NATIONAL UNIVERSITY OF SCIENCE ANDTECHNOLOGY FACULTY OF APPLIED SCIENCESDEPARTMENT OF OPERATIONS RESEARCH AND STATISTICS AN OPTIMAL INVENTORY POLICY FOR PERISHABLEPRODUCTSbyNOBUHLE MUTOMBENI (N01414834Y)SUPERVISOR: MR. H . NAREThis dissertation was submitted to the Department of Operations Research and Statistics of the National University of Science and Technology in partial fulllment of the requirements for the Bachelor of Honors Degree in Operations Research and Statistics , Bulawayo, ZimbabweMAY 2018DeclarationI, Nobuhle Mutombeni , declare that the project which is hereby submitted for the qualica-tion of Bachelor of Science in Operations Research and Statistics at the National Universityof Science and Technology, is my own independent work and has not been handed in beforefor a qualication at/in another University/Faculty/School.

I further declare that all sourcescited or quoted are indicated and acknowledged by means of a comprehensive list of refer-ences. I further cede copyright of the dissertation to the National University of Science andTechnology.Signature….

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….Date: May 2018Copyright c 2018 National University of Science and TechnologyAll rights reservediAbstractThis study compares the alternative time series models that were used to demand.

Two fore-casting models were tted which are the Seasonal Auto Regressive Integrated Moving Aver-age (SARIMA) and the Holt Winter’s or Triple Exponential Method.These models were ttedto the top selling product of Bakers Inn turnover product which is Premium Bread.Dailydemand data was used for the period of January to December 2017.

The performancesof the two models is evaluated using the forecast error methods which are Mean AbsolutePercent Error (MAPE), Root Mean Square Error (RMSE) and the Mean Absolute Deviation(MAD).The study shows that the Holt Winters Method produces better forecasting resultsthan the SARIMA Method.iiDedicationTo Mum, Dad and Lesley T. Love you totally.iiiAcknowledgmentsFirstly I would like to thank the Lord Almighty for all the wisdom and understanding inwriting this project.I would like to convey my sincere appreciation to my supervisor Mr Narefor all his support throughout the project and l will be forever indebted to him for this.

I wouldalso like to thank my dad (Mr J.Mutombeni),my mum (Mrs Mutombeni) and my brothers andsisters for making this project a success.I would like to thank them for the support ,love,motivation and kindness ;words only may not express how l feel but in him there is nodarkness.Also l would like to thank my department of Statistics and Operations Research forallowing me to carry this project .

My fourth acknowledments goes to all my friends for theirsupport,love and motivation.Lastly l would like to thank Innscor Harare for allowing me tocarry this research as a case study to their company especially T.Masundlwane for providingthe data used in the study.God Bless you allivContentsDeclaration iAbstract iDedication iiAcknowledgments iiiTable of Contents vList of Figures viiiList of Tables x1 Introduction 11.1 Introduction . .

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. . . . . . . . . . . . . . . . .31.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.6 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.7 Signicance of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.8 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.9 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.10 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 Literature review 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6v2.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2.1 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.3 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.4 ARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.5 SARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.6 Holt Winters Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132.7 Accuracy measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153 Methodology 163.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.2 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.2.1 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173.2.2 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173.2.3 Steps to create a Pareto Chart . . . . . . . . . . . . . . . . . . . . . . . . .173.3 Box Jenkins Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.4 Components and Fitting of ARIMA model . . . . . . . . . . . . . . . . . . . . . .19 3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193.4.2 Identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193.4.3 Estimation and Diagnostic checks . . . . . . . . . . . . . . . . . . . . . .193.4.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203.5 SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 3.5.1 Assumptions of SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . .223.5.2 Stationarity Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233.5.3 Model identication and estimation . . . . . . . . . . . . . . . . . . . . .233.6 Model tting and Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 3.6.1 Autocorrelation assumption . . . . . . . . . . . . . . . . . . . . . . . . . .243.6.2 Normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243.6.3 Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253.7 Goodness of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253.8 Evaluation of forecasting performance . . . . . . . . . . . . . . . . . . . . . . . .253.8.1 Forecast error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263.8.2 Mean Absolute Percentage Error(MAPE) . . . . . . . . . . . . . . . . . .263.8.3 Root Mean Square Error (RMSE) . . . . . . . . . . . . . . . . . . . . . . .26vi3.8.4 Mean Absolute Deviation (MAD) . . . . . . . . . . . . . . . . . . . . . . .273.8.5 Mean Forecast Error (MFE) . . . . . . . . . . . . . . . . . . . . . . . . . .273.9 Holt Winters Method nTriple Exponential Smoothing . . . . . . . . . . . . . . .274 Data Analysis 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14.2 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14.3 SARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.3.1 Model identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.3.2 Stationery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.3.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54.3.4 Model Fitting and Diagonistic . . . . . . . . . . . . . . . . . . . . . . . . .64.3.5 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84.3.6 Forecasting Perfomance . . . . . . . . . . . . . . . . . . . . . . . . . . . .94.4 Holt Winters Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94.5 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114.5.1 Run’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114.5.2 ACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.5.3 Histogram of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.6 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.6.1 Evaluating Forecasting perfomance . . . . . . . . . . . . . . . . . . . . .144.7 Comparison of the Holt Winters and the SARIMA . . . . . . . . . . . . . . . . .144.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 Conclusion and Recommendations 16 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 5.1.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165.1.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175.1.3 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .185.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Appendix 21viiList of Figures4.1 Pareto chart for the products . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.2 Time Series Plot of sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.3 Trend Analysis Plot of Actual Sales Data . . . . . . . . . . . . . . . . . . . . . .34.4 Autocorrelation of Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34.5 PACF of Actual Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34.6 Unit Root Test for the Differenced Sales . . . . . . . . . . . . . . . . . . . . . . .44.7 Time Series Plot of Differenced Sales . . . . . . . . . . . . . . . . . . . . . . . . .44.8 Trend Analysis for Transformed Sales Data . . . . . . . . . . . . . . . . . . . . .54.9 Autocorrelation for Differenced Sales Data . . . . . . . . . . . . . . . . . . . . .54.10 Final Estimates of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .54.11 ACF for Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64.12 PACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64.13 Modied Box-Pierce (Ljung-Box) Chi-Square Results . . . . . . . . . . . . . . .74.14 Durbin Watson Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74.15 Histogram of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74.16 Jarque Bera Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84.17 Residual vs Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84.18 Accuracy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94.19 Winters Method Additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94.20 Winters Method For Multiplicative . . . . . . . . . . . . . . . . . . . . . . . . . .104.21 Holt Winters Plot for Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104.22 Runs Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.23 ACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.24 Histogram of Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134.25 Accuracy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134.26 Forecasting Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14viii5.1 Pareto Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23ixList of Tables3.1 Behaviour of ACF and PACF of Pure Seasonal ARIMA models . . . . . . . . . .224.1 SARIMA Forecasting Perfomances . . . . . . . . . . . . . . . . . . . . . . . . . .94.2 Holt Winters forecast parameters and errors . . . . . . . . . . . . . . . . . . . .114.3 Holt Winters forecasting Evaluation . . . . . . . . . . . . . . . . . . . . . . . . .144.4 Holt Winters and SARIMA Forecasting Perfomances comparisons . . . . . . .14xChapter 1Introduction 1.1 IntroductionInnscor (Bakers Inn) is one of the company that is trying to keep up with its competitors bynding alternative ways of minimizing cost hence increasing their prot. Bakers Inn sellsperishables and by that it needs a very strong ordering model so that it can withstand today’senvironment. Perishable refers to the items that have an expiration date and such food willgo bad if not eaten or sold in a certain amount of time. The items will be disposed as wastes.Because no company wants its inventory to lose value, business use inventory managementsystems to keep track of the inventory and thereby minimizing wastes. Perishables demandsattention. Much of operations is about being able to match supply and demand. Since thecompany uses average rolling modelling that is they use previous sales to place their ordersand which is more of a deterministic demand model, the researcher will also incorporate thestochastic demand which is modelled as arbitrary probability distribution. Bakers Inn islosing money from overstocking and under stocking perishable products because they do nothave an inventory control that is in intact. The company also uses the First In First Out(FIFO) inventory tracking systems to rotate goods so new arrivals don’t get sold rst beforeolder items with expiring dates that are near but its not consistent. Bakers Inn is nding foralternative methods to solve their forecasting problems that will result in an increase of theirtotal turnovers.1Introduction1.2 Background of studyMuch of the study will be based on Innscor Africa under Fast Foods division (Northern Re-gion) where the researcher was attached. Innscor Fast Foods was the rst establishment ofthe group in 1987, rst shop to open was Chicken Inn situated in the Harare CBD alongSpeke Avenue with a product line of fried chips, chickens and hamburgers, and was followedby Bakers Inn with pies. Over the years due to excellent service delivery and improvedmenu offerings the organization has continued to expand launching more brands and open-ing counters in most strategic areas in Zimbabwe. On top of its local brands the organizationhas diversied into master franchises from South Africa, Nandos and Steers. Innscor Fastfoods have continued to be a powerful generator of free cash, and this has allowed for exten-sive capital investment into new and more efcient technology. In the Innscor Africa groupthe fast foods division boasts of 6451 employees which are 41% of the total employees inthe group. The organization has other partners who they have also joined with, these com-plexes are known as Statutory Shops and the company only incorporates its share of prot inthe books of accounts. Bakers Inn offers customers on the go a wide range of freshly bakedbread, rolls, confectionery and pies. Bakers Inn opened its rst retail outlet in Harare, Zim-babwe, and later expanded a highly successful footprint across Africa, more specically inZimbabwe, Kenya and Zambia. The brand has consistently updated its offering to suit thechanging needs of its customers, and now offers a selection of bread including white, brown,whole-wheat, seed, and low GI. Irresistible treats on offer include chocolate and cream dough-nuts, cakes, mufns, as well as various meat pies, buns and rolls. Inventory management isa widely used concept in most companies here in Zimbabwe but it is not effectively used inour fast foods industry. So the purpose of the study is to apply inventory management. So theeconomic order quantity can be used in determining how much to order to reduce and min-imize stock outs , if the authorities order too much or too less it will result in the companymaking loses from waste and increased holding cost.//1.3 Statement of the ProblemThe current forecasting model at Innscor is a challenge resulting to product stock out, morewaste and loss of turnover. The forecasting method used is the one whereby one uses previ-ous historical demand and calculates the average for the next forecasting period (rolling av-erage method). This method does not take variability into account due to historical demand2Introductionwhich can give inaccurate forecasting results. The company is losing money by overstock-ing products that are perishable, which are thrown away as waste which increases costs tothe company .Also the company loses money and customers through understocking productsthat means placing insufcient orders for the day. To reduce these problems alternative waysand forecasting methods are needed to reduce the companies stock outs and run outs. Thesolution of this question lies in the degree of accuracy one is able to forecast demand with.1.4 Aim of the studyThe aim of the study is to determine and analyse Bakers Inn top selling products from fore-casted demand and sales at different time intervals using Holt Winter and SARIMA model.1.5 Objectives Fit the Holt Winters model and the SARIMA model into daily sales data. Determine the most accurate forecasting model by means of comparing the Root meansquare error and mean absolute percentage error for the product under study. Forecast demand and sales for the product under study.1.6 Research QuestionsA research question determines the methodology, and guides all stages of inquiry, analysis,and reporting. It begins with a research problem, an issue someone would like to know moreabout or a situation that needs to be changed or addressed. The following have been found tobe the research questions for the research projectWhat quantities should be ordered so as to reduce waste, overstocking, holding cost andordering cost for the products. Which is the best forecasting method between Holt Winters and SARIMA that can beused to forecast demand in the fast foods industry.3Introduction1.7 Signicance of StudyThe purpose of this research is to make a comparison between the Holt Winter’s model andthe SARIMA model. The mathematical model chosen is supposed to be a useful tool used bythe company for planning and control of their perishable products so as to reduce waste andminimize stock outs so as to increase prots. The emphasis is on how much to order so as tobalance between sales and demand and enables the shop managers to create more realisticordering schedules.This will also help the company to prepare their budgets more efciently.1.8 LimitationsLimitations are inuences that researchers cannot control. They can be dened as the short-comings, lack of capacity, conditions or inuences that cannot be controlled by researchersthat place restrictions on methodology and conclusions. The following are the limitations forthis study.Time constraints: due to lack of resources (nancial and technological), the researcherfound the research period very short. The model is a one period model only considering current sales.1.9 DelimitationsDelimitations describes the choices or boundaries that have been set for the study beingcarried out.Data to be used is to be picked from a sample which is going to represent brand.Thoughthe reseacher would have loved to cover all Bakers Inn shops in Zimbabwe, the studyonly focuses on Harare Mashonaland Province. For external validity purposes the sample size can be large.1.10 AssumptionsAssumptions in a study are things that are somewhat out of our control, but if they disap-pear the study would become irrelevant. Assumptions are so basic that, without them, the4Introductionresearch problem itself could not exist,biblitex. For this study to be relevant we are assumingthat;Data collected from the company is accurate. Expiry date is the same for all products under study. The tools instruments to be used for collection of data are valid and reliable. Product price remains constant over the period of study.1.11 ConclusionThe research to to investigate the inventory of perishable products considering sales and de-mand is being conducted at Bakers Inn under Northern Region of Zimbabwe. The objectivesof the study were discussed as they are to pave a way in achieving the main aim of this re-search project. The study indicates being of great signicance from the way it was discussed.This chapter ended by stating limitations and delimitations of the study as well as denitionsof terms. In Chapter 2, we will outline the literature review of Holt Winter and the SARIMAmodel, giving a brief discussion of the merits and demerits of these methods. Chapter 3 willgive a description of the mathematical procedures used for forecasting .Chapter 4 gives theanalysis of data and results. Finally Chapter 5 will consist of conclusions.5Chapter 2Literature review 2.1 IntroductionA literature review is an account of what has been published on a topic by accredited scholarsand researchers.