Simulation-based optimization of sampling plans to reduce inspections while mastering the risk exposure in semiconductor manufacturing

Transcription

1 DOI /s x Simulation-based optimization of sampling plans to reduce inspections while mastering the risk exposure in semiconductor manufacturing M hammed Sahnoun Belgacem Bettayeb Samuel-Jean Bassetto Michel Tollenaere Received: 18 February 2014 / Accepted: 5 August 2014 Springer Science+Business Media New York 2014 Abstract Semiconductor manufacturing processes are very long and complex, needing several hundreds of individual steps to produce the final product (chip). In this context, the early detection of process excursions or product defects is very important to avoid massive potential losses. Metrology is thus a key step in the fabrication line. Whereas a 100 % inspection rate would be ideal in theory, the cost of the metrology devices and cycle time losses due to these measurements would completely inhibit such an approach. On another hand, the skipping of some measurements is risky for quality assurance and processing machine reliability. The purpose is to define an optimized quality control plan that reduces the required capacity of control while maintaining enough trust in quality controls. The method adopted by this research is to employ a multi-objective genetic algorithm to define the optimized control plan able to reduce the used metrology capacity without increasing risk level. Early results based on one month of real historical data computation reveal a possible reallocation of controls with a decrease by more than 15 % of metrology capacity while also reducing the risk M. Sahnoun IRISE/CESI, Mont St Aignan, France msahnoun@cesi.fr B. Bettayeb (B) S.-J. Bassetto Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Montreal, Canada belgacem.bettayeb@polymtl.ca S.-J. Bassetto Samuel-jean.bassetto@polymtl.ca M. Tollenaere University of Grenoble Alpes, G-SCOP, Grenoble, France Michel.Tollenaere@grenoble-inpg.fr M. Tollenaere CNRS, G-SCOP, Grenoble, France level on the processing machine (expressed by the wafer at risk (W@R)) by 30 %. Keywords Genetic algorithm Sampling Control plan Wafer at risk Simulation Multi-objective optimization Introduction Semiconductor manufacturers seek to optimize their control plan for reliable production. In this framework, accurate sampling policies have to be elaborated in order to: (1) reduce cycle time losses, and consequently work in process (WIP) at inspection steps (Chien et al. 2012), (2) monitor and reduce process variability, (3) reduce the risk level of processing machines, measured in terms of potential faulty products (Shanoun et al. 2011), and (4) better manage the evolution of technologies in terms of maturity and global mix. The fact remains that, optimized sampling policies are difficult to implement in high-mix semiconductor facilities owing to: (1) permanent changes of the product mix, which sometimes lead to significant fluctuations in the WIP level, (2) a high level of interaction among a number of organizations with a stake in the fab, and (3) a huge number process operations and technologies processed in the same step (Liu et al. 2005; Bassetto and Siadat 2008). Due to the complexity of the semiconductor manufacturing system, it is important to continuously monitor the state of the processing machines and control the quantity of uncertain-quality products. Previous works by the authors were focused on defining an operational index for the evaluation of the risk exposure level (Sahnoun et al. 2010; Shanoun et al. 2011; Bettayeb et al. 2014). This index, called wafer at risk is computed for each processing machine and is expressed by the number of products (wafers) processed

