NATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY FACULTY OF APPLIED SCIENCES DEPARTMENT OF OPERATIONS RESEARCH AND STATISTICS AN OPTIMAL INVENTORY POLICY FOR PERISHABLE PRODUCTS by NOBUHLE MUTOMBENI

NATIONAL UNIVERSITY OF SCIENCE AND
TECHNOLOGY FACULTY OF APPLIED SCIENCES
DEPARTMENT OF OPERATIONS RESEARCH AND STATISTICS AN OPTIMAL INVENTORY POLICY FOR PERISHABLE
PRODUCTS
by
NOBUHLE MUTOMBENI (N01414834Y)
SUPERVISOR: MR. H . NARE
This 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, Zimbabwe
MAY 2018

Declaration
I, 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 University
of Science and Technology, is my own independent work and has not been handed in before
for a qualication at/in another University/Faculty/School. I further declare that all sources
cited 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 and
Technology.
Signature…………………………………………………………
Date: May 2018
Copyright c

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i

Abstract
This 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 tted
to the top selling product of Bakers Inn turnover product which is Premium Bread.Daily
demand data was used for the period of January to December 2017. The performances
of the two models is evaluated using the forecast error methods which are Mean Absolute
Percent Error (MAPE), Root Mean Square Error (RMSE) and the Mean Absolute Deviation
(MAD).The study shows that the Holt Winters Method produces better forecasting results
than the SARIMA Method.
ii

Dedication
To Mum, Dad and Lesley T. Love you totally.
iii

Acknowledgments
Firstly I would like to thank the Lord Almighty for all the wisdom and understanding in
writing this project.I would like to convey my sincere appreciation to my supervisor Mr Nare
for all his support throughout the project and l will be forever indebted to him for this.I would
also like to thank my dad (Mr J.Mutombeni),my mum (Mrs Mutombeni) and my brothers and
sisters 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 no
darkness.Also l would like to thank my department of Statistics and Operations Research for
allowing me to carry this project .My fourth acknowledments goes to all my friends for their
support,love and motivation.Lastly l would like to thank Innscor Harare for allowing me to
carry this research as a case study to their company especially T.Masundlwane for providing
the data used in the study.
God Bless you all
iv

Contents
Declaration i
Abstract i
Dedication ii
Acknowledgments iii
Table of Contents v
List of Figures viii
List of Tables x
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.2 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.3 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.4 Aim of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
1.6 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
1.7 Signicance of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
1.8 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
1.9 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
1.10 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
1.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
2 Literature review 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
v

2.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2.2.1 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2.3 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
2.4 ARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
2.5 SARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
2.6 Holt Winters Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
2.7 Accuracy measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
3 Methodology 16
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
3.2 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
3.2.1 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
3.2.2 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
3.2.3 Steps to create a Pareto Chart . . . . . . . . . . . . . . . . . . . . . . . . .17
3.3 Box Jenkins Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
3.4 Components and Fitting of ARIMA model . . . . . . . . . . . . . . . . . . . . . .19 3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
3.4.2 Identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
3.4.3 Estimation and Diagnostic checks . . . . . . . . . . . . . . . . . . . . . .19
3.4.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
3.5 SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 3.5.1 Assumptions of SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . .22
3.5.2 Stationarity Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.5.3 Model identication and estimation . . . . . . . . . . . . . . . . . . . . .23
3.6 Model tting and Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 3.6.1 Autocorrelation assumption . . . . . . . . . . . . . . . . . . . . . . . . . .24
3.6.2 Normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
3.6.3 Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
3.7 Goodness of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
3.8 Evaluation of forecasting performance . . . . . . . . . . . . . . . . . . . . . . . .25
3.8.1 Forecast error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
3.8.2 Mean Absolute Percentage Error(MAPE) . . . . . . . . . . . . . . . . . .26
3.8.3 Root Mean Square Error (RMSE) . . . . . . . . . . . . . . . . . . . . . . .26
vi

3.8.4 Mean Absolute Deviation (MAD) . . . . . . . . . . . . . . . . . . . . . . .27
3.8.5 Mean Forecast Error (MFE) . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.9 Holt Winters Method nTriple Exponential Smoothing . . . . . . . . . . . . . . .27
4 Data Analysis 1
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
4.2 Pareto Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
4.3 SARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
4.3.1 Model identication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
4.3.2 Stationery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
4.3.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
4.3.4 Model Fitting and Diagonistic . . . . . . . . . . . . . . . . . . . . . . . . .6
4.3.5 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
4.3.6 Forecasting Perfomance . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.4 Holt Winters Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.5 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
4.5.1 Run’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
4.5.2 ACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
4.5.3 Histogram of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
4.6 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
4.6.1 Evaluating Forecasting perfomance . . . . . . . . . . . . . . . . . . . . .14
4.7 Comparison of the Holt Winters and the SARIMA . . . . . . . . . . . . . . . . .14
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
5 Conclusion and Recommendations 16 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 5.1.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
5.1.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
5.1.3 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
Appendix 21
vii

List of Figures
4.1 Pareto chart for the products . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
4.2 Time Series Plot of sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
4.3 Trend Analysis Plot of Actual Sales Data . . . . . . . . . . . . . . . . . . . . . .3
4.4 Autocorrelation of Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
4.5 PACF of Actual Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
4.6 Unit Root Test for the Differenced Sales . . . . . . . . . . . . . . . . . . . . . . .4
4.7 Time Series Plot of Differenced Sales . . . . . . . . . . . . . . . . . . . . . . . . .4
4.8 Trend Analysis for Transformed Sales Data . . . . . . . . . . . . . . . . . . . . .5
4.9 Autocorrelation for Differenced Sales Data . . . . . . . . . . . . . . . . . . . . .5
4.10 Final Estimates of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
4.11 ACF for Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
4.12 PACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
4.13 Modied Box-Pierce (Ljung-Box) Chi-Square Results . . . . . . . . . . . . . . .7
4.14 Durbin Watson Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
4.15 Histogram of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
4.16 Jarque Bera Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
4.17 Residual vs Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
4.18 Accuracy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.19 Winters Method Additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.20 Winters Method For Multiplicative . . . . . . . . . . . . . . . . . . . . . . . . . .10
4.21 Holt Winters Plot for Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
4.22 Runs Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
4.23 ACF of Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
4.24 Histogram of Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
4.25 Accuracy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
4.26 Forecasting Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
viii

