base (e.g. organic solderability preservative, I-Ag, I-Sn surface-treated board) or nickel base (Electroless nickel immersion gold surface-treated board) in the strong heat, and thus a solid interface IMC is generated.^1–4

ANOVA

It is a common statistical model in data analysis, and it mainly explores the relationship between the dependent variable of the continuous data type and the independent variable of the categorical data type.²⁵

Literature review of this study is shown in Table 1.

Table 1.

Literature review of this study.

Title	Main contents	Source
Process and decision-making method	A design optimization method based on reliability for an improved multi-target system is developed to explore the design concept of the conceptual car body under uncertainty.	Duan et al.⁵
	Dimension optimization is combined with a reliability-based design optimization method to improve the performance of the accelerometer.	Teves et al.⁶
	The Six Sigma Method is used in parametric design optimization applications in the aluminum extrusion plant.	Ketan and Nassir⁷
	A parameter design from the integrated production process for the model of the forecasted product is proposed, which takes into account the design, assembly, and process.	Boorla and Howard⁸
	A genetic algorithm based on the genetic algorithm with two different local search strategies is proposed to maximize the module density, and it uses a more general objective function, where tunable parameters can solve the resolution limit.	Mu et al.⁹
	A heuristic method based on simulated annealing (SA) for solving the open location-routing problem is proposed. The goal is to minimize the total cost, including equipment operating costs, vehicle fixed costs, and travel costs.	Yu and Lin¹¹
	A hybrid real coded genetic algorithm with particle swarm optimization and mixed artificial immune algorithm for solving constrained global optimization problems is proposed.	Wu¹²
	The “Analytic Hierarchy Process Method” and the grey relational analysis based on degree of similarity are used for empirical investigation.	Yang et al.¹³
	An enhanced resonance search algorithm is developed, which makes it possible to quickly escape the local optimal solution.	Maheri and Narimani¹⁷
	A powerful feature extraction method is designed for classifying multiple types of signals, identifying valuable features from the original display graph data, and finding the effective classifier of the feature.	Siuly and Li¹⁸
	A flexible process scheduling combinatorial optimization study of non-homogeneous parallel machines minimizes the maximum completion time or manufacturing time.	Dai et al.²³
	The Grey Decision Theory is used for management application basis.	Deng²⁶
	This book describes the decision-making methods that apply to different situations.	Chien²⁷
	It observes a large number of pictures, deals with data quality problems, uses customer background and knowledge to develop appropriate statistical models, and analyzes different management implications.	Hamada and Hamada²⁸
	A multi-objective (minimization of availability and maintenance cost) preventive maintenance scheduling model of the continuous operating system is proposed. A multi-objective genetic algorithm is used to optimize the objective function.	Adhikary et al.²⁹
Discussion on process parameters	This reference presents shear stress and shear creep strain hysteresis loops at the fillet welds, shear stress history, shear creep strain history, and creep strain density range, so as to better understand the thermodynamic behavior of the wafer level chip scale packages of the lead-free solder bumps.	Lau and Lee¹
	The fatigue life of the fine-pitch BGA package under thermal cycling is mainly analyzed.	Lee et al.²
	A detailed drop test and a simulation of the thin-profile fine-pitch BGA and the very-thin-profile fine-pitch BGA package are performed at the board level with the use of an internally developed test program.	Tee et al.³
	A double cantilever beam is configured with the test BGA and the printed circuit boards assembled with the microelectronics package.	Akbari et al.⁴
	As for the scanning probe lithography, it proposes that the anodic oxidation of the atomic force microscope should be used to define the line thickness directly on the surface of the silicon chip. It improves the optical diffraction of conventional optical lithography.	Kuo et al.¹⁰
	In consideration of the uncertain system parameters, the optimal design of the base isolation system used to control seismic vibration is usually the unconditional expected value through minimizing the mean square response of the structure, regardless of the effect due to system parameters’ uncertainty.	Roy and Chakraborty¹⁴
	Due to difficulties in the conceptual design of the compact heat exchanger, it adopts a multi-level, multi-factor, and non-structural fuzzy optimal decision model to select the compact heat exchanger.	Zhou et al.¹⁵
	An application of the harmony search–based algorithm in the optimal detail design of the special seismic moment–reinforced concrete structure under seismic load based on the American standard is introduced.	Akin and Saka¹⁶
	This reference used the global optimization framework based on real coding genetic algorithm (GA) as the common template for designing the optimum proportional-derivative controller to control four different multi-wing chaotic systems.	Das et al.²⁰
	The emulational auxiliary power model and the scalable component model are used to improve the accuracy of the design process by means of trade-off analysis and evaluation methods for multi-objective design problems.	Sarioglu et al.²¹
	In order to improve the solidification speed of soft processing technology, the silicone rubber composite mold material is designed based on multi-objective optimization (MOO), which is contradictory to the target.	Nandi et al.²²
	The reference performed statistical analysis of the realization of microelectronic electro-discharge machining process parameter measurement and acquisition system and noted its effect on the process performance. The linear and non-linear regression methods are used to obtain the most important predictive equation for the micro-electroplating electro-discharge machining process.	D’Urso et al.²⁴
	The choice of processing parameters is important for obtaining the required accuracy for good surface quality in cylindrical grinding. This literature focuses on the optimization of AISI 4140 steel’s continuous and interrupted cylindrical grinding.	Köklü³⁰
	The emulational annealing and the genetic algorithmic computing technology are integrated to find a set of optimal cutting condition values.	Zain et al.³¹
	The main objective is to minimize the annual cost of the hybrid system, the environmental costs, and the loss of power supply probability due to the use of hybrid power generation systems to avoid pollutant emissions.	Shayeghi and Hashemi³²

