How To Write A Demand And Price Model For A Supply Chain

Transcription

1 Proc. Schl. ITE Tokai Univ. vol.3,no,,pp.37-4 Vol.,No.,,pp. - Paper Demand and Price Forecasing Models for Sraegic and Planning Decisions in a Supply Chain by Vichuda WATTANARAT *, Phounsakda PHIMPHAVONG * and Masanobu MATSUMARU * (Received on Ocober 3, and Acceped on January 5, ) Absrac Procuremen and sales forecasing are a fundamenal approach o he esimaion of invenory. The accuracy is an essenial facor for a firm on cos of manufacuring. Forecasing is an imporan for all sraegic and planning decisions in a supply chain. Forecass of produc demand, maerials, labor, financing are useful inpus o scheduling, acquiring resources, and deermining resource requiremen. However, highly volaile and correlaed demand caused inaccuracy. This paper aims o compare real opion mehod and ARIMA model mehod o be able o predic a precise fuure procuremen and sales. To validae he forecasing model, we use he real world daa which are demand obained from one die casing company and copper price. The resul shows ha real opion model is ouperforming ARIMA model quie significanly in case he daa has high volailiy. Keywords: Forecas, Volaile, Demand, Geomeric Brownian moion, ARIMA mehodology. Inroducion Smoohing he demand and supply is he main problem in supply chain managemen. Under supply leads o he loss of poenial revenue, over supply incurs exra carrying cos of he excess produc. Procuremen and invenory conrol are of acive area of research o gain he compeiiveness for he company. The opimal quaniy can enable he company o achieve he cos reducion. To obain he opimaliy in supply chain, we have o deal wih making he decision abou he fuure even, produce o secure he fuure sale. This involves wih forecasing he unforeseen daa for example demand and price. The less deviaion from he real value he less risk we have from doing he operaion. Demand forecasing plays a very crucial role in producion planning of he manufacuring. I is he main inpu for he capaciy planning which links o various chain of supply including invenory, logisic and service level. Demand and price is usually difficul o forecas due o heir uncerainy naure. Basically demand should show an increasing rend over he long erm period. When we look a he shor erm period, we ofen see he unexpeced move eiher going up or down. If we include more realisic possibiliy of he economic facor o ry o undersand and * Graduae suden, Deparmen of managemen sysems engineering * Professor, Deparmen of managemen sysems engineering predic movemen of he demand, we can clearly see from he hisory ha will no always suppose o have an upward rend even in he long erm demand. This case is clearly noiced in he ime of economic crisis. Besides, compeiion and advancemen of echnology also erode he demand of one company o anoher. Taking he example of mobile music indusry like Sony ha loses he marke share o Apple produc. In his paper we compare he new forecasing mehod ha is apply o he forecas of demand and price. There are many ways ha we can adop for forecasing for example linear regression, moving average, ARMA, ARIMA. Here we compare he real opion approach wih he ARIMA model and conclude wha kind of daa is bes suied for hese kind of forecasing. Real opion approach is originaed from he finance sudy. I is used o value he value of uncerainy in an invesmen. Real opion is differen from radiional invesmen crieria like Ne Presen Value (NPV) in ha unexercised choices have also fuure value and should conribue he evaluaion.. Lieraure review Real opion approaches which is based on he diffusion process of he variable for doing he forecas is quie a new field of sudy. The mos common diffusion process ha is used in real opion sudy is Geomeric Brownian Moion Vol. I, 9 37

