Ann Oper Res (2014) 212:201–214 DOI 10.1007/s10479-013-1419-z
Bi-objective burn-in modeling and optimization Zhi-Sheng Ye · Loon-Ching Tang · Min Xie
Published online: 16 July 2013 © Springer Science+Business Media New York 2013
Abstract This study develops a bi-objective method for burn-in decision makings with a view to achieving an optimal trade-off between the cost and the performance measures. Under the proposed method, a manufacturer specifies the relative importance between the cost and the performance measures. Then a single-objective optimal solution can be obtained through optimizing the weighted combination of these two measures. Based on this method, we build a specific model when the performance objective is the survival probability given a mission time. We prove that the optimal burn-in duration is decreasing in the weight assigned to the normalized cost. Then, we develop an algorithm to populate the Pareto frontier in case the manufacturer has no idea about the relative weight. Keywords Cost-based burn-in · Performance-based burn-in · Pareto frontier
1 Introduction To meet a sequence of performance specifications, the design reliability of a new product is often very high. During the post-development stage of the product life cycle, however, the actual reliability usually differs from the designed one due to various quality variations such as non-conforming components that sneak through supplier’s screening, and damage incurred to the components and substrates during assembly, etc. These variations lead to the bathtub or W-shape of the failure rates, as discussed in-depth by Jiang and Murthy (2009). In fact, the bathtub curve has become one of the hallmarks in reliability engineering (cf. Singpurwalla 2006, p. 72). Burn-in is an important engineering approach used to identify
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Z.-S. Ye ( ) · L.-C. Tang · M. Xie Department of Industrial & Systems Engineering, National University of Singapore, Singapore, Singapore e-mail:
[email protected] M. Xie Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China
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and reduce these quality variations. It has been proved effective in dealing with infant mortality, and has been widely used in the semi-conductor industries. Burn-in is often conducted under harsh environments such as high temperature, humility, vibrations and elevated voltage. MIL-STD-883G (2006) specified 6 basic burn-in conditions for microelectronic devices. Latent flaws such as the material defects, software faults and manufacturing defects are precipitated during burn-in. There is a vast amount of literature directing towards this topic. See Kuo et al. (1998) for a review of the related literature and Singpurwalla (2006, Chap. 4) for a Bayesian justification. Burn-in is a cost-intensive procedure because (1) lots of electric power and human resources are consumed, and (2) it prolongs time-to-market of a product. But product reliability can be enhanced after burn-in, leading to less field cost in return. The total book costs for a burnt-in product include both burn-in costs and the tangible field failure costs, e.g., handling and administrative costs. Many burn-in models have been developed to help decide on the optimal burn-in duration by minimizing the total book costs, e.g., see Mi (1996), Cha and Finkelstein (2010a), Yuan and Kuo (2010), among others. Nevertheless, field failures also affect customer satisfactions and lead to intangible costs such as losses of reputation and customer loyalty. These intangible losses have significant effects on product benefits, and even on the overall benefits of the company, as the customer loyalty has great impacts on customer retentions and the first purchase decisions of new customers (cf. Reichheld and Teal 2001). The intangible costs are much more difficult to quantify compared with the book costs. The intangible costs are determined by the performance, especially reliability, of a burnt-in unit. Accordingly, many researchers have proposed a variety of performancebased burn-in models, e.g., Cha and Finkelstein (2010b) and Ye et al. (2011). Optimal burn-in decisions based on the cost and the performance measures are different. For a nontrivial burn-in optimization problem, there does not exist a single burn-in period that simultaneously optimizes both objectives. In order to take into account both measures, some studies transform the intangible losses to a penalty cost by using a penalty factor (e.g., Kuo 1984), and then the total cost consists of the book costs and the penalty cost. However, determination of the penalty factor is difficult and arbitrary in these studies. Alternatively, this study treats these two measures as two separate objectives, under which the bi-objective optimization technique can be applied to find the optimal trade-off between these two measures. Generally speaking, there are two methods to solve a bi-objective model. The first method is called the weighted-sum method (Chanta et al. 2011). It requires the manufacturer to specify relative weights of these two objectives, after which the bi-objective model can be transformed to a single objective one. The best-compromised solution can then be found. On the other hand, when the manufacturer does not have a general idea about the relative importance of these two objectives, Pareto optimal solutions for the problem can be populated. The Pareto optimal solutions help the manufacturer select a best-compromised burn-in duration. We then apply the bi-objective method to build a concrete burn-in model where the performance objective is the survival probability given a mission time. Many burn-in studies have used the survival probability within a mission time as the performance objective. For the cost function, we consider a popular setting called gain due to no failure within the mission time (e.g., Mi 1996; Sheu and Chien 2004 and Cha 2006). Most previous burn-in studies on this setting stressed the problem from the perspective of either cost or performance. Our bi-objective model jointly considers these two objectives, and is flexible in quantifying the penalty of unmet performance. Both the weighted-sum method and the Pareto optimal solutions will be discussed.
