Stochastic demand forecast and inventory management of a seasonal product a supply chain system

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1 Louisiana Sae Universiy LSU Digial Commons LSU Docoral Disseraions Graduae School 008 Sochasic demand forecas and invenory managemen of a seasonal produc a supply chain sysem Mohammad Anwar Ashek Rahman Louisiana Sae Universiy and Agriculural and Mechanical College, mrahma@lsu.edu Follow his and addiional works a: hps://digialcommons.lsu.edu/gradschool_disseraions Par of he Engineering Science and Maerials Commons Recommended Ciaion Rahman, Mohammad Anwar Ashek, "Sochasic demand forecas and invenory managemen of a seasonal produc a supply chain sysem" (008). LSU Docoral Disseraions. 76. hps://digialcommons.lsu.edu/gradschool_disseraions/76 This Disseraion is brough o you for free and open access by he Graduae School a LSU Digial Commons. I has been acceped for inclusion in LSU Docoral Disseraions by an auhorized graduae school edior of LSU Digial Commons. For more informaion, please conacgraded@lsu.edu.

2 STOCHASTIC DEMAND FORECAST AND INVENTORY MANAGEMENT OF A SEASONAL PRODUCT IN A SUPPLY CHAIN SYSTEM A Disseraion Submied o he Graduae Faculy of he Louisiana Sae Universiy and Agriculural and Mechanical College In parial fulfillmen of he Requiremens for he degree of Docor of Philosophy in The Inerdeparmenal Program In Engineering Science by Mohammad Anwar Ashek Rahman B.S., Bangladesh Universiy of Engineering and Technology, 995 M.E., Bangladesh Universiy of Engineering and Technology, 998 M.S., Louisiana Sae Universiy, 003 M.App.Sa., Louisiana Sae Universiy, 007 May, 008

3 ACKNOWLEDGEMENTS I would like o hank my Professor Bhaba R. Sarker, chairman of my docoral commiee, for his invaluable suppor, advice, and encouragemen in bringing his research work o a successful compleion. He has augh me a grea many hings, guided me as o how o deal wih new problems. I would also like o express my graiude o all members of my disseraion commiee, namely Dr. Lawrence Mann Jr. of Indusrial Engineering Deparmen, Dr. Luis Escobar of Experimenal Saisics Deparmen, Dr. Ralph Pike of Chemical Engineering Deparmen and Dr. Guoli Ding of Mahemaics Deparmen for he assisance rendered o me. Their suggesions and commens helped improve he qualiy of his research. I like o convey my special hanks o Dr. Mann for providing suggesions o improve he readabiliy and for his commens. I also feel hankful o Dr. Jack Helms, Graduae School Represenaive, for his useful suggesions regarding various aspecs of my research work. I would like o hank all he people a Louisiana Sae Universiy who welcomed me wih enhusiasm and encouragemen. This is a major milesone in my life for which I wish o hank my parens and family as well. Had I no received heir inspiraion, I migh no have been able o accomplish his work. I would also like o hank all my colleagues and friends for heir sincere suppor during my graduae sudy. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS ii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT.. viii CHAPTER INTRODUCTION Sudy Conex Problem Saemen Research Goals Research Objecives Soluion Approach Scope and Opporuniies Acual Time Series Daa Organizaion of he Disseraion CHAPTER LITERATURE REVIEW Demand Forecass Using Bayesian Procedure Time Series Auoregressive Models....3 Invenory Models Limiaions of he Pas Research Overcoming he Limiaions CHAPTER 3 BAYESIAN FORECASTING MODEL FOR SEASONAL DEMAND Demand Model Formulaion Bayesian Procedure in Demand Model Applicaion of Demand Model Sub-Models Algorihm 3. (Seps o Derive he Prior Values) Bayesian Probabiliy Model (B-P Model) Sample Calculaion (B-P Model) Bayesian Probabiliy Model wih Incomplee Daa (BP-I Model) Forecasing Errors and Model Validiy Summary... 3 CHAPTER 4 THE ARIMA APPROACH TO FORECASTING SEASONAL DEMAND Fundamenal of he ARIMA Approach (F-ARIMA) Difference Operaor o Eliminae Increasing Trend Periodic Difference Operaor o Eliminae Periodic Increase Applicaion of Box-Jenkins Mehodology F-ARIMA Model Idenificaion Parameers Esimaion of F-ARIMA Model iii

5 4..3 Diagnosic Checking and Model Validaion Poin Forecas wih F-ARIMA Model Forecas Resuls by F-ARIMA Model Bayesian Sampling-based ARIMA Model (BS-ARIMA) Bayesian Compuaion a BS-ARIMA Model Forecas Resuls of BS-ARIMA Model from Incomplee Daa Forecas Using Adapive Exponenial Smoohing Technique Forecas Performance and Model Validaion Summary CHAPTER 5 INVENTORY COST REDUCTION USING IMPROVED FORECASTING Model Descripion Procedure o Compue Opimal Invenory Cos Cos Componens o Deermine he Cusomer Service Level Newsvendor Procuremen Model Expedie Cos Facors Urgen Leadime Cos Funcion Cos Minimizaion in Newsvendor Model Numerical Example: (Newsvendor Invenory Policy) Alernae Invenory Policy: (Periodic Review) Selecion of Order Poins Obaining Order Poins by Considering Forecas Errors Numerical Illusraion (Order Poins) Opimal Invenory Cos Opimal Invenory Cos Using Dynamic Programming Deermining Invenory Coss for All Forecass Comparison of Forecasing Mehods Summary CHAPTER 6 CONCLUSION AND FUTURE RESEARCH General Conclusion Fuure Research Direcion Produc Subjec o Obsolescence: New Producs: Producs Sensiive o Economic Condiions: REFERENCES APPENDIX A: BAYESIAN PROBABILITY MODELS APPENDIX B: ARIMA AND BAYESIAN ARIMA MODELS...86 APPENDIX C: INVENTORY COST REDUCTION MODELS VITA. 06 iv