(Rowley and Slack, 2004) denes literature review as a summary of a subject eld that sup-ports the identication of specic research questions. A literature review needs to draw onand evaluate a range of different types of sources including academic and professional jour-nal articles, books, and web-based resources. The literature search helps in the identicationand location of relevant documents and other sources. Search engines can be used to searchweb resources and bibliographic databases. Conceptual frameworks can be a useful tool indeveloping an understanding of a subject area. Creating the literature review involves thestages of: scanning, making notes, structuring the literature review, writing the literaturereview, and building a bibliography The researcher will review literature on sales and de-mand forecasting of inventory products which are perishables for a fast food company.In thischapter the researcher will focus more on two major forecasting methods which are SeasonalAutoregressive Intergrated Moving Average(SARIMA) and the Holt Winters Method or theTripple Exponential Smoothing Technique.6Literature review2.2 Data CollectionData are usually collected through qualitative and quantitative methods. Qualitative ap-proaches aim to address the `how’ and `why’ of a program and tend to use unstructured meth-ods of data collection to fully explore the topic. Qualitative questions are open-ended suchas `why do participants enjoy the program?’ and `How does the program help increase selfesteem for participants?’. Qualitative methods include focus groups, group discussions andinterviews. Quantitative approaches on the other hand address the `what’ of the program.They use a systematic standardised approach and employ methods such as surveys and askquestions such as `what activities did the program run?’ and `what skills do staff need to im-plement the program effectively?This is according to a research done by (Hawe et al., 1990)Qualitative approaches are good for further exploring the effects and unintended conse-quences of a program. They are, however, expensive and time consuming to implement.Additionally the ndings cannot be generalized to participants outside of the program andare only indicative of the group involved.Quantitative approaches have the advantage thatthey are cheaper to implement, are standardized so comparisons can be easily made and thesize of the effect can usually be measured. Quantitative approaches however are limitedin their capacity for the investigation and explanation of similarities and unexpected differ-ences.2.2.1 Pareto AnalysisDendere and Masache (2013) did a research applying Pareto analysis as a quality controltool.The purpose of the study was to map a way to survive in a stiff competition market envi-ronment by focusing efforts on products that are best nancial performers in a grocery retailshop. In doing so, Pareto analysis was used to classify the products according to their salesfrequency contribution. The products that exhibit the largest frequency were chosen as thevital few products and 14 out of 46 were identied. In addition to the sales frequency goalwere 3 more priority goals that had to be considered because high sales do not necessarilymean high prots. That is where goal programming approach came in to strike a balanceamongst the prioritised goals. Finally the number of products reduced to 10 for the opti-mal promotional product mix and they constituted approximately 20% of the total number ofproducts under study. This complies with 80:20 PARETO principle.7Literature reviewAnother study was carried out by Brynjolfsson et al. (2011) which states that many mar-kets have historically been dominated by a small number of best-selling products.Accordingto Brynjolfsson et al. (2011) ,states that the Pareto principle, also known as the 80/20 rule,describes this common pattern of sales concentration. However, information technology ingeneral and Internet markets in particular have the potential to substantially increase thecollective share of niche products, thereby creating a longer tail in the distribution of sales.This paper investigates the Internet’s “long tail” phenomenon. By analyzing data collectedfrom a multichannel retailer, it provides empirical evidence that the Internet channel ex-hibits a signicantly less concentrated sales distribution when compared with traditionalchannels. Previous explanations for this result have focused on differences in product avail-ability between channels. However, demonstration was made that the result survives evenwhen the Internet and traditional channels share exactly the same product availability andprices. Instead,Brynjolfsson et al. (2011) nd that consumers’ usage of Internet search anddiscovery tools, such as recommendation engines, are associated with an increase the shareof niche products.Brynjolfsson et al. (2011) conclude that the Internet’s long tail is not solelydue to the increase in product selection but may also partly reect lower search costs on theInternet.We therefore conlude that if the relationships they uncover persist, the underlyingtrends in technology portend an ongoing shift in the distribution of product sales.2.3 Time Series AnalysisAccording to Osarumwense (2013) a time series is a sequence of ordered data. The orderingrefers generally to time, but other ordering could be envisioned e.g overspace etc. Time seriesanalysis is used to detect patterns of change in statistical information over regular intervalof time. We project these pattern to arrive at an estimate for the future. All statistical fore-cating methods are extrapolatory in nature i.e they involve the projection of past patternsor relationship into the future. Time series can be stationary and non-stationary. However,theory of time series is concerned with stationary time series. A time series is said to bestationary if it has constant mean and variance.Also Wei (2006) did a study that dealt with time domain statistical models and methods onanalyzing time series and their use in applications. It covers fundamental concepts, station-ary and nonstationary models, nonseasonal and seasonal models, intervention and outlier8Literature reviewmodels, transfer function models, regression time series models, vector time series models,and their applications ,discussing the process of time series analysis including model identi-cation, parameter estimation, diagnostic checks, forecasting, and inference.Also discussion ofautoregressive conditional heteroscedasticity model, generalized autoregressive conditionalheteroscedasticity model, and unit roots and cointegration in vector time series processeswere done.2.4 ARIMAEdiger and Akar (2007) did a research on ARIMA forecasting of primary energy demand byfuel a case study of Turkey.Ediger and Akar (2007) stated that forecasting of energy demandin emerging markets is one of the most important policy tools used by the decision makers allover the world. In Turkey, most of the early studies used include various forms of economet-ric modeling. However, since the estimated economic and demographic parameters usuallydeviate from the realizations, time-series forecasting appears to give better results. In thisstudy, we used the Autoregressive Integrated Moving Average (ARIMA) and seasonal ARIMA(SARIMA) methods to estimate the future primary energy demand of Turkey from 2005 to2020. The ARIMA forecasting of the total primary energy demand appears to be more reli-able than the summation of the individual forecasts. The results have shown that the averageannual growth rates of individual energy sources and total primary energy will decrease inall cases except wood and animal–plant remains which will have negative growth rates. Thedecrease in the rate of energy demand may be interpreted that the energy intensity peak willbe achieved in the coming decades. Another interpretation is that any decrease in energydemand will slow down the economic growth during the forecasted period. Rates of changesand reserves in the fossil fuels indicate that inter-fuel substitution should be made leadingto a best mix of the country’s energy system. Based on our ndings we proposed some policyrecommendations.Another study was carried out by Kumar and Jain (2010) on ARIMA forecasting of ambientair pollutants and he argues that In the present study, a stationary stochastic ARMA/ARIMAAutoregressive Moving (Integrated) Average modelling approach has been adapted to fore-cast daily mean ambient air pollutants concentration at an urban trafc site (ITO) of Delhi,India. Suitable variance stabilizing transformation has been applied to each time series inorder to make them covariance stationary in a consistent way. A combination of different9Literature reviewinformation-criterions, namely, AIC (Akaike Information Criterion), HIC (Hannon–Quinn In-formation Criterion), BIC (Bayesian Information criterion), and FPE (Final Prediction Error)in addition to ACF (autocorrelation function) and PACF (partial autocorrelation function) in-spection, has been tried out to obtain suitable orders of autoregressive (p) and moving aver-age (q) parameters for the ARMA(p,q)/ARIMA(p,d,q) models. Forecasting performance of theselected ARMA(p,q)/ARIMA(p,d,q) models has been evaluated on the basis of MAPE (meanabsolute percentage error), MAE (mean absolute error) and RMSE (root mean square error)indicators. For 20 out of sample forecasts, one step (i.e., one