2 Demand and Price Forecasing Models for Sraegic and Planning Decisions in a Supply Chain Times, Times and Times (GBM). GBM is used o replicae he movemen or characer of he objec of sudy in many fields of sudy e.g. paricle movemen in physic, he price of he underlying asse in he financial sudy, waer level in reservoir ec. Huang [] applies real opion in forecasing he demand of producs ha has correlaed o one anoher and ha have high volailiy. The kind of produc ha has such kind of naure is referred o IT produc. In heir research hey forecas he demands of 4 producs ha relae o one anoher. If one produc changes, he volume of oher producs also changes. The GBM model is used o replicae he fuure demand in capaciy sudies []. They sudied capaciy uilizaion over ime assuming ha demand followed a GBM. Yon [3] also sudied abou he capaciy planning. They analyzed in case of demand has high volailiy and machine and process echnology are advanced a a rapid pace, making i difficul o do he capaciy planning in he semiconducor indusry. They assume he demand o follow he GMB. They simulae and compare he resul of he profi in wo difference scenarios. Real opion by mean of GBM has also been used exensively in making invesmen decision as in Dixi and Pindyck [4]. I can be used o find he opimal ime o inves, evaluae opion o pospone invesmen, opion o shu down he business, opion o expand business ec. in which he radiional invesmen crieria canno handle wih well. An auoregressive inegraed moving average (ARIMA) is one of well known model in saaisic and economerics. I is used o sudy he behavior of ime series and predic he fuure ou of he ime series. The simple way o hink abou he ARIMA model is ha i is he combinaion of random walk and rend model. This characerisic is very similar o he GBM and i is a good candidae for he comparison. For a high volaile demand produc, Makridakis and Wheelwrigh [5] exended he ime series analysis by including moving average, regression, exponenial smoohing, and seasonal exponenial smoohing. The ARIMA mehodology is developed by Box and Jenkins [6]. I is a mahemaical model for forecasing a ime series by fiing i wih he daa and using he fied model for forecasing. R.J. Hyndman [7], Pankraz [8] and Marin, R.D., Samarov, A. and W. Vandaele [9] sudy ARIMA model in applicaion. Mecalf & Hasse [] sudy he mean reversion forma equaion, and i has more economic logic han he geomeric Brownian model. Carlos and David [] explore he main sochasic price processes used o model energy spo and forward prices for derivaives valuaion and risk managemen. They apply a Geomeric Brownian moion o be a random walk wih mean reversion. In our research we compare he forecasing mehods which are real opion mehod, ARIMA model for he esimaion of procuremen and sales. We compare our sudy wih he previous sudy in Table. Table Previous Sudy Makridakis and Wheelwrigh (985) Ming-Guan Huang (8) James D. Hamilon ( Book,994) Previous sudy comparison Deails Time series analysis by including moving average, regression, exponenial smoohing, and seasonal exponenial smoohing Real opion approach-based demand forecasing mehod for a range of producs wih highly volaile and correlaed demand Analyze vecor auo regressions, generalized mehod of momens, he economic and saisical consequences of uni roos, ime-varying variances, and nonlinear ime series models The ouline of his sudy is as follow: In secion we explain he mehodologies and how o esimae he fuure value. We se a numerical simulaion and illusrae resuls in secion 3. We prefer o use MATLAB o simulae he simulaion of real opion mehod, Eviews for ARIMA model. Finally, we conclude a summary of our findings. 3. Forecasing Mehodology 3. Geomeric Brownian moion Geomeric Brownian moion is originaed in physics as a diffusion process used o describe he parial moion when i falls. Le be he value of he daa a ime and assume i o follow he Geomeric Brownian Moion, we have he diffusion specify in equaion (). In his equaion here are wo pars. The firs par on he righ hand side is he mean growh of he diffusion and he second par on he righ hand side of he equaion is he volailiy. d = μ + σ () d dz Where dz = ε d is he sandard incremen of Wiener process or he whie noise, μ and σ are drif and variance erms respecively. z has he following properies: Propery : The change in he value of z, dz, over a ime inernal of lengh Δ is proporional o he square roo of Δ 38 Proc. Sch. ITE Tokai Universiy