6 LIST OF TABLES Table 3.: The srucure of finding expeced prior values Table 3.: Projeced demand averaging he sample daa...5 Table 3.3: Prior and poserior parameers derived by B-P model for Table 3.4: Projeced demand averaging he sample daa...7 Table 3.5: Prior and poserior parameers derived by BP-I for Table 3.6: PE, MAD, TS for B-P and BP-I models (unis in million) Table 4.: Seps of F-ARIMA mehodology for ime series modeling...37 Table 4.: AIC and BIC values for various F-ARIMA models Table 4.3: Auocorrelaion check of F-ARIMA residuals.. 4 Table 4.4: Esimaed values of he F-ARIMA parameers..4 Table 4.5: Forecas resuls by F-ARIMA model (0,,)(,,0) (unis in million) Table 4.6: Values of dummy variable for July o December, 004 (unis in million) Table 4.7: Demand Forecas by BS-ARIMA model (unis in million) Table 4.8: Forecas resuls by M-ES model (unis in million) Table 4.9: Forecas by F-ARIMA, BS-ARIMA and M-ES models (unis in million). 5 Table 5.: Uni coss applied o he invenory model Table 5.: Probabiliy of sock ou and corresponding safey facor Table 5.3: Order poin derived by (B-P) Forecasing Model Table 5.4: Order quaniy for sage- by Newsvendor policy (unis in million) Table 5.5: Order quaniy for sage- by Periodic Review policy (unis in million). 65 Table 5.6: Invenory quaniy and procuremen cos by B-P model (unis in million)...66 Table 5.7: Invenory quaniy and replenishmen ime for sage- (unis in million) v

7 Table 5.8: Invenory cos using newsvendor policy for sage-, Table 5.9: Invenory quaniy by newsvendor policy a sage-, 005 (unis in million)...68 Table 5.0: Invenory cos based on periodic review invenory policy Table 5.: Invenory replenishmen quaniy by periodic review a sage-, 005 (unis in million) Table 5.: Invenory cos for each forecasing models and acual demand.. 70 vi

8 LIST OF FIGURES Figure.: Supply-demand flow sysem in a supply chain.. Figure.: A wo-sage invenory model Figure.3: Aciviies of probabiliy disribuion models 5 Figure.4: Aciviies of ime series forecasing models..6 Figure 3.: Variaion of demand daa for apparel produc (Sources: U.S. Deparmen of Commerce, Office of Texiles and Apparel)... 7 Figure 3.: Flow diagram for Bayesian compuaion....5 Figure 3.3: Comparison of forecass wih acual demand Figure 3.4: Error comparison of all models Figure 3.5: Summaries of racking signals of he models.. 30 Figure 4.: Time plo of apparel demand daa series afer differencing Figure 4.: ACF and Parial ACF of differenced demand daa Figure 4.3: Flow char for F-ARIMA esimaion process...4 Figure 4.4: Comparison of forecas and acual demand Figure 4.5: Tracking signal of he F-ARIMA, BS-ARIMA and M-ES models vii

9 ABSTRACT Esimaion of seasonal demand prior o an acive demand season is essenial in supply chain managemen. The business cycle of he seasonal demand is divided ino wo sages: sage-, he slow-demand period, and sage-, he peak-demand period. The focus here is o deermine an appropriae demand forecas for he peak-demand period. In he firs se of forecasing model, a sandard gamma and an inverse gamma prior disribuion are used o forecas demand. The parameers of he prior model are esimaed and updaed based on curren observaion using Bayesian echnique. The forecass are derived for boh complee and incomplee daases. The second se of forecas is derived by ARIMA mehod using Box-Jenkins approaches. A Bayesian ARIMA is proposed o forecas demand from incomplee daase. A parial daase of a seasonal produc, colleced from he US census bureau, is used in he models. Missing values in he daase ofen arise in various siuaions. The models are exended o forecas demand from an incomplee daase by he assumpion ha he original daase conains missing values. The forecas by a muliplicaive exponenial smoohing model is used o compare all he forecas. The performances are esed by several error measures such as relaive errors, mean absolue deviaion, and racking signals. A newsvendor invenory model wih emergency procuremen opions and a periodic review model are sudied o deermine he procuremen quaniy and invenory coss. The invenory cos of each demand forecas relaive o he cos of acual demand is used as he basis o choose an appropriae forecas for he daase. This sudy improves he qualiy of demand forecass and deermines he bes forecas. The resul reveals ha forecasing models using Bayesian ARIMA model and Bayesian probabiliy models perform beer. The flexibiliy in he Bayesian approaches allows wider variabiliy in he model parameers helps o improve demand forecass. These models are paricularly useful when viii

10 pas demand informaion is incomplee or limied o few periods. Furhermore, i was found ha improvemens in demand forecasing can provide beer cos reducions han relying on invenory models. ix

11 CHAPTER INTRODUCTION Demand forecasing includes he predicion, projecion or esimaion of expeced demand of he producs over a specified fuure ime period. The demand of seasonal producs frequenly changes in he markeplace. As soon as he main selling season passes, he excessive invenories of he produc are devalued grealy. On he oher hand, if he produc supplies were relaively shor, a direc sale loss occurs. Therefore, demand planning is considered he firs sep of a supply chain planning process, which provides a coninuous link o manage he invenory posiion and he produc demand. Forecasing is an essenial ool for making sraegic demand planning. In his sudy, a number of demand forecasing models are sudied o predic demand of a seasonal produc for an acive sales period. Forecasing accuracy may be measured using several indicaors, such as relaive error, mean absolue deviaion and racking signals. Afer forecass are derived, he invenory quaniy for a arge business season can be obained based on hese demand forecass. The oal invenory cos of he produc can be deermined using a dynamic opimizaion echnique. This resul can be used as an alernaive measure o decide he bes forecasing model ha provides he minimum invenory cos for he arge period.. Sudy Conex Marke demands of mos producs remain uncerain unil he selling season begins. In mos of oday s business environmen, seasonaliy is an imporan feaure. Many producs have seasonal effecs. The life cycles of hese seasonal producs are shor and he demands are uncerain. I is ofen found ha demand of seasonal producs becomes significan only in he specific period in a year. For example, he demand of winer apparel, fashion goods, Chrismas

