Statistical Approach for Cycle Time Estimation
in Semiconductor Packaging Factories
W. L. Pearn, Yu-Ting Tai, and J. H. Lee
Abstract—In the semiconductor industry, to enhance customer satisfactions and ability of quick responses, the development of cycle time estimation model is very important. Cycle time estima-tion is an essential planning basis, which has many applicaestima-tions, especially on the analyses of performance indexes, capacity plan-ning, and the assignments of due dates. In this paper, we provide a statistical approach for cycle time estimation in semiconductor plastic ball grid array (PBGA) packaging factories. Due to today’s fierce competitive environments in the semiconductor industry, planners involved in PBGA packaging factories need an approach to obtain estimated cycle times with different confidence to ensure the due date assignments more accurately. Therefore, upper confidence bounds of estimated cycle times at various confidence coefficients are also presented in this paper. We demonstrate the applicability of the proposed cycle time estimation model incor-porating the upper confidence bounds by presenting a real-world example taken from a PBGA packaging shop floor in a semicon-ductor packaging factory located in the Science-Based Industrial Park in Hsinchu, Taiwan.
Index Terms—Cycle time estimation, Gamma distribution, plastic ball grid array.
I. INTRODUCTION
T
O increase the customer satisfaction in demand and en-hance the ability of quick response, semiconductor man-ufacturers need to develop a model in order to estimate cycle times fast and accurately. Cycle time estimation is an essential planning basis, which has many applications, especially on the analyses of performance indexes, capacity planning, and the as-signments of due dates in the semiconductor industry. Semi-conductor manufacturing process is comprised of four major processing stages involving wafer fabrication, wafer probing process, integrated circuit (IC) packaging, and final test process. Wafer fabrication and wafer probing processes are usually re-ferred to as the “front-end,” while IC packaging and final test processes are referred to as the “back-end” of production. In this paper, we present a statistical model for cycle time estimation in a plastic ball grid array (PBGA) packaging factory at the ICManuscript received September 02, 2008; revised February 12, 2009. Current version published July 09, 2009.This work was recommended for publication by Associate Editor S. Mason upon evaluation of the reviewers comments.
W. L. Pearn and J. H. Lee are with the Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu 300, Taiwan.
Y. T. Tai is with the Department of Information Management, Kainan Uni-versity, Taoyuan 33857, Taiwan.
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEPM.2009.2022270
packaging processing stage in order to assist the proper assign-ment of due dates and to enhance the ability of quick responses in the whole semiconductor manufacturing process.
In this paper, we focus on constructing a model to be used for cycle time estimation in a plastic ball grid array (PBGA) pack-aging factory. The PBGA packpack-aging processes are increasingly popular because of their efficient mounting real estate, good thermal, and electrical performance [1]. PBGA has emerged as a popular array packaging method since it can include higher input/output (I/O) counts on limited board area than the con-ventional peripheral lead frame packages, such as plastic quad flat packages (PQFPs). The PBGA packaging processes have been applied in extensive applications such as cellular phones, which require high I/O counts on reduced board. Generally, the process of PBGA packaging involves ten major operations: 1) the grinding of the wafer back; 2) the mount of wafer; 3) the sawing of wafer; 4) the bonding of die; 5) the bonding of wire; 6) the molding; 7) the marking; 8) the mounting of ball; 9) the singulation; and 10) the inspection, as shown in Fig. 1. In the process, dies are mounted and bonded by gold wires on sub-strate strips. A subsub-strate strip usually comprises four or eight devices depicted in Fig. 2. In molding operation, dies are encap-sulated as PBGA packages. Unlike leadframe packages, PBGA uses solder balls as the interconnect path from the package to the printed circuit boards. Solder balls are attached to the sub-strate by applying a flux and reflowing the solder. Finally, the individual PBGA devices are cut from the substrate strips in the singulation operation and they are placed in trays for subsequent inspections.
