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A broader view of the economic design of the X-bar chart in thesemiconductor industryB. Baud-Lavignea; S. Bassettoa; B. Penza

a Department of Industrial Engineering, Grenoble Institute of Technology, 38000 Grenoble, France

First published on: 06 November 2009

To cite this Article Baud-Lavigne, B. , Bassetto, S. and Penz, B.(2010) 'A broader view of the economic design of the X-barchart in the semiconductor industry', International Journal of Production Research, 48: 19, 5843 — 5857, First publishedon: 06 November 2009 (iFirst)To link to this Article: DOI: 10.1080/00207540903150593URL: http://dx.doi.org/10.1080/00207540903150593

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International Journal of Production ResearchVol. 48, No. 19, 1 October 2010, 5843–5857

A broader view of the economic design of the X-bar chart in the

semiconductor industry

B. Baud-Lavigne, S. Bassetto* and B. Penz

Department of Industrial Engineering, Grenoble Institute of Technology,46 Avenue Felix Viallet, 38000 Grenoble, France

(Received 13 February 2009; final version received 19 June 2009)

This paper starts from a notice made in the semiconductor industry: a processcontrol system and especially control charts provide information that can beexploited for correlation analyses during process investigations. In this industry,key and costly investigations are made for improving yield and reducing scrap.Daily, engineering teams are working at manufacturing improvements. Withoutprocess data, their work could take much more time and lead to weakimprovements. Nevertheless, design of process control systems and in particularcontrol charts lacks from taking into account this remark as there is no soundapplication to infer an optimal control chart depending on business parameterslike yield, scrap, customer audits, etc. Meetings between several engineering teams(process control, quality, process integration, industrial engineering andproduction) occur frequently to find an affordable quantity of controls for eachoperation. The literature point of view does not provide more recommendationsto take into account the reuse of data into these costly investigations. The paperinvestigates this issue. For this first investigation, work has been focused on thedesign economics of control charts for a simplified process model. The papertranslates this concept into the Lorenzen and Vance’s (1986) model. It simulatesthe design economic of a control chart taking into account this new model andinfers new optimal statistical process control (SPC) set points. An analysis of thisnew link is made in a context of yield improvement, providing reference forknowing optimal quantity and frequency of controls.

Keywords: economic design of control chart; scrap investigation; learning;ramp-up

1. Introduction

The semiconductor industry has to cope with a severe environment, drifting processes, andsensitive products at any process disturbance. Information coming from tools and processmeasurements is a central element for learning how to master manufacturing systems. Foreach new semiconductor technology a learning process follows, enabling the fine tuning ofspecifications, operating modes, control positions, etc. This process is followed to masteroperational risks before increasing the volume of production. Often, it is admitted bymanufacturing teams that during early phases of this learning process, the number ofcontrols have to be high in order to learn faster. However, milestone after milestoneprocesses are expected to be more robust and stable. Progressively, it becomes possible to

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online

� 2010 Taylor & Francis

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Downloaded By: [Bassetto, S.] At: 12:43 10 August 2010

release controls as the process is better mastered. Maintaining a high level of control isthen nonsense (Bergeret et al. 2004) and the pressure to release controls increasesdrastically as the ramp-up process is going on.

Nevertheless, releasing controls remains a highly risky action for managers. As studiedby Gershwin and Kim (2005), and Colledani (2008), it increases the number of wafersproduced between two controls and, by the way, increases the risk of massive scrap. Oftenthe decision of affordable quantity of controls for each operation is taken during anengineering meeting between several teams: process control, quality, process integration,production management. This can lead to sticking the level of control close to a predefinedvalue reasonable in the organisation, often far from an optimum. For example, it is acommon practice in the semiconductor industry to measure over 200 points of control,each wafer during the first phases then to decrease near to 50 points almost each lot and toreach less than 10 points per measured lot every 100 lots, when products are manufacturedin volume. These control rates are not really linked to those provided by classical statisticalprocess control (SPC) design methods (either pure statistical design or economic designmethods), which also fail to take into account business parameters like data reuse or globalcost. There is then an opportunity to develop a decision aided tool to help in balancingoptimal parameters depending on scrap rate. The design of control charts takes care ofexternal parameters, usually considered by managers like scrap rate and the reuse ofquality information.