12 gif producs are higher during specific seasons and hold seasonaliy, rends, or cyclic demand paern. Moreover, fuure demand may no follow he hisorical paern of he pas demand, which may imply differen predicions a differen ime period. Therefore, demand planning for seasonal and shor life producs is considered a vial componen for an effecive business. The mos known forecasing echniques currenly available are based on exrapolaion of hisorical demand daa. For accurae forecasing, i is imporan o esimae he parameers of forecasing models wih he mos recen demand informaion and forecas can hen be updaed as new demand informaion becomes available. If Bayesian mehods are used in forecasing algorihms, he prior knowledge abou he fuure demand and he curren sale informaion can be incorporaed o forecas demand. In business, here are always flows of producs in he invenories since he producs from he sores are demanded consanly. Orders are placed prior o selling season and producs are moved o mee demand. A ypical supply chain srucure is illusraed in Figure., where he producs flow from manufacurers hrough disribuors and reailers o consumers and he demand flows back. The demand forecasing and invenory models can be so consruced ha any member of a supply chain may use he models for forecas processing and invenory deploymen prior o he selling period. Manufacurer Disribuor Reailer Consumer [F] [F] [F] Produc flow [F] = Forecasers Demand flow Figure.: Supply-demand flow sysem in a supply chain

13 . Problem Saemen In his sudy, he problem is o find he demand forecas of a seasonal produc and o find he bes forecas o anicipae he righ demand for a arge selling season. The demand of he seasonal producs increases as he main demand season approaches. Therefore, seasonal demand always occurs in wo sages: slow demand period and busy demand period. For he seasonal produc considered, he business planning horizon is divided ino wo sages: sage-, a prior demand period and sage-, he poserior demand period. The focus is o forecas demand for he sage- period. In he forecasing process, he demand daa is colleced from he pas seasons. Curren sales of he produc are observed a sage- of he forecasing year and he forecass are made prior o he main demand season. The iniial sales a a business cycle sar a ime. Afer demand is observed a sage-, he forecas processing and orders placemen are performed prior o ime. The produc receiving and peak selling coninues hroughou sage- period. The procuremen plans are also anicipaed so ha demand can be delivered on ime during he selling period. The ime-relaed aciviies a differen sages of a business cycle are shown in Figure.. Sage- Collec pas sale daa Forecas for sage- Observe sales a sage- Commi order Sage- Order arrives Targe season begins T Prior model + real-ime daa (Higher demand variabiliy) Poserior disribuion (Lower demand variabiliy) Figure.: A wo-sage invenory model 3

14 .3 Research Goal The goal of his sudy is o forecas demand of a seasonal produc for acive demand periods using various forecasing models and o adop he bes forecasing echnique resuling in minimum errors and invenory coss. The improvemen in demand forecas provides poenial cos reducions and assiss a decision manager o deermine he bes demand planning for he acive demand period of a seasonal produc..4 Research Objecives This sudy is o creae models o predic fuure demand of a seasonal produc for arge sale season using improved forecas informaion so ha demand planning can be performed as precisely as possible wih minimum cos. The forecas analysis focuses he following issues: (a) To forecas seasonal demand of a produc using non-negaive probabiliy disribuion model wih Bayesian echniques, (b) Demand forecas wih ime-series model using auoregressive inegraed moving average (ARIMA), and Bayesian sampling-based ARIMA models. These models are compared wih he forecas derived by muliplicaive exponenial smoohing model, and (c) To find he bes forecas using he invenory models o es he resuls ha provides he minimum invenory cos. The objecives of he above models are illusraed as following. Model I: Demand Forecass using Probabiliy Disribuion involving Bayesian Techniques In his forecasing model, he demand process is described by he probabiliy disribuion where disribuion parameers are unknown. A prior model is seleced o describe he variaion of demand over he periods. The objecive of his model is wo folds: Firs, o predic he unknown parameers of he demand model using he Bayesian approach and o forecas, using hese esimaed parameers. Second, as he pas daa series ofen conains missing values; he 4

15 objecive here is o exend he model o demonsrae he forecasing approach using daa series ha conains missing values. The aciviies of hese models are shown in Figure.3. Probabiliy models Collec Daa No Complee daa Missing daa? Yes Incomplee Daa Esimae model parameers Esimae missing values and model parameers Forecas for sage- period Forecas for sage- period Find beer forecasing model Figure.3: Aciviies of probabiliy disribuion models Model II: Demand Forecass using ARIMA and Bayesian ARIMA Techniques In ime series forecasing models, pas demands can be incorporaed as a variable o find he deerminisic rend of he seasonal demand. This sudy is o predic demand of a seasonal produc using auoregressive inegraed moving average (ARIMA) and Bayesian ARIMA models for he acive demand season. The Bayesian ARIMA model is used here o forecas demand from a daa series ha conains missing values. The forecass compued by hese models are hen compared wih acual demand daa. The aciviies of hese models are demonsraed in Figure.4. A muliplicaive exponenial smoohing model is used as he base 5

16 reference o compare he forecas derived by he probabiliy disribuion models and ime series (ARIMA and Bayesian ARIMA) models. Inpu daa ARIMA model Classical ARIMA Bayesian ARIMA ARIMA Parameers (by max Likelihood mehod) ARIMA parameers (by Bayesian mehod) Forecas for sage- period Forecas for sage- period Find model provides beer forecas Figure.4: Aciviies of ime series forecasing models Model III: Invenory Models Applied o Forecass The objecive of his model is o es he bes forecas hrough he applicaion of invenory models by he resuls ha provides minimum invenory cos. Iinvenory coss are calculaed based on he invenory quaniy required for he arge business season using each if he forecas derived by he above models. The oucome of his model is also o illusrae poenial cos savings uilizing he improved demand forecas..5 Soluion Approach Here, demand forecass are performed using probabiliy disribuion model and ARIMA models. Bayesian saisical echniques are used in he forecasing algorihms, where pas 6