5.1 Pareto Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
ix

List of Tables
3.1 Behaviour of ACF and PACF of Pure Seasonal ARIMA models . . . . . . . . . .22
4.1 SARIMA Forecasting Perfomances . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.2 Holt Winters forecast parameters and errors . . . . . . . . . . . . . . . . . . . .11
4.3 Holt Winters forecasting Evaluation . . . . . . . . . . . . . . . . . . . . . . . . .14
4.4 Holt Winters and SARIMA Forecasting Perfomances comparisons . . . . . . .14
x

Chapter 1
Introduction 1.1 Introduction
Innscor (Bakers Inn) is one of the company that is trying to keep up with its competitors by
nding alternative ways of minimizing cost hence increasing their prot. Bakers Inn sells
perishables and by that it needs a very strong ordering model so that it can withstand today’s
environment. Perishable refers to the items that have an expiration date and such food will
go 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 management
systems to keep track of the inventory and thereby minimizing wastes. Perishables demands
attention. Much of operations is about being able to match supply and demand. Since the
company uses average rolling modelling that is they use previous sales to place their orders
and which is more of a deterministic demand model, the researcher will also incorporate the
stochastic demand which is modelled as arbitrary probability distribution. Bakers Inn is
losing money from overstocking and under stocking perishable products because they do not
have 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 before
older items with expiring dates that are near but its not consistent. Bakers Inn is nding for
alternative methods to solve their forecasting problems that will result in an increase of their
total turnovers.
1

Introduction
1.2 Background of study
Much 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 of
the group in 1987, rst shop to open was Chicken Inn situated in the Harare CBD along
Speke Avenue with a product line of fried chips, chickens and hamburgers, and was followed
by Bakers Inn with pies. Over the years due to excellent service delivery and improved
menu 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 organization
has diversied into master franchises from South Africa, Nandos and Steers. Innscor Fast
foods 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 group
the fast foods division boasts of 6451 employees which are 41% of the total employees in
the 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 in
the books of accounts. Bakers Inn offers customers on the go a wide range of freshly baked
bread, 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 in
Zimbabwe, Kenya and Zambia. The brand has consistently updated its offering to suit the
changing 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 is
a widely used concept in most companies here in Zimbabwe but it is not effectively used in
our fast foods industry. So the purpose of the study is to apply inventory management. So the
economic 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 company
making loses from waste and increased holding cost.//
1.3 Statement of the Problem
The current forecasting model at Innscor is a challenge resulting to product stock out, more
waste 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 demand
2

Introduction
which 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 to
the company .Also the company loses money and customers through understocking products
that means placing insufcient orders for the day. To reduce these problems alternative ways
and forecasting methods are needed to reduce the companies stock outs and run outs. The
solution of this question lies in the degree of accuracy one is able to forecast demand with.
1.4 Aim of the study
The 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 mean
square error and mean absolute percentage error for the product under study.
Forecast demand and sales for the product under study.
1.6 Research Questions
A 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 more
about or a situation that needs to be changed or addressed. The following have been found to
be the research questions for the research project
What quantities should be ordered so as to reduce waste, overstocking, holding cost and
ordering cost for the products.
Which is the best forecasting method between Holt Winters and SARIMA that can be
used to forecast demand in the fast foods industry.
3

Introduction
1.7 Signicance of Study
The purpose of this research is to make a comparison between the Holt Winter’s model and
the SARIMA model. The mathematical model chosen is supposed to be a useful tool used by
the company for planning and control of their perishable products so as to reduce waste and
minimize stock outs so as to increase prots. The emphasis is on how much to order so as to
balance between sales and demand and enables the shop managers to create more realistic
ordering schedules.This will also help the company to prepare their budgets more efciently.
1.8 Limitations
Limitations 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 researchers
that place restrictions on methodology and conclusions. The following are the limitations for
this study.
Time constraints: due to lack of resources (nancial and technological), the researcher
found the research period very short.
The model is a one period model only considering current sales.
1.9 Delimitations
Delimitations describes the choices or boundaries that have been set for the study being
carried out.
Data to be used is to be picked from a sample which is going to represent brand.Though
the reseacher would have loved to cover all Bakers Inn shops in Zimbabwe, the study
only focuses on Harare Mashonaland Province.
For external validity purposes the sample size can be large.
1.10 Assumptions
Assumptions 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, the
4

Introduction
research problem itself could not exist,biblitex. For this study to be relevant we are assuming
that;
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 Conclusion
The 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 objectives
of 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 denitions
of terms. In Chapter 2, we will outline the literature review of Holt Winter and the SARIMA
model, giving a brief discussion of the merits and demerits of these methods. Chapter 3 will
give a description of the mathematical procedures used for forecasting .Chapter 4 gives the
analysis of data and results. Finally Chapter 5 will consist of conclusions.
5

Chapter 2
Literature review 2.1 Introduction
A literature review is an account of what has been published on a topic by accredited scholars
and 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 on
and 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 identication
and location of relevant documents and other sources. Search engines can be used to search
web resources and bibliographic databases. Conceptual frameworks can be a useful tool in
developing an understanding of a subject area. Creating the literature review involves the
stages of: scanning, making notes, structuring the literature review, writing the literature
review, 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 this
chapter the researcher will focus more on two major forecasting methods which are Seasonal
Autoregressive Intergrated Moving Average(SARIMA) and the Holt Winters Method or the
Tripple Exponential Smoothing Technique.
6