BGA: ball grid array.

Research method

Current condition analysis and parameter setting

The current condition data were the calculated process capability index C_pk = 1.34 ( $\geq 1.33$ acceptable) as shown in Table 2, meaning the process capability is unstable and insufficient, and there is a large room for improvement. The experiment is designed according to the ball placement process parameters of the case company, and the six most important controlled factors in the experiment are selected from the process parameters according to the field engineer’s advice, which are substrate design-Pad Open, Profile type, maximum temperature (°C), solder melting time (s), preheating temperature (°C), and solder ball composition, and the levels of the control factors are set as shown in Table 3, and the improved simulated annealing (SA) is used for experimental design.^11,13

Table 2.

Current data (solder ball strength).

634.2	496.7	431.1	559.8	623.3
646.3	492.0	464.5	478.9	524.1
634.3	703.5	426.6	582.5	539.7
⋮	⋮	⋮	⋮	⋮
559.8	665.4	539.0	506.6	588.2

Table 3.

Control factors and levels.

Standard	Level 1	Level 2	Level 3
A. Substrate design—Pad open	0.30 mm ball size, 0.275 mm Pad open	0.20 mm ball size, 0.275 mm Pad open	−
B. Profile type
C. Maximum temperature (°C)	230	235	245
D. Solder melting time (s)	40	60	120
E. Preheating temperature (°C)	140	160	180
F. Solder ball composition	SAC105	SAC305	SAC405

References: Lau and Lee,¹ Lee et al.,² Tee et al.,³ and Akbari et al.⁴

As for the temperature characteristics of the alloy only, the difference is not big, but the slurry interval of SAC105 is larger. This means that when SAC305/405 are completely in the liquid phase, SAC105 is still partially not fully liquified. Therefore, if SAC105 is used in the solder paste, then when the solder ball uses SAC305/405, it will have a higher probability of generating a void. Kim et al.¹⁹ used a sandwich joint test piece to carry out the relevant tensile test on SAC305. The experimental results show that the tensile strength of the test piece is larger than that of the block. It shows that the thickness of the medium metal phase has a considerable correlation with the anti-shear strength of the joint surface, and the thickened medium metal phase will make the breaking strength of the joint surface decrease significantly.^1–4