17 informaion is compiled and knowledge abou he fuure evens is gahered ino a consisen forma o develop he forecasing models. The prior models are used o make inferences abou he unknown fuure demand. The prior models are updaed o he poserior models based on he mos recen demand observaions as hey become available. Thus, he updaed parameers of he forecasing models improve he precision of he forecas. The forecass derived by he parameers of hese poserior models are hen compared wih he forecas obained by he adapive approaches of exponenial smoohing forecasing echniques. The adapive approach of exponenial smoohing echniques is commonly used by he forecasers. A muliplicaive exponenial smoohing echnique is used o serve as he base reference for he forecasing models. Forecass derived by he above models are verified by comparing acual demand of he produc and he forecas accuracy is esed by he resuls of several forecas measuring indicaors such as percenage errors, mean absolue deviaions, racking signals. Once he demand forecas is achieved, an alernaive measure of he forecas is performed hrough he applicaion of invenory models by deermining he oal invenory cos of he produc for he arge business season. A periodic review and an exended newsvendor model wih an emergency procuremen opion are used o find he invenory quaniy based on he demand forecass. The invenory coss are derived by applying he dynamic opimizaion algorihm, which is hen used as a furher basis o compare he forecasing echniques. The bes forecas is seleced as he one ha produces minimum error and invenory cos..6 Scope and Opporuniies Accurae measures of demand uncerainy can be imporan in some applicaions. The forecasing model sudied in his research can be applied o any sale forecass and invenory managemen in a supply chain sysem. The models are especially applicable o forecas sales of 7

18 seasonal producs such as winer jacke, woolen apparel, air condiioner, and Chrismas gifs. Producs wih shor-life cycles are widespread in indusries. The models can also be applied o forecas demand of producs wih shor-life cycles such as fashion apparel, elecronic producs, mobile phones; new producs (any new model elecronic devices such as CD wriers or DVD burner), or basic consumable producs (gasoline, auomobiles, clohing). Forecasing demand and invenory managemen are common in non-indusrial businesses such as ar exhibiion ickes, or airline ickes prior o any special holydays or spors evens. The proposed models can be applied o predic seasonal demand of such non-indusrial businesses..7 Acual Time Series Daa The daase presened in he sudy was colleced from he US corporae business marices hrough US Census Bureau. The daase represens he parial demand of women woolen apparels in he US, supplied a leading apparel manufacuring counry (India) over a ime period from January 996 o December 005. The monhly demand from January 996 o June 005 is used o find he parameers of he forecasing models. Using he models, forecass are made for he period from July o December 005. For he missing values forecasing models, among he one hundred foureen observaions (January 996 o June 005), he demand for he six periods from July o December 004 were considered unavailable and he forecas are made for he period from July o December 005. Daa series presening demand from July o December 005 are used o validae he forecass obained from he models..8 Organizaion of he Disseraion Apar from he inroducion, he sudy is organized as follows. Chaper provides a descripion of he relevan lieraure of demand forecasing and invenory models. In Chaper 3, probabiliy disribuion models are sudied o forecas he seasonal demand of a produc using 8

19 Bayesian approaches. Chaper 4 obains forecas using ime series forecasing mehod. An auoregressive inegraed moving average (ARIMA) model and hen Bayesian ARIMA models are presened. The performances of ARIMA forecasing models and a muliplicaive exponenial smoohing model are also presened in his chaper. In Chaper 5, he invenory coss are deermined using periodic review and newsvendor invenory policies based on he forecass aained by all forecasing models. The bes forecasing model in erms of minimum invenory coss is esablished in his chaper. Chaper 6 summarizes he observaions and conclusions of his research and possible fuure research. 9

20 CHAPTER LITERATURE REVIEW There are hree opics in he lieraure ha are relaed o demand forecas and invenory managemen of seasonal producs. Firs, he lieraure uses probabiliy disribuion and Bayesian approaches o forecas demand - he demand considered here is sochasic, and characerized by he seasonaliy. The second uses he ime series models o forecas demand for he fuure period. The forecass are derived by he esimaed parameers of he model. In he hird, he invenory models are used o deermine he order quaniy and oal invenory cos of he seasonal producs prior o an acive selling period. Invenory cos may demonsrae poenial cos savings due o improved forecas. Following is he lieraure review of he above direcions.. Demand Forecass Using Bayesian Procedure In lieraure, differen aspecs of demand forecasing problems wih unknown demand disribuions and informaion updaes have been sudied. For seasonal demand forecasing, saring from he 990s, a Quick Response (QR) policy was adoped by many researchers. This policy is inended o reduce manufacurers producion ime o respond o reailers order in a quicker way so ha forecas can be improved by collecing more informaion abou he fuure demand. Hammond (990) and Fisher e. al. (994) sudied he QR policy wih ski apparel (ski suis, ski pans, parkas, ec), and showed ha forecas accuracy can be subsanially improved by adoping QR policy. Fisher and Raman (996) developed a forecasing model based on he sale rend using he early sage marke sales daa o reduce he uncerainy of he fuure demand under QR ordering sysem. Iyer and Bergen (997) sudied demand forecas by collecing he demand informaion of a preseason produc o forecas he acual demand of a seasonal produc 0

21 using Bayesian approaches. They proposed ha he demand process of he fashion apparel follows normal disribuion and presened he improvemen of demand forecas due o Bayesian informaion updae in forecasing process. Agrawal and Smih (996) used negaive binomial disribuion (NBD) for he demand model and suggesed ha NBD model provides a beer fi han he normal or Poisson disribued daa. They developed a parameer esimaion mehod for he demand model in which sales are runcaed a a fixed poin. Cachon (000) used he negaive binomial disribuion model o analyze he demand of he fashion goods where i is assumed ha he demand process follows he Poisson disribuion and demand rae varies according o a gamma disribued model. Gallego and Ozer (00) discussed he improvemen of demand forecas using early demand daa for a regular selling season. Lau and Lau (997), Gurnani and Tang (999), Choi e. al. (003) and Choi and Yan (006) all sudied wo-sage demand of a fashion produc under Bayesian approaches. Gurnani and Tang s (999) considered a siuaion in which a reailer can pursue wo orders prior o a arge selling season. In heir model, he forecas was updaed by uilizing marke informaion beween he firs and second orders. Choi e. al. (003) presened a wo-sage newsvendor model including Bayesian demand informaion updaing approach. Their work exended Gurnani and Tang s (999) model by including a cos componen during he second ordering opion. Choi and Yan (006) invesigaed QR policy wih wo Bayesian models considering ha he demand process follows normal disribuions. Their firs model considered he normal disribuion wih an unknown mean and a known variance, while, in he second model boh an unknown mean and an unknown variance were assumed. The forecass are hen are compared for boh models. In ha sudy, he proposed forecasing model is similar o Iyer and Bergen (997) and Choi e. al. (006) bu differen in he following ways: (a) unlike quick response policy, informaion