3 Vichuda WATTANARAT Phounsakda PHIMPHAVONG Masanobu MATSUMARU Times Times Times Times Times where he muliplier is random. In specific dz=z(+ Δ)-z()=εΔ, where ε is a sandard normal random variable. Hence values of dz follow a normal disribuion wih mean and variance equal o he change in ime (Δ) over which dz is measured. Propery : The changes in he value of z for any non-overlapping inervals of ime are independen. Le f=ln and expand f according o he Taylor series we have df d f df = ( d ) + ( d ) +... () d d Then apply Io lemma o (), we have ha Δ ha has power greaer han will approach o zeros quicker when Δ is small. We have df d f df ( d ) + ( d ) d d = (3) Subsiue () ino (3) and rearrange we have ln + ln = μ σ Δ + σε Δ (4) = exp μ σ Δ + σε Δ (5) + Equaion (5) is he discree version of Geomeric Brownian Moion ha we will use o simulae he forecas of demand and price in his sudy. 3. Calibrae he parameer The Geomeric Brownian Moion process can formally be defined as,, he random variable of / + is independen of all values of he variable up o he ime and if in addiion ln( / + ) has a normal disribuion wih mean μ and variance σ. Le r be he difference beween logarihm and - we have ln = r (6) In finance, (6) represen he logarihm of he reurn which is necessary assumpion because he value of he underlying asse ha is assumed o follow Geomeric Brownian Moion should have a value greaer or equal o zero. r also implies he growh rae of he observed daa in each period. In order o esimae he parameer o use in he forecas model as specified by equaion (5) we following equaions E( r ) = ( μ σ ) Δ (7) V ( r r ) = σ Δ (8) r = sr = n r = n n ( r r) ( n ) = = Δ (9) () s σ r () r ˆ σ r sr μ = + = + () Δ Δ Δ 3.3 ARIMA Model In supply chain managemen, An auoregressive inegraed moving average has been used o in many problems ranging from finding opimal order policy, predicing cusomer demand, predicing he price movemen, invenory managemen ec. The auoregressive moving average (ARMA) model was inroduced by Box and Jenkins (976). I consiss of wo pars, an auoregressive (AR) par and a moving average (MA) par. Specifically, he hree ypes of parameers in he model are: he auoregressive parameers (p), he number of differencing passes (d), and moving average parameers (q). Models are summarized as ARIMA (p, d, q). ARIMA combines auo regression-which fis he curren daa poin o a linear funcion of some prior daa poins and moving averages adding ogeher several consecuive daa poins and geing heir mean, and hen using ha o compue esimaions of he nex value. Wih enough elemens regressed and averaged, we can fi an approximaion oo many ime series and wih low discrepancy. We find he bes ARIMA (p,d,q) model condiion by comparing he Akaike informaion crierion (AIC). The smalles AIC is he mos suiable condiion. An ARIMA (p,d,q) model conains hree differen kinds of parameers: () he p AR-parameers; () he q MA-parameers he number of lagged forecas errors in he predicion equaion. (3) d he number of differences. This amouns o a oal of (p,d,q) parameers o be esimaed. These parameers are always esimaed on using he saionary ime series which is saionary wih respec o is variance and mean. To idenify he appropriae ARIMA model for a ime series, we have o idenifying he order(s) of differencing needing o saionary he series and remove he gross feaures of seasonaliy, perhaps in conjuncion wih a variance-sabilizing ransformaion such as logging or Vol. I,

4 Demand and Price Forecasing Models for Sraegic and Planning Decisions in a Supply Chain Times, Times and Times deflaing. An noaion AR(p) refers o he auoregressive model of order p. An AR(p) model is defined as (3) = c + φ + φ φ p p + ε where φ, φ, L L, φ p are he parameers of he model, c is a consan and ε is whie noise. An MA model is a common approach for modeling univariae ime series models. x (4) = μ + ε + θ ε + L + θ q ε q where μ is he mean of he series, he θ,..., θ q are he parameers of he model and he ε, ε,... are whie noise error erms. The value of q is called he order of he MA model. An ARMA consiss of wo pars, an auoregressive (AR) par and a moving average (MA) par. The model is usually hen referred o as he ARMA(p,q) model where p is he order of he auoregressive par and q is he order of he moving average par. = c + φ φ p p (5) + φ ε φ ε + ε An ARIMA model is a generalizaion of an ARMA model. The model is generally referred o as an ARIMA(p,d,q) model where p, d, and q are non-negaive inegers ha refer o he order of he auoregressive, inegraed, and moving average pars of he model. 