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The remainder of the paper is organized as follows. Section 2 reviews the literature on cost-based and performance-based burn-in models, after which the bi-objective burnin problem is formulated. Section 3 investigates properties of the bi-objective burn-in model when the performance objective is the survival probability given a mission time. An algorithm is developed to populate the Pareto frontier. An illustrative example is provided in Sect. 4. Section 5 deals with some concluding comments.
2 Cost and performance objectives in burn-in 2.1 Burn-in to minimize total book costs Typical burn-in costs for each burn-in unit include • Fixed set-up cost; • Burn-in operational cost which is approximately linear in the burn-in duration b; • Disposal cost of a weak unit when the product is non-repairable, or repair cost when the product is repairable. Denote the expected burn-in costs for each unit with a burn-in duration of b by CB (b). After a burnt-in unit is put into field use, unexpected failures of the unit would engender some book costs, e.g., field repair costs or warranty costs. Most existing models consider either maintenance costs or warranty costs. We summarize them as follows. • Minimization of joint burn-in and maintenance costs, e.g., Mi (1996), Pohl and Dietrich (1999), Sheu and Chien (2004), Jiang and Jardine (2007) and Cha and Finkelstein (2010a). After burn-in, a legitimate preventive maintenance schedule is able to further reduce the total costs. Models in this class seek to simultaneously determine the optimal burn-in duration and the maintenance intervals. • Minimization of joint burn-in and warranty costs, e.g., Mi (1999), Yun et al. (2002), Sheu and Chien (2005), Yuan and Kuo (2010), Ye et al. (2012) and Shafiee et al. (2013). Most products are sold with warranty. Breakdown of a burnt-in unit within the warranty period causes warranty claims and transfers to warranty costs in return. We can use burn-in to achieve an optimal balance between the burn-in costs and the warranty costs. Different warranty policies lead to different cost structures. The expected field operation cost for a unit with burnt-in duration b is denoted as CF (b). The objective function of a cost-based model can be expressed as J1 (b) = CB (b) + CF (b).
(1)
The cost-based burn-in models attempt to determine the optimal burn-in scheme through minimizing the expected total cost J1 (b). 2.2 Burn-in to optimize field performance Sometimes, the manufacturer is more concerned with the intangible losses, such as reputation of the brand and the customer loyalty. Unmet performance is associated with the user’s perceived risks and satisfaction with this product, and thus it can be used to represent these losses. The performance objective can be either the survival probability over a pre-specified mission time, the mean residual life (MRL), or percentiles of the residual life. These criteria are summarized and discussed as follows.