2 since the last time that machine was validated or corrected by a quality control. This work focuses on judicious inspection allocation. Specifically, we are seeking to reduce used metrology capacity, while at the same time respecting predefined levels. A sampling plan impacts all the organizations involved in the fab. It directly influences the inspection buffer size (Bulgak et al. 1995) and the metrology time delay (Garvin et al. 2002), the latter being an important parameter in computing the risk level (W@R) (Sahnoun et al. 2012) and the maintenance plan (Chan and Wu 2009). Several studies have been carried out on the risk-based optimization of the sampling plan and the dynamic allocation of inspections in semiconductor fabs (Good and Purdy 2007; Lee 2002). Recently, Dauzère-Pérès et al. (2010) proposed a sampling indicator to choose, in real time, which lot has to be measured. Their sampling heuristic is based on a sum of the ratios between the number of wafers processed and some associated target limits for processing machines. The lot chosen for the metrology is the one that will reduce this indicator the most. Its counterpart, at the planning level, defines target limits for operations and tools, and has been defined by Bettayeb et al. (2012a, b). A target control plan is computed using a two step greedy method that allocates the available metrology capacity to ensure a level of W@R that does not exceed a predefined limit for all processing machines. In more general terms, the sampling problem can be seen as an inspection allocation problem, as metrology capacities are limited. Some five decades ago, an economic algorithm was developed for inspection allocation (Lindsay and Bishop 1964), based on a cost function per unit produced and taking into account the inspection cost and its location in the process. Since their paper was published, the field of inspection allocation has become an area of intense research. We recommend the surveys in Raz (1986) and Tang and Tang (1994) for a complete picture of the field. Several of the more remarkable works on the subject have been of particular importance in the effort to improve production reliability. In Villalobos et al. (1993), for example, a flexible inspection system for serial and multi-stage production systems in the field of printed circuit boards is presented. The authors provide a dynamic programming algorithm to optimize global goals (like costs), while taking into account some local constraints, like inspection tool availability. An interesting case of information-based inspection allocation is presented in Verduzco et al. (2001), in which a cost function is modeled taking into account Type I and Type II errors, and the information gain achieved by each measurement. They also formulate the inspection allocation problem as a Knapsack Problem (KP), and propose a greedy algorithm to solve it. Their simulations reveal that the information-based solution per- Fig. 1 Sampling plan based on operations forms better than static inspections, in terms of classification errors.inrabinowitz and Emmons (1997) and Emmons and Rabinowitz (2002), an inspiring non-linear modeling of the inspection allocation problem is presented, in which the time elapsed since the last inspection is introduced, and the proportion of defect-free items is linked to that time period. This conceptualization is strongly related to the concept of W@R. An interesting model of inspection allocation effort is also presented in Kogan and Raz (2002), where the authors seek to minimize a cost function influenced by inspection costs, as well as multiple failure modes that can affect each production step. They also suggest multiple layers of detection that can be used for every product, at every production stage. An optimal sampling plan can be defined as a highly combinatorial search problem in which the solution space increases exponentially with the number of decision variables. To cope with the combinatorial explosion of these solutions, evolutionary algorithms, like genetic algorithms (GA), can be used. These are techniques capable of evaluating hundreds of thousands of possible solutions and making them converge toward the best alternatives. GAs are highly suitable for sampling plan optimization problems, because of the ease with which codification can be performed to define chromosomes (cf. Codification and population generation section; Fig. 1). In addition, several researchers have shown the effectiveness of GAs for the optimization of sampling plans in various types of industry (Gen and Lin 2013). Rau and Cho (2009) propose a GA modeling for the control plan in reentrant production systems. Zhang et al. (2013) propose a hybrid sampling strategy-based multiobjective evolutionary algorithm to deal with the process planning and scheduling problem. Vinod et al. (2004) use a GA approach to reduce the risk of radiation exposure in the nuclear industry by avoiding unnecessary inspections. Kancev et al. (2011) optimize test intervals for aging equipment by a multi-objective genetic algorithm (MOGA) approach. Linet al. (1998) has designed a random inspection rate for a flexible assembly system based on the GA approach. The works cited above propose several GAs aiming to optimize some operational performances of industrial systems. However, they did not consider more than two objectives and the risk criterion is never combined with other objectives. Moreover, authors did not find any use of GAs

3 to reduce risk exposure in semiconductor manufacturing because of the difficulty to define its corresponding objective function. That is why a simulation-based evaluation of the objective function is used in this paper. Our goal in this paper is to present a MOGA designed to optimize the control plan at the operational level, i.e. guiding real time decisions at the lowest organizational level in the fab. At this level of decision, it is necessary to decide for each produced lot if it has to be inspected or not based on the current sampling rate. MOGA gives to experts the possibility of redesigning the control plan according to some specific needs in terms of metrology capacity usage and risk exposure mastery. This GA will find a near-optimal sampling plan, while at the same time reducing the risk level (W@R) of processing machines and the metrology capacity used. A side-effect is that the metrology time delay is also reduced. In order to process this algorithm, a model of the sampling plan based on processing operations is proposed. In order to perform the MOGA, a sub-simulator has been developed to compute several production system indicators: the W@R for each tool, and the inspection capacity used for each sampling plan. This sub-simulator is used to estimate the impact of sampling plans on fab performances and as an evaluation function for the GA. This paper is organized as follows. Context and problem statement section presents the context of this research and states the problem of interest. The model of sampling plan is formalized in Formalization of the sampling plan design problem section. GA-based sampling plan optimization section presents the MOGA used to optimize the sampling plan, and the simulator used for the evaluation of each chromosome is briefly presented. The results obtained are presented and discussed in Implementation, experiments, and discussion section. Conclusions and perspectives section provides our conclusions and perspectives. Context and problem statement The authors were privileged to participate in the European IMPROVE project, led by STMicroelectronics. This 42- months project ( ) brought together 36 academics and representatives of the semiconductor industry to Implement Manufacturing science solutions to increase equipment productivity and fab performance. This paper is based on a sub workpackage of this project aiming at improving control plans. During this project, strong interactive relationships were formed and joint developments were achieved between the authors and researchers at STMicroelectronics from the 300 mm research and production front-end fab, located in Crolles-France. In this plant several ranges of integrated circuits (IC) are designed, prototyped and produced. They are employed in various applications: medical, defense, automotive and communications, among others. Each IC is made of hundreds of millions of transistors that belong to a technology, which, in the CMOS industry, is labeled according to the key characteristic of a transistor: the width of the oxide gate. During the project, that width ranged from 32 to 90 nm. Process control operations focus on the set-up of the various tools that make up the production system, and it is these operations that are responsible for inspection allocation and the definition of the control plan. The practice in this fab is to rely on internal process experts, who flag operations that can be controlled and define their sampling rates. In order to justify a control plan improvement, a dedicated change management committee has to approve the change based on factual analysis based on a before/after evaluation of some performance indicators related to metrology capacity usage and risk mitigation. Manufacturing data are extracted manually from manufacturing databases, sorted and pretreated. This is a task that often takes more than a working day to complete. The next step is to build a simulation for each sampling plan, in order to compute what-if scenarios. Each of these scenarios retrieves new performance indicator values. This operation takes at least half a day to complete. For the engineer, the main challenge is to choose the sampling plan that provides the most gain and satisfies some predefined constraints. This sampling plan will constitute a sequence of non-zero sampling rates, each of which corresponds to a pairing in the process flow model that can be sampled (technology, operation). The objective is to avoid the possibility of leaving any operation or technology without a control during the production time considered (Sahnoun et al. 2012). This operation is by far the most complex from a combinatory point of view. Sampling rate per (technology,operation) pair ranges from 1 to 100 %. For instance, by considering the processing operations in the Lithography and Etching areas alone, more than 1,200 pairs (technology, operation) have been identified, which generates a huge number of possible sampling plan combinations: 1, The methodology of meta-heuristic optimization is the most suitable for solving problem formulated in this paper, where a knowledge system and optimization should be considered simultaneously (Takahara and Miyamoto 1999). These problem belongs to the multi-objective optimization class of problems (Fonseca and Fleming 1995), and GAs have proven their efficiency in solving this kind of problem (Cheshmehgaz et al. 2012; Simaria and Vilarinho 2004). Because of the complex dynamics and the random behavior of lots in the fab, it is not easy to define the W@R with a mathematical function. Simulation is considered the best way to compute the exact value of W@R, in spite of the considerable computation time involved. For all these reasons and because the GA is easily applicable to the model of the control plan, we propose using MOGA-based simulation to define the optimized sampling plan. The operations that can