4. Empirical analysis Decisions make in advance of uncerain evens. And forecasing is an imporan for all sraegic and planning decisions in a supply chain. Especially, forecass of produc demand, maerials are imporan inpus o scheduling, acquiring resources, and deermining resource requiremen. We use he daa of he medium and small-sized business in real world and check he effeciveness of a suggesed model. This company acceps an order from he upsream company. Because here is a change in his order, his company mus have many socks o avoid lack. As a resul, his company mus risk a burden for a cos o mainain a sock. Therefore, i is imporan ha he company predics quaniy of order from he upsream company. If we predic he volume precisely, his company reduces he invenory cos. The reducion of he invenory cos in he model ha we suggesed had a big influence on he improvemen of corporae earnings. 4. Daa I assumes ha he forces which influenced pas value will coninue o exis in he fuure. In his secion, we follow he forecas mehodology o calculae he numerical value of demand and price. We follow he forecas mehodology o calculae he numerical value of demand and price. For he q q numerical experimen, we use weeks of daa (from November, 4 o December, 8) for esimaion by using Geomeric Brownian moion, ARIMA model and mean Reversion. 4. Empirical resuls of Geomeric Brownian moion In his research, we apply o boh forecasing models o he forecas he demand of one die casing company and he price of copper. We use he daa from November 4 o December 8. The acual demand shows he seasonaliy in he daa series. For he acual price of copper, he daa se shows high volailiy and covers he period of he financial crisis. ()Resuls of demand For he illusraive purpose, we simulae demand rounds for each period by using MATLAB. Figure is a resul from real opion mehod. Real opion mehod provides he forecasing value along wih acual value as well. This mehod can capure demand changes quie accuraely. All he daa were acquired on saisical sofware for economic analysis, called EViews, which is developed by Quaniaive Micro Sofware (QMS). We analyze demand and price daa which are separaed o be numbers of daa. We follow he parameer seing from November 4 o December 8. Demand (kg) Toal Deamnd forecas Toal Demand Toal Demand Forecas Figure ()Resuls of price Demand forecas We ry o find he simulaion resul of mean reversion in price wih imes esimaion. Price (Yen) Figure Price forecas Acual Price Acual Price Forecasing Price Proc. Sch. ITE Tokai Universiy

5 Vichuda WATTANARAT Phounsakda PHIMPHAVONG Masanobu MATSUMARU Times Times Times Times Times 4.3 Empirical resuls of ARIMA model 4.3. Numerical experimen of Demand following real opion mehod We demonsrae a resul of AIC in Table wih he differen value of d in ARIMA (p,d,q) model. The bes ARIMA (p,d,q) parameers for ARIMA model for demand is compued as ARIMA (9,,9) ha AIC is.4. We assume o use ARIMA (9,,9) which offers he bes Akaike informaion crierion for our simulaion. We demonsrae our forecasing resul by using ARIMA (9,,9) in Figure 3 as below. In he beginning ARMA model can produce demand forecas quie well bu in he long run he forecas accuracy is geing lower. Table ARIMA parameer for oal demand Table 3 ARIMA parameer for oal price However real opion approaches does a slighly beer performance han ARIMA model. Demand daa does no show high degree of flucuaion. This is an imporan condiion ha is conribued o he level of accuracy of in each mehod. To verify he precision of forecasing model, we compare he acual and forecas resul again. From Figure 4 and Figure 5, he real opion mehod presens he bes forecas accuracy Demand (Kg) Toal Demand Forecas (ARIMA Model) Acual Demand Forecas Demand Figure 3 Price forecas (ARIMA (9,,9)) The firs sep o simulae he forecasing price by