Cycle time estimation is an essential problem for PBGA packaging factories. In PBGA packaging factories, due to wide applications of PBGA packaging, there is a great proliferation of product types. It should be noted that the number of solder balls is a major characteristic among these various product types. Fig. 3 presents the bottom view and side view of an 8 8 PBGA packaging product. In the PBGA packaging shop floor, a job involves two cassettes comprising 20 substrate strips each, which are clustered according to their product types and processed on identical parallel machines. The job processing time may vary, depending on the product type of the job processed on. Furthermore, to prevent the critical resources from starvation (idle), the CONWIP (constant work in process) control policy is applied in order to maintain the level of WIP constant. In addition, the processing statuses of machines, such as processing, idle, or breakdown, mainly affect the cycle time at each processing operation. Due to the lack of the fast and accuracy cycle time estimation methods in PBGA packaging factories, practitioners often use constant cycle times as bases for due date assignment and scheduling. However, constant
Fig. 1. PBGA packaging process flow.
Fig. 2. Substrate strip is comprised of four PBGA devices.
cycle time is so simplified that inappropriate due dates and schedules may be assigned and constructed. Therefore, the development of a model for cycle time estimation in PBGA packaging factories is difficult but essential.
In this paper, the distribution of cycle time for single opera-tion is first formulated. A two-parameter Gamma analysis char-acterized with different waiting time distributions is used for the
Fig. 3. Example of a PBGA packaging product via bottom and side view im-ages.
cycle time estimation. Subsequently, the combined distribution adopting the reproductive property of the Gamma distribution for multiple operations is also presented. To demonstrate the applicability of the cycle time estimation model with the com-bined distribution, we consider a real-world example taken from a plastic ball grid array factory located in the Science-Based Industrial Park in Hsinchu, Taiwan. The statistical cycle time estimation model can allow us to obtain the upper confidence bounds of cycle time efficiently and further to quickly respond to customer requirements with different levels of customer ser-vice.
This paper is organized as follows. Section II presents a com-prehensive review of conventional cycle time estimation litera-ture. Section III presents the cycle time distribution for single operation. Section IV shows the combined distribution for mul-tiple operations, and Section V gives a real-world example to demonstrate the applicability of cycle time estimation model in a PBGA packaging factory. Finally, Section VI provides the conclusions.
II. CONVENTIONALMETHODS FORCYCLETIMEESTIMATION
In recent years, much research has focused on providing solu-tions to cycle time estimation. Chung and Huang [2] and Backus
et al. [3] provided extensive discussions regarding the methods
of cycle time estimation. Chung and Huang [2] classified the methods for cycle time estimation into analytical, simulation, statistical analysis, and hybrid methods. Moreover, Chang and Liao [4] considered that the tools in soft computing are also widely applied in this field.
For analytical methods, there have been many researchers who have investigated the cycle time estimation problems. Chung and Huang [2] provided an analytical approach to estimate cycle times for wafer fab with engineering lots. Shanthikumar et al. [5] presented a survey regarding the queueing theory for a semiconductor manufacturing system. They provided a novel solution by incorporating a key charac-teristic involving the dependent relationships in the classical queueing theory and expected to point out new directions in queueing model for semiconductor manufacturing systems. Morrison and Martin [6] conducted a comprehensive review regarding the queueing theory applied in cycle time estima-tion. They provided some practical extensions to cycle time approximations for the -queue [6]. They also provided bounds for comparison for with mean cycle time prediction. Moreover, Huang et al. [7] applied analytic approximations for semiconductor wafer fabrication. De Ron and Rooda [8]
described a lumped parameter model for manufacturing lines. They used the Kingman’s equation and considered the basic characteristics of real lines. However, most of those works only focused on one operation, their computational results cannot be extended to whole factory cycle times. Although the methods applied queueing theory are fast in computing time, the accuracy of classical queueing models is less satisfactory than that of simulation, partly because the complex operational behaviors of semiconductor fabs cannot be represented by one single queueing model.
