東海大学紀要情報通信学部
Proc. Schl. ITE Tokai Univ.
vol.3,No2,2010,pp.37-42
Vol. ,No. ,2010,pp. -Paper

Demand and Price Forecasting Models for Strategic and Planning
Decisions in a Supply Chain
by
Vichuda WATTANARAT *1 , Phounsakda PHIMPHAVONG *1 and Masanobu MATSUMARU *2(Received on October 30, 2010 and Accepted on January 25, 2011)

Abstract
Procurement and sales forecasting are a fundamental approach to the estimation of inventory. The accuracy is an
essential factor for a firm on cost of manufacturing. Forecasting is an important for all strategic and planning
decisions in a supply chain. Forecasts of product demand, materials, labor, financing are useful inputs to scheduling,acquiring resources, and determining resource requirement. However, highly volatile and correlated demand caused
inaccuracy. This paper aims to compare real option method and ARIMA model method to be able to predict a precise
future procurement and sales. To validate the forecasting model, we use the real world data which are demandobtained from one die casting company and copper price. The result shows that real option model is outperforming
ARIMA model quite significantly in case the data has high volatility.
Keywords: Forecast, V olatile, Demand, Geometric Brownian motion , ARIMA methodology
predict movement of the demand, we can clearly see from the

1. Introduction

history that will not always suppose to have an upward trend
Smoothing the demand and supply is the main problem ineven in the long term demand. This case is clearly noticed insupply chain management. Under supply leads to the loss ofthe time of economic crisis. Besides, competition andpotential revenue, over supply incurs extra carrying cost ofadvancement of technology also erode the demand of one

the excess product. Procurement and inventory control are ofcompany to another. Taking the example of mobile music

active area of research to gain the competitiveness for theindustry like Sony that loses the market share to Apple

company. The optimal quantity can enable the company toproduct.

achieve the cost reduction. To obtain the optimality in supply

In this paper we compare the new forecasting method thatchain, we have to deal with making the decision about theis apply to the forecast of demand and price. There are many

future event, produce to secure the future sale. This involves

ways that we can adopt for forecasting for example linearwith forecasting the unforeseen data for example demand andregression, moving average, ARMA, ARIMA. Here we

price. The less deviation from the real value the less risk wecompare the real option approach with the ARIMA model andhave from doing the operation.

conclude what kind of data is best suited for these kind of
forecasting.

Demand forecasting plays a very crucial role inproduction planning of the manufacturing. It is the main input

Real option approach is originated from the finance study.

for the capacity planning which links to various chain of

It is used to value the value of uncertainty in an investment.

supply including inventory, logistic and service level.

Real option is different from traditional investment criteria
like Net Present Value (NPV) in that unexercised choices

Demand and price is usually difficult to forecast due tohave also future value and should contribute the evaluation.

their uncertainty nature. Basically demand should show anincreasing trend over the long term period. When we look at
the short term period, we often see the unexpected move

2. Literature review

either going up or down. If we include more realistic
possibility of the economic factor to try to understand andReal option approaches which is based on the diffusion*1 Graduate student, Department of management systems
engineering*2 Professor, Department of management systems
engineering

process of the variable for doing the forecast is quite a newVol. XXXI, 2009

field of study. The most common diffusion process that isused in real option study is Geometric Brownian Motion―1―− 37 −

Demand and Price Forecasting Models for Strategic and Planning Decisions in a Supply Chain

Times, Times and Times(GBM). GBM is used to replicate the movement or charactermodel for the estimation of procurement and sales. Weof the object of study in many fields of study e.g. particle

compare our study with the previous study in Table 1.

movement in physic, the price of the underlying asset in thefinancial study, water level in reservoir etc. Huang [1] applies

Table 1

Previous study comparison

real option in forecasting the demand of products that hascorrelated to one another and that have high volatility. The

Previous Study

kind of product that has such kind of nature is referred to IT
product. In their research they forecast the demands of 4
products that relate to one another. If one product changes,

Makridakis andTime

Wheelwright (1985)

includingtime assuming that demand followed a GBM. Yon [3] also

Ming-Guan Huang (2008)

studied about the capacity planning. They analyzed in case ofandseasonal

approach-based

forecasting

methodfor a range of products withtechnology are advanced at a rapid pace, making it difficult

highly volatile and correlated

to do the capacity planning in the semiconductor industry.

demandThey assume the demand to follow the GMB. They simulate
and compare the result of the profit in two difference
scenarios.