22 abou prior sales was no colleced from he demand of a pre-seasonal produc; (b) due o limied producion capaciy, i may be difficul for manufacurers o apply QR policy o reduce producion lead imes. A disinc beginning and ending of daa collecion and demand forecas period (sage- and sage-) are considered; (c) insead of assuming normal demand process and normal prior models, he proposed model uses non-negaive probabiliy disribuions o model he demand process.. Time Series Auoregressive Models Time series forecasing models are increasingly applied o forecas demand and shor-life produc demand. Under an auoregressive moving average (ARMA) assumpion, Kurawarwala and Masuo (998) esimaed he seasonal variaion of PC producs demand using demand hisory of pre-season producs and validaed he models by checking he forecas performance wih respec o acual demand. Miller and Williams (003) incorporaed seasonal facors in heir model o improve forecasing accuracy while seasonal facors are esimaed from muliplicaive model. Hyndman (004) exended Miller and Williams (003) work by applying various relaionships beween rend and seasonaliy under seasonal auoregressive inegraed moving average (ARIMA) procedure. Forecas from eigh differen combinaion of rend and seasonaliy were compared in he model. The classical approach ARIMA becomes prohibiive, and in many cases i is impossible o deermine a model, when seasonal adjusmen order is high or seasonal adjusmen diagnosics fails o indicae ha ime series is sufficienly saionary afer seasonal adjusmen. In such siuaions, he saic parameers of he classical ARIMA model are considered he main resricion o forecasing high variable seasonal demand. Anoher resricion of he classical ARIMA approach is ha i requires a large number of observaions o deermine he bes fi model for a daa series.

23 In he ARIMA model, if he Bayesian approaches are used, he resricion of he saic values of he parameers is relieved by imposing he probabiliy disribuions o represen he parameers. Alhough he pracices of Bayesian ARIMA models for seasonal forecas are more appropriae, he lieraure on Bayesian mehods applied o ARMA ime series is limied. Mos of he applicaions are resriced o simple models such as auoregressive (AR) processes or forecas demand for a single or wo fuure periods. In recen sudies, de Alba (993) derived an auoregressive model under Bayesian approach o forecas he quarerly GNP of Mexico and he quarerly unemploymen rae for he Unied Saes. Huera and Wes (999) sudied auoregressive models where Markov chain Mone Carlo (MCMC) process is used o forecas from AR processes. McCoy and Sephens (004) exended Huera and Wes s work (999) and proposed ARMA models in which a frequency domain approach is adoped o idenify he periodic behavior of ime series. In ha sudy, firs a classical ARIMA model is developed for a single daase, and he Bayesian mehod is applied o he seleced ARIMA model wih he purpose of forecasing demand from he daase ha conains missing values. In he proposed model, he Bayesian ARIMA is sudied o forecas seasonal demand when here are missing values in he daa series. The Mone Carlo inegraion mehod based on Gibbs sampling algorihm is used for numerical compuaion o derive he model parameers. In he proposed model boh ARIMA and Bayesian ARIMA models are used o forecas demand for an upcoming season..3 Invenory Models In several aricles, Liau and Lau (997), Eppen and Iyer (997), Choi e al. (003, 006), invenory models were sudied o deermine he order quaniy for a lead ime and invenory cos of he seasonal demand. Liao and Shyu (99) firs inroduced he concep of crushing cos 3

24 o variable lead ime for a fixed order quaniy, where crushing cos is he cos ha increases if he procuremen lead ime is reduced. Ben-Daya and Raouf (994) exended Liao and Shyu s (99) work by reaing boh order quaniy and lead ime as he decision variables. The invenory problem involving second ordering opporuniy was sudied by Khouja (996). In his model, he order quaniy is deermined for a single period model wih an emergency supply opion, where he found ha he oal quaniy under emergency supply opion is smaller han ha of he newsvendor model. Liau and Lau (997) sudied he reordering sraegies for a seasonal produc under a newsvendor model where a cusomer receives an order a he beginning of he season and places an addiional order a some poin during he season. They idenified analyical condiions o maximize profis for using he second ordering opporuniy. Eppen and Iyer (997) described an invenory problem of he fashion indusry. They deermined he iniial invenory quaniy for a season and adjused he procuremen quaniy afer informaion updaes using Bayesian echniques. Gurnani and Tang (999), and Choi e al. (003) invesigaed he opimal invenory quaniy for seasonal producs in which a reailer can order wice and he ordering cos a he second ime is a variable. Choi e al. (004) and Tang e. al. (004) sudied muli-sage invenory decisions using he Bayesian process o updae demand informaion in he successive sage. One of he key issues in hese invesigaions is o find he opimal invenory quaniy based on a newsvendor model wih wo supply opions. However, he newsvendor model wih wo supply opion may be exended by including wo addiional cos facors: (i) cusomer waiing ime cos and (ii) expedie shipping cos. In he proposed model, he invenory quaniy is deermined by using an exended newsvendor model along wih a periodic review invenory model based on several forecas daases. The invenory cos of each forecas under each invenory model is used o 4