using ARIMA model is o find he bes AIC as shown in Table 3. We use ARIMA (6,,3) model which provides he bes AIC, 8.7. The round simulaion resul is shown in Figure 4. We can use ARIMA model o forecas he low volailiy price accuraely hrough all period Comparison of demand forecasing When we compare he resul of forecasing beween real opion mehod and ARIMA model, we see ha boh mehod give quie similar degree of accuracy. D e m an d (K g) Toal Demand Forecas Acual Demand ARIMA Real Opion Mean Reversion Figure 4 Demand forecasing resul Comparison of price forecasing The price forecasing resuls from real opion mehod and ARIMA model are very similar as shown in Figure 5. We prefer he real opion model han ARIMA model because real opion model is comparaively give a beer level of accuracy o he acual price. Price (Yen), Toal Price Forecas Acual Price ARIMA Real Opion Mean Reversion Figure 5 Acual and forecas price Vol. I, 9 5 4

6 Demand and Price Forecasing Models for Sraegic and Planning Decisions in a Supply Chain Times, Times and Times This can be explained by he name of he change in price daa. The price daa has a higher level of volailiy comparing he level of volailiy of demand. This explains why Real opions approach ouperforms ARIMA. In shor, we can say ha in case of high volailiy daa Real opions approach is preferred over ARIMA. We show he comparison of he absolue deviaion of demand and price in able 4 and able 5. Table 4 Absolue deviaion of demand Real Opion% AD ARIMA Model%AD.8.68 Table 5 Absolue deviaion of price Real Opion% AD ARIMA Model%AD Conclusion In his sudy we compare he forecasing mehods ha are used o forecas demand and price. We use real opion approach forecasing mehod ha is based on Geomeric Brownian moion for he predicion. This model is good in capuring he rend and he volailiy of he daa, which are he common characerisic of many daa ype especially in finance and supply chain managemen. The model is used o predic in a shor erm period. We highligh and compare he resul wih ARIMA mehod ha is one of he famous models used o forecas ime series. The resul shows ha real opions approach perform beer in boh ses of daa ha we used o es. Especially when he daa has a high volailiy, real opions shows less deviaion han ARIMA model. Besides ha he ime inerval and availabiliy of daa se also have a crucial impac on he accuracy of he forecas. Deseasonaliy of daa and daa segmenaion would probably improve he performance of he forecas and sill remain an ineres of our fuure research. economics, 6. [3] Carlos and David, Mean Revering Processes-Energy Price Processes Used For Derivaives Pricing & Risk Managemen, Commodiies Now, June. [4] Dixi and Pindyck, Invesmen under uncerainly,994 [5] Makridakis,S.,Wheelwrigh,S.C.,995, Forecasing mehods for Managemen, John Wiley & Sons, New York [6] Box George and Jenkins Gwilym (97), Time series analysis: Forecasing and conrol, San Francisco, Holden- Day. [7] Makridikis, S., S.C. Wheelwrigh, and R.J. Hyndman (998), Forecasing: mehods and applicaions, New York, John Wiley & Sons [8] Pankraz, A. (983), Forecasing wih univariae Box Jenkins models: conceps and cases, New York, John Wiley & Sons [9] Marin, R.D., Samarov, A. and W. Vandaele, "Robus Mehods for ARIMA Models", In Time Series Analysis: Theory and Pracice, Proceedings of he Inernaional Conference held a Valencia, Spain, June 98, Anderson, O. D. (Ed.) (98), Amserdam: Norh-Holland Publishing Company, pp [] Mecalf & Hasse (995), "Invesmen under Alernaive Reurn Assumpions Comparing Random Walks and Mean Reversion", Journal of Economic Dynamics and Conrol, vol.9, November 995, pp [] Whi, W., The saionary Disribuion of a Sochasic Clearing Process, Operaions Research 9(), pp (98) Reference [] Ming-Guan Huang (8), Real opions approach-based demand forecasing mehod for a range of profis wih highly volaile and correlaed demand, European journal of operaional research [] Yon-Chun Chou, C.-T Cheng, Feng-Cheng Yang, Yi-Yu Liang, Evaluaion alernaive capaciy sraegies in semiconducor manufacuring under uncerain demand and price scenarios", Inernaional journal of producion 6 4 Proc. Sch. ITE Tokai Universiy