James D. Hamilton

Analyze( Book,1994)

regressions,
methodReal option by mean of GBM has also been used

economic

extensively in making investment decision as in Dixit andvectorauto

generalizedof

moments,
andthestatisticalconsequences of unit roots,

Pindyck [4]. It can be used to find the optimal time to invest,

time-varying variances, andevaluate option to postpone investment, option to shut downnonlinear time series models

the business, option to expand business etc. in which thetraditional investment criteria cannot handle with well.

The outline of this study is as follow: In section 2 we

An autoregressive integrated moving average (ARIMA) is

explain the methodologies and how to estimate the future

one of well known model in statatistic and econometrics. It isvalue. We set a numerical simulation and illustrate results inused to study the behavior of time series and predict thesection 3. We prefer to use MATLAB to simulate thefuture out of the time series. The simple way to think about

simulation of real option method, Eviews for ARIMA model.

the ARIMA model is that it is the combination of random

Finally, we conclude a summary of our findings.

walk and trend model. This characteristic is very similar to
the GBM and it is a good candidate for the comparison.

3. Forecasting Methodology

For a high volatile demand product, Makridakis and

3.1 Geometric Brownian motion

Wheelwright [5] extended the time series analysis byincluding moving average, regression, exponential smoothing,Theaverage,

exponential

Real option
demanddemand has high volatility and machine and processsmoothing.

byexponential smoothing

capacity studies [2]. They studied capacity utilization overexponential

analysismovingsmoothing,

The GBM model is used to replicate the future demand inseasonal

seriesregression,

the volume of other products also changes.

andDetails

Geometric Brownian motion is originated in physics as a

ARIMA

diffusion process used to describe the partial motion when it

methodology is developed by Box and Jenkins [6]. It is afalls. Let Xt be the value of the data at time t and assume it tomathematical model for forecasting a time series by fitting it

follow the Geometric Brownian Motion, we have thewith the data and using the fitted model for forecasting. R.J.

diffusion specify in equation (1). In this equation there are

Hyndman [7], Pankratz [8] and Martin, R.D., Samarov, A.

two parts. The first part on the right hand side is the mean

and W. Vandaele [9] study ARIMA model in application.

growth of the diffusion and the second part on the right hand

Metcalf & Hasset [10] study the mean reversion formatside of the equation is the volatility.

equation, and it has more economic logic than the geometric
Brownian model. Carlos and David [11] explore the maindX t = μX t dt + σX t dz t

stochastic price processes used to model energy spot andforward prices for derivatives valuation and risk management.

(1)

Where dz t = ε t dt is the standard increment of Wiener
process or the white noise, μ and σ are drift and varianceterms respectively. zt has the following 2 properties:
Property 1: The change in the value of z, dz, over a time

They apply a Geometric Brownian motion to be a random
walk with mean reversion. In our research we compare theforecasting methods which are real option method, ARIMA

internal of length Δt is proportional to the square root of Δt―2―− 38 −

Proc. Sch. ITE
Tokai University

Vichuda WATTANARAT・Phounsakda PHIMPHAVONG・Masanobu MATSUMARU

Times Times Times Times Times

where the multiplier is random. In specific dz=z(t+n∑rΔt)-z(t)=εΔt, where ε is a standard normal random variable.

tHence values of dz follow a normal distribution with mean 0

r = t =1nand variance equal to the change in time (Δt) over which dz
is measured.
Property 2: The changes in the value of zt for any