25 demonsrae ha improved forecas resuls in minimum cos. Thus, his sudy differs from he previous models by incorporaing hree objecives: (a) providing order quaniy wih wo supply opions, (b) deriving opimal invenory cos for each forecas and (c) esablishing a basis for comparing demand forecass..4 Limiaions of he Pas Research In mos forecasing problems elegan mahemaical models such as regression analysis, weighed moving average or exponenial smoohing models were developed in which he forecass are performed eiher by exrapolaion or by averaging demand from he pas daa. In hese hisorical daa-driven forecasing models, forecass ofen exhibi he demand rend of he pas periods. Besides, he mahemaical forecasing models do no permi inegraing he subjecive informaion or expers views abou he fuure demand in he forecasing algorihm. They perform badly if he daa series conains mission values. Therefore, forecass derived by pas-daa driven models may lead o a wrong conclusion abou he fuure demand. The demand of seasonal producs varies from season o season, from one business cycle o he nex. In ime series forecasing echniques such as auoregressive models, he parameers of he models are always saic. The saic coefficien of a ime series model canno capure he uncerainy of he fuure demand. The imposiion of saic models implies a fixed relaionship beween he demand of he pas season and he fuure. This may be considered he inflexibiliy of he ime series forecasing models. There exiss a large amoun of lieraure in boh forecasing and invenory models. However, hese wo sreams of research are radiionally separaed. The research in forecasing problems usually ignores he invenory plans, while he research in invenory problems generally presumes ha forecass are given. Very lile work has been 5

26 accomplished on demand forecasing and invenory decision ogeher o deermine he bes forecasing model ha provides minimum invenory cos during an acive demand season.5 Overcoming he Limiaions The forecas of seasonal demand is essenial for invenory planning prior o an acive selling season. In demand forecasing, a single model may no be adequae o represen a paricular demand series for all imes. Furher, he chosen model may have been resriced o a cerain class of ime series. Therefore, a number of forecasing models are sudied o provide wider choices o find he bes demand forecas of a seasonal produc. In he firs forecasing model, forecas by exrapolaion is avoided by using a non-negaive probabiliy disribuion o represen he seasonal demand. The Bayesian approach is applied o updae he parameers of he forecasing model. Thus, he lieraure on forecasing models is exended by using probably disribuion model involving Bayesian echniques. An ARIMA forecasing model is developed as he second model o forecas he seasonal demand. The parameers of he ARIMA model are saic, bu he saic parameers can be enhanced by using he Bayesian echniques. In his sudy, he ARIMA model is exended o Bayesian ARIMA o capure he uncerainy of fuure demand. The use of Bayesian mehods in boh models provided addiional faciliies such as he capaciy o use pre-designed models, communicaing subjecive or prior informaion, forecasing using lile daa or he daa series ha conains missing values. In invenory managemen lieraure, emergency procuremen opion is no always included in procuremen sraegy and invenory cos is no considered as a basis o find he bes demand forecas. In he hird model, a newsvendor model wih emergency procuremen opion is used o deermine he opimal invenory quaniy and cos using several demand forecass where he bes one is chosen by he forecas ha produces minimum invenory cos. 6

27 CHAPTER 3 BAYESIAN FORECASTING MODEL FOR SEASONAL DEMAND Seasonal demand varies grealy during demand seasons. In his chaper he focus is o predic demand of a seasonal produc for an acive demand season using he Bayesian procedure. In he forecasing model, he demand process is described by he probabiliy disribuion model where he sales records of he pas seasons are incorporaed in he forecasing algorihm. Firs, he iniial demand for he arge selling period is esimaed, and he iniial demand is hen updaed using he Bayesian approach. In Bayesian analysis, demand process is viewed in erms of parameers of a probabiliy disribuion and forecas are obained using updaed parameers. Acual demand daa is used in his forecasing model. The daase used in he model is colleced from US census bureau and is he parial demand of women woolen s apparel supplied by an apparel manufacurer counry (India). I is shown in Appendix A. (Table A.). A graphical presenaion of he demand daa is shown in Figure 3.. Figure 3.: Variaion of demand daa for apparel produc (Sources: U.S. Deparmen of Commerce, Office of Texiles and Apparel) 7

28 The daa series presened in Figure 3. includes wo sources of demand variaions, (a) variaion beween he periods wihin a business cycle, (b) variaion beween he business cycles. Due o he higher variabiliy, he demand from he monhs of January o June is considered as slow demand period, (sage-), and demand from he monh of July o December, as he busy periods (sage-). The focus is o forecas demand for he sage-. In many forecasing models, he demand process is described by he normal disribuion, bu he normal disribuion may conain negaive values. As demand quaniy is always a posiive number, i is more pracical o use a non-negaive disribuion. In his sudy, a gamma disribuion is chosen o represen he demand process of seasonal produc. Comparing he maximum likelihood esimaes among a number of non-negaive disribuions under he same parameerized condiion, i is found ha gamma disribuion is he favored model for he seleced daa. The maximum likelihood esimae of he probabiliy disribuions is presened in Appendix A.. A key feaure of he Bayesian analysis is he use of he conjugae prior and poserior disribuion for he exponenial family parameers. A conjugae prior is mahemaically convenien o follow a known poserior disribuion as i belongs o same parameric family. An inverse gamma disribuion is seleced as he conjugae prior for he gamma disribuion (Gelman e. al., 004). Following noaions are used in his model: Y Demand a period, (unis/monh) δ Observed demand rae a sage- (January o June a 005), µ, σ Mean, and sandard deviaion of he demand disribuion model α, β Shape and scale parameer for he prior inverse gamma disribuion model A, B Shape and scale parameer for he poserior inverse gamma disribuion model 8

29 3. Demand Model Formulaion Produc demand is a coninuous process. Y is direcly dependen on ime period, where 0. The shape parameer of he gamma densiy is assumed linear in ime as α (). The gamma densiy wih shape parameer α () > 0 and scale parameer β > 0 is given by α Y ( y) = G( y α, β ) = ( y β ) exp( y β ) f Γ( α ) β (3.) where E( Y ) = (α ) β and Var ( Y ) = ( α ) β, (see Appendix A.3). I is assumed ha he expeced value µ and sandard deviaion σ of he demand model is linear in ime, (Kallen and van Noorwijk, 005). Thus, E ( ) = µ and Var ( Y ) = σ. Using Y he coefficien of variaion, v = σ µ, he parameers of he demand model are given by µ α = =, and (3.) σ v σ β = = µv. (3.3) µ Using Equaion (3.) and (3.3), replacing shape parameer α by /v and scale parameer β by µv in Equaion (3.), he gamma densiy is given by ( ) f y µ = G y v, µ v = Γ( ( v ) y y exp. (3.4) v )( µ v ) µ v µ v If he coefficien of variaion v is remained fixed, he only unknown variable remaining in Equaion (3.5) is he parameer µ. According o Bayesian analysis, a disribuion model is assigned for µ o capure he uncerainy of he fuure demand. An inverse gamma disribuion, which is he conjugae family of he gamma disribuion, is considered as he prior model. The definiion of inverse gamma densiy (IG), ( µ ) ( µ α, β ) f 0 = IG is given by 9

