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### International Journal of Production Research

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### Weighted least-square estimation of demand product mix and its applications to

### semiconductor demand

Argon Chen a; Kyle Yang a; Ziv Hsia a

a Graduate Institute of Industrial Engineering, National Taiwan University, Taipei, Taiwan

First Published:August2008

To cite this Article Chen, Argon, Yang, Kyle and Hsia, Ziv(2008)'Weighted least-square estimation of demand product mix and its applications to semiconductor demand',International Journal of Production Research,46:16,4445 — 4462

To link to this Article: DOI: 10.1080/00207540701244028

URL: http://dx.doi.org/10.1080/00207540701244028

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Vol. 46, No. 16, 15 August 2008, 4445–4462

## Weighted least-square estimation of demand product mix and

## its applications to semiconductor demand

ARGON CHEN*, KYLE YANG and ZIV HSIA

Graduate Institute of Industrial Engineering, National Taiwan University, 1 Roosevelt Rd. Sec. 4, Taipei, Taiwan, 106

(Revision received August 2006)

Estimation of demand product mix is important for effective production plans. Unlike most research in the literature where the product mix is either given or treated as a decision variable in optimization of the production efficiency, this paper focuses on the product mix itself and how to estimate it from the market demand. With more accurate information on the demand product mix, aggregate production plans for product families can be disaggregated into quality detailed plans for individual product items. In this paper, least-square estimates of demand product-mix proportions are first derived. To take into account the effect of the product life cycle, dynamic weighting schemes are then developed to improve the accuracy of the product-mix estimates. For applications, we concentrate particularly on semiconductor demand where new generations of semiconductor products emerge at the pace of every six months, as manifested by the celebrated Moore’s laws. The proposed methodologies will be tested with simulated DRAM demands and actual semiconductor demands of different technology generations.

Keywords: Least-square estimation; Product-mix estimate; Demand disaggregation; Exponential weighting

1. Introduction

Results of demand planning serve as the basis of every planning activity in the supply network and ultimately determine the effectiveness of manufacturing/logistic operations in the network. Because demand forecast at the level of individual product items has become more and more difficult, a common practice of production planning is to plan at the aggregate level first for the product families. The detailed scheduling for individual product items is then determined by disaggregating the aggregate plans. Such a planning practice is commonly known as hierarchical production planning (HPP). A major difficulty of HPP is to uphold the consistency between the aggregate plan and the detailed plan (O¨zdamar et al. 1996, Za¨pfel 1996, O¨zdamar and Yazgac¸ 1999). A coherent detailed plan requires accurate estimation of the individual product demand from the demand of the product family. A demand planning approach is therefore to make a forecast at the aggregate level and then break down (disaggregate) the forecast statistically and/or judgmentally into

*Corresponding author. Email: achen@ntu.edu.tw

International Journal of Production Research

ISSN 0020–7543 print/ISSN 1366–588X online*ß 2008 Taylor & Francis*

http://www.tandf.co.uk/journals DOI: 10.1080/00207540701244028

the individual forecasts. This top-down demand planning approach is known to be an effective means for better forecasting because the aggregated product-family demand is observed to fluctuate less and easier to make a forecast (Grunfeld and Griliches 1960, Schwarzkoph et al. 1988, Ilmakunnas 1990, Kahn 1998, Zotteri et al. 2005). In the literature of supply chain planning, demand aggregation is also known as a ‘risk-pooling’ strategy to reduce demand fluctuation for more effective material/ capacity planning (Simchi-Levi et al. 2000). This approach, however, requires an accurate product-mix prediction to break down the aggregate demand forecast into individual product forecasts. Demand product-mix can be expressed as individual demand proportions in a product family. For example, demand proportions of 64, 128 and 256 M DRAM form the product-mix of the DRAM product family. Thus, a disaggregation method is required for the demand product-mix estimation to break down the aggregate demand. That is, to estimate the demand product-mix, we need to estimate the proportions of individual demands in a product family.