4 be obtained in other ways, such as by Virtual Metrology (VM) (Su et al. 2008; Pan and Tai 2011). However, for others, skipping measurements could increase risk (an increase in which is the number of products processed since the last inspection, cf. Introduction section) because some operations are never controlled or because controls focus only on specific products. It is important to notice that VM is complimentary to our approach, because it also gives the possibility to reduce the number of inspection. However, this method is not always applicable and real measurements are needed to adjust its mathematical model. Let x denote a given sampling plan that is composed of the sequence of sampling rates of the (technology, operation) pairs (Fig. 1). Each sampling plan is characterized by its own sampling plan saturation (SPS x ) value, which is the proportion of the metrology capacity used when applying sampling plan x relative to the metrology capacity needed for a 100 % sampling plan, and is defined as follows: Fig. 2 The approach: a tool to support decision making for sampling plan optimization be sampled are selected by an expert, and the GA is run using simulation as an evaluation function. Populations are generated until the final test on of the best chromosome of the last population has been passed as shown in Fig. 2. Formalization of the sampling plan design problem This section explains how the sampling plan problem has been formalized, providing details of the objectives and constraints to be considered in constructing the best solution. As mentioned in the previous section, in semiconductor manufacturing, any quality control plan design or change has to be negotiated in order to avoid generating undesirable effects on fab systems. This is a major challenge for every quality control planner. As metrology is always considered a necessary evil in manufacturing fabs (Bunday et al. 2007), there is great pressure to reduce the number of wafers sampled for inspection. Moreover, everyone is conscious of the effects of sampling on the stability of the manufacturing system and the investigation capacity. Consequently, the controls are subject to a daily tradeoff, in order to maintain a given level of data. For some products, reducing sampling can release metrology capacity without undesirable consequences. Fortunately, data are sometimes redundant (Shanoun et al. 2011) and/or could SPS x = CAPAx CAPA 100 % (1) where, CAPA x is the metrology capacity used when the sampling plan x is applied, and CAPA 100 % is the metrology capacity needed for a 100 % sampling plan, which is supposed to be equal to the total available capacity. Note that it is possible to define several sampling plans for agivensps. For example, consider a process flow composed of two process operations. If the available metrology capacity allows inspection of the first operation 6 times, and the second one 10 times, an SPS of 50 % can be associated with the following list of feasible sampling plans: ) S 1 = (s1 1 = 1/6, s1 2 = 7/10 ; ) S 2 = (s1 2 = 2/6, s2 2 = 6/10 ; ) S 3 = (s1 3 = 3/6, s3 1 = 5/10 ; ) S 4 = (s1 4 = 4/6, s4 1 = 4/10 ; ) S 5 = (s1 5 = 5/6, s5 1 = 3/10 ; ) S 6 = (s1 6 = 6/6, s6 1 = 2/10. Consequently, each SPS is representative of a class of feasible sampling plans. GA-based sampling plan optimization As mentioned in Context and problem statement section on context, the quality control plan of a complex manufacturing system needs to be adapted to dynamic operational changes in the fab. The use of automatic methods to search for an