In recent years considerable concern has arisen over the simulation methods in cycle time estimation research. Vig and Dooley [9] proposed two methods for flow-time estimation methods. They evaluated relationships between several shop factors and effects on the due-date performance using a simu-lation tool. Vig and Dooley [10] further proposed a flow-time estimation and presented a regression-based approach for setting job-shop due dates. Raghu and Rajendran [11] applied a simulation method to select the best rule for shop floor dispatching and developed a due-date assignment policy for a real-life job shop. Chang [12] developed a cycle time estima-tion approach to provide real-time estimates of the queueing times for the jobs which still wait to perform the remaining operations. He also incorporated this estimated queueing time as essential information to the dispatching heuristics to improve their scheduling performance. However, Backus et al. [3], De Ron and Rooda [8], and Shanthikumar et al. [5] indicated that the most common solution for estimating cycle time in complex processes is simulation; however, the simulation method is time consuming and impractical for complicated manufacturing factories, especially in semiconductor manufacturing systems. In addition, Morrison and Martin [6] indicated that the method of simulation cannot offer closed-form expressions for system metrics. Simulation is used for increasing the understanding of behavior of manufacturing systems. Thus, it is difficult to apply in the realistic shop floor because it needs heavy computation loading.
For statistical analysis methods, Raddon and Grigsby [13] presented a regression model to obtain cycle times. Backus et
al. [3] applied another statistical method, the data-mining
ap-proach, and provided nonlinear predictor variables to estimate factory cycle time. Pearn et al. [14] presented a due-date as-signment model for the semiconductor wafer fabrication under a demand variant environment. They applied the contamination model to tackle the effect of that product mix varies periodi-cally. Backus et al. [3] pointed out that the statistical models can be updated as necessary due to the ability regarding quickly reanalyzing the statistical data. Moreover, in recent years the technologies of soft computing including genetic algorithm, fuzzy, and neural network approaches are applied to estimate cycle times in semiconductor manufacturing processes. Hsu et
al. [15] applied constraint-based genetic algorithm (CBGA) to
conduct the flow time estimation model. The CBGA integrates constraint-based reasoning with genetic algorithm to reveal the rule sets. A filtering mechanism is incorporated in the CBGA to enhance computational efficiency before generating and evalu-ating chromosomes. Chang and Liao [4] presented a flow-time prediction method, which incorporates fuzzy rule bases with
the aid of a self-organizing map (SOM) and genetic algorithm (GA). In addition, Chen [16], [17] applied hybrid fuzzy c-mean and fuzzy back propagation network approaches to estimate cycle time in semiconductor manufacturing processes.
In addition, some research works investigate the hybrid methods to estimate cycle time. Kaplan and Unal [18] combine the simulation and statistical analysis approaches to estimate cycle time. Liao and Wang [19] estimated delivery time using the hybrid method incorporating neural networks and analytical methods. Moreover, Chen [20] presented an intelligent mech-anism which applies hybrid self-organization map and back propagation network in the first part and incorporates a set of fuzzy inference rules to evaluate the achievability of related output time forecast in the second part.
III. CYCLETIMEDISTRIBUTION FORSINGLEOPERATION
In this paper, a statistical approach for the cycle time esti-mation in PBGA packaging process flow is presented. We con-sider a more general and more flexible statistical version of a cycle time estimation model for the PBGA packaging indus-tries. Conventionally, the exponential distribution is commonly used for queuing-time-estimated models; however, unsatisfac-tory results limited their applications in practical factories. On the contrary, Gamma distribution can provide a great flexibility and cover extensive applications due to its two essential parame-ters. Due to Gamma distribution being nonnegative domain and right skewed probability distribution, it is used as the probability model for the estimation of waiting time. For instance, it is used for due date assignment for wafer fabrication [14]. Therefore, a Gamma distribution for the cycle time estimation at single op-eration has been applied in this investigation.
A. Gamma Distribution
The Gamma distribution is denoted as with shape parameter and scale parameter . A random variable is said to have a Gamma distribution with parameters ,
, , if its density function is given by
(1)
where is known as Gamma function.