The solder ball shear strength is a quality characteristic, in which a higher solder ball shear is better, and so it presents a LTB characteristic. However, the high strength may be accompanied by IMC layer embrittlement. The viscoplastic strain energy density is substituted in the model proposed by Darveaux, thus obtaining the fatigue life of TFBGA (LTB).² The literature has paid increasing attention to the warpage (smaller-the-better) and residual stresses (smaller-the-better). Previous studies indicated that the first cause for the warpage of various components of an IC package is the non-uniform warpage resulting from different material coefficient of thermal expansion, or the curing shrinkage of epoxy molding compound, or the non-uniform temperature change.^3,4

According to the specific quality characteristic SN ratio’s significance of the Taguchi method and the specific quality characteristic contribution rate obtained by ANOVA, the cross-over analysis comparison table can be used for judgment, so as to obtain the optimum parameter combination of this part, as shown in Table 4. In terms of significance, according to the contribution rate after weighting, among the ABCDEF factors, the levels reaching the optimum contribution degree are A₁B₂C₂D₁E₂F₂. However, the maximum weighted ratio of solder ball shear strength is 40% in this experiment, and the Factor C solder ball shear strength test has a significant impact. Therefore, aside from considering C₂, C₁ can also be considered. The quality parameters of Factor D are insignificant, but the contribution rate in warpage is the highest, and the residual stresses rank second. Thus, D₁ and D₂ factors shall be considered as well. The four optimum combinations are hence A₁B₂C₁D₁E₂F₂, A₁B₂C₁D₂E₂F₂, A₁B₂C₂D₁E₂F₂, and A₁B₂C₂D₂E₂F₂.

Table 4.

Cross-over analysis of quality characteristics.

	A	B	C	D	E	F
Solder ball shear strength	A₁	B₂	C₂	D₃	E₂	F₁
SN ratio significance	(F = 22.36)*	(F = 9.87)*	(F = 10.21)*		(F = 5.19)*
Contribution rate after weighting (%)	23.82	10.29	9.31	0.93	4.09	1.32
The fatigue life	A₁	B₃	C₂	D₁	E₃	F₂
SN ratio significance	*					(F = 3.29)*
Contribution rate after weighting (%)	12.53	0.91	6.12	1.08	3.91	1.68
Warpage	A₁	B₂	C₁	D₁	E₁	F₁
SN ratio significance	*	*	*
Contribution rate after weighting (%)	6.29	6.26	4.32	6.28	0.87	1.06
Residual stresses	A₂	B₃	C₃	D₂	E₁	F₂
SN ratio significance
Contribution rate after weighting (%)	2.62	5.19	1.23	3.26	1.99	0.87
Optimum parameter level combination	A₁	B₂	C₁ C₂	D₁ D₂	E₂	F₂

F-value is relative efficiency of factors effect in the Taguchi method. F-value < 1: the factor effect is small; F-value > 2: the factor effect is medium; F-value > 4: the factor effect is large.^25,30

Grey relation analysis optimum parameter level combination

The optimum parameter level combination considering the $G_{0 i}$ of grey relation analysis of cost and quality aspects.¹¹

The degree of relationship among sub-systems or elements could be evaluated through grey relational analysis,^26–28 and important influential factors to the development trend are then determined to learn the major features of the system through the following steps:

Step 1. Normalize original data. Normalize by dividing the original data $z_{i} (k)$ with the mean value of the sequence shown in equation (1)

R_{i} (k) = \frac{z_{i} (k)}{\sum_{k = 1}^{N} \frac{z_{i} (k)}{N}}; i = a, . . ., d; k = A, . . ., N

(1)

Step 2. Designate the standard sequence and calculate the difference sequence. Take the mean value as a standard sequence, that is, sequence 0; the difference sequence $Δ_{0 i} (k)$ indicates the absolute difference of elements k between the other sequence i and the standard sequence 0, as expressed in equation (2)

\begin{array}{l} Δ_{0 i} (k) = | R_{0} (k) - R_{i} (k) |; \\ i = 1, 2, 3 \dots; k = A, …, N \end{array}