The aim of this paper is to provide a framework to help this decision process by takinginto account both statistical and informational issues. This paper proposes a two stageapproach towards such a tool:

. The first step enhances the now classical Lorenzen and Vance’s (1986) model. Thecost of one scrap is far more important than out-of-control ones and datacollected from quality measurements is used during scrap analyses for buildingroot causes, curative and preventative action plans. It increases the influencebetween number of points, control frequency, and the necessary time to solvescrap issues.

. The second step consists of discussing the evolution (through the simulation) ofsampling points and rates under the condition of scrap rate evolution.

The paper is structured over five sections. In a first part, the paper goes through aliterature review of economic design of control charts and links with learning. Then, thepaper presents the enhancement to Lorenzen and Vance’s (1986) model. The paperpresents how simulations have been performed and several results. These two partsrepresent the first stage, discussed above. Then the paper goes through the second stage,also aforementioned. The paper ends with perspectives.

2. Literature review

The questioning of interaction between quality and manufacturing operation has beenaddressed recently by Gershwin and Kim (2005), and Colledani (2008). Their studies arethe first investigations of how quality considerations can modify the production control.

The design of control charts involves the selection of three parameters: sampling size(n), control frequency (h), and control limits (L) in order to detect earlier tools andprocesses shifts (Montgomery 2004). Economic design of control charts is a method which

5844 B. Baud-Lavigne et al.


aims at determining these parameters of a control chart in optimising a cost function of

the process monitored. Usually, it considers parameters internal to the monitored

process, such as the cost of sampling and analysing, quality cost and behaviour of the

machine.Classical criteria as statistical properties can be considered as economic variables.

Duncan (1956) proposed the economical optimisation of an �X control chart of a process

which is only shut down during repair. Several studies around this model during the next

40 years have been interested in the other types of control charts, adaptation to specific

situations and resolution methods. A breakthrough has been the generalisation of all these

models by Lorenzen and Vance (1986). It is nowadays a reference in economic design, as it

can be easily implemented and adapted.Optimising the design of control charts in an economic manner affords many

opportunities to take into account external parameters of the process. One example is the

adaptation of the economic design to semiconductor industries by Jang et al. (2000). This

study details cost parameters of the Lorenzen and Vance model, specifically the quality

cost by using opportunity cost of the nonconformities.Several criticisms have been expressed and enhancements have been performed over

economic control chart design. Critics of this type of design highlight a lack of

statistical properties, and the difficulties to evaluate correctly the input parameters

(Woodall 1986). Improvements of the Lorenzen and Vance model have been done by

adding statistical constraints (Saniga 1989), and by proposing more robust implementa-

tion: in using multiple scenarios in the optimisation (Linderman and Choo 2002), or in

expressing the input parameters in ranges of values (Vommi and Seetala 2007).

Extensions of the economic design have also been through adaptive control (Prabhu

et al. 1997). We recommend the overview of Tagaras (1998) for more details about

adaptive control.Another deficiency pointed out by Deming and Edwards (1986), and Woodall (1986) in

the economic design theory is the aim at maintaining the process in its initial situation,

without any emphasis on continuous improvement. Silver (1999) proposed a dynamical

optimisation of the design parameters by reducing the out-of-control frequency at each

cycle during the expected improvement period. This reflects the fact that improvement is

inherent to the process and should be considered as it affects one of the design parameters.