The mean and variance are given, respectively, by
and . Gamma distribution is a nonneg-ative domain and right skewed distribution. The skewness and kurtosis (which are defined as the third and fourth moments of the standardized distribution, respectively) of are and , respectively. The skewness coefficient and the kurtosis coefficient are calculated only by using the shape parameter . This means that the scale parameter cannot af-fect the values of skewness and kurtosis of Gamma distributions. Therefore, we fix in this investigation for the Gamma distribu-tions. Fig. 4 presents several Gamma distributions with different combinations of and . As can be seen in Fig. 4, the Gamma distribution covers a wide class of non-normal applications.
Fig. 4(a)-(c) present graphs of the density for a variety of values of . It should be noted that as becomes large, the density starts to resemble the normal density [21]. To obtain
the maximum likelihood estimators (MLE) of and for the Gamma distribution, we need to solve the following equations simultaneously:
(2)
(3)
Solving the above equation for is rather complicated and there is no explicit close form for the maximum likelihood estimators of . In this paper, therefore, we consider the method of mo-ment estimators to estimate the unknown parameters and . The first two population moments ( and ) of the Gamma distribution with parameters and are
(4) (5) By equating the first two sample moments ( and ) to the corresponding first two population moments, therefore, we can obtain
(6) (7)
From these corresponding sample moments, and are also obtained, where the sample average
and the sample variance are the estimators of and , respectively.
B. Cycle Time Estimation
Like manufacturing processes in other industries, cycle time at single operation in the PBGA packaging process flow equals the process time plus the mean of waiting time. The formula can be expressed as follows:
(8) where is the cycle time of operation , is process time of operation , and is the mean of the fitted Gamma distribution of waiting time of operation in the PBGA packaging process flow.
Consider a small-scaled example for cycle time estimation at single operation in PBGA packaging process flow. The example involves three parallel machines and two different product types, namely, A and B. The various processing times and estimated values regarding the two parameters of waiting time distribu-tions for the two product types are shown in Table I. The process
TABLE I
PROCESSTIMES ANDESTIMATEDPARAMETERS OFWAITINGTIMEDISTRIBUTIONS
time is not affected by the machine processing it, but is depen-dent on job’s product type. The “minute” is used as the unit for process time and waiting time.
Since the waiting time of product type A at this single oper-ation is fitted as Gamma distribution with parameters (21, 2.5), the mean waiting time is 52.5 min. Similarly, since the waiting time of product type B at this operation is fitted as Gamma dis-tribution with parameters (28, 3.2), the mean waiting time is 89.6 min. Therefore, based on (8), the estimated cycle time for product type A is 92.5 min. Similarly, 145.6 min is the value of estimated cycle time of product type B in this operation.
IV. COMBINEDDISTRIBUTION FORMULTIPLEOPERATIONS
In this section, a combined cycle time of the multiple opera-tions in the whole PBGA packaging process flow is developed. Due to today’s fierce competitive environments in the semicon-ductor industry, planner involved in PBGA packaging factories should be capable of providing the estimated cycle times with different confidence to ensure the due date assignments more ac-curately. Therefore, upper confidence bounds of the estimated cycle times at various confidence coefficients are presented.
A. Combined Gamma Distribution
To estimate cycle times of the multiple operations in the whole PBGA packaging process flow, an essential statistical property, reproductive property, of the Gamma distribution is applied; therefore, a combined distribution is applied and a combined cycle time estimated model is then constructed. Gamma distribution has a reproductive property [22]: if are independent random variables each having a Gamma distribution of form
(9) with possibly different values of , but with
common values of , then also has a
distribution of this form, with ,
and with the same values of . Applying this property, let be a sequence of independent distribution of and then the distribution of
is . Fig. 5(a)-(c) presents several Gamma distributions with different values of and the same value of . Fig. 5(d) further depicts the combined Gamma distribution regarding Fig. 5(a)-(c).
Fig. 4. Probability density functions for Gamma distribution with different pa-rameter combinations. (a) Gamma(1,1). (b) Gamma(3,1). (c) Gamma(5,1). (d) Gamma(1,3). (e) Gamma(3,3). (b) Gamma(5,3).
