MC - 231Priority X-Factor Modeling for Differentiated

Manufacturing Service Planning

Shi-Chung Chang Ke-Ju Chenscchang(i,cc.ee.ntu.edu.tw r9392 1 070@,ntu.edu.tw

Dept. of Electrical Engineering, National Taiwan University, Taipei, Taiwan, ROC, 10617Phone: +886 -2 -2362-5187 Fax: +886 -2 -2363-8247

Abstract - This paper addresses the X-factor modeling needsin fab capacity and release rate planning for differentiatedmanufacturing service provision. A priority X-Factorconstrained planning problem is first formulated thatdescribes the relation among profit, release rates ofindividualpriorities, and capacity utilization. Modeling priority X-factors is key to the formulation. We deign a novel M/G/1:PRqueue approximation-based network modeling methodology tocapture in a scalable way how operation priority, productionflow variations, and capacity utilizations may affect individualPXFs of overall fab and tool groups. Numerical studiesdemonstrate the potential applications ofour PXF models.

I. INTRODUCTION

Effective provision of manufacturing services in multiplepriority levels has been one critical aspect to thecompetitiveness of wafer fabs. A customer order with ahigher priority level demands a shorter cycle time than ordersof a lower priority. Wafers of lower priority orders haveelongated cycle times because they need to wait in line untilwafers of higher priority orders finish processing. Machinecapacity loss may occur when processing a high priority orderby a batching machine without requiring a full load policy.Priority mix percentage significantly affects the variations offab performance such as throughput, cycle time, wafer-in-process (WIP) and bottleneck location [1].

Among the many fab performance indices, cycle time has asignificant impact on productivity learning and customerserviceability. There is a basic relationship among capacityutilization (U), throughput (T) and cycle time [2]. The cycletime of a fab increases exponentially with the increase of U/Twhen U/T goes beyond a high level, say, 90%, while it isproportional to U/T at a lower level. To measure and managecycle times, the notion of X-factor (XF), where

XF = cycle time/raw processing time (RPT)has been introduced to provide a sensitive performanceindicator and is standardized across different products [3].

It has been shown that many fab opration problems can beeffectively identified through the analysis of X-factors.Customized X-factor targets can be set for short cycle timemanufacturing (SCM) that not only allow the performancedifferentiation among toolsets of different characteristics butalso guarantee the overall fab objective [3-6].

In production planning of a fab, there are different XF target(XFT) specifications for individual priority levels ofmanufacturing services [7]. The XF of each priority (PXF) isa function of release rates of individual priorities and the totalutilization of the bottleneck tool group, which we shall refer toas a PXF behavior model. Note that different XFs requiredifferent levels of resources and hence lead to different costs

and manufacturing services of different XFTs should bepriced differently. Given a pricing policy, capacity coststructure, and a set of XFT, a PXFT constrained productionplanning decides the priority mix (or wafer release rats) ofproducts in individual manufacturing service priorities forprofit maximization subject to machine capacity and PXFTconstraints. Key to this planning problem is the behaviormodeling of the relationship between PXF and priority mixand capacity utilizations.

Motivated by the problem of PXFT constrained productionplanning. In this paper, we design and develop an M/G/l :PRqueue approximation-based network modeling methodologywith a focus on capturing how operation priority, productionflow variations, and capacity utilizations may affect individualPXFs of a fab. The M/G/1:PR queue model is adopted tomodel the behavior of a service node (tool group). On top ofthe single node model, we derive a PXF contribution theorythat relates PXFs of individual service nodes to the overall fabPXF and provides a novel priority network model. Modelfitting is then adopted to fit the M/G/1 :PR-basedapproximation to empirical fab data. The key idea of fitting isto add a parameter in calculating the mean residual servicetime, which compensates, for each priority, the effect ofarrival process variation to the mean service time at a node.Figure 1 depicts the concept.

The remainder of the paper is organized as follows. Section IIgives a problem formulation of PXFT constrained productionplanning and identify the need for a PXF behavior model.Section III then presents the M/G/l :PR queue approximationand priority contribution theory-based network modelingmethodology. Model analysis and applications are given inSection IV. Finally, Section V concludes the paper.

II. PRIORITY X-FACTOR CONSTRAINED PLANNING

Consider a fab with a given set of prespecified PXFTs forindividual priorities that need to be achieved by fab operations.Prices and WIP costs of individual priorities and capacitycosts are also given. A fab manager can control XFs of thefab, individual stages and priorities at each stage by adjustingpriority mix and utilization levels while maximizing the profit.