1(n − 1)sr =non-overlapping intervals of time are independent.
Let f=ln Xt and expand f according to the Taylor series we

df =

df
1 d2 f(dX t ) +(dX t ) 2 + ......
dX t
2 dX t 2(2)μ=Then apply Ito lemma to (2), we have that Δt that hasn∑ (r − r )

(10)

t

t =1

sr
Δt

(11)

r σˆ 2rs2+=+ r
Δt 2 Δt 2Δt

(12)

σ=

have

(9)

power greater than 2 will approach to zeros quicker when Δt
is small. We havedf =3.3 ARIMA Modeldf
1 d2 f(dX t ) +(dX t ) 2dX t
2 dX t 2

In(3)

supplychain

management,An

autoregressive

integrated moving average has been used to in many
problems ranging from finding optimal order policy,

Substitute (1) into (3) and rearrange we havepredicting customer demand, predicting the price movement,1⎛⎞
ln X t +1 − ln X t = ⎜ μ − σ 2 ⎟Δt + σε Δt
2
⎝
⎠

(4)

⎡⎛
⎤
1
⎞
X t +1 = X t exp ⎢⎜ μ − σ 2 ⎟Δt + σε Δt ⎥
2
⎝
⎠
⎣
⎦

(5)

inventory management etc.
The autoregressive moving average (ARMA) model was
introduced by Box and Jenkins (1976). It consists of twoparts, an autoregressive (AR) part and a moving average(MA) part. Specifically, the three types of parameters in the

Equation (5) is the discrete version of Geometricmodel are: the autoregressive parameters (p), the number ofBrownian Motion that we will use to simulate the forecast ofdifferencing passes (d), and moving average parameters (q).

demand and price in this study.

Models are summarized as ARIMA (p, d, q).

3.2 Calibrate the parameterdata point to a linear function of some prior data points and

ARIMA combines auto regression-which fits the current
moving averages adding together several consecutive data
The Geometric Brownian Motion process can formally bepoints and getting their mean, and then using that to computedefined as Xt ,0≤ t ≤∞ , the random variable of Xt/Xt+1 isestimations of the next value. With enough elements

independent of all values of the variable up to the time t andregressed and averaged, we can fit an approximation too

if in addition ln(Xt/Xt+1) has a normal distribution with mean

many time series and with low discrepancy.
We find the best ARIMA (p,d,q) model condition by2

μt and variance σ t. Let rt be the difference between

comparing the Akaike information criterion (AIC). The

logarithm Xt and Xt-1 we have
⎛ X ⎞ln⎜⎜ t ⎟⎟ = rt
⎝ X t −1 ⎠smallest AIC is the most suitable condition. An ARIMA(p,d,q) model contains three different kinds of parameters:(6)(1) the p AR-parameters; (2) the q MA-parameters thenumber of lagged forecast errors in the prediction equation.

In finance, (6) represent the logarithm of the return which
is necessary assumption because the value of the underlying(3) d the number of differences. This amounts to a total of

asset that is assumed to follow Geometric Brownian Motion(p,d,q) parameters to be estimated. These parameters areshould have a value greater or equal to zero. rt also implies

always estimated on using the stationary time series which isstationary with respect to its variance and mean.

the growth rate of the observed data in each period.

To identify the appropriate ARIMA model for a time

In order to estimate the parameter to use in the forecast

series, we have to identifying the order(s) of differencing

model as specified by equation (5) we following equations
E (rt ) = ( μ − σ 2 2)Δt

(7)

V (rr ) = σ 2 Δt

(8)Vol. XXXI, 2009

needing to stationary the series and remove the gross featuresofseasonality,

perhapsinconjunction

with

a

variance-stabilizing transformation such as logging or―3―− 39 −

Demand and Price Forecasting Models for Strategic and Planning Decisions in a Supply Chain

Times, Times and Times

deflating. An notation AR(p) refers to the autoregressive

numerical experiment, we use 122 weeks of data (frommodel of order p. An AR(p) model is defined as

November, 2004 to December, 2008) for estimation by using

X t = c + φ1 X t −1 + φ 2 1 X t − 2 + ... + φ p X t − p + ε twhere φ 1 , φ 2 , L

L

, φ

p

Geometric Brownian motion, ARIMA model and mean

(13)