30 α + α β β f 0( µ ) = IG( µ α, β ) = exp. (3.5) Γ( α ) µ µ where µ is a posiive random variable. I follows ha µ ~ G ( α, β ) wih shape parameer α > 0 and scale parameer β > 0. The poserior densiy of parameer µ is described in he nex secion. 3. Bayesian Procedure in Demand Model The observed demand variable is y, and he prior disribuion of he parameer µ is f ( ). According o Bayes heorem, he poserior densiy of parameer µ is given by f ( µ, y) f( µ y) = (3.6) f ( y) The join probabiliy f(µ, y) can be expressed by condiioning on µ as f ( µ, y) = f ( y µ ) f0 ( µ ). (3.7a) The marginal densiy funcion of y is given by f ( y) = f ( y µ ) f µ dµ 0 0 ( ). (3.7b) Subsiuing values from Equaions (3.6b) and (3.6c) ino Equaion (3.6a) gives f ( y µ ) f0( µ ) f ( µ y) =. (3.7c) f ( y µ ) f ( µ ) dµ µ The seps o solve Equaion (3.7c) are described in Proposiion 3.. Proposiion 3.: The poserior densiy f ( µ ) may be wrien as y y f ( µ y) = IG µ + α, + β. v v Proof: Proposiion (3.) may be proved by following Equaions (3.7a o 3.7c) in hree seps. The chronology of evens o achieve he poserior densiy of parameer µ is described as he following: 0

31 Sep : Derivaion of join probabiliy disribuion: The join probabiliy f ( y µ ). f0 ( µ ) is expressed condiioning on µ, where f ( y µ ) and f ( ) may be found in Equaions (3.4) and (3.5), respecively. 0 µ f ( y µ ) f0 ( µ ) = Γ µ v v ( ) y µ v ( v ) y exp µ v α β Γ( α) µ + α β exp µ y = v ( v ) Equaion (3.8a) may be simplified as Γ α β Γ( α) µ ( v ) ( v + α ) + y exp µ v + β (3.8a) C f ( y µ ) f0 ( µ ) = exp + B A, (3.8b) µ µ where C = Γ( ( v ) β α y y, A = + α, and B = + β. (3.9) v ) Γ( α) v v v Sep : Derivaion of marginal densiy funcion: The marginal densiy of y is obained by inegraing over µ and using Equaion (3.8b): 0 Changing B µ f ( y µ ) f µ dµ 0 ( ) w = and ( B w ) dw = dµ B = Cµ A exp dµ. (3.0) 0 µ, hen Equaion (3.0) ransforms o 0 A+ w B f ( y µ ) f0 ( µ ) dµ C exp( w) B w 0 = C B A 0 w A exp ( ) w dw dw ( A) = C Γ (3.) A B 0 where w A ( w) dw = Γ( A) exp is he gamma funcion.

32 Sep 3: Derivaion of he poserior densiy using Bayes heorem: Subsiuing he values from Equaions (3.8b) and (3.) ino Equaion (3.7c), he poserior disribuion is given by f ( µ y) = 0 f ( y µ ) f0( µ ) = f ( y µ ) f ( µ ) dµ µ 0 B A Equaion (3.) is an inverse gamma funcion wih parameer A and B, ( A, B) B exp. (3.) µ Γ( A) µ f ( µ A, B) = IG. (3.3) µ Subsiuing values of he poserior parameers A and B from Equaion (3.9) ino Equaion (3.3) yields y f ( µ y) = IG µ α +, β +, (3.4) v v where α and β are he prior parameers, y is demand for period, and v is coefficien of variaion for he sage- period. 3.3 Applicaion of Demand Model The esimae of fuure demand for sage- period in year 005 can be deermined from he he poserior disribuion. The Equaion (3.4) is he poserior inverse gamma densiy wih shape parameer A > 0 and scale parameer B > 0. In Equaion (3.4), he componen, y j, v are known values, which can be obained from pas demand records, bu he parameer values of he prior disribuion α and β are unknown. The values of α and β may be derived hrough he applicaion of coefficien of variaion (v), and he iniial mean demand of each forecas period. The coefficien of variaion v = σ µ, where he poin esimae for µ is y n and an unbiased esimaor of σ is S n for n daa series. For inverse gamma prior disribuion, mean is β ( α ), and variance is β ( α ) ( α ).

33 The coefficien of variaion (v) is given by, β β v = =. (3.5a) ( α ) ( α ) ( α ) ( α ) Afer rearrangemen, Equaion (3.5a) becomes α = v +. (3.5b) Once parameer α is known, parameer β can be esimaed from mean, y = β ( α ) as, β = y ( α ). (3.5c) n From Equaion (3.9), he parameers of he poserior disribuion are as follows A = + α, (3.6a) v y B = + β. (3.6b) v 3.4 Sub-Models In business, here are many insances where marke demand records conain missing values due o naural caasrophe such as hurricane or adverse economical condiions. To demonsrae a forecasing problem wih incomplee daa, a sub-model is presened wih missing value assumpion. The original model may be viewed in wo sub-models: (a) Bayesian probabiliy model (B-P Model), and (b) Bayesian probabiliy model wih incomplee daa (BP-I Model). Boh models are used o forecas demand for sage- (July o December) in 005 using he daa series from January 996 o June 005. The assumpion in BP-I model is ha he demand a sage- (July o December) in 004 is no recorded. Therefore, forecas in BPI model is performed from he daa series ha conains six missing values. To projec he missing values and he iniial demand forecas, an approach is described in he following algorihm. n 3