day) ahead MAPE for carbondioxide(CO),nitrogen monoxide (N O2), nitrogen oxide(NO) and oxygen (O3), have been foundto be 13.6, 12.1, 21.8 and 24.1%, respectively. Given the stochastic nature of air pollutantsdata and in the light of earlier reported studies regarding air pollutants forecasts, the fore-casting performance of the present approach is satisfactory and the suggested forecastingprocedure can be effectively utilized for short term air quality forewarning purposes.Ong et al. (2005) researched on model identication of ARIMA family using genetic algo-rithms.In the research it is said that ARIMA is a popular method to analyze stationary uni-variate time series data. There are usually three main stages to build an ARIMA model,including model identication, model estimation and model checking, of which model iden-tication is the most crucial stage in building ARIMA models. However there is no methodsuitable for both ARIMA and SARIMA that can overcome the problem of local optima. Inthis paper, we provide a genetic algorithms (GA) based model identication to overcome theproblem of local optima, which is suitable for any ARIMA model. Three examples of times se-ries data sets are used for testing the effectiveness of GA, together with a real case of DRAMprice forecasting to illustrate an application in the semiconductor industry. The results showthat the GA-based model identication method can present better solutions, and is suitablefor any ARIMA models.Another study was carried out by Kumar and Vanajakshi (2015) on Short-term trafc owprediction using seasonal ARIMA model with limited input data .Accurate prediction of traf-c ow is an integral component in most of the Intelligent Transportation Systems (ITS)applications. The data driven approach using Box-Jenkins Autoregressive Integrated Mov-ing Average (ARIMA) models reported in most studies demands sound database for modelbuilding. Hence, the applicability of these models remains a question in places where thedata availability could be an issue. The present study tries to overcome the above issue byproposing a prediction scheme using Seasonal ARIMA (SARIMA) model for short term pre-10Literature reviewdiction of trafc ow using only limited input data.2.5 SARIMAVelasquez Henao et al. (2013) carried out a research on the combination of SARIMA and neu-ral network models are a common approach for forecasting nonlinear time series. While theSARIMA methodology was used to capture the linear components in the time series, articialneural networks were applied to forecast the remaining non linearities in the shocks of theSARIMA model. in the research a simple nonlinear time series forecasting model by com-bining the SARIMA model with a multiplicative single neuron using the same inputs as theSARIMA model was proposed. To evaluate the capacity of the new approach, the monthlyelectricity demand in the Colombian energy market was forecasted and compared with theSARIMA and multiplicative single neuron models.However in this research SARIMA andHolt Winters Method will compared and the best forecasting method will be chosen.AlsoSchulze and Prinz (2009) states that SARIMA and Holt Winters models are designed es-pecially to take account of the seasonal behaviour of the daily data to be used.According toSchulze and Prinz (2009) it was seen that the forecasting error measures such as mean squareerror and mean absolute percentage error, the SARIMA-approach yields slightly better val-ues of modelling the container throughput than the exponential smoothing approach.Another researcher Wang et al. (2013) did a research on forecasting with SARIMA and thepurpose was to increase crop production.He states that it is highly difcult to forecast dueto random sequential and seasonal features. In the research,the historical data of time se-ries, it is found that rainfall has a strong autocorrelation of seasonal characteristics in timeseries. Utilizing seasonal periodicity with a Seasonal Autoregressive and Moving Average(SARIMA) methodology the statistical data of precipitation was analysed. The experimentalresults could achieve good prediction tting degree. In this sense, the model is available foractual forecast warning in precipitation. Through the comparison of the model they found theadvantages of forecasting that can make full use of natural rainfall for corresponding areasand save underground water resources. Another reseacher Jeong et al. (2014) did a researchto estimate energy cost budget in educational facility.The aim of was to develop an estimationmodel for determining the AECB in educational facilities using the SARIMA (seasonal au-toregressive integrated moving average) model and the ANN (articial neural network). Thisstudy collected electricity consumption data for 7 years (2005–2011) from 787 educational fa-11Literature reviewcilities. The result of this study showed that the prediction accuracy of the proposed hybridmodel (which was developed by combining SARIMA and ANN) was improved, compared tothe conventional SARIMA model. The MAPE (mean absolute percentage error) of the pro-posed method and conventional method for determining the AECB in educational facilitieswas determined at 0.11–0.24% and 1.23–1.84%, respectively. Namely, it was determined thatthe proposed method was superior to the conventional method. The proposed model couldenable executives and managers in charge of budget planning to accurately determine theAECB in educational facilities. It could be also applied to other types of resources (e.g., waterconsumption or gas consumption) used in educational facilities.Nanthakumar et al. (2012) did a study to forecast the tourism demand for Malaysia fromASEAN countries. The literature on forecasting tourism demand is huge comprising vari-ous types of empirical analysis. Some of the researchers applied cross-sectional data, butmost of the tourism demand forecasting used pure time-series analytical models. One of theimportant time-series modelling used in tourism forecasting is ARIMA modelling,which isspecied based on standard Box-Jenkins method, a famous modelling approach in forecast-ing demand. Many studies have applied this methodology, such as Lee et al. (2008),Song et al.(2003),Du Preez and Witt (2003) just to mention a few .The ARIMA model is proven to be re-liable in modelling and tourism demand forecasting with monthly and quarterly time-series.Another resercherWong et al. (2007) used four types of models, namely seasonal auto-regressiveintegrated moving average model (SARIMA), auto-regressive distributed lag model (ADLM),error correction model (ECM) and vector-autoregressive model (VAR) to forecast tourism de-mand for Hong Kong by residents from ten major origin countries. The empirical results ofthe study shows that forecast combinations do not always outperform the best single forecastswhich have been used frequently in previous studies. Therefore, combination of empiricalmodels can reduce the risk of forecasting failure in practice.Generally, from this study we canconclude that the ARIMA volatility models tend to overestimate demand, and the smoothingmodels are inclined to underestimate the number of future tourist arrivalsAgain Chu (2009) modied ARIMA modelling to fractionally integrated autoregressive mov-ing average (ARFIMA) in forecasting tourism demand. This ARFIMA model is ARMA basedmethods. Three types of univariate models were applied in the study with some modicationin ARMA model to become ARAR and ARFIMA model. The main purpose of this study is toinvestigate the ARMA based models and its usefulness as a forecast generating mechanismfor tourism demand for nine major tourist destinations in the Asia-Pacic region. This studyis different from other tourism forecasting studies published earlier, because we can identify12Literature reviewthe ARMA based models behaviour and the difference between ARFIMA models with otherARMA based modelsAlso Chakhchoukh et al. (2009) did a research on Robust estimation of SARIMA models,Application to short-term load forecasting.The research presents a new robust method toestimate the parameters of a SARIMA model. This method uses robust autocorrelations es-timates based on sample medians coupled with a robust lter cleaner which rejects deviantobservations. The procedure is compared with other robust methods via evaluation of the dif-ferent robustness measures such as maximum bias, breakdown point and inuence function.The asymptotic properties of our method (strong consistency and central limit theorem) areestablished for a gaussian AR process.It is shown that the method improves the French loadforecasting for “normal days” and offers good robustness, easiness and fast execution.In theresearch it is also said that when the data contains deviant observations termed outliers, theclassical estimates of a SARIMA model become unreliable. Thus order selection, parameterestimation, and forecasting can be affected notably. In order to remedy to this drawback,we may resort to a robust statistical estimation or a diagnostic approach. Good diagnosticapproaches achieve robustness via outlier detection and hard rejection, resulting in missingvalues in the time series. By contrast, robust methods accommodate outliers by boundingtheir inuence on the estimates, yielding no missing values. While they are different, thediagnostic and the robust approaches end up to have a similar objective, which is