Gross and Sohl (1990) have investigated and compared twenty-one disaggrega-tion methods to estimate the propordisaggrega-tion of each product item in the product family. Most of the methods have a common root in the literature of forecasting combination (Mahmoud 1984). However, from their empirical study the two most effective methods are based on the simple average of a product item’s share in the family over a period of time (Makridakis and Winkler 1983). One of the simple-average methods (method F in Gross and Sohl 1990) is to calculate the mean product demand and the mean total demand per period first and then calculate the proportion of the mean product demand in the mean total demand. This method is referred to as a mean-proportion estimate:

^ Pi¼ Pn t¼1dit=n Pn t¼1Dt=n , ð1Þ

where ditis the demand of product i at period t; n is the number of historical demand

periods; Dt¼Pki¼1dit is the total demand at period t; k is the number of products in the product family; and ^Piis the mean-proportion estimate of product i. The other simple-average method (method A in Gross and Sohl 1990) is to calculate the proportion of each product for each period first and then take the average of the proportions over a period of time:

^ p i ¼ Pn t¼1dit=Dt n , where ^p

i is referred to as a proportion-mean estimate. These two methods are both provided without theoretical development. One of this paper’s objectives is thus to lay a theoretical foundation for development of the least-square estimates. It is found through our theoretical development that the proportion-mean estimate is actually one of the least-square estimates.

An effective estimate of the demand product-mix should also take into con-sideration characteristics of the market dynamics and the product life cycles. There are three important characteristics that will be considered in our proposed estimators:

1. Product life cycle (PLC) (Brockhoff 1967, Cox 1967, Thorelli and Burnett 1981): fast-paced technology development leads to fast PLC transition (Easingwood 1988).

2. Demand switchover: the demand for new-generation products increasingly replaces the demand for products of old technologies.

3. Variability proportional to volume: the greater the demand volume, the more volatile the demand (Brown 1959, Heath and Jackson 1994).

In figure 1, we simulated the 128 Mb (product 2), 256 Mb (product 1) and 512 Mb (product 3) DRAM demands to resemble these three characteristics. Though we intend to simulate the weekly wafer demand for an IC company, the units of demand and time are not explicitly indicated in figure 1 to reflect the general DRAM demand trend regardless of the market specificity.

It can be seen that the switchover from the demand for phasing-out products to the demand for emerging products leads to a dramatic change in the product-mix. It is difficult to estimate the demand product-mix without considering the effects of PLC and demand switchover. The second objective of this research is then to accommodate these characteristics of dynamic demands in the proposed estimates.

This paper is organized into six sections. Following this introduction, we develop the demand product-mix estimates with least squared errors. The weighted least-square estimates are then derived in section 3. To capture the PLC effect for better product-mix estimation, the dynamic weighting schemes are proposed in section 4. In section 5, the proposed methods are tested with the simulated DRAM demands and the actual semiconductor demands of different technology generations. Finally, some concluding remarks are made in the last section.

2. Least-square estimates of demand product-mix

The first objective of this paper is to find a demand product-mix estimate that best fits the historical demand trends. The sum of squared errors (SSE) is usually used to evaluate the quality of the estimate – the lower the SSE, the better the estimate. A least-square estimate is obtained by minimizing the SSE. We calculate two types

Figure 1. Simulated DRAM demands.

of SSE in this research, one is the sum of squared demand errors (SSEd) and the other is the sum of squared proportion errors (SSEp).

2.1 Least-SSEd demand product-mix estimate

First, we focus on minimizing the demand estimate error. Let pi represent

the expected proportion of product i. Then, the following model describes how the demand of product i is obtained through its proportion in the product family:

dit¼piDtþ"it, ð2Þ

where Pk_{i¼1}pi¼1 and "it is the demand error of product i at period t with

Pk

i¼1"it¼0.