Literature review
2.2 Data Collection
Data 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 such
as `why do participants enjoy the program?’ and `How does the program help increase self
esteem for participants?’. Qualitative methods include focus groups, group discussions and
interviews. 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 ask
questions 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 and
are only indicative of the group involved.Quantitative approaches have the advantage that
they are cheaper to implement, are standardized so comparisons can be easily made and the
size of the effect can usually be measured. Quantitative approaches however are limited
in their capacity for the investigation and explanation of similarities and unexpected differ-
ences.
2.2.1 Pareto Analysis
Dendere and Masache (2013) did a research applying Pareto analysis as a quality control
tool.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 retail
shop. In doing so, Pareto analysis was used to classify the products according to their sales
frequency contribution. The products that exhibit the largest frequency were chosen as the
vital few products and 14 out of 46 were identied. In addition to the sales frequency goal
were 3 more priority goals that had to be considered because high sales do not necessarily
mean high prots. That is where goal programming approach came in to strike a balance
amongst 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 of
products under study. This complies with 80:20 PARETO principle.
7

Literature review
Another 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.According
to 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 in
general and Internet markets in particular have the potential to substantially increase the
collective 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 collected
from a multichannel retailer, it provides empirical evidence that the Internet channel ex-
hibits a signicantly less concentrated sales distribution when compared with traditional
channels. Previous explanations for this result have focused on differences in product avail-
ability between channels. However, demonstration was made that the result survives even
when the Internet and traditional channels share exactly the same product availability and
prices. Instead,Brynjolfsson et al. (2011) nd that consumers’ usage of Internet search and
discovery tools, such as recommendation engines, are associated with an increase the share
of niche products.Brynjolfsson et al. (2011) conclude that the Internet’s long tail is not solely
due to the increase in product selection but may also partly reect lower search costs on the
Internet.We therefore conlude that if the relationships they uncover persist, the underlying
trends in technology portend an ongoing shift in the distribution of product sales.
2.3 Time Series Analysis
According to Osarumwense (2013) a time series is a sequence of ordered data. The ordering
refers generally to time, but other ordering could be envisioned e.g overspace etc. Time series
analysis is used to detect patterns of change in statistical information over regular interval
of 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 patterns
or 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 be
stationary if it has constant mean and variance.
Also Wei (2006) did a study that dealt with time domain statistical models and methods on
analyzing time series and their use in applications. It covers fundamental concepts, station-
ary and nonstationary models, nonseasonal and seasonal models, intervention and outlier
8

Literature review
models, 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 of
autoregressive conditional heteroscedasticity model, generalized autoregressive conditional
heteroscedasticity model, and unit roots and cointegration in vector time series processes
were done.
2.4 ARIMA
Ediger and Akar (2007) did a research on ARIMA forecasting of primary energy demand by
fuel a case study of Turkey.Ediger and Akar (2007) stated that forecasting of energy demand
in emerging markets is one of the most important policy tools used by the decision makers all
over 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 usually
deviate from the realizations, time-series forecasting appears to give better results. In this
study, we used the Autoregressive Integrated Moving Average (ARIMA) and seasonal ARIMA
(SARIMA) methods to estimate the future primary energy demand of Turkey from 2005 to
2020. 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 average
annual growth rates of individual energy sources and total primary energy will decrease in
all cases except wood and animal–plant remains which will have negative growth rates. The
decrease in the rate of energy demand may be interpreted that the energy intensity peak will
be achieved in the coming decades. Another interpretation is that any decrease in energy
demand will slow down the economic growth during the forecasted period. Rates of changes
and reserves in the fossil fuels indicate that inter-fuel substitution should be made leading
to a best mix of the country’s energy system. Based on our ndings we proposed some policy
recommendations.
Another study was carried out by Kumar and Jain (2010) on ARIMA forecasting of ambient
air pollutants and he argues that In the present study, a stationary stochastic ARMA/ARIMA
Autoregressive 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 in
order to make them covariance stationary in a consistent way. A combination of different
9

Literature review
information-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 the
selected ARMA(p,q)/ARIMA(p,d,q) models has been evaluated on the basis of MAPE (mean
absolute 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 carbon
dioxide(CO),nitrogen monoxide (N O
2)
, nitrogen oxide(NO) and oxygen (O
3)
, have been found
to be 13.6, 12.1, 21.8 and 24.1%, respectively. Given the stochastic nature of air pollutants
data 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 forecasting
procedure 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 method
suitable for both ARIMA and SARIMA that can overcome the problem of local optima. In
this paper, we provide a genetic algorithms (GA) based model identication to overcome the
problem 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 DRAM
price forecasting to illustrate an application in the semiconductor industry. The results show
that the GA-based model identication method can present better solutions, and is suitable
for any ARIMA models.
Another study was carried out by Kumar and Vanajakshi (2015) on Short-term trafc ow
prediction 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 model
building. Hence, the applicability of these models remains a question in places where the
data availability could be an issue. The present study tries to overcome the above issue by
proposing a prediction scheme using Seasonal ARIMA (SARIMA) model for short term pre-
10