estimatingin a robust way a model and detecting the outliers.2.6 Holt Winters MethodGoodwin et al. (2010) did a research concerning Holt Winters Method and he stated that manycompanies use the Holt-Winters (HW) method to produce short-term demand forecasts whentheir sales data contain a trend and a seasonal pattern.Goodwin et al. (2010) also outlinedthe uses of this method which are how can to stop the method from being unduly inuencedby sales gures that are unusually high or low (i.e., outliers)? ,checking whether the methodis useful when there are several different seasonal patterns in sales (such as when demandhas hourly, daily, and monthly cycles mixed together)? and how to obtain reliable predictionintervals from the method?.Goodwin et al. (2010) states that the Holt-Winters method wasdesigned to handle data where there is a conventional seasonal cycle across the course of ayear, such as monthly seasonality. However, many series have multiple cycles: the demandfor electricity will have hourly (patterns across the hours of a day), daily (patterns across the13Literature reviewdays of the week), and monthly cycles across the years.Taylor (2003a) went on further to do a research on the Exponential smoothing with a dampedmultiplicative trend.Taylor (2003a) found out that multiplicative trend exponential smooth-ing has received very little attention in the literature. It involves modelling the local slopeby smoothing successive ratios of the local level, and this leads to a forecast function that isthe product of level and growth rate. By contrast, the popular Holt method uses an additivetrend formulation. It has been argued that more real series have multiplicative trends thanadditive. However, even if this is true, it seems likely that the more conservative forecastfunction of the Holt method will be more robust when applied in an automated way to a largebatch of series with different types of trend. In view of the improvements in accuracy seen indampening the Holt method, in this paper we investigate a new damped multiplicative trendapproach. An empirical study, using the monthly time series from the M3-Competition, gaveencouraging results for the new approach at a range of forecast horizons, when compared tothe established exponential smoothing methods.Taylor (2003b) researched on univariate online electricity demand forecasting for lead timesfrom a half-hour-ahead to a day-ahead. A time series of demand recorded at half-hourly inter-vals contains more than one seasonal pattern. A within-day seasonal cycle is apparent fromthe similarity of the demand prole from one day to the next, and a within-week seasonalcycle is evident when one compares the demand on the corresponding day of adjacent weeks.There was a strong appeal in using a forecasting method that were able to capture both sea-sonalities. The multiplicative seasonal ARIMA model has been adapted for this purpose. Inthe paper, the Holt–Winters exponential smoothing formulation was adapted so that it canaccommodate two seasonalities.Correction for residual autocorrelation was done using a sim-ple autoregressive model. The forecasts produced by the new double seasonal Holt–Wintersmethod outperform those from traditional Holt–Winters and from a well-specied multiplica-tive double seasonal ARIMA model.2.7 Accuracy measuresArmstrong and Fildes (1995) proposed the Generalized Forecast Error Second Moment (GFESM)as an improvement to the Mean Square Error in comparing forecasting performance acrossdata series. They based their conclusion on the fact that rankings based on GFESM remainunaltered if the series are linearly transformed. In this paper, we argue that this evalua-14Literature reviewtion ignores other important criteria. Also, their conclusions were illustrated by a simulationstudy whose relationship to real data was not obvious. Thirdly, prior empirical studies showthat the mean square error is an inappropriate measure to serve as a basis for comparison.This undermines the claims made for the GFESM.Also in this research greater weight will beassigned to Mean Absolute Percentage Error (MAPE),the model with the least MAPE valuewill be considered to be the best.MAPE presents problems when it produces values close tozero or equal to zero.These problems can be avoided by using non-negative values.2.8 ConclusionThis chapter demonstrated understanding, and ability to critically evaluate research in theeld,provided evidence that may be used to support your the researchers own ndings,to seewhat has and has not been investigated and to contribute to the eld by moving researchforward. Also this chapter helped to see what came before, and what did and didn’t work forother researchers.15Chapter 3Methodology 3.1 IntroductionThis chapter covers on how the research was conducted to obtain necessary information usedin the research project. It also provides the description of the procedures to be used in con-ducting the research and methods used in data analysis. The researcher will give a briefdescription on the methods to be used which are Seasonal ARIMA and Holt Winter’s methodand these two methods are to be compared. Box-Jenkins methodology was extensively ap-plied for the SARIMA models as the researcher will concentrate more on building SARIMAmodels as they are precise in dealing with data which is seasonal. Seasonality in a time seriesis a regular pattern of changes that repeats over S time periods, where S denes the numberof time periods until the pattern repeats again.3.2 Data collectionData was collected from the Bakers Inn shops and self selection was used as a samplingcriteria to choose amongst the Harare shops.Self sampling is useful when we want to allowevery unit(in this case shops)to take part in the research.There are reasons why a shop iseither chosen or rejected and in this case the researcher chose Reliance because it is the shopMethodology17that has the highest Gross Prot in the Harare region.Also only products that are ordered ona daily basis and have an expiry data of less than seven days were considered to be part ofthe research for efciency.It was seen that products like drink take almost a year to expireand these products are never found to be part of the waste hence will not contribute to theresearch.3.2.1 Pareto AnalysisPareto analysis is the analysis is a problem solving technique that can be used to solve situ-ations that are not evenly distributed.The Pareto analysis is also known as the 80-20 rule orprinciple in this case it means only 20% of products yield 80% of the prots.Pareto’s Principleor the 80-20 Rule helps you to identify and prioritize events and activities that can improveyour productivity and success.It is an analysis using sales as the basis which will be neces-sary to derive the greatest nancial benet from the effort exerted according to( biblex).ThePareto principle makes use of the Pareto distribution.3.2.2 Assumption Independent and identically distributed demand in different time periods.3.2.3 Steps to create a Pareto ChartA Pareto Chart is a type of chart that contains both bars and a line graph where individ-ual values are represented in descending order and the cumulative total is represented bythe line.Basically it is skewed with heavy “slowly decaying” tails where much of the data isexplained in the tails.Create a vertical bar chart with the products on the x-axis and prot on the y-axis. Arrange the bar chart in descending order of cause importance that is, the cause withthe highest count rst. Calculate the cumulative count for each cause in descending order. Calculate the cumulative count percentage for each cause in descending order. Percent-age calculation: I ndividualC auseC ount T otalC ausesC ount100Methodology18Create a second y-axis with percentages descending in increments of 10 from 100% to0%. Plot the cumulative count percentage of each cause on the x-axis. Join the points to form a curve. Draw a line at 80% on the y-axis running parallel to the x-axis. Then drop the line atthe point of intersection with the curve on the x-axis. This point on the x-axis separatesthe important causes on the left (vital few) from the less important causes on the right(trivial).After getting 20% of the products the researcher ranked them according to their percentagecontribution to prot and the top products were chosen.3.3 Box Jenkins ApproachThe Box-Jenkins methodology is a strategy or procedure that can be used to build an ARIMAmodel. Box Jenkins Approach is an iterative procedure for time series forecasting . Accordingto biblex he states that Box Jenkins Approach is subjective in the sense that the results de-pends, to a great degree depends on the analysts experience and background .This approachhas 3 main methods namely identication, estimation and verication . The rst step is toget feel of the data, that is collecting and examining the data graphically and statistically.Thedata is plotted against time and visual inspection will indicate whether it is plausible to as-sume that the process is stationary .This is graphical procedure and if the AutocorrelationFunction (ACF) of the time series values either cuts off or dies down fairly quickly then thetime series is considered stationary .On the other hand , if the ACF of the time series valueseither cuts off or dies down extremely slowly then it should be considered non-stationary .Ingeneral , if the original time series values are