Usually Dt can be modelled as a time series for demand forecasting. Though

forecasting is a very important subject, it is not in our current research scope. In model (2), errors are assumed to cause the period-to-period demand deviations ("it) from its nominal share (piDi) in the total family demand. The sum of squared

demand errors can be calculated as:

SSEd ¼X n t¼1 Xk i¼1 ðditpiDtÞ2: ð3Þ

We want to find the estimate of pi that minimizes SSEd subject to

Pk i¼1pi¼1. That is to minimize: SSEd ¼X n t¼1 Xk i¼1 ðditpiDtÞ2 Xk i¼1 pi1 ! , ð4Þ

where is a Lagrange multiplier (see, for example, chapter 20 of Taha 2002). Since the partial derivatives of (4) with respect to piare:

@SSEd
@pi
¼ 2X
n
t¼1
ditDtþ2pi
Xn
t¼1
D2_{t} , ð5Þ

we obtain the least-SSEd estimate ^piby setting (5) equal to zero:

^ pi¼ Pn t¼1ditDtþ ð=2Þ Pn t¼1D2t : ð6Þ

To satisfyPk_{i¼1}p^i¼1, in (6) has to be 0. Therefore,
^
pi¼
Pn
t¼1ditDt
Pn
t¼1D2t
: ð7Þ
^

pi is the estimate of product i proportion that minimizes the sum of squared differences between Dtp^iand dit. This demand proportion estimate is first proposed

by this research and will be shown to be more accurate, in terms of demand errors, than the two most effective methods in Gross and Sohl’s (1990) study.

2.2 Least-SSEp product-mix estimate

Now, our aim turns to minimize the proportion errors. Let each product i have an observed proportion dit/Dtat period t. The following model is used to describe

how this observed proportion deviates from the expected proportion pi:

dit Dt

¼piþ"it, ð8Þ

where "

it is the proportion error of product i at period t and Pn

t¼1"it ¼0. Unlike model (2), model (8) assumes that errors cause the actual proportion (dit/Dt)

to deviate from its expected value (pi). The sum of squared proportion errors can be

calculated as: SSEp ¼X n t¼1 Xk i¼1 dit Dt pi 2 : ð9Þ

Now we want to find the estimate of pi that minimizes the SSEp subject to

Pk

i¼1pi¼1. That is to minimize:

SSEp ¼X n t¼1 Xk i¼1 dit Dt pi 2 X k i¼1 pi1 ! , ð10Þ

where is the Lagrange multiplier. Since the partial derivatives of (10) with respect to piare: @SSEp @pi ¼ 2X n t¼1 dit Dt þ2npi, ð11Þ

we obtain the least-SSEp estimate ^p

i by setting (11) equal to zero:

^
p_{i} ¼

Pn

t¼1ðdit=DtÞ þ ð=2Þ

n : ð12Þ

To satisfyPk_{i¼1}pi¼1, in (12) has to be 0. Therefore,
^
p_{i} ¼
Pn
t¼1dit=Dt
n : ð13Þ
^
p

i is the proportion estimate that minimizes the sum of squared difference between dit/Dtand ^pi. ^pi is in effect the sample mean of observed proportions, dit/Dt, and

is exactly the same as the proportion-mean estimate mentioned in method A of Gross and Sohl (1990).

3. Weighted least-square estimates

In the previous section, demands are treated equally important in the least-square estimates regardless of their ages. To place varied emphases on demands at different periods of time, weights can be applied to different time periods in (1), (7), and (13) to obtain weighted demand product-mix estimates. First, the mean-proportion estimate in (1) now becomes a ratio of weighted average of historical demands:

^ Pi,nþ1 ¼ Pn t¼1Witdit Pm j¼1 Pn t¼1Wjtdjt , ð14Þ

where ^Pi,nþ1 is the estimate of product i proportion for period n þ 1; and Wit is

the weight applied to product i demand at time t and satisfiesPn_{t¼1}Wit¼1.
Now, we apply the weights to the least-square proportion estimates in (7)
and (13). For the least-SSEd estimate, different weights are given to the squared
estimate errors of different products at different time periods. The weighted sum of
squared demand errors (WSSEd) can be calculated as:

WSSEd ¼X n t¼1 Xk i¼1 witðditpiDtÞ2,

where witis the weight given to the squared demand error of product i at period t.