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• Minimization of the expected number of field failures over a mission time. This measure is directly related to repair costs and warranty costs. Minimizing this measure leads to low book costs. The major concern of this approach is how to model subsequent failures after the first failure. If a complete repair or replacement is adopted, then the system is “good-as-new” after repair and the renewal theory can be used to compute the expected number of field failures. If a minimal repair is assumed, then the system is “bad-as-old” after repair, and the expected number of field failures is equal to the cumulative hazard rate function of a burnt-in unit. For other types of repair and the corresponding modeling techniques, interested readers are referred to Wu and Zuo (2010) for an excellent summary. • Maximization of the survival probability over a mission time, e.g., Mi (1994), Kim and Kuo (2005, 2009). Even when a product is repairable, maximizing the survival probability presents an attractive alternative because it is directly related to customer satisfaction. In fact, this is often the target of product design (Murthy et al. 2009). • Maximization of the percentiles of the residual life, e.g., Ye et al. (2011). This criterion leads to the maximum allowable mission time that the product can offer given a prespecified proportion of failures. • Maximization of the mean residual life (MRL), e.g., Bebbington et al. (2007). The MRL is an important performance index in the context of reliability engineering. If there is no scheduled preventive maintenance, this criterion results in lowest long term average field operation costs. Let J2 (b) be the objective function. A performance-based burn-in model seeks to find an optimal burn-in duration so as to optimize J2 (b). 2.3 Bi-objective burn-in In this study, we propose to treat the cost and the performance objectives as two separate goals. When the manufacturer is able to specify the relative importance of these two objectives, we can use the weighted-sum method to combine these two objectives. When the manufacturer has no idea about the weight, we can populate all Pareto optimal solutions (J1 , J2 ) from which the manufacturer can choose the best compromise afterwards. A plot of all Pareto optimal solutions in the objective space gives a Pareto frontier. The frontier helps visualize the results. However, it is a challenge to populate the Pareto optimal solutions. The weighted-sum method is a straightforward approach to a multiple-objective problem. Therefore, we discuss this method for the bi-objective burn-in problem first. This requires the manufacturer to normalize the objective function before the construction of a specific burn-in model. For the cost objective, we propose using J1 (0) = CF (0) as the normalizing constant. Then 1−J1 (b)/J1 (0) is the relative cost reduction from burn-in. On the other hand, we propose using J2 (0) as the normalizing constant with regard to the performance objective. Then J2 (b)/J2 (0) − 1 is the relative improvement of the performance through burn-in when the performance objective is the-larger-the-better, e.g., the MRL. When this objective is the-smaller-the-better, the relative improvement of the performance is 1 − J2 (b)/J2 (0). After the normalization, weights are assigned to the two normalized objectives to represent their relative importance. Denote w1 ≥ 0 and w2 ≥ 0 as the weights assigned to the cost and the performance objectives, respectively. Here, we consider only convex combinations as it is always possible to normalize w1 and w2 so that w1 + w2 = 1. With these set-ups in place, the bi-objective framework can be constructed. The cost objective is the-smaller-the-better. If the performance objective is the-larger-the-better, the
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Fig. 1 The procedure to construct a specific bi-objective model
bi-objective framework is min J = w1 · J1 (b)/J1 (0) − (1 − w1 ) · J2 (b)/J2 (0) s.t. gi (b) ≤ 0,
i = 1, 2, . . . , n,
(2)
where gi (b) are possible constraints imposed on the burn-in procedure. These constraints may include a time limit on the burn-in duration as well as limited burn-in facilities. If the performance objective is the-smaller-the-better, the bi-objective framework becomes min J = w1 · J1 (b)/J1 (0) + (1 − w1 ) · J2 (b)/J2 (0) s.t. gi (b) ≤ 0,
i = 1, 2, . . . , n.
(3)
The procedure to use this framework is visualized in Fig. 1. It is readily found that when w1 = 0, the model reduces to a purely performance-based one. On the other hand when w1 = 1, the objective function captures the book costs, and the optimal burn-in duration achieves a balance between burn-in costs and costs due to field failures. When 0 < w1 < 1, there is a trade-off between the performance and the costs under the bi-objective framework. An alternative way to understand the framework is to treat the performance objective as a penalty, under which we are able to convert the bi-objective model into a single objective one. Some other studies also take the unmet performance as a penalty. For example, Toyota uses a multiple of six times of the repair cost for a field failure to measure the reputation cost, while the Westinghouse uses a multiple of four times (Balachandran and Radhakrishnan 2005). In effect, it is difficult to quantify the costs for such penalty. Our framework effectively fixes the problem by simply requiring the manufacturers to specify the weights that they would like to assign to these two objectives. When the weights are difficult to assign at the beginning, the Pareto frontier can be provided to help with the decision. The weights w1 and w2 can be determined according to the decision maker’s preference. A weight can be interpreted as the relative worth of that objective when compared to the other objective. The optimal solution depends on a manufacturer’s particular preference structure. Therefore, the solution to a specific bi-objective model is indeed the bestcompromise solution. If the manufacturer believes that these two objectives are equally important, the weights can be set as w1 = w2 = 0.5. However if the manufacturers are not sure about the relative importance, they can perform a pair-wise comparison judgment on
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a set of criteria, e.g., the current flow, the reputation and the customer loyalty, with respect to these two objectives. The analytic hierarchy process (AHP) can be effectively applied to determine the weights (Saaty 1994). The procedure is quite standard and goes beyond the scope of this work, thus it is not discussed here.