(2)

Step 3. Calculate maximal difference $Δ_{max}$ and minimal difference $Δ_{min}$ as expressed in equations (3) and (4)

Δ_{max} = \underset{i, k}{Max} Δ_{0 i} (k)

(3)

Δ_{min} = \underset{i, k}{Max} Δ_{0 i} (k)

(4)

Step 4. Calculate grey relational coefficient: $R_{0 i} (k)$ . The relational coefficient: $R_{0 i} (k)$ is defined below, of which w is the adjustment factor, as shown in equation (5)

R_{0 i} (k) = \frac{Δ_{min} + w Δ_{max}}{Δ_{0 i} (k) + w Δ_{max}}

(5)

Step 5. Calculate the grey relationship $G_{0 i}$ between each sequence and the standard sequence. The grey relationship $G_{0 i}$ is defined as in equation (6)

G_{0 i} = \sum_{k = A}^{N} \frac{R_{0 i} (k)}{N}

(6)

Step 6. Conduct sequencing according to the grey relationship.

Table 5 lists the optimum parameter level combination considering $G_{0 i}$ of grey relation analysis for the cost and quality aspects.

Table 5.

Grey relation analysis.

Divisor $[R_{0 i} (k)]$	A	B	C	D	E	F
Level 1	0.89	0.88	0.76	0.52	0.66	0.83
Level 2	0.63	0.66	0.61	0.69	0.72	0.76
Level 3	−	0.59	0.58	0.59	0.51	0.53
Best level	A₁	B₁	C₁	D₂	E₂	F₁

Artificial neural network setting and execution

The mode is selected as TanhAxon, the lower bound of the learning rate is set as 0.03, the upper bound is 0.2; the lower bound of the initial value of momentum is set as 0.6, the upper limit is 0.9; the number of training cycles is set as 10,000; the number of layers of the artificial neural network is set as 1–9; the performance index is mean squared error (MSE) for determining network quality, as expressed as equation (7)

MSE = \frac{{\sum_{i = 1}^{n} (S_{i} - S_{i}^{'})}^{2}}{n}

(7)

where n is the number of samples, $S_{i}$ is the actual value, and $S_{i}^{'}$ is the prediction value.

The standard criterion is that the MSE value of training and testing is lower the better in the test values; thus, the 6-6-1 artificial neural network is used as the executed network architecture according to Table 6.

Table 6.

Artificial neural network MSE.

Network model	Training	Testing
6-1-1	0.0856	0.0613
6-2-1	0.0923	0.0823
6-3-1	0.0026	0.0734
6-4-1	0.000938	0.0265
6-5-1	0.000834	0.0346
6-6-1	8.76E–09	0.0123
6-7-1	9.28E–09	0.0862
6-8-1	1.263E–08	0.0923
6-9-1	2.234E–08	0.0723

MSE: mean squared error.

Optimum parameter analysis of genetic algorithm

According to the trained neural network model, the model has six input layers, one output layer, six nodes of a hidden layer, the learning rate is 0.15, and the momentum is 0.95. The $G_{0 i}$ value is taken as the target, and the input layer is the input parameter value for gene screening. According to the genetic algorithm principle, the data have approached to convergence in Generation 2000, the minimum value of $G_{0 i}$ of 3000 generations and 4000 generations increases slightly, but it is unnecessary to waste this cost according to practice. This study lays emphasis on $G_{0 i}$ value LTB; thus, the executed standard is defined as 2000 generations, and the results are shown in Table 7.^{15–17,28,29}

Table 7.

Genetic algorithm implementation results.