Reversion.

are the parameters ofthe model, c is a constant and ε t is white noise. An MA

4.2 Empirical results of Geometric Brownian motion

model is a common approach for modeling univariate timeseries models.
(14)

In this research, we apply to both forecasting models towhere μ is the mean of the series, the θ1, ..., θq are thethe forecast the demand of one die casting company and theparameters of the model and the εt, εt−1,... are white noise

price of copper. We use the data from November 2004 toerror terms. The value of q is called the order of the MA

December 2008. The actual demand shows the seasonality inmodel. An ARMA consists of two parts, an autoregressive

the data series. For the actual price of copper, the data set(AR) part and a moving average (MA) part. The model isshows high volatility and covers the period of the financial

usually then referred to as the ARMA(p,q) model where p iscrisis.

x t = μ + ε t + θ 1ε t − 1 + L + θ q ε t − q

the order of the autoregressive part and q is the order of the(1)Results of demand

moving average part.

X t = c + φ 1 X t − 1 + ... + φ p X t − p+ φ 1 ε t − 1 + ... + φ q ε t − q + ε t

For the illustrative purpose, we simulate demand 1000

(15)rounds for each period by using MATLAB. Figure 1 is a

An ARIMA model is a generalization of an ARMA model.

result from real option method. Real option method provides

The model is generally referred to as an ARIMA(p,d,q)

the forecasting value along with actual value as well. Thismodel where p, d, and q are non-negative integers that refermethod can capture demand changes quite accurately.
All the data were acquired on statistical software forto the order of the autoregressive, integrated, and moving

economic analysis, called EViews, which is developed byaverage parts of the model.

Quantitative Micro Software (QMS). We analyze demand andprice data which are separated to be 122 numbers of data.

4. Empirical analysis

We follow the parameter setting from November 2004 to
December 2008.

Decisions make in advance of uncertain events. And
forecasting is an important for all strategic and planning
decisions in a supply chain. Especially, forecasts of productTotal Deamnd forecast

demand, materials are important inputs to scheduling,

140000120000Demand (kg)acquiring resources, and determining resource requirement.
We use the data of the medium and small-sized business inreal world and check the effectiveness of a suggested model.
This company accepts an order from the upstream company.

100000
80000
60000
40000
Total Demand
Total Demand Forecast

20000
0

Because there is a change in this order, this company must1

have many stocks to avoid lack. As a result, this company

12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210
Time (Week)

Figure 1

must risk a burden for a cost to maintain a stock. Therefore, it
is important that the company predicts quantity of order from

Demand forecast(2)Results of pricethe upstream company. If we predict the volume precisely,
this company reduces the inventory cost. The reduction of the

We try to find the simulation result of mean reversion ininventory cost in the model that we suggested had a big

price with 1000 times estimation.

influence on the improvement of corporate earnings.

Actual Price1000

4.1 Data
Price (Yen)800

It assumes that the forces which influenced past value
will continue to exist in the future. In this section, we follow

600
400
200

the forecast methodology to calculate the numerical value ofActual Price

Forecasting Price01

demand and price. We follow the forecast methodology to
calculate the numerical value of demand and price. For the

12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210
Time (Week)

Figure 2―4―− 40 −

Price forecast

Proc. Sch. ITE
Tokai University

Vichuda WATTANARAT・Phounsakda PHIMPHAVONG・Masanobu MATSUMARU

Times Times Times Times Times

4.3 Empirical results of ARIMA model

Table 3

ARIMA parameter for total price4.3.1 Numerical experiment of Demand following real
option method

