Effect of Stress Interaction on Multi-Stress Accelerated Life Test Plan: Assessment Based on Particle Swarm Optimization

Abstract

Sustainability of products that seek to maintain ecosystem balance, such as electric vehicles or solar system inverters, often require extensive testing during their developmental stages in a manner that minimizes wastage and drives creativity. Multi-stress accelerated test planning is often used for these products, their subsystems and components if their in-service failures are driven by multiple stress factors. Multi-stress accelerated life testing (ALT) often expedites time to failure for highly reliable products. Many studies assume model parameters that may not be appropriate for the considered stress factors. Most importantly, the effect stress interaction has on the ALT plan is often ignored, especially for cases where historical data are lacking. To address this gap, in this work, a technique based on a combination of rapid experimental data collection and heuristic-based optimization is proposed for ALT planning. In addition, the effect of stress interaction on the ALT plan was also evaluated. Specifically, the Arrhenius model was used to develop a maximum likelihood mathematical expression for multi-stress factor scenarios with and without interaction. Subsequently, two optimization stages based on particle swarm optimization (PSO) were carried out using time varying inertia weight constants to drive early and late global and local searches, respectively. In the first stage, model parameters were estimated, while, in the second stage, an ALT optimal plan was generated based on a D-optimality criterion. Verification of stress factor interactions was carried out using graphical response analysis. An experiment, designed to investigate electromigration in solder joints under three stress factors (temperature, current density and mechanical load), was used to validate the study. The variation in the choice of Latin hypercube design (LHD) results in disparity in the levels of stress within each stress combination as well as sample allocation. Our results clearly show the need to investigate stress interactions prior to multi-stress acceleration planning.

1. Introduction

Over the years, the world has experienced several industrial revolutions, from the medieval era driven by handcraft through to simple mechanization (first revolution) around the 1780s, to the technical revolution era (second revolution), which saw many mechanical, chemical and electrical inventions. In the 20th century, digitalization (third revolution) driven by the growth in system automation as well as microelectronics and computer development preceded the fourth industrial revolution (Industry 4.0) characterized by information system autonomy enabled through the interconnection of systems. These revolutions relied on the technology of that time and increased job creation as well as provided better working conditions in agreement with Maslow’s hierarchy of need [1] (Figure 1).

However, a key demerit of Industry 4.0 and the revolutions preceding it is the increasing production wastage and reduced collaboration between human and smart systems. Sustainable development seeks to meet the needs of the present without compromising the ability of future generations to meet their own needs through effective human–machine interaction (Industry 5.0) [2]. Reliability evaluations of eco-friendly products, such as electric vehicles or solar system inverters, often require extensive testing to facilitate end user satisfaction. An accelerated life test (ALT) plan is typically used to optimize the number of test samples and conditions during product reliability evaluation. ALT is becoming a mandatory requirement for product reliability estimation [3]. The driving force for ALT adoption is based on the ability to use data or life models derived from higher stress levels to predict life metrics at normal operating conditions [4]. Traditional ALT based on single stress factors is incapable of stimulating the actual stress environments experienced by many products where multiple stresses coexist. Hence, numerous ALT-based studies use multi-stress life models to capture these multiple stresses. [5,6,7] Notably, tests like the highly accelerated life test (HALT) and highly accelerated stress test (HAST) have been used to create aggressive stress environments, but their life modeling has been elusive. Accelerated multi-stress life models can be generally classified into three types: proportional hazard model (PHM), polynomial acceleration model (PAM) and generalized linear logarithmic acceleration model (GLGAM).

The PHM uses a time-dependent basic failure rate function as well as a time-independent positive function as a basis for modeling the failure rate of a product. Elsayed and Zhang [8] implemented a proportional hazard model in their ALT study. PAM is often used when there is a lack of a linear relationship between the transformed stresses and the logarithm of the lifetime characteristic [9]. In GLGAM, the lifetime characteristic of a product is taken as a function of its stress factors. GLGAM is the most used model. In [10], GLGAM was used to evaluate the reliable lifetime of a smart electricity meter. Similarly, GLGAM was used for the optimal design of a step-stress accelerated degradation test with multiple stresses [11,12].

The standard way of developing an ALT plan is to formulate the likelihood function and then derive the Fisher information matrix (FIM) for test planning. Following the derived FIM is the implementation of efficient optimization criteria such as D-optimal design [4]. Other criteria include the asymptotic variance minimization of the expected product lifetime at the product’s use condition (${\mathrm{U}}_{\mathrm{c}}$-optimal design); minimization of the average prediction variance over the design space (I-optimality); minimization of the maximum entry in the diagonal of the hat matrix (G-optimality); minimization of the trace of FIM (A-optimality); and minimization of the average prediction variance over a set of m-specific points (V-optimality) [13]. The choice of optimal criteria is dependent on the optimization goal. For cases involving a precise estimate of model parameters, D-optimality is preferred. The objective in D-optimality is the maximization of the FIM determinant. The larger the determinant, the lower the variation, which leads to a higher joint precision of the estimated parameters.

Literature on optimal ALT designs is vast and a comprehensive review on ALT planning can be found in [14]. Some notable research works in this research area are concisely described. In [15], the maximum likelihood theory for designing optimal ALT plans based on the assumption that the product lifetime follows a Weibull or smallest extreme value distribution was presented. In [16], ALT with Bayesian design criterion was implemented using a sequential design approach. The study described tests at a higher stress condition based on a single stress factor, and a test plan was sequentially determined at a lower stress condition with an additional stress factor. In [17], a quasi-likelihood approach was used to develop the D-optimal ALT test plan with test chamber effects. Similarly, in [17] a three-iterative-step optimization algorithm was implemented using the D-optimal criterion. The D-optimal test plan was obtained via a quasi-likelihood approach. Weaver et al. used a random-effects model to evaluate test plans for degradation studies and unit-to-unit variability [18]. In [19,20], optimal progressive censoring plans were determined from expected FIM. The study’s asymptotic variance–covariance matrix of the maximum likelihood was computed via a progressively type-II censored sample based on Weibull distribution by direct computation. Tse, Ding and Yang [19] provided optimal ALT designs under interval censoring with random removals. Other studies such as [21,22] implemented ALT plans based on s-independent competing risks.

A careful evaluation of past ALT studies described above shows little effort channeled towards the effect of stress interactions on ALT plan optimization. Furthermore, many of the studies are based on assumed model parameters. For instance, in [8], model parameters which do not represent the stress and test conditions were used to generate ALT plans. Similarly, in [6,17], model parameters were used without consideration to the interaction that exists between stress factors. Although an assumed predictor life model which captures interactions for two stress factors was used in [13], it seems not to represent the true interaction model for the example considered in their work. Artificial intelligence techniques, such as genetic algorithms, particle swarm optimization algorithms, etc., popularly used for model parameter estimation as reported in [4,23,24], could facilitate fast and accurate determination of an ALT plan. Furthermore, a test plan generated without consideration to stress factor interactions or coupling common to multi-stress is likely not to be optimal, especially when stress interactions exist. To address this gap, in this paper, an accelerated life test model with and without interaction among stress factors was developed and used to optimize an ALT plan based on two stages of particle swarm optimization algorithm implementation. A case study, involving an experiment, designed to understand electromigration failure mechanisms in solder joints was used to validate the study. Since the optimization goal of this paper is focused on a precise estimate of model parameters as well as optimal stress levels for each factor, D-optimality criterion was adopted. The rest of this paper is organized as follows. In Section 2, a brief description of PSO is provided. The study detailed methodology is provided in Section 3. In Section 4, the results obtained as well as inferences associated with the results are described. Section 5 concludes the study.

2. Particle Swarm Optimization

PSO, initially introduced by Kennedy and Eberhart [25] in 1995, was inspired by social behavior of flocking birds. In nature, a swarm of birds flies through a space following a leader who has the closest position to food. The attraction of PSO to researchers is mainly due to the following:

• Simplicity of implementation.
• Performance variation based on slight modification of its controlling parameters such as inertia weight, cognitive ratio and social ratio, as in [26] and [27].
• Its flexibility to hybridize with other optimization algorithms.