To formulate the XF constrained planning problem, let us firstdefine some notations, where we assume for simplicity ofdiscussions that there is only one type of products in eachpriority.NotationsXF : X-Factor ofj-th priority product at processing step i;x: Mean release rate ofjth priority product;

rij: Mean service time of a jth priority wafer at step i;

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RPT: Average processing time of a wafer at step i;Var[S ]: Variance of service time ofjth priority at step i;

Pi : Utilization ofjth priority product in step i;pj: Per wafer price ofjth priority;

CM: Per wafer manufacturing cost ofjth priority;c': Per wafer and per unit time WIP cost ofjth priority at ith

step.

The revenue rate of manufacturing wafers of one prioritycomes from its offered price and wafer release rate of thepriority, while the WIP cost at a step is proportional to therelease rate and cycle time at the step and the capcity cost isproportional to the release rate. In deciding on the releaserates or mix of individual priorities to maximize one'smanufacturing profit, a fab manager must considerconstraints of capacity and PXF targets. A PXFT constrainedplanning problem is formulated as follows:

Max J = [rRevenuej (Aj) - Costj ({tXF, },2j)]

= [j x Pj (c4 xAjxijxXFXj+ x 2)]

subject to

4', ({j }) < XFTfp ,a

I b

(i,j) uses tool group m

with targets XFTfPU, {XFTp j1,Vj} and capacity {Cm, Vm} given.

In the formulation, function &p21 respresents a model of the

relationship beween priority release rates and XF of the wolefab while function p25 j respresents a model of therelationship beween priority release rates and the XF ofpriorityj at a step s. These functions have to be constructedfor PXFT constrained planning.

III. MODELING PRIORITY X-FACTORS

Exploiting available closed-form results, we first adopt aM/G/l:PR queueing approximation to model the PXF of aservice node (tool group). As the Poisson arrival assumptionof the M/G/1 :PR model may not be a good approximation, themodel needs to be modified for a closer match to fab data.Network relationship among nodes is then needed formodeling fabwide PXF of each priority based on nodalmodels. Our modeling approach is described as follows.

III. I Single Node Model: M/G/I:PR approximation

Key to the fab PXF behavior model is the single-node prioritybehavior modeling. We modified the M/G/i :PR results of [8]by adding a compensation parameter a to account for thevariance of a general, non-Poisson arrival process. Theapproximation model ofXF for priority j is then

X Dp2s_j({4})

1 Z, (Var[Sk] + vdx)2Xk +oUJ,_kt=1 +1,

TI (l-ZpP)(l ZPk)k=1 i=l

where T1 Z-~E riand p, =2Rj xvr.

Figures 3 and 4 depict numerical studies of the properties ofthe modified M/G/l: PR approximation over a two-priorityservice node example, where wafer processing requirementsare the same for both priority. Model data setting refers to thatprovided by ITRS [7]. There is not much difference in PXF ofpriority-I when priority-I mix varies in the low percentagerange of 1-10%. Note that under a fixed mix percentage, thePXF of priority-i increases almost linearly with respect tooverall capacity utilization while the PXF of priority-2increases drastically when utilization is higher than 90%. Thelower priority is much more sensitive to capacity utilizationand priority mix than the higher priority.

1.2 Priority XF Contribution Theory

In [3], D.P. Martin derived a contribution theory thatdescribes the contribution by XFs of individual processingstep/tools to fab XF as following:

XFJab Ze iXFContribution

R P T XZ ~xXF1

j=1 RP Tpfabrs2f( (RPTi, XF, })

where RPTp =zt and Pf21 respresents the relationshipbeween XF at a tool group to fab XF.

In this paper, we exploit the relationship between total queuesize and queue sizes of individual priorities, the relationbetween total cycle time and cycle times of individualpriorities, and Little's formula, to obtain the contribution byXFs of individual priorities to the XF of step i as

m

XI- Y (relative utilization)j FXkjl

WLZ1 m xXflj(WLj)=Dp~25( Xki)

j=l

where WL9= x %Zj

We then obtain the contribution by XFs of individualpriorities to fab XF:

IVF _j= p2f_ j)

stae x q x XFj (WLjj=] RPTfb mEui

j=,

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The Priority XF contribution theory above assumes givenPXF behavior models at individual nodes in a fab. Figure 2shows the relationship among PXFs in the behavior model.

IV. APPLICATIONS OF PXF MODEL TO PLANNMNG

Construction of the PXF behavior model completes the PXFtarget constrained production planning problem. Figures 5gives results of numerical studies of the planning problem.The same two-priority example for Figures 3 and 4 isexamined, where release rate of individual priorities arechanged. Figure 5 shows that profit increases with capacityutilization under different priority mixes (PM). Under thegiven PXFTs, there is a maximum profit level among variousmixes for each utilization level as indicated by a red solidcurve in Figure 5. Our analysis show that the XF of the lowerpriority is very sensitive to slight variation of higher priorityrelase rate. When the bottleneck capacity utilization goesbeyond 82% in this example, the WIP cost of the lowerpriority surpassses the gain in revenue and the maximumprofit drops. We further apply the model to address thefollowing two questions.