Network model	$G_{0 i}$			Generation number
	Maximum	Minimum	Mean
6-6-1	0.71	0.49	0.53	500
	0.78	0.39	0.59	1000
	0.82	0.43	0.69	1500
	0.86	0.46	0.68	2000
	0.83	0.56	0.65	3000
	0.85	0.59	0.63	4000

Optimum parameter analysis of improved SA

The neural network training model is called, the $G_{0 i}$ value is set as the target value, the initial temperature is 160, the stop temperature is 1, and cooling factor $α$ is 0.96. It is obvious that the data have approached to convergence when the Markov Chain is 600; thus, the test stops, and the results are as shown in Tables 8 and 9 and Figure 4.

Table 8.

Summary analysis of SA computed performance.

Parameter	Numerical value
Initial temperature $(T_{0})$ , $S_{0}$	160, 0.42
Final temperature $(T_{i})$ , $S_{i}$	1, 0.69
Cooling factor $α$	0.96
Total number of cooling	38
Optimal strategy temperature layer	26
Optimal strategy target value	0.86

SA: simulated annealing.

Table 9.

Improved SA implementation results.

Network model	$G_{0 i}$			Markov chain
Network model	Maximum	Minimum	Mean	Markov chain
6-6-1	0.62	0.39	0.46	300
	0.66	0.38	0.52	400
	0.71	0.41	0.56	500
	0.87	0.42	0.69	600
	0.68	0.32	0.55	700
	0.59	0.31	0.46	800

SA: simulated annealing.

Figure 4.

Improved SA parameter convergence map.

The SA structure of this study consists of the Metropolis algorithm, sequential Gaussian simulation, and an annealing process, as described below.^11–16,31

The Metropolis algorithm is as follows:

Step 1. When iteration $t = 0$ , an arbitrary feasible solution $S (0) = S$ is generated. The sequential Gaussian simulation generates $S (t)$ .

Step 2. A neighborhood solution is generated according to the state $S$ of current solution $S (t)$ using an effective disturbance mechanism

S' = N [S (t)] \subset S, N [S (t)] \neq Φ (it can contain S or not contain S)

Step 3. Calculate increment $Δ E^{'} = C (S^{'}) - C (S)$ . $C (S^{'})$ as the current solution and $C (S)$ is the value of the iteration.

Step 4. If $Δ E^{'} < 0$ , the solution of the new state is better, $S^{'}$ is accepted as the new current solution.

If $Δ E \geq 0$ , the probability $EXP (- Δ E^{'} / kT)$ is used to judge whether or not to accept $S^{'}$ as new current solution.

A random number u subject to $U (0, 1)$ is generated, if $EXP (- Δ E^{'} / kT) > u$ , and $S^{'}$ is temporarily accepted as new current solution; otherwise it is not, where $k$ is the Boltzmann constant and $T$ is the current temperature.

Step 5. $t = t + 1$ and judge whether the algorithm stops according to the preset convergence criteria (stop search rule), if yes, the computational process is stopped; otherwise proceed to Step 2.

The sequential Gaussian simulation is as follows. The sequential Gaussian simulation is based on the stochastic simulation theory, where the known data points, and all the simulation values in simulated positions, are combined with the conditional probability to simulate the new location point. This simulated new value will be put in the condition data of the next simulation, and the remaining can be deduced accordingly till all unknown point values in the simulation region are simulated. A random variable $Z (x)$ of arbitrary statistical distribution is converted by normal score into a multivariate Gaussian function. The execution steps are described as follows:^{17,18,20,21,32}

Define a random path.

Decide whether to use simple kriging or ordinary kriging and normal variation functions. While the simple kriging is generally used, if the data are large and the local average varies significantly with the region, then ordinary kriging is used. The variation function is selected by cross-validation. The selected kriging method and variation function are used to determine the complementary cumulative distribution function (CCDF) parameter of $Y (x)$ on $x$ coordinate.

The CCDF simulates the $Y (x)$ value of the $m th$ point in the simulation region.

The simulated value obtained in Step 3 is added to the data.

Execute the next point simulation till all the simulated points are simulated.

Convert the simulated normal $y (x)$ value into $z (x)$ value.

Execute simulation once and discuss whether the variation functions of various simulation values conform to the theoretical variation function of the actual value and the distribution of the simulation values in the space.