In PSO, a swarm particle flies in an $h$-dimensional search space seeking an optimal solution. Each particle $i$ possesses a velocity vector $V_i=\left\{v_{i1}, v_{i2}, \dots , v_{id}\right\}$ and position vector $X_i=\left\{x_{i1}, x, \dots , x_{id}\right\}$, where $h$ is the number of dimensions. PSO starts by randomly initializing $V_i$ and $X_i$. Then, after every iteration, the best position that has been found by particle $i$ $P{b}_i$ = {$P{b}_{i1}, P{b}_{i2}, \dots , P{b}_{id}$} and the global best position found by the entire swarm $G{b}_i$ = {$G{b}_{i1}, G{b}_{i2}, \dots , G{b}_{id}$} direct particle $i$ to update its velocity and position using (1) and (2), respectively.

$$v_{id}\left(t+1\right)=v_{id}\left(t\right)+c_1{r}_1\left(P{b}_{id}\left(t\right)-x_{id}\left(t\right)\right)+c_2{r}_2\left(g{b}_{id}\left(t\right)-x_{id}\left(t\right)\right)$$

(1)

$$x_{id}\left(t+1\right)=x_{id}\left(t\right)+v_{id}\left(t+1\right)$$

(2)

where $t$ is the iteration, $c_1$ and $c_2$ are the cognitive and social acceleration coefficients, and $r_1$ and $r_2$ are two uniform random values generated within a [0, 1] interval. Since its first introduction, several PSOs have been studied for binary and continuous problems. In [28], an essential binary particle swarm optimization (EPSO) is proposed based on the idea of omitting the velocity component of PSO. Thus, there is no need to limit the velocity. Queen informants in ant colony optimization (ACO) were applied to the PSO. This resulted in a modified form of EPSO denoted as EPSOq. In [29], a continuous PSO called Multi-swarm Self-adaptive CPSO (MSCPSO) was proposed. MSCPSO population is split into four sub-swarms that allow information sharing among sub-swarms. Cooperative, diversity and self-adaptive strategies were used in MSCPSO to prevent being stuck in local optima and to obtain better solutions. Hybrid PSOs, which combine the strength of PSO and other meta-heuristic algorithms, have also been studied. In [30], gray wolf optimization and particle swarm optimization were combined to solve binary and continuous problems. In another study, PSO was combined with genetic algorithm (GA) for field development optimization. The resultant hybrid algorithm is called genetical swarm optimization (GSO) [31]. In this hybrid algorithm, the population is split into two portions, and it is reconstructed by GA and PSO operations in every iteration.

In [32], PSO topology (particle connection or interaction pattern) was shown to influence its behavior or performance. Experimental results revealed that some topologies perform better than others. Common PSO topologies include star topology, which allows particles to move towards the global optimal [25], ring topology, which allows particles to move towards the local optimal [32], Von Neumann topology, which uses a rectangular matrix to connect a particle to the particles below it [33], and dynamic topology [34]. Other topologies have been studied. For instance, in [35], a unified topology, which combined the star and ring topologies, was proposed. In addition, cluster, pyramid and wheel topologies have been studied. In [36], a comprehensive review on PSO with emphasis on the different topologies is presented. Notwithstanding PSO’s numerous benefits, studies have shown that it suffers from premature and slow convergence. To address this problem, in [37], multiple scale self-adaptive cooperative mutation strategy-based particle swarm optimization algorithm (MSCPSO), which uses multi-scale Gaussian mutations with different standard deviations to promote the capacity of sufficiently searching the whole solution space, was implemented. Memory concept has also been used to prevent premature convergence. In [38], memory was used to store promising historical values, which are later used to avoid premature convergence.

PSO has been applied to solve many optimization problems. In [39], PSO is used to segment medical images to detect brain tumors. In [40], PSO is used for beamforming optimization in IRSs to minimize the transmission power given so that the signal-to-noise ratio (SNR) does not go below a certain threshold. PSO has been widely applied to optimize the performance of electrical power systems including economic dispatch [41], state estimation [42] and power system controllers [43]. Recently, Shuai [44] developed an improved particle swarm optimization algorithm with a specific particle initialization approach (called PIPSO) to solve the global reliability allocation problem (gRRAP). Failure parameter estimations are essential in maintainability and reliability evaluation. In [23] and [24], PSO was used to estimate Weibull and maintainability parameters. These two studies implemented static weight. In PSO, extensive global search (exploration) is required at the early part of the process, while the latter part requires focus at the local search (exploitation). The static weight approach does not meet this requirement. In this study, time-varying inertia weight based on the model presented in [45] was used in the double PSO implementation. In the first stage, model parameters are optimized, while in the second stage, a multi-stress ALT plan was obtained.

3. Materials and Methods

3.1. Assumptions

Although the time to failure of a component or system is expected to vary with respect to the number of stress factors (${\psi }_1, {\psi }_2 ,$ …, ${\psi }_N$) or stress combinations, the failure mechanism is assumed to remain constant and consistent with that experienced at the standard operating condition. Each stress factor can take levels $l_1, l_2 ,$ …, $l_N$. Figure 2 shows hypothetical test stress levels for three-stress factors. The factorial-based design depicted in Figure 2, which shows two stress levels for each stress factor (${\psi }_i; i\le 3)$, serves as a fast way to collect preliminary data essential for parameter optimization purposes. It also helps to remove confounding effects. Considering, an alternative arrangement with the same number of testing points will lead to more stress levels for each stress type. Figure 2, facilitates testing of interactions after data collection as well as allows simultaneous testing utilizing the available equipment in an efficient manner, which can lead to significant savings in test completion time and cost. In this paper, preliminary data collected and used for model parameter estimation were obtained at stress combinations the same as that of Figure 2.

The stresses ${\psi }_{1U}, {\psi }_{2U} , and$ ${\psi }_{3U}$ are the standard operating stress levels corresponding to stresses ${\psi }_1, {\psi }_2 , and$ ${\psi }_3$, respectively. Stress level ${\psi }_{nm}$ ($n=1,2; m=0,1$) corresponds to the type $n$ stress at the $m$ level. We assume the following for the accelerated life test with three stresses:

• Three stresses, ${\psi }_1, {\psi }_2 , and$ ${\psi }_3$, are used in the test, let $\psi =({\psi }_1, {\psi }_2 ,$ ${\psi }_3)$ be the vector of stress levels.
• The life of a solder joint has been generally modeled using Weibull distribution [46]. Hence, it is assumed that the lifetime $\left(t\right)$ of the solder joints used to validate the method described here is independent of itself and follows Weibull distribution. Its probability density function is given as (3).
•  

  $$f\left(t\right)=\frac{\beta }{\zeta }{\left(\frac{t}{\zeta }\right)}^{\beta -1}{e}^{-{\left(\frac{t}{\zeta }\right)}^{\beta }}$$

  (3)

  where $\beta$ and $\zeta$ are the shape and scale parameters under multi-stress combination, respectively.
• The characteristic lifetime of a product subjected to accelerated multi-stress conditions is a function of the accelerated transformed stresses under different multi-stress combinations as given in (4).
•  

  $$\ln \left(\zeta \right)=f\left(\chi \right)$$

  (4)

  where $\chi$ is the transformed multi-stress vector and $\chi =\left({\chi }_1, {\chi }_2, \dots , {\chi }_N\right)$ and ${\chi }_1, {\chi }_2, \dots , {\chi }_r$ are the standardized forms of $1,2, \dots , r$. In this study, temperature (T), current density ($J)$ and mechanical stress ($\sigma$) were considered the predominant stresses acting on the solder joint, and their transformed stress is given as $\left(5\right)$, (6) and (7), respectively.

  $${\chi }_{1i}=\raisebox{1ex}{$\left[log\left(\frac{1}{T_i}\right)-log\left(\frac{1}{T_U}\right)\right]$}\!\left/ \!\raisebox{-1ex}{$\left[log\left(\frac{1}{T_H}\right)-log\left(\frac{1}{T_U}\right)\right]$}\right.$$

  (5)

  $${\chi }_{2i}=\raisebox{1ex}{$\left[log\left(J_i\right)-log\left(J_U\right)\right]$}\!\left/ \!\raisebox{-1ex}{$\left[log\left(J_H\right)-log\left(J_U\right)\right]$}\right.$$

  (6)

  $${\chi }_{3i}=\raisebox{1ex}{$\left[{\sigma }_i-{\sigma }_U\right]$}\!\left/ \!\raisebox{-1ex}{$\left[{\sigma }_H-{\sigma }_U\right]$}\right.$$

  (7)

  where ${\chi }_{1i},{\chi }_{2i} \mathrm{and} {\chi }_{3i}$ are the transformed temperature, current density and stress at stress level $i$. $T_i, J_i \mathrm{and} {\sigma }_i$ are the $i\mathrm{th}$ stress level for temperature, current density and mechanical stress, respectively. $T_U, J_U \mathrm{and} {\sigma }_U$ are the normal operating stresses for temperature, current density and mechanical stress. $T_H, J_H \mathrm{and} {\sigma }_H$ are the highest stress levels for temperature, current density and mechanical stress.
• The shape parameter ($\beta$) remains constant since the failure mechanism of the product is constant irrespective of the combination of stress levels.