5 optimized sampling rate for each technology-operation pair is essential, because of the very large number of combinations that make up the solution space. We maintain that GA could be easily applied to our model to define near-optimal sampling plans. The approach The proposed approach, presented in Fig. 2, consists of providing a decision making support tool that makes it possible to: (1) define different scenarios relative to metrology capacity usage and to risk exposure objectives and constraints; and (2) optimize sampling plans using a MOGA, where the evaluation function is based on simulations that use historical process and metrology data. The approach begins with pretreatment of the historical data. Data are extracted from process and metrology databases covering a predefined period of time. In this step, the actual sampling plan is evaluated through performance indicators. This step also enables rebuilding of the process flow model, the resource capacities/qualifications and the production mix. Based on the performance indicators, experts analyze these results, and may ask the following questions: (1) Is the current sampling plan optimized? (2) Could the metrology capacity used be reduced with limited impact on risk exposure? and (3) What happens if some operations are sampled with lower sampling rate thresholds? These questions are incorporated into the support tool in the form of the constraints and objectives to be taken into account in the optimization process that will use the MOGA. The MOGA is then run to find the sampling plan (SP) that best fits the predefined constraints and objectives. The evaluation of each solution generated by the GA operators is performed using simulation based on the historical data. The best SP obtained by the MOGA is displayed and the expert decides whether or not the result is satisfactory in terms of meeting objectives. If it is not, the expert may choose to modify his initial objectives and constraints, and try again. Objectives and constraints The objectives of the optimization process include two aspects: Metrology capacity allocation and Risk exposure mastery. Metrology capacity allocation Metrology or measurement operations are often perceived as having no value-added, which increases operational costs and slows down the rate of production. Moreover, they generally need expensive metrology resources, which is why metrology capacity needs to be carefully allocated, in order to balance the underlying measurement costs with the objectives of risk exposure reduction. As metrology machines are expensive, ranging from $1 to 3 million, any effective savings realized are welcome. Therefore, the first criterion constituting the objective function is CAPA x, which refers to the metrology capacity used over the entire production period during which sampling plan x is applied. Risk exposure mastery To reduce W@R, several indicators have been computed to add an insurance perspective to the control plan. In order to avoid a scrap crisis, a MaxW@R indicator has been created, which is the maximum number of wafers that can be lost at once in the production system, after a possible drift. Reducing this value has a direct impact on the likelihood of a major scrap event. The second criterion is then MaxW@R x, which is the value of the highest W@R reached over the entire production period during which sampling plan x is applied. It is expressed as follows: Max W@R x = max Max W@R x z z (2) where, z {1,...,Z} denotes the processing machine index; with Z the number of processing machines, and Max W@R x z denotes the maximum value of W@R reached by the processing machine z during the period of time considered. The MeanW@R value is an average value of the risks, in number of products, over the production system considered. It is used to seek W@R improvement, even if there is no relevant MaxW@R peak. The third criterion is then MeanW@R x, which is the mean W@R over the entire production period. It is expressed as follows: MeanW@R x = 1 Z Z MeanW@R z x (3) z=1 where MeanW@R z x denotes the average value of W@R of the processing machine z during the period of time considered. The MeanMaxW@R is the mean maximum loss over the tools. The fourth criterion is then MeanMaxW@R x, which is the mean MaxW@R reached by all the processing machines (denoted z) over the entire production period. It is computed as follows: MeanMaxW@R x = 1 Z Z MaxW@R z x (4) z=1 Each of the presented criterion derived from the W@R is related to a specific phenomenon, which cannot be observed easily by another criterion. For instance, it is possible to have in the same time a small value of meanw@r and a big value of the MaxW@R and vice versa, which have different impacts in terms of potential losses.