non-stationary , performing rst and seconddifferencing transformation on the original data will produce stationary time series values.For regular differencing forecast the equation is given asd(Xt) = (1B)d(Xt)(3.1)When d=1Methodology19(Xt) = (1B)d(Xt) =XtXt 1 (3.2)Once stationarity is rendered then one should identify and estimate the correct ARIMAmodel.3.4 Components and Fitting of ARIMA model3.4.1 OverviewThe ARIMA model divides the pattern of a time series into three components: the autoregres-sive component, p, which describes how observations are related to each other as the resultof being close together in time; the differencing component, d, which is used to make a timeseries stationary and the moving average component, q, which describes outside “shocks” tothe system.3.4.2 IdenticationThe identication steps involve tting the autoregressive component (variable “p”), the mov-ing average component of the ARIMA model (variable “q”), as well any required differingto make the time series stationary or to remove seasonal effects (variable “d”). Together,these user-specied parameters are called the order of ARIMA. The formal specication ofthe model will be ARIMA (p,d,q) when the model is reported.3.4.3 Estimation and Diagnostic checksThe estimation procedure involves using the model with p, d and q orders to t the actual timeseries. A software is used to t the historical time series, while the researcher checks thatthere is no signicant signal from the errors using an ACF for the error residuals, and thatestimated parameters for the autoregressive or moving average components are signicant.If the original model identication is correct , the model requires diagnostics .If the modelfails , the process is repeated until the model satises all assumptions.Methodology203.4.4 ForecastingAfter a model is assured to be stationary, and tted such that there is no information in theresiduals, we can proceed to forecasting. Forecasting assesses the performance of the modelagainst real data. There is an option to split the time series into two parts, using the rstpart to t the model and the second half to check model performance. Usually the utility of aspecic model or the utility of several classes of models to t actual data can be assessed byminimizing a value such as root mean square.3.5 SARIMA ModelAs an extension of the ARIMA method, the SARIMA model not only captures regular dif-ference, autoregressive, and moving average components as the ARIMA model does but alsohandles seasonal behavior of the time series. In the SARIMA model, both seasonal and reg-ular differences are performed to achieve stationarity prior to the t of the ARMA model.A time series is said to be seasonal if there is a sinusoidal or periodic pattern in the seriesand when this happens the SARIMA model inevitably becomes the choice model. A SARIMAmodel is only plausible for stationary time series, where stationarity implies constant mean,variance, and autocorrelation functions over time seasonality in a time series is a regular pat-tern of changes that repeats over S time periods where S denes the number of time periodsuntil the pattern repeats again.The seasonal ARIMA model incorporates both non-seasonaland seasonal factors in a multiplicative model. One shorthand notation for the model isARIMA(p, d, q) (P, D, Q)S, with p = non-seasonal AR order, d = non-seasonal differencing, q= non-seasonal MA order, P = seasonal AR order, D = seasonal differencing, Q = seasonal MAorder, and S = time span of repeating seasonal pattern. Without differencing operations, themodel could be written more formally as(B )s’ (B )(Xt) = Bs (B )Wt (3.3)The non seasonal components are :AR: ‘(B )s= 1 ‘1B ‘pB p(3.4)MA: (B )s= 1 + 1B+ +pB p(3.5)Methodology21The seasonal components are :Seasonal AR: (B )s= 1 1B pB p(3.6)Seasonal MA: (B)s= 1 + 1B+ + pB p(3.7)The multiplicative seasonal autoregressive integrated moving average model or SARIMAmodel is given by (B )p(B h d Dh Xt=(B ) (B ht +c (3.8)The seasonal difference operator is given bys= 1Bs (3.9)The general model is denoted as ARIMA (P,D ,Q)h and are polynomials of order P and Qrespectively and the non-seasonal AR and MA characteristics operators are :(B ) = 1 1B2B 2 P B P(3.10)( B) = 1 + 1B+ 2B 2+ + QB Q(3.11)The seasonal auto-regressive integrated moving average with operators with a seasonal pe-riod s are given as(B s) = 1 1B2B 2s P B P s(3.12)( Bs) = 1 + 1B+ 2B 2s+ + QB Qs(3.13) d Dh Xt= (1B)d(1 Bd)DX t (3.14)Where i andjare constants such that the zeros of equation 3.20 and 3.21 are all outsidethe unit circle for stationarity and invertibility respectively . Equation (3.18) and (3.19) rep-resent the autoregressive and moving average operators respectively for the non-seasonalMethodology22characteristics, while (3.20) and (3.21) represent the autoregressive and moving average op-erators for the seasonal characteristics. The d and D denote the number of non-seasonal andseasonal difference respectively. For a seasonal series ,the time plot reveals the existenceof a seasonal nature in data, and the ACF shows a spike at the seasonal lag. Table belowsummarises the behaviour of the ACF and PACF of Pure Seasonal ARMA modelsTable 3.1: Behaviour of ACF and PACF of Pure Seasonal ARIMA models ACF PACFAR(p) Tails off at lag kh=1,2 Cuts off after lag phMA(q) Cuts off after lag Q Tails off at lags kh=1,2ARIMA(p,q) Tails off at lags kh Tails off at khThe ACF of an MA(q) model cuts off after lag q whereas that of an AR(p) model is a combina-tion of sinusoidals dying off slowly. On the other hand the PACF of an MA(q) model dies offslowly whereas that of an AR(p) model cuts off after lag p. AR and MA models are known toexhibit some duality relationships. These include:A nite order AR model is equivalent to an innite order MA model. A nite order MA model is equivalent to an innite order AR model. The ACF of an AR model exhibits the same behaviour as the PACF of an MA model. The PACF of an AR model exhibits the same behaviour as the ACF of an MA model. The seasonal part of an ARIMA model has the same structure as the non-seasonal part:it may have an AR factor, an MA factor, and/or an order of differencing. In the seasonalpart of the model, all of these factors operate across multiples of lag s (the number ofperiods in a season). A seasonal ARIMA model is classied as an ARIMA(p,d,q)x(P,D,Q) model, where P=numberof seasonal autoregressive (SAR) terms, D=number of seasonal differences, Q=numberof seasonal moving average (SMA) terms3.5.1 Assumptions of SARIMA Model The time series data should be stationary which means that its properties do no dependon time at which the series is observed i.e its mean and variance are constant throughMethodology23time .For practical purposes, it is sufcient to have weak stationary, which means thatthe data is in equilibrium around the mean and the variance remains constant overtime .If a time series data is non-stationary due to its variance not being constant, itoften helps to log-transform the data. Differencing is applied to have a series that isstationary in the mean. Residuals are normally distributed over time .Residuals exhibit homogeneity of varianceover time and have a mean zero. Homoscedasticity ie the series has a constant variance .If the amplitude of the variancearound the mean is great even after differencing, the series is considered heteroscedas-tic .The solution of this problem involves methods such as natural logarithm of data andnormally a log transformation will successfully stabilize the variance of the series.3.5.2 Stationarity Test The rst step is to do a time series plot and examine it for any trend (growth or decline)and seasonality features. Data is collected in months so examining the data acrossmonths to check for seasonal pattern. Also examine the autocorrelation plots of the time series.The ACF is a statistical toolthat measures whether earlier incidence in the series have some relation to later ones For a stationary time series ,the autocorrelations will typically decay rapidly to 0. For anon-stationary time series , the autocorrelations typically decay slowly it at all. For theautocorrelation plots MINITAB and EVIEW 7 for the Augmented Dickey Fuller Test areused as statistical packages . Test for stationarity is essential at this stage and if the data exhibit non stationaryproperty diffencing of the time series data is then applied. The researcher will use 12months of differencing to remove the seasonality component in the data which will givethe given series below;Y t = (1B12)Xt3.5.3 Model identication and estimationPlot the correlograms for the partial autocorrelation functions (PACF) and the autocorrela-tion functions (ACF) of the differenced data to determine the auto regressive order p and theMethodology24moving average q for the differenced data. Then to determine the AR and MA orders ,countthe number of signicant autocorrelations and partial autocorrelations .We then calculate theparameters of the model by making use of the mean square error (MSE) value in the modelselection criteria. The model with the least MSE is selected to be the best.3.6 Model tting and DiagnosticsCheck the statistical signicance of the derived model for adequacy. Consider the residual(error terms)properties from an ARIMA model if they are randomly and normally distributed3.6.1 Autocorrelation assumptionACF and PACF plots for residuals are used to determine whether