3.2. Multi-Stress Acceleration Model

Quasi-physical models such as the Arrhenius model are commonly used to relate lifetime if temperature is a stress factor. Arrhenius-based relationships between the rate of reaction and temperature stress can be expressed as (8).

$$k_r={\gamma }_0{e}^{\left(\frac{{\gamma }_1^{\prime }}{k_{B}T}\right)}$$

(8)

where $k_r$ is the rate of reaction of the product and ${\gamma }_0$ and ${\gamma }_1^{\prime }$ are unknown parameters. $k_B$ is the Boltzmann constant, which is 8.6171 × 10⁻⁵ eV/°C; T is the Kelvin temperature. Deb et al. [47] showed that the rate model under two stresses such as temperature-voltage can be described using (9).

$$k_r={\gamma }_0{e}^{\left(\frac{{\gamma }_1^{\prime }}{k_{B}T}\right)}· e^{\left({\gamma }_{2}V+\frac{{\gamma }_3^{\prime }V}{k_{B}T}\right)}$$

(9)

where $\frac{{\gamma }_2^{\prime }V}{k_{B}T}$ is the interaction effect for the combined stresses, which in this case is voltage ($\mathrm{V})$ combined with temperature. Generally, V can be replaced by any non-thermal stress. Suppose $1/T={\psi }_1$ and $V={\psi }_2$, $\frac{{\gamma }_1^{\prime }}{k_B}={\gamma }_1, \frac{{\gamma }_3^{\prime }}{k_B}={\gamma }_3$, then Equation (9) can be expressed as (10). Given that the characteristic life $L\left(t\right)$ is proportional to the reciprocal of the rate of reaction $k_r$ and the transformation of ${\psi }_1, {\psi }_2 and {\psi }_3$ are ${\chi }_1, {\chi }_2 \mathrm{and} {\chi }_3$, respectively.

$$L\left(t\right)={\gamma }_0{e}^{\left(-{\gamma }_1{\chi }_1\right)}· e^{-\left({\gamma }_2{\chi }_2-{\gamma }_3{\chi }_1{\chi }_2\right)}$$

(10)

Similarly, for three stress factors (8) becomes (9), where ${\gamma }_0, {\gamma }_1, {\gamma }_2, {\gamma }_3, {\gamma }_4, {\gamma }_5, {\gamma }_6, {\gamma }_7$ are the model parameters. If the interaction between stresses is ignored, then (11) becomes (12).

$$L\left(t\right)={\gamma }_0{e}^{\left(-{\gamma }_1{\chi }_1\right)}· e^{\left(-{\gamma }_2{\chi }_2\right)}·e^{\left(-{\gamma }_3{\chi }_3\right)}·e^{\left(-{\gamma }_4{\chi }_1{\chi }_3\right)}·e^{\left(-{\gamma }_5{\chi }_1{\chi }_3\right)}·e^{\left(-{\gamma }_6{\chi }_2{\chi }_3\right)}{e}^{\left(-{\gamma }_7{\chi }_1{\chi }_2{\chi }_3\right)}$$

(11)

$$L\left(t\right)={\gamma }_0{e}^{\left(-{\gamma }_1{\chi }_1\right)}· e^{\left(-{\gamma }_2{\chi }_2\right)}·e^{\left(-{\gamma }_3{\chi }_3\right)}$$

(12)

For $rz$ stress factors, the rate model is given as (13)

$$\begin{gathered} L\left(t\right)={\gamma }_0{\displaystyle \prod }_{s=1}^{rz}{e}^{\left(-{\gamma }_s{\chi }_s\right)}· {\displaystyle \prod }_{\begin{array}{c}1,d=1, d\ne s,s<d\\ u=n+1, n+2, \dots , n+C_2^n \end{array}}^{rz}{e}^{\left(-{\gamma }_u{\chi }_s{\chi }_d\right)}· {\displaystyle \prod }_{\begin{array}{c}s=1,d=1, q=1,d\ne s\ne q, s<d<q\\ c=n+C_2^n+1, n+C_2^n+2, \dots , n+C_2^n+C_3^n \end{array}}^{rz}{e}^{\left(-{\gamma }_c{\chi }_s{\chi }_d{\chi }_q\right)} \\ {\displaystyle \prod }_{\begin{array}{c}s=1,d=1, q=1,d\ne s\ne q\ne , \dots ,\ne N s<d<q<\cdots <z\\ =n+C_2^n+\cdots +C_{n-1}^n+1, n+C_2^n+\cdots +C_{n-1}^n+2, \dots , n+C_2^n+\cdots +C_{n-1}^n+C_n^n \end{array}}^n{e}^{\left({\gamma }_u{\chi }_s{\chi }_d{\chi }_q\cdots {\chi }_z\right)} \end{gathered}$$

(13)

Taking the natural logarithm on both sides of (13) results in (14).

$$\begin{gathered} \ln \left(L\left(t\right)\right)=ln{\gamma }_0+{\displaystyle \sum }_{s=1}^{rz}{e}^{\left({\gamma }_s{\chi }_s\right)}· {\displaystyle \sum }_{\begin{array}{c}s=1,d=1, d\ne s,s<d\\ u=n+1, n+2,\dots , n+C_2^n \end{array}}^{rz}{e}^{\left({\gamma }_u{\chi }_s{\chi }_d\right)} \\ {\displaystyle \sum }_{\begin{array}{c}s=1,d=1, q=1,d\ne s\ne q, s<d<q\\ c=n+C_2^n+1, n+C_2^n+2,\dots , n+C_2^n+C_3^n \end{array}}^{rz}{e}^{\left({\gamma }_c{\chi }_s{\chi }_d{\chi }_q\right)} \\ {\displaystyle \sum }_{\begin{array}{c}s=1,d=1, q=1,d\ne s\ne q\ne , \dots ,\ne N s<d<q<\cdots <z\\ =n+C_2^n+\cdots +C_{n-1}^n+1, n+C_2^n+\cdots +C_{n-1}^n+2, \dots , n+C_2^n+\cdots +C_{n-1}^n+C_n^n \end{array}}^n{e}^{\left({\gamma }_u{\chi }_s{\chi }_d{\chi }_q\cdots {\chi }_z\right)} \end{gathered}$$

(14)

Based on the three stress factors considered in this work, the natural log of the characteristic lifetime for with and without interaction is expressed as (15) and (16), respectively.

$$\begin{gathered} \ln \left(\zeta \left({\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4,{\gamma }_5,{\gamma }_6,{\gamma }_7,\right)\right)=ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3-{\gamma }_4{\chi }_1{\chi }_2-{\gamma }_5{\chi }_1{\chi }_3- \\ {\gamma }_6{\chi }_2{\chi }_3-{\gamma }_7{\chi }_1{\chi }_2{\chi }_3 \end{gathered}$$

(15)

$$\ln \left(\zeta \left({\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\right)\right)=ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3$$

(16)

From (15) and (16), the scale or characteristic parameter estimation basically requires a multi-parameter estimation method. In the next section, a detailed description of our multi-parameter estimation method is described.

3.3. PSO Inspired Parameter Estimation Method for Multi-Stress Accelerated Method

Given that a product is subjected to a multi-stress accelerated life test of $rz$ stress factors and $wz$ stress combinations ($S{C}_1,S{C}_2,\dots , S{C}_{wz}),$ where $S{C}_i=\left( {\chi }_{1i},{\chi }_{2i},\dots {\chi }_{rzi} \right), 1\le i\le wz$ is the $i\mathrm{th}$ stress combination$.$ $O$ specimen is selected randomly in the $i\mathrm{th}$ accelerated stress combination and tested to failure. The specimen time to failure is given as (17)

$$t_{i1},t_{i2}, \dots , t_{iO}, i=1,2, \dots , O$$

(17)

The likelihood function and the log-likelihood function of the product failure under the multi-stress combination ${\mathrm{SC}}_{\mathrm{i}}$ can be expressed as (18) and (19), respectively.

$$L_i={\displaystyle \prod }_{i=1}^o\beta ·{\zeta }_i^{-\beta }·t_{ij}^{\beta -1}·e^{\left(-t_{ij}^{\beta }·{\zeta }_i^{-\beta }\right)}$$

(18)

$$ln{L}_i=ln\left\{{\displaystyle \prod }_{i=1}^o\beta ·{\zeta }_i^{-\beta }·t_{ij}^{\beta -1}·e^{\left(-t_{ij}^{\beta }.{\zeta }_i^{-\beta }\right)}\right\}=Oln\beta -O\beta ln{\zeta }_i+\left(\beta -1\right){\displaystyle \sum }_{i=1}^{O}ln{t}_{ij}-{\zeta }_i^{-\beta }{\displaystyle \sum }_{i=1}^O{t}_{ij}^{\beta }$$

(19)

Therefore, the log-likelihood function for all stress combinations is given as (20).

$$\begin{gathered} L\left(\beta ,\zeta \right)={{\displaystyle \sum }}_{i=1}^{wz}ln{L}_i={{\displaystyle \sum }}_{i=1}^{wz}\left[Oln\beta -O\beta ln{\zeta }_i+\left(\beta -1\right){{\displaystyle \sum }}_{i=1}^{O}ln{t}_{ij}-{\zeta }_i^{-\beta }{{\displaystyle \sum }}_{i=1}^O{t}_{ij}^{\beta }\right] \\ =ln\beta {\displaystyle \sum }_{i=1}^{wz}{O}_i-\beta {\displaystyle \sum }_{i=1}^{wz}{O}_{i}ln{\zeta }_i+\left(\beta -1\right){\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^O{t}_{ij}^{\beta }-{\zeta }_i^{-\beta }{\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^O{t}_{ij}^{\beta } \end{gathered}$$