6 The sampling plan to be optimized can be subject to some constraints that define the solution space from which the optimized sampling plan should be chosen. These constraints concern the variation of the performance indicators relative to those of the initial sampling plan. The following constraints are: Constraint on the metrology capacity used: The user defines his needs with the parameter CAPA c (%) which represents the lower bound of the percentage of reduction of the capacity used. The corresponding constraint on that capacity is defined as follows: CAPA x (%) CAPA c (%) CAPAinit CAPA x CAPA init CAPA x 100 CAPAinit CAPA c CAPA init 100 CAPA c where, CAPA c is the value of the constraint on the capacity used, CAPA init is the capacity used of the initial sampling plan, CAPA x, is the capacity used of the sampling plan x. Constraint on MeanMaxW@R This constraint defines the tolerance related to the mean of MaxW@R z over all the processing machines considered. An acceptable solution under this constraint respects the following relation: MeanMaxW@R x (%) MeanMaxW@R c (%) MeanMaxW@R x MeanMaxW@R c where: MeanMaxW@R x (%) = MeanMaxW@Rx MeanMaxW@R init MeanMaxW@R init 100 MeanMaxW@R c (%) = MeanMaxW@R c MeanMaxW@R init MeanMaxW@R init 100 Mean Max W@R c is the constraint on Mean Max W@R, Max W@R z x is the maximum of W@R reached by processing machine z over the simulated period using sampling plan x, and MaxW@R z init : is the maximum of W@R reached by processing machine z over the simulated period using the initial sampling plan. Constraint on MeanW@R This constraint defines the tolerance regarding the average value of W@R for all processing machines. An acceptable solution under this constraint respects the following relation: MeanW@R x (%) MeanW@R c (%) MeanW@R x MeanW@R c where: MeanW@R x (%) = MeanW@Rx MeanW@R init MeanW@R init 100; MeanW@R c (%) = MeanW@R c MeanW@R init 100; MeanW@R init MeanW@R c is the constraint on the mean of W@R, MeanW@R is the average value of W@R over the simulated period using sampling plan x, and MeanW@R init is the average value of W@R over the simulated period using the initial sampling plan. Constraints on MaxW@R per machine: This constraint defines the tolerance of MaxW@R for each processing machine. An acceptable solution under this constraint respects the following relation: MaxW@R z x (%) MaxW@R c,z(%) z {1,.., Z} MaxW@R z x MaxW@R c,z z {1,.., Z} where: MaxW@R z x (%) = MaxW@Rx z MaxW@Rinit z MaxW@R z init 100; MaxW@R c,z (%) = MaxW@R c,z MaxW@R z init MaxW@R z init 100; MaxW@R c,z is the constraint on the MaxW@R for processing machine z, MaxW@R z x is the maximum of W@R for processing machine z over the simulated period using sampling plan x, and MaxW@R z init is the maximum of W@R for processing machine z over the simulated period using the current sampling plan. Remark For the MOGA, each of the precedent constraints can be either positive or negative, depending on the user s expectations. For instance, a positive Mean Max W@R c (%) implies the user s tolerance to W@R increase. If the user

7 requires reduction, Mean Max c (%) is setted negative. Codification and population generation The implementation of GAs requires codification of the solution space into a finite set of finite-length strings, which are the chromosomes (see Fig. 1). In the proposed GA, each chromosome is a feasible sampling plan where a gene corresponds to the sampling rate of an operation that can be sampled, i.e. a (technology, operation) pair. A gene is coded by a number varying from 1 to 100 %, in increments of 1 %. The size of the chromosomes depends on the selected pairs that can be sampled, which are defined by the expert. At each iteration of the GA, a new population is generated using classical genetic operators with different proportions (selection, mutation, crossover and random generation), except the initial one, which is generated completely randomly. Evaluation by simulation A simulator has been developed to apply each new sampling plan and then evaluate the W@R indicators for each processing machine and the corresponding value of the metrology capacity used, CAPA x (Fig. 3). The input data of the simulation are based on two sets of historical data extracted for the same period of time. The first set contains information about the production system: the lot name, the equipment on which it has been processed, the number of wafers in the lot, the date of processing, the operation name, and the technology. The second set of historical data is related to metrology events, which are characterized by the metrology equipment, the lot name, the inspection date (entrance), the process operation measured, and the associated technology. The initial sampling plan computed from the inputs is denoted by S init = (s1 init,.., si init,.., s init I ), and is characterized by its SPS (Sampling Plan Saturation) indicator: SPS init = CAPAinit CAPA 100 % (5) As illustrated in Fig. 3, the simulation is performed in four steps, which are detailed below. Step 1-Flag potential lots for inspection This step consists of selecting, from the metrology historical data, the subset of measured lots when sampling plan x is applied. It should guarantee the desired sampling rates si x for all the pairs (technology, operation) indexed by i {1,...,I }. For each lot l in the metrology historical data, a partial sampling rate is computed, as follows: s p = N insp + 1 N proc (6) + 1 where, is the index of the operation corresponding to the lot l {1,...,L}, L is the number of lots in the metrology historical data, N insp is the number of inspected lots belonging to operation up to the current lot l, and N proc is the number of processed lots belonging to operation up to the current lot l. This partial sampling rate (s p ) is then compared to sx.if the partial sampling rate is lower than s x,thelotl is flagged and the number of inspected lots relative to operation is increased by 1 (N insp = N insp + 1). Whatever the result of this test, the number of processed lots relative to operation is increased by 1 (N proc = N proc + 1). At the end of this step, the new sampling plan x is presented by the set of flagged lots in the metrology historical data. Step 2-Flag the process data After defining the lots to be inspected, they must be flagged in the historical processing data. In this step, a lot is flagged when it has already been flagged in the inspection data for the same operation. The processing dates of flagged lots are used in the next steps. Fig. 3 Simulation of W@R and τ for a sampling plan x