1.1.X

2.1.X

3.1.X

4.1.X

51.X

6.1.X

7.1X

8.1.X

9.1.X

We demonstrate a result of AIC in Table 2 with the

X.1.1

8.902

8.898

8.898

8.911

8.926

8.939

8.95038.9581

8.970

8.980

different value of d in ARIMA (p,d,q) model. The best

X.1.2

8.892

8.901

8.907

8.921

8.934

8.904

8.9572

8.9713

8.943

8.973

X.1.3

8.897

8.904

8.916

8.93

8.907

8.72

8.9324

8.9479

8.949

8.955

X.1.4

8.9

8.913

8.922

8.936

8.912

8.917

8.8992

8.9239

8.906

8.937

computed as ARIMA (9,1,9) that AIC is 22.4. We assume to

X.1.5

8.908

8.919

8.931

8.91

8.913

8.747

8.7236

8.925

8.927

8.943

use ARIMA (9,1,9) which offers the best Akaike information

X.1.6

8.918

8.928

8.939

8.904

8.919

8.94

8.91388.9272

8.948

8.941

criterion for our simulation.

X.1.7

8.928

8.938

8.912

8.891

8.938

8.933

8.87758.9365

8.96

8.95

ARIMA (p,d,q) parameters for ARIMA model for demand is

X.1.8

We demonstrate our forecasting result by using ARIMA(9,1,9) in Figure 3 as below. In the beginning ARMA model

8.938

10.1.X

8.903

8.92

8.92

8.929

8.927

8.932

8.9493

8.94628.91656

X.1.9

8.927

8.918

8.924

8.919

8.919

8.911

8.93248.95118.95068.93665

X.1.10

8.926

8.938

8.93

8.906

8.922

8.935

8.94368.82778.95198.94281can produce demand forecast quite well but in the long run
However real option approaches does a slightly betterthe forecast accuracy is getting lower.

performance than ARIMA model. Demand data does not
Table 2show high degree of fluctuation. This is an important

ARIMA parameter for total demandcondition that is contributed to the level of accuracy of ineach method.

1.1.X

2.1.X

3.1.X

4.1.X

5.1.X

6.1.X

7.1X

8.1.X

9.1.X

10.1.X

X.1.1

22.53

22.54

22.55

22.44

22.52

22.52

22.55

22.54

22.55

22.55

To verify the precision of forecasting model, we compare

X.1.2

22.52

22.46

22.55

22.45

22.52

22.49

22.56

22.55

22.51

22.54

X.1.3

22.53

22.53

22.56

22.54

22.53

22.49

22.48

22.49

22.51

22.52

the actual and forecast result again. From Figure 4 and Figure

X.1.4

22.55

22.51

22.42

22.44

22.43

22.47

22.46

22.51

22.51

22.51

X.1.5

22.54

22.52

22.43

22.45

22.42

22.41

22.4

22.41

22.44

22.41

X.1.6

22.51

22.52

22.44

22.41

22.43

22.4

22.38

22.43

22.52

22.41

X.1.7

22.52

22.5

22.46

22.41

22.41

22.44

22.44

22.46

22.53

22.46

5, the real option method presents the best forecast accuracy.

Total Demand Forecast

22.53

22.52

22.46

22.41

22.46

22.42

22.48

22.46

22.42

22.43

140000

X.1.9

22.54

22.52

22.45

22.42

22.43

22.43

22.43

22.45

22.4

22.4

120000X.1.10

22.52

22.5

22.52

22.43

22.49

22.5

22.48

22.51

22.43

22.41

100000D e m an d (K g)

X.1.8

60000
40000
20000

Demand (Kg)0
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
Time (Week)

Actual Demand
Forecast Demand

100000Real Option
Mean Reversion

80000

Total Demand Forecast (ARIMA Model)
120000

Actual Demand
ARIMA

80000

Figure 460000

Demand forecasting result

40000

4.3.3 Comparison of price forecasting

20000
0
1

13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205
Time (Week)

The price forecasting results from real option method and
ARIMA model are very similar as shown in Figure 5. We

Figure 3

prefer the real option model than ARIMA model because realPrice forecast (ARIMA (9,1,9))

option model is comparatively give a better level of accuracy
to the actual price.