(20)

Here, the shape parameter $\beta$ is assumed to be a constant, i.e., it will not change over the varying stress conditions. ${\zeta }_i$ is the scale or characteristic parameter of the $i\mathrm{th}$ stress combination, and $t_{ij}$ is the $i\mathrm{th}$ time to failure of the $j\mathrm{th}$ stress combination. For three-stress-factor-based tests, substituting (15) and (16) for (20), the maximum likelihood function expression for a multi-stress acceleration model with and without interaction is expressed as (21) and (22), respectively.

$$\begin{gathered} L\left(\beta ,{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4,{\gamma }_5,{\gamma }_6,{\gamma }_7\right)= \\ ln\beta {\displaystyle \sum }_{i=1}^{wz}{O}_i-\beta {\displaystyle \sum }_{i=1}^{wz}{O}_i{\left[ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3-{\gamma }_4{\chi }_1{\chi }_2-{\gamma }_5{\chi }_1{\chi }_3-{\gamma }_6{\chi }_2{\chi }_3-{\gamma }_7{\chi }_1{\chi }_2{\chi }_3\right]}_i+ \\ \left(\beta -1\right){\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^O{t}_{ij}^{\beta }-{\left[e^{\left(ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3-{\gamma }_4{\chi }_1{\chi }_2-{\gamma }_5{\chi }_1{\chi }_3-{\gamma }_6{\chi }_2{\chi }_3-{\gamma }_7{\chi }_1{\chi }_2{\chi }_3\right)}\right]}_i^{-\beta }{\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^O{t}_{ij}^{\beta } \end{gathered}$$

(21)

$$\begin{gathered} L\left(\beta ,{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\right)=ln\beta {{\displaystyle \sum }}_{i=1}^{wz}{O}_i-\beta {{\displaystyle \sum }}_{i=1}^{wz}{O}_i{\left[ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3\right]}_i+ \\ \left(\beta -1\right){{\displaystyle \sum }}_{j=1}^{wz}{{\displaystyle \sum }}_{i=1}^O{t}_{ij}^{\beta }-{\left[e^{\left(ln{\gamma }_0-{\gamma }_1{\chi }_1-{\gamma }_2{\chi }_2-{\gamma }_3{\chi }_3\right)}\right]}_i^{-\beta }{{\displaystyle \sum }}_{j=1}^{wz}{{\displaystyle \sum }}_{i=1}^O{t}_{ij}^{\beta } \end{gathered}$$

(22)

Using the traditional maximum likelihood estimation method will require differentiation with respect to each parameter to be estimated. For two-parameter Weibull distribution with $rz$ stress factors, the number of parameters is $2^{rz}+1$ and $rz+2$ for with and without interaction. This large number of parameters will result in poor estimation accuracy if the maximum likelihood estimation (MLE) is used. In this paper, PSO is used to solve the multi-parameter estimation problem, by transforming it to a multi-parameter optimization problem. This involves the collection of preliminary failure data. Then, using the collected data to generate more pseudo-lifetime data. The pseudo-lifetime data and stress combination based on Figure 2 is then used to estimate the optimal model parameter via implementation of PSO. The motivation for the used PSO is based on its flexibility, simple structure, good global search and fast speed of convergence. The objective function is the likelihood function (20), while the goal of the optimization is to find the maximum likelihood function value. The parameters to be estimated are ${\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4,{\gamma }_5,{\gamma }_6,{\gamma }_7 and \beta$ if stress interactions are considered and ${\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3 and \beta$ if stress interactions are ignored. These parameters are represented as particles in PSO. The preliminary experimental data collection, pseudo-failure data generation and PSO implementation is briefly described.

3.3.1. Preliminary Data Collection

The data collected to validate the ALT planning method proposed in this study were to facilitate understanding of electromigration failure mechanisms in solder joints under temperature, current density and mechanical stress conditions. A description of the sample preparation and data acquisition approach is provided below.

• Sample preparation

High-temperature insulated magnet wires of 0.2540 mm in diameter were used to form the solder interconnect. The cross-section of the wires was grinded with silicon carbide abrasive paper (400 grit) and then polished with 3 µm and 0.3 µm aluminum oxide papers. The polished surface was washed and cleaned with de-ionized water and isopropyl alcohol (IPA). Flux was then applied at the cross-section of the wire placed on a ceramic plate. Then, a 12 mils (0.305 mm) diameter eutectic lead-tin solder (Pb37Sn63) ball was fixed between the wire ends using Kapton tape, which led to a solder ball connection with two wires. The ceramic plate and the weakly connected solder interconnect were placed in the PUHUI T-962A Infrared IC Heater Desktop Reflow Oven. The oven was programmed with the temperature profile for the eutectic solder. The reflow process was completed in approximately 9 min with a solder joint formed.

• Failure data acquisition setup

Preliminary data acquisition tasks were performed using a built system comprising an Arduino microcontroller, a power supply system, as well as current, voltage, temperature and displacement sensors and a display system (personal computer). The Arduino Mega 2560 microcontroller board consists of 54 I/O pins, 14 PWM and 16 analog outputs, a 16 MHz crystal oscillator and a USB connection. It also supports I2C and SPI communication, which enables accessible communication with many electronic components. The USB cable facilitates the transfer of C++ code as well as monitoring of the I/O of the board using open-source Arduino Software (IDE). The data acquisition system block diagram is shown in Figure 3.

This data acquisition system allows for control of ambient temperature, applied electrical current and applied mechanical load around and on a solder joint sample. A PID loop in an Arduino was used to control resistive heaters embedded inside of ceramic chambers to set and monitor ambient temperature around a given solder sample. Electrical current was applied to the samples using a constant DC current power supply. The voltage and current of the connection were measured with voltage and current sensors, respectively. Measured voltage and current signals were sent to the microcontroller and the resistance of the solder connection was calculated and sent into a PC via serial communication, where the values were recorded on the computer hard drive. Similarly, the strain of the solder was recorded from the signal sent by a displacement sensor. To provide a constant ambient temperature, a closed-loop controller was utilized. The temperature sensor provides the feedback into the control loop by sending the data into the Arduino board. The mechanical load was applied to a given sample by means of a linear sliding rail and pulley mechanism. Figure 4a depicts the mechanical load application schematic diagram, and Figure 4b depicts the physical test setup. More information on this system can be found in [46] and [48].

The mechanical load applied to a solder sample in this testing setup creates tensile stress on the solder joints. To illustrate the interacting effect of stress and electromigration on solder joint failure, the study focuses on the use of preliminary data collected from two levels of each stress factor considered in a study to plan ALT for higher levels of the same stress factors. By elevating the ambient temperature and electrical current, solder samples experienced electromigration failures. By maintaining electrical and temperature stresses and simultaneously applying mechanical loads to the solder joints, a combined failure mode experiment was achieved. This testing setup allowed for time to failure data points to be gathered by monitoring the change in electrical resistance over time with a variety of temperature, electrical current and mechanical loading combinations. The time it took for the wire resistance to raise by 10% was recorded as the failure time. Based on this setup, the time to failure was collected for the seven stress combinations. The stress combination with all stresses at a lower level was ignored as no failure occurred with a reasonable time. Considering this will result in generating model-based failure time censoring. All test combinations considered in this work were tested to failure, which created a completely exact data scenario. For each test stress combination, five samples were tested and the average time to failure was recorded. The parameters ${\psi }_{1U}=75 °\mathrm{C}$, ${\psi }_{2U}=2960.29\left(\frac{\mathrm{A}}{{\mathrm{cm}}^2}\right)$ and ${\psi }_{3U}=0\mathrm{MPa}$ were used as the normal operating stress levels, while the accelerated levels (low and high) for each stress are given in

Table 1. Preliminary stress levels tested.

Stress Factors	Low	High
Temperature (°C)	100	120
Stress (MPa)	98.68	394.71
Current density ($\frac{\mathrm{A}}{{\mathrm{cm}}^2}$)	5920.66	6907.44

. Using the stress transformation expression in (3), (4) and (5) and the stress setting contained in Table 1, the average time to failure for each stress combination shown in

Table 2. Preliminary stress combination.

The Combinations of Multi-Stress Factors	${\mathit{\chi }}_1$	${\mathit{\chi }}_2$	${\mathit{\chi }}_3$	ATF (mins)
SC1	1	1	1	8.48
SC2	1	1	0.25	2944
SC3	1	0.818	1	1645
SC4	0.517	1	1	208
SC5	1	0.818	0.25	4977.46
SC6	0.517	1	0.25	31,444
SC7	0.516	0.818	1	4151

was obtained.