8 Fig. 4 Inspection queue management Step 3-Manage the inspection queues The measurement operations of interest in this study are performed by stand-alone metrology equipments. Each has a separate queue in which the lots are placed before being measured. This step defines the metrology schedule for each sampled lot (flagged in the previous step), according to sampling plan x, without changing the initial dispatching of the sampled lots to the metrology equipments. As the number of flagged lots is less than or equal to the number of measured lots in the initial sampling plan, some lots in the historical metrology data are skipped. These lots are removed and replaced by the new sampled lots in the metrology queues (see Fig. 4). The new measurement dates of these lots are advanced in order to fill the empty slots. These dates should not be earlier than the date of processing plus the transportation time (T t ). This condition is represented in the following equation: D nm (l) D pro (l) + T t (l) (7) where, D nm (l) is the new measurement date of lot l, and D pro (l) is the processing date of lot l. The resulting maximum wait time gain is the sum of the measurement times of all the skipped lots, as illustrated in Fig. 4. At the end of this step, the measurement dates of the lots selected according to sampling plan x are defined. Step 4-Computation of W@R In this step of the simulation, the indicators based on W@R are computed separately for each processing machine. For each process event, i.e. a lot l is processed by the processing machine z at time D proc (l), z is increased incrementally by the number of wafers in lot l. Iflotl is flagged in Step 2, it is saved in a metrology buffer. Then, the algorithm checks for a measurement event in the period of time between the processing date of the current lot (D proc (l)) and the processing date of the next lot (D proc (l + 1)). If a measurement event is found, W@R z is decreased by the value W@R reduction (l ) of the measured lot l and the W@R reduction of all the lots contained in the measurement buffer is decreased by W@R reduction (l ) (see Shanoun et al for more details). This simulation also gives us the opportunity to compute the time delay τ between processing and measurement, in order to study the behavior of lots between these two steps and the variation of τ for each sampling plan. Implementation, experiments, and discussion The implementation of the proposed approach and its subsequent algorithms have been developed with MS Excel VBA This choice was made in order to facilitate case study testing and data exchange with industrial partners. Before launching this tool, historical processing and metrology data should already be loaded into two different sheets. A graphical interface (Fig. 5) allows the user to set the optimization parameters and run the MOGA. This interface is also used to select the (technology, operation) pairs to be considered in the sampling plan, using one of three possible alternatives: Selection of samplable process operations: For each operation selected, all the pairs where this operation is present are added to the set of pairs that can be sampled. Selection of samplable technologies: For each technology selected, all the pairs where this technology is present are added to the set of pairs that can be sampled. Selection of samplable (technology, operation) pairs: In this case, only the selected pairs are added to the set of pairs that can be sampled. Moreover, as the technology mix can be different for each production time period, some operations and/or technologies may not be well represented in the historical metrology data extracted. If sampling is reduced for these pairs, there is a risk that control breaches will be generated or too few data will be used in the Statistical Process Control. That is why there is the option to set a threshold that restricts the selection of operations that can be sampled to those for which the historical data contain a number of measurements greater than or equal to this threshold. The value of the threshold is entered in an input box that appears once the checkbox Use Threshold is activated. After selecting the operations that can be sampled, the user selects and sets the constraints, the criteria to be minimized (objectives), and the MOGA stopping tests. Constraints are not mandatory, but at least one criterion and one stopping test

9 Fig. 5 Graphical interface of the optimization tool developed should be selected in order to start the optimization algorithm. Some additional options, which are outside the scope of this paper, are used to configure the algorithm, depending on the format of the input data or the desired results. MOGA implementation The MOGA we implemented is described in Fig. 6. The basic principle of a GA is to generate successive populations of candidate solutions and to ensure they evolve toward better solutions. The generation of new populations is performed using techniques inspired by natural evolution (such as selection, mutation, and crossover) and they are repeated until a predefined stopping condition is satisfied. Initial sampling characterization Before starting the optimization, an initial simulation is needed to compute the initial W@R (W@R init ) to define the initial sampling plan S init and the number of measurements for each operation N init. This latter represents an upper bound for the corresponding operation of the sampling plan to be optimized. The values of MeanMaxW@R init, MeanW@R init, CAPA init, and MaxW@R init for each processing machine are also deduced from this simulation. Based on the parameter values selected by the user, these Fig. 6 Genetic algorithm