the model meets the as-sumption that residuals are independent.If no signinacant correlations are present then theresiduals are independepent then the model is considered to be appropriate for the set ofdata.Durbin Watson test is used to test for autocorrelation in the error terms .Durbin Watson testlooks at only one type of auto correlation that is rst order autoregressive type of correlationAR(1)process. The test statistic for d is given as; d= 2(1 )We can deduce that1. d= 2 or= 0 there is no auto correlation2. d= 0 or= +1 there is perfect positive auto-correlation 0< d < 2there is some degreeof positive auto-correlation3. d= 4 or= 1there is perfect negative auto-correlation 2< d 0:05 then the residuals are con-sidered as normal.This test also includes the skewness and kutosis of the residuals.If theMethodology25skewness value should be close to 0 and the kurtosis value should be 3 to satisfy the normal-ity assumption.3.6.3 HeteroskedasticityWe use the plot of residuals vs ts to detect if there are any problems in the tted model and italso gives a clear indication of the outlying observations. With the plot of residuals it is easierto see a change of in the variance than with a plot of original data. If all the assumptions aresatised then Gaussian white noise to the error terms of the Seasonal ARIMA is said to besatised.3.7 Goodness of tA goodness-of-t test, in general, refers to measuring how well do the observed data cor-respond to the tted (assumed) model. We will use this concept throughout the researchas a way of checking the model t. Static forecasting on the model is performed to showmeasures of forecast accuracy over the estimation period. The model with the smallest mea-sure of forecast error will be chosen as the one with the most accurate t of the time seriesmodel. Then, some more tests will be performed, such as correlogram of standardized residu-als squared which consists of autocorrelation and partial auto-correlation, test for presentingof conditional heteroskedasticity in the data with, standardized residuals. After an appro-priate ARIMA model has been t , we then examine the goodness of t by means of plottingACF of the errors of the tted model. Most of the sample autocorrelation coefcients of theresiduals are within limits 1.96/ pN where N is the number of observations upon which themodel is based and it shows that the model is a good t.3.8 Evaluation of forecasting performanceThe nal step is to evaluate the forecast performances by our achieved multiplicative seasonalSARIMA model. The evaluation includes the Objective penalty criterion which is a methodof evaluating model accuracy.Methodology263.8.1 Forecast errorThe forecast error is the difference between the observed value and its forecast based on allprevious observations. If the error is denoted as e(t) then the forecast error can be written ase(t) = Yt ^Y t (3.15)where Y(t) are the observations ^Y t is the forecast of Y(t) based on all previous observationsForecast errors can be evaluated using a variety of methods namely Mean Absolute Deviation(MAD), Mean Forecast Error (MFE), Root Mean Square Error (RMSE) and Mean AbsolutePercentage Error(MAPE) of the model under study.3.8.2 Mean Absolute Percentage Error(MAPE)Mean Absolute Percentage Error(MAPE) is the most common measure of forecast error.MAPE functions best when there are no extremes to the data (including zeros).With zerosor near zeros, MAPE can give distorted picture of error.The error near zero item can be in-nitely high causing a distortion to the overall error rate averaged in. For forecasts of itemsthat are at zero or near zero volume Symmetric Mean Absolute Percent Error (SMAPE) is abetter measure. MAPE is the average absolute percent error for each time period or forecastsubtracted from actual divided by actual.M AP E= jActual F orecast j Actual100% N(3.16)The best model is the one with the least MAPE value3.8.3 Root Mean Square Error (RMSE)To construct the RMSE, residuals are needed. Residuals are the difference between the actualvalues and the predicted values.I denoted them by Yt ^Y t.They can be positive or negativeas the predicted value under or over estimates the actual value. Squaring the residuals,averaging the squares, and taking the square root gives us the RMSE. You then use theRMSE as a measure of the spread of the y values about the predicted y value.RM S E=vuut 1N TX1 (Yt ^Y t) 2(3.17)Methodology27where N is the number of forecasted observations3.8.4 Mean Absolute Deviation (MAD)Mean absolute deviation (MAD) of a data set is the average distance between each data valueand the mean. Mean absolute deviation is a way to describe variation in a data set. Meanabsolute deviation helps us get a sense of how “spread out” the values in a data set are. Here’show to calculate the mean absolute deviation.M AD=1 N TX1 jYt ^Y tj(3.18)3.8.5 Mean Forecast Error (MFE)When it is positive, the forecasts have been low in relation to actual demand and when it isnegative, the forecasts have been too high.M F E=1 N TX1 (Yt ^Y t)(3.19)To compare the forecasting capabilities for the two models we therefore plot a graph of thetwo models with the forecasted values together with the actual sales.3.9 Holt Winters Method nTriple Exponential Smooth-ingHolt Winter is a rule of thumb method used for smoothing time series data using the expo-nential window function. Whereas in the simple moving average the past observations areweighted equally, exponential functions are used to assign exponentially decreasing weightsover time. It is an easily learned and easily applied procedure for making some determina-tion based on prior assumptions by the user, such as seasonality.There are two variationsto this method that differ in the nature of the seasonal component. The additive method ispreferred when the seasonal variations are roughly constant through the series, while themultiplicative method is preferred when the seasonal variations are changing proportionalto the level of the series. With the additive method, the seasonal component is expressedin absolute terms in the scale of the observed series, and in the level equation the series isMethodology28seasonally adjusted by subtracting the seasonal component. The raw data sequence is oftenrepresented by xt beginning at timet= 0 , and the output of the exponential smoothing algo-rithm is commonly written as xt, which may be regarded as a best estimate of what the nextvalue of x will be. When the sequence of observations begins at time t= 0 , the simplest formof exponential smoothing is given by the formulas:s 0 =xtst =xt+ (1)st 1; t ;0 (3.20)where is the smoothing factor, and 0; ; 1.For the Triple Exponential Smoothing, suppose we have a sequence of observations At, begin-ning at time t= 0 with a cycle of seasonal change of length L.The method calculates a trend line for the data as well as seasonal indices that weight thevalues in the trend line based on where that time point falls in the cycle of length L.L trepresents the smoothed value of the constant part for time t.b t represents the sequence of best estimates of the linear trend that are superimposed on theseasonal changes.s t is the sequence of seasonal correction factors. ct is the expected proportion of the predictedtrend at any time t in the cycle that the observations take on. As a rule of thumb, a minimumof two full seasons (or 2L periods) of historical data is needed to initialize a set of seasonalfactors.The output of the algorithm is again written as Ft, an estimate of the value of x at time t,based on the raw data up to time t. Triple exponential smoothing with Additive seasonalityis given by the formulasL 0 =AtLevel:Lt=(At+St s) + (1)( Lt 1 +bt 1)(3.21)T rend :bt =(LtLt 1) + (1)bt 1 (3.22)S easonal :St=(AtSt) + (1)bt s (3.23)F orecast :Ft+ m = (AT +bT k) + LT + k 1) =A +1 (3.24)where is the data smoothing factor, 0; ; 1, is the trend smoothing factor, 0; ; 1,Methodology29andis the seasonal change smoothing factor, 0; 0:05 which shows that the resid-uals of this model (in groups of up to 48 values) are independent therefore uncorrelated.6Data AnalysisFigure 4.13: Modied Box-Pierce (Ljung-Box) Chi-Square ResultsFigure 4.14: Durbin Watson Test Results Durbin Watson TestFrom the Durbin Watson test results it can be seen that DW is close to 2 hence we concludethat there is no auto-correlation of the residuals therefore assumption is not violated.Normality AssumptionHistogram of Residuals Figure 4.15: Histogram of ResidualsFig 4. shows thats the histogram is bell shaped indicating normality of residuals and there-fore the selected model meets the assumption of normality.Jarque-Bera Test7Data AnalysisFigure 4.16: Jarque Bera TestFigure 4.17: Residual vs FitsThe test shows that the p-value of 0.205644 is insignicant (p¿0.05). Also the skewness of 0.05which is close to zero and the kurtosis of 2.546 which is close to 3 provides much evidencethat the residuals are normally distributed.The Jarque-Bera value was found to be 3.163221.Heteroskedasticity TestResidual vs Fit plot shows that the residuals bound randomly around the 0 line.This suggeststhe assumption that the relationship is linear is reasonable.The horizontal band formed bythe residuals along the 0 line suggests that the variances of the error terms are consant/e-qual.No one residual stands out from the basic random pattern of residuals. This suggeststhat there are no outliers.4.3.5 