3.3.2. Pseudo-Time-to-Failure Data Generation

Monte Carlo (M-C) simulation was conducted to generate 50 pseudo-times-to-failure for each stress combination, based on proposed time to failure lower and upper bounds. The lower and upper time to failure bounds for each stress combination was estimated using ${\mathrm{ATF}}_{\mathrm{Li}}=\mathrm{ATF}-\mathrm{ATF}/2$ and ${\mathrm{ATF}}_{\mathrm{Ui}}=\mathrm{ATF}+\mathrm{ATF}/2$, respectively, where $\mathrm{ATF}$ is the average time to failure for each test combination. $AT{F}_{Li}$ and $AT{F}_{Ui}$ are the lower and upper average time to failure of the $i\mathrm{th}$ stress combination. M-C simulation implementation is given below:

Set iteration $\left(it=50\right)$

For each stress combination compute

$AT{F}_{Ui}=ATF+ATF/2$

$AT{F}_{Li}=ATF-ATF/2$,

For  $i=1 to it$

$FT\left(i\right)=AT{F}_{Li}+\left(AT{F}_{Ui}-AT{F}_{Li}\right) x rand()$,

End

Sort FT

Here, $it$ is the number of iterations. $rand()$ is a MATLAB code used to generate a random number. $FT\left(i\right)$ is the $i\mathrm{th}$ pseudo-generated failure rate, while FT is all 50 generated pseudo-failure times.

3.3.3. PSO Implementation for Parameter Estimation

In PSO implementation, there is usually a need to balance between accuracy and computational time. The balance is mainly affected by the number of particles and iterations. The greater the number of particles and iterations, the better the convergence as well as accuracy, although computing time correspondingly becomes higher. The implementation of PSO is as follows:

• Set the number of particles and iterations.
• Randomly initialize each particle’s positions and velocities within the constrained space of particles.
• Compute the fitness values of each particle based on the objective function given in (18).
• Compare the current fitness value of each particle with its historical optimal fitness value $\left(\mathrm{pb}\right)$, and redefine the larger value of the two as $\mathrm{pb}$.
• Compare the current fitness value of all particles with its historical optimal fitness value $\left(\mathrm{gb}\right)$, and redefine the larger value of the two as $\mathrm{gb}$.
• Update the position and velocity of the particle in agreement with (23) and (2).

$$v_{id}\left(t+1\right)=w{v}_{id}\left(t\right)+c_1{r}_1\left(P{b}_{id}\left(t\right)-x_{id}\left(t\right)\right)+c_2{r}_2\left(g{b}_{id}\left(t\right)-x_{id}\left(t\right)\right)$$

(23)

where $w$ is the inertia weight. In this work, $w$ is estimated using (24), where $it$ is the maximum number of iterations, $w_{mx}=0.9,$ and $w_{mn}=0.4$; $c_1$ and $c_2$ are the acceleration coefficients. $c_1$= 1.5 and $c_2$= 1.5.

$$w=w_{mx}-\left(w_{mx}-w_{mn}\right)\frac{t}{it},$$

(24)

7.

Display the result if the termination criterion is satisfied.

In this work, 100 particles and 100 iterations were used to search for the optimal parameters without interaction, while 500 particles and 500 iterations were used to search for the optimal model parameters that maximize the log-likelihood function value in the case of interacting stresses.

3.4. Accelerated Test Planning

Based on the parameter estimated from Section 3.3, an ALT plan under $rz$ stress factors and $rz+1$ levels would be determined using an efficient and economical experimental design. The choice of $rz+1$ is in agreement with the recommendation in [4]. Another optimization is required at this stage. In this work, PSO is further used for the ALT optimization. The purpose of this second optimization is the identification of optimal stress levels for each stress factor as well as optimal sample allocation for each stress factor combination. This typically involves experimental design selection, determination of the Fisher information matrix, selection of an optimization criterion and implementation of an optimization technique. The implementation of these activities in this work is briefly described.

3.4.1. Latin Hypercube Design

The experimental design adopted in this work is the Latin hypercube design (LHD). LHD is an $m\times r$z matrix, often denoted as LHD $\left(\mathrm{m}, \mathrm{r}\right)$, where $\mathrm{m}$ rows represent the number of experiments and $rz$ columns represent the stress factors. Compared to full-factorial design which requires $m^r$ experiments, LHD only requires $\mathrm{m}$ experiments. Thus, for the same statistical properties, the sample size, test time and experimental cost would be significantly reduced with LHD design. Although fractional factorial design (FFD) can equally reduce the number of experiments, the selection of an appropriate fraction and allocation of test units is rather a challenge. However, an attractive property of an LHD is that when an $m$-experiment design is projected onto any factor, there are $m$ different levels for each factor. In this study LHD $\left(4, 3\right)$ analysis is conducted.

Table 3. Two LHD considered cases.

No of Experiments	CASE I			CASE II
	$\mathit{r}\mathit{z}$			$\mathit{r}\mathit{z}$
	$\mathit{T}$	$\mathit{j}$	$\mathit{\sigma }$	$\mathit{T}$	$\mathit{j}$	$\mathit{\sigma }$
$z_1$	1	1	4	1	2	3
$z_2$	2	4	3	2	1	4
$z_3$	3	3	2	3	4	2
$z_4$	4	2	1	4	3	1

gives the two LHD cases considered. $z_i , i=1,2,\dots m$ represents the experimental runs, while 1, 2, 3, 4 represents the stress levels.

3.4.2. Fisher Information Matrix

In agreement with the regularity condition [8], the negative s-expectations of the second partial derivatives of the log-likelihood with respect to the unknown parameters in (21) and (22) form the elements of the Fisher information matrix (FIM) for an observation with and without interaction, respectively. The FIM for an observation with and without interaction are given in (25) and (26), respectively. The detailed expression of the derivatives for with and without interaction is given in Appendix A and Appendix B, respectively.

$$F_{i\mathrm{w}}=\left[\begin{array}{ccccccccc}\mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\beta }}^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_0)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_1)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_3)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_4)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_7)\\ & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_1)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_3)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_4)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_7)\\ & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_3)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_4)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_2)\\ & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_3)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_4)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_7)\\ & & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3\mathit{\partial }{\mathit{\gamma }}_4)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3\mathit{\partial }{\mathit{\gamma }}_7)\\ & & & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_4^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_4\mathit{\partial }{\mathit{\gamma }}_5)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_4\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_4\mathit{\partial }{\mathit{\gamma }}_7)\\ & & & & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_5^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_5\mathit{\partial }{\mathit{\gamma }}_6)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_5\mathit{\partial }{\mathit{\gamma }}_7)\\ ⋮& & & & & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_6^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_6\mathit{\partial }{\mathit{\gamma }}_7)\\ symmetry& & & \cdots & & & & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_7^2)\end{array}\right]$$

(25)

$$F_{i\mathrm{w}\mathrm{o}}=\left[\begin{array}{ccccc}\mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\beta }}^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_0)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_1)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }\mathit{\beta }\mathit{\partial }{\mathit{\gamma }}_3)\\ & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_1)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_0\mathit{\partial }{\mathit{\gamma }}_3)\\ & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_1\mathit{\partial }{\mathit{\gamma }}_3)\\ ⋮& & & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2^2)& \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_2\mathit{\partial }{\mathit{\gamma }}_3)\\ symmetry& & & \cdots & \mathit{E}(-{\mathit{\partial }}^2\mathit{L}/\mathit{\partial }{\mathit{\gamma }}_3^2)\end{array}\right]$$

(26)

Given that, $F_{iwo}$ and $F_{iw}$ represents the expected FIM of an observation for the $i\mathrm{th}$ stress combination. The total FIM for Ns-independent stress combinations is $F_w={\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ijw}$ and $F_{wo}={\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ijwo}$ for with and without interaction, where $N$ is the total number of test units available for testing. $P_j$ is the proportion of the $N$ allocated to stress combination $j$. For $r$z stress factors, stress levels less than $rz$ may result in a singular FIM. If cost and time can be justified, $rz+1$ levels are recommended [49]. Based on the fact that more stress levels provide more information, in this work, $rz+1$ was implemented.