10 results are then used to compute the effective values of the constraints: c = ( MeanMax W@R c + 1) MeanMaxW@R init ; MeanW@R c = ( MeanW@R c + 1) MeanW@R init ; CAPA c = ( CAPA c + 1) CAPA init ; and MaxW@R c,z = ( MaxW@R c + 1) MaxW@R init z {1,.., Z}. Initial population generation The initial population is composed of 50 chromosomes, generated randomly using the random function of Excel VBA. Evaluation and validation Each chromosome is evaluated by simulation, as explained in Evaluation by simulation section. In the case where one or more constraints are active, each individual (chromosome) is validated with respect to the constraints. Any chromosome that does not fulfill the constraint will be downgraded toward the evaluation criterion (denoted CR) by increasing its value. CR x = CR x 0 + kx (8) with k x = i=n i=1 Const(i) Const c (i) tr(i) (9) where, CR0 x is the initial evaluation of the criteria (before downgrading) of sampling plan x, CR x is the value of the downgraded criteria of sampling plan x, k x represents the distance between the evaluation of the chromosome and the constraints, n is the number of the constraints considered, Const(i) is the value of the constraint i for the chromosome concerned, Const c (i) is the effective value of the constraint i, is the absolute value function, tr(i) is a Boolean variable equal to 1 if the constraint i is transgressed, and 0 otherwise. Stopping test The GA creates new populations until the condition of the stopping test is satisfied. Three kinds of stopping test are proposed. 1. A stopping test based on the value of the criteria: the user can select one or more conditions on the criteria. The optimization will be stopped when the value of criteria of the best chromosome of a population satisfies all the specified values. 2. A stopping test based on the time of optimization: the algorithm stops when a specified time limit is reached. 3. A stopping test based on the number of generations: the algorithm stops when a specified number of generations is reached. If the user defines a number of conditions, the optimization will stop once the first condition has been satisfied. If the stop condition is not fulfilled, the algorithm goes back to validation, evaluation and evolution. Multi-criteria classification When mono-objective optimization is deployed, the chromosomes are ranked accordingly and the best individuals are selected. When multi-objective optimization is deployed, by selecting two or more criteria for the optimization, the algorithm classifies chromosomes according to each criterion. The best individuals are those with the smallest sum of ranks of all criteria. Figure 7 illustrates the method adopted for multi-criteria classification through a simple example, where 10 chromosomes are classified according to three criteria (C 1, C 2, and C 3 ). First, the chromosomes are evaluated by simulation and classified for each criterion separately. Then, a global classification is performed, based on the best rank for all the criteria. Formation of a new population New individuals are generated using methods of selection, mutation, crossover and random generation. Various trials conducted with the case study datasets led us to the following scheme for evolution methods: Selection 5 % of the new individuals are defined by selection. The best individuals are selected after the classification step. Mutation 12 % of the new population is obtained by the mutation of chromosomes from the old population. In order to explore different regions of the solution space, 60 % of these chromosomes are chosen randomly from the old population. For a local exploration of the solution space, 40 % of the mutated chromosomes are chosen randomly from the group of selected chromosomes (best) chromosomes. The number of genes mutated is defined randomly between 1 and 25 % of the size of the chromosome. Random 9 % of the new individuals are defined randomly (as with the initial population generation step) Crossover 74 % of the new population is obtained by crossover. Chromosomes are crossed in order to create the

11 Fig. 7 Multi-objective classification of a population of 10 chromosomes same number of new chromosomes in the new generation. For each crossover, two chromosomes, called parents, are chosen randomly. The position of the crossover is chosen randomly, and each chromosome is divided into two segments. Two new chromosomes, called children, are made up of these parts. The first child is composed of the first part of the first parent and the second part of the second parent. The second child is composed by the first part of the second parent and the second part of the first parent. Experiments and discussions The proposed approach was tested through a case study using one month of real data provided by STMicroelectronics in Crolles, France. It focused on optimization of the CD measurements used to control the Lithography and Etch workshops. The simulation was executed considering two constraints: (1) 10 % of tolerance in the increase of MeanMaxW@R for all processing machines, and (2) a minimum reduction of 10 % of the capacity used. In other words, an acceptable solution x should satisfy the following relations: MeanMaxW@R x 1.1 MeanMaxW@R init and CAPA x 0.9 CAPA init The goal is then to minimize each of the following criteria: MeanMaxW@R, MeanW@R, CAPA, and the global MaxW@R. The classification of chromosomes is done according to the method explained in the Multi-criteria classification section. Each chromosome used for optimization corresponded to the combined sampling rates of all operations inspected more than 10 times during the time period considered, which means that the chromosomes were composed from 342 genes. Each population was composed of 50 chromosomes, which evolved with the proportions defined above, using various techniques. The optimization was stopped after 300 iterations. Each chromosome was evaluated by simulation, which took about 1 second for a 1 month period. Although the simulation provided a good evaluation of the sampling plan, too much time was required to simulate 300 generations (about 4 h using a computer with a processor Intel(C) Core TM 2Duo CPU 2.40 GHz). Figure 8 presents the evolution of the optimization criteria with iterations. The constraints were satisfied after the 50th iteration, and the criteria continued to improve until the 225th iteration, where they stabilized. Due to the huge size of the chromosomes (342 genes each), the variation in criteria by generation was slow. Improving the results would require many more generations and a great deal more computing power. The selected criteria did not vary in harmony; for instance, the reduction in the capacity used (CAPA) increased Mean Max W@R and vice versa. This made optimization difficult, requiring several iterations. The chromosome of the final result presented in Fig. 9, represents the optimized sampling rate of each selected operation (gene). It is the best solution present in the last generation (generation obtained when the stopping test is verified). The optimized sampling plan corresponds to an SPS equal to 82.9 %. Table 1 summarizes the final results obtained for the four criteria considered. In the init line, the values of the criteria are taken from the initial sampling plan. The second line ( gen ) presents the values obtained after optimization by the