Weheba and Nickerson (2005) consider the economic design of control charts as a reactive

process and have a proactive approach, by adding the cost function of improvement in the

process. Instead of considering improvement as a possibility to reduce costs, the targeted

quality is imposed. The concept of ‘better quality level’ is implemented in their model as

lower process drifts.In the current literature, the fact that controls retrieve data, which is employed to solve

manufacturing issues, is not exploited. However, data coming from tools, process and

product controls are re-employed during problem solving sessions. An example is provided

by a case study on the semiconductor industries by Bassetto and Siadat (2009). This

literature review has not retrieved any article about links between the reuse of process

control data and the design of control charts. This paper investigates this link. Moreover,

considering yield improvement, the paper discusses how quality control equilibriums

(number of points to be measured, sampling rates in particular) will be modified by the

decrease of scrap rate due to learning. Learning, as scrap rate or yield, will be considered

as an exogenous factor of control chart design in this paper.

International Journal of Production Research 5845


3. Description of the model and the cost function

This paper proposes an enhancement of the Lorenzen and Vance’s (1986) economic design

of control charts model, by embedding a link between control plan definition (sampling

rate and number of controls) and duration of scrap investigation. This enhancement comes

from the literature review and the authors experience in the semiconductor industry.The manufacturing process is composed of N stages, as presented in Figure 1, followed

by an �X control chart. It is similar to the model of Colledani (2008), which investigates the

equilibrium between process control actions and line balancing ones.Usually, SPC measurements are employed for monitoring processes drifts. Each out-

of-control is followed by a local action. Most of the time, this action is a predefined

procedure known to be a fast answer at the drift. These kinds of actions remain low cost

and simple to perform, so that any operator can apply them. They are fully part of local

continuous improvement. Long run investigations are usually not allowed within this

frame.At the end of the process, the product is qualified by functional tests. These controls

are employed to sort dices and wafers and to scrap those rejected. When a product is

considered faulty, long run investigations are performed to detect faulty process stages.

Data from the entire production line is used to investigate, through data analysis

algorithms (commonalities, clustering for example). These actions are performed by highly

qualified technicians and engineers. They are time consuming and rely on available process

data. Speeding this stage remains a major stake as it is at the heart of process learning and

global continuous improvement (Sterman 2000).The remainder, the paper aims to determine parameters of each SPC control that will

minimise the costs of the whole model.In order to simplify the mechanism of the model, only the size (n) and the interval

between two sampling (h) will be optimised, given control limit (L). In order to focus the

analysis on the effect of the scrap investigation costs and to constrain the solution space,

L is held at three in our optimisation as presented by Weheba and Nickerson (2005). This

assumption respects the ‘six sigma’ method recommendations. When relaxing this

Figure 1. Modelling of the production line and the quality controls.



assumption, constraints have to be added on the ‘average run length (ARL)’ to ensurestatistical properties of the optimisation, as in the economical-statistical design of Saniga(1989). Besides, emphasis should be put on optimisation methods to maintain a short timeoptimisation.

The function to minimise is:

minni,hi

XNi¼1

fiðni, hiÞ

!þ b

n1h1

,n2h2

, . . . ,nNhN

� �,

where fi is the cost function of the ith SPC control and b is the cost function of the scrapinvestigation.

The cost of the investigation (function b) has been formulated as a function of thequantity of SPC control done during the production. At stage i, if ni is the size of a sampleand 1=hi the frequency of sampling, then ni=hi is the quantity of controls per hour. Eachcontrol is assimilated at one measurement retrieving data. As mentioned before, thisamount of data is stored and used both for local and global improvement. The moreinformation that is available, the more analyses can be driven without any extra test. Timeof investigation is shortened and extra costs are reduced with high quantity of control.

In this model, b is twofold. It is made of a fixed cost C when no information is availableand a function g( ), which decreases the cost when available SPC data increases. As theproblem has been formulated as an optimisation process, C can be suppressed. In theremainder, only g( ) will be taken into account. The function to minimise then becomes:

minni,hi

XNi¼1

fiðni, hiÞ þ bn1h1

,n2h2

, . . . ,nNhN

� �¼ min

ni,hi

XNi¼1

fiðni, hiÞ þ C� gn1h1

,n2h2

, . . . ,nNhN

� �

¼ minni,hi

XNi¼1

fiðni, hiÞ � gn1h1

,n2h2

, . . . ,nNhN

� � !:

3.1 Model studied

For the sake of simplification, a model composed only of the ith stage and the scrapinvestigation has been implemented. It is presented in Figure 2.