3 Model optimization with reliability as the performance objective In order to discuss properties of the proposed bi-objective method, we need to specify the cost function and the performance objective. We assume that a burnt-in system will be put under operation with a mission time τ . The survival probability of this burnt-in unit within the mission time, denoted as R(τ |b), is adopted as the performance objective as it is directly related to customer satisfaction. The cost of a unit consists of the burn-in cost CB (b), which is monotone increasing in the burn-in duration b, and the field operational cost CF (b). We assume that if the unit fails within the mission time τ , a breakdown cost of Cf is incurred. For example, the mission time can be a warranty limit, and then this breakdown cost includes the warranty cost and the administrative cost. The probability that the burnt-in unit fails within τ is 1 − R(τ |b), Therefore, the total cost for the unit is J1 (b) = CB (b) + 1 − R(τ |b) Cf .
(4)
Based on this cost function and the reliability objective, detailed properties of the biobjective model can be investigated. In the bi-objective model, we use R(τ |0) and [1 − R(τ |0)]Cf as normalization constants for the performance and the cost objectives, respectively. 3.1 Bounds for the optimal burn-in duration As w1 varies from 0 to 1, the optimal burn-in duration changes. Denote this optimal duration as bw∗ 1 . Then the optimal burn-in time that maximizes the system reliability is b0∗ while the optimal burn-in time that minimizes the total cost function is b1∗ . Lower and upper bounds for the optimal burn-in time are desirable as they define the range of all possible choices. Some useful results are given in Theorem 1. Theorem 1 Assume all parameters except w1 are fixed. When 0 ≤ w1 ≤ 1, the optimal burnin time bw∗ 1 of the bi-objective model is decreasing in w1 . Moreover, bw∗ 1 is not greater than b0∗ and is not less than b1∗ , i.e., b1∗ ≤ bw∗ 1 ≤ b0∗ for any w1 . Proof When w1 = 0, by substituting (4) into the objective function of the bi-objective model (2), we can rewrite the objective function as J (b|w1 ) =
Cf 1 − w1 CB (b + Cf ) − + R(τ |b). J1 (0) w1 R(τ |0) J1 (0)
(5)
For a specific choice of w1 , denote its optimal burn-in time by bw∗ 1 . For any b < bw∗ 1 , we have J (b|w1 ) ≥ J bw∗ 1 |w1 . (6)
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A useful result shall be proved first. For any b < bw∗ 1 , because the burn-in cost is an increasing function of the burn-in duration b, we can know CB (b) < CB bw∗ 1 . If we suppose R(τ |b) ≥ R τ |bw∗ 1 , we have J (b|w1 ) < J (bw∗ 1 |w1 ) from (5), which contradicts with (6). Therefore for any b < bw∗ 1 , the following relation should always hold: R(τ |b) < R τ |bw∗ 1 .
(7)
Consider another choice w1 with w1 < w1 , J (b|w1 ) can be expressed as 1 − w1 1 − w1 R(τ |b) . J b|w1 = J (b|w1 ) + − w1 w1 R(τ |0)
(8)
Note that the second term on the right hand side is negative. By substituting (6) and (7) into (8), we have that for any b < bw∗ 1 , J b|w1 > J bw∗ 1 |w1 . Therefore, bw∗ ≥ bw∗ 1 . 1
(9)
Inequality (9) implies that bw∗ is decreasing in w1 . When w1 → 1, the model reduces to 1 the case of minimizing the cost function. When w1 → 0, the model reduces to the case of maximizing the survival probability, or equivalently, minimizing the expected number of field failures when minimal repair is assumed. Therefore, bw∗ 1 is not greater than b0∗ and bw∗ 1 is not less than b1∗ . Therefore, the theorem holds. This theorem provides meaningful insights into the burn-in problems. When w1 = 1, the model is purely cost-based. The optimal burn-in duration achieves a balance between the burn-in cost and costs due to field failures. When w1 < 1, there is a trade-off between reliability and costs in the bi-objective model. An alternative way to understand the model is to treat the unmet reliability objective as a cost penalty, under which we are able to convert the bi-objective model into a purely cost-based one. If more weight is assigned to the total costs, a relatively short burn-in duration can be employed since burn-in is costly. This is at the expense of a lower system reliability. If the reliability is important, which is embodied in a small w1 , a longer burn-in duration is desirable to attain a higher screening strength, at the price of higher costs. Remark Given the burnt-in reliability objective and the cost objective in this section, the optimal burn-in duration under the performance-based burn-in model, i.e., w1 = 0 is always greater than that under the cost-based burn-in model, i.e., w1 = 1.