Goodness of FitACF of residuals shows that most of the coefcients of the sample autocorrelation of theresiduals falls within the limit 1:96 349= 0:1049163946 and it indicates that the model is a good8Data Analysist and also all the residual assumptions were met which means that the model is a good t.4.3.6 Forecasting Perfomance Figure 4.18: Accuracy MeasuresTable 4.1: SARIMA Forecasting Perfomances Error MAPE MADValue 11.5384 2.66644.4 Holt Winters MethodA multiplicative and additive plot for Holt winters sales are compared to nd the methodwith the least MAPE.It was seen that the additive method produced the least MAPE and itis therefore used in this research. Figure 4.19: Winters Method Additive9Data AnalysisFigure 4.20: Winters Method For MultiplicativeTrial and error method was used to come up with the best model smoothing parametersand errors. Initially the researcher used 0.2 for all the three smoothing constants that islevel,trend and seasonal components and adjustment thereafter until the best model was ob-tained.The model that produced the least weight was considered to be the best and accordingto table 4.2 the lowest MAPE was found to be 10.514 with 0.6 , 0.01 and0.01 for , and respectively. Figure 4.21: Holt Winters Plot for SalesThe additive exponential smoothing equations are as followsLevel Lt= 0:6( At+St s) + 0:4( Lt 1 +bt 1)(4.1)Trend b1 = 0:1( LtLt 1) + 0:9 + bt 1 (4.2)Seasonal St= 0:2( AtSt) + 0:8 bt s (4.3)10Data AnalysisTable 4.2: Holt Winters forecast parameters and errorsModel MAPE MADA 0.2 0.2 0.2 11.7774 2.3347B 0.2 0.01 0.01 11.1394 2.4455C 0.2 0.001 0.01 12.697 3.1944D 0.3 0.1 0.1 10.582 2.6472E 0.5 0.001 0.01 11.975 2.3817F 0.5 0.0001 0.0001 12.6587 2.4139H 0.6 0.1 0.2 10.514 2.3489I 0.6 0.1 0.3 10.5521 2.3824J 0.6 0.001 0.001 11.7033 2.3602K 0.6 0.0000001 0.00001 12.5522 2.3824L 0.7 0.00000001 0.00000001 13.5426 2.3775M 0.7 0.01 0.01 14.3620 2.3620ForecastFt=Lt 1 +bt 1 +St s (4.4)where s is the number of seasonal periods in a year T is the time period4.5 Model DiagnosticsTo check whether the model assumptions are not violated , some residual tests were carriedout.4.5.1 Run’s TestFig 4. shows that p=0.436 which is greater than 0.05 therefore the residuals are random andthe assumption is not violated.11Data AnalysisFigure 4.22: Runs TestFigure 4.23: ACF of Residuals4.5.2 ACF of ResidualsThe auto correlation of residuals shows that the auto correlations for the in-sample forecasterrors do not exceed the signicance bounds for 1-60 lags.The observed signicant lag at lag1 is due to random error and does not imply that the residuals are not independent.4.5.3 Histogram of residualsThe histogram of residuals shows that the residuals are normally distributed and the as-sumption of normality is met.All the assumptions of the Holt Winter’s Method are met hence the model is a good t.4.6 ForecastingSince the model has a good t we will use it to forecast daily sales and table 4.3 shows theaccuracy measures for the actual and forecasted sales12Data AnalysisFigure 4.24: Histogram of ResidualFigure 4.25: Accuracy Measures13Data Analysis4.6.1 Evaluating Forecasting perfomanceRMSE and MAPE shall be used in evaluating the forecasting perfomanceRM S E =q 1NPT1 (Yt ^Y t) 2= q 128(3654940:604) = 361 :2943Table 4.3: Holt Winters forecasting Evaluation Error MAPE MADValue 10.514 361.29434.7 Comparison of the Holt Winters and the SARIMATable 4.4: Holt Winters and SARIMA Forecasting Perfomances comparisons Error MAPE MADSARIMA 11.5384 2.6664Holt Winters 10.514 2.3489In this research a model with the least MAPE is considered to be better from the other andfrom Table 4.4 it is seen that the Holt Winters Method has the least MAPE of 10.514 henceit is prefered than the SARIMA model.We can now use the Holt Winters Method to forecastfuture sales as shown by gure 4. Figure 4.26: Forecasting SalesThere is a downward trend in the forecasting of future sales.14Data Analysis4.8 ConclusionHolt Winters method was found to be the best forecasting method and is therefore used toforecast sales in the future.A decrease in the sales was seen and it might be a way to reducewaste products , meeting demand at the same time increasing prots.15Chapter 5Conclusion and Recommendations 5.1 IntroductionThis chapters concludes this research project.The conclusions of the study are clearly outlinedand stated as well as answers to the research questions which were stated earlier in the rstchapter of this research.For future studies ,recommendations from the research are going tobe provided.5.1.1 Summary of ResultsThe SARIMA and Holt-Winters forecasting procedures were used to forecast daily sales onemonth.The Pareto analysis of products shows that some products contributes a very low protwhich is almost insignicant and its obviously that these are the same products that arecontinously being ordered and increase the waste cost value.To stabilize variance the BoxCox transformation was applied to the data with = 0 :5 which is the same as the squareroot of the data.To check for stationarity trend analysis and the autocorrelation plot wasused.From the trend analysis it was seen that the the data was not stationary since mostof the lags are signicant.To obtain a stationary series the data was differenced once andtested again for stationarity.The appropriate SARIMA model was found and Auto correlationFunction (ACF) plot was used to check if the data exhibits auto regressive,moving averageor both orders.The ACF plot had one clear spike which clearly suggests a moving averageConclusion and Recommendations17process and then the p values from the nal estimates table and Ljung Box were found tobe signicant p 0:05 for the Ljung Boxrespectively.SARIMA (0;1 ;1)(0 ;0 ;1)7tend out to be best model meeting all the requirementsFor Holt Winters Method trial and error procedure was used, taking note of the results ateach trial and comparisons were made for the MAPE and MAD values to come up with thebest model parameters.The lowest MAPE was found to be 10.514 with 0.6 , 0.01 and0.01 for , and respectively.Residual diagnostic check of the SARIMA and the Holt Winters Method was also performed.Themodels did not violate all the assumptions set in place and it can be concluded that the resid-uals are random hence white noise and therefore the models satises all the assumptions.5.1.2 RecommendationsBakers Inn Harare retail department are therefore recommended to consider using the Mul-tiplicative Holt Winters method to forecast their sales thereby reducing overstocking andunder stocking.The researcher has found out that the Holt Winters Method is the most effec-tive forecasting tool for this company.A decline of the sales is now the managements cause ofconcern to see if this is for a good cause or not.Also all the products that are rarely sold should not be ordered at all or on a daily basis sincethey are being overstocked hence increasing waste.Some products have so much left over therefore the rst in rst out(FIFO) method is rec-ommended to be be taken seriously It is an inventory management that explains the orderin which inventory is purchased and then sold. When a company utilizes the FIFO method,they sell the products that they received rst before selling the products they received last.FIFO is the most popular method of inventory management as it’s easier to use than it’s lastin rst out counterpart and it’s more practical – especially regarding perishable goods.Whena company uses FIFO they are less likely to incur old and outdated inventory that can nolonger be sold. Accountants have to write off what’s called obsolete inventory after a certainamount of time goes by and the product is not used or sold. Because FIFO makes sure thatthe oldest items in stock are used or sold before they are deemed obsolete companies can savemoney (Sponaugle, 2014).Conclusion and Recommendations185.1.3 Suggested Future Work1.Data is diverse and one data set may differ in nature from another.Since its forecasting method has its limitations,larger variety of forecasting methods may be compared.Forexample the ARCH models may be included in the comparative study to carter for datathat is highly volatile.Neural Networks may also be included in the research to carter fornon linear data.This will increase the chances of obtaining a more favorable forecastingmodel for the given data.Intervention analysis(in the presence of promotions) may beincorporated to determine how past sales affected sales and hence how will promotionswill affect future sales.2.An analysis on the factors affecting sales and may be incorporated in the forecasting model.This will allow the research to have a clearer picture of the reasons behind theseasonality and trend factors on the sales data and will allow the organization to makemore informed decisions on how to inuence future sales.3.The Holt Winters Method may give subjective results.The smoothing parameters are not determined in a statistical way hence it is advised to develop a mathematical or sta-tistical and standard appropriate procedures that will determine the smoothing param-eters.This will help increase the accuracy and reability of the Holt Winters Algorithmand also increasing its forecasting power.5.2 ConclusionThis research has compared the forecasting ability of Holt-Winters and SARIMA modelswith respect to their daily demand obtained from the daily sales data. 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