The function to minimise is simplified to:

minn,h

f ðn, hÞ � gn

h

� �� :

Notation and the cost function of the economic design model are taken from Lorenzenand Vance’s (1986) model. This function follows the classical control cycle detailed inFigure 3, the parameters and costs are described in Table 1.

The cycle starts in-control. There is a control before the drift. The duration of thein-control state is 1/�þ s*T0/ARL1. � is the mean time between the last sample before thedrift and the drift. h is the time between samplings. However the control chart cannotimmediately detect the drift. Then h*(ARL2� 1) time passes until the sampling which willdetect the out-of-control. Between the sample and the out-of-control signal, there is a datatreatment duration: n*E. After the out-of-control, T1 is the investigation time, and T2, therepair time. At the end of T2, the issue is fixed and the process is released in-control.



The hourly cost function of a classical economic design is:

f ðn, hÞ ¼CNC þ COoC þ CS

Tcycle:

where:

. Tcycle is the length of a cycle from the production beginning to the occurrence of

an assignable cause, then its detection, and finally its repair:

Tcycle ¼1

�þ

Y:s

ARL1� � þ h:ARL2 þ n:Eþ T1 þ T2;

Figure 3. A control cycle.

Figure 2. Simplified model.



. CNC is the non-conformities costs for a cycle:

CNC ¼C0

�þ C1ð�� þ n:Eþ h:ARL2Þ;

. COoC is the false alarm and detection costs for a cycle:

COoC ¼Y:s

ARL1þW;

. CS is the direct sampling cost for a cycle:

CS ¼ ðaþ b:nÞð1=�� þ n:Eþ h:ARL2Þ

h:

For a detailed explanation of this classical model, the authors recommend reading thepaper of Lorenzen and Vance (1986).

3.2 Expression of the gain function

gð Þ is the gain per hour on the scrap investigation. It is estimated by the following function:

gn

h

� �¼ I:CInv

n

h

� �:

Table 1. List of parameters.

Type Variable Designation

Design variables n Sample sizeh Hours between samplesL Number of standard deviation from control limit

to centre line

Process parameters � 1/mean time process is in-controlD Number of standard deviation slip when out-of-controlC0 Quality cost/hour while in-control (IC)C1 Quality cost/hour while out-of-control (OoC)

Sample parameters a Fixed cost per sampleb Cost per unit sampledE Time to sample and chart on item

OoC parameters Y Cost per false alarmW Cost to locate and repair the assignable causeT0 Expected search time when false alarmT1 Expected time to discover the assignable causeT2 Expected time to repair the process

Calculated variables s Mean number of sample while IC� Mean time between last sample while IC and the drift

ARL1 Mean number of sample between an out-of-controlwhile IC

ARL2 Mean number of sample between an out-of-control samplewhile OoC



where:

. I is the number of investigations per hour. I equals the scrap rate (or mechanicalyield) times number of products per hour;

. CInvðn=hÞ is the expected gain per investigation. It equals the maximum gain timesj ðn=hÞ:

. j ð Þ expresses the impact of the quantity of available information, that is thequantity of control done, on the gain of investigation; j ðn=hÞ 2 ½0, 1�;

. Several functions have been tested, such as linear, logarithmic or sigmoidones. To translate the fact that the value of data is higher when we presenta first order model: j ðn=hÞ ¼ 1� e�Qðn=hÞ, where Q is a constant whichdetermines both the slope of the function and the maximum of profitableinformation.

Even if data is reused for scrap and yield improvement, the learning mechanismremains unknown and will be considered as a black box in this paper. As a consequence,scrap rate will be considered as an exogenous parameter of control charts design.

4. Simulation and results

The case study used to test this model generalises a typical process in the semiconductorindustry (Bassetto 2005) in a 300mm wafer fab. The maximum gain is estimated at 2500Eper investigation, and is reached when controls are about 200 units per hour, that is Q isaround 0.029. We assume that the process drifts to 1 sigma every 24 hours on average.Costs are estimated upon a cost of a machine of 300E/h. C0 is roughly estimated at160E/h; C1 reaches 640E/h.