Fig. 5. Probability density functions for Gamma distribution with different parameter combinations. (a) Gamma(1,1). (b) Gamma(3,1). (c) Gamma(5,1). (d) Gamma(9,1).
B. Combined Cycle Time Estimation
Using the reproductive property of Gamma distribution,
the combined waiting time ( ) is .
Therefore, the combined cycle time ( ) of the whole PBGA packaging process flow can be calculated as
(10)
where is the total number of operations in PBGA packaging process flow, is sum of processing times for each op-eration, and is the mean of the combined Gamma distribution of waiting time.
C. Upper Confidence Bounds for
For an individual job, we can obtain the upper bound of the combined cycle time by taking the integral over the Gamma dis-tribution. Using this method, cycle time equals to its process time plus -percentile waiting time of the combined Gamma distribution. The -percentile waiting time can be obtained by taking the inverse of the cumulative function of the Gamma dis-tribution. However, in many factories, the combined cycle times of the whole PBGA packaging process flow are calculated as where the mean of the combined waiting time is incorporated.
To obtain the upper confidence bounds of the combined cycle times under Gamma distribution, it requires the distribution of which is a scaled Gamma. Finding exact confidence in-terval of directly from the scaled Gamma is rather difficult since the distribution of involves unknown pa-rameters and which have to be estimated. The resulting distribution becomes rather complicated. If we apply the
Cen-tral Limit Theorem, then is
approx-imately distributed as the standard normal distribution, . Consequently, we have
(11)
So the probability that the random upper limit as
(12) is an approximate one-sided confidence interval for the combined waiting time. That is,
provides an upper confidence bound for the combined waiting time with confidence coefficient . Since the of whole PBGA packaging process flow being equal to the total process times plus the confidence interval of the combined waiting time, the upper confidence bound of can be ex-pressed as
TABLE II
200 OBSERVATIONS OFWAITINGTIMES
Fig. 6. Histogram of the 200 observations.
That is, we have an approximate upper confidence bound of
(14) where is the percentile of the standard normal distribution, for which tables are widely available , is the sample size of waiting times, and is the variance of the combined Gamma distribution of waiting time in the PBGA packaging process flow.
V. CYCLE TIME CALCULATION FORPBGA PACKAGINGPROCESS
In this section, we consider a real-world application taken from a PBGA packaging shop floor in a semiconductor pack-aging factory located in the Science-based Industrial Park in Hsinchu, Taiwan and investigate the applicability of the
proposed model. For the example investigated, there are three product types of orders, namely, PBGA658, PBGA596, and PBGA292. An order involves five jobs which must be processed at all the ten operations, at where a set of identical machines are arranged in parallel at each operation in the shop floor. In the factory we investigated, a manufacturing execution system (MES) is applied to enhance the abilities of automation and data collections. To estimate the combined cycle time for the whole process flow, we collect the waiting times regarding the three product types in the shop floor from the MES. Table II displays the 200 observations of waiting times, collected from the historical data, at the wire bonding operation for PBGA596. The “minute” is used as the unit for process time and waiting time. Fig. 6 plots the histogram shown the collected data.
It is evident to conclude the data collected from the PBGA packaging factory is not normal distributed by observing the his-togram in Fig. 6. The historical data indicates that the process is approximated by a Gamma distribution. The maximum-like-lihood estimators (MLE) of and for the Gamma distribution are rather complicated and there is no explicit close form for the MLE of . Therefore, we consider the method of moments. The parameters and of this Gamma process could be estimated from the historical data, giving and .
To obtain the combined cycle time for the whole PBGA pack-aging process flow, we collected the historical data of waiting times for the three product types at the ten operations from the MES applied in the shop floor. Table III shows the order process time and Table IV presents the two essential statistical elements, average ( ) and variance ( ) of the collected data at each operation in the whole PBGA packaging process flow.
Using the method of moment estimators, and , we estimate the parameters for Gamma distributions fitted to of each product type in the PBGA packaging process flow. Table V displays the estimated shape parameters ( ) with fixing the scale parameter ( ) 5, 4.5, and 3 for product type PBGA658, PBGA596, and PBGA292, respectively.