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3.2 The Pareto frontier In practice, if the manufacturer is able to specify the relative weights between the cost and the performance objectives, then w1 in the scalarized model (2) or (3) is known and the optimal burn-in time can be readily computed by optimizing the scalarized objective function. There is only one decision variable, and therefore the optimization can be easily done using most available optimization techniques, e.g., the bisectional search. When the manufacturer is not able to subjectively provide the weight, however, we need to populate the Pareto frontier for the reference of the manufacturer. Two popular methods for populating the Pareto frontier are the random sampling method and the iterative application of the weighted-sum method. Theorem 1 above provides us with a convenient way to populate the Pareto frontier by using the weighted-sum method. First of all, let us discuss the weighted-sum method. It is well known that the solution to the scalarized problem presented in (2) or (3) is Pareto-optimal as long as 0 ≤ w1 ≤ 1. Therefore, we can uniformly discretize the interval [0, 1] using grid points k/n, k = 0, 1, . . . , n, and compute the optimal burn-in duration when w1 takes values at each of the grid points. This requires optimizing the objective function n + 1 times to get a Pareto set with n + 1 non-dominated solutions. With this Pareto set at hand, we can draw the Pareto frontier at the two-dimensional objective plane. However, there is no guarantee that any Pareto-optimal solution can be obtained using a weight w1 ∈ [0, 1], unless both the two objectives are convex functions of the burn-in duration b. In addition, uniformly division of the interval for the weight does not guarantee a uniformly distributed set of Pareto-optimal solutions. Moreover, optimizing the objective function n + 1 times may take a long time. Here, we can use a simple sampling method. To use this method, we first observe the fact that for any Pareto optimal solution, the optimal burn-in duration should be always less than b0∗ , the optimal burn-in time that maximize the system reliability. The result is given in the following theorem. Theorem 2 For the bi-objective burn-in optimization problem where the performance measure is the survival probability given a mission time while the cost function is given by (4), the optimal burn-in duration should always be less than b0∗ , the optimal burn-in time that maximize the system reliability. Proof When b > b0∗ , it is obvious that J2 (b) < J2 (b0∗ ) because a burn-in duration of b0∗ yields the highest reliability, i.e., R(τ |b) < R(τ |b0∗ ). This means that 1 − R(τ |b) Cf > 1 − R τ |b0∗ Cf . On the other hand, the expected burn-in cost is an increasing function of the burn-in duration b, i.e., CB (b) > CB (b1∗ ). Therefore, we have CB (b) + 1 − R(τ |b) Cf > CB b1∗ + 1 − R τ |b0∗ Cf . That is, J1 (b) > J1 (b0∗ ). Therefore, (J1 (b), J2 (b)) is dominated by (J1 (b0∗ ), J2 (b0∗ )). This proves the theorem. Based on this result, we can focus only on the burn-in durations with b ≤ b0∗ . A simple sampling method can thus be developed to find the Pareto-optimal solutions. Obviously, the solution with b = b0∗ is the Pareto optimal solution, and in fact, it is an endpoint of the Pareto frontier curve. A simple algorithm to find the other Pareto optimal solutions is given as follows.
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(a) Find the optimal burn-in duration b0∗ that maximizes the system reliability. (b) Uniformly divide the interval [0, b0∗ ] using n + 1 grid points bk = kb0∗ /n, k = 0, 1, . . . , n. (c) For each grid point of b, evaluate the corresponding expected cost J1 (b) and survival probability J2 (b) to obtain n + 1 candidate solutions Pk = (J1 (bk ), J2 (bk )). Normalize them if necessary. (d) Sort the n + 1 candidate solutions by the cost J1 (b) values in ascending order. Let k = 0. (e) Add Pk = (J1 (bk ), J2 (bk )) to the Pareto set. (f) Look downward at the sorted list and find the first j > k such that J2 (bj ) > J2 (bk ). Set k = j and go back to Step (e). Otherwise if not such j exists, stop the algorithm. Some remarks are readily made. The solutions found by this simple algorithm are all nondominated. In this algorithm, we only need to optimize the objective function (2) or (3) once by setting w1 = 0. Then based on Theorem 2, all optimal Pareto solutions should have a burn-in duration less than b0∗ . Therefore, this algorithm is confined to feasible solutions with b < b0∗ . After obtaining a set of feasible solutions, we sort these points by the cost and then find the non-dominated solutions based on the survival probability. We can easily check that the required computation time is O(n). So the algorithm is expected to be efficient. As can be seen from the example in Sect. 4, this algorithm is able to give evenly distributed points on the Pareto frontier.
