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

2018 National University of Science and Technology

All rights reserved

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