The simulation procedure is as follows:

An initial numerical model is constructed, this model simulates the initial melting of an actual annealing phenomenon, and a random value is allocated to each grid. If the problem to be handled is of co-simulation, that is, the secondary variable can be used, then the initial values can be allocated according to the appropriate condition distribution, as obtained from validation.

The objective function is defined as the measurement of the gap between the wanted spatial characteristics and the implementation result characteristics, for example, the gap between the variation function of the implementation result and the theoretical variation function.

The image is corrected by giving a random location a new value.

The objective function value decreases, and the correction is always accepted. If the objective function value increases, the correction is accepted with a specific probability.

When the probability is accepted with an inappropriate exchange, and the value decreases, the disturbance process continues until a low objective function state is far.

The annealing process is as follows:

Step 1. Select an initial solution $S_{0}$ as the initial current solution, and let $S (0) = S_{0}$ , set the initial temperature $T_{0}$ and iteration $i = 0$ .

Step 2. Let $T = T_{i}$ , and use $T$ and $S_{i}$ to call the Metropolis algorithm, in order to judge whether current solution $S$ is changed to $S_{i}$ .

Step 3. Reduce temperature $T$ according to a specific model, that is, $T_{i} + 1 < T, i = i + 1$ .

Step 4. Check whether the annealing process is finished. If yes, the current solution $S_{i}$ is used as the final solution; otherwise proceed to Step 2.

Validation experiment

According to Table 8, the grey relation analysis, artificial neural network, and the improved SA make up the optimum combination of the validation experiment. Its $G_{0 i}$ is 0.87, better than other combinations, and the $G_{0 i}$ value is 0.87 in the confidence interval, proving that the confirmation experiment results match the experimental data.

A total of 10 confirmatory experiments were conducted in this study. The 95% confidence interval was verified before the $G_{0 i}$ confirmatory experiment. The 95% confidence interval value for this experiment is [0.568, 0.893]. Through the L₁₈ orthogonal array, a higher SN ratio represents better quality (smaller loss), and so the highest SN ratio is the best combination of the parameter level. The variation of the product produced at this parameter level is also at a minimal. The factor effect graph shows the best significant level. From the SN ratio data of the experiment, we can find that the best level of factor combination should be A₁B₂C₂D₂E₃F₃. The confirmatory experiment can verify whether or not the average of the predicted optimal conditions is valid, as shown in Table 10. Therefore, the confidence interval of the expected SN ratio of the confirmatory experiment is [56.28 ± 3.23] = [53.05, 59.51]. Through the above confirmatory experiments, the SN ratio falls within the 95% confidence interval, indicating that the experimental results are successful.

Table 10.

Experimental response value and SN ratio of the L₁₈ orthogonal array.