The WSSEd is then minimized subject to Pk_{i¼1}pi¼1. That is to find estimates of
piminimizing:
WSSEd ¼X
n
t¼1
Xk
i¼1
witðditpiDtÞ2
Xk
i¼1
pi1
!
, ð15Þ

where is a Lagrange multiplier. Setting the partial derivatives of (15) with respect to pito zero: @WSSEd @pi ¼ 2X n t¼1 witditDtþ2pi Xn t¼1 witD2t ¼0,

we obtain the weighted least-SSEd product-mix estimate for period n þ 1:

^ pi,nþ1¼ Pn t¼1witditDtþ ð=2Þ Pn t¼1witD2t , ð16Þ

where should be evaluated such that Pk_{i¼1}p^i¼1 is met. Let bi be

Pn
t¼1witditDt
and aibe
Pn
t¼1witD2t. Then,
¼
2 Qk_{i¼1}ai
Pk
j¼1 bj
Qk
i¼1ai
=aj
Pk
j¼1
Qk
i¼1ai
=aj
: ð17Þ

Similarly, weighted least-SSEp estimate (WSSEp) can be calculated as:

WSSEp ¼X n t¼1 Xk i¼1 w it dit Dt pi 2 , where w

itis the weight given to the squared proportion error of product i at period t.
The WSSEd is then minimized subject to Pk_{i¼1}pi¼1. That is to find estimates of
piminimizing
WSSEp ¼X
n
t¼1
Xk
i¼1
w
it
dit
Dt
pi
2
X
k
i¼1
pi1
!
, ð18Þ

where * is a Lagrange multiplier. Set the partial derivative of (18) to zero:

@WSSEp
@pi
¼ 2X
n
t¼1
w
it
dit
Dt
þ2pi
Xn
t¼1
w
it
_{¼}_{0,}

to obtain the weighted least-SSEp product-mix estimate for product i at period n þ 1:
^
p_{i,nþ1} ¼
Pn
t¼1witðdit=DtÞ þ=2
Pn
t¼1wit
, ð19Þ

where * should be chosen such that Pk_{i¼1}p^

i ¼1 is satisfied. Let ui be
Pn
t¼1witðdit=DtÞand vibe
Pn
t¼1wit. Then,
¼
2 Qk_{i¼1}vi
Pk
j¼1 uj
Qk
i¼1vi
=vj
Pk
j¼1
Qk
i¼1vi
=vj
: ð20Þ

Both weighted least-square estimates, (16) and (19), are first derived by this research. However, how to determine weights with consideration of product life cycle effects remains an issue to be addressed in the following section.

4. Exponential weights and PLC leading indicators

To capture the effects of PLC transition on the product-mix, various weights should
be placed on demands of different ages. When the product demand is on the rise or
decline, more weights should be placed on the most recent demands to reflect the
heavier influence of the recent market transition. On the other hand, when the
product demand is in a steady, mature phase, all demands in the steady phase should
be equally accounted for in the estimate of the product-mix. In order to weigh the
most recent demands more and to be able to adjust the weighting easily for different
demand trends, we propose using the exponential weights for i ¼ 1, . . . , k and
t ¼1, . . . , n:
Wit¼
ið1 iÞnt
1 ð1 iÞn
ð21Þ
wit¼ið1 iÞnt ð22Þ
w_{it}¼_{i}ð1 _{i}Þnt ð23Þ

with single smoothing constants i, i, and i for ^Pi,nþ1, ^pi,nþ1and ^pi,nþ1, respectively. Using i as an example, figure 2 shows how the smoothing constant affects the

weight distribution over different ages of demands.