3.4.3. Optimization Criterion

Let $\theta =\beta , {\gamma }_0,{\gamma }_1,\dots , {\gamma }_k$ be the vector of unknown model parameters and $h\left(\theta \right)$ be a real-valued function, such as the percentile of lifetime distribution, reliability function or Det (FIM) at a defined time and stress level. Let $\widetilde{\theta }=\tilde{\beta }, \widetilde{{\gamma }_0}, \widetilde{{\gamma }_1}\cdots \widetilde{{\gamma }_k}$, and $h\left( \widetilde{\theta }\right)$ be the estimate of $\theta$ and $h\left(\theta \right)$, we used D-optimality. The volume of the asymptotic joint confidence region of the model parameters is proportional to the value of the determinant of the FIM for with and without interaction, respectively. A larger value of the Fisher information matrix determinant corresponds to a higher joint precision of the estimates. Therefore, we chose D-optimality which maximizes the FIM determinant as the optimization criterion. Each ALT plan is characterized by the levels of all stresses, the proportions of test units allocated to each test combination and the total number of test units available. Given that, the normal operating condition as well as the highest levels of each stress beyond which a new failure mode will occur is specified. In addition, the total number of test units are pre-specified. The optimization objective is therefore to determine the stress levels of each stress factor and the proportion of test units allocated to each test condition, such that the determinant will be maximized. This nonlinear optimization problem is formulated as (27) and (28) for with and without interaction.

$$\begin{gathered} Max(Det\left({\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ij\mathrm{w}}\right) \\ \mathrm{s}.\mathrm{t} \\ 0<P_j<1; j=1,2,3,4 \\ {\displaystyle \sum }_{j=1}{P}_j=1 \\ {\psi }_{1D}\le {\psi }_{11}<{\psi }_{12}<{\psi }_{13}<{\psi }_{14}\le {\psi }_{1H} \\ {\psi }_{2D}\le {\psi }_{21}<{\psi }_{22}<{\psi }_{23}<{\psi }_{24}\le {\psi }_{2H} \\ {\psi }_{3D}\le {\psi }_{31}<{\psi }_{32}<{\psi }_{33}<{\psi }_{34}\le {\psi }_{3H} \end{gathered}$$

(27)

$$\begin{gathered} Max(Det\left({\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ij\mathrm{wo}}\right) \\ \mathrm{s}.\mathrm{t} \\ 0<P_j<1;j=1,2,3,4 \\ {\displaystyle \sum }_{j=1}{P}_j=1 \\ {\psi }_{1D}\le {\psi }_{11}<{\psi }_{12}<{\psi }_{13}<{\psi }_{14}\le {\psi }_{1H} \\ {\psi }_{2D}\le {\psi }_{21}<{\psi }_{22}<{\psi }_{23}<{\psi }_{24}\le {\psi }_{2H} \\ {\psi }_{3D}\le {\psi }_{31}<{\psi }_{32}<{\psi }_{33}<{\psi }_{34}\le {\psi }_{3H} \end{gathered}$$

(28)

3.4.4. PSO Implementation for Stress Levels and Sample Allocation Determination

The PSO algorithm described in Section 3.3.3, is re-implemented again. However, the fitness function at this stage is (27) and (28) for with and without interaction, respectively. The number of particles and iterations was set to 500 for the without interaction and 10,000 for the with interaction. The large iteration value used for the with interaction was aimed at ensuring convergence, based on the preliminary test that shows a lack of convergence, even after 5000 iterations. The optimization outputs are ${\psi }_{11},{\psi }_{12},{\psi }_{13},{\psi }_{14}$, ${\psi }_{21},{\psi }_{22},{\psi }_{23},{\psi }_{24}, {\psi }_{31},{\psi }_{32},{\psi }_{33},{\psi }_{34}$ and $N{P}_j$, $j=1, 2, 3,4)$ for with and without interaction.

In general, the method presented in this work comprises three key steps. The first is data acquisition. This step involves the collection of historical or experimental data. In this work, experimental data were collected. The second step involves parameter optimization. In this step, preliminary analysis can be done to ascertain if an interaction exists between stress factors. On the basis of the investigation, if an interaction exists, then a model which captures stress factor interaction is used for the parameter optimization. Otherwise, stress factor interaction is ignored in the selected life estimation model. Figure 5 depicts the ALT planning implementation framework.

3.5. Selection of Multi-Stress ALT Model

In Section 3.2, Section 3.3 and Section 3.4, the multi-stress modeling method as well as PSO implementation, aimed at identifying an ALT plan for stress factors with and without interaction, were described. However, the unanswered question remains: how can the existence of interaction be verified? The answer to that question lies in the test for interaction. If interactions exist among stress factors, the model with interaction is selected. On the other hand, if no interactions exist among stress factors, the model without interaction is selected. In a case where interactions exist between some stress factors, the interaction component of those stress factors are eliminated from the interaction model. That special case is not discussed in this paper. In this section, a simplified approach, which could be used to verify the existence of interaction, is described. Graphical response analysis (GRA) and the analysis of variance ANOVA [50] are commonly used techniques for assessing the presence of stress interaction.

The main objective of GRA is to ascertain the factors and interactions that significantly affect the response as well as help determine the combination of factor levels to achieve the most desirable response. If the stress factors are few (2 or 3), GRA is efficient and easy to implement. For a higher number of stress factors, ANOVA is more efficient [50,51]. Based on the few stress factors considered in this work, only the GRA approach is discussed. GRA implementation, often follows the below steps.

• Identify the experimental stress factors and levels.
• Develop the experimental layout and generate results based on the layout.

  Table 4. Experimental Layout.

  Run	Factor and Interactions			Responses	$\widehat{\mathit{{\rm Y}}}$	Mean
  	A	B	A × B
  1	0	0	0	$y_{11}, y_{12}, \dots , y_{1k}$	${{\rm Y}}_1$	${\overline{y}}_1$
  2	0	1	1	$y_{21}, y_{22}, \dots , y_{2k}$	${{\rm Y}}_2$	${\overline{y}}_2$
  3	1	0	1	$y_{31}, y_{32}, \dots , y_{3k}$	${{\rm Y}}_3$	${\overline{y}}_3$
  4	1	1	0	$y_{41}, y_{42}, \dots , y_{4k}$	${{\rm Y}}_4$	${\overline{y}}_4$

  depicts a hypothetical layout.
• Compute the average responses ${\overline{y}}_i$ at the $i\mathrm{th}$ stress combination using (29), where $n_i$ is the number of responses. For the solder joint evaluation, $n_i$ is the number of test samples. $y_{ik}$ is the responses of the $i\mathrm{th}$ run and $k\mathrm{th}$ sample.

  $${\overline{y}}_i=\frac{1}{n_i} {\displaystyle \sum }_{k=1}^{n_i}{y}_{ik}$$

  (29)
  Estimate the signal-to-noise ratio $\widehat{{\rm Y}}$ based on the quality response characteristics. The signal-to-noise ratio depends on the quality characteristics that drive the evaluation. Commonly used quality characteristics include nominal-the-better, smaller-the-better and larger-the-better. For life-based response, the nominal-the-better quality characteristic is preferred. The $\widehat{{\rm Y}}$ for nominal-the-better quality characteristics is given as (30).

  $${{\rm Y}}_i=10\log \left(\frac{u_i^2}{{\sigma }_i^2}\right)$$

  (30)

  where the variance (${\sigma }_i^2$) of the $i\mathrm{th}$ run is given as ${\sigma }_i^2=\frac{1}{n_i-1}{\displaystyle \sum }_{k=1}^{n_i}{\left(y_{ik}-{\overline{y}}_i\right)}^2$, and the mean ($u_i^2$) is given as $u_i^2=\frac{D_i-{\sigma }_i^2}{n_i}$. $D_i$ is given as $D_i=n_i{\overline{y}}_i{}^2$.
• For two-stress factor problems, construct a two-way table for the average response of the interaction between the two stress factors. For three-factor stress scenarios, with the stress factors represented as A, B and C, generate a two-way table for the interaction between A and B, B and C, A and C and [(A and B) and C]. The two-way interaction table for Table 4 is given as

  Table 5. Average response of the interaction between stress factors.

  Level	${\mathit{A}}_0$	${\mathit{A}}_1$
  ${\mathit{B}}_0$	${{\rm Y}}_1$	${{\rm Y}}_3$
  ${\mathit{B}}_1$	${{\rm Y}}_2$	${{\rm Y}}_4$

  .
• Plot the average response of the interaction between the two stress factors.
• If the average response line of a factor intersects with the average response of another factor, then interactions between the two factors exist. Otherwise, interactions do not exist.
• Repeat Steps 5–6 for all possible factor interactions.

An illustration of the GRA method is briefly presented. Given the two-stress factor experimental layout—the responses and signal-to-noise ratio are shown in

Table 6. An illustration of two-level, two-stress factor experimental layout, responses and signal-to-noise estimates.

Run	Factor and Interactions			Responses			$\widehat{\mathit{{\rm Y}}}$
	A	B	A × B
1	0	0	0	1900	2200	2400	$18.68$
2	0	1	1	1800	2500	2800	$13.20$
3	1	0	1	2800	3600	3800	$15.58$
4	1	1	0	2600	2900	3100	$17.29$

—then the plot of the average responses of the interactions between the factors A and B are given in Figure 6. The intersection indicates interaction between factors A and B, which validates the selection of a multi-stress ALT model with interaction. For a three-stress factor A, B and C scenario, the steps described above are used to check the interactions between A and B; B and C; and A and C. If interactions exist in each of the three combinations, then interactions exist between A,B and C. Hence, the model with interactions is selected.