34 3.4. Algorihm 3. (Seps o Derive he Prior Values) (i) Spli he business cycle ino wo sages, (January o June, as sage-, and July o December, as sage-). (ii) Calculae he average demand for sage- period, and find he demand raio, r j, for sage- period wih respec o average of sage-, ha is, r = j y j (iii) Find he average demand raios for each period a sage-, R = n rj = n, which may be called demand facor. (iv) Projec missing values using demand facor R. (v) Compue iniial esimae of sage- demand (before Bayesian updae) of he forecas {(n+)h} year by muliplying n+ wih he demand facor, R j as ( n+ R ). The purpose in following seps (i) hrough (v) is wo fold: (i) projec values for missing daa, and (ii) projec iniial demand a sage- period of forecas year and his iniial esimae is updaed in Bayesian echniques. The srucure of Algorihm 3. is shown in Table 3.. Table 3.: The srucure of finding expeced prior values Obs. Average Sage- Demand raio, r = j y j ( =,.., n); (j = 7, 8,..., ) ( ) Jul Aug - - Nov Dec r 7, R 8, - - r, r, r 7, r 8, - - r, r, n n r 7,n r 8,n - - r,n r,n R = n j rj = n R 7 R R R Expeced demand for forecas year 005 n+ (n+) R n+ R 7 n+ R n+ R n+ R The flow diagram for he demand esimaion process is shown in Figure 3. 4

35 Sar Inpu daa, y Selec probabiliy densiy for daa Transform parameers α, β ino µ & CV (v) Find conjugae prior disribuion f 0 (µ ) Use Bayes rule f (µ y ) likelihood f(y µ ) Prior f 0 (µ ) Calculae CV (v) from pas daa Inpu y, v, Compue µ 7, µ 8,, µ using Equaion (3.4) Sop Figure 3.: Flow diagram for Bayesian compuaion 3.4. Bayesian Probabiliy Model (B-P Model) Following is he srucure illusraed in Table 3. and Algorihm 3., he iniial projeced demand for he forecas year (before Bayesian updae) is shown in Table 3.. Obs. Table 3.: Projeced demand averaging he sample daa (unis in million) Year Average Sage- Demand raio a sage- periods ( =,.., n); (j = 7, 8,..., ) ( ) Jul Aug Sep Oc Nov Dec (n-) n R = n j rj = n Expeced demand for forecas year by n+ R (n+) n+(=005)

36 The parameric values of he prior and poserior disribuion for each period a sage-, under B-P model are shown in Table 3.3. Table 3.3: Prior and poserior parameers derived by B-P model for 005 Prior parameers (unis in million) Monh Mean y CV(v) α β Poserior parameers (unis in million) Esimaed ŷ A B Mean Jul Aug Sep Oc Nov Dec Sample Calculaion (B-P Model) For he monh of July, he mean, y July =.96, and coefficien of variaion, v = 0.6. Using Equaion (3.5b), (3.5c), he value of (α, β) is given by + α Jul = + = (3.7a) v 0.6 = Jul β Jul = y Jul ( α ) =.96(4.34 ) = 79.9 (million) (3.7b) For he monh of July, he iniial esimae, ŷ July he parameers of he poserior disribuion are deermined as: = 3.4. From Equaions (3.6a) and (3.6b), A B Jul Jul + α = = (3.8a) vjul 0.6 = Jul yˆ Jul = + β Jul vjul 3.4 = = (million). (3.8b) 0.6 The demand for he monh of July in year 005 is esimaed by he parameers (A and B) of poserior disribuion model. The mean of inverse gamma disribuion is given by B ( A ). 6

37 By using he values A jul and B Jul from Equaions (3.8a) and (3.8b), he mean demand for he monh of July is esimaed as 06.5 (80.67 ) =. 59 (million) Bayesian Probabiliy Model wih Incomplee Daa (BP-I Model) Missing daa ofen arise in various seings includes marke sales, indusrial producion, shipmen arrival, new produc rials. The forecas based on missing values can ofen resul in biased and inefficien esimaes. In BP-I model, he projecions of missing values for sage- period in year 004 and he iniial demand esimae for sage- period in year 005 are obained by following Algorihm 3.. The projeced missing values and he iniial demand for he forecas year (before Bayesian updae) are shown in Table 3.4. Table 3.4: Projeced demand averaging he sample daa (unis in million) Obs. Year Average Sage- Sage- y (=7, 8,...,) ( j ) Jul Aug Sep Oc Nov Dec (n-) n ) j j R = r ( n = (n) (004) (n+) (005) Projeced Missing values Acual demand Percenage error Iniial esimae Expeced demand for n-h year by n R % -0.7% 0.07% 0.6% 0.% 0.9% The mean demand, coefficien of variaion, parameers of he prior model and he parameers of he poserior model for each period a sage- in 005 under BP-I model are shown in Table

38 Table 3.5: Prior and poserior parameers derived by BP-I model for 005 Prior parameers (unis in million) Poserior parameers (unis in million) mean CV(v) α β Iniial esimae A B Mean Jul Aug Sep Oc Nov Dec Calculaion procedure o obain he values presened in Table 3.6 is similar o he sample calculaion illusraed in sample calculaion under Secion The graphical presenaion of prior and poserior densiy for boh B-P and BP-I models are shown in Figures (A. and A.) in Appendix A. The resuls and validaion of models are presened in he nex secion. 3.5 Forecasing Errors and Model Validiy In he forecasing procedure, a porion of he daase is used o esimae he parameers of he model; he forecass are hen esed on daa o validae he model. In he analysis, daa poins for he 7 years ( ) are used o produce he forecass for he 8h years (a sage- from July o December, 005). To validae he forecasing models, he forecass are compared wih he original demand for he arge forecas periods. The performance of forecasing models can be achieved by a number of error measure indicaors such as relaive (percenage) errors (PE ), mean absolue deviaion (MAD ) and racking signal (TS ), where index corresponds o a paricular period (monh). Tracking signal is he cumulaive forecas error (running sum) wih respec o MAD for a given ime period. I measures he limi of MAD above or below he acual daa. A comparison beween B-P model and BP-I model wih respec o he percenage error, mean absolue deviaion, and acking signal are shown in Table