How to choose appropriate values of i, i, and _{i} in (21), (22), and (23) becomes

critical for accurate product-mix estimation. i and i can be found such that the

sum of squared demand forecast errors over s periods:

SSFEdðsÞ ¼ X t ¼tsþ1 Xk i¼1 ðp~i,DdiÞ2, ð24Þ

is minimized, where ~pi, is the estimate for product i proportion using historical demands up to period 1 and is used as the forecast for period . ~pi, can be estimated by ^Pi, using (14) or by ^pi, using (16). Similarly, i can be found such that the sum of squared proportion forecast errors over s periods:

SSFEpðsÞ ¼ X t ¼tsþ1 Xk i¼1 ^ p i, di D 2 , ð25Þ is minimized, where ^p

i, is calculated using (19) with historical demands up to 1 and is used as the proportion forecast for period . Determination of smoothing constants is a very time consuming task. Take ias an example and suppose each i

has 99 possible values (i¼0.01–0.99). Then, there are 99k possible (1, 2, . . . , k)

candidates. The time to compute all candidates to search for the best (1, 2, . . . , k)

to minimize (24) requires enormous computing power. The most common method is steepest-descent method (see, for example, chapter 21 of Taha 2000). The search for the smoothing constants can be seen as a k-variable (1k) steepest-descent

search with (24) as the objective function to minimize. Treat (1, 2, . . . , k) as a point

in the k-dimension coordinate system. Here, we simplify the steepest-descent search process to be a process that finds the point minimizing (24) by adjusting each iwith

a fixed step size r. There are three possible movements for each i: adding one step Figure 2. Exponential weight distributions controlled by the smoothing constant. (a) i¼0.1, (b) i¼0.2.

size to i, i.e., iþr, subtracting one step size from i, i.e., ir, or keep the iat the

original position. The direction formed by successive points of representing the most SSE-reduced direction which is called the ‘gradient vector’ while all the other possible directions are called ‘candidate vectors’. Termination of the search occurs at the point where the gradient vector becomes null; i.e., the current point has the smallest SSE value. An example of two-product steepest-descent search for a pair of smoothing constants is shown in figure 3.

The steepest-descent method is still a computation-intensive method. It would be much more efficient if we knew the direction toward the minimum. Taking the effect of changing demand variability into account, we propose a PLC transition leading indicator using the sample one-lag autocorrelation (SAC) statistic:

SACi,t¼

1=sPtðs1Þ_{¼t1} ðdi, d Þðdi,þ1 d Þ
1=sPtðs1Þ_{¼t} ðdi, d Þ2

, ð26Þ

where SACi,tis the sample one-lag autocorrelation of product i calculated at period t

using the demand data set {di,t-sþ1, . . . , di,t}. SAC is a measure of stickiness between

demands of successive periods and thus a measure of demand-trend significance. When the product is at the ‘growth’ or ‘decline’ phase; i.e., the product-mix proportion significantly rises or falls, the value of SAC will become larger because the market trend dominates the product-mix changes. If the product is mature in the

Figure 3. Simplified two-product steepest-descent search process.

market and its product-mix proportion is stable, the value of SAC will become smaller because only the noise dictates its proportion changes. When the value of SAC is large, the value of the smoothing constant should be large as well to weigh the most recent demands more and, thus, to capture the new market trend more effectively. Figure 4 shows the relationship among the SAC trend, the PLC stage, and the expected trend of the smoothing constant.

Sample size s in (26) determines how sensitive the SAC is to the PLC transition and to the demand noise. Figure 5 shows the SAC values with sample sizes s ¼ 15, 25 and 50 for the simulated 256 Mb DRAM demand proportion. Since SAC is in the range of [1, 1] and the demand proportion is in the range of [0, 1], we are able to directly superimpose the 256 Mb demand proportion onto the figure to show how the SAC responds to the proportion changes.