Literature review
diction of trafc ow using only limited input data.
2.5 SARIMA
Velasquez 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 the
SARIMA methodology was used to capture the linear components in the time series, articial
neural networks were applied to forecast the remaining non linearities in the shocks of the
SARIMA 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 the
SARIMA model was proposed. To evaluate the capacity of the new approach, the monthly
electricity demand in the Colombian energy market was forecasted and compared with the
SARIMA and multiplicative single neuron models.However in this research SARIMA and
Holt Winters Method will compared and the best forecasting method will be chosen.Also
Schulze 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 to
Schulze and Prinz (2009) it was seen that the forecasting error measures such as mean square
error 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 the
purpose was to increase crop production.He states that it is highly difcult to forecast due
to 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 time
series. Utilizing seasonal periodicity with a Seasonal Autoregressive and Moving Average
(SARIMA) methodology the statistical data of precipitation was analysed. The experimental
results could achieve good prediction tting degree. In this sense, the model is available for
actual forecast warning in precipitation. Through the comparison of the model they found the
advantages of forecasting that can make full use of natural rainfall for corresponding areas
and save underground water resources. Another reseacher Jeong et al. (2014) did a research
to estimate energy cost budget in educational facility.The aim of was to develop an estimation
model for determining the AECB in educational facilities using the SARIMA (seasonal au-
toregressive integrated moving average) model and the ANN (articial neural network). This
study collected electricity consumption data for 7 years (2005–2011) from 787 educational fa-
11

Literature review
cilities. The result of this study showed that the prediction accuracy of the proposed hybrid
model (which was developed by combining SARIMA and ANN) was improved, compared to
the conventional SARIMA model. The MAPE (mean absolute percentage error) of the pro-
posed method and conventional method for determining the AECB in educational facilities
was determined at 0.11–0.24% and 1.23–1.84%, respectively. Namely, it was determined that
the proposed method was superior to the conventional method. The proposed model could
enable executives and managers in charge of budget planning to accurately determine the
AECB in educational facilities. It could be also applied to other types of resources (e.g., water
consumption or gas consumption) used in educational facilities.
Nanthakumar et al. (2012) did a study to forecast the tourism demand for Malaysia from
ASEAN countries. The literature on forecasting tourism demand is huge comprising vari-
ous types of empirical analysis. Some of the researchers applied cross-sectional data, but
most of the tourism demand forecasting used pure time-series analytical models. One of the
important time-series modelling used in tourism forecasting is ARIMA modelling,which is
specied 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-regressive
integrated 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 of
the study shows that forecast combinations do not always outperform the best single forecasts
which have been used frequently in previous studies. Therefore, combination of empirical
models can reduce the risk of forecasting failure in practice.Generally, from this study we can
conclude that the ARIMA volatility models tend to overestimate demand, and the smoothing
models are inclined to underestimate the number of future tourist arrivals
Again Chu (2009) modied ARIMA modelling to fractionally integrated autoregressive mov-
ing average (ARFIMA) in forecasting tourism demand. This ARFIMA model is ARMA based
methods. Three types of univariate models were applied in the study with some modication
in ARMA model to become ARAR and ARFIMA model. The main purpose of this study is to
investigate the ARMA based models and its usefulness as a forecast generating mechanism
for tourism demand for nine major tourist destinations in the Asia-Pacic region. This study
is different from other tourism forecasting studies published earlier, because we can identify
12

Literature review
the ARMA based models behaviour and the difference between ARFIMA models with other
ARMA based models
Also 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 to
estimate 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 deviant
observations. 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) are
established for a gaussian AR process.It is shown that the method improves the French load
forecasting for “normal days” and offers good robustness, easiness and fast execution.In the
research it is also said that when the data contains deviant observations termed outliers, the
classical estimates of a SARIMA model become unreliable. Thus order selection, parameter
estimation, 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 diagnostic
approaches achieve robustness via outlier detection and hard rejection, resulting in missing
values in the time series. By contrast, robust methods accommodate outliers by bounding
their inuence on the estimates, yielding no missing values. While they are different, the
diagnostic and the robust approaches end up to have a similar objective, which is estimating
in a robust way a model and detecting the outliers.
2.6 Holt Winters Method
Goodwin et al. (2010) did a research concerning Holt Winters Method and he stated that many
companies use the Holt-Winters (HW) method to produce short-term demand forecasts when
their sales data contain a trend and a seasonal pattern.Goodwin et al. (2010) also outlined
the uses of this method which are how can to stop the method from being unduly inuenced
by sales gures that are unusually high or low (i.e., outliers)? ,checking whether the method
is useful when there are several different seasonal patterns in sales (such as when demand
has hourly, daily, and monthly cycles mixed together)? and how to obtain reliable prediction
intervals from the method?.Goodwin et al. (2010) states that the Holt-Winters method was
designed to handle data where there is a conventional seasonal cycle across the course of a
year, such as monthly seasonality. However, many series have multiple cycles: the demand
for electricity will have hourly (patterns across the hours of a day), daily (patterns across the
13

Literature review
days of the week), and monthly cycles across the years.
Taylor (2003a) went on further to do a research on the Exponential smoothing with a damped
multiplicative trend.Taylor (2003a) found out that multiplicative trend exponential smooth-
ing has received very little attention in the literature. It involves modelling the local slope
by smoothing successive ratios of the local level, and this leads to a forecast function that is
the product of level and growth rate. By contrast, the popular Holt method uses an additive
trend formulation. It has been argued that more real series have multiplicative trends than
additive. However, even if this is true, it seems likely that the more conservative forecast
function of the Holt method will be more robust when applied in an automated way to a large
batch of series with different types of trend. In view of the improvements in accuracy seen in
dampening the Holt method, in this paper we investigate a new damped multiplicative trend
approach. An empirical study, using the monthly time series from the M3-Competition, gave
encouraging results for the new approach at a range of forecast horizons, when compared to
the established exponential smoothing methods.
Taylor (2003b) researched on univariate online electricity demand forecasting for lead times
from 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 from
the similarity of the demand prole from one day to the next, and a within-week seasonal
cycle 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. In
the paper, the Holt–Winters exponential smoothing formulation was adapted so that it can
accommodate two seasonalities.Correction for residual autocorrelation was done using a sim-
ple autoregressive model. The forecasts produced by the new double seasonal Holt–Winters
method outperform those from traditional Holt–Winters and from a well-specied multiplica-
tive double seasonal ARIMA model.
2.7 Accuracy measures
Armstrong and Fildes (1995) proposed the Generalized Forecast Error Second Moment (GFESM)
as an improvement to the Mean Square Error in comparing forecasting performance across
data series. They based their conclusion on the fact that rankings based on GFESM remain
unaltered if the series are linearly transformed. In this paper, we argue that this evalua-
14