Sampling considered a fixed cost per sample of 2E (a 1000E test wafer is used about500 times), a time to sample 17 units takes about 5 minutes, i.e., 5 � 10�3min/unit sampled.This leads to a variable cost of 1.4706E (time to sample� 300E/h). The search timefollowing an out-of-control is one hour, whether the process is in-control or not. So thecost of a false alarm is 300E (1 hour� 300E/h). The time to repair the process is 4 hours,and the cost to locate and repair the assignable causes is 2000E (5 hours� 300E/h). Asynthesis of these parameters is presented in Table 2.

The maximum gain over one scrap analysis is estimated at 2500E. The scrap rate (alsonamed S) for this simulation is 50%. The manufacturer processes 12 products per hour.

The optimisation is a systematic enumeration of the possible combinations of n and h,in a few steps: first, a rough step for n and h is used to determine the valuable scale (i.e., n issearched between 1 and 1000 with a step of 10, h between 0 and 20 with a 0.1 step). Thenthe window is narrowed and the step is refined. Thanks to modern computers and the factthat we consider L given, the enumeration is fast and takes less than five seconds for40,000 combinations tested on a Pentium M processor at 2.1GHz with 1GB RAM. Theprogram has been coded in Cþþ with graphic representation using Root (CERN, 2009)libraries, and is available upon request from the authors. The results are represented inFigures 4 and 5 in which the curves represent iso values of f ðn, hÞ � gðn=hÞ depending on nand h. Costs are elevated when:

. n is low, because of false alarm and scrap investigation costs;

. h is low and n is high, because of out-of-control and direct sampling costs.



Table 2. Model parameters.

Variable Case study

Design variables n Optimisedh OptimisedL 3

Process parameters � 1/24D 1C0 160E/hC1 640E/h

Sample parameters a 2Eb 1.4706EE 0.004902E

OoC parameters Y 300EW 2000ET0 1 hT1 1 hT2 4 h

Calculated variables se��h

1� e��h

�1� ð1þ �hÞe��h

�ð1� e��hÞ

ARL1 and ARL2 Calculated following Lorenzenand Vance’s (1986) formulation

Figure 4. Cost per hour of the classical economic design.



The integration of the scrap investigation costs in the economic design of control chartsmodifies the quantity of control optimum. Indeed, in our example, quantity of controlsincreases from 10.85 u/hour to 25.17 u/hour, the interval between two samples decreasesfrom 1.475 hours to 0.715 hour, and the sample size increases from 16 to 17 points. Thisresult shows that: the reuse of SPC data for speeding scrap investigations has a clearimpact on the SPC parameter and tends to increase the number of controls. Thisconnection proves that other sampling plan choices would lead to higher processcontrol cost.

5. Discussions about the use of this model in a ramp-up context

During the ramp up phase of a new product, significant improvement in the processhappens. Preventive actions improve yields, manufacturing stability and robustness.Therefore, production costs and scrap rate are decreasing rapidly (Argote and Epple1990). In the semiconductor context, priority is to reduce ramp-up time and to achieve assoon as possible a good yield (Bohn and Terwiesch 1999), because of the product life-cycle,characterised by decreasing of the prices and short lifetime (Jang et al. 2000). Experienceon the past product and learning curves (Argote and Epple 1990) can provide anestimation of the scrap rate.

The model assumes that S is an exogenous parameter towards C0 and C1. When theprocess is in-control, the monitored tool produces, nevertheless, some scrap. Wafers can bescratched or broken, or contribute to the final yield reduction, without knowing clearlyeach process stage implication. As final scrap rate evolves during the manufacturingramp-up or the technology life-cycle, there is a relation between C0 and S.

Figure 5. Cost per hour with integration of the scrap investigation costs.