It can be seen that the SAC with a large sample size (s ¼ 50) is too slow to reflect the PLC transition while the SAC with a small sample size (s ¼ 15), though responsive to the PLC transition, is too sensitive to the demand noise. The sample

Figure 4. Relationship among , SAC and PLC.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120 140 160 Time

Product-1 proportion Product-1 SAC (Sample size=50)

Product-1 SAC (Sample size=25) Product-1 SAC (Sample size=15)

Figure 5. SAC calculated by different sample sizes.

size of 25, approximately one half of one PLC phase, gives the SAC a very good indication of the PLC transition.

The smoothing constant, in the range of [0, 1], estimated by the steepest-descent search (SDS), SAC, in the range of [1, 1], calculated with s ¼ 25 and the product-mix proportions, in the range of [0, 1], of the simulated DRAM data are directly superimposed together in figure 6. The SDS-estimated smoothing constant goes up when the trend is rising or declining but goes down in the maturity phase. The trend of SAC and the trend of SDS-estimated smoothing constants match very well all

−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120 140 160
Time
Product-1 proportion Product-1* _{α (s=25)}* Product-1 SAC

−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120 140 160
Time
Product-2 proportion Product-2*α (s=25)* Product-2 SAC

−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100 120 140 160
Time
Product-3 proportion Product-3* _{α (s=25)}* Product-3 SAC

0

Figure 6. Smoothing constant estimates and SACi,t(DRAM).

three products at three different PLC stages. SAC is thus a very good leading indicator to determine the changing trend of the smoothing constants. That is, the smoothing constant estimated at period t þ 1 should be higher than that at period t when SACi,tþ1 is higher than SACi,t and vice versa. Using the SAC as a leading

indicator for the search for the best smoothing constants has been proven to cut the computing time to one fifty-fifth of that required by the steepest-descent search.

The smoothing constant estimate of product i at is denoted as ^i,t. The detailed steps are listed below:

1. Given historical demand data from period 1 to period t 1, calculate best-fitted initial values ^i,t (i ¼ 1, . . . , k) using steepest-decent search at t.

2. Calculate the PLC leading indicator SACi,t1for each product.

3. Calculate SSFEd or SSFEp.

4. New observations (period ‘t’) of product demands are available. 5. Calculate new PLC indicators SACi,t(i ¼ 1, . . . , k).

6. Calculate the difference between the two successive indicators, SACi,t-1

and SACi,t, for each product to determine the gradient vector.

7. Move one step from the initial point ^i,t(i ¼ 1, . . . , k) to a new point according to the gradient vector determined in step 5.

8. Recalculate SSFEd or SSFEp.

9. If reduction in SSFEd or SSFEp is found, go back to step 6.

10. Terminate the search and set values of ^i,tþ1 equal to the final point when no further improvement can be found in the direction of the gradient vector. 11. Calculate the product-mix estimate for the next time period with the

smoothing constant estimates ^i,tþ1 (i ¼ 1, . . . , k). 12. Increase the value of t by 1 and go back to step 2.

5. Evaluation of proposed methodologies

To evaluate the performance of dynamic weighted least-SSEd and dynamic weighted least-SSEp product-mix proportion estimates, they are first tested with the simulated DRAM data (figure 1). The proportion estimates will start at the 31st period and end at the 150th period. Demand mean squared error (MSE) and proportion mean squared error (PMSE) are used as the performance measures. They are defined as:

MSE ¼ Ptþn ¼tþ1 Pk i¼1ðp _ i,1DdiÞ2 n k ð27Þ PMSE ¼ Ptþn ¼tþ1 Pk i¼1ðp _ i,1di=DÞ2 n k ð28Þ

where p__{i,} is the proportion estimate for product i proportion at period made by
methodologies developed above and two conventional methods, methods 1 and 2 of
the following:

1. Mean-proportion estimate without dynamic weights:

^ Pi,tþ1¼ Pt ¼tsþ1di =t Pt ¼tsþ1D =t,

where the sample size s is set equal to 25, same as the sample size used in the PLC leading indicator, SAC, for fair comparison;