Literature review
tion ignores other important criteria. Also, their conclusions were illustrated by a simulation
study whose relationship to real data was not obvious. Thirdly, prior empirical studies show
that 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 be
assigned to Mean Absolute Percentage Error (MAPE),the model with the least MAPE value
will be considered to be the best.MAPE presents problems when it produces values close to
zero or equal to zero.These problems can be avoided by using non-negative values.
2.8 Conclusion
This chapter demonstrated understanding, and ability to critically evaluate research in the
eld,provided evidence that may be used to support your the researchers own ndings,to see
what has and has not been investigated and to contribute to the eld by moving research
forward. Also this chapter helped to see what came before, and what did and didn’t work for
other researchers.
15

Chapter 3
Methodology 3.1 Introduction
This chapter covers on how the research was conducted to obtain necessary information used
in 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 brief
description on the methods to be used which are Seasonal ARIMA and Holt Winter’s method
and 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 SARIMA
models as they are precise in dealing with data which is seasonal. Seasonality in a time series
is a regular pattern of changes that repeats over S time periods, where S denes the number
of time periods until the pattern repeats again.
3.2 Data collection
Data was collected from the Bakers Inn shops and self selection was used as a sampling
criteria to choose amongst the Harare shops.Self sampling is useful when we want to allow
every unit(in this case shops)to take part in the research.There are reasons why a shop is
either chosen or rejected and in this case the researcher chose Reliance because it is the shop

Methodology
17that has the highest Gross Prot in the Harare region.Also only products that are ordered on
a daily basis and have an expiry data of less than seven days were considered to be part of
the research for efciency.It was seen that products like drink take almost a year to expire
and these products are never found to be part of the waste hence will not contribute to the
research.
3.2.1 Pareto Analysis
Pareto 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 or
principle in this case it means only 20% of products yield 80% of the prots.Pareto’s Principle
or the 80-20 Rule helps you to identify and prioritize events and activities that can improve
your 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).The
Pareto 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 Chart
A 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 by
the line.Basically it is skewed with heavy “slowly decaying” tails where much of the data is
explained 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 with
the 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 ount

100

Methodology
18
Create a second y-axis with percentages descending in increments of 10 from 100% to
0%.
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 at
the point of intersection with the curve on the x-axis. This point on the x-axis separates
the 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 percentage
contribution to prot and the top products were chosen.
3.3 Box Jenkins Approach
The Box-Jenkins methodology is a strategy or procedure that can be used to build an ARIMA
model. Box Jenkins Approach is an iterative procedure for time series forecasting . According
to 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 approach
has 3 main methods namely identication, estimation and verication . The rst step is to
get feel of the data, that is collecting and examining the data graphically and statistically.The
data 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 Autocorrelation
Function (ACF) of the time series values either cuts off or dies down fairly quickly then the
time series is considered stationary .On the other hand , if the ACF of the time series values
either cuts off or dies down extremely slowly then it should be considered non-stationary .In
general , if the original time series values are non-stationary , performing rst and second
differencing transformation on the original data will produce stationary time series values.
For regular differencing forecast the equation is given as
d(
X
t) = (1
B)d
(X
t)
(3.1)
When d=1

Methodology
19(
X
t) = (1
B)d
(X
t) =
X
t
X
t 1 (3.2)
Once stationarity is rendered then one should identify and estimate the correct ARIMA
model.
3.4 Components and Fitting of ARIMA model
3.4.1 Overview
The 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 result
of being close together in time; the differencing component, d, which is used to make a time
series stationary and the moving average component, q, which describes outside “shocks” to
the system.
3.4.2 Identication
The identication steps involve tting the autoregressive component (variable “p”), the mov-
ing average component of the ARIMA model (variable “q”), as well any required differing
to 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 of
the model will be ARIMA (p,d,q) when the model is reported.
3.4.3 Estimation and Diagnostic checks
The estimation procedure involves using the model with p, d and q orders to t the actual time
series. A software is used to t the historical time series, while the researcher checks that
there is no signicant signal from the errors using an ACF for the error residuals, and that
estimated parameters for the autoregressive or moving average components are signicant.
If the original model identication is correct , the model requires diagnostics .If the model
fails , the process is repeated until the model satises all assumptions.

Methodology
203.4.4 Forecasting
After a model is assured to be stationary, and tted such that there is no information in the
residuals, we can proceed to forecasting. Forecasting assesses the performance of the model
against real data. There is an option to split the time series into two parts, using the rst
part to t the model and the second half to check model performance. Usually the utility of a
specic model or the utility of several classes of models to t actual data can be assessed by
minimizing a value such as root mean square.
3.5 SARIMA Model
As 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 also
handles 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 series
and when this happens the SARIMA model inevitably becomes the choice model. A SARIMA
model 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 periods
until the pattern repeats again.The seasonal ARIMA model incorporates both non-seasonal
and seasonal factors in a multiplicative model. One shorthand notation for the model is
ARIMA(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 MA
order, and S = time span of repeating seasonal pattern. Without differencing operations, the
model could be written more formally as
(B )s
‘ (B )(X
t
) = Bs
(B )W
t (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)