Nevertheless, although it is possible to know broken and scratches costs at each processstage, estimating the true contribution of each operation upon hundreds of them to thefinal yield is hard to perform. That is why learning is considered only through theevolution of S, while C0 and C1 are assumed to be fixed quality costs.

The curves in Figures 6–8 present the evolution of optimal n and h parametersdepending on scrap rate. Figure 6 shows the frequency of control decreases dramatically,with slight peak augmentations when n increases. Figure 7 shows the number of unitssampled is discrete. It increases slightly from 16 when scrap rate is below 28% to 19 above62%. Figure 8 combines the two previous ones. Figure 8 shows the optimal solution for n,h, and the quantity of control (n/h) as a function of the scrap rate: when the scrap rate isnull, optimum values are the classical economic design optimum. The quantity of control

Figure 6. h, as a function of scrap rate.

Figure 7. n, as a function of scrap rate.



considerably increases with the scrap rate. The more scrap there is, the more controls haveto be performed. Moreover, two major phenomena appear on the latter graph:

. The influence of the gain on investigation is strengthened with scrap rate, as moreinvestigations happen, and the derivate of the quantity of control increases withscrap.

. Then, the expression of the gain on investigation weaken the influence of thequantity of control when n/h430.

These two points are related to the gain function. Other gains would have provided othercurves.

As a practical application, by anticipating scrap rate, it is possible to evaluateaccurately the quantity of information needed for both process control and processlearning. It rationalises extensive sampling early in the product life-cycle to achieve fasterlearning.

A simulation of yield learning phenomenon is presented in Figure 9. Yield increasesfollowing a first order learning process from 5 up to 98% in 100 periods: minimum andmaximum yield, yield evolution, and period length should be adjusted to the studiedindustry.

In the semiconductor industry, it is common to find such yield learning curves. Yield isjust assimilated at (100-(scrap rate)). Results presented in Figure 9 can be used as a firsttool for adjusting controls while yield increases, based on an economical approach.

6. Conclusion

In this paper, we presented an adaptation of the economic design of control charts byderiving that SPC measurements are valuable data for scrap analysis. Control charts areused for both monitoring the process locally and gathering information about eachmanufacturing step. Scrap investigations duration are globally decreased by employingdata analysis algorithms, which rely on SPC data.

Figure 8. Evolution of the optimal quantity of control per hour depending on scrap rate.



The paper proposed a model enhancing Lorenzen and Vance’s (1986) design economicmodel. The simulation leads to a new equilibrium of optimal sampling plan. Based on thismodel, the paper also investigated the impact of scrap rate evolution on an optimal controlplan. Several perspectives pursue this enhancement.

Final scrap rate and learning processes, locally and globally are considered as externalvariables of the model. In a next model, they could be considered as endogenousparameters.

Indeed, if (n, h) are modified, little data will be available for investigating scrap issuesand related problems and then it could take more time to investigate. Let t denote adiscrete time period. It is understandable that change of (n,h)t leads to a change of (S)tþ1,which leads to a change of (n, h)tþ1 and so on. (n, h)t is then a function of previous n and hcouples and scraps rate: (n, h)t ¼ �((n, h)t�1, . . . , (n, h)0, (S)t, (S)t�1, . . . , (S)0). It is the samefor the C0 and C1 parameters. During this investigation, the stability of the model shouldbe checked. The convergence toward an attraction pole has to be investigated.

A second investigation to lead is to take into account that process control has severallayers of information (Wonoh and Vachtsevanos 2000, Yang and Sheu 2007): SPC, EPC(engineering process control), run to run, alarms, customer feedbacks, etc. Each of theselayers generates data, which can be used for learning. It can be useful to take into accountthis new configuration to set control of each layer depending of its impact on yieldlearning.

Figure 9. Impact of yield learning curves on process control plans.



A third extension of the paper could be N process stages instead of only one (Lam et al.2005). Stage correlations could be part of this extension. In this extension, the integrationwith WIP (work-in-process) should also be taken into account. Finally a full model:dynamic, multilayer and N process stages could be proposed.

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