2. Proportion-mean, i.e., least-SSEp estimate without dynamic weights:

^
p_{i,tþ1}¼1
t
Xt
¼tsþ1
di
D
,

where the sample size s is also set equal to 25 for fair comparison; 3. Least-SSEd estimate without dynamic weights:

^ pi,tþ1¼ Pt ¼tsþ1diD Pn ¼tsþ1D2 ,

where the sample size s is also set equal to 25;

4. Dynamic weighted mean-proportion estimate in (14); 5. Dynamic weighted least-SSEp estimate in (19); and 6. Dynamic weighted least-SSEd estimate in (16).

Methods 1 and 2 are known to be the two most effective conventional methods in Gross and Sohl (1990). Method 3 is the least-SSEd method without dynamic weights. The smoothing constants of the exponential weights in (14), (16) and (19) are dynamically determined based on the PLC leading indicator, SAC, with a sample size s ¼25.

5.1 Performance evaluation with simulated DRAM demand data

The performances of all product-mix proportion estimates are listed in table 1. As shown in table 1, the two conventional methods (1 and 2) and the least-SSEd method without dynamic weights perform the worst in both PMSE and in MSE. With dynamic weights, the performances of all three product-mix estimation methods improve significantly. Among the dynamic weighted methods, the least-square methods perform better than the mean-proportion method. The least-SSEd estimate performs best in both measures. This reveals that model (2) is the most suitable model to describe the simulated demand data. The least-SSEp estimate does not have the best performance in PMSE in this case, but is very close to the best.

5.2 Performance evaluation with actual semiconductor demand data

Finally, the real semiconductor demands of different technology generations in a certain product group with the same metal-layer number are used to test the product-mix estimation methods. This will give us a comparison to see if the theoretically developed weighted least-square estimates are suitable for actual application.

There are six technology generations and 127 periods of demand data in this case. The demand trends for the six types of technologies are shown in figure 7. Since the demand data is the company’s proprietary information, the demand volume and time are intentionally masked to protect the company’s confidentiality. It is difficult

to see the product demand trends from these raw demand data. Figure 8 shows the product-mix proportions of the six technologies. According to the proportions, demands for Technology 5 and Technology 6 gradually increase after about the 76th period. That indicates that the newly developed technologies are just introduced into the markets. Technology 3 gradually decreases after about the 86th period and represents a matured technology gradually phasing out from the market. The product-mix of different technologies can be seen clearly from this figure. The proportion of Technology 3 demand is much higher than the other five and dominates the total demand until about the 91st period.

Again, demands from period 1 to period 30 are used as the initial demand data. The product-mix estimate will start at period 31 and end at period 127. s is set to 25 periods. The performance comparison of all product-mix estimates is shown in table 2. 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 Technology 1 demand Technology 2 demand Technology 3 demand Technology 4 demand Technology 5 demand Technology 6 demand

Figure 7. Demands of different semiconductor technologies. Table 1. MSE of different product-mix estimates for simulated DRAM

demands.

Methods PMSE MSE

1. Mean-proportion 0.009017259 3 831 961 2. Least-SSEp 0.008947024 3 886 224 3. Least-SSEd 0.009127480 3 793 929 4. Dynamic wt. mean-proportion 0.0010772 528 851 5. Dynamic wt. least-SSEp 0.0009897 471 899 6. Dynamic wt. least-SSEd 0.0009811 466 602

As shown in table 2, the least-SSEp method has the best performance in PMSE and the least-SSEd has the best performance in MSE as expected. With dynamic weights applied to the methods, the performances are further improved by more than 5%. The limited improvement by the dynamic weighting scheme is due to the incomplete product life cycle from which the demand data are sampled. The mean-proportional method has average performances in both PMSE and MSE. Figures 9 and 10 show the estimated product-mix proportions obtained respectively by the dynamic weighted least-SSEp and weighted least-SSEd, versus the actual demand proportions. It can be seen that the proportion estimates by least-SSEp are more responsive to the product-mix changes whereas the least-SSEd method results in 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 Technology 1 demand Technology 2 demand Technology 3 demand Technology 4 demand Technology 5 demand Technology 6 demand

Figure 8. Product-mix proportions of different semiconductor technologies.