Ql: How total utilization should be adjusted with respect toPMchange under given PXF targets?

Optimization tools can be applied to compute optimalcapacity allocations and release rates of individual priorities.A fab manager may also utilize PXF charts for decision-making. For example, consider Figures 2 and 3 as fab modelswith priority-I at 10% and the total capacity utilization at 90%.If there is a new order demanding an increase of priority-1 to20% with bottleneck capacity still at 90% utilization, thenPXFI = 1.34 and PXF2 = 3.58 and profit increases 2.86%from Figure 5. To maintain PXFl<1.3 and PXF2<3.2, the fabmanager may want to reduce the fab utilization to 84% so thatPXF targets are achieved at a price of 3.8% total profitdecrease. In this case, the tradeoff is between the revenue gainfrom increase in high priority orders and the loss in reducinglow priority wafer release. Besides capacity reduction, onemay want to increase the price of priority 1 to recover profitloss in priority 2. So, the model also provides a link forpricing consideration.

V. CONCLUSIONS

In this paper, we have designed and developed an M/G/1 :PRqueue approximation-based network modeling methodology.The PXF models obtained captures how operation priority,production flow variations, and capacity utilizations mayaffect individual PXFs of a fab. We have also demonstratedgood application potential of these models to the planning ofpriority manufacturing services..

Figure 1: PXF model fitting

XF,%age -I L LXFs,age 71Stage 1 Stage T

mi ;EEEEiSS A S..

WVs Fal) XFiT Etsi

Af. ,L. A/+>>

Figure 2: the architecture of the Fab behavior mode

X-Factor of Priority 1 & Utilization

1.8

1.72

1.64

1.56

1.48

1.4

1.32

1.24

1.16

1.081

67.0% 72.0%

- PM=50%--PM=35%& PM=20%

PM=10%--PM=7%

- PM=5%E-PM=3%--PM=1%

77.0% 82.0% 87.0% 92.0% 97.0%

Utilization of system

Figure 3: Relation between PXF and utilization for priority 1

X-

X-Factor of Priority 2 & Utilization

Q2: How tofind cycle time bottleneck oftool groups?

A two-priority, say P1 and P2, experiment is designed forinvestigation. The numbers of operation steps of P1 and P2products are 32 and 60, respectively. There are 12 servicenodes (tool groups). Release processes of the two prioritywafers are Poisson while service time distributions containuniform, erlang-k and exponential distributions. This examplebasically follows that in [9].

Figures 6 and 7 show comparisons of XFs between simulationresults and MJG/l: PR + PXF contribution theory for priorityland 2 under 12 tool groups with different service timedistribution. Table 1 shows the error percentages of each toolgroup. Figure 8 shows XF contributions of each node to fabXF and we can identify that tool group 5 is the cycle timebottleneck but the capacity bottleneck is tool group 3.

28

25-

22-19

16-13

10

7

4-

67.0% 72.0% 77.0% 82.0% 87.0% 92.0% 97.0%Utilization of system

- PM=50%- PM=35%-&-- PM=20%/o

PM=10%-A PM=7%

--- PM=5%iPM=3%- PM=1%

Figure 4: Relation between XF and utilization for priority 2

55

I~~~~~~~~~~~~~~~~~~~~~~~~

T ilT

w

i)ex

x

Profit Curve at each Priority Mix Level

- PM= 1%. PM=3%

PM =-%- PM = 7%

- PM = 10%-PM =20%- PM = imi%

.-XF Target Linit,

200 !ll190180170

a160

10 15004

140 WIM. I?ii13012011010068% 71% 74% 77% 80% 83% 86% 89% 92% 95% 98%

Utflization

Priority 2 X-Factor *-MGSI+PXFCI* Simulation

1412

, 10

W~ 6z4

20

1 2 3 4 5 6 7 8 9 10 11 12

Machine group

Figure 7: Comparison ofX Factor between Simulation andMG1+PXFC for priority 2

Figure 5: Relation between Profitdifferent priority mix

Priority 1 X-Factor

C

aet

and utilization under

* (MG1+PXFC)iSimulation

32.52

1.5

0.50

XF contibution at each MG

0.90.80.7

2 0.6.5

a04u 0.4X 0.3

0.20.10

1 2 3 4 5 6 7 8 9 10 11 12

Machine group

1 2 3 4 5 6 7 8 9 10 11 12

Machine group

Figure 6: Comparison ofX Factor between Simulation andMG1+PXFC for priority 1

Figure 8: XF contributions of each node to fab XF

Figure 9: Cost at each node

ACKNOWLEDGEMENT

* This work was supported in part by the Semiconductor ResearchCorporation and International Semiconductor ManufacturingInitiative under FORCe-II project 1214 and by the National ScienceCouncil, Taiwan, ROC, under grants NSC-93-2213-E-002-043 and94-2213-E-002-015

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