12 Table 1 Results abstract CAPA MeanW@R MaxW@R MeanMaxW@R init 16, gen 13, % 30 % 3.5 % 6.6 % c 10 % 10 % Fig. 8 Convergence of the optimization criteria MOGA. The third line presents the improvements as a percentage, and the last line presents the value of each selected constraint. The capacity is expressed in terms of the number of measurements performed during the observation period. The initial value of 16,600 drops off by 17.1 %, which represents a decrease of 2,792 measurements. As one tool is given for around 1,300 measurements, the improvement saves 2.14 tools. Because a tool can cost between $1 and $3 million, the savings are considerable (the cost drops from $6.84 to $2.14 million). At the same time, the risk of scrapping products was also reduced: on average (second column), a decreases of 30 % from products to 93.7 products. This means that 94 wafers were potentially at risks after MOGA application, instead of 134. This represents a potential savings of 42.2 wafers. With a wafer costing between $2,000 and $10,000, this means that the insurance coverage could be decreased by $84,400, on average, to $422,000. The third criterion was decreased by 3.5 % which represents 25 wafers. This resulted in a potentially massive scrap reduction, valued at between $50,000 and $250,000. The fourth criterion was decreased by 6.6 %, even through the constraint was released by 10 % by the manufacturers. This represents a gain of 23 wafers, or $46,000 $230,000. Operationally, depending on the scrap insurance policy in place to cover wafer scraps, one of the gains mentioned above can be selected to estimate the global gain. For instance, if the scrap insurance policy covers an average loss, then the gain associated with MeanW@R can be chosen. In this case, the MOGA generates savings between $2.14 million + $84,400 = $2,224,400 and $6.84 million + $422,000 = $7,262,000. Globally, the results presented in Table 1 represent very important savings, which can be easily realized by modifying the sampling plan based on the MOGA results. The results regarding risk exposure reduction are detailed in Figs. 10 and 11. Figure 10 shows the percentage reduction in MeanW@R and Max W@R for each processing machine brought about by the optimized sampling plan computed by the MOGA. It shows the significant gap between the optimized solution and the current sampling plan regarding the MeanW@R and Max W@R of some processing machines. In fact, these two indicators were reduced for almost all the processing machines, by on average, 30 and 3.5 %, respectively. This figure also provides analysis from the Fig. 9 Optimized sampling rates (best chromosome)

13 Fig. 10 max and mean per machines using genetic optimization Fig. 11 Reduction of over time workshop point of view. The first 23 processing machines belong to Wor kshop1 (Etching) and the others belong to Wor kshop2 (Photolithography). Results show that the average of is better for Workshop2. This can be explained by the fact that Wor kshop2 processing machines were globally more loaded than those of Wor kshop1, and the reduction in sampling rates is more efficient in terms of risk exposure reduction with an increase in the workload of the processing machines. Figure 11 represents the temporal evolution of the overall (all tools together) Max and with the initial sampling plan compared to the plan computed by the MOGA (Max W@R gen and MeanW@R gen ). The results drawn in this figure confirm the potential improvements regarding risk exposure mastery. Moreover, these improvements are effective over the entire production period considered, especially for the MeanW@R. However, in some periods of time the Max is hardly reduced because of the influence of the scheduling decisions which were not considered in this study. The time delay was not a decision criterion in our algorithm, because it is not the primary concern of the manufac- Fig. 12 Comparison of the distribution of τ in the initialsampling and in genetic sampling turers in this case study. However, it has been demonstrated that variation of the metrology time delay impacts the variation in W@R (Sahnoun et al. 2012). That is why the metrology time delay τ was recorded in these experiments. The observation of τ before and after the optimized sampling plan was obtained by the MOGA confirms this relation. The metrology time delay has been improved, as shown in Fig. 12: the mean and standard deviation of τ are reduced by 40 and 70 %, respectively. Conclusions and perspectives In this paper, a decision support tool has been proposed to optimize sampling plans in complex manufacturing systems. The optimization method uses the MOGA, with which the evaluation function is obtained by simulation using historical

14 data. The objectives optimized are related to risk exposure control and metrology capacity allocation. The results gained in this study show clearly the value of multi-objective optimization. The MOGA provided very interesting results. Very significant savings could have been made by using an optimized sampling plan. The first evaluation shows a range of potential savings between $2.22 and $7.25 million, confirming that optimization based on sampling is a good avenue for improving operational performances. This research also shows that there is an opportunity to integrate the dispatching of lots for the inspection into the simulation process. The study was focused on the implementation of a multi-objectives optimization algorithm to define an optimized sampling plan, without regarding the best algorithm to use. Future works will be focused on the development of an algorithm for the automatic selection of the operations to be sampled, leading to improvement of the chromosomes used in the genetic approach. Moreover, it will be an opportunity to position our MOGA compared to other multi-objective algorithms such as NSGA-II (Nondominated Sorting Genetic Algorithm-II), for example. The aim is to find other metaheuristic algorithms that can improve the results obtained by the MOGA proposed in this paper. Acknowledgments This article describes the research conducted within the context of the IMPROVE and INTEGRATE Projects. It has been finalized with the support of dis544 covery Grants founded by NSERC, under the number RGPIN CRSNG NIP The authors warmly thank STMicroelectronics for providing data and support for their research. Special thanks to the industrial and process control team for their daily involvement and support in helping us achieve sound results in the projects. 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