Methodology
21The 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 SARIMA
model is given by (B )
p(
B h
d
D
h X
t=
(B ) (B h
t +
c (3.8)
The seasonal difference operator is given by
s= 1
B
s (3.9)
The general model is denoted as ARIMA (P,D ,Q)h and are polynomials of order P and Q
respectively and the non-seasonal AR and MA characteristics operators are :
(B ) = 1
1B

2B 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
1B

2B 2
s

P B P s
(3.12)
( Bs
) = 1 + 1B
+
2B 2
s
+ +
QB Qs
(3.13)
d
D
h X
t= (1
B)d
(1 Bd
)D
X t (3.14)
Where
i and

jare constants such that the zeros of equation 3.20 and 3.21 are all outside
the 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-seasonal

Methodology
22characteristics, 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 and
seasonal difference respectively. For a seasonal series ,the time plot reveals the existence
of a seasonal nature in data, and the ACF shows a spike at the seasonal lag. Table below
summarises the behaviour of the ACF and PACF of Pure Seasonal ARMA models
Table 3.1: Behaviour of ACF and PACF of Pure Seasonal ARIMA models ACF PACF
AR(p) Tails off at lag kh=1,2 Cuts off after lag ph
MA(q) Cuts off after lag Q Tails off at lags kh=1,2
ARIMA(p,q) Tails off at lags kh Tails off at kh
The 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 off
slowly whereas that of an AR(p) model cuts off after lag p. AR and MA models are known to
exhibit 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 seasonal
part of the model, all of these factors operate across multiples of lag s (the number of
periods in a season).
A seasonal ARIMA model is classied as an ARIMA(p,d,q)x(P,D,Q) model, where P=number
of seasonal autoregressive (SAR) terms, D=number of seasonal differences, Q=number
of seasonal moving average (SMA) terms
3.5.1 Assumptions of SARIMA Model The time series data should be stationary which means that its properties do no depend
on time at which the series is observed i.e its mean and variance are constant through

Methodology
23time .For practical purposes, it is sufcient to have weak stationary, which means that
the data is in equilibrium around the mean and the variance remains constant over
time .If a time series data is non-stationary due to its variance not being constant, it
often helps to log-transform the data. Differencing is applied to have a series that is
stationary in the mean.
Residuals are normally distributed over time .Residuals exhibit homogeneity of variance
over time and have a mean zero.
Homoscedasticity ie the series has a constant variance .If the amplitude of the variance
around 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 and
normally 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 across
months to check for seasonal pattern.
Also examine the autocorrelation plots of the time series.The ACF is a statistical tool
that 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 a
non-stationary time series , the autocorrelations typically decay slowly it at all. For the
autocorrelation plots MINITAB and EVIEW 7 for the Augmented Dickey Fuller Test are
used as statistical packages .
Test for stationarity is essential at this stage and if the data exhibit non stationary
property diffencing of the time series data is then applied. The researcher will use 12
months of differencing to remove the seasonality component in the data which will give
the given series below;
Y t = (1
B12
)X
t
3.5.3 Model identication and estimation
Plot 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 the

Methodology
24moving average q for the differenced data. Then to determine the AR and MA orders ,count
the number of signicant autocorrelations and partial autocorrelations .We then calculate the
parameters of the model by making use of the mean square error (MSE) value in the model
selection criteria. The model with the least MSE is selected to be the best.
3.6 Model tting and Diagnostics
Check 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 distributed
3.6.1 Autocorrelation assumption
ACF 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 the
residuals are independepent then the model is considered to be appropriate for the set of
data.
Durbin Watson test is used to test for autocorrelation in the error terms .Durbin Watson test
looks at only one type of auto correlation that is rst order autoregressive type of correlation
AR(1)process. The test statistic for d is given as; d= 2(1 )
We can deduce that
1. d= 2 or= 0 there is no auto correlation
2. d= 0 or= +1 there is perfect positive auto-correlation 0< d < 2there is some degree
of positive auto-correlation
3. 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 the

Methodology
25skewness value should be close to 0 and the kurtosis value should be 3 to satisfy the normal-
ity assumption.
3.6.3 Heteroskedasticity
We use the plot of residuals vs ts to detect if there are any problems in the tted model and it
also gives a clear indication of the outlying observations. With the plot of residuals it is easier
to see a change of in the variance than with a plot of original data. If all the assumptions are
satised then Gaussian white noise to the error terms of the Seasonal ARIMA is said to be
satised.
3.7 Goodness of t
A 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 research
as a way of checking the model t. Static forecasting on the model is performed to show
measures 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 series
model. 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 presenting
of 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 plotting
ACF of the errors of the tted model. Most of the sample autocorrelation coefcients of the
residuals are within limits 1.96/ p
N where N is the number of observations upon which the
model is based and it shows that the model is a good t.
3.8 Evaluation of forecasting performance
The nal step is to evaluate the forecast performances by our achieved multiplicative seasonal
SARIMA model. The evaluation includes the Objective penalty criterion which is a method
of evaluating model accuracy.

Methodology
263.8.1 Forecast error
The forecast error is the difference between the observed value and its forecast based on all
previous observations. If the error is denoted as e(t) then the forecast error can be written as
e(t) = Y
t ^
Y t (3.15)
where Y(t) are the observations ^
Y t is the forecast of Y(t) based on all previous observations
Forecast 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 Absolute
Percentage 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 zeros
or 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 items
that are at zero or near zero volume Symmetric Mean Absolute Percent Error (SMAPE) is a
better measure. MAPE is the average absolute percent error for each time period or forecast
subtracted from actual divided by actual.
M AP E= j
Actual F orecast j Actual

100% N
(3.16)
The best model is the one with the least MAPE value
3.8.3 Root Mean Square Error (RMSE)
To construct the RMSE, residuals are needed. Residuals are the difference between the actual
values and the predicted values.I denoted them by Y
t ^
Y t.They can be positive or negative
as 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 the
RMSE as a measure of the spread of the y values about the predicted y value.
RM S E=v
u
u
t 1
N
T
X
1 (
Y
t ^
Y t) 2
(3.17)