The first step to simulate the forecasting price by using
ARIMA model is to find the best AIC as shown in Table 3.
We use ARIMA (6,1,3) model which provides the best AIC,

Total Price Forecast

8.72. The 1000 round simulation result is shown in Figure 4.

1,000

We can use ARIMA model to forecast the low volatility price800
Price (Yen)accurately through all period.
4.3.2 Comparison of demand forecasting

600
400
Actual Price
ARIMA

200

Real Option
Mean Reversion

0

When we compare the result of forecasting between real1

option method and ARIMA model, we see that both method
give quite similar degree of accuracy.

Vol. XXXI, 200915 29 43 57 71 85 99 113 127 141 155 169 183 197 211
Time (Week)

Figure 5―5―− 41 −

Actual and forecast priceDemand and Price Forecasting Models for Strategic and Planning Decisions in a Supply Chain

Times, Times and Times

This can be explained by the name of the change in price data.

economics, 2006.

The price data has a higher level of volatility comparing the[3] Carlos and David, “Mean Reverting Processes-Energy

level of volatility of demand. This explains why Real options

Price Processes Used For Derivatives Pricing & Risk

approach outperforms ARIMA. In short, we can say that inManagement”, Commodities Now, June 2001.

case of high volatility data Real options approach is preferred[4] Dixit and Pindyck, Investment under uncertainly,1994

over ARIMA.

[5] Makridakis,S.,Wheelwright,S.C.,1995, Forecasting

We show the comparison of the absolute deviation of

methods for Management, John Wiley & Sons, New Yorkdemand and price in table 4 and table 5.

[6] Box George and Jenkins Gwilym (1970), Time series
analysis: Forecasting and control, San Francisco, Holden-

Table 4Day.

Absolute deviation of demand[7] Makridikis, S., S.C. Wheelwright, and R.J. Hyndman

Real Option% AD

ARIMA Model%AD

0.182

0.168

Table 5

(1998), Forecasting: methods and applications, New York,John Wiley & Sons
[8] Pankratz, A. (1983), Forecasting with univariate
Box–Jenkins models: concepts and cases, New York, John

Absolute deviation of price

Wiley & Sons
[9] Martin, R.D., Samarov, A. and W. Vandaele, "Robust

Real Option% AD

ARIMA Model%AD

Methods for ARIMA Models", In Time Series Analysis:

0.06

0.6

Theory and Practice 1, Proceedings of the International
Conference held at Valencia, Spain, June 1981, Anderson, O.

5. Conclusion

D. (Ed.) (1982), Amsterdam: North-Holland Publishing
Company, pp.755-756.

In this study we compare the forecasting methods that are[10] Metcalf & Hasset (1995), "Investment under Alternative

used to forecast demand and price. We use real option

Return Assumptions Comparing Random Walks and Mean

approach forecasting method that is based on Geometric

Reversion", Journal of Economic Dynamics and Control,

Brownian motion for the prediction. This model is good invol.19, November 1995, pp.1471-1488

capturing the trend and the volatility of the data, which are[11] Whitt, W., “The stationary Distribution of a Stochastic

the common characteristic of many data type especially in

Clearing Process,” Operations Research 29(2), pp.249-308

finance and supply chain management. The model is used to(1981)

predict in a short term period. We highlight and compare theresult with ARIMA method that is one of the famous models
used to forecast time series. The result shows that real
options approach perform better in both sets of data that weused to test. Especially when the data has a high volatility,
real options shows less deviation than ARIMA model.
Besides that the time interval and availability of data set also
have a crucial impact on the accuracy of the forecast.
Deseasonality of data and data segmentation would probably
improve the performance of the forecast and still remain aninterest of our future research.
Reference[1] Ming-Guan Huang (2008), Real options approach-based
demand forecasting method for a range of profits with highly
volatile and correlated demand, European journal of
operational research[2] Yon-Chun Chou, C.-T Cheng, Feng-Cheng Yang, Yi-Yu
Liang,

Evaluation

alternative

capacity

strategies

insemiconductor manufacturing under uncertain demand andpricescenarios",

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journalofproduction―6―− 42 −

Proc. Sch. ITE
Tokai University