Experiment number	Impact factor						Experimental response value (solder ball strength)										SN	$G_{0 i}$
	A	B	C	D	E	F
1	1	1	1	1	1	1	496.4	608.2	526.1	546.9	599.8	512.9	612.8	536.2	510.9	520.8	54.69	0.775
2	1	1	2	2	2	2	600.2	555.6	533.2	613.9	570.2	560.3	531.2	610.2	570.6	555.6	55.09	0.832
3	1	1	3	3	3	3	586.6	459.5	622.6	512.1	499.1	560.3	524.6	614.6	498.9	583.2	54.63	0.713
4	1	2	1	1	2	2	536.9	469.3	561.4	610.1	540.8	498.6	557.3	617.2	497.3	575.6	54.66	0.768
5	1	2	2	2	3	3	650.8	660.2	660.2	653.2	651.9	631.6	660.6	655.8	650.9	652.4	56.29	0.873
6	1	2	3	3	1	1	585.5	458.4	621.5	511.2	460.2	561.1	522.4	618.4	512.3	500.9	54.43	0.732
7	1	1	1	2	1	3	551.3	512.3	505.4	611.6	499.9	619.8	510.9	493.8	549.2	512.6	54.52	0.742
8	1	1	2	3	2	1	580.2	511.3	610.1	500.6	589.2	509.4	614.8	498.2	512.3	598.3	54.75	0.779
9	1	1	3	1	3	2	490.6	530.8	620.3	540.2	508.1	555.8	512.9	530.1	508.1	520.1	54.46	0.722
10	2	2	1	3	3	2	495.4	536.6	528	542.8	510.7	499.9	501.9	538	610.7	526.3	54.43	0.721
11	2	2	2	1	1	3	608.4	480.1	580.1	501.2	530.9	496.3	522.8	501.3	498.6	581.8	54.41	0.732
12	2	2	3	2	2	1	570.8	560.3	537.8	616.8	577.6	584.3	538.4	499.8	573.5	560.4	54.96	0.799
13	2	1	1	2	3	1	533.2	599.6	580.3	620.2	530.1	518.3	538.1	488.2	506.2	577.2	54.72	0.746
14	2	1	2	3	1	2	512.3	464	559.8	534.5	513.9	508.6	556.8	609.2	514.9	515.6	54.41	0.582
15	2	1	3	1	2	3	468.9	529.2	510.6	500.8	466.1	480.2	510.8	600.9	478.3	482.1	53.96	0.591
16	2	2	1	3	2	3	532.6	518.4	646.3	529.1	537.8	616.9	639.5	530.8	577.8	526.3	54.96	0.812
17	2	2	2	1	3	1	549.6	483.2	588.2	493.5	533.7	612	540.6	500.1	496.3	580.9	54.53	0.749
18	2	2	3	2	1	2	501.6	532.6	519.3	503.9	499.9	511.6	616.9	502.8	499.8	501.3	54.26	0.659

SN: signal to noise.

Results and discussion

According to Table 11, this study is the optimum combination of a validation experiment, the $G_{0 i}$ value is 0.87, $C_{pk} = 2.38$ , which is better than other combinations, and the confidence interval proves that the confirmation experiment results match the experimental data.

Table 11.

Validation experiment judgment value.

Optimum experimental combination	A	B	C	D	E	F	$G_{0 i}$	$C_{pk}$
Taguchi	A₁	B₂	C₂	D₂	E₂	F₃	0.41	1.39
Cross-over analysis table for quality characteristics (1)	A₁	B₂	C₁	D₁	E₂	F₂	0.68	2
Cross-over analysis table for quality characteristics (2)	A₁	B₂	C₁	D₂	E₂	F₂	0.62	1.98
Cross-over analysis table for quality characteristics (3)	A₁	B₂	C₂	D₁	E₂	F₂	0.69	2.03
Cross-over analysis table for quality characteristics (4)	A₁	B₂	C₂	D₂	E₂	F₂	0.61	1.97
Grey relation analysis	A₁	B₁	C₁	D₂	E₂	F₁	0.51	1.48
Integration of neural network and genetic algorithm	A₁	B₂	238(°C)	56 (s)	178 (°C)	F₃	0.86	2.32
Integration of neural network and SA	A₂	B₂	241 (°C)	59 (s)	179 (°C)	F₃	0.69	2.03
Integration of neural network and improved SA	A₁	B₂	236 (°C)	53 (s)	176 (°C)	F₃	0.87	2.38

SA: simulated annealing.

Conclusion and future research

According to the case analysis in this study, substrate design-Pad Open as 0.30 mm ball diameter, 0.275 mm Pad open, Profile type B, maximum temperature 236 (°C), solder melting time 53 (s), preheating temperature 176 (°C), and solder ball component SAC405 can result in the parameter design of the optimal solder ball shear strength of C_pk = 2.38 and $G_{0 i} = 0.87$ , thus reducing the waste costs of extra defectives and improving the process. In the future, as for multiple processes before and after Ball Placement, with an overall consideration of more of its parameters, the decision-making method in this study can be used to determine the multiple parameters of its multiple processes.

Footnotes

Handling Editor: Peter Nielsen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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