Methodology
27where N is the number of forecasted observations
3.8.4 Mean Absolute Deviation (MAD)
Mean absolute deviation (MAD) of a data set is the average distance between each data value
and the mean. Mean absolute deviation is a way to describe variation in a data set. Mean
absolute deviation helps us get a sense of how “spread out” the values in a data set are. Here’s
how to calculate the mean absolute deviation.
M AD=1 N
T
X
1 j
Y
t ^
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 is
negative, the forecasts have been too high.
M F E=1 N
T
X
1 (
Y
t ^
Y t)
(3.19)
To compare the forecasting capabilities for the two models we therefore plot a graph of the
two models with the forecasted values together with the actual sales.
3.9 Holt Winters Method nTriple Exponential Smooth-
ing
Holt 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 are
weighted equally, exponential functions are used to assign exponentially decreasing weights
over 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 variations
to this method that differ in the nature of the seasonal component. The additive method is
preferred when the seasonal variations are roughly constant through the series, while the
multiplicative method is preferred when the seasonal variations are changing proportional
to the level of the series. With the additive method, the seasonal component is expressed
in absolute terms in the scale of the observed series, and in the level equation the series is

Methodology
28seasonally adjusted by subtracting the seasonal component. The raw data sequence is often
represented by x
t beginning at time
t= 0 , and the output of the exponential smoothing algo-
rithm is commonly written as x
t, which may be regarded as a best estimate of what the next
value of x will be. When the sequence of observations begins at time t= 0 , the simplest form
of exponential smoothing is given by the formulas:
s 0 =
x
t
st =
x
t+ (1
)s
t 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 A
t, 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 the
values 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 the
seasonal changes.
s t is the sequence of seasonal correction factors. ct is the expected proportion of the predicted
trend at any time t in the cycle that the observations take on. As a rule of thumb, a minimum
of two full seasons (or 2L periods) of historical data is needed to initialize a set of seasonal
factors.
The output of the algorithm is again written as F
t, an estimate of the value of x at time t,
based on the raw data up to time t. Triple exponential smoothing with Additive seasonality
is given by the formulas
L 0 =
A
t
Level:L
t=
(A
t+
S
t s) + (1
)( L
t 1 +
b
t 1)
(3.21)
T rend :b
t =
(L
t
L
t 1) + (1
)b
t 1 (3.22)
S easonal :S
t=

(A
t
S
t) + (1

)b
t s (3.23)
F orecast :F
t+ m = (
A
T +
b
T k
) + L
T + k 1) =A +1 (3.24)
where is the data smoothing factor, 0; ; 1, is the trend smoothing factor, 0; ; 1,

Methodology
29and

is 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.
6

Data Analysis
Figure 4.13: Modied Box-Pierce (Ljung-Box) Chi-Square Results
Figure 4.14: Durbin Watson Test Results
Durbin Watson Test
From the Durbin Watson test results it can be seen that DW is close to 2 hence we conclude
that there is no auto-correlation of the residuals therefore assumption is not violated.
Normality Assumption
Histogram of Residuals Figure 4.15: Histogram of Residuals
Fig 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 Test
7

Data Analysis
Figure 4.16: Jarque Bera Test
Figure 4.17: Residual vs Fits
The test shows that the p-value of 0.205644 is insignicant (p¿0.05). Also the skewness of 0.05
which is close to zero and the kurtosis of 2.546 which is close to 3 provides much evidence
that the residuals are normally distributed.The Jarque-Bera value was found to be 3.163221
.
Heteroskedasticity Test
Residual vs Fit plot shows that the residuals bound randomly around the 0 line.This suggests
the assumption that the relationship is linear is reasonable.The horizontal band formed by
the 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 suggests
that there are no outliers.
4.3.5 Goodness of Fit
ACF of residuals shows that most of the coefcients of the sample autocorrelation of the
residuals falls within the limit
1:96 349
= 0
:1049163946 and it indicates that the model is a good
8

Data Analysis
t 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 Measures
Table 4.1: SARIMA Forecasting Perfomances Error MAPE MAD
Value 11.5384 2.6664
4.4 Holt Winters Method
A multiplicative and additive plot for Holt winters sales are compared to nd the method
with the least MAPE.It was seen that the additive method produced the least MAPE and it
is therefore used in this research. Figure 4.19: Winters Method Additive
9

Data Analysis
Figure 4.20: Winters Method For Multiplicative
Trial and error method was used to come up with the best model smoothing parameters
and errors. Initially the researcher used 0.2 for all the three smoothing constants that is
level,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 according
to 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 Sales
The additive exponential smoothing equations are as follows
Level Lt= 0
:6( A
t+
S
t s) + 0
:4( L
t 1 +
b
t 1)
(4.1)
Trend b1 = 0
:1( L
t
L
t 1) + 0
:9 + b
t 1 (4.2)
Seasonal St= 0
:2( A
t
S
t) + 0
:8 b
t s (4.3)
10

Data Analysis
Table 4.2: Holt Winters forecast parameters and errors
Model
MAPE MAD
A 0.2 0.2 0.2 11.7774 2.3347
B 0.2 0.01 0.01 11.1394 2.4455
C 0.2 0.001 0.01 12.697 3.1944
D 0.3 0.1 0.1 10.582 2.6472
E 0.5 0.001 0.01 11.975 2.3817
F 0.5 0.0001 0.0001 12.6587 2.4139
H 0.6 0.1 0.2 10.514 2.3489
I 0.6 0.1 0.3 10.5521 2.3824
J 0.6 0.001 0.001 11.7033 2.3602
K 0.6 0.0000001 0.00001 12.5522 2.3824
L 0.7 0.00000001 0.00000001 13.5426 2.3775
M 0.7 0.01 0.01 14.3620 2.3620
Forecast
Ft=
L
t 1 +
b
t 1 +
S
t s (4.4)
where s is the number of seasonal periods in a year T is the time period
4.5 Model Diagnostics
To check whether the model assumptions are not violated , some residual tests were carried
out.
4.5.1 Run’s Test
Fig 4. shows that p=0.436 which is greater than 0.05 therefore the residuals are random and
the assumption is not violated.
11