Based on (10), we can obtain the value of the combined cycle time, 5447.45 min, for PBGA658 in the PBGA packaging
TABLE III
ORDERPROCESSTIMES FOREACHPRODUCTTYPE ATEACHPBGA PACKAGINGOPERATION
TABLE IV
THEAVERAGE(x)ANDVARIANCE(s )OFW TATEACHPROCESSOPERATION
TABLE V
ESTIMATEDSHAPEPARAMETERS(^)FORFITTED
GAMMADISTRIBUTIONS OFW T
factory. Similarly, the value of the combined cycle time for PBGA596 and PBGA292 can be obtained as 4988.37 and 4127.37 minutes, respectively.
Using the reproductive property of the Gamma distribution, the combined cycle time of PBGA658 has Gamma distribu-tion and its corresponding parameters are Gamma (769.09, 5). Therefore, based on (14) for upper confidence bounds calculation, the 95% upper confidence bound of for PBGA658 is 5463.58 min. That is, the combined cycle time for PBGA658 in the PBGA packaging process flow is not greater
than 5463.58 min at 95% confidence. Similarly, 5003.16 and 4138.89 are the 95% upper confidence bounds of for PBGA596 and PBGA292, respectively. We note that the upper confidence bounds can be used as a convenient reference point for assigning due dates and other planning bases in order to help the practitioners to provide an accuracy basis for due date assignment, production planning, and factory performance analysis.
VI. CONCLUSION
In this paper, we considered a statistical approach for cycle time estimation incorporating the upper confidence bounds in semiconductor PBGA packaging factories since the cycle time is an essential basis for production planning and due date assign-ment. We first provide a cycle time estimation model for single operation. Waiting times of each product type are modeled as Gamma distribution. We then present a combined cycle time estimation model which incorporates the reproductive property of Gamma distribution to estimate the whole factory cycle time for the multiple operations in the PBGA packaging shop floors. Moreover, upper confidence bounds at various confidence co-efficients were also provided based on the investigated cycle time estimation model in order to quickly respond to the cus-tomer inquiries regarding due dates and shipping schedules. To demonstrate the applicability of the proposal estimation model, we considered a real-world example taken from a PBGA pack-aging shop floor in a semiconductor factory located in the Sci-ence-Based Industrial Park in Hsinchu, Taiwan. The computa-tional results showed that the cycle time estimation model pro-vided satisfactory values of cycle time. Therefore, we believe that the investigated cycle time estimation model incorporating upper confidence bounds may help industrial practitioners in-volved in PBGA packaging shop floor to estimate cycle time and to provide an accuracy basis for due date assignment, pro-duction planning, and factory performance analysis and to make judicious decisions.
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W. L. Pearn received the Ph.D. degree in operations research from the University of Maryland, College Park.
He is a Professor of Operations Research and Quality Assurance at National Chiao Tung Univer-sity (NCTU), Hsinchu, Taiwan. He was with Bell Laboratories as a member of quality research sci-entists before joining NCTU. His research interests include process capability, network optimization, and production management. His publications have appeared in the Journal of the Royal Statistical
Society, Series C, Journal of Quality Technology, European Journal of Op-erational Research, Journal of the OpOp-erational Research Society, Operations Research Letters, Omega, Networks, International Journal of Productions Research, and others.
Yu-Ting Tai received the Ph.D. degree in industrial engineering and management from National Chiao Tung University, Hsinchu, Taiwan.
She is an Assistant Professor in the Department of Information Management, Kainan University, Taoyuan, Taiwan. Her research interests include scheduling and semiconductor manufacturing man-agement.
J. H. Lee received the M.S. degree in industrial engi-neering and management from National Chiao Tung University, Hsinchu, Taiwan.
She is an Engineer with the ThaiLin Semicon-ductor Corporation, Hsinchu, Taiwan. Her research interests include semiconductor manufacturing management and process capability.