4 An illustrative example To illustrate our bi-objective burn-in method, we analyze a system at the component level to derive its burn-in reliability function and burn-in cost, as parallel to Kim and Kuo (2005, 2009). Consider a 5-component product whose structure is depicted in Fig. 2. Imagine the product as a system with 5 component positions into each of which there is inserted a component. As with Kim and Kuo (2005, 2009), a component position includes the component installed in it as well as the connections that connect the component to the system. Possible defects in the component position include component defects and component connection defects. We assume that lifetimes of all components and connection defect failure times are all independent. A normal component in position i, before assembly, has cumulative probability function (CDF) Fi (t). After assembly, it becomes defective with probability pi , and a defective component has a CDF Hi (t). Therefore, the CDF of the component in position i after assembly, denoted as Fi∗ (t), can be expressed as Fi∗ (t) = (1 − pi )Fi (t) + pi Hi (t).
(10)
The number of component connection defects is assumed to follow a Bernoulli distribution as Ni ∼ Bernoulli(qi ), where qi is a rate parameter. If a defect is introduced, the time till this defect emerges is assumed to follow a Weibull distribution with scale parameter 2 × 103
Fig. 2 Structure of a product with a parallel-series structure
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Table 1 The cost configuration
Parameters
Cr,1
Cr,2
Cr,3
Cr,4
Cr,5
Cd
Cf
C0
Values
$55
$48
$75
$64
$21
$5
$249
$0.03/h
and shape parameter 0.85. We can use burn-in to remove these defects (Zhu and Kuo 2012). During burn-in and field use, we assume that (a) If the component in position i fails during burn-in, it is replaced with a normal one with SF Fi (t), at the cost of Cr,i . When a connection defect is detected, it is perfectly removed with repair cost Cd . The process of failure detection, trouble location, and replacement is assumed to consume no appreciable time. (b) The unit time burn-in operational cost is a constant C0 . (c) After burn-in with duration b, a mission time τ is set for the burnt-in system. (d) Once the system fails during field operation, we assume that repair is not allowed and a breakdown cost of Cf is incurred. Values of these cost parameters used in this example are given in Table 1. After burn-in, the system reliability with the mission time τ can be expressed as
R(τ |b) = 1 − 1 − R1 (τ |b) 1 − R2 (τ |b) R3 (τ |b) 1 − 1 − R4 (τ |b) 1 − R5 (τ |b) , where Ri (τ |b) is the survival probability of component position i after burn-in with duration b, which is given by
Ri,τ (b) = 1 − Fi∗ (τ |b) × 1 − qi G(b + τ ) − G(b) . The function Fi∗ (τ |b) on the right-hand side of the expression is the CDF of the residual life of the component at position i. Due to our assumption that a failed component is replaced with a normal one, the CDF of a replaced component is different from the first one. Therefore, Fi∗ (τ |b) is actually the CDF for the excess life of a delayed renewal process. Computation of Fi∗ (τ |b) can be found in Linz (1985) and Tortorella (2005). The lifetime distribution for the components in position i and the values of pi and qi are given in Table 2. Specifically, the respective PDFs for the exponential (Exp(λ)), Weibull (Weibull(λ, β)), Gamma (Gamma(λ, k)), lognormal (ln N (μ, σ )) and inverse Gaussian (IG (μ, λ)) distributions are given by fE (t) = λ−1 exp(−t/λ), fW (t) = βt β−1 λ−β exp −(t/λ)β , fG (t) = Γ −1 (k)λ−k t k−1 exp(t/λ), √ −1 −1 fln N (t) = t 2πσ 2 exp −(ln t − μ)2 / 2σ 2 and
fIG (t) =
−1 −1 λ 2πt 3 exp −λ(t − μ)2 2μ2 t .