Data Analysis
Figure 4.22: Runs Test
Figure 4.23: ACF of Residuals
4.5.2 ACF of Residuals
The auto correlation of residuals shows that the auto correlations for the in-sample forecast
errors do not exceed the signicance bounds for 1-60 lags.The observed signicant lag at lag
1 is due to random error and does not imply that the residuals are not independent.
4.5.3 Histogram of residuals
The 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 Forecasting
Since the model has a good t we will use it to forecast daily sales and table 4.3 shows the
accuracy measures for the actual and forecasted sales
12

Data Analysis
Figure 4.24: Histogram of Residual
Figure 4.25: Accuracy Measures
13

Data Analysis
4.6.1 Evaluating Forecasting perfomance
RMSE and MAPE shall be used in evaluating the forecasting perfomance
RM S E =q 1
N
P
T
1 (
Y
t ^
Y t) 2
= q 1
28
(3654940
:604) = 361 :2943
Table 4.3: Holt Winters forecasting Evaluation Error MAPE MAD
Value 10.514 361.2943
4.7 Comparison of the Holt Winters and the SARIMA
Table 4.4: Holt Winters and SARIMA Forecasting Perfomances comparisons Error MAPE MAD
SARIMA 11.5384 2.6664
Holt Winters 10.514 2.3489
In this research a model with the least MAPE is considered to be better from the other and
from Table 4.4 it is seen that the Holt Winters Method has the least MAPE of 10.514 hence
it is prefered than the SARIMA model.We can now use the Holt Winters Method to forecast
future sales as shown by gure 4. Figure 4.26: Forecasting Sales
There is a downward trend in the forecasting of future sales.
14

Data Analysis
4.8 Conclusion
Holt Winters method was found to be the best forecasting method and is therefore used to
forecast sales in the future.A decrease in the sales was seen and it might be a way to reduce
waste products , meeting demand at the same time increasing prots.
15

Chapter 5
Conclusion and Recommendations 5.1 Introduction
This chapters concludes this research project.The conclusions of the study are clearly outlined
and stated as well as answers to the research questions which were stated earlier in the rst
chapter of this research.For future studies ,recommendations from the research are going to
be provided.
5.1.1 Summary of Results
The SARIMA and Holt-Winters forecasting procedures were used to forecast daily sales one
month.The Pareto analysis of products shows that some products contributes a very low prot
which is almost insignicant and its obviously that these are the same products that are
continously being ordered and increase the waste cost value.To stabilize variance the Box
Cox transformation was applied to the data with = 0 :5 which is the same as the square
root of the data.To check for stationarity trend analysis and the autocorrelation plot was
used.From the trend analysis it was seen that the the data was not stationary since most
of the lags are signicant.To obtain a stationary series the data was differenced once and
tested again for stationarity.The appropriate SARIMA model was found and Auto correlation
Function (ACF) plot was used to check if the data exhibits auto regressive,moving average
or both orders.The ACF plot had one clear spike which clearly suggests a moving average

Conclusion and Recommendations
17process and then the p values from the nal estimates table and Ljung Box were found to
be signicant p 0:05 for the Ljung Box
respectively.SARIMA (0;1 ;1)(0 ;0 ;1)
7tend out to be best model meeting all the requirements
For Holt Winters Method trial and error procedure was used, taking note of the results at
each trial and comparisons were made for the MAPE and MAD values to come up with the
best 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.The
models 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 Recommendations
Bakers Inn Harare retail department are therefore recommended to consider using the Mul-
tiplicative Holt Winters method to forecast their sales thereby reducing overstocking and
under 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 of
concern 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 since
they 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 order
in 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 last
in rst out counterpart and it’s more practical – especially regarding perishable goods.When
a company uses FIFO they are less likely to incur old and outdated inventory that can no
longer be sold. Accountants have to write off what’s called obsolete inventory after a certain
amount of time goes by and the product is not used or sold. Because FIFO makes sure that
the oldest items in stock are used or sold before they are deemed obsolete companies can save
money (Sponaugle, 2014).

Conclusion and Recommendations
185.1.3 Suggested Future Work
1.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.For
example the ARCH models may be included in the comparative study to carter for data
that is highly volatile.Neural Networks may also be included in the research to carter for
non linear data.This will increase the chances of obtaining a more favorable forecasting
model for the given data.Intervention analysis(in the presence of promotions) may be
incorporated to determine how past sales affected sales and hence how will promotions
will 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 the
seasonality and trend factors on the sales data and will allow the organization to make
more 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 Algorithm
and also increasing its forecasting power.
5.2 Conclusion
This research has compared the forecasting ability of Holt-Winters and SARIMA models
with respect to their daily demand obtained from the daily sales data. The study results
demonstrate that both models are pretty effective; however Holt-Winters model seems to be
a more precise and accurate model. From table 4.4 we found that Holt-Winters model has the
minimum MAD and MAPE values when compared with SARIMA model. The Holt-Winters
model’s relative ease of use makes the model useful in forecasting comprehensive market
trends. The study can be further enhanced by comparing other forecasting techniques with
respect to sales or even prot in order to obtain better accuracy. The results will help the
company to build effective strategy and make reasonable orders for each day to avoid over-
stocking and under stocking.

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Appendix

Appendices
23Figure 5.1: Pareto Calculations