4. Results and Discussions

4.1. Parameter Estimation

In Section 3.3.2, a Monte Carlo simulation used to generate 50 pseudo-times-to-failure under each multi-stress combination (See Table 2) was described. The pseudo-generated lifetime of each multi-stress combination is shown in Figure 7. The Weibull distribution logarithm of pseudo-failure time against the probability of failure plot and the trend lines for each of the stress combinations are shown in Figure 8. The approximately linear nature of the probability plots for each stress condition indicate that the assumed Weibull distribution was acceptable. Using the generated failure time given in Figure 8, the stress combination in Table 2, and mathematical expression of (21) and (22) for with and without interaction, respectively, the PSO described in sub-Section 3.3.3 was implemented. Figure 9 depicts the plot of log-likelihood function value (fitness value) against number of iterations for with and without interaction. For each plot, five simulation runs were carried out. The simulation path of each is clearly shown in Figure 9. The result reveals that convergence occurs after 205 and 48 iterations for with and without interaction. This variation in convergence was due to fewer estimation parameters for the case of without interaction when compared to that with interaction. Upon the convergence, the optimized parameters estimated for both scenarios are shown in

Table 7. Estimated model parameters.

Modeling Scenario	$\tilde{\mathit{\beta }}$	$\widetilde{{\mathit{\gamma }}_0}$	$\widetilde{{\mathit{\gamma }}_1}$	$\widetilde{{\mathit{\gamma }}_2}$	$\widetilde{{\mathit{\gamma }}_3}$	$\widetilde{{\mathit{\gamma }}_4}$	$\widetilde{{\mathit{\gamma }}_5}$	$\widetilde{{\mathit{\gamma }}_6}$	$\widetilde{{\mathit{\gamma }}_7}$
Without interaction	1.5	10	3.218	−1.969	0.176
With interaction	1.5	10	−1.697	−2.239	−5.135	5	−1.346	5	5

. The same value of the shape parameter indicates that the stress interaction did not cause a different failure mechanism. Similarly, the same $\widetilde{{\gamma }_0}$ value, which can be used to estimate the scale parameter at normal operational conditions $\zeta ={\mathrm{e}}^{\widetilde{{\gamma }_0}}={ \mathrm{e}}^{10}$ indicates the solder joint considered can survive for approximately 2.5 years with 62.5% probability. Furthermore, several runs of the PSO generated the same values for each parameter. This indicates lack of estimation variation and a standard error of zero, a key benefit of the PSO method.

4.2. Accelerated Test Plan

Numerical Example

An accelerated life test plan for lead-tin (PbSn) solder interconnects was carried out using three accelerating stresses: temperature, current and mechanical stress. The reliability estimate at the design condition over a 2.5-year period is of interest. Assuming that ${\psi }_{1U}=75 °\mathrm{C}$, ${\psi }_{2U}=2960.29 \frac{\mathrm{A}}{{\mathrm{cm}}^2}$ and ${\psi }_{3U}=0\mathrm{MPa}$ are the operating temperature, current density and mechanical stress, respectively. By engineering judgment, the highest levels (upper bounds) of temperature, current density and mechanical stress are pre-specified as 120 °C, $6907.44 \mathrm{A}/{\mathrm{cm}}^2$ and 394.71 MPa, respectively. The total number of test units placed under test is 50. The minimum number of failures at any test combination is specified as 5, and each stress was evaluated at four stress levels. The test plan that minimizes reliability estimation variation was determined. The test plan was determined based on the steps:

• Generate stress transformation for temperature, current density and mechanical stress using (5), (6) and (7), respectively.
• Generate the log-likelihood second partial derivative s-negative expectation with respect to each parameter in (21) and (22) for with and without interaction to form the FIM for with and without interaction, respectively.
• For each experimental run, randomly generate stress levels in agreement with the LHD cases provided in Table 3 that satisfy the constraint conditions in (31) and (32).
• Randomly generate a number $P_j$ between 0 and 1 for each experimental run and each LHD case which satisfies the constraint conditions.
• Insert the model parameters for with and without interaction given in Table 7, the stress levels obtained from Step 3, and $P_j$ values obtained from Step 4 into the objective functions of (31) and (32) and compute its value.
• Compute the determinant.
• If the determinant of the next iteration is more than the present iteration, the stress levels of each stress factor and $P_j$ values of the next iteration for each experimental run associated with that determinant are reported as the optimum. Otherwise, the stress levels and $P_j$ values of the preceeding iteration are retained as optimum.
• If termination criteria are satisfied, display the stress levels of each stress factor and $P_j$ value of the optimum determinant.

  $$\begin{gathered} Max(Det\left({\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ijw}\right) \\ s.t \\ 0<P_j<1; j=1,2,3,4 \\ {\displaystyle \sum }_{j=1}{P}_j=1 \\ 75 °\mathrm{C}\le {\psi }_{11}<{\psi }_{12}<{\psi }_{13}<{\psi }_{14}\le 120 °\mathrm{C} \\ 2960.29\frac{\mathrm{A}}{\mathrm{c}{\mathrm{m}}^2}\le {\psi }_{21}<{\psi }_{22}<{\psi }_{23}<{\psi }_{24}\le 6907.44\frac{\mathrm{A}}{\mathrm{c}{\mathrm{m}}^2} \\ 0\le {\psi }_{31}<{\psi }_{32}<{\psi }_{33}<{\psi }_{34}\le 394.71\mathrm{MPa} \end{gathered}$$

  (31)

  $$\begin{gathered} Max(Det\left({\displaystyle \sum }_{j=1}^{wz}{\displaystyle \sum }_{i=1}^{N{P}_j}{F}_{ijwo}\right) \\ s.t \\ 0<P_j<1; j=1,2,3,4 \\ {\displaystyle \sum }_{j=1}{P}_j=1 \\ 75 °\mathrm{C}\le {\psi }_{11}<{\psi }_{12}<{\psi }_{13}<{\psi }_{14}\le 120 °\mathrm{C} \\ 2960.29\frac{\mathrm{A}}{\mathrm{c}{\mathrm{m}}^2}\le {\psi }_{21}<{\psi }_{22}<{\psi }_{23}<{\psi }_{24}\le 6907.44\frac{\mathrm{A}}{\mathrm{c}{\mathrm{m}}^2} \\ 0\le {\psi }_{31}<{\psi }_{32}<{\psi }_{33}<{\psi }_{34}\le 394.71\mathrm{MPa} \end{gathered}$$

  (32)

The optimization outputs are ${\psi }_{11},{\psi }_{12},{\psi }_{13},{\psi }_{14}$, ${\psi }_{21},{\psi }_{22},{\psi }_{23},{\psi }_{24}, {\psi }_{31},{\psi }_{32},{\psi }_{33},{\psi }_{34}$ and $N{P}_j$, $j=1, 2, 3,4)$ for with and without interaction. ALT planning framework depicted in Figure 5 was used to solve the optimization problem. The optimized results are provided in Figure 10. The results show three simulated optimal search paths for each of the scenarios considered. Generally, convergence occurred in less than 69 and 47 iterations for the without interaction based on the LHD Case I and II, respectively. On the other hand, over 8000 iterations were required for convergence to occur in both LHD Case I and II with interaction. This was primarily due to more parameters associated with the with-interaction model, which led to a more complex derivative contained in a 9 × 9 matrix. The finding agrees with the study of [21,49], which established that a higher number of search parameters require more iterations. The optimized sample allocation and stress levels for each LHD Case I stress combination are given in

Table 8. LHD Case I optimized sample allocation.

	Without Interaction					With Interaction
Experimental Runs	$\mathit{T} °\mathbf{C}$	$\mathit{j}\left(\mathbf{A}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$	${\mathit{P}}_{\mathit{j}}$	Optimized Sample Allocation	$\mathit{T} °\mathbf{C}$	$\mathit{j}\left(\frac{\mathbf{A}}{\mathbf{c}{\mathbf{m}}^2}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$	${\mathit{P}}_{\mathit{j}}$	Optimized Sample Allocation
${\mathit{z}}_1$	1	2	3	0.40	20	1	2	3	0.48	24
${\mathit{z}}_2$	2	1	4	0.16	8	2	1	4	0.24	12
${\mathit{z}}_3$	3	4	2	0.12	6	3	4	2	0.12	6
${\mathit{z}}_4$	4	3	1	0.32	16	4	3	1	0.16	8

and

Table 9. LHD Case I optimized stress levels.

Stress Factor	Without Interaction					With Interaction
	${\mathit{\psi }}_{\mathit{i}0}$	${\mathit{\psi }}_{\mathit{i}1}$	${\mathit{\psi }}_{\mathit{i}2}$	${\mathit{\psi }}_{\mathit{i}3}$		${\mathit{\psi }}_{\mathit{i}0}$	${\mathit{\psi }}_{\mathit{i}1}$	${\mathit{\psi }}_{\mathit{i}2}$	${\mathit{\psi }}_{\mathit{i}3}$
$\mathit{T} °\mathbf{C}$	75.42	85.76	97.27	119.52		84.97	96.41	108.23	119.52
$\mathit{j}\left(\frac{\mathbf{A}}{\mathbf{c}{\mathbf{m}}^2}\right)$	3651.0	3670.81	5585.16	6848.23		3651.07	4519.44	5583,16	6848.23
$\mathit{\sigma } \left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	48.94	98.68	201.30	390.76		3.95	197.35	296.03	390.76

. Interestingly, although LHD Case I was based on this constraint $({\psi }_{i0}<{\psi }_{i1}<{\psi }_{i2}<{\psi }_{i3})$ condition, the optimized result prescribes current stress levels 1 and 2 to be approximately the same. This indicates that while four stress levels for temperature and load were appropriate, only three stress levels were required in the other to minimize the variation associated with the proposed ALT plan. The sample allocation for each test’s stress combination shows more samples allocated to test combinations with low stress levels for each stress factor and less sample allocation to stress combinations with higher stress levels. This is in agreement with the findings in [6]. Generally, more sample allocation to a stress combination with low stress levels of each stress factor increases the probability of generating sufficient failure samples to facilitate reliability analysis. A similar trend was observed for the LHD Case II optimized sample allocation and stress levels given in

Table 10. LHD Case II optimized sample allocation.