Assume τ = 10 years, i.e., 87600 hours. For each weight w1 , the optimal burn-in duration is determined through a one-dimensional search. Figure 3 depicts the optimal burn-in duration versus the weight. As can be seen from this figure, the optimal burn-in duration
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Table 2 Component lifetime distribution and defect rate qi for the five component positions i
Fi (t)
pi
Hi (t)
qi
1
0.99Exp(8 × 105 ) + 0.01Exp(3 × 105 )
0.03
Exp(4 × 102 )
0.01
2
Weibull(5 × 105 , 1.2)
0
–
0.006
3
Gamma(3 × 105 , 3)
0.05
Gamma(2 × 102 , 0.8)
0.02
4
ln N (11.6, 1.2)
0.04
ln N (6.1, 0.9)
0.015
5
IG(4 × 104 , 1.5 × 105 )
0.02
IG(5 × 102 , 3 × 102 )
0.04
Fig. 3 Optimal burn-in durations versus the weight w1
is decreasing in the weight w1 , which tallies with Theorem 1. Another interesting observation from Fig. 3 is that the curve is rather flat when w1 > 0.2. This means that within this interval, the optimal burn-in decision is not sensitive to the weight. However when w1 is relatively small, a small variation of w1 would lead to significant changes of the decision. These results imply that if the manufacturer deems that the cost is relatively important, then it is not necessary to evaluate the weight accurately. But if they believe that the performance is more important, the relative weight should be estimated with more care. If the manufacturer is able to specify the weight w1 , then the optimal burn-in duration can be chosen based on this figure. However, if the manufacturer has no idea about the weight, we can provide him with the Pareto frontier. We divide the interval [0, 1] using grid points k/n with n = 100. For each w1 = k/n, the optimal burn-in time and the corresponding values of J1 and J2 are determined by optimizing the objective function (5). The combination (J1 , J2 ) is then a non-dominated optimal solution. Figure 4(a) shows the Pareto frontier formed by the 101 non-dominator solutions. From this figure, we can see that the non-dominated solutions located at the left-hand side of the vertical broken line is of special interest because they yield smaller cost and higher reliability compared with those without burn-in. However, these 101 points are not distributed evenly and there are few points on the right tail of the frontier. Therefore, we invoke the procedure developed in Sect. 3.2 to compute the frontier. To use this procedure, we first obtain the optimal burn-in duration when w1 = 0, which yields b0∗ = 1316. Within this upper limit, we evaluate the objective values (J1 (bk ), J2 (bk )) at bk = 10k, k = 0, 1, . . . , 132. By sorting these candidate solutions
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Fig. 4 The Pareto frontier: (a) is determined by using the weighted method and (b) is determined by the sorting method in Sect. 3.2
and picking up the non-dominated solutions, we can form another Pareto frontier, which is shown in Fig. 4(b). As can be seen from this figure, the points in this frontier are more evenly distributed. In addition, this procedure requires less computation time.
5 Conclusions Many cost-based and performance-based burn-in models have been proposed in the literature. Performance- and cost-based burn-in models lay stress on product’s intangible losses and book costs, respectively. A legitimate burn-in strategy should allow the manufacturer to make a trade-off between these two objectives. This study bridges the gap between the cost-based and the performance-based burn-in models by developing a unifying framework for determination of an optimal burn-in duration. An intuitive way to understand this framework is to regard it as a single-objective model by treating the performance losses as a cost penalty. As such, our bi-objective method facilitates quantification of the intangible losses by simply requiring the relative weight between the two objectives. From another point of view, this model allows for incorporation of manufacturers’ preferences in burn-in decision makings. We demonstrated this bi-objective method by considering the survival probability as the performance objective. We showed that the optimal burn-in duration is decreasing in the weight assigned to the cost, given that all other parameters are fixed. Then an algorithm was developed to populate the Pareto frontier. The numerical example using a five-component bridge-structured system demonstrated our methods. Acknowledgements The authors thank the editor and four anonymous reviewers for their critical and constructive comments which have considerably helped in the revision of an earlier version of the paper. This work is partially supported by a grant from City University of Hong Kong (Project No. 9380058).
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