Table 2. MSE of different product-mix estimates for demands of different technologies.

Methods PMSE MSE

1. Mean-proportion 0.0245890253 193 707 314 2. Least-SSEp 0.0198083407 926 967 800 3. Least-SSEd 0.0259923295 186 223 492 4. Dynamic wt. mean-proportion 0.0198953295 198 057 131 5. Dynamic wt. least-SSEp 0.0186601176 1 093 635 664 6. Dynamic wt. least-SSEd 0.0216717924 177 668 486

31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127

31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127

31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127

Technology 1 proportion Dynamic weighted least-SSEp

Technology 2 proportion Dynamic weighted least-SSEp

Technology 3 proportion Dynamic weighted least-SSEp

Technology 4 proportion Dynamic weighted least-SSEp

Technology 5 proportion Dynamic weighted least-SSEp

Technology 6 proportion Dynamic weighted least-SSEp

Figure 9. Proportion estimates with dynamic weighted least-SSEp method.

Technology 1 proportion Dynamic weighted least-SSEd

Technology 2 proportion Dynamic weighted least-SSEd

Technology 3 proportion Dynamic weighted least-SSEd

Technology 4 proportion Dynamic weighted least-SSEd

5 Technology 5 proportion Dynamic weighted least-SSEd

Technology 6 proportion Dynamic weighted least-SSEd

Figure 10. Proportion estimate with dynamic weighted least-SSEd method.

more robust product-mix estimates. This explains why the least-SSEp method performs best in estimating the product-mix proportions while the least-SSEd methods has the best performance in predicting the individual demands by disaggregating the total demand based on its robust estimates of the product-mix proportions.

6. Concluding remarks

The contribution of this research is three-fold. First, we have developed two least-square product-mix estimates. The least-SSEd estimate is the first in the literature and appears to outperform conventional estimates in minimizing the demand estimate errors. The least-SSEp estimate turns out to be the conventional proportion-mean estimate but is first established with theoretical bases by this research. Second, we have extended the least-square estimates to weighted estimates to allow varied emphases on the historical demands with different ages. Third, we have developed a dynamic weighting scheme to capture the PLC effect based on an effective PLC leading indicator. The dynamic weighting scheme is able to automatically adjust the weight distribution over the historical demands and make the weighted least-square estimate adapt to the PLC transition. In this research, the focus is on developing the weighted least-square estimates of the demand product-mix which is subject to the product-mix constraint and differs from the conventional demand forecast techniques. Only the single exponential weighting is used in the weighted scheme. Adoption of the double exponential weighting scheme, as commonly used in the demand trend prediction, is possible to further improve the forecast accuracy. Such an adoption requires extension of (14), (15) and (18) and should be an interesting future research topic. The three developed methodologies have all been tested and validated by the simulated and actual semiconductor demand data sets. The least-SSEd estimate is shown to perform best in estimating the individual product demands while the least-SSEp is best in estimating the product-mix proportions. The conventional mean-proportion estimate appears to fall in-between with a relatively stable performance in both demand and proportion estimations. As expected, estimates devised with the dynamic weighting scheme outperform the equally-weighted estimates, though implementation of the weighted scheme is somehow complicated. With the readily available computing power of modern personal computers, we believe that the superior performance and the adaptive nature of the dynamic weighting scheme are worth the efforts. Even though only applications to semiconductor demands are included in this paper, the methodologies presented are developed with general theoretical bases and can be readily applied to other industry sectors.

Acknowledgements

This research is supported by Semiconductor Research Corporation (SRC) and International SEMATECH under the project contract 879.

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