	Without Interaction					With Interaction
Experimental Runs	$\mathit{T} °\mathbf{C}$	$\mathit{j}\left(\mathbf{A}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$	${\mathit{P}}_{\mathit{j}}$	Optimized Sample Allocation	$\mathbf{T} °\mathbf{C}$	$\mathit{j}\left(\frac{\mathbf{A}}{\mathbf{c}{\mathbf{m}}^2}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$	${\mathit{P}}_{\mathit{j}}$	Optimized Sample Allocation
${\mathit{z}}_1$	1	1	4	0.29	14	1	1	4	0.48	24
${\mathit{z}}_2$	2	4	3	0.145	8	2	4	3	0.24	12
${\mathit{z}}_3$	3	3	2	0.097	5	3	3	2	0.12	6
${\mathit{z}}_4$	4	2	1	0.468	23	4	2	1	0.16	8

and

Table 11. LHD Case II optimized stress levels.

Stress Factor	Without Interaction					With Interaction
	${\mathit{\psi }}_{\mathit{i}0}$	${\mathit{\psi }}_{\mathit{i}1}$	${\mathit{\psi }}_{\mathit{i}2}$	${\mathit{\psi }}_{\mathit{i}3}$		${\mathit{\psi }}_{\mathit{i}0}$	${\mathit{\psi }}_{\mathit{i}1}$	${\mathit{\psi }}_{\mathit{i}2}$	${\mathit{\psi }}_{\mathit{i}3}$
$\mathbf{T} °\mathbf{C}$	75.42	96.82	97.27	119.52		75.42	96.82	108.23	119.52
$\mathit{j}\left(\frac{\mathbf{A}}{\mathbf{c}{\mathbf{m}}^2}\right)$	2980.07	4519.44	4558.91	6848.23		3651.07	4519.44	5585.16	6848.23
$\mathit{\sigma } \left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$	3.95	99.07	296.03	390.76		3.95	197.35	296.03	390.76

, respectively.

The results also show that the presence of stress interaction had an effect on the optimized ALT plan. To illustrate, Table 9 and Table 11 show that the optimized stress levels for each stress factor with stress interaction had a distinct interval from a lower to a higher level. On the other hand, the optimal test plan for without interaction for LHD Case I shows current density for Level 1 and Level 2 to be 3651.0 $\frac{\mathrm{A}}{{\mathrm{cm}}^2}$ and 3670.81 $\frac{\mathrm{A}}{{\mathrm{cm}}^2}$, respectively. Similarly, in LHD Case II, Level 2 and Level 3, were found to be 4519.44 $\frac{\mathrm{A}}{{\mathrm{cm}}^2}$ and 4558.91 $\frac{\mathrm{A}}{{\mathrm{cm}}^2}$. These approximate current density values suggest that current density Levels 1 and 2 for without interaction could be merged as one level for LHD Case I. Similarly, current density Levels 2 and 3 for Case II without interaction could be merged as a single level. Hence for Case I, we adopted the average of the optimized values of current density obtained for Levels 1 and 2 as Level 1, Level 3 as Level 2, and Level 4 as Level 3. Similarly, for Case II, Level 1 remains the average of Level 2, and Level 3 becomes Level 2, while Level 4 becomes Level 3. In summary, while four levels of each stress factor were required for a multi-stress ALT plan necessary to understand the electromigration in solder joints if stress interactions exist, only three levels of current density as well as four levels of both temperature and mechanical stress were required if stress interaction is absent. This finding reveals the need for stress interaction verification prior to implementing multi-stress models in ALT planning especially. This key contribution, to the best of our knowledge, has not been reported before for solder joint electromigration multi-stress ALT planning. Based on the optimized stress levels shown in Table 9 and Table 11, the optimized test plan for LHD Case I and Case I for without interaction is depicted in

Table 12. Optimized stress levels for without interaction for Case I and Case II.

Experimental Runs	Proposed LHD Case I				Optimized Stress Levels
	$\mathbf{T} °\mathbf{C}$	$\mathit{j}\left(\mathbf{A}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$		$\mathbf{T} °\mathbf{C}$	$\mathit{j}\left(\mathbf{A}\right)$	$\mathit{\sigma } (\mathbf{M}\mathbf{P}\mathbf{a})$
${\mathit{z}}_1$	1	1	4		1	1	4
${\mathit{z}}_2$	2	4	3		2	3	3
${\mathit{z}}_3$	3	3	2		3	2	2
${\mathit{z}}_4$	4	2	1		4	1	1
	Proposed LHD Case II				Optimized stress levels
${\mathit{z}}_1$	1	2	3		1	2	3
${\mathit{z}}_2$	2	1	4		2	1	4
${\mathit{z}}_3$	3	4	2		3	3	2
${\mathit{z}}_4$	4	3	1		4	2	1

. The LHD Case I and Case II FIM determinants were compared as depicted in

Table 13. Comparison of the LHD Cases I and II D-optimality criteria values.

Multi-Stress ALT Models	FIM Determinant
	LHD Case I	LHD Case II
Without Interaction	1.68 × 10−18	1.52 × 10−18
With Interaction	8.89 × 10−73	7.84 × 10−73

. From the results, LHD Case I has a higher determinant for both with and without interaction. This indicates lower variation and thus is better from the perspective of an optimized ALT plan compared to Case II.

The technique, presented in this work, simply used the lower and upper bound stresses of each stress factor required to ensure that failure mode is not experienced under normal operating conditions and is avoided during a test. Although the Arrhenius model was used in this study for method validation purposes, any model (empirical-, physical- or physics-based) which correctly describes the life of a product can be used to replace the Arrhenius model. Similarly, other failure distribution such as log normal, gamma or exponential could replace the Weibull distribution used in this work if it best describes the product failure distribution. Following the estimation of model parameters, PSO or any other optimization algorithm can be used to generate the optimal test plan. Most importantly, the study revealed that verification of stress interactions is necessary in multi-stress accelerated life test planning in order to increase the accuracy of the generated optimal ALT plan.

PSO convergence time for multi-stress ALT planning with interaction still requires improvement. This problem occurs due to the lack of population diversity especially in complex multi-stress models. In the future, hybrid PSOs—involving PSO implementation for model parameter estimation and novel metaheuristic optimization algorithms, such as gradient-based optimizer (GBO) [52] for optimal ALT plan identification—will be used to evaluate this convergence performance. In addition, implementation of machine learning-based algorithms, with the view to reduce the convergence time, will also be considered. Despite the extensive efforts of proposing several methods to control PSO constants such as ${\mathrm{c}}_1$, ${\mathrm{c}}_2$ and $\mathrm{w}$ found in literature [30], there is no guarantee that these constants are optimal. This peculiar problem has the potential to affect velocity adjustment. Future study will explore optimizing the parameters prior to implementation. Furthermore, the multi-stress ALT modeling in this study only considered completely exact data scenarios. Inability to generate failure data within a reasonable time at lower levels of all stress conditions indicates the need to extend the work to failure time censoring models also known as Right-Censored Exact Data. Hence, in future work, we will extend this study method to Right-Censored Exact Data as well as Right-Censored Interval Data, while implementing novel meta-heuristic algorithms that address convergence problems in a speedy manner.

5. Conclusions

In this article, a technique was presented to facilitate multi-stress ALT planning. Specifically, the effect of stress factor interaction on the optimal ALT plan was evaluated. The method involves the use of upper and lower bound stress levels for each stress factor considered to generate limited initial failure data. Pseudo-life data are subsequently generated from the failure data based on the Monte Carlo technique. The Arrhenius model was used to develop a characteristic life model. Using the pseudo-life data and the characteristic life model, particle swarm optimization algorithm was used to estimate model parameters. The optimized model parameter, D-optimality criterion, LHD, and a second-stage optimization activity based on PSO were used to estimate the optimal ALT plan. An experimental setup designed to evaluate electromigration in solder joints was used to validate the study. From the study, the following conclusions were reached:

• A multi-stress ALT plan suitable for a scenario with stress interaction may not be suitable if stress interaction does not exist.
• The presence of stress interaction increases model parameters, s-expectation partial derivative complexity as well as exponential raise in convergence time, especially for stage two PSO implementation.
• It is extremely important to verify the presence of stress interaction in order to increase the accuracy of on an optimized ALT plan